=Paper=
{{Paper
|id=Vol-3293/paper8
|storemode=property
|title=Mathematical War Theories in Business: Lanchester's Equations Law, to the Commercial World
|pdfUrl=https://ceur-ws.org/Vol-3293/paper8.pdf
|volume=Vol-3293
|authors=Evangelia-Eleni Priska,Dimitrios Kallivokas,Miltiadis Chalikias
|dblpUrl=https://dblp.org/rec/conf/haicta/PriskaKC22
}}
==Mathematical War Theories in Business: Lanchester's Equations Law, to the Commercial World==
https://ceur-ws.org/Vol-3293/paper8.pdf
Mathematical War Theories in Business: Lanchester’s Equations
Law, to the Commercial World
Evangelia-Eleni Priska 1, Dimitrios Kallivokas 1 and Miltiadis Chalikias 1
1
University of West Attica, 250 Thivon & P. Ralli str, Egaleo, Greece
Abstract
This paper deals with the connection of the most well-known mathematical theory of war,
Lanchester’s equations Law, to the commercial world. In order to evaluate a range of military
conflicts throughout WWII, mathematical models in military operations research and combat
modeling were frequently deployed. One of these, Lanchester’s models describe the outcome
of warfare between two similar forces without backups, for time [0, t).
Keywords 1
Mathematical war theories, Lanchester’s equations law, Decision making, Commercial world,
Industry
1. Introduction - Literature Review
One of the most important issues on the military field is the prediction of a battle outcome with
ultimate goal the prediction of the winner. However, the scientific background was not always sufficient
to bring such a result. Many great military philosophers, historians and generals tried to formulate
principles, with which a Military Force could be led to victory. All those principles were theoretical
and they had the form of rules. For this reason, Mathematical Models and Theories started to be used.
The first attempts, for not so theoretical approaches, started at the beginning of 20th century. At
start, Differential Systems were used, that included mathematical relations in order to describe the loses
and the size of two opponents military forces.
In 1902, American Jehu Valentine Chase, made a first attempt. This attempt took 70 years to be
recognized. Chase studied the result of the battle between the gunboats of his time and he concluded
that an important factor to predict the outcome of a combat was the number of the opponent’s forces.
So, he managed to create a system of 1st Class Ordinary Differential Equation. [1]
In 1905, American B. A. Fiske had published a very simple version of equations, that later were
known as Lanchester’s equation.[2]
Later, in 1914, English engineer Frederick William Lanchester worked with the same purpose and
succeeded to establish mathematical relations between loses and the size of two opponent camps. At
first, his main purpose was to understand the tactics that were applied by the English Navy and
especially the Admiral Horace Nelson. Then he tried to predict the outcome of World War I, that had
already become, and the air fights in general. His model is considered, until these days, very effective
in many kinds of battle fields. In this way, he set the bases for the Mathematical War of Theories with
the Quadratic Law, and that’s why those differential systems, which are known as “Lanchester
Mathematical Combat Models” [1]. They are considered as the most complete and extensive.
Lanchester’s equations have similarities with Richardson law as they both use differential equation
systems [3].
Proceedings of HAICTA 2022, September 22–25, 2022, Athens, Greece
EMAIL: e.priska@gmail.com (A. 1); dimkalliv@yahoo.gr (A. 2); mchalikias@hotmail.com (A. 3)
ORCID: 0000-0003-1482-0926 (A. 3)
©️ 2022 Copyright for this paper by its authors.
Use permitted under Creative Commons License Attribution 4.0 International (CC BY 4.0).
CEUR Workshop Proceedings (CEUR-WS.org)
18
A year later, at 1915, unaware of Lanchester’s efforts, Russian M. Osipov published an article to the
tsarist military newspaper, where he presented his equations on historical events. His equations verified
the equations of Lanchester. Soviets believed that Osipov discovered differential equations before
Lanchester [4].
Later, after the completion of Interwar period and the Word War II, intensive efforts were made so
these models can be verified and extended to more historical battles. For this reason, those models were
revised with the introduction of both deterministic and not deterministic generalizations of differential
equations. So, these relations enriched. The result was the connection among different military
variables, such as the combat duration, surprise attack, nationality, leadership, level of military training,
human loses etc.
Among these efforts were those of Lewis Fry Richardson in 1948. He had formulated the
Mathematical Theory of Competitive Equipment. Through this, Lanchester’s Theory of Mathematical
Models came again to surface. The difference between those two theories is that Lanchester’s theory
describes combat between forces with the same equipment, in quantity and type, while Richardson tried
to generalize this theory in a variety of forces. We could say that the two theories are completing each
other [1].
Three years later, in 1951, P. M. Morse and G. E. Kimball tried to approach, in a deterministic way,
the combats.
J. H. Engel, in 1954, with assistance of the Differential Equation’s Model, managed to calculate the
human loses of American Forces during the battle in Zima. Engel combined the Quality Theory of
differential equations with the Lanchester’ System of differential equations. In this way, he managed to
predict the outcome of the above battle. With this project, he attracted many scientists to study even
more Lanchester’s Theory [1].
