## 4 Methods for Generating a Near-Optimal Subset of Pareto Optimal Solutions
### Revised \(\epsilon\)-Constraint Method
The question is, is it possible that there is a \(\hat{k}\) satisfying \(\hat{k}+1\leq\hat{k}\leq k-1\) and the objective vector of solution \(\mathbf{x}^{*}(\hat{k})\) generated by the revised \(\epsilon\)-constraint method is not an objective vector of any solution generated by the improved revised \(\epsilon\)-constraint method? The answer is no, for the following reasons. [M2(\(k\))\({}^{\prime}\)] is a relaxation of [M2(\(\bar{k}\))] (because \(k\leq k-1<k\)) and they have the same objective function. Corollary 1 states that \(\mathbf{x}^{*}(k)^{\prime}\) is optimal to [M2(\(k\))\({}^{\prime}\)] and \(\mathbf{x}^{*}(\bar{k})\) is optimal to [M2(\(\bar{k}\))]. Therefore, the optimal objective value of [M2(\(k\))\({}^{\prime}\)] is smaller than or equal to that of [M2(\(\bar{k}\))], i.e., \(f_{2}(\mathbf{x}^{*}(k)^{\prime})\leq f_{2}(\mathbf{x}^{*}(\bar{k}))\). Moreover, since we \"let \(\hat{k}\) be the largest integer satisfying \(f_{1}^{\min}+\hat{k}e<f_{1}(\mathbf{x}^{*}(k)^{\prime})\), \(f_{1}^{\min}+(\hat{k}+1)\epsilon\geq f_{1}(\mathbf{x}^{*}(k)^{\prime})\); combined with \(\hat{k}+1\leq k\leq k-1\), we have \(f_{1}(\mathbf{x}^{*}(k)^{\prime})\leq f_{1}^{\min}+\hat{k}\epsilon\), meaning that \(\mathbf{x}^{*}(k)^{\prime}\) is feasible to [M2(\(\bar{k}\))]. Thus, we have \(f_{2}(\mathbf{x}^{*}(k)^{\prime})\geq f_{2}(\mathbf{x}^{*}(\bar{k}))\) (\(\mathbf{x}^{*}(\bar{k})\) is optimal to [M2(\(\bar{k}\))]). Therefore, \(f_{2}(\mathbf{x}^{*}(k)^{\prime})=f_{2}(\mathbf{x}^{*}(\bar{k}))\). Hence, model [M3(\(k\))\({}^{\prime}\)] is identical to model [M3(\(\bar{k}\))]. Therefore, the objective vector of solution \(\mathbf{x}^{*}(\bar{k})\) is the same as the objective vector of solution \(\mathbf{x}^{*}(k)^{\prime}\). The pseudocode for the improved revised \(\epsilon\)-constraint method is shown in Algorithm 3. ```
1:\(f_{1}^{\min}\leftarrow\min_{\mathbf{x}\in X}f_{1}(\mathbf{x})\) // \(f_{1}^{\min}\) represents the minimum objective value of the first dimension. 2:\(f_{2}^{\min}\leftarrow\min_{\mathbf{x}\in X}f_{2}(\mathbf{x})\) // \(f_{2}^{\min}\) represents the minimum objective value of the second dimension. 3:\(f_{1}^{\max}\leftarrow\min f_{1}(\mathbf{x})\) subject to \(f_{2}(\mathbf{x})\leq f_{2}^{\min},\mathbf{x}\in X\) // \(f_{1}^{\max}\) represents the minimum objective value of the first dimension when the second dimension achieves its minimum objective value. 4:\(\epsilon\leftarrow\) a small positive number such that \(f_{1}^{\max}-f_{1}^{\min}\) is an integer multiple of \(\epsilon\) // \(\epsilon\) represents the tolerance. 5:\(k\leftarrow(f_{1}^{\max}-f_{1}^{\min})/\epsilon\)
6:\(\hat{\mathbf{x}}(k)\leftarrow\) null // \(\hat{\mathbf{x}}(k)\) represents an optimal solution obtained by [M2(\(k\))]. 7:\(\mathbf{x}^{*}(k)\leftarrow\) null // \(\mathbf{x}^{*}(k)\) represents an optimal solution obtained by [M3(\(k\))]. 8:\(\Omega\leftarrow\varnothing\) // \(\Omega\) represents a multiset composed of obtained solutions by the improved revised \(\epsilon\)-constraint method. 9:\(\hat{k}\gets\) 0 // \(\hat{k}\) represents the largest integer satisfying \(f_{1}^{\min}+\hat{k}\epsilon<f_{1}(\mathbf{x}^{*}(k))\). 10:while\(\hat{k}\geq 0\)do
11: Solve the model [M2(\(k\))], and obtain an optimal solution \(\mathbf{\hat{x}}(k)\). 12: Solve the model [M3(\(k\))], and obtain an optimal solution \(\mathbf{x}^{*}(k)\). 13:\(\Omega\leftarrow\Omega\cup\{\mathbf{x}^{*}(k)\}\). 14: Calculate the largest integer \(\hat{k}\) satisfying \(f_{1}^{\min}+\hat{k}\epsilon<f_{1}(\mathbf{x}^{*}(k))\). 15:\(k\leftarrow\hat{k}\). 16:endwhile
17:Return \(\Omega\). ```

**Algorithm 3** Improved revised \(\epsilon\)-constraint method

### Augmented \(\epsilon\)-Constraint Method

A deficiency of the improved revised \(\epsilon\)-constraint method is that we must solve two models each time: [M2(\(k\))] and [M3(\(k\))]. Hence, the augmented \(\epsilon\)-constraint method that only needs to solve one model was developed [31]. The augmented \(\epsilon\)-constraint method is elaborated as follows. Define a multiset \(\Omega:=\varnothing\).