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Homogeneous Combat Models

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Military Operations Research

Part of the book series: International Series in Operations Research & Management Science ((ISOR,volume 5))

Abstract

In the beginning of the twentieth century, attempts were made to explain the dynamics of combat through mathematical models. Chase (see Fisk 1905, 1916) seems to be the first to present a model of warfare. Later, simultaneous and independent attempts were made by Lanchester (1914) and Osipov (1915) (see Helmbold 1993) to describe the combat dynamics through differential equations. Due to the logical simplicity, Lanchester models (one may be tempted to call them Chase-Lanchester-Osipov or CLO models; see Bitters 1995) provide an attractive basis for explaining the dynamics of combat and have been used extensively. A brief account of Lanchester type of equations is given in what follows:

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References

  • Allen, Patrick, Gillogly, Jim and Dewar, Jim, Non-Monotonic Effects in Models with Stochastic Thresholds, PHALANX, 16–29, December 1993.

    Google Scholar 

  • Ancker, C.J., Jr., One-on-One Stochastic Duels, Research Monograph, Military Applications Section, Operations Research Society of America, 1982.

    Google Scholar 

  • Ancker, C.J., The Stochastic Duel with Time-Dependent Hit Probabilities, Naval Research Logistics Quarterly, Vol. 31, No. 2, 363–372, 1984.

    Article  Google Scholar 

  • Ancker, C.J., Jr., A Proposed Foundation for a Theory of Combat, Naval Research Logistics, Vol. 42, No. 3, 311–343, 1995.

    Article  Google Scholar 

  • Ancker, C.J., Jr., and Gafarian, A.V., Modem Combat Models: A Critique of Their Foundations, Operations Research Society of America, Baltimore, Maryland, 1992.

    Google Scholar 

  • Ancker, C.J., Jr., and Williams, T., Some Discrete Processes in the Theory of Stochastic Duels, Operations Research, Vol. 13, No. 2,202–216, 1965.

    Article  Google Scholar 

  • Armitage, J.V., Allocation of Sorties in Air Interdiction, Operations Research, Vol. 18, No. 3, 483–496, 1970.

    Article  Google Scholar 

  • Barr, D., Weir, M. and Hoffman, J., An Indicator of Combat Success, Naval Research Logistics, Vol. 40, No. 6, 755–768, 1993.

    Article  Google Scholar 

  • Bhashyam, N., Stochastic Duels with Several Types of Weapons, Defence Science Journal, Vol. 17, 113-118, 1967.

    Google Scholar 

  • Bhashyam, N., Stochastic Duels with Nonrepairable Weapons, Naval Research Logistics Quarterly, Vol. 17, No. 1, 121–129, 1970a.

    Article  Google Scholar 

  • Bhashyam, N., Stochastic Duels with Lethal Dose, Naval Research Logistics Quarterly, Vol. 17, No. 3,397–405, 1970b.

    Article  Google Scholar 

  • Bhashyam, N. and Singh, Naunihal, Stochastic Duels with Varying Single Shot Kill Probabilities, Operations Research, Vol. 15, No.2, 233–244, 1967.

    Article  Google Scholar 

  • Bhat, U.N., A Markov Process Solution for Lanchester Combat Model, Technical Report 82-OR-S, Department of Operations Research and Engineering Management, Southern Methodist University, Dallas, Texas, 1982.

    Google Scholar 

  • Bitters, Davis L., Efficient Concentration of Forces, or How to Fight Outnumbered and Win, Naval Research Logistics, Vol.42, No. 3, 397–418, 1995.

    Article  Google Scholar 

  • Bracken, J., Lanchester Models of the Ardennes Campaign, Naval Research Logistics, Vol. 42, No. 4, 559–577, 1995.

    Article  Google Scholar 

  • Brown, R.H., Theory of Combat; The Probability of Winning, Operations Research, Vol. 11, No. 3, 418–425, 1963.

    Article  Google Scholar 

  • Burden, R.L., Faires, J.D. and Reynolds, A.C., Numerical Analysis, Prindle, Weber & Schmidt, Inc., Boston, Massachusetts, 1978.

    Google Scholar 

  • Chang, S.L. and Menq, J.Y., Modelling a Markov Attrition Process, Defence Science Journal, Vol. 39, No. 2, 211–220, 1989.

    Google Scholar 

  • Choi, H.K., Transformation of Combat Data in Support of Battle Trace, AD-A248 178, 1992.

