assignment and transportation problem in operational research

Operations Research/Transportation and Assignment Problem

The Transportation and Assignment problems deal with assigning sources and jobs to destinations and machines. We will discuss the transportation problem first.

Suppose a company has m factories where it manufactures its product and n outlets from where the product is sold. Transporting the product from a factory to an outlet costs some money which depends on several factors and varies for each choice of factory and outlet. The total amount of the product a particular factory makes is fixed and so is the total amount a particular outlet can store. The problem is to decide how much of the product should be supplied from each factory to each outlet so that the total cost is minimum.

Let us consider an example.

Suppose an auto company has three plants in cities A, B and C and two major distribution centers in D and E. The capacities of the three plants during the next quarter are 1000, 1500 and 1200 cars. The quarterly demands of the two distribution centers are 2300 and 1400 cars. The transportation costs (which depend on the mileage, transport company etc) between the plants and the distribution centers is as follows:

Which plant should supply how many cars to which outlet so that the total cost is minimum?

The problem can be formulated as a LP model:

The whole model is:

subject to,

The problem can now be solved using the simplex method. A convenient procedure is discussed in the next section.

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THE LITERATURE REVIEW FOR ASSIGNMENT AND TRANSPORTATION PROBLEMS.

Operations Research is a logical learning through interdisciplinary collaboration to determine the best usage of restricted assets. In this paper, the importance of Operations research is discussed and the literature of assignment and transportation problem is discussed in detail.

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Rabindra Mondal

A transportation problem basically deals with the problem, which aims to find the best way to fulfil the demand of n demand points using the capacities of m supply points. Here we studied a new method for solving transportation problems with mixed constraints and described the algorithm to find an optimal more-for-less (MFL) solution. The optimal MFL solution procedure is illustrated with numerical example and also computer programming. Though maximum transportation problems in real life have mixed constraints, these problems are not be solved by using general method. The proposed method builds on the initial solution of the transportation problem which is very simple, easy to understand and apply.

Optimization

Hossein Arsham

The family of network optimization problems includes the following prototype models: assignment, critical path, max flow, shortest path, and transportation. Although it is long known that these problems can be modeled as linear programs (LP), this is generally not done. Due to the relative inefficiency and complexity of the simplex methods (primal, dual, and other variations) for network models, these problems are usually treated by one of over 100 specialized algorithms. This leads to several difficulties. The solution algorithms are not unified and each algorithm uses a different strategy to exploit the special structure of a specific problem. Furthermore, small variations in the problem, such as the introduction of side constraints, destroys the special structure and requires modifying andjor restarting the algorithm. Also, these algorithms obtain solution efficiency at the expense of managerial insight, as the final solutions from these algorithms do not have sufficient information to perform postoptimality analysis.Another approach is to adapt the simplex to network optimization problems through network simplex. This provides unification of the various problems but maintains all the inefficiencies of simplex, as well as, most of the network inflexibility to handle changes such as side constraints. Even ordinary sensitivity analysis (OSA), long available in the tabular simplex, has been only recently transferred to network simplex.This paper provides a single unified algorithm for all five network models. The proposed solution algorithm is a variant of the self-dual simplex with a warm start. This algorithm makes available the full power of LP perturbation analysis (PA) extended to handle optimal degeneracy. In contrast to OSA, the proposed PA provides ranges for which the current optimal strategy remains optimal, for simultaneous dependent or independent changes from the nominal values in costs, arc capacities, or suppliesJdemands. The proposed solution algorithm also facilitates incorporation of network structural changes and side constraints. It has the advantage of being computationally practical, easy for managers to understand and use, and provides useful PA information in all cases. Computer implementation issues are discussed and illustrative numerical examples are provided in the Appendix For teaching purposes you may try: Refined Simplex Algorithm for the Classical Transportation Problem with Application to Parametric Analysis, Mathematical and Computer Modelling, 12(8), 1035-1044, 1989. http://home.ubalt.edu/ntsbarsh/KahnRefine.pdf

Journal of Applied Mathematics and Decision Sciences

In a fast changing global market, a manager is concerned with cost uncertainties of the cost matrix in transportation problems (TP) and assignment problems (AP).A time lag between the development and application of the model could cause cost parameters to assume different values when an optimal assignment is implemented. The manager might wish to determine the responsiveness of the current optimal solution to such uncertainties. A desirable tool is to construct a perturbation set (PS) of cost coeffcients which ensures the stability of an optimal solution under such uncertainties. The widely-used methods of solving the TP and AP are the stepping-stone (SS) method and the Hungarian method, respectively. Both methods fail to provide direct information to construct the needed PS. An added difficulty is that these problems might be highly pivotal degenerate. Therefore, the sensitivity results obtained via the available linear programming (LP) software might be misleading. We propose a unified pivotal solution algorithm for both TP and AP. The algorithm is free of pivotal degeneracy, which may cause cycling, and does not require any extra variables such as slack, surplus, or artificial variables used in dual and primal simplex. The algorithm permits higher-order assignment problems and side-constraints. Computational results comparing the proposed algorithm to the closely-related pivotal solution algorithm, the simplex, via the widely-used pack-age Lindo, are provided. The proposed algorithm has the advantage of being computationally practical, being easy to understand, and providing useful information for managers. The results empower the manager to assess and monitor various types of cost uncertainties encountered in real-life situations. Some illustrative numerical examples are also presented."

IJAR Indexing

Assignment problems deal with the question how to assign n objects to m other objects in an injective fashion in the best possible way. An assignment problem is completely specified by its two components the assignments, which represent the underlying combinatorial structure, and the objective function to be optimized, which models \\\\\\\"the best possible way\\\\\\\". The assignment problem refers to another special class of linear programming problem where the objective is to assign a number of resources to an equal number of activities on a one to one basis so as to minimize total costs of performing the tasks at hand or maximize total profit of allocation. In this paper we introduce a new technique to solve assignment problems namely, Divide Row Minima and Subtract Column Minima .For the validity and comparison study we consider an example and solved by using our technique and the existing Hungarian (HA) and matrix ones assignment method(MOA) and compare optimum result shown graphically.

