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"IDA PAPER P-1966

, NI

U 1 MONTE CARLO LAYERED DEFENSE MODEL

LI

Jerome Bracken

i

September 1986

E-3

LL)

SAINSTITUTE FOR DEFENSE ANALYSES1801 N. Beauregard Street, Alexandria, Virginia 22311

8 6 12 03 079 IDA Log No. HO 86931571

The work reported in this document was conducted under IDA'sIndependent Research Program. Its publication does not implyendorsement by the Department of Defense or any other gowern-ment agency, nor should the contents be construed as reflectingthe official position of any government agency.

This paper has been reviewed by IDA to assure that it meets highstandards of thoroughness, objec.ivity, and sound analytical method-ology and that the conclusions stem from the methodology.

This document is unclassified and suitable for public release.

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I WORK UNIT

11. TITLE (beIlude Seomalty CIsIRoffi~oR)

MONTE CARLO LAYERED DEFENSE MO)DEL

12. PERSONAL AUThOR(S).Jerome Bracken

13.TYP O REPORT 3.TM COVERED 14. DAIS OF REPORT (Vtor, Month. Day) )6. PAGE COUNT

FinalTO 1r.86 September 915. SUPPLEMENTARY NOTATION

17. COSATI CODES 10. SUBJECT TERMS_(Cotntnu, on rtfivs I" netaea. mwid Idenhtil byc kI. ýnuinb-o

V The MntearI Layered Defense Mode! includes boos -phase defense, midcourse defense and terminal defense. It accourv

ing fer each RV ftuwugh the succession of layers Is critical, since It is niot possible In advance to know if leakage or exhaustionwill be the source of kill when thi final events involve small ioteger interactions.

Thec Model i§quite flexible. In the boost-mhase layer, the options for battle management/command, control and communica--, .. ~ dons (BM/O'-) include random (decentralized), afficient (centra~ized), and Preferential according to number of RVs on the

missiles. In the midcourse layer, the BM/C3 options include iaridom, efficient, and preferential, in which the attack is sha ' edsuch that it can be handled as well as possible by the terminal def'n"s (in the heuristic sense, not in the guaranteed optimalsense). In th~e terminal lays, a nujmber of different attack and defense situations are represented.The exalnplA presented in the docurnentation gives test results for many combinations of defense force levels And BM10 3

capabilities.

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A'F'II%

I!

IDA PAPER P-1966

MONTE CARLO LAYERED DEFENSE MODEL

Jerome Bracken

September 1986

I DA

4DA -,..

INSTITUTE FOR DEFENSE ANALYSES

IDA Independent Research Program

I

4

PREFACE

This study was conducted as part of the Independent Research Program of theInstitute for Defense Analyses, under which significant issues of general interest to thedefense research community are investigated.

iii,

TABLE OF CONTENTS

PR EF A C E .................................................................. . . ........... iii

SC O P E. ................................................................................................................... 1

KEY A SPECTS ............................................................. . .............. 5

SUBROUTINES. .................................................................... 91. Program MAIN. ..................... ...................................... 92. Subroutine VALUES ............................................................... 143. Subroutine TDTAV ........................................................... 144. Subroutine ABP ..........................A..................................... 155. Subroutine BPD ................................................................ 166. Subroutine MCD .................................................................... 197. Subroutine TD... ........................................................ 248. Subroutine VALSURV ................................................ 27

EXAMPLE .............................................................................. 281. Set Up Targets of Differing Value ............................................ 282. Set Up Terminal Interceptors ......................... . ..... 283. Set Up Attack Allocation .......................................................... 294. Results for 12,950 Terminal Interceptors .................................... 315. Results for 12,950 and 5,950 Terminal Interceptors ....................... 336. Efficient Boost-Phase, Random Midcourse .................................. 367. Random Boost-Phase, Efficient Midcourse .................................. 368. Random Boost-Phase, Hybrid Midcourse .................................. 369. Hybrid Boost-Phase, Hybrid Midcourse .................................... 40

10. Hybrid Boost-Phase, Random Midcourse .................................. 4011. Suppression of Boost-Phase Defense ......................... 4012. Preferential Targeting by Boost-Phase Defenders .............................. 4413. Effects of Decoys ............................................................... 4414. Cencentrated Attack.. .................. .4715. "Expected Value" Versus Monte Carlo Results. .................. 49

FIGURES

Results for 12,950 Terminal Interceptors (Figure 1) .............................. 32Results for 12,950 and 5,950 Terminal Interceptors (Figure 2) ................... 35Efficient Boost-Phase, Random Midcourse (Figure 3) ............................. 37

Random Boost-Phase, Efficient Midcourse (Figure 4) ............................... 38Random Boost-Phase, Hybrid Midcourse (Figure 5) .................................. 39Hybrid Boost-Phase, Hybrid Midcourse (Figure 6) ................................ 41Hybrid Boost-Phase, Random Midcourse (Figure 7) .................................. 42Suppression of Boost-Phase Defense (Figure 8) ....................................... 43Proportional Targeting by Boost-Phase Defense (Figure 9) ................ 45

v

Fý,- t-

Table of Contents (Cont'd)

Effects of Decoys (Figure 10) ............................................................. 46Concentrated Attack (Figure 11) ...................................................... 48Comparison of "Expected Value" and Monte Carlo Results for Pk=.8(Figure 12) .................................................................................. 5 1Comparison of "Expected Value" and Monte Carlo Results for Pk=.9(Figure 13) .................................................................................. 5 2

APPENDIX ADefinitions of Variables ............................................................ A-1

APPENDIX BListing of Computer Program .................................................... B-I

v..i

SCOPE

UThe Monte Carlo Layered Defense Model consists of six major parts. They

are sumrmaeized below.

1. Set up target data base, including the value of each target and the number

of terminal interceptors at each target.

2. Set up, attack allocation. Assign the RVs of all of the attacking missiles

to targets. Equip each missile with heavy and light decoys if desired.

3. Perform boost-phase defense using one of three possible BM/C 3

weapon assignment schemes:

a. Random Assignment: Each boost-phase defender acts independ-

ently of the others. The defender picks a missile target and shoots at

it. Thus each missile may receive from none to many shots. This

" -• may be thought of as unordered fire, fully decentralized BM/C 3, or

informationless fire.

b. Efficient Assignment: Boost-phase defenders are allocated as

I!"•formly as possible across their missile targets. For instance, if

there are 1500 missiles and 1000 boost-phase interceptors, 1000 of

the 1500 missiles, selected at random, receive one shot each. If

there are 1500 missiles and 2000 boost-phase interceptors, 1000

missiles selected at random receive one shot each and the other 500

missiles receive two shots each. This may be thought of as orderedVJ fire, centralized BMIC 3 , or uniform simultaneous m-on-n defense.I~..

c. Proportional Assignment: Boost-phase defenders are allocated to

missiles in proportion to the number of RVs on the missiles.

4. Perform midcourse defens, using one of three possible BM/C 3 weapon

*: assignment schemes:

a. Random Assignment: Analogous to boost-phase defense,

,' discussed above.

-•.

"I=

b. Efficient Assignment: Analogous to boost-phase defense,

discussed above.

c. Preferential Assignment: Midcourse defenders are assigned to RVsto shape the attack so that the RVs reaching targets chosen to bedefended are likely to be killed by the terminal interceptors. Toconserve the limited numbers of midcourse defenders, thealgorithm proceeds as follows: First, all targets receiving less thana specified upper limit of RVs per terminal interceptor are defendedso that the RVs heading for the targets are reduced below aspecified lower limit. Second, the upper limit is increased andanother set of targets is defended. Eventually, the lower limit isreduced if there are enough midcourse interceptors. The algorithmhas not been proven to be optimal, but it is in the spirit of oneknown to be optinmal for adaptive terminal preferential defense.

(Important Note: It turns out that in cases where the number ofmidcourse interceptors is relatively small and damage is moderate,

preferential assignment achieves significant advantages overefficient assignment. However, when the number of midcourseinttAL.ptors is increased and damage becomes very small,

preferential assignment is not superior to efficient assignment).

Decoys are included in the midcourse defense as follows. For any ofthe above three cases, when it is decided to attack an RV with aninterceptor, the RV is examined in the context of all of the objects from

"its missile. The capability of the defense to discriminate RVs from"decoys is represented by input probabilities of (1) missed detectionagainst RVs, (2) false alarm against heavy decoys, and (3) false alarmagainst light deuoys. A random process is used to either (1) attack theRV, (2) fail to attack the RV because of missing a detection, or (3)waste an interceptor on a decoy.

5. Perform termina defens using any of four possible BM/C 3 weapon

assignment schemes:

2

a. Preallocated Fixed-Salvo Assignment: This is an implementation ofthe fixing process associated with the Prim-Read defense. This logic

is consistent with the assumption that at the individual targets theattacks are sequential and of unknown size. The defender decides inadvance to fire a volley of one or more interceptors at each incomingRV. If one o" the interceptors in the volley kills the RV, it does notpenetrate. The number of interceptors scheduled to be fired at all ofthe incoming RVs adds up to the total number defending the target.For instance, if there are 10 interceptors at a target, preallocated tofire at RVs number 1 through 6, such a firing pattern might be3,2,2,1,1,1 with no interceptors fired at RV number 7. Thus RVnumber 7 will surely kill the defended target, and any of the

engagements against RVs 1 through 6 might also result in the

defended target being killed.b. Efficient Assignment: Analogous to boost-phase defense and mid-

course defense, discussed above. Consistent with simultaneousattacks ot known size at the individual targets.

c. Limited Shoot-Look-Shoot: The defender has a certain number ofinterceptors at the target. At the first RV he fires interceptors one-

by-one until either the RV is killed or the upper bound oninterceptors able ro be fired is reached. For instance, there might beup to three shots at each RV. If all three fail to kill the RV there is apenetrator. This doctrine conserves interceptors, unlike the twodoczrines above, for there are no interceptors "wasted" in salvos.

d. Unlimited Shoot-Look-Shoot: Same as limited shoot-look-shoot

above, but no upper bound on number of shots at each RV. Thisdoctrine produces the greatest possible effectiveness of a givennumber of terminal interceptors at a defended target.

