REPORT AFITXTR-EN-88-OV2L_ . I ELECTESOCT 1 11989

NMATHEMATICAL MODELING OF COMBAT ENGAGEMENTS

BY HETEROGENEOUS FORCES

I-

N

J. S. PRZEMIENIECKI

Approved for public release;

distributioln unlimited.

Sept emtber 1988

AIR FORCE INSTITUTE OF TECHNOLOGY

WRIGHT-PATTERSON AIR FORCE BASE, OHIO

REPORT AFIT/TR-EN-88-OV2

MATHEMATICAL MODELING OF COMBAT ENGAGEMENTS

BY HETEROGENEOUS FORCES

3. S. PRZEMIENIECKI

Accesi~o,-"O... ..

NTIS CRA-ti

TIC TAB

Unannou,,, t j

Approved for public release; By-

distribution unlimited. Dsrbt~

Av3 ,uiy C (J eS

I J, '4' ,or

September 1986 0

AIR FORCE INSTITUTE OF TECHNOLOGY

WRIGHT-PATTERSON AIR FORCE BASE, OHIO

REPORT AFIT/TR-EN-88-0V2

MATHEMATICAL MODELIN6 OF C:OMBAT UNGAOLENISB3Y HETEROGiENEOUS FORCES

J. S. Pr~pmjeniecki

'ii- Fo:-ce Institute of Techirrcioo

Wright-Patterscn Air Force Base, Dhio

i BGTRZ{ T

The mathematical modeling of c,,inbat engagements by heterconeous tactical

forres is ciscussed and a new superiarit;, parameter is iitrodLcLEd WhiCh

represents i measure of the effectiveness of the oppoDsinq forcFes. This

parameter is A fiinction of both the ouantitative and qzoalitati\.e 5-trenuths ot

the opposirio forces. It is defined as a produiCt of the- transpose of tthe left

dominant eigervi.,ector of the (go'erolog TMatrji,' of attrItion!- coeffFirientSar the

force Strength vector. This parameter iT- used to establish superiorit,\

c iteria for combat enqagements. Thf- applI Ica3ti1on aof th IS new concept to

heterogeneous forces IS illuIstra3ted for the clme Of a done-on-twor' tactical

The L anichester E-quations ti) 1.1e heI CT (qeoeou" tolicl -n~acia 11 a i Fct

fire combat *c n hp rrsetdSymbol icAl I h

X (MN

a n J s th(e flit I,- of at r It iu ' C cO f t I., t~ Me N o'- L)IL-

matrices denoting the force (weapon) numbers of the Blue and Ped fcr e

respectively. The braces are used for convtroienr,? throughout t , is report t*-

denote column matrices. As discussed I Taylor , se,,-ral autrS r1 ha~e

intrzduced aggregated force parameters to obtain a better insioht into the

effects of varying force size, the weaoon allocation, and the keaucr effecti1..e

firinq rate represented by Eq. (I". This aocroa(h in,,nlpr1 Ps prtiall.

computation of two quantities: the agoreoated force strenths F and F f.- R

the Blue and Red forces, respectively, which may be defined as

MB

F w NR

where v and w are positive vectors representing relativ' .0, of i ndi iduai

weapons while M and N are column matrices representing the force eeapor,'

numbers for the Blue and Red forces, respectivelx. Starting with the

assumptions (3) and (4) , a number of authors used the eigenvector method of

computira v and w (see Section 3). in this method, howe.e , both ectors

could only be determineO up to a constant multiple. . simple sralino

relationship war- oroposed bi Ea-. atd James' as -v -- I. (thei

c lausible scaIlino methods were introduced by different authors. For example,I

Taylor in his hO)ok or Lachester mdels of w-jarfa-e, ment iored th, e oee {thE;3

schemes for .om L ip jt , the scal r fac tos f- the wAapon lu cc to, sV

and w.

This report introduces a new wa, cf detlO min ncl th(-, ellervect, s V -,J w

through a mathematial reoesentatio.7 of the c'cQIlet' anal 't -Ici nuintlrn' anld

the aopi iatior (f tp (ic,i :,an;t 1ef eio .E( tor tI C ts ) s tr.

T

approach leads also to a new measure of Effectiveness 1 X o, its1 0

corresponding nondimensional superiority parameter S , both of which determine

p Dr c,: the direction of the outcome of the battle enolagemeint withw ut the

necessity of solving the differential equations representirg the engagement.

