# A Novel Genetic Algorithm for the Synthetical Sensor-Weapon-Target Assignment Problem
## 2 Problem Analysis and Formulation
### The Improved S-WTA Model  
#### 2.2.1 The Proposed S-WTA Model  
Based on the model (5), the improved S-WTA model is proposed as follows,  
\[\left\{\begin{array}{l}\max\sum\limits_{k=1}^{N}{{{v_{k}}\left({{Q_{k}}{+}{Q_{ k}}\cdot{P_{k}}}\right)}}\\ {{Q_{k}}{=}\left({1-\prod\limits_{i=1}^{M}{{{(1-{q_{ik}})}^{{x_{ik}}}}}}\right) }\\ {{P_{k}}{=}\left({1-\prod\limits_{j=1}^{M}{{{(1-{{({p_{jk}}+\sigma)})}^{{y_{jk}}}}}} }\right)}\\ \sigma=\left\{{ \begin{array}{*{20}{c}{0,\text{if}\sum\limits_{i=1}^{M}{{x_{ ik}}}=0}}\\ {{{(1-{{p_{jk}}})}\cdot tanh(\sum\limits_{i=1}^{M}{{{x_{ik}}}{-1})},\text{if} \sum\limits_{i=1}^{M}{{x_{ik}}}\
eq 0}}\\ {{s.t.}\left\{\begin{array}{*{20}{c}{\sum\limits_{k=1}^{N}{{x_{ik}}}\leq 1,\forall i\in\{1,2,\cdots,M\}}}\\ {\sum\limits_{k=1}^{N}{{y_{ik}}}\leq 1,\forall j\in\{1,2,\cdots,W\}}}\\ {\sum\limits_{i=1}^{M}{{x_{ik}}}\leq{{n^{\prime}}_{k}},\forall k\in\{1,2, \cdots,N\}}\\ {\sum\limits_{j=1}^{N}{{y_{ik}}}\leq{{m^{\prime}}_{k}},\forall k\in\{1,2, \cdots,N\}},\\ \end{array}\right.\end{array} \right. \tag{6}\]  
Figure 3: The dependent S-WTA framework-II. where \(N\) is the number of targets, \(M\) is the number of sensors, \(W\) is the number of weapons, \(Q_{k}\) is the probability of detecting and tracking the \(k\)th target by the sensor, \(P_{k}\) is the probability of destroying the \(k\)th target by the weapon, \(\sigma\) is the gain factor of the kill probability, and the function \(tanh\) is the hyperbolic tangent function. The definitions of \(v_{k}\), \(q_{ik}\), \(p_{jk}\), \(x_{ik}\), and \(y_{jk}\) are shown in Table 1. The first constraint represents that each sensor can detect or track only one target. The second constraint denotes that each weapon can intercept only one target. The third constraint and last constraint limit the sensor cost and the weapon cost for each target, respectively. As presented in Equation (6), the model consists of two parts. The first part \(\sum\limits_{k=1}^{N}v_{k}\cdot Q_{k}\) represents the expected total value of the detected threat. It can be regarded as the benefits of sensor assignment. The second part denotes the expected total value of the eliminated threat. It can be regarded as the benefits of weapon assignment. The sum of these two parts is similar to the framework A, which has the characteristics of the independent S-WTA model. The second part, \(\sum\limits_{k=1}^{N}v_{k}\cdot Q_{k}\cdot P_{k}\), is similar to models (4) and (5), which belong to framework C. It has the characteristics of the dependent S-WTA model. As the proposed model incorporates the characteristics of the framework A and C, the S-WTA problem can be described more completely. Although framework B is also based on the dependent model, it should be noted that frameworks A and B are not integrated in this proposed model. As described in framework B, the sensor assignment and weapon assignment are integrated. As framework B can only reflect the effect of sensor and weapon assignment, it is unable to describe the effect of the sensors on the weapons. In this proposed model, since we incorporate the framework A and C, the two parts involve the detailed information of sensors (i.e., the probability of detecting and tracking the target by sensors). As a result, the interaction between sensors and weapons can be fully reflected. In addition to improve the framework, the second part, which represents the dependent relationship between sensors and weapons, is also improved. As described in models (4) and (5), different sensor assignment schemes only affect the probability of detecting and tracking. However, according to the actual situation of the S-WTA problem, if multiple sensors detect and track a target simultaneously, they will greatly enhance the accuracy of tracking and recognition of the target which are the most important factor of the interception of target by weapons. Therefore, the kill probability of the weapon against the target should be adjusted according to the sensor assignment scheme.