6. Assess damage to targets. For each target, determine if there areone or more penetrating RVs.

There are two significant features of the model available to extend andelaborate on the above, as fe!lows:

3

1. It is possible to suppress the boost-phase defense, either by killing a

certain number of defenders or by permitting a certain rnumber of

missiles not to be engaged by the boost-phase defense.

2. It is possible to perform hybrid combinations of BMIC 3 rules within

boost-phase defense and midcourse defense. Specifically, a certain

portion of the missiles in boost-phase defense, or RVs in midcourse

defense, can be engaged efficiently by a portion of the defense while the

V remainder are engaged randomly by a portion of the defense. This

permits exploration of a partially centralized engagement followed by a

partially decentralized engagement of the survivors.

.4

I.)

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N ,e€ , e 4 , " - ' # % " • " • " " : , -• , - •• ",:,", -' ' ' '.' 'i ''" . ',,•'"'" .• •

KEY ASPECTS

Three key aspects of the qualitative behavior of the model are summarized below.

1. Random Versus Efficient Weapon Allocation in BoLst-Phase and

Experience with the model has shown that there are very significant

effects associated with random versus efficient assignment of boost-

phase defenders. Completely decentralized assignment of boost-phase

defenders leads to many missiles not being attacked, and thus

substantially more RVs being presented to the midcourse defense.

Experience with the model has also shown tmat there are very

significant effects associated with random versus efficient assignment of

midcourse defenders. If midcourse defenders can be efficiently

assigned, at a certain point every single RV is confronted by a one-on-

one engagement, or a two-on-one engagement, and thus there tend to be

few cases in which the terminal defenses are exhausted.

The amount of resources required to achieve near-zero damage is

extremely sensitive to the BM/C 3 logic. Random assignment of boost-

phase and midcourse defenders leads to twice as many boost-phase

cdefenders and midcourse defenders being required to reduce damage to

near-zero levels than does efficient assignment, in a test case.

To achieve efficient defense in boost-phase and midcourse requires

that the defender's weapons be applied to missiles and to RVs

sparingly. First consider boost-phase defense. Assume that each

missile is to be engaged once. Assume that there is no shoot-look-

shoot. If the attack is simultaneous, the centraiized system must allocate

all of the shooters to all of the targets uniformly. This must be donewithin a period of a minute or so. If the attack is sequential, say over

five or ten minutes, the centralized system must allocate the shooters to

the subset of targets presented in accordance with a preallocated

rationing scheme. If one-half of the missiles have been launched, one-half of the defenders should be utilized (or something similar to this).

5

-

Partitioning the boost-phase defenders into subsets should not

preclude efficient, as opposed to random, defense if the subsets are

large enough. Each subset would be responsible for a subset of targets.Difficulties in efficient assignment would occur if the shooters andtargets could not be related one-to-one. Since the shooters orbit out of

range of targets, time-phased assurance that all missile targets receive

about the same. number of shots may be difficult, but should be stressed

as a design parameter of the boost-phase defense.

Now consider midcourse defense, which takes place over a

relatively long period of time. By monitoring each shooter and RV

target, it should be possible to assure that each RV receives about thesame number of shots. With respect to time considerations, efficient

allocation seems easier in midcourse than in boost-phase. Simultan,,,:

attack versus ragged attack does not seem to have significant timing

effects in midcourse defense from an assignment point of view.

The principal problem in midcourse defense seems to be the effect of

large numbers of decoys. If the K-factor, which is basically the number

of standard deviations of the measuring process separating RVs from

decoys, is large, then few RVs will escape detection and few

interceptors will be wasted on decoys. If the K-factor is small, or ifthere are enough light decoys with even a small probability of being

engaged (say 0. 1), then the midcourse interceptors may be exhausted bythe decoys, and the efficient allocation of defenders to RVs will fail.

2. Terminal Defere

The function of the terminal defense is to cope with the RVs which

get through to attack the target. The number of RVs may vary widely,since if there are many missiles shooting at the target there may be any

nuamber from zero to a large number of RVs heading for the target after

the boost-phase defense. The first two types of midcourse defense will

not eliminate this phenomenon. The third type will solve it for some

targets, but othei targets will be heavily attacked since they are "written

off' by the midcourse defense.

6

Performance of the terminal defense in situations where the number of

defenders is just a bit larger than the number of attacking RVs provides

the key difference in effectiveness. For instance, if there are sevendefenders with Pik=.9, and six attackers, a preallocated defense with

firing doctrine 2,2,1,1,1,0 would yield no chance of survival, while

shoot-look-shoot would yield a 50 percent chance of survival. The

Monte Carlo process results in quite different engagements at targets, of

which a significant number are in the regions where firing doctrines

make a big difference.

3. Expected Value Versus Monte Carlo

Another key aspect of the overall performance of the model is in the

difference between an expected value model and a Monte Carlo model.A

typical expected value model makes an assessment of the expected

number of missiles killed by the boost-phase defense and then counts

the expected number of RVs left. It assumes that these RVs are targeted

proportional to the targeting of the missiles from whence they came.Then the expected value model makes an assessment of the number of

RVs killed by the midcourse defense. The RVs surviving are again

assumed to be targeted proportional to the original targeting of themissiles. These RVs are engaged by the terminal defenders,. Typically,

the terminal defenders significantly outnumber the RVs at all of the

targets because the RVs are uniformly allocated to the targets, and thus

only one or two RVs per target are confronted by at least one interceptoreach. The Monte Carlo model, on the other hand, results in a significant

variations in RVs appearing at the attacked targets. This one poprty is

the main reason for the necessity for the Monte Carlo model. Someexperimental results have shown that, when the Pk of boost-phase,

midcourse and terminal defense is .9, the expected percent of the database destroyed in the Monte Carlo Layered Defense Model is 5.1

percent for preallocated terminal defense and 2.1 for efficient terminal

defense, whereas the expected percen of the data base destroyed by an

eApected value calculation is 1.0 percent and .3 percent, respectively.

7•.Y

•_j

For near-perfect defense, or the assured survival mission of SDI, thesedifferences are cnrcial.

8

4

SUBROUTINES

I. Empram MARIN

a. The main indexes are I 1, 12, and 13, where

(1) I1 indexes on the number of boost-phase interceptors

(2) 12 indexes on the number of mich-ourse interceptors(3) 13 indexes on the number of combinations of boost-phase

and midcourse BWiC 3 options

b. For each value of 13, three indicators are+ set:

(1) 131 --indicator for one of three boost-phase defense options(2) 132--indicator for one of three midcourse defense options

(3) 133--indicator for one of four terminal defense options

The options are discussed later. In addition, the indicators IBPMand IMCM are set:

(1) IBPM--indica;or for hybrid boost-phase defense, where 0

means no hybrid and I means hybrid(2) IMCM--indicator for hybrid midcourse defense, where 0 means

no hybrid and 1 means hybrid

c. The long print indicator LPI is set:

(1) LPI--0 means no long printout

(2) LPI-1 means long printout

d. The random number seed IRS for the entire simulation is set- The

seed is in COMMON and gives a new value each time a random

number is generated anywhere in the model.

e. The value target data base is set up by calling Subroutine

VALUES(NV'T). Subroutine VALUES creates array IV(I),I= 1,NVT consisting of the values of targets 1 through NVT. It

provides NVT to MAIN. Array IV is in COMMON.

f. The main loop on the number of replications, irdexed by IMAINfrom I through NUMB, is begun by the following statement:

9

_,• m V_ ,WP . N? - V U€, . W. V W VII T'f Sfl WW- hIF.W;- , rW•, Vr rfl~W f "'W' .,. r - Vr'.. '. V• • ,• . '-•V,. = - .. .......

DO 800O MAIN = 1, NUMB

g. The terminai defense is set up by calling Subroutine TDINV (NVT).

Subroutine TDINV creates array ITD (I), I1INVT consisting of the

number of terminal interceptors at targets 1 through N\'T. Array

ITD is in COMMON.

h. The attack allocation is set up by calling Subroutine ATTALL

(NMJS, MNWT). Subroutine ATTALL creates arrays as follows:

TAT (1,), I= I,NMIS, J= I,MNWT

LAW (I, J), I=1,NMIS, J=I,MrNfWT

IAD(I,J, 1) I=J,NMIS, j 1,MiNWT

IAD(I,J,2) I= 1,NMIS, J= 1 4NUMNWT

NMIS is the total number of missiles and MNWT is the maximuum

number of targets which each missile attacks. For instance, NMIS

woutd be 1500 and MNWT would be 2 when there are 1500

missiles, each attacking no more than two diffeirnt targets.

IAT(Ij) contains the target or targets to which the RVs of missile

I are assigned. For instance, if missile 10 is assigned to targets 100

and 200, then IAT(10,1)=100 and IAT(10,2)=200.

IAW(IJ) contains dhe number of RVs assigned from missile I to

the targets specified by IAT(I,J). In the above example, if missile

10 has 8 RVs, and if four each are assigned to targets 100 and 200,

then LAW (10,1) = 4 and IAW(10,2) = 4.

LAD (I, J, 1) and IAD (1, J, 2) contain the number of decoys from

missile I to the targets specified by [AT (I, J). In the above

example, if missile 10 has 4 heavy decoys and 50 light decoys split

evenly between the two targets then LAD (10, 1, 1) = 2, LAD (10, 1,

2) = 25, IAD (10, 2, 1) = 2 and IAD (10, 2, 2) -= 25.

