At most two threats can be schedules on a WS. i.e one threat can be locked and one can wait in the queue. For a WS j this can be expressed as: 

$$\mathrm{and}$$

 ݈݆݁݀,݇ݑ݄ܵܿ݁݀ ∑ 
$$\sum_{k=1}^{K}S c h e d u l e d_{j,k}\leq2$$
݇ൌ1  2 (10) 
 ݆ܿ݇݁݀,݇ܮ ∑ 
$$\Sigma_{j=1}^{J}\,L o c k e d_{j,k}\,=1$$
Where j ߳ WSi, K  Number of Threats and. WSi shows the weapon set of DAi.and 

݄ܵܿ݁݀ݑ݈݁݀, ൌ ቐ 1 ݂݅ j Ԗ WS୧ ܽ݊݀ ݏݎ݈ܽ, ൌ 1
$$\begin{array}{l}{{i f\;\mathrm{j}\in W S_{\mathrm{i}}\;a n d\;p r o p o s a l_{i,k}=1}}\\ {{a n d\;\mathrm{P o r p o s a l_{j,k}=1}}}\\ {{O t h e r w i s e}}\end{array}$$
$$S c h e d u l e d_{j,k}=\left\{\begin{array}{l l}{{}}&{{1}}&{{k}}\\ {{}}&{{}}\\ {{0}}&{{}}\end{array}\right.$$
as given by equation 12. 
$$Locked_{j,k}=\left\{\begin{array}{rl}1&ifAssign_{j,k}=1\;and\\ k&is\;not\;enqueeued\;at\;j\\ 0&Otherwise\end{array}\right.$$  The outcome each proposal is Boolean, it can either be 1 or 0.  
$$\mathrm{and}$$
 ൌ ,݈ܽݏݎ ە ۔
 ݅ ܣܦ ݂ ݏܹܵ ݏݓ݄ݏ WS୧ ݁ݎ݄݁ݓ WS୧ Ԗ j ݂݅ 1 ۓ݂ݎ ݓ݄݄݅ܿ Porposal୧,୩ is accepted and Capability
݁ݑܸ݈ܽ ݉ݑ݅݊݅݉ܯ ݁݀ݎ݅ݑݍܴ݁  WS୧ ݂ ݔ݊݀݁ܫ ሻݐ݁݊ݏ ݏ݅ ݈ܽݏݎ ݊ሺ ݁ݏ݅ݓݎ݄݁ݐܱ 0 

$$(12)$$

## Iv. Testing And Analysis

Different scenarios were created and tested on a system implementing this solution. Before running actual scenarios, we do the typical battlefield pre-processing simulation tasks that including DA definition, Weapon deployment and pairing with DA and communication server configuration. Communication server is used to make TEWA and simulator communicate through sockets. 

After this battlefield pre-processing, we generate different scenarios to test the optimality of implemented system. These scenarios may range from relaxed K target scenario against J WSs to extremely stressful scenarios where time for scheduling is less or partial information is available to schedule targets. Scenarios can be designed to force starvation or over utilization probability increase explicitly. Cost analysis and damage calculations of this system show that although this algorithm is computation intensive but, even under stressful conditions, it succeeds in coming up with near optimal solution. Table 1 shows summary of few simulations. The main focus of this paper was on proposed model. We shall focus on our results and analysis in our next paper. 

## V. Conclusion

A Novel Two-Staged Dynamic Decision Support based Optimal Threat Evaluation and Defensive Resource Scheduling Algorithm for Multi Air-borne threats is presented that correlates threat and weapon libraries to improve the solution quality. Opportunity, Proximity parameters along with DA characteristic and weapon/target parameters have been 

$$(10)^{\frac{1}{2}}$$
$$(11)$$

explored and used in a most prolific way. For optimality, these parameter weights are kept configurable. This paper explains the main optimization steps required to react to changing complex situations and provide near optimal weapon assignment for a range of scenarios. Proposed algorithm uses a variant of many-to-many Stable Marriage Algorithm (SMA) to solve Threat Evaluation (TE) and Weapon Assignment (WA) 
problem. TE corresponds to Threat Ranking and Threat-Asset pairing while WA corresponds to finding to best WS for each potential target using a flexible dynamic weapon scheduling algorithm, allowing multiple engagements using shoot-lookshoot strategy. Analysis part of this paper shows that this new approach to TEWA computes near-optimal solution for a range of scenarios. 

