methods of solving research problems

Educational resources and simple solutions for your research journey

What is a Research Problem? Characteristics, Types, and Examples

A research problem is a gap in existing knowledge, a contradiction in an established theory, or a real-world challenge that a researcher aims to address in their research. It is at the heart of any scientific inquiry, directing the trajectory of an investigation. The statement of a problem orients the reader to the importance of the topic, sets the problem into a particular context, and defines the relevant parameters, providing the framework for reporting the findings. Therein lies the importance of research problem s.  

The formulation of well-defined research questions is central to addressing a research problem . A research question is a statement made in a question form to provide focus, clarity, and structure to the research endeavor. This helps the researcher design methodologies, collect data, and analyze results in a systematic and coherent manner. A study may have one or more research questions depending on the nature of the study.   

Identifying and addressing a research problem is very important. By starting with a pertinent problem , a scholar can contribute to the accumulation of evidence-based insights, solutions, and scientific progress, thereby advancing the frontier of research. Moreover, the process of formulating research problems and posing pertinent research questions cultivates critical thinking and hones problem-solving skills.   

Table of Contents

What is a Research Problem ?  

Before you conceive of your project, you need to ask yourself “ What is a research problem ?” A research problem definition can be broadly put forward as the primary statement of a knowledge gap or a fundamental challenge in a field, which forms the foundation for research. Conversely, the findings from a research investigation provide solutions to the problem .  

A research problem guides the selection of approaches and methodologies, data collection, and interpretation of results to find answers or solutions. A well-defined problem determines the generation of valuable insights and contributions to the broader intellectual discourse.  

Characteristics of a Research Problem  

Knowing the characteristics of a research problem is instrumental in formulating a research inquiry; take a look at the five key characteristics below:  

Novel : An ideal research problem introduces a fresh perspective, offering something new to the existing body of knowledge. It should contribute original insights and address unresolved matters or essential knowledge.   

Significant : A problem should hold significance in terms of its potential impact on theory, practice, policy, or the understanding of a particular phenomenon. It should be relevant to the field of study, addressing a gap in knowledge, a practical concern, or a theoretical dilemma that holds significance.  

Feasible: A practical research problem allows for the formulation of hypotheses and the design of research methodologies. A feasible research problem is one that can realistically be investigated given the available resources, time, and expertise. It should not be too broad or too narrow to explore effectively, and should be measurable in terms of its variables and outcomes. It should be amenable to investigation through empirical research methods, such as data collection and analysis, to arrive at meaningful conclusions A practical research problem considers budgetary and time constraints, as well as limitations of the problem . These limitations may arise due to constraints in methodology, resources, or the complexity of the problem.  

Clear and specific : A well-defined research problem is clear and specific, leaving no room for ambiguity; it should be easily understandable and precisely articulated. Ensuring specificity in the problem ensures that it is focused, addresses a distinct aspect of the broader topic and is not vague.  

Rooted in evidence: A good research problem leans on trustworthy evidence and data, while dismissing unverifiable information. It must also consider ethical guidelines, ensuring the well-being and rights of any individuals or groups involved in the study.

Types of Research Problems  

Across fields and disciplines, there are different types of research problems . We can broadly categorize them into three types.  

• Theoretical research problems

Theoretical research problems deal with conceptual and intellectual inquiries that may not involve empirical data collection but instead seek to advance our understanding of complex concepts, theories, and phenomena within their respective disciplines. For example, in the social sciences, research problem s may be casuist (relating to the determination of right and wrong in questions of conduct or conscience), difference (comparing or contrasting two or more phenomena), descriptive (aims to describe a situation or state), or relational (investigating characteristics that are related in some way).  

Here are some theoretical research problem examples :   

• Ethical frameworks that can provide coherent justifications for artificial intelligence and machine learning algorithms, especially in contexts involving autonomous decision-making and moral agency.  
• Determining how mathematical models can elucidate the gradual development of complex traits, such as intricate anatomical structures or elaborate behaviors, through successive generations.  
• Applied research problems

Applied or practical research problems focus on addressing real-world challenges and generating practical solutions to improve various aspects of society, technology, health, and the environment.  

Here are some applied research problem examples :   

• Studying the use of precision agriculture techniques to optimize crop yield and minimize resource waste.  
• Designing a more energy-efficient and sustainable transportation system for a city to reduce carbon emissions.  
• Action research problems

Action research problems aim to create positive change within specific contexts by involving stakeholders, implementing interventions, and evaluating outcomes in a collaborative manner.  

Here are some action research problem examples :   

• Partnering with healthcare professionals to identify barriers to patient adherence to medication regimens and devising interventions to address them.  
• Collaborating with a nonprofit organization to evaluate the effectiveness of their programs aimed at providing job training for underserved populations.  

These different types of research problems may give you some ideas when you plan on developing your own.  

How to Define a Research Problem  

You might now ask “ How to define a research problem ?” These are the general steps to follow:   

• Look for a broad problem area: Identify under-explored aspects or areas of concern, or a controversy in your topic of interest. Evaluate the significance of addressing the problem in terms of its potential contribution to the field, practical applications, or theoretical insights.
• Learn more about the problem: Read the literature, starting from historical aspects to the current status and latest updates. Rely on reputable evidence and data. Be sure to consult researchers who work in the relevant field, mentors, and peers. Do not ignore the gray literature on the subject.
• Identify the relevant variables and how they are related: Consider which variables are most important to the study and will help answer the research question. Once this is done, you will need to determine the relationships between these variables and how these relationships affect the research problem . 
• Think of practical aspects : Deliberate on ways that your study can be practical and feasible in terms of time and resources. Discuss practical aspects with researchers in the field and be open to revising the problem based on feedback. Refine the scope of the research problem to make it manageable and specific; consider the resources available, time constraints, and feasibility.
• Formulate the problem statement: Craft a concise problem statement that outlines the specific issue, its relevance, and why it needs further investigation.
• Stick to plans, but be flexible: When defining the problem , plan ahead but adhere to your budget and timeline. At the same time, consider all possibilities and ensure that the problem and question can be modified if needed.

Key Takeaways  

• A research problem concerns an area of interest, a situation necessitating improvement, an obstacle requiring eradication, or a challenge in theory or practical applications.   
• The importance of research problem is that it guides the research and helps advance human understanding and the development of practical solutions.  
• Research problem definition begins with identifying a broad problem area, followed by learning more about the problem, identifying the variables and how they are related, considering practical aspects, and finally developing the problem statement.  
• Different types of research problems include theoretical, applied, and action research problems , and these depend on the discipline and nature of the study.  
• An ideal problem is original, important, feasible, specific, and based on evidence.  

Frequently Asked Questions  

Why is it important to define a research problem?  

Identifying potential issues and gaps as research problems is important for choosing a relevant topic and for determining a well-defined course of one’s research. Pinpointing a problem and formulating research questions can help researchers build their critical thinking, curiosity, and problem-solving abilities.   

How do I identify a research problem?  

Identifying a research problem involves recognizing gaps in existing knowledge, exploring areas of uncertainty, and assessing the significance of addressing these gaps within a specific field of study. This process often involves thorough literature review, discussions with experts, and considering practical implications.  

Can a research problem change during the research process?  

Yes, a research problem can change during the research process. During the course of an investigation a researcher might discover new perspectives, complexities, or insights that prompt a reevaluation of the initial problem. The scope of the problem, unforeseen or unexpected issues, or other limitations might prompt some tweaks. You should be able to adjust the problem to ensure that the study remains relevant and aligned with the evolving understanding of the subject matter.

How does a research problem relate to research questions or hypotheses?  

A research problem sets the stage for the study. Next, research questions refine the direction of investigation by breaking down the broader research problem into manageable components. Research questions are formulated based on the problem , guiding the investigation’s scope and objectives. The hypothesis provides a testable statement to validate or refute within the research process. All three elements are interconnected and work together to guide the research.  

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How to identify and resolve research problems

Updated August 23, 2024

In this article, we’re going to take you through one of the most pertinent parts of conducting research: a research problem (also known as a research problem statement).

When trying to formulate a good research statement, and understand how to solve it for complex projects, it can be difficult to know where to start.

Not only are there multiple perspectives (from stakeholders to project marketers who want answers), you have to consider the particular context of the research topic: is it timely, is it relevant and most importantly of all, is it valuable?

In other words: are you looking at a research worthy problem?

The fact is, a well-defined, precise, and goal-centric research problem will keep your researchers, stakeholders, and business-focused and your results actionable.

And when it works well, it's a powerful tool to identify practical solutions that can drive change and secure buy-in from your workforce.

Free eBook: The ultimate guide to market research

What is a research problem?

In social research methodology and behavioral sciences , a research problem establishes the direction of research, often relating to a specific topic or opportunity for discussion.

For example: climate change and sustainability, analyzing moral dilemmas or wage disparity amongst classes could all be areas that the research problem focuses on.

As well as outlining the topic and/or opportunity, a research problem will explain:

• why the area/issue needs to be addressed,
• why the area/issue is of importance,
• the parameters of the research study
• the research objective
• the reporting framework for the results and
• what the overall benefit of doing so will provide (whether to society as a whole or other researchers and projects).

Having identified the main topic or opportunity for discussion, you can then narrow it down into one or several specific questions that can be scrutinized and answered through the research process.

What are research questions?

Generating research questions underpinning your study usually starts with problems that require further research and understanding while fulfilling the objectives of the study.

A good problem statement begins by asking deeper questions to gain insights about a specific topic.

For example, using the problems above, our questions could be:

"How will climate change policies influence sustainability standards across specific geographies?"

"What measures can be taken to address wage disparity without increasing inflation?"

Developing a research worthy problem is the first step - and one of the most important - in any kind of research.

It’s also a task that will come up again and again because any business research process is cyclical. New questions arise as you iterate and progress through discovering, refining, and improving your products and processes. A research question can also be referred to as a "problem statement".

Note: good research supports multiple perspectives through empirical data. It’s focused on key concepts rather than a broad area, providing readily actionable insight and areas for further research.

Research question or research problem?

As we've highlighted, the terms “research question” and “research problem” are often used interchangeably, becoming a vague or broad proposition for many.

The term "problem statement" is far more representative, but finds little use among academics.

Instead, some researchers think in terms of a single research problem and several research questions that arise from it.

As mentioned above, the questions are lines of inquiry to explore in trying to solve the overarching research problem.

Ultimately, this provides a more meaningful understanding of a topic area.

It may be useful to think of questions and problems as coming out of your business data – that’s the O-data (otherwise known as operational data) like sales figures and website metrics.

What's an example of a research problem?

Your overall research problem could be: "How do we improve sales across EMEA and reduce lost deals?"

This research problem then has a subset of questions, such as:

"Why do sales peak at certain times of the day?"

"Why are customers abandoning their online carts at the point of sale?"

As well as helping you to solve business problems, research problems (and associated questions) help you to think critically about topics and/or issues (business or otherwise). You can also use your old research to aid future research -- a good example is laying the foundation for comparative trend reports or a complex research project.

(Also, if you want to see the bigger picture when it comes to research problems, why not check out our ultimate guide to market research? In it you'll find out: what effective market research looks like, the use cases for market research, carrying out a research study, and how to examine and action research findings).

The research process: why are research problems important?

A research problem has two essential roles in setting your research project on a course for success.

1. They set the scope

The research problem defines what problem or opportunity you’re looking at and what your research goals are. It stops you from getting side-tracked or allowing the scope of research to creep off-course .

Without a strong research problem or problem statement, your team could end up spending resources unnecessarily, or coming up with results that aren’t actionable - or worse, harmful to your business - because the field of study is too broad.

2. They tie your work to business goals and actions

To formulate a research problem in terms of business decisions means you always have clarity on what’s needed to make those decisions. You can show the effects of what you’ve studied using real outcomes.

Then, by focusing your research problem statement on a series of questions tied to business objectives, you can reduce the risk of the research being unactionable or inaccurate.

It's also worth examining research or other scholarly literature (you’ll find plenty of similar, pertinent research online) to see how others have explored specific topics and noting implications that could have for your research.

Four steps to defining your research problem

Image credit: http://myfreeschooltanzania.blogspot.com/2014/11/defining-research-problem.html

1. Observe and identify

Businesses today have so much data that it can be difficult to know which problems to address first. Researchers also have business stakeholders who come to them with problems they would like to have explored. A researcher’s job is to sift through these inputs and discover exactly what higher-level trends and key concepts are worth investing in.

This often means asking questions and doing some initial investigation to decide which avenues to pursue. This could mean gathering interdisciplinary perspectives identifying additional expertise and contextual information.

Sometimes, a small-scale preliminary study might be worth doing to help get a more comprehensive understanding of the business context and needs, and to make sure your research problem addresses the most critical questions.

This could take the form of qualitative research using a few in-depth interviews , an environmental scan, or reviewing relevant literature.

The sales manager of a sportswear company has a problem: sales of trail running shoes are down year-on-year and she isn’t sure why. She approaches the company’s research team for input and they begin asking questions within the company and reviewing their knowledge of the wider market.

2. Review the key factors involved

As a marketing researcher, you must work closely with your team of researchers to define and test the influencing factors and the wider context involved in your study. These might include demographic and economic trends or the business environment affecting the question at hand. This is referred to as a relational research problem.

To do this, you have to identify the factors that will affect the research and begin formulating different methods to control them.

You also need to consider the relationships between factors and the degree of control you have over them. For example, you may be able to control the loading speed of your website but you can’t control the fluctuations of the stock market.

Doing this will help you determine whether the findings of your project will produce enough information to be worth the cost.

You need to determine:

• which factors affect the solution to the research proposal.
• which ones can be controlled and used for the purposes of the company, and to what extent.
• the functional relationships between the factors.
• which ones are critical to the solution of the research study.

The research team at the running shoe company is hard at work. They explore the factors involved and the context of why YoY sales are down for trail shoes, including things like what the company’s competitors are doing, what the weather has been like – affecting outdoor exercise – and the relative spend on marketing for the brand from year to year.

The final factor is within the company’s control, although the first two are not. They check the figures and determine marketing spend has a significant impact on the company.

3. Prioritize

Once you and your research team have a few observations, prioritize them based on their business impact and importance. It may be that you can answer more than one question with a single study, but don’t do it at the risk of losing focus on your overarching research problem.

Questions to ask:

• Who? Who are the people with the problem? Are they end-users, stakeholders, teams within your business? Have you validated the information to see what the scale of the problem is?
• What? What is its nature and what is the supporting evidence?
• Why? What is the business case for solving the problem? How will it help?
• Where? How does the problem manifest and where is it observed?

To help you understand all dimensions, you might want to consider focus groups or preliminary interviews with external (including consumers and existing customers) and internal (salespeople, managers, and other stakeholders) parties to provide what is sometimes much-needed insight into a particular set of questions or problems.

After observing and investigating, the running shoe researchers come up with a few candidate questions, including:

• What is the relationship between US average temperatures and sales of our products year on year?
• At present, how does our customer base rank Competitor X and Competitor Y’s trail running shoe compared to our brand?
• What is the relationship between marketing spend and trail shoe product sales over the last 12 months?

They opt for the final question, because the variables involved are fully within the company’s control, and based on their initial research and stakeholder input, seem the most likely cause of the dive in sales. The research question is specific enough to keep the work on course towards an actionable result, but it allows for a few different avenues to be explored, such as the different budget allocations of offline and online marketing and the kinds of messaging used.

Get feedback from the key teams within your business to make sure everyone is aligned and has the same understanding of the research problem and questions, and the actions you hope to take based on the results. Now is also a good time to demonstrate the ROI of your research and lay out its potential benefits to your stakeholders.

Different groups may have different goals and perspectives on the issue. This step is vital for getting the necessary buy-in and pushing the project forward.

The running shoe company researchers now have everything they need to begin. They call a meeting with the sales manager and consult with the product team, marketing team, and C-suite to make sure everyone is aligned and has bought into the direction of the research topic. They identify and agree that the likely course of action will be a rethink of how marketing resources are allocated, and potentially testing out some new channels and messaging strategies .

Can you explore a broad area and is it practical to do so?

A broader research problem or report can be a great way to bring attention to prevalent issues, societal or otherwise, but are often undertaken by those with the resources to do so.

Take a typical government cybersecurity breach survey, for example. Most of these reports raise awareness of cybercrime, from the day-to-day threats businesses face to what security measures some organizations are taking. What these reports don't do, however, is provide actionable advice - mostly because every organization is different.

The point here is that while some researchers will explore a very complex issue in detail, others will provide only a snapshot to maintain interest and encourage further investigation. The "value" of the data is wholly determined by the recipients of it - and what information you choose to include.

To summarize, it can be practical to undertake a broader research problem, certainly, but it may not be possible to cover everything or provide the detail your audience needs. Likewise, a more systematic investigation of an issue or topic will be more valuable, but you may also find that you cover far less ground.

It's important to think about your research objectives and expected findings before going ahead.

Ensuring your research project is a success

A complex research project can be made significantly easier with clear research objectives, a descriptive research problem, and a central focus. All of which we've outlined in this article.

If you have previous research, even better. Use it as a benchmark

Remember: what separates a good research paper from an average one is actually very simple: valuable, empirical data that explores a prevalent societal or business issue and provides actionable insights.

And we can help.

Sophisticated research made simple with Qualtrics

Trusted by the world's best brands, our platform enables researchers from academic to corporate to tackle the hardest challenges and deliver the results that matter.

Our CoreXM platform supports the methods that define superior research and delivers insights in real-time. It's easy to use (thanks to drag-and-drop functionality) and requires no coding, meaning you'll be capturing data and gleaning insights in no time.

It also excels in flexibility; you can track consumer behavior across segments , benchmark your company versus competitors , carry out complex academic research, and do much more, all from one system.

It's one platform with endless applications, so no matter your research problem, we've got the tools to help you solve it. And if you don't have a team of research experts in-house, our market research team has the practical knowledge and tools to help design the surveys and find the respondents you need.

Of course, you may want to know where to begin with your own market research . If you're struggling, make sure to download our ultimate guide using the link below.

It's got everything you need and there’s always information in our research methods knowledge base.

Scott Smith

Scott Smith, Ph.D. is a contributor to the Qualtrics blog.

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Home » Research Problem – Examples, Types and Guide

Research Problem – Examples, Types and Guide

Table of Contents

Research Problem

Definition:

Research problem is a specific and well-defined issue or question that a researcher seeks to investigate through research. It is the starting point of any research project, as it sets the direction, scope, and purpose of the study.

Types of Research Problems

Types of Research Problems are as follows:

Descriptive problems

These problems involve describing or documenting a particular phenomenon, event, or situation. For example, a researcher might investigate the demographics of a particular population, such as their age, gender, income, and education.

Exploratory problems

These problems are designed to explore a particular topic or issue in depth, often with the goal of generating new ideas or hypotheses. For example, a researcher might explore the factors that contribute to job satisfaction among employees in a particular industry.

Explanatory Problems

These problems seek to explain why a particular phenomenon or event occurs, and they typically involve testing hypotheses or theories. For example, a researcher might investigate the relationship between exercise and mental health, with the goal of determining whether exercise has a causal effect on mental health.

Predictive Problems

These problems involve making predictions or forecasts about future events or trends. For example, a researcher might investigate the factors that predict future success in a particular field or industry.

Evaluative Problems

These problems involve assessing the effectiveness of a particular intervention, program, or policy. For example, a researcher might evaluate the impact of a new teaching method on student learning outcomes.

How to Define a Research Problem

Defining a research problem involves identifying a specific question or issue that a researcher seeks to address through a research study. Here are the steps to follow when defining a research problem:

• Identify a broad research topic : Start by identifying a broad topic that you are interested in researching. This could be based on your personal interests, observations, or gaps in the existing literature.
• Conduct a literature review : Once you have identified a broad topic, conduct a thorough literature review to identify the current state of knowledge in the field. This will help you identify gaps or inconsistencies in the existing research that can be addressed through your study.
• Refine the research question: Based on the gaps or inconsistencies identified in the literature review, refine your research question to a specific, clear, and well-defined problem statement. Your research question should be feasible, relevant, and important to the field of study.
• Develop a hypothesis: Based on the research question, develop a hypothesis that states the expected relationship between variables.
• Define the scope and limitations: Clearly define the scope and limitations of your research problem. This will help you focus your study and ensure that your research objectives are achievable.
• Get feedback: Get feedback from your advisor or colleagues to ensure that your research problem is clear, feasible, and relevant to the field of study.

Components of a Research Problem

The components of a research problem typically include the following:

• Topic : The general subject or area of interest that the research will explore.
• Research Question : A clear and specific question that the research seeks to answer or investigate.
• Objective : A statement that describes the purpose of the research, what it aims to achieve, and the expected outcomes.
• Hypothesis : An educated guess or prediction about the relationship between variables, which is tested during the research.
• Variables : The factors or elements that are being studied, measured, or manipulated in the research.
• Methodology : The overall approach and methods that will be used to conduct the research.
• Scope and Limitations : A description of the boundaries and parameters of the research, including what will be included and excluded, and any potential constraints or limitations.
• Significance: A statement that explains the potential value or impact of the research, its contribution to the field of study, and how it will add to the existing knowledge.

Research Problem Examples

Following are some Research Problem Examples:

Research Problem Examples in Psychology are as follows:

• Exploring the impact of social media on adolescent mental health.
• Investigating the effectiveness of cognitive-behavioral therapy for treating anxiety disorders.
• Studying the impact of prenatal stress on child development outcomes.
• Analyzing the factors that contribute to addiction and relapse in substance abuse treatment.
• Examining the impact of personality traits on romantic relationships.

Research Problem Examples in Sociology are as follows:

• Investigating the relationship between social support and mental health outcomes in marginalized communities.
• Studying the impact of globalization on labor markets and employment opportunities.
• Analyzing the causes and consequences of gentrification in urban neighborhoods.
• Investigating the impact of family structure on social mobility and economic outcomes.
• Examining the effects of social capital on community development and resilience.

Research Problem Examples in Economics are as follows:

• Studying the effects of trade policies on economic growth and development.
• Analyzing the impact of automation and artificial intelligence on labor markets and employment opportunities.
• Investigating the factors that contribute to economic inequality and poverty.
• Examining the impact of fiscal and monetary policies on inflation and economic stability.
• Studying the relationship between education and economic outcomes, such as income and employment.

Political Science

Research Problem Examples in Political Science are as follows:

• Analyzing the causes and consequences of political polarization and partisan behavior.
• Investigating the impact of social movements on political change and policymaking.
• Studying the role of media and communication in shaping public opinion and political discourse.
• Examining the effectiveness of electoral systems in promoting democratic governance and representation.
• Investigating the impact of international organizations and agreements on global governance and security.

Environmental Science

Research Problem Examples in Environmental Science are as follows:

• Studying the impact of air pollution on human health and well-being.
• Investigating the effects of deforestation on climate change and biodiversity loss.
• Analyzing the impact of ocean acidification on marine ecosystems and food webs.
• Studying the relationship between urban development and ecological resilience.
• Examining the effectiveness of environmental policies and regulations in promoting sustainability and conservation.

Research Problem Examples in Education are as follows:

• Investigating the impact of teacher training and professional development on student learning outcomes.
• Studying the effectiveness of technology-enhanced learning in promoting student engagement and achievement.
• Analyzing the factors that contribute to achievement gaps and educational inequality.
• Examining the impact of parental involvement on student motivation and achievement.
• Studying the effectiveness of alternative educational models, such as homeschooling and online learning.

Research Problem Examples in History are as follows:

• Analyzing the social and economic factors that contributed to the rise and fall of ancient civilizations.
• Investigating the impact of colonialism on indigenous societies and cultures.
• Studying the role of religion in shaping political and social movements throughout history.
• Analyzing the impact of the Industrial Revolution on economic and social structures.
• Examining the causes and consequences of global conflicts, such as World War I and II.

Research Problem Examples in Business are as follows:

• Studying the impact of corporate social responsibility on brand reputation and consumer behavior.
• Investigating the effectiveness of leadership development programs in improving organizational performance and employee satisfaction.
• Analyzing the factors that contribute to successful entrepreneurship and small business development.
• Examining the impact of mergers and acquisitions on market competition and consumer welfare.
• Studying the effectiveness of marketing strategies and advertising campaigns in promoting brand awareness and sales.

Research Problem Example for Students

An Example of a Research Problem for Students could be:

“How does social media usage affect the academic performance of high school students?”

This research problem is specific, measurable, and relevant. It is specific because it focuses on a particular area of interest, which is the impact of social media on academic performance. It is measurable because the researcher can collect data on social media usage and academic performance to evaluate the relationship between the two variables. It is relevant because it addresses a current and important issue that affects high school students.

To conduct research on this problem, the researcher could use various methods, such as surveys, interviews, and statistical analysis of academic records. The results of the study could provide insights into the relationship between social media usage and academic performance, which could help educators and parents develop effective strategies for managing social media use among students.

Another example of a research problem for students:

“Does participation in extracurricular activities impact the academic performance of middle school students?”

This research problem is also specific, measurable, and relevant. It is specific because it focuses on a particular type of activity, extracurricular activities, and its impact on academic performance. It is measurable because the researcher can collect data on students’ participation in extracurricular activities and their academic performance to evaluate the relationship between the two variables. It is relevant because extracurricular activities are an essential part of the middle school experience, and their impact on academic performance is a topic of interest to educators and parents.

To conduct research on this problem, the researcher could use surveys, interviews, and academic records analysis. The results of the study could provide insights into the relationship between extracurricular activities and academic performance, which could help educators and parents make informed decisions about the types of activities that are most beneficial for middle school students.

Applications of Research Problem

Applications of Research Problem are as follows:

• Academic research: Research problems are used to guide academic research in various fields, including social sciences, natural sciences, humanities, and engineering. Researchers use research problems to identify gaps in knowledge, address theoretical or practical problems, and explore new areas of study.
• Business research : Research problems are used to guide business research, including market research, consumer behavior research, and organizational research. Researchers use research problems to identify business challenges, explore opportunities, and develop strategies for business growth and success.
• Healthcare research : Research problems are used to guide healthcare research, including medical research, clinical research, and health services research. Researchers use research problems to identify healthcare challenges, develop new treatments and interventions, and improve healthcare delivery and outcomes.
• Public policy research : Research problems are used to guide public policy research, including policy analysis, program evaluation, and policy development. Researchers use research problems to identify social issues, assess the effectiveness of existing policies and programs, and develop new policies and programs to address societal challenges.
• Environmental research : Research problems are used to guide environmental research, including environmental science, ecology, and environmental management. Researchers use research problems to identify environmental challenges, assess the impact of human activities on the environment, and develop sustainable solutions to protect the environment.

Purpose of Research Problems

The purpose of research problems is to identify an area of study that requires further investigation and to formulate a clear, concise and specific research question. A research problem defines the specific issue or problem that needs to be addressed and serves as the foundation for the research project.

Identifying a research problem is important because it helps to establish the direction of the research and sets the stage for the research design, methods, and analysis. It also ensures that the research is relevant and contributes to the existing body of knowledge in the field.

A well-formulated research problem should:

• Clearly define the specific issue or problem that needs to be investigated
• Be specific and narrow enough to be manageable in terms of time, resources, and scope
• Be relevant to the field of study and contribute to the existing body of knowledge
• Be feasible and realistic in terms of available data, resources, and research methods
• Be interesting and intellectually stimulating for the researcher and potential readers or audiences.

Characteristics of Research Problem

The characteristics of a research problem refer to the specific features that a problem must possess to qualify as a suitable research topic. Some of the key characteristics of a research problem are:

• Clarity : A research problem should be clearly defined and stated in a way that it is easily understood by the researcher and other readers. The problem should be specific, unambiguous, and easy to comprehend.
• Relevance : A research problem should be relevant to the field of study, and it should contribute to the existing body of knowledge. The problem should address a gap in knowledge, a theoretical or practical problem, or a real-world issue that requires further investigation.
• Feasibility : A research problem should be feasible in terms of the availability of data, resources, and research methods. It should be realistic and practical to conduct the study within the available time, budget, and resources.
• Novelty : A research problem should be novel or original in some way. It should represent a new or innovative perspective on an existing problem, or it should explore a new area of study or apply an existing theory to a new context.
• Importance : A research problem should be important or significant in terms of its potential impact on the field or society. It should have the potential to produce new knowledge, advance existing theories, or address a pressing societal issue.
• Manageability : A research problem should be manageable in terms of its scope and complexity. It should be specific enough to be investigated within the available time and resources, and it should be broad enough to provide meaningful results.

Advantages of Research Problem

The advantages of a well-defined research problem are as follows:

• Focus : A research problem provides a clear and focused direction for the research study. It ensures that the study stays on track and does not deviate from the research question.
• Clarity : A research problem provides clarity and specificity to the research question. It ensures that the research is not too broad or too narrow and that the research objectives are clearly defined.
• Relevance : A research problem ensures that the research study is relevant to the field of study and contributes to the existing body of knowledge. It addresses gaps in knowledge, theoretical or practical problems, or real-world issues that require further investigation.
• Feasibility : A research problem ensures that the research study is feasible in terms of the availability of data, resources, and research methods. It ensures that the research is realistic and practical to conduct within the available time, budget, and resources.
• Novelty : A research problem ensures that the research study is original and innovative. It represents a new or unique perspective on an existing problem, explores a new area of study, or applies an existing theory to a new context.
• Importance : A research problem ensures that the research study is important and significant in terms of its potential impact on the field or society. It has the potential to produce new knowledge, advance existing theories, or address a pressing societal issue.
• Rigor : A research problem ensures that the research study is rigorous and follows established research methods and practices. It ensures that the research is conducted in a systematic, objective, and unbiased manner.

About the author

Muhammad Hassan

Researcher, Academic Writer, Web developer

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40 problem-solving techniques and processes

All teams and organizations encounter challenges. Approaching those challenges without a structured problem solving process can end up making things worse.

Proven problem solving techniques such as those outlined below can guide your group through a process of identifying problems and challenges , ideating on possible solutions , and then evaluating and implementing the most suitable .

In this post, you'll find problem-solving tools you can use to develop effective solutions. You'll also find some tips for facilitating the problem solving process and solving complex problems.

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What is problem solving?

Problem solving is a process of finding and implementing a solution to a challenge or obstacle. In most contexts, this means going through a problem solving process that begins with identifying the issue, exploring its root causes, ideating and refining possible solutions before implementing and measuring the impact of that solution.

For simple or small problems, it can be tempting to skip straight to implementing what you believe is the right solution. The danger with this approach is that without exploring the true causes of the issue, it might just occur again or your chosen solution may cause other issues.

Particularly in the world of work, good problem solving means using data to back up each step of the process, bringing in new perspectives and effectively measuring the impact of your solution.

Effective problem solving can help ensure that your team or organization is well positioned to overcome challenges, be resilient to change and create innovation. In my experience, problem solving is a combination of skillset, mindset and process, and it’s especially vital for leaders to cultivate this skill.

What is the seven step problem solving process?

A problem solving process is a step-by-step framework from going from discovering a problem all the way through to implementing a solution.

With practice, this framework can become intuitive, and innovative companies tend to have a consistent and ongoing ability to discover and tackle challenges when they come up.

You might see everything from a four step problem solving process through to seven steps. While all these processes cover roughly the same ground, I’ve found a seven step problem solving process is helpful for making all key steps legible.

We’ll outline that process here and then follow with techniques you can use to explore and work on that step of the problem solving process with a group.

The seven-step problem solving process is:

1. Problem identification 

The first stage of any problem solving process is to identify the problem(s) you need to solve. This often looks like using group discussions and activities to help a group surface and effectively articulate the challenges they’re facing and wish to resolve.

Be sure to align with your team on the exact definition and nature of the problem you’re solving. An effective process is one where everyone is pulling in the same direction – ensure clarity and alignment now to help avoid misunderstandings later.

2. Problem analysis and refinement

The process of problem analysis means ensuring that the problem you are seeking to solve is  the   right problem . Choosing the right problem to solve means you are on the right path to creating the right solution.

At this stage, you may look deeper at the problem you identified to try and discover the root cause at the level of people or process. You may also spend some time sourcing data, consulting relevant parties and creating and refining a problem statement.

Problem refinement means adjusting scope or focus of the problem you will be aiming to solve based on what comes up during your analysis. As you analyze data sources, you might discover that the root cause means you need to adjust your problem statement. Alternatively, you might find that your original problem statement is too big to be meaningful approached within your current project.

Remember that the goal of any problem refinement is to help set the stage for effective solution development and deployment. Set the right focus and get buy-in from your team here and you’ll be well positioned to move forward with confidence.

3. Solution generation

Once your group has nailed down the particulars of the problem you wish to solve, you want to encourage a free flow of ideas connecting to solving that problem. This can take the form of problem solving games that encourage creative thinking or techniquess designed to produce working prototypes of possible solutions. 

The key to ensuring the success of this stage of the problem solving process is to encourage quick, creative thinking and create an open space where all ideas are considered. The best solutions can often come from unlikely places and by using problem solving techniques that celebrate invention, you might come up with solution gold. 

4. Solution development

No solution is perfect right out of the gate. It’s important to discuss and develop the solutions your group has come up with over the course of following the previous problem solving steps in order to arrive at the best possible solution. Problem solving games used in this stage involve lots of critical thinking, measuring potential effort and impact, and looking at possible solutions analytically. 

During this stage, you will often ask your team to iterate and improve upon your front-running solutions and develop them further. Remember that problem solving strategies always benefit from a multitude of voices and opinions, and not to let ego get involved when it comes to choosing which solutions to develop and take further.

Finding the best solution is the goal of all problem solving workshops and here is the place to ensure that your solution is well thought out, sufficiently robust and fit for purpose. 

5. Decision making and planning

Nearly there! Once you’ve got a set of possible, you’ll need to make a decision on which to implement. This can be a consensus-based group decision or it might be for a leader or major stakeholder to decide. You’ll find a set of effective decision making methods below.

