D-A163 921 A FUZZY SET APPROACH TO VULNERABILITY NLYSIS(U) ARMY 1/1

BALLISTIC RESEARCH LAB ABERDEEN PROVING GROUND MD

P R SCHLEGEL ET AL. DEC 85 BRL-TR-2697

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Technical Report BRL-TR-2697

4 TITLE (and Subtitl) 5 TYPE OF REPORT a PERIOD COVERED

A FUZ:Y SET APPROACH TO VULNERABILITY ANALYSIS ________________ -'S PERFORMING ORG. REPORT NUMBER e

7. AUTHOR(&) It. CONTRACT OR GRANT 9dUMUER(s)

Palmer R. SchlegelRalph E. ShearMlalcolm S. Tavlor

9. PERFORMING ORGANIZATION NAME AND ADDRESS 10. PROGRAM ELEMENT PROJECT. TASKAREA a WORK UNIT NUMBERSUS Army Ballistic Research Laboratory

ATTN: SLCER-SEAberdeen Proving Ground, N!D 21005-5066 RDT&E 31L1626]8NM143%It. CONTROLLING OFFICE NAME AND ADDRESS12REOTDE

US Army Ballistic Research Laboratory December 1985ATTN: SLCBR-DD-T13NUBROPAEAberdeen Proving Ground, NID 21005-5066 314. MONITORING AGENCY NAME a AOORESS(1I differeut Iroin Con.trollfig Office) IS. SECURITY CLASS. (of tle report;

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I1. SUPPLEMENTARY NOTES

19 K(EY WORDS (Coninue on, reverse side if necessaryj and identify by block number)

Fuzzy setsVulnerability analysisBootstrapFuzy logicV a guen ess20. AUs A CT (csrtor af reverse ai* if ne.eaaf end Ideru If 7 by block nuinbev)

In-this document3-we will illustrateshow vagueness or uncertainty can bemodeled by fuzzy sets and how this uncertainty is reflected in the final measureof vulnerability., 0m emphasis will be on the computation of vulnerable area ofa surrogate system with uncertainty present in the cell probabilities. Theuncertainty in the cell probabilities can be attributed to uncertainty in assign- Wing component kill probabilities, to error in engineering judgement in determiningcritical areas of components and componeiit kill criteria and to unknown svner-

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'gistic effects. Other contributions to uncertainty are of course present, Cfstarting from the target description through a myriad of subjective decisionsand engineering judgements inherent in the vulnerability analysis.

For the purpose of completeness and comparison, a point estimate of thevulnerable area of the surrogate system will also be determined as it iscurrently done in vulnerability analysis. A point and interval estimate of the 14.vulnerable area will be determined by a statistical procedure known as

-,,"bootstrapping. ". Finally, a fuzzy vulnerable area will be calculated and thedifferent approaches compared.

UNCLASSIFIEDSECURITY CLASSIFICATION. Or a1S PAGEH?1-e, Zree t-e-ec

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TABLE OF CONTENTS

Page-

LIST OF IILUSTR?.*TIONS ..................................... 5

*1. INTRODUCTION ............. 7.................................

2. DESC(RIPTION OF A VL'LNER.ABLE AREA COMPUTATION .......

3. ESTIMATION OF VULNERABLE AREA AND PROBABILITYOF KILL .......................................................................... 8

*4. TH E BOOTSTRAP ............................................................... 10

5. BOO0TSTRAP DATA ANALYSIS .............................................. I1

6. NVAGUtENESS IN NVLNTR.ABI LITY ANALYSIS ........................... 13

7.CELL PROBABILITIES AS FUZZYN NUMERS.............................. 13

8. FUZZY ARITHUMETIC ........................................................... 7-

Q. FU*ZZY DA.TA, ANALYSIS ...................................................... i

10. St. MA. ..................Y........................................ ............. 20

*REFERE:NCES.....................................................................23

APPENDIX A. FUZZY SETS AND NUMBERS ............................. 23

*APPENDIX B. A GEOME11"TRIC('INTERZPRETATION OF FUZZYADD)ITION ......................................................... 2

Il-bTRIBl TION LIST ............................................................. 3.3.Accesion For

NTIS CRA&MDriC TAB w_

* By

Ava2 ,c!:i'y Coces

IAvczK:. jjorDizt

34

A -1 1 . .

LIST OF ILLUSTRATIONS

Page

FIGt-RE 1. MULTI-LAYERED TARGET DESCRIPTION WITHENCODED Pk h, VALUES .......................................... 0

FIGURE 2. BOOTSTRAP ESTIATE OF TfW SAMPLINGDISTRIBUTION OF Av ............................................... 12

FIGU'RE 3. MEMBERSHIP FUNCTION FOR FUZZYA .A..................... 21

FIGU'RE 4. MENMERSHIP FUNCTION FOR "IGHLOSS-OF-FUNCTIOM ............................................... 26

FIGURE 5. MEMBERSIUP FUNCTIONS FOR FUZZY P AND P .......... 20

FIGURE 6(a). CASE 1: t < X, + yj .............................................. 30

FIGU-RE 6(b). CASE 2: X2 + Y2< t <X4 + Y4.................................... 31

*FIGURE 6(c). CASE 3: x, + yl< t <x + y2................................... 31

IINTRODUCTION

In defense preparedness studies, decisions requiring evaluation of tile effect ivene;'. ofexisting, and conceptual weapon systems often arise. Such evaluations usually requiresome measure of the vulnerability of the subject systems relative to a specified Scen-rioand ml sion profile. For example, if one is considering which weapon would be .rt

effective against an enemy tank of the 1990O's, a great deal of uncertainty surrould% theS description of this future threat. Current vulnerability model., require as input a1

detailed computer des cription of the system along with a configuration of its criticalcomponents.