The application of this nevi method cf analysis is demonstrated here for the

case of a tactical engagement i nvolving three different weapons: one Flue and

two Red weapons. In order to illustrate the nature of the analytical solutior,

and ,in particular, the oresence of a dominant term in the solution Nlhich is

used to derive the superiority parameter S , a complete discussion of the

general analytical solutior is included .n the Appendix.

F. Larchester Equations for Heteroqeneous Force Combat: Many-on-Many"

Combat Enoaqements.

The two separate governing equations for the heterogeneous force combat

engagements, from Eq.(l), are given by

dM/dt - B5)

dN'dt - AM

where M and N are the matr ices reoresentin-g tre 01 oe And Ped forces ,cl

weapons), respectively. The mathematical model for such eigagemei1ts is

represented as the state variables and N interactinq with each other th,-ou-7h

tre apprcpriate attrition coefficients n and B Of A a- FB e shR,- iP Fi .

I Here the attrition (:ceffic ient ta F into account the I i, tir g r-te

effectiveness and the a llocation- o- c,- weaporns. Thc mac ices A 3 nd B -,e

nonnegative, a proDe, t v whic h wil be tsed in the stb requer,.t auai.s. 1

3

Mi N 0 ~f l

and when this, condi tion is voot satjis~jed, i e. oih en o~ of tI-e f o r c F s

anrihilatE:, %'-whe a z)--tjrcjir M cc N EqsG (5, i-id mut :'

recorf igured by striking oUt the appropriate rows a,-d columns.

Equations (5) and 65) can be corbined iiitu a =,iQle matrix squat ic:.

d)X/dt =X = CX1

where

X ( M N -2

and

If the i> row in contains all zeros th is mr-4ns that the co-rr e5Don-di~

and the X -fo-rce is not beiro fired at. If the jhcolum-n in C conitains all

zeros thEn the X -for-ce is beinq inactive.

The left handed eicenvector q n-f the matrix C aind it , nrresflon-dino

eiqenvalue '. are determrined from

The eioenveitor q car be pa-ti ticned into rowis corrFespcnd] rq, to t~e F 1lne a-c

Red forces, respectively, s o that

q Cc x'

where v and w are the column vectors .niie c and c are corstaot multipliers.1 2

It should be observed that Eq. (1) must represent a set of I -. _ -

equations. ,.oipVe' sets are not admissible in the present analysis because

they represent unrelated (uncoupled, tactical engagements. Examples of

uncoupled and coupled engagemerts are shown in Fig. 2. where arrows indicate

the direction of fire. For these examples, the corresponding matrices of

-trition coefficients are shown by Eqs. (la) and (l1b) where x's indicate

the nonzero coefficients in C.

U -0 -0

-x 0 0 -x ,uncoupled sets ila)0 x 0 0 0

" 0 U - ":0 0 (I -xi

C - 0 0 .'' oupled sets (1b)X 0o 0

The test whether the matrix C represents uncoupled sets of equations is simply

ohether it can be reduced to two or more nonzero diagonal submatrices by

rearranging rows and columns in C, while all other off-diagonal submatrices

are zero. Qtherwise, the matrix C represents coupled sets of equations.

Substituting Eqs. (9) and (10) into Eq. f9),

I T[] [ ]_pT 0 K;: w czw

from which

A W Ac V .132 1

- H ,: v = y*; w

1 2

D

BT

Premultiplyigq Eq.(13) by B and ther substatutinq Eq.(14 ) and pie TltipIi i-,o

Eq.(14) by Ar and then substituting Eq.(i3), the following pair of eQuation,s

is obtained

T T 2A - I c V : 0 15I

R TAT - 2 )c W 0 162

If the dimension, 6' A are nxm and those of B are mxn and if >-, then t h there

2will be n common eiqenvalues .- = Li in Eqs. (15) and (16). In addition, there

will be (m-n) eigenvalues of Eq. (15) equal to zero for which their

corresponding eigenvectors can be obtained from

ATBT O " =7I

If those eigenvaiues were not equal to zero, this would then imply that in

addition to the 2n eiqenvalues k, -/ ., in C there would be additional 2( -- n r

eigenvalues for a total Rn+2(m-n) = 2m which would be greater than the

required nuomber (m+n).

