Method of Controlling Reinforcement
Forces to Win in Combat
Conflict Situations
Nodirbek Karimov
1*
, Rakhmonov Turdimukhammad
1
1
Tashkent University of Applied Sciences, Gavhar Str. 1, Tashkent 100149, Uzbekistan
matematik-uz@mail.ru
https://doi.org/10.5281/zenodo.10471362
Keywords:
Lanchester's quadratic law, opposing sides, reinforcement forces, total reserve forces, piecewise-constant
function, battle of Iwo Jima, winning strategy, coefficient of effectiveness, non-combat losses.
Abstract:
In this article, problems related to the control of reinforcements coming in opposite directions, based on the
Lanchester's extended quadratic law. The rate of receipt of reinforcements is taken as a piecewise constant
function. A successful strategy for managing auxiliary forces arriving to the conflicting parties is described.
On the basis of the model of combat operations, software for the management of auxiliary forces was created.
The results of the battle between Japan and America on the island of Iwo Jima and other conflict situations
were also simulated.
1 INTRODUCTION
In recent years, a number of results have been
achieved in the research of command decision-
making in combat operations based on Lanchester's
quadratic law. Today, on the basis of Lanchester's
quadratic law, the issues of optimal control and
distribution of auxiliary power resources leading to
success are deeply studied. It is worth noting that the
development of decision-making strategies and
optimal strategies for winning according to strengths
has become a focus of research. This article also
proposes methods for controlling auxiliary forces to
win in conflict situations based on Lanchester's
extended quadratic law.
The following works can be cited regarding the
description and simulation of combat actions based
on Lanchester's quadratic law. For example, in the
monograph "Lanchester-type war models" by
J.G.Taylor, the combat actions of opposing forces are
widely covered on the basis of differential equations
and analyzed in depth on the basis of examples [1]. In
his dissertation, Gerardo Minguela Castro empirically
evaluated the battle of Crete, the battles of Iwo Jima
and Kursk, as well as three influential battles of the
Second World War [2].
In [3], J.G.Taylor and G.G.Brown developed
canonical methods for solving Lanchester-type
equations with variable coefficients for modern
warfare.
In [4], C.Sha and J.Zeng show that which side has
the upper hand in the battle, according to the
firepower of weapons, the number of soldiers and
combat equipment on which side has more chances to
win, Lanchester's equations and analyzed based on
the theory of differential games.
Now, let's consider the situation of combat
conflict between two opposing sides based on
Lanchester's quadratic law.
In this article, we present the methods, algorithm
and results of the control of auxiliary forces that come
to the parties to win in combat conflict situations.
2 Description of the Combat Model
with Reinforcement forces
In a conflict situation between two opposing sides
with the same type of forces, we consider the model
of combat operations, taking into account the
incoming reinforcement forces. For example, if
reinforcement forces are added to sides A and B at
speeds ( )
u t
and ( )
t
, respectively, then the rate of
change of their forces is as follows:
1
1 1
1 2
2
2 2
2 1
( )
( )
( )
( )
( )
( )
( )
( )
x t
a x t
b x t
u t
x t
a x t
b x t
t
= −
−
+
= −
−
+
(1)
Where,
1
( )
x t
and
2
( )
x
t
are the number of forces of
the opposing sides at the moment of time
t
;
1, 2
a
- coefficient of non-combat losses of the parties;
1, 2
b
- coefficient of losses due to hostilities of the
parties;
1
10
(0)
x
x
=
and
2
20
(0)
x
x
=
- the number of
initial forces of the parties; ( )
u t
and ( )
t
are the
speed of arrival of reinforcement forces, i.e. control
of the parties.
Next, we consider the speed of arrival of auxiliary
forces in military conflicts as parameters of control of
the parties.
We use the following assumptions and definitions
to solve the above problem. It should be noted that the
auxiliary forces of each side are made up of a limited
number of soldiers and weapons reserves.
Assumption 1.
We assume that the functions
( )
u t
and ( )
t
are, piecewise-constant functions in
the interval
[0, ]
T
.
Definition 1.
