## 2 Knowledge and Tools of Model Preparation
Here, the objective function \(F(x):x\to\mathbb{R}^{m}\) includes \(M\) conflicting object functions with continuous real values \(f_{1}(x),f_{2}(x),\cdots,f_{m}(x)\), and \(\Omega\) is described as  
\[\Omega=\{x\in\mathbb{R}^{n}\Big{|}f_{j}(x)\leq 0,\,j=1,\ldots,m\} \tag{8}\]  
where \(\mathbb{R}^{m}\) represents the target space. The pareto dominance relation of individuals is as follows: if there are decision vectors U and V, and satisfy the following two conditions at the same time, we call U dominance V:  
(i) If and only if \(f_{i}(u)\geq f_{i}(v)\), for every \(i\in\{1,\ldots,m\}\). (ii) There exists at least one index \(j\in\{1,\ldots,m\}\) make \(f_{i}(u)>f_{i}(v)\). In this case V is said to be dominated by U, which can be denoted by \(u\succ v\), and among \(\succ\) are dominant relations. If there is no point \(x\in\Omega\) that makes \(\mathbf{F}(x)\) dominate \(\mathbf{F}(x^{*})\), the point \(x^{*}\in\Omega\) is Pareto optimal. That is, there is only one best set of compromise solutions called the non-dominated (not dominated by all other solutions). The value of Pareto optimization solution in decided space and target space is defined as Pareto solution set (PS) and Pareto frontier (PF). MOEA/D has strong search ability for continuous optimization, combinatorial optimization and PS complex problems. The principle of the algorithm is:  
If a multi-objective optimal problem similar to Equation (7) and a weight vector \(\lambda=(\lambda^{1},\cdots,\lambda^{m})\) are given, and the given weight vector satisfies \(\sum_{i=1}^{m}\lambda^{i}=1\), \(\lambda^{i}\geq 0\), \(i=1,2,\ldots,m\). MOEA/D based on Tchebycheff decomposition uses this weight vector to optimize a MOP into several sub-problems by the following methods. \[\underset{x\in\Omega}{\text{\emph{m}ing}^{lc}}(x\Big{|}\lambda^{i},z^{*})= \underset{x\in\Omega}{\text{\emph{min}}}\underset{1\leq i\leq m}{\text{\emph {max}}}\{\lambda^{i}_{i}\Big{|}f_{i}(x)-z^{*}_{i}\} \tag{9}\]  
where \(z^{*}=(z^{*}_{1},z^{*}_{2},\cdots,z^{*}_{m})\) (_i.e._, \(z^{*}_{i}<\min\Big{\{}f_{i}(x)\big{|}x\in\Omega\Big{\}},1,2,\ldots,m\)) is the ideal point, and \(\lambda^{i}\!\!=(\lambda^{j}_{1},\cdots,\lambda^{j}_{m})^{T}\). By solving multiple sub-problems with different weight vectors in Equation (9), Pareto optimal solution set with good diversity can be obtained [41]. As is known that \(g^{lc}\) is continuous of \(\lambda\), so if \(\lambda^{i}\) is close to \(\lambda^{j}\), the \(g^{lc}(x\big{|}\lambda^{i},z^{*})\) solution must close to the \(g^{lc}(x\big{|}\lambda^{j},z^{*})\) solution. Hence, a useful tool of \(g^{lc}(x\big{|}\lambda^{j},z^{*})\) optimization is information about \(g^{lc}\) with weight vectors near the \(\lambda^{i}\). In the algorithm MOEA/D, the population is made up of the optimal solution of the sub-problem currently found. Each sub-problem maintains a list of neighbors, which preserves sub-problems with weight vectors similar to the sub-problem. Therefore, under the assumption of continuity, two neighbor sub-problems should have similar optimal solutions. In each generation of MOEA/D, each sub-problem is optimized applying only the message of its neighbor sub-problems. As for each generation \(t\), MOEA/D using the Tchebycheff holds [42]:  
(1) A point group \(x^{1},\cdots,x^{N}\in N\), in which the \(x^{i}\) is the present solution for the \(i\)-th sub-problem. (2) \(FV^{1},\cdots,FV^{N}\), where \(FV^{i}=\mathbf{F}(x^{i})\) for each \(i=1,\ldots,N\). (3) The best value \(z_{i}\) for objective \(f_{i}\) now and \(z=(z_{1},\cdots,z_{m})^{T}\). (4) An external population (EP), applying to store non-dominated solutions found during the search. The pseudo code of MOEA/D is described as follows:  
The third is **Fuzzy C-Means (FCM) clustering algorithm**. FCM clustering algorithm divides the sample points in the sample space \(\mathbf{X}=\{x_{1},x_{2},\cdots,x_{n}\}\) into \(c\) (\(c>1\)) classes, and the degree of each sample point \(x_{i}\) belonging to the \(k\)-th (\(1\leq k\leq c\)) class is expressed as \(u_{ik}\).