A Branch and Bound Algorithm for Single Machine Scheduling with Two Stages of Failure Process

Summary & ConclusionsThe objective of the majority of production planning problems is to find the order of jobs on each machine minimizing functions of the jobs’ processing times, such as makespan and flow time. However, production environment is subject to many sources of uncertainty including machine unavailability periods, which may have a major impact on the production plan. Often times, a production line is interrupted due to periodic repair and preventive maintenance. In addition, machines may become unavailable due to unexpected failures. In many industrial settings unexpected machine failures can be potentially costly as a result of its consequences in terms of machine down times, product quality and client satisfaction. In this context, decision makers may want to jointly optimize the order of the jobs on the machine as well as the maintenance operations considering failures in a production environment such that the total expected makespan is minimized.In this paper, we deal with an integrated optimization model for production scheduling and inspection of a single machine. The failure process of the machine follows a two-stage Delay Time Model (DTM), i.e. it starts with an initial defect, which leads to eventual failure if the defect is left unattended. Once a job is interrupted due to failure on the machine, it must be restarted from the beginning when the machine becomes available. To reduce the risk of machine’s breakdown during processing of the jobs, an inspection can be performed prior to start of any job on the machine which exposes down time to the system. We consider the possibility of either minimal repair or replacement of the machine depending on its age at inspection time. We develop a recursive formula to jointly find the optimal inspection policy and production schedule which minimizes the total expected makespan. The problem defined in this paper might be applicable in many industrial and management contexts. Especially when some objective functions such as makespan is of greater importance for the decision makers. We present the application of our proposed model by use of data from a production line consisting of a multiple spindles boring machines which are able to process a number of jobs. We implement a branch and bound algorithm to optimize the model and then evaluate the efficiency of the branch and bound algorithm. The results of the study indicate the optimal solution depends on the input parameters of the model, most specifically, the down time parameters and the distributions of defect arrival and delay time.