## 1 Introduction  
The variations of the manufacturing process have been investigated widely in order to evaluate the capability of the process to meet the desired specifications. In this regard, various techniques have been used, including design of experiments and statistical process control. Process capability analysis has been particularly helpful in this regard [1, 2]. As industrial systems have advanced, process engineers have needed to thoroughly analyze and manage every aspect of their processes [3]. By using process capability analysis, we can assess manufacturing processes and make use of the resulting information to enhance the capabilities of the processes under investigation to meet the required specifications. It is widely accepted that the majority of manufactured goods possess multiple quality characteristics (QCs) that are functionally correlated, indicating that they should be evaluated simultaneously. As a result, the assessment of product quality becomes more intricate with an increase in the number of QCs. Therefore, there is a need for identifying capability indices that can address the capabilities of nonnormal multivariate processes. For instance, [PERSON] [4] conducted a process capability analysis for seven QCs of semiconductor products. There is a significant amount of literature on the topic of multivariate process capability analysis, with several studies proposing different approaches. For example, [PERSON] et al. [5] introduced the first multivariate capability index that employs process regions (PRs) and specification regions (SRs). [PERSON] [6] developed the initial multivariate Cp that utilizes the proportion of nonconformance (PNC). Additionally, [PERSON] et al. [7] conducted a process capability analysis to assess the effectiveness of multivariate quality characteristics (QCs). [PERSON] [8] examined the forms of process regions (PRs) and specification regions (SRs) to create a novel process capability index (PCI). [PERSON] et al. [9] analyzed bivariate process capabilities and produced two indices using the PNC. [PERSON] [10] developed a technique to calculate a multivariate C_pk index. [PERSON] et al. [2] introduced a new index derived from C_p and C_pk using a principal component analysis (PCA) decomposition. There are numerous other studies in the literature concerning multivariate capability analysis [4; 11; 12; 13; 14; 15; 16; 17; 18; 19; 20; 21; 22; 23; 24; 25; 26; 27; 28; 29; 30; 31; 32; 33; 34; 35; 36]. [PERSON] and [PERSON] [37] were the first to utilize principal component analysis (PCA) in process capability analysis. [PERSON] and [PERSON] [28] utilized the g and h multivariate process capability indices for nonnormal data, and their proposed index has been favorably compared to other indices in the literature. However, the calculation of this index involves complex computations. [PERSON] [22] suggested a multivariate process capability index (PCI) that can be employed for both normal and nonnormal quality control (QC) processes. To evaluate multivariate nonnormal processes, he proposes the use of one-sided models to determine the Process Ratio (PR), which is then compared to the Specification Ratio (SR). [PERSON], [PERSON], and [PERSON] [20] built on the work of [PERSON] and [PERSON] [32] to introduce a multivariate nonnormal PCI. They estimated the original probability density function using a weighted standard deviation. [PERSON] [38] defined the conventional PCIs (Cp and Cpk) based on the proportion of nonconforming (PNC). Previous research in this field has primarily focused on PCIs for multivariate normal data, as evidenced by studies [9; 39; 40]. However, there have been some endeavors to expand the application of the proportion of nonconforming (PNC) to nonnormal data. [PERSON] and [PERSON] [41] introduced a multivariate nonnormal PCI by utilizing the PNC concept. To normalize nonnormal data, they applied the root transformation method and subsequently used Monte Carlo simulation to estimate the PNC. [PERSON] et al.