L23_Assignment Problems - Hungarian Algorithm

L23_Assignment Problems - Hungarian Algorithm - The...

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Unformatted text preview: The Assignment Model " The best person for job" is an apt description of the assignment model. The general assignment model with n workers and n jobs is presented below: Jobs 1 2 . n 1 c 11 c 12 c 1n Workers 2 c 21 c 22 c 2n n c n1 c n2 c nn The element c ij is the cost of assigning the worker i to the job j. More meaningfully, it may be thought of as the time taken by the worker i to complete the job j. There is no loss of generality in assuming that the number of workers = the number of jobs. If there are more workers, we may introduce dummy jobs and if there are less workers, we may introduce dummy workers. One important assumption we make is that each worker is assigned to one and only one job. And each job is done by one and only one worker. The assignment model is actually a special case of the transportation model in which the sources are workers, the destinations are jobs, and the number of sources = the number of destinations. Also from the last assumption we get that the availabilities at each source equals 1 and the demand at each destination equals 1. x ij =1 if worker i assigned to job j and = 0 otherwise. Though we can apply the transportation algorithm to solve the assignment model, a special simple algorithm, called the Hungarian algorithm , is used to solve such models. Consider the assignment problem: 4 1 2 3 5 2 4 1 5 1 2 3 The solution is obvious: W1 J2, W2 J4, W3 J1, W4 J3 We note that in any assignment problem, if a constant c is subtracted from the costs of any row ( or column), the optimum solution does not change, but the assignment cost decreases by c . This is because in each row we have to assign 1 to one and only one cell....
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L23_Assignment Problems - Hungarian Algorithm - The...

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