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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Assignment problem
The problem of optimally assigning $ m $ individuals to $ m $ jobs. It can be formulated as a linear programming problem that is a special case of the transport problem :
maximize $ \sum _ {i,j } c _ {ij } x _ {ij } $
$$ \sum _ { j } x _ {ij } = a _ {i} , i = 1 \dots m $$
(origins or supply),
$$ \sum _ { i } x _ {ij } = b _ {j} , j = 1 \dots n $$
(destinations or demand), where $ x _ {ij } \geq 0 $ and $ \sum a _ {i} = \sum b _ {j} $, which is called the balance condition. The assignment problem arises when $ m = n $ and all $ a _ {i} $ and $ b _ {j} $ are $ 1 $.
If all $ a _ {i} $ and $ b _ {j} $ in the transposed problem are integers, then there is an optimal solution for which all $ x _ {ij } $ are integers (Dantzig's theorem on integral solutions of the transport problem).
In the assignment problem, for such a solution $ x _ {ij } $ is either zero or one; $ x _ {ij } = 1 $ means that person $ i $ is assigned to job $ j $; the weight $ c _ {ij } $ is the utility of person $ i $ assigned to job $ j $.
The special structure of the transport problem and the assignment problem makes it possible to use algorithms that are more efficient than the simplex method . Some of these use the Hungarian method (see, e.g., [a5] , [a1] , Chapt. 7), which is based on the König–Egervary theorem (see König theorem ), the method of potentials (see [a1] , [a2] ), the out-of-kilter algorithm (see, e.g., [a3] ) or the transportation simplex method.
In turn, the transportation problem is a special case of the network optimization problem.
A totally different assignment problem is the pole assignment problem in control theory.
| [a1] | D.B. Yudin, E.G. Gol'shtein, "Linear programming" , Israel Program Sci. Transl. (1965) (In Russian) |
| [a2] | R. Frisch, "La résolution des problèmes de programme linéaire par la méthode du potentiel logarithmique" , (1956) pp. 20–23 |
| [a3] | K. Murtz, "Linear and combinatorial programming" , Wiley (1976) |
| [a4] | M. Grötschel, L. Lovász, A. Schrijver, "Geometric algorithms and combinatorial optimization" , Springer (1987) |
| [a5] | C.H. Papadimitriou, K. Steiglitz, "Combinatorial optimization" , Prentice-Hall (1982) |
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Assignment Problem
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The problem of optimally assigning m individuals to m jobs, so that each individual is assigned to one job, and each job is filled by one individual. The problem can be formulated as a linear-programming problem with the objective function measuring the (linear) utility of the assignment as follows:
The problem is a special form of the transportation problem and, as such,...
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Burkard, R., Dell’Amico, M., & Marterllo, S. (2009). Assignment problems . Philadelphia: SIAM.
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Kuhn, H. W. (1995). The Hungarian method for the assignment problem. Naval Research Logistics Quarterly, 2 , 83–97.
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Robert H. Smith School of Business, University of Maryland, College Park, MD, USA
Saul I. Gass
Robert H. Smith School of Business and Institute for Systems Research, University of Maryland, College Park, MD, USA
Michael C. Fu
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(2013). Assignment Problem. In: Gass, S.I., Fu, M.C. (eds) Encyclopedia of Operations Research and Management Science. Springer, Boston, MA. https://doi.org/10.1007/978-1-4419-1153-7_200965
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COMMENTS
The formal definition of the assignment problem (or linear assignment problem) is Given two sets, A and T, together with a weight function C : A × T → R. Find a bijection f : A → T such that the cost function: (, ()) is minimized.
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 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.
The assignment problem is one of the fundamental combinatorial optimization problems in the branch of optimization or operations research in mathematics. In an assignment problem , we must find a maximum matching that has the minimum weight in a weighted bipartite graph .
The assignment problem is defined as finding an assignment that can satisfy all agents on the condition that everyone's reservation payoff is satisfactory. This is an NP-complete problem.
The “assignment problem” is one that can be solved using simple techniques, at least for small problem sizes, and is easy to see how it could be applied to the real world. Assignment Problem Pretend for a moment that you are writing software for a famous ride sharing application.
Assignment Problem is a special type of linear programming problem where the objective is to minimise the cost or time of completing a number of jobs by a number of persons.
Assignment problem. Also known as weighted bipartite matching problem. Bipartite graph. Has two sets of nodes , ⇒ = ∪. And a set of edges connecting them. A matching on a bipartite graph G = (S, T, E) is a subset of edges ∈ ∋ no two edges in are incident to the same node. Nodes 1, 2, 3, 4. Node of are matched or covered.
The assignment problem arises when $ m = n $ and all $ a _ {i} $ and $ b _ {j} $ are $ 1 $. If all $ a _ {i} $ and $ b _ {j} $ in the transposed problem are integers, then there is an optimal solution for which all $ x _ {ij } $ are integers (Dantzig's theorem on integral solutions of the transport problem).
The problem of optimally assigning m individuals to m jobs, so that each individual is assigned to one job, and each job is filled by one individual. The problem can be formulated as a linear-programming problem with the objective function measuring the (linear) utility of the assignment as follows: $$ \begin {array} {l} \text { Maximize}\ \sum ...