# Genetic Multi-Objective Optimization of Sensor Placement for SHM of Composite Structures
## 3 Multi-Objective Optimization
### General Optimization Procedure  
To estimate individual objectives of \(F(\mathbf{x})\), the SHM effectiveness through the estimation of classification performance statistics need to be completed. For this purpose, a classifier should be prepared that takes information on the strain values for the given coordinates according to the sensor's network placement, and returns the information about the health condition of the composite plate. Separately, to avoid using the coordinates of the location of the sensors as design variables, the appropriate method of sensor placement was used internally in this proposed optimization procedure. This allows one to take advantage of the available knowledge on the effectiveness of the selected sensor-placement algorithm used for the composite plate based on the results obtained in [7]. The general optimization procedure is based on the application of a heuristic approach with elitist modification of the NSGA-II algorithm [18; 20; 28; 29; 30; 31]. The algorithm allows approximation of the Pareto front which represents the equivalent trade-off between considered objectives \(f_{1},\ f_{2},\ f_{3}\). In this paper, the optimization framework launch execution of other processes such as OSP and validation of classification. The general scheme of the optimization process is shown in Figure 3.  
The NSGA-II executes the process of multi-objective optimization by defining an initial and any after generation, where individuals represent candidate sensor networks. These sensor networks are generated using the sensor-placement method based on selected design parameters and available strain maps for different shape modes obtained experimentally, according to [7], for the heathy condition of the composite plate. During the process of optimization, some of the parameters are treated as constant parameters, among others, they include the clamping area of the composite plate, where placing sensors is not possible. Another constant parameter is the relative additive noise level used in data augmentation for the purpose of obtaining balanced dataset for learning of the classifiers. In this research, one of the algorithms for sensor placement shown and evaluated in the paper [7] was used. This algorithm is based on the selection of the position of the sensor candidates on the rank defined on the basis of the sum of values of the mode shapes for individual coordinates. Initially, the mode shapes are ordered according to their higher dominance in frequency-response function shown in Table 1. Next, individual strain maps of selected modes are normalized from 0 to 1, assigning 1 to the highest strain value. Then, the sum of the strains of selected modes is performed, and the strain values in the \(x\) and \(y\) direction are organized separately as vectors of values in descending order [7; 8]. Afterwards, based on this order, the sensors are sequentially placed in the coordinates, which corresponds to the highest value in the mentioned vector, taking into account other requirements such as maintaining the required distance between sensors, etc. The effectiveness of this algorithm was compared with other algorithms, among others, with a modified Effective Independence Method [12]. In this comparison, we also took into account the influence of the noise level on the accuracy of damage detection. More details can be found in [7]. In this study, the sensor network obtained using the above algorithm is mainly evaluated using the prepared classifier, which is validated using \(k\)-fold cross-validation. The results of this validation make it possible to compute the objective function \(f_{1},\ f_{2}\), which represents the number of type-I errors and type-II errors. This allows one to compute a multi-objective function for the individuals in the considered generation. The selection process of individuals for the next generation is performed using the tournament selection [23] with a fraction value of 0.8.