## Unbalanced Assignment Problem: Definition, Formulation, and Solution Methods

Table of Contents

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
- Important steps to draw graph
- 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
- Computational aspect of Simplex Method
- 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
- Goal programming model formulation
- Graphical method of goal programming
- The simplex method of goal programming
- Using Excel Solver to Solve Goal Programming Models
- Application areas of goal programming

8 Integer Programming

- Some Integer Programming Formulation Techniques
- Binary Representation of General Integer Variables
- Unimodularity
- Cutting Plane Method
- Branch and Bound Method
- Solver Solution

9 Dynamic Programming

- Dynamic Programming Methodology: An Example
- Definitions and Notations
- Dynamic Programming Applications

10 Non-Linear Programming

- Solution of a Non-linear Programming Problem
- Convex and Concave Functions
- Kuhn-Tucker Conditions for Constrained Optimisation
- Quadratic Programming
- Separable Programming
- NLP Models with Solver

11 Introduction to game theory and its Applications

- Important terms in Game Theory
- Saddle points
- Mixed strategies: Games without saddle points
- 2 x n games
- Exploiting an opponent’s mistakes

12 Monte Carlo Simulation

- Reasons for using simulation
- Monte Carlo simulation
- Limitations of simulation
- Steps in the simulation process
- Some practical applications of simulation
- Two typical examples of hand-computed simulation
- Computer simulation

13 Queueing Models

- Characteristics of a queueing model
- Notations and Symbols
- Statistical methods in queueing
- The M/M/I System
- The M/M/C System
- The M/Ek/I System
- Decision problems in queueing

## A Comparative Analysis of Assignment Problem

- Conference paper
- First Online: 06 June 2023
- Cite this conference paper

- Shahriar Tanvir Alam ORCID: orcid.org/0000-0002-0567-3381 5 ,
- Eshfar Sagor 5 ,
- Tanjeel Ahmed 5 ,
- Tabassum Haque 5 ,
- Md Shoaib Mahmud 5 ,
- Salman Ibrahim 5 ,
- Ononya Shahjahan 5 &
- Mubtasim Rubaet 5

Part of the book series: EAI/Springer Innovations in Communication and Computing ((EAISICC))

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- International Conference on Big Data Innovation for Sustainable Cognitive Computing

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The aim of a supply chain team is to formulate a network layout that minimizes the total cost. In this research, the lowest production cost of the final product has been determined using a generalized plant location model. Furthermore, it is anticipated that units have been set up appropriately so that one unit of input from a source of supply results in one unit of output. The assignment problem is equivalent to distributing a job to the appropriate machine in order to meet customer demand. This study concentrates on reducing the cost of fulfilling the overall customer demand. Many studies have been conducted, and various algorithms have been proposed to achieve the best possible result. The purpose of this study is to present an appropriate model for exploring the solution to the assignment problem using the “Hungarian Method.” To find a feasible output of the assignment problem, this study conducted a detailed case study. The computational results indicate that the “Hungarian Method” provides an optimum solution for both balanced and unbalanced assignment problems. Moreover, decision-makers can use the study’s findings as a reference to mitigate production costs and adopt any sustainable market policy.

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Shahriar Tanvir Alam, Eshfar Sagor, Tanjeel Ahmed, Tabassum Haque, Md Shoaib Mahmud, Salman Ibrahim, Ononya Shahjahan & Mubtasim Rubaet

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Alam, S.T. et al. (2023). A Comparative Analysis of Assignment Problem. In: Haldorai, A., Ramu, A., Mohanram, S. (eds) 5th EAI International Conference on Big Data Innovation for Sustainable Cognitive Computing. BDCC 2022. EAI/Springer Innovations in Communication and Computing. Springer, Cham. https://doi.org/10.1007/978-3-031-28324-6_11

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## Quantitative Techniques: Theory and Problems by P. C. Tulsian, Vishal Pandey

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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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## Solving the Unbalanced Assignment Problem: Simpler Is Better

American Journal of Operations Research

## Related Papers

Dr Avanish Kumar

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:

Malaya Journal of Matematik

DR ANJU KHANDELWAL

International Journal for Research in Applied Science & Engineering Technology (IJRASET)

IJRASET Publication

In this paper a new method is proposed for finding an optimal solution of a wide range of assignment problems, directly. A numerical illustration is established and the optimality of the result yielded by this method is also checked. The most attractive feature of this method is that it requires very simple arithmetical and logical calculations. The method is illustrated through an example.

Hussein Ali Hussein Al-Dallal Al-Saeedi

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.

