MBA Notes

Unbalanced Assignment Problem: Definition, Formulation, and Solution Methods

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Are you familiar with the assignment problem in Operations Research (OR)? This problem deals with assigning tasks to workers in a way that minimizes the total cost or time needed to complete the tasks. But what if the number of tasks and workers is not equal? In this case, we face the Unbalanced Assignment Problem (UAP). This blog will help you understand what the UAP is, how to formulate it, and how to solve it.

What is the Unbalanced Assignment Problem?

The Unbalanced Assignment Problem is an extension of the Assignment Problem in OR, where the number of tasks and workers is not equal. In the UAP, some tasks may remain unassigned, while some workers may not be assigned any task. The objective is still to minimize the total cost or time required to complete the assigned tasks, but the UAP has additional constraints that make it more complex than the traditional assignment problem.

Formulation of the Unbalanced Assignment Problem

To formulate the UAP, we start with a matrix that represents the cost or time required to assign each task to each worker. If the matrix is square, we can use the Hungarian algorithm to solve the problem. But when the matrix is not square, we need to add dummy tasks or workers to balance the matrix. These dummy tasks or workers have zero costs and are used to make the matrix square.

Once we have a square matrix, we can apply the Hungarian algorithm to find the optimal assignment. However, we need to be careful in interpreting the results, as the assignment may include dummy tasks or workers that are not actually assigned to anything.

Solutions for the Unbalanced Assignment Problem

Besides the Hungarian algorithm, there are other methods to solve the UAP, such as the transportation algorithm and the auction algorithm. The transportation algorithm is based on transforming the UAP into a transportation problem, which can be solved with the transportation simplex method. The auction algorithm is an iterative method that simulates a bidding process between the tasks and workers to find the optimal assignment.

In summary, the Unbalanced Assignment Problem is a variant of the traditional Assignment Problem in OR that deals with assigning tasks to workers when the number of tasks and workers is not equal. To solve the UAP, we need to balance the matrix by adding dummy tasks or workers and then apply algorithms such as the Hungarian algorithm, the transportation algorithm, or the auction algorithm. Understanding the UAP can help businesses and organizations optimize their resource allocation and improve their operational efficiency.

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Operations Research

1 Operations Research-An Overview

  • History of O.R.
  • Approach, Techniques and Tools
  • Phases and Processes of O.R. Study
  • Typical Applications of O.R
  • Limitations of Operations Research
  • Models in Operations Research
  • O.R. in real world

2 Linear Programming: Formulation and Graphical Method

  • General formulation of Linear Programming Problem
  • Optimisation Models
  • Basics of Graphic Method
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  • Multiple, Unbounded Solution and Infeasible Problems
  • Solving Linear Programming Graphically Using Computer
  • Application of Linear Programming in Business and Industry

3 Linear Programming-Simplex Method

  • Principle of Simplex Method
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  • Simplex Method with several Decision Variables
  • Two Phase and M-method
  • Multiple Solution, Unbounded Solution and Infeasible Problem
  • Sensitivity Analysis
  • Dual Linear Programming Problem

4 Transportation Problem

  • Basic Feasible Solution of a Transportation Problem
  • Modified Distribution Method
  • Stepping Stone Method
  • Unbalanced Transportation Problem
  • Degenerate Transportation Problem
  • Transhipment Problem
  • Maximisation in a Transportation Problem

5 Assignment Problem

  • Solution of the Assignment Problem
  • Unbalanced Assignment Problem
  • Problem with some Infeasible Assignments
  • Maximisation in an Assignment Problem
  • Crew Assignment Problem

6 Application of Excel Solver to Solve LPP

  • Building Excel model for solving LP: An Illustrative Example

7 Goal Programming

  • Concepts of goal programming
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8 Integer Programming

  • Some Integer Programming Formulation Techniques
  • Binary Representation of General Integer Variables
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9 Dynamic Programming

  • Dynamic Programming Methodology: An Example
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10 Non-Linear Programming

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11 Introduction to game theory and its Applications

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12 Monte Carlo Simulation

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  • Two typical examples of hand-computed simulation
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13 Queueing Models

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UNBALANCED ASSIGNMENT PROBLEM

Unbalanced Assignment problem is an assignment problem where the number of facilities is not equal to the number of jobs. To make unbalanced assignment problem, a balanced one, a dummy facility(s) or a dummy job(s) (as the case may be) is introduced with zero cost or time.

