# Solution Of Assignment Problem It can be stated as follows: given a bipartite graph made up of two partitions V and U, and a set of weighted edges E between the two partitions, the problem requires the selection of a subset of the edges with a maximum sum of weights such that each node or is connected to at most one edge.The problem may also be phrased as a minimization problem by considering, instead of edge weights, a set of non-negative edge costs, , where W is at least as large as the maximum of all the edge weights.An assignment with the smallest possible cost is called an optimal assignment.

It can be stated as follows: given a bipartite graph made up of two partitions V and U, and a set of weighted edges E between the two partitions, the problem requires the selection of a subset of the edges with a maximum sum of weights such that each node or is connected to at most one edge.The problem may also be phrased as a minimization problem by considering, instead of edge weights, a set of non-negative edge costs, , where W is at least as large as the maximum of all the edge weights.An assignment with the smallest possible cost is called an optimal assignment.

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The mathematical formulation of the problem suggests that this is an integer programming problem and is highly degenerated.

All the algorithms developed to find optimal solution of transportation problem are applicable to assignment problem.

Assignment of jobs to workers, contract to contractors undergoing a bidding process, assigning nurses to duty post, or time tabling for teachers in school and many more have become a growing concern to both management and sector leaders alike.

Hungarian algorithm has been the most successful tool for solving such problems.

Step 4: Test for Optimality: i) If the minimum number of covering lines is n, an optimal assignment of zeros is possible and we are finished. and De Causmaecker, P., Eds., Practice and Theory of Automated Timetabling IV, Vol. (2007) Workforce Optimization: Identification and Assignment of Professional Workers Using Constraint Programming. 26th International Symposium on Computer and Information Sciences, London, 26-28 September 2011, 571-576.

ii) If the minimum number of covering lines is less than n, an optimal assignment of zeros is not yet possible. Step 5: Determine the smallest entry not covered by any line. Penalty method (Proposed Heuristic Method) The algorithm for n × n cost matrix using the proposed heuristic method is as shown below. 2740, Lecture Notes in Computer Science, Springer Berlin, Heidelberg, 113-119. IBM Journal of Research and Development, Business Optimization Archive, 51, 263-279.

The authors have proposed a heuristic method for solving assignment problems with less computing time in comparison with Hungarian algorithm that gives comparable results with an added advantage of easy implementation.

The proposed heuristic method is used to compute some bench mark problems. Introduction The Assignment Problem (AP), also known as the maximum weighted bipartite matching problem, is a special type of Linear Programming Problem (LPP), in which the objective is to assign number of jobs to number of workers at a minimum cost (time).

By applying constraint programming, a subfield of artificial intelligence, the authors dealt successfully with the complex constraints encountered in the field and reach near-optimal assignments that take into account all resources and positions in the pool.

Katta and Jay  presented the problem of allocating a set of indivisible objects to agents in a fair and efficient manner. Mathematical Model Let be the cost of assigning the ith resource to the jth task.

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