Decision Support for Rule and Technique Discovery in an Uncertain Environment

Transcription

1 Decision Suppot fo Rule and Technique Discovey in an Uncetain Envionment D. James F. Smith III Naval Reseach Laboatoy, Code 5741 Washington, D.C., Telephone: Fax: Abstact- Fo many applications it is desiable to constuct expet systems to be used as decision aids. It is often essential that the algoithm make decisions at least at the quality level of the best human expets accoding to some measue. The algoithm should not only incopoate human expetise and its elated uncetainty, but also be able to exploit an opponent's uncetainty. In many instances, due to newly emeging paadigms, the necessay human expetise does not exist. Appoaches based on fuzzy logic, fuzzy numbe theoy, evolutionay algoithms, game theoy, and fuzzy linea and nonlinea pogamming have been developed that allow the automatic invesion of ules fom data bases. The ules discoveed ae shown to be equivalent to ules aleady developed by human expets o in the case, whee no human being has expetise, to be oiginal. Along with the theoetical basis, an application to electonic wafae, and elated expeimental esults ae given. Keywods: knowledge discovey, decision suppot, fuzzy logic, genetic algoithms, expet systems, distibuted AI algoithms 1 Intoduction Each yea new paadigms aise. It is necessay to develop algoithms that allow computes to make decisions without human intevention about events govened by these paadigms. Constuction of these algoithms means being able to discove the necessay ules fo opeation unde these new models. These ules must be obtained fom expets o by othe means if no expetise exists. The ules used to constuct algoithms will have an uncetainty associated with them just as ules obtained though othe means, e.g., inteviewing a human expet when such expetise exists. It is valuable to be able to easily quantify this uncetainty. This allows a measue of confidence in the decision pocess to be assigned. In this pape, a mathematical theoy of isk and ewad is developed that allows the constuction of fitness functions fo genetic algoithm (GA) based ule discovey. This method of ule discovey is especially valuable when woking with new situations whee little human expetise exists focing the use of simulations based on physics. The GA pefoms the seach, using a fitness function that is a function of the fuzzy gades of membeship of the ewads and isks fo making such a decision. A theoy of ewad based on measues of effectiveness (MOE) is advanced, as well as, a theoy of isks based on global measues of the enemy s uncetainty and battlespace geomety. The MOE will be expessed as a fuzzy decision tee. The fuzzy decision tee is ceated fom a fuzzy linguistic desciption [1], i.e., a fomal epesentation of the system in tems of fuzzy if-then ules. This is a flexible stuctue that can be alteed o extended, as change necessitates ove time. The fuzzy linguistic desciption builds composite concepts fom simple basic ideas known as oot concepts, using logical connectives: and, o, etc. The oot concepts ae fuzzy sets with associated fuzzy membeship functions. Simila to the way humans evaluate poblems, physical insights and intuition ae used to detemine the fom of the membeship functions. In the following, two competing goups ae discussed and efeed to as the blue and the ed goups. A membe of a goup is efeed to as an agent. If ed is awae of blue's ules o stategies, then ed can constuct counte-stategies. A heavy eliance on a limited set of ules can poduce pedictable stategies with devastating esults. By maximizing ed's uncetainty associated with a paticula blue decision, ed's effot to select existing o invent new counte-stategies ae less effective. A global measue of ed's uncetainty (GMRU) about blue's decisions can be constucted fom blue's oot concepts. By focing the GMRU to be nea o exceed a cetain theshold, the effectiveness of ed's decisions can 80

