## 3 Multiobjective Optimization Approach
### Two-Slope Parameterized ASFs  
We approach the multiobjective mixed-integer problem with a special type of achievement scalarizing functions. In general, the utilization of the achievement scalarizing function (ASF) aims to find a Pareto optimal solution being as close as possible to a so-called reference point \(\mathbf{f}^{R}\). The components \(f_{i}^{R}\), \(i\in I\) include the decision maker's wishes for each objective. This search is done by transforming the multiobjective optimization problem to a certain type of a scalarized problem and then applying some suitable single-objective optimization method. We use here the two-slope parameterized ASF, proposed in [25], which is a generalization of the parameterized ASF [26] and the two-slope ASF [27]. Usually, to find the closest point to the reference point \(\mathbf{f}^{R}\), the distance from \(\mathbf{f}^{R}\) is measured with only one metric. With the parameterization used in the parameterized ASF and the two-slope parameterized ASF, we can combine different metrics such that \(L_{\infty}\) and \(L_{1}\) metrics are the extreme cases. Thus, by systematically producing different Pareto optimal solutions from the same preference information, we can give the decision maker a wider perspective to the range of Pareto optimal solutions. Another benefit of the two-slope parameterized ASF, as well as the two-slope ASF, is that we do not need to test the achievability of the reference point. This is due to the fact that the different weights are used depending on if the reference point is achievable (i.e., the reference point belongs to the image of the feasible solutions in the objective space) or unachievable. The use of different weights is reasonable since the decision-maker usually prefers different solutions if the reference point is achievable or not, as was suggested in [28]. In order to solve the model described in Section 2, we apply the two-slope parameterized ASF. Once the multiobjective problem is converted to the single-objective one, we obtain a scalarized version of the problem (31) in the form [25]  
\[\min_{\mathbf{x}\in X}\max_{ \begin{subarray}{c}I\in I\\ |I|=q\end{subarray} }\Bigg{\{}\sum_{i\in I}\Big{[}\max\{\lambda_{i}^{U}(f_{i}( \mathbf{x})-f_{i}^{R}),0\}+\min\{\lambda_{i}^{A}(f_{i}(\mathbf{x})-f_{i}^{R}),0\}\Big{]} \Bigg{\}}, \tag{32}\]  
where the weighting vectors \(\lambda_{i}^{U},\lambda_{i}^{A}>0\) for all \(i\in I\) are for the unachievable and the achievable reference point, respectively. The parameter \(q\in I\) specifies which metric is used and \(I^{q}\) is a set containing \(q\) integers from the interval \([i,k]\), where \(k\) is the total number of objectives. Then, the maximization is taken over all the sets including \(q\) integers from the interval \([1,k]\). In order to gain the benefits of the parameterization, or in other words, to use more metrics than only \(L_{1}\) (i.e., \(q=k\)) and \(L_{\infty}\) (i.e., \(q=1\)), the problem must contain at least three objectives while the maximum number of different metrics equals the number of the objectives. Next, we are interested to know what can be deduced from the optimal solution of the scalarized problem. As the justification for the use of the two-slope parameterized ASF, we can proof the following results by adapting the proofs from [25]. Theorem 1 ([25]): _For the scalarized problem (32) it holds that:_  
1. _Any optimal solution of the scalarized problem is weakly Pareto optimal for the problem (_31_)._
2. _Among optimal solutions of the scalarized problem, there exists at least one Pareto optimal solution for the problem (_31_)._
3. _If_ \(\mathbf{x}^{*}\) _is a weakly Pareto optimal solution for the problem (_31_), then it is a solution of the scalarized problem (_32_) with_ \(\mathbf{f}^{R}=\mathbf{f}(\mathbf{x}^{*})\)_, and the optimal value is zero._  
(i) Assume that \(\mathbf{x}^{*}\) is an optimal solution of the problem (32) but not a weakly Pareto optimal solution of the problem (31).