PRODUCTION AND OPERATIONS MANAGEMENT …

1 Facility Location: A Robust Optimization ApproachOpher Baron, Joseph MilnerRotman School of MANAGEMENT , University of Toronto, Toronto, Ontario M5S 3E6, NaseraldinDepartment of Industrial Engineering and MANAGEMENT , ORT Braude College of Engineering, Karmiel 21982, Israel, this research, we apply robust optimization (RO) to the problem of locating facilities in a network facing uncertaindemand over multiple periods. We consider a multi-period fixed-charge network location problem for which we find (1)the number of facilities, their location and capacities, (2) the PRODUCTION in each period, and (3) allocation of demand tofacilities.

2 Using the RO approach we formulate the problem to include alternate levels of uncertainty over the periods. Weconsider two models of demand uncertainty: demand within a bounded and symmetric multi-dimensional box, and demandwithin a multi-dimensional ellipsoid. We evaluate the potential benefits of applying the RO approach in our setting using anextensive numerical study. We show that the alternate models of uncertainty lead to very different solution networktopologies, with the model with box uncertainty set opening fewer, larger facilities. Through sample path testing, we showthat both the box and ellipsoidal uncertainty cases can provide small but significant improvements over the solution to theproblem when demand is deterministic and set at its nominal value.

3 For changes in several environmental parameters, weexplore the effects on the solution words:facility location; robust optimization; uncertainty; robust counterpartHistory: Received: November 2008; Accepted: June 2010 by Panos Kouvelis, after 1 IntroductionFacility location problems are often strategic in natureand entail long-term decisions, exposing firms tomany uncertainties during the operational lifetime ofa facility. When solving such problems, a firm mayhave to determine the number of facilities to open,their locations, and their capacities. Because of thehigh fixed costs incurred in changing a network offacilities, a firm may be limited in the frequencyin which it reexamines these strategic decisions.

4 Afterdetermining its facility network, the firm is oftenrelegated to making operational decisions such as de-termining PRODUCTION quantities, service levels, andallocation of supply to demand. Thus, facility locationproblems often have a two-stage approach to solvingthem, location followed by evaluation. They are thusamenable to solution using various techniques ofdecision making under are alternate approaches to solve such prob-lems. In general, as discussed below in our literaturereview, one may either assume some stochastic infor-mation about the possible outcomes or assume thatuncertain information lies within some mathematicalstructure with no distributional information about the former case, one would then optimize someexpectation of the cost or profit of a system.

5 In thelatter case, one often solves for a solution that isrobust to the uncertainty in the problem take this paper considers a facility location problem us-ing a robust optimization (RO) modeling approachwhere demand is uncertain and no probability distri-bution is assumed. We consider the problem where afirm must establish fixed facilities at a charge inan initial period. Further, with a linear cost per unit,the firm must establish the maximum productioncapacity for each facility. In subsequent periods, thefirm observes demand from several nodes and mustdetermine the allocation of demand and PRODUCTION atopen facilities.

6 We allow for PRODUCTION costs at thefacilities and delivery charges (assumed based on dis-tance to the demand nodes). The PRODUCTION of eachfacility in each period is bounded by the maximumproduction capacity initially established. As a result,we allow for lost tackle this problem, we apply the RO approachthat differs from previous work on robust facility loca-tion as discussed in the literature review. This approachwas independently developed by El-Ghaoui and Lebert(1997) and Ben-Tal and Nemirovsky (1998). The key is-sue in RO is how to model the uncertainty. Typically,the uncertainty is confined to a specific mathematicalstructure in order to ensure model tractability.

7 The ROapproach has been applied in recent years in opera-tional, engineering, and financial contexts, , pricing,772 PRODUCTION AND OPERATIONS MANAGEMENTVol. 20, No. 5, September October 2011, pp. 772 785 ISSN1059-1478|EISSN1937-5956|11|2005| PRODUCTION and OPERATIONS MANAGEMENT Societyinventory, truss-design, portfolio selection, and sched-uling. We give a brief tutorial on RO in section paper contributes to the literature by formu-lating robust models for a facility location problem,and by providing insights into the solution consider a profit-maximizing objective where thefirm must trade off establishing sufficient capacitywith uncertain (and possibly low) revenue in thefuture.

8 The mathematical program developed findssolutions that simultaneously address these alterna-tives. We present a tractable formulation that can betested in a consistent manner against an approachwhere uncertainty is ignored. We study the problemthrough extensive numerical testing and provide aroad map to understanding how RO improves systemperformance. In these tests we consider a semi-realisticproblem with 15 nodes and up to 20 periods of de-mand. While realistic problems may be larger than this,alternate approaches, such as stochastic programming,cannot efficiently solve even this limited-size indicate environmental parameters that have great-er influence on the performance.

9 To the best ofour knowledge, this work is the first to apply the ROapproach in the context of facility location paper is organized as follows: In section 2, wereview some of the relevant literature and present theprinciples of RO. In section 3, we describe the setting,model the problem, and discuss our assumptions. Insection 4, we compare the performance of alternateRO models and that of the nominal model in order todemonstrate the expected potential benefits of apply-ing the RO approach to facility location. Finally, insection 5, we highlight the insights gained in thisstudy and propose several future research Literature ReviewIn this section, we briefly review some of the relevantstudies in two streams of research that are related toour problem: facility location and RO.

10 The problem offacility location is an important strategic one andmany models have been developed to approach it. Fortexts on the subject, we refer the reader to Mirchan-dani and Francis (1990), Daskin (1995), and Dreznerand Hamacher (2002). In this paper, we consider avariant of the capacitated fixed-charge multi-periodfacility location problem where the PRODUCTION capac-ity of each facility must be determined beforeobserving demand during the horizon. That is, weconsider the problem under is a considerable body of literature on facilitylocation under uncertainty. The recent review articleby Snyder (2007) describes two types of problems:stochastic location problems and robust location prob-lems.