# Genetic Multi-Objective Optimization of Sensor Placement for SHM of Composite Structures
## 4 Analysis and Discussion  
The analysis of the obtained results indicates that the process of multi-objective optimization based on Pareto front leads to many nondominated solutions, which allows for binding effectiveness of the SHM system with design variables. Based on this information, the designer of the SHM system may decide on the number of sensors in the individual direction, their placement, and the number of modes, simultaneously, while keeping an appropriate level of sensitivity in the system in damage detection as well as an appropriate level of possible false alarms. To facilitate the SHM user to make decisions about the characteristic sensitivity, specificity, and the utilized sensor number, the obtained results of optimization can be further postprocessed to support this selection. A common-sense approach that can be used for that purpose is a Pareto principle that 80% of the effects come from 20% causes. Using this principle, the Pareto front solutions have been divided into the following subset:  
\[\frac{\sum_{z\in A}g(z)}{\sum_{z\in Z}g(z)}\geq 0.8\ for\ g(z)=-\log(z+1)\ and\ g(z_{1})\leq g(z_{2})\leq\cdots\leq g(z_{n}) \tag{9}\]  
where  
\[\left\{z\in A\ \big{|}\ z=f_{1}+f_{2}+\alpha\cdot f_{3}\right\}\text{ or }\{z\in B \ |\ z=\beta\cdot f_{1}+f_{2}\}.\]  
Figure 4: Results of the multi-objective optimization process: (**a**) scatter plot of the results representing approximation of three-dimensional Pareto front in the multi-objective function domain, (**b**) scatter plot of parameter space of design variables, (**c**) a side view Pareto front presenting perspective of type-I error value versus number of sensors, (**d**) a side view Pareto front presenting perspective of type-II error value versus number of sensors. Annotation \(z_{i}\) represents individual solutions of a Pareto front, \(g(z)\) is the negation transformation function, and \(z\in A\) represents the subset of Pareto solutions with a cumulative contribution of the top 20% solutions, when all objective functions \(f_{1},f_{2},f_{3}\) are taken into account, but with limited influence of the number of sensors represented by weight \(\alpha\). Similarly, \(z\in B\) represents the subset of Pareto solutions with cumulative contribution of the top 20% solutions when the sum of type-I and type-II errors are taken into account, with the limited influence of the type-I error represented by weight \(\beta\). In the present study, the following exemplary weights were assumed, \(\alpha=0.2\) and \(\beta=0.5\). These weights can be appropriately estimated for the known costs of sensors and required unbalanced performance toward type-I or type-II errors. The Pareto solutions shown in Figure 4 represented by the magenta-filled circles show the top 20% of solutions for set \(A\), and the green-filled circles show solutions for the weighed sum of the observed type-I and type-II errors represented by set \(B\). The analysis of Figure 4b together with Figure 4c,d of the obtained results indicates that optimal solutions, which belong to the set \(A\) set, require the use of four to six sensors and the first six dominated modes (7 th, 5 th, 6 th, 9 th, 4 th, 10 th). An exemplary sensor network for this subset of possible sensor networks is presented in Figure 5.  
In that case, when set \(B\) is considered, five to eight sensors are necessary to consider with the use of at least first five to seven dominated modes. This can be observed in Figure 4c,d, and also in Figure 6, where the sum of the errors versus the number of sensors are presented. The analysis of the design variables of the solutions belonging to set \(B\), shown in Figure 4b. indicates that for six to seven sensors it requires consideration of at least six modes to effectively recognize the healthy conditions of the composite plate. The solution with the minimal error type-II is presented in Figure 7a.