10

Arays IAT, lAW and LAD are in COMMON. The values of NMIS

and MNWT are. specified in Subroutinc ATTALL and returned to

MAIN.

i. The boost-phase defense is activated against the missiles by

calling Subroutine BPD(NSHT, NTAR, MJfNWT, PK, IN)),

where:

NSFH is the number of boost-phase defenders

NTAR is the number of missile targets (typically setequal to NMIS)

MNWT is the maximum number of value targetsattacked by each missile.

PK is the boost-phase defense single-shot probability of kill.

INID is the defense option, as fellows:

(1) IND = 1 Random allocation of boost-phase defenders. Each

boost-phase defender selects and fires at missile independently of the

other defenders.

(2) IND = 2 Efficient allocation of boost-phase defenders. Boost-

phase defenders are allocated uniformly over missiles.

(3) IND = 3 Proportional allocation of boost-phase defenders.

Boost-phase defenders are allocated to missiles in proportion to the

number of RVs on the missiles.

The indicator for the hybrid boost-phase defense IBPM is queried

before Subroutine BPD is called:

(1) If IBPM = 0, then the number of shots equals NSHT, as

above.

(2) If IBPM = 1, then the boost phase engagement proceeds in two

steps. In the first step a certain number of shoots is assigrned to

efficient defense (IND = 2). In the second step the remaindcr is

assigned to random defense (IND = 1).

| 11

-.•. -

,•- • _ • • =• • _ . • ••= • • , 4t•, ,- i • -, - , ,¢ • 1 .w w-,wrr , l• -r l -•w l r -'i u • , . .. .

j. The midcourse defense is activated against the RVs by calling

Subroutine MCD (NVT, NSHT, NTAR, MNWT, PK, MND), where:

NVT is the number of value targets being protected

NSHT is the number of midcourse defenders

NTAR is the number of missiles from which the RVs arebeing released (typically set equal to NMIS)

MNWT is the maximum number of value targets attacked by eachmissile

PK is the midcourse defense single-shot probability of kill

IND is the defense option as follows:

(1) IND = 1 Random allocation of midcourse defenders. Each

midcourse defender selects and furs at RVs independently of the

other defenders.

(2) IND = 2 Efficient allocation of midcourse defenders. Midcourse

defenders am allocated uniformly over RVs.

(3) IND = 3 Preferential allocation of midcourse defenders.

Assuming knowledge of the complete attack, including the

destination of all RVs, midcourse interceptors are allocated to shape

the attack such that the RVs which remain after the midcourse defense

can be destroyed by the terminal defense, the overall goal being to

preserve as much surviving value as possible.

The indicator for the hybrid midcourse defense IMCM is queried

before Subroutine MCD is called:

(1) If IMCM =0, then the number of shooters =NSHT, as above.

(2) If IMCM = 1, then the midcourse engagement proceeds in two

steps. In the first step a certain number of shooters is assigned to

efficient defense (IND = 2). In the second step the remainder is

assigned to random defense (IND = 1).

k. The terminal defense is activated against RVs arriving at each target

by calling Subroutine TD (NVT, NMIS, MNWT, PK, IN])), where

12

NVT is the number of value targets being protected

NMIS is the number of missiles in the attack

MNWT is the maximum number of value targets attacked by eachmissile

PK is the terminal defense single-shot probability of kill

IND is the defense option, as follows:

(1) IND = 1 Preallocated fixed-salvo defense. For each target, each

arriving RV, sequentially, receives a salvo of a pre-specified number

of interceptors.

(2) IND = 2 Efficient allocation of terminal interceptors. For each

target, interceptors are allocated as unifomily as possible over all

attacking RVs.

(3) IND = 3 Limited shoot-look-shoot. For each target, RVs are

engaged one-by-one. The first RV is shot at by one interceptor, then

another, until either it is killed or the upper limit per RV is reached.

The second RV is engaged similarly. The process terminates when

there are no more RVs or no more interceptors.

(4) IND = 4 Unlimited shoot-look-shoot. Same as (3) above, butno limitation on interceptor engagements per RV.

1. After the terminal defense there are NW(I), I = 1 ,NVT succcessful

penetrators at each target. Where NW(I) = 0 the target is not

destroyed. The terminal defense subroutine TD creates the relative

frequency array of successful penetrators NWRF (I).

m. Value surviving after the attack is computed by Subroutine

VALSURV (NVT, IVSS), which creates the value suni surviving

associated with those targets where NW (1) = 0, 1 = 1, NVT.

n. The average and standard deviation of value damaged for the trials

made thus far (up to the current value of IMAIN) is computed and

printed.

13

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• ; _t • • • • . W:rr .. flr w fl•, f l. W • I• .. W t--' •- - l .• -•. .• ' l•, •, . ', .... ... . .. -.. . .. . . .

o. This is the end of the main loop numbered 8000 on the number of

iterations to be performed.

p. This is thi end of the main loop numbered 9999 on the specification

of the case to be simulated, for a particular number of boost-phase

defenders and midcourse defenders, and for a defense logic for the

three layers.

2. Subroutine VALUES (NV'

Subroutine VALUES (NVT) creates the array of target values IV,

where IV(I) is the value of target I, and returns to MAIN the quantity

NVT, the total number of targets. The content of the subroutine is at the

option of the user.

For instance, if there were 1000 targets with values 1000 through 1,

the contents of the subroutine would be as follows:

NVT= 1000

DO 10 1 = 1, I000

IT= 1001 -1

10 IV(I) = IT

Subroutine VALUES (NVT) of the example, contained in Appendix

B, generates a set of values for 2533 targets.

3. Subrutine TDINV f

Subroutine TDINV (NVT, Terminal Defense Inventory, creates the

array of terminal inventories of interceptors ITD, where r7TD (I) is the

number of terminal interceptors at target I. The content of the

subroutine is at the option of the user.

For instance, if in subroutine VALUES there were created 1000

targets with values 1000 through 1, and if they were to be defended by

one interceptor per ten units of value, truncated to the nearest integer,

the contents of the subroutine would be as follows:

14

1)0 10 1 =1, 1000

10 ITD (I) = IV() / 10

Subroutine TDENV (NVT) of the example, contained in AppendixB, deploys terminal interceptors for 2533 targets equal to one-fourth of

their value, truncated to the nearest integer.

Subroutine ATFALL(NMIS, MNWT), Attack Allocation, contains

the allocation of attackers to targets.

It first sets the number of missiles NMIS and the maximum number

of targets at'tacked by each missile MNWT (equivalently, maximumnumber of targets cross-targeted by any attacking missile),

The subroutine creates arrays IAT (1,J), IAW (IJ), LAD (IJ,K)where I = 1, NMIS, J = 1, •N•WT and K = 1,2.

Consider a simple example of 1000 missiles attacking targets 1

through 1000 with 5 RVs on each missile, 2 heavy decoys on eachmissile and 10 light decoys on each missile. Each missile attacks only

one target. The contents of the subroutine would be as follows:

NMIS = 1000

MNWVT = 1.

DOl10 1 = 1, 1000

IAT (1, 1) 1

IAW (I, 1) = 5

IAAD (1, 1, 1) = 2

10 IAD (I, 1, 2) = 10

Subroutine ATTALL (NMIS, MNWT) of the example, contained inAppendix B, sets up an attack on 500 targets. It allocates 500 ICBMs

with 5 RVs each to the 500 targets, then 500 ICBMs with 5 RVs each to

15

the same 500 targets, then 500 SLBMs with 10 RVs each to the same

500 targets. In the option where decoys are included, the ICBMs have

3 RVs, 4 heavy decoys and 30 light decoys while the SLBMs have 6

RVs, 8 heavy decoys and 30 light decoys.

5. Subroutine BPD (NSHT. NTAR. MNWT. PK. LND

Subroutine BPD (NSHT, NTAR, MNWT, PK, IND), Boost.-Phase

Defense, receives from MAIN the number of boost-phase shooters

NSHT, number of missile targets NTAR, maximum number of targets

attacked by any missile MNWT, probability of kill PK and indicator of

defense option IND.

Its function is to take the array of missiles accounted for in IAT(I, J)

and to kill certain of them. For those killed, the arrays in COMMON

TAT (I, J), LAW (I, J), TAD (I, J, 1) and AD (I, J, 2) are set to zero.

a. Boost-Phase Defense SupIRMUM

The first step is to decide whether or not to suppress the boost-

phase defense, and, if so, how it is to be suppressed. The indicator

IBPDS is consulted, with options as follows:

IBPDS = 0 no defense suppression

IBPDS = 1 reduce number of shooters by NKLD

IBPDS = 2 reduce number of shooters by NSHT *FKLD

IBPDS = 3 do not engage number of missiles NTARS

[BPDS = 4 do not engage fraction of missile°. FTARS

In the beginning of the subroutine the user sets values for 1BPDS,

NKLD, FKLD, NTARS, FTARS. These are used as necessary,

depending on the option indicated by IBPDS.

If IBPDS = 0, 1 or 2, then IBPDST = 0. For these three cases the

number of boost-phase shooters is reduced by 0, NKLD or

NSHT*FKLD,

16

- 4.

If IBPDS = 3 or 4, then IBPDST = 1. The fraction of missiles not

to be engaged is set to FTARS.

b. Random Defense

Consider boost-phase defenders I through NSHT. For each

defender, select a random number A. If IBPDST = 0, the missile

corresponding to that number will be engaged, If IBPDST = 1, themissile corresponding to that number will not be engaged if A <

FTARS.

If the missile is to be engaged, draw a random number to see if itis killed. If it is killed, set IAT, lAW and LAD arrays equal 0 for

that missile.

c. Efficient Defense

(1) Consider the case where boost-phase shooters are fewer than

missile targets (NSHT < NTAR). Set INDTA (I), I = 1, NTAR =

0 to keep track of which targets have not been engaged. Consider

boost-phase defenders 1 through NSHT. If IBPDST = 1, draw a

random number A and do not engage missile if A < FTARS. If a

missile is to be engaged, draw a random number A to see which

missile is to be engaged. Call this IMIS. If INDTA (IMIS) = I

skip this missile for it has already been engaged. Instead, drawanother random number to attempt to find a target. If INDTA

(IMIS) = 0 then set INDTA (IMIS) = 1, draw a random number

and determine if the missile is killed. If it is killed, set IAT, IAW

and LAD to zero for that missile.

