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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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What Is The Assignment Method?

The assignment method is a strategic approach to allocating organizational resources, including tasks and jobs to various departments like people, machines, or teams. It aims to minimize total costs or completion time and gain maximum efficiency, by assigning resources to corresponding units.

The assignment procedure's importance stems from its capacity to optimize resource allocation procedures in a business. Organizations may guarantee that resources are used optimally, reducing waste and increasing productivity by implementing a systematic method. It facilitates the decision-making process for the efficient and economical use of resources by helping to make well-informed choices.

Table of contents

Assignment method explained, methodology, advantages & disadvantages, frequently asked questions (faqs), recommended articles.

• The assignment method strategically allocates resources to tasks, jobs or teams to minimize costs or completion time. It optimizes resource utilization, reduces waste, and improves operational efficiency.
• It involves using methods like complete enumeration, simplex, transportation, or the Hungarian method. It involves using the assignment method of linear programming .
• The   Hungarian assignment method efficiently solves assignment problems by determining optimal assignments using a cost matrix.
• Advantages include structured resource allocation, enhanced resource utilization, and improved operational effectiveness. Limitations include data accuracy requirements and limited flexibility for dynamic changes.

The assignment method in operation research is a strategy for allocating organizational resources to tasks to increase profit via efficiency gains, cost reductions , and improved handling of operations that might create bottlenecks . It is an operations management tool that, by allocating jobs to the appropriate individual, minimizes expenses , time, and effort.

The technique is an essential tool for project management and cost accounting . It assists in allocating indirect expenses , such as overhead, to objects or cost centers according to predetermined standards, such as direct labor hours or required machine hours. The method helps determine the overall cost of every good or service, which helps with pricing, output, and resource distribution decisions. It also guarantees effective work allocation, on-time project completion, and economical use of resources. In short, it solves assignment problems.

Assignment problems involve assigning workers to specific roles, such as office workers or trucks on delivery routes, or determining which machines or products should be used in a plant during a specific period. Transportation problems involve distributing empty freight cars or assigning orders to factories. Allocation problems also involve determining which machines or products should be used to produce a given product or set of products. Unit costs or returns can be independent or interdependent, and if allocations affect subsequent periods, the problem is dynamic, requiring consideration of time in its solution.

The assignment problem can be solved using four methods: The complete enumeration method, the simplex method, the transportation method, and the Hungarian method.

The complete enumeration approach generates a list of potential assignments between resources and activities, from which the best option is chosen based on factors like cost, distance, time, or optimum profit. If the minimum cost, time, or distance for two or more assignments is the same, then this approach offers numerous optimal solutions. However If there are a lot of assignments, it is no longer appropriate for manual calculations. Assignment method calculators, if reliable, can be used for the same.

The simplex method can be solved as a linear programming problem using the simplex algorithm. The transportation method is a special case of the assignment problem. The method is, however, computationally inefficient for solving the assignment problem due to the solution's degeneracy problem.

The Hungarian assignment method problem, developed by mathematician D. Konig, is a faster and more efficient approach to solving assignment problems. It involves determining the cost of making all possible assignments using a matrix. Each problem has a row representing the objects to be assigned and columns representing assigned tasks. The cost matrix is square, and the optimum solution is to have only one assignment in a row or column. This method is a variation of the transportation problem, with the cost matrix being square and the optimum solution being one assignment in a row or column of the cost matrix.

Let us look into a few examples to understand the concept better.

TechLogistics Solutions, an imaginary delivery company, employs the assignment method to optimize the distribution of its delivery trucks. They meticulously consider distance, traffic conditions, and delivery schedules. TechLogistics efficiently allocates trucks to routes through strategic assignments, effectively reducing fuel costs and ensuring punctual deliveries. This method significantly enhances the company's operational efficiency and optimizes the utilization of its delivery resources.

Suppose XYZ Inc., a manufacturing company, is challenged to efficiently assign tasks to its machines (A, B, and C). Using the assignment method, XYZ calculates the cost matrix, reflecting the cost associated with each task assigned to each machine. Leveraging advanced algorithms like the Hungarian method, the company identifies optimal task-machine assignments, minimizing overall costs. This approach enables XYZ to streamline its production processes and enhance cost-effectiveness in manufacturing operations .

Advantages of the assignment method include:

• Resource allocation is carried out in a structured and organized manner.
• Enhancement of resource utilization to achieve optimal outcomes.
• Facilitation of efficient distribution of tasks.
• Improvement in operational effectiveness and productivity.
• Economical allocation of resources.
• Reduction in project completion time.
• Consideration of multiple factors and constraints for informed decision-making.

