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ncHwcAt mpon stei NAVAL FOCrGHACUAK 1 NPS55-77-42 NAVAL POSTGRADUATE SCHOOL Monterey, California A SHORT TABLE OF LANCHESTER-CLIFFORD-SCHLAFLI FUNCTIONS by James G. Taylor and Gerald G. Brown October 1977 FEDDOCS D 208.14/2:NPS-55-77-42

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Page 1: NPS55-77-42 NAVAL POSTGRADUATESCHOOL · 2016-06-24 · Durham,NorthCarolinawithR&DProjectNo.1L161102BH57-05Math(fundedunder MIPR No. ARO 22-77)and partially bythe Office of NavalResearch

ncHwcAt mpon steiNAVAL FOCrGHACUAK1

NPS55-77-42

NAVAL POSTGRADUATE SCHOOL

Monterey, California

A SHORT TABLE OF

LANCHESTER-CLIFFORD-SCHLAFLI FUNCTIONS

by

James G. Taylor

and

Gerald G. Brown

October 1977

FEDDOCSD 208.14/2:NPS-55-77-42

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NAVAL POSTGRADUATE SCHOOLMonterey, California

Rear Admiral Isham Linder Jack R. BorstingSuperintendent Provost

This research has been partially supported by the U.S. Army Research Office,Durham, North Carolina with R&D Project No. 1L161102BH57-05 Math (funded underMIPR No. ARO 22-77) and partially by the Office of Naval Research.

Reproduction of all or part of this report is authorized.

This report was prepared by:

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SECURITY CLASSIFICATION OF THIS PAGE (When Data Entered)

REPORT DOCUMENTATION PAGE1. REPORT NUMBER

NPS55-77-42

2. GOVT ACCESSION NO.

READ INSTRUCTIONSBEFORE COMPLETING FORM

3. RECIPIENT'S CATALOG NUMBER

4 TITLE (and Subtitle)

A Short Table of Lanchester-Clif ford-Schlaf liFunctions

5 TYPE OF REPORT ft PERIOD COVERED

Technical

6. PERFORMING ORG. REPORT NUMBER

7. AUTHOR^.)

James G. Taylor and Gerald G. Brown

6. CONTRACT OR GRANT NUMBERfs;

9. PERFORMING ORGANIZATION NAME AND ADDRESS

Naval Postgraduate SchoolMonterey, Ca. 93940

10. PROGRAM ELEMENT. PROJECT, TASKAREA ft WORK UNIT NUMBERS

11. CONTROLLING OFFICE NAME AND ADDRESS

U.S. Army Research OfficeDurham, North Carolina

12. REPORT DATE

October 197 7

13. NUMBER OF PAGES

63U. MONITORING AGENCY NAME ft ADDRESSf// different from Controlling Office) 15. SECURITY CLASS, (of thle report)

Unclassified15a. DECLASSIFI CATION/ DOWN GRADING

SCHEDULE

16. DISTRIBUTION ST ATEMEN T (ol this Report)

Approved for public release; distribution unlimited

17. DISTRIBUTION STATEMENT (of the abstract entered In Block 20, If different from Reoort)

18. SUPPLEMENTARY NOTES

19. KEY WORDS (Continue on reverse aide if neceaaary and Identify by block number)

Lanchester Theory of Combat Special FunctionsCombat Modelling Deterministic Combat AttritionAttrition Modelling Lanchester-Clif ford-Schlaf li FunctionsCombat Dynamics

20. ABSTRACT (Continue on reverse aide It neceaaary and Identity by block number)

This report contains a reduced set of tables of Lanchester-Clif ford-Schlaf li(LCS) functions. A companion report contains a more extensive (and currentlythe most extensive available) set of tables of the LCS functions. Thesefunctions may be used to analyze Lanchester-type combat between two homo-geneous forces modelled by power attrition-rate coefficients with "no effect.'Theoretical background for the LCS functions is given, as well as a narrativedescription of the physical circumstances under which the associated

DD FORM1 JAN 73 1473 EDITION OF 1 NOV 65 IS OBSOLETE

S/N 0102-014- 6601 |

SECURITY CLASSIFICATION OF THIS PAGE flfhen Data Bntarad)

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.Lt-URITY CLASSIFICATION OF THIS PAGEfWiwi Data Entered)

20. Cont

Lanchester-type combat model may be expected to be applicable. Numericalexamples are given to illustrate the use of the LCS functions for

analyzing "aimed-fire" combat modelled by the power attrition-ratecoefficients with "no offset." Our results and these tabulations allowone to study this particular variable-coefficient combat model almost as

easily and thoroughly as Lanchester's classic constant-coefficient model.

