Lawrence V. Snyder

Facility Location Under Uncertainty: A Review

Plants, distribution centers, and other facilities generally function for years or decades, during which time the environment in which they operate may change substantially. Costs, demands, travel times, and other inputs to classical facility location models may be highly uncertain. This has made the development of models for facility location under uncertainty a high priority for researchers in both the logistics and stochastic/robust optimization communities. Indeed, a large number of the approaches that have been proposed for optimization under uncertainty have been applied to facility location problems. This paper reviews the literature on stochastic and robust facility location models. Our intent is to illustrate both the rich variety of approaches for optimization under uncertainty that have appeared in the literature and their application to facility location problems. In a few instances for which examples in facility location are not available, we provide examples from the more general logistics literature.

The stochastic location model with risk pooling

Abstract In this paper, we present a stochastic version of the location model with risk pooling (LMRP) that optimizes location, inventory, and allocation decisions under random parameters described by discrete scenarios. The goal of our model (called the stochastic LMRP, or SLMRP) is to find solutions that minimize the expected total cost (including location, transportation, and inventory costs) of the system across all scenarios. The location model explicitly handles the economies of scale and risk-pooling effects that result from consolidating inventory sites. The SLMRP framework can also be used to solve multi-commodity and multi-period problems. We present a Lagrangian-relaxation–based exact algorithm for the SLMRP. The Lagrangian subproblem is a non-linear integer program, but it can be solved by a low-order polynomial algorithm. We discuss simple variable-fixing routines that can drastically reduce the size of the problem. We present quantitative and qualitative computational results on problems with up to 150 nodes and 9 scenarios, describing both algorithm performance and solution behavior as key parameters change.

A random-key genetic algorithm for the generalized traveling salesman problem

The generalized traveling salesman problem is a variation of the well-known traveling salesman problem in which the set of nodes is divided into clusters; the objective is to find a minimum-cost tour passing through one node from each cluster. We present an effective heuristic for this problem. The method combines a genetic algorithm (GA) with a local tour improvement heuristic. Solutions are encoded using random keys, which circumvent the feasibility problems encountered when using traditional GA encodings. On a set of 41 standard test problems with symmetric distances and up to 442 nodes, the heuristic found solutions that were optimal in most cases and were within 1% of optimality in all but the largest problems, with computation times generally within 10 seconds. The heuristic is competitive with other heuristics published to date in both solution quality and computation time.

OR/MS Models for Supply Chain Disruptions: A Review

ABSTRACT We review the Operations Research/Management Science (OR/MS) literature on supply chain disruptions in order to take stock of the research to date and to provide an overview of the research questions that have been addressed. We first place disruptions in the context of other forms of supply uncertainty and discuss common modeling approaches. We then discuss 180 scholarly works on the topic, organized into six categories: evaluating supply disruptions; strategic decisions; sourcing decisions; contracts and incentives; inventory; and facility location. We conclude with a discussion of future research directions.

Facility Location in Supply Chain Design

In this chapter we outline the importance of facility location decisions in supply chain design. We begin with a review of classical models including the traditional fixed charge facility location problem. We then summarize more recent research aimed at expanding the context of facility location decisions to incorporate additional features of a supply chain including LTL vehicle routing, inventory management, robustness, and reliability.

Stochastic p-robust location problems

The two most widely considered measures for optimization under uncertainty are minimizing expected cost and minimizing worst-case cost or regret. In this paper, we present a novel robustness measure that combines the two objectives by minimizing the expected cost while bounding the relative regret in each scenario. In particular, the models seek the minimum-expected-cost solution that is p-robust; i.e., whose relative regret is no more than 100p% in each scenario. We present p-robust models based on two classical facility location problems. We solve both problems using variable splitting, with the Lagrangian subproblem reducing to the multiple-choice knapsack problem. For many instances of the problems, finding a feasible solution, and even determining whether the instance is feasible, is difficult; we discuss a mechanism for determining infeasibility. We also show how the algorithm can be used as a heuristic to solve minimax regret versions of the location problems.

