A midcourse thinning decision model for an exoatmospheric missile defence system

Missile defence system: W. W. Lau

A midcourse thinning decision model for an exoatmospheric missile defence system Ivilliam IV. Lau Department o f Management Science, California State University, Fullerton, California 92634, USA [Received November 1978)

The performance of a terminal defence system can be improved by thinning an intense attack of long-range ballistic missiles (ICBM) in the exoatmospheric region (R 1) to avoid overburdening the defence components in the endoatmospheric region. While long-range interceptors used in R1 are more costly and perhaps less reliable than the short-range interceptors, the defence can make several thinning responses to threats en route to some critical targets in R 1 because of the relatively long duration of the threat's midcourse flight. Let t = 1, 2. . . . , N index the tth response. A simple nonlinear programming model is activated before each attempt to estimate grossly the global defence levels committed in the two regions. Then, a ( N - t + 1)-stage decision process can be optimized to obtain the actual defence level for the tth response. Consequences of the response will be observed and parameters updated before the next cycle begins. Numerical examples with 10 ICBMs and four phases are given as illustrations.

Introduction The models developed in this study evolved from the need to thin out a heavy ICBM attack before the threats re-enter the atmosphere. The nature of a Midcourse Defence System (MDS) differs from the currently existing Terminal Defence System (TDS) in that the relatively long duration of tile flight enables the system to have more than one response to the same threat, and then to observe and evaluat~ the consequence before the next group of interceptors are launched. Such is not the case in TDS where there is enough time for the system to respond to a re-entry vehicle only once before the threat impacts on its target although several interceptors may be launched simultaneously to aim at several points on the trajectory path of the same threat. There are two submodels developed here: the gross planning model and the thinning decision model. An alert of an intense attack by some early detection and warning system sensors initiates the process of gross planning, whose objective is to estimate roughly the strategic force levels (interceptors) to be committqd in the two defence systems based on a cost function. The allocated long-range interceptors are used later to estimate the number of long-range interceptors committed in the MDS, while the allocated short-range interceptors measure the level of saturation of 0307-904X/79/030295-05/$02.00 © 1979 IPC Business Press

the TDS, or equivalently, the burden to the TDS should these hostile missiles penetrate the MDS. The estimated burden, when fed back to the MDS in the form of a burden vector/3, is a crucial and sensitive factor in midcourse thinning for determining the actual long-range interceptors launched in each of the several responses on these threats. Background information on the trajectory of a ballistic missile and its defence systems can be found in Adam) Gross planning m o d e l Notation and assumptions Let X = (xl, x 2 . . . . . Xm) be a m * 1 attack vector, whose ith component x i represents the number of hostile missiles en route to the target i. A target is considered here as an area target (e.g., population centre, missile farms) rather than a point target (silos), although it is recognized in the model that an attacker may possess a lethal radius or explosive yield large enough for a single penetrating warhead to be capable of completely annihilating the target. For convenience, we let symbols with upper and lower bars indicate tile MDS parameters and their TDS counterparts respectively. Let [ = ( [ l , f 2 . . . . . C 0 and_/= (_/z,!2,... ,-Ira) be tile resource vectors, where li,_It represent the number of longranged and short-ranged interceptors covering the target i.

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Missile defence system: W. 14/. Lau

U = (u 1, u2 . . . . , urn) gives the utility value for the m targets;H = ( h i , h 2 . . . . , hm) is the relative hardness vector, each component of which represents the fractional damage done when a unit of explosive yield, say a thousandth of a megaton, impacts on that target; r/= (7/1, r/2 . . . . , r/m); r/i is the estimated average yield in a thousandth of a megaton for threats towards target L The parameter can be evaluated by considering the intensity of the attack, the type of missiles, and their expected payload packages. p, _/2are the kill probabilities for long- and short-range interceptors respectively;6, g their costs; and r is the reliability of an attacker. These time-dependent system parameters are estimated periodically throughout the battle. For example, p, p are constantly revised on the basis of the observed perfffrmance of the interceptors, existence of blackouts, weather conditions, and the degradation of sensor performance, etc. }" = ( P l , P 2 . . . . ,-Pro) and _Y= (Yl,Y2 . . . . . Ym) are the decision vectors which represent the all~cated interceptors covering these m targets. Cost function atzd constrahzts A s Y i , y i long- and short-range interceptors are to be launched against x i missiles, r(x i - PYi - P_Y_i)missiles are expected to penetrate the two defence systems to hit target L The expected damage on target i becomes: u l min {1, max[0, hirlir(x i -- PYi -- _P_Yi)]}

