is transportation problem and assignment problem the same

Difference between transportation and assignment problems?

• Engineeringbro
• February 11, 2023
• March 10, 2024
• 3 mins read

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lets understand the 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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Additional Different between Transportation and Assignment Problems are as follows : 

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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Solving Transshipment and Assignment Problems

In a previous blog, we provided an overview of the transportation problem – the problem of finding minimum cost shipping routes for products or resources between supply locations and demand locations. We also provided a little background on the algorithm for solving such problems along with several examples using an implementation in IMSL.

In this blog, we look at two special cases of the transportation problem – the transshipment problem and assignment problem.

What Is the Transshipment Problem?

The transshipment problem is a special case of the transportation problem in which shipping paths can include intermediate points. For example, shipping from Los Angeles to New York via Denver may be less expensive than shipping directly (non-stop) to New York. Intermediate points also arise when different modes of transportation are available at different costs (for example, shipping via truck versus rail) or when products must be stored for a time at intermediate points for special processing or packaging.

A table display helps illustrate the transshipment problem. In the standard table used for a transportation problem, the transshipment or intermediate points are appended to both the demand center columns and the supply center rows. For each of the transshipment points, the supply and demand are set equal to the total supply and the total demand for the problem (assuming the problem is balanced or has been balanced with dummy variables). In the cost matrix, the point-to-point costs are all still required, except that there will be zero values for the cost where the same label exists in both the supply and demand. Arbitrarily large values are used for any paths that aren’t logistically valid, for example, shipping between transshipment points. This table format is illustrated in the following example.

Transshipment Problem Example

Consider an example where wheat is harvested in Nebraska and Colorado and then must be shipped to grain elevators in Kansas City, Omaha, and Des Moines. These transshipment points then send the product along to Chicago, St. Louis, and Cincinnati.

The farms have outputs of 300 units each, while the demand is 200, 100, and 300 respectively at each destination. In this problem statement, the farms cannot ship to the endpoints directly, and the transshipment points cannot ship to each other.

Each of the point-to-point shipping costs appear in the table below. Wherever a path is not valid, a cost of 1000 is applied. In this case, the farms cannot ship directly to the destinations, and the transshipment points are not allowed to transfer between each other. For the transshipment points, which appear as both sources and destinations, there is zero cost to ship to itself.

Using the table format to formulate the problem and provide the point-to-point shipping costs gives:

Table 1: Cost Matrix for Transshipment of Wheat

The problem is balanced, noting that the transshipment points each have a supply and demand equal to the sum of the actual values, 600. The source code to solve this problem using IMSL is given in the next subsection. The solution matrix is shown below for a total cost of $12,400:

Table 2: Solution Matrix for Transshipment of Wheat

The solution indicates that the 300 units from Nebraska are shipped to Des Moines, while all 300 units from Colorado are shipped to Kansas City. Once in Kansas City, all 300 units are sent off to Cincinnati, but at Des Moines, the shipment is split with 200 heading to Chicago and 100 to St. Louis. Notice the 600 units in Omaha indicate that this point is not part of the solution as the site is meeting its (artificial) demand with its own (artificial) supply.

Solving the Transshipment Problem With IMSL

While this example is small enough to solve manually, most problems are too large to do this practically. The C code to solve this problem using IMSL is given below. Note how the inputs (the supply and demand arrays and the cost matrix) are drawn directly from Table 1. Conceptually, larger sized problems are set up in the same way.

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 Olympics. The objective is to build the best (fastest) swimming medley relay team given the four events and the times of five swimmers for each event. That is, the problem is to assign one and only one swimmer to one and only one leg of the medley relay that gives a minimum overall time. Each supply item (a swimmer) provides only one unit of supply, and each destination item (a leg of the race) demands only one unit, as shown in the following table:

Table 3 Cost Matrix for the Assignment of Swimmers

Note that swim times are measured to the tenths of a second (for example 48.1 seconds); to keep everything integer, the times have been multiplied by ten. Due to the linearity of the problem, scaling the cost matrix has no effect on the solution. With unit supply and demand across the board, the resulting solution will be mostly zeros with a value of one located for the best choice of swimmer for each leg of the medley relay.

No special formulation is required when using the IMSL transport function to solve the assignment problem. The source code is shown in the next subsection. The solution matrix is below, giving an optimal best time of 190.1 seconds.

