A project consists of four major jobs for which four contractors have submitted tenders. The following is the cost matrix of assigning:What will be the minimum total cost if contractor 1 cannot be assigned to job A? (Answer up to the nearest integer)Correct answer is '14'. Can you explain this answer?

Question

Answer

In this case, at the position (1, A), the resource cannot be assigned. In such cases the cost of performing that particular activity by a particular source is considered to be very large (Written by M or∞ ), so as to prohibit the entry of this pair of resource-activity into the final solution.

Step 1 : Here, the number of contractors and the number of jobs each equal to 4, so there is no need to add a dummy source. Therefore, the problem is balanced and we move directly onto the next step.

Step 2 : Subtracting the smallest element of each row from every element of the corresponding row, we get the reduced matrix as:

Now mark the column that do not have zero element. Subtracting the smallest element of each column of the reduced matrix from every element of the corresponding column, we get the following reduced matrix.

Step 3 : In the reduced matrix, make assignments in rows and columns having single zeros of the rows and columns, where assignment have been made. We get the following assignment matrix:

Step 4 Initial Iteration : Draw the minimum number of lines to cover all the zeros of the reduced matrix as given in matrix.

Final Iteration : Modify the reduced cost matrix by subtracting element '1' from all the elements not covered by the lines and adding the same at the intersection of two lines and now make the assignment again. This is shown in matrix below:

Since the number of assignments is equal to the order of the matrix, an optimum solution is reached. Also, since there are atleast two zeros for assignment in row 2 and row 4 as well as in column 1 and column 2, there exists an alternative assignment schedule. The optimum solution is :

Assign Contractor 1 to job D, Contractor 2 to job A, Contractor 3 to job C and Contractor 4 to job B or

Assign Contractor 1 to job D, Contractor 2 to job B, Contractor 3 to job C and Contractor 4 to job A.

Total minimum cost will be = 0 + 4 + 4 + 6 = 14

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