Irrelevance of the fractal dimension term in the fractal attrition equation


  • Lesley Richard Kosowski
  • Adrian Pincombe
  • Brandon Pincombe



military operations research, agent based simulation


The modified Fractal Attrition Equation (mFAE) models the casualties produced by Map Aware Non-uniform Automata (MANA), an agent based combat modelling distillation, at each MANA time step. The mFAE has three important differences from the simple early twentieth century models of casualties that used only the numbers on each side to make predictions. Firstly, only those agents within range of the enemy may inflict casualties. Secondly, the detection range is assumed to be greater than the range of weapons and a fitting factor assumed to represent the gap between these two ranges, is introduced. This builds in one of the assumptions of Network Centric Warfare, that you will be able to see your enemies before they can shoot at you. Thirdly, and novelly, a fractal dimension is introduced. We postulate that the important part of the information used in calculating the fractal dimension has already been incorporated into the model through the consideration of range. We test this hypothesis by comparing the outcomes of the mFAE with and without the fractal term on three scenarios: that used by the developers of the mFAE; best practice MANA tactics from the literature; and a rout scenario. When the two models are scaled to fit the MANA casualties there is no significant difference in fit. We conclude that the fractal term in the mFAE is redundant. References
  • J. V. Chase, Sea fights, a mathematical investigation of the effect of superiority of force in. RG 8, Box 109, XTAV(1902), Naval War College Archives.
  • F. W. Lanchester, Aircraft in warfare: the dawn of the fourth arm---No. V, The principle of concentration, Engineering 98: 422--423, (1914).
  • N. J. MacKay, Lanchester combat models, Mathematics Today, 42, pp.170--173, 2006.
  • T. W. Lucas and T. Turkes Fitting Lanchester models to the battles of Kursk and Ardennes. Nav Res Log 51: 95--116 (2004).
  • G. C. McIntosh and M. K. Lauren, Incorporating fractal concepts into equations of attrition for military conflicts, Journal of the Operational Research Society, 2(59): 703--713, (2008). doi:10.1057/palgrave.jors.2602383
  • M. K. Lauren Firepower concentration in cellular automaton combat models---an alternative to Lanchester, Journal of the Operational Research Society, 53: 672--679, (2002). doi:10.1057/palgrave.jors.2601355
  • M. K. Lauren, J. M. Smith, J. Moffat and N. D. Perry Using the fractal attrition equation to construct a metamodel of the MANA cellular automaton combat model, The Technical Cooperation Program, Joint Systems and Analysis Group, Technical Panel 3, November 2005.
  • G. C. McIntosh, D. P. Galligan, M. A. Anderson and M. K. Lauren, MANA (Map Aware Non-uniform Automata) Version 4 User Manual, DTA Technical Note 2007/3 NR 1465.
  • J. Moffat, J. Smith and S. Witty, Emergent behaviour: Theory and experimentation using the mana model, Journal of Applied Mathematics and Decision Sciences---JAMDS , 2006, pp.1--14, 2006. doi:10.1155/JAMDS/2006/54846
  • D. R. Shine, An Exploratory Study of the Army-as-a-System Core Skills: Comparing the Effectiveness of Warfighting Tactics Using MANA, DSTO-TR-1663, (2005), DSTO (Australia).
  • P. Grassberger and I. Procaccia, Characterization of strange attractors, Phys Rev Lett 50(5): 346--349 (1983). doi:10.1103/PhysRevLett.50.346
  • J. G. Eisenhauer, Regression through the Origin, Teaching Statistics, 25(3): 76--80, (2003).





Proceedings Computational Techniques and Applications Conference