ام محمد لا للشات

The problem of finding the initial basic feasible solution of the Transportation Problem has long been studied and is well known to the research scholars of the field. So far three general methods for solving transportation methods are available in literature, namely Northwest, Least Cost and Vogel?s Approximation methods. These methods give only initial feasible solution. However here we discuss a new alternative method which gives Initial feasible solution as well as optimal or nearly optimal solution. In this paper we provide an alternate method to find IBFS (Initial Basic Feasible Solution) and compared the alternate method and the existing IBFS methods using a Graphical User Interface. It is also to be noticed that this method requires lesser number of iterations to reach optimality as compared to other known methods for solving the transportation problem and the solution obtained is as good as obtained by Vogel?s Approximation Method (VAM).

Omega-international Journal of Management Science

Krzysztof Kowalski

nikky kumari

A new method called zero point method is proposed for finding an optimal solution for transportation problems with mixed constraints in a single stage. Using the zero point method, we propose a new method for finding an optimal more-for-less solution for transportation problems with mixed constraints. The optimal more-for-less solution procedure is illustrated with numerical examples. Mathematics Subject Classifications: 90C08 , 90C90

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George Nemhauser

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Assignment Problem

5.1  introduction.

The assignment problem is one of the special type of transportation problem for which more efficient (less-time consuming) solution method has been devised by KUHN (1956) and FLOOD (1956). The justification of the steps leading to the solution is based on theorems proved by Hungarian mathematicians KONEIG (1950) and EGERVARY (1953), hence the method is named Hungarian.

5.2  GENERAL MODEL OF THE ASSIGNMENT PROBLEM

Consider n jobs and n persons. Assume that each job can be done only by one person and the time a person required for completing the i th job (i = 1,2,...n) by the j th person (j = 1,2,...n) is denoted by a real number C ij . On the whole this model deals with the assignment of n candidates to n jobs ...

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Home » Management Science » Transportation and Assignment Models in Operations Research

Transportation and Assignment Models in Operations Research

Transportation and assignment models are special purpose algorithms of the linear programming. The simplex method of Linear Programming Problems(LPP) proves to be inefficient is certain situations like determining optimum assignment of jobs to persons, supply of materials from several supply points to several destinations and the like. More effective solution models have been evolved and these are called assignment and transportation models.

The transportation model is concerned with selecting the routes between supply and demand points in order to minimize costs of transportation subject to constraints of supply at any supply point and demand at any demand point. Assume a company has 4 manufacturing plants with different capacity levels, and 5 regional distribution centres. 4 x 5 = 20 routes are possible. Given the transportation costs per load of each of 20 routes between the manufacturing (supply) plants and the regional distribution (demand) centres, and supply and demand constraints, how many loads can be transported through different routes so as to minimize transportation costs? The answer to this question is obtained easily through the transportation algorithm.

Similarly, how are we to assign different jobs to different persons/machines, given cost of job completion for each pair of job machine/person? The objective is minimizing total cost. This is best solved through assignment algorithm.

Uses of Transportation and Assignment Models in Decision Making

The broad purposes of Transportation and Assignment models in LPP are just mentioned above. Now we have just enumerated the different situations where we can make use of these models.

Transportation model is used in the following:

• To decide the transportation of new materials from various centres to different manufacturing plants. In the case of multi-plant company this is highly useful.
• To decide the transportation of finished goods from different manufacturing plants to the different distribution centres. For a multi-plant-multi-market company this is useful.
• To decide the transportation of finished goods from different manufacturing plants to the different distribution centres. For a multi-plant-multi-market company this is useful. These two are the uses of transportation model. The objective is minimizing transportation cost.

Assignment model is used in the following:

• To decide the assignment of jobs to persons/machines, the assignment model is used.
• To decide the route a traveling executive has to adopt (dealing with the order inn which he/she has to visit different places).
• To decide the order in which different activities performed on one and the same facility be taken up.

In the case of transportation model, the supply quantity may be less or more than the demand. Similarly the assignment model, the number of jobs may be equal to, less or more than the number of machines/persons available. In all these cases the simplex method of LPP can be adopted, but transportation and assignment models are more effective, less time consuming and easier than the LPP.

Related posts:

• Operations Research approach of problem solving
• Introduction to Transportation Problem
• Procedure for finding an optimum solution for transportation problem
• Initial basic feasible solution of a transportation problem
• Introduction to Decision Models
• Transportation Cost Elements
• Modes of Transportation in Logistics
• Factors Affecting Transportation in Logistics

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Transportation Problems ¶

Basic set up ¶.

In the Introduction to MIP, we described the wide range of problems that can be modeled and solved with MIP, and described the setup of some widely used problem models. In this notebook, we introduce the transportation problem, as a special type of MIP problem particularly important for logistic operations. In the literature, the general transportation problem deals with the distribution of any commodity (generally goods) from a group of origins of the supply chain, or source facilities, called sources to the endpoints of the model of the supply chain, the receiving centers, called destinations.

There are different types of goods or services that can be modeled with this model. Sources can be production centers, or warehouses. Destinations can be warehouse, demand regions or end customers, depending on the model.

However, in the general problem formulation, we have \(i \in [1, ..., n]\) source nodes and \(j \in [1, ..., m]\) destination nodes. The objective is to minimize the overall costs of the distribution of goods at a minimum costs, subject to a set of constraints.

Regarding the constraints, let us consider two types of constraints:

supply constraints: These constraints represent the limited capacity of the sources. For instance, production centers may have a limited production capacity, or source warehouses may have a limited storage capacity. Let us note \(s_i\) the source capacity of source node \(i\) .

demand constraints: The demand constraints represent the distribution requirements at the destinations: what are the quantities of the goods required at each destination. Let us note as \(d_j\) the demand of destination node \(j\)

Now, based on this, let us define our set of decision variables as:

\(x_{ij}\) : (Integer) units to transport from source \(i\) to destination \(j\)

Assuming a distribution cost of \(c_{ij}\) for each transported unit yields:

\(\min z = \sum_{i=1}^n{\sum_{j=1}^m{c_{ij}*x_{ij}}}\)

\(\sum_{j=1}^m{x_{ij}} \leq s_i \quad \forall i\)

\(\sum_{i=1}^n{x_{ij}} \geq d_j \quad \forall j\)

That is, the total number of units that leave a source cannot exceed the capacity of the source, and the total number of units that reach a destination must be at least equal to the demand.

