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
| Job | ||||
|---|---|---|---|---|
| Person | 1 | 2 | 3 | 4 |
| A | 20 | 25 | 22 | 28 |
| B | 15 | 18 | 23 | 17 |
| C | 19 | 17 | 21 | 24 |
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.
| Job | ||||
|---|---|---|---|---|
| Person | 1 | 2 | 3 | 4 |
| A | 20 | 25 | 22 | 28 |
| B | 15 | 18 | 23 | 17 |
| C | 19 | 17 | 21 | 24 |
| D (dummy) | 0 | 0 | 0 | 0 |
Now use the Hungarian method to obtain the optimal solution yourself. Ans. = 20 + 17 + 17 + 0 = 54.
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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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A Comparative Analysis of Assignment Problem
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- 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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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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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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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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- DOI: 10.4236/AJOR.2016.64028
- Corpus ID: 124980289
Solving the Unbalanced Assignment Problem: Simpler Is Better
- Nathan K. Betts , Francis J. Vasko
- Published 24 June 2016
- Mathematics
- American Journal of Operations Research
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An amalgamated approach for solving unbalanced assignment problem, modified hungarian method for unbalanced assignment problem with multiple jobs, modified hungarian method for solving balanced fuzzy transportation problems.
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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.
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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Balancing an Unbalanced Assignment problem using optimization techniques
How can I balance the following assignment problem (where machines are to be assigned the jobs in optimal way such that the profit is maximized). Cost matrix is given in the problem.
First step is to convert it into minimization problem by subtracting all the entries in the matrix from maximum value in the matrix.
In next step, We try to balance the unbalanced problem i.e. adding dummy machines in this case. How to do this step? What should be row entries for dummy machines in matrix.
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18 Dec 2019. 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.
Here is the video about unbalanced Assignment problem using Hungarian method,In this video we have seen how to solve unbalanced assignment problem using step...
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 ...
📒⏩Comment Below If This Video Helped You 💯Like 👍 & Share With Your Classmates - ALL THE BEST 🔥Do Visit My Second Channel - https://bit.ly/3rMGcSAThis vi...
Playlist of all my Operations Research videos-http://goo.gl/zAtbi4Today I'll tell how to solve an Unbalanced Assignment Problem in just 6 Easy Steps! Assignm...
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.
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 ...
In Yadaiah and Haragopal [4], they use a different approach to solve the unbalanced assignment problem (see their paper for details). If there are n jobs to be assigned to m machines with n strictly greater than m, then they solve a series of k balanced assignment sub-problems each of size m by m where k is the floor (round down) of n/m. The ...
Abstract. Recently, Yadaiah and Harago pal published in the American Jo urnal of Operations Research a new. approach to solving the u nbalanced assignment problem. They also provide a num erical ...
e minimisation problem.3. The assignment problem wherein the number of rows is not equal to the number of columns is said t. be an unbalanced problem. Such a problem is handled by introducing dummy row(s) if the number of rows is less than the number of columns and dummy column(s) if the number of columns is le.
Tables 2, 3, 4, and 5 present the steps required to determine the appropriate job assignment to the machine. Step 1 By taking the minimum element and subtracting it from all the other elements in each row, the new table will be: Table 2 represents the matrix after completing the 1st step. Table 1 Initial table of a.
The assignment problem consists of finding, in a weighted bipartite graph, a matching of a given size, in which the sum of weights of the edges is minimum. If the numbers of agents and tasks are equal, then the problem is called balanced assignment. Otherwise, it is called unbalanced assignment. [ 1] If the total cost of the assignment for all ...
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 ...
This is the complete recording of the lecture, Module 3 - Lecture 3 - Unbalanced Assignment Problems in MEE1024 - Operations Research.
solving minimisation problems: Step 1:See whether number. f rows are equal to number of columns. If yes, problem is balanced one; if not, then add a Dummy Row or Column to make the problem a balanced one by allotting zero value to each cell of the D. mmy Row or Column, as the case may be.Step 2: Row Subtraction: Subtract the minimum element of.
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.
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. Here only 3 tasks can be assigned to 3 men.
Unbalanced assignment problem, Hungarian method, Optimal solution. Introduction The assignment problem is a combinatorial optimization problem in the field of operations research. It is a special case and completely degenerate form of a transportation problem, which occurs when each supply is 1 and
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 Questions - Free download as PDF File (.pdf), Text File (.txt) or read online for free. The document provides information about an unbalanced assignment problem involving road repair work by 5 contractors. It includes the following details: 1) A city corporation needs to repair 4 roads and has received bids from 5 contractors, with each contractor's time and cost ...
How can I balance the following assignment problem (where machines are to be assigned the jobs in optimal way such that the profit is maximized). Cost matrix is given in the problem. First step is to convert it into minimization problem by subtracting all the entries in the matrix from maximum value in the matrix.
Step-18: Stop. 7.. ConclusionThe present paper suggests a modified method for solving the unbalanced assignment problems. The Hungarian method [1] gives us total assignment cost 870 along with the other three jobs assigned to dummy machine, in other words that these three jobs are ignored for further processing, while, when the original problem is divided in the sub-problems, which are ...
Algorithm & Example-1. Algorithm. Hungarian Method Steps (Rule) Step-1: If number of rows is not equal to number of columns, then add dummy rows or columns with cost 0, to make it a square matrix. Step-2: a. Identify the minimum element in each row and subtract it from each element of that row.