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Difference between transportation and assignment problems?

Transportation problems and assignment problems are two types of linear programming problems that arise in different applications.

The main difference between transportation and assignment problems is in the nature of the decision variables and the constraints.

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Decision Variables:

In a transportation problem, the decision variables represent the flow of goods from sources to destinations. Each variable represents the quantity of goods transported from a source to a destination.

In contrast, in an assignment problem, the decision variables represent the assignment of agents to tasks. Each variable represents whether an agent is assigned to a particular task or not.

Constraints:

In a transportation problem, the constraints ensure that the supply from each source matches the demand at each destination and that the total flow of goods does not exceed the capacity of each source and destination.

In contrast, in an assignment problem, the constraints ensure that each task is assigned to exactly one agent and that each agent is assigned to at most one task.

Objective function:

The objective function in a transportation problem typically involves minimizing the total cost of transportation or maximizing the total profit of transportation.

In an assignment problem, the objective function typically involves minimizing the total cost or maximizing the total benefit of assigning agents to tasks.

In summary,

The transportation problem is concerned with finding the optimal way to transport goods from sources to destinations,

while the assignment problem is concerned with finding the optimal way to assign agents to tasks.

Both problems are important in operations research and have numerous practical applications.

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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:

{\displaystyle x_{ij}}

The whole model is:

subject to,

{\displaystyle x_{11}+x_{12}=1000}

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

difference of transportation problem and assignment problem

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Linear Programming and Its Applications pp 140–184 Cite as

Transportation and Assignment Problems

  • James K. Strayer 2  

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Part of the Undergraduate Texts in Mathematics book series (UTM)

Transportation and assignment problems are traditional examples of linear programming problems. Although these problems are solvable by using the techniques of Chapters 2–4 directly, the solution procedure is cumbersome; hence, we develop much more efficient algorithms for handling these problems. In the case of transportation problems, the algorithm is essentially a disguised form of the dual simplex algorithm of 4§2. Assignment problems, which are special cases of transportation problems, pose difficulties for the transportation algorithm and require the development of an algorithm which takes advantage of the simpler nature of these problems.

  • Assignment Problem
  • Transportation Problem
  • Basic Feasible Solution
  • Unique Optimal Solution
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Department of Mathematics, Lock Haven University, Lock Haven, PA, 17745, USA

James K. Strayer

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© 1989 Springer Science+Business Media New York

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Strayer, J.K. (1989). Transportation and Assignment Problems. In: Linear Programming and Its Applications. Undergraduate Texts in Mathematics. Springer, New York, NY. https://doi.org/10.1007/978-1-4612-1009-2_7

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AllDifferences

Difference Between Assignment and Transportation Model

  • 1.1 Comparison Between Assignment and Transportation Model With Tabular Form
  • 1.2 Comparison Chart
  • 1.3 Similarities
  • 2 More Difference

Comparison Between Assignment and Transportation Model With Tabular Form

The Major Difference Between Assignment and Transportation model is that Assignment model may be regarded as a special case of the transportation model. However, the Transportation algorithm is not very useful to solve this model because of degeneracy.

Assignment Model and Transportation Model Comparison

Comparison Chart

Similarities.

  • Both are special types of linear programming problems.
  • Both have an objective function, structural constraints, and non-negativity constraints. And the relationship between variables and constraints is linear.
  • The coefficients of variables in the solution will be either 1 or zero in both cases.
  • Both are basically minimization problems. For converting them into maximization problems same procedure is used.

More Difference

  • Difference between Lagrangian and Eulerian Approach
  • Difference between Line Standards and End Standards

Balanced and Unbalanced Transportation Problems

The two categories of transportation problems are balanced and unbalanced transportation problems . As we all know, a transportation problem is a type of Linear Programming Problem (LPP) in which items are carried from a set of sources to a set of destinations based on the supply and demand of the sources and destinations, with the goal of minimizing the total transportation cost. It is also known as the Hitchcock problem.

Introduction to Balanced and Unbalanced Transportation Problems

Balanced transportation problem.

The problem is considered to be a balanced transportation problem when both supplies and demands are equal.

Unbalanced Transportation Problem

Unbalanced transportation problem is defined as a situation in which supply and demand are not equal. A dummy row or a dummy column is added to this type of problem, depending on the necessity, to make it a balanced problem. The problem can then be addressed in the same way as the balanced problem.

Methods of Solving Transportation Problems

There are three ways for determining the initial basic feasible solution. They are

1. NorthWest Corner Cell Method.

2. Vogel’s Approximation Method (VAM).

3. Least Call Cell Method.

The following is the basic framework of the balanced transportation problem:

Basic Structure of Balanced Transportation Problem

The destinations D1, D2, D3, and D4 in the above table are where the products/goods will be transported from various sources O1, O2, O3, and O4. The supply from the source Oi is represented by S i . The demand for the destination Dj is d j . If a product is delivered from source Si to destination Dj, then the cost is called C ij .

