A084 997 ARMY ARMAMENT RESEARCH AND DEVELOPMENT COMMAND ASERD--CYC F/S 11RESOURCE EXPENDITURES AND EXPECTED TIME TO OBTAIN BINARY OB.IECT-EVTC(U)MAR 80 L D JOHNSON

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MICROCOPY RESOL I ION TEST CHART

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MEMORANDUM REPORT ARBRL-MR-03006

(Supersedes IMR No. 653)

4 ORESOURCE EXPENDITURES AND EXPECTED

TIME TO OBTAIN BINARY OBJECTIVES

Lawrence D. Johnson

DTICELECTEMarch 1980 JUN 3 1980

'B

US ARMY ARMAMENT RESEARCH AND DEVELOPMENT COMMANDBALLISTIC RESEARCH LABORATORY

ABERDEEN PROVING GROUND, MARYLAND

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MEMORANDUM REPORT ARBRL-MR-03006 72t ) Al9$t914. TITLE (nd Subtitle) S . TYPE OF REPORT & PERIOD COVERED

RESOURCE EXPENDITURES AND EXPECTED TIMETO OBTAIN BINARY OBJECTIVES 6. PERFORMING ORG. REPORT NUMBER

7. AUTHOR(a) 6. CONTRACT OR GRANT NUMBER(a)

Lawrence D. Johnson

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(ATTN: DRDAR-BLB)Aberdeen Proving Ground, MD 21005 1L162618AH80

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IS. SUPPLEMENTARY NOTES

This report supersedes IMR (53 dated Jul 79.

19. KEY WORDS (Continuean revee aide It nacaeaOy and Identiy by block mnumber)

Binary ObjectivesTime to SuccessResource Expeditures

2 L AUSACr (Caen - ,aUm N na s w id? byoFWk k nmer)

Two equations are derived which predict the expected number of attempts and timeto achieve objectives which have only two states: success and failure.

These equations can be used to evaluate the relative merits of strategies asso-ciated with munition mixes and/or methods of sequentially delivering them, e.g.,adjusted fire, change of warhead type, etc.

Several examples are discussed to familiarize the reader with potential applica

tions.

DD IF0" 03 EB9Tnos or I NOV 6S IS OUOLIETIEM 72 ? IINCLARSTFT1RIn .

SECUMY CLASSIFICATION OF THIS PAGE (Mnm Date antere.,

.m...a...........

TABLE OF CONTENTS

Page

I. INTRODUCTION. .. .... ......... ......... ...

II. DERIVATIONS .. ..... ......... ........ ... 5

A. Expected Number of Attempts to Achieve Binary(Objectives. .. .... ......... ...........6

B. Expected Time -to Achieve Binary Objectives .. .. ...... 10

C. Median Values. .. ........ ......... .... 11

III. EXAM4PLES. .. .... ......... ......... .... 12

El. Constant Probabilities. .. ......... ....... 12

E2. Constant Probabilities within Subsequence .. .. ...... 13

E3. Intradependent Attempts .. .. ........ ........1

IV. SUMMARY .. ..... ........ ......... ..... 17

DISTRIBUTION LIST .. .................. .... 19

ACCESION f hteSctr

UNANNOUNCED 03

.................................... .... ...........

3

I. INTRODUCTION

This report describes the derivation and utilization of two equa-tions which predict the number of attempts and the time which can beexpected to achieve objectives which have but two states, success andfailure. This report refers to these types of objectives as binary.

The equations are merely generalizations of the solution to thesimplist of situations where the expected number of attempts, E is

N'2equal to the inverse of the probability of success of each attempt, P;i.e., EN = 1/P. As will be seen, this degenerate case is premised ontotal independence of attempts and a highly restrictive condition thatthe probability of success of each attempt is constant and equal to allothers.

The equations derived herein allow for the analysis of a much broaderclass of problems, including interdependence and varying probabilities

4 of success. However, the condition that the objective of an attempt bebinary is still required. Since binary objectives are common to systems9 ~ analyses, this requirement is not overly restrictive. For example theobjectives assumed are nearly all duel and battlefield simulations of tankencounters are binary since the tanks are either assumed killed* orunaffected by an individual threat attempt.

