## 3 Multiobjective Optimization Approach
### Multiobjective Interactive Method Utilizing the Two-Slope Parameterized ASFs  
In the following, we state an outline of the multiobjective interactive solution approach utilizing the two-slope parameterized ASFs (MITSPA) applying reference point based preference information. The general framework of interactive methods is usually similar: firstly, some range for Pareto optimal solutions is given the decision-maker, secondly, the decision-maker provides some preference information, thirdly, some solutions are presented for the decision-maker, and fourthly, the decision-maker express his/her opinion on the solutions and modify the preference information as a base for the new solutions. The process is stopped when the decision-maker is satisfied with the solution. The main differences in various interactive methods can be found in the ways the preference information is given and which solvers are applied (see e.g., [32; 33]). Similar approaches to ours in terms of the utilization of scalarization functions and the reference point as preference information are proposed, for instance, in [32; 33; 34; 35; 36; 37; 38]. Compared with these, in our case with the two-slope parameterized ASFs, we can systematically produce different Pareto optimal solutions to obtain a reasonably distributed selection of Pareto set. Multiobjective interactive method utilizing the two-slope parameterized ASFs (MITSPA)  
1. [leftmargin=*]
2. Give the ideal vector \(\mathbf{f}^{id}\), the nadir vector \(\mathbf{f}^{mad}\), and/or some Pareto optimal solution \(\mathbf{f}_{0}\) to the decision maker in order to illustrate the Pareto set. 3. Set the iteration counter \(h=1\) and select the maximum number of iterations \(h_{\text{max}}\). Ask the decision maker to provide the reference point \(\mathbf{f}_{h}^{R}\) and the number of solutions \(s\in\{1,\dots,k\}\) presented for each reference point. Initialize the positive coefficients \(\mathbf{\lambda}^{U}\) and \(\mathbf{\lambda}^{A}\). 4. Update the coefficients \(\mathbf{\lambda}^{U}\) and \(\mathbf{\lambda}^{A}\) if needed. Solve the problem (32) with the augmentation term (33) with the current reference point \(\mathbf{f}_{h}^{R}\). 5. Present \(s\) solutions to the decision maker and ask the decision maker to select the most preferable solution among them as the current solution \(\mathbf{f}_{h}\) and go to Step 5 or if more solutions for the current reference point \(\mathbf{f}_{h}^{R}\) are needed go to Step 4. 6. Present supplementary solutions to the decision maker. Ask the decision maker to select the most preferable solution among the previous \(s\) solutions and the supplementary solutions as the current solution \(\mathbf{f}_{h}\) and go to Step 5. 7. If \(h=h_{\text{max}}\) or the decision maker is satisfied with the current solution \(\mathbf{f}_{h}\), stop with the current solution as the final solution \(\mathbf{f}^{*}\). Otherwise, ask the decision maker to specify the new reference point \(\mathbf{f}_{h+1}^{R}\) as the current reference point, set \(h=h+1\), and go to Step 2. Some remarks about the above algorithm are in order. Step 0 consists of the illustration of the Pareto set. Some Pareto optimal solutions for the decision-maker to start with can be calculated, for example, by using the two-slope parameterized ASF (32) with an ideal vector as a reference point or by applying some suitable no-preference method like descent methods [39; 40; 41; 42; 43]. In Step 3, \(s\in[1,k]\) solutions are presented to the decision-maker, where the \(k\) is the number of objectives. As mentioned, with the two-slope parameterized ASF we are able to solve as many different solutions as there are objectives. If the number of objectives is high, it facilitates the task of the decision-maker if only some of these solutions are presented. However, if the decision-maker is willing to see more solutionsfrom the same reference point, this is enabled in step 4.