2 be significantly educed and with it blue s isk. Note this appoach to isk analysis does not assume ed has full knowledge of blue s oot concepts. The measue of isk quantifies ed s minimum uncetainty if he had full knowledge of blue s oot concepts. Section biefly intoduces the ideas of fuzzy set theoy, fuzzy logic, and fuzzy decision tees. Section 3 intoduces global measues of uncetainty, the fuzzy entopy, and a fitness function fo ule discovey. Section 4 povides a simple ule discovey example fom the theoy of electonic attack (EA). Section 5 gives computational esults fo the example in section 4. Section 6 intoduces techniques based on fuzzy linea and nonlinea pogamming fo augmenting the pocedue of the pevious sections. Section 7 discusses an altenative, but elated technique fo ule and stategy discovey based on game theoy that uses a technique fom section 6. Finally, section 8 finishes with a summay. A bief intoduction to fuzzy sets and logic Fuzzy logic is a mathematical fomalism that attempts to imitate the way humans make decisions. Though the concept of the gade of membeship, fuzzy set theoy and fuzzy logic allow a simple mathematical expession of uncetainty. The decision tee of classical atificial intelligence povides a gaphical epesentation of expetise that is easily adapted by adding o puning limbs. Finally, the fuzzy decision tee, a fuzzy logic extension of this concept, allows easy incopoation of uncetainty as well as a gaphical codification of expetise. This section will develop the basic concepts of fuzzy sets and fuzzy logic. Fuzzy decision tees ae intoduced and a MOE tee povided as an example.. Fuzzy decision tees The paticula appoach to fuzzy logic used hee is the fuzzy decision tee. The fuzzy decision tee is an extension of the classical atificial intelligence concept of decision tees. The nodes of the tee of degee one, the leaf nodes, ae labeled with what ae efeed to as oot concepts. Nodes of degee geate than unity ae labeled with composite concepts, i.e., concepts constucted fom the oot concepts using and, o, not, bounded sums, etc. [1,3-5]. Each oot concept has a fuzzy membeship function assigned to it. The membeship functions fo composite concepts ae constucted fom those assigned to the oot concepts using fuzzy logic connectives and modifies. Figue 1 povides an example of a fuzzy decision tee. This is a subtee of a fuzzy decision tee used to detemine the measue of effectiveness fo a blue technique allocation poblem. The MOE tee and the technique allocation poblem ae discussed in moe detail in section 4. In Figue 1, the oot concepts ae "false taget" (FT) and "disupt. The composite concepts ae "deny" and "MOE". A vetex with a line with a diamond on it epesents a bounded sum; and a vetex without a line, the logical connective "o". DISRUPT DENY FT MOE FT DISRUPT.1 Fuzzy set theoy This subsection povides a basic intoduction to the ideas of fuzzy set theoy. Fuzzy set theoy allows an object to have patial membeship in moe than one set. It does this though the intoduction of a function known as the membeship function, which maps fom the complete set of objects X into a set known as membeship space. Moe fomally, the definition of a fuzzy set [] is If X is a collection of objects denoted geneically by x then a fuzzy set A in X is a set of odeed pais: A = {( x, µ A( x)) x X} (1) µ A ( x) is called the membeship function o gade of membeship (also degee of compatibility o degee of tuth) of x in A which maps X to the membeship space M. Figue 1. Fuzzy decision tee epesenting EA technique measue of effectiveness 3 Using fuzzy measues of uncetainty in fitness functions Fo each of the oot concepts in a fuzzy decision tee, thee is an associated fuzzy set and an associated membeship function. The membeship functions associated with oot concepts can be combined to give fuzzy membeship functions fo composite concepts, global measues of uncetainty o measues of effectiveness. 81

3 3.1 Uncetainty measue fo the fuzzy decision tee A fuzzy decision tee can povide vaious types of quantitative infomation. By pogessing up the tee, oot concept membeship functions can be combined using logical connectives and modifies to give a fuzzy gade of membeship fo the uppemost composite concept, i.e., the composite concept at the top of the tee. They can also povide a measue of global uncetainty. This global uncetainty is the fuzzy entopy taken ove the oot concepts of the decision tee. If a given input esults in the fuzzy gade of membeship fo a paticula oot concept being nea unity, while the fuzzy gades of membeship fo all othe oot concepts ae nea zeo, the global uncetainty is nea zeo. Altenatively, if the gades of membeship of all of the oot concepts ae intemediate between zeo and one, i.e., nea 0.5, the global uncetainty can be high. This uncetainty can be intepeted as an uncetainty in ed s peception of blue's techniques being used at a given time. It is to blue's advantage to use combinations of techniques that ae both effective and also have a high GMRU, educing the pobability ed will engage in successful countemeasues. Fo a fuzzy decision tee with m oot concepts A i, i=1,,3,, m the total entopy, h Total, is defined as h ( ) m Total d A i i= 1 Fo the fuzzy decision tee of Figue 1, this educes to with and ( FT ) d ( disupt ) () h Total = d + (3) ( A) H ( A) H ( A ) C d = + (4) H n ( A) = K A ( xi ) ln ( µ A ( xi )) i= 1 µ (5) whee n is the numbe of elements in the suppot of A and K is a positive constant [,6]. 