SECURITY CLASSIFICATION OF THIS PAGEfWhen Dmtm Entered)

Page 5: NPS55-77-42 NAVAL POSTGRADUATESCHOOL · 2016-06-24 · Durham,NorthCarolinawithR&DProjectNo.1L161102BH57-05Math(fundedunder MIPR No. ARO 22-77)and partially bythe Office of NavalResearch

A SHORT TABLE OF LANCHESTER-CLI FFORD-SCHLAFLI FUNCTIONS

by

James G. TaylorDepartment of Operations Research

and

Gerald G. BrownDepartments of Operations Research

and Computer Science

Naval Postgraduate SchoolMonterey, California

This research was partially supported by the U. S. Army Research Office,Durham, R&D Project No. 1L161102BH57-05 Math (funded under MirR No.

ARO 22-77) and partially by the Office of Naval Research.

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TABLE OF CONTENTS

PAGE

1. Introduction 1

2. Variable-Coefficient Lanchester-Type Equations of

Modern Warfare 2

3. Combat Modelled with Power Attrition-Rate Coefficients 5

4. Lanchester-Clifford-Schlafli (LCS) Functions 12

5. Use of LCS Functions for Analyzing Combat 18

6. Tabulation of LCS Functions 21

7

.

Numerical Examples 22

8 Final Remarks 32

References 35

Appendix: Tabulation of LCS Functions F (x), H, (x) , anda 1-a

T (x) for 11 Fractional Values of a "if,a

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1. Introduction

Lanchester-type* differential-equation combat models are an important

tool for analyzing many important problems of military operations research.

In such a combat model, a so-called attrition-rate coefficient represents

the fire effectiveness of a particular weapon-system type against a particular

target type, i.e. the weapon-system type's effective firepower against such

a target. Time-dependent attrition-rate coefficients are used to model

temporal variations in firepower on the battlefield. Thus, we see that

time-dependent attrition-rate coefficients are important (and, in fact,

essential [4-6]) for the quantitative analysis of hypouhetical combat.

Militarily realistic computer-based Lanchester-type models of quite

complex military systems have been developed for almost the entire spectrum

of combat operations, from combat between battalion-sized units to theater-

level operations. Nevertheless, a simple combat model may yield a clearer

understanding of significant interrelationships that are difficult to per-

ceive in a more complex model, and such insights can subsequently provide

valuable guidance for more detailed computerized investigations. In this

report we consider such a simplified variable-coefficient Lanchester-type

model of combat between two homogeneous forces.

For this variable-coefficient Lanchester-type model of combat

between two homogeneous forces, different functional forms for the attrition-

rate coefficients lead to different mathematical functions being involved

in representing and computing the force-level trajectories. In a previous

paper [5] we have discussed the plausibility of the hypothesis that except

for the special case of a constant ratio of attrition-rate coefficients,

*So-called after pioneering work of F. W. Lanchester [3]

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the solutions to such differential equations cannot be represented in term

of "elementary" functions of analysis. Thus, new transcendental functions

arise in the study of combat modelled with time-dependent attrition-rate

coefficients. In particular, we have previously introduced [5-6] so-called

Lanchester-Clif ford-Schlaf li (LCS) functions for analyzing combat modelled

with power attrition-rate coefficients with "no offset" (see Section 3 below).

In the Appendix to this report is contained a reduced set of tables

for the LCS functions: it contains tables of five-decimal-place values of

the hyperbolic-like LCS functions F (x) , tL (x) , and T (x) (see Section 4yr a 1-a a

below) for 11 fractional values of a (see Section 6 below) . A companion

report [8] contains the most extensive set of tables currently available.

The main body of this report provides the theoretical and modelling background

for the use of these tables. In particular, we examine a model of a constant-

speed attack on a static defensive position and show how associated range-

dependent kill rates give rise to time-dependent attrition-rate coefficients

with "no offset." Numerical computations are presented to illustrate the

use of the LCS functions for analyzing such "aimed-fire" combat. As a con-

sequence of the availability of these tables, one can now study this variable-

coefficient combat model almost as easily and thoroughly as Lanchester's

classic constant-coefficient model.

2 . Variable-Coefficient Lanchester-Type Equations of Modern Warfare .

We consider combat between two homogeneous forces modelled by the

following variable-coefficient Lanchester-type [3] (see [4,5]) equations

of modern warfare

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dxdt— = -a(t)y

dt-b(t)x

with x(0) = x ,

with y(0) = yQ,

(2.1)

where t = denotes the time at which the battle begins, x(t) and y(t)

denote the numbers of X and Y at time t, and a(t) and b(t) denote

time-dependent Lanchester attrition-rate coefficients, which represent the

effectiveness of each side's fire. These coefficients depend on variables

such as force separation, tactical posture of targets, rate of target acqui-

sition, firing rate, etc. (sef> [4-7] for further details). Variable attrition-

rate coefficients are used to model temporal variations in firepower on the

battlefield. In any analysis of combat, moreover, we should use the above

equations (2.1) only for x and y > and, for example, set dx/dt =

when x = 0, since negative force levels have no physical meaning.

