A Multi-objective Economic Statistical Design of the CUSUM chart: NSGA II Approach

Abstract

This paper presents an approach for the economic statistical design of the Cumulative Sum (CUSUM) control chart in a multi-objective optimization framework. The proposed methodology integrates economic considerations with statistical aspects to optimize the design parameters like the sample size ($n$), sampling interval ($h$), and decision interval ($H$) of the CUSUM chart. The Non-dominated Sorting Genetic Algorithm II (NSGA II) is employed to solve the multi-objective optimization problem, aiming to minimize both the average cost per cycle ($C_E$) and the out-of-control Average Run Length ($ARL_\delta$) simultaneously. The effectiveness of the proposed approach is demonstrated through a numerical example by determining the optimized CUSUM chart parameters using NSGA II. Additionally, sensitivity analysis is conducted to assess the impact of variations in input parameters. The corresponding results indicate that the proposed methodology significantly reduces the expected cost per cycle by about 43\% when compared to the findings of the article by M. Lee in the year 2011. A more extensive comparison with respect to both $C_E$ and $ARL_\delta$ has also been provided for justifying the methodology proposed in this article. This highlights the practical relevance and potential of this study for the right application of the technique of the CUSUM chart for process control purposes in industries.