Procurement allocation planning with multiple suppliers under competition

This paper examines a procurement planning scenario in which a firm wishes to obtain some quantity of a good from a set of capacitated suppliers whose products are interchangeable. The total cost to acquire the good from a supplier is a nondecreasing concave function of the amount of the product purchased, as a result of economies of scale and/or bulk discounts offered by the supplier. The problem of minimising the total cost incurred in obtaining the required amount of the good is NP-hard in itself, but further complicating the situation is the existence of a competing firm (or set of firms) with its own demand for the same input or good. The existence of such competition leads to a situation in which the first firm (the leader) must consider the actions of their competitor(s) (whom we model as a single follower) when minimising procurement cost, because the followers procurement amounts decrease the suppliers available capacity levels. To mitigate the effects of the followers actions, the leader can protect any supplier at some cost (e.g. by signing a contract in which the supplier guarantees some level of capacity). Therefore, our problem is a three-stage game in which the leader first chooses which suppliers to protect, the follower satisfies its demand, and the leader satisfies its demand from the remaining capacity. We model this problem as a three-stage mixed-integer programme, and propose algorithms for its optimal solution via reformulation and cutting-plane techniques.