search theory

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Professional Paper 49/ April 1986

SEARCH THEORY

by

Henry R. Richardson

ADivision of Budson IniueA

w4401 Ford Anue Pos~t Office Box 16268 -Ale -andrza, Virg ra 22302-0268 * 703) 824-20(y)

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Cleared for Public Release. Distribution Unlimited.

The ideas expressed in this paper are those of the author.The paper does not necessarily represent the views of theCenter for Naval Analyses.


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Professional Paper 449 / April 1986SE

Search Theoryby

Henry R. RichardsonNaval Warfare OoerationsDivision

A LVv:,wr ,,f Hud.-Y, I'l,fltc

4AENME.FOR6.NVAL.ANA(YS(S-4401 r-trd Avrime/t - Pt?.,t Office &x 16268q Ahlemidria. Virginia 2230240268 - (703) 824-2000


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SEARCH THEORY

Henry R. RichardsonCenter fo r Naval Analyses

4401 Ford AvenueP.O. Box 16268Alexandria, Virginia 22301-0268

April 14, 1986

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SEXRCH THEORY

Search theoryI came into being during World War II with the work ofB. 0. Koopman and his colleagues in the Antisubmarine Warfare OperationsResearch Group (ASWORG). ASWORG was directed by P. M. Morse andreported to Admiral Ernest King, Chief of Naval Operations and Commanderin Chief, U. S. Fleet. Inspired by Morse, many of th e fundamentalconcepts of search theory such as sweep width and sweep rate had beenestablis..ed by Spring of 1942. Since that time, search theory has grownto be a major discipline within the field of Operations Research. Itsapplications range from deep-ocean search fo r submerged objects to deep-space sarveillance fo r artificial satellites.

The reader interested in the origins of search theory should con-stilt Morse [29] and Koopman [24]. Early use of search theory to developNaval tact ics is discussed in Morse and Kimball [30]; excerpts from thiswork appear in [31].

For a modern account of search theory, the reader should consultStone [41]. This book p:ovides a rigorous development of the theory andincludes an excellent bibliography and notes on previous research.Was.aburn's monograph [521 is also recommended.

I. This arl-cle, in essential ly th e same form, was originally preparedfo r the Kotz-Johnson Encyclopedia of Statistical Sciences copywrited byJohn Wiley and Sons.


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",Since World War II, the principles of search theory have beenapplied successfully in numerous important operations. These includethe 1966 search for a lost H-bomb in the Mediterranean near Palomares,Spain, the 1968 search fo r the lost nuclear submarine Scorpion near theAzoresY-[3-6 and the 1974 underwater search fo r unexploded ordnanceduring clearance of the Suez Canal. The U. S. Coast Guard employs

/search theory in its open ocean search and rescue planning)[-3-91. Searchtheory is also used in astronomy /447-, and in radar search for satel-lites j341. Numerous additional applications, including those toindustry, medicine, and mineral exploration, are discussed in the pro-ceedings 4151 of the 1979 NATO Advanced Research Institute on SearchTheory and Applications. Applications to biology and-)

and a-, -pp-o 1-U'machine maltiLenance and inspection isdescribed inaf-1-------a .'i .....

Further references to the literature are provided in the firstsection/ This is followed in the second section by an illustration ofhow search theory can be used to solve an optimal search problem.

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1. REVIEW OF SEARCH THEORY LITERATURE

Work in search theory can be classified, at least in part, accord-ing to the assumptions made about measures of effectiveness, targetmotion, and the way in vhich search effort is characterized. Thisreview is organized according to these criteria.

Measures of Effectiveness

Among the many measures of effectiveness that are used in searchanalysis, the most common are:

probability of detection,

expected time to detection,

probability of correctly estimating target "whereabouts," and

entropy ef the posterior target location probability distri-bution.

Usually the objective of an optimal search is to maximize probaoll-ity of detection with some constraint imposed on th e amount of searcheffort available. For a stationary target, it is shown in Stone [411

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that waen the detection functiot. is concave or the search space andsearch effort are continuous, a p' L which max'mizes probability ofdetection in each of successive incremen.ts of search effort (incre--mentally optimal) will also be optimal for the total effort contatned inthe increments (totally optimal).