Once your group has reached consensus and selected a solution, there are some additional actions that also need to be decided upon. You’ll want to work on allocating ownership of the project, figure out who will do what, how the success of the solution will be measured and decide the next course of action.

Set clear accountabilities, actions, timeframes, and follow-ups for your chosen solution. Make these decisions and set clear next-steps in the problem solving workshop so that everyone is aligned and you can move forward effectively as a group. 

Ensuring that you plan for the roll-out of a solution is one of the most important problem solving steps. Without adequate planning or oversight, it can prove impossible to measure success or iterate further if the problem was not solved. 

6. Solution implementation 

This is what we were waiting for! All problem solving processes have the end goal of implementing an effective and impactful solution that your group has confidence in.

Project management and communication skills are key here – your solution may need to adjust when out in the wild or you might discover new challenges along the way. For some solutions, you might also implement a test with a small group and monitor results before rolling it out to an entire company.

You should have a clear owner for your solution who will oversee the plans you made together and help ensure they’re put into place. This person will often coordinate the implementation team and set-up processes to measure the efficacy of your solution too.

7. Solution evaluation 

So you and your team developed a great solution to a problem and have a gut feeling it’s been solved. Work done, right? Wrong. All problem solving strategies benefit from evaluation, consideration, and feedback.

You might find that the solution does not work for everyone, might create new problems, or is potentially so successful that you will want to roll it out to larger teams or as part of other initiatives. 

None of that is possible without taking the time to evaluate the success of the solution you developed in your problem solving model and adjust if necessary.

Remember that the problem solving process is often iterative and it can be common to not solve complex issues on the first try. Even when this is the case, you and your team will have generated learning that will be important for future problem solving workshops or in other parts of the organization. 

It’s also worth underlining how important record keeping is throughout the problem solving process. If a solution didn’t work, you need to have the data and records to see why that was the case. If you go back to the drawing board, notes from the previous workshop can help save time.

What does an effective problem solving process look like?

Every effective problem solving process begins with an agenda . In our experience, a well-structured problem solving workshop is one of the best methods for successfully guiding a group from exploring a problem to implementing a solution.

The format of a workshop ensures that you can get buy-in from your group, encourage free-thinking and solution exploration before making a decision on what to implement following the session.

This Design Sprint 2.0 template is an effective problem solving process from top agency AJ&Smart. It’s a great format for the entire problem solving process, with four-days of workshops designed to surface issues, explore solutions and even test a solution.

Check it for an example of how you might structure and run a problem solving process and feel free to copy and adjust it your needs!

For a shorter process you can run in a single afternoon, this remote problem solving agenda will guide you effectively in just a couple of hours.

Whatever the length of your workshop, by using SessionLab, it’s easy to go from an idea to a complete agenda . Start by dragging and dropping your core problem solving activities into place . Add timings, breaks and necessary materials before sharing your agenda with your colleagues.

The resulting agenda will be your guide to an effective and productive problem solving session that will also help you stay organized on the day!

Complete problem-solving methods

In this section, we’ll look at in-depth problem-solving methods that provide a complete end-to-end process for developing effective solutions. These will help guide your team from the discovery and definition of a problem through to delivering the right solution.

If you’re looking for an all-encompassing method or problem-solving model, these processes are a great place to start. They’ll ask your team to challenge preconceived ideas and adopt a mindset for solving problems more effectively.

Six Thinking Hats

Individual approaches to solving a problem can be very different based on what team or role an individual holds. It can be easy for existing biases or perspectives to find their way into the mix, or for internal politics to direct a conversation.

Six Thinking Hats is a classic method for identifying the problems that need to be solved and enables your team to consider them from different angles, whether that is by focusing on facts and data, creative solutions, or by considering why a particular solution might not work.

Like all problem-solving frameworks, Six Thinking Hats is effective at helping teams remove roadblocks from a conversation or discussion and come to terms with all the aspects necessary to solve complex problems.

The Six Thinking Hats   #creative thinking   #meeting facilitation   #problem solving   #issue resolution   #idea generation   #conflict resolution   The Six Thinking Hats are used by individuals and groups to separate out conflicting styles of thinking. They enable and encourage a group of people to think constructively together in exploring and implementing change, rather than using argument to fight over who is right and who is wrong.

Lightning Decision Jam

Featured courtesy of Jonathan Courtney of AJ&Smart Berlin, Lightning Decision Jam is one of those strategies that should be in every facilitation toolbox. Exploring problems and finding solutions is often creative in nature, though as with any creative process, there is the potential to lose focus and get lost.

Unstructured discussions might get you there in the end, but it’s much more effective to use a method that creates a clear process and team focus.

In Lightning Decision Jam, participants are invited to begin by writing challenges, concerns, or mistakes on post-its without discussing them before then being invited by the moderator to present them to the group.

From there, the team vote on which problems to solve and are guided through steps that will allow them to reframe those problems, create solutions and then decide what to execute on. 

By deciding the problems that need to be solved as a team before moving on, this group process is great for ensuring the whole team is aligned and can take ownership over the next stages. 

Lightning Decision Jam (LDJ)   #action   #decision making   #problem solving   #issue analysis   #innovation   #design   #remote-friendly   It doesn’t matter where you work and what your job role is, if you work with other people together as a team, you will always encounter the same challenges: Unclear goals and miscommunication that cause busy work and overtime Unstructured meetings that leave attendants tired, confused and without clear outcomes. Frustration builds up because internal challenges to productivity are not addressed Sudden changes in priorities lead to a loss of focus and momentum Muddled compromise takes the place of clear decision- making, leaving everybody to come up with their own interpretation. In short, a lack of structure leads to a waste of time and effort, projects that drag on for too long and frustrated, burnt out teams. AJ&Smart has worked with some of the most innovative, productive companies in the world. What sets their teams apart from others is not better tools, bigger talent or more beautiful offices. The secret sauce to becoming a more productive, more creative and happier team is simple: Replace all open discussion or brainstorming with a structured process that leads to more ideas, clearer decisions and better outcomes. When a good process provides guardrails and a clear path to follow, it becomes easier to come up with ideas, make decisions and solve problems. This is why AJ&Smart created Lightning Decision Jam (LDJ). It’s a simple and short, but powerful group exercise that can be run either in-person, in the same room, or remotely with distributed teams.

Problem Definition Process

While problems can be complex, the problem-solving methods you use to identify and solve those problems can often be simple in design. 

By taking the time to truly identify and define a problem before asking the group to reframe the challenge as an opportunity, this method is a great way to enable change.

Begin by identifying a focus question and exploring the ways in which it manifests before splitting into five teams who will each consider the problem using a different method: escape, reversal, exaggeration, distortion or wishful. Teams develop a problem objective and create ideas in line with their method before then feeding them back to the group.

This method is great for enabling in-depth discussions while also creating space for finding creative solutions too!

Problem Definition   #problem solving   #idea generation   #creativity   #online   #remote-friendly   A problem solving technique to define a problem, challenge or opportunity and to generate ideas.

The 5 Whys 

Sometimes, a group needs to go further with their strategies and analyze the root cause at the heart of organizational issues. An RCA or root cause analysis is the process of identifying what is at the heart of business problems or recurring challenges. 

The 5 Whys is a simple and effective method of helping a group go find the root cause of any problem or challenge and conduct analysis that will deliver results. 

By beginning with the creation of a problem statement and going through five stages to refine it, The 5 Whys provides everything you need to truly discover the cause of an issue.

The 5 Whys   #hyperisland   #innovation   This simple and powerful method is useful for getting to the core of a problem or challenge. As the title suggests, the group defines a problems, then asks the question “why” five times, often using the resulting explanation as a starting point for creative problem solving.

World Cafe is a simple but powerful facilitation technique to help bigger groups to focus their energy and attention on solving complex problems.

World Cafe enables this approach by creating a relaxed atmosphere where participants are able to self-organize and explore topics relevant and important to them which are themed around a central problem-solving purpose. Create the right atmosphere by modeling your space after a cafe and after guiding the group through the method, let them take the lead!

Making problem-solving a part of your organization’s culture in the long term can be a difficult undertaking. More approachable formats like World Cafe can be especially effective in bringing people unfamiliar with workshops into the fold. 

World Cafe   #hyperisland   #innovation   #issue analysis   World Café is a simple yet powerful method, originated by Juanita Brown, for enabling meaningful conversations driven completely by participants and the topics that are relevant and important to them. Facilitators create a cafe-style space and provide simple guidelines. Participants then self-organize and explore a set of relevant topics or questions for conversation.

Discovery & Action Dialogue (DAD)

One of the best approaches is to create a safe space for a group to share and discover practices and behaviors that can help them find their own solutions.

With DAD, you can help a group choose which problems they wish to solve and which approaches they will take to do so. It’s great at helping remove resistance to change and can help get buy-in at every level too!

This process of enabling frontline ownership is great in ensuring follow-through and is one of the methods you will want in your toolbox as a facilitator.

Discovery & Action Dialogue (DAD)   #idea generation   #liberating structures   #action   #issue analysis   #remote-friendly   DADs make it easy for a group or community to discover practices and behaviors that enable some individuals (without access to special resources and facing the same constraints) to find better solutions than their peers to common problems. These are called positive deviant (PD) behaviors and practices. DADs make it possible for people in the group, unit, or community to discover by themselves these PD practices. DADs also create favorable conditions for stimulating participants’ creativity in spaces where they can feel safe to invent new and more effective practices. Resistance to change evaporates as participants are unleashed to choose freely which practices they will adopt or try and which problems they will tackle. DADs make it possible to achieve frontline ownership of solutions.

Design Sprint 2.0

Want to see how a team can solve big problems and move forward with prototyping and testing solutions in a few days? The Design Sprint 2.0 template from Jake Knapp, author of Sprint, is a complete agenda for a with proven results.

Developing the right agenda can involve difficult but necessary planning. Ensuring all the correct steps are followed can also be stressful or time-consuming depending on your level of experience.

Use this complete 4-day workshop template if you are finding there is no obvious solution to your challenge and want to focus your team around a specific problem that might require a shortcut to launching a minimum viable product or waiting for the organization-wide implementation of a solution.

Open space technology

Open space technology- developed by Harrison Owen – creates a space where large groups are invited to take ownership of their problem solving and lead individual sessions. Open space technology is a great format when you have a great deal of expertise and insight in the room and want to allow for different takes and approaches on a particular theme or problem you need to be solved.

Start by bringing your participants together to align around a central theme and focus their efforts. Explain the ground rules to help guide the problem-solving process and then invite members to identify any issue connecting to the central theme that they are interested in and are prepared to take responsibility for.

Once participants have decided on their approach to the core theme, they write their issue on a piece of paper, announce it to the group, pick a session time and place, and post the paper on the wall. As the wall fills up with sessions, the group is then invited to join the sessions that interest them the most and which they can contribute to, then you’re ready to begin!

Everyone joins the problem-solving group they’ve signed up to, record the discussion and if appropriate, findings can then be shared with the rest of the group afterward.

Open Space Technology   #action plan   #idea generation   #problem solving   #issue analysis   #large group   #online   #remote-friendly   Open Space is a methodology for large groups to create their agenda discerning important topics for discussion, suitable for conferences, community gatherings and whole system facilitation

Techniques to identify and analyze problems

Using a problem-solving method to help a team identify and analyze a problem can be a quick and effective addition to any workshop or meeting.

While further actions are always necessary, you can generate momentum and alignment easily, and these activities are a great place to get started.

We’ve put together this list of techniques to help you and your team with problem identification, analysis, and discussion that sets the foundation for developing effective solutions.

Let’s take a look!

Fishbone Analysis

Organizational or team challenges are rarely simple, and it’s important to remember that one problem can be an indication of something that goes deeper and may require further consideration to be solved.

Fishbone Analysis helps groups to dig deeper and understand the origins of a problem. It’s a great example of a root cause analysis method that is simple for everyone on a team to get their head around. 

Participants in this activity are asked to annotate a diagram of a fish, first adding the problem or issue to be worked on at the head of a fish before then brainstorming the root causes of the problem and adding them as bones on the fish. 

Using abstractions such as a diagram of a fish can really help a team break out of their regular thinking and develop a creative approach.

Fishbone Analysis   #problem solving   ##root cause analysis   #decision making   #online facilitation   A process to help identify and understand the origins of problems, issues or observations.

Problem Tree 

Encouraging visual thinking can be an essential part of many strategies. By simply reframing and clarifying problems, a group can move towards developing a problem solving model that works for them. 

In Problem Tree, groups are asked to first brainstorm a list of problems – these can be design problems, team problems or larger business problems – and then organize them into a hierarchy. The hierarchy could be from most important to least important or abstract to practical, though the key thing with problem solving games that involve this aspect is that your group has some way of managing and sorting all the issues that are raised.

Once you have a list of problems that need to be solved and have organized them accordingly, you’re then well-positioned for the next problem solving steps.

Problem tree   #define intentions   #create   #design   #issue analysis   A problem tree is a tool to clarify the hierarchy of problems addressed by the team within a design project; it represents high level problems or related sublevel problems.

SWOT Analysis

Chances are you’ve heard of the SWOT Analysis before. This problem-solving method focuses on identifying strengths, weaknesses, opportunities, and threats is a tried and tested method for both individuals and teams.

Start by creating a desired end state or outcome and bare this in mind – any process solving model is made more effective by knowing what you are moving towards. Create a quadrant made up of the four categories of a SWOT analysis and ask participants to generate ideas based on each of those quadrants.

Once you have those ideas assembled in their quadrants, cluster them together based on their affinity with other ideas. These clusters are then used to facilitate group conversations and move things forward. 

SWOT analysis   #gamestorming   #problem solving   #action   #meeting facilitation   The SWOT Analysis is a long-standing technique of looking at what we have, with respect to the desired end state, as well as what we could improve on. It gives us an opportunity to gauge approaching opportunities and dangers, and assess the seriousness of the conditions that affect our future. When we understand those conditions, we can influence what comes next.

Agreement-Certainty Matrix

Not every problem-solving approach is right for every challenge, and deciding on the right method for the challenge at hand is a key part of being an effective team.

The Agreement Certainty matrix helps teams align on the nature of the challenges facing them. By sorting problems from simple to chaotic, your team can understand what methods are suitable for each problem and what they can do to ensure effective results. 

If you are already using Liberating Structures techniques as part of your problem-solving strategy, the Agreement-Certainty Matrix can be an invaluable addition to your process. We’ve found it particularly if you are having issues with recurring problems in your organization and want to go deeper in understanding the root cause. 

Agreement-Certainty Matrix   #issue analysis   #liberating structures   #problem solving   You can help individuals or groups avoid the frequent mistake of trying to solve a problem with methods that are not adapted to the nature of their challenge. The combination of two questions makes it possible to easily sort challenges into four categories: simple, complicated, complex , and chaotic .  A problem is simple when it can be solved reliably with practices that are easy to duplicate.  It is complicated when experts are required to devise a sophisticated solution that will yield the desired results predictably.  A problem is complex when there are several valid ways to proceed but outcomes are not predictable in detail.  Chaotic is when the context is too turbulent to identify a path forward.  A loose analogy may be used to describe these differences: simple is like following a recipe, complicated like sending a rocket to the moon, complex like raising a child, and chaotic is like the game “Pin the Tail on the Donkey.”  The Liberating Structures Matching Matrix in Chapter 5 can be used as the first step to clarify the nature of a challenge and avoid the mismatches between problems and solutions that are frequently at the root of chronic, recurring problems.

Organizing and charting a team’s progress can be important in ensuring its success. SQUID (Sequential Question and Insight Diagram) is a great model that allows a team to effectively switch between giving questions and answers and develop the skills they need to stay on track throughout the process. 

Begin with two different colored sticky notes – one for questions and one for answers – and with your central topic (the head of the squid) on the board. Ask the group to first come up with a series of questions connected to their best guess of how to approach the topic. Ask the group to come up with answers to those questions, fix them to the board and connect them with a line. After some discussion, go back to question mode by responding to the generated answers or other points on the board.

It’s rewarding to see a diagram grow throughout the exercise, and a completed SQUID can provide a visual resource for future effort and as an example for other teams.

SQUID   #gamestorming   #project planning   #issue analysis   #problem solving   When exploring an information space, it’s important for a group to know where they are at any given time. By using SQUID, a group charts out the territory as they go and can navigate accordingly. SQUID stands for Sequential Question and Insight Diagram.

To continue with our nautical theme, Speed Boat is a short and sweet activity that can help a team quickly identify what employees, clients or service users might have a problem with and analyze what might be standing in the way of achieving a solution.

Methods that allow for a group to make observations, have insights and obtain those eureka moments quickly are invaluable when trying to solve complex problems.

In Speed Boat, the approach is to first consider what anchors and challenges might be holding an organization (or boat) back. Bonus points if you are able to identify any sharks in the water and develop ideas that can also deal with competitors!   

Speed Boat   #gamestorming   #problem solving   #action   Speedboat is a short and sweet way to identify what your employees or clients don’t like about your product/service or what’s standing in the way of a desired goal.

The Journalistic Six

Some of the most effective ways of solving problems is by encouraging teams to be more inclusive and diverse in their thinking.

Based on the six key questions journalism students are taught to answer in articles and news stories, The Journalistic Six helps create teams to see the whole picture. By using who, what, when, where, why, and how to facilitate the conversation and encourage creative thinking, your team can make sure that the problem identification and problem analysis stages of the are covered exhaustively and thoughtfully. Reporter’s notebook and dictaphone optional.

The Journalistic Six – Who What When Where Why How   #idea generation   #issue analysis   #problem solving   #online   #creative thinking   #remote-friendly   A questioning method for generating, explaining, investigating ideas.

Individual and group perspectives are incredibly important, but what happens if people are set in their minds and need a change of perspective in order to approach a problem more effectively?

Flip It is a method we love because it is both simple to understand and run, and allows groups to understand how their perspectives and biases are formed. 

Participants in Flip It are first invited to consider concerns, issues, or problems from a perspective of fear and write them on a flip chart. Then, the group is asked to consider those same issues from a perspective of hope and flip their understanding.  

No problem and solution is free from existing bias and by changing perspectives with Flip It, you can then develop a problem solving model quickly and effectively.

Flip It!   #gamestorming   #problem solving   #action   Often, a change in a problem or situation comes simply from a change in our perspectives. Flip It! is a quick game designed to show players that perspectives are made, not born.

LEGO Challenge

Now for an activity that is a little out of the (toy) box. LEGO Serious Play is a facilitation methodology that can be used to improve creative thinking and problem-solving skills. 

The LEGO Challenge includes giving each member of the team an assignment that is hidden from the rest of the group while they create a structure without speaking.

What the LEGO challenge brings to the table is a fun working example of working with stakeholders who might not be on the same page to solve problems. Also, it’s LEGO! Who doesn’t love LEGO! 

LEGO Challenge   #hyperisland   #team   A team-building activity in which groups must work together to build a structure out of LEGO, but each individual has a secret “assignment” which makes the collaborative process more challenging. It emphasizes group communication, leadership dynamics, conflict, cooperation, patience and problem solving strategy.

What, So What, Now What?

If not carefully managed, the problem identification and problem analysis stages of the problem-solving process can actually create more problems and misunderstandings.

The What, So What, Now What? problem-solving activity is designed to help collect insights and move forward while also eliminating the possibility of disagreement when it comes to identifying, clarifying, and analyzing organizational or work problems. 

Facilitation is all about bringing groups together so that might work on a shared goal and the best problem-solving strategies ensure that teams are aligned in purpose, if not initially in opinion or insight.

Throughout the three steps of this game, you give everyone on a team to reflect on a problem by asking what happened, why it is important, and what actions should then be taken. 

This can be a great activity for bringing our individual perceptions about a problem or challenge and contextualizing it in a larger group setting. This is one of the most important problem-solving skills you can bring to your organization.

W³ – What, So What, Now What?   #issue analysis   #innovation   #liberating structures   You can help groups reflect on a shared experience in a way that builds understanding and spurs coordinated action while avoiding unproductive conflict. It is possible for every voice to be heard while simultaneously sifting for insights and shaping new direction. Progressing in stages makes this practical—from collecting facts about What Happened to making sense of these facts with So What and finally to what actions logically follow with Now What . The shared progression eliminates most of the misunderstandings that otherwise fuel disagreements about what to do. Voila!

Journalists  

Problem analysis can be one of the most important and decisive stages of all problem-solving tools. Sometimes, a team can become bogged down in the details and are unable to move forward.

Journalists is an activity that can avoid a group from getting stuck in the problem identification or problem analysis stages of the process.

In Journalists, the group is invited to draft the front page of a fictional newspaper and figure out what stories deserve to be on the cover and what headlines those stories will have. By reframing how your problems and challenges are approached, you can help a team move productively through the process and be better prepared for the steps to follow.

Journalists   #vision   #big picture   #issue analysis   #remote-friendly   This is an exercise to use when the group gets stuck in details and struggles to see the big picture. Also good for defining a vision.

Problem-solving techniques for brainstorming solutions

Now you have the context and background of the problem you are trying to solving, now comes the time to start ideating and thinking about how you’ll solve the issue.

Here, you’ll want to encourage creative, free thinking and speed. Get as many ideas out as possible and explore different perspectives so you have the raw material for the next step.

Looking at a problem from a new angle can be one of the most effective ways of creating an effective solution. TRIZ is a problem-solving tool that asks the group to consider what they must not do in order to solve a challenge.

By reversing the discussion, new topics and taboo subjects often emerge, allowing the group to think more deeply and create ideas that confront the status quo in a safe and meaningful way. If you’re working on a problem that you’ve tried to solve before, TRIZ is a great problem-solving method to help your team get unblocked.

Making Space with TRIZ   #issue analysis   #liberating structures   #issue resolution   You can clear space for innovation by helping a group let go of what it knows (but rarely admits) limits its success and by inviting creative destruction. TRIZ makes it possible to challenge sacred cows safely and encourages heretical thinking. The question “What must we stop doing to make progress on our deepest purpose?” induces seriously fun yet very courageous conversations. Since laughter often erupts, issues that are otherwise taboo get a chance to be aired and confronted. With creative destruction come opportunities for renewal as local action and innovation rush in to fill the vacuum. Whoosh!

Mindspin  

Brainstorming is part of the bread and butter of the problem-solving process and all problem-solving strategies benefit from getting ideas out and challenging a team to generate solutions quickly. 

With Mindspin, participants are encouraged not only to generate ideas but to do so under time constraints and by slamming down cards and passing them on. By doing multiple rounds, your team can begin with a free generation of possible solutions before moving on to developing those solutions and encouraging further ideation. 

This is one of our favorite problem-solving activities and can be great for keeping the energy up throughout the workshop. Remember the importance of helping people become engaged in the process – energizing problem-solving techniques like Mindspin can help ensure your team stays engaged and happy, even when the problems they’re coming together to solve are complex. 

MindSpin   #teampedia   #idea generation   #problem solving   #action   A fast and loud method to enhance brainstorming within a team. Since this activity has more than round ideas that are repetitive can be ruled out leaving more creative and innovative answers to the challenge.

The Creativity Dice

One of the most useful problem solving skills you can teach your team is of approaching challenges with creativity, flexibility, and openness. Games like The Creativity Dice allow teams to overcome the potential hurdle of too much linear thinking and approach the process with a sense of fun and speed. 

In The Creativity Dice, participants are organized around a topic and roll a dice to determine what they will work on for a period of 3 minutes at a time. They might roll a 3 and work on investigating factual information on the chosen topic. They might roll a 1 and work on identifying the specific goals, standards, or criteria for the session.

Encouraging rapid work and iteration while asking participants to be flexible are great skills to cultivate. Having a stage for idea incubation in this game is also important. Moments of pause can help ensure the ideas that are put forward are the most suitable. 

The Creativity Dice   #creativity   #problem solving   #thiagi   #issue analysis   Too much linear thinking is hazardous to creative problem solving. To be creative, you should approach the problem (or the opportunity) from different points of view. You should leave a thought hanging in mid-air and move to another. This skipping around prevents premature closure and lets your brain incubate one line of thought while you consciously pursue another.

Idea and Concept Development

Brainstorming without structure can quickly become chaotic or frustrating. In a problem-solving context, having an ideation framework to follow can help ensure your team is both creative and disciplined.

In this method, you’ll find an idea generation process that encourages your group to brainstorm effectively before developing their ideas and begin clustering them together. By using concepts such as Yes and…, more is more and postponing judgement, you can create the ideal conditions for brainstorming with ease.

Idea & Concept Development   #hyperisland   #innovation   #idea generation   Ideation and Concept Development is a process for groups to work creatively and collaboratively to generate creative ideas. It’s a general approach that can be adapted and customized to suit many different scenarios. It includes basic principles for idea generation and several steps for groups to work with. It also includes steps for idea selection and development.

Problem-solving techniques for developing and refining solutions 

The success of any problem-solving process can be measured by the solutions it produces. After you’ve defined the issue, explored existing ideas, and ideated, it’s time to develop and refine your ideas in order to bring them closer to a solution that actually solves the problem.

Use these problem-solving techniques when you want to help your team think through their ideas and refine them as part of your problem solving process.

Improved Solutions

After a team has successfully identified a problem and come up with a few solutions, it can be tempting to call the work of the problem-solving process complete. That said, the first solution is not necessarily the best, and by including a further review and reflection activity into your problem-solving model, you can ensure your group reaches the best possible result. 

One of a number of problem-solving games from Thiagi Group, Improved Solutions helps you go the extra mile and develop suggested solutions with close consideration and peer review. By supporting the discussion of several problems at once and by shifting team roles throughout, this problem-solving technique is a dynamic way of finding the best solution. 

Improved Solutions   #creativity   #thiagi   #problem solving   #action   #team   You can improve any solution by objectively reviewing its strengths and weaknesses and making suitable adjustments. In this creativity framegame, you improve the solutions to several problems. To maintain objective detachment, you deal with a different problem during each of six rounds and assume different roles (problem owner, consultant, basher, booster, enhancer, and evaluator) during each round. At the conclusion of the activity, each player ends up with two solutions to her problem.

Four Step Sketch

Creative thinking and visual ideation does not need to be confined to the opening stages of your problem-solving strategies. Exercises that include sketching and prototyping on paper can be effective at the solution finding and development stage of the process, and can be great for keeping a team engaged. 

By going from simple notes to a crazy 8s round that involves rapidly sketching 8 variations on their ideas before then producing a final solution sketch, the group is able to iterate quickly and visually. Problem-solving techniques like Four-Step Sketch are great if you have a group of different thinkers and want to change things up from a more textual or discussion-based approach.

Four-Step Sketch   #design sprint   #innovation   #idea generation   #remote-friendly   The four-step sketch is an exercise that helps people to create well-formed concepts through a structured process that includes: Review key information Start design work on paper,  Consider multiple variations , Create a detailed solution . This exercise is preceded by a set of other activities allowing the group to clarify the challenge they want to solve. See how the Four Step Sketch exercise fits into a Design Sprint

Ensuring that everyone in a group is able to contribute to a discussion is vital during any problem solving process. Not only does this ensure all bases are covered, but its then easier to get buy-in and accountability when people have been able to contribute to the process.

1-2-4-All is a tried and tested facilitation technique where participants are asked to first brainstorm on a topic on their own. Next, they discuss and share ideas in a pair before moving into a small group. Those groups are then asked to present the best idea from their discussion to the rest of the team.

This method can be used in many different contexts effectively, though I find it particularly shines in the idea development stage of the process. Giving each participant time to concretize their ideas and develop them in progressively larger groups can create a great space for both innovation and psychological safety.

1-2-4-All   #idea generation   #liberating structures   #issue analysis   With this facilitation technique you can immediately include everyone regardless of how large the group is. You can generate better ideas and more of them faster than ever before. You can tap the know-how and imagination that is distributed widely in places not known in advance. Open, generative conversation unfolds. Ideas and solutions are sifted in rapid fashion. Most importantly, participants own the ideas, so follow-up and implementation is simplified. No buy-in strategies needed! Simple and elegant!

15% Solutions

Some problems are simpler than others and with the right problem-solving activities, you can empower people to take immediate actions that can help create organizational change. 

Part of the liberating structures toolkit, 15% solutions is a problem-solving technique that focuses on finding and implementing solutions quickly. A process of iterating and making small changes quickly can help generate momentum and an appetite for solving complex problems.

Problem-solving strategies can live and die on whether people are onboard. Getting some quick wins is a great way of getting people behind the process.   

It can be extremely empowering for a team to realize that problem-solving techniques can be deployed quickly and easily and delineate between things they can positively impact and those things they cannot change. 

15% Solutions   #action   #liberating structures   #remote-friendly   You can reveal the actions, however small, that everyone can do immediately. At a minimum, these will create momentum, and that may make a BIG difference.  15% Solutions show that there is no reason to wait around, feel powerless, or fearful. They help people pick it up a level. They get individuals and the group to focus on what is within their discretion instead of what they cannot change.  With a very simple question, you can flip the conversation to what can be done and find solutions to big problems that are often distributed widely in places not known in advance. Shifting a few grains of sand may trigger a landslide and change the whole landscape.

Problem-solving techniques for making decisions and planning

After your group is happy with the possible solutions you’ve developed, now comes the time to choose which to implement. There’s more than one way to make a decision and the best option is often dependant on the needs and set-up of your group.

Sometimes, it’s the case that you’ll want to vote as a group on what is likely to be the most impactful solution. Other times, it might be down to a decision maker or major stakeholder to make the final decision. Whatever your process, here’s some techniques you can use to help you make a decision during your problem solving process.

How-Now-Wow Matrix

The problem-solving process is often creative, as complex problems usually require a change of thinking and creative response in order to find the best solutions. While it’s common for the first stages to encourage creative thinking, groups can often gravitate to familiar solutions when it comes to the end of the process. 

When selecting solutions, you don’t want to lose your creative energy! The How-Now-Wow Matrix from Gamestorming is a great problem-solving activity that enables a group to stay creative and think out of the box when it comes to selecting the right solution for a given problem.

Problem-solving techniques that encourage creative thinking and the ideation and selection of new solutions can be the most effective in organisational change. Give the How-Now-Wow Matrix a go, and not just for how pleasant it is to say out loud. 

How-Now-Wow Matrix   #gamestorming   #idea generation   #remote-friendly   When people want to develop new ideas, they most often think out of the box in the brainstorming or divergent phase. However, when it comes to convergence, people often end up picking ideas that are most familiar to them. This is called a ‘creative paradox’ or a ‘creadox’. The How-Now-Wow matrix is an idea selection tool that breaks the creadox by forcing people to weigh each idea on 2 parameters.

Impact and Effort Matrix

All problem-solving techniques hope to not only find solutions to a given problem or challenge but to find the best solution. When it comes to finding a solution, groups are invited to put on their decision-making hats and really think about how a proposed idea would work in practice. 

The Impact and Effort Matrix is one of the problem-solving techniques that fall into this camp, empowering participants to first generate ideas and then categorize them into a 2×2 matrix based on impact and effort.

Activities that invite critical thinking while remaining simple are invaluable. Use the Impact and Effort Matrix to move from ideation and towards evaluating potential solutions before then committing to them. 

Impact and Effort Matrix   #gamestorming   #decision making   #action   #remote-friendly   In this decision-making exercise, possible actions are mapped based on two factors: effort required to implement and potential impact. Categorizing ideas along these lines is a useful technique in decision making, as it obliges contributors to balance and evaluate suggested actions before committing to them.

If you’ve followed each of the problem-solving steps with your group successfully, you should move towards the end of your process with heaps of possible solutions developed with a specific problem in mind. But how do you help a group go from ideation to putting a solution into action? 

Dotmocracy – or Dot Voting -is a tried and tested method of helping a team in the problem-solving process make decisions and put actions in place with a degree of oversight and consensus. 

One of the problem-solving techniques that should be in every facilitator’s toolbox, Dot Voting is fast and effective and can help identify the most popular and best solutions and help bring a group to a decision effectively. 

Dotmocracy   #action   #decision making   #group prioritization   #hyperisland   #remote-friendly   Dotmocracy is a simple method for group prioritization or decision-making. It is not an activity on its own, but a method to use in processes where prioritization or decision-making is the aim. The method supports a group to quickly see which options are most popular or relevant. The options or ideas are written on post-its and stuck up on a wall for the whole group to see. Each person votes for the options they think are the strongest, and that information is used to inform a decision.

Straddling the gap between decision making and planning, MoSCoW is a simple and effective method that allows a group team to easily prioritize a set of possible options.

Use this method in a problem solving process by collecting and summarizing all your possible solutions and then categorize them into 4 sections: “Must have”, “Should have”, “Could have”, or “Would like but won‘t get”.

This method is particularly useful when its less about choosing one possible solution and more about prioritorizing which to do first and which may not fit in the scope of your project. In my experience, complex challenges often require multiple small fixes, and this method can be a great way to move from a pile of things you’d all like to do to a structured plan.

MoSCoW   #define intentions   #create   #design   #action   #remote-friendly   MoSCoW is a method that allows the team to prioritize the different features that they will work on. Features are then categorized into “Must have”, “Should have”, “Could have”, or “Would like but won‘t get”. To be used at the beginning of a timeslot (for example during Sprint planning) and when planning is needed.

When it comes to managing the rollout of a solution, clarity and accountability are key factors in ensuring the success of the project. The RAACI chart is a simple but effective model for setting roles and responsibilities as part of a planning session.

Start by listing each person involved in the project and put them into the following groups in order to make it clear who is responsible for what during the rollout of your solution.

• Responsibility  (Which person and/or team will be taking action?)
• Authority  (At what “point” must the responsible person check in before going further?)
• Accountability  (Who must the responsible person check in with?)
• Consultation  (Who must be consulted by the responsible person before decisions are made?)
• Information  (Who must be informed of decisions, once made?)

Ensure this information is easily accessible and use it to inform who does what and who is looped into discussions and kept up to date.