Mathematically, we assume that the attrition-rate coefficients a(t)

and b(t) are defined, positive, and continuous for t < t < + °° with

t _< 0. We also assume that a(t) and b(t) £ L(t ,T) for any finite

T _> t . We further take a(t) and b(t) to be given in the form

a(t) = k g(t) ,a

and b(t) = kbh(t)

, (2.2)

whe re k and k, are positive constants chosen so that a(t)/b(t) = k /k,a b a b

when g(t) E h(t). We introduce the combat-intensity parameter A and

the relative-fire-effectiveness parameter X defined byR

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X T= /k k, , and X_ = k /k, . (2.3)lab R a b

From our assumptions about a(t) and b(t), it follows that, for example,T

a(t) f. L(t ,T) implies / a(t)dt = + «>

.

tQ

The X force level as a function of time may be represented

as [5,6]

x(t) = X() {CY(0)C

x(t) - S

y(0)S

x(t)} - y

Q/A^ (C

x(0)S

x(t) - S

x(0)C

x(t)}, (2.4)

where the hypeibolic-like general Lanchester functions (GLF) C (t) and S (t)X X

are linearly-independent solutions to the X force-level equation

2 ( )

d x \ 1 da f d_x

.2 " a(t) dt dtat i '

- a(t)b(t)x = , (2.5)

with initial conditions

W = 1 W =° '

U/a(t )} dCx/dt(t

Q) = , {l/a(t

Q)} dS

x/dt(t

Q) = 1//A^ .

(2.6)

Here t denotes the largest finite time at which a(t) or b(t) ceases

to be defined, positive, or continuous. The Y force level as a function

of time is given by a similar expression, with C (t) and S (t) being

analogously defined for the corresponding Y force-level equation.

It is sometimes convenient to introduce the new independent variable

t defined by

4

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t

t - / /a(s)b(s) ds . (2.7)

It is readily seen that the transformation t = i(t) is well defined and

invertible. Let us denote t(0) as t_. We observe that t <_ implies

that t_ _> 0. If we denote the "average intensity of combat" as

/a(t)b(t) , then

/a(t)b(t) t - {(1/t) J /a(s)b(s) ds}t = t - tq

. (2.8)

The substitution (2.7) transforms (2.5) into

A{I,{£ (nR(T) }^..x=0, (2.9)

dt

with initial conditions

x(tq

) = xQ

, and U//R(t )} dx/di^) = -y ,

where R(t) = a(t)/b(t).

3. Combat Modelled with Power Attrition-Rate Coefficients .

The above equations (2.1) basically apply to "aimed-fire" combat

when target-acquisition times do not depend on the numbers of targets available

(see [5,6] for further details). A large class of tactical situations of

interest can be modelled with the following general power attrition-rate

coefficients [5-7]

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a(t) = k (t + C)M

, and b(t) = k, (t + C + A)V

, (3.1)a t>

where A and C >_ 0. We will call A the offset parameter , since it allows

us to model (with \\ and v _> 0) battles between opposing weapon systems

with different maximum effective ranges (see [5,6]). We will call C the

starting parameter , since it allows us to model (again, with \i and v _> 0)

battles that begin within the maximum effective ranges of the two opposing

systems. We observe that for the general power attrition-rate coefficients

(3.1) we have t = -C, and u and v must be > -1 in order that a(t)

and b(t) 6 L(tQ,T).

The above nomenclature is motivated and possible applications of our

work are indicated by considering S. Bonder's model of the constant-speed

attack on a static defensive position (see [4-7] for further details)

£ = -a(r)y, and & = -B(r)x , (3.2)at at

where r denotes the range between opposing forces, and a(r) and B(r)

denote range-dependent attrition-rate coefficients. Range is related to time

by

r(t) = RQ

- vt, (3.3)

where R denotes the opening range of battle and v > denotes the

constant attack speed. For example, let us consider the constant-speed attack

of a homogeneous Y force against the static defensive position of a homo-

geneous X force. Figure 1 diagrammatically portrays this situation.

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>

H 1

(U CT3 PioE II

X ,->

u urt ^-",

u (-1

4-J

Cfl

s*.

T3 J20)

01 CCX cu

CO >1 Hu 60CC3 CO

(J •HCO

c «

o •->

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4->

w ra

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o ex01

CTj

o^ cu

M urt 5-1

•H oQ fn

iH

0)

MDM•H^

Page 16: NPS55-77-42 NAVAL POSTGRADUATESCHOOL · 2016-06-24 · Durham,NorthCarolinawithR&DProjectNo.1L161102BH57-05Math(fundedunder MIPR No. ARO 22-77)and partially bythe Office of NavalResearch

The basic idea is that force separation, i.e. range between the opposing

forces, changes over time, and the fire effectiveness of, for example,

a single Y firer, denoted as ot(r), depends on this force separation.