Section II describes the actual derivation of the equations where-as Section III describes, by example, potential applications of the

* equations.

Section IV is a short summary intended to alert the reader to theadvantages and restrictions of the equations discussed in the previoussections.

II. DERIVATIONS

This section describes the derivation of two equations predictingthe expected number of attempts and the expected time to achieve binaryobjectives.

The first derivation described is that for predicting the expectednumber of attempts to success. This will be in considerably more detailthan the time to success derivation, which can be viewed as a simpleexcursion from the first. As an aside, an algorithm is provided whichresults in the median number of attempts required and median time tosuccess. This is useful information in that it allows the analyst todetermine the 50% point of a given strategy.

Mobility, fire pow~er, and/or catastrophic kill.

A. Expected Number of Attempts to Achieve Binary Objectives

This is handled as a standard expected value problem which, bydefinition, merely sums all the statistically possible values weightedby the probability that the values will occur, i.e.:

(1) EN = iP(i)V.( 1

where EN expected value of attempts

i a the number of attempts to success

P(i) = the probability that the "ith" attempt will be thefirst to succeed.

Note the P(i) is just the probability that the ith attempt succeedsgiven that all (i - 1) attempts failed multiplied by the probabilitythat all (i - 1) attempts would fail. If we define

P(i) = the probability that at least one success would resultin "i" attempts,

then [1 - P(i)] is the probability that all i attempts failed. Since

P(i) = P(i - 1) + P(i)

it follows that

(2) P(i) = P(i) - P(i - I)

This will prove to be a very important relation since it uses thesimplest of probability calculations and its form will allow the con-traction of several series later in the derivation.

Associated with each attempt is a probability that the attempt willsucceed given that all previous attempts failed. Often these proba-bilities are related to immediately preceding or subsequent attempts.For example, if an artillery piece had a firing strategy consisting ofthree rounds to register and the rest for effect, tbc first threeattempts would be related in a different manner than the remainder.

To assure that these similarities can be used, the series ofEquation (1) are rewritten as a series of series, i.e.,

N1 NI+N 2(3) E N : ii)+ , i0(i) + . .

i=l i=N +1

li t ............. ... .. .. .. .. . .... .. .. .. . . . . ... . .. ..1

where N. is the number of attempts which have been identified ashaving some similarities in their probabilities of success with otherattempts in the "ith" subsequence. Since one can always set N. = 1, noloss of generality occurs. 1

It will prove convenient to further focus on the individual sub-sequences. Therefore, define

PK(i) = the probability that at least one success would result ifthe first "i" attempts were made in the "Kth" subsequencegiven that all attempts prior to the "K" subsequencefailed.

To provide a glimpse of where this is leading, consider two subsequencesconsisting of three and five attempts, respectively; i.e., the firstthree attempts are potentially related as are the last five attempts.Then

A(6) = [(P2 (3) - P2 (2)]* [ - P1 (3)]

Note that the first bracketed term uses the relationship of Equation (2)whereas the second bracketed term is merely the probability that allattempts in the first subsequence failed to succeed.

Returning to Equation (3) with our newly defined term and a slightrewrite to clarify the range of indices

N1 N2

(4) EN = iPl(i) + [i + N1]*[P2 (i) - P2 (i - 1)]*1 - Pl(N 1)]Si=l i=l

N3+ [i + N1 + N 2*[P(i) - P3 (i - 1)]*[1 - PI(N 1 )]*[1 - PI(N2 )]i=1

....

We define

KSK N. which is the total number of attempts prior to the "j+l"

i=l subsequence

1K Cl - P1 (N)]* ... *[ - PK(NK)] which is the probability of

failing through SK attempts. Note: 11 = 1.

Then Equation (4) can be written as

NK

(5) EN = ' X [i + SK_1]*[PK() - PK~' - l)]*TK-1VK i=l

Equations (1) through (5) require that all statistically possiblevalues of i be included. However, often physical constraints/inclina-tions limit the number of attempts possible. For example, a tankcontaining 40 rounds cannot make more than 40 attempts even though thereis a finite probability associated with success after 40 attempts.

