[American Institute of Aeronautics and Astronautics Modeling and Simulation Technologies Conference...

Copyright© 1997, American Institute of Aeronautics and Astronautics, Inc.

A97-37243AIAA-97-3523

THE TISES MANY-ON-MANY ENGAGEMENT PLANNING ALGORITHM

James R. Ashburn (Nichols Research Corporation, Huntsville, Alabama 35802)Dawn Horn (THAAD Project Office, Huntsville, Alabama)

A missile defense many-on-many engagement planning algorithm designed for the THAADIntegrated System Effectiveness Simulation (TISES) is described. The algorithm represents aneffort to fulfill as many of the features envisioned by the THAAD community for the objectivesystem as possible. This paper describes the basis for the algorithm's objective functions, theconstraints imposed on the problem, and the various defense objectives that may be represented. Itaddresses issues such as inventory-limited defense, overlapping asset damage, and drawdownstrategies. The paper concludes with discussions of "adaptive method-of-fire," the strengths andweaknesses of predictive ("look-ahead") engagement planning, software implementation issues, andan integrated approach to engagement generation and selection.

DISTRIBUTION STATEMENT A. Approved for public release; distribution is unlimited.

IntroductionThe many-on-many engagement planning problem is

among the most well known battle managementproblems facing missile defense systems. Engagementplanning software must seek to fulfill defense objectiveswithin a large set of constraints while minimizingresource utilization. To the extent the driving factorscan be regarded as random, engagement planning is astochastic control problem, not unl ike that of aninterceptor nulling a predicted miss distance.1'2 In thecase of engagement planning, the parameters to becontrolled include the final (i.e., foreseeable end-of-battle) states of the threats, defended assets, andinterceptor inventories. As a control problem, thebattle manager must periodically reevaluate theperceived state of the threat and the current response tothat threat and determine the best changes to theresponse to achieve the desired balance between

References'The Look-Ahead Battle Planner, John J. Shaw, Mark

A. Gerber. Ronald M. James, Major Anthony P.Rizzuto, Proceedings of the 1993 Summer ComputerSimulation Conference, sponsored by The Society forComputer Simulation, P. O Box 17900, San Diego,CA 92177, pp. 717-726.

"To the extent the external stimuli are not random orat least not so regarded, such as the actions of an enemyin coordinating an attack, the engagement planningproblem is a game theory problem. The implicationsof this fact appear to be well beyond the scope of theengagement planning art as it exists today. Personalcommunica t ion , Gary Makowski, TecMasters,Incorporated, 30 May 1996.

fulfilling defense objectives and conserving resources fordetecting and negating future threat objects.

The algorithm described herein was developed forTISES, a high-fidelity, discrete-event, end-to-endmissile defense system simulation, to address thecapabilities envisioned for the THAAD objectivesystem.

Among the capabilities envisioned for the objectivesystem was the probabilistic treatment of projectedinterceptor inventories. This simple application of themean-value approximation is often referred to as"fractional rounds" by much of the THAAD BM/C4Icommunity and was a major driver in the selection of anapproach to engagement planning in TISES. Becausethe idea of probabilistic treatment of resources hadalready been documented by AlphaTech in a publisheddescription of their Look-Ahead Battle Planner (LABP),predictive (or "look-ahead") planning was thereforeadopted as a pattern for the TISES algorithm.

PurposeWhereas the purpose of the LABP was to "project

how the situation will look at some future time if thedefense executes a specified CoA [Course of Action],"the algorithm described herein seeks to use theseprojections as the basis for an objective function thatcan simplify or augment preplanned rules, priorities,and constraints, such as those in the NMD CoAs or theTMD Engagement Subplans. The original goal was todevelop an algorithm that required that the operator ofthe system specify only the performance objective whilethe algorithm determined the response that best achievedit within the limitations of the system. All other inputsto the algorithm were presumed to represent measurablephysical capabilities of the system.

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The focus of this paper is to describe the basis forthe TISES many-on-many algorithm and to highlightits relative merits, and identify its potential weaknesses.The rigors of the mathematicl basis for the algorithmsare summarized to comply with security restrictions.Only a brief discussion of software implementationissues will be provided. No sample results will bepresented is this exposition.

TerminologySome discussion of the terminology used in this

paper is beneficial at this point. First, the terms"engagement" and "shot" are often used interchangeablyin reference to the utilization of a single interceptoragainst a threat object. Depending upon the context,these terms may also include the expenditure of radarresources necessary to support the interceptor in flight.

The terms "non-interactive" and "interactive"constraints must also be defined. Non-interactiveconstraints are timing and geometry restrictions that areimposed on an engagement independent of all otherengagements. Examples inc lude kinematicaccessibility, sun/moon/earth limb exclusion angles,and time of flight limits. Interactive constraints, as thename suggests, involve dependent interactions betweenengagements usually associated with timing andresource limitations. Examples include interceptorinventory at a launcher and maximum launch rates.

