# A Novel Genetic Algorithm for the Synthetical Sensor-Weapon-Target Assignment Problem
## 2 Problem Analysis and Formulation
### Existing Work of the S-WTA Problem
#### 2.1.2 Framework B: The Dependent S-WTA Framework-I  
In addition to the independent model, [PERSON] and [PERSON] [19] also proposed a dependent model. Equation (2) shows the dependent model. As this model is more realistic of the current engagement, it is used frequently [19; 20; 21]. \begin{table}
\begin{tabular}{c p{284.5 pt}} \hline \(v_{k}\): & The threat value of the \(k\)th target. \\ \(a_{ik}\): & The benefit of assigning \(i\)th sensor to \(k\)th target. \\ \(b_{jk}\): & The benefit of assigning \(j\)th weapon to \(k\)th target. \\ \(c_{\bar{r}k}\): & The benefit of assigning \(i^{\prime}\)th sensor-weapon pair to \(k\)th target. \\ \(h_{ijk}\): & The benefit of assigning \(i\)th sensor and \(j\)th weapon to \(k\)th target. \\ \(p_{jk}\): & The kill probability that the weapon \(j\) against target \(k\). \\ \(q_{ik}\): & The detection and tracking probability that the sensor \(i\) against target \(k\). \\ \(p_{ijk}\): & The kill probability that the weapon \(j\) against target \(k\) under the detection and tracking of sensor \(i\). \\ \(d_{ik}\): & The sensor detection decision variable (if the \(i\)th sensor detects \(k\)th target \(d_{ik}=1\); otherwise, \(d_{ik}=0\)). \\ \(g_{ij}\): & The sensor guidance decision variable (if the \(i\)th sensor guides the \(j\)th weapon \(g_{ij}=1\); otherwise, \(g_{ij}=0\)). \\ \(x_{ik}\): & The sensor assignment decision variable (if the \(i\)th sensor is assigned to \(k\)th target \(x_{ik}=1\); otherwise, \(x_{ik}=0\)). \\ \(y_{jk}\): & The weapon assignment decision variable (if the \(j\)th weapon is assigned to \(k\)th target \(y_{jk}=1\); otherwise, \(y_{jk}=0\)). \\ \(z_{\bar{r}k}\): & The sensor-weapon assignment decision variable (if the \(i^{\prime}\)th sensor–weapon pair is assigned to \(k\)th target \(z_{\bar{r}k}=1\); otherwise, \(z_{\bar{r}k}=0\), \(z_{\bar{r}k}=x_{ik}\cdot y_{jk}\)). \\ \(u_{ijk}\): & The sensor and weapon platform assignment decision variable (if the \(i\)th sensor and \(j\)th weapon platform are assigned to \(k\)th target \(u_{ijk}=1\); otherwise, \(u_{ijk}=0\)). \\ \hline \end{tabular}
\end{table} 
Table 1: Variables declaration. \[\max\,\sum\limits_{k=1}^{N}\sum\limits_{i^{\prime}=1}^{R}{(c_{i^{\prime}k}\cdot z _{i^{\prime}k})}, \tag{2}\]  
where \(R\) is the number of the sensor-weapon pair. In this paper, we assume that there are \(M\) sensors and \(W\) weapons, so \(R=M\cdot W\). The constraints of this formulation are the same as the model (1). As the decision variable in this model is the assignment of the sensor-weapon pair, the objective function of this formulation is the total benefit of the assignment of the sensor-weapon pair. Therefore, sensor assignment will affect the benefit of the assignment of weapons. [PERSON] et al. [22] and [PERSON] et al. [24] present a similar dependent S-WTA model to model (2), but reformulate the problem with a different method of calculating the benefits. The model is shown as  
\[\left\{ \begin{array}{l}\min\,\sum\limits_{i=1}^{M}\prod\limits_{j=1}^{W}{ \left(1-\sum\limits_{k=1}^{N}{h_{ijk}\cdot u_{ijk}}\right)}\\ s.t.\,\,\left\{\begin{array}{l}\sum\limits_{j=1}^{N}\sum\limits_{k=1}^{N}{u_ {ijk}}{=1},\forall i\in\{1,2,\cdots,M\}\\ \sum\limits_{i=1}^{M}\sum\limits_{j=1}^{N}{u_{ijk}}\leq{n_{j}},\forall j\in\{1,2,\cdots,W\}\\ \sum\limits_{i=1}^{M}\sum\limits_{j=1}^{N}{u_{ijk}}={m_{k}},\forall k\in\{1,2,\cdots,N\},\end{array}\right.\end{array} \right. \tag{3}\]  
where \(n_{j}\) is the number of weapons in \(j\)th weapon platform in which all the weapons have the same kill probability to a specific target and \(m_{k}\) is the number of weapons of intercepting \(k\)th target. As the variables and structure of model (2) and (3) are very similar, these two models can be regarded as the same kind of dependent S-WTA model. These dependent S-WTA models describe another S-WTA framework. As we divide the existing dependent S-WTA formulations into two types, the frameworks of the dependent S-WTA models have two forms.