ABSTRACTSince 1965, the United States Air Force hasrelied on mathematical programming for theplanning of conventional air-to-groundmunitions. The centerpiece of this planningeffort is HEAVY AITACK, a theater-levelmodel employing large-scale nonlinear pro­gramming to load weapons onto aircraft andassign sorties to targets. The single-periodobjecti\'e is to maximize the expecteddestroyed target value over the forecastweather states by assigning sorties which usethe best delivery tactics in each weather statewith available aircraft and weapons stocks.Over multiple periods, HEAVY ArrACKaccounts for differences between targets inregeneration rate, value, and ease of damageassessment, and evaluates aircraft attritionand remaining weapons stocks, mountingthe best sorties possible with the remainingresources. In 1988 approximately $2 billionworth of weapons were purchased withguidance from HEAVY AITACK; additionalexpenditures of $5.2 billion are beingplanned for 1994-99.

In 199G---91, media coverage of DesertStorm made the focus of HEAVY ArrACKapparent to millions of viewers.

As many arrows, loosed severalways, _ so maya thousandactions, once afoot, end in onepurpose, and be all well bornewithout defeat.

Shakespeare (Henry Vi

Legislative and Executive branches of theU.s. Government, and, ultimately, to U.s.taxpayers. USAF is unique amongst the mili­tary services in the extent to which it relies onmathematical programming to accomplishthis. In 1988 approximately $2 billion worthof weapons were purchased, and expendi­tures of $5.2 billion are already planned for1994-99 (e.g., Department of Defense[1993]) with guidance from HEAVYAITACK, the main subject of this paper.HEAVY AITACK is one of the major appli­cations of mathematical programming in theUnited States. In this article we review thehistory of the system, describe its current use,and project near-term enhancements basedon current research.

BACKGROUNDThe USAF is interested in optimizationbecause aircraft are flexible weapon systems;depending on how an aircraft is loaded withweapons, it can more or less effidently attacka variety of targets. Given a collection of sev­eral types of aircraft (say ai aircraft sorties oftype i; i = 1,...,,4) to be used in attacking a col­lection of targets (say I' targets of type j, j =1,...,n the problem o( assigning aircraft totargets naturally arises. Perhaps the simplestformulation of the problem would be to letEij be the average number of targets of type jkilled by a sortie of type i, xi' the number ofsorties of type i assigned to Vargets of type j,and then solve program LPl:(LPl)

SortieOptimizationandMunitionsPlanning

Gerald G. BrownOperations ResearchNaval Postgraduate SchoolMonterey, CA 93943

Dennis M. CoulterHeadquarters, USAFPentagonWashington, DC 20330

Alan R. WashburnOperations ResearchNaval Postgraduate SchoolMonterey, CA 93943

TLXij~ai; i=I,... ,Aj=1

Xij~ 0,

where Y' (a variable) is the average numberof targeh of type j killed and Vj (an input) isthe subjectively-assessed value of a target oftype j. The meaning of the objective functionis "average value of targets killed".

Although (LPl) is a good starting point for

INTRODUCTIONThe United States Air Force (USAF) bases itsair-to-ground munitions planning on theprojected need for weapons in fighting a pro­tracted war. Sufficient stocks of suchweapons must be in place in, or trans­portable to, a theater of operations for timelyuse in combat. In order to determine therequired stores of such weapons, some eval­uation of their use in hypothesized theater­level conflict is required.

Over the past 25 yedrs, the USAF has pio­nt'ered in the modeling and optimization ofthe end effects of the procurement, stockpil­ing, and combat use of conventional air-to­ground munitions; the goal is to provideguid,mce credible to military planners, to the

,\'lilil<,ry Opa,'lioIlS R<search, Slimmer 1994

T

max: L"jYjx,y j=1

AS.t. LEjjXij=Yj; j=I, ... ,T

i=1

(1)

Pa~< 13

SORTIE OPTIMIZATION

this exposition, USAF has not to our knowledgeever actually used it. The difficulty is that solu­tions to such formulations tend to be very extremein nature. Each aircraft type is entirely assigned toa single target type (i is assigned to j if j maximizesVlij); it is even conceivable that all aircraft typesmight be assigned to the same target type.Although this kind of solution might be reason­able in a target-rich environm~ntwhere sorties arehopelessly outnwnbered by targets, it is neitherrealistic nor acceptable in general, even merely forplanning purposes.

