w a game- theoretic model for defense of an oceanic ... · pdf filemighthave: inacon...

ABSTRACT

We develop a game-theoretic modelcalled BASTION to guide theemployment of antisubmarine

warfare (ASW) platforms such as shipsand aircraft that are defending a stationaryoceanic bastion from attack by hostile sub-marines The model is an example of a two-person zero-sum game with some additionalvariables (ship locations) that are under thecontrol of the maximizing defender butknown to the minimizing attacker The at-tacker knowing the ship locations but notthe locations of other platforms such asaircraft must select a path to the bastionThe probability of detecting the attacker asit follows this path is the objective sharedby the opponents

INTRODUCTIONThis paper develops a new planning aid

for defensive ASW called BASTION We en-visage a Blue battle-group commander witha set of ASW platformsmdashthese will typicallycomprise a group of surface ships subma-rines fixed-wing aircraft and helicoptersmdashthat are available to search for enemysubmarines The commander must employhis forces to protect an HVU (High ValueUnit typically an aircraft carrier) from en-emy (Red) submarines The HVU and possi-bly some other ships are assumed to lie ina bastion of protected cells and the objectof the Red attackers is to penetrate to thebastion without being detected by Blue

The Submarine ThreatModern submarines both nuclear (SSN)

and diesel-electric (SSK) pose a significantthreat to the US Navy Technological inno-vation since World War II has improvedthe capabilities of submarines to the pointwhere Keegan (1986) questions the viabilityof surface ships in the face of submarine at-tacks Air-independent propulsion now al-lows SSKs to operate submerged for weeksat a time compared to only hours in WorldWar II SSNs are even more effective thanSSKs and improved weapons systems havegreatly enhanced the lethality of both typesAs a result even a single submarine pos-sesses the stealth and firepower necessaryto put warships at risk

More than 40 countries currently oper-ate submarines including several countrieswith large fleets (Benedict 2006) Holland(1991) explains the effect submarine attacksmight have lsquolsquoIn a conflict with less than a su-perpower public or political patience willrun thin concerning losses or delays by sub-marines The magnitude of the political ca-tastrophe arising from the torpedoing ofan aircraft carrier in a limited conflict canhardly be overestimatedrsquorsquo

Antisubmarine Warfare Platformsand Systems

The US Navy uses multiple ASW plat-forms including surface ships aircraft ofvarious kinds and SSNs

A surface shiprsquos primary ASW sensor isits sonar either hull-mounted or towedSurface ships also employ radars that arecapable of detecting surfaced submarinesor even submarine periscopes or mastsHowever the locations of surface ships areusually detectable by submarines well inadvance of any detection by the ships InBASTION we assume that the locations ofall ships are known by Red when planningan attacking submarinersquos path to the bastion

All aircraft enjoy the advantage of beingmuch faster than submarines Aircraft car-riers operate MH-60 helicopters that in-clude ASW among their many missionsThe SH-60 helicopter operated from cruisersdestroyers and frigates can drop expend-able sonobuoys and can also dip a sonarsensor into the ocean Land-based maritimepatrol aircraft (MPA) also employ sono-buoys but in significantly larger numbersand are equipped with a surface search ra-dar Every manned aircraft is equipped withhuman eyeballs which remain one of themost effective ASW sensors In BASTIONwe assume that all aircraft operations are in-herently stealthy to Red submarines exceptfor the complicating feature that Red knowsthe locations of the bastion (the source ofMH-60 sorties) and the ship hosts of theSH-60s

A battle group typically includes one ortwo Blue SSNs These lsquolsquodirect supportrsquorsquo sub-marines often provide the most effective so-nar search capability due to their quietnessand their ability to vary search depth to ad-just to the oceanrsquos local acoustic conditionsTo prevent conflicts with other Blue platformsBlue submarines typically have exclusive

A Game-TheoreticModel forDefense ofan OceanicBastionAgainstSubmarines

Dr Gerald BrownJeff KlineAdam ThomasDr Alan Washburn andDr Kevin Wood

Operations ResearchDepartment NavalPostgraduate School

gbrownnpsedujeklinenpseduathomasnpseduawashburnnpsedukwoodnpsedu

APPLICATION AREASStrategic OperationsLittoral WarfareRegional Sea Control

OR METHODSLinear ProgrammingNetwork MethodsStochastic Processes

Military Operations Research V16 N4 2011 doi 1057111082598316425 Page 25Military Operations Research V16 N4 2011 doi 1057111082598316425 Page 25

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4 TITLE AND SUBTITLE A Game-Theoretic Model for Defense of an Oceanic Bastion Against Submarines

5a CONTRACT NUMBER

5b GRANT NUMBER

5c PROGRAM ELEMENT NUMBER

6 AUTHOR(S) 5d PROJECT NUMBER

5e TASK NUMBER

5f WORK UNIT NUMBER

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8 PERFORMING ORGANIZATIONREPORT NUMBER

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11 SPONSORMONITORrsquoS REPORT NUMBER(S)

12 DISTRIBUTIONAVAILABILITY STATEMENT Approved for public release distribution unlimited

13 SUPPLEMENTARY NOTES

14 ABSTRACT We develop a game-theoreticmodel called BASTION to guide the employment of antisubmarine warfare(ASW) platforms such as ships and aircraft that are defending a stationary oceanic bastion from attack byhostile submarines The model is an example of a twoperson zero-sum game with some additional variables(ship locations) that are under the control of the maximizing defender but known to the minimizingattacker The attacker knowing the ship locations but not the locations of other platforms such as aircraftmust select a path to the bastion The probability of detecting the attacker as it follows this path is theobjective shared by the opponents

15 SUBJECT TERMS

16 SECURITY CLASSIFICATION OF 17 LIMITATION OF ABSTRACT Same as

Report (SAR)

18 NUMBEROF PAGES

16

19a NAME OFRESPONSIBLE PERSON

a REPORT unclassified

b ABSTRACT unclassified

c THIS PAGE unclassified

Standard Form 298 (Rev 8-98) Prescribed by ANSI Std Z39-18

use of certain regions known as Submarine Op-erating Areas (SOAs) and do not leave those re-gions without pressing reasons The locations ofthese SOAs are assumed known to the attackerbut not the disposition of the SSNs within theSOAs Red can infer the location of the SOAmore or less from the locations of previoussub-versus-sub engagements and the lack of ac-tivity in the area by other Blue forces

Current Planning Tools for ASWMissions

The US Navy currently employs a numberof tools for planning ASW missions

The Personal Computer-Based Interactive Mul-tisensor Analysis Trainer (PCIMAT) is the pre-mier ocean acoustic analysis and planningtool available on all ASW platforms (SPAWAR2008) Although employable from a standalonelaptop computer an implementation of PCIMAT(the Sonar Tactical Decision Aid or STDA) inte-grates its capabilities with a platformrsquos entirefire-control system Among other outputs STDAoffers a mission-planning module that providesa graphical representation of a platformrsquos effec-tive coverage area

The ASW Screen Planner Tactical Decision Aid(SWDG 2004) aids the planning of ASW screensfor a battle group in transit The planner spec-ifies a threat submarine from a database andproposes assignments of available ASW plat-forms to sectors surrounding the battle groupnot including MPA The decision aid then calcu-lates the probability of detecting the enemy sub-marine in each sector if it transits that sector ina straight line The planner manually assignsplatforms to sectors until he creates a solutionwith an acceptable probability of detection

The Active System Performance Estimate Com-puter Tool (ASPECT) aids MPA in maximizingthe effectiveness of active sonobuoy search(FAST 2006) The planner manually specifiesseveral sonobuoy patterns and describes howRed can be expected to move ASPECT thensimulates approximately 500 submarine tracksin the search area and reports the resulting sam-ple detection probability for each pattern