In many cases of tactical interest, we may model the fire effective-

ness of, for example, the Y weapon system (as a function of range) with

a(r) = <

ao i

1r

Rfor < r < R

,— — a

for R < r ,a —

(3.4)

where R denotes the maximum effective range of the Y weapon system and

y _> 0. Here u is used to model the range dependency of Y's attrition-rate

coefficient (see Figure 2). We model 3(r) similarly, with corresponding

quantities R and v being analogous to R and y above.

If we use (3.3) to eliminate range r from (3.4), we obtain

f dx , ,.

dt~= "a(t)y

'

(3.5)

dtb(t)x ,

where the time-dependent attrition-rate coefficients a(t) and b(t) are

given by (3.1). It follows that the offset and starting parameters are given

by

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e

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R - R \ / R - R.

A = I

M

v-

», and C = ^7- •/, (3.6)

and that

ka

=°o(lH •

and*b

= Bo (^J

• (3 - 7)

We observe that A and C > if and only if R„ > R > R rt . By consider-—3— a —

ing (3.6) and Figure 3, the reader should have no trouble in understanding

our terminology for A and C.

When the offset parameter is equal to zero (i.e. A = 0) , then the

coefficients (3.1) reduce to

a(t) = k (t + C)y

, and b(t) = kb(t + C)

V. (3.8)

We will refer to (3.8) as power attrition-rate coefficients with "no offset ."

As we have seen above in Bonder's constant-speed attack model, these

coefficients model, for example, combat between weapon systems with the same

maximum effective range so that there is no "offset" in the "reaching out"

of the weapon systems against each other in combat (again, see Figure 3).

For these coefficients (3.8), the transformed X force-level equation (2.9)

becomes

H + (^r) dT- x =° » (3 - 9)

dx

with initial conditions

10

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U*q-i ta | _/ 4 \

2q_1

x(TQ

) = xQ ,

andj(j) d7(

T=T" "y AR \ v + v + 2 /

Here

«-(7fvT7] -

<=(^W (t + c >

(v+u+2)/2.

and

r /2A

I \p (p + v+2)/2

Ty + v + 2J

C ' (3 ' 12)

Let us observe that < q < 1 when \i and v > -1 . Furthermore, q > 1/2

if and only if dR/dt < 0, i.e. R(t) is a strictly decreasing function of

time.

A. Lanchester-Clifford-Schlafli (LCS) Functions .

Consider the function F (x) defined by the power series

k=0

For a j* 0, -1, -2,... the radius of convergence for Fa(x) is infinite

by the ratio test for convergence of power series [2]. Hence, F (z) is

an entire function of the complex variable z = x + iy, with an essential

12

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singularity at the point at infinity. Now consider the function H (x)

defined by the infinite series

, 2(k+u)

Observing that

H (x) = (l/a)(x/2)2a

F ..(x) , (A. 3)a a+1

we see that for a > the infinite series (4.2) is uniformly convergent

on compact subsets of the complex plane. From (4.3) one can readily deduce

the recursive relation

( ,/9

x2 )

F (x) = F ^(x) + ,l\, I F _,_(x) . (4.4)a a+1 ( a (a + 1) I a+2

We will call the functions F (x) and H (x) Lanchester-Clif ford-Schlaf lia a —

(LCS) functions (see Note 10 on pp. 66-67 of [5]). Other properties are

readily deduced and are given in Table I.

The function F (x) satisfies the linear second-order ordinary

differential equation

d2F . . dF

dx

with initial conditions

J 3

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Table I. Properties of the LCS Functions F (x) and H Cx)a a

1. dFa/dx = (x/2)

1_2aHa(x)

2. dHa/dx = (x/2)

2a"1Fa(x)

3. F (x)F1

(x)-H (x)H. (x) = 1 Vxa 1-a a 1-a

where a is not an integer (including zero)

4. F (x=0) = 1a

5. H (x=0) =0 for a >a

6. dF /dx(x=0) =a

7. {(x/2)1 2a

dH /dx} . = 1a x=0

8. F /^(x) = cosh x

9. H. ,_(x) = sir.h x

14

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dF

F (0) = 1, and -^ (0) = ,

while H (x) satisfiesa

d2H , dH

_^_ (

2£ll) " H =0, (A. 6),2 x dx a

dx

with initial conditions

l-2a dH )

(?) T±} =1Ha(0) =0 , and /v -, ^

(J x=0

Thus, {F , H, } is a fundamental system of solutions toa 1-a

dV(M)« p .

