Two Effects of Firepower: Attrition and Suppression

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1995 Two Effects of Firepower: Attrition and Suppression

Hughes, Wayne P., Jr.

Military Operations Research http://hdl.handle.net/10945/38442 ABSTRACT • Military force, or combat power, is a real phenomenon, the results of which are Anew basis for the quantitative study observed by its effects on the enemy in of ground combat is introduced that battle. argues the inadequacy of attrition • The observable effects of combat Two Effects models and the need to incorporate the power are not merely physical (producing effects of suppression of the enemy with fire- casualties) but also mental (persuading the of Firepower: power. A quantitative approach to suppres- enemy of our superiority) and spiritual sion of enemy fire is offered. Then an (diminishing enemy morale and will to con- Attrition and analysis shows that the effect of own fire in tinue fighting.) suppressing enemy fire will, in suitable, fre- For purposes of this paper, we will look Suppression quent circumstances, reverse the conclusions carefully only at the most measurable mani- derived from the Lanchester square law, so festation of combat power's mental effect, that the squared term determining the victor which is suppression of enemy actions, Wayne P. Hughes, Jr., FS is unit firepower instead of the numbers of specifically his return of fire. Its spiritual units engaged. effect to diminish enemy morale plays no Senior Lecturer A fundamental aim of physical science part in the computations that follow, but Department of Operations Research is to describe its processes with dynamic may be seen to be an unquantified bonus. Naval Postgraduate School models, mathematical if possible. The same Let us begin with a fresh look at the Monterey, California 93943 aim, often implicit, is true of descriptions of Lanchester square law from the perspective combat phenomena in military operations of combat science. As it was conceived and research. Models of military operations are employed by Chase [1902], Fiske [1905], necessarily more abstract and approximate Osipov [1915] and Lanchester [1915] the than those of physical science. This is espe- square law describes combat as a purely cially so because the scientific study of com- physical phenomenon: bat operations is complicated by human presence. The problem, of course, is to [Unit fire's physical effect in casualties reduce the enormous effects of "human fac- imposed/minute] tors," to an understandable construct, or paradigm. An excellent statement of the problem is in Davis and Blumenthal's mono- graph, The Base-of-Sand Problem: A White [# of physical units firing] Paper on the State of Military Combat Modeling [1991]. For more than a decade The Military [Fewer enemy units] Conflict Institute (TMCI) has been address- ing the problem via a theory of combat that Starting with Osipov, who reached his derives from six axioms. A goal of TMCI is own unique conclusion as to the appropriate to express the theory in a study, thus far relationships, there have been many objec- unpublished.1 As something of a status tions that the classical square law does not report, I undertook a Naval Postgraduate fit the historical battle casualty data for ground combat. The most commonly cited School research paper to digest what seemed reason is (properly enough) that the law can- to be the essence of TMCI's work. It is enti- not apply when the required conditions are tled "Combat Science: An Organizing not present in the battle. The conditions are Study" [1993]. The present paper is an expo- that each participant must be able to fire at sition of a small but important consequence each participant on the opposing side; and of the two endeavors. incapacitated opponents must be known at We may come at our subject with the once, so that fire is distributed only against following question which arises from the active opponents. Patently these conditions remarkable results of the Gulf War: are seldom fully met, with not-so-obvious What analytical proposition helps to explain the evidence that much superior effects on the square law's applicability. combat power when properly applied will There are probably three additional result in disproportionately small losses to major reasons, each related to the other two. the winner while he achieves his objective? One is insufficient attention to defender The relevant principles from Combat advantage. Unit fire effectiveness will nor- Science are: mally be greater on the defender side until