Next, the terms "perceived" and "projected"situations must be clarified. The perceived situationincludes the current perception of the states of theobjects such as assets, threat objects, and defenseresources, including those currently being employed oravailable for employment against the threat. Associatedwith each perceived state of an object is a projected statecorresponding to an estimate of its condition at theforeseeable end of the battle.

The terms "existing," "candidate," and "planned"engagements must also be defined. Existingengagements are those which are currently beingexecuted. To be more precise, they include allengagements currently committed against a threat forwhich kill assessment has not been completed.Candidate engagements represent feasible potentialengagements against a given threat; they are"candidates" for execution. Each is feasible in the sensethat it can be executed if no other candidate engagementsare executed. To qualify as a candidate, the engagementmust first fu l f i l l all non-interactive constraints.Second, the candidate engagement must not violate anyinteractive constraints associated with any existingengagements. Planned engagements are those candidateengagements that are expected to be subsequentlyexecuted contingent upon the perception that the

associated threat is still alive at the required committime. Only the planned engagements that requireimmediate commit are executed in a given planningcycle; the remainder are used only for estimating theoutcome of the battle.

Next, the terms "execution," "kill quality hit," and"kill" must be clarified. The terms "execution" and"commit decision" are effectively synonymous. Theuse of the term "execution" is not meant to implysuccessful completion of an engagement, only acommit decision and an attempt to perform anengagement. The term "kill quality hit" is used todistinguish a hit of sufficient nature to have killed athreat (independent of the status of the threat before thehit) from a kill (a hit that renders a live threat inactive).

The Mean-Value ApproximationProjecting the possible outcomes of a battle

involves combinations and permutations of manyconditional events. For a problem of even modest size,enumeration or even sampling of the possible outcomesis impractical. Consequently, the mean-valueapproximation is employed, hereinafter referred to asMVA. MVA replaces enumerations of possibleoutcomes with the associated expected (or mean)outcome. Shaw, et al. provide an excellent discussionof this concept.

Some examples of the implications of MVA areworthy of mention. First, perceived states that aremeasured as discrete quantities, such as interceptorinventory, have continuous projected state counterparts.For example, consider a launcher with ten interceptorsand a planned shoot-look-shoot method of fire against athreat, each with an expected 90% chance of success.The projected remaining inventory is then 8.9 since theprobability the second interceptor will be used is only10%. Second, no distinction is made between binarydiscrete states (e.g., operational/not operational) andtheir continuous counterparts (e.g., X% operational).For example, one threat that has a 90% chance ofdestroying an asset and a 10% chance of doing nodamage to the asset is indistinguishable from a secondthreat that will always destroy exactly 90% of the asset.

The Simplified ApproachTo formulate an initial solution to the engagement

planning problem, this author has attempted todecompose the problem into the greatest number ofserial processes without sacrificing the quality of thesolution. In other words, a decision to integrate anytwo processes is based upon its necessity to retainingthe quality of the solution; runtime and memory are notpriorities in this simplified approach. Theseconsiderations are discussed in a subsequent section on

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the integrated approach. The processes that have beenidentified are as follows:

• Situation Maintenance• Threat Assessment• Candidate Engagement Generation• Candidate Engagement Selection• Engagement Task Message Generation

The first process, Situation Maintenance, is requiredto insure the most up-to-date representation of thesituation as it is perceived by the battle manager. Theprocess operates upon the perceived situation data andnotes state transitions that occur as a result of thepassage of time. Namely, these consist of assumedstate transitions as a result of overdue messages (i.e.,incoming data, either periodic or scheduled, which hasfailed to arrive at its expected time). These include thedetermination of stale tracks (e.g., overdue periodic trackreport), presumed non-operational defense resources(e.g., overdue periodic status report), and presumedfailed tasks (e.g., overdue scheduled reports such asinterceptor status reports and kill assessment reports).Many of the messages incoming to a battle manager fallinto this category of "anticipated" messages.

The second process, Threat Assessment, determineswhich threat objects are eligible for engagement. If athreat value-based objective is selected, threat objects areassigned values consistent with their damage potential.In the case of an asset value-based defense objective,threat objects must be associated with the assets theyare likely to damage. A mean damage estimate for eachthreat/asset combination is also required.

The third process, Candidate EngagementGeneration, calculates a set of feasible engagements forall threat/launcher/supporting radar combinations tosome sufficient time granularity. This process can beequated to identification of the solution space for theoptimization problem that is to be solved in the nextprocess. A good rule of thumb is to generate at leastone engagement commit opportunity per planning cyclewithin non-interactive constraints for eachthreat/launcher/radar combination. First, this insuresthat at least one engagement opportunity for eachthreat/launcher/radar combination is available forimmediate commit if feasible so that engagementopportunities are not deferred indefinitely. Second, itinsures reasonable temporal packing for shoot-look-shoot and salvo engagements. Alleviating thecomputational burden of generating a potentially largenumber of engagements is discussed in a later section.

