Cost sharing methods for capacity restricted cooperative purchasing situations

This paper analyzes capacity restricted cooperative purchasing (CRCP) situations in which a group of cooperating purchasers face two suppliers with limited supply capacity. To minimize the total purchasing costs, we show that two extreme policies have to be compared: order everything at one supplier and the possible remainder at the other. Interestingly, as order quantities increase, various policy switches can occur. To find suitable cost allocations of the total purchasing costs, we model a CRCP-situation as a cost sharing problem. As increasing order quantities also imply concavity breaks due to a forced change in supplier, the corresponding cost function is piecewise concave. For cost sharing problems with concave cost functions, we show that the serial cost sharing mechanism satisfies two desirable properties, unit cost monotonicity (UCM) and monotonic vulnerability for the absence of the smallest player (MOVASP). However, these properties are lost in the setting of piecewise concave cost functions. We develop a new context specific class of piecewise serial rules based on claims rules. We show that the proportional rule is the only claims rule for which the corresponding piecewise serial rule satisfies UCM. Moreover, the piecewise serial rule corresponding to the constrained equal losses rule satisfies MOVASP.