3. Ceating the fitness function The seach is conducted using a standad genetic algoithm [3-5]; theefoe a fitness function must be fomulated to measue the effectiveness of a paticula solution. Continuing with the example fom Figue 1, the output of the fuzzy decision tee is a measue of effectiveness fo the technique combination. Coupling this MOE with its measue of uncetainty, denoted by h Total, can be useful in competitive scenaios. If the measue of uncetainty is lage, and the MOE is maximized, not only will the seach etun an effective solution, but also a spead of techniques. If just the MOE is used as a fitness function, the EA techniques selected may have little uncetainty associated with them, i.e., though a knowledge of physics o othe a pioi infomation ed may be capable of e-constucting blue's MOE tee and detemining which technique combination blue will use. This gives ed oppotunities to constuct counte-techniques to cicumvent the pocedue implemented by blue. A fitness function that allows the MOE, µ E to be maximized while focing the GMRU to be nea o geate than a theshold τ is f = µ p( ). (6a) whee p ( h) = E h Total e e ( h τ ) h0 ( h τ ) h0 + 1 (6b) whee h is the uncetainty, τ is the theshold fo h, and h 0 is a smoothing constant. The expession in (6b) educes to a Heaviside step function in the limit h 0 appoaches zeo. Note (6b) is used instead of a Heaviside step function to allow some uncetainty in the theshold τ. 4 Intoduction to EA with an example decision tee An aea that can benefit fom the GA based seach descibed above is the electonic wafae (EW) community due to the amount of uncetainty associated with vaious solutions and the amount of expetise associated with the poblem. In this pape electonic attack (EA) efes to the active use of electonic techniques to neutalize enemy equipment such as ada [7]. Povided in this section will be a bief intoduction to EA and how vaious EA techniques ae epesented mathematically. This section will conclude with a subsection discussing an initial fuzzy decision tee used as a measue of effectiveness fo EA techniques. 4.1 Objectives of EA techniques EA is defined as actions taken to pevent o educe the enemy s effective use of the electomagnetic spectum. The key featues of EA ae jamming, disupting, and deceiving. The objectives of most EA systems ae to deny the enemy the infomation he seeks, to suound his etun with so much false data that the tue infomation cannot be extacted, o to supply so much false data that the infomation handling capacity of the victim system is swamped [7]. These objectives povide the infomation necessay fo the constuction of a fuzzy MOE. It will 8

4 pove useful to moe pecisely specify what an EA technique is. 4. Mathematical model of EA techniques An EA technique, i.e., a method of jamming an enemy senso o ceating a false taget is oughly detemined by the bandwidth (BW), delay time, duty cycle, and powe of the electomagnetic waves used to implement the technique. This model, although appoximate, is effective enough to epesent a paticula EA technique. It also povides a simple method of epesenting chomosomes fo a GA. 4.3 An example decision tee Fom the objectives given in section 4.1, ules can be witten down and used to epesent the oveall effectiveness of a paticula EA technique. Two main ules that can be seen immediately elate to the ability of a technique to "disupt" an opponent and the ability to intoduce "false tagets" (FT) [7] o othe eoneous data into an opponent s system. Thee is also a platfom s capability to "deny" infomation to the opponent, which is typically accomplished by a combination of disuptive and false data techniques. The fuzzy gade of membeship fo the oot concept "disupt" is assigned by fist defining a measue of how well a ed ada opeato can see the blue platfom on his ada sceen. This measue is the fuzzy gade of membeship fo the concept "sees-a-taget." A