The fourth process, Candidate EngagementSelection, is the focus of this paper; it is the processwhich contains the many-on-many algorithm. This

process must select from the set of candidateengagements a set of planned engagements expected tobest fulfill the defense objectives. These plannedengagements include both those that may require animmediate commit decision along with those whosecommit decision must be deferred. The latter includethose whose commit decision may be conditioned uponthe outcome of other engagements.

Separation of the third and fourth processes intoserial operations is the identifying characteristic of thesimplified approach. In a subsequent section, anintegrated approach to these processes is introduced.

The fifth and final process, Generate EngagementTask Messages, selects from the set of plannedengagements those requiring an immediate commitdecision and generates and transmits the messagesnecessary to initiate them.

Selection of an Optimization MethodSelection of an appropriate optimization method was

based upon the following reasoning. Becauseindependent probabilities are never additive, linearmethods are dismissed immediately. Because of thecomplex nature of the constraints imposed upon feasiblesolutions to the many-on-many problem, all non-general optimization methods are precluded. Because ofthe combinatorial size of the problem and the speedwith which it must execute, only the most rudimentarygeneral optimization method remains viable, namelymaximum marginal return (hereinafter referred to asMMR). Finally, because of the convoluted nature ofthe objective function and the constraints, only asuboptimal solution can be expected.

Admittedly, MMR has some potentially seriousdeficiencies when faced with problems such as estimatedkill probabilities that increase with the advancement oftime and overlapping asset damage. However, theseproblems are not insurmountable and proposedsolutions are discussed in later sections.

The Inputs and OutputsThe basic TISES many-on-many algorithm operates

upon assets, threats, engagements, launchers, andlauncher sites. These objects are identified by thefollowing indices:

i = assetj = threatk = engagement1 = launcherm = launcher site

* MMR is an optimal method only in the case of anobjective function with no local optima.

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Examples of associations between these objects are asfollows:

Jk = threat j associated with engagement kLk = launcher 1 associated with engagement k{K,} = set of engagements associated with launcher 1{KE |} = set of existing engagements associated with

threat j{Lm} = set of launcher s associated with site m

Note the shorthand notation used for set intersections.For example,

= {KE}n{Kj}

In the following, inputs and outputs associatedstrictly with leakage, threat-value, or defense-valuedefense criteria are identified by (L), (T), and (A),respectively. Summary outputs include the following:

A = projected total surviving asset value (A)A' = linearized projected total surviving asset

value (A)B = projected total remaining interceptor inventoryB1 = projected total remaining interceptor valueN = projected total number of threats negated (L)H = projected total negated threat value (T)

The inputs for each asset i are as follows:

Aj = current value of asset i (A)

The corresponding outputs are

Vj = projected surviving fraction of asset i (A)v'| = linearized projected surviving fraction of

asset i (A)

The inputs for each threat j are the following:

FJ = nominal damage capability of threat j (T)FJJ = mean fraction of asset i destroyed by threat j if

not engaged (A){KEj} = set of existing engagements against

threat j{Kcj} = set of candidate engagements against

threat j

Convention would normally dictate a set of decisionvariables associated with the candidate engagements. Inlieu of such decision variables, an initially empty set ofplanned engagements is introduced. Equivalent totoggling a decision variable, a candidate engagement ismoved to the set of planned engagements.Consequently, the threat outputs are as follows:

I = projected probability of leakage of threat jJ = set of planned engagements against

threat j

The inputs for each engagement k consist of

Lk = launcher associated with engagement kPQIE* - probability of kill quality hit given execution

of engagement k (i.e., the single shot killprobability pSSKp,k)

tk = commit time of engagement ktk = launch time of engagement kTk = intercept time of engagement k

In the case of existing engagements, pSSKP is ideally thecurrent estimate, appropriately adjusted to account forengagement performance up to the current time. Forexample, it may be increased for successful boosterseparation or decreased for low remaining divert fuel.The outputs for each engagement are

PE k = probability of execution of engagement kPQJC = probability of kill quality hit of engagement kPKiQ.k = probability of kill given kill quality hit of

engagement kPKA = probability of kill of engagement k

The inputs for each launcher 1 are the following:

W, - current interceptor inventory at launcher 1M! = launcher site m associated with launcher 1

Note that existing engagements have already beenconsidered in determination of the current launcherinventories; the current inventory equates to thecurrently remaining available inventory. The launcheroutputs are as follows:

w, = projected remaining inventory of launcher 1w'| = projected remaining interceptor value at

launcher 1

The outputs for each launcher site m are

wm = projected remaining interceptor inventory atlauncher site m

w'm = projected remaining interceptor value atlauncher site m

Inputs which describe system timing include thefollowing:

tc = current timeD = nominal shoot-look-shoot delayATmin = minimum time between launches at a

launcher

The nominal shoot-look-shoot delay is the delaybetween kill assessment and subsequent commit forplanned shoot-look-shoot engagements.