Moreover, fo r stationary targets, it is often theoreticallypossible to construct a "unifornly optimal" search plan. This is a planfor which probability of detection is maximized at each moment duringits period of application. Tf a uniformly optimal search plan exists,then it will (1) maximize probability of detection over any period ofapplication (i.e., be totally and incrementally optimal) and (2) mini-r .ze the expected time to detection. An example ot such a plan(originally due to Koopman) is given in the last section.

In a "whereabouts" search, the objective is to correctly estimatethe target's location in a collection of cells given a constraint onsearch cost. The searcher mz, succeed either by finding the targetduring search or by correctly guessing the target's location aftersearch. These searches were first studied systematically by Kadane (see[21]). In many cases of interest, Kadane shows that the optimal where-abouts search consists of an optimal detection search among all cellsexclusive of the cell with the highest prior target location probabil-ity. If the search fails to find the target, then one guesses that i-is in the excluded highest probability cell.

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More recently, Kadane and Stone [22] have considered whereaboutssearch in the context of moving targets. They show that the optimalwhereabouts search plan may be found by solving a finite number ofoptimal detection search problems, one for each cell in the grid.

Consideration of entropy as a measure of effectiveness is useful incertain situations and can be used to draw a distinction tetween searchand surveillance. For certain stationary target detection search prob-lems with an exponential detection function, Barker [5] has shown thatthe search plan which maximizes the entropy of the posterior targetlocation probability distribution conditionted upon search failure is thesame az the search plan which maximizes probability of detection.

In a surveillance search, the objectives are usually more complexthan in a detection search. For example, one may wish to correctlyestimate target location at the end of a period of search in order totake some further action. In this case, detection before the end of theperiod can contribute to success but does not in i tself constitutesuccoss. More general problems of this type are discussed by Tierneyand Kadane [49], and they obtain necessary conditions fo r optimalitywhen target motion is Markovian. In surveillance problems involvingmoving targets and false contacts where the time of terminal action maynot be known in advance, Richardson [38] suggests allocating searcheffort to minimize expected entropy 'maximize information).

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Im

Among other measures of effectiveness which are used in search,those based upon minimax criteria are of particular interest. Corwin[8] considers search as a statistical game and seeks estimates fortarget location. Alpern [31, G&l [14], and Isaacs [19] consider gamesin which minimax strategies are sought for a moving targeL seeking toavoid a moving searcher.

Target Motion

Assumptions about target motion have a considerable influence onthe characteristics of search plans and the difficulty of computation.Until rcccntly a ll ,ut the -. -imi..... rch. p^roblms invol-ving targetmotion were intractable from the poiat of view of mathematical optimiza-tion. Vesults were usually obtained by considering transformationswhich would convert the problem into an equivalent stationary targetproblem (e.g., see Stone and Richardson [42], Stone [43], and Pursiheimo[35]. Representative early work on search with Markcvian target motionis given in Pollock [33], Dobbie [12), and McCabe [281,. Hellman [16]investigates the effect of search upon targets whose motion is a diffu-sion process.

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The first computationally practical solution to the optimal searchproblem fo r stochastic target motion involving a large number of cellsand time periods is due to Brown [6]. For exponential detection func-tions, he found necessary and sufficient conditions fo r discrete timeand space search plans, and provided an iterative method fo r optimizingsearch fo r targets whose motion is described by mixtures cf discretetime and space Markov chains. Washburn [51] extended Brown's necessaryconditions to the case of discrete search effort . Washburn [531 alsoprovides a useful bound on Mow close a plan is to the optimal plan.

Very general treatments of moving target search are provided byStone [44] and by Stroaquist and Stone [46] allowing efficient numericalsolution in a wide class of practical moving target problems; theseinclude, fo r example, non-Markovian motion and non-exponential detectionfunctions.

The existence of optimal search plans fo r moving targets is no t tobe taken fo r granted. L. K. Arnold has shown that there are cases whereno allocation function satisfies the necessary conditions given in[46]. In his examples, there appear to be optimal plans, but theyconcentrate effort on sets of measure zero and ari outside the class ofsearch allocation functions usually considered. He also shows th eexistence of optimal plans whenever th e search density is constrained tobe bounded.