One method of circumventing this problem is to assume that what-ever strategy was used during the admissible (physically possible)number of attempts would be used again and again, i.e., is cyclic. Inthe tank example, this is akin to assuming it will reload and engage inthe same manner.

Employing this method, Equation (5) becomes

t NK(6) EN = F [i SK. + iS] (P(i) - PK(i - 1)]*K-l*(ft)J

j=O K=l i=l

It" a the total number of subsequences considered in the admissiblesequence.

Note that St is the total number of admissible attempts.

We are now ready to simplify the equation into a form that justifiesthe expenditure of time to this point in the derivation. To accomplishthis, the following relationships are used:

L "i . . .. ..... "" ........ ..... ..... . . '

t -1i=0

(8) (fl)J nt(l - ldj =0

N K

(9) lp(' [ PK( K'- 1)] K PKK

N K(10) L i*P(ii 1)] N NPK(NK -

where X =K F

Inserting these relationships into Equation (6) results in

(11 t T=- [SK*PK (N K)~ - K + ]tP N )1EN F, K~ L 1 t - (1-_ .-

A final simplification is made by including the following observa-tions:

(12) P K(N K) " K- = 11K-1 ]'

t(13) F, (nK-1 -1 1 R

K= 1

t t(14) 'r SK (f1 - 11 K -S Stllt + L NfK K-

K=l K=l

These, when inserted into Equation (1), precipitate the desired equation.

(15) EN t. 1 - 1 ~ (NK ~K

N E (1 11 K 9

This equation contains a single form of hit probability, P (i), which isusually the simplest to construct, is finite and if the subsequences arechosen with discretion, allows for maximum utilization of intra-sequence simulation.

B. Expected Time to Achieve Binary Objectives

This problem is handled in the same way as the preceding deri-vation. Therefore, many of the intervening steps will be skipped.Using the same nomenclature, we begin by constructing the basic functionto be analyzed, i.e.,

NK

(16) ET =1 [i - 1) AK + aK + TKI][PK(i) - PK(i - K-1V i=lK

where AK the time between. attempts in the "K" subsequence

aK the time between the SK= attempt and the first attemptK of subsequence "K" 1

T K the time to complete SK attempts

It should be noted that there is an implicit assumption made whendefining AK, i.e., that the intra-subsequence time intervals areconstant. This could greatly influence how the sequence is sectioned,since this condition must hold if the following derivation is to bevalid. Again, since N. can be set equal to unity, generality is notlost, but caution is avised.

Circumventing the finite attempt constraint in a similar manner aspreviously discussed, Equation (16) becomes

t NK(17) ET = O I [= i-l)A * + T (j*T)l "

j=0 K=l i=lK K K-

[PK(i) - PK(i - 1)] *n *(flJd

Employing the relationships depicted by Equations (7), (8), (9), and(10), this can be reduced to

t (rKPK (N K) - A K*IK rt11tp K (N K)(18) ET= L nK-I I[ (1- f )

K=l 11 -1i

10

io'

9

with the aid of Equation (12) and the observation that

t tL TK(K 1

11 K = -ftTt + 2 11 K-ITKK=1 K=1

where TK = TK - TK- 1

Equation (18) takes the form

t IK_1(19) ET = 1 -t( - AKIK)

K=I t

which has the same basic benefits as does Equation (15), i.e., simpleand finite.

Equations (15) and (19)'represent the average number of attemptsrequired and the average time to successfully achieve binary objectives.This should not be confused with the median number of attempts andassociated time to achieve the same.

C. Median Values

Median values are those for which 50% of the time, success wouldhave been achieved in attempts/time less than or equal to the value.Although median values are of interest in systems analyses, they containless information than expected values and therefore are less useful.However, since the mathematical framework is already at hand, the follow-ing algorithm can be used to determine the median values at the sametime one is solving Equations (15) and (19).

