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Frederick Lanchester, a British mathematician, tried to apply mathematical analysis to warfare. Mathematics in the form of operational research or logistics has some very practical applications for the military. Lanchester was interested in more abstract analysis of warfare. For example, there is the Principle of Concentration that says that the best strategy is to concentrate the whole of a belligerent forces on a definite objective. Lanchester's analysis provides a justification of that principle.
Lanchester developed his analysis from a set of differentinal equations. Let n_{1} and n_{2} be the numerical strengths of two military forces. These are the number of fighting units. They could be the number of infantry soldiers or the the number of tanks in a tank battle. The rate of casualities for the two forces are then:
where c_{1} and c_{2} are coefficients that reflect the effectiveness of of the units of forces 1 and 2, respectively.
Lanchester then asked the question of what condition determines the fighting strength of two forces. He argued that the two strengths are equal if both suffer the same proportional losses; i.e.,
This condition, combined with the differential equations, then implies that
Thus the fighting strengths of the two forces are equal when the products of the squares of the numerical strengths times the coefficients of effectiveness are equal. In other words, the strength of a fighting force is equal to the product of the square of numerical strength times the effectiveness of an individual fighting unit, c_{i}n_{i}^{2}.
This justifies the Principle of Concentration. In other terms, there are economies of scale in military strength.
Lanchester illustrates the implications of this deduction by considering the case in which a machine gunner has the effectiveness of sixteen riflemen. He then asks how many machine gunners would be required to replace 1000 riflemen. By his calculation the number is
Lanchester also considers alternate fighting conditions. Suppose firepower is directed at positions rather than individual soldiers or other fighting units. The casaulities would then be proportional to the density of the force as well as the rate of fire. Thus,
where a_{i} is the area over which force i is deployed.
The above two equations reduce to
Hence equality of strength exists when
Thus an application of the previous analysis indicates that under these conditions the strength of a force is proportional to its numerical size rather than the square of the numerical size. But the analysis also indicates that its strenghth is proportional to the area of its deployment.
In other conditions, such as the defense of a narrow pass, the strength of a force may have little to do with its numerical strength. This would be the situation in mountainous terrain.
In summary, Lanchester's analysis of warfare indicates that the strength of a fighting force is of the form
where D reflects the dimensionality of the fighting situation. For fighting units targeting fighting units D=2; for fighting units targeting areas D=1 and for a narrow pass D=0.
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