RAACI   #roles and responsibility   #teamwork   #project management   Clarifying roles and responsibilities, levels of autonomy/latitude in decision making, and levels of engagement among diverse stakeholders.

Problem-solving warm-up activities

All facilitators know that warm-ups and icebreakers are useful for any workshop or group process. Problem-solving workshops are no different.

Use these problem-solving techniques to warm up a group and prepare them for the rest of the process. Activating your group by tapping into some of the top problem-solving skills can be one of the best ways to see great outcomes from your session.

Check-in / Check-out

Solid processes are planned from beginning to end, and the best facilitators know that setting the tone and establishing a safe, open environment can be integral to a successful problem-solving process. Check-in / Check-out is a great way to begin and/or bookend a problem-solving workshop. Checking in to a session emphasizes that everyone will be seen, heard, and expected to contribute. 

If you are running a series of meetings, setting a consistent pattern of checking in and checking out can really help your team get into a groove. We recommend this opening-closing activity for small to medium-sized groups though it can work with large groups if they’re disciplined!

Check-in / Check-out   #team   #opening   #closing   #hyperisland   #remote-friendly   Either checking-in or checking-out is a simple way for a team to open or close a process, symbolically and in a collaborative way. Checking-in/out invites each member in a group to be present, seen and heard, and to express a reflection or a feeling. Checking-in emphasizes presence, focus and group commitment; checking-out emphasizes reflection and symbolic closure.

Doodling Together  

Thinking creatively and not being afraid to make suggestions are important problem-solving skills for any group or team, and warming up by encouraging these behaviors is a great way to start. 

Doodling Together is one of our favorite creative ice breaker games – it’s quick, effective, and fun and can make all following problem-solving steps easier by encouraging a group to collaborate visually. By passing cards and adding additional items as they go, the workshop group gets into a groove of co-creation and idea development that is crucial to finding solutions to problems. 

Doodling Together   #collaboration   #creativity   #teamwork   #fun   #team   #visual methods   #energiser   #icebreaker   #remote-friendly   Create wild, weird and often funny postcards together & establish a group’s creative confidence.

Show and Tell

You might remember some version of Show and Tell from being a kid in school and it’s a great problem-solving activity to kick off a session.

Asking participants to prepare a little something before a workshop by bringing an object for show and tell can help them warm up before the session has even begun! Games that include a physical object can also help encourage early engagement before moving onto more big-picture thinking.

By asking your participants to tell stories about why they chose to bring a particular item to the group, you can help teams see things from new perspectives and see both differences and similarities in the way they approach a topic. Great groundwork for approaching a problem-solving process as a team! 

Show and Tell   #gamestorming   #action   #opening   #meeting facilitation   Show and Tell taps into the power of metaphors to reveal players’ underlying assumptions and associations around a topic The aim of the game is to get a deeper understanding of stakeholders’ perspectives on anything—a new project, an organizational restructuring, a shift in the company’s vision or team dynamic.

Constellations

Who doesn’t love stars? Constellations is a great warm-up activity for any workshop as it gets people up off their feet, energized, and ready to engage in new ways with established topics. It’s also great for showing existing beliefs, biases, and patterns that can come into play as part of your session.

Using warm-up games that help build trust and connection while also allowing for non-verbal responses can be great for easing people into the problem-solving process and encouraging engagement from everyone in the group. Constellations is great in large spaces that allow for movement and is definitely a practical exercise to allow the group to see patterns that are otherwise invisible. 

Constellations   #trust   #connection   #opening   #coaching   #patterns   #system   Individuals express their response to a statement or idea by standing closer or further from a central object. Used with teams to reveal system, hidden patterns, perspectives.

Draw a Tree

Problem-solving games that help raise group awareness through a central, unifying metaphor can be effective ways to warm-up a group in any problem-solving model.

Draw a Tree is a simple warm-up activity you can use in any group and which can provide a quick jolt of energy. Start by asking your participants to draw a tree in just 45 seconds – they can choose whether it will be abstract or realistic. 

Once the timer is up, ask the group how many people included the roots of the tree and use this as a means to discuss how we can ignore important parts of any system simply because they are not visible.

All problem-solving strategies are made more effective by thinking of problems critically and by exposing things that may not normally come to light. Warm-up games like Draw a Tree are great in that they quickly demonstrate some key problem-solving skills in an accessible and effective way.

Draw a Tree   #thiagi   #opening   #perspectives   #remote-friendly   With this game you can raise awarness about being more mindful, and aware of the environment we live in.

Closing activities for a problem-solving process

Each step of the problem-solving workshop benefits from an intelligent deployment of activities, games, and techniques. Bringing your session to an effective close helps ensure that solutions are followed through on and that you also celebrate what has been achieved.

Here are some problem-solving activities you can use to effectively close a workshop or meeting and ensure the great work you’ve done can continue afterward.

One Breath Feedback

Maintaining attention and focus during the closing stages of a problem-solving workshop can be tricky and so being concise when giving feedback can be important. It’s easy to incur “death by feedback” should some team members go on for too long sharing their perspectives in a quick feedback round. 

One Breath Feedback is a great closing activity for workshops. You give everyone an opportunity to provide feedback on what they’ve done but only in the space of a single breath. This keeps feedback short and to the point and means that everyone is encouraged to provide the most important piece of feedback to them. 

One breath feedback   #closing   #feedback   #action   This is a feedback round in just one breath that excels in maintaining attention: each participants is able to speak during just one breath … for most people that’s around 20 to 25 seconds … unless of course you’ve been a deep sea diver in which case you’ll be able to do it for longer.

Who What When Matrix 

Matrices feature as part of many effective problem-solving strategies and with good reason. They are easily recognizable, simple to use, and generate results.

The Who What When Matrix is a great tool to use when closing your problem-solving session by attributing a who, what and when to the actions and solutions you have decided upon. The resulting matrix is a simple, easy-to-follow way of ensuring your team can move forward. 

Great solutions can’t be enacted without action and ownership. Your problem-solving process should include a stage for allocating tasks to individuals or teams and creating a realistic timeframe for those solutions to be implemented or checked out. Use this method to keep the solution implementation process clear and simple for all involved. 

Who/What/When Matrix   #gamestorming   #action   #project planning   With Who/What/When matrix, you can connect people with clear actions they have defined and have committed to.

Response cards

Group discussion can comprise the bulk of most problem-solving activities and by the end of the process, you might find that your team is talked out! 

Providing a means for your team to give feedback with short written notes can ensure everyone is head and can contribute without the need to stand up and talk. Depending on the needs of the group, giving an alternative can help ensure everyone can contribute to your problem-solving model in the way that makes the most sense for them.

Response Cards is a great way to close a workshop if you are looking for a gentle warm-down and want to get some swift discussion around some of the feedback that is raised. 

Response Cards   #debriefing   #closing   #structured sharing   #questions and answers   #thiagi   #action   It can be hard to involve everyone during a closing of a session. Some might stay in the background or get unheard because of louder participants. However, with the use of Response Cards, everyone will be involved in providing feedback or clarify questions at the end of a session.

Tips for effective problem solving

Problem-solving activities are only one part of the puzzle. While a great method can help unlock your team’s ability to solve problems, without a thoughtful approach and strong facilitation the solutions may not be fit for purpose.

Let’s take a look at some problem-solving tips you can apply to any process to help it be a success!

Clearly define the problem

Jumping straight to solutions can be tempting, though without first clearly articulating a problem, the solution might not be the right one. Many of the problem-solving activities below include sections where the problem is explored and clearly defined before moving on.

This is a vital part of the problem-solving process and taking the time to fully define an issue can save time and effort later. A clear definition helps identify irrelevant information and it also ensures that your team sets off on the right track.

Don’t jump to conclusions

It’s easy for groups to exhibit cognitive bias or have preconceived ideas about both problems and potential solutions. Be sure to back up any problem statements or potential solutions with facts, research, and adequate forethought.

The best techniques ask participants to be methodical and challenge preconceived notions. Make sure you give the group enough time and space to collect relevant information and consider the problem in a new way. By approaching the process with a clear, rational mindset, you’ll often find that better solutions are more forthcoming.  

Try different approaches  

Problems come in all shapes and sizes and so too should the methods you use to solve them. If you find that one approach isn’t yielding results and your team isn’t finding different solutions, try mixing it up. You’ll be surprised at how using a new creative activity can unblock your team and generate great solutions.

Don’t take it personally 

Depending on the nature of your team or organizational problems, it’s easy for conversations to get heated. While it’s good for participants to be engaged in the discussions, ensure that emotions don’t run too high and that blame isn’t thrown around while finding solutions.

You’re all in it together, and even if your team or area is seeing problems, that isn’t necessarily a disparagement of you personally. Using facilitation skills to manage group dynamics is one effective method of helping conversations be more constructive.

Get the right people in the room

Your problem-solving method is often only as effective as the group using it. Getting the right people on the job and managing the number of people present is important too!

If the group is too small, you may not get enough different perspectives to effectively solve a problem. If the group is too large, you can go round and round during the ideation stages.

Creating the right group makeup is also important in ensuring you have the necessary expertise and skillset to both identify and follow up on potential solutions. Carefully consider who to include at each stage to help ensure your problem-solving method is followed and positioned for success.

Create psychologically safe spaces for discussion

Identifying a problem accurately also requires that all members of a group are able to contribute their views in an open and safe manner.

It can be tough for people to stand up and contribute if the problems or challenges are emotive or personal in nature. Try and create a psychologically safe space for these kinds of discussions and where possible, create regular opportunities for challenges to be brought up organically.

Document everything

The best solutions can take refinement, iteration, and reflection to come out. Get into a habit of documenting your process in order to keep all the learnings from the session and to allow ideas to mature and develop. Many of the methods below involve the creation of documents or shared resources. Be sure to keep and share these so everyone can benefit from the work done!

Bring a facilitator 

Facilitation is all about making group processes easier. With a subject as potentially emotive and important as problem-solving, having an impartial third party in the form of a facilitator can make all the difference in finding great solutions and keeping the process moving. Consider bringing a facilitator to your problem-solving session to get better results and generate meaningful solutions!

Develop your problem-solving skills

It takes time and practice to be an effective problem solver. While some roles or participants might more naturally gravitate towards problem-solving, it can take development and planning to help everyone create better solutions.

You might develop a training program, run a problem-solving workshop or simply ask your team to practice using the techniques below. Check out our post on problem-solving skills to see how you and your group can develop the right mental process and be more resilient to issues too!

Design a great agenda

Workshops are a great format for solving problems. With the right approach, you can focus a group and help them find the solutions to their own problems. But designing a process can be time-consuming and finding the right activities can be difficult.

Check out our workshop planning guide to level-up your agenda design and start running more effective workshops. Need inspiration? Check out templates designed by expert facilitators to help you kickstart your process!

Save time and effort creating an effective problem solving process

A structured problem solving process is a surefire way of solving tough problems, discovering creative solutions and driving organizational change. But how can you design for successful outcomes?

With SessionLab, it’s easy to design engaging workshops that deliver results. Drag, drop and reorder blocks  to build your agenda. When you make changes or update your agenda, your session  timing   adjusts automatically , saving you time on manual adjustments.

Collaborating with stakeholders or clients? Share your agenda with a single click and collaborate in real-time. No more sending documents back and forth over email.

Explore  how to use SessionLab  to design effective problem solving workshops or  watch this five minute video  to see the planner in action!

Over to you

The problem-solving process can often be as complicated and multifaceted as the problems they are set-up to solve. With the right problem-solving techniques and a mix of exercises designed to guide discussion and generate purposeful ideas, we hope we’ve given you the tools to find the best solutions as simply and easily as possible.

Is there a problem-solving technique that you are missing here? Do you have a favorite activity or method you use when facilitating? Let us know in the comments below, we’d love to hear from you! 

James Smart is Head of Content at SessionLab. He’s also a creative facilitator who has run workshops and designed courses for establishments like the National Centre for Writing, UK. He especially enjoys working with young people and empowering others in their creative practice.

thank you very much for these excellent techniques

Certainly wonderful article, very detailed. Shared!

Your list of techniques for problem solving can be helpfully extended by adding TRIZ to the list of techniques. TRIZ has 40 problem solving techniques derived from methods inventros and patent holders used to get new patents. About 10-12 are general approaches. many organization sponsor classes in TRIZ that are used to solve business problems or general organiztational problems. You can take a look at TRIZ and dwonload a free internet booklet to see if you feel it shound be included per your selection process.

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• How to Define a Research Problem | Ideas & Examples

How to Define a Research Problem | Ideas & Examples

Published on November 2, 2022 by Shona McCombes and Tegan George. Revised on May 31, 2023.

A research problem is a specific issue or gap in existing knowledge that you aim to address in your research. You may choose to look for practical problems aimed at contributing to change, or theoretical problems aimed at expanding knowledge.

Some research will do both of these things, but usually the research problem focuses on one or the other. The type of research problem you choose depends on your broad topic of interest and the type of research you think will fit best.

This article helps you identify and refine a research problem. When writing your research proposal or introduction , formulate it as a problem statement and/or research questions .

Table of contents

Why is the research problem important, step 1: identify a broad problem area, step 2: learn more about the problem, other interesting articles, frequently asked questions about research problems.

Having an interesting topic isn’t a strong enough basis for academic research. Without a well-defined research problem, you are likely to end up with an unfocused and unmanageable project.

You might end up repeating what other people have already said, trying to say too much, or doing research without a clear purpose and justification. You need a clear problem in order to do research that contributes new and relevant insights.

Whether you’re planning your thesis , starting a research paper , or writing a research proposal , the research problem is the first step towards knowing exactly what you’ll do and why.

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As you read about your topic, look for under-explored aspects or areas of concern, conflict, or controversy. Your goal is to find a gap that your research project can fill.

Practical research problems

If you are doing practical research, you can identify a problem by reading reports, following up on previous research, or talking to people who work in the relevant field or organization. You might look for:

• Issues with performance or efficiency
• Processes that could be improved
• Areas of concern among practitioners
• Difficulties faced by specific groups of people

Examples of practical research problems

Voter turnout in New England has been decreasing, in contrast to the rest of the country.

The HR department of a local chain of restaurants has a high staff turnover rate.

A non-profit organization faces a funding gap that means some of its programs will have to be cut.

Theoretical research problems

If you are doing theoretical research, you can identify a research problem by reading existing research, theory, and debates on your topic to find a gap in what is currently known about it. You might look for:

• A phenomenon or context that has not been closely studied
• A contradiction between two or more perspectives
• A situation or relationship that is not well understood
• A troubling question that has yet to be resolved

Examples of theoretical research problems

The effects of long-term Vitamin D deficiency on cardiovascular health are not well understood.

The relationship between gender, race, and income inequality has yet to be closely studied in the context of the millennial gig economy.

Historians of Scottish nationalism disagree about the role of the British Empire in the development of Scotland’s national identity.

Next, you have to find out what is already known about the problem, and pinpoint the exact aspect that your research will address.

Context and background

• Who does the problem affect?
• Is it a newly-discovered problem, or a well-established one?
• What research has already been done?
• What, if any, solutions have been proposed?
• What are the current debates about the problem? What is missing from these debates?

Specificity and relevance

• What particular place, time, and/or group of people will you focus on?
• What aspects will you not be able to tackle?
• What will the consequences be if the problem is not resolved?

Example of a specific research problem

A local non-profit organization focused on alleviating food insecurity has always fundraised from its existing support base. It lacks understanding of how best to target potential new donors. To be able to continue its work, the organization requires research into more effective fundraising strategies.

Once you have narrowed down your research problem, the next step is to formulate a problem statement , as well as your research questions or hypotheses .

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If you want to know more about the research process , methodology , research bias , or statistics , make sure to check out some of our other articles with explanations and examples.

Methodology

• Sampling methods
• Simple random sampling
• Stratified sampling
• Cluster sampling
• Likert scales
• Reproducibility

 Statistics

• Null hypothesis
• Statistical power
• Probability distribution
• Effect size
• Poisson distribution

Research bias

• Optimism bias
• Cognitive bias
• Implicit bias
• Hawthorne effect
• Anchoring bias
• Explicit bias

All research questions should be:

• Focused on a single problem or issue
• Researchable using primary and/or secondary sources
• Feasible to answer within the timeframe and practical constraints
• Specific enough to answer thoroughly
• Complex enough to develop the answer over the space of a paper or thesis
• Relevant to your field of study and/or society more broadly

Research questions anchor your whole project, so it’s important to spend some time refining them.

In general, they should be:

• Focused and researchable
• Answerable using credible sources
• Complex and arguable
• Feasible and specific
• Relevant and original

Your research objectives indicate how you’ll try to address your research problem and should be specific:

A research aim is a broad statement indicating the general purpose of your research project. It should appear in your introduction at the end of your problem statement , before your research objectives.

Research objectives are more specific than your research aim. They indicate the specific ways you’ll address the overarching aim.

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• A 5-Step Problem-Solving Strategy (https://academic.cuesta.edu/acasupp/as/407.htm) Appreciate the Complexities Involved in Decision-Making & Problem Solving
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• Analyses of Information Systems Students' Applications of Two Holistic Problem Solving Methodologies. by Musa, P. F., Edmondson, V., & Munchus, G. Journal of Information Systems Education. Winter2005, Vol. 16 Issue 4, p391-408. 18p.
• Breakthrough problem solving with action learning: Concepts and cases by Marquardt, M. J., & Yeo, R. K Publication Date: 2012 eBooks on EBSCOhost
• Decision making and problem solving strategies by Adair, John Publication Date: 2010 Ebrary
• Diagnostic Quality Problem Solving: A Conceptual Framework and Six Strategies. by DE MAST, JEROEN Quality Management Journal. 2013, Vol. 20 Issue 4, p21-36. 16p.
• Problem posing based on investigation activities by university students. by Ponte, J., & Henriques, A. Educational Studies in Mathematics; May2013, Vol. 83 Issue 1, p145-156, 12p, 1 Diagram
• Problem Solving for Teams : Make Consensus More Achievable by Pokras, Sandy Publication Date: 2010 eBooks on EBSCOhost
• Scenario Visualization : An Evolutionary Account of Creative Problem Solving by Arp, Robert Publication Date: 2008 eBooks on EBSCOhost
• Solving Everyday Problems with the Scientific Method : Thinking Like a Scientist by Mak, Don K., Angela T. Mak, and Anthony B. Mak. Publication Date: 2009 eBooks on EBSCOhost
• Solving Problems with Design Thinking : 10 Stories of What Works by Liedtka, J., Bennett, K. B., & King, A. Publication Date: 2013 eBooks on EBSCOhost
• Toward a creative problem-solving methodology with knowledge provision by Zhu, Z., Nagalingam, S., & Hsu, H. Applied Artificial Intelligence. Oct2011, Vol. 25 Issue 9, p836-881. 46p.

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• Table Of Contents

Three essential characteristics of operations research are a systems orientation, the use of interdisciplinary teams, and the application of scientific method to the conditions under which the research is conducted.

The systems approach to problems recognizes that the behaviour of any part of a system has some effect on the behaviour of the system as a whole. Even if the individual components are performing well, however, the system as a whole is not necessarily performing equally well. For example, assembling the best of each type of automobile part, regardless of make, does not necessarily result in a good automobile or even one that will run, because the parts may not fit together. It is the interaction between parts, and not the actions of any single part, that determines how well a system performs.

Thus, operations research attempts to evaluate the effect of changes in any part of a system on the performance of the system as a whole and to search for causes of a problem that arises in one part of a system in other parts or in the interrelationships between parts. In industry, a production problem may be approached by a change in marketing policy. For example, if a factory fabricates a few profitable products in large quantities and many less profitable items in small quantities, long efficient production runs of high-volume, high-profit items may have to be interrupted for short runs of low-volume, low-profit items. An operations researcher might propose reducing the sales of the less profitable items and increasing those of the profitable items by placing salesmen on an incentive system that especially compensates them for selling particular items.

Scientific and technological disciplines have proliferated rapidly in the last 100 years. The proliferation, resulting from the enormous increase in scientific knowledge, has provided science with a filing system that permits a systematic classification of knowledge. This classification system is helpful in solving many problems by identifying the proper discipline to appeal to for a solution. Difficulties arise when more complex problems, such as those arising in large organized systems, are encountered. It is then necessary to find a means of bringing together diverse disciplinary points of view. Furthermore, since methods differ among disciplines, the use of interdisciplinary teams makes available a much larger arsenal of research techniques and tools than would otherwise be available. Hence, operations research may be characterized by rather unusual combinations of disciplines on research teams and by the use of varied research procedures.

Until the 20th century, laboratory experiments were the principal and almost the only method of conducting scientific research. But large systems such as are studied in operations research cannot be brought into laboratories. Furthermore, even if systems could be brought into the laboratory, what would be learned would not necessarily apply to their behaviour in their natural environment , as shown by early experience with radar. Experiments on systems and subsystems conducted in their natural environment (“operational experiments”) are possible as a result of the experimental methods developed by the British statistician R.A. Fisher in 1923–24. For practical or even ethical reasons, however, it is seldom possible to experiment on large organized systems as a whole in their natural environments . This results in an apparent dilemma: to gain understanding of complex systems experimentation seems to be necessary, but it cannot usually be carried out. This difficulty is solved by the use of models , representations of the system under study. Provided the model is good, experiments (called “simulations”) can be conducted on it, or other methods can be used to obtain useful results.

Phases of operations research

To formulate an operations research problem, a suitable measure of performance must be devised, various possible courses of action defined (that is, controlled variables and the constraints upon them), and relevant uncontrolled variables identified. To devise a measure of performance, objectives are identified and defined, and then quantified. If objectives cannot be quantified or expressed in rigorous (usually mathematical) terms, most operations research techniques cannot be applied. For example, a business manager may have the acquisitive objective of introducing a new product and making it profitable within one year. The identified objective is profit in one year, which is defined as receipts less costs, and would probably be quantified in terms of sales. In the real world, conditions may change with time. Thus, though a given objective is identified at the beginning of the period, change and reformulation are frequently necessary.

Detailed knowledge of how the system under study actually operates and of its environment is essential. Such knowledge is normally acquired through an analysis of the system , a four-step process that involves determining whose needs or desires the organization tries to satisfy; how these are communicated to the organization; how information on needs and desires penetrates the organization; and what action is taken, how it is controlled, and what the time and resource requirements of these actions are. This information can usually be represented graphically in a flowchart, which enables researchers to identify the variables that affect system performance.

Once the objectives, the decision makers, their courses of action, and the uncontrolled variables have been identified and defined, a measure of performance can be developed and selection can be made of a quantitative function of this measure to be used as a criterion for the best solution.

The type of decision criterion that is appropriate to a problem depends on the state of knowledge regarding possible outcomes. Certainty describes a situation in which each course of action is believed to result in one particular outcome. Risk is a situation in which, for each course of action, alternative outcomes are possible, the probabilities of which are known or can be estimated. Uncertainty describes a situation in which, for each course of action, probabilities cannot be assigned to the possible outcomes.

In risk situations, which are the most common in practice, the objective normally is to maximize expected (long-run average) net gain or gross gain for specified costs, or to minimize costs for specified benefits. A business, for example, seeks to maximize expected profits or minimize expected costs. Other objectives, not necessarily related, may be sought; for example, an economic planner may wish to maintain full employment without inflation; or different groups within an organization may have to compromise their differing objectives, as when an army and a navy, for example, must cooperate in matters of defense.

In approaching uncertain situations one may attempt either to maximize the minimum gain or minimize the maximum loss that results from a choice; this is the “minimax” approach. Alternatively, one may weigh the possible outcomes to reflect one’s optimism or pessimism and then apply the minimax principle. A third approach, “minimax regret,” attempts to minimize the maximum deviation from the outcome that would have been selected if a state of certainty had existed before the choice had been made.

Each identified variable should be defined in terms of the conditions under which, and research operations by which, questions concerning its value ought to be answered; this includes identifying the scale used in measuring the variable.

A model is a simplified representation of the real world and, as such, includes only those variables relevant to the problem at hand. A model of freely falling bodies, for example, does not refer to the colour, texture, or shape of the body involved. Furthermore, a model may not include all relevant variables because a small percentage of these may account for most of the phenomenon to be explained. Many of the simplifications used produce some error in predictions derived from the model, but these can often be kept small compared to the magnitude of the improvement in operations that can be extracted from them. Most operations research models are symbolic models because symbols represent properties of the system. The earliest models were physical representations such as model ships, airplanes, tow tanks, and wind tunnels. Physical models are usually fairly easy to construct, but only for relatively simple objects or systems, and are usually difficult to change.

The next step beyond the physical model is the graph , easier to construct and manipulate but more abstract. Since graphic representation of more than three variables is difficult, symbolic models came into use. There is no limit to the number of variables that can be included in a symbolic model, and such models are easier to construct and manipulate than physical models.

Symbolic models are completely abstract. When the symbols in a model are defined, the model is given content or meaning. This has important consequences. Symbolic models of systems of very different content often reveal similar structure. Hence, most systems and problems arising in them can be fruitfully classified in terms of relatively few structures. Furthermore, since methods of extracting solutions from models depend only on their structure, some methods can be used to solve a wide variety of problems from a contextual point of view. Finally, a system that has the same structure as another, however different the two may be in content, can be used as a model of the other. Such a model is called an analogue . By use of such models much of what is known about the first system can be applied to the second.

Despite the obvious advantages of symbolic models there are many cases in which physical models are still useful, as in testing physical structures and mechanisms; the same is true for graphic models. Physical and graphic models are frequently used in the preliminary phases of constructing symbolic models of systems.

Operations research models represent the causal relationship between the controlled and uncontrolled variables and system performance; they must therefore be explanatory, not merely descriptive. Only explanatory models can provide the requisite means to manipulate the system to produce desired changes in performance.

Operations research analysis is directed toward establishing cause -and-effect relations. Though experiments with actual operations of all or part of a system are often useful, these are not the only way to analyze cause and effect. There are four patterns of model construction, only two of which involve experimentation: inspection, use of analogues , operational analysis, and operational experiments. They are considered here in order of increasing complexity.

In some cases the system and its problem are relatively simple and can be grasped either by inspection or from discussion with persons familiar with it. In general, only low-level and repetitive operating problems, those in which human behaviour plays a minor role, can be so treated.

When the researcher finds it difficult to represent the structure of a system symbolically, it is sometimes possible to establish a similarity, if not an identity, with another system whose structure is better known and easier to manipulate. It may then be possible to use either the analogous system itself or a symbolic model of it as a model of the problem system. For example, an equation derived from the kinetic theory of gases has been used as a model of the movement of trains between two classification yards. Hydraulic analogues of economies and electronic analogues of automotive traffic have been constructed with which experimentation could be carried out to determine the effects of manipulation of controllable variables. Thus, analogues may be constructed as well as found in existing systems.

In some cases analysis of actual operations of a system may reveal its causal structure. Data on operations are analyzed to yield an explanatory hypothesis , which is tested by analysis of operating data. Such testing may lead to revision of the hypothesis. The cycle is continued until a satisfactory explanatory model is developed.

For example, an analysis of the cars stopping at urban automotive service stations located at intersections of two streets revealed that almost all came from four of the 16 possible routes through the intersection (four ways of entering times four ways of leaving). Examination of the percentage of cars in each route that stopped for service suggested that this percentage was related to the amount of time lost by stopping. Data were then collected on time lost by cars in each route. This revealed a close inverse relationship between the percentage stopping and time lost. But the relationship was not linear; that is, the increases in one were not proportional to increases in the other. It was then found that perceived lost time exceeded actual lost time, and the relationship between the percentage of cars stopping and perceived lost time was close and linear. The hypothesis was systematically tested and verified and a model constructed that related the number of cars stopping at service stations to the amount of traffic in each route through its intersection and to characteristics of the station that affect the time required to get service.

In situations where it is not possible to isolate the effects of individual variables by analysis of operating data, it may be necessary to resort to operational experiments to determine which variables are relevant and how they affect system performance.

Such is the case, for example, in attempts to quantify the effects of advertising (amount, timing, and media used) upon sales of a consumer product. Advertising by the producer is only one of many controlled and uncontrolled variables affecting sales. Hence, in many cases its effect can only be isolated and measured by controlled experiments in the field.

The same is true in determining how the size, shape, weight, and price of a food product affect its sales. In this case laboratory experiments on samples of consumers can be used in preliminary stages, but field experiments are eventually necessary. Experiments do not yield explanatory theories, however. They can only be used to test explanatory hypotheses formulated before designing the experiment and to suggest additional hypotheses to be tested.

It is sometimes necessary to modify an otherwise acceptable model because it is not possible or practical to find the numerical values of the variables that appear in it. For example, a model to be used in guiding the selection of research projects may contain such variables as “the probability of success of the project,” “expected cost of the project,” and its “expected yield.” But none of these may be calculable with any reliability.

Models not only assist in solving problems but also are useful in formulating them; that is, models can be used as guides to explore the structure of a problem and to reveal possible courses of action that might otherwise be missed. In many cases the course of action revealed by such application of a model is so obviously superior to previously considered possibilities that justification of its choice is hardly required.

In some cases the model of a problem may be either too complicated or too large to solve. It is frequently possible to divide the model into individually solvable parts and to take the output of one model as an input to another. Since the models are likely to be interdependent, several repetitions of this process may be necessary.

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One way is the wrong way to do math. Here’s the right way.

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Research by Ed School psychologist reinforces case for stressing multiple problem-solving paths over memorization

There’s never just one way to solve a math problem, says Jon R. Star , a psychologist and professor of education at the Harvard Graduate School of Education. With researchers from Vanderbilt University, Star found that teaching students multiple ways to solve math problems instead of using a single method improves teaching and learning. In an interview with the Gazette, Star, a former math teacher, outlined the research and explained how anyone, with the right instruction, can develop a knack for numbers.

Jon R. Star

GAZETTE: What is the most common misconception about math learning?

STAR: That you’re either a math person or you’re not a math person — that some people are just born with math smarts, and they can do math, and other people are just not, and there’s not much you can do about it.

GAZETTE: What does science say about the process of learning math?

STAR: One thing we know from psychology about the learning process is that the act of reaching into your brain, grabbing some knowledge, pulling it out, chewing on it, talking about it, and putting it back helps you learn. Psychologists call this elaborative encoding. The more times you can do that process — putting knowledge in, getting it out, elaborating on it, putting it back in — the more you will have learned, remembered, and understood the material. We’re trying to get math teachers to help students engage in that process of elaborative encoding.

GAZETTE: How did you learn math yourself?

STAR: Learning math should involve some sense-making. It’s necessary that we listen to what our teacher tells us about the math and try to make sense of it in our minds. Math learning is not about pouring the words directly from the teacher’s mouth into the students’ ears and brains. That’s not the way it works. I think that’s how I learned math. But that’s not how I hope students learn math and that’s not how I hope teachers think about the teaching of math. Teachers should teach math in a way that encourages students to engage in sense-making and not merely to memorize or internalize exactly what the teacher says or does.

GAZETTE: Tell us about the teaching method described in the research.

STAR: One of the strategies that some teachers may use when teaching math is to show students how to solve problems and expect that the student is going to end up using the same method that the teacher showed. But there are many ways to solve math problems; there’s never just one way.

The strategy we developed asks that teachers compare two ways for solving a problem, side by side, and that they follow an instructional routine to lead a discussion to help students understand the difference between the two methods. That discussion is really the heart of this routine because it is fundamentally about sharing reasoning: Teachers ask students to explain why a strategy works, and students must dig into their heads and try to say what they understand. And listening to other people’s reasoning reinforces the process of learning.

GAZETTE: Why is this strategy an improvement over just learning a single method?

STAR: We think that learning multiple strategies for solving problems deepens students’ understanding of the content. There is a direct benefit to learning through comparing multiple methods, but there are also other types of benefits to students’ motivation. In this process, students come to see math a little differently — not just as a set of problems, each of which has exactly one way to solve it that you must memorize, but rather, as a terrain where there are always decisions to be made and multiple strategies that one might need to justify or debate. Because that is what math is.

For teachers, this can also be empowering because they are interested in increasing their students’ understanding, and we’ve given them a set of tools that can help them do that and potentially make the class more interesting as well. It’s important to note, too, that this approach is not something that we invented. In this case, what we’re asking teachers to do is something that they do a little bit of already. Every high school math teacher, for certain topics, is teaching students multiple strategies. It’s built into the curriculum. All that we’re saying is, first, you should do it more because it’s a good thing, and second, when you do it, this is a certain way that we found to be especially effective, both in terms of the visual materials and the pedagogy. It’s not a big stretch for most teachers. Conversations around ways to teach math for the past 30 or 40 years, and perhaps longer, have been emphasizing the use of multiple strategies.

GAZETTE: What are the potential challenges for math teachers to put this in practice?

STAR: If we want teachers to introduce students to multiple ways to solve problems, we must recognize that that is a lot of information for students and teachers. There is a concern that there could be information overload, and that’s very legitimate. Also, a well-intentioned teacher might take our strategy too far. A teacher might say something like, “Well, if comparing two strategies is good, then why don’t I compare three or four or five?” Not that that’s impossible to do well. But the visual materials you would have to design to help students manage that information overload are quite challenging. We don’t recommend that.

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Module 5: Thinking and Analysis

Problem-solving with critical thinking, learning outcomes.

• Describe how critical thinking skills can be used in problem-solving

Most of us face problems that we must solve every day. While some problems are more complex than others, we can apply critical thinking skills to every problem by asking questions like, what information am I missing? Why and how is it important? What are the contributing factors that lead to the problem? What resources are available to solve the problem? These questions are just the start of being able to think of innovative and effective solutions. Read through the following critical thinking, problem-solving process to identify steps you are already familiar with as well as opportunities to build a more critical approach to solving problems.

Problem-Solving Process

Step 1: define the problem.

Albert Einstein once said, “If I had an hour to solve a problem, I’d spend 55 minutes thinking about the problem and five minutes thinking about solutions.”

