Abstract
In this research, we try to improve Bracken's and Chen's work to significantly better fit our extended Lanchester model into the Ardennes Campaign live data. Essentially, we adopt the concepts of the tactical factor variable and the shift time variable to improve the original Lanchester's model. Moreover, we use the Lanchester square law model instead of Lanchester linear law model to reflect the fact that the Ardennes Campaign was not an indirect-fire but a direct-fire combat. According to our numerical experimental result, we improved Bracken's work by 39.26%, and Chen's work by 19.51%. The contribution of this research is that we propose a much better qualitative analysis model for the explanation of modern combat.
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Appendix A: Proof of Proposition 1
Appendix A: Proof of Proposition 1
First we compute ∂SSE k /∂a, ∂SSE k /∂b and ∂SSE k /∂d
Second we solve the critical points of SSE k (a, b, d).
By Equations (A.1), (A.2) and (A.3), with notation of g(i, k), we know that
If we substitute the expressions for a and b from Equations (A.4) and (A.5) into Equation (A.6), then we get an equation in d only. That equation can be written as
We now consider the positive roots of Equation (A.7). Let f(d)=∑j=06θ
j
d2j. From θ6>0, we have that 
. On the other hand, we know that f(0)=θ0<0. Since f(d) is a continuous function, we deduce that f(d) has positive roots.
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Hung, CY., Yang, G., Deng, P. et al. Fitting Lanchester's square law to the Ardennes Campaign. J Oper Res Soc 56, 942–946 (2005). https://doi.org/10.1057/palgrave.jors.2601878
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DOI: https://doi.org/10.1057/palgrave.jors.2601878