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Search Effort

Search effort may be either discrete (looks, scans, etc.) or con-tinuous (time, track length, etc.).

In problems involving discrete search effort , the target is usuallyconsidered to be located in one of several cells or boxes. The searchconsists of specifying a sequence of looks in the cells. Each cell hasa prior probability of containing th e target. A detection function bis specified, where b(j,k) is th e conditional probability of detecting

th e target on or before th e kth look in cell J, given that th etarget is located in cell J. A cost function c is also specified,where c(J,k) is the cost of performing k looks in cell J. An earlysolution to this problem fo r independent glimpses and unitorm cost isgiven by Chew [7]. In this case,

b(j,k) - b(j,k-1) -a i(1-a )k-l

fo r all j and fo r 7 0; c(j,k) - k fo r all j and k > 0. Addi-t ional important results have been obtained by Matula [27], Blackwell(see Matula [27]), Kadane [20], and Wegener [54], [55), and [56].

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Kadane's result [201 is particularly interesting since he uses avariant of the Neyman-Pearson lemma to obtain an optimal plan for thegeneral case where b(j,k) - b(j,k-l) is a decreasing function of kfor all J.

In problems involving continuous effort, the target ma y be locatedin Euclidean n-space or in cells as in the case of discrete search. Inthe former case it is assumed that the search effort is "infinitelydivisible" in the sense that it may be allocated as finely as necessaryover the entire search space. The search problem was originallyexpressed in this form by Koopman (see [231). The continuous effortcase will be considered in greater detail in the remainder of this

Just as with discrete search effort, there is a detection func-tion b, where b(x,z) is the probability of detecting the targetwith z amount of effort applied to the point ), given the target islocated at x. If 7 is a cell index, then z represents the amountof time or track length allocated to the cell. If x is a point inEuclidean n--space, then z is a density as will be made clear in thenext section.

SKoopman's original solution [23] to the search pr, made use ofan exponential function for b of the form

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b(x,z) 1 - exp (-Kz) ,

where K is a positive constant which may depend upon x. DeGuenin[10] considered a more general class of detection functions now referredto as "regular" (see 141]). Dobbie [11] considered sequential searchwith a concave detection function. Richardson and Belkin [37] havetreated a special type of regular detection function obtained when theparameter K in the exponential effectiveness function is a randomvariable. Such functions occur when sensor capabilities are uncer-

tain. Tatsuno and Kisi [48] address similar problems. Stone has con-sidered very geneial detection functions and has collected the resultsin [41].

For differentiable detection functions, Wagner [50] obtained sufti-cient conditions for an optimal search plan with continuous effort usinga non-linear functional version of the Neyman-Pearson lemma.

Remarks

Search theory remains a field of active research in spite of theconsiderable advances made since its inception more than forty yearsago. A review of the current status of search theory in terms of prac-tical applications is given in 145]. Many problems remain to be solved,particularly in cases involving multiple targets and false targets.Also systematic methods are needed for constructing tht prior target

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location probability distribution frou someclmes conflicting subjectiveopinion. More work is also needed on problems where it is essential totake exact account of search track continuity or the switching cost ofmoving from one region to another. These problems remain intractablealthough some recent progress has been made (see, e.g., [541 and [25]).

Browne's innovative solution to an important class of moving targetsearch problems has removed an impediment to progress in this area. Aainteresting extension of Brown's results occurs in the context of searchand rescue operations carried out at sea. Here the target's state maybe unknown. That is, the target ma y be the survivor of a mishap and maybe aboard the distress vessel, afloat in a life raft, or alone in thewater supported only by a life jacket. The detectability of the targetvaries with these alternatives as does the drift motion and the expectedtime of survival of the target. A solution to this important problem isgiven in [9] under very general assumptions.

Unfortunately, Brown's approach applies only to the case wheresearch effort is continuous. Efficient algorithms are still required inthe case where search effort is discrete.

