
$$(x-x_{0})^{2}+\ (y-y_{0})^{2}=\ r^{2}$$

set of DAs against a set of threats k, and allows for multiple target assignment per DA. The model is kept flexile to handle unknown type of threats. 

B. Threat Evaluation Model Suppose the defense has i numbers of DAs to be guarded against potential threats. Where each DA has its own set of one or more type of WSs; suppose the total number of WSs is j. 

Assume DAs can be attacked by k number of threats of any kind and capability in any order. The main objective of this process is to rank the targets according to their opportunity index and assign each target to the most suitable DA. 

We model DAs with circles and threats with line segments. 

Using circle line intercept equations, we compute the points of intersection between each threats and DA. Let the DA be shown by equation 1. 

 ሺݔ െ ݔሻଶ  ሺݕ െ ݕሻଶ ൌ ݎଶ (1) 
Where x0, y0 are center points of DA and r shows the radius of circle representing DAi. 

Extending threat velocity vector we get a line represented by equation 2. 

 y = mx + c (2) 
To assign threats to a DA, we need to calculate proximity parameters, intent parameters and capability parameters of threats and DAs. Threat ordering is important because we want to neutralize the most threatening targets as early as possible. 

After assigning initial threat index, TE processes all identified threats to calculate refined threat index based on opportunity (intent and capability) and proximity parameters. This refined threat index specifies the order in which threats should be processed for WA. Proposed solution uses Defended Asset 
(DA) based TE by applying (one-to-one) stable marriage algorithm (SMA) with weighted proposals between threats and DAs. For each threat, list of matching capable DAs is searched on the basis of kill capabilities (index), status of DA (Free to Fire, On Hold, and Freeze) and priorities of DAs. So, a threat k, proposes to only those DAs that have capability high enough to neutralize it. Each proposed DA, looks for proposal acceptance feasibility and responds accordingly. Each DA has its own vulnerability index, priority and other parameters. Weights are assigned to each DA on the basis of DA priority and threat related parameters like Time to DA (TDA), heading, velocity etc. For each threat k, the proposal with the maximum weight is processed first. If situation allows DAi to accept this proposal, threat is assigned to proposed DA, else the next proposal highest weighted proposal is sent. Mostly TDA calculations are based on constant speed that is quite unrealistic. Our proposed algorithm caters for this deficiency by calculating speed and velocity along different axis at different time stamps, making it scale up its efficiency for scenarios modeling track maneuvering. Once a threat is assigned to DA, WA process can start. 

 Proximity parameters are closely related to intent parameters. So we calculate earliest DA point of intersection 
(POI) for each threat and DA that has Kill probability high enough to neutralize assigned threat. Kill probability K.P of a DAi is shown by (3) where i = {1,2,3….. NDA}; NDA= Number of DAs. 

 

$$\left(\prod_{\mathrm{k=1}}^{\mathrm{K}}\left(1-\left(\left(\mathrm{W_{I}*ll_{k}+W_{cl}*cl_{k}+\mathrm{W_{L}*Load_{j}}}\right)*C_{\mathrm{j,k}}\right)^{B_{i,j}}\right)\right)\tag{3}$$

Where 

$$K\ {\stackrel{d w^{\prime}}{=}}\ \,N u m b e r\,o f\,T h r e a t s$$
$$W_{I}\ {\stackrel{\mathrm{def}}{=}}\ W e i g h t\ A s s i g$$

WI  Weight Assigned to Intent Parameters 

$I_k\equiv I_l$
IIk  *Intent index of threat k* 

$$W_{C l}\triangleq W e i g l$$

WCI  *Weight Assigned to Capability Parameters* CIk  Capability index of threat k 

$\sigma_{\mathrm{T}}$a. 
WL  Weight Assigned to Load Parameter 

$$L o a d_{k}\ \stackrel{d e r}{=}\ \,L o a d\ \ o n\ \ W S_{j};\ j\ \in\ \{1,2,\ldots\ N_{W S}\};\ N_{W S}=$$
$${\mathrm{tion~1~as:}}$$
$$(1)$$
.$ \quad\mathcal{Y}_{00}\quad\quad\cdot$  points of DA and z above. 
Number of WSs € DAi. 

 To calculate POIs we expand equation 1 as: 

$$x^{2}+\ x_{o}^{2}-2x x_{0}+\ y^{2}+\ y_{o}^{2}-2y y_{0}=\ r^{2}$$
$$\mathrm{have,}$$

Putting (2) in (4) we have, 

$$x^{2}+\ x_{o}^{2}-2xx_{0}+\ (mx+c)^{2}+\ y_{0}^{2}-2(mx+c)y_{0}=\ r^{2}\tag{5}$$  Solve (5) and 
Solving (5) we have 

$\left(\frac{1}{2}\right)+\left(\frac{1}{2}\right)+\left(\frac{1}{2}\right)+\left(\frac{1}{2}\right)+\left(\frac{1}{2}\right)$ 2. 
* [16] A. A. K.  
ݔଶሺ1  ݉ଶሻ  ݔ൫2ሺ݉ܿ െ ݔ െ ݕሻ൯  ሺݔݕ  ଶଶ  ܿଶ െ ݎଶ െ 2ݕܿሻ ൌ 0 

$$\quad(4)$$
.$2\,\,\,(5)$  . 
$$\left(7\right)$$
$$(8)$$

(6) 
We can solve (6) using quadratic equation as shown below 

 െܾ േ √ܾଶ െ 4ܽܿ
2ܽ ሺ7ሻ
Where 
$$a=(1+m^{2})$$
$$b=2(m c-x_{0}-y_{0})$$
$$c=\,x_{o}^{2}+\,y_{0}^{2}+c^{2}-r^{2}-2y_{0}c$$
$$\mathrm{Time}={\frac{\mathrm{Distance}}{\mathrm{Speed}}}$$
This POI calculation is done for each threat DA pair. After calculating POIs we calculate time to DA using (8) 

 Time = ୈ୧ୱ୲ୟ୬ୡୣ
S୮ୣୣୢ (8) 
Where Distance is calculated using Euclidian (9) 
$$\sqrt{\sum_{i=1}^{n}(\mathrm{POI_{i}}-\mathrm{\TP_{i}})^{2}}\qquad\qquad\qquad(9)$$
$$\operatorname{tan}\,(9)$$
Where TP= target Position and i $\in$[1, 2].  
The target Tn that ahs lesser time to DAi and is headed towards that DAi will have higher intent value for DAi as compared to a threat Tm that has lesser time to DAi at a time stamp t0. Figure 5 shows block diagram of proposed model. 