The Stochastic Location-Inventory Network Design Model with Risk Pooling

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(n 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.