a Fuzzy Set By a Crisp One 1.1 Fuzzy Methods Usually Result in a Fuzzy Recommendation

On Approximation of Fuzzy Sets by Crisp Sets:From Continuous Control-Oriented Defuzzification To Discrete Decision MakingHung T.Nguyen Witold Pedrycz Vladik KreinovichMathematics Dept.Dept.of Electrical&puter Science New Mexico State University Computer Engineering University of TexasLas Cruces,NM88003,USA University of Alberta El Paso,TX79968,USA hunguyen@ Edmonton T9G2G7Canada vladik@pedrycz@ee.ualberta.caAbstractIn this paper,we show that the necessity to make crisp decisions inuncertain(fuzzy)situations leads to the necessity to“approximate”fuzzy sets by crisp sets.We show that seemingly natural approximationideas–such as using-cut for a given–often do not work,and wedescribe new approximations which not only work,but which are optimalin some reasonable sense.Keywords:fuzzy sets,decision making,“shadowed”fuzzy sets,approximating fuzzy sets by crisp sets1Why There Is a Practical Need To“Ap-proximate”a Fuzzy Set By a Crisp One 1.1Fuzzy Methods Usually Result in a FuzzyRecommendationFuzzy logic and fuzzy set theory enable us to use ex-perts’uncertain(“fuzzy”)knowledge in decision mak-ing.The corresponding methods usually generalize known methods of decision making which are based on the crisp(non-fuzzy)knowledge about the environment. These crisp methods enable us to come up with a crisp decision:e.g.,whether we should build a plant or not. When we generalize these methods to fuzzy knowledge, as a result,we usually get not a crisp decision,but rather a fuzzy decision:e.g.,a decision may be“most probably it is better to build a plant”.Formally,in the simplest(“yes”-“no”)decision situa-tion,a crisp decision procedure means that for every in-put,we decide whether to choose a positive alternative or a negative alternative.Such a crisp decision can be described by a set of all the inputs for which the decision is positive,or,alternatively,by the charac-teristic function of this set,i.e.,by a function for which if and if.Similarly,as a result of a fuzzy decision making pro-cedure,for every input,we generate a degreeto which choosing a positive alternative is reasonable.Thus,a fuzzy decision making procedure produces a fuzzy subset of the set of all inputs,a sub-set which is characterized by the membership function .1.2In Many Practical Application,the FuzzyRecommendation is All We NeedIn many practical situations(e.g.,in medicine),the main objective of the fuzzy decision making procedure is to help a human decision maker(e.g.,a medical doc-tor).The system presents the value to the human decision maker,and she takes this value into account when making a decision.1.3In Automated Decision Making,We Need aCrisp Decision(Which Best Approximatesthe Fuzzy Recommendation)However,in some practical problems,there is no time for a human operator to make a decision: e.g., when we plan a robotic mission to a distant planet,the robot must make urgent decisions for which we cannotwait minutes and hours for the signal to travel to Earth and back.In such situations,we must,for every possible input,either choose a positive alternative,or choose a negative one.In other words,we must end up with a crisp set.This crisp set should reflect–as close as possible–the fuzzy set produced by the fuzzy decision making system.Thus,we face a problem offinding,for every fuzzy set,a crisp set which is,in some reasonable set,the best approximation for.2The Best Approximation Should Be an -CutFor each input,the only information that we have about the quality of a positive alternative is the degree .Thus,whether we select the positive alternative or not depends only on the value of.When,we select the positive alternative ;when,we select the negative alternative .If for some alternative,we decided to select the positive alternative,then for every alternative for which,it is even more reasonable to se-lect.Similarly,if for some,we selected,then for all alternatives with,we should also select.We can describe the corresponding decision making if we use the greatest lower bound of the set of all values for which we decided on: for all inputs for which,we select(i.e.,);andfor all inputs for which,we select(i.e.,).For the values for which,we may have or.In other words,the crisp set corresponding to the characteristic function is an -cut of the original fuzzy set.Thus,the above-described informal approximation problem can be described as follows:given a fuzzy set ,find for which its-cut is,in some reasonable set,“the closest”to the original fuzzy set.3The Simplest Approach–Using the Same For All Fuzzy Sets–And Its DrawbacksIn view of the above reformulation of the problem of automatic decision making as a problem of choosing an-cut it may seem reasonable to just select a certain degree(e.g.,or)and use this same value of for all fuzzy sets.The main drawback of this approach is that it contra-dicts to the intuitive idea that if two fuzzy sets are close,then the corresponding crisp decision sets should also be close.Indeed,for each,let us consider a membership function for which for ,for all,for,and is linear on and on .Then,if we pick a value and select(i.e., )when,then for a sequence of membership functions,we havefor all,but for the limit mem-bership function,we have for all.Thus,for large,the fuzzy sets and are