Joint Planning of Production and Inspection of Parallel Machines with Two-phase of Failure

Abstract Utilizing available information on random machine breakdowns is highly effective in making joint production and inspection decisions. The current paper studies the joint planning of production schedule and inspection of identical parallel machines to process n jobs with different processing times. Each machine's failure is modeled using the two-stage delay time concept. The first stage is a defect arrival which does not cause a machine stoppage, but if it is not removed, may cause the machine failure. A machine's lifetime is divided into two stages in such a failure process: defect arrival and failure arrival stages. The defects and failures of the identical parallel machines follow non-linear and time-dependent distributions. Inspection activities are intended to identify potential defects in the machines and subsequently removing them prior to eventual failures. The objective is to establish the best jobs’ schedule of the machines and the best sequence of inspection decisions that minimize the expected makespan. For this problem, unequal defect and failure arrival rates for the machines impose some complications to the expected makespan calculation. Therefore, obtaining an integrated scheduling and inspection plan requires considering all possible combinations of the jobs order and inspections on the machines that further increase the complexity of the algorithm implementation. We propose a branch and bound (B&B) algorithm that restricts the search time and space and demonstrates the effectiveness of integrating inspection and maintenance operations with the jobs sequence to minimize the expected makespan. We propose an upper bound and develop a lower bound to assign the maximum number of jobs on the empty capacity of the machines. We analyze the impact of the model's parameters on the results of the B&B algorithm and validate the algorithm's performance by comparing the results with the analytical solutions for small-sized problems for which analytical solution exists.