Often, when we first hear of or learn about a problem, we do not have all the information. If we immediately try to find a solution without having a thorough understanding of the problem, then we may only be solving a part of the problem.  This is called a “band-aid fix,” or when a symptom is addressed, but not the actual problem. While these band-aid fixes may provide temporary relief, if the actual problem is not addressed soon, then the problem will continue and likely get worse. Therefore, the first step when using critical thinking to solve problems is to identify the problem. The goal during this step is to gather enough research to determine how widespread the problem is, its nature, and its importance.

Step 2: Analyze the Causes

This step is used to uncover assumptions and underlying problems that are at the root of the problem. This step is important since you will need to ensure that whatever solution is chosen addresses the actual cause, or causes, of the problem.

Asking “why” questions to uncover root causes

A common way to uncover root causes is by asking why questions. When we are given an answer to a why question, we will often need to question that answer itself. Thus the process of asking “why” is an  iterative process —meaning that it is a process that we can repeatedly apply. When we stop asking why questions depends on what information we need and that can differ depending on what the goals are. For a better understanding, see the example below:

Problem: The lamp does not turn on.

• Why doesn’t the lamp turn on? The fuse is blown.
• Why is the fuse blown? There was overloaded circuit.
• Why was the circuit overloaded? The hair dryer was on.

If one is simply a homeowner or tenant, then it might be enough to simply know that if the hair dryer is on, the circuit will overload and turn off.  However, one can always ask further why questions, depending on what the goal is. For example, suppose someone wants to know if all hair dryers overload circuits or just this one. We might continue thus:

• Why did this hair dryer overload the circuit? Because hair dryers in general require a lot of electricity.

But now suppose we are an electrical engineer and are interested in designing a more environmentally friendly hair dryer. In that case, we might ask further:

• Why do hair dryers require so much energy?

As you can see from this example, what counts as a root cause depends on context and interests. The homeowner will not necessarily be interested in asking the further why questions whereas others might be.

Step 3: Generate Solutions

The goal of this step is to generate as many solutions as possible. In order to do so, brainstorm as many ideas as possible, no matter how outrageous or ineffective the idea might seem at the time. During your brainstorming session, it is important to generate solutions freely without editing or evaluating any of the ideas. The more solutions that you can generate, the more innovative and effective your ultimate solution might become upon later review.

You might find that setting a timer for fifteen to thirty minutes will help you to creatively push past the point when you think you are done. Another method might be to set a target for how many ideas you will generate. You might also consider using categories to trigger ideas. If you are brainstorming with a group, consider brainstorming individually for a while and then also brainstorming together as ideas can build from one idea to the next.

Step 4: Select a Solution

Once the brainstorming session is complete, then it is time to evaluate the solutions and select the more effective one.  Here you will consider how each solution will address the causes determined in step 2. It is also helpful to develop the criteria you will use when evaluating each solution, for instance, cost, time, difficulty level, resources needed, etc. Once your criteria for evaluation is established, then consider ranking each criterion by importance since some solutions might meet all criteria, but not to equally effective degrees.

In addition to evaluating by criteria, ensure that you consider possibilities and consequences of all serious contenders to address any drawbacks to a solution. Lastly, ensure that the solutions are actually feasible.

Step 6: Put Solution into Action

While many problem-solving models stop at simply selecting a solution, in order to actually solve a problem, the solution must be put into action. Here, you take responsibility to create, communicate, and execute the plan with detailed organizational logistics by addressing who will be responsible for what, when, and how.

Step 7: Evaluate progress

The final step when employing critical thinking to problem-solving is to evaluate the progress of the solution. Since critical thinking demands open-mindedness, analysis, and a willingness to change one’s mind, it is important to monitor how well the solution has actually solved the problem in order to determine if any course correction is needed.

While we solve problems every day, following the process to apply more critical thinking approaches in each step by considering what information might be missing; analyzing the problem and causes; remaining open-minded while brainstorming solutions; and providing criteria for, evaluating, and monitoring solutions can help you to become a better problem-solver and strengthen your critical thinking skills.

iterative process: one that can be repeatedly applied

• Problem solving. Authored by : Anne Fleischer. Provided by : Lumen Learning. License : CC BY: Attribution
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Metaheuristics for Solving Global and Engineering Optimization Problems: Review, Applications, Open Issues and Challenges

• Review article
• Open access
• Published: 21 August 2024

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• Essam H. Houssein   ORCID: orcid.org/0000-0002-8127-7233 1 ,
• Mahmoud Khalaf Saeed 1 ,
• Gang Hu 2 , 3 &
• Mustafa M. Al-Sayed 1  

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The greatest and fastest advances in the computing world today require researchers to develop new problem-solving techniques capable of providing an optimal global solution considering a set of aspects and restrictions. Due to the superiority of the metaheuristic Algorithms (MAs) in solving different classes of problems and providing promising results, MAs need to be studied. Numerous studies of MAs algorithms in different fields exist, but in this study, a comprehensive review of MAs, its nature, types, applications, and open issues are introduced in detail. Specifically, we introduce the metaheuristics' advantages over other techniques. To obtain an entire view about MAs, different classifications based on different aspects (i.e., inspiration source, number of search agents, the updating mechanisms followed by search agents in updating their positions, and the number of primary parameters of the algorithms) are presented in detail, along with the optimization problems including both structure and different types. The application area occupies a lot of research, so in this study, the most widely used applications of MAs are presented. Finally, a great effort of this research is directed to discuss the different open issues and challenges of MAs, which help upcoming researchers to know the future directions of this active field. Overall, this study helps existing researchers understand the basic information of the metaheuristic field in addition to directing newcomers to the active areas and problems that need to be addressed in the future.

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1 Introduction

As the world moves towards competition in all fields, people need to best use the limited resources to score a better result and thus achieve a better place in the competition. In this context, optimization is strongly needed. Optimization is a process of picking up the optimal values of the optimization problem's parameters from a given set of values to achieve the desired output, which specifically means output minimization or maximization. In other words, we need to obtain the best optimal solution under a set of limitations and constraints by tuning the parameters of the problem to be addressed. As mentioned in [ 1 ], the optimization process includes a set of steps which starts with formulating the problem to be in the form of an optimization problem, constructing the objectives (cost or fitness) function, determining the decision variables and the restrictions on these variables, then simplifying the reality of the problem by generating the mathematical model that represents the problem. Finally, the problem solver seeks to generate the most acceptable solution by maximizing or minimizing the value of the objective function.

Stochastic optimization algorithms are the most promising type under the umbrella of optimization, which can be classified as heuristic Algorithms (HAs) and metaheuristic Algorithms (MAs). In simple words, stochastic optimization is the general class of algorithms that depends on the random nature in the process of getting the optimal or near-optimal solution. HAs are iterative algorithms, iterating many times seeking a better solution than the solution obtained previously. HAs are used to find a feasible and reasonable solution but may not be the optimal one. In addition, HAs do not provide any evidence of the optimality of the solution obtained. A set of issues can be found in HAs, such as being problem-dependent algorithms specifically designed for a particular problem [ 2 ]. Another challenge in HAs is immeasurable success as there is no information about how close the obtained solution is to the optimal. Finally, there is a dilemma in measuring the computational time. Disclosing these gaffes and achieving a trade-off between solution quality and computational time is a main purpose of the appearance of metaheuristics [ 3 ]. As it is used to solve different types of problems, metaheuristics are the most preferable type of these algorithms. Metaheuristics were introduced for the first time by Glover in [ 4 ].

1.1 Understanding Optimization: What and Why?

In this section, we will help researchers understand the fascinating world of optimization. First, we will present examples of what optimization is, then we will try to answer the question, "Why do we study optimization?" Finally, this section will clarify how to optimize anything you have.

What is Optimization? In simple words, optimization is the art of perfectionism—how to perfectly make something in the best way. Optimization answers the question: how to obtain the best solution for a problem while applying a set of limitations? Maximizing profit, minimizing mass, pollution minimization, noise reduction, and drag reduction are all practical examples that can be achieved by using optimization. In most cases, optimization helps in the design process as a replacement for the traditional approach, which depends mainly on trials or humans. To clarify the simplicity and practical power of optimization in the design process, a diagrammatic view of how optimization methods help in the design process is presented in Fig.  1 .

The optimization methods as a replacement of traditional method in the design process

Why Optimization? People may ask why we study optimization. In most cases, we do not have the opportunity to physically perform trials; instead, we use optimization to simulate a solution for a specific problem to see what the result of this trial would be. Hence, we can decide whether this trial is applicable or not. People may benefit from applying optimization in the industry to achieve a better position in competition under limited resources.

1.2 Paper Structure

The rest of this paper is structured as follows: various metaheuristic taxonomies and the development process are illustrated in Sects.  2 and 3 . Taxonomies of optimization problems based on many criteria and their performance assessment are introduced in Sects.  4 and 5 . The applications of metaheuristic algorithms (MAs) in different fields are presented in Sect.  6 . The open issues and challenges of MAs and the observations from the experiment are introduced in Sect.  7 . Finally, the research review is concluded in Sect.  8 . The outline of the article is illustrated in Fig.  2 .

The outline of the article

2 Metaheuristics Optimization Algorithms Taxonomies

Due to the rapid growth of the optimization field, many metaheuristic (MA) algorithms have been proposed recently. These algorithms need to be classified according to four main taxonomies: inspiration source, number of search agents, the mechanisms followed in the optimization process, and solution updating, in addition to the number of parameters included in the algorithm. In this section, these new algorithms will be classified.

2.1 Taxonomy According to the Inspiration Source

This is the most familiar and oldest classification of metaheuristic algorithms (MAs) and is suitable for studying the subcategory of MAs, which are nature-inspired metaheuristic algorithms. In general, by including the source of inspiration in the calculation, different studies use different classifications according to the inspiration, as illustrated in Table  1 . In this study, Fig.  3 shows a more comprehensive taxonomy for MAs.

The proposed classification of MAs based on the source of inspiration

As follows, the subcategories of the source of inspiration for MAs shown in Fig.  3 are illustrated in detail.

Swarm Intelligence (SI) is a self-organized system of collaborative behavior. SI has a set of characteristics, such as good communication skills between individuals, the ability to share information among its individuals, and the ability to learn from doing (adaptable beings). On the other hand, organisms do not have the ability to defend themselves against predators; they need to be in a swarm to perform the search or attack process for food. Mimicking the behavior of beings that live in flocks or herds seeking to hunt for prey or find food is the main inspiration for SI algorithms [ 8 ]. One of the most famous algorithms in this category is Particle Swarm Optimization (PSO) [ 9 ], which is inspired by mimicking the intelligent behavior of a flock of birds. Monkey Search Optimization (MSO) [ 10 ] is another example of SI algorithms that simulate the tree climbing process during the food discovery process. Hunting strategy and hierarchy-based leadership are the inspiration for Grey Wolf Optimizer (GWO) [ 11 ], Ant Colony Optimization (ACO) [ 12 ], Cuckoo Search (CS) [ 13 ], Ant Lion Optimizer (ALO) [ 14 ], and Honey Badger Algorithm (HBA) [ 15 ], which are well-known instances of SI algorithms.

Evolutionary Algorithms (EA) simulate the behavior of evolution, including recombination, mutation, crossover, and selection. EA begins by generating a random population; this population is then evaluated to choose the most fit individuals to contribute to the next generation. After several iterations, the population evolves to find the optimal solution. Genetic Algorithm (GA) [ 16 ] is the oldest algorithm in this class, mimicking Charles Darwin's theory of natural evolution. Other well-known EA methods include Differential Evolution (DE) [ 17 ], Genetic Programming (GP) [ 18 ], Coronavirus Disease Optimization Algorithm (COVIDOA) [ 19 ], Liver Cancer Algorithm [ 20 ], and Red Deer Algorithm (RDA) [ 21 ].

Human-based Algorithms (HA) is the main inspiration for this category. Mimicking the learning process between teachers and students led to the introduction of Teaching Learning-Based Optimization (TLBO) [ 22 ]. Tabu Search (TS) [ 23 ] enhances the search process through long and short memory. Other well-known HA algorithms include Group Leader Optimization Algorithm (GLO) [ 24 ], Stock Exchange Trading Optimization (SETO) [ 25 ], and Social Emotional Optimization Algorithm (SEOA) [ 26 ].

Physics-based Algorithms (PhA) are inspired by the physical laws or simulating a physical phenomenon such as gravitation, Big Bang, black hole, galaxy, and field. In other words, the physical rules are used in the process of generating new solutions. The most popular instances of this class are the Gravitational Search Algorithm (GSA) [ 27 ], Big Bang Big Chain (BBBC) [ 28 ], Heat Transfer Search [ 29 ], Henry Gas Solubility Optimization (HGSO) [ 30 ], Archimedes Optimization Algorithm [ 31 ], and Light Spectrum Optimizer (LSO) [ 32 ], which are some of the most common algorithms in the PhA category.

Chemistry based algorithms (ChAs) are algorithms that concentrate on the principle of chemical reactions such as molecular reaction, Brownian motion, molecular radiation. A list of algorithms that fall into this category are Chemical Reaction Optimization (CRO) [ 33 ], Gases Brownian Motion Optimization (GBMO) [ 34 ], Artificial Chemical Process (ACP) [ 35 ], Ions Motion Optimization Algorithm (IMOA) [ 36 ], and Thermal Exchange Optimization (TEO) [ 37 ], are common instances of the ChA category.

Math-based algorithms (MathA) Math-based optimization algorithms are algorithms that can be inspired from the mathematical theorems, concept and rules. Some algorithms fall into this group including; Gaussian Swarm Optimization (GSO) [ 38 ], Sine Cosine Algorithm (SCA) [ 39 ], Lévy flight distribution [ 40 ], Exponential Distribution Optimizer (EDO) [ 41 ], and Golden Sine Algorithm (GSA) [ 42 ], are common instances of the MathA category.

Plant-based Algorithms (PlA) The PLAs is relays on the simulation of the intelligent behavior of the plants. Specifically, a set of concepts in plant nature is used to inspire new metaheuristic optimization algorithms such as the flower flow pollination process, the phenomenon of colonization of invasive weeds in nature, the ecology and weed biology. Some algorithms fall into this group including; Flower Pollination Algorithm (FPA) [ 43 ], Invasive Weed Optimization (IWO) [ 44 ], Paddy Field Algorithm (PFA) [ 45 ], Artificial Plant Optimization Algorithm (APOA) [ 46 ], Plant Growth Optimization (PGO) [ 47 ], Root Growth Algorithm (RGA) [ 48 ], Rooted Tree Optimization (RTO) [ 49 ] are common instances of the PlA category.

Sports and Game based Algorithms (SpGA) Depending in the information and rules applied in the sports and gaming, a set of optimization algorithms can be inspired from team game strategies used in football, Basketball, and volleyball, Ludo Game. Ludo Game-Based Swarm Intelligence (LGSI) [ 50 ], Team Game Algorithm (TGA) [ 51 ], Football game algorithm (FGA) [ 52 ], World Cup Optimization (WCO) [ 53 ], Soccer League Competition (SLC) algorithm [ 54 ], and League championship algorithm (LCA) [ 55 ] are common instances of SpGA algorithms.

Miscellaneous The rest of metaheuristics optimization algorithms can be collected to be belongs to the miscellaneous class, the purpose of using the term miscellaneous is the miscellaneous ideas such as politics, Artificial thoughts, atmosphere, trade and other topics. Work occurring in clouds such as cloud movement, spread, and creation is the basic idea behind the inspiration of the Atmosphere Cloud Model Optimization Algorithm (ACMO) [ 56 ], the exchange of information in the stock market occurs, and is the basic motivation behind the Exchange Market Algorithm (EMA) [ 57 ]. The Grenade Explosion Method (GEM) [ 58 ], Passing Vehicle Search (PVS) [ 59 ], Small World Optimization (SWO) [ 60 ], Yin-Yang Pair Optimization (YYPO) algorithm [ 61 ], Political Optimizer (PO) [ 62 ], and the Great Deluge Algorithm (GDA) [ 63 ] are other examples of this category.

2.2 Taxonomy According to the Number of Search Agents

The classification according to the source of inspiration is the most familiar and is usually introduced in studies to summarize the concept of classification. However, this classification is not enough to tackle the classification process, as it does not provide any information about the internal mathematical structure or programming ideas of the algorithms. Hence a new angle of classification is used. Meta-heuristics can be categorized based on the number of search agents seeking to find the optimal into two groups of single-solution-based MAs (SMAs), and population-based MAs (PMAs). The following two paragraphs provide more information about each group. Figure  4 is a clarification view of this taxonomy.

The classification of MAs based on the number of search agents

Single-solution based MAs (SMAs) SMAs is also called Trajectory‑based algorithms (TAs) as the algorithms in this class depends on single trajectory nature in its work. In other words, in each iteration, the solution is directed to a single trajectory. The optimization procedure (searching about the optimal solution) of SMAs is started with single solution (from one search agent), later, and in the subsequent iterations, the solution is refined with the aim of achieving the optimal solution. We can say that the algorithm generates a single path to the optimal solution over the course of the iteration. For SMAs, the Simulated Annealing (SA) [ 64 ] is one of the familiar algorithms. where a single search agent moves through the design or search space of the problem being tackled. Over the course of iteration, a better solution or moves is accepted to participate in determining the optimal solution while the weak movements and solution are more likely to participate in the optimization process. Applying these actions guarantee generating an optimal path through the search space with a great probability of achieving a global optimal solution. Hill climbing (HC) reviewed in [ 65 ], Tabu Search (TS) [ 23 ], Great Deluge Algorithm (GDA) [ 63 ], Iterated Local Search (ILS) [ 66 ], and Greedy Randomized Adaptive Search Procedures (GRASP) [ 67 ] are some instances of this class.

Population-based MAs (PMAs) In contrast, and taking advantage of sharing information among agents, Collaborative work and data remembering, the PMAs is introduced. First, we can say that more than one agent is superior to a single agent in achieving the optimal solution. Specifically, a great number of search agents work together to extensively explore the search space, so we can call PMAs explorative-based algorithms. The optimization procedure starts with employing a population of search agents positioned at many distinct positions in the search space, and over the course of iterations, the population uses the advantage of sharing information to better achieve the best global solution. In simple words, a set of lines is drawn in the search space to extensively search the search space in order to obtain the best optimal solution achieved by all search agents. One of the oldest and widely used algorithms in PMAs is the Genetic Algorithm (GA), Chemical reaction optimization (CRO), Particle Swarm Optimization (PSO), Archimedes Optimization Algorithm (AOA), Sine Cosine Algorithm (SCA), Exponential Distribution Optimizer (EDO), Grey Wolf Optimizer (GWO), Ant Colony Optimization (ACO) and Honey Badger Algorithm (HBA) are some instances from this category.

In general, no class is totally better than the other where PMAs escape from the local optima dilemma in contrast to SMAs, also SMAs consume less computational time than PMAs, for a Itr number of iterations, the SMAs perform a lower number of objective function evaluation which equals 1  ×  Itr while the PMAs perform N  ×  Itr evaluation of the objective function. N here stands for the number of search agents employed by the algorithm to obtain the optimal solution. But overall, the scientists prefer to use the PMAs as it has a greater probability of achieving global optimal solution in a considerable amount of time.

2.3 Taxonomy According to Updating Mechanisms

However, the classification according to the number of search agents provides information about the internal structure of the algorithm, but it cannot be treated as a uniform classification due to the few algorithms belonging to one group while the remainder (majority) falls under the other group. In this context, we need to provide a different classification angle to achieve an acceptable degree of uniform classification. According to the most important step of any algorithm, which is the solution update process. From this prospective MAs can be classified as solution creation-based algorithms (SCBAs) and differential vector movement-based algorithms (DVMs) [ 68 ]. In the following paragraph, we introduce a simple classification based on the behavior of the algorithms. Figure  5 is a clarification view of this taxonomy.

The classification of MAs based on population update mechanisms

Solution Creation Based Algorithms (SCs) In SCs, A set of parent solutions are merged to generate the new solution, in other words no single solution is used to create the fresh solution. Furthermore, the SCs can be categorized into two subcategories which are combination-based algorithms and stigmergy-based algorithms. In combination-based algorithms several solutions are combined or crossover-ed. Genetic Algorithm (GA), Gene Expression (GE), Harmony Search (HS), Bee Colony Optimization (BCO), Cuckoo Search (CS), Dolphin Search (DS) are some examples of this subcategory. On the other hand, in strategy-based solutions different solutions are indirectly coordinated by intermediate structure to generate new solutions. Ant Colony Optimization (ACO), Termite Hill Algorithm (THA), River Formation Dynamics (RFD), Intelligence Water Drops Algorithm (IWDA), Water-Flow Optimization Algorithm (WFOA), and Virtual Bees Algorithm (VBA) are some examples of the second subcategory.

Differential Vector Movement Based Algorithms (DVMs) Applying the mutation or shifting operation on the algorithm in order to generate a new solution is called Differential Vector Movement method. The fresh generated solution needs to be fitted to the previous one to participate in the next iteration of the optimization procedure. In this context, DVMs is categorized into three subcategories. In the first subcategory, the whole population's solution is used to generate the new solution, such operation occurs in Firefly Algorithm (FA) Gravitational Search Algorithm (GSA), Central Force Optimization (CFO), Human Group Formation (HGF) and Charged System Search (CSS). In the second sub-category, a small number of solutions (neighbourhoods) in population is employed to generate a new solution such as Artificial chemical process (ACP), Thermal Exchange Optimization (PSO), Group Search Optimizer (ALO), and Group Search Optimizer (GWO). In the last sub-category, only the relevant (best/worst) solutions are employed to generate the new solution such as Differential Evolution (DE), Artificial Bee Colony (ABC), Particle Swarm Optimization (PSO), Ant Lion Optimizer (ALO), and Grey Wolf Optimizer (GWO).

2.4 Taxonomy According to Number of Parameters

To deeply consider the internal configuration of the algorithm for this type of classification. Tuning the parameter of the algorithm plays a vital role in the performance of the algorithm when solving a specific problem. As mentioned in [ 1 ], it is a complicated task to choose the best values of the parameter that scores a better solution. Furthermore, the parameters can enhance the robustness and flexibility of the MAs if they are adjusted correctly. The optimization problem plays a vital role in defining the values of parameters. From a complexity perspective, the complexity of an algorithm is affected by the number of parameters. In this context and taking into account the importance of the parameters, this classification is introduced. Kanchan Rajwar et al. in [ 68 ] first classify the MAs according to the number of primary parameters employed in the MAs as illustrated in Fig.  6 .

The classification of MAs according to the number of primary parameters

The number of parameters changes from one algorithm to another, which can be 0, 1, 3, 4, etc. For simplicity we will consider four main groups holding algorithm parameter numbers up to 3 and the rest fall into the miscellaneous group. The following paragraphs provide a detailed explanation of the five groups in this classification.

Zero‑parameter-based algorithms (ZPAs): The ZPAs do not have any parameter in their internal configuration so it also called Free‑parameter-based algorithms. The absence of parameters in ZPAs gives the user the opportunity to easily adapt the algorithm to be utilized in different optimization problems. Hence, the algorithms belong to this group considered as flexible, adaptive, and easy-to-use algorithms. Teaching–Learning-Based Optimization (TLBO) [ 22 ], Black Hole Algorithm (BH) [ 69 ], Multi-Particle Collision Algorithm (M-PCA) [ 70 ], Symbiosis Organisms Search (SOS) [ 71 ], Vortex Search Optimization (VS) [ 72 ], Forensic-Based Investigation (FBI) [ 73 ], and Lightning Attachment Procedure Optimization (LAPO) [ 74 ] are some examples of ZPAs.

Single‑parameter-based algorithms (SPAs): SPAs is the type of algorithms that own a single primary parameters in their internal configuration. So, it also is called monoparameter-based algorithms. Mostly, this single parameter has the ability to change the amount of exploration and exploitation that occurred in the algorithm. For example, in the Artificial Bee Colony (ABC) algorithm the single parameter Limit is used to determine the amount of food source left [ 75 ], in the Salp Swarm Algorithm (SSA) c1 is the parameter used to achieve a better balance between explorative and exploitative capabilities [ 76 ], and in Harris Hawks Optimizer (HHO) [ 77 ] the switch between soft and hard besiege is achieved by the magnitude value parameter E . Cuckoo Search (CS), Killer Whale Algorithm (KWO), and Social Group Optimization (SGO) are another example of this group.

Two‑parameter-based algorithms (TPAs): In TPAs only two primary parameters exist in the internal structure of the algorithm. For example, in the Grey Wolf Optimizer (GWO), the two primary parameters a and c must be adjusted. The a is adjusted to be equal to 2 to 0 , allowing the algorithm to perform a smooth transition from exploration and exploitation while the c parameter is used to allow the algorithm to reach distinct locations around the optimal agent relative to the current location, In the Marine Predators Algorithm (MPA), P and FADs are the two primary control parameters. To overstate the predator or prey move, P is adjusted, while FADs is used to manage exploration behavior. Finally, in the Whale Optimization Algorithm (WOA) the two primary parameters A and C need to be modified to perform the exploration-to-exploitation transition and to allow the algorithm to explore several positions around the optimal agent relative to the present location. Differential Evolution (DE), Simulated Annealing (SA), Grasshopper Optimization Algorithm (GOA), Political Optimizer (PO), and Artificial Chemical Reaction Optimization Algorithm (ACROA) are just a few instances of TPAs.

Three‑parameter-based algorithms (TrPAs): In TPAs only three primary parameters exist in the internal structure of the algorithm. For example, the mutation rate mr , the crossover rate cr , and the new population selection criterion are the three parameters used in the Genetic Algorithm (GA) to allow the algorithm to escape from the local optima, improve the accuracy of the solution, and generate a most fit new generation, respectively. The randomization, attractiveness, and absorption are the three parameters included in the Firefly Algorithm (FA) to manage the execution of the algorithm and the random walks of fireflies. Finally, the distance bandwidth (BW), the harmony memory considering rate (HMCR), and the pitch adjusting rate (PAR) are the three primary parameters used in Harmony Search (HS) to increase the opportunity of achieving a global search and improve the local search problem. Squirrel Search Algorithm (SSA), Krill Herd (KH), Spring Search Algorithm (SSA), Artificial Algae Algorithm (AAA), Gases Brownian Motion Optimization (GBMO), Hurricane-Based Optimization Algorithm (HOA), Orca Optimization Algorithm (OOA), Social Spider Algorithm (SSA), Water Cycle Algorithm (WCA), Equilibrium Optimizer (EO), Parasitism Predation Algorithm (PPA), and Heap-Based Optimizer (HBO) are few instances of this group.

Miscellaneous: The rest of algorithms that own over three parameters in their internal configuration fall under the category of the miscellaneous group. It is not easy to cover all three-parameter algorithms. so, only three subgroups are introduced. the first subgroup is the four parameter-based algorithms such as Ant Colony Optimization (ACO), Sine Cosine Algorithm (SSA), Archimedes Optimization Algorithm (AOA), and Gravitational Search Algorithm (GSA). The second subgroup holds algorithms that employed five primary parameters in their internal structure such as Particle Swarm Optimization (PSO), Cheetah Chase Algorithm (CCA) and Farmland Fertility Algorithm (FFA). The last subgroup is algorithms with more than five primary parameters in their internal configuration. Biogeography-Based Optimization (BBO) with six parameters, Henry Gas Solubility Optimization (HGSO) with twelve primary parameters and the Camel Algorithm (CA) with seven } primary parameters are the most familiar algorithms in this subgroup. Cheetah Chase Algorithm (CCA), Exchange Market Algorithm (EMA), and Forest Optimization Algorithm (FOA) are also instances of this subgroup.

In general algorithms with few parameter-based MAs are easy to be adapted and hence the applicability of these algorithms to handle any optimization problem will increase and, on the other hand, large parameter-based MAs cause a disability of these algorithms to handle the optimization problems, as we encounter a problem in adapting all of their parameters to be suited for problem being tackled. hence the applicability will be decreed.

2.5 Metaheuristic Algorithms Merits

The MAs have a priority to be studied by the researcher than HAs, as they have four characteristics [ 78 ], which can be summarized as follows.

Metaheuristics simplicity It is painless to inspire a MAs as we can use a natural concept, physical phenomena or an animal behavior in the inspiration process. Utilizing the merit of simplicity, the researchers Seize the opportunity to make an extension in the metaheuristics works as they develop a new method by mimicking a natural idea, use the ability of search enhancement techniques to boost the performance of an existing algorithm, or even take the advantages in two metaheuristics algorithms and generate a new metaheuristics algorithm by applying a hybridization process. Furthermore, simplicity encourages computer scientists and other researchers to easily study the existing MAs and then apply them to solve a wide range of problems.

Metaheuristics flexibility In the other techniques there is a need to modify the structure of the algorithm to be matched with the problem being solved, unlike these techniques metaheuristics flexibility virtue allows the researchers to easily apply the MAs on any problem as the MAs have the capability of treating the problem as black box, in other words it need the input(s), output(s) of a problem on hand. No effort is used in modifying the structure; all effort is directed towards formulating the problem being solved in the form of an optimization problem.

Metaheuristics stochastic nature Computing the derivation of the search space of the problem is a necessity for the gradient-based optimization techniques to achieve an optimal solution. Dissimilar to these techniques, the preponderance of MAs is considered as a derivative-free mechanism when applying the process of optimization, specifically the MAs follow a stochastic nature during the search process as they start the optimization process by employing a set of search agents to generate random solutions without computing the derivative of the search space. The collaborative work of these search agents allows the algorithm to get the optimal solution. This merit allows researchers to easily use the MAs algorithms to perfectly tackle compound, expensive, and difficult problems that suffer from the trouble of obtaining the derivative information.

According to the previous features, the research community has increased, and researchers from different fields and application areas have been using the metaheuristic optimization algorithm in their work. About 4,476 documents founded in the Scopus have used the word metaheuristics in the last decade. Figure  7 a is introduced to visualize the distribution of research studies according to the subject area, while Fig.  7 b is used to depict the number of studies generated in each year of the previous decade.

Scopus statistics from 2014 to 2023

3 Development Process of Metaheuristic Optimization Algorithms

The simplicity merit of MAs allows researchers to easily develop a large number of algorithms in different application areas. To develop a new metaheuristic algorithm, a researcher can follow one of the following development processes according to the type of algorithm that is being developed, and some processes can also be used together.

Develop a new optimization algorithm The most of work for developing an optimization algorithm done by inspire the main idea of the algorithm from a different metaphors or concepts. These metaphors or concepts are mainly a simulation of rules or processes in different disciplines such as Chemistry, Physics, Biology, Psychology, Computation, Maths, and Human. Figure  3 is used to visualize a different source of inspiration with examples in each category. In general, most metaheuristics have been designed to mimic the system of living and survival of beings such as animals, birds, and insects, in addition to mimicking natural evolution. Insects (specifically, bees and ants) are the most popular metaphor for the development of a new optimization method by researchers.

Develop a new optimization algorithm from existing one One of the most popular ways to develop a new optimization method is to benefit from the operators of a specific algorithm in enhancing the structure of another algorithm. In simple words, the operators of other algorithms can emerge into the basic structures of the algorithm to boost the performance of the previously developed algorithm, and hence use it in solving different types of problems and issues. There are many enhancement operators used in the field; one of the most used ones is opposition-based learning (OBL). OBL is a machine learning mechanism that is used to increase the performance of the optimization algorithm by considering the opposite position of the solution in the search space. Specifically, two values are computed, the main and opposite positions, according to the objective function value, one of the two values maintained in the optimization process, and the other discarded. Taking into account only the best values, the optimization process became more accurate and a high level of performance is achieved. The orthogonal learning (OL) strategy is another example of an operator used as an enhancement strategy for MAs. The OL strategy mainly improves the exploitation capabilities. For example, the OL strategy was used to improve the Archimedes optimization algorithm, the cuckoo search algorithm and the artificial bee colony optimization algorithm, respectively. Enhanced solution quality (ESQ) is another mechanism used in the MA enhancement process. The ESQ was used to improve the performance of the reptile search algorithm and the Harris Hawks optimization (HHO) algorithm, respectively. Finally, the Local Escaping Operator (LEO) is used to develop an optimized version of the MPA called the enhanced marine predator algorithm (EMPA).

Hybridizing two or more optimization algorithm As a trial for enhancing the performance and applicability of the optimization method, researchers can benefit from hybridizing two or more optimization algorithms together in order to take the main strengths of each algorithm. The idea behind hybridization is to choose one algorithm better in exploration capabilities and another better in exploitation capabilities. Many challenges are encountered when we develop a new algorithm using the hybridization process, such as how to select the algorithm and how to merge them together, and is the new algorithm better than each one separately?

As shown in the previous paragraphs, there is a different development process for developing a new optimization method, although there is a set of limitations that must be considered during the development process such as the difficulty of transforming all the concepts with details into a mathematical form, how the algorithm totally manages the change in information about the source of inspiration, in addition to how people with low familiarity with the inspiration sources develop new methods.

3.1 Criteria for Comparative Algorithms

To gauge the effectiveness of newly developed algorithms, it is crucial for research to present the process of comparing them with existing algorithms. This should include a discussion of the selection criteria for comparative algorithms and the methodology used for comparison. The selection criteria for comparative algorithms depend mainly on the nature of the algorithm and the development process followed in developing the algorithm. In all cases, comparative algorithms should contain common criteria, which are state-of-the-art algorithms, newly developed algorithms, CEC winner algorithms, and high-performance algorithms. Specifically in case of developing the algorithm using the inspiration of a phenomenon process, the comparative algorithms list must contain algorithms with the same inspiration source or concept if there exist in addition to the common criteria algorithms. In case of developing an algorithm using the restructure method (i.e., merging a new operator or strategy), the comparative algorithms must contain the basic algorithm, algorithms developed using the same strategy if exists, algorithms that contain the strategy itself, in addition to the common criteria algorithms. In the case of developing algorithms using the hybridization process, the comparative algorithm list must contain the two basic algorithms that participate in the hybridization process, in addition to the common criteria algorithms.

