# Fuzzy Evaluation Model of Process Improvement Capability with Costs Consideration
## 1 Introduction
[1] regarded the expected value of the Taguchi loss function as a basis and considered the corresponding cost of the process improvement to propose a process improvement capability index as follows:  
\[C_{PIM}=\frac{1}{3\sqrt{\delta^{2}+R\gamma^{2}}}. \tag{3}\]  
Figure 1 is a schematic diagram of the Taguchi loss function \(L(\text{x})=k(\text{x}-T)^{2}\), the expected loss (\(E[L(X)]=k(\mu-T)^{2}+\sigma^{2}\)) with tolerances \(0\pm 1\) for processes A, B and C, respectively as follows:  
Process A\(\sim\)\(N\left(0,(1/6)^{2}\right)\): \(E[L(X)]=k\left(\sigma^{2}+(1/6)^{2}\right)=4k/144\)  
Process B\(\sim\)\(N\left((1/4),(1/6)^{2}\right)\): \(E[L(X)]=k\left((1/4)^{2}+(1/6)^{2}\right)=13k/144\)  
Process C\(\sim\)\(N\left((1/2),(1/3)^{2}\right)\): \(E[L(X)]=k\left((1/2)^{2}+(1/6)^{2}\right)=52k/144\)  
In the equation, \(R\) is the ratio of the improvement cost (\(C_{2}\)) of process precision to the improvement cost (\(C_{1}\)) of process accuracy, that is \(R=C_{1}/C_{2}\). [PERSON] et al. [1] believe that the main reason for a lack of accuracy is the deviation of machine parameters. Therefore, to solve this problem, what we just need to do is to adjust the machine parameters to improve the quality, so the required investment cost is quite low. Additionally, a lack of accuracy may result from the high turnover rate, poor homogeneity caused by unstable material supply, or old machinery. Since the cost of improvement is relatively high, the value \(R\) is usually less than 1.0. Besides, [PERSON] et al. [1] claimed that the Taguchi capability index can be used to measure and evaluate the original supplier's process capability. When the supplier's process capability is insufficient, we can assess whether to improve its accuracy or precision based on the cost of process improvement. Next, the process improvement capability index can be employed to evaluate the supplier's process improvement capability. By means of these two indexes, not only capable suppliers but also potential suppliers can be selected. Also, they can assist suppliers with long-term partnerships to reduce their process improvement costs as well as improve their process quality, resulting in a win-win situation for both sides, and thus, the entire supply chain's competitiveness can be improved. Subsequently, given the cost considerations and the need for timely, quick responses, many studies have also pointed out that the sample size of sampling testing is usually not large. At the same time, the confidence interval-based fuzzy testing method can be applied because this method, which incorporates experts and their accumulated experiences, can not only reduce the risk of misjudgment due to sampling errors but also improve the testing accuracy [15; 16; 17; 18]. In view of this, this study proposes a fuzzy testing method based on the upper confidence limit to evaluate whether the supplier's process improvement ability is above the average level. Thus, this study applies the process improvement capability index to construct an evaluation model of the supplier's process improvement capability, which can be provided to the industry for application. In the second section, this study introduces the process improvement capability index, applies the methods of [PERSON] [19] and [PERSON] et al. [20], takes the process improvement capability index as the objective function, regards the joint confidence region of accuracy index and precision index as a constraint, and obtains the upper confidence limit of the process improvement capability index through mathematical planning. Next, in the third section, a fuzzy membership function is constructed according to the upper confidence limit of the index. After applying the fuzzy membership function testing method, a fuzzy testing method is proposed to evaluate whether the supplier's process improvement capability is above the average level. In the fourth section, this study takes an application example and proposes fuzzy evaluation rules to illustrate how to apply the fuzzy testing method proposed in the third section.