The function
[ ,
]
f
B t T
is
called piecewise-continuous if, [ ,
]
t T
is divided by
1
2
1
,
, ...,
n
t t
t
−
points
such
that
in
each
1
)
(
,
k
k
t
t
−
interval the function
f
is continuous
and has one-sided limits
1
0
)
lim
(
k
t
t
f t
−
→
+
and
0
)
lim
(
k
t
t
f t
→ −
. The value of the function
f
at
k
t
points
is
not
important.
Where,
1
2
1
0
n
n
T
t
t
t
t
t
−
=
=
.
Definition 2.
The function
f
is called
piecewise-constant in the section [ ,
]
t T
, if it is
piecewise-continuous in every section
1
)
(
,
k
k
t
t
−
and invariant in every interval of continuity.
Case 1.
Assumption 2.
Let
0
X
and
0
Y
denote the total
number of auxiliary forces coming to the opposite
sides, and let them satisfy the following conditions:
0
0
0
0
,
0
,
0
( )
( )
0,
0,
,
0
,
0
( )
( )
0,
0,
u
u
X
d if
t
d if
t
T
d
u t
or
u t
X
if t
T
if t
d
Y
k if
t
k if
t
T
k
t
or
t
Y
if t
T
if t
k
=
=
=
=
(2)
Assumption 3.
We assume that the number of
auxiliary forces is added to the opposite sides at a
constant speed until it reaches the total number of
reserves:
0
0
( )
u
T
u t dt
X
=
and
0
0
( )
T
t dt
Y
=
(3)
where
u
T
,
T
are the total deployment times of the
auxiliary forces for the respective sides.
There are two possible situations to consider here.
Case 1a.
The total time to spend the auxiliary
forces can be longer than the time to finish the battle.
In this case, all reserve forces are not fully used (
u
T
T
as well as
T
T
).
Case 1b.
The total time spent by the auxiliary
forces can be less than or equal to the end of the battle.
In this case, all reserve forces are fully used (
u
T
T
as well as
T
T
).
Definition 2.
The winning strategy of party
X
is
the control ( )
u t
for which
u
T
T
(or
u
T
T
) holds
for inequality
1
2
( )
,
( )
0.
T
M
T
x
x
=
(4)
Case 2.
Auxiliary forces join the conflicting parties either
at random times or continuously throughout the battle
as shown in Figure 1 below.
Figure. 1. Schematic of Lanchester's Augmented
Quadratic Model Including Auxiliary Forces and
Non-Combat Losses
3
RESULTS AND DISCUSSIONS
Now, let's see how Lanchester's square law
applies to combat. Based on the above assumptions
and conditions, we implement the model (1) in
practical examples.
Problem 1.
As an example, let's take the 36-day
battle between America and Japan on the island of
Iwo Jima, which lasted from February 19 to March
26, 1945, [5, 6, 7]. In this case, we consider side A as
the Americans, and side B as the Japanese. For the
simulation of the Battle of Iwo Jima, the parameters
in model (1) are equal to
1
2
( )
0
a
a
t
=
=
=
. In [5],
J.H.Engel estimated the coefficients of efficiency (the
rate of losses of the parties due to combat action) for
the Americans and the Japanese as follows.
Table 1. Efficiency coefficients of Americans and
Japanese
Efficiency coefficient of the
Japanese
0,05440
Efficiency coefficient of the
Americans
0,01060
These coefficients of combat effectiveness
correspond to values
1
0.0106
b
=
and
2
0.0544
b
=
for model (1). Also, in this battle, the American side
received reinforcements twice, and the Japanese did
not. Below is the amount of reinforcements that have
arrived on the American side.
0,
0
1
0,
1
2
6000,
2
3
0,
3
4
0,
4
5
13000, 5
6
0,
6
36
( )
t
t
t
t
t
t
t
u t
=
(5)
During this battle, the Americans landed two
reinforcements: 6,000 troops on the third day of the
battle and 13,000 on the sixth day. So, according to
definition 2, the management chosen by the
Americans is a winning strategy. Below is the
simulation result of Problem 1.
