# Optimal Assignment Problem An assignment problem can be easily solved by applying Hungarian method which consists of two phases.In the first phase, row reductions and column reductions are carried out.

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Result: The optimal solution: Conclusion: it is optimal to assign Person 1 to task 2, Person 2 to Task 3 and Person 3 to Task 1.

Summary: The objective of the Quadratic Assignment Problem (QAP) is to assign \(n\) facilities to \(n\) locations in such a way as to minimize the assignment cost.

In the second phase, the solution is optimized on iterative basis.

In a given problem, if the number of rows is not equal to the number of columns and vice versa, then add a dummy row or a dummy column.

The column-wise reduction matrix is shown in the following table.

Take the smallest element of the matrix that is not covered by single line, which is 3. Now, draw minimum number of lines to cover all the zeros and check for optimality. Select a row that has a single zero and assign by squaring it.

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 1, the smallest value is 0, column 2 is 4, column 3 is 3 and column 4 is 0.

Strike off the remaining zeros in that column or row, and repeat the same for other assignments also.

If there is no single zero allocation, it means multiple numbers of solutions exist.

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