Rules of engagement inputs include the following:

Rmin = minimum rate of return

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AI = launcher inflation constantAT = rate of return decay time constantADm - deflation constant for launcher site m

Pg.k ~ PojE.k Pe.kPK,k = PKIQ.k PQ,k

(1)(2)

Pe.k

Assuming perfect kill assessment, the probability that aplanned engagement will be executed is equal to theprobability that the threat is still alive at the committime for that engagement. For planned engagementsrequiring an immediate commit decision and existingengagements, the probability of execution must beunity. These conditions can be written as follows:

V pK k. if k e {Kp} and immediate commit not requiredk-6{KE. j}u{KP. j}.j=Jk

The ModelFor each engagement, three events are defined with

which probabilities are associated— execution, killqual i ty hi t , and kill , as defined earlier. Therelationships between the engagement probabilities andtheir conditional counterparts can be written as,

T,,<t,-D

if k e {Kp} and immediate commit required

i f ke{K E }

(3)

If single shot kill probabilities (SSKPs) are assumed to be uncorrelated, then the probability of execution canalternatively be written as follows:

TTO'PoiEk") if k e {KP} and immediate commit not requiredks{KE i}u{KP ,} , j=Jk

TV<t t -D

PE.k = if k e {Kp} and immediate commit required

i fke{K E }

(4)

To avoid calculations of probabilities outside the range[0.0, 1.0] due to round-off errors, the form in the latterequation is preferred.

The probability of kill quality hit given execution issimply the SSKP, again neglecting correlations,

PQIE.k = PsSKP.k (5)

Given the equations above, estimates of the projectedlauncher inventories can be determined. These estimatesare denoted by the following:

w, = projected remaining inventory of launcher 1

Utilizing MVA, the projected remaining interceptorinventory for a launcher is simply

The probability of kill given kill quality hit is simplythe probability that the threat is alive at intercept giventhat it was alive at commit. This can be written as ke K

PE,k (8)

PKIQ.k (6)k'6{KEi}u{KP ,}.j=Jk

<k-D<Tk .<Tk

Again, assuming uncorrelated SSKPs, this can also bewritten as follows:

The projected total remaining inventory,

B = projected total remaining interceptor inventory

is simply

(9)

PKIQ.k (7)

tk-D<Tk-<Tk

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Interactive ConstraintsBefore addressing defense objectives and the objective

function, a discussion of interactive constraints isfitting. The most prominent of these constraints is theinterceptor inventory at each launcher. Consistent withMVA, the projected remaining inventory must beconstrained as follows:

w,>0 for each launcher 1 (10)

Note that MVA does not guarantee that adequateinventory will be available for all planned shots. It is,however, an approximation that tends to becomeincreasingly accurate in proportion to the size of theproblem as it grows.

In addition to inventory constraints, timingconstraints may also be present. Consider the exampleof a launch rate constraint at each launcher. Again,using MVA, this may be represented as follows:

PE,k')Al:min if Lk = Lk'

and J Jk.ATmin i fL k =L k ,

andJ k =J k .

(H)

(12)

where

Aimin = minimum time between launches at alauncher

The first of these two inequalities approximatesengagements on different threats as uncorrelated. Thelast inequality assumes that engagements on the samethreat sufficiently closely spaced to violate a launch rateconstraint have perfectly correlated probabilities ofexecution; that is, commit decisions are based onexactly the same kill assessment data.

Another constraint worthy of mention is radarloading. While the current implementation of theTISES many-on-many algorithm does not address radarloading, predictive planning in general providesestimates of the probabilities that radar support will berequired. Because these probabilities may be quite lowfor follow-on shots, failure to adjust expected loadingaccordingly can easily negate any advantages ofconsidering radar loading.

Defense ObjectivesGiven the model of engagement probabilities above,

objective functions consistent with a number of defenseobjectives can be formulated. In this section, three areconsidered: minimize leakage, maximize threat valuenegated, and maximize asset value saved.

For a leakage-based defense, the goal is to minimizetotal leakage for any given expenditure of interceptors.

The objective function to be optimized in this case is atrade between the number of threats negated and thenumber of interceptors remaining. To maximize theobjective function for a given expenditure ofinterceptors in the context of MMR, one must select(one at a time) from the set of candidates theengagement that maximizes the objective functionquantity. The process continues until a minimum rateof return can no longer be achieved. Note that theminimum rate of return describes the aggressiveness ofthe defense response; it defines when the expectedmarginal decrease in leakage is of more value than thatof the expended interceptor. If set properly, it willinsure that not all of the interceptors are expended onthe first threats to be detected and become eligible forengagement.

As an alternative to a leakage-based defense, boththreat value-based and asset-based defenses can similarlybe represented.

In regard to the latter two strategies, it is importantto recognize thatsince threats may damage multipleassets and assets may be damaged by multiple threats,no general simplified form exists. In addition, note thatin each of the three defense criteria cases, the minimumrate of return parameter must be adjusted appropriately.In the case of leakage-based defense, the units are threatsper interceptor. For threat value-based defense, the unitsare threat value per interceptor (where the value scale isentirely arbitrary). For asset value-based defense, theunits are asset value per interceptor (where, again, thevalue scale is arbitrary).