Present met hods, o unrbltaaysspvide a point estimate; e.g.. a

probability of kill or a vulnerable area, with no corresponding measure of uncertaintyNthat is inherent in (tie e-stimate. Attempts to model this uncertainty stochastically havebeen made by Dotterweich and Taylor and Bodt. 2

In this document, wve will illustrate ho-w vagueness or uncerlaintv can be modi ledby fuzzy- sets and how this uncertainty is reflected in the final measure of vulnerabilitv.Our emphasis will be on the computation of vulnerable area of a surrogate systemi "ithl

*uncertainty present in the cell probabilities. The uncertainty in the cell probabillii'scan be attributed to uncertainty in assigning component kill probabilities, to error 19

*engfI neeri ng judgement in determining critical areas of components and component killcriteria and to unknown synergistic effe-ts. Other contributions to uncertainty are of3 course present. strigfrom the target deSCription through a myriad of subjectiVedecisions and engineering judgements inherent in the vulnerability analysis.

For purpose of completeness and comparison, a point estimate of ithe vulner~ibarea of the surrogate syistem will also be determined as it i-s currently' done inlvulnerability analvyis. A point and interval estimate of the v ulneralle ara ill ief

d teried by a statistical procedure known asbootst rapping." Finaliy, a fuzz\*Vulnerable area A4ill be calculated and the different approaches compared.

2. DESCRIPTlON OF A VULNER.ABLE AREAX COMPUTA*TION

Thle models currently us;ed for computing vulnerable area of armol(redt vthilou-.*require a numbler of detailed inputs, including a geometric description of the tnrtt

spec .I yin the locatio-n, thi'ckness, and obliquity of armor plates andi the location anidsize of major compi mnents. For a specified direct ion of attack, impact lcal n, o)n lietarget are selected by superimposing a rectangular grid and choosing onc point orimpact within eachi grid cell. Then, at each point of impact, a ray is traced through the

E. J. Dotte rweich, "A Storhastic Approach to 1'ulnfrability Anatymqs ARI3RL- TJ?.0257h. Bal~eiclResearcha Laboratory. JP'.

M S Taylor and 1? .4 flodt. "Construction of Appro.Timate Confidrner Iv;!rraol, for Prcba~hiy. cf-h ai

6ia to Ilootrtrap.- AR1IIIL. TJ?0257v, Baili.1tic Rre arch Laboratory, 19, 4

target model in the attack direction and a list of the components, barriers., eniptspaces, and other structures encountered along the ray is compiled.

For each component encountered along the ray, a fragment mass and velocitvtogether with an estimate of fragment shape is used to predict a bole area in the

component. A kill criteria in the form of a hole of minimum cross-sectional area in asensitive region" of the component is required in order to regard the component as

inoperable. If the predicted hole area exceeds the minimum hole area, then aprobability of kill Pk I h,, is assigned to the jth component encountered by the ith ray. InI

. determining probability of kill given a hit on a component, the vulnerability anilva-- estimales the -presented area- of the component and the presented area of the se.'-iltI% v'-

region of the component. The estimated probability of kill is taken as the ratio of IIw"-pre-cnted area of the sensitive region to the presented area of the component. (Anoverall component kill probability is obtained by summing the sensitive area fronseveral aspects and dividing by the corresponding total component presented area.)

*If several components are encountered along a ray, and determined to be killed.then the probabilities Pklh,, are combined under an independence assumption to

I- calculate a probability of kill. Pklh,. associated with the ith cell. This probability isthen multiplied by the area of the cell. A i, to yield a "cell vulnerable area". cellvulnerable areas are then summed over all the cells in the grid to obtain an estimatedtarget vulnerable area

A = Pk Ih.Ai (2.1)

for the given attack aspect. Hlafer and Hafer 3 and Nail 4 provide further detailsconcerning computation of vulnerable area.

3. ESTIMATION OF VLNERABLE AREA AND PROB.BILITY OF KILL

Consider the conceptual item of military hard',are illustrated in Figure 1. Withoutloss of generality, the component edges are assumed to be parallel with thesupe rimpodse 4x4 rectangular grid. The numbers in the ith cell represent theprobatiliy Pkh, that the target will be killed should a prescribed round of ammunitionfired from a fixed range and orientation impact in that location.

The vulnerable area A, for the target is calculated by

Av Ed~ Pk hAi A Pk kh, (3.1)

-. T. F. Ilaffr and .4 S lafrr, " 1ulnrraoblly .4naly'is for Surfacr Tar t (4AST)," ARBRL- TR-6..'i."Baliptit Ro, Parch Laboratory, 1979

4 C. L Nail. I'ulnerabfiti Anail/yis for Surfart Tarpffs IVAST), Rrniion I," Computer Scifnr r. "Corporation Rerort CSC. TR- f- 740. 9c..

I--

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

i.__: / .[; :, d 2..' i.;-....-..-.: .- :. : ,..:.. ... . . . . . ..." '" "": .... ".. .. . .. ..... . " : ....,*:-'. .,2:"" A.... . ."<......., -

9K

.3 3

3

IL 5

K r5

FIGUtRE 1. MU"LTI-LA)TERED TARGET DESCRIPTION WITH ENCODED Pki hVALUES(Vacant cells correspond to = 0b

%%here i is indexed over the cells of the grid. In this example, the cell aroas A, aroidentical and equal to A. The overall probability of kill I'k Is

uvhre A~ is the total presented area of the target. For the conceptual target sliov.n i*Figure 1. the exact value-, of vulnerable area and probability of kill are A,= 105.1 and..

1% =0-41.

