Optimal and heuristic repairable stocking and expediting in a fluctuating demand environment

We consider a single stock point for a repairable item. The repairable item is a critical component that is used in a fleet of technical systems such as trains, planes or manufacturing equipment. A number of spare repairables is purchased at the same time as the technical systems they support. Demand for those items is a Markov modulated Poisson process of which the underlying Markov process can be observed. Backorders occur when demand for a ready-for-use item cannot be fulfilled immediately. Since backorders render a system unavailable for use, there is a penalty per backorder per unit time. Upon failure, defective items are sent to a repair shop that offers the possibility of expediting repair. Expedited repairs have shorter lead times than regular repairs but are also more costly. For this system, two important decisions have to be taken: How many spare repairables to purchase initially and when to expedite repairs. We formulate the decision to use regular or expedited repair as a Markov decision process and characterize the optimal repair expediting policy for the infinite horizon average and discounted cost criteria. We find that the optimal policy may take two forms. The first form is to never expedite repair. The second form is a type of threshold policy. We provide necessary and sufficient closed-form conditions that determine what form is optimal. We also propose a heuristic repair expediting policy which we call the world driven threshold (WDT) policy. This policy is optimal in special cases and shares essential characteristics with the optimal policy otherwise. Because of its simpler structure, the WDT policy is fit for use in practice. We show how to compute optimal repairable stocking decisions in combination with either the optimal or a good WDT expediting policy. In a numerical study, we show that the WDT heuristic performs very close to optimal with an optimality gap below 0.76% for all instances in our test bed. We also compare it to more naive heuristics that do not explicitly use information regarding demand fluctuations and find that the WDT heuristic outperforms these naive heuristics by 11.85% on average and as much as 63.67% in some cases. This shows there is great value in leveraging knowledge about demand fluctuations in making repair expediting decisions.