Redesigning Benders Decomposition for Large-Scale Facility Location

Abstract

T he uncapacitated facility location (UFL) problem is one of the most famous and most studied problems in the operations research literature. Given a set of potential facility locations and a set of customers, the goal is to find a subset of facility locations to open and to allocate each customer to open facilities so that the facility opening plus customer allocation costs are minimized. In our setting, for each customer the allocation cost is assumed to be a linear or separable convex quadratic function. Motivated by recent UFL applications in business analytics, we revise approaches that work on a projected decision space and hence are intrinsically more scalable for large-scale input data. Our working hypothesis is that many of the exact (decomposition) approaches that were proposed decades ago and discarded soon after need to be redesigned to take advantage of the new hardware and software technologies. To this end, we "thin out" the classical models from the literature and use (generalized) Benders cuts to replace a huge number of allocation variables by a small number of continuous variables that model the customer allocation cost directly. Our results show that Benders decomposition allows for a significant boost in the performance of a mixed-integer programming solver. We report the optimal solution of a large set of previously unsolved benchmark instances widely used in the available literature. In particular, dramatic speedups are achieved for UFL problems with separable quadratic allocation costs, which turn out to be much easier than their linear counterparts when our approach is used.

A node-based layered graph approach for the Steiner tree problem with revenues, budget and hop-constraints

“…In this paper, we proposed a new extended formulation based on a layered graph for hop-constrained spanning/Steiner tree problems. Our formulation follows a "thinning out" idea proposed in [10,11]: instead of using variables associated with arcs of the layered graph, our new model projects them out and relies only on variables associated to the nodes of the layered graph. Thus, the resulting MIP formulation is considerably smaller than the ones considered in previous literature, which allowed us to tackle instances based on larger graphs and/or hop-limits.…”

“…However, we show that better bounds can be obtained by imposing an exponential number of subtour elimination constraints on G. Our model provides a good compromise between quality of obtained LP-bounds and the size of the underlying model. This approach of "thinning out" MIP models has been recently exploited in [10,11] for solving Steiner trees and facility location problems, respectively. Note, however, that except from the high-level idea of deriving sparser MIP models to deal with large-scale instances, there are no direct similarities between the model presented in this paper and those studied in [10,11].…”

Unit Commitment With Gas Network Awareness

“…The extreme points and rays of these subproblems can be used to find the (feasibility and optimality) Benders cuts of Problem (11). The solution method also uses the acceleration schemes from [25], [26] which normalize the rayŝ y and perturbẑ p . The solution method also obtains feasible solutions periodically (e.g., every 30 iterations) heuristically by turning off violated generators.…”

A Redesigned Benders Decomposition Approach for Large-Scale In-Transit Freight Consolidation Operations

“…Both papers establish that the in-transit freight consolidation problem is NP complete and hence researchers have been focusing on developing heuristic approaches to scale up as well as Our model builds on the model proposed in Croxton et al (2001) along with the redesigned Benders decomposition approach proposed in Fischetti et al (2016). We include two linear cost structures that correspond to shipment from shipper to the consolidation point and from the consolidation point to the customer respectively and a time constraint on each shipment in addition to the constraints accounted for by Croxton et al (2001).…”