The Operational Route Planner (ORP) modelsthe area search problem seeking to identify

routes for search platforms that are most likelyto lead to the detection of an SSK in a specifiedarea (Kierstead and DelBalzo 2003 Wagner As-sociates 2008) ORP simulates the SSKrsquos actionsbased on a probabilistic description of enemybehavior The planner specifies patrol-regionassignments for available search platformsand ORP heuristically optimizes search plansfor multiple ASW searchers using a geneticalgorithm

None of these currently employed systemssynchronously coordinates the actions of mul-tiple ASW platforms in defense of a bastionBASTION is intended to do that

TWO BACK-OF-THE-ENVELOPEMODELS

Throughout the development of BASTIONwe have wrestled with whether ASW searchshould be modeled as exhaustive or randomThese are the only two viable alternatives be-cause our optimization ambitions for BASTIONdictate a simple analytic model of some kind Tomake the issues clear in this section we outlinetwo simple models one based on exhaustivesearch and one based on random search The ex-act formulation of BASTION together with theassumptions behind it are the subject of thenext section

For the moment assume there are onlythree searching platform types ships subsand MPA (helicopters are a special case becausethey are based on ships so we omit them for themoment) Red knows all ship locations Bluesubs operate within an SOA that is known toRed but otherwise Red does not know their lo-cations To prevent interference and even fratri-cide Blue insists that only Blue SSNs operatewithin the SOA and that Blue SSNs operatenowhere else The bastion is assumed to havethe shape of a circular disc with radius r andthe surrounding waters are assumed to behomogeneous

Exhaustive Search ModelThis is a one-dimensional model in which

Blue attempts to make the perimeter of the

A GAME-THEORETIC MODEL FOR DEFENSE OF AN OCEANIC BASTION AGAINSTSUBMARINES

Page 26 Military Operations Research V16 N4 2011

bastion a barrier that Red subs cannot penetratewithout being detected Define the followingfour quantities

bull D frac14 length that can be guarded by shipsbull S frac14 length that can be guarded by SSNsbull A frac14 length that can be guarded by aircraft

andbull B frac14 length of bastion perimeter (2pr)

The general idea is that each platform typeis given responsibility for part of the barrier IfD 1 S 1 A is larger than B then the entire perim-eter can be guarded and the detection probabil-ity is 1 Otherwise because all points guardedby ships are known to Red the SSNs and MPAmust do their best to cover the remaining perim-eter length B 2 D Suppose that Blue chooses theSOA to have length L with L S Because Redknows where the SOA is located but not the lo-cations of the submarines within it the detec-tion probability will be SL if Red penetratesthe SOA or A(B 2 D 2 L) if Red penetratesthe MPA part of the barrier (Red will avoid theship part because it is completely covered) Blueshould choose L (his only strategic decision inthis model) to make the smaller of these twoquantities as large as possible As a result thedetection probability is (S 1 A)(B 2 D) withRed being indifferent between the submarineand MPA parts of the barrier Note that Bluepays no penalty for having to reveal the SOAto Red as long as he chooses the size of theSOA judiciously the payoff is the same as ifthe Blue sub length S was simply added to theMPA length A in the first place Note also thatBlue does pay a penalty for having to revealthe ship locations because the detection proba-bility is smaller than (D 1 S 1 A)B

Our main objections to this model are

bull Most ASW sensors are not of the cookie cut-ter type There is no distance R such that de-tection is certain within R and nondetectioncertain at longer ranges

bull ASW platforms particularly aircraft cannotbe assumed to be present all the time espe-cially in the face of enemy actions that wouldaccompany a war in which Red attackers areattempting to attack Blue HVUs There aremany reasons for this one of which is thatso-called lsquolsquoASW platformsrsquorsquo have other mis-

sions besides searching for Red submarinesOne of them is to engage and kill Red subma-rines a separate function

Considerations such as these have classi-cally led to the assumption that search is effec-tively lsquolsquorandomrsquorsquo rather than lsquolsquoexhaustiversquorsquo(Koopman 1980) Because this assumption is es-sential to BASTION we next outline a randomsearch model for the current scenario

Random Search ModelASW sensors operate in two dimensions

rather than one For continuously moving sen-sors such as eyeballs or passive sonars thisobservation has led to the definition of sweep-width W as the effective width of a cleared stripIf the platform moves at speed V the rate of cov-ering area is VW If the platform is present onlyf of the time the average rate of covering area isreduced to fVW If the target is located some-where within area A and if the sensor searchesrandomly within A then the rate of detectionis l frac14 fVWA To be precise detections areassumed to be a Poisson process with rate l Ifmultiple platforms are present and all aresearching independently at random then thedetection rates can be summed These consider-ations lead to characterizing each platform typewith its total rate of clearing area rather thana guardable distance as in the exhaustive modelDefine

bull D frac14 total sweep rate of shipsbull S frac14 total sweep rate of Blue submarines andbull F frac14 total sweep rate of aircraft

In the abstract random search amounts todistributing confetti over an area in the hopethat some flake will cover the point that repre-sents the target Assume that the area is a ringsurrounding the bastion that extends from r tosome larger distance R that Blue controls (seeFigure 1) and that the ring is divided into threesegments one for each of the three separatetypes of Blue platform The segment areasshould be selected to be proportional to sweeprates mdashwe omit the proofmdash and the net effectof this observation is that we might as well sim-ply add up the three sweep rates to get the totalsweep rate C frac14 D 1 S 1 F


Military Operations Research V16 N4 2011 Page 27Military Operations Research V16 N4 2011 Page 27

Bluersquos plan is essentially to distribute con-fetti at rate C over a ring that has area p(R2 ndash r2)If Redrsquos penetration speed is U he will spenda time (R ndash r)U in this ring before arriving atthe bastion The resulting probability of detec-tion is just the probability of at least one eventin a Poisson process

P 5 1 2 exp 2CethR 2 rTHORN

pUethR22 r

2THORN

5 1 2 exp 2C

pUethR 1 rTHORN

The outer radius R is under Bluersquos controlbut must be at least r in order to keep the ringseparate from the bastion The best value isR frac14 r and the resulting detection probability isP 5 1 2 expeth2 C=U

B THORN where B is the same barrierlength defined earlier In the limit Blue distrib-utes confetti inside a vanishingly small ring Wemight insist that R be larger than r by an amountthat is roughly the detection radius of the Bluesensors involved Nonetheless the essential factis that Blue wants to defend a thin ring aroundthe bastion

In this model Bluersquos only strategic concernis to make sure that each platform type isassigned a part of the ring that is proportionalto its sweep rate Note that ships pay no penaltyfor being visible to Red because their sweeprate is just one of the terms in the sum definingC This is a significant problem with the randomsearch model because ship visibility has impor-tant consequences in reality (see discussion underthe heading Ships in the next section) Although

BASTION is essentially a generalization of thismodel it makes an exception for ships

DEVELOPMENT OF BASTIONThe US Navy does its bastion planning on

rectangular lsquolsquoFour-Whiskeyrsquorsquo (4W) grids withcells whose sides are typically 5 or 10 nm Plat-forms are assigned to cover cells or groups ofcells but not partial cells BASTION adopts thisrectangular reference system Certain cells donot need to be defended because they are im-passable to Red attackers (hereafter lsquolsquolandrsquorsquo) orare part of the bastion All other cells constitutethe traversable set C and the SOA is a subset ofC Figure 2 shows a typical categorization inBASTION

A Red attacker is assumed to start outsidethe grid and must choose a path to the bastionconsisting of adjacent traversable cells possiblyincluding diagonal moves between cells thatshare a corner Redrsquos object is to get to the bas-tion without being detected Bluersquos object is todetect Red before he gets to the bastion so wehave a two-person zero-sum game The situa-tion is assumed to be stochastically stationarywith no time limit within which Red must act

The subscripts i and j will index cells in thegrid We define the area of cell i to be Ai typi-cally but not necessarily independent of i

For modeling purposes each traversablecell i in C is connected to each adjacent travers-able cell j by a directed arc (i j) in a network

