# Genetic Multi-Objective Optimization of Sensor Placement for SHM of Composite Structures
## 3 Multi-Objective Optimization  
### OSP Problem  
The primary goal of OSP is to localize the positions of the sensors in order to collect relevant information about the analyzed structure. This information, together with the available knowledge about the observed structure, allows for estimating its health condition which is, in turn, one of the main objectives of an SHM system. Usually, the process of determining sensor placement can be a part of the design of an SHM system [20]. This is often preceded by the evaluation of the structural characteristic and dynamics of the structure and also the selection of appropriate sensor types. The selection of the sensor type and the evaluation of the dynamic characteristic of the composite plate was a part of the research in [7], and is based on the application of strain gauges oriented horizontally or vertically to monitor modal strain of the composite plates. Usually, the next step comprises the necessity of sensor placement. This optimal placement strongly depends on the objective function of the unconstrained or constrained optimization process. There are many objective functions widely used for that purpose, e.g., a function based on Fisher matrix indices [24; 25], MAC [20; 24], condition number [24], modal kinetic-energy-based index [26], and many others, e.g., [9]. Most of them are mainly based on the assumption of widely understood criteria of independency, which allow one to select these locations of the sensors with the minimum redundancy of the captured data. In addition to that, from the perspective of the effectiveness of the whole SHM system, including the OSP problem, information on the health condition of the analyzed structure is the primary outcome of any SHM system. Taking the above into account, the spatial placement of the sensor network can be based on the final evaluation of the SHM system, such as the effectiveness of a properly recognized health condition of the structure. This allows one to establish adequate input to design parameters of the optimization process such as number of sensors, number of analyzed modes, etc., and to link them directly to the final effectiveness of the SHM system instead of using intermediate criteria such as statistical independency criteria, mutual information criteria, and any others [20; 24; 25; 26]. In this study, the minimalization of the type-I and type-II errors were taken separately as the main criteria for the evaluation of the effectiveness of the SHM system [10; 23; 27]. The minimalization of type-I error allows for increasing the specificity of the system leading to a lower number of false alarms. The minimalization of the type-II error allows one to enhance the sensitivity of the system in accurate detection of the health condition minimizing any misses in the condition recognition. Keeping both of these criteria in balance requires a trade-off and is crucial for designing SHM systems. For instance, too-high sensitivity may lead to low specificity with a high number of false alarms, which may lead to information overload of the human operator of the SHM system. On the other hand, maintaining high specificity may lead to poor performance in the recognition of all faulty states of the structure. In the paper [23], it was shown that both sensitivity and specificity are often opposite counterparts in classification problems. Both types of errors are also naturally opposite to the minimalization criteria of the number of sensors used. Usually more relevantly selected and placed sensors lead to minimization of the type-I and type-II errors, but are more costly and require more data to analyze. In many cases, the designer of the SHM system can be interested in solutions with enhanced sensitivity rather than false-alarm avoidance. In this case, a weighed objective function in multi-objective optimization allows for analysis of a specific region of the Pareto front, biasing the search towards a solution that optimizes weighed objective function.