(2) Now consider the case where boost-phase shooters are equal

to missile targets (NSHT=NTAR). Do not bother to set INDTA

(I), I = 1, NTAR to 0, since all targets will be engaged unless

defense is suppressed.

For boost-phase defenders 1 through NSHT, if IBPDST = 1,

draw a random number A and do not engage missile if A .•FIARS. If missile is to be engaged, draw a random number A to

17

see if it is killed. If so, set LAT, IAW and LAD to zero for that

missile.

(3) Now consider the case where boost-phase shooters are greater

than number of missile targets (NSI-IT > NTAR). Set INDTA (I), I

= 1, NTAR equal to 0 to keep track of targets which have been

engaged.

A fraction P1 of the targets will receive IRP1 shots and a fraction

(1 - P1) of the targets will receive IR shots, where

R SHT/TAR

IR=R

FIR= IR

P1 = R -FIR

IRPI = [R + 1

For example, if SHT = 1400 and TAR 1000, then .4 of the

targets will receive 2 shots and .6 of the targets will receive 1 shot.

First, consider the NTP1 targets which will receive IRP1 shots.

Check to see if IBPDST= 1. If so, and if A < FTARS, a missile is

not attacked. Otherwise, pick a missile IMIS at random. If

-INDTA(IMIS)=0 it can be attacked. Set INDTA(I)= 1 and shoot

IRP1 shots at the missile. If it is killed, set IAT, IAW and LAD to

zero for that missile.

Second, consider the remainder of the targets. If INDTA(I)=0 and

defense not suppressed, shoot IR shots at the missile. If it is

killed, set IAT, LAW and LAD to zero for that missile.

d. Defense in Proportion to Number of RVs

Count the total number of RVs FIAWT. Compute the desired

shots per warhead SPW.

1

- - - - - - -

(1) Consider the case where shooters are less than targets (NSI-I"

< NTAR). Set INDTA(I)=O, I= 1, NTAR to 0 to keep track of

targets which have been engaged.

Pick a missile at random and count the number of RVs on it

FIAWMT. The number of shots to be fired at that missile is

SPvW*FIAWMT, rounded to the next higher or lower integer (using

a random number based on the number of shooters and targets).Shoot up to that many shots at the missile and if it is killed set IAT,

LAW and lAD to zero.

(2) Consider the case where shooters are equal to or greater than

targets (NSHT Ž NTAR). There are two options within this case.

If at least one shot per missile (complete coverage) is not required

ICMPLT=O) then fire shots at missiles in accordance with their

number of RVS. In this case a missile may not be engaged it it has

fewer RVs than the cutoff.

Alternatively, pass through all of the missiles, firing one shot at

each. Then, utilizing the remainder, fire in proportion to the

originally computed shots per warhead.

6. Sbroutine MCD (NVT.NSHT.NFAR.J[NW-r.PK.1NDJ

Subroutine MCD (NVT, NSHT, NTAR, MNWT, PK, 1ND)),

Midcourse Defense, receives from MAIN the number of value targets

NVT, number of midcourse shooters NSHT, number of RV targets

NTAR, maximum number of value targets attacked by any missile

-MNWT, probability of kill PK and indicator of defense option IND.

The indicator IDECOYS is set at the beginning of MCD. If

fDECOYS=O there are no decoys. If IDECOYS= 1 there are decoys.

Associated with decoys are parameters as follows:

PMD = probability of missed detection

PFA I probability of false alarm against decoys of"type 1

19

PFA2 =probability of false alarm against decoys of

type 2.

'The array IAWT (IJ) is created. It is set equal to LAW (I,J). The

array IAWT will contain the actua number of RVs as the subroutine

progresses. In making decisions some of the options will consult the

array LAW (IJ), the RVs entering the subroutine, while others will

consult the array IAWT (IJ), dte current status of the simulation.

a. No Decoys. Random Defense

For shots 1 through NSHT, select an RV by drawing a random

number, pilcking a missile and seeing if there were any RVs on that

missile before it entered subroutine: MCD. This is done by

examining the array LAW (I, J).

If there is one or more RV associated with the missile, shoot at

the RV. If it is killed, reduce the actual number lin IAWT (I, J)

down as far as zero.

In this option RVs can be killed even if already dead, since there

"is no coordination.

b. Lo Decoys. Efficient Defense

First, consider shooters equal to or less than targets (NSHT ,5

NTAR). For each shot, select a missile. If that missile actually

contains an RV as indicated by IAWT (I, J), shoot at it, and if

"successful, reduce the actual RVs on the missile by 1. If that

missile does not contain an RV, go on to consider another missile

by again drawing a random number.

Next, consider shooters greater than targets (NSHT > NTAR).

In this case plan to shoot at a fraction P1 of the targets IRP1 shots

and at a fraction (l-P1) of the targets IR shots, where, as in the

boost-phase defense,

R = SHTiTAR

SIR !R

20

FIR = JR

PI = R-FIR

IRP1 = IR+i

For each missile, determine whether or not it has any RVs. If it

does, count the total IAWTT. For each of the IAWTI', by

drawing a random number, decide if it will receive IRP1 or IR

shots. Shoot up to the desired number of shots at the RV, If it iskilled, reduce IAWT (I, J) by 1.

c. No..Decoys. Preferential Defense

For this cavt Subroutine MCD3 (NVT, NSHT, NTAR,

MNWT, PK) is called.

The number of RVs assigned to each target is computed as

NWT (IMAIN), IMAIN= 1, NVT. For each target indexed by

IMAIN, the indicator INDI (IMAIN, JT)=T indicates the number

of the Ith missile shooting at it and the indicator INDJ (IMAIN,

MT) = J indicates the Jth group of RVs from the Ith missile. In the

example there ame three missiles shooting at each target, and 2533

targets, so the dimensions are INDI (2533,3) and INDJ (2533,3).

Next, there are seven iterations wherein the midcourse defense

attempts to shape the entire attack so that the zermnnal defenders

can cope with it. Upper bounds RATSU and lower boundsRATSL are set for the quantity

RAT = N WT(IMAIN)/ITD(IMAIM)

or ratio of attackers to terminal defenders. The least-attacked

targets are dcfended first (as measured by an upper bound',

reducing the RVs until they ar-. belcv a certain lower bound.

Then another set of controls is esmblished, and so on. The

algorithm is summarized as follows:

21

-= N

Count number of RVs assigned to each target from all surviving

missiles. At each target let X=RVs,/Terminal Defenders. Until

midcourse defense is exhausted:

(1) For all targets where X < 1 kill sufficient RVs to achieve

X<.8

(2) For all targets where X <2 kill sufficient RVs to achieve

X•.8

(3) For all targets where X •3 kill sufficient RVs to achieve

X<:.8

(4) For all targets where X •4 kill sufficient RVs to achieve

X:5.8

(5) For all targets where X < U* kill sufficient RVs to achieve

X 5.8

(6) For all targets where X : .8 kill sufficient RVs to achieve

X <.6

(7) For all targets where X : .6 kill sufficient RVs to achieve

X = 0,

where U* is the input upper bound (currently 10).

As RVs are killed they are subtracted from IAWT.

d. Dccoys. Random Defense

For shots I through NSHT, an RV is selected by drawing a

random number, picking a missile and seeing if there were any

RVs on that missile before it entered subroutine MCD. This is

done by examining IAW(I, J). If there is an RV in IAW (IMIS,

J) the subroutine MCDWD (Midcourse Defense with Decoys) is

called, with calling sequence MCDWD (IhD,IAWT,IMIS, J,

22

MNWT, PMD, PFA1, PFA2, PK, INDSHT). The subroutine

decreases IAWT (MIS, J) if an RV is killed and sets

INDSHT=O when no mnidcourse defender expended

INDS-r= 1 when midcourse defender expended (eitheragainst real target or false target).

The first indicator in the calling sequence, IN), is set to 0

for random defense and I for efficient defense.

The subroutine MCDWD operates as follows. The total

number of objects NOBJT on a missile is counted, made up of

the components:

NOBJI = number of RVs

NOBJ2 = number of heavy decoys

NOBJ2 = number of light decoys

where

NOBJT = NOBJ I + NOBJ2 + NOBJ3.

Each RV is txeated es though it comes from a cluster

characterized by the initial mix of objects as above, except

when:

IND= 1, NOBJ 1 is computed from IAWT.

Ua4IND=0, NOBJI is computed from IAW

The group of objects totaling NOBJT is processed as

follows. A random number is drawn and an object chosen

which matches it. (For instance, if there are 4 RVs, 2 heavy

decoys and 4 liglit decoys and the random number = .05 the

object is an RV, while if the random number = .91 the object is

a light decoy.

Another random number A is chosen. If the object is an RVand if A .< PMD (probability of missed detection) it is not shot

23

t3

at. A defender is not wasted but neither is an RV engaged. If

A > PMD the RV is engaged by drawing a random number A

and if A < PK the RV is killed.

If the object is a decoy PFA is set equal to PFA 1 or PFA2,

depending on the type. A random number is chosen. If A _

PFA there is a fdlse alarm and midcourse defender is wasted.

e. Decoys. Efficient Defens

This portion of the program i3 analogous to the no decoys,

efficient defense portion discussed in b. above. However, the

subroutine MCDWD is called.

Before entering MCDWD, IND. 1, so the number of RVs is

computed from IAWT rather than LAW as in the above

discussion. Otherwise, the procedure is the same as discussed

in d. above.

f. Dtcoys. Preferential Defense

Not currently coded.