close(as close as possible), but the resulting crisp functions are as far away as pos-sible:for,we have while .Similarly,if we pick a value and select(i.e., )when,then for a sequence of membership functions,we havefor all,but for the limit mem-bership function,we have for all.Thus,for large,the fuzzy sets and are close(as close as possible), but the resulting crisp functions are as far away as pos-sible:for,we have while .In other words:intuitively,for the transition from to, close functions must turn into close ones,i.e.,this transition must be continuous in some reasonable sense,butif wefix the same for all fuzzy sets,we cannot have this continuity.Due to this continuity requirement,we cannot choose the same for all fuzzy sets;we must select based on.4A Similar Problem Occurs In Fuzzy Control,Where It Is Solved By Using DefuzzificationA similar problem of transforming the fuzzy recom-mendation into a crisp decision exists in another appli-cation area of fuzzy logic and fuzzy set theory:namely, in fuzzy ly,in fuzzy control,for every in-put and for every possible control,we generate the degree to which this particular control value is reasonable for a given input.Then,to generate the actual control which the automated fuzzy controller will apply for a given input,we use a defuzzification procedure,e.g.,a centroid defuzzification5Why Cannot We Use Known Defuzzi-fication Procedures In Fuzzy Decision Making?5.1Centroid Defuzzification Cannot Be Usedfor Discrete Decision MakingAtfirst glance,it may seem that we can apply the above formula(1)to decision making as well.This im-pression is,however,erroneous:in control,the set of possible decision is con-tinuous,so averaging described by the formula(1) makes sense;on the other hand,in decision making,we only have two possible decisions:0or1,or;in this case,averaging leads as described by the for-mula(1)leads to a value intermediate between0 and1,and we still face the same problem:how to transform this fuzzy recommendation into a crisp decision.5.2A Similar Problem Occurs in Fuzzy Con-trolThis problem is not exclusive for fuzzy decision mak-ing,it also sometimes occurs in fuzzy control as well. For example,J.Yen and his collaborators have consid-ered a reasonable situation in which a car is going to-wards an obstacle on an empty road[4,5,6].To avoid this obstacle,it should either swerve to the left,or to the right.This swerve can be described by a turning angle .Due to the symmetry of of the situation,the result-ing membership function is symmetric with respect to changing to and,hence,the formula(1)leads to .In other words,the car should go smack into the obstacle.This crisp recommendation makes no sense, so J.Yen and his coauthors described modifications of the formula(1)which enable us to avoid this counterin-tuitive recommendation.5.3In Discrete Decision Making,the ProblemIs Even More Serious Than in Fuzzy Con-trolIn fuzzy control,this problem is rather rare(actually, it was unnoticed for thefirst decade of fuzzy control),so it is OK to either ignore it,or use some hacked ad hoc tools to solve it.In contrast,in fuzzy decision making, this problem is always present,so we better be able to solve it in the best possible manner.6How We Can Solve This ProblemA natural way tofind the best crisp approximation to a fuzzy set is to formalize the notion of“closeness”(“metric”)between two fuzzy sets,and then to select a crisp set which is the closest(in this sense)to the given fuzzy set.What is the natural metric on the set of all fuzzy sets? 7A Natural Metric on the Class of All Fuzzy Sets:First Idea7.1From an Intuitive Idea to a Formula for aDistance Between Two Fuzzy Sets What does the value mean?One way to assign a numerical value to the degree to which has a certain property is to poll several experts;then,for every ,we can define as the ratio, where:is the total number of experts whom we polled, andis the total number of experts who believe that the given value satisfies the given property. To get the entire membership function,we should ask the experts about several values.With this procedure in mind,it is natural to charac-terize the difference between two fuzzy sets and by the total number of experts who disagree on these two sets.For each value,the difference between the values and means that different num-ber of experts believe that the corresponding properties are true for:these numbers areand.Thus,the number of ex-perts who disagree on this value is equal to at least.The total number of disagreements can be thus estimated as the sum over all,i.e.,asThe more values we take,the more accurate the re-sulting description of the membership functions.When the values get close,the sum(2)becomes close to the corresponding integral sum:Thus,the smaller,the closer the left-and right-hand sides.7.2Resulting Formula for a Distance BetweenTwo Fuzzy SetsWe are not interested in the absolute values of the distances,only in which pairs are closer and which are more distant.Thus,for our purposes,it is suf-ficient to multiply all the values by a constant and consider the scaled distance7.3The Approximation Problem Formulatedin Terms of This DistanceFor this distance(3),we can formulate the above problem:given the fuzzy set,find the crisp set (for which for all)for which7.4Solving the Corresponding ApproximationProblemThis