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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. 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.

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). 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. However, due to its highly degeneracy nature, a specially designed algorithm widely known as Hungarian method proposed by Kuhn [

Franses and Gerhard [

Toroslu and Arslanoglu [

Naveh et al. [

Katta and Jay [

Bogomolnaia and Moulin [

Zhang and Bard [

Suppose we have n resources to which we want to assign to n tasks on a one-to-one basis. Suppose also that we know the cost of assigning a given resource to a given task. We wish to find an optimal assignment?one which minimizes total cost.

Let

An assignment is a set of n entry positions in the cost matrix, no two of which lie in the same row or column. The sum of the n entries of an assignment is its cost. An assignment with the smallest possible cost is called an optimal assignment. The assignment problem can be written mathematically as:

Minimize

The following algorithm applies the above theorem to a given n × n cost matrix to find an optimal assignment.

Step 1: Subtract the smallest entry in each row from all the entries of its row.

Step 2: Subtract the smallest entry in each column from all the entries of its column.

Step 3: Draw lines through appropriate rows and columns so that all the zero entries of the cost matrix are covered and the minimum number of such lines is used.

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.

ii) If the minimum number of covering lines is less than n, an optimal assignment of zeros is not yet possible. In that case, proceed to Step 5.

Step 5: Determine the smallest entry not covered by any line. Subtract this entry from each uncovered row, and then add it to each covered column. Return to Step 3.

The algorithm for n × n cost matrix using the proposed heuristic method is as shown below.

・ Compute row/column penalties by subtracting the least entry from the next least;

・ Select the two maximum penalties from the row or column penalties. Use the maximum penalties to select the least cost among these penalties.

・ Cross-out the row and column on which the minimum cost is located

・ If there is a tie of the minimum cost from the same row or the same column, choose the cost of the third maximum penalty

・ Break ties arbitrary when there is a tie on the cost from maximum row and column penalty

・ If the least cost emanating from maximum penalty has its corresponding row or column penalty as zero, choose the least cost from the next maximum penalty.

・ Repeat bullet two

・ Stopping criteria: stop when all rows and columns have been crossed out.

The following examples give the optimal results from the two methods and their computing time. The two algorithms were coded using C++ on

Test computation from a randomly generated numbers with our proposed heuristic method gave comparable results compared with that of Hungarian method. However, the computational time and convergence rate of the proposed method showed much less time than that of the Hungarian method.

Amponsah, S.K., Otoo, D., Salhi, S. and Quayson, E. (2016) Proposed Heuristic Method for Solving Assignment Problems. American Journal of Operations Research, 6, 436-441. http://dx.doi.org/10.4236/ajor.2016.66040