Figure 1 Blue surrounds the bastion with a ringwhere he searches randomly

Figure 2 The grid is a rectangle of cells with lati-tude and longitude coordinates Land is crosshatchedand nearly surrounds a white area that Red can tra-verse The bastion is horizontally marked whereasthe SOA is vertically marked Red enters at the leftborder



model i being the tail and j being the head of thearc The arcs are the feasible moves for theattacking Red submarine Each arc has associ-ated with it a time tij that represents the amountof time required for Red to transit between i andj These transit times are inversely proportionalto the attackerrsquos assumed constant speed an in-put to BASTION

We let Redrsquos transit begin at an artificialstart cell i1 that lies outside of the grid and con-nects to all cells through which Red can enter (tra-versable cells on the gridrsquos border) Redrsquos transitends at an artificial terminal cell i2 that repre-sents all cells in the bastion The detection ratein the start and terminal cells is 0 by definition

We denote the full network model as (C A)where C denotes the set of traversable cells andA denotes the set of arcs We denote Redrsquos paththrough the network as y a vector of arcs inwhich the head of each arc is the same as the tailof its successor The tail of the first arc in y is i1and the head of the last is i2 We refer to the col-lection of all of Bluersquos ASWassignments as x andto the resulting detection rate in cell i as si (x)

When a Red attacker is present in cell i weassume that si (x) is the rate of a Poisson processof detections even though some of Bluersquos assets(aircraft in particular) may operate on a sched-ule that is more regular than Poisson In this weare relying on the tendency of point processes tobecome Poisson in the presence of complica-tions For example the sum of many indepen-dent stationary processes tends to becomePoisson (Khinchin 1960) and thinning a station-ary process tends to have the same effect A pro-cess of aircraft flights is lsquolsquothinnedrsquorsquo when aircraftare randomly assigned to patrol in specific cellsBlue is motivated to choose those cells ran-domly because he is playing a game withRedmdashas long as Red cannot predict the se-quence of chosen cells it is not particularly im-portant if Red can predict the sequence oftakeoffs and landings In short Bluersquos searchfor Red attackers is assumed to be lsquolsquorandomrsquorsquo

Let

zethx yTHORN5XethijTHORN2y

tijethsiethxTHORN1 sjethxTHORNTHORN=2 (1)

If we assume that half of Redrsquos time in tran-siting from i to j is spent in each cell then z(x y)is the average number of times that Red is

detected on his path from i1 to i2 We will referto this quantity as the lsquolsquopressurersquorsquo that Bluersquos de-fensive efforts place on Red as he attempts topenetrate through to the bastion We take thispressure to be the mean of a Poisson random var-iable so the probability of (at least one) detectionis 1 ndash exp(ndashz(x y)) Blue chooses x to maximizethis payoff while Red chooses y to minimize it

The total search rate si(x) in cell i is itself thesum of several components that are determinedby the various platforms under Bluersquos commandThere are four platform types in BASTION sub-marines (Blue SSNs) land-based MPA ship-based helicopters and the ships themselvesEach platform type is discussed separately be-low The entire Bastion model will then be spec-ified in detail

SubmarinesBlue submarines operating within an SOA

are the easiest platform to model at least if theirsearch is by passive sonar Let RSi be the detec-tion rate of a submarine in cell i normallyobtained by multiplying the speed of the Bluesub by its sweepwidth and dividing by the cellrsquosarea and let xsi be the average number of sub-marines patrolling in cell i Then the total sub-marine detection rate in cell i is RSixsi If thereare NS Blue submarines in direct support thenthe allocation variables are subject to the con-straint

Pi2SOA xsi 5 NS

BASTION has two modes of operation Inthe free mode SOA is C and all platforms canoperate anywhere In the constrained modea specific SOA within C is set by the plannerand only SSNs can be located there

Land-Based Aircraft (MPA)An MPA squadron usually describes its ca-

pability in terms of a sortie generation rate GAthe average number of sorties per hour thatcan be launched by the squadron (a sortie iswhatever happens between a takeoff and thefollowing landing) If there is a differencebetween lsquolsquosurgersquorsquo and lsquolsquosustainrsquorsquo sortie ratesthen the appropriate one here because of thelong time horizon envisioned in employing



BASTION is lsquolsquosustainrsquorsquo Let BAjk be the amountof area searched in cell j by a sortie assigned tocell k divided by the area of cell j The interpre-tation of this dimensionless number is lsquolsquothe av-erage number of detections if Red is located incell j when a sortie is assigned to cell krsquorsquo Ouridea here is that BAkk will be determined pri-marily by sonar search considerations but thataircraft have a significant eyeball and radarsearch capability for exposed periscopes andsurfaced attackers while transiting over cellsother than k The value for BAjk will typicallybe a strong function of cell index k with cellsfar away from the squadronrsquos base having rela-tively small values Determining BAjk from morefundamental parameters is a significant task thatwe defer to Appendix A taking BAjk as given forthe moment Let xak be the number of MPA sor-ties per hour assigned to cell k The MPA detec-tion rate in cell j is then

Pk BAjkxak and the

applicable constraint isP

k xak 5 GA where bothsums are over all cells possibly excepting SOAif aircraft are forbidden to search in that region

We use the term lsquolsquosquadronrsquorsquo to mean allMPA operating from a given base and will de-scribe BASTION in the sequel as if only a singlesquadron were involved If multiple squadronsare actually available then each would be mod-eled as a separate asset

HelicoptersMH-60 helicopters operate from aircraft

carriers typically located within the bastionwhereas SH-60 helicopters operate from otherships located outside the bastion The effect ofhelicopter search is similar to that of MPA ex-cept for the mobility of their bases

To some extent Red can predict helicopteroperations Consider the operations of MH-60sfrom within the bastion Might Red reason thatbecause the aircraft carrier is currently locatedin the south edge of the bastion cells near thenorth edge should currently be safe from the at-tentions of MH-60s Answering the question inthe affirmative would require introducing anunwelcome time dimension to BASTION be-cause the track of the aircraft carrier(s) wouldhave to be an input The movements of aircraftcarriers when launching and recovering aircraftare notoriously unpredictable even to Blue and

Red would have to consider more than the cur-rent locations in planning his penetration Wetherefore assume that carrier movements withinthe bastion are unknown to Red except forremaining within the bastion and that Bluetakes advantage of this ignorance to maximizethe on-station time of MH-60 flights The com-putational effect of this in BASTION is that anMH-60 will always have its on-station endur-ance calculated as if it were both launched andrecovered from the most favorable point in thebastion Except for this flexibility MH-60s arehandled like MPA In the case of SH-60s we as-sume that the location of their base (a ship lo-cated outside the bastion) is known to Red Forall helicopters we next calculate the area coveredby one sortie and proceed as with MPA The sor-tie generation rate GM for MH-60s should beonly those sorties allocated to ASW search

In the case of SH-60s additional constraintsare necessary to enforce the idea that a sortie can-not begin in cell i unless the helicopterrsquos host oc-cupies that cell (see formulation below) SH-60stypically search using a dipping sonar an activedevice that in principle provides information toRed about the helicopterrsquos location and thereforea basis on which to predict and avoid subsequentdip locations However because helicopters oper-ate at much higher speeds than submarines theeffect of this information is ignored in BASTION

ShipsShips are the most difficult ASW platform to

model All other platforms are sufficiently fastor well hidden that Red cannot predict their ex-act locations when planning his route to the bas-tion but this is not true for ships Ships revealtheir locations in several ways they are largesurface vessels and therefore subject to obser-vation by eyeball and radar they operate radarsthemselves which are subject to intercept theirengines are noisy which permits submarines todetermine bearing at considerable distance Fi-nally ships sometimes operate powerful activesonars the signals from which can be intercep-ted at long distances by submarines For thesereasons ships are the only platforms whose lo-cations are assumed to be known to Red beforeplanning his penetration route This is accom-plished in BASTION by requiring the variables



xfmi denoting the presence of ship m in cell i tobe either 0 or 1 (binary) This necessity forcesBASTION to be a Mixed Integer Program(MIP) rather than a linear program