    Google Scholar 

  • Clark, G.M., The Combat Analysis Model, Ph.D. Thesis in Industrial Engineering, OSU, UMIC #69-15905, 1969.

    Google Scholar 

  • Clausewitz, K.V., On War, Barnes and Noble, New York, 1956.

    Google Scholar 

  • Cooper, Gerry, Non-Monotonicity and Other Combat Modeling Ailments, PHALANX, June 1994

    Google Scholar 

  • Cooper, Gerry, Non-Monotonicity Revisited, PHALANX, September 1994

    Google Scholar 

  • Craig, J.D., The Effect of Uncertainty on Lanchester-Type Equation of Combat, MS Thesis in Operations Research, NPGS, DTIC AD A017 550, 1975.

    Google Scholar 

  • Daly, F., The UK Lanchester Study Group, Omega, Vol.13, No.2, 131–133, 1985.

    Article  Google Scholar 

  • Davis, Paul K., Dynamic Instability, PHALANX, December 1992

    Google Scholar 

  • Dewar, J. A., Gillogly, J.J. and Juncosa, M.L., Non-Monotonicity, Chaos and Combat Models, R-3995-RC, The RAND Corporation, Santa Monica, 1991.

    Google Scholar 

  • Dietchman, S.J., A Lanchester Model of Guerrilla Warfare, Operations Research, Vol.10, 818–827, 1962.

    Article  Google Scholar 

  • Dockery, J.T. and Chiatti, S., Application of Catastrophe Theory to the Problems of Military Analysis, European Journal of Operational Research, Vol. 24, No. 1, 46–53, 1986.

    Article  Google Scholar 

  • Dockery, J.T. and Santoro, R.T., Lanchester Revisited: Progress in Modeling C2 in Combat, Signal, 41–48, July 1988.

    Google Scholar 

  • Dockery, J.T. and Woodcock, A.E.R., The Military Landscape — Mathematical Models of Combat, Woodhead Publishing Limited, Cambridge, England, 1993.

    Google Scholar 

  • Draper, N.R. and Smith, H., Applied Regression Analysis, John Wiley, New York, 1981.

    Google Scholar 

  • Dupuy, T. N., Can We Rely Upon Computer Combat Simulations?, Armed Forces Journal International, 58–60, August 1987.

    Google Scholar 

  • Engel, J.H., A Verification of Lanchester’s Law, Operations Research, Vol.2, 163–171, 1953.

    Article  Google Scholar 

  • Fawett, C.D. and Jones, C.G., Effectiveness Determination of Bombers Penetrating through an Air-to-Air Defense, Operations Research, Vol. 18, No. 3, 516–525, 1970.

    Article  Google Scholar 

  • Fisk, B., American Naval Policy, US Naval Institute, Vol. 113, 1–80, 1905.

    Google Scholar 

  • Fisk, B., Lt J.V. Chase’s, Force on Force Effectiveness Model for Battle Lines, Appendix C, in The Navy as a Fighting Machine, Naval Institute Press, Annapolis, MD, 1916 (Revised 1988).

    Google Scholar 

  • Gafarian, A.V. and Ancker, C.J., Jr., The Two-on-Two Stochastic Duel, Naval Research Logistics Quarterly, Vol. 31, No. 2, 309–324, 1984.

    Article  Google Scholar 

  • Gafarian, A.V. and Manion, K.R., Some Two-on-One Homogeneous Stochastic Combat, Naval Research Logistics, Vol. 36, No. 6, 721–764, 1989.

    Article  Google Scholar 

  • Griffiths, J.D., Marshall, S.A., Weale, T.G. and William, J.E., Air-Naval Combat, Journal of the Operational Research Society, Vol. 42, No. 4, 289–300, 1991.

    Google Scholar 

  • Hajeck, O., Pursuit Games, Academic Press, London, 1975.

    Google Scholar 

  • Hardeck, W. and Hilden, H., A Stochastic Extension of Lanchester Theory (Translated Version), Department of Mathematics, ZOR Trier, 55 series, Treverer Strasle, West Germany, 1967.

    Google Scholar 

  • Hartley, D.S., III, Can the Square Law be Validated ?, K/DSRD-57, Martin Marietta Energy Systems, Inc., Oak Ridge Gaseous Diffusion Plant, March 1989.

    Google Scholar 

  • Hartley, D.S., III and Kruse, K.L., Historical Support for a Mixed Law Lanchesterian Attrition Model: Helmbold’s Ratio, K/DSRD-113, Martin Marietta Energy Systems, Inc., Oak Ridge Gaseous Diffusion Plant, November 1989.

