Optimal production run for a process having multilevel tool wear

This paper considers the problem of selecting the most economical production run for a continuous process having multi-tool machines. Each tool may have different but similar wear characteristics. The process produces a certain type of product having two measurable quality control characteristics which jointly determine the effectiveness of the product. Simultaneous gradual changes in process means and process variances are experienced as the time passes due to tools wearing out. The process is also subject to the occurrence of an assignable cause which could shift the process means. This assignable cause, however, will not affect the process variances. Thus, at any point in time, a total change in the product quality may result from drift or shift or both. The process is in control at the beginning of the production run and stopped at the end when all the tools are changed or reset. An expected total cost per unit is developed which consists of the cost of resetting the tools, the cost of defective items, the loss of production cost due to shutdown, and the sampling cost. An optimal production run is determined by employing a one-dimensional Fibonacci search technique on the expected total cost function. A graphical method is also provided to determine the optimal production run. Numerical examples are provided to demonstrate the applications of the model.