Target selection in lanchester combat: Heterogeneous forces and time‐dependent attrition‐rate coefficients

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: 84%

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

Computers & Operations Research

Brown

1978

Self Cite

“…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%

“…the difference between the military worths of friendly and enemy forces). Taylor [30][31][32][33]40] has studied how the optimal fire-distribution policy depends on the assignment of these linear utilities. In other words, he examined the sensitivity of the optima& combat policy to parametric variations in the assigned linear utilities for survivors.…”

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

confidence: 99%

“…Computing the h t two time derivatives of the switching function we see that on a singular subarc (see [32] for a further discussion) we have [8,211 with the singular control given.by (33) .4s=&l(cl+c2~.…”

Section: '@mentioning

confidence: 99%

“…In other words, he examined the sensitivity of the optimal combat policy to parametric variations in the assigned linear utilities for survivors. It has been shown that the n-versusone firedistribution problems studied in [30][31][32][33] all have quite simple solutions when enemy survivors are valued in direct proportion to their kill capabilities (as measured by their Lanchester attrition-rate coefficients [34, 381 against the (homogeneous) friendly forces). [29] have suggested that a appropriate payoff, or objective function (in our terminology, criterion .functional) for the quantitative evaluation of combat strategies is the loss ratio (calculated possibly using weighting factors for heterogeneous forces).…”

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

confidence: 99%

See 2 more Smart Citations

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

Naval Research Logistics

Brown

1978

Self Cite

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.

Target selection in Lanchester combat: Linear‐law attrition process

Naval Research Logistics

1973

Self Cite

We develop the solution to a simple problem of target selection in Lanchester combat against two enemy force types each of which undergoes a "linear-law" attrition process.In addition to the Pontryagin maximum principle, the theory of singular extremals is required to solve this problem. Our major contribution is to show how to synthesize the optimal target selection policies from the basic optimality conditions. This solution Synthesis methodology is applicable to more general dynamic (tactical) allocation problems. For constant attrition-rate coefficients we show that whether or not changes can occur in target priorities depends solely on how survivors are valued and is independent of the type of attrition process.