Note that the model has a feasible solution only if:

\(\sum_{i=1}^n{s_i} \geq \sum_{j=1}^m{d_j}\)

That, is, the problem is only feasible if the overall capacity is sufficient to meet the overall demand.

It is important to note that we may allow ourselves less slack by making any of the three inequalities above equalities.

Sourcing design ¶

In the basic set up above, it is assumed that it is possible to transport units from any source node to any destination. This implies that the design of the distribution network is known. But this may not be always the case. For instance, we may have different possible locations for source warehouse, each with different associated costs. In this case, our decision variables are not only the number of units to transport, but also determine from which subset of source nodes they must be transported. Thus, we can introduce a new set of decision variables:

\(Y_i\) : (Binary) determines whether units to any destination may be delivered from source node \(i\)

We need to introduce new logical constraints to ensure that we do not deliver any units from source nodes that are not selected:

\(\sum_{j=1}^m{x_{ij}} \leq M*Y_i \quad \forall i\)

Where M is a large number. Since \(Y_i\) is binary, this constraint forces that, in case that it is equal to zero, no units are sourced from source node \(i\) .

This logical constraint can be merged with the source capacity constraint as:

\(\sum_{j=1}^m{x_{ij}} \leq s_i*Y_i \quad \forall i\)

This constraint takes into account the logical constraint and the capacity constraint. Since both constraint are equivalent, but the former is more restrictive and renders the first one irrelevant.

We can now take into account in the objective function fixed costs that do not depend on the number of units. For instance, let us consider that the source warehouses have a fixed operation costs \(f_i\) . The problem in the basic set up becomes:

\(\min z = \sum_{i=1}^n{\sum_{j=1}^m{c_{ij}*x_{ij}}} + \sum_{i=1}^n{f_i*Y_i}\)

Single source ¶

The single source constraint imposes that all the demand of a destination node is transported from the same origin. With this constraint, now the decision variables \(x_{ij}\) become binary:

\(X_{ij}\) : (Binary) determines if source \(i\) delivers all demand to destination \(j\) {1 if yes, 0 otherwise}

The demand constraint now becomes:

\(\sum_{i=1}^n{d_j*X_{ij}} = d_j \quad \forall j\)

Note that we now multiply each binary decision variable times the demand of each destination. We have changed the type of the constraint from \(\geq\) to \(=\) to avoid any slack. Although this change is optional, if not applied, the slack will be a multiple of the demand. Note that this is exactly the same requirement:

\(\sum_{i=1}^n{X_{ij}} = 1 \quad \forall j\)

We just divided by \(d_j\) the right hand side and the left hand side.

Also, we need to factor this into the objective function as:

\(\min z = \sum_{i=1}^n{\sum_{j=1}^m{c_{ij}*d_j*X_{ij}}}\)

Since \(X_{ij}\) is binary, and only equal to 1 for a destination node.

Network design ¶

In the set up above, we assume that the cost per unit from sources to destinations is fixed. In many cases however, this cost depends to a great extent on the design considerations taken in the distribution network between sources and destination. Similar to what we did with the sourcing design, we may be interested on evaluating the impact in the cost of the design of the distribution network, for instance, the location of intermediate nodes or junctions between the source nodes and destination nodes. In literature, this is known as the transshipment problem in the literature. Let us come back to the original problem, and assume that the source nodes are production plants, the destination nodes are retailer regions, and that we have intermediate warehouses that we can use to optimise costs. Let us assume that the production plants are fixed (i.e. the sourcing design is known), but that we can select a subset of warehouses from a set of candidate locations.

First, let us modify the indices so that the distribution stages (from production plants through warehouses to regions) follow an alphabetical order:

\(i\) : Production plants \(i \in [1, ..., n]\)

\(j\) : Possible warehouse locations \(j \in [1, ..., m]\)

\(k\) : Retailer regions \(k \in [1, ..., l]\)

Let us use the following notation for the coefficients of our problem:

\(d_k\) : yearly demand in retail region k

\(a_{ij}\) : cost of transporting 1 unit from plant i to warehouse j

\(b_{jk}\) : cost of transporting 1 unit from warehouse j to retailer region k

\(F_{j}\) : yearly operation costs of warehouse j

\(c_i\) : yearly production capacity of plant i

Since we can decide the location of the warehouses, and the units to transport at each stage, the decision variables now become:

\(Y_{j}\) : (Binary) { 1 if a warehouse is placed in location j } {0 otherwise}

\(s_{ij}\) : integer units transported from plant i to warehouse j

\(t_{jk}\) : integer units transported from warehouse j to region k

The objective function is to minimise costs:

\(\min z = \sum_{i=1}^{n}{\sum_{j=1}^{m}{a_{ij}*s_{ij}}} + \sum_{j=1}^{m}{\sum_{k=1}^{l}{b_{jk}*t_{jk}}} + \sum_{j=1}^{m}{F_j*Y_j}\)

Now, for the constraints, clearly, we need to satisfy all the constraints. The units sourced from each production plant cannot exceed the production capacity:

\(\sum_{j=1}^{m}{s_{ij}} \leq c_i \quad \forall i\)

We also need to ensure that the demand at each region is satisfied:

\(\sum_{j=1}^{m}{t_{jk}} \geq d_k \quad \forall k\)

Now, we need to ensure the material flow, that is, that the unit that leave a warehouse are not less than the units that enter the same warehouse:

\(\sum_{i=1}^{n}{s_{ij}} \geq \sum_{k=1}^{l}{t_{jk}} \quad \forall j\)

And that a warehouse is built if it supplies to any region:

\(\sum_{k=1}^{l}{t_{jk}} \leq M*Y_j \quad \forall j\)

Known variations ¶

Combination of single-source in design constraints ¶.