In Practice. th'e values Pk h, are unknown, and estimates 1 k Ih, are required to,olam thle approximat ions

AP-

Dectermnining valid Pk h. estimates is a persistent problem because of the difficulty in

m~deling the underlying, damag-e mechanisms, and is an additional source orunccrtainly. The bootstra1p procedure to be introduced in Section 4 assumes thevirilix idlial i, are accurat ely represented, and does not attempt to assign a

C'MI-TI17n of error due to this uncertainty. This problem is however addressod in

a vct hn 7 '1( heecll I~ h,'s arc considered to be fuzzy numbers.

4. THE BOOTSTRAP

The bootstrap isa technique for data analysis whose goal is to extract informationIfrom a set of dat a through repeated inspection. An expository article by Diaconis and* -Efron. an intermediate-level paper by Efron and Gong. 6or a more advanced paper by

Erron serve a-s an introduction to the bootstrap.

Suppose %%e want to estimate some attribute 0 of a population X = xj 'E

%%here the Index set I may be finite or infinite. To obtain an estimate 0 of 0 a randomnn

sample jx,) is taken from X and a statistic 6(x1 . x , ) is evaluated. For example.

if 0 =p (the population mean), then estimates

rP. Diaconil' and B. Efron, Computu.Ilnteneirc Afcthods in Statieticp. S-ifntific Arririan, (19 s'.1Ir-

B. Efron and G. G;ong, .4 Lr'i.urely Look at the Boote.trop, the Jockkniff. and Cro,P Volidaticn, Th,Amrnrrion Siatiptirian, 87fJ919.S6-4c

r B. Efron. Bootstrap Methods: Another Look at the Jackknife, .4nnals of Statistics. 7f 1979. 1*J-f

1\ xi or 0()= median (x,) (11.

art, commonlv used; if 0 = 2 (the population variance), then

n(n - 1) (1.2)

is an appropriate stati stic. If random samples {xj are repeatedly drawn from X

and a statistic 0) evaluated, a distinct value of 0(' may be determined for eachsamldh dra\on. These values are from the so-called sampling distribution of 0 ) )"is itself a random variable under this sampling plan.

KnoNledge of the sampling distribution of O( ) permits questions of accuracy to 1w Lans'-ered, confidence intervals to be constructed and hypotheses to be tested. Uiidernppropriate as.,umptions, and for some functional forms of 0'1 , the samplildistribution may be determined analytically; however, in most instances it cannot.Furthermore, the option of repeatedly drawing random samples to construct anempirical approximation to the sampling distribution for 0 is usually not available.

n

Typically, only a single sample of data {xi) exists for analysis.

The essence of the bootstrap procedure for the application here is as follows: If

samples, with replacement from {xj} are taken to generate 'bootstrap" samrplen

X, and 0"-- (x......x) is determined a large number of times, the

diktribulion of b(,otstrapped p provides an approximation to the unknown samplingdistribution of 0 . (This procedure provides a Monte Carlo approximation to thedistribution of the nonparametric maximum likelihood estimate of 0.-

5. BOOTSTRAP DATA ANALYSIS

A random point of impact and its corresponding Pk h, were determined for each16

cell on the target in Figure 1. These sixteen values X = {Wk I h.) form the data

base for all sub!-equent analyses- conventional, bootstrap, and fuzzy. In this instance.the sixteen values formed the population that was sampled with replacement todetermine bootstrap estimates A" of the vulnerable area A,. Estimate A: was basedon a sample equal to the number of cells in the overlaid grid. One thousand samples ofsixteen were drawn from X, and one thousand estimates A, computed. Theapproximation to the distribution of A, is shown in Figure 2.

i-i

K..:... .. .. -. : .. ..... .. . . .. .. . . . . .. .. . . °.* . *.. .. :. . _ : ., . _ . : , :: :; ) (: . . .: :: ;

,--5

40

AT

FG1 C L P-F T TAF UTIlATF OF THE SAMPLING E1ZRIED'\0 OFA

The histogram in Figure 2 is a bootstrap estimate of the sampling distribution Of

A..The median of this distribution, which we will take as an estimate of A,~ is1A, 132.8 (compared to the true Vulnerable area A, = 105.1). An approximate

central 90%- confidence interval for A., based on the bootstrap is [9.3 163.6].

In this example, the median of the bootstrap distribution is considerably larg-ertban the true vulnerable area. This is because the sample X which forms the basis for allthe bootstrap calculations estimates A, to be 133.2; i.e., the procedure operates on asample providing a biased estimate of A., . The bootstrap has no way to compensate for

* an unknown initial bias, and in this instance led to a bootstrap estimate A , 132.8.SThi.s bias has no effect on the estimate of variance of A, since the error is in locatiOn

* The location error can be controlled in this application by using a finer grid on thetarget.

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

6. VAG(;ENESS IN VULNERABILITY ANALYSIS

As indicated in Section 2, to perform a vulnerability assessment of a given system,it is often necessary to develop an elaborate computer description. Information aboutthe system may be varied in form and in abundance, ranging from intelligence data tofullv documented commercial products. The system may be foreign or, in someinstances, conceptual. One of the most subjective parts of the assessment involves whatthe vulnerability analyst may refer to as "engineering judgement." Knowkhdge of acomponent's function, for example, may cause the analyst to predict its size andlocation in a system for which little information is available. The analyst may be calledupon to judge armor thickness, material and thickness of component structures, and tospecify kill criteria for individual components, as well as their sensitive regions. Theanalyst assesses the effect of different types and degrees of damage to a system and it"components and infers the systems "loss of function." Given damage to a component orsubsystem, the analyst may infer a resulting loss of mobility, firepower, etc. Themeaning of some of these terms and appropriate measures for their quantification arevague or imprecise; even vulnerability analysts differ as to the precise meanings andinterpretations.

