Explicit Analytical Expression for a Lanchester Attrition-Rate Coefficient for Bonder and Farrell’s m-Period Target-Engagement Policy

Working Paper #5, DTRA Project, July 9, 2001The purpose of this working paper is to give an explicit analytical expression for a Lanche s- ter-type attrition-rate coefficient for direct-fire combat in a heterogeneous-target environment with serial acquisition of targets for Bonder and Farrell’s m-period target-acquisition policy1. It develops this result (its main result) from Taylor’s [2001d] new important general result (that does not depend on the target-engagement policy of a firer type or even on the particulars of the target-acquisition process) for a Lanchester attrition-rate coefficient for serial acquisition by developing explicit ana- lytical expressions for the two key intermediate quantities on which the coefficient depends: namely, (1) expected time to acquire a target that will be engaged, (2) next-target-type-to-be-engaged probability. An analytical expression for the former quantity (the expect value) was recently developed by one of the authors (Taylor [2001e]), while the paper at hand develops such an expression for the latter probability. These two new important intermediate results have allowed us to develop the explicit analytical expression for a Lanchester attrition-rate coefficient for Bonder and Farrell’s target- acquisition policy via Taylor’s general expression for direct-fire combat in a heterogeneous-target environment with serial acquisition of targets. These analytical results are then verified against simulation results

Modeling of combat operations

Introduction/purpose: The goal of the research in this paper is to present and evaluate the method of modeling operations by aggregating forces by simulating the battle process with Lanchester's equations. This method is the software basis of a certain number of programs used in NATO, in war simulations, and in the planning and analysis of operations. Its value is in understanding the consequences of decisions made with outcomes and results of combat actions. Methods: The case study of the well-known Operation Desert Storm gathered the necessary data on operational parameters and the way forces are used in battles. The obtained data were transformed into operational variables of the combat model using the force aggregation method, whose simulation was carried out using the method of differential Lanchester's equations (quadratic law). Results: By simulating the modeled operation, the parameters of the outcome of the conflict were obtained with numerical indicators of success, consumption of resources, etc. The results were analyzed and a certain correlation with the parameters of the real operation was determined, which enables the validation of the model. Conclusion: The partial validity of the model describing the conflict on a practical historical example from a case study was confirmed. There are objective limitations in the application of modeling of military operations and optimization of the use of forces. The value of this method is the possibility of a reliable strategic assessment of the adversary's military power at the strategic level

The attack of a target with the simultaneous use of air and artillery.

The purpose of this report is to assess the feasibility of attacking a target with the simultaneous use of air and artillery. A method for generating circular error probability as a function of release altitude is presented. Techniques for determining probabilities of kill for the air attack system, artillery system, and for the combined air-artillery attack system are examined. From the probability of kill information and from the rate of fire (delivery) of the systems, expected time to target destruction calculations are developed. The restrictions that allow the use of the combined air-artillery attack system are presented, as well as a discussion of the advantages and disadvantages of this system of attack.http://archive.org/details/thettackoftarget1094516495Major, United States Marine CorpsApproved for public release; distribution is unlimited

Application of differential games to problems of military conflict: Tactical allocation problems, Part I

The mathematical theory of deterministic optimal control/differential games is applied to the study of some tactical allocation problems for combat described by Lanchester-type equations of warfare. A solution procedure is devised for terminal control attrition games. H. K. Weiss' supporting weapon system game is solved and several extensions considered. A sequence of one-sided dynamic allocation problems is considered to study the dependence of optimal allocation policies upon model form. The solution is developed for variable coefficient Lanchester-type equations when the ratio of attrition rates is constant. Several versions of Bellman's continuous stochastic gold-mining problem are solved by the Pontryagin maximum principle, and their relationship to the attrition problems is discussed. A new dynamic kill potential is developed. Several problems from continuous review deterministic inventory theory are solved by the maximum principle.The Office of Naval Researchhttp://archive.org/details/applicationofdif00taylN

Lanchester-Type Models of Warfare, Volume I

The Twentieth Century has been characterized by innumerable attempts to use the Scientific Method as a basis for policy planning in national and international affairs. The emergence of the field of operations research (OR) out of attempts of scientists in the Western Democracies to apply the Scientific Method to military problems during World War II is well known. Since World War II there has been a dramatic growth in both the interest in and use of OR and systems-analysis techniques for such purposes within the U.S. defense establish- ment, especially since the beginning of the so-called McNamara Era of defense planning. A concomitant trend has been an equally dramatic increase in both the number and variety of mathematical models used to support these analytical activities

NA

This thesis presents an operational Lanchester-type model of small-unit amphibious operations. This relatively simple model has been developed to demonstrate the basics of model building to the beginning student interested in amphibious warfare. The model is a time sequenced, deterministic, force-on-force combat model that is implemented on a digital computer. A brief discussion of considerations for modeling amphibious operations is given. The details of the model are presented for a specific amphibious-warfare scenario. Additionally, a computer terrain-contour-line plotting program is provided to assist the combat modeler to fit a parameterized-terrain to real terrain.http://archive.org/details/operationallanch00parkNANARepublic of South Korea Marine Corps authorApproved for public release; distribution is unlimited

Zur aggregierten Abbildung elementarer Gefechtsprozesse in hochauflösenden Gefechtssimulationssystemen

