## 2 Methodology  
### Definition for Multi-Objective Optimization (MOO) Problem  
The MOO problem is generally regarded as a mathematical optimization problem with multiple objective functions to be optimized simultaneously [31]. Mixture proportion optimization involves determining specifications and contents of raw materials. Generally, some objectives are used to evaluate the mechanical and pavement performances, such as strength, stability, and so on. With the diversification of additives and modifiers, competing objectives, such as additive contents and performances are introduced when designing a proportion for road materials. At present, the mixture proportion optimization methods involving only a single mechanical performance have gradually lost recognition. Many researchers tried to construct multi-objective mathematical models to obtain better mixture proportion schemes. However, as far as we know, there are few papers reported to investigate a general optimization model of asphalt mixture proportion. Thus, the MOO problem for mixture proportion is a complex combinatorial optimization problem with more challenges.  
Figure 1: The study structure of metaheuristic optimization based on the machine learning (ML) model.  
MOO problem could be defined to search the optimal set of input variables for minimizing and/or maximizing multiple objectives simultaneously. A typical MOO mathematical formulation can be expressed as:  
\[\begin{split}\min/\max\quad\mathbf{F}(\mathbf{x})&=[f_{ 1}(\mathbf{x}),f_{2}(\mathbf{x}),\ldots,f_{m}(\mathbf{x})]^{T}\\ &\qquad\qquad\qquad\qquad\qquad s.t.\\ g_{jmin}\leq g_{j}(\mathbf{x})&\leq g_{jmax}(j=1,2, \ldots,p)\end{split} \tag{1}\]  
where \(\mathbf{x}\) is the vector of design variables, \(\mathbf{x}\in[x_{1},\ x_{2},\ \ldots,\ x_{\cdots}]^{T}\); \(\mathbf{F}(\mathbf{x})=[f_{1}(\mathbf{x}),f_{2}(\mathbf{x}),\ldots,f_{m}( \mathbf{x})]^{T}\) denotes the objective function vector containing \(m\) objective functions; \(g_{j}(\mathbf{x})\) and \(p\) represent the constraint function and the number of constraints, respectively.  
In general, a solution to the MOO problem in Equation (1) is known as a Pareto optimal solution without degrading some objective values, which is considered equally efficient. Thus, from different viewpoints, a set of Pareto optimal solutions could be found according to the goals, that could satisfy the subjective preferences. Due to multiple Pareto optimal solutions for MOO problems, these usually require a lot of evaluations. Therefore, some methods are used to convert the original MOO problem into a single-objective optimization problem by a weighted-average method as follows [32]:  
\[\mathbf{F}(\mathbf{x})=\sum_{q=1}^{m}\omega_{q}\frac{f_{q}}{f_{q}*} \tag{2}\]  
where \(\omega_{q}\) is the \(q\)-th weight coefficient; \(f_{q}*\) is the normalized factor to the \(q\)-th objective function.  
However, the above weighted-average method has some disadvantages. The optimal solution by the method relies on a suitable weight coefficient [33]. Moreover, the scaling coefficients are often chosen based on the ideal optimal solutions of each single optimization problem which can be rather time-consuming with increase in the number of objectives. It cannot find certain Pareto-optimal solutions in the case of a nonconvex objective space. For the above reasons, this study aims to investigate a novel MOO method by combining the GPR based on ML methods with PSO based on metaheuristic algorithms. The GPR method is utilized as an optimizer for the objective functions, and the PSO algorithm is adopted to search the optimal solution for MOO problems.