Contributions to Lanchester combat theory

Abstract

The purpose of the dissertation is to review and supplement existing Lanchester theory and to develop a model which may be used to describe and explain a number of real processes. The processes which have been given most emphasis are in the field of military conflict, but this does not preclude the application to other areas.

The model considered has the unusual feature of replacement of forces in the field from reserves, which at least for some conflict applications, adds one further dimension of realism. Both sides deploy only a constant fraction of their initial strengths actually in the field, the remainder being held in reserve and used to replace casualties. This phase continues until only one side can replace and finally there may follow a phase which is in classical Lanchester tradition with no replacement possible.

The form of the model, expressed deterministically is dm/dt = - µmn - δn, dn/dt = - λmn - γn, where m and n are the number of fighting units actively fighting at time t andµ, o, A and y are attrition coefficients. The form of this model is such that the Linear and Mixed laws arise as special cases.

Using the deterministic method of analysis, expressions are obtained for the determination of the victor, the number of survivors for each side, the duration of the battle and the number of survivors as a function of time. These conditions and expressions are basically simple in form.

This simplicity of form does not extend however to the results of the stochastic analysis. A new approach is used to obtain the fundamental distributions which has the advantage over previous methods in that the moments of the distribution of the duration of the battle are obtainable. This approach is developed not only for this particular model but also for a general class of models. Expressions are obtained for the victory probabilities, distribution of survivors, the generating function of the distribution of survivors and the moments of the distribution of the duration of the battle. Approximations, for the specific model, are obtained for the victory probabilities and the distribution of survivors.

The general conclusion concerning a comparison between the two methods of analysis is that, where appropriate, the deterministic approach is increasingly valid as an approximation to the stochastic approach as the number of units available to each side increases.

A discussion of tactics reveals that both antagonists have a great deal of control over the duration of the battle but that the A force alone controls the victory probabilities and the distribution of survivors once the initial number of units and their effectiveness (i.e. the attrition coefficients) have been fixed.