Let us now explore the process of solving the balanced transportation problem using one of the ways known as the NorthWest Corner Method in this article.

Solving Balanced Transportation problem by Northwest Corner Method

Consider this scenario:

Balanced Transportation Problem -1

With three sources (O1, O2, and O3) and four destinations (D1, D2, D3, and D4), what is the best way to solve this problem? The supply for the sources O1, O2, and O3 are 300, 400, and 500, respectively. Demands for the destination D1, D2, D3, and D4 are 250, 350, 400, and 200, respectively.

The starting point for the North West Corner technique is (O1, D1), which is the table’s northwest corner. The cost of transportation is calculated for each value in the cell. As indicated in the diagram, compare the demand for column D1 with the supply from source O1 and assign a minimum of two to the cell (O1, D1).

Column D1’s demand has been met, hence the entire column will be canceled. The supply from the source O1 is still 300 – 250 = 50.

Balanced Transportation Problem - 2

Analyze the northwest corner, i.e. (O1, D2), of the remaining table, excluding column D1, and assign the lowest among the supply for the appropriate column and rows. Because the supply from O1 is 50 and the demand for D2 is 350, allocate 50 to the cell (O1, D2).

Now, row O1 is canceled because the supply from row O1 has been completed. Hence, the demand for Column D2 has become 350 – 50 = 50.

Balanced Transportation Problem - 3

The northwest corner cell in the remaining table is (O2, D2). The shortest supply from source O2 (400) and the demand for column D2 (300) is 300, thus putting 300 in the cell (O2, D2). Because the demand for column D2 has been met, the column can be deleted, and the remaining supply from source O2 is 400 – 300 = 100.

Balanced Transportation Problem - 4

Again, find the northwest corner of the table, i.e. (O2, D3), and compare the O2 supply (i.e. 100) to the D2 demand (i.e. 400) and assign the smaller (i.e. 100) to the cell (O2, D2). Row O2 has been canceled because the supply from O2 has been completed. Column D3 has a leftover demand of 400 – 100 = 300.

Balanced Transportation Problem -5

Continuing in the same manner, the final cell values will be:

Balanced Transportation Problem - 6

It should be observed that the demand for the relevant columns and rows is equal in the last remaining cell, which was cell (O3, D4). In this situation, the supply from O3 was 200, and the demand for D4 was 200, therefore this cell was assigned to it. Nothing was left for any row or column at the end.

To achieve the basic solution, multiply the allotted value by the respective cell value (i.e. the cost) and add them all together.

I.e., (250 × 3) + (50 × 1) + (300 × 6) + (100 × 5) + (300 × 3) + (200 × 2) = 4400.

Solving Unbalanced Transportation Problem

An unbalanced transportation problem is provided below. Because the sum of all the supplies, O1, O2, O3, and O4, does not equal the sum of all the demands, D1, D2, D3, D4, and D5, the situation is unbalanced.

Unbalanced Transportation Problem - 1

The idea of a dummy row or dummy column will be applied in this type of scenario. Because the supply is more than the demand in this situation, a fake demand column will be inserted, with a demand of (total supply – total demand), i.e. 117 – 95 = 22, as seen in the image below. A fake supply row would have been introduced if demand was greater than supply.

Unbalanced Transportation Problem - 2

Now this problem has been changed to a balanced transportation problem, and it can be addressed using any of the ways listed below to solve a balanced transportation problem, such as the northwest corner method mentioned earlier.

Frequently Asked Questions on Balanced and Unbalanced Transportation Problems

What is meant by balanced and unbalanced transportation problems.

The problem is referred to as a balanced transportation problem when both supplies and demands are equal. Unbalanced transportation is defined as a situation where supply and demand are not equal.

What is called a transportation problem?

The transportation problem is a type of Linear Programming Problem in which commodities are carried from a set of sources to a set of destinations while taking into account the supply and demand of the sources and destinations, respectively, in order to reduce the total cost of transportation.

What are the different methods to solve transportation problems?

The following are three approaches to solve the transportation issue:

  • NorthWest Corner Cell Method.
  • Least Call Cell Method.
  • Vogel’s Approximation Method (VAM).

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  1. Difference Between Transportation Problem and Assignment Problem

    A transportation problem is a Linear Programming Problem that deals with identifying an optimal solution for transportation and allocating resources to various destinations and from one site to another while keeping the expenditure to a minimum.