Any combinations of the variations above can be found in a problem instance. For instance, we might be interested in evaluating our network design with a single source constraint on intermediate nodes. Note that when we introduce a single source constraint in our problem, the decision variable that models the transport of units becomes binary. For instance, in the network design problem above, the decision variables Any combinations of the variations above can be found in a problem instance. For instance, we might be interested in evaluating our network design with a single source constraint on intermediate nodes. Note that when we introduce a single source constraint in our problem, the decision variable that models the transport of units becomes binary. For instance, in the network design problem above, the decision variables \(t_{jk}\) become:

\(T_{jk}\) : (Binary) {1 units from warehouse j transported to region k, 0 otherwise}

Since the variable are now binary, the single source constraint modifies the demand constraint, which now becomes:

\(\sum_{j=1}^n{T_{jk}} = 1 \quad \forall k\)

Now, since all \(T_{jk}\) are binary, the logical constraint that rules the construction of warehouses can be written as:

\(T_{jk} \leq Y_{j} \quad \forall j, \forall k\)

Other indices ¶

We may also introduce other indices in the problem, most common indices are:

Type of product : We can have different costs or demands depending on the type of product

Periods : The costs, or the demands can be dynamic and depend on a discrete time period, like the year, the month, or the day, so we might be interested in optimising the distribution network for a set of periods.

When we introduce a new index into the model, the size of the problem (number of coefficients and decision variables) will grow accordingly.

Transportation Modelling and Operations Research: A Fruitful Connection

Cite this chapter.

• Philippe L. Toint 5  

Part of the book series: NATO ASI Series ((NATO ASI F,volume 166))

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The purpose of this paper is twofold. It first aims at introducing the subject of transportation modelling and therefore at setting the stage for the other contributions of this volume. At the same time, it also aims at pointing out the many connections between transportation modelling and operations research, and the rich and insightful nature of these connections.

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Toint, P.L. (1998). Transportation Modelling and Operations Research: A Fruitful Connection. In: Labbé, M., Laporte, G., Tanczos, K., Toint, P. (eds) Operations Research and Decision Aid Methodologies in Traffic and Transportation Management. NATO ASI Series, vol 166. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-03514-6_1

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In this chapter, we present the family of transportation models and demonstrate how SAS/OR® can be applied to solve transportation, assignment, and transshipment problems to optimality. The problem formulations are described first. Then, various SAS/OR® procedures are applied to tackle the problems with the aid of examples. Following that, result analyses are carried out. After this chapter, the reader will be more familiar with SAS/OR® and the applications of its procedures.

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Solving a multi-objective solid transportation problem: a comparative study of alternative methods for decision-making

• Mohamed H. Abdelati   ORCID: orcid.org/0000-0002-5034-7323 1 ,
• Ali M. Abd-El-Tawwab 1 ,
• Elsayed Elsayed M. Ellimony 2 &
• M Rabie 1  

Journal of Engineering and Applied Science volume  70 , Article number:  82 ( 2023 ) Cite this article

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The transportation problem in operations research aims to minimize costs by optimizing the allocation of goods from multiple sources to destinations, considering supply, demand, and transportation constraints. This paper applies the multi-dimensional solid transportation problem approach to a private sector company in Egypt, aiming to determine the ideal allocation of their truck fleet.

In order to provide decision-makers with a comprehensive set of options to reduce fuel consumption costs during transportation or minimize total transportation time, a multi-objective approach is employed. The study explores the best compromise solution by leveraging three multi-objective approaches: the Zimmermann Programming Technique, Global Criteria Method, and Minimum Distance Method. Optimal solutions are derived for time and fuel consumption objectives, offering decision-makers a broad range to make informed decisions for the company and the flexibility to adapt them as needed.

Lingo codes are developed to facilitate the identification of the best compromise solution using different methods. Furthermore, non-dominated extreme points are established based on the weights assigned to the different objectives. This approach expands the potential ranges for enhancing the transfer problem, yielding more comprehensive solutions.

This research contributes to the field by addressing the transportation problem practically and applying a multi-objective approach to support decision-making. The findings provide valuable insights for optimizing the distribution of the truck fleet, reducing fuel consumption costs, and improving overall transportation efficiency.

Introduction

The field of operations research has identified the transportation problem as an optimization issue of significant interest [ 1 , 2 ]. This problem concerns determining the optimal approach to allocate a given set of goods that come from particular sources to the designated destinations to minimize the overall transportation costs [ 3 ]. The transportation problem finds applications in various areas, including logistics planning, distribution network design, and supply chain management. Solving this problem relies on the assumption that the supply and demand of goods are known, as well as the transportation cost for each source–destination pairing [ 4 , 5 ].

Solving the transportation problem means finding the right quantities of goods to be transported from the sources to the destinations, given the supply and demand restrictions. The ultimate goal is to minimize the total transportation cost, which is the sum of the cost for each shipment [ 6 ]. Various optimization algorithms have been developed for this problem, such as the North-West Corner Method, the Least Cost Method, and Vogel’s Approximation Method [ 7 ].

A solid transportation problem (STP) is a related transportation problem that centers around a single commodity, which can be stored at interim points [ 8 ]. These interim points, known as transshipment points, act as origins and destinations. The STP involves determining the most efficient means of transporting the commodity from the sources to the destinations, while minimizing transportation costs by going through the transshipment points. The STP has real-world applications in container shipping, air cargo transportation, and oil and gas pipeline transportation [ 9 , 10 ].

Multi-dimensional solid transportation problem (MDSTP) represents a variation on the STP, incorporating multiple commodities that vary in properties such as volume, weight, and hazard level [ 11 ]. The MDSTP aims to identify the best way to transport each commodity from the sources to the destinations, taking into account the capacity restrictions of transshipment points and hazardous commodity regulations [ 12 ]. The MDSTP is more complex than the STP and requires specific algorithms and models for its resolution.

Solving the STP and MDSTP requires identifying the most effective routing of commodities and considering the storage capacity of transshipment points. The goal is to minimize total transportation costs while satisfying supply and demand constraints and hazardous material regulations. Solutions to these problems include the Network Simplex Method, Branch and Bound Method, and Genetic Algorithm [ 13 ]. Solving the STP and MDSTP contributes valuable insights into the design and operation of transportation systems and supports improved sustainability and efficiency.

