## Assignment Problem: Meaning, Methods and Variations | Operations Research

After reading this article you will learn about:- 1. Meaning of Assignment Problem 2. Definition of Assignment Problem 3. Mathematical Formulation 4. Hungarian Method 5. Variations.

## Meaning of Assignment Problem:

An assignment problem is a particular case of transportation problem where the objective is to assign a number of resources to an equal number of activities so as to minimise total cost or maximize total profit of allocation.

The problem of assignment arises because available resources such as men, machines etc. have varying degrees of efficiency for performing different activities, therefore, cost, profit or loss of performing the different activities is different.

Thus, the problem is “How should the assignments be made so as to optimize the given objective”. Some of the problem where the assignment technique may be useful are assignment of workers to machines, salesman to different sales areas.

## Definition of Assignment Problem:

Suppose there are n jobs to be performed and n persons are available for doing these jobs. Assume that each person can do each job at a term, though with varying degree of efficiency, let c ij be the cost if the i-th person is assigned to the j-th job. The problem is to find an assignment (which job should be assigned to which person one on-one basis) So that the total cost of performing all jobs is minimum, problem of this kind are known as assignment problem.

The assignment problem can be stated in the form of n x n cost matrix C real members as given in the following table:

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Assignment problem in linear programming : introduction and assignment model.

Assignment problem is a special type of linear programming problem which deals with the allocation of the various resources to the various activities on one to one basis. It does it in such a way that the cost or time involved in the process is minimum and profit or sale is maximum. Though there problems can be solved by simplex method or by transportation method but assignment model gives a simpler approach for these problems.

In a factory, a supervisor may have six workers available and six jobs to fire. He will have to take decision regarding which job should be given to which worker. Problem forms one to one basis. This is an assignment problem.

## 1. Assignment Model :

Suppose there are n facilitates and n jobs it is clear that in this case, there will be n assignments. Each facility or say worker can perform each job, one at a time. But there should be certain procedure by which assignment should be made so that the profit is maximized or the cost or time is minimized.

In the table, Co ij is defined as the cost when j th job is assigned to i th worker. It maybe noted here that this is a special case of transportation problem when the number of rows is equal to number of columns.

## Mathematical Formulation:

Any basic feasible solution of an Assignment problem consists (2n – 1) variables of which the (n – 1) variables are zero, n is number of jobs or number of facilities. Due to this high degeneracy, if we solve the problem by usual transportation method, it will be a complex and time consuming work. Thus a separate technique is derived for it. Before going to the absolute method it is very important to formulate the problem.

Suppose x jj is a variable which is defined as

1 if the i th job is assigned to j th machine or facility

0 if the i th job is not assigned to j th machine or facility.

Now as the problem forms one to one basis or one job is to be assigned to one facility or machine.

The total assignment cost will be given by

The above definition can be developed into mathematical model as follows:

Determine x ij > 0 (i, j = 1,2, 3…n) in order to

Subjected to constraints

and x ij is either zero or one.

## Method to solve Problem (Hungarian Technique):

Consider the objective function of minimization type. Following steps are involved in solving this Assignment problem,

1. Locate the smallest cost element in each row of the given cost table starting with the first row. Now, this smallest element is subtracted form each element of that row. So, we will be getting at least one zero in each row of this new table.

2. Having constructed the table (as by step-1) take the columns of the table. Starting from first column locate the smallest cost element in each column. Now subtract this smallest element from each element of that column. Having performed the step 1 and step 2, we will be getting at least one zero in each column in the reduced cost table.

3. Now, the assignments are made for the reduced table in following manner.

(i) Rows are examined successively, until the row with exactly single (one) zero is found. Assignment is made to this single zero by putting square □ around it and in the corresponding column, all other zeros are crossed out (x) because these will not be used to make any other assignment in this column. Step is conducted for each row.

(ii) Step 3 (i) in now performed on the columns as follow:- columns are examined successively till a column with exactly one zero is found. Now , assignment is made to this single zero by putting the square around it and at the same time, all other zeros in the corresponding rows are crossed out (x) step is conducted for each column.

(iii) Step 3, (i) and 3 (ii) are repeated till all the zeros are either marked or crossed out. Now, if the number of marked zeros or the assignments made are equal to number of rows or columns, optimum solution has been achieved. There will be exactly single assignment in each or columns without any assignment. In this case, we will go to step 4.

4. At this stage, draw the minimum number of lines (horizontal and vertical) necessary to cover all zeros in the matrix obtained in step 3, Following procedure is adopted:

(iii) Now tick mark all the rows that are not already marked and that have assignment in the marked columns.

(iv) All the steps i.e. (4(i), 4(ii), 4(iii) are repeated until no more rows or columns can be marked.

(v) Now draw straight lines which pass through all the un marked rows and marked columns. It can also be noticed that in an n x n matrix, always less than ‘n’ lines will cover all the zeros if there is no solution among them.

5. In step 4, if the number of lines drawn are equal to n or the number of rows, then it is the optimum solution if not, then go to step 6.

6. Select the smallest element among all the uncovered elements. Now, this element is subtracted from all the uncovered elements and added to the element which lies at the intersection of two lines. This is the matrix for fresh assignments.

7. Repeat the procedure from step (3) until the number of assignments becomes equal to the number of rows or number of columns.

## Related Articles:

• Two Phase Methods of Problem Solving in Linear Programming: First and Second Phase
• Linear Programming: Applications, Definitions and Problems

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## What Is the Assignment Method?

The assignment method is a way of allocating organizational resources in which each resource is assigned to a particular task. The resource could be monetary, personnel , or technological.

## Understanding the Assignment Method

The assignment method is used to determine what resources are assigned to which department, machine, or center of operation in the production process. The goal is to assign resources in such a way to enhance production efficiency, control costs, and maximize profits.

The assignment method has various applications in maximizing resources, including:

• Allocating the proper number of employees to a machine or task
• Allocating a machine or a manufacturing plant and the number of jobs that a given machine or factory can produce
• Assigning a number of salespersons to a given territory or territories
• Assigning new computers, laptops, and other expensive high-tech devices to the areas that need them the most while lower priority departments would get the older models

Companies can make budgeting decisions using the assignment method since it can help determine the amount of capital or money needed for each area of the company. Allocating money or resources can be done by analyzing the past performance of an employee, project, or department to determine the most efficient approach.

Regardless of the resource being allocated or the task to be accomplished, the goal is to assign resources to maximize the profit produced by the task or project.

## Example of Assignment Method

A bank is allocating its sales force to grow its mortgage lending business. The bank has over 50 branches in New York but only ten in Chicago. Each branch has a staff that is used to bring in new clients.

The bank's management team decides to perform an analysis using the assignment method to determine where their newly-hired salespeople should be allocated. Given the past performance results in the Chicago area, the bank has produced fewer new clients than in New York. The fewer new clients are the result of having a small market presence in Chicago.

As a result, the management decides to allocate the new hires to the New York region, where it has a greater market share to maximize new client growth and, ultimately, revenue.

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## Difference Between Transportation Problem and Assignment Problem

In this article, we will learn what is transportation problem, what is assignment problem, and the difference between them 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 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 have 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 variable and constraints are 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!!

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.

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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