In spite of their locations being known toRed ships can still be effective ASW search plat-forms BASTION currently models a shiprsquos de-tection rate as a Gaussian plume of the formlexp(ndash(rR)b) where r is the distance from theship l is the maximum possible detection rateR is a relaxation distance and b is a shaping con-stant The connection between the (l R b) pa-rameters and more fundamental parameterssuch as ship speed and sonar range is tenuousespecially for a platform whose location isknown by the target that it seeks Washburn(2010a) is relevant but not definitive By makingl and b large one can implement the idea thatthe ship controls a disk with radius R that Redwill not challenge This could easily be general-ized to model noncircular regions of controlsuch as those provided by PCIMAT or STDA

Let Smij be the detection rate in cell j of shipm located in cell i obtained by substituting rel-evant parameters into the Gaussian plumemodel The total rate of detection in cell j dueto ships is then

Pmi Smijxfmi Note that ships

are the only platforms modeled individuallyall other platforms are modeled as aggregationsof identical units The resulting proliferation ofship indices is necessitated by Redrsquos informa-tion advantage but is also useful because shipsdiffer significantly from one another The factthat xfmi is required to be binary usually meansthat the detection rate will be wastefully highnear the shiprsquos location but Blue cannot avoidthis because the shiprsquos location is known toRed

After accumulating the sweep rates from allplatforms in cell i we have the total detectionrate si(x) as a linear function of x for all i andtherefore (using (1)) the detection probability1 ndash exp(ndashz(x y)) We are dealing with a two-person zero-sum game in which the payofffunction is a concave function of x while x (ex-cept for the integer variables representing shiplocations) is constrained to lie in a compact setTherefore (Washburn 2003) Blue has an optimalpure strategy and the value of the game is maxx

miny (1 ndash exp(ndashz(x y))) Although the detectionprobability is our ultimate concern the pressure

z(x y) itself will suffice for an objective functionbecause the transformation from pressure to de-tection probability is strictly increasing The ob-ject is therefore to find maxx miny z(x y) thelargest pressure that Blue can guarantee againstthe worst-case path This value and Bluersquos opti-mal pure strategy x can be obtained by solvinga MIP with variables (xv) where v is the maxi-mal pressure If we let X represent the set of fea-sible Blue defensive allocations (we will specifyX in detail later) and Y represent the finite set offeasible Red paths this program can be com-pactly expressed as MIP0

max v

subject to zethx yTHORN2 v $ 0 for all y 2 Y

x 2 X

The difficulty with MIP0 is that the set Ydoes not scale wellmdashthe number of feasiblepaths increases rapidly with the number of cellsIt would be better if there were a constraint forevery arc in A rather than one for every feasiblepath in Y The first step in finding such a revisionis to define variables uij that indicate whetherarc (i j) is in Redrsquos path (uij frac14 1) or not(uij frac14 0) In terms of those variables zethx yTHORN5PethijTHORN2A tijethsiethxTHORN1 sjethxTHORNTHORN=2

uij This expression

replaces the reference to Y in MIP0 with a ref-erence to A Following standard techniques(Fulkerson and Harding 1977) MIP0 can thenbe shown to be equivalent to MIP1 with vari-ables (x v) MIP1 is

max vi2

subject to vj vi 1 tijethsiethxTHORN1 sjethxTHORNTHORN=2

for all ethi jTHORN 2 A and for all x 2 X

In MIP1 there is one variable vi for every celli in C and one constraint for every arc in A in ad-dition to the variables x and the set X that con-strains them It is permissible to take vi 1 5 0in which case vi can be interpreted as lsquolsquothe min-imal average number of detections up to and in-cluding cell i on any path that starts in cell i1given xrsquorsquo Variable vi 2 is the desired maximalpressure and the detection probability is1 2 exp 2 vi

2eth THORN Note that Redrsquos path is not ex-plicitly represented in MIP1

In moving from MIP0 to MIP1 we haveexploited the fact that z(x y) is a sum and this



analytic form is the case because we haveassumed that various search processes are inde-pendent of each other Without these indepen-dence assumptions we would have to retreatto a model like MIP0 where the paths areenumerated

The precise specification of X is in the sum-mary formulation below In that formulationlsquolsquocellrsquorsquo means traversable cell in C unless artifi-cial cells are explicitly included in the com-ments All variables are nonnegative and realexcept for xfmi

BASTION formulation

Indices and sets [cardinality]

i 1 and i 2 Artificial start and end cells for Redrsquos pathi j k 2 C Traversable cells [500]m Ships [10]ethi jTHORN 2 A Directed arcs assumed sufficient to permit at least one path from i 1 to i 2

SOA Traversable cells that constitutes the SOA a subset of CNSOA Traversable cells that can be occupied by non-submarines a subset of C

Data [units]Smij Detection rate in cell j of ship m located in cell i [hr] i 2 NSOA j 2 CGHm SH-60 sortie rate by ship m [hr]GM MH-60 sortie rate from the bastion [hr]GA MPA sortie rate [hr]BHmijk Average detections in cell j by a SH-60 sortie to cell k from ship m located in cell i

i k 2 NSOA j 2 CBMjk Average detections in cell j by a MH-60 sortie to cell k i k 2 NSOA j 2 CBAjk Average detections in cell j by an MPA sortie to cell k k 2 NSOA j 2 CNS Number of SSNs in direct supportRSj Detection rate of an SSN in cell j [hr] j 2 SOAtij Time required for an attacker to move from i to j [hr] artificial cells included

Variables [units]xfmi 1 if ship m is located at cell i 0 otherwise i 2 NSOAxhmik Rate of SH-60 search sorties to cell k from ship m in cell i i k 2 NSOA[hr]xmk Rate of MH-60 search sorties to cell k k 2 NSOA[hr]xak Rate of MPA sorties to cell k 2 NSOA[hr]xsj Average number of SSNs searching in cell j j 2 SOAsj Total detection rate in cell j [hr]vi Average detections up to cell i on a path that starts in i1 artificial cells included

Constraints fdual variablesgXi2NSOA

xfmi 5 1 Each ship m selects a cell to occupy

Xk2NSOA

xhmik 5 GHmxfmi Helicopter sortie generation limits for all mand i fdhmig

Xi2NSOA

xai 5 GA MPA sortie generation limit fdagX

i2SOA

xsi 5 NS Overall SSN population limit fdsg

sj 5X

i2NSOAm

Smijxfmi 1X

ik2NSOAm

BHmijkxhmik 1X

k2NSOA

BMjkxmk 1X

k2NSOA

BAjkxak 1RSjxsj for all j

vj vi 1 tijethsi 1 sjTHORN=2 for allethi jTHORN 2 A Including artificial cells f2yijgvi 1 5 0 Path starts out with no detections



Objective

Maximize vi 2 the average number of detec-tions up to the bastion

All dual variables referenced above shouldbe thought of as the dual variables of a linearprogram with the integer variables fixed at theiroptimal values Thus dhmi is of no interest forcells i that ship m does not occupy because itrepresents the incremental value of SH-60 sor-ties from a cell where there is no place to takeoff or land a concept of little interest to helicop-ter pilots If ship m occupies cell i then dhmi isthe incremental value of sorties from that shipThe dual variables of the value constraints areof interest because yij can be interpreted as theprobability that Red includes arc (i j) in hispath In principle an optimal path selectionstrategy for Red could be derived from them(Ahuja Magnanti and Orlin 1993 Washburnand Wood 1995) Although we have scant inter-est in that mixed strategy Redrsquos probabilities arestill of diagnostic value

EXAMPLES AND COMPUTATIONALRESULTS

Standard scenarioAll examples considered in this section are

variations of the standard scenario describedbelow The platform details will be completelyspecified here only for Blue submarines (see Ap-pendices A and B for the rest)

