On coalition formation in a non-convex multi-agent inventory problem

A multi-agent inventory problem is a situation in which several agents face individual inventory problems and can coordinate their orders to reduce costs. This paper analyses a multi-agent inventory problem in which each agent faces a continuous-review inventory problem, with a deterministic linear demand, no holding costs and a limited capacity warehouse. In the case under study, shortages are allowed as follows. Goods are bought from an external supplier and then they are stored in each agent’s warehouse. These stored goods usually satisfy demand. However, each agent may alternative produce their own goods, which are less costly but of lower quality. When a shortage of the purchased goods occurs, demand is satisfied with the produced goods until a new order arrives. The problem under study herein arises in a farming community, and is a variation of a problem addressed by Fiestras-Janeiro et al. (2015). However, the existence of two acquisition costs makes it substantially different from Fiestras-Janeiro et al.’s problem and significantly complicates its analysis since the resulting cost functions may now be non-convex. This paper establishes the optimal inventory policies for our problem and obtains a stable order structure when agents allocate the joint costs using a proportional rule. In addition, it illustrates the performance of our model and results in an example.