A Price-Directed Approach to Stochastic Inventory/Routing

We consider a new approach to stochastic inventory/routing that approximates the future costs of current actions using optimal dual prices of a linear program. We obtain two such linear programs by formulating the control problem as a Markov decision process and then replacing the optimal value function with the sum of single-customer inventory value functions. The resulting approximation yields statewise lower bounds on optimal infinite-horizon discounted costs. We present a linear program that takes into account inventory dynamics and economics in allocating transportation costs for stochastic inventory routing. On test instances we find that these allocations do not introduce any error in the value function approximations relative to the best approximations that can be achieved without them. Also, unlike other approaches, we do not restrict the set of allowable vehicle itineraries in any way. Instead, we develop an efficient algorithm to both generate and eliminate itineraries during solution of the linear programs and control policy. In simulation experiments, the price-directed policy outperforms other policies from the literature.