In all cases we employ a 26 3 26 grid ofcells although only part of the grid will beshown in the figures below The cell width is Lfrac14 10 nm There are NS frac14 2 submarines in directsupport each of which has a speed of V frac14 5 ktand a sweepwidth of W frac14 5 nm Redrsquos speedis assumed to be Ufrac14 5 kt We assume that Bluersquosvelocity will at most times be perpendicular toRedrsquos so we take the relative speed between thetwo platforms to be

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiU2 1 V2p

Direct supportsubmarines are assumed to be available for searchonly 90 of the time so the detection rate of a Bluesubmarine patrolling in an arbitrary cell i is

RSi 5 eth09THORNffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiU

21 V

2p

W=L2

5 03182=hr

There is one MPA squadron generating sor-ties at the rate of 01hour 24 hours per day Agiven MPA sortie will be most effective in thenorthwest part of the grid because the base islocated in that direction MPA and helicoptersorties are assumed effective only in the cellsto which they are assigned that is the possibil-ity of Bluersquos detecting Red while Blue is in tran-sit is ignored

There are two ships named s5 and s6 eachof which has an effective detection range ofabout 15 nm not considering its helicoptersThe model of ship effectiveness used is in Ap-pendix B a lsquolsquoGaussian plumersquorsquo that decreasesgradually over the effective range A ship isthe only platform capable of detection in cellsother than the cell it occupies

Verification ExamplesIn our first example we assume that only

submarines are available to defend the bastionFigure 3 shows the bastion as a single cell sur-rounded by eight cells with p or q written ineach except that the p and q symbols are skip-ped in two cells containing arrows

The back-of-the-envelope random searchmodel introduced earlier suggests that the idealway to protect the bastion is to form a ringaround it as narrow as possible the eight illus-trated pq cells Let q be the average numberof SSNs patrolling in each of the four cornercells and define p similarly for the other fourThe corner cells require less pressure than theothers because Red has to spend more time inthem on his way to the bastion To penetratethrough a p cell as illustrated with arrows in

Figure 3 The bastion is a single cell surrounded byeight cells that Blue defends Two possible paths forRed are illustrated with arrows



Figure 3 Red must spend LU hours To pene-trate through a q cell Red must spend 05 LUon the way in and 05

ffiffiffi2p

L=U on the way outa total of 05eth1 1

ffiffiffi2pTHORNL=U hours (taking the diag-

onal route into the q cell would unnecessarily in-crease the transit time but Red has no choicebut to take the diagonal route out) To equalizethe pressure on all Red routes into the bastionthe ratio of p to q should therefore bep=q 5 05eth1 1

ffiffiffi2pTHORN ffi 12 Because we also have

4p 1 4q frac14 NS frac14 2 we can solve these two equa-tions for p and q The maximized minimal aver-age number of detections will then be z frac14p(RSi)LU With parameters as specified in thedefault scenario this quantity is 01740

Figure 4 shows a formulation and solutionof this problem in BASTION The problem isshown as one of protecting a bastion located ina large bay but the only purpose of all the landis to limit the size of the MIP which would bemuch larger if all of the lsquolsquolandrsquorsquo were convertedto traversable cells As long as Blue does notchoose to exert pressure in any cell borderingland as is the case in the solution shown bythe 1 marks the configuration is effectivelyopen ocean The BASTION objective functionagrees exactly with the analysis given abovebut the solution does notmdashsubmarines patrolin four unanticipated additional cells BASTIONhas discovered an alternative optimal solutionwhere Red could move into a p cell diagonallyand still be detected exactly z times just as hecould using the two routes portrayed in Figure

3 Because the achieved z is the same as in the an-alytic solution given above we take this as evi-dence that BASTION performs as intended

Consider next a scenario (not illustrated) inwhich the only available platforms are MPAand all cells are land except for a 7 3 2 set lo-cated near the western border The seven cellsactually on the border are traversable whereasthe seven cells just to the east are the bastionThe optimal solution has the MPA presence inthe seven traversable cells increasing from northto south which at first seems odd because MPAsorties are more effective in the north (the MPAbase is in the northwest) However this ten-dency to search mostly where one is least ef-ficient is a known characteristic of gametheoretic solutions to search problems so wetake this to be further evidence of verity By em-phasizing cells in the south the MPA make Redindifferent among seven paths to the bastion

SOA Selection ExampleBASTION can be used as an aid in locating

an SOA Figure 5 shows an example where thebastion has been located north of a continentand south of two islands hoping to take advan-tage of considerable land in the proximity TheSOA has been located between the continentand the eastern island in the hope that otherforces will seal off the western approach andthe strait between the islands The optimal solu-tion is also partially shown in Figure 5 using s5and s6 to locate the two ships and 1 to indicateactivity by MPA and helicopters

The solution may appear odd Note thatRed can pass between the islands without fearof detection by aircraft as long as he stays tothe west and that there appears to be a largegap between s6 and the north edge of the conti-nent The gap is illusory however because s6 iscapable of long-range detections and Red gainsnothing by passing between the islands

Bluersquos strategy might be called lsquolsquoprotect thewest with ships and make Red pay for getting tothe east of the bastionrsquorsquo The MPA activity alongthe north edge of the continent protects againstRedrsquos making an end run around s6 to get eastThe submarine activity within the SOA alsoblocks access from the east The maximized ob-jective function is 03213

Figure 4 BASTION finds an optimal solution forthe same situation as in Figure 3 but with unexpectedpositive allocations in the cells labeled



There is a strategic question in situationssuch as that depicted in Figure 5 Should oneseal off the region by protecting the variousgaps between land masses or just retreat to sur-rounding the bastion with search activity as onewould in the open ocean The pictured SOAalmost enforces the former viewpoint becausesubmarines are dedicated to protecting one ofthe gaps To explore which viewpoint is cor-rect one can simply delete the SOA and runBASTION in its free mode Doing so in this casereveals a solution in which all search platformsretreat to surrounding the bastion with an in-crease in the objective function to 03545 Al-

though this increase would probably not belarge enough to make Blue feel comfortable itis still a significant improvement The free solu-tion has submarines and other platforms beingjointly active in some cells but an objective nearlyas large can be found by locating the SOA approx-imately where the submarines are located in thefree solution Figure 6 shows the result The objec-tive function decreases slightly to 03420 Notethat the SOA is not a connected region in Figure6 Making it connected would enforce an addi-tional decrease in the objective function

We have found various graphics to be of usein debugging and understanding some of the

Figure 6 A better albeit unconnected SOA than the one shown in Figure 5 Cell boundaries omitted for clarity

Figure 5 The SOA is located in the east expecting to block access from that direction The optimal solution hastwo ships located as pictured with aircraft activity shown by 1 marks



plans produced by BASTION One of them isshown in Figure 7 a plot for the same scenarioshown in Figure 6 that shows vi in the variouscells As Red approaches the bastion regardlessof the path chosen he must pay a higher andhigher price to achieve proximity

Computational CommentsThe most computationally stressful situa-

tion would seem to be when there is no landthe bastion consists of a single central celland there is no SOA to limit the activities ofany platform Such a case generates an instancewith 915302 decision variables 1350 of thesebinary and 6548 constraints On a LenovoT510 laptop problem generation in GAMS(GAMS 2008) requires about six minutes andoptimization with CPLEX 122 (ILOG 2007)MIP to a 01 integrality gap requires about15 GB of memory and three minutes Howeverit turns out that there are problems with fewerdecision variables that are much more difficultespecially if we insist on a pure optimal (ie 0gap) solution as we have done for the examplesreported above We have found examples re-quiring hours rather than minutes includingthe just-discussed SOA examples

CAVEATS AND EXTENSIONSWe have made many assumptions and

approximations in the course of developing

BASTION some of which are at odds withreality In this section we describe some ofthese difficulties suggest remedies and alsosuggest how BASTION might be modified orextended

