New facets for the two-stage uncapacitated facility location polytope

Abstract

Location, Two-stage, Set packing, Facet, Combinatorial optimization,

When centers can fail: A close second opportunity

“…Two-stage location problems (see, e.g., [21]) and hub location problems [8,9], among others, have been formulated by means of binary variables associated with the possible routes from each origin through the plants/hubs to the final destination. In our case, a route is given by three elements: site i, reference center j and backup center k. Then, for our first formulation we introduce binary variables y j : to indicate whether there is a center in site j A A, x ijk : to indicate whether the reference center of i is j and the backup center of i (closest facility to j that is not j) is k only defined for i; j; k A A with ja k, i a k and d ij r d ik ), and the auxiliary continuous variable z, to account for the largest total assignment cost.…”

“…Each such constraint forces a z-variable to take value 1 if a center is established at site j and there is no pair (a, b) of centers with a not further from i than j, a closer to i than b and giving a total cost δ iab strictly smaller than Δ ℓ . Finally, constraints (21) and (22) force variables to take consistent values. As in previous formulations we next collect some useful observations:…”

“…One can derive from [31] that the set of binary solutions to P j y j ¼ p and y jk ¼ y j y k , k 4 j coincides with the set of binary solutions to (21) and (22), y jk Z y j þ y k À 1, k 4 j and X…”

“…Note that variables y jk with j ak only appear in one term of constraints (32) that is required to cope correctly with situations where different centers can be set at the same distance from a given site i, and in the linking constraints (21) and (22) required to force them to take values consistent with those of variables y j . Thus, if there are no ties in the distances from any site i to the others, this formulation can be sensibly reduced.…”

Semi-Lagrangian relaxation applied to the uncapacitated facility location problem

“…Some of the most successful methods are: Tabu search in [16,33], a combination of Tabu search, Genetic Algorithm and Greedy Randomized Adaptative Search in [26] and Hybrid Multistart heuristic in [32]. Other contributions have investigated enhanced versions of the UFL problem, as the two-stage UFL [28] or have investigated new computational techniques applied to the UFL problem, as parallel interior point methods [12].…”