# Multi-Objective Optimization of Spatially Truss Structures Based on Node Movement
## 2 MOEA Algorithm and Improvement
### Improvement to MOEA
#### 2.2.2 Pareto Dominate Relations  
When there are multiple targets, there are usually solutions that cannot be easily compared. This solution is usually the Pareto optimal solution. The definition of decision space and target space are given here.  
Definition (1): definition of decision space:  
\[\Omega=\Big{\{}x\in R^{n}\big{|}g_{i}\geq 0,\,h_{j}(x)=0\,;\,(i=1,2,\ldots,\,k \,;\,j=1,2,\ldots,\,k)\,\Big{\}}, \tag{4}\]  
Definition (2): definition of target space:  
\[\Pi=\{f(x)|x\in\Omega\}, \tag{5}\]  
The objective function _f(x)_ maps \(\Omega\) to the set \(\Pi\).  
Definition (3): definition of dominate relations:  
Let \(x_{a}\) and \(x_{b}\) be two different solutions. If \(x_{a}\) dominates \(x_{b}\), the two conditions of Equations (6) and (7) must be satisfied.  
For all subgoals, \(x_{a}\) is no worse than \(x_{b}\):  
\[\forall i\in\{1,2,\ldots,m\}\ \ f_{i}(x_{a})\leq f_{i}(x_{b}). \tag{6}\]  
At least there are subgoals that make \(x_{a}\) better than \(x_{b}\):  
\[\exists j\in\{1,2,\ldots,m\}\ \ f_{j}(x_{a})<f_{j}(x_{b}), \tag{7}\]  
where \(m\) is the number of subgoals; then \(x_{a}>x_{b}\) \"\(\sim\)\"represents dominate relations.  
The distribution of each individual in the population is shown by a Pareto strength value, indicating the number of individual \(i\) in the evolutionary population \(P\). The calculation formula is shown as Equation (8):  
\[S(i)=\Big{|}\Big{\{}j\big{|}\,j\in P\,,\,\,i>j\Big{\}}\,\Big{|}. \tag{8}\]  
Based on the Pareto strength value, the fitness value of an individual represents the sum of the strength values of the dominant individual in the evolutionary population. The calculation formula is shown as Equation (9):  
\[R(i)=\sum_{j\in P\,,\,j>i}\,S(j). \tag{9}\]  
If there is the same original fitness value between individuals, the density information of individuals is combined into the final fitness. The density information here refers to the application of the adjacent individual strategy, that is, the distance between individual \(i\) and other individuals in the evolutionary population \(P\), then calculated and arranged in increasing order. The calculation formula is shown as Equation (10):  
\[D(i)=1/(\delta_{i}^{k}+2), \tag{10}\]  
where \(\delta_{i}^{k}\) where is the distance between individual \(i\) and the \(k\)th adjacent individual.  
When the distance between individual \(i\) and the \(k\)th adjacent individual is larger, the individual distribution is better; the smaller the fitness value, the more likely the individual is to be selected for the next evolution.  
Considering these factors, the C# language was adopted to compile the MOEA optimization program based on the Pareto strength value order, that each individual in the population is distributed with a strength value, representing the number of individuals dominated by individual in the evolutionary population. The fitness value of an individual represents the sum of the strength values of the dominant individual in the evolutionary population. The smaller the fitness value, the more likely the individual is to be selected for the next evolution.  
The ANSYS scripting language APDL was used to write the analytical procedure for the truss finite element in this study. During the counting process, two objective functions (weight and maximum displacement of the structure) were calculated by the finite element method, and this method adopted an alternative pattern to update in real time by way of transferring the value to the optimization module through the connector.