fuzzy set theory

Advanced Review

Fuzzy set theoryH.-J. Zimmermann∗

Since its inception in 1965, the theory of fuzzy sets has advanced in a varietyof ways and in many disciplines. Applications of this theory can be found, forexample, in artificial intelligence, computer science, medicine, control engineering,decision theory, expert systems, logic, management science, operations research,pattern recognition, and robotics. Mathematical developments have advanced toa very high standard and are still forthcoming to day. In this review, the basicmathematical framework of fuzzy set theory will be described, as well as themost important applications of this theory to other theories and techniques. Since1992 fuzzy set theory, the theory of neural nets and the area of evolutionaryprogramming have become known under the name of ‘computational intelligence’or ‘soft computing’. The relationship between these areas has naturally becomeparticularly close. In this review, however, we will focus primarily on fuzzy settheory. Applications of fuzzy set theory to real problems are abound. Somereferences will be given. To describe even a part of them would certainly exceedthe scope of this review. 2010 John Wiley & Sons, Inc. WIREs Comp Stat 2010 2 317–332

Most of our traditional tools for formal modeling,reasoning, and computing are crisp, determin-

istic, and precise in character. Crisp means dichoto-mous, that is, yes-or-no type rather than more-or-lesstype. In traditional dual logic, for instance, a state-ment can be true or false—and nothing in between.In set theory, an element can either belong to a setor not; in optimization a solution can be feasibleor not. Precision assumes that the parameters of amodel represent exactly the real system that has beenmodeled. This, generally, also implies that the modelis unequivocal, that is, that it contains no ambigui-ties. Certainty eventually indicates that we assume thestructures and parameters of the model to be definitelyknown and that there are no doubts about their val-ues or their occurrence. Unluckily these assumptionsand beliefs are not justified if it is important, that themodel describes well reality (which is neither crispnor certain). In addition, the complete description ofa real system would often require far more detaileddata than a human being could ever recognize simul-taneously, process, and understand. This situation hasalready been recognized by thinkers in the past. In1923, the philosopher B. Russell referred to the firstpoint when he wrote: ‘All traditional logic habitually

∗Correspondence to: [email protected]

RWTH Aachen, Korneliusstr. 5, 52076 Aachen, Germany

DOI: 10.1002/wics.82

assumes that precise symbols are being employed. Itis, therefore, not applicable to this terrestrial life butonly to an imagined celestial existence’.1 L. Zadehreferred to the second point, when he wrote: ‘As thecomplexity of a system increases, our ability to makeprecise and yet significant statements about its behav-ior diminishes until a threshold is reached beyondwhich precision and significance (or relevance) becomealmost mutually exclusive characteristics’.2 For a longtime, probability theory and statistics have been thepredominant theories and tools to model uncertaintiesof reality.3,4 They are based—as all formal theo-ries—on certain axiomatic assumptions, which arehardly ever tested, when these theories are applied toreality. In the meantime more than 20 other ‘uncer-tainty theories’ have been developed,5 which partlycontradict each other and partly complement eachother. Fuzzy set theory—formally speaking—is oneof these theories, which was initially intended to bean extension of dual logic and/or classical set theory.During the last decades, it has been developed in thedirection of a powerful ‘fuzzy’ mathematics. Whenit is used, however, as a tool to model reality betterthan traditional theories, then an empirical validationis very desirable.6 In the following sections, the for-mal theory is described and some of the attemptsto verify the theory empirically in reality will besummarized.

Volume 2, May/June 2010 2010 John Wiley & Sons, Inc. 317

Advanced Review www.wiley.com/wires/compstats

HISTORY

A short historical review may be useful to betterunderstand the character and motivation of thistheory. The first publications in fuzzy set theoryby Zadeh7 and Goguen8 show the intention of theauthors to generalize the classical notion of a setand a proposition to accommodate fuzziness in thesense that it is contained in human language, that is,in human judgment, evaluation, and decisions. Zadehwrites: ‘The notion of a fuzzy set provides a convenientpoint of departure for the construction of a conceptualframework which parallels in many respects theframework used in the case of ordinary sets, butis more general than the latter and, potentially, mayprove to have a much wider scope of applicability,particularly in the fields of pattern classification andinformation processing. Essentially, such a frameworkprovides a natural way of dealing with problems inwhich the source of imprecision is the absence ofsharply defined criteria of class membership ratherthan the presence of random variables’.7 ‘Imprecision’here is meant in the sense of vagueness rather thanthe lack of knowledge about the value of a parameter(as in tolerance analysis). Fuzzy set theory provides astrict mathematical framework (there is nothing fuzzyabout fuzzy set theory!) in which vague conceptualphenomena can be precisely and rigorously studied.It can also be considered as a modeling language,well suited for situations in which fuzzy relations,criteria, and phenomena exist. The acceptance ofthis theory grew slowly in the 1960s and 1970s ofthe last century. In the second half of the 1970s,however, the first successful practical applications inthe control of technological processes via fuzzy rule-based systems, called fuzzy control (heating systems,cement factories, etc.), boosted the interest in this areaconsiderably. Successful applications, particularly inJapan, in washing machines, video cameras, cranes,subway trains, and so on triggered further interestand research in the 1980s so that in 1984 alreadyapproximately 4000 publications existed and in 2000more than 30,000. Roughly speaking, fuzzy set theoryduring the last decades has developed along twolines:

1. As a formal theory that, when maturing, becamemore sophisticated and specified and wasenlarged by original ideas and concepts as wellas by ‘embracing’ classical mathematical areas,such as algebra,9,10 graph theory,11 topology,and so on by generalizing or ‘fuzzifying’ them.This development is still ongoing.

2. As an application-oriented ‘fuzzy technology’,that is, as a tool for modeling, problem solving,and data mining that has been proven superior toexisting methods in many cases and as attractive‘add-on’ to classical approaches in othercases.

In 1992, in three simultaneous conferencesin Europe, Japan, and the United States, thethree areas of fuzzy set theory, neural nets, andevolutionary computing (genetic algorithms) joinedforces and are henceforth known under ‘compu-tational intelligence’.8,12,13 In a similar way, theterm ‘soft computing’ is used for a number ofapproaches that deal essentially with uncertainty andimprecision.

MATHEMATICAL THEORY ANDEMPIRICAL EVIDENCE

Basic Definitions and OperationsThe axiomatic bases of fuzzy set theory are manifold.Gottwald offers a good review.14–16 We shallconcentrate on the elements of the theory itself:

Definition 1 If X is a collection of objects denotedgenerically by x, then a fuzzy set A in X is a set ofordered pairs:

A = {(x, µA(x)|x ∈ X)

}(1)

µA(x) is called the membership function (generalizedcharacteristic function) which maps X to themembership space M. Its range is the subset ofnonnegative real numbers whose supremum is finite.For sup µA(x) = 1: normalized fuzzy set.

In Definition 1, the membership function of thefuzzy set is a crisp (real-valued) function. Zadeh alsodefined fuzzy sets in which the membership functionsthemselves are fuzzy sets. Those sets can be defined asfollows:

Definition 2 A type m fuzzy set is a fuzzy set whosemembership values are type m − 1, m > 1, fuzzy setson [0, 1].

Because the termination of the fuzzification onstage r ≤ m seems arbitrary or difficult to justify,Hirota17 defined a fuzzy set the membership functionof which is pointwise a probability distribution: theprobabilistic set.

318 2010 John Wiley & Sons, Inc. Volume 2, May/June 2010

WIREs Computational Statistics Fuzzy set theory

Definition 3 A probabilistic set17 A on X is definedby defining function µA,

µA : X × � � (x, ω) → µA(x, ω) ∈ �C (2)

and (�C, BC) = [0, 1] are Borel sets.

Definition 4 A linguistic variable2 is characterizedby a quintuple (x, T(x), U, G, M), in which x is thename of the variable, T(x) (or simply T) denotes theterm set of x, that is, the set of names of linguisticvalues of x. Each of these values is a fuzzy variable,denoted generically by X and ranging over a universeof discourse U, which is associated with the basevariable u; G is a syntactic rule (which usually has theform of a grammar) for generating the name, X, ofvalues of x. M is a semantic rule for associating witheach X its meaning. M(X) is a fuzzy subset of U. Aparticular X, that is, a name generated by G, is calleda term (e.g., Figure 1).

Of course, the fuzzy sets in Figure 1 representingthe values of the linguistic variable ‘AGE’ may moreappropriately be continuous fuzzy sets.

Definition 5 A fuzzy number M is a convex normalizedfuzzy set M of the real line R such that

1. it exists exactly one x0 ∈ R µM(x0) = 1 (x0 iscalled the mean value of M).

2. µM(x) is piecewise continuous.

Dubois and Prade9 defined LR-fuzzy numbers asfollows:

Definition 6 A fuzzy number M is of LR-type if thereexist reference functions L (for left) and R (for right),and scalars α > 0, β > 0, with

µM(x) = L(

m − xα

)for x ≤ m (3)

= R(

x − mβ

)for x ≤ m (4)

(see Figure 2)

Some other useful definitions are the following:

Definition 7 The support of a fuzzy set A, S(A) is thecrisp set of all x ∈ X such that µA(x) > 0.