The matrix products A B and B A in Eqs. (15) and (I) are nonnegative

since all A and B are either positive or zero. Therefore, according to the

Frobenius-Perron theorem (Ref. 7, p. 193.1, Eqs. (15) and (16) ha,.e a real

2

nonnogative eiqenvalue 12 which is either equal to or exceeds the moduli of

an other eigenvalue. Also to this maximal eigenvalue there co -responds an

eigenvector with all nornegative elements, which will be denoted as c v and

c w fo)r Eqls. i15) and (16), respective I,. Since c , 3 and c 2 W e2'1 1 1 2 1

submatrices of the left Pi oenvector of C it. fnllows fr,-,,T, Eo. 1C) ,ha t

q (c v cw1 1

which can be determined directly frot,, Eq. (9). Whe- ct-!uuting v and w IfcmI I

Eqs. (15) and (16) any suitable scaling method can be used. For example the

largest element can be made equal to unity, but the actUal scaling between v

and w is accomplished through the fartors c and c to be determined late.1 1 2

Transposing Eqs. (13) and (14 with v v I and w = w1 1

wTA T- cwA ?.c l9V-I c2 1 1 1 1

c v TB - w 201 1 1 2 1

Next, premultiplying the above equations by

e = I I ... I) ( )21

and eliminating the following relationship bett.ee- I and c is obtained:

" T

Sw TAc- wT$-S- T1 ,I

I I 2

SI wr TAe wTeJ

- I I

The negative sign in Eq. (23) is selected to satisfy Eqs. (19) and (2f-) in

Ahich A. P, v , and w are dil ronnegative matrices and I Is a :os tiI..(

rumbcr. Coincidentlv, Eq. (23) i of the same form suQoested I. T7. o

without Dnoof for the so-called "summed result" in interpretir g fhe acqreqate

fore strernqths. F LLatior, (23 provides a rational eatirh'j t,et .eF, v

7

and w wher these eiqenvectors are calculated fro- Eqn (' qA : .I

It should be noted that i, practice the jeft dpnjmi,.f ejG,- 5. - a '

its corresponding eigen.ector q I ca. Qe 'onta-,d ,irec',1 fr - -.. . atle

than from Eqs. (15), 16), and '29). Computer programs ich ns WATL- a-e

available which car be conenientlv ,sed to find thF dc. iao.-t ece .a .

and the domirart left peioepecto, q1"

Because of the relationship CA.281 , the qeneral ied left ei~er,.e tW,

are orthogonal to the right eigenvectcrs p. ,.is . thcu:,nal it w : t.

e'oressed as

tj 1 = 1 fol -

0 for i X J,

This property is used next to derive a scalar parameter from the .clution of

the governing equations by premultiplvirng X bi the transpose oy tne dcnl,.a ,t

left eigenvector q ,corresponding to the largest eiqenv,,.. I . K,-

resulting parameter is a measure of merit for the combat enqaQemFrt.

ConseQuently, employing Eqs. (A.25) dId (A.14b) and noting that fo, the

notation used ir APpendix A 1 =" and qJ11

[ _ r xp( x t YD 1 1T

[ ' I p x . . F e x | t /i t T X

B |

0

1 T ~T

T T

L O tj I L)

£ utsti +j t rQ rovi Ea. 18 for- q inito Eq. E', and ob, eI-\1 iiathat X ~

It t, l f c II t ha3t

F X I [c V'E cwT10 1 1 2 1

fVT M . wTN 1

2' 1

V TBe~ v T~ e1 2

c [_M w TNI ~t T T wN

w Ac. w -I I

,,j *rrp k~ :-.n. EC-r-C e C be a(:r c ne. 8k,-r Ausi& V 0ri w C) e ot h

I ) I i

1, 2nor7 ;end ie The tom~ - d en r N o, a bo Itiie h maa u

t e, ir he 4- om t e Iniedom1n tn-n te n I LI te c (11'.efeeltm h

eigenvalue, or if other eigenvalues are close to the dominant eioerva)je, rI!E

use of qTX as a measure of merit mCAv not be appropriate. rnse e t .0

important to check the distibut'on of ei e!-. .alues befoe e i.i, t i

measure of merit. In most practical cases, however, the I's are

separated. For example, for equal coefficients in both A a,-id B i e. A

and B =b for all ard j where a and b are constantsi the eicev..,es -,

t I

) ,0,0 .... ,0, ,- . m...and q..£

-- /mb/na.. .r,-th-,.;) where m and n refer to the number of B1je and Red tC_&es.

