## 4 Methods for Generating a Near-Optimal Subset of Pareto Optimal Solutions
### Revised \(\epsilon\)-Constraint Method
\(K:=(f_{1}^{\max}-f_{1}^{\min})/\epsilon\). Set \(k\gets K\). We solve the following single-objective optimization model:

\[\text{[M4($k$)]:~{}}\min\left\{f_{2}(\mathbf{x})+\epsilon\frac{f_{1}(\mathbf{x})-f_{1} ^{\min}}{f_{1}^{\max}-f_{1}^{\min}}\right\} \tag{7}\]

subject to

Constraints (3)-(4),and let \(\dot{x}(k)\) be an optimal solution obtained and \(\Omega\leftarrow\Omega\cup\{\dot{x}(k)\}\). Let \(\hat{k}\) be the largest integer satisfying \(f_{1}^{\min}+\hat{k}e<f_{1}(\dot{x}(k))\). If \(\hat{k}<0\), return \(\Omega\) and stop. Otherwise, set \(k\leftarrow\hat{k}\) and repeat the above procedure of solving [M4(\(k\))]. Then, Propositions 5 and 6 are shown below. **Proposition 5**. : \(\dot{x}(k)\) _is a Pareto optimal solution._

Proof. : Suppose there is an \(x^{\#}\in X\) such that \(f_{1}(x^{\#})<f_{1}(\dot{x}(k))\) and \(f_{2}(x^{\#})\leq f_{2}(\dot{x}(k))\) or such that \(f_{1}(x^{\#})\leq f_{1}(\dot{x}(k))\) and \(f_{2}(x^{\#})<f_{2}(\dot{x}(k))\). Evidently, \(x^{\#}\) is feasible to [M4(\(k\))] and has a smaller objective value than that of \(\dot{x}(k)\), which contradicts the optimality of \(\dot{x}(k)\). **Proposition 6**. : _Let \(\Phi\) be the full set of Pareto optimal solutions. Then, \(\Omega\) is a \"good representation\" of \(\Phi\) in the sense that for any \(\bar{x}\in\Phi\), there exists an \(\dot{x}\in\Omega\) satisfying \(f_{1}(\dot{x})<f_{1}(\bar{x})+\epsilon\) and \(f_{2}(\dot{x})\leq f_{2}(\bar{x})+\epsilon\)._

Proof. : Similar to the equivalence between the revised \(\epsilon\)-constraint method and the improved revised \(\epsilon\)-constraint method, we can modify the augmented \(\epsilon\)-constraint method by enumerating all values of \(k=0,\ldots,K\). Next, we consider this modified version. For any \(\bar{x}\in\Phi\), let \(k\) be an integer such that \(f_{1}^{\min}+(k-1)e<f_{1}(\bar{x})\leq f_{1}^{\min}+ke\); evidently, \(k\in\{0,\ldots,K\}\). We now prove that \(\dot{x}(k)\) satisfies \(f_{1}(\dot{x}(k))<f_{1}(\bar{x})+\epsilon\) and \(f_{2}(\dot{x}(k))\leq f_{2}(\bar{x})+\epsilon\). Since \(\dot{x}(k)\) is optimal, and thus, feasible to [M4(\(k\))], \(f_{1}(\dot{x}(k))\leq f_{1}^{\min}+ke\). Combined with \(f_{1}^{\min}+(k-1)e<f_{1}(\bar{x})\), we have \(f_{1}(\dot{x}(k))<f_{1}(\bar{x})+\epsilon\). Evidently, \(\bar{x}\) is feasible to [M4(\(k\))]. Since \(\dot{x}(k)\) is optimal to [M4(\(k\))], \(f_{2}(\dot{x}(k))+e(f_{1}(\dot{x}(k))-f_{1}^{\min})/(f_{1}^{\max}-f_{1}^{\min}) \leq f_{2}(\bar{x})+\epsilon(f_{1}(\bar{x})-f_{1}^{\min})/(f_{1}^{\max}-f_{1}^ {\min})/(f_{1}^{\max}-f_{1}^{\min})\), which implies that \(f_{2}(\dot{x}(k))\leq f_{2}(\bar{x})+\epsilon\) because \((f_{1}(\bar{x})-f_{1}^{\min})/(f_{1}^{\max}-f_{1}^{\min})\in[0,1]\). Proposition 6 immediately implies Corollary 3 as follows. **Corollary 3**. : _If both \(f_{1}(x)\) and \(f_{2}(x)\) only take integer values (e.g., integer coefficients multiplied by integer decision variables), we can set \(\epsilon\) to be a number less than 1 (e.g., 0.9) and use the augmented \(\epsilon\)-constraint method to obtain a subset of Pareto optimal solutions whose objective vectors form the full set of nondominated objective vectors for the problem._

The pseudocode for the augmented \(\epsilon\)-constraint method is shown in Algorithm 4. ```
1:\(f_{1}^{\min}\leftarrow\min_{x\in X}f_{1}(x)\) // \(f_{1}^{\min}\) represents the minimum objective value of the first dimension. 2:\(f_{2}^{\min}\leftarrow\min_{x\in X}f_{2}(x)\) // \(f_{2}^{\min}\) represents the minimum objective value of the second dimension. 3:\(f_{1}^{\max}\leftarrow\min f_{1}(x)\) subject to \(f_{2}(x)\leq f_{2}^{\min},x\in X\) // \(f_{1}^{\max}\) 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.