, (4.7), ^ x axdx

with Wronskian W(F ,11, ) = (x/2) . It follows that the GLF for thea 1-a

X and Y force-level equations for combat modelled with the attrition-rate

coefficients (3.8) are given by

X \2q-lCy

(t) F (t(t)), S y (t) =1. . - H (i(t)) , (4.8)

X q X \u + v + 2 / p

Cy(t) - F

p(x(t», S

Y(t) -

(h+ 1 + 2 )

^ Hq(r(t» , (4.9)

where p = 1-q. If we define

15

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then

T (x) = H (x)/F (x) , (4.10)

Sv (t) / X \2q-l H ( T (t))

Tx(t) - 60 - Utvtt turn' C4 -U)

or

/ x \2q-i

where T (t) denotes a hyperbolic-like GLF, which corresponds to theA.

hyperbolic tangent. Observing that for u, v > -1, lim t(t) = + °°,

we see that T (x) is a strictly increasing function of x on the interval

[0, + «) and

< T (x) <F^ °*

for < x < + -, (4.13)— a r(a; —

with

x-*+ °°

since by the results of Taylor and Comstock [7] the parity-condition parameter

Q* = Q*(u, v, C = 0) is given by

tff.Tx(t) QM.A, 0) (^Vr)

2'"1

7^ «- 15 >

We recall that Taylor and Comstock [7] have introduced the so-

called parity-condition parameter Q* as the value (or range of such values)

for the initial condition Q to the initial-value problem

16

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7r - - ~h a(t)EY

with VV = 3•

dT= -/^b(t)E- with E-( t() ) = Q ,

(A. 16)

such that E (t;Q*) and E (t;Q*) > for all t >_ t . In other words,

Q* is the value of Q in (4.16) ahove such that neither E nor EA.

ever become zero. In this case, both E (t;Q*) and E (t;Q*) are positive,

strictly decreasing functions, similar to decreasing exponentials. Thus,

we may call Q* "the Y equivalent of an X force of unit strength,"

since the forces are "at parity," with neither force being annihilated in

finite time. Taylor and Comstock have shown that for either a(t) £ L(0,+ <")

or b(t) £ L(0, + °°) , then Q* is unique and given by

sx(t)

1

£. v^= ^ • (4 - 17)

The significance of the parity-condition parameter Q* is that it allows

us to predict force annihilation as the following theorem shows.

THEOREM 1 (Taylor and Comstock [7]): Assume that either

a(t) £ L(0, +») or b(t) £ L(0, +~). Then the X force

will be annihilated in finite time if and only if

Xr~ fV 0) " Q* S X

(0)]< /a r (q*c

y(0) - s

Y(o) /

'(4<18)

i /

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5. Use of LCS Functions for Analyzing Combat .

The Lanchester-Clifford-Schlafli (LCS) functions F (x) and H (x)a a

are useful for analyzing "aimed-fire" combat (see Section 3 above) modelled

with the power attrition-rate coefficients with "no offset" (3.8), which

we rewrite here as

a(t) = k (t + C)u

, and b(t) = k^(t + C)V

. (5.1)

In other words, the LCS functions arise in solving the differential combat

model (2.1) with attrition -rate coefficients (5.1). In order that both

a(t) and b(t) £ L(t~,T), we must have u and v > -1 . Military situations

modelled by these equations have been discussed in Section 3 above, e.g.

combat between two weapon systems with the same maximum effective range.

For such combat, the LCS functions may be used to

(1) compute force-level declines,

(2) predict force annihilation,

and (3) predict the time of force annihilation.

Let us now see how the LCS functions may be used to obtain the

above information about force-level declines and force-annihilation pre-

diction. According to (2.4), (4.8), and (4.9) above, the X force level

is given by

x(t) = xn {F (t_) F (i(t)) - H (t.) H (i(t))}p q q p

- * ^ L + Xa )

2 " 1

<yvyT(c)) -yvy T(t)))-

(5 - 2)

18

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where q is given by (3.10), p = 1-q, and T (t) is given by (3.11), which

we rewrite as

2A1 1 / ^ (V'+^+2)/2 /c „,i(t —

+ v + 2J (t + C)

v, (5.3)

The time to annihilate the X force* is determined by x(t ) - 0, and3

it follows that

q a / X \q-p

x„H (t.) + yn AT ————

y

F (t )OqO R\p+v+2/ q0

where from (4.10)

T (-i(t)) = H (i(t))/F (t(O) , (5.5)q p q

and we recall that p + q = 1. It follows that the time to annihilate X,

X_,

t , is given bya

If we multiply the first equation of (2.1) by y, the second by x, add,

and integrate, we obtain

t

x(t) y(t) = x y • {a(s) y2(s) + b(s) x

2(s)}ds ,

which shows that x(t) and y(t) can have at most one finite zero. Hence,if x(tx ) = 0, then we know that y(t) > for all t > 0.

19

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^-T-^T-1

a \ q

V,"o» +'o ^ IttVT:

q-pH (O

2 / p

q-p

OVV +y ^ U + Vf 2/Fq

(V(5.6)

Taylor and Comstock [7] have shown that T (t) is strictly increasing andq

satisfies (see also (A. 12) above)

< T (t) < r(P )/r(q) ,— q(5.7)

where p = 1-q. It follows that in order for X to be annihilated in finite

time, the right-hand side of (5.4) must be less than T(p)/r(q). Let us

observe that for t = -C = 0, (5.4) simplifies to

p-q

T (t(O) =q a

^0^ p + v + 2(5.8)

Thus, we have proved the following theorem concerning force-

annihilation prediction.