respectivelv (see Appendi ). It is inter esting to obsei vn that fc, e

multiple eigenvalue cases, the dominant eigenvector 1-1 contains zero elements

which means that some force strengths are not included in q X

To reducE Eq. (26) into a nondimensional parameter, the ecuation car be

normalizei with respect to c v TM by introducing a nondimens)onal soecicl,I 1 1 0

pa.- ameter S such that

S q qTx 'C v T M

T T WI T

and =:~ i~ WC ~ zw

1 01 1 1It)

T Tc vMwAe w v M

10r T Tv fie v 0 - w N

1 1 01 1 1 El)

The nndimern5icnal parameters S, S 1 , a n d T' ire all icndenendient h t Ctn

I)= 0 '

of t hP arhi trarv m tiIt I, IV, no factors witt v Ind w . he :-j;:nI I

parameter S is a scalar quantity measuring the contributions of t;-e domi,,at

ter-m in the solution for the force levels at any given time. It should also

be noted that

0 0

S =0for- 1 = 1 ,3n

o 0

The parameters X or S can be used as measures of effectiveness to1 0 0

dete-rmir e the outcome of a Qarticular tactical engaqement vithcut actually

sclvino the differential equations subject to the restrictive conditions

discussed earlier for the measure of merit q X Thus as in the case of -3

"one-un-one" engagement the Blues forces wir the engagement wher 1' 1 or

h rS w. ' , hiie the Red forces wi-, -.ihen . ' I or wen 5 : O. When ' t I0) 0 0 0

or S 0, both sides reduce their strengths proportionately.

it should be noted that as in the case of a one-or,-one' engagement the

norma ized force ratio U for the general case consists of a product of t.uC

component= -epresent i ng the qualitative and quantitative ratios. The• '' T 1' 2

Qualitative ratio is C )(AeL- we)k cTp-v/( V , II- and the quantitati.e rat 1cI I I I

i s iVT M 3,w TN1 0 I

-. 4qgo-eqa ed Forcu- Stiejt~th

T: ,-edicj he mat,,emat ical rem en ,tat ia- of t.' ii , a

ePrgaemer t t:. :i " ne--on-c ?" representat ion the ag', eqated foce .tt enc t t

r,ra, be ntr dr .d ioto Eq .. (5) arid . Premo 1 t p I,n. Eq ( q h, c v ac-+

i 1

I.Eq. 5~ ~t, c w the flir. ng diffeet ml equation are &,tai --,d:2 1

ii

cvT vTBN Pc wNTN (31J 11 2 1

11 21

c w N = C w AM :: c v 322 1 2 1 1 1

where the overdots represent differentiation with respect to time and the

right sides of these equations, with positive constants .: and ' rae at th-is

point been introduced arbitrarily. Introducing the aggregated force strengths

F and F from Eqs. (3) and (4) asB Tt

F H = v H 33)

F = TN = -c WTN 34)R 2 1

where

v = cv (35)

w = -c wN2 1

Hence, substituting Eq. (33) and (34) into Eqs. (31) and (32)

F -,: F 1:37)U R

F (39)R B

whict. are the governing equations for a "one-on-one" engagement between the

acgi-eqated forces F and F ; hovie,.e, a ord of caution is appr-opriate here.B R

The solutions, for F and F are or, l appro imations to the e act solutionsB R

because they d- not incorporate the condition that one of the M o N can Le

reduced to zerc indeppndently of others. I n the aogreoated soltion of Eqs.

(37) ind (2a) all components of F or F ae uduced to :,- n sinultaneously.

5ince M and N are ai t,tra, in ,n nude for the I !oht oi lps of Fq. 31

1L

and (32) to be valid the following relations must be true:

TT _T 3

-C v TB = (?c W or -8 C v =C w 3)1 1 2 1 1 2 1

-c w A .-*c v or -A CW Vc (40)2 1 2 2 1i

Solving fcr v and w ,I

A T B T c = -('3A = .-c v11 2 1 1

ATBT - cf?1)C v" = 0 4l)11I

TT T

BTA c w -. _?BT c v .-c.,3c W2(1 1 1 2 1

(BTAT - 4I)c w = 42

2T1

Hence, comparing Eqs. (41) and (42) with Eqs. (15) and (16)

= (43)

Similarly, by comparing Eqs. (39) and (20) and Eqs. (40) and (19)

-,= C-' = .(q4)144

which is the same result derived from the scaling method assumed by Holter5

and Anderson. 6

The above analysis has demonstrated that a unified theory for the

analysis of Lanchester equations for heterogeneous forres enQaged in a direct

fire combat can be developed. This is accomplished by correlating the

quantities v w ' and C , used previousl, by other authors, to the left

13

dominant eigenvector of the governing matrix of attrition coefficients and its

dominant eigenvalue. The theory provides for a proper scaling of the

eigenvectors v and w and leads directly to the measure of effectiveness qT X1 I I L

or the superiority parameter S . Although the use of the left dominant0

eigenvector introduced here for the combat analysis is new, similar approach

has been used in the past in other areas. For example the use of the left

dominant eigenvector in the cohort population model that leads to the total

reproductive value of the population as described by Luenberger (Ref.7,

p. 185-).