A frequent compromise between threat value andasset value-based defenses that is worthy of mention is athreat value-based defense where the threat value is afunction of its expected asset damage independent ofother threats. Overlapping damage is assumed small orsimply neglected.

Of course, other constraints may be formulated anddefense objectives defined in addition to or in lieu ofthose described above. The predictive model is simplyoffered as a foundation upon which such can be built.

The Value of TimeOne key objective that any many-on-many algorithm

must achieve with some level of efficiency is topreserve the number of available shoot-look-shootengagement opportunities. To accomplish this, thefirst shot against a threat must be taken as early aspossible and subsequent shots must be committed soonafter kill assessment for the preceding shot is complete.Obviously, timing constraints and contention forinterceptors make this a potentially difficult problem.For example, to achieve the maximum number ofshoot-look-shoot opportunities, multiple threats may

502American Institute of Aeronautics and Astronautics


contend for the last interceptor at a given launcher.Alternatively, sufficient inventory may be available butlaunch rate constraints require that shoot-look-shootopportunities on one or more of the threats be forfeited.It is therefore apparent that the number of shoot-look-shoot opportunities for a given threat cannot bedetermined absolutely prior to solving the many-on-many problem. In other words, the assignment andscheduling problems are inseparable.

Because MMR is a suboptimal optimizationalgorithm, the objective function must be designed tocompel the algorithm to select from among the earliestengagements and work forwards in time. To inducesuch "top-down" allocation in the simplified approach,the SSKPs must be constrained such that twoconditions are met. First, the SSKP for the set ofcandidate engagements against a given threat should be afunction only of the threat type and the intercept time.Second, it should be a non-increasing function ofintercept time with the following exception— zeroSSKP regions may be superimposed without limit aslong as the non-zero portions of the function fulfill thenon-increasing requirement.

SSKPs that increase with time represent a difficultproblem for any optimization method— how muchtime can be safely traded for increased SSKP withoutsacrificing a shoot-look-shoot opportunity? Becausesuch determinations are made for each threatindependently, they represent the most optimisticestimates. Therefore, the question persists. If toomuch time is given up, then a simple launch rateconstraint or any other interactive constraint could costa shoot-look-shoot opportunity. As mentioned earlier,inventory constraints may automatically preclude theexpected number of opportunities. Without a moresophisticated optimization method, the best engagementfor a given shoot-look-shoot opportunity must beassumed to correspond to the earliest feasible intercept.Remaining within the confines of MMR for now,representing early engagements as better engagementscan be achieved by constraining the SSKP to non-increasing functions of the intercept time as describedabove. If SSKP is a strictly decreasing function oftime, no change to the algorithm as it has been definedis required. However, let us consider here an SSKPdependent only on threat type and therefore constantwith respect to time (a common approximation). Inthis case, the rate of return calculations must bemodified to "tip the scales" ever so slightly in favor ofearlier engagements. One simple solution is tointroduce a new algorithm control parameter whichwhen multiplied by the rate of return expressionsrepresents an "opportunity cost" associated with theadvancement of time. In the case of SSKPs dependent

only on threat type, the time constant may simply beset very large.

In addition to the cost of lost shoot-look-shootopportunities, the term may also be assumed to accountfor the opportunity cost associated with the risk ofsystem saturation as a result of postponingengagements in the face of a growing threat raid size.Note, however, that a potential cost associated withengaging early always exists; interceptors may beexpended on low-value engagements that could havebeen withheld for higher value engagements againstsubsequently detected threats. However, the minimumrate of return parameter described earlier is theappropriate means by which a balance can be achieved.

Inventory-Limited DefenseIn a previous example, the problem of preserving

shoot-look-shoot opportunities was complicated by aninventory-limited situation. However, even in theabsence of shoot-look-shoot opportunities, inventory-limited defense remains a problem. In effect, localoptima are created as a result of finite interceptorresources bounding the solution space. Consequently,no simple solutions to this problem exist within theconfines of simplistic methods such as MMR.

Responses to the inventory-limited problem fall intofour categories. First, use more sophisticated generaloptimization methods. Current computer processorlimitations make the prospects for this approachdiscouraging. Second, force fit the many-on-manyproblem to linear methods. Unfortunately, thisapproach often addresses the inventory-limited problemat the expense of interactive timing and geometryconstraints, overlapping asset damage, etc. Third,augment MMR with heuristics specific to addressinginventory-limited situations. Unfortunately, mostheuristics are at best only slightly less near-sighted thanMMR. Therefore, they cannot consistently recognizethe situations they are designed to remedy. As a result,improved inventory-limited performance is achieved atthe expense of performance in the absence of inventorylimitations. Finally, accept the risks associated withMMR. This author contends that such a response tothe problem is worthy of consideration. Whilegedankenexperiments that exploit the limitations ofMMR can be easily conceived, formulating realisticthreat scenarios against realistic defense deploymentsthat can similarly exploit such weaknesses is lesstrivial. In any case, a reasonable defense deploymentwill usually mitigate any significant advantage.