The (crisp) set of elements that belong tothe fuzzy set A at least to the degree α is called

the α-level set:

Aα = {x ∈ X|µA ≥ α

}(5)

A′α = {

x ∈ X|µA > α}

is called strong α-level set orstrong α-cut.

Definition 8 A fuzzy set A is convex if

µA(λx1 + (1 − λ)x2) ≥ min{µA(x1), µA(x2)

},

x1, x2 ∈ X, λ ∈ [0, 1]. (6)

Alternatively, a fuzzy set is convex if all α-level setsare convex.

Definition 9 For a finite fuzzy set A, the cardinality|A| is defined as

|A| =∑x∈X

µA(x). (7)

||A|| = |A||X| is called the relative cardinality of A.

A variety of definitions exist for fuzzy measuresand measures of fuzziness. The interested reader isreferred to Refs 18–21.

Operations on Fuzzy SetsIn his first publication, Zadeh7 defined the followingoperations for fuzzy sets as generalization of crisp setsand of crisp statements (the reader should realize thatthe set theoretic operations intersection, union andcomplement correspond to the logical operators and,inclusive or and negation):

Definition 10 Intersection (logical and): the member-ship function of the intersection of two fuzzy sets Aand B is defined as:

µA∩B(X) = Min(µA(X), µB(X))∀x ∈ X (8)

Definition 11 Union (exclusive or): the membershipfunction of the union is defined as:

µA∪B(X) = Max(µA(X), µB(X))∀x ∈ X (9)

Definition 12 Complement (negation): the member-ship function of the complement is defined as:

µA(X) = 1 − µA(X)∀x ∈ X (10)



20 25 30 35 40 45 50

Base variable

55 60 65 70 75 80 Age

Age

OldYoungVery young

Linguistic variable

Values of age

Very old

FIGURE 1 | Linguistic variable ‘Age’.2

1

5 10

FIGURE 2 | LR-representation of a fuzzy number.

These definitions were later extended. The‘logical and’ (intersection) can also be modeled asa t-norm18,21–28 and the ‘inclusive or’ (union) as at-conorm. Both types are monotonic, commutative,and associative.

Typical dual pairs of nonparameterized t-normsand t-conorms are compiled below:

(1a) drastic product:

tw(µA(x), µB(x)

)=

{min

{µA(x), µB(x)

}if max

{µA(x), µB(x)

} = 1

0 otherwise(11)

(1b) drastic sum:

sw(µA(x), µB(x)

)=

max{µA(x), µB(x)

}if min

{µA(x), µB(x)

} = 0

1 otherwise

(12)

(2a) bounded difference:

t1(µA(x), µB(x)) = max{0, µA(x) + µB(x) − 1

}(13)

(2b) bounded sum:

s1(µA(x), µB(x)) = min{1, µA(x) + µB(x)

}(14)

(3a) Einstein product:

t1.5(µA(x), µB(x))

= µA(x) · µB(x)

2 − [µA(x) + µB(x) − µA(x) · µB(x)

](15)

(3b) Einstein sum:

s1.5(µA(x), µB(x)) = µA(x) + µB(x)1 + µA(x) · µB(x)

(16)

(4a) Hamacher product:

t2.5(µA(x), µB(x))

= µA(x) · µB(x)µA(x) + µB(x) − µA(x) · µB(x)

(17)

(4b) Hamacher sum:

s2.5(µA(x), µB(x))

= µA(x) + µB(x) − 2µA(x) · µB(x)1 − µA(x) · µB(x)

(18)



(5a) minimum:

t3(µA(x), µB(x)) = min{µA(x), µB(x)

}(19)

(5b) maximum:

s3(µA(x), µB(x)) = max{µA(x), µB(x)

}.

(20)

These operators are ordered as follows:

tw ≤ t1 ≤ t1.5 ≤ t2 ≤ t2.5 ≤ t3 (21)

s3 ≤ s2.5 ≤ s2 ≤ s1.5 ≤ s1 ≤ sw. (22)

Also a number of parameterized operators weredefined, such as:

1. Hamacher union:

µA∪B(x) = (γ ′ − 1)µB(x) + µA(x) + µB(x)1 + γ ′µA(x)µB(x)

(23)

2. Yager intersection:

µA∩B(x) = 1 − min{1,

((1 − µA(x))

p

+(1 − µB(x))p)1/p

}, p ≥ 1 (24)

3. Yager union:

µA∩B(x) = min{1,

(µA(x)p + µB(x)p)1/p

}, p ≥ 1

(25)

4. Dubois and Prade intersection:

µA∩B(x) = µA(x) · µB(x)

max{µA(x), µB(x), α

} , α ∈ [0, 1]

(26)

5. Union:

µA∪B(x) =µA(x) + µB(x) − µA(x) · µB(x)−

min{µA(x), µB(x), (1 − α)

}max

{(1 − µA(x)), (1 − µB(x)), α

} .

