An integrated multi-depot location- inventory-routing problem for logistics distribution system planning of a chain enterprise

We consider an integrated location-inventory-routing problem where the decision marker needs to realize the integrated decisions of facility location, inventory control, and vehicle routing for a logistics distribution system owned to a chain enterprise. This distribution system consists of a center warehouse, multiple potential distribution centers (DCs) and multiple retail stores. Retailer stores in the system face random demand, and receive distribution service from DCs in the same distribution period. The objective is to minimize the total system operation cost that includes facility location cost, inventory cost and transportation cost. We show that this problem can be formulated as a nonlinear mixed integrated programming model, for which we proposed a nested lagrangian relaxation-based solution algorithm. An outer Lagrangian relaxation embedded in subgradient optimization decompose the parent problem into two sub-problems. The first sub-problem is a distribution network design with risk pooling effect problem, along with a solution approach based on Lagrangian relaxation. The second one resembles a capacitated and degree constrained minimum spanning forest problem, which is tackled with an augmented lagrangian relation. We present computational results for several instances of the problem with sizes ranging from 50 to 150 stores. The results indicate that this method performs well in terms of the solution quality and run time consumed.