Spare parts inventory control under a fixed-term contract with a long-down constraint

We are interested in service contracts for spare parts. We introduce a new performance measure, XLD (extreme long down), that limits the number of deliveries that are later than an agreed threshold during the contract period. We consider a single item, single location stockpoint serving multiple systems where demand is satisfied in an alternative way if the stockpoint is out of stock. Using a finite horizon Markov decision process, we derive the optimal spare parts inventory policy for meeting the contract at minimum costs. We prove that a state-dependent and time-dependent base stock policy is optimal. We also prove that the optimal base stock level is non-increasing in the number of allowed extreme long downs in the remaining contract period and that the optimal base stock level decreases at most by one unit per step. We formulate three heuristics for our finite horizon problem. These heuristics are a translation of heuristics commonly used for other service measures than the XLD measure. We assess their performance in comparison to the optimal policy. The results of our numerical study show that Heuristic 3, a myopic heuristic, is the best performing with an average optimality gap of 4.5% and an optimality gap lower than 5% in 81% of the instances. The maximum optimality gap is very high for all three heuristics, showing that important savings can be made by taking into account the actual contract performance and the remaining contract duration in stocking decisions.