Collaborative transportation can significantly reduce transportation costs as well as greenhouse gas emissions. However, allocating the cost to the collaborating companies remains difficult. We consider the cost-allocation problem, which arises when companies, each with multiple delivery locations, collaborate by consolidating demand and combining delivery routes. We model the corresponding cost-allocation problem as a cooperative game: the joint network vehicle routing game (JNVRG). We propose a row generation algorithm to either determine a core allocation for the JNVRG or show that no such allocation exists. In this approach, we encounter a row generation subproblem, which we model as a new variant of a vehicle routing problem with profits. Moreover, we propose two main acceleration strategies for the row generation algorithm. First, we generate rows by relaxing the row generation subproblem, exploiting the tight linear programming (LP) bounds for our formulation of the row generation subproblem. Secondly, we propose to also solve the row generation subproblem heuristically and to only solve it to optimality when the heuristic fails. We demonstrate the effectiveness of the proposed row generation algorithm and the acceleration strategies by means of numerical experiments for both the JNVRG as well as the traditional vehicle routing game, which is a special case of the JNVRG. We create instances based on benchmark instances of the capacitated vehicle routing problem from the literature. We are able to either determine a core allocation or show that no core allocation exists, for instances ranging from 5 companies with a total of 79 delivery locations to 53 companies with a total of 53 delivery locations.

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Transportation Science
Department of Econometrics