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the attack is nearly consumated. This advantage forces remaining at any later time t; and dA/dt comes from the defender's superior posture. and dB/dt are the rate of losses of A and B When identical unit effectiveness is (wrongly) respectively at time t. assumed for attacker and defender the results A solution to equations [1] for end time T is will appear to show a less-than-square-law effect the state equation: for the attacker's numerical advantage. The second reason is that a battle is usually a [A(0)2 - A(T)2] = ß [B(0V - B(T)2] (2) episodic, such that different elements of a force play predominant roles at different phases of the Equation [2] shows the well known square battle. A battle might involve disproportionate law phenomenon, in which A will win a fight losses upon the attacker during his assault and when the weaker side is annihilated: disproportionate losses upon the defender during his attempt to disengage. Different weapons aA(0)2>ßB(0)2 produce casualties more and less effectively during different phases. Historical battle data and vice-versa. The classic conclusion from the rarely specify casualty production in this way square law is that the "fighting strength" and so the episodic effects are disregarded out of (Lanchester's term) of a force increases in direct necessity. proportion to unit effectiveness and in propor- The third reason is shortage of data with tion to the square of the number of forces which to measure the effect of suppression of engaged. enemy actions from firepower.2 Since suppres- Notably, the casualty-producing effect of fire sion produces no casualties and is a transitory is regarded as a constant. There is ample evi- phenomenon that disappears when the battle is dence that unit effectiveness, a or ß, is not even over, the effect of its presence is overlooked (or is approximately constant but is highly variable. merely implicit) in almost all combat models, This is true when one is using casualty data including high resolution simulations. In taken battle after battle or day after day during a Chapter 18 of Understanding War, T. Dupuy campaign: see for example, D. Hartley [1990, pp. began his fine discussion of suppression with: 3-4; and 1991, p. 3]. It is true when one is using "There is probably no obscurity of combat casualty data taken from different places on a requiring clarification and understanding more battlefield: see G. Kuhn [1989], for example. It is urgently than that of suppression" [1987]. true during different phases of a battle: see The heart of the problem is not a refusal to R. Helmbold [1979], for example.3 Even though acknowledge the importance of suppression but data by phase is seldom available, the variability the lack of an analytical approach that describes of fire effectiveness is readily appreciated by the phenomenon and its importance. We wish reflecting upon a battle's sequence: for example, here to develop a quantifiable model of it. Then an artillery preparation phase, an armored we will fulfill the purpose of this paper, which is assault phase, and an exploitation phase after an to show how an appreciation of the cognitive enemy position is overrun. influence of fire can reverse present conceptions If a (t) and ß (t) are also variables, then the of the value of numbers relative to the unit fire- Lanchester square law equations must be writ- power of a fighting force. ten: We begin with the common form of the dA dB Lanchester square law: = -a(t)A(t) dt = -ß(t)B(t) dt (3) dA dB a A{t) (1) dt -ßB(t) dt Two important reasons that a (t) and ß (t) vary are: first, a change in range between the fighting units as the battle unfolds; and second, where a and ß are the constant unit effectiveness the suppressive effect of each side's fire on the coefficients for A and B respectively, measured in other's fire. For two forces fixed in place with no kills per shooter per unit of time. A(0) and B(0) change in range, the suppression suffered by A are the initial force strength; A(t) and B(t) are the will be in consequence of the volume and

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accuracy of fire by B, and vice versa. Let us pos- We may easily explore circumstances in tulate a constant of proportionality, g, such that which the volume and precision of fire have a g/3(t)B(t) is the time-rate at which each A-shoot- much greater effect to suppress enemy fire than er's fire is curtailed (measured in volume or to impose casualties. Define b(t) as the rate of fire accuracy or both). Similarly define a value, h, in shots by each unit B, and a(t) as the rate of fire such that ha(t)A(t) is the rate that each B-shoot- by each A. Define s as the rate of reduction of fire er's effectiveness is diminished by A's fire.4 We imposed on the other side per shot fired, such write: that: da dß da db ■ha(t)A(t) (4) Tt=-sBKt) -ä = -sAa(t) (5)

The total effect of firepower represented by With equations [5] we measure combat equations [3] and [4] is twofold. Fire will reduce power wholly by its effect on the volume and both the numbers of enemy forces remaining accuracy of the enemy's return fire during the and the lethal effectiveness of the remaining course of the battle. The model, in its stylized forces. Old equations [3] and new equations [4], purity, says that a battle is won by the side that however, have dimensions incompatible with sustains its own fire and suppresses that of the each other. After reconciling the dimensions (to enemy. Casualties are an unmeasured by- killing power lost per unit of time), it is possible product. to retain all quantities and proceed to explore the The state equation for [5] is entirely analo- relationships between the four equations. There gous to equation [2]: remains, of course, the serious problem of finding data, even rough data, for inputs. A[a(0)2 - a(T)2] = B[b(0)2 - b(T)2] (6)

