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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

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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

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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

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Daily Personnel Casualties

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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.”

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Daily On-hand Tanks

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Daily Tank Losses

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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)

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

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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

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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.”

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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)

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Plot of Cumulative Findings

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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)

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

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

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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

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

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Kursk ACUD Basic Lanchester Models Lanchester LawWeightspqdR2R2 SquareBracken’s LinearBracken’s LogarithmicBracken’s Optimum fitBracken’s SquareManpower LinearManpower LogarithmicManpower Optimum fitManpower

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Kursk ACUD, Different Weights

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

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

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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

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Ardennes Basic Lanchester Models Bracken’s formulation (10 days) Lanchester LawWeightspqdR2R2 SquareBracken’s LinearBracken’s LogarithmicBracken’s Optimum fitBracken’s.91– SquareManpower LinearManpower111.06–.226 LogarithmicManpower Optimum fitManpower.15–

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

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

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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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