measue is defined to detemine how "isolated" the pixels of maximum amplitude ae on the ed ada sceen. The fuzzy gade of membeship of the oot concept "disupt" is defined to be the degee to which the point of maximum amplitude unde the measue "sees-a-taget" is "not isolated." The concept "not isolated" is the fuzzy negation of isolated. A blue platfom is said to have ceated a false taget though EA techniques if it can make a taget appea on the ed ada opeato's sceen that does not coespond to a eal taget. Obviously, it is desiable that the false taget be some distance away fom any eal blue taget. The fuzzy gade of membeship fo the oot concept "FT" is geneally defined as a function of the distance between the eal taget and the false taget. The concept of deny epesents the ability of the EA technique to deny ange infomation to a ed platfom. The oot concepts "disupt" and "FT" ae combined using a bounded sum [1,]. This is denoted in Figue 1 by a line with a diamond on it. These thee concepts ae connected togethe, using the fuzzy "o" opeato as shown in Figue 1. Togethe these concepts fom the fuzzy decision tee that epesents the measue of effectiveness fo an EA technique. The global uncetainty, i.e., ed's uncetainty about the techniques that make up blue's solution is given by (3). 5 Expeimental esults fom an example decision tee The fist tests conducted on the decision tee descibed in the pevious section wee meant to be contol scenaios whee some fom of tuth could be attained fom the EA liteatue [7]. Fom these tests, some measue of validity fo the MOE could be established. Thee wee also compaison tests conducted to see the effect of the uncetainty measue on the spead of techniques fo multiple platfoms. It was hypothesized that the GRMU would make it moe difficult fo ed to distinguish techniques fom etuns elated to othe souces. Subsection 5.1 discusses an EA escot mission and the associated scenaio tested. Subsection 5. descibes EA technique solutions obtained using the uncetainty measue. Examples ae pesented with and without the uncetainty measue, so that the effect of the uncetainty measue can be established. 5.1 The EA escot mission and esults of scenaio based testing The escot mission poblem consists of a goup of platfoms within the lethal ange of a paticula electonic suveillance device, while anothe platfom(s) lies just outside the egion being scanned by the device at a paticula time. As time passes, since the platfom is within lethal ange, eventually the goups of platfoms will exchange oles and thus exchange techniques. In this paticula mission, both goups emit a low intensity signal, the goup within the scanning ange usually poduces some sot of false data, while the goup outside the scanning ange emits a noise signatue [7]. Roles ae patially detemined by EA stategy and by the physics constaining the eception of the signal by the suveillance device. In the test case fo this mission type, two platfoms wee placed within the detection ange of a paticula hostile ada system. Both platfoms wee placed at simila anges fom the ada, with the fist platfom within the main scanning beam of the ada, and the othe slightly off beaing of the scanning beam. The esulting technique solution yields signal amplitude maxima at anges of two,10, and 13 nautical miles (nmi) that ae indicative of false tagets". In this paticula test, the fist platfom executed a false taget technique, while the second platfom executed what is known as a spot noise technique [7]. In this example, the solution selected by the GA seach was consistent with a known solution found in the open liteatue [7]. 83

5 5. Effect of uncetainty on technique selection In section 5.1, the fitness of each technique was detemined using only the MOE tee shown in Figue 1. This section will show expeimental esults taken fom data in which the fitness of the technique was calculated using the fomula in equation (6a). As stated peviously, incopoating uncetainty ceates technique sets that ed would find difficult to penetate with countemeasues. When the fitness function of (6a) that incopoates uncetainty was applied to the escot mission scenaio of section 5.1 then a technique solution consisting of six false tagets with spot noise was found. As shown in Figue, the FT s appea at two, fou, 18, 34, 40, and 46 nmi, hence they exhibit geate vaiation in spacing, while at the same time being distibuted ove a geate ange than the example of section 