In the field of operations research, two critical research areas are the multi-objective transportation problem (MOTP) and the multi-objective solid transportation problem (MOSTP) [ 14 ]. The MOTP aims to optimize the transportation of goods from multiple sources to various destinations by considering multiple objectives, including minimizing cost, transportation time, and environmental impacts. The MOSTP, on the other hand, focuses on the transportation of solid materials, such as minerals or ores, and involves dealing with multiple competing objectives, such as cost, time, and quality of service. These problems are essential in logistics and supply chain management, where decision-makers must make optimal transportation plans by considering multiple objectives. Researchers and practitioners often employ optimization techniques, such as mathematical programming, heuristics, and meta-heuristics, to address these challenges efficiently [ 15 ].

Efficient transportation planning is essential for moving goods from their source to the destination. This process involves booking different types of vehicles and minimizing the total transportation time and cost is a crucial factor to consider. Various challenges can affect the optimal transportation policy, such as the weight and volume of products, the availability of specific vehicles, and other uncertain parameters. In this regard, several studies have proposed different approaches to solve the problem of multi-objective solid transportation under uncertainty. One such study by Kar et al. [ 16 ] used fuzzy parameters to account for uncertain transportation costs and time, and two methods were employed to solve the problem, namely the Zimmermann Method and the Global Criteria Method.

Similarly, Mirmohseni et al. [ 17 ] proposed a fuzzy interactive probabilistic programming approach, while Kakran et al. [ 18 ] addressed a multi-objective capacitated solid transportation problem with uncertain zigzag variables. Additionally, Chen et al. [ 19 ] investigated an uncertain bicriteria solid transportation problem by using uncertainty theory properties to transform the models into deterministic equivalents, proposing two models, namely the expected value goal programming and chance-constrained goal programming models [ 20 ]. These studies have contributed to developing different approaches using fuzzy programming, uncertainty theory, and related concepts to solve multi-objective solid transportation problems with uncertain parameters.

This paper presents a case study carried out on a private sector company in Egypt intending to ascertain the minimum number of trucks required to fulfill the decision-makers’ objectives of transporting the company’s fleet of trucks from multiple sources to various destinations. This objective is complicated by the diversity of truck types and transported products, as well as the decision-makers’ multiple priorities, specifically the cost of fuel consumption and the timeliness of truck arrival.

In contrast to previous research on the transportation problem, this paper introduces a novel approach that combines the multi-dimensional solid transportation problem framework with a multi-objective optimization technique. Building upon previous studies, which often focused on single-objective solutions and overlooked specific constraints, our research critically analyzes the limitations of these approaches. We identify the need for comprehensive solutions that account for the complexities of diverse truck fleets and transported products, as well as the decision-makers’ multiple priorities. By explicitly addressing these shortcomings, our primary goal is to determine the minimum number of trucks required to fulfill the decision-makers’ objectives, while simultaneously optimizing fuel consumption and transportation timeliness. Through this novel approach, we contribute significantly to the field by advancing the understanding of the transportation problem and providing potential applications in various domains. Our research not only offers practical solutions for real-world scenarios but also demonstrates the potential for improving transportation efficiency and cost-effectiveness in other industries or contexts. The following sections will present a comparative analysis of the proposed work, highlighting the advancements and novelty introduced by our approach.

Methods/experimental

This study uses a case study from Egypt to find the optimal distribution of a private sector company’s truck fleet under various optimization and multi-objective conditions. Specifically, the study aims to optimize the distribution of a private sector company’s truck fleet by solving a multi-objective solid transportation problem (MOSTP) and comparing three different methods for decision-making.

Design and setting

This study uses a case study design in a private sector company in Egypt. The study focuses on distributing the company’s truck fleet to transport products from factories to distribution centers.

Participants or materials

The participants in this study are the transportation planners and managers of the private sector company in Egypt. The materials used in this study include data on the truck fleet, sources, destinations, and products.

Processes and methodologies

The study employs the multi-objective multi-dimensional solid transportation problem (MOMDSTP) to determine the optimal solution for the company’s truck fleet distribution, considering two competing objectives: fuel consumption cost and total shipping time. The MOMDSTP considers the number and types of trucks, sources, destinations, and products and considers the supply and demand constraints.

To solve the MOMDSTP, three decision-making methods are employed: Zimmermann Programming Technique, Global Criteria Method, and Minimum Distance Method. The first two methods directly yield the best compromise solution (BCS), whereas the last method generates non-dominated extreme points by assigning different weights to each objective. Lingo software is used to obtain the optimal solutions for fuel consumption cost and time and the BCS and solutions with different weights for both objectives.

Ethics approval and consent

This study does not involve human participants, data, or tissue, nor does it involve animals. Therefore, ethics approval and consent are not applicable.

Statistical analysis

Statistical analysis is not conducted in this study. However, the MOMDSTP model and three well-established decision-making methods are employed to derive the optimal distribution of the company’s truck fleet under various optimization and multi-objective conditions.

In summary, this study uses a case study design to find the optimal distribution of a private sector company’s truck fleet under various optimization and multi-objective conditions. The study employs the MOMDSTP and three methods for decision-making, and data on the truck fleet, sources, destinations, and products are used as materials. Ethics approval and consent are not applicable, and statistical analysis is not performed.

Multi-objective transportation problem

The multi-objective optimization problem is a complex issue that demands diverse approaches to determine the most satisfactory solution. Prevalent techniques employed in this domain include the Weighted Sum Method, Minimum Distance Method, Zimmermann Programming Technique, and Global Criteria Method. Each method offers its own benefits and limitations, and the selection of a specific method depends on the nature of the problem and the preferences of the decision-makers [ 21 ].

This section discusses various methodologies employed to identify the most optimal solution(s) for the multi-objective multi-dimensional solid transportation problem (MOMDSTP), which is utilized as the basis for the case study. These methodologies encompass the Minimum Distance Method (MDM), the Zimmermann Programming Technique, and the Global Criteria Method [ 22 ].

