A joint location-inventory model

We consider a joint location-inventory problem involving a single supplier and multiple retailers. Associated with each retailer is some variable demand. Due to this variability, some amount of safety stock must be maintained to achieve suitable service levels. However, risk-pooling benefits may be achieved by allowing some retailers to serve as distribution centers (and therefore inventory storage locations) for other retailers. The problem is to determine which retailers should serve as distribution centers and how to allocate the other retailers to the distribution centers. We formulate this problem as a nonlinear integer-programming model. We then restructure this model into a set-covering integer-programming model. The pricing problem that must be solved as part of the column generation algorithm for the set-covering model involves a nonlinear term in the retailer-distribution-center allocation terms. We show that this pricing problem can (theoretically) be solved efficiently, in general, and we show how to, solve it practically in two important cases. We present computational results on several instances of sizes ranging from 33 to 150 retailers. In all cases, the lower bound from the linear-programming relaxation to the set-covering. model gives the optimal solution.