Pricing strategies for the site-dependent vehicle routing problem

The vehicle pricing game (VPG) which addresses the vehicles’ viewpoints within a vehicle routing problem (VRP) is introduced. Each vehicle acts as a player who demands a price per kilometer. That is, vehicles represent decentralized actors in transport systems, e.g., carriers under subcontracts. Based on these prices, a VRP is solved and profits are generated. Which price should a vehicle choose to maximize its own profit, considering the competition among vehicles? To answer this question, site dependencies leading to inhomogeneous vehicles are included. More detailed, skill-levels, e.g., relating to the size of a vehicle, are used to indicate a vehicle’s ability to carry out particular services. Moreover, penalty options are added. The VPG serves as an element of a vertical collaboration in a transport scenario and thus provides decision support for cooperative models. Theoretical results for the VPG are provided for a particular case of a two-player ring network game, for which the full set of equilibria is described and their uniqueness is discussed. It is shown that the uniqueness of the higher-skilled vehicle’s payoff is guaranteed even for multiple equilibria. The competition ratio is defined; it restricts a vehicle’s price to keep its competitiveness. Moreover, the acceptance ratio gives a lower bound on prices such that a loss of market share is still accepted. Experimental results are provided for general networks including the analysis of penalty options. It is demonstrated that strict site dependencies by tendency lead to monopolistic structures. In addition, particular penalty types show a positive effect regarding load imbalances caused by universally skilled vehicles.