Existence of coordinating transshipment prices in a two-location inventory model

We consider a two-location production/inventory model where each location makes production decisions and is subject to uncertain capacity. Each location optimizes its own profits. Transshipment (at a cost) is allowed from one location to another. We focus on the question of whether one can globally set a pair of coordinating transshipment prices, i.e., payments that each party has to make to the other for the transshipped goods, that induce the local decision makers to make inventory and transshipment decisions that are globally optimal. A recent paper suggests, for a special case of our model, that there always exists a unique pair of coordinating transshipment prices. We demonstrate through a counterexample that this statement is not correct and derive sufficient and necessary conditions under which it would hold. We show that in some conditions, coordinating prices may exist for only a narrow range of problem parameters and explore conditions when this can happen. Finally, we study the effects of demand and capacity variability on the magnitude of coordinating transshipment prices.