## 4 Control Chart Performance Measures  
In situations where processes are stabilized, i.e., free of special causes, \(\text{ARL}_{0}\) is defined as the main performance measure. The \(\text{ARL}_{0}\) reveals the number of samples required until there is a false alarm. This implies that a control chart indicates when a sample is out of control while it is supposedly in control [27]. According to [20], \(\text{ARL}_{0}\) is the inverse of a type I error (\(\alpha\)) as in Equation (26):  
\[\text{ARL}_{0}=\frac{1}{\alpha} \tag{26}\]  
Another important measure concerning the performance analysis of control charts is the standard deviation of the run length (SDRL). For a better performance analysis of control charts, [28; 29] state that it is important to use a performance measure associated with dispersion coupled with ARL, since \(\text{ARL}\) presents high variability. The value of \(\text{SDRL}_{0}\), i.e., the expected value for a process free of special causes, can be calculated as in Equation (27):  
\[\text{SDRL}_{0}=\frac{\sqrt{1-\alpha}}{\alpha} \tag{27}\]  
From simulated RL values, it is possible to perform other statistical analyses, such as RL percentiles. Proposed by [30], RL percentiles are used to complement the ARL and SDRL analysis, since it is capable of demonstrating variation in false alarms given by control charts under study. According to [31], an exclusive use of ARL as a performance measure may not provide a complete picture of chart performance, since both in-control and out-of-control distributions are largely skewed. The RL percentile is a robust measure of control chart performance. The 100 th percentile (\(0<\text{q}<1\)) can be given by Equation (28):  
\[1-\beta^{m}=\frac{\text{q}}{100} \tag{28}\]  
In order to compare the proposed control charts to traditional control charts, the control factor (k) of each chart must be initially adjusted individually so that when the process is under control, the two charts reach approximate values of \(\text{ARL}_{0}\). Once \(\text{ARL}_{0}\) values are very close, the difference in \(\delta\) standard deviations of the process mean and the displacement of \(\lambda\) in the process range are simulated so as to compare ARL values found through a traditional control chart with those of the proposed fuzzy control chart. The one with the lowest ARL value is the most efficient for a given situation. Therefore, the most efficient graphs should reach high \(\text{ARL}_{0}\) values, i.e., few false alarms and low ARL values, so as to quickly detect variations in the process mean and range [32]. Figure 7: IT2 TFN R control chart example. The type-2 fuzzy mean \(R=\left(R_{1}^{U},R_{1}^{L},R_{2}^{U},R_{3}^{3},R_{3}^{U}\right)\) is transformed into range \(\text{RD}_{\text{THT}}\). The red lines correspond to the upper and lower control limits, respectively, while the dashed green line is the center line. Thus, according to [33; 34], control charts mainly aim to monitor any random cause changing the value of the process mean in control \(\mu_{0}\) to \(\mu_{1}\) = \(\mu_{0}\)\(\pm\)\(\delta\sigma\) and/or those increasing the standard deviation in control \(\sigma_{0}\) to \(\sigma_{1}\) = \(\lambda\sigma_{0}\), where \(\lambda\) is the magnitude of variability increase. The fuzzy sets used in control charts by [8] allowed performing a fuzzification method using random variables, and the performance of type-1 fuzzy \(\overline{X}\)-R control charts was measured using the average run length (ARL) and the extra quadratic loss (EQL). On the other hand, [9] evaluated the performance of exponentially weighted moving average (EWMA) control charts using ARLs for traditional and fuzzy control charts. Considering a process whose mean is 0 and standard deviation equals 1, triangular fuzzy numbers are subsequently found through a fuzzification method. In this work, the ARL, SDRL and RL percentile values are simulated as performance measures.