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what is unbalanced assignment problem example

Assignment Problem: Meaning, Methods and Variations | Operations Research

what is unbalanced assignment problem example

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:

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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:

what is unbalanced assignment problem example

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What Is An Unbalanced Assignment Problem? How It Is Solved With Hungarian Method

what is unbalanced assignment problem example

An assignment problem is a combination of optimization problems which occurs in the field of operations research. It is a special case and is the degenerate form of a transportation problem when each supply is 1 and each demand is 1. When a particular work is assigned to an agent or it involves assigning a number of tasks to an equal number of agents and these agents can be people, machines, vehicles and plants, etc. and the objective is to increase the total profit or minimize the total cost. The process can be applicable in different fields like assigning personnel to offices, machines to jobs, teachers to classes or salesmen to sales territories. Before we study more on this topic, we want to inform you that we have assignment services where you get assignments on more than 180+ subjects through top-notch experts at GotoAssignmentHelp platform. We provide assignments, essays, thesis help , and many more so join now on GotoAssignmentHelp platform to avail our services.

The assignment problem being a special case of transportation problem can be solved by transportation technique or simplex method (North-West corner rule, least cost method, Vogel’s approximation method, etc.). The most common and popular method to solve assignment problems is the Hungarian method which was developed and published by H.W. Kuhn. The Hungarian method was so named because the algorithm was based on the works of two Hungarian mathematicians: D. Konig and J. Egervary. Later the algorithm was reviewed and observed by James Munkres and was found to be strongly polynomial. The simplex method for linear programming was also followed to solve the assignment problem. The unbalanced assignment problem was solved by Kore’s new method that was proposed by him which was based on generating “ones” instead of “zeros” in the matrix. Many algorithms for solving cost minimization and profit maximizing assignment problems was investigated by Abdur Rashid. Sometimes, in assignment problems, the number of agents is not equal to the number of tasks and these problems are known as unbalanced assignment problems. Then the matrix will not be a square matrix and with these types of sums, the traditional method was to consider costs in such rows or columns as zero. After forming a balanced equation, the problem could be solved by the Hungarian method. If you find assignments and their numerical difficulties, the experts of GotoAssignmentHelp can help you solve them with the best tricks.

As the assignment problem is a special case of transportation problem it can be formulated as a linear programming problem where you have to take n tasks and solve them with n agents. Each agent has to complete only one task through this solving method. Here one should take the first set of constraints as an agent and the second set of constraints as a task. From many formulations, it is pointed out that the assignment problem is a 0-1 integer programming problem. You have to make a matrix of n x n type with a i = b j = 1. If you have problems with assignments, homework , thesis, case studies, avail our expert help from GotoAssignmentHelp.

Hungarian method

In hungarian method you have to solve by the following steps:.

Step 1: You have to determine the total opportunity cost matrix: the minimum element from each row of the given matrix should be subtracted from the elements of the respective rows and then continue the same with the column. The minimum element from each column of the matrix should be subtracted from the elements of the respective columns. Through this process, you obtain a total opportunity cost matrix.

Step 2: The minimum number of horizontal and vertical lines should be drawn to cover all the zeros obtained in the total opportunity cost matrix. If the order of the matrix is maintained then an optimal assignment can be made by the usual procedure so that the total opportunity cost involved is zero. And if the number of lines drawn is less than the order of the payoff matrix follows the next step.

Step 3: The uncovered element in the current total opportunity cost matrix should be the smallest. Then this smallest element should be subtracted from all the uncovered elements and be added at the intersection of the vertical and horizontal lines. Thus, you obtain a total opportunity cost matrix.

Step 4: repeat process 2 and 3 until you get total opportunity cost zero.

There are various numerical illustrations which you will find on our site, GotoAssignmentHelp for understanding the theory better so if you are having problems in assignment help , essays and homework you can reach us. We have a group of experts who are highly qualified and take urgent assignments so that you can relax. We also offer these assignments at affordable prices.

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what is unbalanced assignment problem example

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Unbalanced Assignment Problems

Whenever the cost matrix of an assignment problem is not a square matrix, that is, whenever the number of sources is not equal to the number of destinations, the assignment problem is called an unbalanced assignment problem. In such problems, dummy rows (or columns) are added in the matrix so as to complete it to form a square matrix. The dummy rows or columns will contain all costs elements as zeroes. The Hungarian method may be used to solve the problem.