7. iY-&tZiMTDNYT. NMIS. 1MWT. PK. _

Subroutine TD (NVT, NMIS, NP4WT, PK IND), Terminal

Defense, receives from MAIN the number of value targets NWT,

numb,-r of missiles in the attack NMIS, maximrum number of value

targets attacked by any missile MNWT, probabii~ty of kill PK, and

indicator of defense option IND.

The number of warheads arriving at each target is ca!.nulaied fer

all targets, namely NW (IMAIN), IMAIN = 1, NVT. Then for all

targets one of four .4ttrition methods is utilizcxI in detenrL'dng the

number of warheads surviving after the terrr.inal defrise, or the

final status of NW (IMAIN), IMAIN = 1, NVT.

a- Ilat =ed-Salvo Terrinal DefenseA particular doctrine is chosen for the targets defended by

each number of interceptors. In the example, there are options

24

for 2, 5 and 10 interceptors at each target. Their firing

doctrines are, respectively,

2 interceptors - 1, 1

5 interceptors -- 2, 1, 1, 11

10 interceptors-- 2, 2, 2, 1, 1, 1, 1

For instance, with five interceptors, 2 are fired at the first

RV, 1 at the second, 1 at the third, 1 at the fourth, and none

thereafter. This defense should be conceptualized as fitting a

situation where the attack is sequential and of size unknown to

the defender. For 10 interceptors, the "price" of the target is 8,

which surely kills it. The defense is balanced so that the

attacker has about the same expected value per RV up to an

attack size of 8 RVs. This preallocated fixed-salvo defense is

analogous to the "Prim-Read" defense.

For each target IMAIN = I, NVT, the number of warheads

NWT destined for it is determined. The number of

engagements JENGS is chosen corresponding to the firing

doctrine associated with the number of interceptors defendingthe target. For RV 1 through NWT the number of interceptors

fired is ;hosen from the fiing doctrine. As RVs are killed they

are subtracted from NW (IMAIN).

b. Efficient Terminal Dfnse

At each target indexed by IMAIN = 1, NVT, there are NWV

(MAIN) RVs arriving. There are IfD (IMAIN) terminal

interceptors. Set NWT = NW (•,'IAIN) and 1TDT = ITD

(IMAIN).If ITDT .• NWT, there are equal to or fewer interceptors than

RVs. A one-on-one engagement is simulated against ITDT of

the RVs. The number of successful RVs is noted.If •TD'I'> NWT there are more interceptors than RVs. As in

the efficient boost-phase defense and efficient midcourse

defense previously discussed, there are IRPI defenders shot at

25

* a fraction P 1 of the RVs and IR defenders shot at a fraction (1-

P1) of the RVs.

Each RV killed causes NW (YMAIN) to be decremented by

one. The final number of successful RVs is the remainder in

NW (IMA1N) after the terminal defense.

c. Limited Shoot-Look-Shoot Terminal Defense

In this case the defender need not fire salvos at RVs. He can

shoot an interceptor, observe if the RV is killed, shoot again if

necessary, and so on, up to a certain limit per incoming RV.

Thus he can substantially conserve defenders, and, if the

upper limit on number of shots is reasonably high, have a high

probability of kill against each RV.

The program is set to have the upper limit of the truncation

limit of the minimum of Value/5 and 2. In the example, the

value of the attacked target is either 20 or 10, so there are

effectively 4 or 2 shots as an upper limit.

The number of terminal defenders is ITDT. The number of

warheads heading for the target is NWT. The maximum

number of shots is MAXDEF. For RVs 1 through NWT,

interceptors are tired one by one up to a maximum per target of

MAXDEF. If an interceptor is successful, as determined by

the random number drawn being equal to or less than PK, then

NWOMAIN) is decremented by one. As each interceptor is

f'red ITDT is decremented by one. The process terminates

when no more RVs are coming or when ITDT is 0. The final

number of successful RVs is the remainder in NW(IMAIN).

d. Unlimited Shoot -Look-Shoot Terminal DefenseThe logic for this defense is exactly the same as c. above,

except there is no upper limit on the number of shots taken at

any particular RV. There is no check of MAXDEF as in the

above discussion.

26

e. l -ati.e Frequency Array of Successful Penetrators

At the end of Subroutine TD a relative frequency array of

successful penetrators is created. The array is denoted

NWRF.

In the example there may be any number from 0 through 20

successful penetrators, so the relative frequency array NWRF

has 21 elements,

8. Subroutine VYLSURV (NVT. IVSS)

Subroutine VALSURV (NVT, IVSS) receives from MAIN the

number of value target NVT and returns the value sum surviving

IVSS. It simply sums the values IV(I), I = 1, NVT associated

with the targets receiving zero successful ptnetrators (those with

NW (1) --0, I= 1, NVT).

27

EXAMPLE

This example is presented in order to give the ,•ader information on the level of

detail at which the model is used and of the sensitivities of the outcome to variations inBMIC 3 options and defensive resources. Most of the features of the program are exercised

and illustrated.

1. Set up Targets of Differing Value

SThe target data base is composed of the array IV of targets of differing

value. As an example, consider the following list of targets. There are eleven

conceptual types of targets, with value per target as shown and total value as

shown.

ValueIyLoTg= Number of Targets PS I Total Value

SLBM Base 3 200 600C3 Installation 10 100 1000

ABM Base 10ý 50 500

Bomber Base 50 20 1000

Air Defense Base 20 20 400

CONUS M~litary Inst. 200 20 4000

Theater Military Inst. 200 20 4000C3 Installation 40 10 400

CONUS Econ. Inst. 500 10 5000

Theater Econ. Inst. 500 10 5000

ICBM m 5

2,533 26,900

The target data base illustrated consists of 2533 targets with a total value of

26,900.

The subroutine VALUES creates the target data base.

2. Set Up Terminal Intecpt

For each target, a certain number of terminal interceptors is deployed in

array rTD. As an example, for each target in the data base given above,

28

interceptors per target are deployed equal to half of the value per target (rounded

down to the next lower integer). The result is as follows:

Number Value Interceptors Total1= of Tag o Largcts Em J~et E Tacr~get h==rp=~

SLBM Base 3 200 100 300

C3 Installation 10 100 50 500

ABM Bases 10 50 25 250

Bomber Bases 50 20 10 500

Air Defense Bases 20 20 10 200

CONUS Military Inst. 200 20 10 200

Theater Military Inst. 200 20 10 2000

C3 Installations 40 10 5 200

CONUS Econ. Inst. 500 10 5 2500

Theater Econ. Inst. 500 10 5 2500

ICBMs 5 2

2,533 12,950

There are 12,950 terminal interceptors assigned to the 2533 targets. The

subroutine TDINV creates the terminal interceptors as signments in array lTD.

3. Set UtL Attack Allocation

The example attack allocation consists of attacking the first 500 targets

defended by fewer than 25 interceptors, namely targets 24 through 523. Each

target is attacked by the RVs from three missiles. Target 24, for instance, is

attacked by missile number 1 with 5 RVs, by missile 501 with 5 RVs, and by

missile number 1001 with 10 RVs. The attack is as follows:

i 24 5

5"00 5"23 5501 24 5

29

it

1000 523 51001 24 10

1500 .523 10

The above example includes cross-targeting in the assignment of more

than one missile to each target. The computer program, however, is more

flexible in the sense that the logic permits each RV to be cross-targeted. For

instance, missile 1 could logically fire 1 RV each at targets 24, 25, 26, 27 and

28. (The program dimensions are currently limited to the targeting at most two

targets by each missile, but the logic is completely flexible).

From the point of view of the targets, the above allocation results in the

following assignment of RVs:

Terminal AttackingTarget Value Interceptors RVs

24 20 10 20

493 20 10 20494 10 5 20

523 10 5 20

The salient point is that each target receives 20 RVs, ftom three

missiles. Since the boost-phase defense kills missiles before they release RVs

each target can be attacked by 0, 5, 10, 15 or 20 RVs, depending on which

missiles (if any) are destroyed.

Another interpretation worth keeping in mind is that the example attack

allocation might be conceptualized as assigning 1000 ICBNvs with 5 RVs each,

plus 500 SLBMs with 10 RVs each, to 500 relatively high value targets.

30

4. Resultsfor 12.950 Terminal Inter-c ors

Figure 1 presents, on the left side, results for 1000 boost-phase

interceptors and, on the right side, results for 2000 boost-phase interceptors.

Units of value destroyed are presented as a function of number of midcourseinterceptors for a wide variety of BM/C 3 options.

On tihe left side, graph A,A,A is for random allocation of boost-phase,

random allocation of midcourse, and preallocated fixed-salvo defense at each

target. It yields the most damage for the attack of 10,000 RVs. Graph

AAD, with unlimited shoot-look-shoot, does quite a bit better, and damage is

relatively small with 5000 midcourse interceptors.

Graphs B,B,A and B,B,D display much more defense effectiveness than

A,A,A and A,AD. In particular, B,B,D with 3000 interceptors yields

relatively small damage. Note that B,BD as compared with B,B,A, and

A,AD as compared with A,A,A are significantly better. The inventory of

12,950 terminal defenders with unlimited shoot-look-shoot is much more

effective than when these same terminal defenders are limited to a preallocated

fixed-salvo f'ing doctrine.

Finally, on the left side, consider B,CD. It is the best defense with

1000 interceptors but does not do much better than B,B,D with 3000

interceptors. (There will be more discussion of preferential midcourse defense

in the following section).

Turning to the right side of Egure 1, note that 2000 efficient boost-

phase interceptors (B,BA and BB,D) achieve near- zero damage with as few

as 1000 efficient midcourse interceptors. Confronting 1500 missiles with

2000 bnvt phase interceptor shots efficiently allocated, and confronting the

remaining RVs with 1000 midcourse interceptors efficiently allocated, results

in expected damage of zero for B,BD and a very small amount for B,B,A.