Table 4 Solution Matrix for the Assignment of Swimmers

With this small example it’s easy to pick out a few characteristics of the solution. First, we see that swimmer E, while not the slowest in any event, is also not the fastest in any event and is not selected for the medley. Swimmer D, who is the slowest swimmer in some events, is nevertheless selected for the backstroke event because other, faster swimmers in that event are needed for other legs. In real life, the coach might average a swimmer’s last 5 times and might also consider the variation in those times. How consistently fast is a swimmer in any given event?

The idea of incorporating variation or additional uncertainties extends to any problem with respect to costs. If there is known or estimable variation in costs, how is it accounted for in the optimal solution? Similarly, how would we account for variation or uncertainties in the supplies and/or demands? These are ideas we will be exploring in future blogs on the transportation problem.

Solving the Assignment Problem With IMSL

The C code to solve this problem definition using the IMSL transport algorithm is as follows:

Final Thoughts

In this blog, we looked at the transshipment and assignment problems and described how they can be understood and solved as transportation problems.

If you missed our first blog on this topic, you can read about it here: Solving the Transportation Problem .

Easily Solve the Transshipment and Assignment Problems With IMSL

Transshipment and assignment problems along with traditional transportation problems are easily solved using the transportation algorithm included in IMSL.

Whether you’re working in C/C++, Fortran, Java, or Python, you can evaluate the IMSL library for your application free. Try it today via the link below.

TRY IMSL FREE

Transportation and Assignment Problems

Cite this chapter.

• James K. Strayer 2  

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

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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.

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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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Difference Between Transportation Problem and Assignment Problem

The transportation problem in operational research aims to find the most economical way of transporting goods from multiple sources to multiple destinations. On the other hand, the assignment problem focuses on assigning tasks, jobs, or resources one-to-one. Both of these problems are usually solved through linear programming techniques. The transportation problem is commonly approached through simplex methods, and the assignment problem is addressed using specific algorithms like the Hungarian method. In this article, we will learn the difference between transportation problems and assignment problems with the help of examples.

Transportation Problems and Assignment Problems are types of Linear Programming Problems. Transportation Problem deals with the optimal distribution of goods or resources from multiple sources to multiple destinations. While Assignment Problem deals with allocating tasks, jobs, or resources one-to-one.

These LPP methods are used for cost minimization, resource allocation, supply chain management, workforce planning, facility location, time management, and decision-making support.

This article will briefly discuss the difference between transportation problems and assignment problems based on different parameters.

So, let’s explore the article.

Table of Content

• Transportation Problem vs Assignment Problem
• Transportation Problem
• Assignment Problem
• Key Differences and Similarities

What is the Difference Between Transportation and Assignment Problems?

What is a 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.

In simple words, the main objective of the Transportation problem is to deliver (from the source to the destination) the resources at the minimum cost.

What are the different types of transportation problems?

• Balanced: A transportation problem in which total supply equals total demand, i.e., Total Supply = Total Demand .
• Unbalanced: A transportation problem in which total supply doesn’t equal total demand, i.e., Total Supply != Total Demand .

What are the different methods to solve the transportation problem?

The initial feasible solution can be found by any of the three methods:

• Northwest Corner Method (NWC)
• Least Corner Method (LCM)
• Vogel’s Approximation Method (VAM)

Once you find the initial feasible solution, use the Modified Distribution Method (MODI method) or the u-v method to find the optimal solution.

What is an Assignment Problem?

An Assignment Problem is a special type of Transportation Problem in Operational Research that deals with assigning n origins (workers or instances) to n destinations (jobs or machines). The goal of the assignment problem is to determine the minimum cost of the assignment.

Each origin must be assigned to one and only one destination and each destination must be assigned to one and only one origin.

The solution to the assignment method can be found using the Hungarian Method.

Note: Transportation methods can be used to find the solutions to assignment problems, but assignment methods can’t be used to find the solutions to transportation problems.

What are the Key Differences and Similarities Between Transportation and Assignment Problems?

• Assignment Problem is a special type of transportation problem.
• Both are minimization problems having an objective function and structural and non-negative constraints.
• The relationship between variables and constraints is linear.