Equation [6] leads to a radically new "law" DEVELOPMENT of combat in which the quantity of fire (measured by its suppressive effect) is the squared Let us proceed, now, to develop our princi- term. Now side A will win a battle in which the pal point. As remarked above, casualties are a permanent consequence of fire, but suppres- enemy's capacity to fire back is "annihilated" sion's effect is only transitory. We usually say (fully suppressed) if:

combat forces are "pinned down" under fire, but 2 2 not permanently affected. Arguably the changes Aa(0) >Bb(0) in a and ß will be much greater during a battle than the changes in force size caused by attrition. When the cognitive effects (on mind and In a typical land battle, casualties will be less will to win) predominate over casualties inflicted than 10%.5 See among many examples, R. because the accuracy and intensity of fire cause McQuie, "Battle Outcomes: Casualty Rates as a the enemy's accuracy and intensity to deterio- Measure of Defeat" [1987]. rate, under those circumstances effective unit As McQuie says, something other than casu- firepower is more valuable than numbers en- alties must cause the losing side to break off gaged. The losing side will discontinue the battle action in a typical battle. In many, perhaps most, because it is being dominated by the enemy's fire, battles mission success is obtained by the cumu- will over time be reduced to impotence, and will lative cognitive effects of fire on the enemy. The eventually be destroyed if it does not concede losing side's forces curtail their maneuvers and the enemy's military objective. Under such cir- fire to avoid destruction. The losing side is also cumstances the traditional Lanchester square increasingly dispirited, with further loss of fight- law conclusion favoring numbers over quality is ing effectiveness. Or the losing commander reversed, and unit firepower is more influential thinks that the situation can only go from bad to than numbers of units. worse and withdraws to fight afresh from what he hopes will be a more favorable posture.

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A NUMERICAL EXAMPLE INTERPRETATION OF THE As a numerical example, let A(0) = 10 units SUPPRESSION MODEL of force, and B(0) = 5 units. Let the initial An interpretation of the suppression compu- destruction rate of A be a (0) = 1 kill of B per A tation is interesting in several regards. unit per hour; and of B, ß (0) = 2 kills of A per B First, by supposition B inflicts no casualties, unit per hour. The two sides are equal at the out- and so all of A may withdraw and live to fight set in the sense that another day. It is possible that A's force is permanently disorganized and demoralized, but that is a A = ß B = 10 kills per hour. not inherent in the model. Second, the battle is won by superior unit If the Lanchester square law applies, unit firepower, even though at its outset the total rate fire is constant and continuous, with a = 1 and of fire and fire effectiveness of both sides were ß = 2, and equation [2] obtains. equal; unit effectiveness is more influential than the number of units. • A will win a fight to the finish, and when Third, B wins by forcing A to concede B's the B force is destroyed, A(T) = 7.07 units, so 71% tactical objective or else suffer complete suppres- of the A force survives. Such is the power of sion and, by presumption, destruction. Mission numbers under Lanchester square law conditions. accomplishment is a good way to decide who • If B senses defeat after one of its 5 units won the battle, but it is not the usual way in (20%) is lost and successfully ends the battle, analysis, which is to compare casualties. Prob- A(T) = 9.05 when it is over. A has lost nearly one ably B's organization and morale will be unit to B's superior fire, which is almost as much stronger after the battle than before it, but again, as B lost. A's long term cumulative advantage of the model does not tell us. numbers has not had time to take effect. Fourth, we should not conclude that A's sit- Next assume that the effect of suppression uation is hopeless in future battles. Now that we on enemy fire works much faster and is far more know the significance of treating a and b (or a important than attrition. Equation [6] applies, so and ß) as time-dependent variables, we may that in a formal sense neither side loses units. anticipate battlefield conditions in which A finds A = 10 and B = 5 are constants, and only the vol- a stronger position with an improved firepower ume and accuracy of each side's fire is affected ratio and so is able to exploit its numerical by the other's. Let the initial firing rate of A be advantage. a(0) = 10 rounds per minute per shoooter and let Fifth, observe that the victory went to the each round suppress enemy fire at s = .01 rounds side whose fire dominated on the battlefield. In a per minute per enemy unit. Correspondingly, let formal sense this was true by postulation, for we b(0) = 20 rounds per minute with the same value assumed that attrition played no part. Never- of s. The two sides' initial volumes of fire are theless it is useful to look at how and why the equal in that Aa(0) = Bb(0) = 100 rounds per lethal potential of B's superior fire was decisive. minute. It was in part because B's fire attenuated A's fire • At time T when B has totally suppressed more rapidly, and in part because after A is A's fire, each unit of B has an unsuppressed rate reduced to impotence he must surrender or face of fire remaining of b(T) = 1.414 rounds per destruction. In war the activation and effective minute, or 71% of the initial rate. We may pre- employment of superior firepower is the central sume A to be impotent and at the mercy of B. cause of victory, whether or not casualties deter- • If A senses that B's volume of fire is domi- mine the outcome. (Our model ignores move- nant after his fire has been curtailed by 20%, he ment for the sake of analytical simplicity, but it is may end the battle and withdraw as he is able. safe to say battlefield movement unsupported by At the time the battle is over, a(T) = 8 rounds per covering, suppressive fire is a rare occurrence. minute per shooter, and b(T) = 18.1 at its end, Tactical maneuvering is achieved by an astute which is 90.5% of B's initial firepower. blend of covering fire and movement.)