Overlapping Asset DamageA classic problem associated with the application of

MMR to engagement planning is that of overlappingdamage. Consider the case of two threat objects



attacking a single asset. Each is likely to completelydestroy the asset. Because MMR as it has been appliedabove considers only one additional planned engagementat a time, the rate of return estimate realized byengaging either of the threats individually is zero; thealgorithm is incapable of looking far enough ahead torealize the return associated with engaging both.Consequently, the objective function must beaugmented to eliminate (i.e., smooth out) local optima.The following approach is suggested.

The total estimated damage on each asset is dividedamong all of the threats targeting it in proportion totheir damage capabilities independent of the otherthreats. This quantity can be identified as the "linearizeddamage."

To maximize surviving asset value, a newexpression is substituted for the previously defined assetvalue-based rate of return equation. Using the new formof the rate of return equation and assuming high killprobabilities, half of the asset will appear to be saved ifeither one of the threats are engaged. If the minimumrate of return is sufficiently low, an engagement will beselected for one of the threats. On the next pass, thenonlinear damage equations will dominate and MMRwill determine that the entire asset may be saved if thesecond threat is engaged. Because MMR normallyallocates in order of decreasing rate of return, these twoengagements would tend to be selected on consecutivepasses since the second engagement would yield ahigher rate of return than the first engagement. Thecombination of linear and nonlinear functions willtherefore compel the algorithm to complete theengagements necessary to defend an asset once the firstengagements have been selected.

Drawdown StrategiesIn addition to the primary defense objectives

introduced earlier, namely leakage, threat value, andasset value, secondary defense objectives may also exist.Foremost among them is the manner in whichinterceptors are expended across launchers and launchersites— drawdown strategies. In effect, the value of aninterceptor may vary from launcher to launcher and siteto site. Two key objectives in the category ofdrawdown strategies are balancing the drawdown ofinterceptors across launcher sites to improve the defenseposture against future threats and depleting launcherswith the lowest inventory to facilitate reloads.

Because secondary objectives tend to be highlysubjective, mathematical representations may varyconsiderably. For this work, several requirements wereself-imposed. First of all, the necessary properties of anobjective function must be preserved— it must be asingle-valued function of the current configuration of

the decision variables; it must not depend upon the pathtaken to arrive at the current configuration.Consequently, it is reversible. Second, the marginalvalue of the last differential unit of interceptors at a siteis unity. In this way, if the control parameters are setsuch that the interceptor values are constant (i.e.,unity), then the objective functions revert to the theiroriginal forms.

Consider first the case of balancing drawdown acrosssites. To reflect this objective in the value of theinterceptors, let the interceptor value at each launcherdecrease according to a set of user-defined parameters.The relative values of the deflation constant for aspecific launcher site (AD) define the desired relativebalance of interceptors across sites. The values can thenbe scaled up or down as a set to control the importanceof balancing relative to the primary objectives. Verylarge values essentially disable balancing altogether;small values cause interceptors at inventory-rich sites tobe relatively inexpensive from a rate of returnstandpoint. A critical consideration in establishing thevalues of these parameters is that the relativeimportance of balancing may be set to the point thatshot opportunities may be sacrificed.

Similarly, the case of depleting launchers with thelowest inventories to enable reloads can be represented.Initially, this will be treated independently of sitebalancing. In this case, because interceptor valuesshould increase with increasing inventory, the user-defined parameter controlling this behavior is labeled aninflation constant.

Now consider the problem of combining balancingand depletion factors. This can be accomplished bysimply replacing the interceptor inventory at a launcherwith the inventory value at a launcher in the sitebalancing equations.

Similar to the balancing case, large values of thedepletion parameter (AI) diminish the relativeimportance of depleting nearly empty launchers;whereas small values enhance its importance,potentially to the point that shoot-look-shootopportunities may be sacrificed.

With both balancing and depletion cases, theinterplay between the time constant introduced earlierand the deflation and inflation constants introduced herewill determine the relative influence of these factors.Simple rules of thumb can be formulated based uponthe relative spacing of launchers and sites, typicalclosing speeds, and so forth to determine initial settingsfor these parameters. As noted previously, thealgorithm has no explicit mechanisms for insuringconservation of shoot-look-shoot opportunities.Consequently, the balancing and depletion parametersshould be set to insure that these factors are weak



relative to the time constant factor so. that these effectsoccur only at the expense of small amounts of time.

