## 3 Problem Formulation
### TOPSIS for Solving MOP Problems  
[PERSON] and [PERSON] [34] proposed TOPSIS (technique for order preference by similarity to ideal solution). Usually, the solutions based on the positive ideal solution (PIS) are different from the solutions based on the negative ideal solution (NIS). Therefore, both \(PIS(x^{*})\) and \(NIS(x^{*})\) can be used to normalize the distance family as follows [7; 37; 38]:  
\[d_{p}=\left\{\sum_{j=1}^{k}w_{j}^{p}\left[\frac{z_{j}^{+}-z_{j}(x)}{z_{j}^{+}- z_{j}^{-}}\right]^{p}\right\}^{1/p},\text{where }p\geq 1 \tag{22}\]  
where \(z_{j}^{+}\) and \(z_{j}^{-}\) are the maximum and minimum values of the function \(z_{j}(x)\), respectively. They include the positive ideal solution (PIS) and the negative ideal solution (NIS) for solving multi-attribute decision making (MADM) problems. Using a similar concept, the principle of TOPSIS for MODM is to determine the optimal solution such that it minimizes the distance to PIS and maximizes the distance to NIS. Suppose there is the following MOP problem:  
\[max/\min\qquad[z_{1}(x),z_{2}(x),\ldots,z_{k}(x)] \tag{23}\]  
\[\text{s.t.}X=\left\{x\in X|g_{i}(x)\leq,=,\geq b_{i},\right. \tag{24}\]  
\(\forall i=1,\ldots,m,x\geq 0,x\in R^{k}\}\)  
where  
\(z_{j}(x)=\) benefit objective for maximization \(j\in J\)  
\(z_{i}(x)=\) cost objective for minimization \(i\in I\)  
The ideal and anti-ideal values of a maximization objective are calculated using (25) under the problem constraints. The same values of a minimization objective are computed from (26) under the constraints of the problem. \[ \begin{gathered} z^{+}=\left\{\underset{x\in X}{\text{mazz}}_{j} (x),j=1,2,\ldots,k\right\},\\ z^{-}=\left\{\underset{x\in X}{\text{mizz}}_{j}(x),j=1,2,\ldots,k \right\}\qquad\text{or}\end{gathered}  \tag{25}\]  
\[ \begin{gathered} z^{+}=\left\{\underset{x\in X}{\text{mizz}}_{j} (x),i=1,2,\ldots,k\right\},\\ z^{-}=\left\{\underset{x\in X}{\text{mazz}}_{i}(x),i=1,2,\ldots,k \right\}\end{gathered}  \tag{26}\]where \(j\in J\), \(i\in I\), \(z^{+}=\left\{z_{1}^{+},\cdots,z_{k}^{+}\right\}\), and \(z^{-}=\left\{z_{1}^{-},\cdots,z_{k}^{-}\right\}\) is a set of individual positive (negative) ideal solutions and is a point in the k-dimensional objective functional space. Using the PIS and the NIS, the following distance functions are obtained:  
\[d_{p}^{PIS}=\left\{\sum_{j\in k_{1}}w_{j}^{p}\left[\frac{z_{j}^{+}-z_{j}(x)}{z_ {j}^{+}-z_{j}^{-}}\right]^{p}+\sum_{i\in k_{2}}w_{i}^{p}\left[\frac{z_{i}(x)-z_ {i}^{+}}{z_{i}^{-}-z_{i}^{+}}\right]^{p}\right\}^{1/p} \tag{27}\]  
and  
\[d_{p}^{NIS}=\left\{\sum_{j\in k_{1}}w_{j}^{p}\left[\frac{z_{j}(x)-z_{j}^{-}}{z_ {j}^{+}-z_{j}^{-}}\right]^{p}+\sum_{i\in k_{2}}w_{i}^{p}\left[\frac{z_{i}^{-}- z_{i}(x)}{z_{i}^{-}-z_{i}^{+}}\right]^{p}\right\}^{1/p} \tag{28}\]  
where \(w_{j}\), \(w_{i}\), \(j=1,2,\cdots,k_{1}\), and \(i=1,2,\cdots,k_{2}\) are the relative weights of the objectives, \(k_{1}\) denotes the number of maximizing (or minimizing) objective functions, \(k_{2}\) represents the number of minimizing (or maximizing) objective functions and \(p=1,\ldots,\infty\). To obtain a compromise solution, the MOP problem is transformed into the following bi-objective problem with two justifiable goals:  
\[min\ \ d_{p}^{PIS}(x) \tag{29}\]  
\[max\ \ d_{p}^{NIS}(x) \tag{30}\]  
\[\text{s.t.}\ \ \ \ X=\{x\in X|g_{i}(x)\leq,=,\geq b_{i}, \tag{31}\]  
\(\forall i=1,\ldots,m,x\geq 0,x\in R^{k}\}\),  
where \(p=1,2,\ldots,\infty\),  
Next, the information from the PIS and NIS payoff tables was used to transform (27) and (28). Thus, \(d_{p}^{PIS}\) and \(d_{p}^{NIS}\) in (36) and (37) were acquired. \[\mu_{1}(x_{k})=\left\{ \begin{array}{ll}1&\text{if }d_{p}^{PIS}(x_{k})> \left(d_{p}^{PIS}\right)^{+}\\ 1-\frac{d_{p}^{PIS}(x_{k})-\left(d_{p}^{PIS}\right)^{+}}{\left(d_{p}^{PIS} \right)^{-}-\left(d_{p}^{PIS}\right)^{+}}\text{ if }\left(d_{p}^{PIS}\right)^{-} \leq d_{p}^{PIS}(x_{k})\leq\left(d_{p}^{PIS}\right)^{+}\\ 0&\text{if }d_{p}^{PIS}(x_{k})<\left(d_{p}^{PIS}\right)^{-}\end{array} \right. \tag{34}\]  
\[\mu_{2}(x_{k})=\left\{