The design of a CUSUM control chart for LR-fuzzy data

Quality characteristics are often imperfect (incomplete, censored, vague or partially unknown) in representing the quality information about products or services. Such imperfectness sometimes may be well described in vague, imprecise or linguistic way. In practice the LR-fuzzy number data are frequently recommended to be used in above cases. LR-fuzzy number itself can be generated with method of Cheng based on expert´s evaluations on products or services quality. We propose a fuzzy Cumulative Sum (CUSUM) control chart, in which the possibility distribution determined by the membership function of the fuzzy test statistic is employed. LR-fuzzy data are viewed as a fuzzy random variables with approximately normally distributed center and two approximately χ² distributed spreads. Using Feng´s measure of a distance between two fuzzy numbers as a measure of variability and a modified hypothesis testing methods, a fuzzy decision rule and a method for the calculation of the average run length (ARL) for the chart are proposed.