At many places in a vulnerability analysis, the analyst is required to act uponinformation which is vague or imprecise and, in some cases, where information is notavailable at all. Some of the uncertainty present in the process is not due torandomness and therefore probability theory may not provide a completely satisfactorymodeling tool. The theory of fuzzy sets provides a rationale for dealing with sources of .

uncertainty Mhich are non-random. A brief introduction to fuzz) sets is given inAppendix A. In the following section we illustrate the application of fuzzy sets tovulnerable area calculation.

7. CELL PRO.31AILITIES AS FUZZY NUMBERS

The vulnerable area model VAST requires as part of its input data a collection ofrays corresponding to fragment trajectories for given angles of attack. These rays arerepresented by lists of components, armor and barriers that would be encountered bythe fragments. Along each ray or trajectory the mass and velocity of the fragment as itexits obstaclhs it encounters is computed. For an initial fragment mass, velocity andshape at the target surface, the residual mass and velocity are calculated from the so-called TIIOR 8 equations. As the fragment encounters a particular component, a holearea is estimated in that component.

*."

Project TIIORIThe Rsistanre of Varioup Af~talic Mat~rial.q to Perforation bt Stel Fra.",.r:..'.Empirical Rdlationehipp for Fragment RePidual Velocity and Rfcidual Weight," Projat TiOR T.Rfport No. 47. Balli.otic .4nallyie Laboratory, Inet. for Coopfrairt' Re.eearch, The JohnP ilc,-.':Ini'treitl. April [19f1j.

°-*. . . . . . . . ... . . . . . .

As indicated previously, there is a great deal of vagueness and subjectivity involvedin the target description, material specification and thickness, component I), h,evaluation and kill criteria. Methods currently in use for vulnerability assessnifnt.including the bootstrap procedure discussed in Section 4, do not take these uncertaintie,into account. Assumptions regarding the fragment shape, barrier material and thicknessdictate that the hole estimate is actually a fuzzy number.

Incorporating fuzziness into both the kill criteria and the component conditionalkill probability results in the following fuzzy implication: "If the cross-sectional holearea in the component is about cm2 or greater, then the component conditional killprobability Pk lh, is 13, where B is now a fuzzy number. Thus, we have a fuzzy holearea estimate and a fuzzy implication relating kill criteria with kill probability. If these,were all "crisp," as assumed in the current method for vulnerability assessment, one

need only compare the predicted hole area with that of the kill criteria. If the predictedhole area is greater than the required value, then a probability of kill is assigned. This isdescribed by the syllogism

If hisA, then pisBh is Ap is B

lowevr. when A and B are fuzzy and we have h is A*, then we must use thecompositional rule of inference or the generali:ed modus ponens of fu::y logic to obtainthe conclusion. This can be represented by the syllogism

If h is A, then p is Bh is A*p is BZ.

The resulting fuzzy component conditional kill probabilities, when combined, result in afuzzy cell probability and finally a fuzzy vulnerable area estimate, reflecting theuncertainty in the process.

Zadeh 19 defined "If A then B" as a special case of the fuzzy implication -if A thenB else C.- Ntationally, we represent If A then B else C as ( AX3 )U ( -AXC ) %%herc"-" indicates nvgation and X is the cartesian product of the two fuzzy • sets. If C is th'empty set, then If A then 13 is written AXB ). Recalling that the cartesian product o)ftwo fuzzy sets A and 13 is given by

pAil (x, y) min{iPA (x), PB (Y)}( (7.1)

L. A. Zadth, "Ouline of a N, " Approach to the 4nalysis of Complfx Sy tfrnis and Dteision Pro.,,..

IEEE Trans. on Syettemp, Man and Cyberneticsr, SMl ('S(1). (197,?/.

]&ibid.

1-4

* . .-... °.

.1. . . . '.".* *,""~'

iecan derive a relation between the bole area and the kill probability, namely

PR (h, p) =min~pl 1 (h), lpp,(p)} (7.2)__

her(' R 1 1 X P k .-

Example: If If = 0.1/h,1 + 0.5/,, + 0.7/hl3 + 1/h 4 where hi1 < hi n

J)' h, 1 I/P1 + O.-V/P- + 1/P3 + O.)/4 + OVIP5 ,then Iti (h, p) is given by

pg (h. p)

p Pi P2 P3 P4 PSh _ _ _ _ _ _ __ _ _ _ _ _

h, 0.1 0.1 0.1 0.1 0.1

1:11 0.1 0.5 0.15 0.5 0.1

h3 0.1 0.5 0.7 0.5 0.1

h4 0.1 0.5 1.0 0.5 0.1

Furthermore. if we assume that the hole estimate 1i, is given by

III = 0.3/hi + 1/h2, + 0.5/b 3 + 0.1/h 4 .

then we would like to combine this estimate with that given by the fuzzy implication toinfer something about Pk Ih,. To proceed further, we need to invoke the composit ional

r ule of inference.

Given universes, U and V, a fuzzN set A on U and a fuzzy relation R on UXV,Characterized by PA~ (u) and PR (u, v) respectively, the compositional rule of inferencestates that the result "4ill be a fuzzy set B on V' defined by

P19 (v) =U [PA (u) n PR (u, v)J

and written as 13 = A o R?.

............................................ 6

('ontinuing the above example, we have

w~ith B denoting the fuzzy set resulting from this composition. Formally,

Pn (p,) = u{ ,iu, (h1 ) nlPiR (hi, pl)} (7.1)

= U 10.3nl~l o.11ofnlo.1.o. 5 n o.1,O.I n o.1 )

= (0{.1, 0.1, 0.1, 0.1)y-

=0.1.

Similarly. PB (P2) = PB (Ns) = i (P4) =0.5, nd PB (P5) 0.1; hence III o R B .

