# A Novel Genetic Algorithm for the Synthetical Sensor-Weapon-Target Assignment Problem
## 2 Problem Analysis and Formulation
### The Improved S-WTA Model
#### 2.2.2 Behaviors of the Proposed Gain Factor \(\sigma\)  
**(1) Validity in the proposed S-WTA model**  
As the S-WTA problem is a complicated practical application problem, the gain factor \(\sigma\) should be designed appropriately to satisfy the constraints of the S-WTA problem. The validity of the designed gain factor \(\sigma\) is analyzed as follows. As the kill probability \(p_{jk}\) is in [0, 1], \(p_{jk}+\sigma\) should be limited in [0, 1]. Therefore, \(\sigma\) should be limited in [0, 1 - \(p_{jk}\)]. According to the model of the designed gain factor \(\sigma\) in (6), if \(\sum\limits_{i=1}^{M}x_{ik}\
eq 0\), then the hyperbolic tangent function \(tanh(\sum\limits_{i=1}^{M}x_{ik}-1)\) is in [0, 1]. As a result, \(\sigma\) is in [0, 1 - \(p_{jk}\)], so \(p_{jk}\) can meet the constraint of the S-WTA problem. In conclusion, the proposed gain factor \(\sigma\) can ensure that the proposed model can meet the constraints of the S-WTA problem. It is a valid and feasible variable in S-WTA model. **(2) Efficiency in the proposed S-WTA model**  
The gain factor \(\sigma\) is proposed to adjust the kill probability, which can improve the dependent relationship between the sensors and weapons. With a certain kill probability \(p_{jk}\), the gain factor \(\sigma\) is changed according to the sensor assignment decision variable \(x_{ik}\). Therefore, it is necessary to analysis the efficiency of \(\sigma\) to study its effects on the proposed S-WTA model under different decision variables. As shown in model (6), the gain factor \(\sigma\) is divided into two cases according to the value of \(\sum\limits_{i=1}^{M}x_{ik}\), which is the number of sensors assigned to the \(k\)th target. If \(\sum\limits_{i=1}^{M}x_{ik}=0\), then \(\sigma=0\). It means that if no sensors are assigned to the target, the target will not be attacked. As a result, as the weapons are not affected by the sensors, the kill probability \(p_{jk}\) is not enhanced. If \(\sum\limits_{i=1}^{M}x_{ik}\
eq 0\), one or more sensors are assigned to the \(k\)th target, and the kill probability \(p_{jk}\) is enhanced by \(\sigma\). As the kill probability \(p_{jk}\) is a constant, the efficiency of \(\sigma\) is depend on the hyperbolic tangent function \(\tanh(\sum\limits_{i=1}^{M}x_{ik}-1)\). To analyze the efficiency of \(\sigma\) conveniently, we figured out the value of \(\sigma\) with \(\sum\limits_{i=1}^{M}x_{ik}\) in [1, 5]. It is shown in Table 2.  
As shown in Table 2, when \(\sum\limits_{i=1}^{M}x_{ik}=1\), then \(\sigma=0\). Therefore, the kill probability \(p_{jk}\) is not enhanced because one sensor is assigned to the target; it is the prerequisite for weapons to be able to attack the target. When \(\sum\limits_{i=1}^{M}x_{ik}>1\), \(\sigma\) increases as the variable \(\sum\limits_{i=1}^{M}x_{ik}\) increases. It should be noted \begin{table}
\begin{tabular}{c c c c c c} \hline \hline \multirow{2}{*}{**Function**} & \multicolumn{5}{c}{**Variable**} \\ \cline{2-6}  & \(\sum\limits_{i=1}^{M}x_{ik}=1\) & \(\sum\limits_{i=1}^{M}x_{ik}=2\) & \(\sum\limits_{i=1}^{M}x_{ik}=3\) & \(\sum\limits_{i=1}^{M}x_{ik}=4\) & \(\sum\limits_{i=1}^{M}x_{ik}=5\) \\ \hline \(tanh(\sum\limits_{i=1}^{M}x_{ik}-1)\) & 0 & 0.7616 & 0.964 & 0.9951 & 0.9993 \\ \(\sigma\) & 0 & 0.7616(1-\(p_{jk}\)) & 0.964(1-\(p_{jk}\)) & 0.9951(1-\(p_{jk}\)) & 0.9993(1-\(p_{jk}\)) \\ \hline \hline \end{tabular}
\end{table} 
Table 2: The value of \(\sigma\) with \(\sum\limits_{i=1}^{M}x_{ik}\) in [1, 5]. that the smaller the variable \(\sum\limits_{i=1}^{M}x_{ik}\), the larger \(\sigma\) increases when the variable \(\sum\limits_{i=1}^{M}x_{ik}\) increases. This is because in the S-WTA problem the effect of tracking and detection of the sensors on the target depends on the number of sensors assigned to the target. When there are few sensors assigned to the target, increasing the number of sensors can significantly improve the detection and tracking accuracy of the target, so as to significantly improve the kill probability to the target.