The disadvantages of the assignment method are as follows:

• Dependence on accurate and up-to-date data for effective decision-making.
• Complexity when dealing with resource allocation on a large scale.
• Subjectivity is involved in assigning values to the resource-requirement matrix.
• Limited flexibility in accommodating dynamic changes or unforeseen circumstances.
• Applicable primarily to quantitative tasks, with limitations in addressing qualitative aspects.

Johnson's rule is an operations research method that aims to estimate the optimal sequence of jobs in two work centers to reduce makespan. It optimizes the overall efficiency of the process. In contrast, the assignment method is useful for resource allocation, matching resources to specific tasks or requirements to optimize efficiency.

The study assignment method refers to the process of allocating students to specific courses or study programs based on their preferences, skills etc. It involves matching students with appropriate courses or programs to ensure optimal utilization of educational resources and meet individual student needs.

The assignment method is frequently employed when there is a requirement to allocate restricted resources like personnel, equipment, or budget to particular tasks or projects. It aids in enhancing resource utilization operational efficiency, and enables informed decision-making regarding resource allocation considering various factors and constraints.

This article has been a guide to what is Assignment Method. Here, we explain its methodologies, examples, advantages, & disadvantages. You may also find some useful articles here -

• Cost Allocation Methods
• Propensity Score Matching
• Regression Discontinuity Design

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Ampl A Modeling Language for Mathematical Programming

Robert fourer, david m. gay, brian w. kernighan, transportation and assignment models - all with video answers.

Chapter Questions

This transportation model, which deals with finding a least cost shipping schedule, comes from Dantzig's Linear Programming and Extensions. A company has plants in Seattle and San Diego, with capacities 350 and 600 cases per week respectively. It has customers in New York, Chicago, and Topeka, which order 325,300 , and 275 cases per week. The distances involved are: $\begin{array}{lccc} & \text { New York } & \text { Chicago } & \text { Topeka } \\ \text { Seattle } & 2500 & 1700 & 1800 \\ \text { San Diego } & 2500 & 1800 & 1400\end{array}$ The shipping cost is $$\$ 90$$ per case per thousand miles. Formulate this model in AMPL and solve it to determine the minimum cost and the amounts to be shipped.

A small manufacturing operation produces six kinds of parts, using three machines. For the coming month, a certain number of each part is needed, and a certain number of parts can be accommodated on each machine; to complicate matters, it does not cost the same amount to make the same part on different machines. Specifically, the costs and related values are as follows: $$ \begin{array}{cccccccc} & {}{}{\text { Part }} \\ \text { Machine } & 1 & 2 & 3 & 4 & 5 & 6 & \text { Capacity } \\ 1 & 3 & 3 & 2 & 5 & 2 & 1 & 80 \\ 2 & 4 & 1 & 1 & 2 & 2 & 1 & 30 \\ 3 & 2 & 2 & 5 & 1 & 1 & 2 & 160 \\ \text { Required } & 10 & 40 & 60 & 20 & 20 & 30 & \end{array} $$ (a) Using the model in Figure 3-1a, create a file of data statements for this problem; treat the machines as the origins, and the parts as the destinations. How many of each part should be produced on each machine, so as to minimize total cost? (b) If the capacity of machine 2 is increased to 50 , the manufacturer may be able to reduce the total cost of production somewhat. What small change to the model is necessary to analyze this situation? How much is the total cost reduced, and in what respects does the production plan change? (c) Now suppose that the capacities are given in hours, rather than in numbers of parts, and that it takes a somewhat different number of hours to make the same part on different machines: $$ \begin{aligned} &\text { Part }\\ &\begin{array}{cccccccc} \text { Machine } & 1 & 2 & 3 & 4 & 5 & 6 & \text { Capacity } \\ 1 & 1.3 & 1.3 & 1.2 & 1.5 & 1.2 & 1.1 & 50 \\ 2 & 1.4 & 1.1 & 1.1 & 1.2 & 1.2 & 1.1 & 90 \\ 3 & 1.2 & 1.2 & 1.5 & 1.1 & 1.1 & 1.2 & 175 \end{array} \end{aligned} $$ Modify the supply constraint so that it limits total time of production at each "origin" rather than the total quantity of production. How is the new optimal solution different? On which machines is all available time used? (d) Solve the preceding problem again, but with the objective function changed to minimize total machine-hours rather than total cost.

