Approximations and empirics for stochastic war equations
ABSTRACT The article develops a theorem which shows that the Lanchester linear war equations are not in general equal to the Kolmogorov linear war equations. The latter are time-consuming to solve, and speed is important when a large number of simulations must be run to examine a large parameter space. Run times are provided, where time is a scarce factor in warfare. Four time efficient approximations are presented in the form of ordinary differential equations for the expected sizes and variances of each group, and the covariance, accounting for reinforcement and withdrawal of forces. The approximations are compared with “exact” Monte Carlo simulations and empirics from the WWII Ardennes campaign. The band spanned out by plus versus minus the incremented standard deviations captures some of the scatter in the empirics, but not all. With stochastically varying combat effectiveness coefficients, a substantial part of the scatter in the empirics is contained. The model is used to forecast possible futures. The implications of increasing the combat effectiveness coefficient governing the size of the Allied force, and injecting reinforcement to the German force during the Campaign, are evaluated, with variance assessments. © 2005 Wiley Periodicals, Inc. Naval Research Logistics, 2005.
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ABSTRACT: This article constructs a foundation for warfare at the individual level, where agents in two groups fire and absorb shots according to a non-stationary Poisson process. We determine for generalized forms of warfare the conditional and unconditional point probabilities of a certain number of agents in each group through time, and the conditional and unconditional expected sizes and variances. Conditional variables are especially useful in modern warfare since these allow for updated intelligence. We determine the conditions for discrepancies between the stochastic version and the associated Lanchester model. Correspondence is demonstrated for square warfare for large groups where the probability that a group goes extinct is negligible. For linear warfare equivalence occurs for the conditional case, whereas for the unconditional case correspondence arises at the limit where the covariance of the group sizes approaches zero. Finally the stochastic model is tested against newly released empirics for the Ardennes Campaign during World War II.European Journal of Operational Research. 01/2002;
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ABSTRACT: A detailed data base of the Ardennes campaign of World War II (December 15, 1944 through January 16, 1945) has recently been developed. The present article formulates four Lanchester models of the campaign and estimates their parameters for these data. Two-sided time histories of warfare on battles and campaigns are very rare, so Lanchester models have seldom been validated with historical data. The models are homogeneous in that tanks, armored personnel carriers, artillery, and manpower are weighted to yield a measure of strength of the Allied and German forces. This weighting is utilized for combat power and for losses. The models treat combat forces in the campaign (including infantry, armor, and artillery manpower) and total forces in the campaign (including both combat manpower and support manpower.) Four models are presented. Two models have five parameters (Allied individual effectiveness, German individual effectiveness, exponent of shooting force, exponent of target force, and a tactical parameter reflecting which side is defending and attacking.) The other two models remove the tactical parameter, which is not generally known prior to warfare, and estimate the other parameters without the tactical parameter. The main results of the research are (a) the Lanchester linear model fits the Ardennes campaign data in all four cases, and (b) when combat forces are considered Allied individual effectiveness is greater than German individual effectiveness, whereas when total forces are considered Allied and German individual effectiveness is the same. The interpretation of the latter result is that the two sides had essentially the same individual capabilities but were organized differently—the Allies chose to have more manpower in the support forces, which yielded greater individual capabilities in the combat forces. The overall superiority of the Allies in the campaign led to the attrition to the Allies being a smaller portion of their forces. © 1995 John Wiley & Sons, Inc.†Naval Research Logistics 05/1995; 42(4):559 - 577. · 0.69 Impact Factor
- Naval Research Logistics - NAV RES LOG. 01/1998; 45(1):1-22.