Price-directed replenishment of subsets: methodology and its application to inventory routing

The idea of price-directed control is to use an operating policy that exploits optimal dual prices from a mathematical programming relaxation of the underlying control problem. We apply it to the problem of replenishing inventory to subsets of products/locations, such as in the distribution of industrial gases, so as to minimize long-run time average replenishment costs. Given a marginal value for each product/location, whenever there is a stockout the dispatcher compares the total value of each feasible replenishment with its cost, and chooses one that maximizes the surplus. We derive this operating policy using a linear functional approximation to the optimal value of a semi-Markov decision process on continuous spaces. This approximation also leads to a math program whose optimal dual prices yield values and whose optimal objective value gives a lower bound on system performance. We use duality theory to show that optimal prices satisfy several structural properties and can be interpreted as estimates of lowest achievable marginal costs. On real-world instances, the price-directed policy achieves superior, near optimal performance as compared with other approaches.