In 1961, R. L. Helmbold published some contributions on Deterministic Battle Models. Those
contributions came up after studying and analyzing 92 combats. Two years later, he extended his studies
to even more. From those studies, he concluded that the ratio of enemy’s size and victory are connected
[5],[6].
Eight years after Engel, in 1962, D. Willard studied 1500 combats that took place between 1618 and
1905. From those combats, only 1000 could contribute to study ‘s analysis. From his analysis, he came
up to a generalization of the homogeneous differential equations that contained both Lanchester’s linear
model and quadratic model as well. After his analysis, D. Willard concluded that quadratic model
should not be used on large scale [7]. Eighteen years later, in 1972, R. W. Samz came up to the same
conclusions using less information than Engel [8].
In 1977, fifteen years after Willard’s work, J. Fain studied and accomplished statistical analysis in
60 combats during World War II. She tried to apply methodology of Willard’s study on small duration
combats. Fain did not use the number of weapons that were used by each military force but only parts
into which it was divided. After that, she concluded that the shorter the battle duration is, the better the
data description could be done by the specific model [9],[10].
In 1985, for first time T. N. Dupuy referred to a Deterministic combat model. Through this model,
he introduced a new methodology. With this study, he stated that the size of opposing force could be
measured in different ways, depending on the division that could be achieved at the total weapons units
of each force [11].
A decade after Fain’s study, around 1988, Hartley studied different battles and concluded that only
the Lanchester deterministic model is capable to efficiently describe the results of the historic recorded
battles. Also, Hartley suggested a theory about Casualties. In 2001, D. S. Hartley III published a study
relevant to the prediction of the surprise attacks and the battle’s duration. [12].
2. Methodology of Lanchester’s Models and Applications in Business
In the initial Lanchester’s combat models, at time t = 0, two forces of equal martial aptitude, R(t)
and G(t), were considered to commence a battle. R(t) neutralizes g soldiers, while G(t) neutralizes r
soldiers [13]. The numbers g and r stand for R and G force efficiency coefficients, respectively [14]. In
the initial state the below equations are applied:
19
𝑑𝐺
= −𝑔𝑅
{ 𝑑𝑡 (1)
𝑑𝑅
= −𝑟𝐺
𝑑𝑡
In other versions of the above model, it is possible to add additional variables, which for example,
will describe how the number of soldiers, of the forces involved during the battle, increase or decrease
[15][16].
Chalikias & Skordoulis [16] applied Lanchester’s combat models to purchase type of cola in Greece,
where they considered the accessible products to be the variables at time t for the two firms. Businesses
A and B compete against each other and are the only ones (duopoly). It is assumed that the technology
is the same for the two businesses and cannot be changed.
They used financial data from the period 2003 to 2007, to evaluate the coefficients of the model and
compared them with real values.
Before them, Chintagunta και Vilcassim [17], utilized Lanchester’s model in order to find out the
results of promoting in duopoly. The same year, Erickson [18] used statistics to analyze the promoting
techniques of two firms. In 1997, Fruchter & Kalish [19] utilized Lanchester’s model to depict the
elements of a duopoly.
In 2016, [20] extended the model to a 3x3 differential equation system for local mobile
communications firms. The solution was:
𝑐 𝑏 𝑐
𝑥(𝑡) = −𝑐1 𝑒 −𝑡 − 𝑐2 𝑒 −𝑡 + 𝑐3 𝑒 2𝑡
𝑎 𝑎 𝑎
−𝑡 2𝑡
𝑐 (2)
𝑦(𝑡) = 𝑐2 𝑒 + 𝑐3 𝑒
𝑏
{ 𝑧(𝑡) = 𝑐1 𝑒 −𝑡 + 𝑐3 𝑒 2𝑡
In order to examine the best fitting results on real data, they applied suitable statistical analyses.
Later, the same research team expanded more the model to a 4x4 differential equation system for
Greek banks, [21],[22] as bank sector is a really competitive sector and that bank competition is closely
connected to the financial stability. They chose the four banks due to the fact that these banks gather
more than 97% of the market share.
The research team used the data set of banks’ profit between 2009 and 2015.
For the solution of the above system, it was used the Eigenvalue – Eigenvector method. The
homogeneous linear system was converted in the matrix form:
𝑑𝑋
=𝐴·𝑋 (3)
𝑑𝑡
The previous research team along with Triantafyllou and Kallivokas [23] examined the case of
healthcare companies in Athens Stock Exchange. They examined a nxn differential equation system.