Figure 2. Simulation result of the Battle of Iwo Jima
In all, about 28,000 American soldiers were killed
in this battle, including about 8,000 killed, while the
Japanese lost more than 20,000 soldiers killed and
about 1,000 captured. Also, the cost of taking the
island of Iwo Jima was more than the Americans
expected [6].
Problem 2.
A battle ensued between two sides, A
and B, with 1,500 soldiers each. The efficiency
coefficients of the parties are 0.03, and the
coefficients of non-combat losses are 0.003. Also,
each side has 600 reserve or auxiliary forces. The two
opposing sides manage and distribute their auxiliary
forces as follows. Let these auxiliary forces reinforce
both sides from the 15th day of the battle. If side A
receives 300 soldiers on the 15th day of the battle,
then the remaining 300 soldiers join on the 75th day
of the battle, and side B receives 10 soldiers
continuously from the 15th day of the battle until the
75th day. , then how many days will the battle last and
which side will win?
Numerical
solution.
The
mathematical
expression of the available reserve forces on side A
can be written as:
1
2
2
1
2
1
2
1
0
0
1
0
( )
( )
( )
( )
( )
u t dt
u t dt
u t
u t
u t dt
X
+
=
=
=
=
here ( )
u t
is the speed of arrival of auxiliary forces to
A, i.e. equal to 300.
For side B:
120
0
20
( )
t dt
Y
=
here, ( )
t
is
the speed of arrival of auxiliary forces to
side B is equal to 10. The meaning of this problem is
reflected in Figure 1 above. Considering the speed of
arrival of auxiliary forces to opposing parties as a
control
function,
the
control
algorithm
is
implemented as follows:
Table 2. Algorithm of control functions of
parties
int ControlU(int t) {
int u;
if (t == 15)
u = 300;
else if (t == 75)
u = 300;
else
u = 0;
return u;
}
int ControlV(int t) {
int v;
if (t>=15 && t<=75)
v = 10;
else
v = 0;
return v;
}
Now,
based
on
the
initial
conditions
1
2
(0)
1500,
(0)
1500
x
x
=
=
and combat and non-
combat losses
1, 2
1, 2
0.003
0.03
,
a
b
=
=
, we find the
numerical and graphical solutions of problem 2 using
the modified Euler method.
Figure 3. Dynamics of changing the forces of
warring parties
This battle will last 95 days based on the initial
conditions above, resulting in Side A winning and
448 soldiers surviving. According to Definition 2, the
strategy chosen by Party A to control the auxiliary
forces is the winning strategy.
Of course, the situation may change if one or more
parameters change. For example, if side B
continuously reinforces 10 soldiers from day 5 of the
battle until day 65, then side B wins in 100 days and
350 soldiers survive. In this case, the strategy chosen
by B to control the auxiliary forces is the winning
strategy.
Figure 4. Dynamics of changing the forces of
warring parties
Figure 4 above shows the dynamics of changes in
the power of parties A and B, where it can be seen
that the power of each party decreases according to
the exponential law. Also, the arrival of auxiliary
forces will increase the strength of the sides, and as a
result, the strength of the B side will suffer serious
losses from a certain point in time, while the strength
of the
A side will rapidly decrease and eventually
disappear completely.
4 CONCLUSIONS
In this work, two cases of introducing auxiliary
forces into the model of combat operations and
controlling them were seen. Also provided is a
scheme of the combat operations model written with
auxiliary forces. The modified Euler method and
programming technologies were used to find the
numerical solutions of the model (1).
The battle of Iwo Jima was simulated based on
J.H.Engel [5]. Based on the model of combat
operations, an algorithm and software for the
construction of control functions for the distribution
of auxiliary forces of opposing parties were created.
Also, the software created can calculate the
amount of forces left on each side, how long the
conflict will last, which side will win, combat and
non-combat loss ratios, taking into account the arrival
of reinforcements in combat situations. allows you to
determine how it affects the outcome of the battle and
simulate various military conflicts.
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1980.
Volume
1.
606
p.
https://apps.dtic.mil/sti/citations/ADA090842
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(España). Programa de Doctorado en Ingeniería de
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