Assignment problem maximum

operational research in topic assignment problem

1. assignment PROBLEM- MAXIMUM There are problems where certain facilities have to be assigned to a number of jobs so as to maximize the overall performance of the assignment. The problem can be converted into a minimization problem in the following ways and then Hungarian method can be used for its solution. Change the signs of all values given in the table. Select the highest element in the entire assignment table and subtract all the elements of the table from the highest element.

2. Example :  A marketing manager has five salesmen and sales districts. Considering the capabilities of the salesmen and the nature of districts, the marketing manager estimates that sales per month (in hundred rupees) for each salesman in each district would be as follows. Find the assignment of salesmen to districts that will result in maximum sales.

3. Step 1 : The given maximization problem is converted into minimization problem by subtracting from the highest sales value (i.e., 41) with all elements of the given table. Max value

4.   Hungarian METHOD Step 2 : Reduce the matrix by selecting the smallest value in each row and subtracting from other values in that corresponding row. the smallest value: row 1, is 1,row 2 is 1,row 3 is 0, row 4 is 0 and row 5 is 1. 

5.   Hungarian METHOD Step 3 : Reduce the new matrix given in the following table by selecting the smallest value in each column and subtract from other values in that corresponding column. In column A, the smallest value is 0, column B is 2, column C is 0, column D is 5 and column E is 0. 

6.   Hungarian METHOD Step 4 : Draw minimum number of lines possible to cover all the zeros in the matrix given in Table Check whether number of lines drawn is equal to the order of the matrix, i.e., 3 ≠ 5. Therefore optimally is not reached .

7.   Hungarian METHOD Step 4 :  Take the smallest element of the matrix that is not covered by single line , which is 3 . Subtract 3 from all other values that are not covered and add 3 at the intersection of lines. Leave the values which are covered by single line.  4 Select the least uncovered element , i.e., 4 and subtract it from other uncovered elements , add it to the elements at intersection of line and leave the elements that are covered with single line unchanged ,

8.   Hungarian METHOD Now, number of lines drawn = Order of matrix, hence optimality is reached.

9.   Hungarian METHOD There are two alternative assignments due to presence of zero elements in cells (4, C), (4, D), (5, C) and (5, D).