Heuristic and exact algorithms for the quadratic assignment problem
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The quadratic assignment problem is a mathematical model arising from the facility location problem. As a classical combinatorial optimization problem, the quadratic assignment problem is N P-complete and includes the traveling salesman problem as a special case. Extensive research has been done on this problem for over 3 decades. However, the quadratic assignment problem still remains computationally very difficult to solve. To date, it is generally impractical to find the optimal solutions of instances of the quadratic assignment problem of sizes greater than 15. Furthermore, there is unlikely to be any polynomial-time $\epsilon$-approximate algorithm for the quadratic assignment problem.
In this thesis, the focus is on the following aspects of the quadratic assignment problem: heuristic algorithms, lower bounds, exact algorithms, test problem generation, and parallel algorithms. First, the classical lower bounding techniques are studied and it is shown that the Gilmore-Lawler bound is hard to improve. New lower bounding techniques are proposed as a generalization of the classical Gilmore-Lawler lower bound by means of optimal reduction of flow and distance matrices. A branch-and-bound exact algorithm is proposed based on the new lower bounding techniques. Then the local search algorithms and genetic algorithms for the quadratic assignment problem are studied. Neighborhood design principles are proposed and a new local search algorithm is proposed based on a new neighborhood structure. Comparison with other neighborhood structures is made both theoretically and computationally. A new genetic algorithm is proposed that incorporates a new crossover operator and is shown to be effective when combined with local hill-climbing. Due to the lack of test problems with known optimal solutions in the literature, a new class of test problem generators was proposed to generate test problems very efficiently. The proposed algorithms in the thesis were implemented in Fortran on IBM 3090 and Sun workstations and were tested on many test problems known in the literature and randomly generated problems.