(27)

Finally, a class of ‘averaging operators’29 weredefined, which do not have the mathematical prop-erties of the t-norms and t-conorms. A justificationfor those operators will be given in Section EmpiricalEvidence. Two of the best known are the followingtwo:

Definition 13 The compensatory and operator isdefined as follows:

µAi,comp(x) =

(m∏

i=1

µi(x)

)(1−γ ) (1 −

m∏i=1

(1 − µi(x))

)γ

x ∈ X, ≤ γ ≤ 1. (28)

In effect, each convex combination of at-norm with the respective t-conorm could be usedas averaging operator.

Definition 14 An OWA-Operator30 is defined asfollows:

µOWA(x) =∑

j

wjµj(x) (29)

where w = {w1, . . . , wn} is a vector of weights wi withwi ∈ [0, 1] and

∑i wi = 1.

µj(x) is the jth largest membership value foran element x for which the (aggregated) degree ofmembership shall be determined. The rationale behindthis operator is again the observation that for an ‘and’aggregation (modeled i.e., by the min-operator) thesmallest degree of membership is crucial, whereas foran ‘or’ aggregation (modeled by ‘max’) the largestdegree of membership of an element in all fuzzy setsis to be aggregated. Therefore, a basic aspect of thisoperator is the re-ordering step. In particular, thedegree of membership of an element in a fuzzy setis not associated with a particular weight. Rather aweight is associated with a particular ordered positionof a degree of membership in the ordered set ofrelevant degrees of membership (Table 1).

The Extension PrincipleOne of the most basic concepts of fuzzy set theorythat can be used to generalize crisp mathematicalconcepts to fuzzy sets is the extension principle. Inits elementary form, it was already implied in Zadeh’sfirst contribution. In the meantime, modifications havebeen suggested. Following Zadeh and Dubois andPrade,2,10 we define the extension principle as follows:

Definition 15 Let X be a Cartesian product ofuniverses X = X1 × · · · × Xr, and A1, . . . , Ar be rfuzzy sets in X1, . . . , Xr, respectively. f is a mappingfrom X to a universe Y, y = f (x1, . . . , xr). Then theextension principle allows us to define a fuzzy set B inY by

B = {(y, µB(y))|y = f (x1, . . . , xr), (x1, . . . , xr) ∈ X

}(30)



TABLE 1 Classification of Aggregation Operators

Intersection operators Union operators

References t-norms Averaging operators t-conorms

Nonparameterized

Zadeh 1965 Minimum Maximum

Algebraic product Algebraic sum

Giles 1976 Bounded sum Bounded difference

Hamacher 1978 Hamacher product Hamacher sum

Mizumoto 1982 Einstein product Einstein sum

Dubois and Prade1980,1982

Drastic product Drastic sum

Dubois and Prade1984

Arithmetic meanGeometric mean

Silvert 1979 Symmetric summation

Symmetric difference

Parameterized families

Hamacher 1978 Hamacher intersectionoperators

Hamacher union operators

Yager 1980 Yager intersection operators Yager union operators

Dubois and Prade1980, 1982, 1984

Dubois intersection operators Dubois union operators

Werners 1984 ‘fuzzy and’, ‘fuzzy or’

Zimmermann andZysno 1984

‘compensatory and’, convex comb.of maximum and minimum , oralgebraic product and algebraicsum

Yager 1988 OWA operators

where

µB(y) =

sup(x1,...,xr)∈f−1(y) min{µA1

(x1), . . . , µAr(xr)}

if f−1(y) �= �

0 otherwise(31)

where f−1 is the inverse of f.For r = 1, the extension principle, of course,

reduces to

B = f (A) = {(y, µB(y))|y = f (x), x ∈ X

}(32)

where

µB(y) ={

supx∈f−1(y) min{µA(x)} if f−1(y) �= �

0 otherwise(33)

Fuzzy Relations and GraphsFuzzy relations are fuzzy sets in product space.31,32

Definition 16 Let X, Y ⊆ R be universal sets, then

R = {((x, y), µR(x, y)

) |(x, y) ⊆ X × Y}

(34)

is called a fuzzy relation on X × Y.

