## 3 Overview and Insights on Bi-Objective Optimization  
Bi-objective optimization problems aim to find a set of Pareto optimal solutions that balance two trade-off objectives, providing scientific guidance for decision makers. For ease of elaboration in the subsequent content, a typical bi-objective optimization model (1) with both objectives to be minimized is introduced.  
\[\begin{cases}\min_{x\in X}f_{1}(\mathbf{x})\\ \min_{x\in X}f_{2}(\mathbf{x}),\end{cases} \tag{1}\]  
where \(X\) is a nonempty compact set and both \(f_{1}(\mathbf{x})\) and \(f_{2}(\mathbf{x})\) are lower semi-continuous over \(X\). \((f_{1}(\mathbf{x}),f_{2}(\mathbf{x}))\) is an objective vector and it is a feasible objective vector if \(\mathbf{x}\in X\). The set of all feasible objective vectors represented by \(\mathbb{Y}=\{(f_{1}(\mathbf{x}),f_{2}(\mathbf{x}))|\mathbf{x}\in X\}\) is the feasible region in the criteria space. An objective vector \(\bar{\mathbf{y}}\in\mathbb{Y}\) is nondominated if there does not exist another \(\mathbf{y}\in\mathbb{Y}\) such that \(y_{i}\leq\bar{y}_{i}\) for each \(i\in\{1,2\}\) and \(y_{i}<\bar{y}_{i}\) for at least one \(i\). A Pareto optimal solution is a feasible solution with a nondominated objective vector. In other words, the Pareto optimal solutions are the solutions that cannot be improved in one objective function without deteriorating their performance in at least one of the rest [31]. The set of Pareto optimal solutions, denoted by \(\Phi=\{\mathbf{x}\in X|(f_{1}(\mathbf{x}),f_{2}(\mathbf{x}))\in\mathbb{Y}\}\), is called the Pareto frontier.  
A natural approach for multi-objective optimization is to generate the whole efficient frontier and then decide about the preferred solution in a higher-level decision-making process. However, we find that generating a partial efficient frontier is sufficient to deal with the multi-objective optimization problem. There are three reasons that we generate only a subset of Pareto optimal solutions. First, the full set of Pareto optimal solutions is often too large, e.g., it may contain 10,000 Pareto optimal solutions. It will then be hard for the decision makers to choose one solution. Second, the full set of Pareto optimal solutionsmay contain an infinite number of solutions, or, even if the cardinality of the full set (i.e., the number of elements in the full set) is finite, it may be computationally challenging to generate all solutions in the full set. Third, some Pareto optimal solutions have the same objective vector and we only need one of them.  
It is worth noting that the subset of Pareto optimal solutions generated is not arbitrary. In fact, we aim to generate a subset of Pareto optimal solutions whose resulting set of nondominated objective vectors is the same as, or at least a good approximation of, the full set of nondominated objective vectors for the problem. For instance, in a bi-objective optimization problem that aims to minimize transportation time and fuel consumption, suppose that the full set of nondominated objective vectors is composed of \((38,275)\), \((38.1,274.5)\), \((38.2,274)\), \((50,220)\), and \((50.1,215)\), where the first element of each objective vector represents the hours required for transporting cargo, and the second element represents the fuel consumption in liters. Since several nondominated objective vectors are close in value, a subset consisting of Pareto optimal solutions with objective vectors \((38.2,274)\) and \((50.1,215)\) can be used as a good approximation of the full set of Pareto optimal solutions.