Staying with the right side of Figure 1, consider the random defenses

A,A,A and A,AD. The former requires 5000 midcoarse interceptors to

achieve near-zero damage, while the lauter achieves this v~itb 3000 midcourse

interceptors.

31

rb Y

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z o 0L Cz m 4 c 'k5 42C

LAJ~ ~~ La O0a0 jC- 3c

U -. Qj 4d j 0-C. 04 1" orC

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M w 2 M U U.~ V

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00 cc L3 C. cc uF KgoQc

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32-

Comparing the left with the right sides of Figure 1, note that 1000 boost-phaseinterceptors and 3000 midcourse interceptors with B,BD does almost as well

as 2000 boost-phase interceptors and 5000 midcourse interceptors with A,A,A.

The latter needs twice the number of boost-phase interceptors and almost twice

the number of midcourse interceptors to achieve the same effectiveness. Thisillustrates the importance of BM/C 3 capabilities for weapon allocation.

5. &,, its for 12.950 and 5.950 Terminal InterceptorsThe next case is to let the number of terminal interceptors per target be equal

to one-fourth of the value per target (rounded down to the next lower integer).

The result is as follows:

Number Value Interceptors Totalof., ..Tair e,.t Pe [rg e T g FgL .Ta=g lntercepto.rs

SLBM Base 3 200 50 150

C3 Installation 10 100 25 250

ABM Base 10 50 12 120

Bomber Base 50 20 5 250

Air Defense Base 20 20 5 100

CONUS Military Inst. 200 20 5 1000

Theater Military Inst. 200 20 5 1000

C3 Installation 40 10 2 80

CONUS Econ. Inst. 500 10 2 1000

Theater Econ. Inst. 500 10 2 1000

ICBMs IWO 5 1 10QQQ

2,533 5,950

There are thus 5,950 as opposed to 12,950 terminal interceptors.

Figure 2 presents the results. First, on the left side, consider the solid linesfor A,A,A and A,A,D. The two cases are quite close together, whereas in Figure 1

33

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they had been far apart (as they are in the dashed lines of Figure 2). When there

are fewer terminal interceptors, whether they are preallocated or shoot-look-shoot

does not yield so much difference in payoff.

Second, on the left side, consider the solid lines for B,B,A and B,B,D.

There is a dramatic impact between 3000 and 4000 midcourse interceptors. The

payoff from taking on all of the RVs surviving the boost-phase defense is immense.When there are 1000 boost-phase interceptors with Pk of 0.9, there are roughly 600

sttrviving missiles of which 400 have 5 RVs and 200 have 10 RVs, or 4000 RVs.

If these can all be engaged the number of survivors is roughly 400, distributed

(quite non-uniformly) across the 500 attacked tar;;ets. Both types of terminal

defenses are effective, with the unrmited shoot-lock-shoot doing better. The main

point, however, is that hlving tnough efficient midcourse defenders to engage all

of the RVs makes a tremendous difference.

Third, on the left side, consider the solid line for B,C,D. Up to 3000

midcourse defenders preferential midcourse defense does quite a bit better than

efficient midcourse defense (B,BX)). Beyond this the algorithm does not beat the

uniform defense. If the defender were. to observe the attack size, he could simply

switch to uniform defense if he knew the attack size exceeded 4000.

In general, the preferential defense B,CD is a difficult optimization

problem. The algorithm described previously is indcpendent of attack size. A

better algorithm would solve the large-scale integer assignment of shooters to

targets knowing both numbers of shooters and targets, and knowing the tenrinai

defense capabilities. However, the present example makes the point that

preferential midcourse defense is much bett-r than efficient wridcourse defense

when the number of midcourse defenders is limited.

Fourth, consider the right side of Figure 2. As before, if there are 2000

efficient boost.-phase interceptors and 1000 or more efficient midcourse interceptors

(B,BA or B,BD) there is very little damage for the solid or dashed lines. Halving

the numbers of terminal interceptors is not a problem. However, if both the boost-

phase and midcourse interceptors are random (A,A,A and A,A,D) the solid lines are

significantly worse than the dashed ones.