There are two direct methods of embellishing(LPl) so that the solutions are not so extreme. Thesimpler is to add the constraints Yj ::; t· (note that(LPl) does not involve the data tj at alll to preventthe possibility of killing more targets on averagethan are known to exist. Call the resulting linearprogram (LP2). In (LP2), sorties may be assignedto targets other than their favorite type if thefavorite type is exhausted. The Theater AttackModel (TAM) discussed by Might [1987J is a lin­ear generalization of (LP2) where variables haveadditional subscripts corresponding to weather,weapons, etc., so descendants of (LP2) are well­represented amongst contemporary military plan­ning models. However, even though (LP2)accomplishes the goal of cutting off the extremesolutions while introducing minimal complica­tion to (LPl), it was not the method originally cho­sen by USAF.

The other direct method of fixing (LPl) is tomake the objective ftmction nonlinear to reflect theidea of decreasing returns as y' is increased. Anearly USAF model, SABER Mrt, was identical to(LPl) except that the objective ftmction was

TL, Vi t/ l -exP(-Yj/tj)) (2)j=1

This objective ftmction might be justified byarguing that the "fog of war" will cause the statis­tics of the number of times a particular target oftype j is killed (say, X-) to obey the Poisson distri­bution (Blackett [1961]). Since the expected valueof Xj is Yjl1i' it follows from the Poisson assump­tion that the probability that any target of type1is/lot killed is

Equation (2) then follows directly. The objectiveftmction still has the meaning "average value oftargets killed", just as in (LPl) and (LP2). Ofcourse the new mathematical program (call it(NLPl)) is of a more difficult type; the constraintsare linear, but the objective ftmction is not.

Since 1 - exp(-x) ::; x for x ? 0, (NLPl) willalways have a smaller optimized objective ftmc­tion than (LP2). (LP2) essentially incorporates theasswnption that sorties can be coordinated so thattargets that have already been killed will not befurther attacked. This kind of coordination isasswned to be impossible in (NLPl), with theattendant possibility of wastage due to overkill.The two programs correspond to extremeasswnptions about the kind of command and con­trol that can be exercised in battle.

(NLPl) was a nontrivial optimization problemwhen it was formulated in the 1960's. An earlyattempt at a solution involved the asswnptionthat all targets were to be attacked by one aircrafttype. This problem has the same mathematicalform as the corresponding Search Theory prob­lem of allocating random search effort to a collec­tion of cells (Chames and Cooper [19581), so anefficient solution technique was available by thetime the Munitions Planning Branch was formed.However, there were obvious problems withassigning each aircraft type as if none of the othersexisted, so the desirability of solving the joint opti­mization problem where all aircraft types are con­sidered simultaneously was quickly recognized.

In the early 1970's, the Directorate of DefenseProgram Analysis and Evaluation (DDPA&E)funded a mathematical programming-basedscheme for assigning aircraft to targets that incor­porated the best features of SABER M1X and twoother systems that were then in use. The resultingformulation (NLP2) is reported in Clasen, Graves,and Lu [1974J. The (NLP2) objective ftmction issimilar to that of SABER fl,UX except for the incor­poration of an additional parameter ']'

T

L,v/j<I-exp(-cIYjltl))hl (3)

1=1

Note that (3) is the same.1S (2) if (I = 1, and that(3) is the same as (I) in the limit as Ct' approachesll. The parameter Cj thus bridges he situation

Military Operations Research. Slimmer 1994

SORTIE OPTIMIZATION

where command and control is impossible (thePoisson case Cj = 1) and where it is perfect (the lin­ear case where Cj approaches 0). However, a pre­cise meaning for Cj has never been given. Clasen,Gra\'es, and Lu say only that "DDPA&E suggest­ed this function to us. It is similar to the objectivefunction of the SABER MIX methodology." Thelack of a physical meaning has proven to be trou­blesome for subsequent generations of USAF offi­cers (see Embellishments) required to estimate c·for various classes of targets, and various levels otengagement "fog" in the theater.

(NLP2) also incorporates constraints on Yj thatensure that not more than t· targets of type j arekilled, on average. Thus (iP2) and (NLPl) areboth special cases.

HEAVY ATIACKThe HEAVY ATIACK model currently in use bythe USAF Weapons Division is still, at its heart,the Clasen, Graves, and Lu model (NLP2), butnow larger and embellished with additional typesof linear constraints. HEAVY ATIACK considersa sequence of allocation problems, with eachproblem corresponding to one time period in awar that is projected to be several time periodslong. Targets that survive the attacks of one peri­od are still available in the next, together withreinforcements and also with targets that werekilled in previous periods but have since beenrepaired (reconstituted), possibly with differentvalues. The optimization is done myopically, withthe objective in each period being to kill as muchtarget value as possible without regard to theeffect on future periods. The myopic feature isanalytically convenient, since it permits the analy­sis of a sequence of small problems rather thanone large one, but it is also realistic in the sensethat the actual policy for assigning aircraft to tar­gets (which must be distinguished from the value­based method in HEAVY ATTACK) is ajoint-service process that does not include the ideaof "saving targets for the future". There may be anelement of making virtue out of what was once anecessity here, but still thdt is the justification usu­ally given for myopia.