Ship DifficultiesAll platforms engaged with defending the

bastion will have missions other than ASWbut the conflict between objectives is likely tobe greatest for ships which have importantroles in command and control power projec-tion surface defense and air defense Ship loca-tions that are optimal for ASW may very welllimit the shiprsquos effectiveness in other roles Theonly remedy that retains BASTIONrsquos role asa strictly ASW tool is to allow the planner to lo-cate the ships as he wishes using BASTION toplan the employment of other assets to fill inthe ASW capability around whatever the shipsare able to offer This is easily done and evenhas the benefit of reducing BASTION to a linearprogram rather than a MIP BASTIONrsquos abilityto optimize ship locations still serves to quantifythe sacrifice that ASW must make to the otherroles

Another difficulty is that we have notfound a simple model that can quantify a shiprsquosdetection rate in terms of more fundamentalquantities such as speed and detection rangewhether through the Gaussian plume modelor any other We are still looking

Figure 7 This figure shows 100vi rounded to the nearest integer The bastion has 34 written in it because theobjective function is 03420 The other cells show how detections accumulate as Red approaches the bastion Cellboundaries omitted for clarity



MPA DifficultiesWe earlier argued that si(x) should be taken to

be the mean of a Poisson process but there aresome features of ASW that make the assumptionproblematic Chief among these is the lsquolsquopulseproblemrsquorsquo that arises when platforms concentrateASW effort in time and space This is especiallytrue of MPA When the cell dimension L is smalla single MPA sortie can quickly cover an area thatis much larger than that of the sortiersquos assignedcell If Red is unfortunate enough to be in the cellat the same time as the sortie he will surely bedetected but otherwise Blue will wish that hehad some way of spreading all that confetti overmultiple cells to avoid overcoverage Except forships BASTION analytically permits Blue to doexactly that by using infinitely divisible allocationvariables but how can BASTIONrsquos results aboutsortie allocation be used to provide practicalguidance to MPA We suggest two possibilities

One way to lsquolsquofixrsquorsquo the pulse problem is to letthe MPA squadron itself solve it The squadronis not given a grid showing the rate of flying sor-ties to various cells which would be the directoutput from BASTION Instead the BASTIONoutput to the squadron is a grid showing thefraction of area covered that is in each cell sym-bolically faj [

Pk BAjkxak=

Pjk BAjkxak together

with advice to lsquolsquoFly as many sorties as youcan and make it so that cell j gets a fraction faj

of the total area searchedrsquorsquo Note that the numer-ator of this fraction is the only occurrence of thevariables xak in the objective function This less-specific advice leaves it to the squadron to for-mulate missions and fly sorties The underlyingassumption is that the squadron will find someway of (nearly) retaining the efficiency of thesingle-cell sorties used in BASTION while si-multaneously avoiding overcoverage As longas this assumption is correct BASTION can bea useful operational tool without dealing di-rectly with MPA tactics

Another way to fix the pulse problem is byassigning platforms to missions rather thancells a mission being a sequence of cells to-gether with a detailed program of activity ineach visited cell In this manner a large pulseof covered area can be spread out over enoughcells to prevent overcoverage in any given cellThis method has the advantage that the activi-

ties suggested by the optimal solution will al-ways be reasonable because only reasonableactivities (missions) are considered in the firstplace It is certainly implementablemdashone merelyhas to let the index i on xai refer to a larger set ofmissions than those that are named for a singlecell We have experimented with this success-fully (Thomas 2008 Pfeiff 2009) however it mustbe employed carefully The number of missionsis potentially enormous and also BASTIONhas a tendency to prefer missions that overcovera small number of cells whether those missionsare realistic or not Indeed if the mission set in-cludes all single-cell missions then BASTIONwill exclude all others in its optimal solution(see Appendix C for a proof of this) Thus caremust be exercised to provide a rich set of mis-sions all of which avoid overcoverage Becauseovercoverage is a gradual phenomenon ratherthan a sudden one and given BASTIONrsquos ten-dencies construction of a good multicell missionset is problematic For the moment BASTIONmanipulates only single-cell missions

SOA (SSN) DifficultiesBASTION currently models SSNs as a class

just like aircraft but there are good reasons formodeling them individually Like ships theytend to be present in small numbers and havemuch lower speeds than aircraft We have ex-perimented with individual SSNs and ratherthan require SOAs as input to recommend SOAsby specific assignment of these SSNs to cells Wecan optimally partition search cells into a numberof SOAs Each SOA can be planned to be patrolledby some number of SSNs (say one or two each)these patrols can be planned to invest some mini-mum and maximum search pressure in each celland the diameter of each SOA can be limited Em-pirically these embellishments do not noticeablyadd computational effort to the optimization Sim-ilar means could be employed to shape patrolareas for MPA but we have not implemented this

Multiple SSKsThroughout we have referred to lsquolsquothersquorsquo pen-

etrating Red SSK If there are actually several RedSSKs the same analysis applies as long as thepenetration attempts are well separated in time



However Red has every motive to locally over-whelm Bluersquos defenses by making multiple si-multaneous penetration attempts The fractionof successful penetrations in that case couldsubstantially exceed the detection probabilitypredicted by BASTION We view the task ofmodifying BASTION to account for cooperativeRed penetration tactics as formidable and haveno plans to do so

If Red possesses multiple types of subma-rines then Blue should plan against the worstof them

The 4W Grid May Be anUnsupportable Historical Artifact

The square search regions of US Navy 4Wgrids may have been historically easy to specifyby boundary coordinates but they complicateestimates of search effectiveness that tend tobe circular rather than rectilinear The tessel-lated hexagonal region used by many land war-fare models appeals here

Given the introduction of new long-rangesubmarine-launched antiship weapons we antici-pate the operational necessity to increase the num-ber of cells that have traditionally been employedto enable patrol of much greater sea-space

Scaling Up for Larger Numbers of(Unmanned) Search Platforms

Future bastion planning problems may in-clude scores of high-endurance unmanned heli-copters rather than just a few manned ones andperhaps more unmanned autonomous subma-rine searchers As long as such platforms aremodeled collectively as all aircraft are currentlymodeled in BASTION this expansion presentsno computational difficulty

If such unmanned submarine searchers areviewed as expendable it is also possible to includethe decision whether each should search passively(and covertly) or actively (revealing itself at somerisk but with much enhanced search effective-ness) Thomas (2008) pursues this embellish-ment for SSN searchers BASTIONrsquos treatmentof SSNs does not include this option implicitlyassuming that an SSN will not search in a man-ner that would reveal its location

SUMMARYBASTIONrsquos function is to help plan the de-

fense of a stationary oceanic segment from attackby submarines Blue first locates his ships andthen the rest of the Blue forces engage in a two-person zero-sum game with Red The principalsummary statistic is BASTIONrsquos objective functionthe probability that an optimally arranged Blue de-fense will detect an optimally operated Red sub-marine before it can penetrate to the bastion

Bluersquos problem has aspects of an assign-ment problem in which the various platformsattempt to accomplish an overall mission whileeach platform type does what it is efficient atBASTION can partition the action space pro-viding guidance to each platform type abouthow to best operate cooperatively with the otherplatforms This function is particularly impor-tant for Blue submarines where BASTION canbe of use in locating and sizing an SOA

The capabilities of modern computers andsoftware permit the solution of realisticallyscaled problems in reasonably quick responsetimes as we have shown by example

REFERENCESAhuja R Magnanti T and Orlin J 1993

Network Flows Theory Algorithms and Appli-cations Prentice Hall Englewood Cliffs NJ

Benedict J 2006 lsquolsquoASW Lecture to Naval Post-graduate Schoolrsquorsquo presented at MennekenLecture Naval Postgraduate School MontereyCA 16 November

FAST (Fleet Assistance and Support Team)2006 ASPECT Userrsquos Guide Naval Air Sys-tems Command

Fulkerson D and Harding G 1977 lsquolsquoMaxi-mizing the Minimum Source-Sink Path Subjectto a Budget Constraintrsquorsquo Mathematical Pro-gramming 13 pp 116-118

GAMS 2008 On-line Documentation httpwwwgamscomdocsdocumenthtm accessed25 May 2011

