## 2 Problem Description
### Solving the Proposed Model (Modified Epsilon-Constrained Approach)  
In multi-objective problems, there is generally no single optimal solution that simultaneously optimizes all objective functions. Therefore, the concept of optimality is replaced with Pareto optimality or efficiency. Pareto optimal solutions (efficient, non-dominated) are solutions that cannot improve one objective function without worsening the performance of at least one other objective function ([PERSON], 2016). The Pareto set is a set of Pareto optimal solutions ([PERSON] et al., 2012). According to [PERSON] and [PERSON] (2012), there are three main approaches for solving multi-objective optimization problems based on decision maker preference elicitation: a priori methods, a posteriori or generative methods, and interactive methods. In a priori methods, the decision maker provides their preferences before the process begins. This method has the drawback of reducing preferences determined by the decision maker. Posteriori methods involve optimizing all objective functions simultaneously ([PERSON] et al., 2012). In this approach, the Pareto set is first generated. Then, at the end of the search process, the decision maker selects a preferred Pareto set. In interactive methods, the decision maker engages in the computation stages instead of expressing preferences in the dialogue stages. In this method, after several iterations, the process usually converges to a more desirable solution. The decision maker sequentially guides the search towards the preferred solution based on their responses. To solve this three-objective model, the epsilon-constrained method for exact solution is used. In this method, one of the objective functions is kept as the primary objective, and the other objective functions are transformed into inequality constraints based on minimization or maximization of the primary objective function as follows:  
\[\max ax \rightarrow ax\geq\varepsilon \tag{17}\] \[\min ax \rightarrow ax \leq\varepsilon \tag{18}\]  
In this model, the first constraint, which maximizes the covered demand, is kept as the objective function, while the second and third objectives, which minimize the cost and the amount of changes and maximize the greenness, are considered as constraints and are transformed as follows:  
\[\sum\
olimits_{l,j,t}F_{j,t}^{l}x_{j,t}^{l}+\sum\
olimits_{j}r_{j}(\varphi) \leq\varepsilon_{1} \tag{19}\]  
\[\sum\
olimits_{l,j,t}G_{j,t}^{l}x_{j,t}^{l}\geq\varepsilon_{2} \tag{20}\]  
As a result, the first objective function is modified as follows:  
\[Min/Max\bigg{(}f_{1}(x)+\vartheta\times\bigg{(}\frac{s_{2}}{r_{2}}+\frac{s_{ 3}}{r_{3}}+\cdots+\frac{s_{i}}{r_{i}}\cdots+\frac{s_{n}}{r_{n}}\bigg{)}\bigg{)} \tag{21}\]  
As a result, the first objective function is modified as follows:  
\[\max(\sum\
olimits_{i}h_{i}z_{i}^{0}+\vartheta\times\bigg{(}\frac{s_{2}}{r_{ 2}}+\frac{s_{3}}{r_{3}}\bigg{)}) \tag{22}\]  
\[\sum\
olimits_{l,j,t}F_{j,t}^{l}x_{j,t}^{l}+\sum\
olimits_{j}r_{j}(\varphi) -s_{2}=\varepsilon_{1} \tag{23}\]  
\[\sum\
olimits_{l,j,t}G_{j,t}^{l}x_{j,t}^{l}+s_{3}=\varepsilon_{2} \tag{24}\]  
\(s_{i}\in R^{+}\). According to the above equation, the Pareto optimal solutions are obtained, where r\({}_{i}\) is the range of the \(i\)th objective function, \(\vartheta\) is a small number between 0.001 and 0.000001, and S\({}_{i}\) is a non-negative auxiliary variable. First, the values of \(NIS_{fi}\) (worst value) and \(PIS_{fi}\)(best value) for each objective function are determined. Then, the value of the domain of the ith objective function is calculated according to the following equation:  
\[r_{i}=PIS_{fi}-NIS_{fi} \tag{25}\]  
Afterward, \(r_{i}\) is divided into equal intervals \(l_{i}\). Then, \(l_{i}+1\) points are obtained, and according to the following equation, the values of epsilon are determined based on these points (Grid points). In this method, the model must be solved for all obtained epsilons, where \(\eta\) is the number of grid points obtained according to the equation.