optimization problem is easy to solve:namely, the minimized integral is,in effect,the sumIn this sum,we have unknowns, and each term depends only on one of these un-knowns.Thus,the sum takes the smallest possible value when each of these terms takes the smallest possible value,i.e.,when for each,we select,as,the value0or1which is the closest to.One can easily see that when,this closest value is,and when,this closest value is.Thus,as a solution to the above optimization problem,we get an-cut with the same value for all fuzzy sets.7.5Drawbacks of This SolutionWe have already observed that this is not a very intu-itive choice.Thus,if we want to avoid this choice,we cannot use the distance(3),we must use some a sophis-ticated formula.8A Natural Metric on the Class of All Fuzzy Sets:Second Idea8.1New IdeaBy definition,the distance between the two sets isequal to0when these sets are equal,and grows larger and larger as the two sets become different.Thus,thedistance can be viewed as a degree of difference(in-equality)between the two fuzzy sets.In the above text,we tried to directly formalize thisnotion of inequality,and we got a counterintuitive result.Let us now,instead,try to formalize inequality indi-rectly.It is known that the two sets and are equalif and only if and.Correspondingly, the two sets and are different if and only if eitheror.Thus,instead of trying to directly formalize the degree to which two fuzzy sets and are different,let usfirst formalize the degrees to which or,and then use an“or”-operation(e.g., )to combine these two degrees into a degree with which.8.2From Idea to FormulaTo characterize the degree to which,we willuse the same polling idea as ly,when,this means that for every,the number of experts believe that has a property cannot be larger than the number of experts who believe that has the property.Thus,if,then the expert opinions about this value are consistent with the hypothesis that.If, then clearly some experts disagree that.The number of experts who contribute to the belief that the property is violated for a given is equal toThe total number of experts’opinions which contribute to the belief that is therefore equal toSimilarly to the above text,we can conclude that the de-gree to which can be characterized by an integralSimilarly,and therefore,the degree with which two fuzzy sets are different can be characterized as8.3This Formula Can Be SimplifiedWe are interested in the values for the case when is a characteristic function of a crisp set, i.e.,when for every,either or.In this case,both maximized expressions(4a),(4b)in the formula(4c)can be drastically simplified.Let us start with thefirst one.When,i.e.,when,we have,and therefore,.Thus,in comput-ing thefirst integral(4a),it is sufficient to consider only the values,i.e.,the values from the complement to the set.When,i.e.,when,then we have.Therefore,thefirst integral(4a)from the formula(4c) takes the formSimilarly,the second integral(4b)takes the form8.4Resulting Formulation of the Approxima-tion ProblemThus,the above optimization problem takes the fol-lowing form:given a fuzzy set,find an-cut for which8.5Solving the Resulting ApproximationProblemWhen the set increases,thefirst integral in the formula(5)decreases(because its integration domain shrinks),and the second integral increases.So: If thefirst integral is smaller than the second integral,then the value of the maximum is equal to the value of the second integral:.We can decrease a little bit;then:–the value of the second integral slightly de-creases:,while–the value of thefirst integral slightly in-creases,to.For a sufficiently small increase in,we still have .Thus,for the new set,we haveand therefore,.Hence,the minimum in(5)cannot be attained when.Similarly,when,we can slightly increaseand get a new set for which.Hence,the minimum in(5)cannot be attained when.Thus,the minimum is attained when,i.e.,whenComment.This metric wasfirst proposed in[3];in that paper,for simple fuzzy sets,explicit formulas are given for the optimal crisp approximation.9Possibility of“Undecided”9.1PossibilityIn the above text,we assumed that for every input ,we have to make a certain decision:either chose or.In reality,in many practical situations,we can select a third option:claiming that there is not enough information for making a meaningful decision,and fur-ther tests are necessary.9.2Towards a Formalization of This Possibil-ityFor one of such inputs,it may later turn out that it is best to choose,i.e.,that.It may also later turn out that it is best to choose,i.e.,that. It may also turn out that the best strategy is picking with a certain probability.This possibility can be described by saying that is also a possible value of.In other words,for such,the set of possible values of is the entire interval.9.3Formal Definition:the Notion of a“Shad-owed”SetTo take this possibility into consideration,the authors of[3]introduce a new notion of a shadowed set.This set is characterized by a characteristic function which maps every input into either0(meaning that we choose) or1(meaning that we choose),or the interval (meaning that we do not know what to choose).The set of inputs for which is called the shadow of the shadowed set.9.4Describing Distance Between ShadowedFuzzy Sets:First IdeaHow can we describe the distance for such sets?The fact