OPTIMAL ENGAGEMENT POLICIES

1. 1st Annual Israel Multinational BMD Conference & Exhibition OPTIMAL ENGAGEMENT POLICIES Presented By: Dr. N. Israeli WALES, Ltd., Israel

2. OPTIMAL ENGAGEMENT POLICIES PRESENTATION TOPICS • Introduction • Threat Definition • Optimality Criteria • Dynamic Programming • Optimal Policy Equations • Three Examples: • Example 1: Calculation Walkthrough • Example 2: Solvable Case • Example 3: A Computationally Untractable Threat Definition and Possible Approximation Schemes • Roadmap to Practicality • Summary and Next Steps

3. OPTIMAL ENGAGEMENT POLICIES INTRODUCTION • Interceptors are a primary defense system resource whose management greatly determines the overall system performance • When engaging an attacking TBM one faces the dilemma of choosing between: • Allocating many interceptors to the current threat to ensure its destruction • Saving interceptors for possible future threats • A key difficulty here is the uncertainty regarding the materialization of future threats • The above dilemma becomes more pronounced as defense systems become richer in terms of interceptor types and the number of available interception opportunities

4. OPTIMAL ENGAGEMENT POLICIES INTRODUCTION (Cont.) • In this work we pose threat engagement as an optimization problem and seek optimal engagement policies • By an engagement policy we mean the number and types of interceptors launched at attacking TBMs, as a function of prior intelligence, TBM type and damage capability, attack history, remaining interceptors stock, etc. • For the sake of simplicity we consider cases where: • Attacking TBMs appear and are engaged in a sequential order • All allocated interceptors are used, namely, no Shoot-Look-Shoot capability • Generalizations of our work to salvo engagement and Shoot-Look-Shoot capability are possible

5. OPTIMAL ENGAGEMENT POLICIES INTRODUCTION (Cont.) • For each appearing TBM the defense makes an engagement decision based on: • The type of the current threat and its damage inflicting capability • The current available interceptors stock and their effectiveness against the current threat • The attack history • Possible attack continuations TBM type 1 TBM type 2 TBM type 2 TBM type 1 ??? ??? ???

6. OPTIMAL ENGAGEMENT POLICIES SCOPE AND RESERVATIONS • Our work is based on the assumption that it is possible to quantify the benefit of negating ballistic threats • Quantifying this benefit should reflect decision makers’ priorities and is thus a difficult political, economical and strategic problem • This problem is outside the scope of the current work. We will not attempt to derive such priorities and only use representative numbers • We do however recognize that this is a difficult issue and therefore do not claim that our method provides “The Correct” engagement policies • Rather, we wish to provide decision makers with a tool that allows them to explore the implications of their priorities in terms of engagement policies, interceptor consumption, etc.

7. OPTIMAL ENGAGEMENT POLICIES THREAT DEFINITION • In order to speak about optimality we must have a threat definition that represents our knowledge regarding possible future attacks • Policies can be evaluated (and therefore optimized) only with respect to a threat definition • Naturally, the more precise the threat definition, the more effective the optimal engagement policy • Since we never have perfect intelligence, threat definitions are probability functions that specify the chances of different attacks to materialize

8. OPTIMAL ENGAGEMENT POLICIES OPTIMALITY CRITERIA • Given a threat definition we wish to find the optimal policy of spending our interceptor stock • The optimality criterion depends on what we want to achieve • In this work we focus on minimizing the total attack expected damage • Generalization of our method to other optimization criteria is certainly possible

9. OPTIMAL ENGAGEMENT POLICIES DYNAMIC PROGRAMMING • Dynamic Programming (DP) is an approach developed to solve sequential, or multi stage decision problems • It reduces the runtime of algorithms by dividing the decision problem into overlapping sub problems • The essence of DP is Richard Bellman’s principle of optimality: • An optimal policy has the property that whatever the initial state and the initial decisions are, the remaining decisions must constitute an optimal policy with regard to the state resulting from the first decision

10. OPTIMAL ENGAGEMENT POLICIES FINDING AN OPTIMAL ENGAGEMENT POLICY USING DP • DP recursion equation: expected damage in current stage expected damage in future stages Notation • The optimal engagement policy  is a graph that tells us how to engage in different situations • Note that the dependence of  on the attack history comes from its dependence on the conditional materialization probability of the next threat • If this probability is a simple function of attack history the optimal policy graph will be small • Complex conditional probabilities may lead to prohibitively large policy graphs

11. OPTIMAL ENGAGEMENT POLICIES EXAMPLE 1 – CALCULATION WALKTHROUGH • Threat definition: one type of TBM, damage D=1, attack size uniformly distributed in [1,N]: • Conditional materialization probability of the n’th threat • Interceptor stock: one type of interceptor with probability of kill Pk, survival probability of the attacking TBM Ps=1-Pk • DP recursion equation reduces to: With boundary conditions