* While calculating R K required in Equations (15) and (19) testfor

9

(1 K - O.5)*( K1 - 0.5) < 0

o Having found K such that the preceding inequality is satisfied,monitor the calculations used in IK to find i such that

lPK(i) - [1 - K ]I* PK(i - ) -E 1J} 1 0

* The value of i satisfying this inequality when added to Srepresents the median number of attempts to success, and Netime associated with making i + SK attempts represents themedian time to success.

1].

III. EXAMPLES

This section discusses several example applications of Equations(15) and (19). As will become evident the examples become progressivelymore complex.

El. Constant Probabilities. Every so often, the sun shines, flowersgrow, and the problem at hand is one where each attempt has a constantindependent equal probability of success. A tank with no fixed orvariable bias attempting to hit another tank, and a man trying to "flipheads" on a coin represent two such situations.

In any case, let "7" represent the probability of success associatedwith each attempt, then it can easily be shown that

P K(N K) 1 0 (1 N K

NK

1 (1 - P)KIPK NK P

S K

11 fK (1P)

Substitution into Equation (15) results in

t KN N [1- (1-P

N ~K K P

which by Equations (12) (13) can be reduced to

(20) En

which is not the most surprising result ever derived. Now assume thatthe time between events is constant, i.e., A. A, a. A. Then,

i i

TK (N l)A + A

12

and Equation (19) becomes

t 1K1 N K 1-P(21) EK-I (I P(1 ET = -1 - riH (NK - 1)A + A - [ K -

A

Implicit in this example is the assumption that the time betweenstart and the first attempt is A. One can eliminate that by assuming1 = O, a i# =A where upon, for large SK

(22) ET 1

44For a specific example where the probability of success associatedwith each attempt is 0.2 and the time between attempts is 10 s

EN - 0 = 5 attempts

E (-I - 1)*10 = 40 s.T 0.2 -

Using the algorithm of Section II, it is easily shown that for thisexample, the median number of attempts is 3 and median time is 20 s.

E2. Constant Probabilities within Subsequence. Sometimes the proba-bility of success is constant for each attempt in a subsequence, butassumes a different value for each subsequence. A firepower system withno bias delivering two distinct types of rounds to destroy a target issuch an animal.

For this example, assume PK is the probability of success for eachattempt in subsequence "K." Then

PK(NK) =1 -(1 - K]

N

K= NK - (K

K N.SK

= 1 Pi)

13

Substitution with Equation (15) leads to

N

(23) E- T K-1 KK=l (1 11T [ K

As a specific example assume that a firepower system has two roundsof type A and two rounds of type B ammunition. If A has a probabilityof success of 0.4 and B, 0.3, which sequence of delivery is best--ABAB,BABA, AABB, or BBAA? Equation (23) provides the following results bysetting t = 4 and N. = 1:

1

E ABAB = 2.84N

E BABA = 2.94N

EN: AABB = 2.7dN

EN: BBAA = 3.0

Thus sequence AABB is a marginally better strategy if delivery ofresource depletion per success is the criteria. Not surprising!

On the other hand, if time to success were important and if therewere an additional time burden associated with type A rounds, the resultis not so obvious. Assume that type A, being bulky, requires 7 s toload and fire whereas type B requires only 5 s. Then,

ET: ABAB = 17.0 s

ET: BABA = 16.8 s

ET: AABB = 17.1 s

E BBAA = 16.8 s

which leads to the conclusion that both BABA and BBAA are the betterstrategies.

However, if one assumes that the first round is already chambered(i.e., a 1 = 0) one could conclude that ABAB is the best strategy since

ET: ABAB = 10 s

ET: BABA = 11.8 s

ET: AABB = 10.1 s

ET BBAA = 11.8 s

14

As this example illustrates, it is rather important to use the equationspecifically applicable to the measure of interest.

E3. Intradependent Attempts. Unfortunately, it is often the case thatthe probability of success of each attempt in a particular sequence isimplicitly dependent on the success or failure of previous attempts inthe subsequence. This is the case with most firepower systems which areplagued with both round-to-round and occasion-to-occasion errors. It isthe latter which causes intradependence whereas changing munitionsand/or aimpoints causes the interdependence of attempts.

