## 2 Process Capability Analysis Chart with Asymmetric Tolerances  
As mentioned above, the process capability index is an accessible and useful tool that the industry applies to gauge the process quality level of each quality characteristic for products. [PERSON] et al. [25] came up with a process capability index named \(C_{pm}\) on the basis of the Taguchi loss function. In the index, the denominator is the expectation of the Taguchi loss function so that the process loss can be fully depicted. Further, this index is one of the tools that the manufacturing industry comprehensively applies to the process quality evaluation, defined as follows:  
\[C_{pm}=\frac{d}{3\sqrt{\sigma^{2}+(\mu-T)^{2}}}, \tag{1}\]  
where \(\mu\) refers to the process mean, \(\sigma\) represents the process standard deviation, d = (USL \(-\) LSL)/2 means the half of the specification interval, T = (USL \(+\) LSL)/2 means the midpoint of the specification limits, USL stands for upper specification limit, and LSL stands for lower specification limit. Plenty of studies have suggested that an unequal relationship is formed between the index \(C_{pm}\) and the process yield as the value of the process capability index \(C_{pm}\) becomes large enough [26; 27]. According to [PERSON] et al. [28], \(C_{pm}\) has an unequal relationship with the process yield (_Yield_%), such that \(Yield\%\geq 2\Phi(3C_{pm})-1\), where \(\Phi(\cdot)\) symbolizes the cumulative distribution function of the standard normal distribution. Thus, since \(C_{pm}\) is only suitable for assessing symmetric tolerances, not only can \(C_{pm}\) depict the process loss, but it also can fully mirror the process yield. As mentioned earlier, practically, the nominal-the-best quality characteristics of component products processed by machine tools usually belong to asymmetrical tolerances. Thus, [PERSON] et al. [29] put forward a process capability index called \(C_{pm}\) with asymmetric tolerances. As noted by some studies, a process has asymmetric tolerances when the upper tolerance \(d_{U}=\) USL \(-\) T is unequal to the lower tolerance \(d_{L}=\) T \(-\) LSL [30; 31; 32]. This asymmetric tolerance index can be shown below:  
\[C_{pm}^{\prime\prime}=\frac{d^{*}}{3\sqrt{\sigma^{2}+A^{2}}}, \tag{2}\]where \(A=Max\{(\mu-T)d^{*}/d_{U},(T-\mu)d^{*}/d_{L}\}\) and \(d^{*}=Min\{d_{U},d_{L}\}\). Apparently, when the preset target value T = (USL + LSL)/2 = M (symmetric case), then d* = d, A = \(\mid\mu-\text{T}\mid\), and the asymmetric tolerance index \(C_{pk}^{{}^{\prime\prime}}\) will drop to the original index \(C_{pk}\). In the definition, factor A guarantees that \(C_{pk}^{{}^{\prime\prime}}\) can reach its maximal value at \(\mu=\text{T}\) (on-target process) no matter if the tolerance is symmetric (T = M) or asymmetric (T \(\
eq\) M). Let \(\gamma=\sigma/d^{*}\) and  
\[C_{pm}^{{}^{\prime\prime}}=\frac{d^{*}}{3\sqrt{\sigma^{2}+A^{2}}}, \tag{3}\]  
where \(\delta^{\prime}=(\mu-T)/d^{*}\), \(d_{1}=d^{*}/d_{U}\), and \(d_{2}=d^{*}/d_{L}\). Then  
\[C_{pm}^{{}^{\prime\prime}}=\frac{1}{3\sqrt{\gamma^{2}+\delta^{2}}}. \tag{4}\]  
When \(\delta=0\), it means that the process average \(\mu\) is just located at the target value T. When \(\delta>0\), it shows that the process is shifted to the right. For example, when \(\delta=1/4\), it indicates that the process average \(\mu\) is shifted to the right by \(1/4\) tolerance from the target value T. When \(\delta<0\), then the process is shifted to the left. For instance, when \(\delta=-1/2\), then the process average \(\mu\) is shifted to the left by \(1/2\) tolerance from the target value T.  
As mentioned earlier, usually multiple important quality characteristics exist in a product after it is processed. When the process quality of all quality characteristics attains the required quality level, then the product's process quality can be ensured to meet customer demand. Without losing generality, it is assumed that there are k quality characteristics of the product in this paper. Based on [PERSON] et al.