A Semidefinite Programming Model for the Facility Layout Problem
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The continuous facility layout problem consists of arranging a set of facilities so that no pair overlaps and the total sum of the pairwise connection costs (proportional to the center-to-center rectilinear distance) is minimized. This thesis presents a completely mixed integer semidefinite programming (MISDP) model for the continuous facility layout problem. To begin we describe the problem in detail; discuss the conditions required for a feasible layout; and define quaternary variables. These variables are the basis of the MISDP model. We prove that the model is an exact formulation and a distinction is made between the constraints that semidefinite programming (SDP) optimization software can solve and those that must be relaxed. The latter are called exactness constraints and three possible exactness constraints are shown to be equivalent. The main contribution of this thesis is the theoretical development of a MISDP model that is based on quaternary, as oppose to binary, variables; nevertheless preliminary computational results will be presented for problems with 5 to 20 facilities. The optimal solution is found for problems with 5 and 6 facilities, confirming the validity of the model; and the potential of the model is revealed as a new upper bound is found for an 11-facility problem.