Adaptive Method of FireThe reader may note that no heuristics, rule sets, or

other logic specific to achieving particular methods offire have appeared explicitly in the algorithm design.The question may arise concerning whether or not thealgorithm can exploit such strategies as shoot-look-shoot or whether or not it "understands" the basis forreserving salvos for the last shot opportunity. Theanswer is traceable to the origins of these methods offire— the statistical merits upon which the strategieswere originally founded. These statistical foundationsare those that have been represented in the modeldescribed above. Because the algorithm is built uponthe mathematics and not the discrete strategies derivedfrom them, its inclination to exploit these strategies isinherent. Therefore, while "shoot-look-shoot last-shotsalvo" or (S-L-)mS" is the preferred strategy, thealgorithm is not limited to it.

For example, consider a case of two launcher sites, aforward site with 50 interceptors and a rear site with asingle interceptor. Consider also a set of 25 inboundthreats such that the only available shoot-look-shootopportunities are those where the first shots are fromthe forward site and the follow-on shots from the rearsite. Assume also a constant SSKP of 0.8, long timeconstant, weak balancing and depletion, and a rate ofreturn threshold sufficiently low that all interceptorsshould be expended in any case. Ignoring inventoryconstraints, the preferred method of fire would be shoot-look-salvo-n. However, inventory constraints dictateshoot-two-look-shoot-one as the optimal method of fireacross all the threats (25*2 = 50 shots from the forwardlauncher and 25*[ 1-0.8] = 1 shot from the rear launcherbased upon MVA). Many rule set or heuristic-basedalgorithms would preclude such seemingly unusualmethods of fire; the algorithm described above wouldattempt to execute such a strategy. In short, attemptsto specify or limit methods of fire at any time prior toassignment of weapons are not without risk.

Advantages of Predictive PlanningThe advantages of predictive engagement planning

are significant. Predictive algorithms tend to be lesssusceptable to off-nominal situations. They aregenerally less biased towards certain types of situationsor scenarios. Their "first principles" foundation makethem receptive to additional features, particularly thoseof a probabilistic nature. Consider these examples:

- Imperfect Kill Assessment- Correlated Kill Assessments- Correlated Kill Probabilities.

Imperfect kil l assessment includes both theclassification of a kill as a "no kill" and a "no kill" as akill. Correlated kill assessments include the possibilityof a non-lethal hit adversely influencing the killassessment accuracy of subsequent shots. Correlatedkill probabilities may represent one of the mostsignificant and potentially serious obstacles to effectiveengagement planning. To date, most if not all of theproposed algorithms have assumed that SSKPs areuncorrelated. As a result, these algorithms have aninherently optimistic predilection. Factors that maycontribute to correlated failures are innumerable—erroneous classification, typing, or discrimination;underestimated track errors; radar biases; underestimatedlauncher alignment errors; poor uplink data (a particularproblem for salvos); and so forth. The correlations maybe linked to a threat, radar, launcher, etc. and may varywith time. Temporal correlations in uplink qualityrepresent a factor that, if considered, would tend tofurther enhance the merits of shoot-look-shoot methodsof fire over salvos (assuming a common supportingradar). Predictive many-on-many algorithms have thepotential to accurately account for these effects.

A predictive algorithm also possesses advantageswith respect to interactive constraints of a geometricalnature such as intercept flashes in the fields of view ofother interceptors. Some countermeasures may also fallinto this category. The probability of these eventsoccurring is generally high only for the first shot on athreat. Failure to factor in the diminished probabilityfor subsequent shots may lead to excessively cautiousplanning. The risks inherent in ignoring these eventsfor subsequent shots may be equally serious.

Finally, one of the most attractive advantages of apredictive algorithm is the minimum of operator inputsrequired for robust behavior over a wide spectrum ofsituations. The control parameters for the algorithmdescribed above are summarized in the following table:

SymbolRmin

AT

AD

AI

NameMinimum rate ofreturn on investmentof resourcesRate of return decaytime constantDeflation constantfor site mLauncher inflationconstant

Controls...Aggressiveness,exchange rate

Value of time

Site balancing

Launcher depletion forreloads

Software ImplementationGiven the potential computational intensity of the

many-on-many problem, the manner in which the abovemodel is implemented in software is important. Certain



principles of probability may be exploited to acceleratethe computations. The most notable is causality—events can only be influenced by preceding events andcan only influence subsequent events. This principleinsures no circular dependencies. As candidateengagements are moved in or out of the set of plannedengagements, the effects of preceding engagements onthe engagement under consideration are evaluatedfollowed by the effects of that engagement onsubsequent engagements. Taking advantage of the factthat only one engagement is moved at a time, manysum and product equations can be replaced with formsrepresenting the changes to these quantities, requiringonly that the values be appropriately initialized andretained between calculations. Notionally, these maytake forms such as

Q<-Q+AQ<-Q-AQ<-Q*AQ<-Q/A

(13)(14)(15)(16)

The Integrated ApproachOne goal of the simplified approach was to assess

the quality of the basic predictive many-on-manyalgorithm somewhat independently from processing andmemory considerations. In the simplified form, arelatively exhaustive set of candidate engagements iscalculated first, in effect, mapping the entire solutionspace for a single threat independent of other threats.This set of engagements is then passed to the many-on-many algorithm. This is the method currentlyrepresented in TISES; a more computationally efficientapproach (the integrated approach, as it shall be called)is described below.