(I)

The defence may at best achieve zero leakage regardless of defence levels. Since 'negative penetration' is meaningless, the 'maximum' operator is necessary. Likewise, the offensive may achieve no more than a complete destruction of target i of value ui, the 'minimum' operator thus appears in the above expression. Adding the interceptor costs (~Pi + ~_Y_i), we have the total cost function and the model for the system: m

z(x, } ' , Z )

=

E

ui min {I, max [0, hiTlir(xi - PPi

i=l

For a particular phase t, an N - t + I stage optimization process is activated. Within each stage n, as n goes backwards from stage = N - t + l , N - t . . . . . 2, a (x~{t) + 1)* 1 dummy decision vector: O(t, n) = [d0(t, n), d r (t, it) . . . . . dxi(t)(t , 11)]

will be obtained, whose ]th component di(t, n) represents the least-burden decision assuming that ] hostile missiles have entered the nth stage. For n = 1, only one optimiza. tion is necessary since it is known with complete certainty that] = xi(t ). In fact d * ( t ) = dxi(t)(t, 1) is the number of interceptors actually launched in this phase. After d * ( t ) interceptors have been fired and their performance observed, the kill probability, the number of surviving threats, the inventory for resource and the burden vector are then revised with the latest information. The new estimates on these parameters become contributory factors for the next d * ( t +1). The cycle thus repeats until t = N, after which, the TDS will take over the responsibility for defending those remaining missiles who have successfully penetrated the MDS. Transitional probability Suppose dhOz) interceptors are launched to intercept h ICBMs. The observed remaining threats, k, can be h again (no hit), or h - 1, h - 2 . . . . . h --dh(n). Let Phk denote the transitional probability that the state (remaining ICBMs) changes from h to k, Phk is a function of the decision dh(n) and the kill probability p(t). In fact: Phk = Phk(dt, OZ), p(t)) = 0 = (d,,(n)] #(t),,_k(l \h -k/

for k > h _p(t))a,,(n)_(,,_k )

for k = h - d h O 0 . . . . . h 3htltistage decision process The total expected cost function within each phase consists of two components: the interceptor cost 6 and

~~¢nt

- PYi)] + 6~i + _a_Yi}

with the following constraints: Yi, Yi are non.negative integers Vi i

2i <.ii,

T z,
(2)

y i<~[i V i

T h i n n i n g decision m o d e l Overview x] hostile ICBMs have been observed on their way to a critical target i. Because of the relatively long duration of a midcourse flight, it is advantageous to carryout the thinning mission in several shoot-then-observe cycles rather than to launch the entire allocated Yi interceptors in one response. As it takes time to launch an interceptor and to observe its consequences, we assume that there are N such cycles possible throughout the threats' midcourse flight and define each oycle as a phase indexed by t.The objective of the thinning decision model is to decide on the number of interceptors launched at each phase, denoted by d*(t), to thin out the number of surviving missiles in rids phase, denoted by x,(t).

296

A p p l . M a t h . M o d e l l i n g , 1979, V o l 3, A u g u s t

(3)

Guidance control

r , ( ~ l - - ~

center

.~111

0(~1

I

~

1

Figure I N = 4. four midcourse thinning responses possible. Threats enter first phase, four-stage process necessary to obtain d*(1)

Figure 2 xi(2) remaining threats now enter phase 2. A three-stage decision process is necessary to obtain d * (21

Missile defence s/stem: 14/. IV. Lau the estimated burden to the TDS. Because of the probabilistic nature of the problem, we present the objective cost function by stages. We start with stage N - t + 1 and go backwards until stage 1, where it is known for sure that there are xi(t ) ICBMs entering that stage.