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2. SOLUTION TO AN OPTIMAL SEARCH PROBLEM

This section shows how eptimal search theory can be applied to animportant class of search problems where the prior (i.e., before search)probability distr ibution fo r target location is normal and the detectionfunction b (see section 1) is exponential. Many search problems wh!choccur in practice are of this form.

Definitions

Let X denote the plane, and let A be some arbitrary region ofinterest. Then the prior probability that th e target's location X isin A is given by

Pr [z E A] - fA p(x)dx

where

p(X) = 72 exp (-[x[2/202)

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is the circular normal density function, with mean at the origin andvariance a2 in both oordinate directions. The distance of x fromthe origin is denoted by 1xl.

Let F be the class of non-negative functions defined on X withfinite integral. By definition, this is the class of search allocationfunctions, and fo r feF

fA f(x) dx

is the amount of search effort placed in region A.

We will assume that the unit ef search effort is "time" and thusthe "cost" C associated with the search is the total amount of timeconsumed. In this case c(x,z) - z, and the cost functional C isdefined by

C[f] - JX c(x, f(x)) dx - fX f(x) dx

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The measure of effectiveness will be probability of detection.Hence, the "effectiveness functional" D is assumed to have the form

D[fJ - fX b(x, f(x)) p(x) dx (2)

where b is the detection function. As mentioned earlier, the detec-tion function is assumed to be exponential, and hence, for R > 0

b(x,z) - I - exp (-Rz) . (3)

The coefficient R is called the "sweep rate" and measures the rate atwhich search is carried out (see, e.g., [24]).

In order to understand (2), suppose fot the moment that searchallocation function f is constant over A, zero outside of A, andcorresponds to a finite amount of search time T. Then for xEA

f(x) - T/area (A)

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since by definition

T - f X f (z) dx = f (zo) f,. dx - f (xO) area (A)

for any xOEA.

The detection functional cnn then be written

D[f) - fX b(z, f(x)) p(x) dx

' fA fl - exp(-RT/area (A))j p(x) dx

11 exp (-RT/area (A))j fA p(x) dx

11 exp (-RT/area (A))l Pr[ICA]

This is the so-called "random-search formula."


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Sufficient Conditions for Otimal Search Plans

Using Lagrange multipl'.ers in th e fashion introduced by Everett[13), one can find sufficient conditions fo r optimal search plans fotstationary targets that provide efficient methods for computing theseplans. Define th e pointwise Lagrangian I as follows:

(x, z, X) - p(x)b(x, z) -Ac (x, z) fo r all xEX, (4)

z>0andX>O0

If we have an allocation f*x which maximizes the pointwise Lagrangianfo r some value of X > 0, i.e.,

X(x, f*X(x),X) > k(x,z,X) fo r all xrX and z >.0 (5)

then we can show that D[f*X] is optimal fo r its cost C[f*Xl, i.e.,

D[f*Xl] > D[f] fo r any feF sucl that C[f] < C[f*X] . (6)

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Equation (6) says that plan f*x maximizes the detection probabili tyover all plans using effort C[f*XJ or less.

To show that (6) is true, we suppose feF and C[f] < C[f*X].Since f(x) > 0, (5) yields

p(x)b(z,f*,(x))-Ac(x,f*X(x)) > p(rx)b(x,f(x))-Xc(xf(x)) fo r xnX (7)

Integrating both sides of (7) over X, we obtain

D[f*Xl - XC[f*g] > D[f] - AC[f]

which, along with X > 0 and C[f] < C[f*Xl, implies

D[f*X] - D[f] > X (C[f*X] - C[f]) > 0 .

This proves (6) and hence that f*X is an optimal search plan for itscost C[f*xI.

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The wa y to calculate optimal search plans is to choose a X > 0and find f*X to maximize the pointwise Lagrangian for this X. Foreach xcX, finding fX(x) is a one-dimensional optimization problem.If the detection function is well-behaved (e.g., exponential), then onecan solve fo r f*X analytically. Since Cif*X] is usually a decreas-ing function of X, one can use a computer to perform a binary searchto find the value of A which yields cost C[f*X] equal to the amountof search effort available. The resulting f*X is the optimal searchplan.

