# Genetic Multi-Objective Optimization of Sensor Placement for SHM of Composite Structures
## 3 Multi-Objective Optimization
### OSP Problem
In this study, for the purpose of not losing the wider diversity of Pareto front solutions, no weights have been considered, leaving the designer to select results from the wider range of possible solutions and its design parameters. Taking all of these multiple conflicting objectives into account, the multi-objective optimal-sensor-placement problem can be defined as follows, where the objective functions can be represented as  
\[f_{1}(x_{1},x_{2},x_{3})=\frac{FP(x_{1},x_{2},x_{3})}{TN(x_{1},x_{2},x_{3})+FP(x _{1},x_{2},x_{3})}, \tag{2}\]  
\[f_{2}(x_{1},x_{2},x_{3})=\frac{FN(x_{1},x_{2},x_{3})}{TP(x_{1},x_{2},x_{3})+FN(x _{1},x_{2},x_{3})}, \tag{3}\]  
\[f_{3}(x_{1})=n_{x}(x_{1})+n_{y}(x_{1}), \tag{4}\]  
where, \(f_{1},\ f_{2},\ f_{3}\) are objective functions associated with the minimalization of type-I error, type-II error, and the minimalization of the total sensor used, respectively. The vector of \(\mathbf{x}=[x_{1}.x_{2},\ x_{3}]\) is a vector of design variables, where \(x_{1}\) represents the total number of sensors, \(x_{2}\) is number of used shape modes, and \(x_{3}\) indicates the required minimal distance between the sensors measuring the strain displacements in the same direction, because the modal modes, especially for the healthy condition, are symmetrical. The minimum required distance allows for maintaining the spread of the sensors at the surface of the plate avoiding placing them in the same quadrant. _FP_, _TN_, _FN_, and _TP_ correspond to false-positive, true-negative, false-negative, and true-positive cases, respectively [25]. Because the proposed SHM is a classification-based information system, these parameters are evaluated using classification statistics obtained using \(k\)-fold validation based on classification-error statistics [9; 25]. The \(n_{x},n_{y}\) corresponds to the number of strain sensors used for measuring along the \(x\) and \(y\) direction, respectively. Finally, multi-objective optimization is defined to minimize  
\[F(\mathbf{x})=\min([f_{1},\ f_{2},\ f_{3}]), \tag{5}\]  
subject to  
\[3\leq x_{1}\leq 10\ ;\ \ 2\leq x_{2}\leq 7\ ;\ 120\ \leq x_{3}\leq 200,\]  
\[d\big{(}sx_{i}(x,y),sx_{j}(x,y)\big{)}\geq x_{3};\ d\big{(}sy_{i}(x,y),sy_{j}( x,y)\big{)}\geq x_{3};\ for\ i\
eq j,\]  
where \(sx_{i}(x,y)\), \(sy_{i}(x,y)\) are the positions of the \(i\)-th and \(j\)-th strain gauges oriented horizontally, \(sx_{i}\) and vertically, \(sy_{i}\), at the coordinates \((x,y)\), and \(d\) is the Euclidean distance between the considered sensors. The range from 2 up to 10 sensors was used to maintain the consistency of this paper with previous studies [7]. The lower limit of 2 sensors requires the use of at least one sensor in every X or Y direction. The upper boundary has been selected based on the limited number of channels in our measurement path presented in [8]. The number of modes was also selected according to the number of modes used in previous research [7]. The upper boundary of the number of modes was limited to seven most-dominated modes in the frequency-response function; therefore, the 7 th, 5 th, 6 th, 9 th, 4 th, 10 th, and 8 th mode from Table 1. were taken into account. The last mode from this set, the 8 th mode, has about 10% of the energy of the highest mode. Other modes in the considered band rate have less than or equal to 5% of the energy in comparison to the most dominated mode. The value of minimum required distance was established based on the occurrence of the symmetric modes' shapes of which the strain values are the same (for the healthy condition of the plate) in four imaginary quadrants of the plate visible in Figure 1a. To force the placing sensor in separate imaginary quadrants, this value of 120 mm was selected as the low boundary. A similar approach was used in the study [7]. On the other hand, the upper limit of the required distance between the sensors was also limited to avoid considering unfeasible solutions where a large number of sensors with the same orientation could not be placed in the limited plate size.