Optimal Commitment of Forces in Some Lanchester-Type Combat Models

This paper shows that by considering the instantaneous casualty-exchange ratio one can determine whether it is beneficial for the victor to commit as many forces as possible to battle initially in Lanchester-type combat between two homogeneous forces. It considers the initial-commitment decision as a one-sided static optimization problem and examines this nonlinear 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 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 may often 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.