3.2 Novelty Claims of Metaphor-Based Methods

The different ways of developing an optimization algorithm and the simplicity merit of the metaheuristic allow researchers to easily develop a large number of MAs. But a question must be asked here: Does this inspiration convey a novelty? In this section, we will present a set of claims and myths in the inspiration process of the metaheuristic optimization algorithms. As introduced in [ 79 ] a six widely used algorithms have been analyzed to prove that all components of the six (grey wolf, moth-flame, whale, firefly, bat and ant lion) are equivalent to a component of well-known techniques such as evolutionary algorithms and particle swarm optimization. Hence the authors called these algorithms misleading or tricky optimization algorithms, as they were inspired by bestial or duplicated metaphors and did not bring any novelty or useful principles in the metaheuristics field. We will present what considerations must be taken when developing a new novel algorithm and how to judge about the novelty of the new proposed algorithm in the field of metaheuristics.

Recently, a large number of publications have developed self-proclaimed or novel metaphor-based methods, but it is not obvious why they used them and what the novelty ideas are behind these methods. The set of all negative points, criticizes about novelty claims of various metaphor-based methods, can be introduced in the following points:

The metaphor-based methods redefine a well-known concept in the field of optimization and deliver it as a new concept or under new terminologies.

weak translation of the metaphors into a mathematical model or equations, and the model cannot be used totally to reflect the metaphors correctly. Finally, the proposed algorithm does not translate the mathematical model obtained from the metaphor correctly.

There is a myth in introducing the motivations behind the use of metaphor where instead of delivering the motivations as a sound or scientific basis they use accurate motivations such as a new metaphor "has never been used before" or a new mathematical model "has never been appeared in the past". Additionally, there is no concentration on the optimization process itself and how this process is employed to introduce effective design choices.

Instead of applying the evaluations of the proposed algorithms mainly on the state-of-the-art problems, the authors of these methods depend on the comparison with other algorithms or experimental analysis of low complexity problems in evaluating the performance or applicability of the proposed algorithm.

To prevent these negative points, the authors must apply two metrics analyses of the proposed algorithm before naming it as a "novel", which are:

Usefulness: in which the author must clearly introduce what are the useful ideas that come from the metaphor and how this metaphor helps in solving the optimization problems.

Novelty: When proposing a new method in the field of metaheuristics, was this new metaphor novel used to convey ideas?

4 Optimization Problems Overview

Achieving an acceptable solution is the main goal of any algorithm. Due to the rapid expansion of the complexity of the problem, scientists need to develop new methods that can cover this rapid extension. In this context, scientists are working to formulate any problem as an optimization problem to be easily tackled by optimization algorithms, as they provide better solutions than other traditional methods. In different fields, a great number of problems are formulated as an optimization problem, such as genetic algorithms used to automatically find and classify solitary lung nodules \cite{de2014automatic}, perform a classification for web pages, mining the web content, and dynamic organizing of the web content by ant colony optimization [ 80 ], In [ 81 ] Hussein et al., use the HHO to discover and design the drug through chemical descriptor selection and chemical compound activities. Applying HHO in microchannel heat sinks to minimize entropy generation [ 82 ], COVID-19 prediction [ 83 ], finally applying image segmentation and thresholding in [ 84 , 85 ].

4.1 Basic Structure of Optimization Problems

In this section, we will try to support the readers who may not be familiar with optimization methods with the basic definitions and terms related to the optimization field. The process of solving an optimization problem using a metaheuristic algorithm starts with identifying the real-world problem, after that we move to the problem description stage in which we define the characteristics of the problem, determining the functional requirement in addition to analysis of nonfunctional requirements. After completing the problem description stage, we move to the research stage, in which the researcher first concentrates on how to mathematically formulate the problem in a mathematical form. To formulate the problem, we need to determine the design variables and parameters, formulating the objective function, determining the basic constraints on the variables, analyzing the complexity of the problem, and finally justifying the use of a metaheuristic algorithm. In the following paragraphs, the three main components which exist in any optimization problem are the objective function, the decision variables, and a set of constraints on these variables are discussed in detail.

Optimization model: Every system can be considered as a set of inputs producing one or more input, The system uses the set of constraints to minimize the number of inputs, in other words, we consider only the inputs that obey the constraint (i.e., valid inputs) and discard the other which does not match with the constraints (i.e., invalid inputs). The system performs processing on the valid inputs to produce the optimal solution, which can be evaluated using the objective function to obtain the minimum or maximum output value. In fact, the optimization algorithm will not find the optimal values of constraints; instead, it uses the constraints to produce the optimal solutions and construct the feasible solution area. The feasible solution area can be considered as the area which contains an infinite number of feasible solutions and one or more can be classified as the optimal one.

Formulating the optimization model as an optimization problem: When we solve the problem using the optimization algorithm, we look for all possible combinations of inputs. For example, if we have 3 inputs each with 10 discrete values, then we get 1000 combinations of inputs. The initial test to solve and evaluate the input is to use brute-force techniques. The brute force techniques will do better to obtain the optimal solution, but what about the large sized problems. Certainly, we will find a big problem in handling these problems using the brute-force techniques; hence searching all possible combinations for most real-world problems is impossible.

To avoid confusion for non-familiar people with the area of optimization, in this study, we will introduce the basic and most frequent terminologies used in the field:

The search space: it is the area in which all possible combinations of inputs are located.

The search landscape: it is the set of all possible combinations of inputs with their corresponding objective values.

Decision variables: it is the unknown quantities that need to be determined by assigning values to them. It is also known as the design variables. All possible values that can be assigned to these variables are named variable scope or domain. It can be mathematically as X i where i = 1,2,3…N.

The objective function: This is the equation of the decision variables. In which all the decision variables exist with different parameters. It is used to judge the quality of the solution obtained for the problem being handled. In other words, after calculating the values of the decision variables, we substitute them in the objective function to obtain the objective value. The minimum objective value is the optimal solution for minimization problems, and the maximum is the optimal solution for the maximization problem.

Mathematically the single objective optimization problem can be formulated as Eq. ( 1 ) while the Multi objective optimization problem can be formulated as Eq. ( 2 ).

The optimization problems can be categorized in different ways. Categorizing optimization problems is an important step in choosing the algorithm that provides the optimal solution. It is not easy to introduce a rigorous or comprehensive taxonomy for optimization problems. This is due to the multiplicity of the classification term. But due to the important role of this taxonomy, in this paper we present a simplified and summarized version of the available taxonomies, illustrated in Fig.  8 . In the following subsections, the different subcategories of the optimization problem are discussed in detail.

Optimization problem taxonomy

4.2 Taxonomy According to the Objective Function

In terms of the number of objectives, there are two types. If the number of objectives is greater than one, the problem is called a multi-objective optimization problem; otherwise, the problem is named a single-objective optimization problem. Usually, real-world optimization problems are multi-objective. For example, if we need to design a table, we will consider two objectives, for example, minimizing the weight and the price of the table.

Single-objective optimization Only one global optimal solution exists in single-objective optimization. The objective function only considers one objective; therefore, the best optimal solution can be easily determined by comparing the obtained solutions using basic comparison operators < , > , ≤ , ≥ , and = , the nature of this type allows the algorithm to easily tackle optimization problems. Without loss of generality, Eq. ( 1 ) is used to determine the mathematical structure of a single-objective optimization problem.

where the problem decision variable is symbolized by n , m and P exist to represent the number of inequality and equality constraints, respectively. For the i th variable, ub i and lb i are used to represent the upper and lower boundaries, respectively.

Multi-objective optimization In contrast to single objective optimization, A set (more than one) of objectives need to be optimized simultaneously in the multi-objective optimization problems. Usually, these objectives are a conflict with each other, so most of work in this type is paid to achieving a trade-off between these objectives. The set of solutions in this type is called a Pareto optimal solution. The Pareto optimal dominance is employed to compare the solutions obtained in order to determine the optimal solutions. Extra storage is needed to hold the Pareto optimal solutions. Without loss of generality, Eq. ( 2 ) is used to determine the mathematical structure of a single objective optimization problem.

where the problem decision variable is symbolized by n , m and P exist to represent the number of inequality and equality constraints respectively. For the variable i th , U i and L i are used to represent the upper and lower limits, respectively. The number of objectives is denoted by o , and the g i and h i are the i th inequality and equality constraints, respectively. In general, the clash among objectives enforces the problem designer to consider more than one criterion in the comparison of obtained solutions and therefore the classical comparison operator does not perform better, instead, the Pareto dominance Eq. ( 3 ) is used to define the best optimal solutions.

Here the two solutions are represented by the vectors x and y . The x is said to dominate y denoted as (x  ≤  y) if x has at least one better value in all objectives.

4.3 Taxonomy According to Function Form

From another angle, classification can be done according to function form. If we have a real-world optimization problem, the constraints are linear qualities and inequalities and the objective function formed as linear then the problem is said to be a linear optimization problem. In nonlinear optimization, one or both of the objective functions and constraints are nonlinear, and this is the realistic and complex one [ 86 ].

4.4 Taxonomy According to the Design Variable

According to the nature of the design variables, we can present three different types of optimization problems, as detailed in the following points.

Discrete optimization problems In discrete optimization problems the values of the design variables are discrete and in which there is a finite set of values. The shortest path problem and the minimum spanning tree problem are two instances of this type. For more details, we can mention that the discrete optimization consists of integer programming and combinatorial optimization. Integer programming deals with the formulation and solution of discrete integers (or binary integers) valued in the design variables. On the other hand, combinatorial optimization emphasizes the combinatorial origin, formulation, or solution of a problem. Mainly it seeks to achieve pairs (i.e., Assignments, groupings, orderings) of discrete and finite values under the influence of specific constraints. These pairs involve a component of solutions of potential combinatorial problem solutions [ 87 ]. In Bioinformatics, Artificial intelligence and other fields combinatorial optimization can be applied such as identifying propositional formula models or defining the 3D structure of protein, finding the shortest path in graphs, the travelling salesman problem, the knapsack problem in addition to the pin packing problem, the quadratic assignment problem which has been tackled in this study.

Continuous optimization problems In continuous optimization problems, A range of values is assigned to the design variables, so every design variable has an infinite set of values. These problems have two types constrained continuous optimization problems which there is a constant on the variables. For unconstrained continuous optimization problems there is an absence of these constraints maximization the general yield for differential amplifiers, optimization of the mechanical system of shock absorption are two examples of this type [ 88 ].

Mixed discrete–Continuous optimization problems In many problems a design variable has a mixture of discrete and continuous values, in this case we call the problem mixed discrete–continuous type. This type is the most widely used one, where numerous real-world problems are complex and possess a mixed quantitative and qualitative input. In [ 89 ], a set of instances is addressed using black-box optimization techniques.

4.5 Taxonomy According to Constraints

Furthermore, the classification can be according to the restrictions on the design variables.

Unconstrained Optimization Problem If there are no constraints on the design variables we call this type as unconstrained optimization problem, the unconstrained optimization can be viewed as iterative methods stating with initial estimation for the optimal solution then a set of iteration is used to reach for the optimal solution. Usually, the solutions were reduced iteratively to an optimal solution. In [ 90 ], Fletcher and Roger mentioned that the unconstrained optimization methods differ according to how much information the user provides, such as the gradient method, the second derivative method, and the non-derivative method.

Constrained optimization problem If there is one or more constraints, the optimization problem falls under the constrained optimization problem class. Furthermore, there are two subclasses of this type. The first subclass is equality constraint problem in which the values of the design variables are restricted to be equal to the specific value. The second subclass is inequality-constrained problem the design variables are restricted to greater / smaller than a specific value. From a formulation perspective, every equality constraint can be mathematically transformed to two inequality constraints. For example, ϕ(x) = 0 is equivalent to ϕ(x) < 0 and ϕ(x) > 0 [ 86 ]. Mainly, the constrained optimization covers three types of optimizations which are network optimization, bound constraints optimization, and the global optimization. Global optimization includes one of the most widely used problems, which is engineering design optimization problems.

5 Performance Assessment of Optimization Algorithms

First of all, we must refer to an important term, the efficiency of the algorithm, which means how the algorithm responds against finding the optimal solutions for the problem to be solved. Achieving an optimal solution is not the only purpose of a good optimization algorithm; instead, the algorithm must be high quality and achieve a better situation in the applicability process on different classes of problems. To judge the quality and applicability of the algorithm, the algorithm must be compared against a set of qualitative and quantitative measures. The good quality algorithm performs better and achieves better results when tested against qualitative and quantitative measures. In this section, we will present the whole assessment environment used to test the quality and applicability of the algorithm.

CEC Test suite CEC stands for Congress on evolutionary computation. Mainly the CEC holds a different class of problems, which may be uni-model, multi-modal, fixed-dimension multi-modal, and composite. CEC is usually used to test the performance of the algorithm and its ability to solve different classes of problems. In the art of optimization almost all studies perform the CEC function as a fitness function to test algorithm's performance itself and to compare the algorithm's performance against other algorithms.

Statistical Measures In this metric, the Best, Worst, Mean, and standard deviation are computed to the obtained solutions to judge about the quality of all solutions obtained together. The best solution is the one with a minimum value of fitness function in minimization optimization and the opposite is right for maximization optimization. The Worst is the solution which has a maximum value of fitness function on the minimization optimization and the opposite is right for maximization optimization. while the mean is used to compute the average value of all obtained solutions (obtained from executing the algorithm many times), and the small value of the mean means that the algorithm is doing better. Finally, the standard deviation or STD is the statistical measure that gives the reader insight into the differences among the obtained solutions, and the algorithm with small STD value is also better than the other with large value. There is also an important statistical measure, which is capable of measuring the whole performance of algorithms for any number of functions. This measure uses the mean rank sum value to rank the algorithms. Ranking these values in ascending order enables us to say that the algorithm with the lowest value is the best among all algorithms participating in comparison for all functions together.

Convergence curve Drawing a relation between the solutions scored by the algorithm and the number of iterations or number of function evaluations is the primary goal of the convergence curve. To summarize the behavior of the algorithm, the convergence curve is drawn to judge the speed of the algorithm in reaching the global optimal solution. For the minimization problem and to compare the performance of many algorithms. The lower convergence curve is better than the upper one. Also, we can compute how fast the algorithm converges towards the optimal solution through the rate of convergence measure.

Diversity Diversity measure is one of the measures related to the algorithm’s convergence behavior. In simple words, diversity means how the search agents of the algorithm are distributed in the search space. A high diversity value of the algorithm can be translated into a great exploration ability of the algorithm, and a low value can be translated into a great exploitation ability of the algorithm. Hence the diversity values of the algorithm must be smoothly transited from high value in the first iterations of the algorithm to low value in the rest of iterations of the algorithm. In this context, we can say that the good diversity of the algorithm leads to avoiding premature convergence and achieve a good speed in achieving the optimal solution hence score a high level of efficiency.

Trajectory diagram In order to test the behavior of a specific agent of the algorithm over the curse of iterations the trajectory diagram is used. The fluctuations of the curve are an indication of the better performance of that agent and its ability to explore and exploit the search space better.

Search history diagram To visualize the history of positions scored by the search agent during the process of optimization, the search history curve is drawn.

Exploration and exploitation The exploration and exploitation (EXPL-EXPT) curves are used to visualize the exploitative and explorative capabilities of the algorithm. Usually, the overlaps between the two curves exist to tell us about the shifting between exploration and exploitation, and therefore an EXPL-EXPT balance.

Real-world problems To test the ability of the algorithm in solving different classes of problem the real-world problem is tackled. Engineering design problems are the most widely used problems as many algorithms use the (pressure vessel, welded beam, 15/3/25/52-bar truss system, tension/compression spring…etc.) classical design problems to quiz the algorithm performance.

Operation platforms Alongside the previous measures, the algorithm quality can be affected by the environment setup in which the algorithm is executed. The good environment in both software and hardware capabilities leads to good behavior of the algorithm. In this context, we must mention that when we compare more than one algorithm to judge which is better, we must execute the algorithms in the same environment to achieve a fair comparison.

6 Metaheuristics Applications

As mentioned above, MAs have a great degree of applicability, as they operate better in solving different problems that involve a computation time restriction, a high-dimensional problem, and other kinds of problems. Specifically, MAs are capable of dealing with different classes of optimization problems in different fields. In the following subsections, the applicability of MAs in some of these fields are illustrated in detail.

6.1 IEEE Congress on Evolutionary Computation (IEEE CEC)

CEC stands for Congress on evolutionary computation. Mainly the CEC holds a different class of problems, which may be uni-model, multi-modal, fixed-dimension multimodal, and composite. CEC is usually used to test the performance of the algorithm and is considered as an indication of the capabilities of the algorithm to solve different classes of problems. In the art of optimization, almost all studies perform the CEC function as fitness functions to test the algorithm's performance itself and to compare the algorithm's performance against other algorithms. Almost a different version of the CEC test suite is introduced every year. Tables 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 , 10 , 11 , and 12 are presented below to provide the reader with the basic information on each version of the CEC benchmark function and how metaheuristic algorithms are applied to solve these benchmark functions.

6.2 Engineering Design Problems

It is easy to provide an optimal design for a simple problem that contains a small number of design variables with a small range of values. In contrast to complex problems with many components, algorithms consume a huge amount of time to develop an optimal design. For example, the mechanical problem with different components and multiple objectives and constraints. Another example for complex problems is the engineering design problems in which the design process starts with exploiting the experience of designers to guess an optimal design for any problem, but this is not the optimal direction. In order to treat this poor thinking, we need systematic work that guarantees achieving an optimal design that is better than any other human design. Automatic techniques or, in other words, metaheuristics algorithms (MAs) are used to effectively diversify the search space with large parameters, minimizing the cost, and improving the product life cycle. Mainly, the MAs tune the parameters of the problem to produce the best optimal values of the design variables, hence achieving the optimal design. Table 13 is used to highlight the work of single-objective metaheuristics optimization algorithms in solving engineering design problems; also, Table  14 is introduced to clarify the tries of multi-objective optimization algorithms in tackling engineering design problems.

6.3 NP‑Hard Problems

In NP‑hard problems the NP stands for nondeterministic polynomial time where the nondeterministic refers to nondeterministic Turing machines which apply the idea of brute-force search method. On the other hand, the polynomial is used to refer to the amount of time required to apply the quick search to get the single solution of the deterministic algorithm, or the time consumed by the nondeterministic Turing machines to perform extensive search. P is the set of all decision problems solvable in polynomial time. Specifically, the decision problem has two answers YES and No. Consequently, if all YES answers are checked in polynomial time, then the problem belongs to set of NP problems; on the other hand, co-NP is used for NO answer. If the polynomial-time solution obtained for a specific problem leads to a polynomial-time solution for all problems in the NP in this case the problem is said to be NP-hard. Also, a problem is NP-Complete iff it is NP-Hard, and it is in NP itself. Due to the high computational complexity, the exhaustive search methods do not have the ability of getting the best solution.

Quadratic Assignment Problem (QAP) As mentioned in [ 195 ] the QAP is NP-hard, as the polynomial time is not sufficient to obtain the approximate solution from optimal solution. QAP was first introduced by Koopmans and Beckmann in 1957 [ 196 ] as an extension of the linear assignment problem. QAP is considered a combinatorial optimization problem that has been considered and tackled by many research studies in the last three decades. However, the good results obtained in these studies but until now the QAP is not well solved as there is no exact algorithm capable of solving problems with more than 20 input sizes in a reasonable amount of computational time [ 197 ]. In QAP we seek to locate the facilities in its appropriate location under the condition that it is an exact-one-to-exact-one problem, that is, each site can only grasp only one facility and each facility must be placed in only one site where the distances between facilities and sites are determined. The main optimization goal of QAP is to minimize the distance and flow between each pair of facilities delegated to their relevant sites. Recently, QAP is addressed by many books, studies and reviews, as listed in Table  15 . From another angle, there are several problems that are considered as special types of QAP (Table  16 ).

The Bin Packing Problem (BPP) The Bin Packing Problem (BPP) is one of most familiar combinatorial problems that is considered as strongly NP-hard problem [ 218 ]. In BPP we need to pack a set of items m into n bins with the aim of minimizing the number of bins required to hold all items. The BPP can be mathematically formulated as in Eq.  4 [ 219 ].

where capacity of the bin y j is symbolized by C yj .

The BPP benchmark data sets consist of three different types that are commonly classified as Easy, Medium, and Hard class as mentioned in [ 220 ]. Also, the BPP appears in one, two, three, and multi-dimensional form (Table  17 ).

Travelling Salesman Problem (TSP) One of the most familiar combinatorial optimization NP-Hard is the TSP in which we need to minimize the route as possible consumed to visit all cites precisely once and return to the initial city given a list of cities and distances among them. For example, in the TSP of 20 city, we have a huge number of feasible solutions (approx.1.22 × 1017). Guess how much time is required to perform this task using exhaustive search? the answer is very long. Therefore, exhaustive searches have disabilities in tackling such problems. The use of MAs destroys this disability, as it was used to find near optimal solutions in a reasonable amount of time [ 233 ]. Vehicle routing problems (VRPs) is the general form of TSP and is a multi-objective real-world problem tackled by many MAs such as genetic algorithm (GA), particle swarm optimization (PSO) and colony optimization (ACO) as in [ 234 ].

Job Shop Scheduling (JSS) JSS is a NP-Hard problem in which the algorithm seeks to consume a polynomial time to solve it. In JSS we need to process a finite set of jobs using a limited set of machines. JSS is a general type of scheduling problem. JSS is addressed by many MAs in [ 235 , 236 , 237 ], and [ 238 ].

6.4 Medical Science

Most of medical activities (i.e., Diagnosing, imaging, treatment, and monitoring) depends in its work on the computer or electronic device that is operate using an algorithm-based software [ 68 ]. Several researchers have used GAs for edge detection of images acquired using different imaging modalities, including magnetic resonance imaging, CT, and ultrasound [ 239 ]. In [ 240 ], Pereira et al., applied a set of computational tools for mammogram segmentation to improve the detection of breast cancer using GA combined with wavelet analysis to allow the detection and segmentation of suspicious areas with 95% sensitivity. GA has been applied for feature selection to identify a region of interest in mammograms as normal or containing a mass [ 241 ]. Also GA is combined with a support vector machine to differentiate benign and malignant breast tumours in ultrasound images [ 242 ], GA is combined with diversity index to discover lung nodules by developing an automatic threshold clustering method [ 80 ]. In [ 243 ] electroencephalography signals were used to detect hypoglycemia in patients with type 1 diabetes. Depending on neural networks in conjunction with ant colony optimization (ACO) and particle swarm optimization (PSO) Suganthi and Madheswaran use a more advanced computer-aided decision support system and mammogram to group tumours and detect breast cancer stages as described in [ 244 ]. Based on artificial bee colony (ABC) algorithm Kockanat and et al., Develop a technique for demonising images using 2D impulse response digital filter as illustrated in [ 245 ].

6.5 Robotics

Robotics is a vital active research field that owns some challenges that needs to be optimized such as task performance, Decrease the robotics cost, achieve a better reliability, in addition to minimize the unit complexity over other traditional robot systems. In this context, metaheuristics can be used to tune machine learning methods to enhance the collaborative behavior of robotics. One of the most active problems in robotics is the redundant humanoid manipulator issue. The complexity of this problem comes from the existence of multiple number of degrees of freedom and complex joint structure. This problem causes difficulty in achieving an inverse kinematics solution. Scientists make an effort to formulate this problem as a minimization problem, hence the MAs can perform better in solving this problem. In [ 246 ], the multilayer perceptron neural network is trained by the exploitative and explorative capabilities of the bee’s algorithm to learn the inverse kinematics of a robot manipulator arm. To conquer the problem of multi-solution, the GA is used to achieve a global optimal solution for inverse kinematics of 7-DOF (seven degree of freedom) manipulator [ 247 ]. Also, the inverse kinematics of the seven-degree-of-freedom (7-DOF) manipulator is perfectly tackled by the particle swarm optimization algorithm (PSO) by exploiting the strong intelligent scene and collaborative behavior among particles [ 248 ]. Biogeography-based optimization (BBO) is hybrid with differential evolution (DE) and uses the merits of the hybrid migration operator and the adapted Gaussian mutation operator to solve the inverse kinematics problem of the 8-DOF redundant humanoid manipulator [ 249 ].

6.6 Finance

Metaheuristics algorithms can be one of the most promising techniques used to solve different types of problem that occur in the finance and banking activities. In the following points, we will introduce a list of the most familiar problems and how the metaheuristics used to solve these problems.

Portfolio optimization and selection problem (POSP) in this problem, investors seek to assign optimal weights to the assets of the portfolio to achieve a minimal risk of investment. In [ 250 ], the authors provide a survey to solve POSP using metaheuristics and examples. Furthermore, the three GA, TS, and SA metaheuristic algorithms are used to solve POSP. The authors of [ 251 ] use the PSO algorithm to solve the POSP version with a cardinality constraint.

Index tracking problem (ITP) The ITP is a trading strategy that can depend mainly on two processes (hold and buy). In ITP we want to simulate the behavior of the index of the stock market using a minimum number of stocks. In other words, the ITP is developed to passively simulate the performance of the stock market index. For the specific German index, the authors of [ 252 ] use the SA to minimize tracking errors. The combinatorial search is hybrid with the DE for solving the ITP. The authors of [ 253 ] compare the performance of GA with quadratic programming and propose a solution approach to minimize the returns on the index using data from the FTSE100 index. Finally, in [ 254 ] the authors conducted a set of experiments to solve a special type of ITP and noticed that there was an improvement in an index.

Options pricing problem (OPP) Speculative activities are one of the most familiar tasks in financial markets, and the option can be one of the tools for speculative activities. Due to the fast dynamic motion of the financial market, it is difficult to guess the price of the option using traditional methods, so metaheuristic algorithms can be a promising choice in that case. In order to find parameters that achieve consistency between the model and market prices, Gilli and Schumann [ 255 ] use the PSO and DE to study the pricing of the calibration option. Finally, the authors of [ 256 ] have shown that the pricing of option operations can be enhanced compared to the traditional binomial lattice method when we use the ACO algorithm.

6.7 Telecommunications Networks

The recently needs for developing complex and large computer systems lead to an urgent demand for designing and developing high quality and more extensive network design and routing techniques and optimally solving problems in such an area. Also, we can notice that most problems in telecommunications are complex and hard to solve using traditional techniques and approximate algorithms, so there is urgent need to employ metaheuristic algorithms to solve network design and routing problems. A set of nodes (i.e., computers, databases, equipment, or radio transmitters) can be connected together using a transmission link (i.e., optical fiber, copper cable, radio, or satellite links) to construct communication networks. Under a set of constraints such as reliability, throughput, delay and link capacity, we seek to achieve a minimum cost of configurations as an objective function for these networks, and many problems can be appeared such as number of nodes, number of routing paths, the frequency assignment, and the capacity installation. A large number of studies using metaheuristics in solving telecommunications problems such as Kim et al. [ 257 ] employ a SA algorithm in the mobile radio system to allocate the nominal cells of channels. To minimize the installation cost and maximize traffic, the authors in [ 258 ] use the tabu search algorithm with randomized greedy procedures to find the location of the base stations of the universal mobile-based communication system. Specifically, good approximate solutions for large and medium-sized instances are obtained by the randomized greedy procedures, and these solutions were improved by using the tabu search algorithm. Finally, a new metaheuristic algorithm developed based on the Genetic Algorithm and Ant System was proposed to achieve better and efficient solutions for real-life transportation network design problems in large real networks located in two different places (Canada, city of Winnipeg) [ 259 ].

6.8 Food Manufacturing Industry

Recently, the metaheuristics can be considered as one of the most widely used efficient decision-making techniques that can be used to solve problems in different disciplines. In this section, we will present brief information about using metaheuristics in one of these disciplines, which is the food manufacturing industry. Specifically, metaheuristics can be applied in many food processes such as thermal drying, fermentation, and distillation. In [ 260 ], the authors develop a new hybrid method based on artificial bee colony (ABC) and the record-to-record travel algorithm (RRT) for Optimizing the Traceability in the Food Industry. The proposed method is employed to solve and provide the optimal minimal solution for the batch dispersion manufacturing problem. The hybrid RRT-ABC is used in the French food industry to carry out real-world experiments (that is, sausage manufacturing) to obtain high-performance results compared to traditional methods. The Artificial Bee Colony Algorithm (ABCA) used in the development of a delivery route optimization method to achieve a fresh food distribution without decreasing the quality of the food [ 261 ]. Finally, in [ 262 ], the Simulated Annealing (SA) is hybrid with the Virus Colony Search Algorithm (VCS) to improve the quality of the result of a sustainable Closed-Loop Supply Chain Network (CLSCN) design in the olive industry.

7 Open Issues and Challenges

However, the good features and abilities of the MAs in solving a wide range of problems, like other techniques, suffers from a set of problems in the following points, we will refer to these problems.

The stochastic nature and near optimal solution As we know, generating an optimal solution is one of the main features of deterministic algorithms such as simplex method. On the contrary to that, the metaheuristics algorithms (as it is a stochastic algorithm in nature) does not guarantee optimality of the obtained solution, but it provides an acceptable solution. This is one of the significant disadvantages of MAs. It is worth mentioning that the deterministic methods (unlike the stochastic methods) face difficulty when dealing with high-complex problems (that is, high-dimensionality and non-differentiable problems). Practically, when we decide to use one of the previous two methods, we choose to gain something and give the other.

The scale-ability and expensive computational cost Practically, the MAs score great promising results in solving problems in different natures such as discrete, continuous and combinatorial problems that contain a large number of decision variables. However, when solving large-scale global optimization problems (LSGOP) the MAs consume an expensive amount of computational cost. This scalability, challenge is one of the most important challenges that researchers must consider in the future due to the great growth in the size of the optimization problems when dealing with high-dimensional machine learning and large-scale engineering problems. In this context, many strategies are developed by the researchers to cover this problem such as the parallelization, approximation and surrogate modelling, hybridization of local search and memetic algorithms, decomposing the big problems into sub-problems, and befit from the sampling techniques.

The weakness of theoretical and mathematical analysis In most sciences such as chemistry, Biology, physics and others, the mathematical analysis of a method can be computed accurately to specify how much the method costs in terms of computational cost. Unlike those sciences, in metaheuristics we encounter a challenge in computing the exact computational cost of the algorithm, the reason behind this difficulty is from mathematical perspective it is difficult to analyze why the metaheuristics algorithms are so successful. Also, researchers need to pay attention to solving problems in determining the convergence analysis of many metaheuristics’ optimization algorithms. Finally, researchers also need to develop innovative methods that allow researchers to easily analyze and compute the algorithm's cost in the case of modification and scaling up the algorithm.

Intensification and diversification trade-off The algorithm's degree of effectiveness is measured by the ability of the algorithm to transit smoothly between the exploration (that is, explore as much as possible the feasible area) and the exploitation (that is, achieving good steps towards the optimal solution's area) stages. Achieving a high degree of intensification and diversification balance is one of the most important challenges or issues in most MAs. However, some algorithms achieve an acceptable degree of trade-off between exploration and exploitation; the vast majority of MAs need to address this challenge by scoring a high level of global diversification and local intensification [ 263 ].

Large-scale real-world problem formulation Nowadays the vast majority of problems in recent fields such as data science and big data analysis tasks are considered as large-scale real-world problem (LSRP) that is due to the large number of problem components and problem dimensions. Formulating a large-scale real-world problem (LSRP) is one of the crucial issues in metaheuristic algorithms. The issue comes from the large number of optimization variables (decision variables) included in the problem, how these variables interact with each other, how much the variables or components are related to each other, and what is the effect of one variable on the other variables. Also, it is worth mentioning that the large number of variables is translated as the problem size, which affects the computational cost of the algorithm that deals with this problem.

The limitations of the No-Free-Lunch theorem One of the most fundamental theories in the field of optimization is the No-Free-Lunch theorem [ 264 ] which states that there is no universal optimizer for all kinds of problems that is the algorithm may do better in some kinds of problems and do no better for the other kinds. We cannot generalize this theory, as it has been proved for the type of single-objective optimization, but it does not hold yet for problems with continuous and infinite domains in addition to multi-objective optimization [ 265 , 266 ]. In this context, the researchers in the field of metaheuristics must answer how to apply the NFL in terms of several dimensions?

Comparing different algorithms Comparing similar algorithms through the absolute value of the objective function or number of function evaluations is a possible task. On the other hand, we encounter a problem in comparing different algorithms with different objectives through a formal theoretical analysis. Practically no fair/honest or rigorous comparisons exist in this field [ 267 ].

Parameter tuning and control The algorithm's parameter plays the most vital role in determining the performance of any optimization algorithms. The algorithm's designer can change the performance of the algorithms by applying the parameter tuning process of the algorithm. Specifically, we can say that poor tuning leads to poor performance, and the opposite is true. As mentioned in [ 268 ], it is practically not an easy task to tune the algorithm parameter and control it by changing its values. Another point we must refer to is that, for well-tuned parameters, there are no clear reasons for unchanging the values of these parameters during the optimization process. Until now, the process of parameter tuning has been implemented by applying parametric tests, while parameter control can be implemented stochastically in which the values of the parameters are picked randomly within a prespecified range. Therefore, there is an urgent need to develop automatic systematic methods to control and tune the parameters. The authors in [ 269 ] and [ 270 ] proposed a self-tuning method as a trial to encounter problems of parameter tuning and control, but with this trial, the computational cost is still expensive. Based on the previous notes, there is an urgent need to develop an automatic method that applies an adaptive change of the parameters in addition to less effect on the computational cost of the algorithm.