Zimmermann Programming Technique

The Zimmermann Programming Technique (ZPT) is a multi-objective optimization approach that was developed by Professor Hans-Joachim Zimmermann in the late 1970s. This technique addresses complex problems with multiple competing objectives that cannot be optimized simultaneously. Additionally, it incorporates the concept of an “aspiration level,” representing the minimum acceptable level for each objective. The aspiration level ensures that the solution obtained is satisfactory for each objective. If the solution does not meet the aspiration level for any objective, the weights are adjusted, and the optimization process is iterated until a satisfactory solution is obtained.

A key advantage of ZPT is its ability to incorporate decision-makers’ preferences and judgments into the decision-making process. The weights assigned to each objective are based on the decision-maker’s preferences, and the aspiration levels reflect their judgments about what constitutes an acceptable level for each objective [ 23 ].

The Zimmermann Programming Technique empowers decision-makers to incorporate multiple objectives and achieve a balanced solution. By assigning weights to objectives, a trade-off can be made to find a compromise that meets various criteria. For example, this technique can optimize cost, delivery time, and customer satisfaction in supply chain management [ 24 ]. However, the interpretation of results may require careful consideration, and computational intensity can increase with larger-scale and complex problems.

 In order to obtain the solution, each objective is considered at a time to get the lower and upper bounds for that objective. Let for objective, and are the lower (min) and upper (max) bounds. The membership functions of the first and second objective functions can be generated based on the following formula [ 25 ]:

Next, the fuzzy linear programming problem is formulated using the max–min operator as follows:

Maximize min \({\mu }_{k}\left({F}_{k}\left(x\right)\right)\)   

Subject to \({g}_{i}\left(x\right) \left\{ \le ,= , \ge \right\}{b}_{i}\mathrm{ where }\;i = 1, 2, 3, ..., m.\)   

Moreover, x ≥ 0.

Global Criteria Method

The Global Criteria Method is a multi-objective optimization method that aims to identify the set of ideal solutions based on predetermined criteria. This method involves defining a set of decision rules that assess the feasibility and optimality of the solutions based on the objectives and constraints [ 26 ]. By applying decision rules, solutions that fail to meet the predetermined criteria are eliminated, and the remaining solutions are ranked [ 27 ].

The Global Criteria Method assesses overall system performance, aiding decision-makers in selecting solutions that excel in all objectives. However, it may face challenges when dealing with conflicting objectives [ 28 ]. Furthermore, it has the potential to overlook specific details, and the choice of aggregation function or criteria can impact the results by favoring specific solutions or objectives.

Let us consider the following ideal solutions:

f 1* represents the ideal solution for the first objective function,

f 2* represents the ideal solution for the second objective function, and

n 1* represents the ideal solution for the nth objective function.

Objective function formula:

Minimize the objective function F  =  \(\sum_{k=1}^{n}{(\frac{{f}_{k}\left({x}^{*}\right)-{f}_{k}(x)}{{f}_{k}({x}^{*})})}^{p}\)

Subject to the constraints: g i ( x ) \(\le\) 0, i  = 1, 2,.., m

The function fk( x ) can depend on variables x 1 , x 2 , …, x n .

Minimum Distance Method

The Minimum Distance Method (MDM) is a novel distance-based model that utilizes the goal programming weighted method. The model aims to minimize the distances between the ideal objectives and the feasible objective space, leading to an optimal compromise solution for the multi-objective linear programming problem (MOLPP) [ 29 ]. To solve MOLPP, the proposed model breaks it down into a series of single objective subproblems, with the objectives transformed into constraints. To further enhance the compromise solution, priorities can be defined using weights, and a criterion is provided to determine the best compromise solution. A significant advantage of this approach is its ability to obtain a compromise solution without any specific preference or for various preferences.

The Minimum Distance Method prioritizes solutions that closely resemble the ideal or utopian solution, assisting decision-makers in ranking and identifying high-performing solutions. It relies on a known and achievable ideal solution, and its sensitivity to the chosen reference point can influence results. However, it does not provide a comprehensive trade-off solution, focusing solely on proximity to the ideal point [ 30 ].

The mathematical formulation for MDM for MOLP is as follows:

The formulation for multi-objective linear programming (MOLP) based on the minimum distance method is referred to[ 31 ]. It is possible to derive the multi-objective transportation problem with two objective functions using this method and its corresponding formula.

Subject to the following constraints:

f * 1 , f * 2 : the obtained ideal objective values by solving single objective STP.

w 1 , w 2 : weights for objective1 and objective2 respectively.

f 1, f 2: the objective values for another efficient solution.

d : general deviational variable for all objectives.

\({{c}_{ij}^{1}, c}_{ij}^{2}\) : the unit cost for objectives 1 and 2 from source i to destination j .

\({{x}_{ij}^{1}, x}_{ij}^{2}\) : the amount to be shipped when optimizing for objectives 1 and 2 from source i to destination j .

Mathematical model for STP

The transportation problem (TP) involves finding the best method to ship a specific product from a defined set of sources to a designated set of destinations, while adhering to specific constraints. In this case, the objective function and constraint sets take into account three-dimensional characteristics instead of solely focusing on the source and destination [ 32 ]. Specifically, the TP considers various modes of transportation, such as ships, freight trains, cargo aircraft, and trucks, which can be used to represent the problem in three dimensions When considering a single mode of transportation, the TP transforms into a solid transportation problem (STP), which can be mathematically formulated as follows:

The mathematical form of the solid transportation problem is given by [ 33 ]:

Subject to:

Z = the objective function to be minimized

m = the number of sources in the STP

n = the number of destinations in the STP

p = the number of different modes of transportation in the STP

x ijk represents the quantity of product transported from source i to destination j using conveyance k

c ijk = the unit transportation cost for each mode of transportation in the STP

a i = the amount of products available at source i

b j = the demand for the product at destination j

e k = the maximum amount of product that can be transported using conveyance k

The determination of the size of the fleet for each type of truck that is dispatched daily from each factory to all destinations for the transportation of various products is expressed formally as follows:

z ik denotes the number of trucks of type k that are dispatched daily from factory i .