Example :  A company has five machines that are used for four jobs. Each job can be assigned to one and only one machine. The cost of each job on each machine is given in the following Table.

Unbalanced Maximization Assignment problem example

Assignment Problem

topic 5.1.png

Solution:  Convert the 4 × 5 matrix into a square matrix by adding a dummy row D5.

Dummy Row D5 Added

topic 5.2.png

Row-wise Reduction of the Matrix

topic 5.3.png

Column-wise reduction is not necessary since all columns contain a single zero. Now, draw minimum number of lines to cover all the zeros, as shown in Table.

All Zeros in the Matrix Covered

topic 5.4.png

Number of lines drawn ≠ Order of matrix. Hence not optimal.

Select the least uncovered element, i.e., 1, subtract it from other uncovered elements, add to the elements at intersection of lines and leave the elements that are covered with single line unchanged as shown in Table.

Subtracted or Added to Elements

topic 5.5.png

Again Added or Subtracted 1 from Elements

topic 5.6

Number of lines drawn = Order of matrix. Hence optimality is reached. Now assign the jobs to machines, as shown in Table.

Assigning Jobs to Machines

topic 5.7.png

Example :  In a plant layout, four different machines M1, M2, M3 and M4 are to be erected in a machine shop. There are five vacant areas A, B, C, D and E. Because of limited space, Machine M2 cannot be erected at area C and Machine M4 cannot be erected at area A. The cost of erection of machines is given in the Table.

topic 5.9.png

Find the optimal assignment plan.

Solution:  As the given matrix is not balanced, add a dummy row D5 with zero cost values. Assign a high cost H for (M2, C) and (M4, A). While selecting the lowest cost element neglect the high cost assigned H, as shown in Table below.

topic 5.10

– Row-wise reduction of the matrix is shown in Table.

Matrix Reduced Row-wise

topic 5.11.png

Note:  Column-wise reduction is not necessary, as each column has at least one single zero. Now, draw minimum number of lines to cover all the zeros, see Table.

Lines Drawn to Cover all Zeros

topic 5.12.png

Number of lines drawn ≠ Order of matrix. Hence not Optimal. Select the smallest uncovered element, in this case 1. Subtract 1 from all other uncovered element and add 1 with the elements at the intersection. The element covered by single line remains unchanged. These changes are shown in Table. Now try to draw minimum number of lines to cover all the zeros.

Added or Subtracted 1 from Elements

topic 5.13.png

Now number of lines drawn = Order of matrix, hence optimality is reached. Optimal assignment of machines to areas are shown in Table.

Optimal Assignment

topic 5.14.png

Hence, the optimal solution is:

topic 5.15.png

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Hungarian Method for Unbalanced Assignment Problem-examples

In this article we will study the step by step procedure to solve unbalanced assignment problem using Hungarian method .

Unbalanced Assignment Problem Example 1

The coach of an age group swim team needs to assign swimmers to a 200-yard medley relay team to send to the Junior Olympics. Since most of his best swimmers are very fast in more than one stroke, it is not clear which swimmer should be assigned to each of the four strokes. The five fastest swimmers and the best time (in seconds) they have achieved in each of the strokes (for 50 yards) are

The coach wishes to determine how to assign four swimmers to the four different strokes to minimize the sum of the corresponding best times.

a. Fomulate this problem as an assignment problem. b. Obtain an optimal solution.

In the given problem there are 4 type of strokes and 5 Swimmers. The problem can be formulated as $4\times 5$ assignment problem with $c_{ij}$ = time taken by the $j^{th}$ swimmer using $i^{th}$ type of stroke.

$$ \begin{equation*} x_{ij}=\left\{ \begin{array}{ll} 1, & \hbox{if $j^{th}$ swimmer is assigned to $i^{th}$ stroke;} \\ 0, & \hbox{otherwise.} \end{array} \right. \end{equation*} $$

The five fastest swimmers and the best time (in seconds) they have achieved in each of the strokes (for 50 yards) are

As the table is not a square matrix, add a dummy row with best times = 0.

In first row smallest is 32.9, second row is 33.1, third row is 28.5, fourth row is 26.4 and fifth row is 0.

Subtract the minimum of each row of the above cost matrix, from all the elements of respective rows. The modified matrix is as follows:

Assignment Problem

The minimum of each column of the modified matrix is 0.