The Effect of Supply Disruptions on Supply Chain Design Decisions

We study an integrated supply chain design problem that determines the locations of retailers and the assignments of customers to retailers to minimize the expected costs of location, transportation, and inventory. The system is subject to random supply disruptions that may occur at either the supplier or the retailers. Analytical and numerical studies reveal the effects of these disruptions on retailer locations and customer allocations. In addition, we demonstrate numerically that the cost savings from considering supply disruptions at the supply chain design phase (rather than at the tactical or operational phase) are usually significant.

Infinite-Horizon Models for Inventory Control under Yield Uncertainty and Disruptions

We consider a firm facing supply chain risk in two forms: disruptions and yield uncertainty. We demonstrate the importance of analyzing a sufficiently long time horizon when modeling inventory systems subject to supply disruptions. Several previous papers have used single-period newsboy-style models to study supply disruptions, and we show that such models underestimate the risk of supply disruptions and generate sub-optimal solutions. We consider one case where a firms only sourcing option is an unreliable supplier subject to disruptions and yield uncertainty, and a second case where a second, reliable (but more expensive) supplier is available. We develop models for both cases to determine the optimal order and reserve quantities. We then compare these results to those found when a single-period approximation is used. We demonstrate that a single-period approximation causes increases in cost, under-utilizes the unreliable supplier, and distorts the order quantities that should be placed with the reliable supplier in the two-supplier case. Moreover, using a single-period model can lead to selecting the wrong strategy for mitigating supply risk.

Supply disruptions with time-dependent parameters

We consider a firm that faces random demand and receives shipments from a single supplier who faces random supply. The suppliers availability may be affected by events such as storms, strikes, machine breakdowns, and congestion due to orders from its other customers. In our model, we consider a dynamic environment: the probability of disruption, as well as the demand intensity, can be time dependent. We model this problem as a two-dimensional non-homogeneous continuous-time Markov chain (CTMC), which we solve numerically to obtain the total cost under various ordering policies. We propose several such policies, some of which are time dependent while others are not. The key question we address is: How much improvement in cost is gained by using time-varying ordering policies rather than stationary ones? We compare the proposed policies under various cost, demand, and disruption parameters in an extensive numerical study. In addition, motivated by the fact that disruptions are low-probability events whose non-stationary probabilities may be difficult to estimate, we investigate the robustness of the time-dependent policies to errors in the supply parameters. We also briefly investigate sensitivity to the repair-duration distribution. We find that non-stationary policies can provide an effective balance of optimality (low cost) and robustness (low sensitivity to errors).

Models for Reliable Supply Chain Network Design

Recent examples of disruptions in the news suggest a strong geographical dimension to supply chain disruptions, and to their effects. For example: The west-coast port lockout in 2002 strangled U.S. retailers’ supply lines while east-coast ports were essentially unaffected (Greenhouse 2002) The foot-and-mouth disease scare in the U.K. in 2001 caused the U.S. to ban imports of British meat (Marquis and McNeil 2001). The suspension of the license of the Chiron plant in Liverpool, England reduced the U.S. supply of the influenza vaccine by nearly 50% during the 2004/5 flu season (Pollack 2004). In the U.S. Gulf Coast region in 2005, Hurricane Katrina idled facilities situated at all levels of the supply chain, including production (e.g., coffee; Barrionuevo and Deutsch 2005), processing (oil refining; Mouawad 2005), warehousing (lumber storage; Reuters 2005), transit (banana imports; Barrionuevo and Deutsch 2005), and retail (groceries and home-repair; Fox 2005, Leonard 2005). These facilities were located in or near New Orleans but were integral parts of global supply chains. These examples highlight the need for supply chain design models that account for the spatial nature of both supply chains and their operation.