THEOREM 2: Consider combat between two homogeneous forces

modelled by (2.1) with power attrition-rate coefficients

(5.1). Assume that u and v > -1 and that the above

equations hold for all time. Then the X force will be

annihilated in finite time if and only if

20

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I ., - /

y.-P

>) Vp (y + y /A

R \ v + v + 2 / W|

< r(p) |x Hq(T ) + y

Q,T

R ( v + l\ 2 )Fq(T )

|, (5.9)

where q = (\) + 1) / (ij + v + 2) and p = 1-q . For t =

(i.e. C = so that t = 0) , X will be annihilated in finite

time if and only if

x. „, , / x >q -p

-g <^ •*: +1

+ . i (5.io)y I (q) R \ p + v + 2 '

6 . Tabulation of LCS Functions .

This report contains a reduced set of tables of the Lanchester-Clif ford-

Schlafli functions. The Appendix contains tables of f ive-decimal-place values

of the hyperbolic-like LCS functions F (x) , FL (x) , and T (x) for variousa 1-a a

values of the argument x, namely x = 0.00 (0.01) 2.00 (0.1) 10.0, and

a = 1/2, 1/3, 2/3, 1/4, 3/4, 1/5, 2/5, 3/5, 4/5, 3/7, and 4/7. These values

of the index a correspond to u, v = 0, 1, 2, and 3 in (3.8) and allow one

to analyze, for example, a basic spectrum of range capabilities for weapon

systems in the constant-speed-attack model of Section 3. These tables have

been calculated by the recursive means given in Section 8 of [5], A more

extensive tabulation (namely, for a = 1/2, 1/3, 2/3, 1/4, 3/4, 1/5, 2/5,

3/5, 4/5, 2/7, 3/7, 4/7, 5/7, 4/9, 5/9, 3/11, 5/11, 6/11, 8/11, 5/13, 8/13,

5/17, 12/17, 5/21, and 16/21 corresponding to u , \; = 0, 1/4, 1/2, 1, 1 -|, 2, 3)

.'1

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is to be found in a companion report [8]. This companion report contains

the most extensive set of tables of the Lanchester-Clif ford-Schlaf li

functions currently available.

A representative tabulation of the hyperbolic-like LCS functions

F (x) , H. (x) , and T (x) is given in, for example, Tables 8A and 8B ofa 1-a a

the Appendix for a = 3/5. The values of the argument x are the same as those

used for the tabulation of the hyperbolic functions by Abramowitz and Stegun

[1]. We observe from Table 8B and (A. 13) that the limiting value of

T (x) as x -> + °° (here a = 3/5) is quickly reached, with three-decimal-a

place accuracy already attained for x = A. 5. Moreover, the use of these

tables (specifically, Tables 8A and 8B of the Appendix) for combat analysis

is illustrated in the next section.

7. Numerical Examples

In this section we examine a couple of numerical examples to show

some of the insights that may be gained into the dynamics of combat between

two homogeneous forces from our results (see also [6]). These examples

illustrate the use of the LCS functions F (x) , H 1 (x) , and T (x) fora ' 1-a ' a

analyzing "aimed-f ire" combat modelled with the power attrition-rate coefficients

with "no offset" (5.1). As in [4-7], we consider S. Bonder's model (3.2)

for the constant-speed attack against a static defensive position. We will

focus on the use of the LCS functions for predicting force annihilation,

since the computing of force-level trajectories with Lanchester functions

is adequately handled elsewhere (see [4-5]),

22

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Let us accordingly consider the constant-speed attack of a homogeneous

Y force against the static defensive position of a homogeneous X force

(see Section 3 above for further modelling details, especially Figure 1).

For our numerical computations, we assume that the fire effectiveness of the

Y weapon system varies linearly with range, i.e.

a(r) - <

V rfor < r < R ,— a

fcr R < r,a —

(7.1)

and that the fire effectiveness of the X weapon system varies quadratically

with range, i.e.

6(r)

1 -^ for < r < R

for R n< r

,

(7.2)

with R = R , i.e. both weapon systems have the same maximum effective range.