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Sixth, these models are formalisms, whether One should not go so far as to disregard basic equations [1], [3] or [5] are used. For one attrition. But even if one regards casualties as the thing, casualties will occur on both sides. For available and hence utilitarian measure, never- another, the reduction of fire on the losing side theless the Lanchester square law does not fol- caused by the winner's fire will usually not go low with its conclusion that numerical ad- all the way to zero. There is a point of diminish- vantage is more valuable than unit firepower. ing returns in the suppression effect of fire Indeed, in recent research to demonstrate the towards attenuating enemy fire. utility of his "defender's advantage parameter," Seventh, as a reminder, ground combat is R. Helmbold [1995] shows that numerical advan- our subject. Naval and air combat take their own tage, which is the most important determinant in form. In the judgment of the author, attrition is the square law, is among the worst predictors of the essential phenomenon, and suppression as a victory in 83 historical battles. One should driving cause of sea and air battle outcomes is regard attrition and suppression both as impor- rare. tant. This analysis suggests, to say the least, that their source—firepower—is more influential that numbers engaged. But the answer we offer here to the question SUMMARY AND CONCLUSIONS posed in the beginning is that sharply superior combat power must impose and exploit suppres- Let us summarize the major points of the sion when the goal is a relatively bloodless victo- quantitative analysis: ry. The best example in the past 50 years of • The coefficients of unit effectiveness victory by suppression is the German blitzkrieg ought to be regarded as variables in time. of 1940 in France. The blitzkrieg had many of the • Fire volume and unit effectiveness will properties of the Gulf War. No attrition model often be far more variable than the change due to can explain the blitzkrieg phenomenon, which casualties, in which case they have a greater was achieved by an intensive, local suppression immediate effect on the outcome than attrition. of the defenders' ability to resist during a • When unit firepower is diminished more German armored-mechanized breakthrough. rapidly than the surviving number of units (due The breakthrough was followed by extensive to the effectiveness of enemy fire) then the unit demoralization of the French defenders when firepower advantage has the square law effect, they faced an enemy in their rear. Blitzkrieg not numerical advantage. (a.k.a. lightning war or maneuver warfare) It is quite reasonable to expect that the sup- worked on the minds and spirits of the enemy to pression effect of superior combat power will reduce his combat power to near zero. frequently be greater than the attrition effect. It is in the nature of a successful campaign of Referring again to McQuie, at the end of 80 bat- maneuver that few battles are fought and the tles in and after World War II, the median casual- casualties that attract the interest of military his- ties were 4% at the time the attacker abandoned torians and analysts are largely absent. To under- his objective and 8% when the defender con- stand the full value of very superior combat ceded defeat. McQuie cites the decisive effect of power one must look for Sherlock Holmes' dog maneuver (or by implication the inability to that didn't bark: mission attainment with the maneuver) as the salient reason for termination near-absence of combat and casualties. Perusal by the loser in the majority of the battles. We of battles analyzed quantitatively reveal few if have not examined the suppression of maneu- any for the entire 1940 German campaign in ver, though it is possible to do so. In any case, France. For example, Combat History Analysis McQuie's data suggest that casualties are seldom Study Effort: CHASE, by R. Helmbold [1986] lists the determinant of battle outcome. They also only one, Sedan-Meuse River of 13 May 1940. suggest that when the attacker's success results Another example is the Japanese defeat of the in a rout or surrender of the enemy, casualties British on the Malay Peninsula in 1941 in a bril- may be principally in the form of prisoners taken liant campaign of maneuver with never a battle instead of bodies slain. worthy of the history books. These are examples

