## 3 Methodology
### Calculation of Evaluation Criteria Weights Based on Rough BWM and Entropy Weight Method
#### 3.2.1 Calculation of Subjective Weights of Evaluation Criteria Using Rough BWM  
Step 1. Let the set of CV evaluation criteria be \(\{a_{1},a_{2},\ldots,a_{n}\}\). The importance of each criterion is assessed via expert questionnaires [45], and the results of this assessment are adopted to identify the best criterion \(c_{B}\) and worst criterion \(c_{W}\). Step 2. Based on the 9-point scale, by determining the preference of the optimal criterion over other criterions, a best-to-others vector \(H^{R}{}_{B}=(h^{k}{}_{B1},\)\(h^{k}{}_{B2},\ldots,\)\(h^{k}{}_{Bn})\) is constructed. Similarly, the preference of other criteria for the worst criterion is evaluated, and an others-to-worst vector \(H^{k}{}_{W}=(h^{k}{}_{1W},\)\(h^{k}{}_{2W},\)\(\ldots,\)\(h^{k}{}_{nW})\) is constructed, where \(h^{k}{}_{Wj}\) represents the preference of DM \(k\) (\(1\leq k\leq m\)) for other criteria \(j\) (\(1\leq j\leq n\)) compared to the worst criterion. Step 3. Rough comparison vectors are constructed for all of the DMs [10; 38]. Suppose that \(R^{k}{}_{B1}=\{h^{1}{}_{B1},\)\(h^{2}{}_{B1}\ldots,\)\(h^{m}{}_{B1}\}\), where any preference assessment relative to Criterion 1 is \(h_{p}\in h^{k}{}_{B1}\), \(\forall X\in R^{k}{}_{B1}\). The lower (\(\underline{Apr}\big{(}h_{p}\big{)}\)) and upper (\(\overline{Apr}\big{(}h_{p}\big{)}\)) approximations can then be defined as:  
\[\frac{Apr(h_{p})}{Apr(h_{p})}=\cup\Big{\{}X\in R^{k}{}_{B1}/R^{k}{}_{B1}(X)\leq h _{p}\Big{\}} \tag{1}\]  
Then, based on Equation (2), the rough number \(RN(h_{p})\) is defined by its lower limit (\(\underline{Lim}\big{(}h_{p}\big{)}\)) and the upper limit (\(\underline{Lim}\big{(}h_{p}\big{)}\)), respectively. \begin{table}
\begin{tabular}{c c c c c} \hline \hline \multirow{2}{*}{**CV**} & \multirow{2}{*}{**Description**} & \multicolumn{3}{c}{**Evaluation Criteria**} \\ \cline{3-5}  & & **Label** & **Criterion** & **Category** \\ \hline \multirow{2}{*}{Reliable Value} & Security, service efficiency, the ability to proactively & P1 & Product system reliability & Benefit \\  & provide services, and the prevention and correction & P2 & Digital controlling and \\  & of errors. & & smartness & Benefit \\ \hline \multirow{2}{*}{Environmental Value} & The effects on the environment (including product & P3 & Beneficial effects on the \\  & scrap, recycling, environmental grade, etc.) & P4 & Product operating conditions & Benefit \\ \hline \multirow{2}{*}{Flexible Value} & Custom design, customer participation & P5 & Interactive customization & Benefit \\  & and feedback & P6 & Plan communication and \\  & & modification & Benefit \\ \hline \multirow{2}{*}{Social Value} & Positive social image (including product quality, & P7 & Community feeling & Benefit \\  & user experience and personnel performance, etc.) & P8 & Convenience and safety & Benefit \\ \hline \multirow{2}{*}{Economic Value} & Cost and resource utilization of product system & P9 & Ability to improve the & Benefit \\  & design (including maintenance, hardware & product durability & Benefit \\  & restoration, delivery, etc.) & P10 & Total cost for PSS & Cost \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Description of the PSS evaluation criteria for CV (Modified from [PERSON] et al. [12]). \[ \begin{array}{c}\underline{Lim}(h_{p})=\frac{1}{Q_{L}}\sum R_{Bi}(X)\Big{|}Y \in\underline{Apr}(h_{p})\\ \overline{Lim}(h_{p})=\frac{1}{Q_{U}}\sum R_{Bi}(X)\Big{|}Y\in\overline{Apr}(h_ {p})\\ RN(h_{p})=[\underline{Lim}(h_{p}),\overline{Lim}(h_{p})]\end{array}  \tag{2}\]  
Equations (1) and (2) are used to construct rough comparison vectors, \(RN(h_{Bj})\) and \(RN(h_{jW})\), for criterion \(j\), as expressed in Equation (3). \[ \begin{array}{c}RN(h_{Bj})=\left\{[h_{Bj}^{1},\overline{h_{Bj}^{1}}],[h_{Bj} ^{2},\overline{h_{Bj}^{2}}],\cdots,[h_{Bj}^{m},\overline{h_{Bj}^{m}}]\right\} \\ RN(h_{jW})=\left\{[h_{\underline{jW}}^{1},\overline{h_{jW}^{1}}],[h_{ \underline{jW}}^{2},\overline{h_{jW}^{2}}],\cdots,[h_{\underline{jW}}^{m}, \overline{h_{jW}^{m}}]\right\}\end{array}