ILOG 2007 CPLEX 11 Solver Manual httpwwwgamscomdddocssolverscplexpdfaccessed 25 May 2011

Khinchin A 1960 Mathematical Methods in theTheory of Queueing Hafner New York chapter 5



Kierstead D and DelBalzo D 2003 lsquolsquoA GeneticAlgorithm Approach for Planning SearchPaths in Complicated Environmentsrsquorsquo MilitaryOperations Research 8(2) pp 45-59

Koopman B 1980 Search and Screening PergamonPress New York

Pfeiff DM 2009 lsquolsquoOptimizing Employment ofSearch Platforms to Counter Self-PropelledSemi-Submersiblesrsquorsquo MS Thesis in OperationsResearch Naval Postgraduate School

SPAWAR 2008 PCIMAT Version 70 UserrsquosManual SPAWAR Systems Center San Diego

SWDG (Surface Warfare Development Group)2004 ASW Screen Planner TDA User ManualmdashTACMEMO SWDG 3-213-04 SWDG NorfolkVA

Thomas A J 2008 lsquolsquoTri-level Optimization forAnti-Submarine Warfare Mission PlanningrsquorsquoMS Thesis in Operations Research NavalPostgraduate School

Wagner Associates Inc 2008 ORP AlgorithmDescription httpwwwwagnercomtechnologiesmissionplanningORPaccessed 25 May 2011

Washburn A and Wood K 1995 lsquolsquoTwo-PersonZero-Sum Games for Network InterdictionrsquorsquoOperations Research 43(2) pp 243-241

Washburn A 2003 Two-Person Zero-Sum Games3rd ed Institute for Operations Research andthe Management Sciences Linthicum MDp 85

Washburn A 2010a lsquolsquoBarrier Gamesrsquorsquo MilitaryOperations Research 15(3) pp 31-41

Washburn A 2010b lsquolsquoA Multistatic SonobuoyTheoryrsquorsquo Naval Postgraduate School TechnicalReport NPS-OR-10-005

APPENDIX A DETERMINING THEAREA COVERED BY A GIVEN MPASORTIE

We assume that MPA search using fields ofmultistatic sonobuoys where some buoys aresources and some are receivers Sonobuoys re-main functional for some time after they are acti-vated so there are two modes of detection asonobuoy might detect a submarine as soon asit is dropped (mode 1) or the moving submarinemight run into it later (mode 2) Sources and re-

ceivers need not be colocated but if they arethe detection radius determines d one of the in-puts This radius will typically depend on the cellin which the buoys are located Other requiredinputs are listed below with [nominal values]

bull d frac14 detection radius [2 nm]bull Dfrac14 distance from MPA base to cell [1000 nm]bull V frac14MPA transit speed [330 kt]bull U frac14 submarine transit speed [5 kt]bull T frac14MPA endurance [12 hr]bull t frac14 time between sonobuoy fields while on

station [1 hr]bull s frac14 number of sources in a field [8] andbull r frac14 number of receivers in a field [32]

Washburn (2010b) shows that the equivalentarea covered by a multistatic field containing ssources and r receivers is a 5 08pd2

ffiffiffiffirsp

[1609nm2] Because there is a new sonobuoy field laidevery t hours as long as the MPA is on stationthe mode 1 area covered by one sortie is A1 frac14a(T ndash 2DV)t [9553 nm2] To account for mode2 detections we assume that the useful life ofa sonobuoy is t the same as the time between so-nobuoy fields and that the effective diameter ofthe region covered is 2

ffiffiffiffiffiffiffiffia=p

p[7155 nm] The

area covered by type 2 detections is then theproduct of the length of time on station the sub-marine speed and the effective diameterA2 5 ethT 2 2D=VTHORN2U


p[4250 nm2]

Note that the mode 2 area is proportional tothe submarine speed whereas the mode 1 area isindependent of submarine speed A Red subma-rine that is aware of an aircraft overhead butdoes not know exactly what the aircraft is doingwould be well advised to slow down Becausethe submarinersquos state of awareness is unlikelyto be known with any accuracy by the Blue plan-ner this makes the calculation of the mode 2 areaproblematic A conservative planner might ig-nore the possibility of mode 2 detections

APPENDIX B DETAILS OF THESTANDARD SCENARIO

The standard scenario used for generatingour examples uses the MPA parameters of Ap-pendix A except that the distance D is not al-ways 1000 nm Instead D is the distance tothe relevant cells from an MPA base located



800 nm west and 900 nm north of the NW cor-ner of the grid The MPA sortie generation rateis 24day

The cell dimension is L frac14 10 nm and Redrsquosspeed is 5 kt

Each ship has four parameters three for theGaussian plume plus an SH-60 sortie generationrate GH Table 1 shows the data

Table 1 Ship Parameters

Ship l R b GH

s1 2hour 10 nm 2 15days2 1hour 20 nm 2 18day

MH-60 helicopters are assumed to carry 10monostatic sonobuoys with a detection radiusof 1 nm They transit at 150 kt and have an en-durance of 3 hours As long as the center of thedestination cell is within range an MH-60 cancover an area of 10p(1 nm)2 Because the cell di-mension is 10 nm the average number of timesa Red submarine in the chosen cell will bedetected is 0314 The bastion generates MH-60sorties at the rate of 16day

SH-60 helicopters search by dipping a sonarwith a range of 05 nm Dips occur every 05 hourwhile the aircraft is on station Endurance is4 hours and transit speed is 150 kt so the areacovered by a sortie to range r is p(05)2(4 ndash 2r150)(05)

Blue SSNs are as described in the lsquolsquoDevelop-ment of BASTIONrsquorsquo section

APPENDIX C PROOF THAT SIMPLESORTIES ARE DOMINANT

The text claims that simple sorties are domi-nant when a given rate of sortie generation mustbe split among missions that include simple sor-ties It is equivalent to prove that a given sortie isdominated by a probabilistic mixture of simplesorties Assume then that an aircraft is locatedat point 0 with endurance T A collection of m

additional points i is given with dij being thetime required to fly from i to j For a given flightthe aircraft must decide which points (0 ij 0) to visit and the amount of time xi to spendmonitoring each point in the flight subject to theconstraint that

Pi j di j 1

Pi xi T Mixed strat-

egies are permissibleTheorem Assume that the triangle inequal-

ity holds dij dik 1 dkj i j k Then there existsa mixture of flights visiting a single cell (simpleflights) that will dominate any flight visitingmultiple cells in the sense of spending at leastas much time in every cell on the average

Proof For a simple flight visiting cell i letthe time available for search in cell i be Xi frac14T ndash d0i ndash di0 which without loss of generalitywe can assume positive in every cell Now con-sider a flight that visits points (0 1 n 0) inthat order spending a time xi at point i Let

x [Xn

i 5 1

xi

If the flight is feasible we must have T $ d 1 xwhere d [ d01 1 d12 1 1 dindash1i 1 dij11 1 1

dn21n 1 dn0 is the total time spent in transit Wewill show that this flight is dominated by a par-ticular mixture of simple flights in the sensethat the mixture spends at least as much timeat each of the n points on the average This istrivial if x frac14 0 so assume x 0

For 1 i n we have after multiple ap-plications of the triangle inequality d0i d01 1

d12 1 1 di21i and di0 dii11 1 1 dn21n 1

dn0Therefore d0i 1 di0 d and because Tfrac14Xi 1

d0i 1 di0 we have Xi $ x Let K 5Pn

i 5 1 xi=XiThen K

Pni 5 1 xi=x 5 1 Now let the probability

that a simple flight of type i is used be yi withyifrac14 xi(KXi) It is a simple matter to confirm that(y1 yn) is a probability distribution But theexpected time spent at point i by this mixedstrategy is yiXi and K 1 so the mixture spendsat least xi at point i on the average Because i isarbitrary this completes the proof n