Exampl.e

In the case of bivariate normal target distribution (1) and expo-nential detection function (3), we can compute C[f*XI and theoptimal plan explicitly (see [23]). The result is that for T amountof search time, the optimal allocation function f* (dropping thesubscript X which depends on T) is given by

I (r02 - I x ), I rx 0, otherwise,

where all search is confined to a disk of radius ro defined by

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r2 =o -RT/w

The probability of detection corresponding to f* is

r2D~f*J -1I -( + o ) exp (-r2o 2o2)

and the expected time to detection F is

T 6 w 02/ R

Thus, for a given amount of search effort T, the optimal searchplan concentrates search in the disk of radius ro which then expandsas more effort becomes available. It can be shown that the optimalsearch plan can be approximated by a succession of expanding and over-lapping coverages. 1he fact that search is repeated in the high proba-bility areas is typical of optimal search in a great many situations.

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Note that probability of detection depends upor. the ratio ro/OanJ increases to one as search t -fort Increases without bound. Theexpected time to detection is finite and varies directly with a2 andinversely with the sweep rate R.

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REFERENCES

[1] Axhlswede, R. and Wegener, I. (1979). Suchprobleme. Tuebner,Germany.

[2] Allais, I. M. (1957). Manag..Sci. 3, 285-347.[31 Alpern, S. (1974). Differential Games and Control Theory.181-200.[4] Arkin, V. I. (1964). Theory of Probability and Appl. 9, 674-680.(5] Barker, W. H. II (1977). Operat. Res. 25, 304-314.[61 Brown, S. S. (1980). Operat. Res. 28 , 1275-1289.[7, Chew, M. C. (1967). Ann. Math. Statist. 38, 494-502.

[8] Corwin, T. L. (1981). J. Appl. Prob. 18, 167-180.191 Discenza, J. H. and Stone, L. D. (1980). Operat. Res. 29,309-323.[101 DeGuenin, J. (1961). Operat. Res. 9, 1-7.[11] Dobbie, J. M. (1963). Naval Res. Logist. Quart. 4, 323-334.[12] Dobbie, J. M. (1974). Operat._Res. 22, 79-92.[13] Everett, H. (1963). Operat. Res. I1, 399-417.[14] Gal, S. (1972). Israel J. Math. 12, 32-45.[151 Haley, K. B. and Stone, L. D. (1980). Search Theory and Applica-

tions. Plenum Press, New York.[16] Hellman, 0. B. (1970). Ann. Math. Statist. 41 , 1717-1724.[17] Hoffman, G. (1983). Behav. Ecol. Sociobol. 13, 81-92.[18] Hoffman, G. (1983). Behav. Ecol. Sociobol. 13, 93-106.[19] Isaacs, R. (1965). Differential Games. Wiley, New York.

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120] Kadane, J. B. (1968). J. Math. Anal. Appl. 22, 156-171.[211 Kadane, J. B. (1971). Oerat. Res. 19, 894-904.[22] Kadane, J. B. and Stone, L. D. (1981)..Operat. Res. 29,

1154-1166.[23] Koopman, B. 0. (1957). Operat. Res. 5, 613-626.[24] Koopman, B. 0. (1980). Search and Screening, General Principles

and Historical Applications. Pergammon Press, New York.[25] L3soner, U. and Wegener, I. (1982). Operat. Res. 7, 426-439.[261 Mangel, M. (1982). Operat. Res. 30, 216-222.[27] Matula, D. (1964). tAmer. Math. Monthly. 71 , 15-21.[28] McCabe, B. J. (1974). J. Appl. Prob. II, 86-93.[29] Morse, P. M. (1977). In at the Beginnings: A Physicist's Lite.

MIT Press, Cambridge.[30] Morse, P. M. and Kimball, G. E. (1946). Operations EvaluationGroup Report 54. Ceater for Naval Aualyses.[31] Newman, J.R . (1956). The World of Mathematics. Simon & Schuster,

New York.[32] Ozan, T. 1., Chang-Tong Ng, and Saatcioglu, 0. (1976). Int. J.