In this article, we have briefly discussed the transportation problem, the assignment problem, and the difference between both based on different parameters. Hope you will like the article.

Happy Learning!!

Related Reads

What is the key differences and similarities between Transporation problem and assignment problem?

Transportation problem deals with the optimal distribution of goods or resources from multiple sources to multiple destinations, whereas assignment problem deals with allocating tasks, jobs, or resources one-to-one. Assignment Problem is a special type of transportation problem. Both transport and assignment problems are Linear Programming Problems. Both are minimization problems having an objective function and structural and non-negative constraints. The relationship between variable and constraints are linear.

How are transportation and assignment problems solved?

Both problems are typically solved using linear programming methods. The transportation problem often employs methods like the simplex method, while the assignment problem can use algorithms like the Hungarian method.

Can the assignment problem be considered a type of transportation problem?

Yes, the assignment problem is often seen as a special case of the transportation problem, where the objective is to optimally match elements of one set to another, such as workers to tasks.

Vikram has a Postgraduate degree in Applied Mathematics, with a keen interest in Data Science and Machine Learning. He has experience of 2+ years in content creation in Mathematics, Statistics, Data Science, and Mac... Read Full Bio

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• Introduction

Transportation Problem Explained and how to solve it?

• Transportation Problem
• Balanced Problem
• Unbalanced Problem

Contributed by: Patrick

Operations Research (OR) is a state of art approach used for problem-solving and decision making. OR helps any organization to achieve their best performance under the given constraints or circumstances. The prominent OR techniques are,

• Linear programming
• Goal programming
• Integer programming
• Dynamic programming
• Network programming

One of the problems the organizations face is the transportation problem. It originally means the problem of transporting/shipping the commodities from the industry to the destinations with the least possible cost while satisfying the supply and demand limits.  It is a special class of linear programming technique that was designed for models with linear objective and constraint functions. Their application can be extended to other areas of operation, including

• Scheduling and Time management
• Network optimization
• Inventory management
• Enterprise resource planning
• Process planning
• Routing optimization  

The notations of the representation are:

m sources and n destinations

(i , j) joining source (i) and destination (j)

c ij 🡪  transportation cost per unit

x ij 🡪  amount shipped

a i   🡪 the amount of supply at source (i)

b j   🡪 the amount of demand at destination (j)

Transportation problem works in a way of minimizing the cost function. Here, the cost function is the amount of money spent to the logistics provider for transporting the commodities from production or supplier place to the demand place. Many factors decide the cost of transport. It includes the distance between the two locations, the path followed, mode of transport, the number of units that are transported, the speed of transport, etc. So, the focus here is to transport the commodities with minimum transportation cost without any compromise in supply and demand. The transportation problem is an extension of linear programming technique because the transportation costs are formulated as a linear function to the supply capacity and demand. Check out the course on transportation analytics .

Transportation problem exists in two forms. 

• Balanced 

It is the case where the total supply equals the total demand.

It is the case where either the demand is greater than the supply, or vice versa.

In most cases, the problems take a balanced form. It is because usually, the production units work, taking the inventory and the demand into consideration. Overproduction increases the inventory cost whereas under production is challenged by the demand. Hence the trade-off should be carefully examined. Whereas, the unbalanced form exists in a situation where there is an unprecedented increase or decrease in demand.

Let us understand this in a much simpler way with the help of a basic example. 

Let us assume that there is a leading global automotive supplier company named JIM. JIM has it’s production plants in many countries and supplies products to all the top automotive makers in the world. For instance, let’s consider that there are three plants in India at places M, N, and O. The capacity of the plants is 700, 300, 550 per day. The plant supplies four customers A, B, C, and D, whose demand is 650, 200, 450, 250 per day. The cost of transport per unit per km in INR and the distance between each source and destination in Kms are given in the tables below.

Here, the objective is to determine the unknown while satisfying all the supply and demand restrictions. The cost of shipping from a source to a destination is directly proportional to the number of units shipped.

Many sophisticated programming languages have evolved to solve OR problems in a much simpler and easier way. But the significance of Microsoft Excel cannot be compromised and devalued at any time. It also provides us with a greater understanding of the problem than others. Hence we will use Excel to solve the problem.

It is always better to formulate the working procedure in steps that it helps in better understanding and prevents from committing any error.