Subtract the minimum of each column of the modified matrix, from all the elements of respective columns. The modified matrix is as follows:

Draw the minimum number of horizontal and vertical line to cover all the zeros in the modified matrix.

Assignment Problem

The minimum number of lines = 2, which is less than the order of assignment problem (i.e. 5). Hence the optimal assignment is not possible.

The smallest element in the matrix, not covered by the lines is 0.9. Subtract 0.9 from all the uncovered elements and add 0.9 at the intersection of horizontal and vertical lines. And obtain the second modified matrix.

Assignment Problem

Draw the minimum number of horizontal and vertical line to cover all the zeros in the above modified matrix.

Assignment Problem

The minimum number of lines = 3, which is less than the order of assignment problem (i.e. 5). Hence the optimal assignment is not possible.

The smallest element in the matrix, not covered by the lines is 0.7. Subtract 0.7 from all the uncovered elements and add 0.7 at the intersection of horizontal and vertical lines. And obtain the second modified matrix.

Assignment Problem

The minimum number of lines = 4, which is less than the order of assignment problem (i.e. 5). Hence the optimal assignment is not possible.

The smallest element in the matrix, not covered by the lines is 1.2. Subtract 1.2 from all the uncovered elements and add 1.2 at the intersection of horizontal and vertical lines. And obtain the second modified matrix.

Assignment Problem

The minimum number of lines = 5, which is equal to the order of assignment problem (i.e. 5). Hence the optimal assignment using Hungarian method is possible.

Examine the row successively until a row-wise exactly single zero is found, mark this zero by $\square$ to make the assignment and mark cross $(\times)$ over all zeros in that column. Continue in this manner until all the rows have been examined. Repeat the same procedure for columns also.

Repeat the Step 6 successively we have

Assignment Problem

From the above table it is clear that in each row and in each column exactly one marked $\square$ zero is obtained. The optimal assignment is

Backstroke $\to$ David

Breaststroke $\to$ Tony

Butterfly $\to$ Chris

Freestyle $\to$ Carl

Dummy $\to$ Ken

The optimal best time for each swimmer is as follows:

Thus the optimal (minimum) total best time for five swimmers is

$\text{Minimum total Best time }= 33.8+34.7+28.5+29.2+0 =126.2$ seconds.

In this tutorial, you learned about step by step procedure of Hungarian Algorithm to solve unbalanced assignment problem.

To learn more about Assignment problems please refer to the following tutorials:

Assignment Problems

Read step by step solution

Balanced assignment problem using Hungarian method .

Assignment problem of maximization type using Hungarian method .

Assignment problem with restrictions using Hungarian method

Let me know in the comments if you have any questions on Hungarian Algorithm to solve Unbalanced Assignment problems and your thought on this article.

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Procedure, Example Solved Problem | Operations Research - Solution of assignment problems (Hungarian Method) | 12th Business Maths and Statistics : Chapter 10 : Operations Research

Chapter: 12th business maths and statistics : chapter 10 : operations research.

Solution of assignment problems (Hungarian Method)

First check whether the number of rows is equal to the numbers of columns, if it is so, the assignment problem is said to be balanced.

Step :1 Choose the least element in each row and subtract it from all the elements of that row.

Step :2 Choose the least element in each column and subtract it from all the elements of that column. Step 2 has to be performed from the table obtained in step 1.

Step:3 Check whether there is atleast one zero in each row and each column and make an assignment as follows.

what is unbalanced assignment problem example

Step :4 If each row and each column contains exactly one assignment, then the solution is optimal.

Example 10.7

Solve the following assignment problem. Cell values represent cost of assigning job A, B, C and D to the machines I, II, III and IV.

what is unbalanced assignment problem example

Here the number of rows and columns are equal.

∴ The given assignment problem is balanced. Now let us find the solution.

Step 1: Select a smallest element in each row and subtract this from all the elements in its row.

what is unbalanced assignment problem example

Look for atleast one zero in each row and each column.Otherwise go to step 2.

Step 2: Select the smallest element in each column and subtract this from all the elements in its column.

what is unbalanced assignment problem example

Since each row and column contains atleast one zero, assignments can be made.

