1. OPTIMAL ENGAGEMENT POLICIES
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. SCOPE AND RESERVATIONS
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. FINDING AN OPTIMAL ENGAGEMENT POLICY USING DP
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. EXAMPLE 1 – CALCULATION WALKTHROUGH
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. 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
I=0
=2.5
=0
I=1
=1.7
=1
I=2
=1.1
I=3
=0.7
Engagement option
Expected
Damage
shoot 0
1+P(4)*(4,2)=1.02
shoot 1
0.2+P(4)*(4,1)=0.3
shoot 2
0.04+P(4)* (4,0)=0.54
Engagement option
Expected
Damage
shoot 0
1+P(3)*(3,3)=1.093
shoot 1
0.2+P(3)*(3,2)=0.4
shoot 2
0.04+P(3)*(3,1)=0.507
shoot 3
0.008+P(3)*(3,0)=1.008
Engagement option
Expected
Damage
shoot 0
1+P(3)*(3,2)=1.2
shoot 1
0.2+P(3)*(3,1)=0.666
shoot 2
0.04+P(3)*(3,0)=1.04
Engagement option
Expected
Damage
shoot 0
1+P(3)*(3,1)=1.466
shoot 1
0.2+P(3)*(3,0)=1.2
Engagement option
Expected
Damage
shoot 0
1+P(4)*(4,3)=1.004
shoot 1
0.2+P(4)*(4,2)=0.22
shoot 2
0.04+P(4)*(4,1)=0.14
shoot 3
0.008+P(4)*(4,0)=0.508
Engagement option
Expected
Damage
shoot 0
1+P(4)*(4,1)=1.1
shoot 1
0.2+P(4)*(4,0)=0.7
2nd
TBM
I=0
=2
=0
I=1
=1.2
I=2
=0.666
I=3
=0.4
=1
=1
=1
3rd
TBM
I=0
=1.5
=0
I=1
=0.7
I=2
=0.3
I=3
=0.14
=1
=1
=2
4th
TBM
I=0
=1
=0
I=1
=0.2
=1
I=2
=0.04
=2
I=3
=0.008
=3

13. EXAMPLE 2 – A SOLVABLE CASE
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 1 2 a
0.8
0.6 b
0.5
0.9
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 1 2 a
0.8
0.6 b
0.5
0.9
2nd
TBM
5a, 2b
6a, 1b
Interceptor
type
(2a,0b)
(1a,1b)
Pk matrix
(2a,0b)
(1a,1b)
(2a,0b)
(1a,1b)
3a, 2b
4a, 1b
5a, 0b
3rd
TBM
(2a,0b)
(0a,2b)
(2a,0b)
(1a,1b)
(2a,0b)
(3a,0b)
4th
TBM
1a, 2b
2a, 1b
3a, 0b
2a, 0b
(1a,0b)
(0a,2b)
(1a,0b)
(1a,1b)
(2a,0b)
(1a,0b)
(1a,0b)
(2a,0b)
5th
TBM
0a, 2b
1a, 1b
1a, 0b
2a, 0b
0a, 0b
(0a,2b)
(1a,1b)
(1a,0b)
(2a,0b)
(0a,0b)

15. EXAMPLE 2 (Cont.) – POLICY TOPOLOGY
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
TBM index
a int. stock
b int. stock
TBM index
a int. stock
b int. stock
Interceptor a
engagement
TBM index
a int. stock
b int. stock
TBM index
a int. stock
b int. stock
Interceptor b
engagement

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

17. OPTIMAL ENGAGEMENT POLICIES
EXAMPLE 2 (Cont.)
N=100 and initial interceptor stocks between 0 to 150 a’s and 0 to 40 b’s
Threat expected damage under optimal engagement policy
b Int. stock
Damage
Comparison with fixed policies
a Int. stock
Initial interceptors stock
Best fixed engagement policy
Expected damage of optimal policy
Eng. of type 1 TBM
Eng. of type 2 TBM
Expected damage
150 a, 40 b
2 a, 0 b
1 a, 1 b
4.4
2.75
75 a, 20 b
1 a, 0 b
16.2
14.25
50 a, 12 b
33.52
27.03
30 a, 8 b
0 a, 0 b
49.95
42.87

18. EXAMPLE 3 – UNTRACTABLE THREAT DEFINITION
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
TBM Type
Expected Damage
Attack size
(Uniformly distributed) 1
50-100 2 3 4
20-40 10 5 20 6
100

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