that means that later on,the actual (unknown)value of may turn out to be any value from the interval.Thus,a shadowed set is not a single membership function,but a whole set of mem-bership functions.Normally,a distance between a point and a set is defined as the smallest possible distance be-tween this points and points from this set:Thus,if wefix a formula for the distance between two membership functions(e.g.,the formula(3)),then we can naturally define a distance between a membership function and a shadowed set as the smallest possible distance between and different functions:9.5Minimizing the Corresponding DistanceSimilarly to the above discussion of the formula(3), the minimum is attained when the termis the smallest for every.When,then the minimum of this term is attained when, and this minimum is equal to0.Thus,in computing the expression(6),it is sufficient to take into consideration only the values for which or.For such values,we terms get the following values: when,we have;when,we have.Thus,9.6Solution to the Corresponding Optimiza-tion Problem and Its DrawbacksThis expression makes sense,but its minimization does not:its minimum is attained(and equal to0)when for every,we have,i.e.,when the shadow coincides with the entire universe of discourse.9.7Re-Defining Distance Between ShadowedSets:From a New Idea to a Definition Since our goal is to make a decision,such a super-cautious shadowed set,in which the answer is always“I do not know”,does not make much sense.It therefore makes sense to require that the shadowed set is close to the original fuzzy set and the shadow is not too large. This ideal requirement is not satisfied if either the shad-owed set is not equal to the original fuzzy set or the shadow is large.Thus,if we use to describe“or”, we can take,as a degree to which this ideal is not sat-isfied,the maximum of the degree to which the shadowed set is different from and the sizeof the shadow.In other words,we want tofind a shad-owed set for which9.8Solving the Corresponding OptimizationProblemWhen we increase the shadow,thefirst of the max-imized terms decreases,the second increases,and vice versa.Thus,similarly to the non-shadowed case,we can conclude that the minimum is attained when the two terms are equal,i.e.,whenComment.This metric was alsofirst proposed in[3];in that paper,for simple fuzzy sets,explicit formulas are given for the optimal shadowed approximation.10ConclusionIn decision making problems under fuzzy uncer-tainty,traditional fuzzy technique leads to a fuzzy rec-ommendation like“it is somewhat reasonable to make a decision”.This fuzzy recommendation can be de-scribed by a fuzzy set(membership function),which as-signs,to every possible situation,the degree to which the decision is reasonable in this particular sit-uation.Such a fuzzy representation is of great help to a human decision maker(e.g.,to a medical doctor),but if we want to produce an automatic decision making sys-tem,we must transform the fuzzy recommendationinto a crisp recommendation,which,for every, either recommends to make the decision(), or to make the opposite decision().This crisp recommendation can be therefore described as a crisp set(for which is a characteristic function).We would like to make this crisp recommendation to reflect the original fuzzy recommendation in the best pos-sible way.In other words,we would like to select a crisp set which is,in some reasonable sense,the optimal ap-proximation to the original fuzzy set.In this paper, we show that the optimal approximation is the-cut for whichIf we allow the possibility of not making any deci-sion in some situations,then we arrive at the neces-sity to approximate the original fuzzy set by a“shad-owed”set,i.e.,a set in which a characteristic function can take three possible values:“yes”(),“no”(),and“undecided”().The opti-mal approximation is given by a combination of-cuts which satisfies the formula(9). AcknowledgmentsThis work was supported in part by NASA un-der cooperative agreement NCC5-209,by NSF grants No.DUE-9750858and CDA-9522207,by the United Space Alliance,grant No.NAS9-20000(PWO C0C67713A6),by the Future Aerospace Science and Technology Program(FAST)Center for Structural In-tegrity of Aerospace Systems,effort sponsored by the Air Force Office of Scientific Research,Air Force Ma-teriel Command,USAF,under grant number F49620-95-1-0518,and by the National Security Agency under Grant No.MDA904-98-1-0561.References[1]Klir,G.,and Yuan,B.,Fuzzy Sets and Fuzzy Logic:Theory and Applications,Prentice Hall,Upper Sad-dle River,NJ,1995.[2]Nguyen,H.T.,and E.A.Walker,E.A.,First Coursein Fuzzy Logic,CRC Press,Boca Raton,FL,1999.[3]Pedrycz,W.,and Vukovich,G.,“Representationand processing of fuzzy sets in the setting of granu-lar computing”,2000(to appear).[4]Yen,J.and Pfluger,N.,“Path planning and execu-tion using fuzzy logic”,In:AIAA Guidance,Navi-gation and Control Conference,New Orleans,LA, 1991,3,1691–1698.[5]Yen,J.and Pfluger,N.,“Designing an adaptive pathexecution system”,IEEE International Conference on Systems,Man and Cybernetics,Charlottesville, VA,1991.[6]Yen,J.,Pfluger,N.,and Langari,R.“A defuzzifica-tion strategy for a fuzzy logic controller employing prohibitive information in command formulation”, Proceedings of IEEE International Conference on Fuzzy Systems,San Diego,CA,March1992.。