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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
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.
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A Short Table of Lanchester-Clif ford-Schlaf liFunctions
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James G. Taylor and Gerald G. Brown
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Naval Postgraduate SchoolMonterey, Ca. 93940
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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
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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)
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.
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
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]
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
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
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
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]
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.
>
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 •->
u lj
CO U•-
^1 -
01 cT3 oc •H
4->
w ra
^U-l rt
o ex01
CTj
o^ cu
M urt 5-1
•H oQ fn
•
iH
0)
MDM•H^
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
e
<x>
co
31
J
ijBQJ
re re «4-1 en Mc aj a) >. re ^y0/ <j M 03 Hc re re X 1 re C
m re MTj re QJ .'.
ex re • aX H O re. CJ —re re OJ re
g•H > li •J CJ
a j ^ •H l< .re re
-C 4-i o H re:
u -re
CJa CJ re
C re lt-1 ii *J • en
o m 're re
re /-> tfl--—
-
ai c 11
>-s *j B ^—
'
U •-re
M en 3 8 o / s rev—
'
>. B re re re
W H CJ 00X "3 re re
re re CN re re
4J o B <—
v
re re
c a en >>— 4-1
(U re OJ • re re
•H CD JC tn OJ re BU 5 H »-i re o•H cu H •H o>+H 'J 4-J IW 4-1 mU-< x. • ai re r j
QJ u H E Vh reC |H re
CJ
Ooo re.
re
li
CJ in o U-l re cu GJ W r
J ere
cfl M H QJ
re re O II >-J re
I re S3 CU M re
c Vl i—
j
o X o re
o CtS B >+H
•H OJ re X4-i > •
re o reH •H 4-1 CO re 4-1
re 4- C re X CJ O+-i u re N re
U GJ u X) QJ re
cfl IH en QJ fc"4-' -re
14-J re J-l •H re
en QJ 4-J in•"" u re H>H B Cu 4-J |H
3 XI -re H re
B I—
1
re re(-4-< H o CO re re 4 J
ore
x: •H v—' 4J
re
CD B CJ B CO XCJ to CJ OJ re
C a: re aj •H 4J U-l
QJ -C re en 4J re oT3 n u >> rH re
c v. re re
CJ .re re re wQ. 1-1 u CJ en r-H re
CU H CJ .re re •H re
Q ^ N *j CJ rei re
0)
re
re
ere
•H
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
(X) <J*
o 1
.1 ».0*.-
r?**-k. a h~
QC 9< oO
CVJ
o
o T3 jjjo a O
«ni-i
QJ
03 cCO
ro 3 QJ
o 43 a CO
a. ^< <as CO
>• •
o • w T3 >4-1 rH nj c ^~^
O T3cfl
aOO
a QJ as
W4J
OO i
en
roS-< H c OSQJ O QJ
^—
'
u z Tl CO
a> '
—
->nj ii
ri QJ
(-i h T3- <u • 03 >'j
H 4«! 4-1
O ex, CJ CO o >.y""v 03 B c 43O 60 u QJ QJ
lO c *J > OJ C i—
i
•H rt CO 'J *
CSJ W ^ CO > >- T3 CO •H •H -^03 CD 0.0 -"V4-J CO c QJ oCO CX o rH CO e«S
co ex 4-J •H i
QJ i co u aO 42 u CJ 03 H cs
O u C u 43 OJ s—
'
^1 1^ 03 4-J
O to T3 1-1
CO >H14-1
O11J ii
OJ CO GO QJ
"3
|X|
o*— U XI 60 03
CD <: C C CX >-.
E M 03 03
(-1 4-J
43
m •H QJ aO ^""'
D r-
1
X 60 CO QJ
U H C '4-4 >oIf)
1) QJ •H 4-4 •H
<?B
oOJ
43CQJ
O 60
—
-
S-i E •u CX QJ CO-' o 43 -H
a>CX CO m H
4J QJ Men 4-1 a 43 QJ
c a) ai CO H • 4-J
o co •H OJ ro '1)
IH U 60 B
o er U-l •H c •n3
o '4-i 03 CI • S-i
oo
'-M U •—
>
03
QJ QJ CQ CXX OJ • OSU a > >, *. 60
•H rH a CM QJ 4-1 QJ as •Ho 4J CJ > ^—
'
4-1
«j OJ H B uc •*j '4-1 4-J 3 03
. LM CJ E 4-1
c •H c OJ OJ H CO
U o CX C
o A H E CO •H QJ
C aj 3 QJ E x:
lO cd H E V-J HH M •H Vo. 4-J X »
X •u 03 OQ o •
U4 ra E as cjS -t
rn
'1!
1-4
60•H
11
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
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
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
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
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
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 /
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
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
^-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
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
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
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
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
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-
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
— < 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
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
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^
^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
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
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
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
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
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
APPFNDIX' Tabulation of the LCS Functions F (x) , H (x) , and T (x) fori ii i i_niy ± /\
.
a ^_a a
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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