In other words, p = 1 in (3.4) and v = 2 for £(r). We consider a battle

modelled by the input data given in Table II. In terms of time as the

independent variable, the attrition-rate coefficients (7.1) and (7.2) become

via (3.3)

a(t) = k (t + C) and b(t) = kb(t + cr , (7.3)

23

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Table II. Input Data for Numerical Examples

u *» 1, v = 2

an= 0.06 X casualtles/minute/Y firer

6 n= 0.6 Y casualties/minute/X firer

R = R = 2000 metersa 6

v = 5 miles/hour

24

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where R a = R g,

rRc-'

R„ V

„ / V \2

a \ 3/

From the input data given in Table II, we compute the parameter values shown

in Table III, since the transformed X force-level equation is given by (3.9)

with q = (v + l)/(u + v + 2), p = 1-q, m = 1, and v - 2. Thus, the X force

level may be computed with F (1) and H. (t) with a = q = 3/5. Foree-a 1-a ^

annihilation prediction involves the limiting value of T (t) = H (t)/F (t)1-a a

as x ->+ ro. From Table 8B of the Appendix and Table III, we note the predicted

agreement between r(l-a)/r(a) and the limiting value of T (x) as x ^ + °°

a

[recall (4.13)] for a = q = 3/5. We now consider two cases: (T) R = 2000

meters, and (II) R = 1250 meters.

When R = 2000 meters (see Figure 3 of [A]), we have C = and

1 n = 0. The maximum time that the battle can last is t = R„/v = 14.91 minutes,max

since at this time the attackers reach their final objective, i.e. the

defender's position (again, see Figure 1). We now consider the qualitative

behavior of the u = 1, v = 2 force-level trajectory shown in Figure 3 of

[4]. Theorem 2 tells us that the X force can be annihilated in finite time

if and only if

A. V q-p

.^Pl sr, \ J , (7.3)

y r(q) R \ u + v + 2

where q = 3/5 and p = 1-q. Using the numerical values in Table III, we

compute from (7.3) that the X force can be annihilated in finite time if

and only if

2-

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Table III. Parameter Values for Numerical Examples

k = 4.0233 x 10~ X casualties/ (minute) y/Y firer

a

k, = 2.6979 x io~ Y casualties/ (minute)V/X firer

b

p = 2/5, q = 3/5

r(p)/r(q) = 1.48951

A =

26

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— < 0.420 . (7.4)yo

When the X force can be annihilated, its annihilation time is given by

(5.8), which we rewrite here as

where

T d(t*)) = —^_q a a—

y \

^^(^Tr) t<,,+"2)/2

•' ™

Thus, for the numerical values given in Table III, the time of annihilation

of the X force is given by

T (i(tX)) = 3.544 -9-

, (7.7)q a y

with q = 3/5. We will now illustrate further computations for x = 10 and

y = 30. From (7.4) we see that the X force can be annihilated in finite

Xtime (but we must verify that t < t ). In this case (7.7) becomesJ a — max

T (x(tX)) = 1.18122 . (7.8)

q a

X X -1We must now determine x(t ) such that i(t ) = T (1.18122) by using inter-

a a q

polatlon methods and Tables 8A and 8B . From Table 8A, we find

21

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T (t) = 1.18172 for = 1.01q

T (t) = 1.17630 for = 1.00q

so that using linear interpolation, we obtain

T (tX

) = 1.009, (7.9)

a

whence use of (7.6) yields

tX

= 14.24 minutes, (7.10)a

which is less than t =14.91 minutes so that the defending X force ismax

indeed annihilated before the attacking Y force reaches its final objective.

Since r(t) = R - vt, we find that force separation at the instant of

annihilation of the X force is

rX

= 89.8 meters . (7.11)a

Further results may be computed in a similar fashion and are given in

Table IV.

When R = 1250 meters (see Figure 3 of [5]), we have C = 5.5923

minutes, t„ = 0.0975, and t = R^/v = 9.32 minutes. In this casemax

Theorem 2 tells us that the X force can be annihilated in finite time

if and only if

28

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Table IV. Annihilation of the X Force as a Function

of the Initial Force Ratio for R = 2000 meters

(xQ/y ) r

a(minutes) r (meters)

a

0.333 14.24 89.8

0.250 11.61 443.2

0.200 10.19 633.2

2^

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^7 "V

"R U + v + 2

h \«r(P ) (W "T^Wlr(q) {yv-f^vv}

(7.12)

with q = 3/5 and p = 1-q. Using linear interpolation, we obtain from

Tables 7A and 8A of the Appendix that for the numerical values of Table III

F (O = 1.006 ,

P

F (t.) = 1.004,

q o

H (t ) = 0.044 ,

q o

H (t ) = 0.223,

P o

(7.13)

so that (7.12) says that the X force can be annihilated if and only if

— < 0.382 .

y(7.14)

When the X force can be annihilated, its annihilation time is given by

(5.4), which we rewrite here as

'0

/r \m + v + 2

p-q

F (t_) + H (t )p p

T (t(0) =q a

o R

Fq

( ^0 ) +

y AR

\i + v + 2

p-q

h d n )

q o

(7.15)

whence for the data of Table III

Y 3.565u_ + 0.223T (r(t

X)) =

0.156u + 1.004 '(7.16)

30

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where u = x /y„. Let us also record here that (3.11) yields

(r , ~i \2/(}l+V+2){ "

+ " +2lT)

- C . (7.17)

We will again illustrate further computations for x = 10 and y = 30.