The general solution of this system is:
𝑛−1
𝑎𝑛−𝑗+1 −𝑡 𝑎𝑛
𝑥1 (𝑡) = − (∑ 𝑐𝑗 ) 𝑒 + 𝑐𝑛 𝑒 (𝑛−1)𝑡
𝑎1 𝑎1
𝑗=1 (4)
𝑎
𝑥𝑗 (𝑡) = 𝑐𝑛−𝑗+1 𝑒 −𝑡 + 𝑐𝑛 𝑎𝑛 𝑒 (𝑛−1)𝑡 , j = 2, …, n-1
𝑗
𝑥𝑛 (𝑡) = 𝑐1 𝑒 −𝑡 + 𝑐𝑛 𝑒 (𝑛−1)𝑡
For the set of the examined stock data, it has been estimated the function of time, in order for the
above solution to fit in real data. There were used different functions of time for every stock due to
different monotony of every stock.
20
In order to examine the best fitting results, they applied suitable statistical analyses (Wilcoxon test
due to the fact that normality was not satisfied).
It must be mentioned that except Lanchester’s and Richardson models many business applications
have to do with probabilistic war theories [25][26].
3. Further Ideas
Moreover, generalized Lanchester’s model showed up to portray the advancement of the fight
conducted between military powers that use the same weapons with no help or working misfortune.
The rate of misfortune of each side is relative to the number of remaining units of weapon of the rival
side.
If there are two rival armed military forces [14] then we can assume that force 1 and force 2 have
(1) (2) (3) (𝛭 )
𝑀1 and 𝑀2 , respectively, different sorts of weapons 𝛰𝛴1 , 𝛰𝛴1 , 𝛰𝛴1 , … . , 𝛰𝛴1 1 and
(1) (2) (3) (𝛭 )
𝛰𝛴2 , 𝛰𝛴2 , 𝛰𝛴2 , … . , 𝛰𝛴2 2 .
This conducts to a system that can be described from the expression: 𝑋̇ = 𝛩 𝛸 [14].
4. Research Results and Discussion
From the above applications of Lanchester’s model, it was determined that such theory can be
applied to modern businesses, due to the fact that firms compete with one another such as a warfare on
a battlefield. In this case, we count their market share instead of the battles win [27].
As a result, models of mathematical war theories can be employed in cases involving modern
enterprises, after necessary adjustments and the adoption of relevant theoretical conditions.
5. References
[1] Wrigge, S., Fransen, A., & Wigg, L. (1995). The Lanchester Theory of Combat and Some Related
Subjects. A Bibliography 1900 - 1993. FOA.
[2] Fiske, B. A. (1905). American Naval Policy (Vol 113).
[3] Chalikias, M., & Skordoulis, M. (2014). Implementation of Richardson’s Arms Race Model,
Applied Mathematical Sciences, 8(81), 4013-4023.
[4] Osipov, M. (1991). The Influence of the Numerical Strength of Engaged Forces on Their
Casualties. Translated from Russian by Helmbold, R.L & Rehm, A.S.
[5] Helmbold, R. L. (1961). Lanchester Parameters for Some Battles of the Last Two Hundred Years.
Virginia: Combat Opns Res Group.
[6] Helmbold, R. L. (July 1961). Historical Data and Lanchester's Theory of Combat. Virginia:
Combat Opns Res Group.
[7] Willard, D. (1962, November). Lanchester as a Force in History: An Analysis of Land Battles of
the Years 1618 - 1905. Res Analysis Corp, p. 37.
[8] Samz, R. W. (1972, January - February). Some Comments on Engel's "A Verification of
Lanchester's Law. Opns Res, Vol 20, p. 49 - 52.
[9] Fain, J. B. (1977). The Lanchester Equations and Historical Warfare: An Analysis of Sixty World
War II Land Engagements (Vol. 1). History, Numbers and War.
[10] Fain, W. W., Fain, J. B., Feldman, L., & Simon, S. (1970). Validation of Combat Models against
Historical Data. Center for Naval Analyses.
[11] Dupuy, T. (1987). Understanding War. History and Theory of Combat. Paragon Publishers, p. 312.
[12] Hartley III, D. S. (1988). Non - Validation of Lanchester's Original Equations in the Inchon - Seoul
Campaign of the Korean War. Tennessee: Oak Ridge National Lab.
[13] McKay, N. (2006). Lanchester combat models. Math Today, 42, pp. 170-178.
[14] Daras, N. (2001). Differential equations and mathematical theories of war (Τόμ. Volume II).
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[15] Lanchester, F. W. (1914). Aircraft in Warfare: The Dawn of the Fourth Arm - Vol V. The Principle
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[17] Chintagunta, P., & Vilcassim, N. (1992). An empirical investigation of advertising strategies in a
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[19] Fruchter, G., & Kalish, S. (1997). Closed-loop advertising strategies in a duopoly. Manag Sci,
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[21] Chalikias, M., Lalou, P., & Skordoulis, M. (2016b). Modeling a bank data set using differential
equations: the case of the Greek banking sector. Proceedings of 5th International Symposium and
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[22] Chalikias, M., Lalou, P., Skordoulis, M., Papadopoulos, P. & Fatouros, S. (2020). Bank oligopoly
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[24] Lalou, P., Chalikias, M., Skordoulis, M., Papadopoulos, P., & Fatouros, S. (2016, June). A
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