The lack of big data applicability Dealing with big data and developing a big data algorithm has turned into an urgent demand today as the data volume has increased dramatically with the help of automatic data collection methods. In this context, we noticed that there is no more concentration on the application of metaheuristics on big data in the current literature. There are no more studies on how to benefit from applying metaheuristics along with big data algorithms. Consequently, in this review, we inform the researchers to spend more effort and trials in developing new reliable methodologies and algorithms to solve big data problems with the help of metaheuristics.

The lack of machine learning and metaheuristics combination One of the most powerful and influential methods for making a decision and performing a predictions task is the machine learning (ML). Recently, very helpful results have been achieved by the ML techniques. So, researchers in the metaheuristics field must pay an attention to methods that benefit from the ML techniques in optimizing the work of current MAs algorithms or developing a new ML-based metaheuristics algorithms. The following points may be helpful and promising with regard to this point.

Using the new advances in reinforcement, ensemble, and deep learning in applying an automatic choice of specific problems to be handled by existing and new optimization algorithms [ 271 ].

Benefit from the capabilities of machine learning techniques in optimizing the work of the optimization field by generating an automatic model for representing the optimization problems, adjusting the analysis techniques for analyzing the search space, in addition to beating large and complex problems by decomposing them into smaller size problems [ 272 ]. In another prospective, we can use the ML capabilities in applying automatic configurations of the algorithms by allowing the ML algorithms to choose the appropriate values for the algorithm's operators, especially for metaheuristic algorithms due to a large number of parameters [ 273 ].

Shortened the gap between the metaheuristic’s algorithms and the problem domain knowledge Treating the problem as a black box is a double-edged weapon. However, this can be considered as a strength of the metaheuristic’s algorithms over other algorithms, but it also a challenge. Considering and integrating the domain knowledge of the problem with the designed algorithm will dramatically increase its performance. For example, a problem-orientated research direction can be obtained by designing the algorithm's operator and search mechanisms based on the characteristics of the problem which also can be benefit in reducing the complexity of the algorithm by considering the optimality conditions of the problem being considered [ 273 ].

In summary, the following observations from the experiment are:

Apply the MAs on parallel computing and combine the metaheuristic techniques with the modern parallel computing technologies to generate a powerful method matched with the future generation of computing.

Exploit the benefits of artificial intelligence and machine learning techniques to provide new algorithms that have the ability to automatically adjust the parameters and automatically analyze the algorithms.

Developing new methods directed towards strengthens the ability of MAs in addressing the large-scale global optimization (LSGO) problems.

A great effort must be paid for the hybridization process to allow the algorithms to use the Powers of many algorithms, also generating intelligent techniques that can provide the researcher with insights about what algorithms best suited to be hybridize together?

7.1 Emerging Technologies

After discussing the open issues and challenges, we see that there is much future work in the field of metaheuristics, a set of guidelines must be declared to help the future researcher in the field to address these challenges. In this section, the guidelines used to dive deeper into potential future research directions are introduced. Specifically, we will concentrate on two emerging technologies which are machine learning and quantum computing and how these technologies enhance the optimization process.

7.1.1 Quantum-Inspired Metaheuristics

Metaheuristics can be employed to obtain a global optimal solution for a wide range of different problems in different computational aspects. These methods can benefit from the concept of quantum computing (QC) to enhance the solutions obtained. Hybridizing the quantum computing with the metaheuristics will produce a quantum-inspired metaheuristic algorithm (QIMAs). QIMAs can be considered as an alternative approach to classical optimization methods for solving the optimization algorithm [ 274 ]. The main idea behind the QIMAs is to better use the quantum computing principles with the metaheuristics in order to boost the performance of the classical optimization algorithms by scoring a higher-performing results than traditional metaheuristic algorithms. Specifically, the use of QC in metaheuristics will accelerate convergence, enhance exploration, enhance exploitation, and provide a good balance between the two capabilities of the algorithm. The most promising merit that affects the performance of the algorithm is the parallel processing feature in QC [ 275 ]. Finally, QIMAs can be used in different disciplines such as engineering and science.

7.1.2 Intelligent Optimization

In this section we will introduce a new type of optimization that is considered as one of the most promising topics in the future of the metaheuristic field. Intelligent optimization (IO) is developed as a test to intelligently adjust the set of inputs and their values to achieve an optimal output(s). In other words, IO cost minimal consumption in determining and choosing the optimal solution among all possible solutions of the problem. The importance of using the IO is dramatically increased when solving complex and NP-hard problems in which the selection of the optimal solution through an exhaustive search is considered impossible or practically difficult. In addition, IO can be used as an important solution for the time-consuming problem of many optimization algorithms. IO can be used in all steps of the optimization process, such as defining the problem, handling, and formulating the objective function(s) and constraints.

7.1.3 Hybrid Metaheuristics and Mathematical Programming

In the last years, hybrid optimization algorithms have achieved promising results compared to classical optimization algorithms. The main aim behind the hybrid metaheuristics is to provide a reliable and high-performance solutions for the large and complex problems. One of the most widely used combinations is hybrid metaheuristics with mathematical programming approaches. This combination will increase the quality of the solution, as it benefits from the two methods in determining an exact solution in a reasonable amount of time. The following points define the mathematical programming approaches that can be used with metaheuristics to increase the quality of the solutions obtained [ 276 ].

Enumerative algorithms: in this approach we can use one of the well-known tree search methods such as dynamic programming and branch and bound. These methods follow the divide-and-conquer strategy where the search space can be divided into smaller search spaces, and then in each sub area we apply the optimization separately. By applying this strategy, the quality of the solution will increase, and the time consumed will decrease.

Decomposition and Relaxation methods: in this approach we can decompose the large problem using the Bender’s decomposition method or apply the Lagrangian relaxation method to convert the problem into smaller problems.

Pricing and Cutting plane algorithms: in this approach, we prune using polyhedral combinatorics.

8 Conclusion

In this review, a comprehensive study of metaheuristic algorithms is introduced that involves defining the concept of optimization. Studying the appearance of metaheuristic term. Introducing an explanation of the features of the MAs more than other techniques; Different taxonomies of the MAs according to different aspects such as inspiration source, number of search agents, population updating mechanisms, and number of parameters. Studying the metrics used in the Performance Evaluation of the algorithm. A great effort is paid to clarify the optimization problem in detail, concentrating on different classification techniques, and, moreover, the study reviews the use of metaheuristics in different application areas such as engineering design problems, NP hard problems, medical science, and robotics. Finally, we introduce some of the issues that exist in the MAs literature and the future directions of this important field.

Data Availability

Data sharing is not applicable to this article as no data sets were generated or analyzed during the current study.

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Houssein, E.H., Saeed, M.K., Hu, G. et al. Metaheuristics for Solving Global and Engineering Optimization Problems: Review, Applications, Open Issues and Challenges. Arch Computat Methods Eng (2024). https://doi.org/10.1007/s11831-024-10168-6

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How Georgia Tech is Using AI to Solve Sustainability Problems

Aug 21, 2024 —.

L to R, T to B: Josiah Hester, Peng Chen, Yongsheng Chen, Rosemarie Santa González, and Joe Bozeman.

- Written by Benjamin Wright -

As Georgia Tech establishes itself as a national leader in AI research and education , some researchers on campus are putting AI to work to help meet sustainability goals in a range of areas including climate change adaptation and mitigation, urban farming, food distribution, and life cycle assessments while also focusing on ways to make sure AI is used ethically and equitably.

Josiah Hester, interim associate director for Community-Engaged Research in the Brook Byers Institute for Sustainable Systems (BBISS), sees these projects as wins from both a research standpoint and for the local, national, and global communities they could affect.

“These faculty exemplify Georgia Tech's commitment to serving and partnering with communities in our research,” he says. “Sustainability is one of the most pressing issues of our time. AI gives us new tools to build more resilient communities, but the complexities and nuances in applying this emerging suite of technologies can only be solved by community members and researchers working closely together to bridge the gap. This approach to AI for sustainability strengthens the bonds between our university and our communities and makes lasting impacts due to community buy-in.”

Flood Monitoring and Carbon Storage

Peng Chen , assistant professor in the School of Computational Science and Engineering in the College of Computing, focuses on computational mathematics, data science, scientific machine learning, and parallel computing. Chen is combining these areas of expertise to develop algorithms to assist in practical applications such as flood monitoring and carbon dioxide capture and storage.

He is currently working on a National Science Foundation (NSF) project with colleagues in Georgia Tech’s School of City and Regional Planning and from the University of South Florida to develop flood models in the St. Petersburg, Florida area. As a low-lying state with more than 8,400 miles of coastline, Florida is one of the states most at risk from sea level rise and flooding caused by extreme weather events sparked by climate change.

Chen’s novel approach to flood monitoring takes existing high-resolution hydrological and hydrographical mapping and uses machine learning to incorporate real-time updates from social media users and existing traffic cameras to run rapid, low-cost simulations using deep neural networks. Current flood monitoring software is resource and time-intensive. Chen’s goal is to produce live modeling that can be used to warn residents and allocate emergency response resources as conditions change. That information would be available to the general public through a portal his team is working on.

“This project focuses on one particular community in Florida,” Chen says, “but we hope this methodology will be transferable to other locations and situations affected by climate change.”

In addition to the flood-monitoring project in Florida, Chen and his colleagues are developing new methods to improve the reliability and cost-effectiveness of storing carbon dioxide in underground rock formations. The process is plagued with uncertainty about the porosity of the bedrock, the optimal distribution of monitoring wells, and the rate at which carbon dioxide can be injected without over-pressurizing the bedrock, leading to collapse. The new simulations are fast, inexpensive, and minimize the risk of failure, which also decreases the cost of construction.

“Traditional high-fidelity simulation using supercomputers takes hours and lots of resources,” says Chen. “Now we can run these simulations in under one minute using AI models without sacrificing accuracy. Even when you factor in AI training costs, this is a huge savings in time and financial resources.”

Flood monitoring and carbon capture are passion projects for Chen, who sees an opportunity to use artificial intelligence to increase the pace and decrease the cost of problem-solving.

“I’m very excited about the possibility of solving grand challenges in the sustainability area with AI and machine learning models,” he says. “Engineering problems are full of uncertainty, but by using this technology, we can characterize the uncertainty in new ways and propagate it throughout our predictions to optimize designs and maximize performance.”

Urban Farming and Optimization

Yongsheng Chen works at the intersection of food, energy, and water. As the Bonnie W. and Charles W. Moorman Professor in the School of Civil and Environmental Engineering and director of the Nutrients, Energy, and Water Center for Agriculture Technology , Chen is focused on making urban agriculture technologically feasible, financially viable, and, most importantly, sustainable. To do that he’s leveraging AI to speed up the design process and optimize farming and harvesting operations.

Chen’s closed-loop hydroponic system uses anaerobically treated wastewater for fertilization and irrigation by extracting and repurposing nutrients as fertilizer before filtering the water through polymeric membranes with nano-scale pores. Advancing filtration and purification processes depends on finding the right membrane materials to selectively separate contaminants, including antibiotics and per- and polyfluoroalkyl substances (PFAS). Chen and his team are using AI and machine learning to guide membrane material selection and fabrication to make contaminant separation as efficient as possible. Similarly, AI and machine learning are assisting in developing carbon capture materials such as ionic liquids that can retain carbon dioxide generated during wastewater treatment and redirect it to hydroponics systems, boosting food productivity.

“A fundamental angle of our research is that we do not see municipal wastewater as waste,” explains Chen. “It is a resource we can treat and recover components from to supply irrigation, fertilizer, and biogas, all while reducing the amount of energy used in conventional wastewater treatment methods.”

In addition to aiding in materials development, which reduces design time and production costs, Chen is using machine learning to optimize the growing cycle of produce, maximizing nutritional value. His USDA-funded vertical farm uses autonomous robots to measure critical cultivation parameters and take pictures without destroying plants. This data helps determine optimum environmental conditions, fertilizer supply, and harvest timing, resulting in a faster-growing, optimally nutritious plant with less fertilizer waste and lower emissions.

Chen’s work has received considerable federal funding. As the Urban Resilience and Sustainability Thrust Leader within the NSF-funded AI Institute for Advances in Optimization ( AI4OPT ), he has received additional funding to foster international collaboration in digital agriculture with colleagues across the United States and in Japan, Australia, and India.

Optimizing Food Distribution

At the other end of the agricultural spectrum is postdoc Rosemarie Santa González in the H. Milton Stewart School of Industrial and Systems Engineering, who is conducting her research under the supervision of Professor Chelsea White and Professor Pascal Van Hentenryck , the director of Georgia Tech’s AI Hub as well as the director of AI4OPT.

Santa González is working with the Wisconsin Food Hub Cooperative to help traditional farmers get their products into the hands of consumers as efficiently as possible to reduce hunger and food waste. Preventing food waste is a priority for both the EPA and USDA. Current estimates are that 30 to 40% of the food produced in the United States ends up in landfills, which is a waste of resources on both the production end in the form of land, water, and chemical use, as well as a waste of resources when it comes to disposing of it, not to mention the impact of the greenhouses gases when wasted food decays.

To tackle this problem, Santa González and the Wisconsin Food Hub are helping small-scale farmers access refrigeration facilities and distribution chains. As part of her research, she is helping to develop AI tools that can optimize the logistics of the small-scale farmer supply chain while also making local consumers in underserved areas aware of what’s available so food doesn’t end up in landfills.

“This solution has to be accessible,” she says. “Not just in the sense that the food is accessible, but that the tools we are providing to them are accessible. The end users have to understand the tools and be able to use them. It has to be sustainable as a resource.”

Making AI accessible to people in the community is a core goal of the NSF’s AI Institute for Intelligent Cyberinfrastructure with Computational Learning in the Environment (ICICLE), one of the partners involved with the project.

“A large segment of the population we are working with, which includes historically marginalized communities, has a negative reaction to AI. They think of machines taking over, or data being stolen. Our goal is to democratize AI in these decision-support tools as we work toward the UN Sustainable Development Goal of Zero Hunger. There is so much power in these tools to solve complex problems that have very real results. More people will be fed and less food will spoil before it gets to people’s homes.”

Santa González hopes the tools they are building can be packaged and customized for food co-ops everywhere.

AI and Equity

Like Santa González, Joe Bozeman III is also focused on the ethical and sustainable deployment of AI and machine learning, especially among marginalized communities. The assistant professor in the School of Civil and Environmental Engineering is an industrial ecologist committed to fostering equitable climate change adaptation and mitigation strategies. His Social Equity and Environmental Engineering Lab works to make sure researchers understand the consequences of decisions before they move from academic concepts to policy decisions, particularly those that rely on data sets involving people and communities.

“With the administration of big data, there is a human tendency to assume that more data means everything is being captured, but that's not necessarily true,” he cautions. “More data could mean we're just capturing more of the data that already exists, while new research shows that we’re not including information from marginalized communities that have historically not been brought into the decision-making process. That includes underrepresented minorities, rural populations, people with disabilities, and neurodivergent people who may not interface with data collection tools.”

Bozeman is concerned that overlooking marginalized communities in data sets will result in decisions that at best ignore them and at worst cause them direct harm.

“Our lab doesn't wait for the negative harms to occur before we start talking about them,” explains Bozeman, who holds a courtesy appointment in the School of Public Policy. “Our lab forecasts what those harms will be so decision-makers and engineers can develop technologies that consider these things.”

He focuses on urbanization, the food-energy-water nexus, and the circular economy. He has found that much of the research in those areas is conducted in a vacuum without consideration for human engagement and the impact it could have when implemented.

Bozeman is lobbying for built-in tools and safeguards to mitigate the potential for harm from researchers using AI without appropriate consideration. He already sees a disconnect between the academic world and the public. Bridging that trust gap will require ethical uses of AI.

“We have to start rigorously including their voices in our decision-making to begin gaining trust with the public again. And with that trust, we can all start moving toward sustainable development. If we don't do that, I don't care how good our engineering solutions are, we're going to miss the boat entirely on bringing along the majority of the population.”

BBISS Support

Moving forward, Hester is excited about the impact the Brooks Byers Institute for Sustainable Systems can have on AI and sustainability research through a variety of support mechanisms.

“BBISS continues to invest in faculty development and training in community-driven research strategies, including the Energy Equity, Environmental Justice, and Community Engagement Faculty Fellows Program (with the Center for Sustainable Communities Research and Education ), while empowering multidisciplinary teams to work together to solve grand engineering challenges with AI by supporting the AI+Climate Faculty Interest Group, as well as partnering with and providing administrative support for community-driven research projects.”

Brent Verrill , Research Communications Program Manager, BBISS

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Operation Reserch Models Problems Solving Methods

Introduction:

Management deals with reality that is at once complex, dynamic, and multifaceted. It is neither possible nor desirable, to consider each and every element of reality before deciding the courses of action. It is impossible because of time available to decide the courses of action and the resources, which are limited in nature. More over in many cases, it will be impossible for a manager to conduct experiment in real environment.

Methods of Solving Operations Research Problems

There are three methods of solving an operations research problem. They are:

1.                   Analytical method,

2.                   Iterative Method,

3.                   The Monte-Carlo Technique.

Analytical Method : When we use mathematical techniques such as differential calculus, probability theory etc. to find the solution of a given operations research model, the method of solving is known as analytical method and the solution is known as analytical solution. Examples are problems of inventory models. This method evaluates alternative policies

efficiently.

Iterative Method (Numerical Methods) : This is trial and error method. When we have large number of variables, and we cannot use classical methods successfully, we use iterative process. First, we set a trial solution and then go on changing the solution under a given set of conditions, until no more modification is possible. The characteristics of this method is

that the trial and error method used is laborious, tedious, time consuming and costly. The solution we get may not be accurate one and is approximate one. Many a time we find that

after certain number of iterations, the solution cannot be improved and we have to accept it

as the expected optimal solution.

Monte-Carlo Method : This method is based on random sampling of variable's values from a distribution of the variable. This uses sampling technique. A table of random numbers must be available to solve the problems. In fact it is a simulation process.

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• Published: 19 August 2024

A many-objective evolutionary algorithm based on three states for solving many-objective optimization problem

• Jiale Zhao 1 , 4 ,
• Huijie Zhang 3 ,
• Huanhuan Yu 2 , 4 ,
• Hansheng Fei 2 , 4 ,
• Xiangdang Huang 2 , 4 &
• Qiuling Yang 2 , 4  

Scientific Reports volume  14 , Article number:  19140 ( 2024 ) Cite this article

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In recent years, researchers have taken the many-objective optimization algorithm, which can optimize 5, 8, 10, 15, 20 objective functions simultaneously, as a new research topic. However, the current research on many-objective optimization technology also encounters some challenges. For example: Pareto resistance phenomenon, difficult diversity maintenance. Based on the above problems, this paper proposes a many-objective evolutionary algorithm based on three states (MOEA/TS). Firstly, a feature extraction operator is proposed. It can extract the features of the high-quality solution set, and then assist the evolution of the current individual. Secondly, based on Pareto front layer, the concept of “individual importance degree” is proposed. The importance degree of an individual can reflect the importance of the individual in the same Pareto front layer, so as to further distinguish the advantages and disadvantages of different individuals in the same front layer. Then, a repulsion field method is proposed. The diversity of the population in the objective space is maintained by the repulsion field, so that the population can be evenly distributed on the real Pareto front. Finally, a new concurrent algorithm framework is designed. In the algorithm framework, the algorithm is divided into three states, and each state focuses on a specific task. The population can switch freely among these three states according to its own evolution. The MOEA/TS algorithm is compared with 7 advanced many-objective optimization algorithms. The experimental results show that the MOEA/TS algorithm is more competitive in many-objective optimization problems.

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Hybrid selection based multi/many-objective evolutionary algorithm

A new optimization algorithm to solve multi-objective problems

A multi-population multi-stage adaptive weighted large-scale multi-objective optimization algorithm framework

Introduction.

In reality, many optimization problems involve multiple conflicting objectives, such as the design of urban public transport routes 1 , production scheduling 2 , securities portfolio management 3 and so on. These types of optimization problems are called multi-objective optimization problems (MOPs). This means that there is no one solution to make all the objectives reach the optimum simultaneously, that is, the optimization of one objective may lead to the deterioration of other objectives 4 , 5 . Consequently, the solutions of MOPs are usually a set of compromise solutions that weigh all objectives. The definition of MOPs is as follows:

Among them, \(f\left(x\right)\) is the m-dimensional objective vector, which contains m conflicting objective functions; \({f}_{i}\left(x\right)\) represents the i-th objective function; x represents the n-dimensional decision variable; \(\Omega \) represents decision space; R m represents the objective space.

In the field of multi-objective optimization, problems with 2 or 3 optimization objectives are called general multi-objective optimization problems (GMOPs). Problems with more than 3 optimization objectives are called many-objective optimization problems (MaOPs) 6 , 7 , 8 . GMOPs aren’t the focus of our attention, as there have been many reports about GMOPs 9 , 10 . On the contrary, MaOPs are the focus of our attention, as there are still some challenges to be solved. The fundamental difference between GMOPs and MaOPs is the number of optimization objectives. Assuming that the number of optimization objectives is m, the probability that one individual dominates another is \(1/{2}^{m-1}\) in theory 11 , 12 . This means that with the increase of the number of optimization objectives, traditional Pareto dominance will fail, Pareto resistance will occur, and most multi-objective optimization algorithms will lose selection pressure in terms of convergence.

In recent years, with the research and exploration of MaOPs, many-objective optimization technology has been developed to a certain extent, and basically 4 mainstream many-objective optimization algorithms have been formed 13 . The first is many-objective optimization algorithm based on dominance. The algorithm modifies the definition of traditional Pareto domination by domination relaxation technique to enhance the selection pressure of the algorithm in terms of convergence. \(\alpha\) -Dominance, \(\upepsilon \) -Dominance and Cone \(\upepsilon \) -Dominance are all common domination relaxation techniques. Compared with traditional Pareto dominance, the effectiveness of dominance relaxation technology has been reported in many works. Therefore, dominance relaxation technology has been widely used to solve MaOPs. However, the current domination relaxation technique also faces two problems: (1) With the increase of the number of optimization objectives, the effect of the domination relaxation technique is getting worse and worse; (2) The domination relaxation technique tends to make the population converge to a certain sub-region of the real Pareto front (PF).

The second is many-objective optimization algorithm based on index. The algorithm guides the selection and evolution of the population by integrating convergence and diversity into one index (such as IGD, HV). Its representative work includes: HypE, MaOEA/IGD, SMS-EMOA. However, the algorithm faces some problems when it is used to solve MaOPs, such as complex index calculation, difficult selection of reference point or reference PF.

The third is many-objective optimization algorithm based on decomposition. The algorithm transforms MaOPs into several single-objective optimization sub-problems through an aggregation function, and then drives the individuals in the neighborhood to update by neighborhood strategy, finally realizes the evolution of the whole population. Its representative work includes: MOEA/D, MOEA/D-D, MOEA/D-DU. However, many-objective optimization algorithm based on decomposition is only suitable for MaOPs with regular PF (such as the DTLZ1 problem). When dealing with MaOPs with irregular PF, many-objective optimization algorithm based on decomposition often performs poorly.

The fourth is many-objective optimization algorithm based on hybrid strategy. The algorithm adopts different search strategies in different environments (different stages or different sub-populations), and uses the advantages of their respective search strategies to deal with complex MaOPs. Its representative work includes: AHM, eMOFEOA, CPSO. In many reports, many-objective optimization algorithm based on hybrid strategy is more suitable for solving MaOPs.

According to the above analysis, this paper considers using the many-objective optimization algorithm based on hybrid strategy, and further proposes the many-objective evolutionary algorithm based on three states (MOEA/TS). The innovations and contributions of this paper are as follows: (1) A feature extraction operator is proposed. The feature extraction operator is a feature extractor, which can extract the features of the high-quality solution set, and then assist the evolution of the current individual. (2) Based on the Pareto front layer, the concept of “individual importance degree” is proposed. The importance degree of an individual can reflect the importance of the individual in the same Pareto front layer, so as to further distinguish the advantages and disadvantages of different individuals in the same front layer, and effectively solve the phenomenon of Pareto resistance. (3) A repulsion field method is proposed. The repulsion field is used to maintain the diversity of the population in the objective space, so that the population can be evenly distributed on the real PF. (4) Design a new concurrent algorithm framework. In the framework, the algorithm is divided into three states, and each state focuses on a specific task. The population can freely switch among these three states according to its own evolution.

The remainder of this paper is organized as follows: Sect. " Preparatory work " introduces the basic definition, related work and research motivation. Sect. " Basic definition " introduces each part of the MOEA/TS algorithm in detail. Sect. " Related work " introduces the test results of MOEA/TS algorithm and 7 advanced many-objective optimization algorithms on various test problems, and then analyzes and summarizes them according to the test results. Sect. " Many-objective optimization algorithm based on dominance "summarizes this article and looks forward to future work.

Preparatory work

Basic definition.

In this section, we will introduce some basic definitions related to many-objective optimization technology.

Definition of dominance: if solution x isn’t worse than solution y in all objectives and solution x is better than solution y in at least one objective, it is said that x dominates y . That is, if \(\forall i\in \left\{\text{1,2},3,...,m\right\}\) satisfies \({f}_{i}\left(x\right)\le {f}_{i}\left(y\right)\) and \(\exists j\in \left\{\text{1,2},3,...,m\right\}\) satisfies \({f}_{j}\left(x\right)<{f}_{j}\left(y\right)\) , it is said that x dominates y .

Definition of non-dominated solution: if there are no solutions that can dominate x in the decision space, then x is called a Pareto optimal solution or a non-dominated solution. That is, if \(\nexists {x}^{*}\in \Omega \) makes x* dominate x , then x is called a Pareto optimal solution or a non-dominated solution.

Definition of Pareto optimal solution set: the set composed of Pareto optimal solutions is called the Pareto optimal solution set (PS). The mathematical description of PS is as follows:

Definition of Pareto front: the mapping of PS in the objective space is called Pareto front (PF). The mathematical description of PF is as follows:

The goal of the many-objective optimization technology is to find a set of non-dominated solutions that are close to the real PF (convergence) and make them well distributed on the real PF (diversity).

Related work

In recent years, many scholars have conducted in-depth research and exploration in the many-objective optimization technology.

Many-objective optimization algorithm based on dominance

Considering the limitations of Pareto dominance relationship in high-dimensional objective space, Zhou et al 14 proposed a many-objective optimization algorithm based on dominance relation selection. Firstly, they introduced an angle domination relationship with higher selection pressure based on the traditional Pareto domination relationship, and designed a new dominance selection strategy. Additionally, they proposed an angle-based individual distribution method to ensure even population distribution in the objective space. The algorithm shows strong competitiveness in solving MaOPs. Wang et al 15 believed that as the number of objectives increased, the traditional dominance relationship would become invalid. Therefore, they proposed a modified dominance relation. That is, they used penalty-based adaptive matrix regions to assist the traditional dominance relationship. Further, for MaOPs with irregular Pareto fronts, they introduced a population-based adaptive adjustment method to replace the predefined weight vector. On this basis, for MaOPs, they developed a many-objective optimization algorithm based on modified dominance relation and adaptive adjustment method. Zhang et al 16 believed that the current many-objective optimization algorithms focused too much on convergence, which would cause the population to converge to a certain sub-region of the real Pareto front. In order to solve this problem, they proposed a many-objective optimization algorithm based on double distance domination. In this algorithm, double distance can not only measure the convergence of the algorithm to adapt to different Pareto fronts, but also combine angle-based niche technology to emphasize the diversity of the algorithm. In addition, they also designed a special mutation operator. This operator can generate high-quality individuals in sparse areas to improve the diversity of the algorithm.

Many-objective optimization algorithm based on index

Aiming at the high complexity problem of hypervolume computation, Shang et al 17 proposed a new multi-objective evolutionary algorithm (MOEA) based on R2 index, namely the R2HCA-EMOA algorithm. The core idea of this algorithm is to use R2 index variables to approximate the contribution of hypervolume. The basic framework of the proposed algorithm is similar to that of SMS-EMOA. In order to improve the calculation efficiency of the algorithm, the utility tensor structure is introduced to calculate R2 index variables. In addition, the normalization mechanism is incorporated into the R2HCA-EMOA algorithm to improve its performance. Zhang et al 18 believed that the loss of selection pressure was the core reason for the poor performance of the algorithm. In order to solve this problem, they proposed a many-objective optimization algorithm based on fitness evaluation and hierarchical grouping. The fitness evaluation method combined the convergence measure based on the cos function and the diversity measure based on angle to create the selection pressure of convergence and diversity. In order to further strengthen the selection pressure, they proposed a hierarchical grouping strategy. Firstly, individuals are divided into different layers by front index, and then individuals in the same layer are divided into different groups by R2 index. Although some indexes can approximate the contribution of HV, However, Nan et al 19 believed that the key of performance evaluation was to find the worst solution rather than accurately approaching the HV value of each solution. In order to improve the ability to identify the worst solution, they proposed a two-stage R2 index evaluation method. In the first stage, the R2 indexes of all individuals are roughly evaluated to select some candidate solutions. In the second stage, these candidate solutions are accurately evaluated. Finally, they proposed a many-objective optimization algorithm based on the two-stage R2 index.

Many-objective optimization algorithm based on decomposition

In order to balance the convergence and diversity of the decomposition-based algorithm and reduce its dependence on the real PF direction, Wu et al 20 developed a many-objective optimization algorithm based on antagonistic decomposition method. This method utilizes the complementary characteristics of different sub-problems in a single example. Specifically, two populations are co-evolved by two sub-problems with different contours and opposite search directions. In order to avoid allocating redundant computing resources to the same area of PF, two populations are matched into one-to-one pairing according to their working areas on PF. In mating selection, each solution pair can only contribute one parent at most. In order to improve the performance of decomposition-based algorithms, Fan et al 21 proposed a differential multi-objective optimization algorithm based on decomposition. Firstly, they designed a neighborhood intimacy factor to improve the diversity of the algorithm based on the characteristics of neighborhood search. Then, they introduced a Gaussian mutation operator with dynamic step size to enhance the algorithm’s ability to escape from local optimal regions and improve convergence. Finally, they combined a difference strategy with the decomposition-based multi-objective optimization algorithm to further strengthen its evolutionary ability. Peng et al 22 believed that data dimensionality reduction could be applied to the objective space. Based on this consideration, they proposed a many-objective optimization algorithm based on projection. Firstly, they used the idea of data dimensionality reduction and spatial decomposition to divide the objective space into projection plane and free dimension. Then, a double elite strategy was used to maintain the balance between convergence and diversity of the algorithm. Finally, the algorithm based on decomposition was used as the algorithm of free dimension to solve MaOPs.

Many-objective optimization algorithm based on hybrid strategy

Aiming at convergence problem and diversity problem of the algorithm, Sun et al 23 proposed a many-objective optimization algorithm based on two independent stages. The algorithm deals with convergence and diversity problems in two independent and successive stages. Firstly, they introduced a non-dominated dynamic weight aggregation method, which is capable of identifying the Pareto optimal solutions of MaOPs. Then, they used these solutions to learn the Pareto optimal subspace in order to solve the convergence problem. Finally, the diversity problem was solved by using reference lines in the Pareto optimal subspace. Considering the advantages of the multi-objective and multi-population (MPMO) framework in solving MaOPs, Yang et al 24 proposed an algorithm based on the MPMO framework. The algorithm adopts the deviation sorting (BS) method to solve MaOPs, so as to obtain good convergence and diversity. In terms of convergence, the BS method is applied to each population in the MPMO framework, and the effect of non-dominant sorting is enhanced by the optimization objectives of the corresponding population. In terms of diversity, the maintenance method based on reference vector is used to save the diversity solutions. Aiming at the five-objective job shop scheduling problem (JSSP), Liu et al 25 proposed a new genetic algorithm based on the MPMO framework. Firstly, five populations are used to optimize five objectives, respectively. Secondly, in order to prevent each population from focusing only on its corresponding single objective, an archive sharing technology (AST) is proposed to store the elite solutions collected from five populations, so that the population can obtain the optimization information of other objectives from the archive. Thirdly, the archive updating strategy (AUS) is proposed to further improve the quality of the solutions in the archive.

Research motivation

Based on the related work, we believe that there are still the following problems in the current many-objective optimization technology:

(1) The diversity and convergence of the algorithm are difficult to balance. Most algorithms can’t coordinate the balance between them well, and they either emphasize convergence or diversity too much, which leads to poor quality of the non-dominated solution set.

(2) It is difficult to maintain the convergence of the algorithm. When the number of optimization objectives is large, the algorithm will produce Pareto resistance, and the traditional Pareto dominance may fail.

(3) It is difficult to maintain the diversity of the algorithm. Especially when the real PF is complex or the latitude of the objective space is high, individuals may have the clustering effect, and the population may not be evenly distributed on the real PF.

(4) The evolution efficiency of the algorithm is low. The traditional evolution operators have strong randomness and low evolution efficiency, and aren’t suitable for dealing with MaOPs.

Therefore, solving these problems and providing a good many-objective optimization algorithm constitute the research motivation of this paper.

For problem 1, some work attempts to separate the convergence optimization and diversity optimization of the algorithm, thus designing a concurrent algorithm architecture. Concurrent algorithm architecture means that only one of convergence or diversity is considered in one iteration instead of considering both convergence and diversity simultaneously. In order to solve GMOPs, Professor Ye Tian 26 tried to design a concurrent algorithm architecture and proposed the MSEA algorithm, and the experimental results were satisfactory. Therefore, it seems to be a feasible path to solve MaOPs by using concurrent algorithm architecture. However, recent research 23 shows that in MaOPs, the concurrent algorithm architecture seems to be unstable, and the experimental results fluctuate greatly (such as MaOEA/IT algorithm). Because when the algorithm only considers the convergence of the population, it often affects the diversity of the population; Similarly, when the algorithm only considers the diversity of the population, it often affects the convergence of the population. If a coordination intermediary can be added to the concurrent algorithm architecture to alleviate the contradiction between diversity and convergence, the concurrent algorithm architecture will become stable and its superiority will be truly reflected. Based on this motivation, this paper proposes a new concurrent algorithm framework. In the new algorithm framework, the algorithm is divided into three states, namely, convergence maintenance state, diversity maintenance state and coordination state. Each state focuses on a specific task. That is, the convergence maintenance state is responsible for improving the population convergence; Diversity maintenance state is responsible for improving population diversity; the coordination state is responsible for coordinating the contradiction between diversity and convergence. The population can freely switch among these three states according to its own evolution.

