Cooperative lot sizing with transshipments and scarce capacities: solutions and fair cost allocations

In this article, we study the cooperative capacitated lot sizing problem where the available resources of the players may be used in common and payable transshipments may take place to transport items from one player to the other. Two research questions are attacked. On the one hand, there is the hard-to-solve optimisation problem of deciding which player shall produce what amount of items at what point of time, how many items shall be kept in stock when, and when transport of what quantities shall take place among which players. The objective is to minimise the total cost of the supply chain taking into account thorny side constraints due to capacity limits. On the other hand, there is the hard-to-solve problem of distributing cost shares among the players such that the cost allocation is stable and fair. Both questions are discussed in detail and answers are given by proposing mathematical programming approaches.