Mathematical Programming with Stochastic Equilibrium Constraints applied to Optimal Last-mile Delivery Services

In e-commerce business, the delivery of products is a crucial part for the success of an e-shop. An efficient delivery system should offer various services and predict customers behaviour. The latter are influenced by the price of a delivery service, but also by its quality (perceived through congestion effect induced by customers' choices). In this study, we introduce a bi-level model to optimize a delivery system. At the upper level, the provider control services' tariffs. At the lower level, users react by choosing their delivery service according to a disutility function which incorporates the provider tariff and the congestion effects. We model the customers' reaction using stochastic user equilibrium (SUE). We also present a sensitivity analysis for the SUE that gives explicit expression of the derivatives of customers distribution with respect to services' tariffs. Based on a local search that exploits the derivatives information, a new heuristic algorithm for a delivery services pricing problem is developed and compared to others existing methods.