    Google Scholar 

  • Hartley, D.S., III, The Constraint Model of Attrition, K/DSRD-114/R1, Martin Marietta Energy Systems, Oak Ridge Gaseous Diffusion Plant, November 1989.

    Google Scholar 

  • Hartley, D.S., III, Historical Validation of an Attrition Model, K/DSRD-115, Martin Marietta Energy Systems, Oak Ridge Gaseous Diffusion Plant, May 1990.

    Google Scholar 

  • Hartley, D.S., III, Predicting Combat Effects, K/DSRD-412, Martin Marietta Energy Systems, Oak Ridge Gaseous Diffusion Plant, August 1991.

    Google Scholar 

  • Harltey D.S., III, A Mathematical Model of Attrition Data, Naval Research Logistics, Vol.42, No.4, 585–607, 1995.

    Article  Google Scholar 

  • Hartley, D.S., III and Helmbold, R.L., Validating Lanchester’s Square Law and Other Attrition Models, Naval Research Logistics, Vol.42, No. 4, 609–633, 1995.

    Article  Google Scholar 

  • Hawkins, C.F., Modeling the Breakpoint Phenomenon, Signal, 37–41, July 1989.

    Google Scholar 

  • Helmbold, R.L., Historical Data and Lanchester Theory of Combat, CORG-SP-128, July 1961.

    Google Scholar 

  • Helmbold, R.L., A Modification of Lanchester’s Equations, Journal of the Operations Research Society of America, Vol. 13, 857–859, 1965.

    Google Scholar 

  • Helmbold, R.L., Decision in Battle: Breakpoint Hypothesis and Engagement Termination Data, R-772-PR, The RAND Corporation, Santa Monica, California, June 1971.

    Google Scholar 

  • Helmbold, R.L., Combat History Analysis Study Effort (CHASE) Progress Report for the Period August 1984-June 1985, CAA-TP-86-2, US Army Concepts Analysis Agency, Bethesda, Maryland, August 1986.

    Google Scholar 

  • Helmbold, R.L., A Compilation of Data on Rates of Advance in Land Combat Operations, CAA-RP-90-04, U.S. Army Concepts Analysis Agency, February 1990a.

    Google Scholar 

  • Helmbold, R.L., Rates of Advance in Historical Land Combat Operations, CAA-RP-90-1, US Army Concepts Analysis Agency, June 1990b.

    Google Scholar 

  • Helmbold, R.L., Osipov: The ‘Russian Lanchester’, European Journal of Operational Research, Vol. 65, No. 2, 278–288, 1993.

    Article  Google Scholar 

  • Helmbold, R.L., The Constant Fallacy: A Persistent Logical Flaw in Applications of Lanchester’s Equations, European Journal of Operational Research, Vol. 75, No. 3, 647–658, 1994.

    Article  Google Scholar 

  • Huber, Reiner K. and Tolk, Andreas, Non-Monotonicity Effects in Combat Models and Tactical Decision Modeling, PHALANX, September 1994.

    Google Scholar 

  • Hughes Jr., Wayne P., A Salvo Model of Warships in Missile Combat Used to Evaluate Their Staying Power, Naval Research Logistics, Vol.42, No. 2, 267–290, 1995.

    Article  Google Scholar 

  • Isaacs, R., Differential Games, John Wiley, New York, 1969.

    Google Scholar 

  • Jaiswal, N.K., Probability Analysis of Combat Models, Annals of Operations Research, Vol. 9,561–573, 1987.

    Article  Google Scholar 

  • Jaiswal, N.K. and Bhashyam, N., Stochastic Duels with Flight-Time and Replenishment, Opsearch, Vol. 3, 169–185, 1966.

    Google Scholar 

  • Jaiswal, N.K. and Nagabhushana, B.S., Combat Modeling with Spatial Effects, Reserve Deployment and Termination Decision Rules, Computers & Operations Research, Vol. 21, No. 6, 615–628, 1994.

    Article  Google Scholar 

  • Jaiswal, N.K. and Nagabhushana, B.S., Termination Decision Rules in Combat Attrition Models, Naval Research Logistics, Vol. 42, No. 3, 419–434, 1995a.

    Article  Google Scholar 

  • Jaiswal, N.K. and Nagabhushana, B.S., Measure of Combat Success through Historical Data, Proceedings of APORS’94 Conference,(Eds.) Fushimi, M. and Tone, K., World Scientific, Singapore, 227–234, 1995b.