Beta to read 038

12 15

Stage N

ThreatJ

4

0

0

1

1

2 3 4 5 6 7 8 9 10

2 3 4 5 6 0 0 0 0

Stage n = N - t + I. For convenience and simplicity, define djO 0 = di(t, n), and ¢Oz) = c)i(t , n). I f k out o f j ICBMs survive the diOz ) interception attempts in this stage, the total expected cost COt) is the sum of the interceptor cost 6diOz ) and the expected burden to the TDS, Pjk * ilK(t). As k ranges from ] - diOz) to j, we have, assuming various j missiles entering this final stage: i

Oj(n)=gMjO0+

~. pik(djQz),p(t))*13x(t) k=i-dj(n)

(4)

where Pik(dj(n), p(t)) is given by equation (3) and dj(n) = O, 1,2 . . . . . j. Since no more interceptors should be launched than remaining threats in each of the MTS responses, for each j, diO 0 can be chosen such that ¢i(n) is minimized. A set of minimized costs (~TOz),j = o, I . . . . . x~(t) + 1 can then be computed and saved for the next stage. Other stages. For the ( N - t)th stage, the expected cost function assumes the same expression as given by equation (4), except that the burden 13k(t) is replaced by the 4 ~ , ( N - t + 1) obtained in the previous stage. Again, the least-cost decision d i ( N - t) can be determined f o r j = 0, 1,2, . . . , x~{t), and their corresponding minimized costs 4~7(N-t) computed and saved for the next stage. In general, for n = N - t , N - t - 1 . . . . . 2, we choose diOz ) to minimize ~b~(n):

{

q~(n) = min ~d/(n) +

aj(n)

~

k=j --ai(n)

pik(di(n ),

17

18 19 20 20 20

O p t . d e c D * Min. cost

For stage 4 min. cost vector is 0 2.5 5.5 8.625 11.6875 14.5955 17.298 19 20 20 20 Stage N

Threat J

Opt. dec D* Min. cost

3

0

0

1

1

2 3 4 5 6 7 8 9 10

2 3 4 5 6 7 8 9 0

0 2.5 4.5 6.60983 8.63281

!

j = O, 1,2 . . . . . xi(t) = o,

I, 2 . . . . . i

J ~

Threat J

Stage N

Finally, for stage n = 1, dxi(t)(t, 1) = dx~'(t)(1) is the decision that would minimize: 6di(1) +

0 1 2 3 4 5 6 7 8 9 10

2

pik(di(l),p(t))*qSk(2))

k=J-aiO ) where j = Xl{t ). The decision dx*.(t)(1) is the output of the thinning decision model for this phase. It Is the number of mferceptors actually launched by the guidance control centre. t

,

0 2.25

4.5 6.60938 8.63281 10.6295 12.6217 14.6116 16.6083 18.6117 20

Forstage 3 min. costvectoris

p(t))* ¢~(n + I)1 djO0

0 2.5 5.5 8.625 11.6785 14.5955 17.298 19 20 20 20

.

Numerical example Suppose i0 ICBMs (remaining) are on their way to hit a critical target. There is enough time for four separate attempts on them before they re-enter the atmosphere. Let the cost of a long-range interceptor be 1 and its current estimated reliability (kill probability) be 0.5. The current burden to the TDS, as estimated from the present resource status at TDS and the utility value of the critical target, is expressed in 13= (0, 3, 8, 12, 15, 17, 18, 19, 20, 20, 20) whose ith component represerrts the burden to the TDS if ( / + 1) ICBMs penetrate the MDS. The objective here is to determine the number of interceptors launched. After a four-stage decision process is finished, no interception will be attempted as demonstrated below.

