Optimal unrestricted dynamic stochastic scheduling with partial losses of work due to breakdowns

In this article, we discuss the optimal scheduling problem of the recently introduced model for partial loss due to machine breakdowns, which fills up a significant gap in the existing literature. More specifically, we consider the problem of processing a number of jobs with arbitrary random processing times by a machine subject to general stochastic breakdowns, where each breakdown may cause an uncertain loss of the work achieved on the job being processed. The objective is to maximize the expected weighted discounted reward of completing the jobs in the class of unrestricted dynamic policies. We obtain the optimal dynamic polices using multi-armed bandit process methodology, which are characterized by a set of Gittins indices as solutions to a system of integral equations. Optimal solutions for a number of problems with specific loss patterns are derived. Application of the theory to the classical no-loss model is also discussed which leads to new results.