    Google Scholar 

  • Jaiswal, N.K., Sangeeta, Y. and Gaur, S.C., Stochastic Analysis of Combat Models under Different Termination Decision Rules, European Journal of Operational Research, Vol. 83, No. 3, 530–546, 1995c.

    Article  Google Scholar 

  • Jennings, N., The Mathematics of Battle — Stochastic ‘Linear Law’ Battle, DOAE Memorandum 7316, Ministry of Defence, UK, 1973.

    Google Scholar 

  • Kikuta, K., A Note on the One-against-Many Battle, Operations Research, Vol. 1, 952–956, 1983.

    Article  Google Scholar 

  • Kress, M., The Many-on-One Stochastic Duel, Naval Research Logistics, Vol. 34, No. 5, 713–720, 1987.

    Article  Google Scholar 

  • Kress, M., A Many-on-Many Stochastic Duel Model for a Mountain Battle, Naval Research Logistics, Vol.39, No. 4, 437–446, 1992.

    Article  Google Scholar 

  • Kress, M., A Two-on-One Stochastic Duel with Maneuvering and Fire Allocation Tactics, Naval Research Logistics, Vol. 38, No. 3, 303–313, 1991.

    Article  Google Scholar 

  • Kwon, Tai Young and Bai, Do Sun, Stochastic Duels with Multiple Hits and Limited Ammunition Supply, Naval Research Logistics Quarterly, Vol. 30, No. 2, 347–358, 1983.

    Article  Google Scholar 

  • Lanchester, F.W., The Principle of Concentration, Engineering, October 1914.

    Google Scholar 

  • Lepingwell, J.W.R., The Laws of Combat ? Lanchester Re-examined, International Security, Vol.12, No.1, 89–139, 1987.

    Article  Google Scholar 

  • Louer, Philip E., More on Non-Linear Effects …, PHALANX, March 1993.

    Google Scholar 

  • Morse, Philip M. and Kimball, George E., Methods of Operations Research, The MIT Press, Cambridge, Massachusetts and John Wiley, New York, 1951.

    Google Scholar 

  • Nagabhushanam, A. and Jain, G.C., Stochastic Duels with Damage, Operations Research, Vol. 20, No. 2, 350–356, 1972.

    Article  Google Scholar 

  • Osipov, M, The Influence of Numerical Strength of Engaged Forces on Their Casualties, (in Russian), Military Collection Nos. 6-10, 1915. (English translation by Robert L Helmbold and Allan S. Rehm published as US Army Concepts Analysis Research Paper CAA-RP-91-2, September 1991) (Also published in Naval Research Logistics, Vol. 42, No.3, 435-490, 1995).

    Google Scholar 

  • Palmore, Dynamic Instability in Combat Models, PHALANX, December 1992.

    Google Scholar 

  • Peterson, R., On the ‘Logarithmic Law’ of Attrition and its Application to Tank Combat, Operations Research, Vol. 15, 557–558, 1967.

    Article  Google Scholar 

  • Protopopescu, V., Santoro, R.T., Dockery, I, Cox, R.L. and Barnes, J.M., Combat Modeling with Partial Differential Equations, Report No. ORNL/TM-10636, Oak Ridge National Laboratory, 1987.

    Google Scholar 

  • Protopopescu, V., Santoro, R.T. and Dockery, J., Combat Modeling with Partial Differential Equations, European Journal of Operational Research, Vol. 38, No. 2, 178–183, 1989.

    Article  Google Scholar 

  • Przemieniecki, J.S., Mathematical Models in Defense Analyses, (Second Edition), AIAA Education Series, Air Force Institute of Technology, Wright Patterson Air Force Base, Ohio, U.S.A., 1994.

    Google Scholar 

  • Quade, E.S., A Preliminary Model for an Air Battle, RM-318, RAND Corporation, Santa Monica, CA, September 1975.

    Google Scholar 

  • Santoro, R.T., Azmy, Y.Y. and Protopopescu, V., Oak Ridge Cross Validates Modeling Versus Wargaming, Signal, 49–52, July 1990.

    Google Scholar 

  • Sivazlian, B.D., Aircraft Sortie Effectiveness Model, Naval Research Logistics, Vol. 36, No. 2, 127–137, 1989.

    Article  Google Scholar 

  • Sivazlian, B.D., Mission Effectiveness Analysis of an Aircraft Attacking Passive Targets, European Journal of Operational Research, Vol. 44, No. 1, 47–59, 1990.