Steps to be followed to solve the problem:

• Create a transportation matrix (define decision variables)
• Define the objective function
• Formulate the constraints
• Solve using LP method 

Creating a transportation matrix:

A transportation matrix is a way of understanding the maximum possibilities the shipment can be done. It is also known as decision variables because these are the variables of interest that we will change to achieve the objective, that is, minimizing the cost function.

Define the objective function: 

An objective function is our target variable. It is the cost function, that is, the total cost incurred for transporting. It is known as an objective function because our interest here is to minimize the cost of transporting while satisfying all the supply and demand restrictions.

The objective function is the total cost. It is obtained by the sum product of the cost per unit per km and the decision variables (highlighted in red), as the total cost is directly proportional to the sum product of the number of units shipped and cost of transport per unit per Km.

The column “Total shipped” is the sum of the columns A, B, C, and D for respective rows and the row “Total Demand” is the sum of rows M, N, and O for the respective columns. These two columns are introduced to satisfy the constraints of the amount of supply and demand while solving the cost function. 

Formulate the constraints:

The constraints are formulated concerning the demand and supply for respective rows and columns. The importance of these constraints is to ensure they satisfy all the supply and demand restrictions.

For example, the fourth constraint, x ma + x na + x oa = 650 is used to ensure that the number of units coming from plants M, N, and O to customer A should not go below or above the demand that A has. Similarly the first constraint x ma + x mb + x mc + x md  = 700 will ensure that the capacity of the plant M will not go below or above the given capacity hence, the plant can be utilized to its fullest potential without compromising the inventory. 

Solve using LP method:

The simplest and most effective method to solve is using solver. The input parameters are fed as stated below and proceed to solve. 

This is the best-optimized cost function, and there is no possibility to achieve lesser cost than this having the same constraints.

From the solved solution, it is seen that plant M ships 100 units to customer A, 350 units to C and 250 units to D. But why nothing to customer B? And a similar trend can be seen for other plants as well. 

What could be the reason for this? Yes, you guessed it right! It is because some other plants ship at a profitable rate to a customer than others and as a result, you can find few plants supplying zero units to certain customers. 

So, when will these zero unit suppliers get profitable and can supply to those customers? Wait! Don’t panic. Excel has got away for it too. After proceeding to solve, there appears a dialogue box in which select the sensitivity report and click OK. You will get a wonderful sensitivity report which gives details of the opportunity cost or worthiness of the resource.

Basic explanation for the report variables,

Cell: The cell ID in the excel

Name: The supplier customer pairing

Final value: Number of units shipped (after solving)

Reduced cost: How much should the transportation cost per unit per km should be reduced to make the zero supplying plant profitable and start supplying

Objective coefficient: Current transportation cost per unit per Km for each supplier customer pair

Allowable Increase: It tells us the maximum cost of the current transportation cost per unit per Km can be increased which doesn’t make any changes to the solution

Allowable Decrease: It tells how much maximum the current transportation cost per unit per Km can be lowered which doesn’t make any changes to the solution

Here, look into the first row of the sensitivity report. Plant M supplies to customer A. Here, the transportation cost per unit per Km is ₹14 and 100 units are shipped to customer A. In this case, the transportation cost can increase a maximum of ₹6, and can lower to a maximum of ₹1. For any value within this range, there will not be any change in the final solution. 

Now, something interesting. Look at the second row. Between MB, there is not a single unit supplied to customer B from plant M. The current shipping cost is ₹22 and to make this pair profitable and start a business, the cost should come down by ₹6 per unit per Km. Whereas, there is no possibility of increasing the cost by even a rupee. If the shipping cost for this pair comes down to ₹16, we can expect a business to begin between them, and the final solution changes accordingly.

The above example is a balanced type problem where the supply equals the demand. In case of an unbalanced type, a dummy variable is added with either a supplier or a customer based on how the imbalance occurs.

Thus, the transportation problem in Excel not only solves the problem but also helps us to understand how the model works and what can be changed, and to what extent to modify the solution which in turn helps to determine the cost and an optimal supplier. 

If you found this helpful, and wish to learn more such concepts, head over to Great Learning Academy and enroll in the free online courses today.

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Table of contents

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:

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:

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.

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.

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.

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.

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

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.

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.

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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