Cross-evaluation values and their applications to profit or cost allocation in collaborative logistics

Collaborative logistics, realized by collaboration among shippers or carriers, can significantly reduce logistics costs in urban distribution and intercity transportation. One important issue for such collaboration is cost or profit allocation among its participants, which is usually addressed in the framework of cooperative games. In this paper, a new family of values called cross-evaluation values are introduced for cooperative games with nonempty core, where each value allocates an amount of payoff to each player according to the average of its contributions to the grand coalition evaluated by itself and all other players. These values are fair in the sense that all players apply the same rule to their self evaluation and the same rule to their peer evaluations. They have important properties such as efficiency, equal treatment of equals, null player, and weak coalitional monotonicity. We prove that the Alexia value is a cross-evaluation value for all balanced games and the Shapley value is a cross-evaluation value for all convex games. Moreover, by adopting marginalism attitude and egalitarianism attitude for peer evaluation respectively, we propose two new cross-evaluation values and apply them to a cost game in shipper collaboration and a profit game in carrier collaboration. Numerical experiments and a case study demonstrate advantages of the two new allocation methods over existing methods and provide managerial insights for the choice of an allocation method for such games.