C k represents the capacity of truck k in terms of the number of pallets it can transport.

x ijk denotes a binary decision variable that is set to one if truck k is dispatched from factory i to destination j to transport product p , and zero otherwise. The summation is performed over all destinations j and all products p .

This case study focuses on an Egyptian manufacturing company that produces over 70,000 pallets of various water and carbonated products daily. The company has 25 main distribution centers and eight factories located in different industrial cities in Egypt. The company’s transportation fleet consists of hundreds of trucks with varying capacities that are used to transport products from factories to distribution centers. The trucks have been classified into three types (type A, type B, and type C) based on their capacities. The company produces three different types of products that are packaged in pallets. It was observed that the sizes and weights of the pallets are consistent across all product types The main objective of this case study is to determine the minimum number of each truck type required in the manufacturer’s garage to minimize fuel consumption costs and reduce product delivery time.

The problem was addressed by analyzing the benefits of diversifying trucks and implementing the solid transport method. Subsequently, the problem was resolved while considering the capacities of the sources and the requirements of the destinations. The scenario involved shipping products using a single type of truck, and the fuel consumption costs were calculated accordingly. The first objective was to reduce the cost of fuel consumption on the one-way journey from the factories to the distribution centers. The second objective was to reduce the time of arrival of the products to the destinations. The time was calculated based on the average speed of the trucks in the company’s records, which varies depending on the weight and size of the transported goods.

To address the multiple objectives and the uncertainty in supply and demand, an approach was adopted to determine the minimum number of trucks required at each factory. This approach involved determining the maximum number of trucks of each type that should be present in each factory under all previous conditions. The study emphasizes the significance of achieving a balance between reducing transportation costs and time while ensuring trucks are capable of accommodating quantities of any size, thus avoiding underutilization.

Figure  1 presents the mean daily output, measured in pallets, for each factory across three distinct product types. Additionally, Fig.  2 displays the average daily demand, measured in pallets, for the distribution centers of the same three product types.

No. of pallets in each source

No. of pallets in each destination

Results and discussion

As a result of the case study, the single objective problems of time and fuel consumption cost have been solved. The next step is to prepare a model for the multi-objective multi-dimensional solid transportation problem. Prior to commencing, it is necessary to determine the upper and lower bounds for each objective.

Assuming the first objective is fuel consumption cost and the second objective is time, we calculate the upper and lower bounds as follows:

The lower bound for the first objective, “cost,” is generated from the optimal solution for its single-objective model, denoted as Z 1 ( x 1 ), and equals 70,165.50 L.E.

The lower bound for the second objective, “time,” is generated from the optimal solution for its single-objective model, denoted as Z 2 ( x 2 ), and equals 87,280 min.

The upper bound for the first objective is obtained by multiplying c ijkp for the second objective by x ijkp for the first objective. The resulting value is denoted as Z 1 ( x 2 ) and equals 73,027.50 L.E.

The upper bound for the second objective is obtained by multiplying t ijkp for the first objective by x ijkp for the second objective. The resulting value is denoted as Z 2 ( x 1 ) and equals 88,286.50 min.

As such, the aspiration levels for each objective are defined from the above values by evaluating the maximum and minimum value of each objective.

The aspiration level for the first objective, denoted as F 1, ranges between 70,165.50 and 73,027.50, i.e., 70,165.50 <  =  F 1 <  = 730,27.50.

The aspiration level for the second objective, denoted as F 2, ranges between 87,280 and 88,286.50, i.e., 87,280 <  =  F 2 <  = 88,286.50.

The objective function for the multi-objective multidimensional solid transport problem was determined using the Zimmermann Programming Technique, Global Criteria Method, and Minimum Distance Method. The first two methods directly provided the best compromise solution (BCS), while the last method generated non-dominated extreme points by assigning different weights to each objective and finding the BCS from them. The best compromise solution was obtained using the Lingo software [ 34 ]. Table 1 and Fig.  3 present the objective values for the optimal solutions of fuel consumption cost and time, the best compromise solution, and solutions with different weights for both objectives. Figure  4 illustrates the minimum required number of each type of truck for daily transportation of various products from sources to destinations.

Objective value in different cases

Ideal distribution of the company’s truck fleet

The primary objective of the case study is to determine the minimum number of trucks of each type required daily at each garage for transporting products from factories to distribution centers. The minimum number of trucks needs to be flexible, allowing decision-makers to make various choices, such as minimizing fuel consumption cost, delivery time, or achieving the best compromise between different objectives. To determine the minimum number of required trucks, we compare all the previously studied cases and select the largest number that satisfies the condition: min Zik (should be set) = max Zik (from different cases). Due to the discrepancy between the truck capacity and the quantity of products to be transported, the required number of trucks may have decimal places. In such cases, the fraction is rounded to the nearest whole number. For example, if the quantity of items from a location requires one and a half trucks, two trucks of the specified type are transported on the first day, one and a half trucks are distributed, and half a truck remains in stock at the distribution center. On the next day, only one truck is transferred to the same distribution center, along with the semi-truck left over from the previous day, and so on. This solution may be preferable to transporting trucks that are not at full capacity. Table 2 and Fig.  5 depict the ideal distribution of the company’s truck fleet under various optimization and multi-objective conditions.

Min. No. of trucks should be set for different cases

Conclusions

In conclusion, this research paper addresses the critical issue of optimizing transportation within the context of logistics and supply chain management, specifically focusing on the methods known as the solid transportation problem (STP) and the multi-dimensional solid transportation problem (MDSTP). The study presents a case study conducted on a private sector company in Egypt to determine the optimal distribution of its truck fleet under different optimization and multi-objective conditions.

The research utilizes the multi-objective multi-dimensional solid transportation problem (MOMDSTP) to identify the best compromise solution, taking into account fuel consumption costs and total shipping time. Three decision-making methods, namely the Zimmermann Programming Technique, the Global Criteria Method, and the Minimum Distance Method, are employed to derive optimal solutions for the objectives.

