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
- 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
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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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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:
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Now use the Hungarian method to obtain the optimal solution yourself. Ans. = 20 + 17 + 17 + 0 = 54.
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International Symposium on Combinatorial Optimization
ISCO 2022: Combinatorial Optimization pp 172–186 Cite as
Nash Balanced Assignment Problem
- Minh Hieu Nguyen 11 ,
- Mourad Baiou 11 &
- Viet Hung Nguyen 11
- Conference paper
- First Online: 21 November 2022
334 Accesses
2 Citations
Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 13526))
In this paper, we consider a variant of the classic Assignment Problem (AP), called the Balanced Assignment Problem (BAP) [ 2 ]. The BAP seeks to find an assignment solution which has the smallest value of max-min distance : the difference between the maximum assignment cost and the minimum one. However, by minimizing only the max-min distance, the total cost of the BAP solution is neglected and it may lead to an inefficient solution in terms of total cost. Hence, we propose a fair way based on Nash equilibrium [ 1 , 3 , 4 ] to inject the total cost into the objective function of the BAP for finding assignment solutions having a better trade-off between the two objectives: the first aims at minimizing the total cost and the second aims at minimizing the max-min distance. For this purpose, we introduce the concept of Nash Fairness (NF) solutions based on the definition of proportional-fair scheduling adapted in the context of the AP: a transfer of utilities between the total cost and the max-min distance is considered to be fair if the percentage increase in the total cost is smaller than the percentage decrease in the max-min distance and vice versa.
We first show the existence of a NF solution for the AP which is exactly the optimal solution minimizing the product of the total cost and the max-min distance. However, finding such a solution may be difficult as it requires to minimize a concave function. The main result of this paper is to show that finding all NF solutions can be done in polynomial time. For that, we propose a Newton-based iterative algorithm converging to NF solutions in polynomial time. It consists in optimizing a sequence of linear combinations of the two objective based on Weighted Sum Method [ 5 ]. Computational results on various instances of the AP are presented and commented.
- Combinatorial optimization
- Balanced assignment problem
- Proportional fairness
- Proportional-fair scheduling
- Weighted sum method
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INP Clermont Auvergne, Univ Clermont Auvergne, Mines Saint-Etienne, CNRS, UMR 6158 LIMOS, 1 Rue de la Chebarde, Aubiere Cedex, France
Minh Hieu Nguyen, Mourad Baiou & Viet Hung Nguyen
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Correspondence to Viet Hung Nguyen .
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ESSEC Business School of Paris, Cergy Pontoise Cedex, France
Ivana Ljubić
IBM TJ Watson Research Center, Yorktown Heights, NY, USA
Francisco Barahona
Georgia Institute of Technology, Atlanta, GA, USA
Santanu S. Dey
Université Paris-Dauphine, Paris, France
A. Ridha Mahjoub
Proposition 1 . There may be more than one NF solution for the AP.
Let us illustrate this by an instance of the AP having the following cost matrix
By verifying all feasible assignment solutions in this instance, we obtain easily three assignment solutions \((1-1, 2-2, 3-3), (1-2, 2-3, 3-1)\) , \((1-3, 2-2, 3-1)\) and \((1-3, 2-1, 3-2)\) corresponding to 4 NF solutions (280, 36), (320, 32), (340, 30) and (364, 28). Note that \(i-j\) where \(1 \le i,j \le 3\) represents the assignment between worker i and job j in the solution of this instance. \(\square \)
We recall below the proofs of some recent results that we have published in [ 10 ]. They are needed to prove the new results presented in this paper.
Theorem 2 [ 10 ] . \((P^{*},Q^{*}) = {{\,\mathrm{arg\,min}\,}}_{(P,Q) \in \mathcal {S}} PQ\) is a NF solution.
Obviously, there always exists a solution \((P^{*},Q^{*}) \in \mathcal {S}\) such that
Now \(\forall (P',Q') \in \mathcal {S}\) we have \(P'Q' \ge P^{*}Q^{*}\) . Then
The first inequality holds by the Cauchy-Schwarz inequality.
Hence, \((P^{*},Q^{*})\) is a NF solution. \(\square \)
Theorem 3 [ 10 ] . \((P^{*},Q^{*}) \in \mathcal {S}\) is a NF solution if and only if \((P^{*},Q^{*})\) is an optimal solution of \(\mathcal {P(\alpha ^{*})}\) where \(\alpha ^{*} = \frac{Q^{*}}{P^{*}}\) .
Firstly, let \((P^{*},Q^{*})\) be a NF solution and \(\alpha ^{*} = \frac{Q^{*}}{P^{*}}\) . We will show that \((P^{*},Q^{*})\) is an optimal solution of \(\mathcal {P(\alpha ^{*})}\) .
Since \((P^{*},Q^{*})\) is a NF solution, we have
Since \(\alpha ^{*} = \frac{Q^{*}}{P^{*}}\) , we have \(\alpha ^{*}P^{*}+Q^{*} = 2Q^{*}\) .