Sultana Rafi

The assignment problem is a particular type of linear programming problem. In this paper, we analyzed the standard and existing proposed methods. After studying these methods, we proposed a new alternative method for solving the assignment problem. We examined the newly proposed method by a couple of numerical examples and compare this result with the standard method. The main characteristic of this newly proposed method is that it constructed a very easy logical and arithmetical algorithm. Here we point out some advantages and limitations of the new proposed method. Programming code for the newly proposed method has been added in this paper.

Ranjan Kumar Mondal

Thecloudcomputingpresentsatypeofassignmentsandsystemswhichoccupydistributedresources toexecutearoleinadistributedway.Cloudcomputingmakeuseoftheonlinesystemsonthewebto assisttheimplementationofcomplicatedassignments;thatneedhuge-scalecomputation.Itwassaid withtheintentionofinourlivingworld;wecanfinditchallengingtobalanceworkloadsofcloud computingamongassignments(jobsortasks)andsystems(machinesornodes),sothemajorityofthe timewehavetopromoteaconditiontounbalancedassignmentproblems(unequaltaskallocations). The present article submits a new technique to solve the unequal task allocation problems. The techniqueisofferedinanalgorithmicmodelandputintopracticeontheseveralgroupsofinputto investigatethepresentationandusefulnessoftheworks.Anevaluationispreparedwiththepresented approach.Itmakessurethattheproposedapproachprovidesabetteroutcomebycomparingwith someotherexistingalgorithms.

Industrial Engineering Journal

Shridhar Mhalsekar

Journal of Advances in Mathematics and Computer Science

Hudu Mohammed

Assignment problem is an important area in Operation Research and is also discussed in real physical world. In this paper an attempt has been made to solve the assignment problem using a new Method called the Penalty method. We discuss a numerical example by using the new Method and compare it with standard existing method which is the Hungarian method. We compare the optimal solution of the new Method and the Hungarian method. The new method is a simple procedure, easy to apply for solving assignment problem.

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## Unbalanced Assignment Problem

In the previous section, the number of persons and the number of jobs were assumed to be the same. In this section, we remove this assumption and consider a situation where the number of persons is not equal to the number of jobs . In all such cases, fictitious rows and/or columns are added in the matrix to make it a square matrix.

- Maximization Problem
- Multiple Optimal Solutions

## Example: Unbalanced Assignment Problem

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:

Use Horizontal Scrollbar to View Full Table Calculation.

Now use the Hungarian method to obtain the optimal solution yourself. Ans. = 20 + 17 + 17 + 0 = 54.

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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

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

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

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

Again Added or Subtracted 1 from Elements

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

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.

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.

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

Matrix Reduced Row-wise

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

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

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

Optimal Assignment

Hence, the optimal solution is:

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## 4. Unbalanced Assignment Problem

## 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:

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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:

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## Solving an Assignment Problem

This section presents an example that shows how to solve an assignment problem using both the MIP solver and the CP-SAT solver.

In the example there are five workers (numbered 0-4) and four tasks (numbered 0-3). Note that there is one more worker than in the example in the Overview .

The costs of assigning workers to tasks are shown in the following table.

The problem is to assign each worker to at most one task, with no two workers performing the same task, while minimizing the total cost. Since there are more workers than tasks, one worker will not be assigned a task.

## MIP solution

The following sections describe how to solve the problem using the MPSolver wrapper .

## Import the libraries

The following code imports the required libraries.

## Create the data

The following code creates the data for the problem.

The costs array corresponds to the table of costs for assigning workers to tasks, shown above.

## Declare the MIP solver

The following code declares the MIP solver.

## Create the variables

The following code creates binary integer variables for the problem.

## Create the constraints

Create the objective function.

The following code creates the objective function for the problem.

The value of the objective function is the total cost over all variables that are assigned the value 1 by the solver.

## Invoke the solver

The following code invokes the solver.

## Print the solution

The following code prints the solution to the problem.

Here is the output of the program.

## Complete programs

Here are the complete programs for the MIP solution.

## CP SAT solution

The following sections describe how to solve the problem using the CP-SAT solver.

## Declare the model

The following code declares the CP-SAT model.

The following code sets up the data for the problem.

The following code creates the constraints for the problem.

Here are the complete programs for the CP-SAT solution.

Except as otherwise noted, the content of this page is licensed under the Creative Commons Attribution 4.0 License , and code samples are licensed under the Apache 2.0 License . For details, see the Google Developers Site Policies . Java is a registered trademark of Oracle and/or its affiliates.

Last updated 2023-01-02 UTC.

## 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.

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.

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.

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.

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

Step 3 (Assignment):

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

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.

∴ The given assignment problem is balanced.