Report Documentation Page Form ApprovedOMB No 0704-0188

Public reporting burden for the collection of information is estimated to average 1 hour per response including the time for reviewing instructions searching existing data sources gathering andmaintaining the data needed and completing and reviewing the collection of information Send comments regarding this burden estimate or any other aspect of this collection of informationincluding suggestions for reducing this burden to Washington Headquarters Services Directorate for Information Operations and Reports 1215 Jefferson Davis Highway Suite 1204 ArlingtonVA 22202-4302 Respondents should be aware that notwithstanding any other provision of law no person shall be subject to a penalty for failing to comply with a collection of information if itdoes not display a currently valid OMB control number

1 REPORT DATE 2011 2 REPORT TYPE

3 DATES COVERED 00-00-2011 to 00-00-2011

4 TITLE AND SUBTITLE A Game-Theoretic Model for Defense of an Oceanic Bastion Against Submarines

5a CONTRACT NUMBER

5b GRANT NUMBER

5c PROGRAM ELEMENT NUMBER

6 AUTHOR(S) 5d PROJECT NUMBER

5e TASK NUMBER

5f WORK UNIT NUMBER

7 PERFORMING ORGANIZATION NAME(S) AND ADDRESS(ES) Naval Postgraduate SchoolOperations Research DepartmentMontereyCA93943

8 PERFORMING ORGANIZATIONREPORT NUMBER

9 SPONSORINGMONITORING AGENCY NAME(S) AND ADDRESS(ES) 10 SPONSORMONITORrsquoS ACRONYM(S)

11 SPONSORMONITORrsquoS REPORT NUMBER(S)

12 DISTRIBUTIONAVAILABILITY STATEMENT Approved for public release distribution unlimited

13 SUPPLEMENTARY NOTES

14 ABSTRACT We develop a game-theoreticmodel called BASTION to guide the employment of antisubmarine warfare(ASW) platforms such as ships and aircraft that are defending a stationary oceanic bastion from attack byhostile submarines The model is an example of a twoperson zero-sum game with some additional variables(ship locations) that are under the control of the maximizing defender but known to the minimizingattacker The attacker knowing the ship locations but not the locations of other platforms such as aircraftmust select a path to the bastion The probability of detecting the attacker as it follows this path is theobjective shared by the opponents

15 SUBJECT TERMS

16 SECURITY CLASSIFICATION OF 17 LIMITATION OF ABSTRACT Same as

Report (SAR)

18 NUMBEROF PAGES

16

19a NAME OFRESPONSIBLE PERSON

a REPORT unclassified

b ABSTRACT unclassified

c THIS PAGE unclassified

Standard Form 298 (Rev 8-98) Prescribed by ANSI Std Z39-18
















6 Military Operations Research V16 N4 2011


















pUethR22 r

2THORN

5 1 2 exp 2C

pUethR 1 rTHORN


















Let








Pi2SOA xsi 5 NS







Pk BAjkxak and the





















max v


x 2 X


uij This expression


max vi2










BASTION formulation









Xk2NSOA


Xi2NSOA


i2SOA


sj 5X

i2NSOAm

Smijxfmi 1X

ik2NSOAm

BHmijkxhmik 1X

k2NSOA

BMjkxmk 1X

k2NSOA





Objective










21 V

2p

W=L2

5 03182=hr










ffiffiffi2p






































Pk BAjkxak=



















































ffiffiffiffirsp



p[7155 nm] The



p[4250 nm2]










Ship l R b GH








Pi j di j 1





x [Xn

i 5 1

xi







i 5 1 xi=XiThen K






































pUethR22 r

2THORN

5 1 2 exp 2C

pUethR 1 rTHORN


















Let








Pi2SOA xsi 5 NS







Pk BAjkxak and the





















max v


x 2 X


uij This expression


max vi2










BASTION formulation









Xk2NSOA


Xi2NSOA


i2SOA


sj 5X

i2NSOAm

Smijxfmi 1X

ik2NSOAm

BHmijkxhmik 1X

k2NSOA

BMjkxmk 1X

k2NSOA





Objective










21 V

2p

W=L2

5 03182=hr










ffiffiffi2p






































Pk BAjkxak=



















































ffiffiffiffirsp



p[7155 nm] The



p[4250 nm2]










Ship l R b GH








Pi j di j 1





x [Xn

i 5 1

xi







i 5 1 xi=XiThen K









Let








Pi2SOA xsi 5 NS







Pk BAjkxak and the





















max v


x 2 X


uij This expression


max vi2










BASTION formulation









Xk2NSOA


Xi2NSOA


i2SOA


sj 5X

i2NSOAm

Smijxfmi 1X

ik2NSOAm

BHmijkxhmik 1X

k2NSOA

BMjkxmk 1X

k2NSOA





Objective










21 V

2p

W=L2

5 03182=hr










ffiffiffi2p






































Pk BAjkxak=



















































ffiffiffiffirsp



p[7155 nm] The



p[4250 nm2]










Ship l R b GH








Pi j di j 1





x [Xn

i 5 1

xi







i 5 1 xi=XiThen K






Pk BAjkxak and the





















max v


x 2 X


uij This expression


max vi2










BASTION formulation









Xk2NSOA


Xi2NSOA


i2SOA


sj 5X

i2NSOAm

Smijxfmi 1X

ik2NSOAm

BHmijkxhmik 1X

k2NSOA

BMjkxmk 1X

k2NSOA





Objective










21 V

2p

W=L2

5 03182=hr










ffiffiffi2p






































Pk BAjkxak=



















































ffiffiffiffirsp



p[7155 nm] The



p[4250 nm2]










Ship l R b GH








Pi j di j 1





x [Xn

i 5 1

xi







i 5 1 xi=XiThen K













max v


x 2 X


uij This expression


max vi2










BASTION formulation









Xk2NSOA


Xi2NSOA


i2SOA


sj 5X

i2NSOAm

Smijxfmi 1X

ik2NSOAm

BHmijkxhmik 1X

k2NSOA

BMjkxmk 1X

k2NSOA





Objective










21 V

2p

W=L2

5 03182=hr










ffiffiffi2p






































Pk BAjkxak=



















































ffiffiffiffirsp



p[7155 nm] The



p[4250 nm2]










Ship l R b GH








Pi j di j 1





x [Xn

i 5 1

xi







i 5 1 xi=XiThen K







BASTION formulation









Xk2NSOA


Xi2NSOA


i2SOA


sj 5X

i2NSOAm

Smijxfmi 1X

ik2NSOAm

BHmijkxhmik 1X

k2NSOA

BMjkxmk 1X

k2NSOA





Objective










21 V

2p

W=L2

5 03182=hr










ffiffiffi2p






































Pk BAjkxak=



















































ffiffiffiffirsp



p[7155 nm] The



p[4250 nm2]










Ship l R b GH








Pi j di j 1





x [Xn

i 5 1

xi







i 5 1 xi=XiThen K





Objective










21 V

2p

W=L2

5 03182=hr










ffiffiffi2p






































Pk BAjkxak=



















































ffiffiffiffirsp



p[7155 nm] The



p[4250 nm2]










Ship l R b GH








Pi j di j 1





x [Xn

i 5 1

xi







i 5 1 xi=XiThen K






ffiffiffi2p






































Pk BAjkxak=



















































ffiffiffiffirsp



p[7155 nm] The



p[4250 nm2]










Ship l R b GH








Pi j di j 1





x [Xn

i 5 1

xi







i 5 1 xi=XiThen K



























Pk BAjkxak=



















































ffiffiffiffirsp



p[7155 nm] The



p[4250 nm2]










Ship l R b GH








Pi j di j 1





x [Xn

i 5 1

xi







i 5 1 xi=XiThen K












































ffiffiffiffirsp



p[7155 nm] The



p[4250 nm2]










Ship l R b GH








Pi j di j 1





x [Xn

i 5 1

xi







i 5 1 xi=XiThen K









Ship l R b GH








Pi j di j 1





x [Xn

i 5 1

xi







i 5 1 xi=XiThen K





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