35

~~~~~~~~~~~~' Ye- f ~ ~ .[ ~ I ~ . ~ ~ U2~ Z '~

In particular, A,A,D is much worse, for unlimited shoot-look-shoot with

fewer terminal interceptors is negated by cxhaustion at quite a few targets. Where

3000 midcourse interceptors result in near-zero damage with the dashed line,

A,AD, 5000 midcourse interceptors are required with the dashed line A,A,D.

6. cfijjn, Blost-Phase. Random Midcourse

The right side of Figure 3 presents results for efficient boost-phase defense

followed by random midcourse defense. The dotted lines are the new results.

There are few enough missiles surviving the efficient boost-phase defense

that random midcourse defense does not result in significantly more damage than

efficient midcourse defense.

7. Random Boost-Phase. EMice Midcourse.

The right side of Figurer 4 presents results for random boost-phase followed

by efficient midcoutse defense. The dotted lines are the new results.

Note that 2000 rawidom boost-phase defenders let through so many RVs that

iO000 efficient midcourse interceptors do not do much good.

However, as the numbers of efficient midcourse interceptors increases from

2000 to 3000 the attick is sufficiently diluted so that just a few targets are

destroyed. This mid,;ourie defense situation is exactly analogous to the left side of

the figure increasing from 3000 to 4000 efficient midcourse interceptors.

8. Random

The right side of Figure 5 presents results for random boost-phase defense

followed by a combination of efficient midcourse defense and random rmdcourse

aeftnse. The dotted lines are the results.

Hybrid defense is defined as one-half efficient defense followed by one-halfrandom defense. (It is denoted by B'!= 1i2B, 1/2A). Thus, the same number of

RVs after boost-phase defense as in Figure 4 is confronted by a partially efficient

and partially random midcourse defense.

3 6

a yca

o -- C* CAI /U wC

- - -C. b. .

o w C,-ac / / Cc' .3 LlC- C

c 0 A - LUQ .4Lu a ,.~C.3

- I-mi CSL rC uL A"C =CU.,E 0" , A cc ~ J I IJ LJ01 cLL. = -W a aL z e ILL;

IN C.- I-

CIP-

V3~~~; aca* cc g

Om, C/

/, IS C"I~ S'4J = CS ' I '

us Wu c , r w On .r m4C

!2 s z/ -. ./AL-., U

A 4 "=O atV/

W -L -f/ 71 ,/ V -

/ ,U L,- -c; cc U.

cm C. IS.(.'0

C.)m

_______~ ~~~~~~~ =___ ... L . ........a~~u a 6a a aI a a 0a

U~AO~1S~U ~11V :10 CIOf

37I

00 UU

S( u

z U3 W. C/= mLU-,c I-Ca ILE/ 5 ' z 0

I, L_ Uw n LI9

C3 U3OI1I lVA) SIf-C a3L aww mC)c

C=C

0-/ / 0 0

/j -ca w 1-i W 1- 0-4

X I 4= S C5 1@ =C

U I. -U r- cc -n -W

a3OI~ 3ni0 40 S(0

@4 ~c v0 ~ nC'LAJi3 Z-1V a: W,~f

p 38

CJD

i: uJI I-.

w C02CD Z c . - ?t..A c L0

z cc CD1 C

I.- = 0 - 9 2 .- C, 9 Co Cc

- A 'C 0 CL

In a Lu a L"c

cIm L,

cm uCgo I.-

/M/d

co0

0~ 0

O~AOCC ~ -cc1V dO cc;N

39, c

G3-tIc

The general behavior is for the units of value destroyed to decrease more

gradually as a function of number of midcourse defenders. The namber ofmidcourse defenders to yield low damage increases from 3000 with efficient

defense (A,B,A and A,BXD) to 4000 with hybrid defense (A,B',A and A,B',D).

9. Hybrid Boost-Phase. Hybrid Midm

The right side of Figure 6 presents results for hybrid boost-phase defense

followed by hybrid midcourse defense.

In this case, a hybrid boost-phase defense is composed of 1000 efficientboost-phase defenders and 1000 random boost-phase defenders. The 1000

efficient defenders engage a subset of the 1500 missiles and the .1000 randomdefenders engage the remaining missiles (roughly 600 missiles). Thus the numberof RVs proceeding to be confronted by the midcourse defense is more than in theefficient boost-phase case but less than in the random boost-phase case.

As expected, the effects of adding hybrid midcourse int-erceptors aresomewhat gradual, with value destroyed being fairly low with 3000 defenders.

The gradual effects of the hybrid midcourse defense can be conirasted with thesharp effects of the efficient midcourse defense (see, for example, B',B',A andB',B',D versus A,B,A and AB,D).

10. Hybrid Boost-Phase. Random Midcourse

The right side of Figure 7 presents the last excursion of this type.

Hybrid boost-phase defense is followed by random midcourse defense.The effects of adding random midcourse defenders are more gradual than addinghybrid midcourse defenders.

11. Suppression of Boos t-hase Defense

The right side of Figure 8 presents results for two types of suppression of

boost-phase defenses.

In the first type of suppression boost-phase defenders are reduced by one-third, or from 2000 to 1333. Not all of the missiles are engaged. When the

40

0m10

'U,

I- I 3

x z 0 CD V3-04 C#CD: Wu 0 0 0zCLC.

0.CD cc

uj~ :jcnz cc wa

Lu -- Eý

M C tMIC` CC 0A 06 IJCLZ -

C0 =

co C2L~ -a 06

cz~~ Sa. po o n9

Z - I- J

IC CW C2. CD 0ý 0 ,~xoaC) CD. C2 0 0D0

OW~U 0 cz P - to CS..?, C'J -4

= - auA0U1SM 31fl1VA .40 SIM~f

zo.cl 0 U, I- C=

c'l'

uCa (5

co 2c'.FE~ w -L" 2-

41i ci 0

CD C.3 4C-2 =~

LLI-

oZ o

-cc'Cl~~~m 11 cm C I

cc C o c C

C" to l) C.

moi nlAJ l/

/4

I- aI%a~ i.- 0I

z C2 = r, -3c za C,~ Z aCa

aU P1 raI.3C L4 -CCDC

41I C-

aj M* /, 02 a

Ui~c,-c*U -

Z acI m-

C.3a ~ a ~ a C

jLUn a~ conI I5_ _ _

4c -K dc 9= OO-

00 ~ ~ u -idý l

I I cc160 W/Ito

Ia C,/

3 LO 0

U I"

U,.

ii =

/ -4l,

/ /,. ~ --- -

03OI~ anV *0 SIa I 0a

~ *~ ': a 42

LU

C) 'LL IA.

ci W cc~ CD hE

-z 0 CD~ 40: DC

ci QC U.i -C .a L

us~~~~ ~ ~ ~ *** .* < * Qa J- :,C : -C2cia CSi W CD ui..

9314 LA, ... 1.- *2 ujdCD

r~ g; -**C* -r. LL a Ew

0 :c =:0 JE 0: 0 .

14 ~ ~ i0: 0m: zL

aC ca LLa ,

go t2 cm

-J -l 0

ce a

QE'i Aicica C1 J

I =' p mCDC

ciz

vic

/ U

W. =

ui Cci

c ia o. I.-c

434

.0,~ '01 '.40 .oo

number of efficient midcours;e defenders is sufficient to engage all of the surviving

RVs, units, of value destroyed is reduced to a small amount. Again, the behavior of

results is the same as in previous examples with sufficient numbers of midcourse

defenders.

In the second type of' suppression one-third of the missiles, at random, are

not engaged., The number of efficient midcourse defenders required io result in

small damage is larger than in the first case.

12. referential, Targeting by Boost-Phase Defenders

The left side of Figure 9 presents results for boost-phase defensive firing

proportional to number of RVs on the missiles being attacked.

Two cases, C,B,A and C,BD, are presented. They can be compared with

B,B,A and B,BD, and are seen to have less damage.

The reason why the return to the defender is not greeter for propo:rtional

targeting of boost-phase defenders is that the attacking missiles have either 5 or 10RVs each. If tthe muix of RVs were from 1 to 20 per missile, or something quite

skewed, the improvement for the defender would be much greater.

13. Ef~fectssff c~ j

The left side of Figure 10 shows the effects of three decoy situations.

To begin, replace two of each five RVs by four heavy decoys, resulting in a

total attack consisting of 6000 RVs and 8000 decoys. In all cases, assume that the

BM/C3 system of the defense is B,B,A.

In the first example, K = 2.5, which can be interpreted as meaning that the

inherent error of the measuring device is such that RVs and decoys yield

observations w.4hich are two and one-half standard deviations apart. A detection

threshold is set resulting in PMD =. 1 and PFA =. 1 (probability of missed

detection against an RV equal. 1 and probability of false alarm against a decoy

equal. 1). In this case, the resulting units of value desvroyerl as a function of

number of midcourse interceptors is lower than the baseline curve B,B,A. So this

is not a good attacker option.

44

C2

U. a A UAU)-K = LAJ A,0u

2c z Cz?a 5 =c VXa~ C2a 0~u 3 D u ,Z4c.

S p S J P: c -2 Co. ,

b. C I.d Kc L -C G2 / ;

C'C am (?) =j0

I-- Z L-L2 )t C

/ CaL 6L CL ==U

e /r4., A ao

00C30

~ a ~ a w ~ a a aa aP

a , CI--Cc

a oe

rc, $*.

U, co C2 . C2 CDc C3

C) z

caa

no C*J -= wU3O1SU3l1l OSIu

C45

C= -=l

:I. C2 . C

3F,4 10 w D 0 b.2

;::~~ I.Ii 3P-C

0 Uo ca /* /3wWGo -i W -S :J Q9 go S N O O C 2 "

W 0 f;c

us 49 =0Li2!9'c 9 I

U.a C, U aau 3uw ac U. mGn it z~ Am'

on =04 10 a a

wCl ql -

ru a

P. - ao

ft~,/ .* C, '1010 2 , S

I.- W2- U,ac 0I-

-J j 233=

-CC -

c.9 a // IL cm C4 0m

z zp.m W

AO.L3 CD1V wI 1vi0 0 =C

!2 LLgc LL0

SJ.I I

OW ca46

W,"7,, • r = 'W'!W , = r:, ¸ -• ' . . .. : -- • -, -=, * F• , ¶ =: Y *V• , ,uVmwl- -¶' =¶W W W W~ WDV -- ' ' r - ' " * ' ' fl •~

In the second example, K = 1.0, and the detection threshold yields PMD =.3 and PFA = .3. This is much worse for the defense than K = 2.5. But, still, it

is not a good attacker option.

In the third example, however, assume that 30 light decoys can be addedwith PFA =. 1 (smaller than the PFA = .3 for heavy decoys). The damage is

smaller than B,B,A until the number of midcourse interceptors is greater than3000, but then it is not reduced to near-zero. Even with 5000 interceptors, there is

significant damage.

The third example is a striidng demonstration of the difficulty ir to the

defense of dealing with decoys. Near-zero damage becomes very difficult to

attain.

14. Concentrated Attack

This example is very different from those previously discussed, for in all ofthose cases the attack was held constant. Now, the effects of concentrating on a

smaller set of targets are investigated.

Allocate all of the missiles to the 50 targets 24 through 73. The maximumpossible amount of value destroyed is 1000 units as compared to 9700 units

previously.

Both the left and right sides of Figure 11 show results of the new attack.

On the left side the BM/C 3 assumption is B,B,D while on the right side it is

A,B,D.

On the left side, for the original attack, 4000 interceptors yields near-zerodamage while for the near attack even 6000 interceptors result in significantdamage. This is because there are 200 RVs initially aimed at each target, and theprocess is sufficiently "lumpy" that at a few targets there are enough RVsremaining to exhaust the terminal defense.

On the right side, for the original attack 3000 efficient interceptors wereenough to significantly blunt the attack. Here 4000 are required.

47

ta

CD~~ I C) C,

PC i/~ -C3 C2aJ #to Ct Cu .Q,0wL

IN .5L . (A D LaiC

C.: LU Lana t CA w

A.J.

C , I: go

gi cc;

ruz ._ _ _ Ai

O30L3 in1AJOSIf

ca fs e48

IE is desirable to compare the results of "expected value" calculations with

Monte Carlo calculations in oroer to illustrate the large differences which can

occur under certain circumstances. This is done below for two cases.

