A Mathematical Formulation of the Quadratic Assignment Problem

Mathematically, we can formulate the problem by defining two n by n matrices: a flow matrix F whose (i,j) element represents the flow between facilities i and j, and a distance matrix D whose (i,j) element represents the distance between locations i and j. We represent an assignment by the vector p, which is a permutation of the numbers {1, 2, ... , n }. p(j) is the location to which facility j is assigned.

With these definitions, the QAP can be written as

The most effective algorithms for optimally solving quadratic assignment problems are based on Branch and Bound.

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Acknowledgments

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