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of what Sun Tzu told us long ago is the true maneuver warfare are involved because one measure of successful generalship [1963, p. 77]. effect of suppressive fire is to "pin down" the We recall what Clausewitz said in the same vein: enemy with one element of force (notably it is not the attacker but the defender who initi- artillery and close air support), while another ates a battle; the defender may choose to con- achieves a penetration (notably armor), or cede the opponent his aim instead of fighting maneuvers on a flank (notably mechanized or [1976, p. 377]. other mobile forces). One may well ask why American military b. The Lanchester square law form has been tacticians should not long ago have perceived the basis of this discourse. There is no reason to the quite reasonable, if not obvious, relationships think the square law is "right," except that described above. It is probably a proclivity to aimed fire is the image of the hypothetical en- leap at once to the "practical:" an ad hoc per- gagements described. I have been somewhat spective and disinterest in general theory Since vague about the extent to which suppressive fire 1950 the practical problem under study has been is aimed or fire-hosed. What we do know from "how to fight outnumbered and win," tied to the Osipov, Helmbold, Hartley and others is that the NATO central front in Europe. The advantage of square law advantage of numbers has rarely superior firepower, its potential for reducing been achieved when measured by attrition. I casualties, and the deeper effects of the phenom- think operations analysts of ground combat ena of combat power were not ours to exploit should wish to investigate whether historical until the Gulf War. results comport with combat models better after The best military leaders have always suppression is introduced. I do not know sought victory by domination and control, not whether to suggest starting with the square law, bloodshed. We analysts do a disservice if our the linear law, or something in between. Their studies cannot carry the quantitative analysis of instincts will be better than mine. battle outcomes beyond the measurement of c. For the purpose of modeling maneuver casualties. The U. S. government seems to be in warfare, a more direct approach is to extend the the process of casting aside the means to apply equations to include the suppression of move- overwhelming combat power in the next major ment as well as fire. On one hand, this is appeal- regional conflict. It would be most timely to give ing because we know that fire inhibits enemy some attention to suppression as a root cause movement, yet combat models do not reflect it and dominance as the vital effect in determining as a function of the accuracy and volume of the highly favorable outcome of the ground cam- enemy fire. On the other hand, one discovers a paign in the Gulf War, and many others. dimensionality problem and must wrestle with inputs in terms of the deceleration rate, which has units of meters/minute-enemy shot or the FUTURE RESEARCH equivalent. If we analysts are serious about help- ing to explore the operations and tactics of The Army and Marine Corps espouse maneuver warfare, it seems incumbent on us to "maneuver warfare." The thoughtful reader will try, by showing the connection between move- see the possibility of extending the model herein ment and supporting (suppressing) fire in in a number of ways that lead to a better under- maneuver warfare. For what it is worth, it is cer- standing of maneuver's advantage, and the rela- tainly possible to write equations in which fire tionship between fire and movement. has all three effects: casualties, suppression of a. The first and simplest is to work out the fire, and suppression of movement. mathematics when both attrition and suppres- d. Another approach would be to regard sion result from fire; in other words, when fire the effect of opening fire as a step function. In reduces enemy numbers and return fire simulta- such a model, the instant fire is opened, enemy neously. Some parametric analysis of the rela- return fire (or motion) is diminished to a new, tionships will then be possible, but eventually lower level. This is more or less implicit now in the more difficult task of obtaining numerical the inputs for the coefficients of unit effective- values will have to be confronted. Clues to ness. The new feature would be to scale the

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effectiveness of B's fire (or motion) inversely AXIOM 5. In combat, DOMINATION of with the accuracy and intensity of A's fire. To my the opposing military force is the ultimate knowledge, this has never been attempted. The means of achieving an objective. conceptualization is not a big step, but the scal- ing factors would come only with great difficulty. AXIOM 6. UNCERTAINTY is inherent in combat.