For problem 2, some scholars try to modify the definition of traditional Pareto dominance by using dominance relaxation technology to enhance the selection pressure of the algorithm in terms of convergence. However, with the increase of the number of optimization objectives, the effect of dominance relaxation technology is getting worse and worse. They only focus on the modification of Pareto domination definition, but ignore the difference between objective values. If we can distinguish the importance of different individuals by using the difference between the objective values, we can further create the selection pressure of the algorithm in terms of convergence, and finally Pareto resistance will be eliminated. Therefore, based on Pareto front layer, this paper proposes the concept of “individual importance degree”. The importance degree of an individual can reflect the importance of the individual in the same Pareto front layer, so as to further distinguish the advantages and disadvantages of different individuals in the same front layer, and effectively solve the phenomenon of Pareto resistance. Obviously, compared with domination relaxation technique, individual importance degree has greater advantages.

For problem 3, the traditional diversity maintenance technology isn’t suitable for high-dimensional objective space. For instance: the niche method, the density evaluation method, and the weight vector method. In the field of microphysics, when the distance between particles is too close, repulsion will push the particles away from their neighbors. On the contrary, when the distance between particles is too great, the repulsion will decrease and the particles tend to be close to the neighboring particles. This way makes the distribution of particles present a state of mutual coordination. Based on the characteristics of particle distribution, a repulsion field method is proposed in this paper. The repulsion field is used to maintain the diversity of the population in the objective space, so that the population can be evenly distributed on the real PF.

For problem 4, traditional evolution operators aren’t suitable for dealing with MaOPs. Because traditional evolution operators have strong randomness and low evolution efficiency. For instance: the binary crossover operator, the polynomial mutation operator, and the differential evolution operator. In principal component analysis, the decomposition of the covariance matrix and correlation matrix is a very important step. By decomposing the covariance matrix or the correlation matrix, we can obtain a set of orthogonal bases. These orthogonal bases are the most important features of the original data 27 . Therefore, this paper designs a feature extraction operator based on Cholesky decomposition 28 . The feature extraction operator can be understood as a feature extractor. It can extract the features of the high-quality solution set, and then assist the evolution of the current individual. Obviously, compared with traditional evolution operators, the feature extraction operator has higher evolution efficiency.

MOEA/TS algorithm

Feature extraction operator.

The feature extraction operator is a feature extractor, which can extract the features of the high-quality solution set, and then assist the evolution of the current individual. The workflow of the feature extraction operator is shown in Fig.  1 .

The workflow of feature extraction operator.

In the first step, W high-quality solutions \({x}^{1},{x}^{2},{x}^{3},...,{x}^{W}\) are selected from the population. These W solutions will form the high-quality solution set S.

In the second step, calculate the mean \(\overline{x}\) and covariance matrix A of the high-quality solution set S:

Among them, \({x}^{i}={\left({x}_{1}^{i},{x}_{2}^{i},{x}_{3}^{i},...,{x}_{n}^{i}\right)}^{T}, i\in (1,...,W); {x}_{j}={\left({x}_{j}^{1},{x}_{j}^{2},{x}_{j}^{3},...,{x}_{j}^{W}\right)}^{T}, j\in (1,...,n)\)

In the third step, Cholesky decomposition is performed on the covariance matrix A. That is, the covariance matrix A is decomposed into the product of the lower triangular matrix and the transposition of the lower triangular matrix. Assuming that the lower triangular matrix is L, there is

Through formula \(A=L*{L}^{T}\) , we can calculate \({a}_{11}={l}_{11}^{2}\) , that is, \({l}_{11}=\sqrt{{a}_{11}}\) . Then, according to \({a}_{i1}={l}_{i1}*{l}_{11}\) , we can get \({l}_{i1}={a}_{i1}/{l}_{11}\) , so we can get the first column element of matrix L .

Assuming that we have calculated the first k-1 column elements of the matrix L. Through

In this way, we can solve the k-th column element of matrix L through the first k-1 column elements of matrix L. Then, we can solve matrix L by recursion.

In the fourth step, the sampling vector \(s={\left({s}_{1},...,{s}_{n}\right)}^{T}\) is generated by Gaussian distribution \(N\left(\text{0,0.7}\right)\) . Then, a feature solution is generated.

Among them, \({x}^{feature}={\left({x}_{1}^{feature},...,{x}_{n}^{feature}\right)}^{T}\)

It should be noted that the standard deviation std is an important parameter of the Gaussian distribution. In this paper, the standard deviation std is set to 0.7. The parameter analysis verifies that 0.7 is a reasonable standard deviation. For more details on parameter analysis, please browse the experiment chapter (Parameter sensitivity analysis section).

In the fifth step, assuming that the selected individual is \({x}^{i}({x}_{1}^{i},...,{x}_{n}^{i})\) . Based on binary crossover operator 29 and feature solution, the formula of generating offspring individual is as follows:

Among them, \({c({c}_{1},...,{c}_{n})}^{T}\) is the offspring individual. \({\beta }_{k}\) is dynamically determined by the feature factor \(\mu \) :

Among them, \(rand\) is used to generate a random number between 0 and 1; \(r and i(\text{0,1})\) is used to generate 0 or 1 randomly.

For the design principle of formula ( 15 ), please browse the Supplementary Information Document.

In the sixth step, the individual \(c\) is detected and repaired. When some components in individual \(c\) exceed the upper bound or lower bound, these components need to be repaired. The repair formula is as follows:

Among them, \({c}_{i}^{u}\) , \({c}_{i}^{l}\) represent the upper bound and lower bound of the i-th component of individual \(c\) , respectively. \({{c}{\prime}({c}_{1}{\prime},...,{c}_{n}{\prime})}^{T}\) represents the repaired individual.

Individual importance degree based on the Pareto front layer

When the number of optimization objectives is large, the algorithm will produce Pareto resistance, and the traditional Pareto dominance may fail. Some scholars try to modify the definition of traditional Pareto dominance by using dominance relaxation technology to enhance the selection pressure of the algorithm in terms of convergence. However, with the increase of the number of optimization objectives, the effect of dominance relaxation technology is getting worse and worse. They only focus on the modification of Pareto domination definition, but ignore the difference between objective values. Figure  2 shows 4 non-dominant individuals. Among them, individual B is the closest to Origin O, individual C is second, individual A is third, and individual D is the farthest from Origin O. This means that in the population, individual B is the most important, individual C is the second most important, individual A is the third most important, and individual D is the least important. In addition, we can also find from Fig.  2 that there is a significant difference between the objective values of individual B and the objective values of other individuals, that is, \(\sum_{X\in \left\{A,C,D\right\}}\sum_{i=1}^{2}{f}_{i}(B)-{f}_{i}(X)\) is the smallest. This shows that there is a special relationship between the importance of individuals and the difference of the objective values. Based on this discovery, if we can distinguish the importance of different individuals by using the difference between the objective values, we can further create the selection pressure of the algorithm in terms of convergence, and finally Pareto resistance will be eliminated. Therefore, based on Pareto front layer, we propose the concept of “individual importance degree”.

Schematic diagram of individual importance.

Assuming that there are n solutions in a certain Pareto front layer, the objective function values of these solutions are normalized to [0,1] based on the maximum and minimum values of each objective function. \({f}_{k}{\prime}({x}^{i})\) represents the k-th normalized objective function value of individual \({x}^{i}\) .

Define the Pareto dominance function

The trend of the Pareto dominance function is shown in Fig.  3 .

The trend of the Pareto dominance function.

Pareto dominance function can be used to reflect the dominance degree among different individuals. For example, \(PDF({f}_{k}{\prime}({x}^{i})-{f}_{k}{\prime}({x}^{j}))\) represents the dominance degree of individual \({x}^{i}\) to individual \({x}^{j}\) on the k-th objective function; \(PDF({f}_{k}{\prime}({x}^{j})-{f}_{k}{\prime}({x}^{i}))\) represents the dominance degree of individual \({x}^{j}\) to individual \({x}^{i}\) on the k-th objective function; Obviously, the greater the dominance degree, the better one individual is than another on one objective function. Therefore, on one objective function, the dominance degree of one individual to another can be expressed as:

On this basis, the dominance degree of one individual to another can be expressed as:

Further, the importance degree of one individual to another can be expressed as:

Importance degree can indicate the importance of one individual to another. The greater the importance degree, the more important one individual is than another.

Since a certain Pareto front layer has n solutions, each solution needs to be compared with other n-1 solutions, so as to construct n-1 competing pairs. Assuming that an individual is \({x}^{i}\) , then the n-1 competing pairs are \(\left({x}^{i},{x}^{1}\right),\left({x}^{i},{x}^{2}\right),...,\left({x}^{i},{x}^{j}\right),...,\left({x}^{i},{x}^{n}\right)\) , respectively (note: \(i\ne j\) ). Thus, the importance degree of individual \({x}^{i}\) to the other n-1 individuals is \(Imp\left({x}^{i},{x}^{1}\right),Imp\left({x}^{i},{x}^{2}\right),...,Imp\left({x}^{i},{x}^{j}\right),...,Imp\left({x}^{i},{x}^{n}\right)\) , respectively (note: \(i\ne j\) ).

Finally, the importance degree of the individual \({x}^{i}\) can be expressed as:

The importance degree of one individual can reflect the importance of the individual in the same Pareto front layer, so as to further distinguish the advantages and disadvantages of different individuals in the same front layer. The greater the importance degree of one individual, the more important it is in the same Pareto front layer.

Figure 4 shows the use of individual importance degree. Firstly, based on a certain Pareto front layer, the competition pools and competition pairs are constructed. Then, the individual importance degree of different individuals is calculated by Formula ( 24 ). Finally, the importance of different individuals in the same Pareto front layer is obtained.

The use of individual importance degree.

Taking the two-objective optimization problem as an example, it is assumed that there are 4 non-dominant individuals. They are \(A\left(\text{17,5}\right)\) , \(B\left(\text{9,7}\right)\) , \(C\left(\text{7,15}\right)\) and \(D\left(\text{5,25}\right)\) , respectively. It means that these 4 individuals belong to the first non-dominant layer, and their advantages and disadvantages can’t be compared by the non-dominant rank. The distribution of 4 individuals in the objective space is shown in Fig.  5 (a).

The distribution of 4 individuals.

In order to better compare the advantages and disadvantages of these 4 individuals, we use the individual importance degree to deal with these 4 individuals. Firstly, the objective function values of these 4 individuals are normalized to [0,1]. After normalization, the coordinates of these 4 individuals are \(A\left(\text{1,0}\right)\) , \(B\left(\text{0.333,0.1}\right)\) , \(C\left(\text{0.167,0.5}\right)\) and \(D\left(\text{0,1}\right)\) , respectively. The distribution of 4 individuals in the normalized objective space is shown in Fig.  5 (b). Next, according to Fig.  4 , the competition pools and competition pairs are constructed. Then, according to formula ( 22 ) and formula ( 23 ), the \(P\left({x}^{i},{x}^{j}\right)\) and \(Imp\left({x}^{i},{x}^{j}\right)\) of each competition pair are calculated. The calculation results are shown in Table 1 . Finally, according to the formula ( 24 ), the importance degree of these 4 individuals is 0.1918, 0.9488, 0.6673 and 0.1921, respectively. The results show that individual B is the most important, individual C is the second most important, individual D is the third most important, and individual A is the least important. This result is consistent with the intuitive perception that we get from Fig.  5 (b). Based on the above example, we believe that the concept of individual importance degree and related process are effective and can achieve the desired goals.

Repulsion field method

In the field of microphysics, when the distance between particles is too close, repulsion will push the particles away from their neighbors. On the contrary, when the distance between particles is too great, the repulsion will decrease and the particles tend to be close to the neighboring particles. This way makes the distribution of particles present a state of mutual coordination (As shown in Fig.  6 ). Based on the characteristics of particle distribution, a repulsion field method is proposed in this paper. The repulsion field is used to maintain the diversity of the population in the objective space, so that the population can be evenly distributed on the real PF.

The uniform distribution of microscopic particles.

Firstly, according to the maximum and minimum values of each objective function, the objective function values of all solutions in the population are normalized to [0,1]. \({f}_{k}{\prime}({x}^{i})\) represents the k-th normalized objective function value of individual \({x}^{i}\) .

Then, a repulsion potential field with repulsion radius r is constructed around each individual. Assuming that a repulsion potential field has been constructed for individual \({x}^{i}\) , then all individuals within the repulsion potential field will be subject to the repulsion potential from individual \({x}^{i}\) . The magnitude of the repulsion potential depends on the distance between other individuals and individual \({x}^{i}\) . When other individuals are outside the repulsion potential field of individual \({x}^{i}\) , the repulsion potential is 0. When other individuals are within the repulsion potential field of individual \({x}^{i}\) , the closer the other individuals are to individual \({x}^{i}\) , the greater the repulsion potential that they obtain. Assuming that there is individual \({x}^{j}\) , then the repulsion potential that individual \({x}^{j}\) obtains is

Among them, \(\rho \) is the gain coefficient of the repulsion potential field, usually set to 1; \(r\) is the radius of the repulsion potential field; \(dis\left({x}^{j},{x}^{i}\right)\) represents the euclidean distance between individual \({x}^{j}\) and individual \({x}^{i}\) in the objective space. The formula is as follows:

Further, the repulsion \(Rep\left({x}^{j},{x}^{i}\right)\) that individual \({x}^{j}\) obtains is the negative gradient of the repulsion potential \(Repfield\left({x}^{j},{x}^{i}\right)\) . The formula is as follows:

It means that when \(dis\left({x}^{j},{x}^{i}\right)\le r\) , the smaller \(dis\left({x}^{j},{x}^{i}\right)\) is, the larger \(Rep\left({x}^{j},{x}^{i}\right)\) is. when \(dis\left({x}^{j},{x}^{i}\right)>r\) , \(Rep\left({x}^{j},{x}^{i}\right)=0\) .

Based on the repulsion potential field, the total repulsion potential that individual \({x}^{j}\) obtains is

Finally, the total repulsion that individual \({x}^{j}\) obtains is

It should be noted that the repulsion potential and repulsion proposed in this paper are both vectors. It means that repulsion potential and repulsion have both magnitude and direction. The addition of different repulsion is the vector synthesis of repulsion, rather than the pure numerical addition. This is also an obvious feature that the repulsion field method is different from other scalar function methods (such as niche method). Figure 7 shows the vector synthesis process of repulsion in a two-dimensional space environment. Among them, F SUM is the total repulsion that individual A obtains; F BA is the repulsion generated by individual B to individual A; F CA is the repulsion generated by individual C to individual A.

The vector synthesis process of repulsion.

In the repulsion field method, the individual with large repulsion usually means that the individual is located in the multiple repulsion potential field that other individuals construct. It indicates that the individual is located in a dense area in the objective space and is close to other individuals. Therefore, individuals with large repulsion aren’t conducive to maintaining population diversity. Naturally, we hope that individuals with large repulsion can move away from dense areas in the objective space along the direction of repulsion. Based on this idea, firstly, we need to find some individuals closest to the direction of the repulsion to construct a high-quality solution set. Then, the feature extraction operator is used to extract the location features of the high-quality solution set. Finally, based on these features, individuals with large repulsion can evolve along the direction of repulsion. As shown in Fig.  8 , individual D and individual E are the individuals closest to the direction of repulsion. The feature extraction operator is used to extract the position features of these two individuals. Based on these features, individual A evolves into individual A*, which is far away from the previous dense area.

Individual A is far away from the dense area.

It should be noted that the feature extraction operator has the randomness caused by Gaussian sampling. Therefore, the evolution of individuals also has a certain degree of randomness.

Framework of MOEA/TS algorithm

The framework of the MOEA/TS algorithm is shown in Fig.  9 . Firstly, the relevant parameters of the algorithm are initialized; secondly, judge which state the algorithm is in. If the algorithm is in the convergence maintenance state, the following steps are adopted to improve the convergence of the algorithm: (1) Randomly select the parent individual. (2) Use feature extraction operator to generate offspring individuals. (3) If the offspring individual is superior to the individual with the worst convergence in the population, the worst individual is replaced by the offspring individual. If the algorithm is in the diversity maintenance state, the following steps are adopted to improve the diversity of the algorithm: (1) Select the individual with the worst diversity in the population. (2) Use feature extraction operator to generate offspring individuals. (3) If the offspring individual is superior to the individual with the worst diversity in the population, the worst individual is replaced by the offspring individual. If the algorithm is in the coordination state, the following steps are adopted to coordinate the convergence and diversity of the algorithm: (1) Randomly select the parent individual. (2) Use the Gaussian mutation operator to generate offspring individuals. (3) If the offspring individual is superior to the parent individual in convergence and diversity, the parent individual is replaced by the offspring individual. Then, it is judged whether the algorithm has completed the i-th iteration. If the algorithm doesn’t complete the i-th iteration, the corresponding maintenance step or coordination step is re-executed. If the algorithm completes the i-th iteration, the current state of the algorithm is updated. Finally, it is judged whether the algorithm ends. If the algorithm doesn’t end, the corresponding maintenance step or coordination step is performed according to the current state of the algorithm. If the algorithm is finished, the population is output.

The framework of MOEA/TS algorithm.

Description of MOEA/TS algorithm

Main framework.

This section describes the main framework of the MOEA/TS algorithm. The pseudo-code of the main framework of the MOEA/TS algorithm is shown in Algorithm 1. The main steps include: in line (1), initializing population P, repulsion field radius r, and state value (state=1 means that the algorithm is in convergence maintenance state, state=2 means that the algorithm is in diversity maintenance state, and state=3 means that the algorithm is in coordination state.); In line (2), the Front value, Imp value and Rep value of each solution in population P are calculated (The Front value is calculated by the fast non-dominated sorting method.); In line (3), it is judged whether the algorithm meets the termination condition (The termination condition is usually the maximum iterations.); In line (4), the count value is initialized. The count value is used to count the number of updated solutions in the i-th iteration; in lines (5)-(11), according to the current state of the algorithm, the update way of the population is selected. When state=1, the convergence of the population is updated. When state=2, the diversity of the population is updated. When state=3, the convergence and diversity of the population are coordinated; in line (12), the state value of the algorithm is updated according to the current state of the algorithm and the count value.

Main framework.

Convergence maintenance

This section mainly describes the convergence maintenance of the population. The pseudo-code of convergence maintenance is shown in Algorithm 2. The main steps include: in line (1), the algorithm enters the i-th iteration; In line (2), one parent individual is randomly selected from population P; In lines (3)–(4), based on Front, Imp, the high-quality solution set S is constructed; In line (5), the feature extraction operator is used to extract the features of the high-quality solution set S, and then assist the evolution of the parent individual; In line (6), the individual with the worst convergence in the population is found; In lines (7)–(13), if the offspring individual is superior to the individual with the worst convergence in the population, the worst individual is replaced by the offspring individual and flag is marked as 1. If the offspring individual is inferior to the individual with the worst convergence in the population, the flag is marked as 0. Among them, the flag value is used to indicate whether the population P has changed (flag=0 means that the population P hasn’t changed, flag=1 means that the population P has changed.); In lines (14)–(16), it is judged whether flag equals 1. If flag equals 1, the count value is updated, and the Front value, Imp value and Rep value of each solution in population P are updated.

Convergence maintenance.

Diversity maintenance

This section mainly describes the diversity maintenance of the population. The pseudo-code of diversity maintenance is shown in Algorithm 3. The main steps include: in line (1), the algorithm enters the i-th iteration; In line (2), the individual with the worst diversity in the population is found; In line (3), according to the direction of total repulsion that the worst individual obtains, the distance \({dis}_{j}\) can be calculated; In line (4), based on \({dis}_{j}\) , the high-quality solution set S is constructed; In line (5), the feature extraction operator is used to extract the features of the high-quality solution set S, and then assist the evolution of the worst individual; In lines (6)–(12), if the offspring individual is superior to the individual with the worst diversity in the population, the worst individual is replaced by the offspring individual and flag is marked as 1. If the offspring individual is inferior to the individual with the worst diversity in the population, the flag is marked as 0; In lines (13)–(15), it is judged whether flag equals 1. If flag equals 1, the count value is updated, and the Front value, Imp value and Rep value of each solution in population P are updated.

Diversity maintenance.

Coordination of convergence and diversity

This section mainly describes the coordination of convergence and diversity of the population. The pseudo-code of coordination of convergence and diversity is shown in Algorithm 4. The main steps include: in line (1), the algorithm enters the i-th iteration; In line (2), one parent individual is randomly selected from population P; In line (3), based on the parent individual, the Gaussian mutation operator is used to generate the offspring solution; In lines (4)–(10), if the offspring individual is superior to the parent individual in convergence and diversity, the parent individual is replaced by the offspring individual and flag is marked as 1. If the offspring individual is inferior to the parent individual in convergence and diversity, the flag is marked as 0; In lines (11)–(13), it is judged whether flag equals 1. If flag equals 1, the count value is updated, and the Front value, Imp value and Rep value of each solution in population P are updated.

Coordination.

This section mainly describes the feature extraction operator. The pseudo-code of the feature extraction operator is shown in Algorithm 5. The main steps include: in line (1), the features \(\overline{x},L\) of the high-quality solution set S are extracted by formula ( 4 ) and formula ( 13 ); In line (2), the sampling vector \(s={\left({s}_{1},...,{s}_{n}\right)}^{T}\) is generated by the Gaussian distribution \(N\left(\text{0,0.7}\right)\) ; In line (3), based on \(\text{s},\overline{x},L\) , the feature solution \({x}^{feature}\) is generated by formula ( 14 ); In line (4), based on parent, \({x}^{feature}\) , the offspring solution O is generated by formula ( 15 ).

Feature extraction.

Update of algorithm state

In this paper, the algorithm state is further updated according to the current state of the algorithm and the stability of the population. The pseudo-code of the update of the algorithm state is shown in Algorithm 6. When the algorithm is in the convergence maintenance state and the number of updated solutions in the i-th iteration is less than or equal to 5%*N, it is considered that the population tends to be stable in terms of convergence, then the algorithm turns to the diversity maintenance state; When the algorithm is in the diversity maintenance state and the number of updated solutions in the i-th iteration is less than or equal to 5%*N, it is considered that the population tends to be stable in terms of diversity, then the algorithm turns to the coordination state; When the algorithm is in the coordination state and the number of updated solutions in the i-th iteration is less than or equal to 5%*N, it is considered that the population tends to be stable in terms of coordination, then the algorithm turns to the convergence maintenance state. It should be noted that the threshold value T is a key parameter in measuring whether the population tends to be stable or not. In this paper, the threshold value T is set to 5%. The parameter analysis verifies that 5% is a reasonable threshold. For more details on parameter analysis, please browse the experiment chapter (Parameter sensitivity analysis section).

Determination state.

Computational complexity of one iteration of MOEA/TS algorithm

Assuming that the size of the population is N, the number of the objective function is m, the dimension of the decision variable is n, and the size of the high-quality solution set is W, then the computational complexity of Rep is O(mN2), the computational complexity of Front is O(mN2), and the computational complexity of Imp is O(mN2). The core steps of the feature extraction operator (Algorithm 5) include the construction of the covariance matrix and Cholesky decomposition. The computational complexity of covariance matrix construction is O(Wn2) and the computational complexity of Cholesky decomposition is O(n3/6). Therefore, the computational complexity of the feature extraction operator (Algorithm 5) is O(Wn2+n3/6). The core steps of convergence maintenance (Algorithm 2) include population ranking, feature extraction operator, selection of the worst individual, and updating of Front, Imp and Rep. Their computational complexity is O(N2), O(Wn2+n3/6), O(N), O(mN2), O(mN2), O(mN2), respectively. Therefore, the computational complexity of convergence maintenance (Algorithm 2) is O(N(N2+Wn2+n3/6+N+3mN2)). The core steps of diversity maintenance (Algorithm 3) include selection of the worst individual, distance calculation, population ranking, feature extraction operator, and updating of Front, Imp and Rep. Their computational complexity is O(N), O(nN), O(N2), O(Wn2+n3/6), O(mN2), O(mN2), O(mN2), respectively. Therefore, the computational complexity of diversity maintenance (Algorithm 3) is O(N(N+nN+N2+Wn2+n3/6+3mN2)). The core steps of coordination of convergence and diversity (Algorithm 4) include the Gaussian mutation operator, and updating of Front, Imp and Rep. Their computational complexity is O(n), O(mN2), O(mN2), O(mN2), respectively. Therefore, the computational complexity of coordination of convergence and diversity (Algorithm 4) is O(N(n+3mN2)). The computational complexity of Determination State (Algorithm 6) is O(1). Based on the above computational complexity analysis, the computational complexity of one iteration of the MOEA/TS algorithm is max{O(N(N2+Wn2+n3/6+N+3mN2)), O(N(N+nN+N2+Wn2+n3/6+3mN2)), O(N(n+3mN2))}+O(1)≈max{O(NWn2+Nn3+mN3), O(NWn2+Nn3+mN3), O(mN3)}= O(NWn2+Nn3+mN3). In this paper, N>>max{W, n, m}. Therefore, the computational complexity of the MOEA/TS algorithm is O(mN3). As a reference algorithm, the computational complexity of the NSGA-III algorithm is O(mN2). The computational complexity of the MOEA/TS algorithm is an order of magnitude higher than that of the NSGA-III algorithm. This shows that the MOEA/TS algorithm is an expensive many-objective optimization algorithm.

It should be noted that although MOEA/TS algorithm has a higher computational complexity. But compared with the NSGA-III algorithm, the MOEA/TS algorithm also has greater advantages. In terms of convergence optimization, the NSGA-III algorithm adopts the traditional definition of Pareto domination. Obviously, the traditional definition can’t solve the problem of Pareto resistance. MOEA/TS algorithm uses the concept of “individual importance degree”. Individual importance degree can solve the problem of Pareto resistance. In terms of diversity optimization, the NSGA-III algorithm uses predefined reference points. The predefined reference points can’t solve the problem that the population can't be evenly distributed on the real PF in the high-dimensional objective space. MOEA/TS algorithm uses the repulsion field method. The repulsion field method can solve the problem that the population can’t be evenly distributed on the real PF in the high-dimensional objective space. In terms of algorithm architecture, the NSGA-III algorithm adopts the serial algorithm architecture. The serial algorithm architecture is difficult to coordinate the convergence optimization and diversity optimization of the algorithm. MOEA/TS algorithm adopts the concurrent algorithm architecture. The concurrent algorithm architecture can coordinate the convergence optimization and diversity optimization of the algorithm. In terms of operators, the NSGA-III algorithm uses the traditional binary crossover operator and polynomial mutation operator. The evolutionary ability of these two operators is weak. MOEA/TS algorithm uses feature extraction operator. Feature extraction operator has strong evolutionary ability. Therefore, the MOEA/TS algorithm has better performance than the NSGA-III algorithm. The comparison results support our conclusion. For the comparison results of these two algorithms, please browse Supplementary Information Document.

Experimental results and analysis

Experimental settings, configuration of experimental software and hardware.

The hardware and software configurations of the experiment are shown in Table 2 . Among them, PlatEMO 30 is a professional many-objective optimization experiment platform. The platform includes multiple test function sets and many-objective optimization algorithms.

Test function

The test functions used in the experiment include: DTLZ test function set (DTLZ1-7), MAF test function set (MAF1-6) and WFG test function set (WFG1-9). Literature 31 describes the characteristics of related test functions. The parameter settings of the related test functions are shown in Table 3 .

Comparison algorithm

In order to verify the performance of MOEA/TS algorithm in the many-objective optimization field, this paper compares MOEA/TS algorithm with 7 advanced many-objective optimization algorithms. These 7 many-objective optimization algorithms include: VMEF 32 , BiGE-BEW 33 , MOEA/DG 34 , MOEA/D 35 , LSMaODE 36 , MaOEA/IT 23 and MaOEA/IGD 37 .

For all test cases, Wilcoxon rank sum test at 5% significance level 38 is used to compare the significance of the difference between the MOEA/TS algorithm and the comparison algorithms. The symbol “+” indicates that the comparison algorithms are significantly better than the MOEA/TS algorithm; the symbol “-“indicates that the comparison algorithms are significantly inferior to the MOEA/TS algorithm. The symbol “=” indicates that there is no significant difference between the MOEA/TS algorithm and the comparison algorithms.

Performance evaluation

In the aspect of performance evaluation, this paper uses inverted generational distance plus (IGD+) and hypervolume (HV) 39 to measure the performance of many-objective optimization algorithm. The smaller the IGD+ value that the algorithm obtains, the better the performance of the algorithm. The larger the HV value that the algorithm obtains, the better the performance of the algorithm.

In order to facilitate observation, we provide the normalized HV value of each algorithm relative to the best HV result. This normalization makes all the results lie in the range [0,1], and 1 represents the best value.

Considering the length of the paper, we only show the IGD+ values of different algorithms in the experiment chapter. For the HV values of different algorithms, please browse the Supplementary Information Document.

Parameter setting

In terms of algorithm parameters, according to some existing parameter research results 13 , 40 , the feature factor \(\mu \) is set to 20 in this paper. According to the parameter sensitivity analysis, the number of high-quality solutions W is set to 9 in this paper. The parameter sensitivity analysis of W is detailed in the subsequent chapters.

The algorithm parameters of the 7 comparison algorithms are determined according to the best parameters provided by the corresponding literature.

Performance comparison under benchmark test functions

Performance comparison under dtlz test function set.

In this paper, each algorithm is executed 30 times to get the average data as shown in Table 4 . As can be seen from Table 4 , MOEA/TS algorithm wins the first place in 15 test cases; BiGE-BEW algorithm wins the first place in 5 test cases; MOEA/D algorithm wins the first place in 15 test cases. In the 35 test cases, the number of MOEA/TS algorithm is significantly superior to VMEF algorithm, BiGE-BEW algorithm, MOEA/DG algorithm, MOEA/D algorithm, LSMaODE algorithm, MaOEA/IT algorithm and MaOEA/IGD algorithm is 21, 27, 25, 16, 32, 35 and 31, respectively. The number of MOEA/TS algorithm is significantly inferior to VMEF algorithm, BiGE-BEW algorithm, MOEA/DG algorithm, MOEA/D algorithm, LSMaODE algorithm, MaOEA/IT algorithm and MaOEA/IGD algorithm is 6, 5, 5, 15, 1, 0 and 0, respectively. Statistically, the number of MOEA/TS algorithm is similar to VMEF algorithm, BiGE-BEW algorithm, MOEA/DG algorithm, MOEA/D algorithm, LSMaODE algorithm, MaOEA/IT algorithm and MaOEA/IGD algorithm is 8, 3, 5, 4, 2, 0 and 4, respectively. Therefore, in the DTLZ test function set, MOEA/TS algorithm and MOEA/D algorithm have the best performance. The performance of VMEF algorithm, MOEA/DG algorithm, BiGE-BEW algorithm and LSMaODE algorithm decreases in turn. The performance of MaOEA/IGD algorithm and MaOEA/IT algorithm is similar and the worst.

Based on Table 4 , we further analyze the performance of these algorithms. In the DTLZ test function set, MOEA/TS algorithm performs poorly on DTLZ1, DTLZ5 and DTLZ6 test functions. The possible reasons are that the DTLZ1 test function has multiple local optima, and the DTLZ5 and DTLZ6 test functions have a narrow convergence curve. In the DTLZ1 test function, although the repulsion field method of the MOEA/TS algorithm makes the population widely distributed. However, its population distribution isn’t uniform and regular. The population distribution of some algorithms using predefined weight vectors is uniform and regular. In the DTLZ5 and DTLZ6 test functions, the coordination mechanism of MOEA/TS algorithm fails. The narrow convergence curve makes the population more concentrated, but the repulsion field method will disperse the population. The coordination mechanism is difficult to play a role.

The real Pareto front of DTLZ test function set is regular and the function complexity isn’t high. Therefore, algorithms with better diversity may be more popular. MOEA/D algorithm uses predefined weight vectors to maintain diversity and aggregation functions to maintain convergence. Therefore, it has good performance. VMEF algorithm uses different convergence ranking methods to deal with different test problems. Therefore, VMEF algorithm is good in convergence and poor in diversity. Based on the convergence measure and diversity measure, BiGE-BEW algorithm transforms the many-objective optimization problem into a two-objective optimization problem. In theory, the algorithm should perform well. However, there are defects in its convergence and diversity measurement formula. Finally, the experimental results of the algorithm aren’t as good as the expected results. MOEA/DG algorithm still uses the traditional dominance relationship to maintain the convergence of external archives. Therefore, MOEA/DG algorithm is poor in convergence and good in diversity. LSMaODE algorithm divides the population into two subpopulations and uses different strategies to optimize them. Because the real Pareto front of DTLZ test function set isn’t complex, the advantage of this multi-population algorithm architecture isn’t obvious. Therefore, compared with other algorithms, its performance is mediocre. MaOEA/IT algorithm optimizes convergence and diversity through two independent phases. However, the algorithm's performance is always poor because it doesn’t alleviate the contradiction between convergence and diversity. The reference Pareto front of MaOEA/IGD algorithm is poor. Therefore, the algorithm’s performance is always poor.

