Economic design of max charts using Taguchi's loss function

Recent researches have shown that a single control chart, Max chart, simplifies and performs as effectively as joint X over bar and S charts for monitoring both the process mean and standard deviation. This paper develops an economic design of Max chart based on a unified model and embellishes it with Taguchi's quality loss function. A simplified algorithm is applied to determine the optimal settings of three control chart parameters: the sample size, the sampling interval and the control limit. A numerical example is presented to illustrate its application. A fractional factorial experiment is conducted to study the sensitivity of the input parameters on the optimal designs of Max chart and joint X over bar and S charts. Statistical and economical properties including in-control average time to signal (ATS), out-of-control ATS, adjusted average time to signal (AATS) and minimum expected cost per hour of the two charts are compared. The results show that the two charts operate with approximately equal cost but different false alarm rate and detecting speed for the assignable cause without and with consideration of process failure mechanism. Critical input parameters that cause these differences are then investigated. This paper concludes that Max chart can be a good alternative to joint X over bar and S charts from not only statistical and simplified properties but also economical considerations.