$$\begin{array}{l}{{s o j\;\;D A\;t}}\\ {{1\;C a p a b i l i t y}}\\ {{u m\;V a l u e}}\\ {{s e n t)}}\end{array}$$

| TABLE I.                                                                                                                                                                                                                                                    | SUMMARY OF FEW SIMULATIONS                                                                                                                                                                                                                                                                                                 |                                                                                                                                                                                                                                                                             |    |
|-------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------|----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------|-----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------|----|
| TE                                                                                                                                                                                                                                                          | Number of Threats is                                                                                                                                                                                                                                                                                                       | Number of Threats                                                                                                                                                                                                                                                           | Relaxed one-toone case    |
| WA                                                                                                                                                                                                                                                          | less                                                                                                                                                                                                                                                                                                                       | is greater                                                                                                                                                                                                                                                                  |    |
| K= 50 ;I= 10 ; All  threats {T1, T2 …  T50}were  allocated to best  candidate DA  within few  seconds.  For a smooth  deplyment, all  DAs were engaged  in TEWA. Few  DAs were found  idle when  deployemnt is  introduced with  unbalanced WS  assignment. | K= 10 ; I= 10 ;  All threats {T1,  T2 … T10}were  allocated to best  candidate DA  within few milliseconds.  For a smooth  deplyment, all  DAs were  engaged in  TEWA. Few DAs  were found idle  when deployemnt  is introduced with  unbalanced WS  assignment.                                                                                                                                                                                                                                                                                                                            |                                                                                                                                                                                                                                                                             |    |
| K= 5 ; I= 10 ; All  threats {T1, T2 …  T50}were allocated  to best candidate DA  within few milliseconds. Some DAs  were idle, some had  balanced amount of  load.                                                                                                                                                                                                                                                             |                                                                                                                                                                                                                                                                                                                            |                                                                                                                                                                                                                                                                             |    |
| TE                                                                                                                                                                                                                                                          | J= 10 ; K= 50 ; All  threats {T1, T2 …  T50}were allocated  to best candidate  WS within few  seconds. The WS  locked one target  at a time and  allowed to have  one in its  associated queue.  Few targets stayed  in a queue.  System shifted to  preferential mode,  maximizing the  total survival  value of each DA. | J= 10 ; K= 50 ;  All threats {T1, T2 … T50}were  allocated to best  candidate WS  within few  seconds. The WS  locked one target  at a time and  allowed to have  one in its  associated queue.  System continued  to stay in the  same mode it was  already executing  in. |    |
| J= 10 ; K= 5 ; All  threats {T1, T2 …  T50}were allocated  to best candidate WS  within few milliseconds. The WS  locked one target at  a time. Few WS  stayed idle. No  resouce conflict  found.  System executed in  subtractive mode,  minimizing the  expected survival  time of each Threat.                                                                                                                                                                                                                                                             |                                                                                                                                                                                                                                                                                                                            |                                                                                                                                                                                                                                                                             |    |
| WA                                                                                                                                                                                                                                                          |                                                                                                                                                                                                                                                                                                                            |                                                                                                                                                                                                                                                                             |    |

## References

[1] Patrik. A Hosein and M. Athans, 1990, "An Asymptotic result for the Multi-stage Weapon Target Allocation Problem", Proceedings of the 29th IEEE Conference on Decision and Control, vol.1, pp. 240-245. 

[2] M. K. Allouche, 2006. "A pattern recognition approach to threat stabilization," tech. rep., DRDC Valcartier. 

[3] M. K. Allouche, 2005. "Real-time use of Kohonen's self-organizing maps for threat stabilization, " in Proceedings of the 8th International Conference on Information Fusion, vol. 6, pp. 153–163. 