Step 3 (Assignment):

what is unbalanced assignment problem example

Thus all the four assignments have been made. The optimal assignment schedule and total cost is

what is unbalanced assignment problem example

The optimal assignment (minimum) cost

Example 10.8

Consider the problem of assigning five jobs to five persons. The assignment costs are given as follows. Determine the optimum assignment schedule.

what is unbalanced assignment problem example

∴ The given assignment problem is balanced.

Now let us find the solution.

The cost matrix of the given assignment problem is

what is unbalanced assignment problem example

Column 3 contains no zero. Go to Step 2.

what is unbalanced assignment problem example

Thus all the five assignments have been made. The Optimal assignment schedule and total cost is

what is unbalanced assignment problem example

The optimal assignment (minimum) cost = ` 9

Example 10.9

Solve the following assignment problem.

what is unbalanced assignment problem example

Since the number of columns is less than the number of rows, given assignment problem is unbalanced one. To balance it , introduce a dummy column with all the entries zero. The revised assignment problem is

what is unbalanced assignment problem example

Here only 3 tasks can be assigned to 3 men.

Step 1: is not necessary, since each row contains zero entry. Go to Step 2.

what is unbalanced assignment problem example

Step 3 (Assignment) :

what is unbalanced assignment problem example

Since each row and each columncontains exactly one assignment,all the three men have been assigned a task. But task S is not assigned to any Man. The optimal assignment schedule and total cost is

what is unbalanced assignment problem example

The optimal assignment (minimum) cost = ₹ 35

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Assignment Problem: Maximization

There are problems where certain facilities have to be assigned to a number of jobs, so as to maximize the overall performance of the assignment.

The Hungarian Method can also solve such assignment problems , as it is easy to obtain an equivalent minimization problem by converting every number in the matrix to an opportunity loss.

The conversion is accomplished by subtracting all the elements of the given matrix from the highest element. It turns out that minimizing opportunity loss produces the same assignment solution as the original maximization problem.

  • Unbalanced Assignment Problem
  • Multiple Optimal Solutions

Example: Maximization In An Assignment Problem

At the head office of www.universalteacherpublications.com there are five registration counters. Five persons are available for service.

How should the counters be assigned to persons so as to maximize the profit ?

Here, the highest value is 62. So we subtract each value from 62. The conversion is shown in the following table.

On small screens, scroll horizontally to view full calculation

Now the above problem can be easily solved by Hungarian method . After applying steps 1 to 3 of the Hungarian method, we get the following matrix.

Draw the minimum number of vertical and horizontal lines necessary to cover all the zeros in the reduced matrix.

Select the smallest element from all the uncovered elements, i.e., 4. Subtract this element from all the uncovered elements and add it to the elements, which lie at the intersection of two lines. Thus, we obtain another reduced matrix for fresh assignment. Repeating step 3, we obtain a solution which is shown in the following table.

Final Table: Maximization Problem

Use Horizontal Scrollbar to View Full Table Calculation

The total cost of assignment = 1C + 2E + 3A + 4D + 5B

Substituting values from original table: 40 + 36 + 40 + 36 + 62 = 214.

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What is Balanced or Unbalanced Assignment problem?

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The Assignment problem can be Balanced or Unbalanced problem.

A Balanced problem means the no. of rows and no. of columns in the problem are equal. E. g. if the problem contains 4 workers and 4 jobs, then it is balanced.

Where as, an Unbalanced problem means the no. of rows and no. of columns are not equal. E. g. if the problem contains 4 workers and 3 jobs it is not balanced. Then first we need to balance the problem by taking a Dummy job (imaginary job).

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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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Unbalanced Assignment Problem by Using Modified Approach

Profile image of Trisna Darmawansyah

— The assignment problem is one of the main problems while assigning task to the worker. It is an important problem in mathematics and is also discuss in real physical world. It is a combinatorial optimization problem in the field of operational research. In a normal case of transportation problem where the objective is to assign the available resources to the activity going on so as to get the minimum cost or maximize total benefits of allocation. In this paper we proposed modified assignment model for the solution of assignment problem. Here in this paper with the help of numerical examples or problem is solved to show its efficiency and also its comparison with Hungarian method is shown. A new cost is achieved by using unbalanced assignment problem.

