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Quadratic assignment problem
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<p>The quadratic assignment problem (QAP) is one of the fundamental <a href="/facts/Combinatorial_optimization/PTcloS79">combinatorial optimization</a> problems in the branch of <a href="/facts/Optimization_(mathematics)/oRn8Iv5I">optimization</a> or <a href="/facts/Operations_research/H5yJkV5n">operations research</a> in <a href="/facts/Mathematics/pxTouaz4">mathematics</a>, from the category of the <a href="/facts/Optimal_facility_location/ssbU6ww2">facilities location</a> problems first introduced by Koopmans and Beckmann. </p><p>The problem models the following real-life problem: </p> There are a set of <i>n</i> facilities and a set of <i>n</i> locations. For each pair of locations, a <i>distance</i> is specified and for each pair of facilities a <i>weight</i> or <i>flow</i> is specified (e.g., the amount of supplies transported between the two facilities). The problem is to assign all facilities to different locations with the goal of minimizing the sum of the distances multiplied by the corresponding flows. <p>Intuitively, the cost function encourages facilities with high flows between each other to be placed close together. </p><p>The problem statement resembles that of the <a href="/facts/Assignment_problem/dIMpM0aW">assignment problem</a>, except that the <a href="/facts/Loss_function/xv5ozuhl">cost function</a> is expressed in terms of quadratic inequalities, hence the name. </p>