Given a set of players and the cost of each possible coalition, the question we address is which coalitions should be formed. We formulate mixed integer linear programming models for this problem, considering core stability and strong equilibrium. The objective function looks for minimizing the total cost allocated among the players. Concerned about the difficulties of managing large coalitions in practice, we also study the effect of a maximum cardinality constraint per coalition. We test the models in two applications. One is in collaborative forest transportation and the other one in inventory of spare parts for oil operations. In these situations, collaboration opportunities involving significant savings exist, but for several reasons, it may be better to group the players in different sub-coalitions rather than in the grand coalition. The models we propose are thus relevant for deciding how to partition the set of players. We also prove that if the strong equilibrium model is feasible, its optimal cost is equal to the optimal cost of the core stability model and, consequently, a coalition structure that solves one problem also solves the other problem. We present results that illustrate this property. We also present results where the core stability problem is feasible and the strong equilibrium problem is infeasible. Setting an upper bound on the maximum cardinality of the coalitions, allows us to study the marginal savings of enlarging the cardinality of the coalitions. We find that the marginal savings of allowing one more player significantly decreases as the bound increases. 
Operations research models for coalition structure in collaborative logistics