On the treatment of force‐level constraints in times‐equential combat problems

Computers & Operations Research

Brown

1978

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Scope and purpose-This paper presents new computational methods that facilitate digital-computer analysis of some important military operations research problems. Lanchester-type combat models [l] are deterministic differentialequation models of combat attrition in which the state variables are typically the numbers of the different weapon-system types. Even though combat between two military forces is a complex random process, such deterministic combat models are commonly used for computational reasons in defense-planning studies, for example, to assess the relative importance of various weapon-system and force-level parameters. A so-called attrition-rate coefficient in such a combat model represents the fire effectiveness of a particular weapon-system type against a particular target type, and time-dependent attrition-rate coefficients are used to model temporal variations in fire effectiveness when, for example, the range between firers and targets changes appreciably during battle. For such a variable-coefficient Lanchester-type combat model that is a generalization of Lanchester's[Z] classic "square-law" model, we present a simple numerical procedure for determining the so-called parity-condition parameter, which is "the enemy force equivalent of a friendly force of unit strength" and may be used to predict battle outcome in specific engagements. These results allow one not only to predict battle outcome but also to tradeoff quality vs quantity of two opposing weapon systems.Abstract-This paper presents a simple numerical procedure for determining the parity-condition parameter for deterministic Lanchester-type combat between two homogeneous forces. Deterministic dierentidequation combat models are commonly used in parametric studies for computational reasons, since they give essentially the same results for the mean course of combat as do corresponding stochastic attrition models. The combat studied in this paper is modelled by Lanchester-type equations of modern warfare with timedependent attrition-rate coefficients. Previous research has generalized Lanchester's classic "square law" to such variable-coefficient combat. It has shown that the prediction of battle outcome (in particular, force annihilation) without having to spend the time and effort of computing force-level trajectories depends on a single parameter, the so-called parity-condition parameter, which is "the enemy force equivalent of a friendly force of unit strength" and depends on only the attrition-rate coefficients. Unfortunately, previous research did not show generally how to determine this parameter. We present general theoretical considerations for its numerical noniterative determination. This general theory is applied to an important class of attrition-rate coefficients (offset power-rate coefficients). Our results allow one to study such variable-coefficient combat models almost as easily and thoroughly as Lanchester's classic constant-coefficient model.

Section: Introductionmentioning

confidence: 86%

Section: Theoremmentioning

confidence: 99%

Numerical determination of the parity-condition parameter for lanchester-type equations of modern warfare

Computers & Operations Research

Brown

1978

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“…The only systematic examinations of the irduences of the nature of the criterion function on the structure of optimal time-sequential firedistribution strategies known to the authors are those of Taylor [30][31][32][33]35,37,40,43]. In [31] and [40], however, the influences of the nature of the targettype attrition process on the structure of optimal firedistribution policies were examined.…”

Section: Previous Work On the Structure Of Optimal Fire-distribution mentioning

confidence: 99%

“…Rather than explicitly constructing extremals and determining domains of controllability [30,35,401, it is more convenient to show that the return (i.e. value of the criterion functional) corresponding to certain extremals dominates that from others.…”

Section: Synthesis Of Extremalsmentioning

confidence: 99%

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An examination of the effects of the criterion functional on optimal fire‐support policies

Naval Research Logistics

Brown

1978

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This paper examines the dependence of the structure of optimal time-sequential fire-support policies on the quantification of military objectives by considering four specific problems, each corresponding to a different quantiiication of objectives (i.e. criterion functional). We consider the optimal time-sequential allocation of supporting fires during the "approach to contact" of friendly infantry against enemy defensive positions. The combat dynamics are modelled by deterministic Lanchester-type equations of warfare, and the optimal fire-support policy for each one-sided combat optimization p~oblem is developed via optimal control theory. The problems are all nonconvex, and local optima are a particular di5culty in one of them. For the same combat dynamics, the splitting of supporting &es between t*o enemy forces in any optimal policy (i.e. the optiality of singular subarcs) is shown to depend only on whether the terminal payoff reflects the objective of attaining an "overall" military advantage or a "local" one. Additionally, switching times for changes in the ranking of target priorities are shown to be different (sometimes significantly) when the decision criterion is the difference and the ratio of the military worths (computed accolding to linear utilities) of total infantry survivors and also the difference and the ratio of the military worths of the combatants' total infantry losses. Thus, the optimal fie-support policy for this attack scenario is shown to be significantly influenced by the quantification of military objectives.

Some simple victory‐prediction conditions for lanchester‐type combat between two homogeneous forces with supporting fires

Naval Research Logistics

1979

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a y l o r ~e~a r t f n e~~f qf ~~e r a t i o~s Research ABSTRACT This paper develops new "simple" victory"prediction conditions for a linear Lanchester-type model o f combat between two homogeneous forces with superimposed effects o f supporting fires not subject to attrition. These simple victory-prediction conditions involve only the initial conditions o f battle and certain assumptions about the nature o f temporal variations in the attrition-ratecoefficients. They are developed for a fixed-force-ratio-breakpoint battle by studying the force-ratio equation for the linear combat model. A n important consideration is shown to be required for developing such simple victoryprediction conditions: victory is not guaranteed i n a fixed-force-ratiobreakpoint battle even when the force ratio i\ always changing to the advantage o f one o f the combatants. One must specify additional conditions to hold for the cumulative fire effectivenesses o f the primary weapon systems in order to develop correct victory-prediction conditions. The inadequacy o f previous victory-prediction results is explained by examining (for the linear combat model without the supporting fires) new "exact" victory-prediction conditions, which show that even the range of possible battle outcomes may be significantly different for variable-coefficient and constant-coefficients models. INT~ODUCTIONEven though combat between two military forces is a complex random process (see Note 1 on p. 65 of Taylor and Brown [211), as a consequence of F.W. Lanchester's [lo] pioneering 1914 work, from about the end of World War I1 military operations analysts have used simplified deterministic differential-equation models to develop insights into the dynamics of combat 11-3, 25-27], Today, Lanchester-~ype complex system models, which rely on modern digitai-computer technology for their implementation (see, for example, Bonder and Honig [3]), have been developed for various levels of combat, from combat between battalion-sized units [4] to theater-level operations [5,71 ([20] for further references). Nevertheless, a simple combat model may yield an understanding of important relations that are difficult to perceive in a more complex model, and such insights can provide valuable guidance for higher-resolution computerized investigations (see Bonder and Farrell f21 and Weiss 1271). In this paper we will develop new victory-prediction conditions for several such simplified Lanchester-type models of