If 11, = 1H, and both are non-fuzzy, then

H- OR = Pklh.~ (7.5)

in agreement with the classical modus ponens. With 1I, 34 11 nnd fuzzy, v.i'11 l;v afuzzy version of the classical modus ponens with the important difference thant III anld 11can be different. Thus, in the above example, we have a hole estimate %whik-h ikdifferent, in general, from the required kill criterion.

Similar results can be obtained for each component encountered along the fra-m-ent*trajectory. These kill probabilities must be ultimately aggregated into a cell kill

probability. As mentioned in Section 2, current methods include combining compone'ntkills as independent events; however, the problem of aggregation of these probabilliies-

* remains open. Some methods of aggregating fuzzy sets such as union and intersectionare presented in Appendix A. Zimmerman and Zysno 11 have investigated several othermethods. For vulnerable area calculations, aggregation of the fuzzy sets requires furtherresearch; therefore, in the following we shall assume that somec form of aggregation has

* been done, and the resulting probability of kill associated with each cell, Pk jh,- is afuzzy- number.

ILL

11HJ. Zimmrrman and P. Zysno, "Latent Connectives in Humnan Dercion Mlaking," Fu::y SO.- adSj-oerwm 0(9501, 85,

8. FUZZY AITJIMETIC

In this section, we consider Pk I to be a fuzzY numhvr and A to be :I crilInumber; to simplify the notation we will denote i = Pk , h lqu:ition (3.1) %ill Ihn IwevialualMd by fuzzy arithmetic to obtain a fuzzy- number

A," A V I. 1. ,

Let i, (x) and j'~lY) be normalized membership functions of P, and P',, %iIhsupport in 10.1]. The definition of the sum of two fuzzy numbers P, + P' , gil ,, ))I

" , r w(t) = max [ min (lp, (x), pp (v))1. --x+v=t L-

We will choose our class of membership functions to be continuous and symmetric abuta point x0 such that

monotone nondecreasing x < x3- monotone nonincreasing x > x0 .

In particular, consider the following two functions:

0 X < n N.J

Xo < X < X3 ,

and

0 y<y ,p. (y) = g(y) yI < Y<v, ( Y

Y2 < Y !5 Y3

where the functions it , (x) and lp, (y) are symmetric about x3 and , respect iv1

Dubois and Prade 12 show how to operate on fuzzy numbers of this form in an eflicitntmanner. In Appendix B a geometric interpretation useful in evaluating (8.2) for ourspecific choice of membership function is detailed.

The product of two fuzzy numbers A and P is defined as

SpA. pt) - max [min (PA (, p (y))] . (SB)

1'D. Dutoip and I1 pradf, "Operations on Fu:y \'umbere," Int. J. Syptrms Srif nre. 9(6j, (iv S* ti Y

17'

.. . . ..... ,- " - * , '.°;W- "" -,.L-.--.*'-----..

In thk instance, A is coiis*h.d a crisp number, which has ats i's membership fun'tiri

(1 x'-APA (x- 0 otherwise.

Since. for a gixen t. PA () 0 for x -4 A, the only value undr consideration is x - A.Th is. imlies

P A (t) max [mi (1, pp (y,}] (8.8)Ay

t

A

0. FUZZY DATA ANA.LYSIS

16We "%ll now fuzzifv the data base P I introduced in Section 5 for the

bootstrap procedure and calculate the vulnerable area A, using fuzzy arithmetic. In

choosing the membership functions pp,, Ii=l, ,16, we make use of the fact that the

variance of the estimate Pk h, is Pklh,(l - k h) although no staiastical properties willbe invoked in the ensuing computation.

Let A, " 1/2 Pkh, (I - Pkh, I • The membership function for Pi was taken to be

-0 X Pk Ih,- 2A

lip ,(x) -- Li(x) PkIh. 2Ai < x <_ PkI h,- 61 .1)Pklh, Ai < x < Pk h,

where Lj(x) is the line determined by the points (Pk h,- 2Ai, 0) and (Pk h,- Ai. I)alo, recall that ipix1 is symmetric about Pkh,

For example, the fuzzification of P1 1 = 0.6 has as its membership function

0 x < 0.36

pp,,(xj = 25x/3 - 3 0.36 < x < 0.48 (9.2)

1 0.48 < x < 0. 60

16 .

From Dubois and Prade or Appendix B the fuzzy number P = Pi is given by

the membership function

,-'° p

% -i IE ..g..I u ..- ~~-w.,--

I °

I..0 x<aPP(x) I(X a<x<b(.3b<x--_ c

A here16

a (i-I h, 2A )

b -- (Pkh.-10

16c = P Pklh,,

and (x) is a line segment determined by the points (a,O) and (b,l).

Finally, A, the product of the crisp number A and the fuzzy number P has as itsmembership function

0 x_< Aa

PA,(X)- Lx) Aa < x < Ab (0.4)I Ab < x < Ac "

where L(x) = l(x/A) on (Aa, Ab). The membership function for A, for the example inSection 3 is given by

0 x < 93.7

p,(x) - .05Ix - 4.78 g3.7 < x < 113.4 (9.5)1 113.4 < x < 133.2

as illustrated in Figure 3.

Since the function PA,(X) is symmetric about x = 133.2, values of vulnerable areain the interval (113.4, 153.0), where the membership function takes on value one, areequally acceptable. The vulnerable area obtained from conventional calculation on thesixteen random shotlines is at the midpoint of this interval, while the true vulnerablearea for this example. A,. - 105.1, has from (9.5) a membership value of 0.58 .

' Since the fuzzy vulnerable area depends on the cell probabilities, the fuz7V* vulnerable area is dependent on the grid size. Grid size impacts the bootstrap and

conventional calculations as well. In this example, if the number of cells is increased bya factor of four, the interval of k'alues for which IA,(x) 1 has as its midpoint the truevalue.

: ** * . . . ... -.-

r...

