# An Interface to Monitor Process Variability Using the Binomial ATTRIVAR SS Control Chart
## 2 Chart B-ATTRIVAR SS \(S^{2}\)  
After sample collection, all \(n\) items of the B-ATTRIVAR SS \(S^{2}\) chart are inspected based on attribute, similarly to the \(np_{S^{2}}\) from [9]. These inspections make use of some device (such as a go/no-go gauge, for example) configured using the discriminant control limits \(LDL\) (lower discriminant limit) and \(UDL\) (upper discriminant limit). Each item outside these limits is classified as nonconforming. After classifying all items in the sample, \(Y_{D}\) is recorded as the number of rejected items. If \(Y_{D}\geq CL_{Y}\) (attribute control limit), the process is defined as being out of control. Conversely, it is assessed whether \(Y_{D}<W_{Y}\) (warning attribute control limit). If so, it is found that the process is in control. If otherwise, the variable inspection process is initiated using the same sample.  
The stage of variable inspection is performed similarly to that in the classic \(S^{2}\) chart described by [2,4], in which quality is measured based on all \(n\) items in the sample, and variance \(S^{2}\) is calculated. If \(S^{2}\) is outside the acceptance interval defined by \(LCL\) (variable lower control limit) and \(UCL\) (variable upper control limit), the process is defined as being in control. Otherwise, it is out of control. This entire process is depicted in Figure 1.  
It is worth mentioning that it is common to use only the upper control limit to inspect variable \(S^{2}\), as a process should have the least possible variability, as mentioned by [20], given that this strategy makes the chart more effective at detecting increases in process variance. In such a case, \(UCL\) is denoted as \(CL_{S^{2}}\).  
In this paper, both the attribute and variable inspection limits are optimized to achieve acceptable performance values determined by \(ARL_{0}\) and \(ARL_{1}\), or by \(\alpha\) and \(\beta\), considering that \(ARL\) refers to the average run length or the average number of samples collected until the chart signals for the first time, in addition to the fact that [4]:  
\[ARL=\frac{1}{(Probability\ of\ sample\ rejection\ by\ the\ chart)} \tag{1}\]  
\[ARL_{0}=\frac{1}{\alpha} \tag{2}\]  
\[ARL_{1}=\frac{1}{(1-\beta)} \tag{3}\]  
where \(\alpha\) is the probability of type I error, \(\beta\) is the probability of type II error, the index \"0\" indicates a process in control, and index \"1\" indicates a process being out of control.  
The performance of the B-ATTRIVAR SS \(S^{2}\) chart (mixed inspection) was compared to that of \(S^{2}\) (variable inspection) and np (attribute inspection) charts, as in Table 1, which shows simulated \(ARL\) values with sample sizes of 5 (\(n=5\)) by varying the standard deviation of simulated samples through \(\lambda\) in each row. The results demonstrate a similar performance of the studied chart to that of the np chart. Nevertheless, it is still worth mentioning its advantages over the np charts, mainly on account of the fact that the quality characteristic measurements required in the stage of variable inspection of the B-ATTRIVAR SS \(S^{2}\) are capable of providing valuable information to enhance the process analysis and make improvements. Although the performance of the ATTRIVAR proved inferior to that of the \(S^{2}\) chart, it still offers notable advantage over it, i.e., its operational simplicity [14].  
Figure 1: Flowchart of the B-ATTRIVAR SS \(S^{2}\) Chart.