Die vorliegende Arbeit befasst sich mit dem Problem der Aggregation in hochauflösenden Gefechtssimulationssystemen und dabei insbesondere mit Möglichkeiten zur aggregierten Abbildung der elementaren Gefechtsprozesse Bewegung, Zielaufklärung sowie aktive- und passive Abnutzung durch Flachfeuerwaffensysteme. In einem ersten Teil werden vor dem Hintergrund des veränderten Aufgabenspektrums moderner Streitkräfte Vorteile aggregierter und nicht aggregierter Simulationssysteme verglichen und theoretische Ansätze zur Kopplung entsprechender Modelle erläutert. Im Anschluss daran wird eine Auswahl existierender Aggregationsansätze vorgestellt und ihre Vor- sowie Nachteile dargelegt. Betrachtet werden die bekannten Modelle von Lanchester mit einigen ihrer wichtigsten Erweiterungen, die Verwendung zellulärer Automaten und Markov-Prozesse sowie sogenannte Netzwerke stochastischer Kleingefechte. Als Ergebnis der Betrachtung wird festgestellt, dass diese Ansätze aus bestimmten Gründen in hohem Maß szenarabhängig sind und deshalb die geforderte Flexibilität in der Abbildung und Verarbeitung wechselnder militärischer Szenarien nicht erbringen können. Aufgrund dieser Feststellung wird im zweiten Teil der Arbeit das Leitschützenprinzip als neuer, alternativer Aggregationsansatz vorgestellt. Dieser fasst Gruppen homogener Einzelsysteme in aggregierten Gefechtselementen zusammen und bildet die elementaren Gefechtsprozesse mit Hilfe geringfügig modifizierter Algorithmen nicht aggregierter Modelle ab. Auf eine Verwendung aggregierter Modellparameter wird im Rahmen des Leitschützenprinzips bewusst verzichtet, da diese in den vorgestellten herkömmlichen Aggregationsansätzen als Ursache für die festgestellte Szenarabhängigkeit identifiziert werden können. Am Beispiel des neu entwickelten Experimentalsystems SAMBA (Simple Attrition Model for variably Aggregated Forces) wird eine prototypische Umsetzung des Leitschützenprinzips für das Flachfeuergefecht von Infanteriekräften vorgestellt und anschließend im Rahmen einer Vielzahl unterschiedlicher Szenarien systematisch das Rechenzeitverhalten sowie die Abbildungsgüte des Ansatzes untersucht. Die Ergebnisse dieser Rechenexperimente zeigen, dass das Leitschützenprinzip grundsätzlich in der Lage ist, bei deutlich verbessertem Rechenzeitverhalten eine flexible Aggregation zu ermöglichen. Bei sinnvoller Bildung der aggregierten Gefechtselemente kommt es zu keinen signifikanten Verfälschungen, so dass Konsistenz zwischen den Ergebnissen der aggregierten und nicht aggregierten Abbildung herrscht. Das Leitschützenprinzip erweist sich somit gegenüber bisherigen Ansätzen als besser geeignet zur aggregierten Abbildung - besonders von Infanteriekräften - in hochauflösenden Gefechtssimulationssystemen.This theses reflects on the general problem of aggregation and deaggregation in high resolution combat simulation systems and presents a new method for the aggregated modelling of dismounted infantry in such systems. Preceded by an introductory description of the changed relevance of modelling and simulation tools for the planning and execution of operations other than war, different methods for multi-variable-resolution modelling including their specific advantages and limits are presented. The subsequent chapter describes existing methods for the aggregated modelling of dismounted infantry forces in high resolution combat simulation systems. For various methods - like for example the well known Lanchestrian Model of Warfare and Markov-Processes - the use of aggregated model parameters, without a directly measurable equivalent in military reality, can be identified as their main shortcoming. In case of even small variations in the modelled scenario, these aggregated model parameters usually have to be redetermined by time costly analytical processes or the use of non-aggregated simulation systems. Therefore existing methods for aggregation have to be judged as lacking in the flexibility needed to adequately support military personal. As an result of this conclusion, a new method for the aggregated modelling of dismounted infantry forces is developed. The so called "Leitschützenprinzip" combines homogenous individual combat systems into aggregated combat elements. To model the basic combat processes, like movement, target acquisition and shooting, of this aggregated combat elements, normally non-aggregated algorithms are slightly modified and used in a event-discrete simulation - without the need for aggregated model parameters. The experimental combat simulation system SAMBA (Simple Attrition Model for variably Aggregated Forces) serves as an example for the implementation of this new aggregation method and is used to systematically examine a large number of different scenarios. As a result of these tests it can be concluded, that the newly developed "Leitschützenprinzip" is able to provide a method for variable aggregation and the flexible modelling of frequently changing infantry scenarios. While distinctly reducing the time needed to simulate a given scenario, the new aggregation method is at the same time able to preserve consistence between the results of aggregated and deaggregated simulation

A mathematical theory for variable-coefficient Lanchester-type equations of 'modern warfare'

A mathematical theory is developed for the analytic solution to deterministic Lanchester-type "square-law" attrition equations for combat between two homogeneous forces with temporal variations in system effectiveness (as expressed by the Lanchester attrition-rate coefficient). Particular attention is given to solution in terms of tabulated functions. For this purpose Lanchester functions are Introduced and their mathematical properties that facilitate solution given. The above theory is applied to the following cases: (1) lethality of each side's fire proportional to a power of time, and (2) lethality of each side's fire linear with time but a nonconstant ratio of these. By considering the force-ratio equation , the classical Lanchester square law is generalized to variable-coefficient cases in which it provides a "local" condition of "winning."Prepared for: Office of Naval Research, Arlington, Virginiahttp://archive.org/details/mathematicaltheo00taylN