From (7. 14) we see that the X force can be annihilated in finite time (but

yaeain we must investigate whether or not t < t ). In this case (7.16)b a — max

becomes

T (i(tX)) = 1.33651

, (7.18)q a

whence Table 8A of the Appendix and linear interpolation yield

T (tX

) = 1.397, (7.19)

a

so that by (7.17)

tX

= 10.63 minutes . (7.20)a

Since t = R„/v = 9.32 minutes < t , we see that the attacking Y forcemax a

overruns the defender's position before annihilation of the X force occurs.

Thus, the battle ends with x,- = x(t^) > and y,. > at t r = tf f f f max

9.32 minutes. Corresponding to t = 9.32 minutes is x = 1.1318, and

then Table 8A of the Appendix yields

3 J

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F (t r= 1.1318) = 1.589 , H (1.1318) = 1.973 , (7.21)

q f p

whence via (2.4), (A. 8), (4.9), and (7.13) we obtain

x = x(t ) = x(r = 0) = 1.35 . (7.22)

Some further numerical results are given in Table V. Again, these parametric

results should be contrasted with the single" u = 1, v = 2 force-level

trajectory shown in Figure 3 of [5],

8. Final Remarks

In the previous section above, we have seen how the LCS functions

allow one to conveniently obtain much valuable information about the model

(2.1) with power attrition-rate coefficients (3.8) without having to explicitly

compute the entire force-level trajectories. Previously we were limited to

computing only force-level trajectories (see [4-5]). With the availability

of these tabulations of LCS functions (see the Appendix of this report and

[8]), we can now tell who is going to be annihilated and when this event will

happen without having to compute the trajectories. Not only did we answer

questions about the qualitative behavior of the model (e.g. force annihilation)

for specific values of, for example, initial force levels but also for a

range of values of the initial force ratio (i.e. parametric analysis of

model behavior)

.

32

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Table V. Annihilation of the X Force as a Function

of the Initial Force Ratio for R = 1250 meters

(xo/y

o}

Xt

a(minutes)

Xra

(meters

)

0.333 10.63

7.56

j.

0.250 235.9

0.200 6.17 422.8

+t =9.32 minutes and x^ = x(r=0) =1.35max f

3 3

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The results of this report may be used for other parametric

analyses, e.g. parametric dependence of battle outcome on attrition-rate

coefficients. Thus, the contents of this report allow one to develop

important insights into the dynamics of combat between two homogeneous

forces with temporal variations in fire effectiveness. With the avail-

ability of tabulations of the LCS functions, one can now analyze such

combat modelled by the power attrition-rate coefficients (3.8) with some-

what the same facility as he can for the constant-coefficient case and thus

aid in parametric analyses. For further discussions of the significance

of such results for military operations research, the redder is directed

to [6] and [7].

34

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REFERENCES

1. M. ABRAMOWITZ and I. STEGUN (Editors), Handbook of MathematicalFunctions , National Bureau of Standards Applied Mathematics Series,No. 55, Washington, D.C., 1964.

2. K. KNOPP, Infinite Sequences and Series , Dover, New York, 1956.

3. F. W. LANCHESTER, "Aircraft in warfare: The Dawn of the Fourth Arm-No. V., The Principle of Concentration," Engineering 98 , 422-423 (1914)

(reprinted on pp. 2138-2148 of The World of Mathematics , J. Newman(Editor), Simon and Schuster, New York, 1956) .

4. J. G. TAYLOR, "Solving Lanchester-Type Equations for 'Modern Warfare'with Variable Cof f icients ,

" Opns. Res . 22, 756-770 (1974).

5. J. G. TAYLOR and G. G. BROWN, "Canonical Methods in the Solution of

Variable-Coefficient Lanchester-Type Equations of Modern Warfare,"Opns. Res . 24, 44-69 (1976).

6. J. G. TAYLOR and G. G. BROWN, "Further Canonical Methods in the

Solution of Variable-Coefficient Lanchester-Type Equations of ModernWarfare," NPS55-77-27, Naval Postgraduate School, Monterey, California,June 1977.

7. J. G. TAYLOR and C. COMSTOCK, "Force-Annihilation Conditions for

Variable-Coefficient Lanchester-Type Equations of Modern Warfare,"Naval Res. Log. Quart . _24, 349-371 (1977).

8. J. G. TAYLOR and G. G. BROWN, "A Table of Lanchester-Clif ford-

Schlafli Functions," NPS55-77-39, Naval Postgraduate School, Monterey,California, October 1977.

3 5

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APPFNDIX' Tabulation of the LCS Functions F (x) , H (x) , and T (x) fori ii i i_niy ± /\

.

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a = 1/2, 1/3, 2/3, 1/4, 3/4, 1/5, 2/5, 3/5, 4/5, 3/7, and 4/7.

36

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