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
335 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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Bertsimas, D., Farias, V.F., Trichakis, N.: The price of fairness. Oper. Res. January–February 59 (1), 17–31 (2011)
MathSciNet MATH Google Scholar
Martello, S., Pulleyblank, W.R., Toth, P., De Werra, D.: Balanced optimization problems. Oper. Res. Lett. 3 (5), 275–278 (1984)
Article MathSciNet MATH Google Scholar
Kelly, F.P., Maullo, A.K., Tan, D.K.H.: Rate control for communication networks: shadow prices, proportional fairness and stability. J. Oper. Res. Soc. 49 (3), 237–252 (1997). https://doi.org/10.1057/palgrave.jors.2600523
Article Google Scholar
Ogryczak, W., Luss, H., Pioro, M., Nace, D., Tomaszewski, A.: Fair optimization and networks: a survey. J. Appl. Math. 2014 , 1–26 (2014)
Marler, R.T., Arora, J.S.: The weighted sum method for multi-objective optimization: new insights. Struct. Multi. Optim. 41 (6), 853–862 (2010)
Heller, I., Tompkins, C.B.: An extension of a theorem of Dantzig’s. Ann. Math. Stud. (38), 247–254 (1956)
Google Scholar
Kuhn, H.W.: The Hungarian method for assignment problem. Naval Res. Logist. Q. 2 (1–2), 83–97 (1955)
Martello, S.: Most and least uniform spanning trees. Discrete Appl. Math. 15 (2), 181–197 (1986)
Beasley, J.E.: Linear programming on Clay supercomputer. J. Oper. Res. Soc. 41 , 133–139 (1990)
Nguyen, M.H, Baiou, M., Nguyen, V.H., Vo, T.Q.T.: Nash fairness solutions for balanced TSP. In: International Network Optimization Conference (INOC2022) (2022)
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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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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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To make an unbalanced assignment problem balanced, what are added with all entries as zeroes? A. Dummy rows: B. Dummy columns: C. Both A and B: D. Dummy entries: Answer» C. Both A and B View all MCQs in. Operations Research Discussion No comments yet Login to comment Related MCQs. To make an unbalanced assignment problem balanced, what are ...
Identify the correct statement. a. an assignment problem may require the introduction of both dummy row and dummy column. b. an assignment problem with m rows and n columns will involves a total of m x n possible assignments. c. an unbalanced assignment is one where the number of rows is more than, or less than,the number of columns. d.
Dep 2 - Depreciation MCQs; M.Sc., Applied Psychology CBCS 2012 13; M.Sc., Applied Psychology IC 2012 13; Related documents. Mphil-Phd Psychology 2014-15; ... To make an unbalanced assignment problem balanced, what are added with all entries as zeroes? a) Dummy rows b) Dummy columns c) Both A and B d) Dummy entries ...
📒⏩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...
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 ...
Here we discuss how to convert an unbalanced assignment problem into a balanced one and further solve it.
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: A dummy job or facility is an imaginary job/facility with zero cost or time introduced to make an unbalanced assignment problem balanced ...
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 ...
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 ...
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.
An unbalanced assignment problem occurs when the number of tasks and the number of agents are not equal. To make it balanced, we need to add either dummy rows or dummy columns with all entries as zeros, depending on whether there are more tasks or more agents. Step 2/2 So, the correct answer is (c) Any one from (a) and (b).
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.
In this video, we will learn how to balance an unbalanced assignment problem and solve it using Hungarian method.
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.
An Alternative Approach for Solving Unbalanced Assignment Problems Abdur Rashid Department of Mathematics, Jahangirnagar University, Savar, Dhaka-1342, Bangladesh. Abstract 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.
To make an unbalanced assignment problem balanced, what are added with all entries as zeroes? Before starting to solve the problem, it should be balanced. If not then make it balanced by ‐‐‐‐‐ ‐‐‐‐‐‐ column incase demand is less than supply or by adding ‐‐‐‐‐‐‐‐‐‐‐‐ raw incase supply is less than ...
The Balanced Assignment Problem (BAP) is a variant of the classic AP where instead of minimizing the total cost, we minimize the max-min distance which is the difference between the maximum assignment cost and the minimum one in the assignment solution. In [ 2 ], the authors proposed an efficient threshold-based algorithm to solve the BAP in ...
Before applying Hungarian method, form a balanced / square matrix..Add dummy rows ..Add dummy columns ..
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 proposed approach is quite simple ...
To make an unbalanced assignment problem balanced, what are added with all entries as zeroes? To make an unbalanced assignment problem balanced, what are added with all entries as zeroes? If the total supply is less than the total demand, a dummy source (row) is included in the cost matrix with ‐‐‐‐‐‐‐‐‐‐‐‐‐
The assignment problem is one of the fundamental combinatorial optimization problems in the branch of optimization or operations research in mathematics.The ...