Related Papers

archana pandey

Assignment problems arise in different situation where we have to find an optimal way to assign n-objects to mother objects in an injective fashion. The assignment problems are a well studied topic in combinatorial optimization. These problems find numerous application in production planning, telecommunication VLSI design, economic etc. The assignment problems is a special case of Transportation problem. Depending on the objective we want to optimize, we obtain the typical assignment problems. Assignment problem is an important subject discussed in real physical world we endeavor in this paper to introduce a new approach to assignment problem namely, matrix ones assignment method or MOA-method for solving wide range of problem. An example using matrix ones assignment methods and the existing Hungarian method have been solved and compared it graphically. Also some of the variations and some special cases in assignment problem and its applications have been discussed in the paper.

what is unbalanced assignment problem example

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International Journal of Mathematical Archive ( …

Surapati Pramanik, Ph. D.

This paper presents solution methodology for assignment problem with fuzzy cost. The fuzzy costs are considered as trapezoidal fuzzy numbers. Ranking method (introduced by S. Abbasbandy and T. Hajjary, 2009) has been used for ranking the trapezoidal fuzzy ...

IOSR Journals publish within 3 days

In this paper a ground reality the entries of the cost matrix are not always crisp. In many application this parameters are uncertain and this uncertain parameters are represented by interval. In this contribution we proposed a new Interval Hungarian Method to solve the mid values of each interval in the cost matrix with four different cases of the assignment problem and finally we conclude that same optimum solution.

IAEME Publication

This paper presents multi-objective assignment problem with traveling time and territory control of the mail carrier from the central post office Bandung in delivering the package to the destination location, where all the objectives are optimized by using Hungarian method. Sensitivity analysis against data changes that may occur was also conducted. The sampled data in this study are the territory control and traveling time of 10 mail carriers who will be assigned to deliver mail package to 10 post office delivery centers in Bandung. The result of this research is the combination of traveling time and territory control optimal from 10 mail carriers as follows: mail carrier 1 to Soreang, mail carrier 2 to Dayeuh Kolot, mail carrier 3 to Ujung Berung, mail carrier 4 to Padalarang, mail carrier 5 to Situ Saeur, mail carrier 6 to Cipedes, mail carrier 7 to Cimahi, mail carrier 8 to Asia-Afrika, mail carrier 9 to Cikutra, mail carrier 10 to Cikeruh. Based on this result, manager of the central post office Bandung can make optimal decisions to assign tasks to their mail carriers.

Bhausaheb G Kore

In this paper I have proposed a new approach to solve an unbalanced assignment problem (UBAP). This approach includes two parts. First is to obtain an initial basic feasible solution (IBFS) and second part is to test optimality of an IBFS. I have proposed two new methods Row Penalty Assignment Method (RPAM) and Column Penalty Assignment Method (CPAM) to obtain an IBFS of an UBAP. Also I have proposed a new method Non-basic Smallest Effectiveness Method (NBSEM) to test optimality of an IBFS. We can solve an assignment problem of maximization type using this new approach in opposite sense. By this new approach, we achieve the goal with less number of computations and steps. Further we illustrate the new approach by suitable examples. INTRODUCTION The assignment problem is a special case of the transportation problem where the resources are being allocated to the activities on a one-to-one basis. Thus, each resource (e.g. an employee, machine or time slot) is to be assigned uniquely to a particular activity (e.g. a task, site or event). In assignment problems, supply in each row represents the availability of a resource such as a man, machine, vehicle, product, salesman, etc. and demand in each column represents different activities to be performed such as jobs, routes, factories, areas, etc. for each of which only one man or vehicle or product or salesman respectively is required. Entries in the square being costs, times or distances. The assignment method is a special linear programming technique for solving problems like choosing the right man for the right job when more than one choice is possible and when each man can perform all of the jobs. The ultimate objective is to assign a number of tasks to an equal number of facilities at minimum cost (or maximum profit) or some other specific goal. Let there be 'm' resources and 'n' activities. Let c ij be the effectiveness (in terms of cost, profit, time, etc.) of assigning resource i to activity j (i = 1, 2, …., m; j = 1, 2,…., n). Let x ij = 0, if resource i is not assigned to activity j and x ij = 1, if resource i is assigned to activity j. Then the objective is to determine x ij 's that will optimize the total effectiveness (Z) satisfying all the resource constraints and activity constraints. 1. Mathematical Formulation Let number of rows = m and number of columns = n. If m = n then an AP is said to be BAP otherwise it is said to be UBAP. A) Case 1: If m < n then mathematically the UBAP can be stated as follows:

Trisna Darmawansyah

Different situations give rise to the assignment problem, where we must discover an optimal way to assign 'n' objects to 'm' in an bijective function. We have, in this research, propose the possibility of solving exactly the Linear Assignment Problem with a method that would be more efficient than the Hungarian method of exact solution. This method is based on applying a series of pairwise interchanges of assignments to a starting heuristically generated feasible solution, wherein each pairwise interchange is guaranteed to improve the objective function value of the feasible solution.It seems that our algorithm finds the optimal solution which is the same as one found by the Hungarian method, but in much simpler. 7980 M. Khalid et al.