This exercise deals with generalizations of the transportation model and data of Figure 3-1. (a) Add two parameters, supply_pct and demand_pct, to represent the maximum fraction of a mill's supply that may be sent to any one factory, and the maximum fraction of a factory's demand that may be satisfied by any one mill. Incorporate these parameters into the model of Figure 3-la. Solve for the case in which no more than $50 \%$ of a mill's supply may be sent to any one factory, and no more than $85 \%$ of a factory's demand may be satisfied by any one mill. How does this change the minimum cost and the optimal amounts shipped? (b) Suppose that the rolling mills do not produce their own slabs, but instead obtain slabs from two other plants, where the following numbers of tons are to be made available: $$ \begin{array}{ll} \text { MIDTWN } & 2700 \\ \text { HAMLTN } & 4200 \end{array} $$ The cost per ton of shipping a slab from a plant to a mill is as follows: $$ \begin{array}{lccc} & \text { GARY } & \text { CLEV } & \text { PITT } \\ \text { MIDTWN } & 12 & 8 & 17 \\ \text { HAMLTN } & 10 & 5 & 13 \end{array} $$ All other data values are the same as before, but with supply_pct reinterpreted as the maximum fraction of a plant's supply that may be sent to any one mill.

Formulate this situation as an AMPL model. You will need two indexed collections of variables, one for the shipments from plants to mills, and one for the shipments from mills to factories. Shipments from each mill will have to equal supply, and shipments to each factory will have to equal demand as before; also, shipments out of each mill will have to equal shipments in. Solve the resulting linear program. What are the shipment amounts in the minimum-cost solution? (c) In addition to the differences in shipping costs, there may be different costs of production at the plants and mills. Explain how production costs could be incorporated into the model. (d) When slabs are rolled, some fraction of the steel is lost as scrap. Assuming that this fraction may be different at each mill, revise the model to take scrap loss into account. (e) In reality, scrap is not really lost, but is sold for recycling. Make a further change to the model to account for the value of the scrap produced at each mill.

This exercise considers variations on the assignment problem introduced in Section 3.3 . (a) Try reordering the list of members of DEST in the data (Figure 3-2), and solving again. Find a reordering that causes your solver to report a different optimal assignment. (b) An assignment that gives even one person a very low-ranked office may be unacceptable, even if the total of the rankings is optimized. In particular, our solution gives one individual her sixth choice; to rule this out, change all preferences of six or larger in the cost data to 99 , so that they will become very unattractive. (You'll learn more convenient features for doing the same thing in later chapters, but this crude approach will work for now.) Solve the assignment problem again, and verify that the result is an equally good assignment in which no one gets worse than fifth choice. Now apply the same approach to try to give everyone no worse than fourth choice. What do you find? (c) Suppose now that offices $\mathrm{C} 118, \mathrm{C} 250$ and $\mathrm{C} 251$ become unavailable, and you have to put two people each into $\mathrm{C} 138, \mathrm{C} 140$ and $\mathrm{C} 246$. Add 20 to each ranking for these three offices, to reflect the fact that anyone would prefer a private office to a shared one. What other modifications to the model and data would be necessary to handle this situation? What optimal assignment do you get?

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Transportation and Assignment Models

The linear programs in Chapters 1 and 2 are all examples of classical ''activity'' models. In such models the variables and constraints deal with distinctly different kinds of activities-tons of steel produced versus hours of mill time used, or packages of food bought versus percentages of nutrients supplied. To use these models you must supply coefficients like tons per hour or percentages per package that convert a unit of activity in the variables to the corresponding amount of activity in the constraints. This chapter addresses a significantly different but equally common kind of model, in which something is shipped or assigned, but not converted. The resulting constraints, which reflect both limitations on availability and requirements for delivery, have an especially simple form. We begin by describing the so-called transportation problem, in which a single good is to be shipped from several origins to several destinations at minimum overall cost. This problem gives rise to the simplest kind of linear program for minimum-cost flows. We then generalize to a transportation model, an essential step if we are to manage all the data, variables and constraints effectively. As with the diet model, the power of the transportation model lies in its adaptability. We continue by considering some other interpretations of the ''flow'' from origins to destinations, and work through one particular interpretation in which the variables represent assignments rather than shipments. The transportation model is only the most elementary kind of minimum-cost flow model. More general models are often best expressed as networks, in which nodes-some of which may be origins or destinations-are connected by arcs that carry flows of some kind. AMPL offers convenient features for describing network flow models, including node and arc declarations that specify network structure directly. Network models and the relevant AMPL features are the topic of Chapter 15. 43