  2. PDF CHAPTER 15 TRANSPORTATION AND ASSIGNMENT PROBLEMS

    Learning objectives After completing this chapter, you should be able to 1. Describe the characteristics of transportation problems. 2. Formulate a spreadsheet model for a transportation problem from a description of the problem. 3. Do the same for some variants of transportation problems. 4.

  3. PDF Module 4: Transportation Problem and Assignment problem

    Types of Transportation problems: Balanced: When both supplies and demands are equal then the problem is said to be a balanced transportation problem. Unbalanced: When the supply and demand are not equal then it is said to be an unbalanced transportation problem.

  4. Difference between transportation and assignment problems?

    In summary, The transportation problem is concerned with finding the optimal way to transport goods from sources to destinations, while the assignment problem is concerned with finding the optimal way to assign agents to tasks. Both problems are important in operations research and have numerous practical applications.

  5. PDF Transportation and Assignment Problems

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    44 9 · Transportation and Assignment Problems m ij i,j i P k:(i,k)∈Em ik −b i j P k:(j,k)∈Em jk −b j 0 c ij Figure 9.1: Representation of flow conservation constraints by a transportation problem We now construct a transportation problem as follows. For every vertex i ∈ V, we add a sink vertex with demand P km ik−b i. For every ...

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  9. Transportation and Assignment Problems

    Abstract. Transportation and assignment problems are traditional examples of linear programming problems. Although these problems are solvable by using the techniques of Chapters 2-4 directly, the solution procedure is cumbersome; hence, we develop much more efficient algorithms for handling these problems.

  10. Transportation, Transshipment, and Assignment Problems

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  12. Transportation Problem: Definition, Formulation, and Types

    Difference between Transportation and Assignment Problem What is the Transportation Problem? A transportation problem is a Linear Programming Problem that deals with identifying an optimal solution for transportation and allocating resources to various destinations and from one site to another while keeping the expenditure to a minimum.

  13. PDF Transportation, and Assignment Problems

    1. Transportation Problem (TP) Distributing any commodity from any group of supply centers, called sources, to any group of receiving centers, called destinations, in such a way as to minimize the total distribution cost (shipping cost). 1.

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    Assignment Problem. Transportation Problem (i) Assignment means allocating various jobs to various people in the organization. Assignment should be done in such a way that the overall processing time is less, overall efficiency is high, overall productivity is high, etc. (i) A transportation problem is concerned with transportation method or ...

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    Subscribe 410 views 10 months ago Operation Research In this video, we discuss the introduction of an Assignment problem and the mathematical representation of the Assignment problem. Link...

  16. Solving Transshipment and Assignment Problems

    What Is the Assignment Problem? The assignment problem is another special case of the transportation problem. This type of problem arises when assigning workers to different tasks or, as illustrated below, assigning athletes to different legs of a relay. Assignment Problem Example. Consider the example of a swimming relay team in the Summer ...

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    Definition of the Transportation Problem. Properties of the A Matrix. Representation of a Nonbasic Vector in Terms of the Basic Vectors. The Simplex Method for Transportation Problems. Illustrative Examples and a Note on Degeneracy. The Simplex Tableau Associated with a Transportation Tableau. The Assignment Problem: (Kuhn's) Hungarian Algorithm

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  19. What is the difference between Assignment Problem and Transportation

    2 Answers +1 vote answered Apr 25, 2023 by MonaliAgarwal (16.9k points) selected May 9, 2023 by faiz Best answer +2 votes answered Aug 26, 2020 by Vijay01 (50.7k points) The assignment problem is a special case of the transportation problem. The differences are given below. ← Prev Question Next Question → Find MCQs & Mock Test

  20. PDF 4 UNIT FOUR: Transportation and Assignment problems

    Figure 8: Constructing a transportation problem 4.3.2 Mathematical model of a transportation problem Before we discuss the solution of transportation problems we will introduce the notation used to describe the transportation problem and show that it can be formulated as a linear programming problem. We use the following notation; x

  21. Difference Between Assignment and Transportation Model

    The Major Difference Between Assignment and Transportation model is that Assignment model may be regarded as a special case of the transportation model. However, the Transportation algorithm is not very useful to solve this model because of degeneracy. Comparison Chart Similarities Both are special types of linear programming problems.

  22. Difference Between Transportation Problem and Assignment ...

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  23. Balanced and Unbalanced Transportation Problems

    Unbalanced Transportation Problem. Unbalanced transportation problem is defined as a situation in which supply and demand are not equal. A dummy row or a dummy column is added to this type of problem, depending on the necessity, to make it a balanced problem. The problem can then be addressed in the same way as the balanced problem.