    Article  Google Scholar 

  • Smith, D.G., The Probability Distribution of the Number of Survivors in a Two-Sided Combat Situation, Operational Research Quarterly, Vol. 16, No. 4, 429–437, 1965.

    Article  Google Scholar 

  • Snow, R.N., Contribution to Lanchester Attrition Theory, Report RA-15078, Project RAND, Douglas Aircraft Company, Inc., Santa Monica, CA, 1948.

    Google Scholar 

  • Taylor, J.G., Approximate Solution (with Error Bounds) to a Nonlinear, Nonautonomous Second-Order Differential Equation, Journal of the Franklin Institute, Vol. 306, No. 2, 195–208, 1978.

    Article  Google Scholar 

  • Taylor, J.G., Prediction of Zero Points of Solutions to Lanchester-Type Differential Combat Equations for Modern Warfare, SIAM J. Appl. Math., Vol. 36, No. 3, 438–456, 1979.

    Article  Google Scholar 

  • Taylor, J.G., Simple-Approximate Battle-Outcome-Prediction Conditions for Variable-Coefficient Lanchester-Type Equations of Modern Warfare, Naval Research Logistics Quarterly, Vol. 30, No. 1, 113–131, 1983a.

    Article  Google Scholar 

  • Taylor, J.G., Lanchester Models of Warfare, Vols. I and II, Operations Research Society of America, Military Applications Section, Virginia, 1983b.

    Google Scholar 

  • Taylor, J.G. and Brown, Gerald, G., Numerical Determination of the Parity-Condition Parameter for Lanchester-Type Equations of Modern Warfare, Computers & Operations Research, Vol. 5, 227–242, 1978.

    Article  Google Scholar 

  • Taylor, J.G. and Brown, Gerald, G., Annihilation Prediction for Lanchester-Type Models of Modern Warfare, Operations Research, Vol. 31, No. 4, 752–771, 1983.

    Article  Google Scholar 

  • Taylor, J.G. and Comstock C., Force Annihilation Conditions for Variable Coefficient Lanchester Type Equations of Modern Warfare, Naval Research Logistics Quarterly, Vol. 24, No. 2, 349–371, 1977.

    Article  Google Scholar 

  • Thorn, R., Topological Models in Biology, Topology, Vol. 8, 313–335, 1969.

    Article  Google Scholar 

  • Tsai, Harrison K. and Ellenbogen, J.C., Bounded Potentially Pathological Nonlinear Behavior in Combat Models and Simulations, PHALANX, December 1992.

    Google Scholar 

  • Vincent, Thomas L., Sticht, Dougles, J. and Peng, Willy Y., Aircraft Missile Avoidance, Operations Research, Vol.24, No.3, 420–437, 1976.

    Article  Google Scholar 

  • Wallis, P., Recent Developments in Lanchester Theory, Operational Research Quaterly, Vol. 19, 191–195, 1968.

    Article  Google Scholar 

  • Wand, K., Humble, S. and Wilson, R.J.T., Explicit Modeling of Detection Within a Stochastic Duel, Naval Research Logistics, Vol. 41, No. 4, 431–450, 1993.

    Article  Google Scholar 

  • Wand, K. and Wilson, R.J.T., MATADOR: An Analysis Model of Stochastic Tank Duel, Journal of the Operational Research Society, Vol. 38, No. 6, 545–551, 1987.

    Google Scholar 

  • Weale, T.G., The Mathematics of Battle — A Bivariate Probability Distribution, DOAE Memorandum 7129, Ministry of Defence, UK, 1971.

    Google Scholar 

  • Weale, T.G., The Mathematics of Battle — A Moments of Distribution of Battle States, DOAE Memorandum 7130, Ministry of Defence, UK, 1972.

    Google Scholar 

  • Williams, T. and Ancker, C.J. Jr., Stochastic Duels, Operations Research, Vol. 11, No. 5, 803–817, 1963.

    Article  Google Scholar 

  • Woodcock, A.E.R., Cobb, L. and Dockery, J.T., Models of Combat with Embedded C2. IV: The Decision Space, International CIS Journal, Vol. 3, 5–31, 1989.

    Google Scholar 

  • Yang, J. and Gafarian, A.V., A Fast Approximation of Homogeneous Stochastic Combat, Naval Research Logistics, Vol.42, No.3, 503–533, 1995.

    Article  Google Scholar 

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Jaiswal, N.K. (1997). Homogeneous Combat Models. In: Military Operations Research. International Series in Operations Research & Management Science, vol 5. Springer, Boston, MA. https://doi.org/10.1007/978-1-4615-6275-7_9

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