Of Horseless Carriages, Flying Machines, and Operations Research

Abstract

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Force‐annihilation conditions for variable‐coefficient lanchester‐type equations of modern warfare

“…Besides satisfying (16) and (17), Ex+ and Ex-are linearly independent solutions to the X force-level equation…”

“…In the initial conditions of (16) and ( It follows that the solution to (11) with initial conditions (2) is given by…”

“…whrrr f -0 denotes the time a t which the battle begins, z(t) and y(t) denote the numbers of X and P at time t , and a and b are nonnegative constants which are today called Lunchester attrition-rate coefficients and represent each side's fire effectiveness. Lanchester (see McCloskey [16] for his influence on operations research) considered this model (1) in order to provide insight into the dynamics of combat under "modern conditions" and justify the principle of concentration. t We will nccordinglj refer t o (1) as Lunchester's equations of modern warfare.…”

“…We want the solutions Ex+ and Ey+ to (16) and Exand Eyto (17) to satisfj-the given initial conditions, to be continuous, and to act like exponentialst for general a(t) and b ( f ) . Accordingly, we must further restrict a(t) and b(t) slightly.…”

“…In the initial conditions of (16) and (17), b=max(toX, toY), where toX denotes the largest finite singular point on the t-axis for (18) (see Taylor and Brown [%I; also p. 69 of Ince [ll]),tt. Since Ex+ and Exare linearly independent, we may use them to construct the general solution to (18).…”