Dividing two sides of ( 6 ) by \(P^{*} > 0\) we obtain
So we deduce from ( 7 )
Hence, \((P^{*},Q^{*})\) is an optimal solution of \(\mathcal {P}(\alpha ^{*})\) .
Now suppose \(\alpha ^{*} = \frac{Q^{*}}{P^{*}}\) and \((P^{*},Q^{*})\) is an optimal solution of \(\mathcal {P}(\alpha ^{*})\) , we show that \((P^{*},Q^{*})\) is a NF solution.
If \((P^{*},Q^{*})\) is not a NF solution, there exists a solution \((P',Q') \in \mathcal {S}\) such that
We have then
which contradicts the optimality of \((P^{*},Q^{*})\) . \(\square \)
Lemma 3 [ 10 ] . Let \(\alpha , \alpha ' \in \mathbb {R}_+\) and \((P_{\alpha }, Q_{\alpha })\) , \((P_{\alpha '}, Q_{\alpha '})\) be the optimal solutions of \(\mathcal {P(\alpha )}\) and \(\mathcal {P(\alpha ')}\) respectively, if \(\alpha \le \alpha '\) then \(P_{\alpha } \ge P_{\alpha '}\) and \(Q_{\alpha } \le Q_{\alpha '}\) .
The optimality of \((P_{\alpha }, Q_{\alpha })\) and \((P_{\alpha '}, Q_{\alpha '})\) gives
By adding both sides of ( 8a ) and ( 8b ), we obtain \((\alpha - \alpha ') (P_{\alpha } - P_{\alpha '}) \le 0\) . Since \(\alpha \le \alpha '\) , it follows that \(P_{\alpha } \ge P_{\alpha '}\) .
On the other hand, inequality ( 8a ) implies \(Q_{\alpha '} - Q_{\alpha } \ge \alpha (P_{\alpha } - P_{\alpha '}) \ge 0\) that leads to \(Q_{\alpha } \le Q_{\alpha '}\) . \(\square \)
Lemma 4 [ 10 ] . During the execution of Procedure Find ( \(\alpha _{0})\) in Algorithm 1 , \(\alpha _{i} \in [0,1], \, \forall i \ge 1\) . Moreover, if \(T_{0} \ge 0\) then the sequence \(\{\alpha _i\}\) is non-increasing and \(T_{i} \ge 0, \, \forall i \ge 0\) . Otherwise, if \(T_{0} \le 0\) then the sequence \(\{\alpha _i\}\) is non-decreasing and \(T_{i} \le 0, \, \forall i \ge 0\) .
Since \(P \ge Q \ge 0, \, \forall (P, Q) \in \mathcal {S}\) , it follows that \(\alpha _{i+1} = \frac{Q_i}{P_i} \in [0,1], \, \forall i \ge 0\) .
We first consider \(T_{0} \ge 0\) . We proof \(\alpha _i \ge \alpha _{i+1}, \, \forall i \ge 0\) by induction on i . For \(i = 0\) , we have \(T_{0} = \alpha _{0} P_{0} - Q_{0} = P_{0}(\alpha _{0}-\alpha _{1}) \ge 0\) , it follows that \(\alpha _{0} \ge \alpha _{1}\) . Suppose that our hypothesis is true until \(i = k \ge 0\) , we will prove that it is also true with \(i = k+1\) .
Indeed, we have
The inductive hypothesis gives \(\alpha _k \ge \alpha _{k+1}\) that implies \(P_{k+1} \ge P_k > 0\) and \(Q_{k} \ge Q_{k+1} \ge 0\) according to Lemma 3 . It leads to \(Q_{k}P_{k+1} - P_{k}Q_{k+1} \ge 0\) and then \(\alpha _{k+1} - \alpha _{k+2} \ge 0\) .
Hence, we have \(\alpha _{i} \ge \alpha _{i+1}, \, \forall i \ge 0\) .
Consequently, \(T_{i} = \alpha _{i}P_{i} - Q_{i} = P_{i}(\alpha _{i}-\alpha _{i+1}) \ge 0, \, \forall i \ge 0\) .
Similarly, if \(T_{0} \le 0\) we obtain that the sequence \(\{\alpha _i\}\) is non-decreasing and \(T_{i} \le 0, \, \forall i \ge 0\) . That concludes the proof. \(\square \)
Lemma 5 [ 10 ] . From each \(\alpha _{0} \in [0,1]\) , Procedure Find \((\alpha _{0})\) converges to a coefficient \(\alpha _{k} \in \mathcal {C}_{0}\) satisfying \(\alpha _{k}\) is the unique element \(\in \mathcal {C}_{0}\) between \(\alpha _{0}\) and \(\alpha _{k}\) .
