## 2 Background
### MOLP and STEM  
A general formulation of the MOLP problem is given in model 2. \[ \begin{array}{c}\rm{Max}\ \ G(u)=(g_{1}(u),\ldots,g_{k}(u))=\left(C_{1}^{T}u, \ldots,C_{k}^{T}u\right)\\ \rm{s.t.}\ u\in W=\{u\in R_{+}^{v}\,|\,Au\,\leq b\}\end{array}  \tag{2}\]  
where \(G(u)=(g_{1}(u),\ldots,g_{k}(u))\) represents the objective function vector. Linear objective functions are denoted by \(C_{1}^{T}u\) where u = (u\({}_{1}\),..., u\({}_{v}\)) is the decision-making vector. The symbol T is a transposed vector. \(g_{i}(u)\), \(C_{i}\), and \(c_{i}\) are the \(l^{\rm th}\) objective function, the vector of decision-making variable coefficients in the \(l^{\rm th}\) objective function, and the coefficients of \(l^{\rm th}\) objective function namely \(C_{i}\), \(l=1\),..., \(k\), per n existing variables, respectively. W is the feasible region of the MOLP problem and k is the number of the objective function. The decision-making variable multiples matrix is denoted by A, while b represents the right-hand side vector of the constraints. Au \(\leq b\) is the constraints of the feasible region and \(R_{+}^{v}\), representing the Euclidean space comprising all nonnegative vectors in a v-dimensional space. In this MOLP problem, the \(l^{\rm th}\) objective function is formulated as \(C_{l}^{T}u=c_{l1}u_{1}+c_{l2}u_{2}+\cdots+c_{vl}u_{v}\), \(1=1,\ldots,k\). The vector u\({}^{v}\in W\) is considered as an efficient (non-dominated) solution, if there does not exist another u \(\in W\), such that \(g_{l}(u)\geq g_{l}(u*)\) for all l and \(g_{l}(u)>g_{l}(u*)\) for at least one l.  
STEM is an IM that can be used to solve MOLP problems. It works based on the obtained information from DM preferences and reduces the feasible region, step by step. STEM relies on DMs information to identify feasible and efficient solutions during the procedure. STEM includes following steps [27; 30]:  
* building-up the pay-off table  
Objective functions should be optimized separately as follows (cf. model 3 and Table 1):  
\[ \begin{array}{c}\rm{Max}\ \ g_{i}(u)=C_{i}^{T}u\\ \rm{s.t.}\ u\in W=\{u\in R_{+}^{v}\,|\,Au\,\leq b\}\end{array} \]The diagonal elements, represented by \(g_{1}^{*}\), are the optimal solutions for the single \(g_{1}(u)=C_{1}^{T}(u)\), \(l=1,\ldots,k\) problem obtained through the solving of model 3. \(z_{\text{dl}}\) values are the results for \(\text{d}^{\text{th}}\) objective function, computed upon the optimal solution obtained for \(\text{l}^{\text{th}}\) objective function, \(l=1,\ldots,k\), \(d=1,\ldots,k\), \(d\
eq l\). 1. Calculation Phase  
The computation of coefficients \(\beta_{1}\), \(l=1\ldots,k\), is the cornerstone to compute the relative importance (Equation (4)) of each distance from the optimal objective function value. Suppose that \(\pi_{1}\) denotes the relative importance of the distance between objective functions and their optimal values. Although these coefficients are locally meaningful, they cannot capture the overall importance, unlike other utility models. It is therefore necessary to solve model 5, where the solution obtained in the \(\text{p}^{\text{th}}\) iteration is denoted by \(\text{G}^{\text{P}}=(g_{1}(u^{\text{P}}),g_{2}(u^{\text{P}}),\ldots,g_{k}(u^{ \text{P}}))\). \[\pi_{1}=\left\{ \begin{array}{l}\left(\frac{\left(g_{1}^{*}-g_{1}^{\text{min} }\right)}{g_{1}^{*}}\right)\left[\frac{1}{\sqrt{\sum_{d-1}^{*}(C_{\text{dl}}) ^{2}}}\right]\text{ if }g_{1}^{*}>0\\ \\ \left(\frac{g_{1}^{\text{min}}-g_{1}^{*}}{g_{1}^{*}}\right)\left[\frac{1}{ \sqrt{\sum_{d-1}^{*}(C_{\text{dl}})^{2}}}\right]\text{ if }g_{1}^{*}\leq 0\end{array} \right. \tag{4}\]  
and \(\beta_{1}=\frac{\pi_{1}}{\sum_{d-1}^{k}\pi_{d}}\), \(l=1,\ldots,k\). Model 5 is given as follows:  
\[\text{Min }h\]  
\[\text{s.t }h\geq(g_{1}^{*}-g_{1}(u)).\beta_{1}1=1,\ldots,k,\]  
\[\text{u}\in W^{\text{P}},\,h\geq 0.\,\text{Where }W^{\text{P}}=\{u|\text{Au}\leq b,u\geq 0 \}\cup W^{\text{P}-1} \tag{5}\]  
W\({}^{\text{P}}\) represents the feasible region in the \(\text{p}^{\text{th}}\) iteration.