A Profit-Maximization Location-Routing-Pricing Problem: A Branch-and-Price Algorithm

Abstract

This paper for the first time considers a profit-maximization location-routing problem with price-sensitive demands. The problem determines the location of facilities, the allocation of vehicles and customers to established facilities, and the pricing and routing decisions in order to maximize the total profit of serving customers. A mixed-integer linear programming model is presented, which can only be used to solve small-size instances with commercial optimization solvers. Then, the model is reformulated as a set-packing model and solved by an efficient branch-and-price algorithm for large-size instances. The proposed algorithm can also be used to solve the more basic problems such as location-routing with profit and price-inelastic demands or vehicle routing with profit and price-sensitive demands, which has not been considered by any research earlier. The column-generation procedure is developed based on a new variant of the elementary shortest path problem with resource constraints where demands are price dependent. Our numerical study indicates the substantial advantage of the integrated model. The proposed model can be used to design the distribution networks of online shopping systems in which delivered pricing is influenced by the last mile delivery.