ENDNOTES The status of TMCI theory of combat may be had from the author at the Naval Postgraduate 1. To TMCI it seemed desirable to call their School, Monterey, California 93943, or Dr. cornerstones Axioms, whether traced from an Donald S. Marshall, Executive Director of The innate comprehension of truth or derived from Military Conflict Institute, 12 Fairfield Street, common experience. The six axioms are thought Salem, Massachusetts 01970. to be the fewest possible, so obvious as to be indisputable, and so essential that no structure of 2. A source of some rare exceptions is theory or model should proceed without cog- MORS' two-volume Proceedings of the Second nizance of them. Root definitions must, of Mini-Symposium on the topic, Human Behavior course, accompany them: and Performance as Essential Ingredients in Realistic Modeling of Combat—MORIMOC II. Among the Definition 1. A MILITARY FORCE is a set of noteworthy papers are: elements which are activated for combat. a. David Rowland, "Assessment of Combat AXIOM 1. COMBAT occurs by deadly Performance with Small Arms" interactions between military forces. b. Charles L. Frame, Brian R. McEnany and Definition 2. COMMAND is the function Kurt A. Kadivko, "Combat Operational Data which organizes, motivates, makes decisions Analysis: An Examination World War II regarding, and directs the activities of its force. Suppression Data"

AXIOM 2. In combat each COMMAND c. George Schecter, James C. Richards and seeks to achieve a goal, called its mission, Henry A. Romberg, "Tactical Deterrent Effects which has perceived value. Model"

Definition 3. COMBAT POTENTIAL is the d. Trevor N. Dupuy, "The Fundamental latent capacity of a military force to achieve Information Base For Modeling Human results in combat. Behavior In Combat"

AXIOM 3. COMBAT POTENTIAL is 3. On page 2 of this early paper by one of embodied in military forces. our most prolific and perceptive analysts of historical ground combat, Helmbold writes: Definition 4. COMBAT POWER is the real- "[Daniel] Willard's approach is typical of those ized capability of a military force at an instant of which attempt to infer the form of Lanchester's time to achieve results in combat. equations from an analysis of a large number of battles for which the initial and final strengths on AXIOM 4. Command activates its combat both sides are known, without depending on potential to create COMBAT POWER in fur- any information about the details of their attri- therance of a mission. tion histories. A finding that this approach is fully justified would be of capital importance for Definition 5. DOMINATION is the condition the development of a theory of combat, because wherein one military force imposes its will on data on initial and final strengths are available the other. for hundreds of battles. . .they at least are much

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more readily available than data on attrition his- D. S. Hartley III, Confirming the Lanchestrian tories" [1979]. In the preceding paragraph Linear-Logarithmic Model of Attrition, Report Helmbold has just finished taking note of the K/DSRD-263, Oak Ridge, TN, Martin Maritta famous effort by J. Engel to validate the Lan- Energy Systems, Inc. December 1990. chester square law's shape as it unfolded during the extended Battle of Iwo Jima, so he is clearly D. S. Hartley III, "Validating Lanchester's Square concerned with the attrition history occuring Law and Other Attrition Models," Naval Research within a battle, or short campaign. Logistics Journal, June 1995.