Hogpodrat Pangkum

Assignment problem is a special case of Transportation problem. It is actually a minimizing model that assigns numbers of people with equal number of jobs, henceforth, minimizing the corresponding costs. In this paper an introduction is given to " New Improved Ones Assignment " which is a path to making an assignment problem. Earlier H. Gamel also brought to light the drawbacks of One assignment method. Our improvement to the Ones assignment method, leads to comparatively brief computation time and more convenient and strong codes. It also overcomes the drawbacks as mentioned previously

IJAR Indexing

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

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IMAGES

  1. UNBALANCED ASSIGNMENT PROBLEM EXAMPLE NO. 1 BY DR KUNAL KHATRI #

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  2. Unbalanced Assignment Problem

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  3. Unbalanced Assignment Problem ~ Part 2

    what is unbalanced assignment problem example

  4. Unbalanced Assignment Problems

    what is unbalanced assignment problem example

  5. MEANING OF UNBALANCED ASSIGNMENT PROBLEM WITH EXAMPLE BY DR KUNAL

    what is unbalanced assignment problem example

  6. Lecture 19 Assignment problem : Unbalanced and maximal Assignment Problems

    what is unbalanced assignment problem example

VIDEO

  1. Assignment Problem (Unbalanced)

  2. unbalanced transportation problem in least cost method in tamil

  3. Unbalanced Assignment Problem with Multiple Solutions or assignments

  4. Assignment Problem 03

  5. Unbalanced Transportation Problem in Assignment Chapter By Puneet Kumar RPIIT CAMPUS

  6. Unbalanced Assignment Problem

COMMENTS

  1. Assignment problems

    (a) Multiple Optimal solution (b) The problem is unbalanced (c) Maximization problem (d) Balanced problem If the given matrix is not a square matrix, the assignment problem is called an In such type of problems, add dummy row (s) or column (s) with the cost elements as zero to convert the matrix as a square matrix.

  2. Unbalanced Assignment Problem: Definition, Formulation, and Solution

    In summary, the Unbalanced Assignment Problem is a variant of the traditional Assignment Problem in OR that deals with assigning tasks to workers when the number of tasks and workers is not equal.

  3. Assignment problem

    Formal definition The formal definition of the assignment problem (or linear assignment problem) is Given two sets, A and T, together with a weight function C : A × T → R. Find a bijection f : A → T such that the cost function : is minimized.

  4. Unbalanced Assignment Problem

    To make unbalanced assignment problem, a balanced one, a dummy facility (s) or a dummy job (s) (as the case may be) is introduced with zero cost or time. Get Quantitative Techniques: Theory and Problems now with the O'Reilly learning platform.

  5. Unbalanced Assignment Problem

    Solution Since the number of persons is less than the number of jobs, we introduce a dummy person (D) with zero values. The revised assignment problem is given below: Table Now use the Hungarian method to obtain the optimal solution yourself. Ans. = 20 + 17 + 17 + 0 = 54. Share and Recommend Operations Research Simplified Back Next

  6. Solving the Unbalanced Assignment Problem: Simpler Is Better

    In this short paper, we will show that solving this same example from the Yadaiah and Haragopal paper by using a simple textbook formulation to balance the problem and then solve it with the...

  7. PDF UNIT -2 Chapter: II ASSIGNMENT PROBLEM

    Example I: A manager has four persons (i.e. facilities) available for four separate jobs (i.e. jobs) and the cost of assigning (i.e. effectiveness) each job to each person is given. His objective is to assign each person to one and only one job in such a way that the total cost of assignment is minimised. Example II:

  8. Solving the Unbalanced Assignment Problem: Simpler Is Better

    This paper suggests an exhaustive search approach to the unbalanced assignment problem through partial bound optimization and the devised technique, computational algorithm of the approach and its implementation, and the efficiency of the proposed techniques are given. Expand 5 PDF

  9. Modified Hungarian method for unbalanced assignment problem with

    A numerical example is taken to demonstrate the performance and efficiency of the proposed method. The obtained result is then compared with several existing methods to show the superiority of our algorithm. ... Unbalanced assignment problem is a particular sub class of the transportation problem where the objective is to find the optimum ...

