## 3 Problem Formulation
### TOPSIS for Solving MOP Problems
\begin{array}{ll}1&\text{if }d_{p}^{NIS}(x_{k})> \left(d_{p}^{NIS}\right)^{+}\\ 1-\frac{\left(d_{p}^{NIS}\right)^{+}-d_{p}^{NIS}(x_{k})}{\left(d_{p}^{NIS} \right)^{+}-\left(d_{p}^{NIS}\right)}\text{ if }\left(d_{p}^{NIS}\right)^{-} \leq d_{p}^{NIS}(x_{k})\leq\left(d_{p}^{NIS}\right)^{+}\\ 0&\text{if }d_{p}^{NIS}(x_{k})<\left(d_{p}^{NIS}\right)^{-}\end{array} \right. \tag{35}\]  
\[d_{p}^{PIS}=\left\{ \begin{array}{ll}\left(\sum_{i=1}^{2}\sum_{t=1}^{6}w_{1}^ {p}\times[(9,500,000-Z_{1})/(9,500,000-1,677,200)]^{p}+\right.\\ \left.\sum_{i=1}^{2}\sum_{t=1}^{6}w_{2}^{p}\times[(Z_{2}-1,642,785)/(8,494,955 -1,642,785)]^{p}+\right\}\\ \sum_{i=1}^{2}\sum_{t=1}^{6}w_{3}^{p}\times[(Z_{3}-0)/(33,930-0)]^{p})\end{array} \right\} ^{1/p} \tag{36}\]  
\[d_{p}^{NIS}=\left\{ \begin{array}{ll}\left(\sum_{i=1}^{2}\sum_{t=1}^{6}w_{1} ^{p}\times[(Z_{1}-1,677,200)/(9,500,000-1,677,200)]^{p}+\right.\\ \left.\sum_{i=1}^{2}\sum_{t=1}^{6}w_{2}^{p}\times[(8,494,955-Z_{2})/(8,494,95 -1,642,785)]^{p}+\right.\\ \left.\sum_{i=1}^{2}\sum_{t=1}^{6}w_{3}^{p}\times[(33,930-Z_{3})/(33,930-0)]^{ p})\end{array} \right\}^{1/p} \tag{37}\]  
To obtain numerical solutions in (29)-(31), the weights of \(d_{p}^{PIS}\) and \(d_{p}^{NIS}\) are set as \(w_{1}=0.4\), \(w_{2}=0.3\), and \(w_{3}=0.3\). According to [PERSON] et al. 's study [39], increasing in \(p\) increases the influence of objectives with large weights on the measurement distances. Thus, the value of \(p\) is related to the preferences of decision-makers. In this study, \(p=1\) was set to obtain the problem as either concave or convex. If the preference for the \(p\) value that is significantly different from \(1\) can be efficiently solved with multiple local optima, then the other \(p\) values can be set in the future. Table 7 showed the PIS Payoff Table for this case. Finally, the membership functions \(\mu_{1}(x)\) and \(\mu_{2}(x)\) of (29)-(31) can be used to represent the satisfactory level of bi-objective functions and use the max-min operation to control the corresponding model. This results in the following formulation can be obtained:  
\[max \gamma\] (38) s.t. \[\left(d_{1}^{PIS}(x)-1.35858\right)/(2.4302)\geq\gamma\] \[\left(1.43973-d_{1}^{NIS}(x)\right)/(-0.05474)\geq\gamma \tag{39}\] \[x\in X\]  
When the results of \(d_{1}^{PIS}\) and \(d_{1}^{NIS}\) are calculated, the problems of (38) and (39) can be solved. In Table 7, the maximum satisfactory level (\(\gamma_{max}\)) is \(0.7120\) to achieve the \begin{table}
\begin{tabular}{c c c c} \hline \hline  & \(Z_{1}\) & \(Z_{2}\) & \(Z_{3}\) \\ \hline Min & \(Z_{1}\) & 1,677,200 \({}^{-}\) & 1,654,480 & 3318 \\ Max & \(Z_{2}\) & 9,500,000 & 8,494,955 \({}^{-}\) & 18,389 \\ Max & \(Z_{3}\) & 7,776,000 & 5,183,500 & 33,930 \({}^{-}\) \\ \hline \hline \end{tabular}  
* NIS: \(z^{-}\) = (1,677,200, 8,494,955, 33,930).  
\end{table} 
Table 6: NIS Payoff Table. \begin{table}
\begin{tabular}{c c c} \hline \hline  & \(d_{1}^{PIS}\) & \(d_{1}^{NIS}\) \\ \hline \(Mind_{1}^{PIS}\) & 1.35858 & 1.49444 \\ \(Maxd_{1}^{NIS}\) & 3.78877 & 1.43973 \\ \hline \hline \end{tabular}  
* NIS: \(z^{-}\) = (1,677,200, 8,494,955, 33,930).  
\end{table} 
Table 7: PIS Payoff Table. compromise solution for the production plan in Table 8. From the system of Equations (36), (37) and (3), three values \(Z_{1}\), \(Z_{2}\), \(Z_{3}\) are found. The sales revenue (\(Z_{1}\)), production costs (\(Z_{2}\)), and repair costs (\(Z_{3}\)) are NTS7,662,900, NTS4,779,525, and NTS10,458, respectively.