Optimal job scheduling and inspection of a machine with delayed failure

In this paper, we consider a single machine, which must process n jobs in sequence. The machine's failure process follows the two-stage Delay Time Model, i.e. it starts with an initial defect, and will lead to eventual failure if the defect is left unattended. An inspection may be performed before starting a job to detect a defect. We improve the machine's maintenance decision making process by considering the possibility of performing minimal repair or replacement at any event time with regard to the age of the machine. This assumption affects the complexity of the problems in terms of executing loops in MATLAB excessively. The objective is to find the optimal inspection policy and the jobs sequence, which minimise the total expected makespan. We will develop two models and derive their corresponding recursive formulas. For the optimisation of the first model, we will combine the Genetic Algorithm with the recursive equations to jointly optimise the job sequence and inspection policy. In the second model, due to cumbersome recursive equations, we will adopt a simulation algorithm to obtain the required expected values in the objective function. We will provide numerical examples to present the application of the models, and study the influence of various input parameters on the best-obtained policies. We conduct extensive computational experiments on randomly generated problems with different configurations to evaluate the efficiency of models.