5.1. The moe complex spacing, 44 nmi spead fom fist to last FT, and geate numbe of false tagets all seve to confuse an enemy ada opeato. The fitness function helped the GA to find a solution that was highly uncetain, i.e., confusing fo the ed ada opeato. The geate the ed ada opeato s confusion the less likely a missile will be fied, hence the fitness function with the uncetainty constaint, (6a), has poduced a bette technique solution than the one without the uncetainty constaint. nomalized powe etun (mw) distance fom platfom (nmi) Figue. Multiple false tagets esponse to escot mission scenaio The measue of global uncetainty built into the fitness function in (6a) has inceased ed's uncetainty to the point, he may not fie a missile, o if he does the pobability that he fies at a eal taget is vey low. Thus, ed's effectiveness has been geatly educed by his inceased uncetainty. By unning the GA many times, technique sets ae found that have simila levels of effectiveness and high GMRU's. By selecting diffeent multi-platfom technique sets fo the blue foce at diffeent times ed's effectiveness in implementing o inventing counte-measues is significantly educed. The pocedues used hee can be extended to n esponding platfoms. As the opponent scans ove the diffeent platfoms, the uncetainty measue will esult in the platfoms making up the blue goup making technique selections that change in time and as such ae hade fo ed to pedict. The ability of this GA based pocedue to einvent known techniques is inteesting. The tue value of this algoithm esides in its ability to ceate new techniques, ules and stategies fo multi-platfom engagements fo cases in which no human expetise exists. The pocedue descibed above poduces ules, and associated multi-platfom technique sets that ae effective accoding to a MOE and difficult to pedict o intepet by the enemy as measued by the GMRU. Howeve, this ule selection pocedue does not fully define the stategy fo using each ule since it does not povide a pobability assignment fo the ule s use at a given time. One can always make assignments of pobabilities to define stategies. Fo example an appoach analogous to the pinciple of indiffeence [8] could be assumed, i.e., the pobability fo using each ule could be taken as the same ove the entie collection of ules. This would pove to be unsatisfactoy fo a lage collection of technique sets. The technique set descibed in this section should have a much highe pobability of use than some technique combinations that aise fom this pocedue. This can be emedied by using game theoy as descibed in section 7 to calculate the pobabilities necessay fo using each ule so that blue can play the optimal game with the highest expected payoff. 6 Minimum EA powe allocation in the face of uncetainty In the pevious section a genetic algoithm detemined all technique paametes including powe level. Uncetainty was exploited using infomation theoetic entopies. This section pesents two powe allocation algoithms that can be used in conjunction with the GA aleady discussed to poduce a significant eduction in seach time, with the game theoy appoach of section 7, o as stand-alone eal-time algoithms. The two powe allocation algoithm pesented in this section will allocate EA powe levels fo a collection n blue netwoked agents engaged in conflict with a single ed agent. The esouce manage (RM) itself assigns the n blue agents to deal with the one ed agent, so effectively the RM can deal with m ed agents vs. n blue agents. The powe allocation algoithms equie the assignment of a lage numbe of paametes associated with each blue agent and each blue agent's best estimate of the ed agents paametes. These paametes can be a significant souce of uncetainty. Fo easons detailed below, it is desiable to use no moe powe than is necessay, but uncetainty must be taken into 84