4. Numerical Example: "One-On-Two' Tactical Erqement

The proposed method of analysis of heterogeneous forces combat is

illustrated here for the case of a tactical engagement of one Blue force and

two Red forces as shown in Fig. 3. For this example the equations describing

the engagement have been assumed as

XI 0 -3/2 -- I FX -4/3 0 0 X

X -2 0 o L X

It can be verified easily that for this case

X = (X X × ) = M N N )1 2 3 1 1 2

q = ( 1 -34 --1/2)

T "I 1

1 I

4and = -

o 3 (X +(2/3)X2 3

In Figs. 4, 5 and 6 the variation of X X and X wIt1- the1 2 3

nondimensional time - is shown for X (20C, 1I0 i00}, (125 100 100}, arJ0

(100 100 100) which correspond to I = 1.6, 1.0, and 0.8, respectively. 1heU

solutions for X were obtained by numerical integration. For Case 1 (Fin.

in which 1 1.6, X = 0 when T = 0.7 and at that time the , lion of X0 3 1

allocated to fight X is reallocated against X , requirirg that the equations

must be reformulated. SUbsequently the X -force is reduced quickly to zero bv2

the X -force. As the battle progresses, the superiority parameters S showsI

rapid increase (see Fig. 7 ). Here the initial positive value of S is an

indication of the superiority of the Blue force. For Case 2 in which I = 1.00

the opposing forces diminish their strengths gradually without any side

gainirg a clear advantage (see Fig. 5). Here S = 0. Case 3 for which 1 -

0.8 represents a rapid demise of the X - force (see Fig. 6). For this case S

starts with a negative value which diminishes as the battle progresses (see

Fig. 7). Tne initial negative value of S is an indication of the inferiority

of the Blue force.

Sirce the dominant left eigenvector is the key parameter the piopused

measure of merit S . the eigenvectors q are shown as examples in Appendix BIiI

for the cases of m-on-n" engagements for which all attrition coefficients

are equal.

5. Conclusions

Thp premultip]ication of the solution ,.actor X k tM N for the ergagino

forces by the transpose of the dominant left eigenvector qI of the matri: of

attrition coefficients helps to clarify the meaning cf the eigen\ector methodc

It

of determining the relative values of heterogeneous forces in a combat

engagement. Although, the eigen~ector methods have been arouTnd fo- a long

time, problems existed in deciding on the best method of scaling the resulting!

eigenvectors. The present report demonstrates clearly the meanirg of the

individual eigenvectors v and w as components of the dominant left1 1

eigenvector of the governing matrix of attrition coefficients. This in turn,

allows for a proper determination of the relative scaling of eigenvectors for

the two opposing forces and it leads to the nondimensional superiority

parameter S which can be used as a measure of merit and an indicator of theo

final outcome of the tattle engagement.

The practical utility of the proposed measure of merit can be

demonstrated by obtaining numerical solutions to heterogeneous force

engagements while at the same time computing the parameter S introduced in

this paper. The sign of the parameter S and the extent to which it is either

increasing or decreasing can be used as a measure of superiority of the Blue

forces in relation to the Red forces. Although the method is valid only for

constant attrition coefficients, it could also be of some value for cases with

variable coefficients where the instantaneous values of S could be used to

indicate the general trends as the battle engagement progresses.

16

APPENDI X

Analytical Solution of Lanchester Equations

a. General Solution of X = CX

The general solution of the fundamental equation

X = CX A1

can be obtained as a product solution of the form

X : Pp

where P is the matrix of p.etea-,L£ &zed right eigenvectors and * is the column

matrix whose elements are function of time. Substituting Eq.(A2) into (Al)

X = P;= cP

Hence-" P-1CPp = 4p (A3)

where (

1) (2)

and J P- CP (Aba)

j a (A6b)

is the Jordan canonical form of the matrix C, modified to take into the

account the dimensionally of the matrix elements. The governing eqtatICO

matrix C, its eigenvalues, ard the matrix J are of the dimension I 'time. Also

for reasons explained later J is taken herp as the lovier trianoular matri -x

instead of the conventional upper triangular matrix.