10.6295 12.6217 14.6116 16.6083 18.6117 20

Opt. d e c D * Min. cost 0 I 2 3 4 5 6 7 8 8 0

0 2.125 4.1875 6.19336 8.18945 10.1889 12.1885 14.1883 16.1879 18.1894 20

For stage 2 min. cost vector is 0

2.125

4.1875

StageN

Threat J

1

10

6.19336 8.18945 14.1883 16.1879

10.1889 18.1894

12.1885 20

O p t . d e c D * Min. cost 0

20

Case 2: Different burden vector. If we are given the same data as in Case 1 except that the burden vector has slightly higher components: / 3 = ( 0 , 3 , 8 , 12, 15, 17, 18,21,23, 24,25) The least-cost decision would be to launch 8 interceptors

Appl. Math. Modelling, 1979, Vol 3, August

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Missile defence system: W. W. Lau

in this phase with an expected cost of 20.0753. The output computation is summarized below: Beta to be read 0 3 8 12 15 StageN

Threat J

4

0

17 18 21

23 24 25

O p t . d e c D * Min.cost 0

1

1

2 3 4 5 6 7 8 9 10

2 3 4 5 6 7 8 0 0

10 threats. Suppose only three of them successfully hit their targets. The remaining seven threats now enter the new phase and there are only three separate responses possible before they re-enter the atmosphere. Suppose p, the kill probability, remains unchanged while the burden to the TDS increases from the latest assessment of the environment and resource status by the TDS: /3' = (0, 5, 10, 14, 16, 18, 20, 22, 24, 25, 25)

0 2.5 5.5 8.625 11.6875 14.5955 17.298 19.7968 22.1267 24 25

A three-stage optimization process is activated and the decision is to launch seven: Beta to read 0 5 10 StageN

ThreatJ

3

0

0

1

1

2 3 4 5 6 7

2 3 4 5 6 7

For stage 4 min. cost vector is 0

2.5

5.5

8.625

StageN

ThteatJ

3

0

11.6875

14.5955 22.1267

17.298 24

19.7968 25

Opt.deeD* Min. cost 0

0

1

1

2 3 4 5 6 7 8 9 10

2 3 4 5 6 7 8 9 10

2.25

4.5 6.60938 8.63281 10.6295 12.6217 14.6179 16.6168 18.6203 20.6196

For stage 3 min. cost vector is 0 2.25 4.5 6.60938 8.63281 10.6295 12.6217 14.6179 16.6168 18.6203 20.6196 Stage N

Threat J

2

0 1 2 3 4 5 6 7 8 9 10

Opt. deeD* Min. cost 0 I 2 3 4 5 6 7 8 8 8

0 2.125 4.1875 6.19336 8.18945 10.1889 12.1885 14.1884 16.1879 18.1895 20.1888

For stage 2 min. cost vector is 0

2.125

4.1875

StageN

ThreatJ

1

10

6.19336 8.18945 14.1884 16.1879

10.1889 12.1885 18.1895 20.1888

Opt. d e e D * Min. cost 8

20.0753

Case 3. Next phase. Case 3 extends the result of case 2. As decided, 8 interceptors were launched against these

298

Appl. Math. Modelling, 1979, Vol 3, August

14 16 18 20 22 Opt.deeD* Min. cost 0 3.5 7 10.375 13.5 16.3457 18.9388 21.3359

StageN

ThreatJ

Opt.deeD* Min. cost

2

0

0

0

1

1

2.75

2 3 4 5 6 7

2 3 4 5 6 7

StageN

ThreatJ

opt.deeD*

1

7

7

5.125 7.25 9.26563 11.2569 13.2481 15.2451 Min. cost 14.466

Case 4. If the reliability of a long-range interceptor is currently estimated at p = 0.3 instead of 0.5 and everything else being the same as in case 3, the decision would have been launching no interceptors during this phase: Beta to be read 0 5

10 14 16 18 20 22 Opt. decD* Min. cost

StageN

Threat J

3

0

0

1

I

2 3 4 5 6 7

2 3 0 0 0 0

0 4.5 9 13.157 16 18 20 22

Opt.deeD* Min. cost

Stage N

Threat J

2

0

0

0

1

I

4.15

2 3 4 5 6 7

2 3 4 0 0 0

8.153 11.8927 15.2076 18 20 22

Missile defence system: 14/. W. Lau

StageN

ThreatJ

1

7

Opt. d e e D * Min. cost 0

22

Case 5. Continuing from case 4, we let the entire seven threats enter the next phase. The new burden vector is given based on a current assessment of the TDS:

/3T = (0, 3, 13, 14, 15, 16, 17, 27) A two-stage decision process is then activated and the optimal decision is to launch three interceptors. Beta to read 0 3 13

14 15 16

17 27

Stage N

ThreatJ

2

0

0

1

0

2 3 4 5 6 7

2 0 0 0 0 4

StageN

ThreatJ

1

7

O p t . d e e D * Min. cost 0 3

9.63 14 15 16 I7 22.9609

O p t . d e e D * Min. cost 3

21.7971

As demonstrated in the previous numerical examples, the decision variable is sensitive to such estimated values of the systems parameters in each phase: the burden vector /3 and the interceptor reliability #. As/3 increases, MDS tends to launch more interceptors and as p decreases, it will launch less. As sensor performance improves, the estimate on p and/3 become more accurate and the cost ¢ becomes more realistic. A decision is therefore as good as the information (estimates on these parameters) fed to the decision-maker (the thinning decision model).

Concluding remarks

A moment's reflection on the total cost function in the section on tl~e gross planning model reveals that a target with high utility value will receive a bigger share of defenders. The tradeoff is between the utility value of a target and that of the interceptors. The most difficult part of implementing this model is to assign these utility' values to different potential targets. How does one equate the value of a small town, say, with that of the interceptors? How does one objectively derive the necessary guidelines to evaluate the target values? Furthermore, who should set up these guidelines? As evidenced in the two mathematical models, the decision variables are sensitive to many parameters,/3, p, N, ~, xi, tti, etc. Their estimates undoubtedly depend on resource status and the precision of their estimates depends much on sensor performance (radars, early warning system components). Each parameter itself merits further analysis. Perhaps a large-scaled simulation study is the most effective way to study the interaction and sensitivity of these parameters. As can be expected, the implementation o f a large realtime mathematical programming would impose stringent requirements on the capability of a computer. However, it should be pointed out that the probabilities corresponding to different decisions may be stored in core to allow speedy table lookup. Furthermore, the time requirement does not impose as tight a constraint in the MDS as in the TDS. Some numerical procedureswhich are not acceptable in the current TDS because of lengthy computation may find their places in the future MDS. Besides, the predicted capabilities o f super computers in the 1980s indicate that the implementation of two large defence systems are entirely feasible. Acknowledgement This work is a development from one of the author's research projects originally sponsored by an independent research and development fund from TRW, Defense and Space Systems Group, Redondo Beach, California. It has been reviewed and cleared by TRW for open presentation and publication.

The main objectives of an ICBM midcourse defence system are to:

References

(a), alleviate the workload of a saturated Terminal Defence System; (b), extend the time interval for the defenc~ to make several responses on the same threats; (c), use the most current information to obtain the most effective decisions and (d), increase the interaction and communication between the two defence .systems. The mathematical models developed in this paper aim to accomplish these objectives. However, the implementation of such a system may require further in-depth study in areas su~ested or implied in the remarks below:

1 2 3 4

5 6 7

Adams, B. 'Ballistic missile defence', American Elsevier, New York, 1971 Bellman,R. E. and Dreyfus, S. E. 'Applied Dynamic Programming', Princeton University Press, Princeton, 1962 Duncan, R.L.hzdus. AppLMath. Rev. 1964,6,111 Howard, R. A. 'Dynamic Programming and Markov Processes', MIT Press, Cambridge, 1960 Howard, R. A. 'Dynamic Probabilistie Systems', Vol. 11, Wiley, New York, 1971 Layno, S. E. Oper. Res. 1971, 19, 1505 Lau, W. W. Indep. Res. and Dev. Rep., Department of Modelling and Simulation, TRW Defense and Space Systems Group, Redond 0 Beach, California, April, 1976

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