Now let us find the solution.

The cost matrix of the given assignment problem is

Column 3 contains no zero. Go to Step 2.

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

The optimal assignment (minimum) cost = ` 9

Example 10.9

Solve the following assignment problem.

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

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.

Step 3 (Assignment) :

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

The optimal assignment (minimum) cost = ₹ 35

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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:

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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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 ...

research is to develop an optimal result for an assignment problem and to provide an example of how to handle a balanced and unbalanced assignment problem. The literature review is summarized in Sect. 2 of this paper. The proposed methodology is explained in Sect. 3, whereas the conclusion and future recommendations for ...

The typical textbook solution to the balanced assignment problem is then found using Kuhn's [3] Hungarian method. Problems in which there are more jobs than machines and more than one job can be ...

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. Get Quantitative Techniques: Theory and Problems now with the O ...

For unbalanced assignment problem where the number of jobs is more than the number of machines, the existing approaches assign some jobs to a dummy machine which means these jobs are left without ...

Recently, Yadaiah and Haragopal published in the American Journal of Operations Research a new approach to solving the unbalanced assignment problem. They also provide a numerical example which they solve with their approach and get a cost of 1550 which they claim is optimum. This approach might be of interest; however, their approach does not guarantee the optimal solution.

Worked example of assigning tasks to an unequal number of workers using the Hungarian method. The assignment problem is a fundamental combinatorial optimization problem. In its most general form, the problem is as follows: The problem instance has a number of agents and a number of tasks.Any agent can be assigned to perform any task, incurring some cost that may vary depending on the agent ...

In this paper I have proposed a new approach to solve an unbalanced assignment problem (UBAP). This approach includes two parts. ... 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 ...

Example: 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.

Several problems of management have a structure identical with the assignment problem. For example: Example I: A manager has four persons (i.e. facilities) available for four separate jobs (i.e. jobs) and the cost ... Unbalanced Assignment problem is an assignment problem where the number of facilities is not equal to the number of jobs. To ...

Unbalanced Maximization Assignment problem example. Assignment Problem. 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. Column-wise reduction is not necessary since all columns contain a single zero.

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. Let's go through the steps of the Hungarian method with the help of a solved example.

Unbalanced assignment problem, Hungarian method, Optimal solution. Introduction The assignment problem is a combinatorial optimization problem in the ... Example 1: There are four jobs to be assigned to five machines. Only one job can be assigned to one machine. The amount of time in hours required

The Unbalanced Assignment Problem Alternative Optimal Solutions Restriction on Assignments 5.4 Travelling Salesman Problem 5.5 Summary ... Example 1: A computer centre has four expert programmers and needs to develop four application programmes. The head of the computer centre,

Unbalanced Assignment Problems If the number of rows and columns are not equal then such type of problems are called as unbalanced assignment problems. Example A company has 4 machines on which to do 3 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 Assignment Problem If number of rows is not equal to number of columns then it is called Unbalanced Assignment Problem. So to solve this problem, we have to add dummy rows or columns with cost 0, to make it a square matrix. Example Find Solution of Assignment problem using Hungarian method (MIN case)

Unbalanced Assignment Problem: Any assignment problem is said to be unbalanced if the cost matrix is not a square matrix, i.e. the no of rows and the no of columns are not equal. ... Example 4: (Airline Crew Assignment). An airline, that operates seven days a week, has a time table shown below, crews must have a minimum layover of 6 hours ...

Unit 8: Assignment Problem - Unbalanced. When an assignment problem has more than one solution, then it is Notes (a) Multiple Optimal solution (b) The problem is unbalanced (c) Maximization problem (d) Balanced problem. 8 Unbalanced Assignment Problem. If the given matrix is not a square matrix, the assignment problem is called an unbalanced ...

Note: Thus this method can be used not only for balanced but also for unbalanced assignment problems. Example 3. (Maximization and balanced assignment problem) Consider the following assignment problem [5], where the total profit is to be maximized. A company has 5 jobs to be done.

This section presents an example that shows how to solve an assignment problem using both the MIP solver and the CP-SAT solver. Example. In the example there are five workers (numbered 0-4) and four tasks (numbered 0-3). Note that there is one more worker than in the example in the Overview.

The optimal assignment (minimum) cost = ` 9. Example 10.9. Solve the following assignment problem. Solution: 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

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.

A novel framework that can solve linear assignment problems employing methodologies derived from the field of statistical physics is proposed and it is demonstrated that when β values are sufficiently large, the exact solution to the assignment problem can be derived by rounding off the elements of the computed assignment matrix to the nearest integer. Linear assignment problems hold a ...