Assume one is analyzing a firepower system which has an errorbudget such that random, variable bias, and fixed bias error distri-butions are known for the range in question. Also, assume that theconditional kill probability can be reasonably estimated for the targetsof interest. Using the following definitions, we set up the solution.Let:

x,y spacial variables indicating the impact point ofthe projectile in the plane of the target. Theplane may be defined as vertical or horizontaldepending on weapon type, e.g., tank, artillery.

nx ny variable describing the system bias consisting of botha fixed and variable bias2 2

a R Io 2 the variance of the random errors, in the x and yx y direction

2nx 2 2 the variance of the variable biases

x y, i the fixed bias in the x and y direction

Ki (x,y) the conditional probability that the target will bedefeated if the projectile impacts at point x,y in the"ith" sequence.

Assume that all error sources have normal distributions. Furtherdefine

(x - nx ) 2 ( y n

exp 2 ( +

[2 2a R aRx

xy

pi(nxny) f ff K(x,y)p(x,y:n xny) dx dy

~15

which is the probability of defeating a target with an attempt givenspecific values for n r,n. Note that for many munitions against hardtargets, such as tanks, he bounds of this integration are limited tothe projection of the target on the target plane. This occurs becauseKi(x,y) = 0 when the projectile misses these target types.

(24) P K(N K) ff [1 P K (nX, ny)] N K P(Ix2T ] I nxd

-K X y

= -(N K NxKny:nx, y:1 d xP dyx n

(NK) =P NK . K (nIxny ~xn:Uy nn dn

When these are substituted into Equation (13), the expected numberof rounds used and time to defeat the target are easily calculated.

As a specific example, assume we are interested in whether adjustingfire is advantageous in a system whose error budget is

aR = aR = 0.6 mradx y

a = a = 0.4 mradn x n y

1x = Py = 0.0 mrad.x

As a baseline case, we assume no adjustment and decree that if thetarget is not hit in 10 rounds, firing ceases, i.e., ST = 10. Therange is arbitrarily chosen to be 1500 m and 3000 m, tie target is a23x23 panel and the objective is to hit the panel.

Under these conditions

1500 m 3000 m

P1 (10) = 0.99 = 0.805

*1(10) = 7.92 = 4.799

10 - *1(10)

P1 (10)

E =2.1 =6.5n

16

Now we will evaluate the option of adjusting fire after each roundassuming that it can be done perfectly. Since the sample size islimited, much of the error observed and corrected for is random ratherthan the bias which we wish to eliminate. It can be shown that thestandard deviation of the bias error reduces from its value to the valueof the random error divided by the square root of the sample size.Since the strategy in this example is to use information only on thepreceding round, the sample is 1.

Under these conditions

4..'1500 m 3000 m

P 1(1) =0.51 =0.164

~(1) =0 =0

P i>1 (1) =0.402 = 0.122

EI 2.2 = 7.7

Hence, in this example, closed-loop fire control actually degrades-the system even under the assumption that everything is done perfectly,

i.e., zero measurement errors.

* IV. SUMMARY

This final section merely reiterates the caveats annunciated* previously and summarizes the results. The caveats noted are

9 Binary Objectives. For Equations (15) and (19) to be validthe attempts being analyzed can have only two outcomes, totalsuccess or total failure. Although this represents a largeclass of problems, there are also many for which this condi-tion does not hold. For example, consider a pugilist who isattempting to KO his opponent. Although an individual blowfails, it may sufficiently condition the opponent to enable alesser blow to be successful. Thus his attempt, althoughnot successful, did affect the objective, i.e., was not atotal failure.

o Expected Versus Median Values. Equations (15) and (19) arenot median values and therefore caution is advised in inter-preting their results. Median values can be simultaneouslyfound by use of the algorithm described in Section II.

9 Expected Attempts Versus Time. As examples of Section IIIillustrate, depending on the measure considered important,use of Equation (15) versus (19) can lead to quite differentconclusions. Thus caution is advised in inferring the expectedtime from expected attempts and vice versa.

17

This report discussed the derivation to two equations which when usedwith discretion may significantly reduce the complexity and timeassociated with determining the expected value of resources depletionand time to success associated with achieving binary objectives.

18

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