HEA \"Y ATIACK depends on a separate pro­gram, SELECTOR, for the sortie effectivenessinputs Eii' SELECTOR is needed not because

Milit'lry Opemtiolls Rese,lrcfI. Slimmer /994

effectiveness coefficients would otherwise be lack­ing, but rather because there are too many ofthem. The Joint Munitions Effectiveness Manual(fMEM, e.g. Joint Technical Coordinating Group/Munitions Effectiveness [1980]) shows how to tab­ulate

average number oftargets of type j killedby a sortie of type iusing tactic t in weathertypew.

SELECTOR's role is essentially to get rid of thelast two subscripts. The method for doing this isimportant, since it is often the case that the mosteffective tactics are associated with expensivemunitions or high attrition to the delivering air­craft. SELECTOR adopts only the most cost-€ffec­tive tactic: literally the tactic that maximizes theratio of sortie cost (including the cost of weaponsused and expected attrition) to Ei'!w' Let this tac­tic be t*(i, j, w), and let Eij"w' be ~e effectivenesswhen that tactic is usea. The coefficients Eijrequired by HEAVY ATTACK for each period areobtained by simply averaging Eij"w over whatev­er w~ather distribution is appropriate in the areaof the supposed conflict; the natural notationwould be Eij'" but we will drop the two dots to beconsistent with earlier usage. t (I, j, w) may changefrom period to period as stocks of the requisiteweapons become exhausted. The weather distrib­ution may also change from period to period, sothe same is true of Eij. Crawford [1989) concludesthat usage of HEAVY ATTACK with Eij comput­ed in this manner biases weapons purchasestowards cheap but inefficient weapons. He rea­sons that the real "cost" of attrition considerablyexceeds the aircraft flyaway cost used by USAF indetermining "bang-per-buck."

The inputs to HEAVY ATIACK are deter­mined once each year for each potential theater ofoperations. The Weapons Division hosts an annu­al theater planning session attended by mid-leveloperational, intelligence, logistics, and planningstaff officers. The goal is to produce a realistic cur­rent requirement for theater weapons stocks.Attendees must travel great distances to partici­pate in these sessions, and can only be expected toremain briefly in residence before retuming to exi­gent duty. Cadre analysts (e.g., Coulter) are

Page 15

SORTIE OPTIMIZATION

responsible for care and feeding of HEAVYATTACK, and thus must interpret the proposalsof the theater planning group and endeavor toprovide compelling scenarios for their evaluation.Once HEAVY ATTACK has determined sortie-to­target allocations for each period of the war,munition requirements can be recovered byrecalling the function t(i, j, w) determined bySELECTOR and doing the appropriate account­ing. The resulting estimates of theater weaponsstocks requirements are used to justify the aggre­gate USAF annual weapons buy request. Thesebudget requests are exposed to exhaustive scruti­ny by USAF, and subsequently by other levels ofreview. Budget revisions by higher authority arereconciled with mission requirements with thehelp of HEAVY ATTACK.

EMBELLISHMENTSHEAVY ATTACK as first formulated by Clasen,Graves, and Lu in 1974 was a nonlinear programwith at most 10 sortie types and 45 target types;even with their new method, solving problems ofthat size required quite a while on contemporarycomputers. Enriching the model in any mannerthat would have increased run times was out ofthe question, however nervous its users mighthave been about such things as the SELEC­TOR/HEAVY ATTACK system's myopiCapproach to optimization over time, fractionalsortie assignments, and the suboptimizationimplicit in SELECTOR's preprocessing of theJMEM effectiveness data.

However, computers and computer softwarehave improved substantially since 1974. Lord[1982J reports mainframe solution times mea­sured in seconds, rather than minutes, upon com­pleting the installation of the X-system solver (e.g.,INSIGHT [199OJ, and Brown and Olson [1994]).Bausch and Brown [1988] describe a prototypiCimplementation of HEAVY ATTACK on anR0386-based IBM-eompatible microcomputer, anuncommon feat at the time. In 1991 more power­ful 80486 machines were configured for produc­tion use in various environments, implementedwith mainframe-compatible software (SiliconValley Software [1991 D, and shipped to users asHEAVY ATTACK machines. Wallace [19921exploits this new capability by designing andimplementing a prototypic graphical user inter-

face (GUl) which is especially useful for compar­ing outputs from multiple scenarios. Bradley, etal. [1992] give an unclassified demonstration ofthis unified hardware and software decision sup­port system: sortie optimization for 25 aircrafttypes, 90 weapon types, and 100 target types,problems with hundreds of constraints and thou­sands of variables for each of 6 time periods,requires about two minutes from SELECTORinput to final output. Washburn [1989] describes anew method tailored to the HEAVY ATTACKproblem that solves (NLP2) with 13 sortie typesand 61 target types in about two seconds on an80386 machine. Plainly, the computation timeconsiderations that drove the original (NLP2) for­mulation have now been substantially relaxed, tothe point where more computationally stressfulreformulations can be considered.