  10. Assignment Problem: Meaning, Methods and Variations

    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.

  11. What Is An Unbalanced Assignment Problem? How It Is Solved With

    admin - January 17, 2022 0 An assignment problem is a combination of optimization problems which occurs in the field of operations research. It is a special case and is the degenerate form of a transportation problem when each supply is 1 and each demand is 1.

  12. Unbalanced Assignment Problems

    Unbalanced Maximization Assignment problem example Solution:Convert the 4 × 5 matrix into a square matrix by adding a dummy row D5. Dummy Row D5 Added Row-wise Reduction of the Matrix All Zeros in the Matrix Covered Subtracted or Added to Elements Again Added or Subtracted 1 from Elements Assigning Jobs to Machines Example : Solution:

  13. Solving the Unbalanced Assignment Problem: Simpler Is Better

    The assignment problem is a standard topic discussed in operations research textbooks (See for example, Hillier and Lieberman or Winston ). A typical presentation requires that n jobs must be assigned to n machines such that each machine gets exactly one job assigned to it.

  14. Unbalanced Assignment Problem

    Download PDF containing solution to the same problem which is explained in the video from link https://drive.google.com/file/d/11K1ESmjnJQee2z6MJfkBF8kbFpFlK...

  15. An Alternative Approach for Solving Unbalanced Assignment Problems

    Numerical examples are provided to demonstrate the efficiency and validity of the proposed approach. The results of computational experiments on a large number of test problems reveal that the proposed method is capable of assigning a number of tasks to an equal number of agents optimally.

  16. Hungarian Method for Unbalanced Assignment Problem-examples

    Unbalanced Assignment Problem Example 1 The coach of an age group swim team needs to assign swimmers to a 200-yard medley relay team to send to the Junior Olympics. Since most of his best swimmers are very fast in more than one stroke, it is not clear which swimmer should be assigned to each of the four strokes.

  17. Modified Hungarian method for unbalanced assignment problem with

    A numerical example is taken to demonstrate the performance and efficiency of the proposed method. The obtained result is then compared with several existing methods to show the superiority of our algorithm. ... Unbalanced assignment problem is a particular sub class of the transportation problem where the objective is to find the optimum ...

  18. Solution of assignment problems (Hungarian Method)

    Example 10.7 Solve the following assignment problem. Cell values represent cost of assigning job A, B, C and D to the machines I, II, III and IV. Solution: Here the number of rows and columns are equal. ∴ The given assignment problem is balanced. Now let us find the solution.

  19. Assignment Problem, Maximization Example, Hungarian Method

    Here, the highest value is 62. So we subtract each value from 62. The conversion is shown in the following table. Now the above problem can be easily solved by Hungarian method. After applying steps 1 to 3 of the Hungarian method, we get the following matrix. Draw the minimum number of vertical and horizontal lines necessary to cover all the ...

  20. What is Balanced or Unbalanced Assignment problem?

    The Assignment problem can be Balanced or Unbalanced problem.. A Balanced problem means the no. of rows and no. of columns in the problem are equal. E. g. if the problem contains 4 workers and 4 jobs, then it is balanced. Where as, an Unbalanced problem means the no. of rows and no. of columns are not equal.E. g. if the problem contains 4 workers and 3 jobs it is not balanced.

  21. Hungarian Method

    Hungarian Method The Hungarian method is a computational optimization technique that addresses the assignment problem in polynomial time and foreshadows following primal-dual alternatives. In 1955, Harold Kuhn used the term "Hungarian method" to honour two Hungarian mathematicians, Dénes Kőnig and Jenő Egerváry.

  22. Balanced and Unbalanced Transportation Problems

    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

  23. Unbalanced Assignment Problem by Using Modified Approach

    Here in this paper with the help of numerical examples or problem is solved to show its efficiency and also its comparison with Hungarian method is shown. A new cost is achieved by using unbalanced assignment problem. Keywords— Unbalanced Assignment problem, Optimization, Hungarian method, proposed method. I.