Shapley value-based cost allocation in the cooperative traveling salesman problem under rolling horizon planning

This paper examines cooperation among salesmen from a game-theoretic perspective for the case of the traveling salesman problem with rolling horizon. The rolling horizon planning is modeled using stochastic demands. The goal is to determine a cost allocation for this problem using the Shapley value. The game-theoretic formulation of the problem is provided and structural properties are investigated. To compute the value of the characteristic function of this game, an optimization problem formulated as a stochastic dynamic program is introduced. Due to high computational effort of stochastic dynamic programming and calculation of the Shapley value, a tailor-made solution procedure is developed. This procedure combines methods of approximate dynamic programming to find an approximate value of the characteristic function and an estimation algorithm for the Shapley value. The developed computational study tests the computational performance of the solution procedure.