Operations-Research-Spektrum

, Volume 1, Issue 3, pp 199–205 | Cite as

Dependence of the parity-condition parameter on the combat-intensity parameter for Lanchester-type equations of modern warfare

  • J. G. Taylor
Theoretical Paper

Summary

This paper studies the parametric dependence of battle outcome (in particular, force annihilation) for combat between two homogeneous forces modelled by Lanchester-type equations of modern warfare with time-dependent attrition-rate coefficients. Force-annihilation prediction has been shown to depend on a so-called parity-condition parameter, which depends on only the attrition-rate coefficients. New important results are given on how the parity-condition parameter depends on the intensity of combat and the relative fire effectiveness of the combatants. Previous analytical results of the author are shown to apply to a much wider class of attrition-rate coefficients. These new results allow a wide class of Lanchester-type equations of modern warfare with temporal variations in fire effectiveness to be studied almost as easily as Lanchester's classic constant-coefficient model.

Keywords

Temporal Variation Wide Class Parametric Dependence Modern Warfare Fire Effectiveness 

Zusammenfassung

Diese Arbeit untersucht die Parameter-Abhängigkeit von Ergebnissen kriegerischer Auseinandersetzungen zweier homogener Gegner (insbesondere Vernichtung eingesetzter Kräfte), die mit Gleichungen der Lanchester-Theorie mit zeitabhängigen Verlustraten modelliert werden. Es wurde gezeigt, daß die Vorhersage der Vernichtung von Kräften von sogenannten Paritätsbedingungen nur von Verlustraten abhängen. Neue wichtige Ergebnisse geben Auskunft, wie diese Paritätsbedingungen von der Kampfintensität und der relativen Feuereffektivität der Gegner abhängen. Bisherige analytische Ergebnisse des Autors können Anwendung in einer viel umfangreicheren Art von Verlustraten finden. Diese neuen Ergebnisse erlauben es, daß eine umfangreiche Art von Gleichungen der LanchesterTheorie moderner Kriegsführung mit zeitlicher Feuereffektivitätsveränderung beinahe so leicht untersucht werden können, wie das klassische Modell von Lanchester mit konstanter Verlustrate.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Bieberbach L (1965) Theorie der gewöhnlichen Differentialgleichungen, 2. Auflage. Springer-Verlag, Berlin Heidelberg New YorkGoogle Scholar
  2. 2.
    Bonder S (1965) A theory for weapon systems analysis. In: Proceedings of the Fourth Annual US Army Operations Research Symposium, Redstone Arsenal, Alabama, pp 111–128Google Scholar
  3. 3.
    Bonder S (1967) The Lanchester attritionrate coefficient. Oper Res 15:221–232Google Scholar
  4. 4.
    Bonder S (1970) The mean Lanchester attrition rate. Oper Res 18: 179–181Google Scholar
  5. 5.
    Bonder S (1971) Systems analysis: A purely intellectual activity. Mil Rev 51, no. 2: 14–23Google Scholar
  6. 6.
    Bonder S, Farrell RL (eds) (1970) Development of models for defense systems planning. Report No. SRL 2147 TR 70-2 (U), Systems Research Laboratory, The University of Michigan, Ann Arbor, MichiganGoogle Scholar
  7. 7.
    Bostwick SP, Brandi FX, Burnham CA, Hurt JJ (1974) The interface between DYNTACS-X and Bonder-IUA. In: Proceedings of the Thirteenth Annual U.S. Army Operations Research Symposium, Fort Lee, Virginia, pp 494-502Google Scholar
  8. 8.
    Clark GM (1969) The combat analysis model. Ph. D. Thesis. The Ohio State University, Columbus, OhioGoogle Scholar
  9. 9.
    Dolansky L (1964) Present state of the Lanchester theory of combat. Oper Res 12: 344–358Google Scholar
  10. 10.
    Farrell RL (1975) VECTOR 1 and BATTLE: Two versions of a high-resolution ground and air theater campaign model. In: Huber RK, Jones LF, Reine E (eds) Military strategy and tactics. Plenum Press, New York, pp 233–241Google Scholar
  11. 11.
    Huber RK, Jones LF, Reine E (eds) (1975) Military strategy and tactics. Plenum Press, New YorkGoogle Scholar
  12. 12.
    Kamke E (1944) Differentialgleichungen, Lösungsmethoden und Lösungen, Band 1, Gewöhnliche Differentialgleichungen, 3. Aufl Akademische Verlagsgesellschaft, Leipzig (reprinted: 1971 Chelsea, New York)Google Scholar
  13. 13.
    Lanchester FW (1914) Aircraft in warfare: The dawn of the fourth arm-No. V, the principle of concentration. Engineering 98:422–423 (reprinted: 1956) in: Newman J (ed) The world of mathematics, Vol. IV.: Simon and Schuster, New York, pp 2138–2148Google Scholar
  14. 14.
    Schwarz HA (1872) Über diejenigen Fälle, in welchen die Gaussische hypergeometrische Reihe eine algebraische Function ihres vierten Elementes darstellt. J. Reine Angew. Math. 75:292–335 (= Ges. Abh. Bd. 2, S. 211–259)Google Scholar
  15. 15.
    Taylor JG (1974) Solving Lanchester-type equations for “modern warfare“ with variable coefficients. Oper Res 22:756–770Google Scholar
  16. 16.
    Taylor JG (1979) Recent developments in the Lanchester theory of combat. In: Haley KB (ed) Operational Research '78. Proceedings of the Eighth IFORS International Conference on Operational Research. North-Holland, Amsterdam, pp 773–806Google Scholar
  17. 17.
    Taylor JG (1979) Attrition modelling. In: Huber RK, Niemeyer K, Hofmann HW (eds) Operations-analytische Spiele für die Verteidigung. R. Oldenbourg, München, pp 139–189Google Scholar
  18. 18.
    Taylor JG (1979) Prediction of zero points of solutions to Lanchester-type differential combat equations for modern warfare. SIAM J Appl Math 36:438–456Google Scholar
  19. 19.
    Taylor JG (1980) Force-on-force attrition modelling. Military Applications Section of Operations Research Society of America, Arlington, VirginiaGoogle Scholar
  20. 20.
    Taylor JG, Brown GG (1976) Canonical methods in the solution of variable-coefficient Lanchester-type equations of modern warfare. Oper Res 24:44–69Google Scholar
  21. 21.
    Taylor JG, Comstock C (1977) Force-annihilation conditions for variable-coefficient Lanchester-type equations of modern warfare. Nav Res Log Qu 24: 349–371Google Scholar
  22. 22.
    Taylor JG, Parry SH (1975) Force-ratio considerations for some Lanchester-type models of warfare. Oper Res 23:522 to 533Google Scholar
  23. 23.
    Wallis PR (1968) Recent developments in Lanchester theory. Oper Res Qu 19: 191–195Google Scholar
  24. 24.
    Weiss HK (1957) Lanchester-type models of warfare. In: Davies M, Eddison RJ, Page T (eds) Proceedings of the First International Conference on Operational Research. Operations Research Society of America, Baltimore, pp 82–98Google Scholar
  25. 25.
    Weiss HK (1959) Some differential games of tactical interest and the value of a supporting weapon system. Oper Res 7:180–196Google Scholar

Copyright information

© Springer-Verlag 1980

Authors and Affiliations

  • J. G. Taylor
    • 1
  1. 1.Department of Operations ResearchNaval Postgraduate SchoolMontereyUSA

Personalised recommendations