Furthermore, the interval of values for which the membership function equals on(-depends upon the interval of values for which the individual celi probabilities havemembership values equal to one. We assumed in (0.1) a specific form for the fuzzy cellprobabilities. The vulnerability analyst may be able to sharpen these bounds, and in sodoing, sharpen the fuzzy vulnerability estimates.

-1o. SUMMARY

In this report, we have attempted to make the point that at many places invulnerability analysis there is a margin for error that comes about, not fromrandomness, but from a basic inability to model uncertainty. We have considered, for aspecific example, two new approaches for vulnerability analysis: (i) a fuzzy set approachand (ii) a statistical resampling plan known as the bootstrap.

The membership function jAz lx) in (0.5) for the fuzzy vulnerable area calculation isillustrated in Figure 3. The fu::y procedure seeks to quantify the uncertainty inherent ina ulnerable area calculation due to the uncertainty in the cell Pk h, estimates. Thisuncertainty stems from the assignment of fuzzy kill probabilities to the individualcomponents and from the method of aggregation into a cell probability of kill. It hasthe distinct advantage, however, of clearly delineating the inherent subjectivity.

The point estimate of A, provided by the current method of vulnerability analysis,Av --- 133.2. is indicated in Figure 3. The fuzzy vulnerable area is symmetric about A,but considers values in the interval (113.4, 153.0), corresponding to values of A, forwhich the membership function equals one, to be equally acceptable. This interval has "c

- the appearance of a statistical confidence interval, but does not convey the sameinformation, since it has no corresponding probability level attached.

As discussed in detail in Section 5, the sampling distribution of A, provided by thebootstrap has a median of 132.8 and an approximate central 0O confidence interval of[09.3, 163.6]. The bootstrap procedure seeks to quantify uncertaintV inherent in avulnerable area calculation due to the rariability in the sampling procedure responsiblefor selection of the cell Pk I h, estimates. The bootstrap takes the cell Pk Ih, sgi en. andmakes no attempt to model uncertainty in the values themselves. 5

I- I

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

.7

FIUR 3 IMESI UCTO O UYA

I eIII1 1 "l~lmo , thIai1,1I' n t es m ln sh m h tslcst eei k

th ni, iV (f Iio ell agt gi, a d lk l dilo n sure. Th

The AcaL et -imiate, s subj e tow errrodued to thutauntermnin the ellIX

esInecti th theiabilOtv in the seslin scemethaisegt the cellatvi I i( I)())(- nl

riplriein a svn'i ftI~ h rea ofr a ion uld su t boostapin n fil wa!y nu brc.

Th fuzzyrc etipra n. fotrmted fuzy set( appy rachio and the bootstrap are 1)(ort kilol

purabilit %as iat b\a the aresin prodedt wthout\ fude in the lOtreo

* Te fu iti im xi-to iilan forme by ohe muposit ion andvs teg. fuzz rcc ilflJ( (nent 'rl

NlIzu mot o. 13

'A1 AI::umolo. "Fu::y iem oing Under .eu- Compcitiional Iiulf.* (,f Inferm-ncr 7Kyibrnfiet., .10 7-117.

T

REFERENCES

1. E. J. Dotterweich, "A Stochastic Approach to Vulnerability Anialyse.- Afllsl'l-TR1-02578, Ballistic Research Laboratory, 108-1.

*2. M. S. Taylor and 1). A. Bodt, "Construction of Approximate Confidence lnler\wik1for 1'robabil it v-of-K ill via the Blootstrap," A.RBIZLTR-02570, Ballistic lResv;rchia. Laboratory, 1981.

* 3. T. F. Hiafer and A. S. Hafer, "Vulnerability Analysis ror Surface Target., (VAS])

* ARBRL-TR-021 51, Ballistic Research Laboratory, 1979.

*4. C. L. .Nail, -Vulnerability Analysis for Surface Targets (VAST), RviJo 1.'

K Computer Sciences Corporation Report CSC7-TR-82-5740, 1082.

5. P. Diaconis and B. Efron, "Computer-intensive Methods in Statist ics,- Scie'ntific

American, (19S3), M1-130.

G. B. Efron and G. Gong, "A Leisurely Look at the Bootstrap, the Jackknife, andCross-Validation," The American Statistician, 37(1983), 3W-48.

7. B. Efron, "b~ootstrap Methods: Mnother Look at the Jackknife," Aninals of

Stat istics, 1(1979), 1-26.

S. -The Resistance of Various Metaffic Materials to Perforation by Stec) Fragment,.:Empirical Relationships for Fragment Residual Velocity and Residual VeIglit.7Project THOR Tech. Report No. 47, Ballistic Analysis Laboratory. Inst. f,)rCooperative Research, The Johns Hopkins University, April (1901).

9. L. A. Zadeh, "Outline of a New Approach to the Analysis of Complex Sysicemc IndK Decision Processes,- IEEE Trans. on Svst ems, Man and Cybernetics. SM(-3t I).

(1973).

10. L. A. Zadeh, "Outline of a New Approach to the Analysis of Complex System, 1!1,Decision Processes," IEEE Trans. on Systems, Man and Cybernetics. S\(-3( I).(1973).

11. Il-J. Zimmerman and P. 7vsno, "Latent Connectives, in Human Decilon 1KIinc

Fuzzy Sets and Systems, 4(1980). 37-51.

12. D. Dubois and HI. Prade, "Operations on Fuzzy Numbers,- Int. J. S~stems Sciencc.9(6)(1978), 613-626.

13. N1. Mizumoto, "Fuzzy Reasoning Under New Compositional Rules of Inference.

Kybernetes, 12(1985), 107-117.

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

14. L. A. Zadvb, "Fuzzy Sets," Information and Control, 8(1967). 3-5.

p 15. 1). 1. M3ockley, "The Role of Fuzzy Sets in Civil En~gineering," Fuzzy Scis aiS~stvnis, 2(1919), 267-278.

