A semidefinite optimization approach for the single-row layout problem with unequal dimensions

Abstract

The facility layout problem is concerned with the arrangement of a given number of rectangular facilities so as to minimize the total cost associated with the (known or projected) interactions between them. We consider the one-dimensional space-allocation problem (ODSAP), also known as the single-row facility layout problem, which consists in finding an optimal linear placement of facilities with varying dimensions on a straight line. We construct a semidefinite programming (SDP) relaxation providing a lower bound on the optimal value of the ODSAP. To the best of our knowledge, this is the first non-trivial global lower bound for the ODSAP in the published literature. This SDP approach implicitly takes into account the natural symmetry of the problem and, unlike other algorithms in the literature, does not require the use of any explicit symmetry-breaking constraints. Furthermore, the structure of the SDP relaxation suggests a simple heuristic procedure which extracts a feasible solution to the ODSAP from the optimal matrix solution to the SDP relaxation. Computational results show that this heuristic yields a solution which is consistently within a few percentage points of the global optimal solution.

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

“…Our objective in this section is to explore in more detail than was done in [6] the relationship between R n and Π n . More specifically, we prove that the function f : R n → Π n defined by f (ρ) = (π 1 , .…”

“…One noteworthy aspect of the SDPbased approach is that it implicitly accounts for these symmetries, and thus does not require the use of additional explicit symmetry-breaking constraints. The quadratic formulation of the SRFLP proposed in [6] is obtained as follows. For a given permutation π of [n], for each pair of integers ij with 1 ≤ i < j ≤ n, define a binary ±1 variable such that…”

“…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).…”

“…This is motivated by the fact that the results in [6] for instances with up to 80 facilities were obtained using the spectral bundle solver SB [16,17], and while this solver is able to handle very large SDP problems, a major drawback is that its convergence slows down significantly after several hours of computation. As a consequence, instances with 80 facilities were the largest for which bounds could be obtained in reasonable time in [6], and some of the optimality gaps were greater than 10%. Our contribution is the use of a new matrix-based formulation of the SRFLP that yields an SDP relaxation that can be solved using the interior-point method solver CSDP.…”

“…In Section 2, we prove the validity of the bijection between the feasible set of the quadratic formulation and the set of all permutations. In Section 3, we introduce the new matrix-based formulation of the SRFLP, and prove its validity by showing it is equivalent to the SDP-based formulation in [6]. In Section 4, we report preliminary computational results showing that using the SDP relaxation obtained from the new formulation, a primal-dual interior-point method and an acceptable amount of computational effort, we consistently obtain global lower bounds and feasible layouts with a global optimality gap of 5% or less for instances with up to 100 departments.…”