However, SELECTOR/HEAVY ATTACK's longand successful lifetime as a planning tool makes itdifficult to consider any substantial reformulation,even now that computational cost is negligible.Generations of Air Force officers have learned tocope with the idiosyncrasies, assumptions, anddata requirements. Inter-organizational relation­ships have evolved to provide inputs and inter­pret outputs. "If it ain't broke don't fix it."

So the Weapons Division is naturally attractedmost by changes that merely affect computationalefficiency or ease of use. For example, linear sideconstraints can now be used to insure flyable mix­tures of sorties. Graphical user interfaces areexpensive to design and develop, but are invalu­able for quick, reliable formulation of input sce­narios and interpretation of their output. Further,although HEAVY ATTACK can run on manyhardware platforms, and employs the fastestlarge-scale optimizer in our experience, a 486 PCis now the favored host due to its low cost, conve­nience, and portability. In early 1993, theater con­ferences were completed for the first time withoutrequiring the physical attendance of the theaterplanners.

Less fa\'ored, but sometimes essential, arechanges that produce the same output quantitiesfrom the same input quantities. For example, avariety of changes have been made to the targetreconstitution model. These changes do notrequire new inputs nor change the meaning of theoutputs, although the output quantities are ofcourse affected.

Least favored are reformulations th,1t require

Alilil<"y Operatiol/s R<?s,wcil, SlIlIllller 1994

SORTIE OPTIMIZATION

an essentially different way of looking at things.Recently, constraints on weapon usage by periodhave been added. The basic idea of SELEC­TOR/HEAVY ATTACK is to buy whateverweapons are required to fight a cost-€ffective war,so it would seem illogical to include constraints onthe usage of a particular weapon. The trouble isthat certain weapons (ACM-65A/B Maverick air­to-ground missiles, for example) are no longer inproduction but still quite effective. The only real­istic way to handle such weapons is to constraintheir usage to the size of the current stockpile.

Note that the current system includes no bud­get constraints, even though it is a principallybudgetary tool. SELECTOR utilizes cost inputs indetermining the most cost-€ffective tactic, butweapon usage is not actually constrained by anybudget. The HEAVY ATTACK output can there­fore be interpreted as the classic military "require­ments" for weapons with some implied budgetlevel B. The idea that B should be an input, ratherthan an output, requires a fundamentally differ­ent view of the problem.

We continue to pursue enhancements andreformulations. Boger and Washburn [1985Jdescribe an alternate nonlinear objective functionwhere the par~eterCj has a physical interpreta­tion. They also descriOe how the organization ofcomputations in SELECTOR could lead tooverutilization of weather-specialized weaponsand weather robust aircraft. Wirths [1989] devel­ops several prototypic reformulations using theCAMS/ MINOS system (e.g., Murtagh andSaunders [1984J, Brooke, et aI. [1992]). Amongstother things he derives a differential equation forwhich (3) is a solution, shows that the impact ofusing a linear objective function is not as great ashad previously been thought, and asserts that themyopic approach to optimization over time ispossibly of more concern. Utilizing a linear objec­tive function would of course open up a greatmany other possibilitil'S for reformulation, includ­ing the adoption of the conceptually simpler, butlarger TAM (Might [1987]), which includes sub­scripts for weapons, weather, and, according toJackson 11988J, one-dimensional sortie range andtime period too. However. TMvl utilizes budgetconstraints, and our computational experiencewith TA~l, as well as that of Jackson [19881. showsthat it is H'ry time consuming to soh'e in spite ofits line.1r objecti,·" tlilKtion.