* 16. D. 1. Illbwkley, The Nnliirr' of Struet'iral Design and S.ifety, Ellis 11orwoodl iitti

17i. J. Tr. 1'. Yao, "Damage Assessment of Existing Struciur'es," Jouiinal of the.Engineering Mechanics Division, ASCE, 106(EM 1), Paper 15653, (19F80), 7ck5-79)9.

18. M. Mizumoto and K. Tanaka, "Some Properties Of Fuzzy Numbers3," l~m.it1177% Sett Thenry ind .Ar212iitiona, N1. M. Gupta, R. K. I'Lagade, It. 1(. Yayrr

Editors, -Nort h-hiolland, (1079).

10. D. Dubois and Hf. Prade, "Operations on Fuzzy Numbers," Int. J. Systems ScinIe."'

9(6), (1978) 613-626.

20. D. Dubois and If. Prade, Fuzzy Sets and Systems: Thoory -and nicP,-

Academic Press, (l08O).

24

Appendix A. FVi'ZY SE1'-; AND) N1 \Il I:l?1

pZ:(Idv'.1 notI ig Ilh-t nuit(an C of o!hjccl, encountcred In I Iiv rv:1l %%-l nir irprecivcly (left i((l. Ilr'lj)(N'l Ihe ConCePIt Of a fU7Nv .e tO i)MMIdl SiV-li Cl '11,1N fl. (I-

* all rcal numriu- w~hich are *'muih greater thin one, Or t he (+i-.. of ",(-.erc% (killV ehiclv ,- for vvahttjle. do not Cohi1 ' it til (+I"-;~. i the 11,11:1 thai 4.Iteuiva:Ilit.'.Ie/set met hi iulo lo dd oh prim~' for vultivralhy anil%-.' ;and( dmnififv i'u~2

.itpro% ide., a mcchajri i % %hich sujbjvctive IIr'Or1ittv)1 can lie qui"int ifited "1nd (.1' r.;101on. mit'h resiillt reflect jg the degree of imprec;ision in input inrorrinioin nrid diii It,

- ~clomely rclitedl field. BlOckley I and Ya o I' hVesCcesu appliedfi ''

* prOblems in ci\vi niteering andI str uctural damige.

* The fundanient al not ion Of a ftuZ7\ set is the follo~wing:

If V. (the universe' of discourse) is a collect i or objects, then aI fu--.z/ .sI A\ 1, 1~ ,fordered pairs

A PA { Wx )ix here it..% (x ) ,called a mem~bership functi'on, quanitifies the grade, of member'.ltiii ,f 'MtA. The function p.. (x) maps x into a memnbership space \1 which is usualix taken I,, I,the unit interval [o,11.

Example: U Aut hor,, of t lik report: A. The set or bald invii

A = (N Aeom, 0. 1). (Palmner. 0.7) (Ralphi. 0.6)).

Fobr -crisp- sets. i.e., ordinary sets, the membership funct ion reduces ti ih-* characteristic function

PA(x =(0, ot hvrwNIse

FuZ7V sets need not be finite fets: for example, in vulneral1,1lir studies vehIclc riki r"1loss of function" values in the unit interval [0.1]. One p0o';ibIlit y f(or 1 f1177\ .

ml charactcri7ing -highi loss of fun ction- Is dhllii tvtd in Figure 41.

5.~~14 A. Zadeiu. "Ft::y SOtP," Informateion and Control. H19C.,. ! -c,

* 1~D 1. I?!orklfy. - h e~ Ifc't of Fu::.q S' le in C'ivil En;:nf r rn;7 I't::y Sf t.-- ar~d ms.'ur f'i ' .

*1) 1 Ifo rkir TA 'Mfur, of Stru rdural1 p i'n an d .%ft ti. Vlti: lorto od I d. (1'.9

' J T P 112. "Dariac'~ Apf.rnfnli of Ein.'ii~ig Strutu~rs.- Jcu~rt:al of 16Dii EnMfr -: .!.Deizcn, AS(E 1,17F14 )iPhaPi , r /109i. 73.LX?

06LOF -

L FIGURE 4 MIDMERSHIP FLNCTION FOR ' HIGH LOS-c-OF-F L'NCTION'

* We now give several definitions for properties of fuzzy sets which are obviousextensions of corresponding definitions for crisp sets.

A fuzzy set A is empty if and only if

AA (x) =0 for all x U U.

T'wo fuzzy sets A and B are equal if and only if

PAXW=PB(XW for all x U

The complement of a fuzzy set A, denoted by A', is defined by

PA M) 1PA W for all xtU .

The union of two fuzzy sets A and B, with membership functions PA and PDis a fuzzy set C whose membership function is given by

PC (x = max PA(X), PB(X W X EU

alternatively written

PC W)=PA W)U PB W, X U

where Udenotes the maximum operator.

The intersectilon of two fuzzy sets A PHd B is a fuzzy set C given by

PC W)= PA (x)fln Px)1, xifu

where n denotes the minimum operator.

The support of a fuzzy set A is a crisp set S(A), given by

S(A)= X XPA (X) >01.

A fuzzy set A is said to he normal if and OL~y if

sUP PUA Wxx

The cartesian product AX13 of fuzzy sets A and B from universes U and V,respectively, is defined by

PA (X, Y)=rin (PAW), IB (Y)), X U, YV.-

A fu-:y number A is a fuzzy subset of the real line R.