Prod. Res. 14, 85-98.[331 Pollock, S. M. (1970). _perat. Res. 18, 883-903.[34] Posner, E. C. (1963). IEEE Trans. Inf. Theory. IT-9, 157-168.[35] Pursiheimo, U. (1977). SIAM J. of Appl. Math. 32 , 105-114.136] Richardson, H. R. and Stone, L. D. (1970). Naral Res. Logist.

Quart. 18, 141-157.[371 Richardsor, H. R. and Belkin, B. (1972). Operat. Res. 20,764-784.

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(38] Richardson, H. R. (1973). ASW Information Processing and OptimalSurveillance in a False Target Environment. Daniel H. Wagner,Associates, Report to the Office of Naval Research (AD A002254).

[j9j Richardson, H. R. and Discenza, J. H. (1980). Naval Res. Logist.Quart. 27, 659-680.(40] Stewart, T. J. (1979). Comp. Operat. Res. 6, 129-140.(41] Stine, L. D. (1975). Theory of Optimal Search. Academic Press,

New York.[42] Scone, L. D. and Richardson, H. R. (1974). SIAM J. Appl. Math.

27, 239-255.[43] Stone, L. D. (1977). SIAM J. of Appl. Math. 33, 456-468.[44] Stone, L. D. (1979). Math. Operat. Res. 4, 431-440.[45] Stone, L. D. (1982). Operat. Res. 31 , 207-233.[46] Stromquist, W. R. and Stone, L. D. (1981). Math. Operat. Res. 6,

518-529.14/J Taff- L. Ge (,194). J, Icarue,[48] Tatsuno K. and Kisi, T. (1982). J. Operat. Res. Soc. 23, 363-j75.[49] Tierney, L. and Kadane, J. B. (1983). Operat. Res. 31, 72-738.[50] Wagner, D. H. (1969). SIAM Rev. 11, 52-65.[51] Washburn, A. R.(1980). Naval Res. Logist. Quart. 27, 315-322.[52] Washburn, A. R. (1981). Search and Detection. Military Applica--

tions Section, ORSA, Arlington, Virginia.[53] Washburn, A. R. (1981). Operat. Res. 29, 1227-1230.154] Wegener, 1. (1980). Math. Operat. Res. 5, 373-80.[55] Wegener, I. (1981). Naval Res. Logist. Quart. 28, 533-543.j56] Wegener, I. (1982). Naval Res. Logist. Ouart. 29 , 203-212.

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CNA PROFESSIONAL PAPER INDEX'PP 4072 PP 427

Laird, Robbin F. The French Strategic Dilemma, 22 pp., Cavalluzzo, Linda C. OpTetnpo and Training Effectveness,Nov 1984 19 pp., Dec 1984. (Presented at the Symposium on TrainingEffectiveness, NATO Defense Research Group, Brussels,

PP415 7-9 January 1986)Mizrahi, Maurice M. Can Authortative Studies Be Trusted?2 pp., Jun 1984 PP 428

Matthes, Greg, Cdr., USN and Evanovich, Pester Force Levela,PP 416 Readines6, and Capability,24 pp., Nov 1D84. IFresented at the

Jondrow, James M.. and Levy. Robert A. The Displacementof ORSA-TIMS 26-28 November Meeting, Wrahington, D.C.)Local Spending for Polluton Control by Federal ConstructionGrants, 6 pp., Jun 1984 (Reprinted from American Economic PP 429Review, May 1984) Perla, Peter P. and Barrett, Raymond T. LCdr.. USN.

Wargaming and Its Uses, 13 pp., Nov 1984. (Published in thePP 418 Naval War College Review, XXXVIII, No. 5 /Sequence 311,

Reslock, Patricia A. The Care and Feeding of Magnetic Tapes, Septamber.October 198517 pp..Jul 1984 PP 4OPP 420 Goldberg, Matthew S. The Relationship Between Mater,alWeiss, Kenneth G. The War for the Falklands: A Chronorogy. Failures And FlightHours: StatisicalCon.stderations,18 pp.,32 pp., Aug 1982 Jan 1985

PP 422 PP 431Quester, Aline, and Marcus, Alan. An Evaluation of The McConnell, James M. A PossibleChange in Soviet Views on theEffectiveness of Classroom ana On the Job Training, 35 pp., Prospects fo r Anti-.Submarine Warfare, 1 9 pp., Jar, 1985Dec 1984. (Presented at the Symposium on Training

