Designing the distribution network for an integrated supply chain
Journal of Industrial and Management Optimization
We consider an integrated distribution network design problem in which all the retailers face uncertain demand. The risk-pooling benefit is achieved by allowing some of the retailers to operate as distribution centers (DCs) with commitment in service level. The target is to minimize the expected total cost resulted from the DC location, transportation, and inventory. We formulate it as a two-stage nonlinear discrete stochastic optimization problem. The first stage decides which retailers to be
... ch retailers to be selected as DCs and the second stage deals with the costs of DCretailer assignment, transportation, and inventory. In the literature, the similar models require the demands of all retailers in each scenario to have their variances identically proportional to their means. In this paper, we remove this restriction. We formulate the problem by using a setcovering model, and solve the problem by a column generation approach. With a variable fixing technique, we are able to efficiently solve problems of moderate-size (up to one hundred retailers and nine scenarios). The solution technique exploits only the concavity of the risk-pooling cost structure and can therefore be used in solving more general problems. • which retailers to be assigned to a given DC in each scenario; • how much safety stock to maintain at a given DC so as to protect against the uncertainty in the demands in each scenario; • how much working inventory to keep at a given DC so as to satisfy the demands of the served retailers in each scenario. These requirements are often encountered in practice, e.g., in the ALKO Inc. case from a widely used textbook on supply chain management by Chopra and Meindl (2001) . Daskin et al. (2002) also mentioned the possible application of this type of models in e-commerce. Shen (2000) , Shen et al. (2003) and Daskin et al. (2002) discuss a single-echelon supply chain network design problem. They use Lagrangian-relaxation and column generation algorithms to handle a nonlinear integer optimization problem. They show that their model can be solved efficiently when the demand at each DC is Poisson or deterministic. Shu, Teo and Shen (2005) and Shu (2003) extended the model to arbitrary demand. The papers of Teo and Shu (2004) and Romeijn, Shu and Teo (2005) considered infinite horizon multi-echelon network design problems in both deterministic and stochastic demand settings. However, to the best of our knowledge, very few research works in this area jointly consider location, inventory, and transportation costs. Another important feature of our model is to properly incorporate the uncertainties into the choice of the DCs. In the literature of facility location, it is a common practice to consider the possible change in the parameters by a scenario based approach. The pioneering work includes Sheppard (1974), Berman and Krass (2001), and Owen and Daskin (1998), which developed into the study of the stochastic facility location problem. The main concern of those works are the location cost and the distribution cost. As a result, the inventory related cost are often ignored or simplified. Snyder et al. (2005) provided a stochastic version for the location model with risk pooling by Shen (2000) , Shen et al. (2003) and Daskin et al. (2002) . A Lagrangian-relaxation based algorithm is adopted and a variable fixing approach is used to reduce the problem size. Similar to the deterministic version, however, there is a rather restrictive assumption on the demand pattern, i.e., the demand should either be deterministic or have a variance identically proportional to its mean for all the retailers. This paper removes the restrictive demand pattern requirement of Snyder et al. (2005) . We reformulate the problem as a set-covering model, solve the pricing problem in O(n 2 log n) time, and speed up the column generation process using the variable fixing routines.