6 account so sufficient powe is made available. Fo the analysis consideed below thee will be one blue platfom in the main beam of a ed victim ada and n-1 blue platfoms in the sidelobes of the ed ada. The single blue platfom in the main lobe of the ed ada will be efeed to as a self-sceening agent. Quantities elated to a self-sceening agent will cay ss subscipts. The othe n-1 blue platfoms will be efeed to as a suppot agents, with elated mathematical quantities caying a su subscipt. The moe geneal poblem, powe allocation fo n blue platfoms vesus m ed platfoms will be consideed in a futue publication. 6.1 Fuzzy linea and nonlinea pogamming appoaches to powe allocation Intinsic to EA technique allocation is detemining the amount of powe each blue platfom should diect at a ed platfom. It is desiable to use as little powe as is possible while taking into account the uncetainties undelying the conflict. If too much powe is used against one ed platfom thee might not be sufficient powe to use against anothe ed platfom. Excessive use of powe by blue may allow an enemy to home in on blue s position possibly without use of othe sensos. Also, excessive powe usage duing application of EA techniques may intefee with othe systems. Finally, if powe is used in an intelligent fashion ed may misintepet blue's EA activity as a natual phenomena o a system failue giving blue an additional advantage. It can be shown easily using the techniques of [9] that the following system of inequalities (7) and (8) goven the powe allocation poblem Π Y 1 (7) whee Π Π min, j and Πmax, j min, j Π j Π max,j (8) ae the minimum and maximum values of the j th component fo fo j= 0 to n-1. Let the powe vecto be defined as Π ρ ρ, K ρ (9) with ρ 1,ss, the powe output fo the self-sceening blue agent and ρ i,su, the powe output fo the i th suppot blue agent fo i=1 to n-1. The components of the vecto Y in (7) and defined in (10-13) consist of many factos epesenting the paametes govening the opeation of the senso and EA systems on the blue platfoms and the ed ada. The paametes chaacteizing the ed ada ae confined to the facto c R, defined in (13). Y λ G j G ( (10) 4 π L j B j LaRssoc ss R o ) ( ) 1, ss, 1, su, n 1, su λ G j G, ji Yi ( ) (11) 4 π L j B j La, i Ri cr c R Pτ 3 ( 4π) R 4 ss pw 1 su, i Y ( Yo, Y1, K, Yn ) f τg q T σλ L D ( n, P fa, P The subscipts "ss" and " su,i" in (10,11) efe to the selfsceening agent and the i th suppot agent. The appeaance of these subscipts on a backet is a shothand implying that each of the quantities in the backet ae labeled with the elevant subscipt. The paametes appeaing in equations (10-13) ae L j, the loss, excluding atmospheic, fo the espective agent; B j, the bandwidth of the espective agent; G, the ada antenna gain in the diection of the ssagent; L a, the atmospheic loss fo the ss-agent; R sso, the ange of the ss-agent; G j, the agent antenna gain in the diection of the ada; G,ji, the ada antenna gain in the diection of the i th suppot agent; La,i, the atmospheic loss fo the i th suppot agent; R i, the ange of the i th suppot agent; τ pw, the pulse width of the victim ada; f, PRF of the victim ada; τ, the time on taget of the victim ada; L 1, the loss facto of the victim ada; σ, the coss section of self-sceening agent; P, the peak powe of the victim ada; G T, the gain of the ada tansmitte; P D, the minimum pobability of detection equied by the victim ada to detect blue; P fa, the pobability of false alam fo the victim ada; n, the numbe of pulses integated by the victim ada; q, the numbe of the Sweling model [9] appopiate to how victim the ada "sees" blue; D q (n, P fa, P D ), the detectability facto [9]; T s, the themal noise tempeatue of the ada; λ, the ed ada's wavelength; and k, Boltzman's constant. All anges ae measued with espect to the ed victim ada. To detemine the appopiate powe level each blue beam of each blue platfom must use, the blue metaagent powe output must be elated to the minimum estimated powe eceived by the ed agent that is equied fo a paticula type of detection. Blue may want ed to conclude that he is only seeing noise bon of some natual pocess. The powe deliveed to ed should pobably only be lage than a cetain minimum detection theshold. If blue desies that ed detect ultimately what is a false taget then blue's powe output should be much lage. Inequalities (7) and (8) specify the constaints on powe. The set of powe vectos that satisfy each inequality defines a closed half space in R n, the space of n- tuples of eal numbes. The intesection of the closed halfspaces defines a bound polyhedal convex set. To extact a unique solution a linea function in powe f ( Π) = µ Π (14) D kt ) S (1) (13) 85