E........................-.,...s... Hcaarh -u, Brzs.s PP 432'7-9 January 1985) Marcum, Alan J. and Curran, Lawrence E., Cdr., USN. The Useof Flight Simulators in Measuring and Improving Tramtnz,gPP 423 Effecuvenees, 29 pp., Jan 1985 (Presented at the Symposium

Dismukes, N. Bradford, and Weiss, Kenneth G. MARE on Training Effectiveness, NATO Defense Research Group,MOSSO: The Mediterranean Theater, 26 pp., Nov 1984. Bruasehl,7-9Januzry1985)(Presented at the Seapower Conference, Washington, D.C.,26-27 November 1984) PP 433

Quester, Aline 0. and Lockman, Robert F. The All VolunteerPP 424 Force: Outlook for the Eightiesand Nineties, 20 pp.. Mar 1984.Berg, Dr. Robert M.. The CNA Ordnance Programming Model (To be published in Armed Forces andSoctety, 1985)and MethodoaLgy, 27 pp., Oct 1984. (Presented at the ORSA-MAS/MDRS Symposium, Washington, Aug 1984) PP 435

Levine, Daniel B. and Jondrow, James M. Readiness orPP 425 Resources: Which Comes Firt 12 pp., Mar 1985

Horowitz, Stanely A. , and Angier, Bruce N. Costs and Benefitsof Tn'sningand Experience. 18 pp., Jan 1985. (Presented at the PP 436Symposium on Training Effectiveness, NATO Defense Goldborg, Matthew S. Loggt Specification Tests UsingResearch Group, Brussels, 7-9 January 1985) GroupedData, 26 pp., Jan 1985

1. CNA Professional Papers with an AD number may be obtained from the National Technical Irformation Service, U.S. Departmentof Commerce, Springfield, Virginia 22151. Other papers are available from the Management Information Gffice, Center for NavalAnalyses, 4401 Ford Avenue, Alerandria, Virginia 22302-0268. An index of selected publications is also available on request. Theindex includes a listing of profesaional papers, with abstracts, issued from 1969 to December 1983).2. Listings for Professional Papers issued prior to PP 407 can be found in Index of Selected Publications through December 1983),March 1984.


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CNA PROFESSIONAL PAPER INDEX (Continued)PP 43 8 PP 445

Fletcher, Jean W. Supply Problems in the Naval Reserve, Kober, Stanley, Strategic Defense, Deterrence, and Arms14 pp., Feb 1986. (Presented at the Third Anautal Mobilization Control, 23 pp., Aug 1686. (Published in The WashingtonConference. Induatzs College of the Armed Forces, National Quarterly, Winter 1986)Defense University)

PP 440PP 440 Mayberry, Paul W. and Maier, Milton H. , Towards JustifyingBell. Jr., Thomas D. The Center for Naval Analyses Past, Enlistment Standards: Linking Input Characteristics o JobPrgunt, GndFutre, 12pp.,Aug 1985 Performance, 11 pp., Oct 1986. (Paper to be presented at 1986

American Psychological Asociation symposium entitled "SettingPP 441 Standards in Performance Measutrement".)Schneiter, George R. IlmpiCotwns of the Strategic Defense

lIn"ate for the ABM Treaty, 13 pp., Feb 1986. (Published in PP 448Sur.-val,September/Octaber 1985) Cymrot, Donald J., Miliary Reuremtnt and Social Security: A

ComparatiueArialysis, 28 pp.,Oct 1986PP 442

Berg. Robert. Dennis, Richard, and Jondrow, James. Price PP 449Analysis and the Effects of Competition, 23 pp., Sep 1985. Richardson, HenryR.,SearchTheory, 13pp.,Apr1986(Presented at the Association for Public Policy Analysis andManagement-The Annual Rosearch Conference, ShorehamHotel, Washington, D.C., 25 October 1985)

PP 443FitzGerald, Mary C., Marshal Ogarkou on Modern War:1977-1986, 65 pp., Mar 1986

search theory

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