A typical term in Eq.(A3) is of the foim

p+ = . ., )-

1 2

1-7

1 2 -

where J '" is essentially one of thc three types, or combinations thereof, as

8

discussed oelow.

Tygpe ( t..

P C-. (LI )

when the algebraic and geometric multiplicity of the eigenvalue . is equal

to one.

Iype C L>

when the algebraic and geometric multiplicity of the eigenvalue X<'is equal

to k with k-_2 and the corresponding eigenvectors are linearly independent.

Ty pe L- t ):

1 0 0 0 ... 0 0

0 1 1 0 ... 0 C)1 0

- CI1 C) (A12)

0 0 0 0 ... 1 0

0 0 0 C) . .. 1 I k.:

when the aloebraic multiplicity of Lte eiqenvalue ,' exceeds it; comet1ric

multiplicity. The expression for J't" has been modified he ve ,from its

standard textbook, form in which only the diagonal terms are equal to . to

ensure proper dimensionally of all its elements. The eioenvalues ]n

two different J"'s can, be the same. Also the 7",v va, ap:ear i

combination with Typef C A and _. z When k=I the Ir,- ,eduCes to C ., .i

18

It is easy to verify that the solution to Eq.(A7) w ith J given b,

Eq(AI2) (i.e. Type ,t tt)) can be obtained by solving first for .< and then

substituting it into the equation for <z This process is repeated until

the solution for ? is found. These solutions can be written corcisel\ as

r )t t t)zV exp t) 2 H + .H 2+ (A13a)

21

[ A X \ t)2 zA;0i~ .t H - 12 + J Al

= exp(...t)tI + tH + ' +

exp (k .ut)t) D(L A 1 3c

where

X t t) 2 t)k-i' I+_ H +--- + .. + H ( lLa)

I 2 (k-l)

I (Al4b)

, t 0

k 212 . tI

0. 't ):3/6 0 (1. t)2 2 t.I ).t

and.,f~, ) PPr P> . t)

is the column matrix of constants of inteqration. The H--matrices are given b.

H 0

0 Ij 0 -

etc.

19

which finally leads to

kH = 0 (Als

Substituting Eq.(Al3c) into (A4) and (A2), it follow s that the general

solution of X = CX can be written as

X =RV = P [ exp( .t),' Z A P-4

where

exp(..,t) J . [ exp(\ 't) n ' iD exp(. 't (A2C;

ff = { - < ,2) (A211

At time t=C, X=X and therefore from Eq.(A19)0

X = Pa kA22>0

since exp(O)=! and E-I for t=O. Hence

id:P X = QTX AP3,0 0

where the inatrix Q is defined as

Q (P ) (A24)

Hence from Eqs.(A21) and (A24)

X = P [ exp. t i ] T Xo Q'2

If the matrix C is not defective, i.e. its geometric multiplicity is equal to

the algebraic multiplicity for all eigenvalues " then £D.-I and the

solution for X simplifies to

X =P f exp(o. t) J &X (-26

b. Right and Left Generalized Eiqenvectors

From Eqs.(A6) and (A24) it follows that

C11= P j H2-

Q =I I (APB)

PQ T

20

and QC = P- C

= p I CPP

= 30 T ,A

Equation (A30) implies that Q is the matrix of the ,e:'ercd.L;.J lef.

eigenvectors, while Eqs.(A28) and (A29) indicate orthogonality between the

right and left -e:-er iLUed eigenvectors.

c. Computation of Generalized Right and Left Eigenvectors

Introducing the individual generalized eiqenvectors into P and Q stic

that

1) (2) ( .L

and Q q q ...q ... ] 32)

it can be demonstrated that the application of Eqs.(A27) and (A30) for J' of

the TLpe ,) and order kxk leads to the following relations:

-<I )$ = "p2 or (C-,.I) PIL 0

I'C- .T Pz = >p or (C-. ,I) Pz 0

)2 )P3 CP2 0

(A33)

p p or (C-.I) 2 pk 0

-'l )p 0 or C-..I)p - 0

and 1 T .C-kI) 0 or q (C 0£ 1

T T T 22(C-)'i = ' or q zC-\I> = 0

q(C-),I )q or (:C 0) CA34

q Tq q T 0 . -

T TT.q C- qI ( C 0:.I l

where for simplicity all superscript', i havP been omilttcd and the subh: t pts

21

I through k refer to the eigenvectors compriirn the calumns i-, p ar qI

Since the analysis of the relative strenQths in combat engagements ii,,ol ,--

the calculation of the dominant left eiqenvector q it is clea( t ,ot te

selection of J as the lower triangular matrix was mo:e :onve. iei t hec ause ,

the case of defective matrice, C it resulteo in a simple p-pessin,

T C-'.I = 01

T I

or ((- ,.1 )q, 0

for the calculation of q1.

d. Examples of the Dominant Left Eioen,ect!,I

The dominant left igerivectors are i lustrated he'e fcr 5Ee, p1 ,

engagements fo)r which all attrition coefficients are eQual 'o a corstart

The number of engaging types of weapons is indicated b., the ;ut,s cts rM,.

with each matrix C. All eigenvalues of C are shown, with the drT, ir, ant

eigcnvalue(s) listed first. The dominant eigervector q Ihas bee, t, al Ze i

on ts largest element for the BIue forces.

-r c, : ,",. = -,, -/ab; - -=- I/ha

, ,, -L ]-- b. -t / Vt, -

0 -b,.-,1 -a (0

C 0 ...- bC, . -bo

. .q - t A1 , 1 1

22

C F c) K) -b -b] . V aF. O, O, -2babC) 0 -b -bI-a~ ='q ( 1 1 -lba -Vb/a

- 2,2',

C 0--b : . 6a-, , &, -b-, -VlabC' -b - b

-a -a -a -

-b - -L '

' -b -b -b1' , 0-b -b-b = ( 1 1 -/b/a -/,'a

-a --a -~ C' 0 ' C

-a -a -a ' C' 0-a - a -a C' C C

For tr e -era1 case c-f m Blue forces or n Red forces Im-on- with eq,,

coefficierts it, A ard B

1 ' 11 V nniab

| -- f i ..... "n... Tt?'a Vm'b ' n-ta . t...*

The 4cllowing two examoie v, ith some 2ero attrition Ceffi-icie -

represe.nt cases of defective matrl-es 7 with repeated eeaies.

b C C -b -b' .. - /,h -Vab, - Vab0 0 0 -bt

-a ) ' C, a C I -V/b'aI -a(23

C' -a C' ' 2.2.

23

= 0 o-h --b-bh " " = Y 'ab, , 0, --Vab, -a"bS0 0 0 () 0 -b = 0 ,, 0 - -,a

D 0 0 0 0-b 0 1 10 0 :

-a 0 0 () 0 0

o0 -a C)0 0 0

e. Computation of the Dominant Left Eigenvectors and tWier Eie-,al es

Ir addition to the dc.minant eigenvalue ' the matri' CT contains also a-I

eiqenvalue equal to -1. . Eoncequetly, because there are toc eigerale _ withI

Pqual absolute ,,alues the standard power method of computinq the larqest

eigenvalue and its corresponding eigenector can not be appliedz hoveopr, the

power method with a shift of origin car be used to f2nd both , aed 'i

Startirq with Eq. C.35) fo r

T

(C -, P=0.I

and adding .qI to both sides, where -: is a constant ,.4..

Henc e,

when e

1

In this way the shift of the origin for thp eigen,,alups a-qd - n a,

I

resulted in 1 1 " 1 being greater than, -)' "'V I; therpff . , t',p n."a-,i, d :,:,.,e

method can be applied to Eq. (o. , to find the" c nd tht f , i-.,rsiti O +: en rae ha '*: tee,}'-*V~~

References

ITaylor, J. G., Lwcahes te f:de! .ct h u:'fcae, Vol. II. Operations Reseacch

Society of America, Arlington, VA, March 1983, p.567.

2 Dare, D. P. and James, B. A. P., -The Derivation of Some Parameters for a

Corps/Division Model from a Battle Group Model," Defense Operational Analysis

Establishment, West Byfleet, UK, Rept. M7120, July 1971.

3Spudich, J., "The Relative Kill Productivity Exchange Ratio Technique,

Booze-Allen Applied Research, Inc., Combined Arms Research Office, Fort

Leavenworth, KS, also in Table E of Appendix II to Anne> L, "Cost-

Effectiveness Evaluation to Tank, Anti-Tank Assault Weapons Requirements

Study, Phase III (TATAWS I)," U.S. Army Combat Development Command, Fort

Belvoir, YA, AD 500 635, December 1968.