A fuzzy number A is said to be eonvez, if for any real numbers x, y, z cwith x < y < z

PA (Y) PA (x) n 11A (Z)

An ci-let'el see of a fU7zy number A is a crisp set A. defined as

A. ={fXIPA Wx) 0), 0 < 0 < .I

The fuzzy set A is commonly %vritten, when U is finite, as

A PA (XI)/XI+ + PA (Xn)/X PA (Xi)/Xi

%here denotes the union of fuzzy sets; and when U is not finite as

A =f PA Wx x.U

dam

The algL'brai" properties of fuzzy numbers uner the tour arithmt ic ,,:r. ,,fuzzy addition, subtraction, mult iplicat ion and di ision i a'e boet n :. 'li I IMizunioto and Tanaka 1 and )ubois and Plrade. F For exaleh,, if A and iB are f,,numbers Nvith nwmlwhrship funct ions PA and P13, then the sum r, A and B is giveni by

A+ z) = U (PA (X) "u (.) )x + y - z

= u wp.: (x) n ti, (z -x) )x

(I~ccalI that U and N represent the maximum and ninimum operators.)

E.ramplc If 1' = I0. , 2 ... }, A = 0.2/1 + 1/2 + 0.3/3 and

B - 0.1/2 + 1/3 + 0.2/4. then z can take tho values 3, 4, 5, G and 7 and

PA.+D(.t)= max l{p l) m (3), PA1(2) PB (2)}

a (10.2n ,. 1 0.nI} = mx (0.2, 0.1}

- 0.2.

Fuzzy addition is clearhy more c imhvrson than crisp aditio hu, ,.\,r.summation mav be readilh implemented on the computer.

1M. Ai:urnoto and K. Traaa, "Somf Proprhfu of F'u::y Nu,'i r< , 04 Fn:.-y S, 1 VI.Application,-, 3t. .11. Gupta. It' A'. Ita~odf. R. R. Ya-fr. Edddcrp. orth-Illirpd 11,9 7)).

D. DuboiP and If. Pradf, vp. cit.

0D. Duboz.- and 11. I'ade, Fu::y Stt nnd SyIf rwv. Thf ory and .prlica!iwtr-. .tcd- t,:;c P'r! i'.. J-

2S

rI

",1

-. J .

Appendix B. A GEOMETRIC INTERPRETATION OF FUZZY ADDITION

to orde2r to determine the sum of P, and P2, we will use a geometric interpretationto valate(8.2). On an xy-coordinate system, graph pp, (x); for the membership

function PP2 (y), graph the independent variable along the y-axis as shown in Figure 5.For a given t, consider the line x + y = t; the values that satisfy the equation of this '

line will be used to evaluate p(p, + pj (t).

74

7 B

XI I24X

FIGURE 6 MEIMPERSHIP FUNCTIONS FOR FUZZY P1 AND1 P2

If t < x, + yj (Figure 6(a)), then either pp, (x) =0 or p 2 y)=0; this implies

min (P) W /jP (y)] 0. (B. 1)

From (9.2), p~1,, + P (t 0. By symmetry, an analogous result holds for t > x5 + vs.

Now consider t in the interval X2 + Y2 < t < X4 + Y4. Then the line x + y = tpasses through the rectangle determined by the line segments x') x4 and Y2 y4 (Figure6(b)). At every point (X,y) in the rectangle, and in particular, at every point on the linex + y = t within the rectangle, the value of both membership functions is one;

therefore,1(,P() for X2 +Y 2 < t < X4 +y 4 . (B.2)

Finally, consider t in the interval x, + yj < t < X2 + Y2. On the line x + y=t(Figure 6(c)), as x takes on values zero to X2, PP,(xW is nondecreasing from zero to one:simultaneously, the values of pp,(y), y =t - x, are nonincreasing. Thus,

min Wx, P,2 (Y)]=g~(x,13)

until 11I~ (x) ppa (y), then

mm ) p /sa(y)l = I (y)(.)

Therefore,

P(p. + Pj Mt = pp, (x () (B3. 5)

where x* (t) satisfies the equation

P" (x*) = p, (t - x'). (8.6)

A similar argument holds for X4 + y4 :5 t < X5 + YIS.

Suppose f (x) anu' g (y) have the form f (x) = a~x + b, and g (Y) 8 2Y + bq.Then the x* that satisfies f (x*) =g (t - x*) is given by

6= t + (D.7)

Thus, the sum of two fuzzy numbers from the class (8.4-5) maintains the same analyticstructure; that is, line segmecnts are invariant. However, the support is now contained in10,2]. This argument extends naturally to the sum of n fuzzy numbers.

LIZr123 Z4 Z§

x*Yot X14I

FIGURE 6(a) CASE) I<Z + Yj

%, . *.. 9 ~ Wj ~ CTr. ~74 - - -- -. -* - -- r .- -

t.

< Z24IF

*Z 3 X2 X 4 .

FIGUR E (c) CASES 3 Z + Pa< * 2 2+

31

p ~ ~ ~ ~ ~ ~ ~ ~ ~ A N * .* ~ pp ~ * ~ ~ p*

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This Laboratory undertakes a continuing effort to improve the quality of thereports it publishes. Your comments/answers to the items/questions below willaid us in our efforts.

1. BRL Report Number Date of Report______

2. Date Report Received_____

3. Does this report satisfy a need? (Comment on purpose, related project, orother area of interest for which the report will be used.)

4. How specifically, is the report being used? (Information source, designdata, procedure, source of ideas, etc.)_ _ _ _ _ _

5. Has the information in this report led to any quantitative savings as faras man-hours or dollars saved, operating costs avoided or efficiencies achieved,etc? If so, please elaborate.___ ___

6. General Comments. What do you think should be changed to improve future

reports? (Indicate changes to organization, technical content, format, etc.)

Name j.

CURRENT Organization

ADDRESS Address

City, State, Zip

7. If indicating a Change of Address or Address Correction, please provide theNew or Correct Address in Block 6 above and the Old or Incorrect address below.

Name

OLD OrganizationADDRESS

Address

City, State, Zip

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