In both cases the numbers of defenders are as follows:

Boost-Phase 1500

Midcourse 2000

Terminal 5950

The expected value calculation procedure follows. Assume that 1500efficiently allocated ooost-phase defenders confrm)nt 1500 missiles, and the Pk is

.8. The expected number of missiles surviving is 300, comprised of 200 missileswith 5 RVs and 100 missiles with 10 RVs, for a total of 2000 RVs. Assume that

2000 efficiently allocated midcourse defenders confvornt the 2000 RVs. The

expected number of survivors is 400.

Now assume that 400 RVs are uniformly distributed over the 500 targets.

Since there are 470 targets of value 20 and 30 targets of value 10 assume that 376

of the first type and 24 of the second type are. attacked by one R V each. With

preallocated fixed-salvo defease of 2 interceptors, the probability of penetration is

.04, so 376 x 20 x .04 = 300.8 units of value are killed at the higher-value

targets. Similarly, 24 ;z 10 x .2 = 48 units of value are killed at the lower-value

targets. The total is 348.8. With efficiently allocated terminal defense the

equivalent figures are 376 x 20 x .00032 = 2.406 and 24 x 10 x .04 = 9.6. The

total is 12.006.

Rumiing the Monte Carlo model for 30 trials yields, for the two cases

above, the foliowing results:

M~aaS~apk £andrd eviaionoLSaix't

Prealocued Terminal Defense 1 184 286

Efficient Terminal Defense 844 344

The inference about the process from the sample is as follows:

49

Standard Deviation ofEstimate cf Population

Preailocated Terminal Defense 1184 53

Efficient Termimal Defense 844 64

Summarizing the expected Monte Carlo v"ade calculations the following is

obtained:

Standard Deviation ofEstimate of Estunate of Estimate ofMean with Mean with Mean withExpected Value Monte Carlo Monte CarloC&lý Cd~ altmalatiam

Preallocated Terminal Defenbe 349 1134 28

Efficient Terminal Defense 12 844 54

Performing exactly the same analysis with jrobability of kill equal .9 yields

the foldowring (-10 Mon'e Carlo trials):

Standard Deviation ofEstimate of Estimate of Estimate ofMe,&a with Mean with Mean withExpected Value Monte Carlo Monte Carlo

Preallocated Terminal Defense 25 246 28

Efficient Terminal Etfense 1 126 20

Figure 12 and 13 Fhow the calculations leading to the above results.

N

50

00o b L~ 1

+

E~C =I- ~co

0j x

= 0j

.W .4 A a C

LU elz

C0 0o WLL

06 U. IDA

U-JLaiLU UA C= X w

+n GO C* L)

LU LLA i 0. U

LUWZcLMU

'00 -Jc z 0

U.~U LL Co I- Mt 1 -

LU w Lu L ZL U LL LL M UCý

LU UJ = C-3 LULLý LL C2 I

L&A L. C ýr- T-5C

z Icco

w'

Not ~

uj0uLL m

'~ + + cc

CD + L5zlc-

Lo z z

CD IF I+ Q. ~

LUý X C2

~~IC I - C x x -

""r 2 :& w wr-

LU J-E C) 01"LUW 2" I rUJLLIUZ '~~a~ 0

Cf. = zr =L LU cm oft m~ .

uj wa I-CUcn9

00 In ~ L

J1 LU COI-I u*j LIJZ CD r

LU zo -<,.LO~~ Ch c~jco o

LO~C -J =I-- t7 C-

a: E

wL LL U.CO tLU +I CD CD oc C2 CLu~~~ WlLI LzL ! v) x uj Lx CM co

LA C3I- 0 =- -cJvr -L& u

COO LI 3 =-jx10, CS cm I= n L&I C-

E uw cCd-)W

I--LUI

007

52

APPENDIX A

DEFINITIONS OF VARIABLES

DEFINITIONS OF VARIABLES

Danic

AVES Average value sum surviving, completed cumulatively as of eachtrial. Computed and used in MAIN.

AVED Average value sum destroyed, computed cumulatively as of eachtrial. Computed and used in MAIN.

F!AWT Sum of total number of RVs in array IAW(I,J). Computed andused in Subroutine BPD.

FKLD Fraction of boost-phase defenders killed in boost-phase defensesuppression. Set by user at beginning of Subroutine BPD, ifdesired.

FTARS Fraction of missile targets which cannot be attacked by boost-phase defenders. Set by user at beginning of Subroutine B11D, ifdesired.

IAD(I,J,K) Number of decoys of type K in the Jth group of RVs carried bymissile I. In COMMON.

IAT(I,J) Target aimed at by all of the RVs in the Jth group of RVs carriedby missile I. In COMMON.

IAW(I,J) Number of RVs in the Jth group of RVs carried by missile I. InCOMMON.

IAWMT Sum of total number of warheads on a particular missile.Computed and used in Subroutine BPD.

LAWS Sum of total number of RVs in array IAW(IJ). Computed andused in MLAIN.

IAWT Sum of total number of RVs in array IAW(IJ). Computed andused in Subroutine BPD.

IAWT(IJ) Temporary array of number of RVs in the Jth group of RVscarried by missile I. Computed and used in Subroutines MCD,MCD3, MCDWD.

IBPDS Indicator for boost-phase defense suppression. If IBPDS=O, noboost-phase suppression. If IBPDS=1,2,3 or 4,boost-phasesuppression of different types. Set by user at beginning ofSubroutine BPD.

A-1

IBPDST Working variable indicator for boost-phase defense suppression.May be set at 0 or 1. Set and used within Subroutine BPD.

IBPM Indicator for hybrid boost..phase defense. If IBPM=0, nohybrid. If IBPM=1, hybrid. Used in MAIN.

ICMPLT Indicator denoting whether boostiers receive complete coverage(i.e., whether all boosters necessarily receive at least one shot ifpossible). if ICMPLT--O, no complete coverage. If ICMPLT= 1,complete coverage. Computed and used in Subroutine BPD.

IFRAC Indicator for fraction of defense which is efficient or random.IFRAC= t denotes efficient portion and IFRAC=2 denotesrandom portion. Used in MAIN.

SIMCM Indicator for hybrid midcourse defense. If IMCM=0, no hybrid.If IMCM= 1, hybrid. Used in MAIN.

INDI(IJ) Indicator for the Ith target and the Jth missile aimed at that target,giving the number of the missile. Computed and used inSubroutine MCD3.

INDJ(I,J) Indicator for the Ith target and the Jth missile aimed at that target,giving the value ef J. Computed and used in Subroutine MCD3.

INDTA(I) Indicator for whether or not Ith missile has been assigned to aboost-phase defense shooter. If INDTA(I)=O, it has not. IfINDTA(I)=1, it has. Computed and used in Subroutine BPD.

IRS Seed of the random number generator, changed each time a

random number is generated. In COMMON.

ITD(I) Number of terminal defenders at target I. In COMMON.

ITDS Sum of total number of terminal defend=-a. Computed and usedin MAIN.

IV(I) Value of target 1. In COMMON.

IVS Sum of total number of units of value. Computed and used inMAIN.

IVSD Value sum destroyed. Computed and used in MAIN.

IVSS Value sum surviving. Computed in Subroutine VALSURV.Used in MAIN.

A-2

U

JPR(I) Number of shots taken by preallocated preferential defense withtotal number of interceptors I. Specified and used in SubroutineTD.

KPR(I,J) Number of interceptors fired at the Jth incoming RV when thetotal number of interceptors is I. in a preallocated preferentialdefense. Specified and used in Subroutine TD.

LPI Long printout indicator. If LPI=O, no long printout. If LPI= I.,long printout. In COMMON.

NKLD Number of boost-phase defenders killed in boost-phase defensesuppression. Set by user at beginning of Subroutine BPD, ifdesired.

NOBJ1 Number of objects of type 1, namely RVs. Computed and usedin Subroutine MCDWD.

NOBJ2 Number of objects of type 2, namely heavy decoys. Computedand used in Subroutine MCDWD.

NOBJ3 Number of objects of type 3, namely light decoys. Computedand used in Subroutine MCDWD.

NOBJT Number of objects total. Computed and used in SubroutineMCDWD.

NTARS Number of missile targets which cannot be attacked by boost-phase defenders. Set by user at beginning of Subroutine BPD, ifdesired.

NVT Number of value targets. Created in VALUES. Passed in callingsequences of subroutines.

NW(I) Number of RVs headed for target I. In COMMON,

NWRF(1) Frequency distribution of number of targets receiving Ipenetrators at the end of the simulation. In COMMION.

NWRMP Maximum number of RVs which can hit a target, plus 1. InCOMMON.

NWT(I) Temporary array of number of RVs headed for target I.Computed and used in Subroutine MCD3.

PFA1 Probability of false alarm against heavy decoys, where midcoursedefender mistakenly calls a decoy an RV. Specified inSubroutine MCD and used in Subroutine MCDWD.

A-3

PFA2 Probability of false almm against light decoys, where midcoursedefender mistakenly calls a decoy an RV. Specified inSubroutine MCD and used in Subroutine MCDWD.

PMD Probability oil missed detection, where midcourse defender doesnot detect an RV. Specified in Subroutine MCD and used inSubroutine MCDWD.

RAT Ratio of attacking RVs to terminal interceptors. Computed andused in Subrcoutine MCD3.

RATSL Lower bound on ratio of attacking RV- to terminal interceptorsfor particular iteration of preferential defense algorithm.Computed and used in Subroutine MCD3.

RATSU Upper bound on ratio of attacking RVs to terminal interceptorsfor particular iteration of preferential defense. algorithm.Computed and used in Subroutine MCD3.

SD Standard deviation of value sum surviving, computedcumulativaly as of each trial. Computed and used in MAIN.

VSS(I) Working variablk to preserve value surviving at the end of the Ithtrial of the simulation. Used in MAIN.

A-4

APPENDIX B

LISTING OF COMPUTER PROGRAM

a

WN

-y -Cp-

L'Si

Ix, -

~CoW LL

041.4

U - I0 , tr % o- CL

x C. 0 A

-0 *P. .. -,

'r. -' -- i i.0 - -jtO-x~4a U;. Z wX..~ Z a. ** Z

= cý .- C- -C0 .ý C5 =1a.0 01- 0ýaNa.a a . Y. .~. 1 'I. a x. x. 0 0 -

pa. .Li- W. a. L. W. LA. ý 66 U.a 0- LL 0ý U- LaL La -. -l LL I -k d ý 0 I..)0

c. X~'- X CC Cl 0J N~ 4 N .r.n I. 6r -CID, 0hC-ý0 a 03 0 M f- C, 00 C

Lr C. rý M, f-0 r?' 0 m

f . . . . . . . . . . . C. 5 C cCC

B-1

tI

00

zz

N CC

4),,

Co >

6w -

o --

> =

er' L) Cc , >

L, -J

oj I r - W72-

C5 a

o ~ ýa - .

W 4 ý P -Q -C -6If 4V1 , 11 JUa, D ý " -1 ' Wý0 ''e,1 0Q - 6F, ý t - ; ý,M C VX V'X W X ý 0 9 0 -- 0 0 , C o

C,0 , :, Cýa ) C <I c a Q0 ) 0C50 ' ýýDC,0 0(ý0 ý0 3 C ( 0 - -- 4- -ý 10 1 ý ý C =0Q )K= ý ><ýCP t aC C C -- 0C) C?0Cý1 C , -- C QC)4 ýcCý( 0C,

z 19

*c 07

L'I

0-

Z, Z

z C

0 zi

IIU- w D- w <

-' or, u 3 -'

LL iJ V 3

z zC -.

0, 14) 0 C

uJw mid -ý -4 - - Zz z 1 z z

Ya- a- 5-x' _ w ý t- -2 .j 19 In - R73

C, r L ýA) 0'* '0 'o .00 0 4 z x S, C>

c Z) Cl 0 C5 Jj*,

0.

1.

j V,

C.C,

2.p,

4%B-4

cc0,

zz-

o y-

CNo 1 - 0w C

A 9CL4 a c ','- . % t. o f - Z 1

7a -0 n,"

IdIa - N zC 0 4) 0,0Cv

C5 C, 0 Q4 n- 0 . a r>

U.

IL

II

(3-

74..

040

C -K

~w

4.V

LiW

M - -.

ci 0 90 X~-C z7z

-6.4 x' 0- 11 1. it

x 0 0 :it- -. -

L& = = z %- -JM LL = vi V) x- 0' 11 CW u.4 "I L

-~l -l -C- - 0X . ~Z :I - ~ -. - tZZ -z zý z z

- Z - -' -l F-I4 -0 -

6- 4.ý t- D Ic, CL C. C'-) 0 r14 N, 4 j w14 w 0 0 'a Z -4 , 04 rJ I ý6l

no -. % % w 4.0 tZL 0 0 ::: %. .4 % , . '4% tz 4. 4 .ý -k IL -j I 0 4/4/ 1 0 4

-~~~~D -a -A -W --- SI~J 1. C

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.MONTE CARLO LAYERED DEFENSE MODEL

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