An Effectively Improved Statistically Constrained Economic Model for Designing MCUSUM Control Charts

The statistically constrained economic design of multivariate cumulative sum, MCUSUM, control charts involves determining the main parameters of the charts such that while the implementation cost of the chart is minimized, the desired statistical performances of the chart are maintained. The average run length when the process is in control (ARL) and the average run length while the process goes to an out-of-control state (ARL) are the two main statistical performances of the MCUSUM charts. In this paper, the main MCUSUM parameters (the reference value k, the control limit H, the sample size n, and the sampling interval h) are determined using a statistically constrained economic model. The cost function of the model is the Lorenzen-Vance function that needs to be minimized while an upper bound on ARL and a lower bound for ARL are satisfied. The statistically constrained economic model is extended for two different implementation situations. In the first situation, intangible external costs are incorporated to the Lorenzen-Vance function using a multivariate Taguchi loss function. In the second situation, a nonlinear constraint on the average wasted products (AWP) that is obtained by multiplying n by ARL is employed to improve the efficiency of the solutions obtained. Finally, a genetic algorithm is developed to solve the extended model. The results of the application of the proposed methodology show that without a significant increase on the cost of the optimum solution an effective MCUSUM chart can be obtained with desired statistical performances and loWA WPs.

Keywords:

Multivariate CUSUM , Control Chart , Statistically Constrained design: Genetic Algorithm