The findings of this study make a significant contribution to the development of approaches for solving multi-objective solid transportation problems with uncertain parameters. The research addresses the complexities of diverse truck fleets and transported products by incorporating fuzzy programming, uncertainty theory, and related concepts. It critically examines the limitations of previous approaches that often focused solely on single-objective solutions and overlooked specific constraints.

The primary objective of this research is to determine the minimum number of trucks required to fulfill decision-makers objectives while optimizing fuel consumption and transportation timeliness. The proposed approach combines the framework of the multi-dimensional solid transportation problem with a multi-objective optimization technique, offering comprehensive solutions for decision-makers with multiple priorities.

This study provides practical solutions for real-world transportation scenarios and demonstrates the potential for enhancing transportation efficiency and cost-effectiveness in various industries or contexts. The comparative analysis of the proposed work highlights the advancements and novelty introduced by the approach, emphasizing its significant contributions to the field of transportation problem research.

Future research should explore additional dimensions of the multi-objective solid transportation problem and incorporate other decision-making methods or optimization techniques. Additionally, incorporating uncertainty analysis and sensitivity analysis can enhance the robustness and reliability of the proposed solutions. Investigating the applicability of the approach in diverse geographical contexts or industries would yield further insights and broaden the potential applications of the research findings.

Availability of data and materials

The data that support the findings of this study are available from the company but restrictions apply to the availability of these data, which were used under license for the current study, and so are not publicly available. Data are however available from the authors upon reasonable request. Please note that some data has been mentioned in the form of charts as agreed with the company.

Abbreviations

Solid transportation problem

Multi-objective solid transportation problems

Multi-dimensional solid transportation problem

Multi-objective multi-dimensional solid transportation problem

Best compromise solution

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Automotive and Tractor Engineering Department, Minia University, Minia, Egypt

Mohamed H. Abdelati, Ali M. Abd-El-Tawwab & M Rabie

Automotive and Tractor Engineering Department, Helwan University, Mataria, Egypt

Elsayed Elsayed M. Ellimony

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MHA designed the research study, conducted data collection, analyzed the data, contributed to the writing of the paper, and reviewed and edited the final manuscript. AMA contributed to the design of the research study, conducted a literature review, analyzed the data, contributed to the writing of the paper, and reviewed and edited the final manuscript. EEME contributed to the design of the research study, conducted data collection, analyzed the data, contributed to the writing of the paper, and reviewed and edited the final manuscript.MR contributed to the design of the research study, conducted programming using Lingo software and others, analyzed the data, contributed to the writing of the paper, and reviewed and edited the final manuscript. All authors have read and approved the manuscript.

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Abdelati, M.H., Abd-El-Tawwab, A.M., Ellimony, E.E.M. et al. Solving a multi-objective solid transportation problem: a comparative study of alternative methods for decision-making. J. Eng. Appl. Sci. 70 , 82 (2023). https://doi.org/10.1186/s44147-023-00247-z

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Received : 19 April 2023

Accepted : 27 June 2023

Published : 20 July 2023

DOI : https://doi.org/10.1186/s44147-023-00247-z

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Transportation and Assignment Problems in Operations Research

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Questions and Answers

What is the main objective of the transportation problem.

• Increase the number of facilities
• Minimize the total cost (correct)
• Maximize the total cost
• Balance the distribution of resources

During which major event was the Transportation Problem first significantly applied?

• Industrial Revolution
• World War II (correct)
• Great Depression

What is a facility in the context of the Transportation Problem?

• An organizational unit producing goods or services (correct)
• A city or town
• An individual customer
• A transportation company

Which problem is widely encountered in supply chain management and logistics planning?

<p>Assignment Problem</p> Signup and view all the answers

What is the main purpose of the Assignment Problem?

<p>Minimize total cost</p> Signup and view all the answers

Who formulated the Assignment Problem?

<p>János von Neumann</p> Signup and view all the answers

Which algorithm was originally used by János von Neumann to solve the Assignment Problem?

<p>Hungarian Algorithm</p> Signup and view all the answers

How is the Transportation Problem related to the Assignment Problem?

<p>They are fundamentally different and unrelated problems</p> Signup and view all the answers

Which method is widely used today to solve the Assignment Problem?

<p>Hungarian algorithm</p> Signup and view all the answers

Study Notes

Operations research.

Operations research is an interdisciplinary field that deals with mathematical modeling and statistical analysis of real-world problems. It involves the application of methods from various fields such as mathematics, economics, computer science, statistics, and engineering to create procedures and models that help make better decisions during wartime and peacetime. Operational researchers work in a wide range of industries including transportation, health care, manufacturing, telecommunications, defense, finance, and energy.

Two prominent areas within operations research are the Transportation Problem and the Assignment problem. These problems are often encountered in supply chain management and logistics planning.

Transportation Problem

The Transportation Problem is a classic optimization problem in which resources are distributed over several facilities to minimize the total cost. In this context, a facility refers to any organizational unit that produces goods or services; it could be a factory, warehouse, office, store, or even a group of stores. The problem involves minimizing the total cost of shipping goods from suppliers to customers.

The first major application of the Transportation Problem was in the U.S. Army during World War II when the problem of shipping goods from stockpiles to the ports needed to be solved. The first published account of the transportation problem was an article by George Dantzig in 1951. Dantzig's article provided the first algorithmic solution for the problem and also showed how to solve a general linear programming problem using the simplex method.

Assignment Problem

The Assignment Problem, also known as the Matching Problem, is another optimization problem in operations research. It is a combinatorial optimization problem in which there are a number of agents and a number of jobs, each agent can do only one job and each job can be done by only one agent. The assignment problem is to assign jobs to agents so that the total cost is minimized.

The Assignment Problem was formulated by the Hungarian mathematician János von Neumann in 1947. Neumann's original approach did not use linear programming but rather a method known as the Hungarian algorithm, also called the Kuhn–Munkres algorithm. The Hungarian algorithm is now widely used to solve the assignment problem and other problems related to the transportation problem.

In summary, operations research is a valuable tool for decision-making in various industries. The Transportation Problem and the Assignment Problem are two important applications of operations research in the field of logistics and supply chain management.

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