4. A more thorough description would back- D. S. Hartley III, Predicting Combat Effectiveness, track to equations [3] to explain that functions a Report K/DSRD-412, Oak Ridge TN, Martin and ß are really composites of a firing rate term Marietta Energy Systems, Inc. August 1991. in shots fired/minute, and an accuracy term, in hits per shot. Then in equations [4] we would R. L. Helmbold, "Some Observations on the have same firing rate value as in equations [3] Choice of Exponent in Lanchester's Law," plus a suppression term, in shots surpressed per copyrighted manuscript, June 1979. shot fired. R. L. Helmbold, Combat History Analysis Study 5. In memorable and decisive battles, casual- Effort (CHASE): Progress Report For the Period ties have been as much as 25-35%, but such bat- August 1984-June 1985, Bethesda, Md. U. S. Army tles are usually episodic, comprising separate Concepts Analysis Agency, CAA-TP-86-2, phases in which large changes of a and ß in August 1986. effect define the phases. R. L. Helmbold, "Combat Analysis: Defender's Advantage Parameter, Some Empirical Evidence," PHALANX, March 1995.

REFERENCES G. W S. Kuhn, Ground Forces Casualty Rate J. V. Chase, "A Mathematical Investigation of the Patterns: The Empirical Evidence, Report Effect of Superiority of Force in Combats Upon FP703TR1, Bethesda, MD, Logistics the Sea," Newport RI, Naval War College Management Institute, September, 1989, and archives, 1902; reprinted in its entirety as Ground Forces Battle Casualty Rate Patterns: Appendix C of B. A. Fiske [1988]. Current Rate Projections Compared to the Empirical Evidence, Report FP703TR2, May, 1990. Carl von Clausewitz, On War, edited and translated by Michael Howard and Peter Paret, F. W Lanchester, "Aircraft in Warfare: The Dawn Princeton, NJ, Princeton University Press, 1976. of the Fourth Arm, Number V, The Principle of Concentration," Engineering 98,1914 (Reprinted P. K. Davis and D. Blumenthal, "The Base-of- pp. 2138-2148 of The World of Mathematics, J. Sand Problem: A White Paper on the State of Newman, editor, New York, Simon and Shuster, Military Combat Modeling," Santa Monica, 1956). RAND, N-3148-OSD/DARPA, 1991. R. McQuie, "Battle Outcomes: Casualty Rates T. N. Dupuy, Understanding War: History and as a Measure of Defeat," Army Magazine, Theory of Combat, New York, Paragon House, November 1987. 1987.

B. A. Fiske, "American Naval Policy," Naval Institute Proceedings, April 1905.

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Military Operations Research Society, originally in the Russian Journal, Military Proceedings of the second mini-symposium on Collection, June-October 1915; Translated and Human Behavior and Performance as Essential annotated by R. L. Helmbold and A. S. Rehm in Ingredients in Realistic Modeling of Combat - research paper CAA-RP-91-2, Bethesda, Md, MORIMOCII, two volumes, Alexandria VA, U. S. Army Concepts Analysis Agency, 1991. MORS, May 1989. Sun Tzu, The Art of War, Translated with an M. Osipov, "The Influence of the Numerical introduction by S. B. Griffith, New York, Oxford Strength of Engaged Forces on Their Casualties," University Press, 1963.

EIGHTEENTH SYMPOSIUM ON MATHEMATICAL PROGRAMMING .The WITH DATA PERTURBATIONS 23-24 MAY 1996 UnÄty CALL FOR PAPERS 'WASHINGTON DC*

The EIGHTEENTH Symposium on Mathematical Programming with Data Perturbations will be held at George Washington University's Marvin Center on 23-24 May 1996. This symposium is designed to bring together practitioners who use mathematical programming optimization models and deal with questions of sensitivity analysis, with researchers who are developing techniques applicable to these problems.

CONTRIBUTED papers in mathematical programming are solicited in the following areas:

(1) Sensitivity and stability analysis results and their applications. (2) Solution methods for problems involving implicitly defined problem functions. (3) Solution methods for problems involving deterministic or stochastic parameter changes. (4) Solution approximation techniques and error analysis.

"CLINICAL" presentations that describe problems in sensitivity or stability analysis encountered in applications are also invited.

ABSTRACTS of papers intended for presentation at the Symposium should be sent in triplicate to Professor Anthony V. Fiacco. Abstracts should provide a good technical summary of key results, avoid the use of mathematical symbols and references, not exceed 500 words, and include a title and the name and full mailing address of each author. The deadline for submission of abstracts is 17 March 1996.

APPROXIMATELY 30 minutes will be allocated for the presentation of each paper. A blackboard and overhead projector will be available.

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