4Howes, D. R. and Thrall, R. M., "Theory of Linear Weights for Heterogeneous

Combat Forces," Thrall and Associates Rept. to U.S. Army Strategy and Tactics

Analysis Group (STAG), Houston, TX, AD 759 279, May 1972, pp. 21-47, also in

Naval Research Logistics Quarterly. Vol. &0, No.4, December 1973, 645-659.

tW. H. Holter, "A Method for Determining Individual and Combined

Effectivenesn, Measures Utilizing the Results of a High-Resolution Combat

Simulation Node!," Proceedings of the Twelfth Annual U.S. Armi Opeiations

Pesearch Symposium, 1973, pp. 18-196.

& onderson, L. B., "Antipotential Potential.' Institute fo, Defense

Analvses, Arlington, VA, Rept. N-845, April 1979.

25

Luenberger, D. 1., 1ntrcdluc torn [c'-..Cm LC St.,, Wiley,New YoI-k, Ir.l

pp. 83-84, 185, 193.

aYoung, D. M. and Gregory, R. T., A .rueyn Numer.c."_ fr.t.en;-c .:,Voi. M t,

Addison-Wesley , Reading, MA, 1973, pp. 7?5-752.

Young, D. M. and Gregory, R. T., A _iue o t NuerK.- ,rt/les c-. ,ol.

II, Addison-Wesley , Reading, MA, 1973, pp. 915-918.

FIGURES

Fig. 1 Model of heterogeneous force engagements (Many-on-manv).

Fig. 2 Example of uncoupled and coupled tactical engagements.

Fig. 3 Numerical example: one-on-two engagement.

;71q. 4 ,U,,,e ' .L- eAcffO)ip (C0se I', Variation of force strengths X , . X

with the nondimensional time 7 = ,.'t for ,1 = 1.b and X { 200 100 100 ,.C )

Fig. 5 Numerical example Case 2): Variation of force strenqths X, , X ,

with the non'dimensional time r =.t for '1 = 1.0 and X ( 125 100 100

Fig. 6 Numerical example (Case 3): Variation of force strengths Y , X , X 3

with the nondimensional time ; t for '1 0.8 and X { 10 100 1.70

Fig. 7 Numerical examples (CaseF, 1, 2, and 3): Variation of the superirit',

parameter S with the nondimensional time for ,1 C 1.o. 1.0, and

o.S.

26

&f)

C)

IiI0

0

77--

<4

Lil)

<227

BLUE RED

a) Uncoupled engagements b) Coupled engagements

Fig. 2 Examples of uncoupled and coupled tactical engagements.

28

A

A1

Fig. 3 Numerical example: one-on-two engagement.

29

ea

-) II

0

E

C)cC)zSHJLD9'dlS9DEO-

30z

00

* 0

- 0

0

LIPZ 0

IIW

6 - 0-

-~ 0

z zo

CDE

CA~

SHIDNHIS 0HO-

31-

100

z

so-

00 0.2 0.4 0.6 0.8 1.0 1.2

NONDIMENSIONAL TIME T=X)

Fig. 6 Numerical example (Case 3): variation of the force strengths X , X ,and Xwith the nondimensiona1 time tr=>)t for C=0.8 and Y,,=(100 100 100).

32

Cl,

(D

0

00)

< 'II

0

0o (A

zz

O z6 E

E E

60

S ~I3M'VlIVd AIIH1OI gdnjS

33

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11. TITLE (Include Security Classification)MATHEMATICAL MODELING OF COMBAT ENGAGEMENTS BY HETEROGENEOUS FORCES

12. PERSONAL AUTHOR(S)PRZEMIENIECKI, J. S.

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FIELD GROUP SUB-GROUP Military modelingLanchester Models of Combat

19. ABSTRACT (Continue on reverse if necessary and identify by block number)The mathematical modeling of combat engagements by heterogeneous tactical forces is

discussed and a new method of deriving aggregate force strengths is proposed. The analysis

leads naturally to the concept of the superiority parameter which represents a measure of

the effectiveness of the opposing forces. This parameter is a function of both the

quantitative and qualitative strengths of the opposing forces. Both the aggregate force

strengths and the superiority parameter are obtained through the application of the left

dominant eigenvector of the governing matrix of the attrition coefficients. The appli-

cation of these new concepts to heterogeneous forces is illustrated for the case of a

'one-on-two" tactical engagement.

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