Performance comparison under MAF test function set

In this paper, each algorithm is executed 30 times to get the average data as shown in Table 5 . As can be seen from Table 5 , MOEA/TS algorithm wins the first place in 10 test cases; BiGE-BEW algorithm wins the first place in 8 test cases; MOEA/DG algorithm wins the first place in 2 test cases; MOEA/D algorithm wins the first place in 5 test cases; LSMaODE algorithm wins the first place in 5 test cases. In the 30 test cases, the number of MOEA/TS algorithm is significantly superior to VMEF algorithm, BiGE-BEW algorithm, MOEA/DG algorithm, MOEA/D algorithm, LSMaODE algorithm, MaOEA/IT algorithm and MaOEA/IGD algorithm is 22, 18, 25, 21, 20, 27 and 30, respectively. The number of MOEA/TS algorithm is significantly inferior to VMEF algorithm, BiGE-BEW algorithm, MOEA/DG algorithm, MOEA/D algorithm, LSMaODE algorithm, MaOEA/IT algorithm and MaOEA/IGD algorithm is 6, 11, 2, 5, 9, 1 and 0, respectively. Statistically, the number of MOEA/TS algorithm is similar to VMEF algorithm, BiGE-BEW algorithm, MOEA/DG algorithm, MOEA/D algorithm, LSMaODE algorithm, MaOEA/IT algorithm and MaOEA/IGD algorithm is 2, 1, 3, 4, 1, 2 and 0, respectively. Therefore, in the MAF test function set, MOEA/TS algorithm has the best performance. The performance of BiGE-BEW algorithm, LSMaODE algorithm, VMEF algorithm, MOEA/D algorithm, MOEA/DG algorithm and MaOEA/IT algorithm decreases in turn. The performance of MaOEA/IGD algorithm is the worst.

Based on Table 5 , we further analyze the performance of these algorithms. In the MAF test function set, MOEA/TS algorithm performs poorly on MAF2 and MAF3 test functions. The possible reasons are that the MAF2 test function greatly increases the difficulty of convergence on the basis of the DTLZ2 test function, and the MAF3 test function has a convex Pareto front and many local fronts. In the MAF2 test function, although the MOEA/TS algorithm can recognize the advantage and disadvantage of different individuals in the same front layer, the evolutionary efficiency of the MOEA/TS algorithm isn’t ideal. In other words, after the algorithm is finished, the population still has the large evolution potential in convergence. In the MAF3 test function, MOEA/TS algorithm can effectively deal with the convex Pareto front. However, MOEA/TS algorithm is difficult to deal with multiple local fronts because feature extraction operator of MOEA/TS algorithm is difficult to extract features of multiple local fronts.

MAF test function set is the variety of DTLZ test function set. It adds a lot of characteristics to the DTLZ test function set. For example, degenerate, convex, concave, partial, multimodal, deceptive, et al. Therefore, the MAF test function set is more difficult in terms of convergence and diversity. Based on the convergence measure and diversity measure, BiGE-BEW algorithm transforms the many-objective optimization problem into a two-objective optimization problem. Although there are some defects in its diversity and convergence measurement formula, BiGE-BEW algorithm shows good performance in convergence when dealing with more complex MaOPs. VMEF algorithm uses different convergence ranking methods to deal with different test problems. However, the complex Pareto fronts and diversified characteristics still pose a great challenge to VMEF algorithm. Therefore, the performance of VMEF algorithm is mediocre. MOEA/DG algorithm still uses the traditional dominance relationship to maintain the convergence of external archives. Therefore, MOEA/DG algorithm is poor in convergence. MOEA/D algorithm uses predefined weight vectors to maintain diversity and aggregation functions to maintain convergence. MOEA/D algorithm can easily deal with the DTLZ test function set. However, its performance isn’t ideal when dealing with more complex MAF test function set. Surprisingly, LSMaODE algorithm shows good performance. We speculate that the possible reason is that the real Pareto front of the MAF test function set is complex, and then the advantages of multi-population algorithm architecture can be reflected. MaOEA/IT algorithm optimizes convergence and diversity through two independent phases. However, the algorithm’s performance is always poor because it doesn’t alleviate the contradiction between convergence and diversity. The reference Pareto front of MaOEA/IGD algorithm is poor. Therefore, the algorithm’s performance is always poor.

Performance comparison under WFG test function set

In this paper, each algorithm is executed 30 times to get the average data as shown in Table 6 . As can be seen from Table 6 , MOEA/TS algorithm wins the first place in 27 test cases; VMEF algorithm wins the first place in 8 test cases; BiGE-BEW algorithm wins the first place in 6 test cases; MOEA/DG algorithm wins the first place in 1 test case; LSMaODE algorithm wins the first place in 3 test cases. In the 45 test cases, the number of MOEA/TS algorithm is significantly superior to VMEF algorithm, BiGE-BEW algorithm, MOEA/DG algorithm, MOEA/D algorithm, LSMaODE algorithm, MaOEA/IT algorithm and MaOEA/IGD algorithm is 26, 29, 42, 45, 39, 45 and 43, respectively. The number of MOEA/TS algorithm is significantly inferior to VMEF algorithm, BiGE-BEW algorithm, MOEA/DG algorithm, MOEA/D algorithm, LSMaODE algorithm, MaOEA/IT algorithm and MaOEA/IGD algorithm is 10, 9, 3, 0, 3, 0 and 0, respectively. Statistically, the number of MOEA/TS algorithm is similar to VMEF algorithm, BiGE-BEW algorithm, MOEA/DG algorithm, MOEA/D algorithm, LSMaODE algorithm, MaOEA/IT algorithm and MaOEA/IGD algorithm is 9, 7, 0, 0, 3, 0 and 2, respectively. Therefore, in the WFG test function set, MOEA/TS algorithm has the best performance. The performance of VMEF algorithm, BiGE-BEW algorithm, LSMaODE algorithm and MOEA/DG algorithm decreases in turn. The performance of MaOEA/IGD algorithm, MOEA/D algorithm and MaOEA/IT algorithm is similar and the worst.

Based on Table 6 , we further analyze the performance of these algorithms. MOEA/TS algorithm performs well in all WFG test functions. The possible reason is that the problem characteristics of the WFG test function set are bias, fraud and degradation. The WFG test function set is more difficult than the DTLZ test function set. However, the problem characteristics of the WFG test function set don’t include multiple local fronts (From the previous analysis, we know that MOEA/TS algorithm isn’t good at dealing with multiple local fronts.). MOEA/TS algorithm can deal with these problem characteristics. Therefore, MOEA/TS algorithm performs well in all WFG test functions. It should be noted that the WFG3 test function has a narrow convergence curve, but the performance of MOEA/TS algorithm is still the best. This is an interesting phenomenon. Because from the previous analysis, we know that MOEA/TS algorithm isn’t good at dealing with test functions with narrow convergence curves (such as DTLZ5 and DTLZ6 test functions). Based on the convergence difficulty of the WFG test function set, we speculate that the performance of the other 7 algorithms is worse, thus highlighting the performance of MOEA/TS algorithm.

Compared with the DTLZ test function set, the MAF test function set is more difficult in terms of convergence and diversity. VMEF algorithm uses different convergence ranking methods to deal with different test problems. This approach helps VMEF algorithm to deal with different problem characteristics. Therefore, the performance of VMEF algorithm is good. Based on the convergence measure and diversity measure, BiGE-BEW algorithm transforms the many-objective optimization problem into a two-objective optimization problem. Although there are some defects in its diversity and convergence measurement formula, BiGE-BEW algorithm shows good performance in convergence when dealing with more complex MaOPs. MOEA/DG algorithm still uses the traditional dominance relationship to maintain the convergence of external archives. Therefore, MOEA/DG algorithm is poor in convergence. MOEA/D algorithm uses predefined weight vectors to maintain diversity and aggregation functions to maintain convergence. This approach isn’t suitable for dealing with test functions with bias characteristic. Therefore, the performance of MOEA/D algorithm is the worst. LSMaODE algorithm divides the population into two subpopulations and uses different strategies to optimize them. Because most WFG test functions have bias characteristic, LSMaODE algorithm doesn’t consider the bias problem. Therefore, the performance of LSMaODE algorithm is mediocre. MaOEA/IT algorithm optimizes convergence and diversity through two independent phases. However, the algorithm’s performance is always poor because it doesn’t alleviate the contradiction between convergence and diversity. The reference Pareto front of MaOEA/IGD algorithm is poor. Therefore, the algorithm’s performance is always poor.

Comparison and analysis

By synthesizing Tables 4 , 5 , 6 , we can obtain the data shown in Table 7 . As can be seen from Tables 4 , 5 , 6 , MOEA/TS algorithm wins the first place in 52 test cases; VMEF algorithm wins the first place in 8 test cases; BiGE-BEW algorithm wins the first place in 19 test cases; MOEA/DG algorithm wins the first place in 3 test cases; MOEA/D algorithm wins the first place in 20 test cases; LSMaODE algorithm wins first place in 8 test cases. As can be seen from Table 7 , in the 110 test cases, the number of MOEA/TS algorithm is significantly superior to VMEF algorithm, BiGE-BEW algorithm, MOEA/DG algorithm, MOEA/D algorithm, LSMaODE algorithm, MaOEA/IT algorithm and MaOEA/IGD algorithm is 69, 74, 92, 82, 91, 107 and 104, respectively. The number of MOEA/TS algorithm is significantly inferior to VMEF algorithm, BiGE-BEW algorithm, MOEA/DG algorithm, MOEA/D algorithm, LSMaODE algorithm, MaOEA/IT algorithm and MaOEA/IGD algorithm is 22, 25, 10, 20, 13, 1 and 0, respectively. Statistically, the number of MOEA/TS algorithm is similar to VMEF algorithm, BiGE-BEW algorithm, MOEA/DG algorithm, MOEA/D algorithm, LSMaODE algorithm, MaOEA/IT algorithm and MaOEA/IGD algorithm is 19, 11, 8, 8, 6, 2 and 6, respectively. Based on the above data, we can get the following conclusions: MOEA/TS algorithm has the best performance; the performance of BiGE-BEW algorithm, VMEF algorithm, MOEA/D algorithm, LSMaODE algorithm, MOEA/DG algorithm and MaOEA/IT algorithm decreases in turn. MaOEA/IGD algorithm has the worst performance.

In addition to the above conclusions, we can also observe 3 interesting phenomena:

(1) In the MAF test function set and WFG test function set, MOEA/TS algorithm has no competitors. However, in the DTLZ test function set, MOEA/TS algorithm and MOEA/D algorithm are competitors, and they have similar performance. This is because most DTLZ test functions have regular PF, while most MAF test functions and WFG test functions have more complex PF. It can be seen from Sect. " Introduction " that MOEA/D algorithm is suitable for MaOPs with regular PF. Therefore, in the DTLZ test function set, MOEA/D algorithm can compete with MOEA/TS algorithm. In the MAF test functions and WFG test functions, only MOEA/TS algorithm shows excellent performance.

(2) The performance of MOEA/TS algorithm is better on the test cases with 10 objectives, 15 objectives and 20 objectives. The performance of MOEA/TS algorithm is relatively ordinary on the test cases with 5 objectives and 8 objectives. This is because when the number of optimization objectives is small, most many-objective optimization algorithms perform well. Compared with other many-objective optimization algorithms, the advantages of MOEA/TS algorithm aren’t obvious. However, with the increase of the number of optimization objectives, the performance of other many-objective optimization algorithms becomes worse and worse. In contrast, the performance of MOEA/TS algorithm isn’t significantly affected. Therefore, compared with other many-objective optimization algorithms, MOEA/TS algorithm has obvious advantages. This shows that MOEA/TS algorithm is more suitable for solving MaOPs with more than 10 objectives.

(3) Without considering MOEA/TS algorithm, MOEA/D algorithm has the best performance in the DTLZ test function set. BiGE-BEW algorithm has the best performance in the MAF test function set. VMEF algorithm has the best performance in the WFG test function set. This shows that different many-objective optimization algorithms are suitable for different test function sets. However, MOEA/TS algorithm can show excellent performance on three test function sets. This indicates that MOEA/TS algorithm has strong universality and applicability.

Distribution diagram of solutions in the objective space

In order to describe the distribution of solutions in the high-dimensional objective space more intuitively, this paper draws the distribution diagram of solutions in the objective space. Considering the length of the paper, it is unrealistic to show the distribution diagrams of all test functions. Therefore, this section only shows the distribution diagrams of 3 representative test cases. These 3 test cases are DTLZ2 test case with 20 objectives, MAF1 test case with 15 objectives and WFG3 test case with 10 objectives, respectively.

Figure 10 shows the distribution diagrams of each algorithm on DTLZ2 test case with 20 objectives. It can be seen from Fig.  10 that distribution diagrams of MOEA/TS algorithm, BiGE-BEW algorithm, MOEA/DG algorithm and MOEA/D algorithm are similar, which indicates that these 4 algorithms are excellent in convergence and diversity; VMEF algorithm and LSMaODE algorithm are good in diversity, but poor in convergence; MaOEA/IT algorithm and MaOEA/IGD algorithm are very poor in convergence and diversity.

Distribution diagrams of each algorithm on DTLZ2 test case with 20 objectives.

Figure 11 shows the distribution diagrams of each algorithm on MAF1 test case with 15 objectives. It can be seen from Fig.  11 that MOEA/TS algorithm and VMEF algorithm are good in convergence, but poor in diversity; BiGE-BEW algorithm and LSMaODE algorithm are good in diversity, but poor in convergence. MOEA/DG algorithm, MOEA/D algorithm, MaOEA/IT algorithm and MaOEA/IGD algorithm are very bad in convergence and diversity.

Distribution diagrams of each algorithm on MAF1 test case with 15 objectives.

Figure 12 shows the distribution diagrams of each algorithm on WFG3 test case with 10 objectives. It can be seen from Fig.  12 that MOEA/TS algorithm has the best convergence and diversity; LSMaODE algorithm is also excellent, only slightly worse than MOEA/TS algorithm in terms of diversity; BiGE-BEW algorithm and MOEA/DG algorithm are good in diversity, but poor in convergence. VMEF algorithm is good in convergence, but poor in diversity. MOEA/D algorithm, MaOEA/IT algorithm and MaOEA/IGD algorithm are very bad in convergence and diversity.

Distribution diagrams of each algorithm on WFG3 test case with 10 objectives.

Evolution curve analysis of the algorithm

This section takes DTLZ2 test case with 20 objectives, MAF1 test case with 15 objectives and WFG3 test case with 10 objectives as examples to display the evolution curves of 8 algorithms (as shown in Figs.  13 , 14 , 15 ).

Evolution curve of each algorithm on DTLZ2 test case with 20 objectives.

In Figure 13 , in terms of the final IGD+ value of the algorithm, MOEA/TS algorithm has the smallest IGD+ value, while the IGD+ values of MOEA/DG algorithm, BiGE-BEW algorithm, MOEA/D algorithm, LSMaODE algorithm, VMEF algorithm and MaOEA/IGD algorithm successively increase, and MaOEA/IT algorithm has the largest IGD+ value. This shows that MOEA/TS algorithm has the best convergence and diversity within the specified number of iterations. In terms of the evolution of the algorithm, the final IGD+ values of all algorithms are smaller than the initial IGD+ values. This shows that all algorithms have strong evolution ability, especially MOEA/TS algorithm has the strongest evolution ability. In terms of algorithm fluctuation, MaOEA/IT algorithm fluctuates greatly. This shows that MaOEA/IT algorithm isn’t stable. Based on the above analysis, we believe that MOEA/TS algorithm has the best comprehensive performance on DTLZ2 test case with 20 objectives, and is suitable for solving DTLZ2 test problem with 20 objectives.

In Figure 14 , in terms of the final IGD+ value of the algorithm, MOEA/TS algorithm has the smallest IGD+ value, while the IGD+ values of BiGE-BEW algorithm, VMEF algorithm, LSMaODE algorithm, MaOEA/IGD algorithm, MOEA/D algorithm and MOEA/DG algorithm successively increase, and MaOEA/IT algorithm has the largest IGD+ value. This shows that MOEA/TS algorithm has the best convergence and diversity within the specified number of iterations. In terms of the evolution of the algorithm, the final IGD+ values of all algorithms are smaller than the initial IGD+ values. This shows that all algorithms have strong evolution ability, especially MOEA/TS algorithm has the strongest evolution ability. In terms of algorithm fluctuation, MaOEA/IT algorithm fluctuates greatly. This shows that MaOEA/IT algorithm isn’t stable. Based on the above analysis, we believe that MOEA/TS algorithm has the best comprehensive performance on MAF1 test case with 15 objectives, and is suitable for solving MAF1 test problem with 15 objectives.

Evolution curve of each algorithm on MAF1 test case with 15 objectives.

In Fig.  15 , in terms of the final IGD+ value of the algorithm, MOEA/TS algorithm has the smallest IGD+ value, while the IGD+ values of LSMaODE algorithm, MOEA/DG algorithm, VMEF algorithm, BiGE-BEW algorithm, MaOEA/IGD algorithm and MOEA/D algorithm successively increase, and MaOEA/IT algorithm has the largest IGD+ value. This shows that MOEA/TS algorithm has the best convergence and diversity within the specified number of iterations. In terms of the evolution of the algorithm, the final IGD+ values of the MaOEA/IT algorithm, VMEF algorithm, MaOEA/IGD algorithm, BiGE-BEW algorithm and VMEF algorithm are all greater than the initial IGD+ values. This shows that the performance of these 5 algorithms deteriorates during evolution, and they aren’t suitable for dealing with WFG3 test problem with 10 objectives. The initial IGD+ value of MOEA/DG algorithm is close to the final IGD+ value, and the IGD+ value of MOEA/DG algorithm fluctuates little during the evolution. This shows that MOEA/DG algorithm is insensitive to evolution. Only the final IGD+ values of LSMaODE algorithm and MOEA/TS algorithm are less than the initial IGD+ values. This shows that LSMaODE algorithm and MOEA/TS algorithm have strong evolution ability, especially MOEA/TS algorithm has the strongest evolution ability. In terms of algorithm fluctuation, MOEA/D algorithm, MaOEA/IT algorithm and MaOEA/IGD algorithm have greater fluctuation. This shows that these 3 algorithms aren’t stable. Based on the above analysis, we believe that MOEA/TS algorithm has the best comprehensive performance on WFG3 test case with 10 objectives, and is suitable for solving WFG3 test problem with 10 objectives.

Evolution curve of each algorithm on WFG3 test case with 10 objectives.

In addition, we can also observe an interesting phenomenon from Fig.  13 to Fig.  15 : the IGD+ values of some algorithms sometimes increase significantly with the increase of iterations. That is, the performance of some algorithms sometimes deteriorates seriously with the increase of iterations. The reasons for this phenomenon may include three aspects: (1) The algorithm doesn’t adopt the elite preservation strategy. Some high-quality solutions may gradually disappear; (2) Due to the complexity of the optimization problems, the evolutionary direction of the population may be misled by some pseudo-elite individuals; (3) The convergence optimization and diversity optimization of the algorithm aren’t coordinated. The optimization of convergence may affect the optimization of diversity or the optimization of diversity may affect the optimization of convergence. It can be seen from the pseudo-code of the algorithm in Section 3.5 that the MOEA/TS algorithm proposed in this paper considers the above three aspects. Therefore, MOEA/TS algorithm can effectively alleviate this phenomenon.

Effectiveness verification of innovation part

In order to verify the effectiveness of the innovative parts, 4 variants are designed in this section. As follows:

MOEA/TS-1 algorithm: The feature extraction operator in MOEA/TS algorithm is changed to the binary crossover operator and polynomial mutation operator;

MOEA/TS-2 algorithm: The repulsion field method in MOEA/TS algorithm is removed;

MOEA/TS-3 algorithm: The concurrent architecture in MOEA/TS algorithm is changed to serial architecture;

MOEA/TS-4 algorithm: The individual importance degree in MOEA/TS algorithm is removed.

This paper takes WFG test function set (45 test cases) as samples, and then verifies the performance of 5 algorithms. In this paper, 5 algorithms are executed 30 times to get the average data as shown in Table 8 . As can be seen from Table 8 , MOEA/TS algorithm wins the first place in 24 test cases; MOEA/TS-1 algorithm wins the first place in 13 test cases; MOEA/TS-2 algorithm wins the first place in 7 test cases; MOEA/TS-3 algorithm wins the first place in 1 test case. In the 45 test cases, the number of MOEA/TS algorithm is significantly superior to MOEA/TS-1 algorithm, MOEA/TS-2 algorithm, MOEA/TS-3 algorithm and MOEA/TS-4 algorithm is 21, 30, 40 and 45, respectively. The number of MOEA/TS algorithm is significantly inferior to MOEA/TS-1 algorithm, MOEA/TS-2 algorithm, MOEA/TS-3 algorithm and MOEA/TS-4 algorithm is 11, 6, 0 and 0, respectively. Statistically, the number of MOEA/TS algorithm is similar to MOEA/TS-1 algorithm, MOEA/TS-2 algorithm, MOEA/TS-3 algorithm and MOEA/TS-4 algorithm is 13, 9, 5 and 0, respectively. The average ranking of MOEA/TS algorithm is about 1.64; the average ranking of MOEA/TS-1 algorithm is about 2.02; the average ranking of MOEA/TS-2 algorithm is about 2.62; the average ranking of MOEA/TS-3 algorithm is about 3.71; the average ranking of MOEA/TS-4 algorithm is 5.

Therefore, we think that the 4 innovative parts of MOEA/TS algorithm are necessary and indispensable. The lack of any innovative parts will seriously affect the performance of MOEA/TS algorithm. This shows that our innovations are effective. In addition, based on the above data, we can also find that “individual importance degree” has the greatest influence on the algorithm; the algorithm architecture ranks second; the repulsion field method ranks third; the feature extraction operator ranks fourth.

Ablation experiment of selection approach

In the feature extraction operator, we select W high-quality solutions. To prove the effectiveness of this selection approach over random selection, the ablation experiment will be performed in this sect. " Introduction " variant is designed in this section. As follows:

MOEA/TS-5 algorithm: W solutions are randomly selected in the feature extraction operator.

This paper takes WFG test function set (45 test cases) as samples, and then verifies the performance of 2 algorithms. In this paper, 2 algorithms are executed 30 times to get the average data as shown in Table 9 . As can be seen from Table 9 , MOEA/TS algorithm wins the first place in 45 test cases. In the 45 test cases, the number of MOEA/TS algorithm is significantly superior to MOEA/TS-5 algorithm is 42. The number of MOEA/TS algorithm is significantly inferior to MOEA/TS-5 algorithm is 0. Statistically, the number of MOEA/TS algorithm is similar to MOEA/TS-5 algorithm is 3. Therefore, we believe that the performance of MOEA/TS algorithm is better than MOEA/TS-5 algorithm in the WFG test function set. It proves that the selection approach that we use is better than random selection in the feature extraction operator.

In addition, the performance of MOEA/TS-5 algorithm isn’t as good as that of MOEA/TS-1 algorithm. It means that the performance of the feature extraction operator based on random selection is even worse than that of some classical operators. The possible reason is that the randomly selected solution set will cause the feature extraction operator to extract many bad features. These bad features hinder individual evolution, which makes the convergence maintenance state and diversity maintenance state of MOEA/TS algorithm fail for a long time, and only the coordination state can play some role. The architecture of the MOEA/TS algorithm is undermined by some bad features.

Parameter sensitivity analysis.

The algorithm parameters analyzed in this paper are mainly the number of high-quality solutions W, threshold value T, standard deviation std. Due to the high complexity of the WFG3 test case with 10 objectives, it is difficult for the population of each algorithm to cover the real Pareto front, so this paper considers the WFG3 test case with 10 objectives as the main function of parameter analysis.

The initial value and value range of each parameter are shown in Table 10 .

As shown in Fig.  16 , when \(W<9\) , the IGD + value of the algorithm decreases significantly with the increase of W . It means that when \(W<9\) , the performance of the feature extraction operator is greatly improved with the increase of W . This is because the features extracted by the feature extraction operator are closer to the ideal situation. When \(W=9\) , the IGD + value of the algorithm is minimum. This shows that when \(W=9\) , the feature extraction operator performs best. When \(W>9\) , the IGD + value of the algorithm increases slowly. It means that when \(W>9\) , the performance of the feature extraction operator deteriorates gradually with the increase of W . This is because some features are over-extracted by feature extraction operators. Therefore, for WFG3 test case with 10 objectives, \(W=9\) is the best parameter selection.

The corresponding relationship between IGD + value and W.

As shown in Fig.  17 , when \(T<5\%\) , the IGD + value of the algorithm decreases significantly with the increase of T . This is because if the threshold value T is too small, the algorithm will remain in the same state for a long time, and it is difficult to be adjusted to other states. Convergence and diversity of algorithm will also be difficult to balance. This situation will be improved with the increase of T . When \(T=5\%\) , the IGD + value of the algorithm is minimum. This shows that when \(T=5\%\) , the algorithm has the best performance. When \(T>5\%\) , the IGD + value of the algorithm increases gradually with the increase of T . This is because if the threshold value T is too large, the algorithm’s state will be adjusted frequently. Even if the population isn’t stable in one state (convergence, diversity, coordination), the algorithm will also be adjusted to other states. This isn’t conducive to improving the convergence and the diversity of the algorithm. The efficiency of the algorithm will also be affected. Therefore, for WFG3 test case with 10 objectives, \(T=5\%\) is the best parameter selection.

The corresponding relationship between IGD + value and T.

As shown in Fig.  18 , when \(std<0.7\) , the IGD + value of the algorithm decreases significantly with the increase of std . This is because if std is too small, the results of Gaussian sampling are too concentrated in the middle region, and the randomness of the sampling vector is weak, which isn’t conducive to the use of features and generation of diversified feature solutions. When \(std=0.7\) , the IGD + value of the algorithm is minimum. This shows that when \(std=0.7\) , the feature extraction operator performs best. When \(std>0.7\) , the IGD + value of the algorithm increases significantly with the increase of std . This is because if the std is too large, the result of Gaussian sampling is too scattered, the randomness of the sampling vector is strong, some components are easy to exceed the upper bound or lower bound, and some features are easy to be eliminated by the repair operator. Therefore, for WFG3 test case with 10 objectives, \(std=0.7\) is the best parameter selection.

The corresponding relationship between IGD + value and std.

Based on the above analysis of algorithm parameters, we think \(W=9, T=5\%, std=0.7\) are the best parameter combinations in WFG3 test case with 10 objectives. Further, we test the performance of the above parameter combinations in more test cases. The experimental results show that the above parameter combinations perform well in most test cases. Therefore, this paper sets the number of high-quality solutions \(W\) , the threshold value \(T\) and the standard deviation \(std\) to 9, 5% and 0.7, respectively.

Practical problem testing

This section mainly explores the performance of MOEA/TS algorithm in practical problems. The practical problem selected in this section is the industrial internet optimization problem based on the blockchain provided in reference 40 .

The industrial internet can support effective control of the physical world through a large amount of industrial data, but data security has always been a challenge due to various interconnections and accesses. Blockchain technology supports the security and privacy protection of industrial internet data with its trusted and reliable security mechanism. Fragmentation technology can help improve the overall throughput and scalability of the blockchain network. However, due to the uneven distribution of malicious nodes, the effectiveness of fragmentation is still challenging. In addition, the conflict between multiple industrial network indicators is also a problem we have to consider. Therefore, the industrial internet optimization problem based on blockchain is an important research problem.

In this section, the industrial internet optimization problem based on blockchain has the following 4 optimization objectives:

(1) Minimizing the shard invalidation probability (SIP);

(2) Minimizing the transmission delay (TD);

(3) Maximizing the throughput (TP);

(4) Minimizing the load of Malicious Nodes (LMN).

The research background of the industrial internet based on blockchain and the calculation formulas of these 4 objectives are detailed in reference 40 .

In this section, we set the population size to 220, the number of iterations to 300, and the number of function evaluations to 66000. We still use inverted generational distance plus (IGD+) to measure the performance of many-objective optimization algorithms. However, the real PF of the practical problem is unknown. Therefore, we run these algorithms many times to obtain the different non-dominated solution sets. The non-dominated union set of the different non-dominated solution sets is considered as the real PF. The relevant parameters of these algorithms are shown in Section 4.1.

In this section, each algorithm is executed 30 times to get the data as shown in Table 11 . As can be seen from Table 11 , MOEA/TS algorithm has absolute advantages. The performance of BiGE-BEW algorithm and MOEA/DG algorithm is good and similar. The performance of VMEF algorithm and MOEA/D algorithm in practical problems is obviously not as good as that in benchmark test functions. This is because the real PF of the practical problem is more complex. The performance of LSMaODE algorithm is close to that of MOEA/D algorithm. The performance of MaOEA/IT algorithm and MaOEA/IGD algorithm is the worst. Based on the above observations and analysis, we believe that MOEA/TS algorithm still has excellent performance and strong applicability in practical problems.

Considering that the solutions obtained by the many-objective optimization algorithms are the population, it is unrealistic to compare different network indicators of different algorithms intuitively. However, in practical applications, we only need to make choices according to the specific needs or preferences of users or enterprises. In this section, we first select the individuals with the largest throughput in each algorithm, and then compare the MOEA/TS algorithm with other algorithms on the basis of ensuring the maximum throughput. The network indicators obtained by these 8 algorithms are shown in Table 12 . As can be seen from Table 12 , in terms of SIP and TP, MOEA/TS algorithm has the best performance; In terms of TD, MOEA/TS algorithm ranks second; In terms of LMN, MOEA/TS algorithm ranks third. Therefore, we believe that the MOEA/TS algorithm has the best comprehensive performance in the industrial internet optimization problem based on blockchain, and various network indicators are at the forefront.

Based on the experimental analysis from Section 4.2 to Section 4.8, we can obtain the following conclusions:

(1) In the benchmark test cases, MOEA/TS algorithm is superior to the other 7 advanced many-objective optimization algorithms.

(2) MOEA/TS algorithm is more suitable for dealing with the MaOPs with more than 10 objectives.

(3) MOEA/TS algorithm can show excellent performance in different test function sets, and has strong universality and applicability.

(4) MOEA/TS algorithm has the best convergence and diversity, the strongest evolution ability and the fastest convergence speed.

(5) The 4 innovative parts of MOEA/TS algorithm are necessary and indispensable. The lack of any innovative parts will seriously affect the performance of MOEA/TS algorithm.

(6) MOEA/TS algorithm still has excellent performance and strong applicability in practical problems.

Summary and future work

Aiming at some difficulties in the many-objective optimization field, this paper proposes a many-objective evolutionary algorithm based on three states (MOEA/TS). Firstly, a feature extraction operator is proposed. The feature extraction operator is a feature extractor, which can extract the features of the high-quality solution set, and then assist the evolution of the current individual. Secondly, in terms of convergence maintenance, this paper doesn’t consider using domination relaxation technique, because the current domination relaxation technique still faces some problems. Based on Pareto front layer, this paper proposes the concept of “individual importance degree”. The importance degree of an individual can reflect the importance of the individual in the same Pareto front layer, so as to further distinguish the advantages and disadvantages of different individuals in the same front layer, and effectively solve the phenomenon of Pareto resistance. Then, in terms of diversity maintenance, this paper considers maintaining the diversity of the population in the objective space by repulsion field, so that the population can be evenly distributed on the real PF. Finally, a new concurrent algorithm framework is designed. In the framework, the algorithm is divided into three states, namely, convergence maintenance state, diversity maintenance state and coordination state. Each state focuses on a specific task. That is, the convergence maintenance state is responsible for improving the convergence of population; Diversity maintenance state is responsible for improving the diversity of population; the coordination state is responsible for coordinating the contradiction between diversity and convergence. The population can freely switch among these three states according to its own evolution. The experimental results show that MOEA/TS algorithm is superior to the other 7 advanced many-objective optimization algorithms. In addition, the effectiveness of the innovation parts is further verified.

However, MOEA/TS algorithm also has obvious defects: MOEA/TS algorithm isn’t good at dealing with test problems with narrow convergence curves or multiple local fronts. Therefore, in the future, we will further improve MOEA/TS algorithm, so that MOEA/TS algorithm can deal with test problems with narrow convergence curve or multiple local fronts. In addition, constrained MOPs and high-dimensional MOPs are also the focus of our future research.

Data availability

All data generated or analysed during this study are included in this published article.

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Acknowledgements

This work was supported in part by the National Natural Science Foundation of China under Grant 62362026 and 62162021; in part by the specific research fund of The Innovation Platform for Academicians of Hainan Province under grant YSPTZX202314; in part by the Key Project of Hainan Province under Grant ZDYF2023GXJS158.

National Natural Science Foundation of China, 62362026, 62362026, Specific research fund of The Innovation Platform for Academicians of Hainan Province, YSPTZX202314, YSPTZX202314, Key Project of Hainan Province, ZDYF2023GXJS158, ZDYF2023GXJS158.

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Conceptualization, J.L.Z., H.J.Z. and X.D.H.; Methodology, J.L.Z., H.J.Z. and X.D.H.; Software, J.L.Z., H.J.Z., H.H.Y. and H.S.F.; Validation, H.H.Y. and H.S.F.; Formal analysis, J.L.Z. and H.J.Z.; Investigation, H.H.Y. and H.S.F.; Resources, Q.L.Y.; Data curation, H.H.Y. and H.S.F.; Writing-original draft, J.L.Z.; Writing-review&editing, J.L.Z., H.J.Z., H.H.Y. and H.S.F.; Visualization, J.L.Z. and H.J.Z.; Supervision, H.H.Y. and H.S.F.; Project administration, J.L.Z. and Q.L.Y.; Funding acquisition, Q.L.Y.; All authors have read and agreed to the published version of the manuscript.

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Zhao, J., Zhang, H., Yu, H. et al. A many-objective evolutionary algorithm based on three states for solving many-objective optimization problem. Sci Rep 14 , 19140 (2024). https://doi.org/10.1038/s41598-024-70145-8

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