Cost Sharing Methods for Capacity Restricted Cooperative Purchasing Situations

This paper analyzes cooperative purchasing situations with two suppliers with limited supply capacity. In these capacity restricted cooperative purchasing (CRCP) situations we consider a group of cooperating purchasers for whom it turns out to be optimal as a group to or- der as much as possible at one supplier and the possible remainder at the other. To find suitable cost allocations of the total purchasing costs, a CRCP-situation is modeled as a cost sharing problem, in which the corresponding cost function is piecewise concave and the intervals of concavity are determined by the restricted capacity of the suppliers. It is seen that in the setting with piecewise concave cost functions, standard cost sharing mechanisms lose their axiomatic appeal, e.g. the serial cost sharing mechanism will neither satisfy unit cost monotonicity (UCM) nor monotonic vulnerability for the absence of the smallest player (MOVASP). In fact, it is shown that these two properties are incompatible now. We in- troduce a new context specific class of piecewise serial rules, in which the vector of order quantities is divided into separate vectors for the different intervals of concavity, using a bankruptcy rule and subsequently apply the serial rule to each interval. We show that the proportional rule is the only bankruptcy rule for which the corresponding piecewise serial rule satisfies UCM. Moreover, we show that the piecewise serial rule corresponding to the constrained equal losses rule satisfies MOVASP.