# A Novel Genetic Algorithm for the Synthetical Sensor-Weapon-Target Assignment Problem
## 2 Problem Analysis and Formulation
### Existing Work of the S-WTA Problem
#### 2.1.3 Framework C: The Dependent S-WTA Framework-II  
[PERSON] et al. [27] propose another dependent S-WTA model. In this model, some complex factors are introduced, such as the detection and guidance. The formulation is described as:  
\[\left\{\begin{array}{l}\max\sum\limits_{k=1}^{N}v_{k}\cdot\left(1-\prod\limits _{j=1}^{W}\prod\limits_{i=1}^{M}\left(1-p_{ijk}\cdot g_{ij}\cdot d_{ik}\right) \right)\\ s.t.\left\{\begin{array}{l}\sum\limits_{i=1}^{M}\xi_{ij}\leq 1,\forall j\in\{1,2, \cdots,W\}\\ \sum\limits_{k=1}^{N}\xi_{ij}\cdot d_{ik}\leq 1,\forall i\in\{1,2,\cdots,M\},\forall j\in\{1,2,\cdots,W\}\\ \sum\limits_{k=1}^{N}d_{ik}\leq 1,\forall i\in\{1,2,\cdots,M\}.\end{array}\right.\end{array}\right. \tag{4}\]  
In addition to the above three constrains, there are another three complex constrains that indicate the conditions of the detection, guidance, and interception. The details of these constraints can be seen in the literature [27]. As the objective function of this formulation is the expected total value of the eliminated threat, this formulation is similar to the model of WTA problem. The assignment of sensors can direct influence the survivability of the target.  
[PERSON] et al. [28] take into account the detection probability of the sensors to establish the dependent S-WTA model. The model is shown as  
\[\left\{\begin{array}{l}\max\sum\limits_{k=1}^{N}v_{k}\left(1-\prod\limits_{i =1}^{M}\left(1-q_{ik}\right)^{x_{ik}}\right)\left(1-\prod\limits_{j=1}^{W} \left(1-p_{jk}\right)^{y_{jk}}\right)\\ s.t.\left\{\begin{array}{l}\sum\limits_{k=1}^{N}x_{ik}\leq 1,\forall i\in\{1,2, \cdots,M\}\\ \sum\limits_{k=1}^{N}y_{jk}\leq 1,\forall j\in\{1,2,\cdots,W\}\\ \sum\limits_{i=1}^{M}x_{ik}\leq n^{\prime}_{k},\forall k\in\{1,2,\cdots,N\}\\ \sum\limits_{j=1}^{W}y_{ik}\leq m^{\prime}_{k},\forall k\in\{1,2,\cdots,N\}, \end{array}\right.\end{array}\right. \tag{5}\]  
where \(n^{\prime}_{k}\) is the sensor cost of target \(k\) and \(m^{\prime}_{k}\) is the weapon cost of target \(k\). In this model, the assignment of sensors can affect the detection probability of the target. It can fully reflect the role of sensors in the whole engagement. As the objective function is the same as Equation (4), these two models can be regarded as another same kind of dependent S-WTA model.  
These S-WTA models indicate the other independent S-WTA framework. It is the second form of the dependent S-WTA models, which is summarized and shown in Figure 3.  
Figure 2: The dependent S-WTA framework-I.  
As shown in Figure 3, sensors are assigned to detect and track the targets. Then, the information achieved by sensors is shared to the weapon. Afterwards, weapons are launched to intercept the target. Therefore, in this framework, partial information of the sensor and weapon is shared. In addition, as described in models (4) and (5), the objective functions of these formulations are the expected total value of the eliminated threat. Therefore, in this framework, only weapons can impact on the battlefield situation. As a result, since the sensor assignment schemes would affect the weapon assignment schemes, the effectiveness of weapon assignment can be fully analyzed. However, the effectiveness of weapon assignment has received so much attention that the effectiveness of sensor assignment has been neglected.  
According to the above analysis, all frameworks have their own advantages and disadvantages. Because of the different frameworks focus on the different aspects of the sensor assignment and the weapon assignment, some details of the sensors and weapons are neglected. Therefore, it is necessary to design a relatively comprehensive S-WTA model and framework to fully describe the S-WTA problem.