Related Papers

sachin gupta

Models of networks have appeared in several chapters, notably in the transportation problems in Chapter 3. We now return to the formulation of these models, and AMPL's features for handling them. Figure 15-1 shows the sort of diagram commonly used to describe a network problem. A circle represents a node of the network, and an arrow denotes an arc running from one node to another. A flow of some kind travels from node to node along the arcs, in the directions of the arrows. An endless variety of models involve optimization over such networks. Many cannot be expressed in any straightforward algebraic way or are very difficult to solve. Our discussion starts with a particular class of network optimization models in which the decision variables represent the amounts of flow on the arcs, and the constraints are limited to two kinds: simple bounds on the flows, and conservation of flow at the nodes. Models restricted in this way give rise to the problems known as network linear programs. They are especially easy to describe and solve, yet are widely applicable. Some of their benefits extend to certain generalizations of the network flow form, which we also touch upon. We begin with minimum-cost transshipment models, which are the largest and most intuitive source of network linear programs, and then proceed to other well-known cases: maximum flow, shortest path, transportation and assignment models. Examples are initially given in terms of standard AMPL variables and constraints, defined in var and subject to declarations. In later sections, we introduce node and arc declarations that permit models to be described more directly in terms of their network structure. The last section discusses formulating network models so that the resulting linear programs can be solved most efficiently. 15.1 Minimum-cost transshipment models As a concrete example, imagine that the nodes and arcs in Figure 15-1 represent cities and intercity transportation links. A manufacturing plant at the city marked PITT will

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Robert Fourer

Transportation Research Part B: Methodological

M. Grazia Speranza

Diego Klabjan

The Transportation Problem is the special class of Linear Programming Problem. It arises when the situation in which a commodity is shipped from sources to destinations. The main object is to determine the amounts shipped from each sources to each destinations which minimize the total shipping cost while satisfying both supply criteria and demand requirements. In this paper, we are giving the idea about to finding the Initial Basic Feasible solution as well as the optimal solution or near to the optimal solution of a Transportation problem using the method known as " An Alternate Approach to find an optimal Solution of a Transportation Problem ". An Algorithm provided here, concentrate at unoccupied cells and proceeds further. Also, the numerical examples are provided to explain the proposed algorithm. However, the above method gives a step by step development of the solution procedure for finding an optimal solution.

Mollah Mesbahuddin Ahmed

Industries require planning in transporting their products from production centres to the users end with minimal transporting cost to maximize profit. This process is known as Transportation Problem which is used to analyze and minimize transportation cost. This problem is well discussed in operation research for its wide application in various fields, such as scheduling, personnel assignment, product mix problems and many others, so that this problem is really not confined to transportation or distribution only. In the solution procedure of a transportation problem, finding an initial basic feasible solution is the prerequisite to obtain the optimal solution. Again, development is a continuous and endless process to find the best among the bests. The growing complexity of management calls for development of sound methods and techniques for solution of the problems. Considering these factors, this research aims to propose an algorithm " Incessant Allocation Method " to obtain an initial basic feasible solution for the transportation problems. Several numbers of numerical problems are also solved to justify the method. Obtained results show that the proposed algorithm is effective in solving transportation problems.

Taesung Hwang

Đào Thanh Duy

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• Published: 20 July 2023

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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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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This chapter begins with a brief overview of traffic assignment in transportation systems. Section 3.1 introduces the assignment problem in transportation as the distribution of traffic in a network considering the demand between locations and the transport supply of the network. Four trip assignment models relevant to transportation are presented and characterized. Section 3.2 covers traffic assignment to uncongested networks based on the assumption that cost does not depend on traffic flow. Section 3.3 introduces the topic of traffic assignment and congested models based on assumptions from traffic flow modeling, e.g., each vehicle is traveling at the legal velocity, v , and each vehicle driver is following the preceding vehicle at a legal safe velocity. Section 3.4 covers the important topic of equilibrium assignment which can be expressed by the so-called fixed-point models where origin to destination (O-D) demands are fixed, representing systems of nonlinear equations or variational inequalities. Equilibrium models are also used to predict traffic patterns in transportation networks that are subject to congestion phenomena. Section 3.5 presents the topic of multiclass assignment, which is based on the assumption that travel demand can be allocated as a number of distinct classes which share behavioral characteristics. In Sect. 3.6, dynamic traffic assignment is introduced which allows the simultaneous determination of a traveler’s choice of departure time and path. With this approach, phenomenon such as peak spreading in response to congestion dynamics or time-varying tolls can be directly analyzed. In Sect. 3.7, transportation network synthesis is introduced which focuses on the modification of a transportation road network to fit a required demand. Section 3.8 covers a case study involving a diverging diamond interchange (DDI), an interchange in which the two directions of traffic on a nonfreeway road cross to the opposite side on both sides of a freeway overpass. The DDI requires traffic on the freeway overpass (or underpass) to briefly drive on the opposite side of the road. Section 3.9 contains comprehensive questions from the transportation system area. A final section includes references and suggestions for further reading.

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Dietmar P. F. Möller

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Möller, D.P.F. (2014). Traffic Assignments to Transportation Networks. In: Introduction to Transportation Analysis, Modeling and Simulation. Simulation Foundations, Methods and Applications. Springer, London. https://doi.org/10.1007/978-1-4471-5637-6_3

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Published : 12 August 2014

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