Optimal commitment of forces in some Lanchester-type combat models

This paper shows that one can determine whether or not it is beneficial for the victor to initially commit as many forces as possible to battle in Lanchester-type combat between two homogeneous forces by considering the instantaneous casualty-exchange ratio. It considers the initial-commitment decision as a one-sided static optimization problem and examines this non-linear program for each of three decision criteria (victor's losses, loss ratio, and loss difference) and for each of two different battle-termination conditions (given force-level breakpoint and given force-ratio breakpoint). The paper's main contribution is to show how to determine the sign of the partial derivative of the decision criterion with respect to the victor's initial force level for general combat dynamics without explicitly solving the Lanchester-type combat equations. Consequently, the victor's optimal initial-commitment decision many times may be determined from how the instantaneous casualty-exchange ratio varies with changes in the victor's force level and time. Convexity of the instantaneous casualty-exchange ratio is shown to imply convexity of the decision criterion so that conditions of decreasing marginal returns may be identified also without solving the combat equations. The optimal initial-commitment decision is shown to be sensitive to the decision criterion for fixed force-ratio breakpoint battles. (Author)