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...

The design of the distribution system is a strategic issue for almost every company. The problem of locating facilities and allocating customers covers the core topics of distribution system design. Model formulations and solution algorithms which address the issue vary widely in terms of fundamental assumptions, mathematical complexity and computational performance. This paper reviews some of the contributions to the current state-of-the-art. In particular, continuous location models, network location models, mixed-integer programming models, and applications are summarized.

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. This research was supported by NSF Grants DMI-9634750 and DMI-9812915. This support is gratefully acknowledged.

Increased uncertainty is one of the characteristics of product recovery networks. In particular the strategic design of their logistic infrastructure has to take uncertain information into account. In this paper we present stochastic programming based approaches by which a deterministic location model for product recovery network design may be extended to explicitly account for the uncertainties.

The stochastic location-inventory network design model with risk pooling considers the distribution network design problem in which all the retailers face uncertain demand. The riskpooling benefits can be achieved by allowing some of the retailers to operate as distribution centers (DCs) for other retailers to commit a given service level. The target is to minimize the expected total cost of DC location, DC-retailer assignment, transportation, and inventory. We formulate it as a two-stage nonlinear discrete optimization problem. The first stage is to decide which DCs to open, and the second stage deals with DC-retailer assignment and inventory decisions. Snyder et al. (2003) described a similar problem, and solved it using a Lagrangianrelaxation-based algorithm. However, their approach requires the demand for all retailers in each scenario has a variance identically proportional to the mean demand. Here we remove this requirement. We formulate the problem by using a set-covering model, and propose primal-dual O(n2 log n) algorithms to address the nonlinear discrete pricing problem in a column generation framework. With the variable fixing technique, we are able to efficiently solve problems of moderate-size (up to one hundred retailers and nine scenarios). Our solution technique exploits only the concavity property of risk-pooling cost structure and hence can be used for a wide range of other concave cost minimization problems. Moreover, our framework is also applicable to the problems with multiple commodities or multiple periods. 1

This paper presents a generic stochastic model for the design of networks comprising both supply and return channels, organized in a closed loop system. Such situations are typical for manufacturing/re-manufacturing type of systems in reverse logistics.

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