CONCLUSIONWe chose HEAVY ATTACK for this expositorypaper for some reasons not yet discussed. HEAVYATTACK has for some time been a favorite class­room example at the Naval Postgraduate School.HEAVY ATTACK is simple to explain and under­stand without resorting to excessive mathematics,can be used in hands-on homework and modifiedfor experiments by students, yet exhibits all thefeatures, man-machine interaction, and richness adecision support system should have. The systemhas been in use for many years, and its remarkablelongevity and direct influence on billion-dollardecisions automatically enhance student interestand warrant study of its design and application.Even issues of client and analyst psychology, theinfluence of politics on decision making, and tech­niques for preserving run-to-run solution persis­tence and comparison of optimized results can behighlighted. Being a nonlinear optimizationmodel, it also provides rich collateral mathemati­cal material such as characterization of concavity,numerical analysis, function approximation,aggregation, and proofs of convergence.

When the problem description for HEAVYATTACK is given as a homework formulationexercise, students immediately construct detailedlinear models: we call this approach the "Xsu~ajpt­

fimwrytlling" method. Asked to provide answers tothe problem under time pressure, students quick­ly c1iscover the large size and data appetite of theirmodels, and devise reasonable aggnegation strate­gies sometimes reminiscent of HEAVY ATTACK.Required to interpret their answers to this prob­lem, students face many of the paradoxes inherentin modeling.

HEAVY ATTACK is an important member of astandard set of models we use to test new opti­mization techniques. The fact that we have alwaysmaintained nonlinear optimization capability inall our systems has derived in part from consider­ation of this application. We admit some profes­sional satisfaction that HEAVY ATTACK hasevol\"ed, with many cohort models, from a daunt­ing computational feat to a keystroke quick appli­cation, even for a microcomputer.

SORTIE OPTIMIZATION

ACKNOWLEDGMENTSThe Air Force Office of Scientific Research and theOffice of Naval Research have supported ourbasic mathematical research in optimization andencouraged its application. The Naval Post­graduate School promotes research on importantproblems even when there is no direct benefit forthe Navy. Insight, Inc. of Alexandria, Virginia, hasprovided optimization software for HEAVYAITACK. We thank the Air Force for the oppor­tunity to publish this account in the unclassifiedliterature.

REFERENCESBausch, D. and Brown, G. 1988, "A PC

Environment for Large-Scale Prograrnming,"OR/MS Today, 15, 3 Oune), pp. 20-25.

Blackett, P. 1%2, Studies ofWar, Hill & Wang, NewYork.

Boger, D. and Washburn, A. 1985, "Not~ fromthe Stockpile Seminar:' NPS55-85-D14, NavalPostgraduate School.

Bradley, G., Buckingham, L, Brown, G., andWallace, D. 1992, "New Tools for OptimizingUSAF Sortie Allocation Planning:' ORSA/T1MS, Orlando, Florida (April 26).

Brooke, A, Kendrick, D., and Meeraus, A 1992,GAMS: A User's Guide, Release 2.25, TheScientific Press, South San Francisco, California.

Brown, G. and Olson, M., 1994, "DynamicFactorization in Large-Scale Optimization,"Matht'matiaIl Programming (to appear) .

Chames, A. and Cooper, W. 1958, "The Theory ofSearch: Optimum Distribution of Search Effort:'Mallag,went Science,S, pp. 44-50.

Clasen, R, Graves, G., and Lu, J. 1974, "SortieAllocation by a Nonlinear Programming Modelfor Determining Munitions Mix," RANDCorporation Report R-I-II-DDPAE, March.

Crawford, G. 1989, "The Air Force's MunitionsRequirements Process (The Nonnuclear Con­sumables Annual Analysis)", RAND Corpora­tioll Report N-2821-P,L ~1ay 31.

Department of Defense 1993, "Program Objecti,-eMemorandum for FY 199.J---1999:' Washing­ton, D.C.

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INSIGHT, Inc. 1990, "XS(F) Mathematical Pro­gramming System:' Copyright 1990, Alex­andri~, Virginia.

Jackson, J. 1988, "A Taxonomy of AdvancedLinear Programming Techniques," MastersThesis in Operations Research, Air Force Insti­tute of Technology.

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Lord, P. 1982, "An Examination of the UnitedStates Air Force Optimal Nonnuclear MunitionsProcurement Model," Masters Thesis in Opera­tions Research, Naval Postgraduate School.

Might, R 1987, "Decision Support for Aircraft andMunitions Procurement:' Illterfaces, 17, 5,September, pp. 5~2.

Murtagh, B. and Saunders, M. 1984, "MINOS ­Version 5.0," Department of Operations Re­search, Stanford University.

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Wallace, D. 1992, "Analysis Tools for UnitedStates Air Force Sortie Optimization and Muni­tions Planning," Masters Thesis in OperationsResearch, Naval Postgraduate School.

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Wirths, K. 1989, "A Nonlinear ProgrammingModel for Optimized Sortie Allocation,"Masters Thesis in Operations Research, NavalPostgraduate School.

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