7 is defined on and minimized ove the polyhedal convex set specified by ( 7-8) The elements of the powe coefficient vecto µ K in (14) ae fuzzy gades of membeship eflecting the impotance conseving the beam s powe. Minimization of a linea function ove a polyhedal convex set defines a linea pogamming poblem. Since the polyhedal convex set is bound, the Fundamental Exteme Point Theoem [10] guaantees that the minimum of the linea function (14) occus at a cone point. A simplex algoithm is used to apidly seach the cone points fo the coect solution. The physical intepetation of the linea function in (14) is that it is the total weighted powe sum of the individual blue beams with the weights eflecting the impotance of conseving the beam s powe. It is of couse useful to minimize powe as discussed above. Although physical, the use of a linea function is abitay, it is made to obtain an unique solution. Thee is nothing to exclude the selection of a nonlinea function as shown below. A natual nonlinea function in powe to minimize is f ( Π) The function in (16) can be minimized ove the polyhedal convex set defined by (7-8) though a staightfowad application of the Cauchy-Bunykovskii- Swatz Theoem [11] yielding the following powe vecto components as a solution to the nonlinea minimization poblem whee it is assumed that ( µ o, µ ) =, n 1 n 1 j= 0 ( Π / Π ) j max, j 1( ) Π min, j =0 fo j= 0 to n-1. (15) n j Y j max,j Π = Π Yk Π (17) max, k k = 0 Note the solution given by (17) is closed fom and exact up to model assumptions. Futhemoe, it is computationally much faste than the linea pogamming appoach and uses vey little memoy. The linea pogamming appoach has the advantage it is much easie genealized to moe complicated scenaios. It should be noted that both powe allocation algoithms allow the detemination of powe levels each blue agent uses without a fixed cental commande, i.e., they contibute to the selfoganizing aspect of the RM. The linea pogamming appoach can be easily extended to an existing fuzzy linea pogamming appoach, e.g., that of Negoita [1]. This allows immediate incopoation of uncetainty. Also, the linea (16) pogamming o fuzzy linea pogamming appoaches can be easily extended to handle n blue platfoms vs. m ed platfoms. This is not necessay fo the RM, which aleady assigns a numbe of the available blue platfoms to each ed agent. It does howeve povide an altenate appoach allowing compaison and simulation. The nonlinea appoach can also be extended to the n blue agents vs. m ed agent poblem, but the extension is less staightfowad. In both the linea and nonlinea pogamming appoaches, fuzzy numbes [1] ae used to quantify the uncetainty in the paametes of the ight-hand-side of expession (10,11,13). This has poven vey effective [13]. When the uncetainty in the paametes in (10,11,13) is small, the fuzzy linea pogamming appoach is significantly bette than the fuzzy nonlinea appoach. 7 Game theoetic appoach to ule and stategy discovey A global MOE like the one descibed above is being investigated as a component of anothe ule and stategy discovey pocedue. The diffeent combinations of blue techniques and ed esponses can be enumeated in a physical simulation envionment though a seach algoithm, genetic algoithm based o othewise. This allows cisp if-then statements to be witten. Each if-then statement can have a MOE value assigned to it. The MOE gives a measue of the ewad obtained by using multiplatfom EA techniques. When a genetic algoithm is used the MOE can be used as a fitness function. In section 3, the fuzzy entopy was used as a measue of the isk associated with using multi-platfom EA techniques. This isk aises fom the ability of an opposing agent to extact enough infomation fom blue s attempts at infomation denial to get a good estimate of blue s position. The game-theoetic appoach uses a moe sophisticated appoach to isk, based on expetise fomulated as a fuzzy decision tee. The isk fuzzy decision tee will be the subject of a futue publication. A payoff matix element is defined as a linea combination of the isk and ewad measues fo blue and ed. The payoff matix elements ae intepeted as foming a payoff matix in the sense of game theoy [10]. The game theoy poblem can be solved to give pobabilities fo optimally playing the game. Ove a long peiod of time by playing the optimal game, blue can anticipate the highest expected payoff. The outcome of such a pocess yields cisp if-then statements and associated pobabilities fo each set of blue platfom techniques and ed platfom esponses. The esulting data base of multi-platfom ifthen ules and associated pobabilities can then be data mined using a genetic algoithm to ceate a fuzzy linguistic desciption. This appoach is a cuent topic of eseach and is yielding pomising esults. 86

