# Fitting Lanchester Models to the Battles of Kursk (and Ardennes) by Tom Lucas, Turker Turkes, Ramazon Gozel, John Dinges Naval Postgraduate School, Monterey.

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Fitting Lanchester Models to the Battles of Kursk (and Ardennes) by Tom Lucas, Turker Turkes, Ramazon Gozel, John Dinges Naval Postgraduate School, Monterey CA Dr. Frederick W. Lanchester

Outline Overview of the Battle of Kursk Some Previous Research on Fitting Lanchester Equations to Battle Data Lanchester and the Battle of Kursk (and Ardennes)  More details are in the students’ theses and two summary papers

The Battle of Kursk “This attack is of decisive importance. It must succeed, and it must do so rapidly and convincingly. It must secure for us the initiative for this spring and summer. The victory of Kursk must be a blazing torch to the world.”— Hitler, 2 July 1943

Daily Personnel Casualties

Cumulative Personnel Casualties Field Marshal Erich von Manstein: “The last German offensive in the East ended in a fiasco, even though the enemy…suffered four times their losses.”

Daily On-hand Tanks

Daily Tank Losses

Outline Overview of the Battle of Kursk Some Previous Research on Fitting Lanchester Equations to Battle Data Lanchester and the Battle of Kursk (and Ardennes)

Lanchester’s equations: The most important model of combat? Dr. Frederick W. Lanchester

Bracken’s Generalized Homogeneous Lanchester Model If p = 1 and q = 0  Square law If p = 1 and q = 1  Linear law If p = 0 and q = 1  Logarithmic law If p = 1 and q = 0  Square law If p = 1 and q = 1  Linear law If p = 0 and q = 1  Logarithmic law

Validation Efforts: Before and After Battle Data Lots of efforts –Osipov (1915), Willard (1962), Weiss (1966), Fain (1977), Peterson (1967) Contradictory findings –Schneider (1985): “At best, one can say that the results of these studies have been contradictory” Hartley’s (2001) conclusions –“The homogeneous linear-logarithmic (p =.45, q =.75) Lanchestrian attrition model is validated to the extent possible with current initial and final size data.” –“Two-sided daily attrition data on a large number of battles are needed to absolutely confirm these results.”

Validation Efforts: Time-phased Battle Data Iwo Jima –Engle (1954) Inchon-Seoul –Busse (1971), Hartley and Helmbold (1995) Ardennes –Bracken (1995), Fricker (1998), Wiper et al. (2000)

Plot of Cumulative Findings

Outline Overview of the battle of Kursk Some Previous Research on Fitting Lanchester Equations to Battle Data Lanchester and the Battle of Kursk (and Ardennes)

Pair-wise Scatter Plots on Combat Power and Combat Power Losses

Three Kursk Data Sets: Manpower Averages as a Function of Unit Status

Approach to Finding Best-fitting p and q Find best-fitting response surface as a function of p and q –Given p and q, it turns out to be relatively easy to find a, b, and d to minimize SSR –For a fixed d, solve for a and b by regression through the origin –Through a one-dimensional line search on d, we find the d, and associated a and b, that minimize SSR for the given p and q

Kursk ACUD Surface (Bracken’s Weights) R 2 = 0.237 * With d (= 1.028), R 2 = 0.238

Kursk ACUD Basic Lanchester Models Lanchester LawWeightspqdR2R2 SquareBracken’s101.09.081 LinearBracken’s111.02.131 LogarithmicBracken’s011.02.085 Optimum fitBracken’s5.871.011.03.238 SquareManpower101.11.074 LinearManpower111.04.116 LogarithmicManpower011.04.086 Optimum fitManpower7.743.41.86.234

Kursk ACUD, Different Weights

Kursk ACUD Change Points: Four Different d’s  8 parameters with 14 days of data  * = different SST

Kursk Manpower Fits as a Function of Contact Status The best-fitting Lanchester law is p = 1.156 and q = 1.000, with an R 2 of.624. x x xx x x

Assessing the Model Fit A comparison of the estimated losses (Soviets = SLest and Germans = GLest) with the actual losses (Soviets = SLact and Germans = GLact) for the best fitting constant attrition coefficient Lanchester linear law using the FCUD data

Ardennes Basic Lanchester Models Bracken’s formulation (10 days) Lanchester LawWeightspqdR2R2 SquareBracken’s101.14.367 LinearBracken’s111.17.291 LogarithmicBracken’s011.23.330 Optimum fitBracken’s.91–.611.12.381 SquareManpower101.04.079 LinearManpower111.06–.226 LogarithmicManpower011.10.025 Optimum fitManpower.15–.901.05.280

Ardennes 10 Day Surface R 2 = 0.380  Square law fits best  p* =.9, q* = -.6, d* = 1.125

Fricker’s 32 day Surface R 2 = 0.500  Logarithmic law fits best  p* = -.2, q* = 5.0, d* = 1.23

Conclusions Much better fits are obtained with Fighting Units only The data provide no conclusive differentiation among the basic Lanchester models (though Linear and Logarithmic better than Square) Much more of the variation in casualties is explained by the phases of the battle

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