This monograph is a comprehensive treatise on Lanchester-type models of warfare, i. Its goal is to provide an introduction to and current- state-of-the-art overview of Lanchester-type models of warfare as well as a comprehensive and unified in-depth treatment of them. Both deterministic as well as stochastic models are considered. Such models have been widely used in the United States and elsewhere for the modeling of force-on-force attrition over the complete spectrum of combat operations, from combat between platoon-sized units through theater-level air-ground combat.

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Lanchester's laws are mathematical formulae for calculating the relative strengths of military forces. The Lanchester equations are differential equations describing the time dependence of two armies' strengths A and B as a function of time, with the function depending only on A and B.

Osipov independently devised a series of differential equations to demonstrate the power relationships between opposing forces. Among these are what is known as Lanchester's Linear Law for ancient combat and Lanchester's Square Law for modern combat with long-range weapons such as firearms. For ancient combat, between phalanxes of soldiers with spears , say, one soldier could only ever fight exactly one other soldier at a time. If each soldier kills, and is killed by, exactly one other, then the number of soldiers remaining at the end of the battle is simply the difference between the larger army and the smaller, assuming identical weapons.

The linear law also applies to unaimed fire into an enemy-occupied area. The rate of attrition depends on the density of the available targets in the target area as well as the number of weapons shooting.

If two forces, occupying the same land area and using the same weapons, shoot randomly into the same target area, they will both suffer the same rate and number of casualties, until the smaller force is eventually eliminated: the greater probability of any one shot hitting the larger force is balanced by the greater number of shots directed at the smaller force. With firearms engaging each other directly with aimed shooting from a distance, they can attack multiple targets and can receive fire from multiple directions.

The rate of attrition now depends only on the number of weapons shooting. Lanchester determined that the power of such a force is proportional not to the number of units it has, but to the square of the number of units. This is known as Lanchester's square law. More precisely, the law specifies the casualties a shooting force will inflict over a period of time, relative to those inflicted by the opposing force. In its basic form, the law is only useful to predict outcomes and casualties by attrition.

It does not apply to whole armies, where tactical deployment means not all troops will be engaged all the time. It only works where each unit soldier, ship, etc.

For this reason, the law does not apply to machine guns, artillery, or nuclear weapons. The law requires an assumption that casualties accumulate over time: it does not work in situations in which opposing troops kill each other instantly, either by shooting simultaneously or by one side getting off the first shot and inflicting multiple casualties.

Note that Lanchester's square law does not apply to technological force, only numerical force; so it requires an N-squared-fold increase in quality to compensate for an N-fold decrease in quantity.

Suppose that two armies, Red and Blue are engaging each other in combat. Red is shooting a continuous stream of bullets at Blue. Meanwhile, Blue is shooting a continuous stream of bullets at Red. Let symbol A represent the number of soldiers in the Red force at the beginning of the battle. Lanchester's square law calculates the number of soldiers lost on each side using the following pair of equations.

A negative value indicates the loss of soldiers. Lanchester's equations are related to the more recent salvo combat model equations, with two main differences. First, Lanchester's original equations form a continuous time model, whereas the basic salvo equations form a discrete time model. In a gun battle, bullets or shells are typically fired in large quantities. Each round has a relatively low chance of hitting its target, and does a relatively small amount of damage.

Therefore, Lanchester's equations model gunfire as a stream of firepower that continuously weakens the enemy force over time. By comparison, cruise missiles typically are fired in relatively small quantities.

Each one has a high probability of hitting its target, and carries a relatively powerful warhead. Therefore, it makes more sense to model them as a discrete pulse or salvo of firepower in a discrete time model. Second, Lanchester's equations include only offensive firepower, whereas the salvo equations also include defensive firepower.

Given their small size and large number, it is not practical to intercept bullets and shells in a gun battle. By comparison, cruise missiles can be intercepted shot down by surface-to-air missiles and anti-aircraft guns. So it is important to include such active defenses in a missile combat model.

Lanchester's laws have been used to model historical battles for research purposes. Examples include Pickett's Charge of Confederate infantry against Union infantry during the Battle of Gettysburg , [4] and the Battle of Britain between the British and German air forces.

In modern warfare, to take into account that to some extent both linear and the square apply often, an exponent of 1. From Wikipedia, the free encyclopedia. Part of a series on War History. Prehistoric Ancient Post-classical Early modern Late modern industrial fourth-gen. Blitzkrieg Deep operation Maneuver Operational manoeuvre group.

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## Lanchester Models of Warfare, Volume I

Lanchester's laws are mathematical formulae for calculating the relative strengths of military forces. The Lanchester equations are differential equations describing the time dependence of two armies' strengths A and B as a function of time, with the function depending only on A and B. Osipov independently devised a series of differential equations to demonstrate the power relationships between opposing forces. Among these are what is known as Lanchester's Linear Law for ancient combat and Lanchester's Square Law for modern combat with long-range weapons such as firearms. For ancient combat, between phalanxes of soldiers with spears , say, one soldier could only ever fight exactly one other soldier at a time.

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## Lanchester's laws

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