12. I=0 =2.5 I=1 =1.7 I=2 =1.1 I=3 =0.7 =1 =1 =0 =1 I=0 =2 =0 I=0 =1.5 =0 I=0 =1 I=1 =0.2 I=2 =0.04 I=3 =0.008 =0 =2 =3 =1 OPTIMAL ENGAGEMENT POLICIES EXAMPLE 1 (Cont.) • Solving for N=4, I1 =3, Pk=0.8, Materializing probabilities: P(2)=3/4, P(3)=2/3, P(4)=1/2 1st TBM 2nd TBM I=1 =1.2 I=2 =0.666 I=3 =0.4 =1 =1 =1 3rd TBM I=1 =0.7 I=2 =0.3 I=3 =0.14 =1 =1 =2 4th TBM

13. OPTIMAL ENGAGEMENT POLICIES EXAMPLE 2 – A SOLVABLE CASE • Assessed Threat definition: • Two types of TBMs (T=1,2) with damage D(T=1)=1 and D(T=2)=4 • Joint attack size uniformly distributed in [1,N] • Probability of a threat to be of types 1 and 2 is 0.7 and 0.3 respectively • Conditional materialization probability of the n’th threat is thus • Interceptors: • Two types of interceptors (a, b) • Probability of kill is given by the matrix • No more than 6 engagements per TBM • Solution complexity: • Policy size • Running time TBM type Interceptor type Maximal attack size Interceptors initial stocks Number of possible engagements of a single TBM

14. OPTIMAL ENGAGEMENT POLICIES EXAMPLE 2 (Cont.) • Policy graph for N=5 and initial interceptor stock (7a,2b) 1st TBM 7a, 2b TBM type TBM type 1 Prob. 0.7 D=1 TBM type 2 Prob. 0.3 D=4 6a, 1b 2nd TBM 5a, 2b Interceptor type (2a,0b) (1a,1b) Pkmatrix (2a,0b) (1a,1b) (2a,0b) (1a,1b) 3a, 2b 4a, 1b 5a, 0b 3rd TBM (2a,0b) (2a,0b) (1a,1b) (2a,0b) (3a,0b) (0a,2b) 4th TBM 1a, 2b 2a, 1b 3a, 0b 2a, 0b (1a,0b) (1a,0b) (0a,2b) (1a,0b) (1a,1b) (2a,0b) (1a,0b) (2a,0b) 0a, 2b 1a, 1b 1a, 0b 2a, 0b 0a, 0b 5th TBM (0a,2b) (1a,1b) (1a,0b) (2a,0b) (0a,0b)

15. b int. stock b int. stock b int. stock b int. stock TBM index TBM index TBM index TBM index a int. stock a int. stock a int. stock a int. stock OPTIMAL ENGAGEMENT POLICIES EXAMPLE 2 (Cont.) – POLICY TOPOLOGY • Policy for N=100 and interceptor stocks between 0 to 150 a’s, and 0 to 40 b’s Policy for type 1 TBM Policy for type 2 TBM Shoot 0 Shoot 1 Shoot 2 Shoot 3 Shoot 4 Shoot 5 Shoot 6 Interceptor a engagement Interceptor b engagement

16. 70% type 1 TBMs 30% type 2 TBMs 90% type 1 TBMs 10% type 2 TBMs 20% type 1 TBMs 80% type 2 TBMs TBM index TBM index OPTIMAL ENGAGEMENT POLICIES EXAMPLE 2 (Cont.) • N=100 and initial interceptor stock of 150 a’s and 40 b’s • Example engagement sequences: type 1 TBM type 2 TBM Int. a eng. Int. b eng. Stock a Stock b TBM index Expected damage TBM index

17. OPTIMAL ENGAGEMENT POLICIES EXAMPLE 2 (Cont.) Threat expected damage under optimal engagement policy • N=100 and initial interceptor stocks between 0 to 150 a’s and 0 to 40 b’s b Int. stock Damage Comparison with fixed policies a Int. stock

18. OPTIMAL ENGAGEMENT POLICIES EXAMPLE 3 – UNTRACTABLE THREAT DEFINITION • Consider the following threat definition: • The optimal policy for say, the 100th TBM should be a function of the number of TBMs of each type that composed the first 99 attacking threats. Different response as the different types approach depletion • The problem is that the number of possible partitions of the first 99 TBMs to the five types is huge • Accordingly, the optimal policy will be prohibitively complex

19. OPTIMAL ENGAGEMENT POLICIES EXAMPLE 3 (Cont.) Possible approximation scheme: • Approximate the previous detailed threat definition by a simpler one • Generate a large ensemble of attacks from the original threat definition and calculate the materialization probability of each threat type as a function of the joint attack size • With this simpler function the problem reduces to one similar to Example 2 • Note however that information is lost here. In particular, we may assign a finite probability for a TBM to appear even though the enemy has already fired the entire ORBAT of the relevant type Prob. n

20. OPTIMAL ENGAGEMENT POLICIES SUMMARY • TBM attack engagement has been posed as an optimal sequential decision problem • We showed that for certain types of threat definitions the optimization problem can easily be solved with Dynamic Programming • The optimal policy for these solvable cases is simple in certain parts of the attack phase space (early attack stages with lots of interceptors) but tends to bifurcate with diminishing interceptor stock • Some types of threat definitions imply complex attack materialization probabilities which lead to prohibitively large policy graphs • In these complex cases we suggest approximate solutions