Computing Globally Optimal Solutions for Single-Row Layout Problems Using Semidefinite Programming and Cutting Planes

Abstract

This paper is concerned with the single-row facility layout problem (SRFLP). A globally optimal solution to the SRFLP is a linear placement of rectangular facilities with varying lengths that achieves the minimum total cost associated with the (known or projected) interactions between them. We demonstrate that the combination of a semidefinite programming relaxation with cutting planes is able to compute globally optimal layouts for large SRFLPs with up to thirty departments. In particular, we report the globally optimal solutions for two sets of SRFLPs previously studied in the literature, some of which have remained unsolved since 1988.

Provably near-optimal solutions for very large single-row facility layout problems

“…[13]. This SDP relaxation, used with a simple scheme to add violated triangle inequalities as cutting planes, was used in [7] to solve SRFLPs with up to 30 facilities to global optimality. Its main limitation from a computational perspective is that it has O(n 3 ) linear constraints; this limits the size of instances that can be tackled with it.…”

“…(X ij,jk − X ij,ik − X ik,jk ) = −(n − 2) for all pairs i < j Suppose X is feasible for (7). Then the constraints diag (X) = e and rank (X) = 1 together imply that X ij,k = ±1 for all entries of X.…”

“…Significant progress was achieved recently through the first ever polyhedral study of the SRFLP carried out by Amaral and Letchford [4]. Anjos and Vannelli [7] considered the semidefinite programming relaxation in [6] and tightened it using triangle inequalities as cutting planes. Their method routinely found optimal solutions for SRFLP instances with up to 30 facilities (although requiring several dozen hours on a 2.0GHz Dual Opteron with 16Gb of RAM).…”

“…In this study they were able to provide optimal solutions for SRFLPs that had remained unsolved since 1988. At the time of writing, the current state-of-the-art in terms of globally optimal methods is the cutting-plane algorithm of Amaral [2] which found optimal solutions for all SRFLP instances tested in [7] but which performed much faster than the method in [7]. Optimal solutions for new instances with up to 35 facilities were also computed by Amaral [2].…”

Semidefinite Programming and Constraint Programming

“…There exists a vast literature on explicit symmetry breaking in constraint programming and integer programming, see Gent et al [2006] and Margot [2010] for recent surveys. In contrast, certain types of symmetry breaking are implicitly accounted for in semidefinite relaxations; see for example Anjos and Vannelli [2008]. These complementary approaches could perhaps be combined very effectively.…”