## 2 Related Backgrounds
### Multiobjective Evolutionary Algorithm Based on Decomposition  
MOEA/D [17] is a new framework for solving multi-objective optimization problems. It decomposes a multiobjective optimization problem into a number of scalar optimization subproblems and optimizes them simultaneously. In MOEA/D, the weighted sum approach, the Tchebycheff approach, and the boundary intersection approach are used to convert the problem of approximation of the \(PF\) into a number of scalar optimization problems. The description of MOEA/D is as follows. ```
Input: MOP, the multiobjective optimization problem; \(NP\), population size; \(GEN\), maximum generations; \(FES\), the maximal number of function evaluations; \(NB\), the number of weight vectors in the neighborhood of each weight vector; Output: X, the non-dominated population; Approximation to the PF, \(\{\mathrm{F}(x^{1})\),...,\(\mathrm{F}(x^{N})\}\)
1: Randomly initial population \(\mathrm{X_{n}}\)
2: Compute the Euclidian distances between any two weight vectors and define neighborhood of each subproblem. For the _ith_ subproblem, set \(\mathrm{B(i)}=\{\mathrm{i}_{1},\mathrm{i}_{2},\ldots\mathrm{i}_{m}\}\), where \(\lambda^{i1}\), \(\lambda^{i2}\),..., \(\lambda^{im}\) are the \(m\) closest weight vectors to \(\lambda^{i}\)
3: Initialize z=(z\({}_{1}\),..., z\({}_{m}\))\({}^{T}\) by problem-specific method
4: while \(t<FES\) do
5: for \(i=1\):\(NP\) do
6: Produce \(y\) by DE-mutation strategies
7: polynomial mutation
8: Evaluation \(y\)
9: for each \(j=1\) to m do
10: if z\({}_{j}\)\(>\) f\({}_{j}(y)\) then z\({}_{j}\)\(=\) f\({}_{j}(y)\)
11: endfor
12: for each j \(\in\) B(i) do
13: if g\({}^{\mathrm{e}}(y|\lambda^{j}\),z) \(<\) g\({}^{\mathrm{e}}(x^{i}|\lambda^{j}\),z)then
14: x\({}^{i}\)\(=\)\(y\) and F(x\({}^{i}\)) \(=\) F(y)
15: endif
16: endfor
17: endfor
18: endwhile
```  
**Algorithm 1**The procedure of MOEA/D  
### Multiobjective Evolutionary Algorithm Based on Decomposition  
MOEA/D [17] is a new framework for solving multi-objective optimization problems. It decomposes a multiobjective optimization problem into a number of scalar optimization subproblems and optimizes them simultaneously. In MOEA/D, the weighted sum approach, the Tchebycheff approach, and the boundary intersection approach are used to convert the problem of approximation of the \(PF\) into a number of scalar optimization problems. The description of MOEA/D is as follows. ```
Input: MOP, the multiobjective optimization problem; \(NP\), population size; \(GEN\), maximum generations; \(FES\), maximum number of function evaluations; \(NB\), the number of weight vectors in the neighborhood of each weight vector; Output: X, the non-dominated population; Approximation to the PF, \(\{\mathrm{F}(x^{1})\),...,\(\mathrm{F}(x^{N})\}\)
1: Randomly initial population \(\mathrm{X_{n}}\)
2: Compute the Euclidian distances between any two weight vectors and define neighborhood of each subproblem. For the _ith_ subproblem, set \(\mathrm{B(i)}=\{\mathrm{i}_{1},\mathrm{i}_{2},\ldots\mathrm{i}_{m}\}\), where \(\lambda^{i1}\), \(\lambda^{i2}\),..., \(\lambda^{im}\) are the \(m\) closest weight vectors to \(\lambda^{i}\)
3: Initialize z=(z\({}_{1}\),..., z\({}_{m}\))\({}^{T}\) by problem-specific method
4: while \(t<FES\) do
5: for \(i=1\):\(NP\) do
6: Produce \(y\) by DE-mutation strategies
7: polynomial mutation
8: Evaluation \(y\)
9: for each \(j=1\) to m do
10: if z\({}_{j}\)\(>\) f\({}_{j}(y)\) then z\({}_{j}\)\(=\) f\({}_{j}(y)\)
11: endfor
12: for each j \(\in\) B(i) do
13: if g\({}^{\mathrm{e}}(y|\lambda^{j}\),z) \(<\) g\({}^{\mathrm{e}}(x^{i}|\lambda^{j}\),z)then
14: x\({}^{i}\)\(=\)\(y\) and F(x\

## 3 Frameworks of MOEA/D-CPDE

In MOEA/D, the new individual is produced by combining differential mutation with polynomial mutation, which is conducive to maintain the population diversity and improve the exploration ability. However, the convergence speed of MOEA/D may be slower. Inspired by the cloud model, a new evolution mechanism is proposed for solving the problem.