8 When game theoy is applied to a scenaio like that descibed in section 5.1 and 5., many diffeent combinations of EA techniques distibuted ove the blue platfoms give the same value fo the MOE. By incopoating geate sophistication into the isk calculation, game theoy detemines that the optimal stategies involve blue using pue noise appoximately 18% of the time and noise combined with multiple closely spaced false tagets 8% of the time. Fo this calculation, the fuzzy linea pogamming vesion of powe allocation was used. It should be emphasized that the esults ae specific to the type of missile assumed when doing the ed isk and ewad calculations. 8 Summay A method fo incopoating uncetainty into a seach algoithm, employing a genetic algoithm has been discussed and is cuently being tested. The measue of effectiveness fo a paticula solution is calculated using a fuzzy decision tee and the uncetainty associated with that decision is calculated using a fuzzy entopy that is a function of the oot concepts fo that decision tee. The method has been applied to the EA banch of the electonic wafae field. Expeimental esults have shown this method to be effective and consistent with the known EA esponse techniques found in the open liteatue. An augmented appoach based on fuzzy logic, genetic algoithms and game theoy fo automatic stategy discovey is discussed. Two new eal-time algoithms fo allocating EA powe distibuted ove n blue agents ae intoduced. Uncetain paametes ae taken into account using fuzzy numbe theoy. The powe allocation algoithms can be used in combination with the genetic algoithm based pocedues fo technique discovey, with the game theoy appoach, o as stand-alone powe allocation algoithms called fom the RM. The fuzzy linea pogamming based powe allocation algoithm is used in conjunction with game theoy to discove pobabilities chaacteizing stategies fo EA technique use. Examples ae povided. Finally, the eal value of the genetic algoithm and game theoy based knowledge discovey algoithms is thei ability to detemine new multi-agent techniques, ules, and stategies fo paadigms fo which no human expetise exists. 9 Acknowledgements This wok was sponsoed by the Office of Naval Reseach. The authos would also like to acknowledge M. Robet D. Rhyne II, M. Edwad Khouy, M. Robet Xande, D. Joseph Lawence III, and Ms. Kistin Fishe. 10 Refeences [1] L.H. Tsoukalas and R.E. Uhig, Fuzzy and Neual Appoaches in Engineeing, Chapte 5, John Wiley and Sons, New Yok, 1997 [] H. J. Zimmeman, Fuzzy Set Theoy and its Applications, Chaptes 1; 4, Kluwe Academic Publishes Goup, Boston, [3] J.F. Smith, III and R. Rhyne, II, A Resouce Manage fo Distibuted Resouces: Fuzzy Decision Tees and Genetic Optimization, Poceeding of the Intenational Confeence on Atificial Intelligence, IC-AI 99, Las Vegas, Nevada, USA, June 8-July 1, 1999, pp , CSREA Pess. [4] James F. Smith III and Robet D. Rhyne II, A Fuzzy Logic Resouce Manage and Undelying Data Mining Techniques, Fusion000: Poceedings of the 3 d Intenational Confeence on Infomation Fusion, R. Reynaud, Pais, Fance, July 10-13, 000, Vol. II, pp. WEB1-3 WEB1-9, Intenational Society of Infomation Fusion. [5] J.F. Smith, III and R. Rhyne, II, Genetic Algoithm Based Optimization of a Fuzzy Logic Resouce Manage: Data Mining and Co-evolution, Poceeding of the Intenational Confeence on Atificial Intelligence, IC- AI 000, H. Aabnia, Vol. I, pp 41-48, Las Vegas, Nevada, USA, June 000, CSREA Pess. [6] Hans Bandeme and Siegfied Gottwald, Fuzzy Sets, Fuzzy Logic, Fuzzy Methods with Applications, pg. 165, John Wiley and Sons, Chicheste, [7] D. C. Schlehe, Electonic Wafae in the Infomation Age, Chapte 1, pg. 9-16, Atech House, Boston, [8] David L. Hall, Mathematical Techniques of Multisenso Data Fusion, p. 175, Atech House, Boston, 199. [9] D. C. Schlehe, Intoduction to Electonic Wafae, Chapte 6, Atech House, Boston, [10] A. M. Glicksman, An Intoduction to Linea Pogamming and the Theoy of Games, p. 5, Dove Publications, Inc., Mineola, NY, [11] R. G. Batle, The Elements of Real Analysis, p. 56, John Wiley and Sons, New Yok, [1] T. Teano, K. Asai, M. Sugeno, Fuzzy System Theoy and Its Applications, Chapte 7, Academic Pess, Inc., Boston, 199. [13] J.F. Smith, III and R.D. Rhyne, II, Fuzzy Logic Resouce Manage: Multi-Agent Fuzzy Rules, Self- Oganization and Validation, Fusion00: Poceedings of the 5 th Intenational Confeence on Infomation Fusion, Intenational Society of Infomation Fusion, to be published