As a consequence of Lemma 4 , Procedure \(\textit{Find}(\alpha _{0})\) converges to a coefficient \(\alpha _{k} \in [0,1], \forall \alpha _{0} \in [0,1]\) .
By the stopping criteria of Procedure Find \((\alpha _{0})\) , when \(T_{k} = \alpha _{k} P_{k} - Q_{k} = 0\) we obtain \(\alpha _{k} \in C_{0}\) and \((P_{k},Q_{k})\) is a NF solution. (Theorem 3 )
If \(T_{0} = 0\) then obviously \(\alpha _{k} = \alpha _{0}\) . We consider \(T_{0} > 0\) and the sequence \(\{\alpha _i\}\) is now non-negative, non-increasing. We will show that \([\alpha _{k},\alpha _{0}] \cap \mathcal {C}_{0} = \alpha _{k}\) .
Suppose that we have \(\alpha \in (\alpha _{k},\alpha _{0}]\) and \(\alpha \in \mathcal {C}_{0}\) corresponding to a NF solution ( P , Q ). Then there exists \(1 \le i \le k\) such that \(\alpha \in (\alpha _{i}, \alpha _{i-1}]\) . Since \(\alpha \le \alpha _{i-1}\) , \(P \ge P_{i-1}\) and \(Q \le Q_{i-1}\) due to Lemma 3 . Thus, we get
By the definitions of \(\alpha \) and \(\alpha _{i}\) , inequality ( 9 ) is equivalent to \(\alpha \le \alpha _{i}\) which leads to a contradiction.
By repeating the same argument for \(T_{0} < 0\) , we also have a contradiction. \(\square \)
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Nguyen, M.H., Baiou, M., Nguyen, V.H. (2022). Nash Balanced Assignment Problem. In: Ljubić, I., Barahona, F., Dey, S.S., Mahjoub, A.R. (eds) Combinatorial Optimization. ISCO 2022. Lecture Notes in Computer Science, vol 13526. Springer, Cham. https://doi.org/10.1007/978-3-031-18530-4_13
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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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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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IMAGES
VIDEO
COMMENTS
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 ...
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 ...
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. Dummy Job/Facility: ...
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.
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 Assignment Problem (AP) is a fundamental combinatorial optimization problem. It can be formally defined as follows. Given a set n workers, a set of n jobs and a \(n \times n\) cost matrix whose elements are positive representing the assignment of any worker to any job, the AP aims at finding an one-to-one worker-job assignment (i.e., a bipartite perfect matching) that minimizes certain ...
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, 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
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.
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 ...
Nash Balanced Assignment Problem 173 the assignment of any worker to any job, the AP aims at finding an one-to-one worker-job assignment (i.e., a bipartite perfect matching) that minimizes certain objective functions. In the classic AP, we seek to find an assignment solution minimizing the total cost. It is a well-known optimization problem ...
10 Feb 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.
Objectives. After studying this unit, you should be able to: formulate an assignment problem; determine the optimal solutions of assignment problems using the Hungarian method; obtain the solutions for special cases of assignment problems, i.e, maximisation problem, unbalanced assignment problem, alternative optimal solutions and restriction on ...
The assignment problem involves assigning jobs to operators to minimize total processing time. The Hungarian method consists of two phases: row and column reduction, and optimization of the problem. The balanced assignment problem has an equal number of rows and columns, while the unbalanced problem requires adding dummy rows or columns.
Variations of the Assignment Problems: 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. To make it balanced we add a dummy row or dummy column with all the entries is zero. Example 3:
A heuristic algorithm for solving an unbalanced single-objective assignment problem without transforming it to a balanced one is developed using the concept of penalty.
A few solutions for addressing assignment problems in an Ant colony environment are proposed in the literature, such as ACO for the cell assignment problem in PCS networks (2006), Improved Ant Colony Algorithm for Solving Assignment Problem [6], and so on. In this paper, we research the balanced and unbalanced assignment problems, with a few
balanced and unbalanced problems without using Dummy Cells as in Hungarian Method. And also it can be applied for both maximization and minimization problems. To show the efficiency of this method I'll consider some numerical examples for balanced and unbalanced assignment problems, and also for minimization and maximization assignment problems.
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-task assignment. It is required to perform as many tasks as possible by assigning at most one ...
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. This paper is devoted to present a new approach to make an unbalanced assignment problem into a balanced one and a comparison is carried out with the existing methods.
The fuzzy salesman and areas by Hungarian unbalanced assignment problems is proposed and formulated to the crisp assignment problem in the linear programming problem form and solved by using Hungarian method and using Robust's ranking method for the fuzzy numbers. In this paper, we proposed the fuzzy salesman and areas by Hungarian unbalanced assignment problems.
It is the detailed note about Assignment problems - unbalanced operations research notes unit assignment problem unbalanced contents objectives introduction. Skip to document ... the concept city represents, for example, customers, soldering points, or DNA fragments, and the concept distance represents travelling times or cost, or a similarity ...