The processes of generating and selecting from theset of candidate engagements account for the bulk of theengagement p l ann ing computations. Thecomputational burden of both processes scales with thenumber of candidate engagements. It is thereforeapparent that if candidate engagements can be generatedselectively, the time required for both processes can bereduced substantially. However, selective generationrequires integration of the generation and selectionprocesses. In effect, as the selection process continues,searching through the solution space as defined by thecandidate engagements, additional candidates are addedrepresenting regions of the solution space adjacent tothose already explored. The solution space is mappedon an as-needed basis. The result is not unlike branch-and-bound techniques.

The net effect of this integrated approach is thatfewer engagements are generated and fewer are consideredby the many-on-many process. A third advantagerelevant to the real-time environment is that if time

expires for the engagement planning process, it is morelikely to exit with at least some desirable engagementsselected for execution. A fourth advantage is that firstcommits, salvos, and shoot-look-shoot shots can bemore efficiently packed in time. In the simplifiedapproach, the engagement timing precision is limitedby the time granularity of the candidate engagement set,typically on the order of the planning cycle time.

Integration of the generation and selection processesrequires the creation of some additional supportingalgorithms. Most significant is an algorithm that cangenerate an engagement for a given threat and launcherwith the earliest intercept time corresponding to acommit time greater than and, assuming non-interactiveconstraints are not violated, within some tolerance of aspecified time. The specified time may be the planningend time, the commit time corresponding to the launchtime of another engagement (plus some launch spacingif the same launcher), or the kill assessment time ofanother engagement (plus some delay associated withthe planning cycle). These three correspond to earliestengagements, salvo engagements, and shoot-look-shootengagements. If non-interactive constraints introduce a"hole" in the batUespace, then the earliest feasibleintercept within some tolerance after the hole isreturned. If no feasible engagements exist, then noengagement is returned and no error condition isrequired. Because launch time may be a very complexfunction of intercept time for a given threat/launchercombination, this algorithm is non-trivial. Thealgorithm must also maintain records of battlespaceholes (by threat, launcher, radar, etc., as required) toprevent redundant traverses of infeasible regions.

The integrated approach consists of three main partsorganized as follows:

1) Generate initial set of candidate engagementsLoop until rate of return threshold can no longer be

met:2) Find candidate engagement with best rate of

return3) Add best engagement to planned engagements

and generate additional candidates as neededEnd loop

The algorithm is shown in greater detail in Figure 1.Italics indicate those parts of the algorithm unique tothe integrated approach.

Because launcher inventory is treated as a continuousquantity and is never allowed to be negative, "non-empty" in the context of the algorithm may best bedefined as greater than some small positive number inorder to prevent futile attempts at expending every last"fraction" of an interceptor from a given launcher.



1) Generate initial set of candidate engagements:loop over Threats

Generate Candidate Engagement (Earliest Intercept)from each non-empty launcher

loop over Existing Engagements on this ThreatGenerate Candidate Engagement (Next SLS) fromeach non-empty launcher

end loopend loop

loop — main2) Find candidate engagement with best rate of

return:Best rate of return = - large numberloop over Candidate Engagements

Move to Planned Engagements (temporarily) anddetermine rate of return

if Launcher Inventory > 0.0 and Rate of return >maximum (Best rate of return, rate of returnthreshold) thenMark Candidate Engagement as incumbent best

end ifMove back to Candidate Engagements

end loop3) Add best engagement to planned engagements and

generate additional candidates:if Best rate of return > rate of return threshold then

- a qualifying engagement is foundif Interactive Timing and Geometry Constraints

violated thenGenerate Candidate Engagement (Next Salvo)

from this launcher if not emptyDelete best engagement

elseMove best engagement to Planned Engagements

(permanently)Generate Candidate Engagement (Next Salvo)

from each non-empty launcherGenerate Candidate Engagement (Next SLS) from

each non-empty launcherend if

elseexit main loop

end ifend loop — main __

AcknowledgmentsThe TISES effort for which the algorithm described

herein was developed is currently sponsored by theTHAAD Project Office System Engineering Divisionunder the auspices of the THAAD Software IndependentValidation and Verification contract number DASG60-94-C-0016. Past sponsorship has been provided by theGround Based Element System SimulationDevelopment contract number DASG60-92-C-0173.The author wishes to thank Mr. Tony Cosby, Mr.Doug Engle, Ms. Dawn Horn, and Mr. Rich Mullin ofTPO System Engineering Division for their support ofthis work. The author also wishes to thank PhilColvert and Gary Makowski of Tec-Masters,Incorporated and Daniel Dean and Anthony Glover ofColeman Research Corporation for their valuablesuggestions and thorough critique of this paper.

Figure 1. The Integrated Algorithm.


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