# Optimal Assignment Problem The overall measure of performance is the total cost of the assignment, so the objective is to minimize this quantity. To make the model easier to understand, name the following ranges. Total Cost equals the sumproduct of Cost and Assignment. The result should be consistent with the picture below. With this formulation, it becomes easy to analyze any trial solution. We shall describe next how the Excel Solver can be used to quickly find the optimal solution. You have the choice of typing the range names or clicking on the cells in the spreadsheet. Write down the assignment results and find the minimum cost/time.

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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.

The assignment cost is the sum, over all pairs, of the flow between a pair of facilities multiplied by the distance between their assigned locations.

The quadratic assignment problem (QAP) was introduced by Koopmans and Beckman in 1957 in the context of locating "indivisible economic activities".

In row A, the smallest value is 13, row B is 15, row C is 17 and row D is 12.

The row wise reduced matrix is shown in table below.

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.

Subtract 3 from all other values that are not covered and add 3 at the intersection of lines. Here in table minimum number of lines drawn is 4 which are equal to the order of matrix. Strike off remaining zeros if any in that row or column.

Formulate the Model | Trial and Error | Solve the Model Use the solver in Excel to find the assignment of persons to tasks that minimizes the total cost.

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