Example 1 Let X = Y = R and R := ‘considerablylarger than’.The membership function of the fuzzy relation, whichis, of course, a fuzzy set on X × Y can then be

µR(x, y) =

0 for x ≤ y(x−y)10y for y < x ≤ 11y

1 for x > 11y(35)



A different membership function for this relationcould be

µR(x, y) ={

0 for x ≤ y(1 + (y − x)−2)−1 for x > y.

(36)

For discrete supports, fuzzy relations can alsobe defined by matrices.21 Quite a number of differentkinds of fuzzy relations have been defined. As anexample, we will here only define one kind: similarityrelations33:

Definition 17 A fuzzy relation that is reflexive,symmetric, and transitive is called a similarity relation.

A fuzzy relation R(x0x) is called

reflexive if µR(x, x) = 1symmetric if µR(x, y) = µR(y, x)

(37)

max − mintransitive

}if µR◦R ≤ µR(x, y). (38)

Definition 18 Max-min composition: let R1(x, y),(x, y) ∈ X × Y and R2(y, z), (y, z) ∈ Y × Z be twofuzzy set relations. The max-min compositionR1 max − min R2 is then the fuzzy set

R1◦R2 ={[

(x, z), maxy

{min

{µR1

(x, y), µR2(y, z)

}}]|x ∈ X, y ∈ Y, z ∈ Z

}.

(39)

µR1◦R2is again the membership function of a fuzzy

relation on fuzzy sets.

The ‘counterpart’ of similarity relations areorder relations or preference relations21 whichare further differentiated in fuzzy preorders, fuzzytotal orders, or fuzzy strict total orders, whichare particularly often used in multicriteria analysisand preference theory. Fuzzy relations can also beconsidered as representing fuzzy graphs.11 Let theelements of fuzzy relations, as defined in Definition16 be the nodes of a fuzzy graph that is representedby this fuzzy relation. The degrees of membershipof the elements of the related fuzzy sets define the‘strength’ of or the flow in the respective nodes ofthe graph, whereas the degrees of membership of thecorresponding pairs in the relation are the ‘flows’ orthe ‘capacities’ of the edges. Fuzzy graph theory hasin the meantime become an extended area of (fuzzy)mathematics.

Fuzzy AnalysisA fuzzy function is a generalization of the concept of aclassical function. A classical function f is a mapping(correspondence) from the domain D of definition ofthe function into a space S; f (D) ⊆ S is called the rangeof f . Different features of the classical concept of afunction can be considered fuzzy rather than crisp.Therefore, different ‘degrees’ of fuzzification of theclassical notion of a function are conceivable:

1. There can be a crisp mapping from a fuzzy setthat carries along the fuzziness of the domainand, therefore, generates a fuzzy set. The imageof a crisp argument would again be crisp.

2. The mapping itself can be fuzzy, thus blurringthe image of a crisp argument. This is normallycalled a fuzzy function. Dubois and Prade callthis ‘fuzzifying function’.10

3. Ordinary functions can have properties or beconstrained by fuzzy constraints.

Particularly for Case 2, definitions for classicalnotions of analysis, such as, extrema of fuzzyfunctions, integration of fuzzy functions, integrationof fuzzy functions aver a crisp interval, integrationof a crisp function over a fuzzy interval, fuzzydifferentiation, and so on have been suggested.9,10,34,35

It would exceed the scope of this survey to describeeven a larger part of them in more detail.

Empirical EvidenceSo far fuzzy sets and their extensions and operationswere considered as formal concepts that need no proofby reality. If, however, fuzzy concepts are used, forexample, to model human language, then one hasto make sure that the concepts really model, what ahuman says or thinks. One has, with other words, to‘extract’ thoughts from the human brain and comparethem with the modeling tools and concepts. Thisis a problem of psycho-linguistics.6,36 Bellman andZadeh37 suggested in their paper that the ‘and’ bywhich in a decision objective functions and constraintsare combined can be modeled by the intersection ofthe respective fuzzy sets and mathematically modeledby ‘min’ or by the product.

The interpretation of a decision as the intersec-tion of fuzzy sets implies no positive compensation(trade-off) between the degrees of membership of thefuzzy sets in question, if either the minimum or theproduct is used as an operator. Each of them yieldsdegrees of membership of the resulting fuzzy set (deci-sion) which are on or below the lowest degree of



membership of all intersecting fuzzy sets. This is alsotrue if any other t-norm is used to model the inter-section or operators, which are no t-norms but mapbelow the min-operator. It should be noted that, infact, there are decision situations in which such a‘negative compensation’ (i.e., mapping below the min-imum) is appropriate. The interpretation of a decisionas the union of fuzzy sets, using the max-operator,leads to the maximum degree of membership achievedby any of the fuzzy sets representing objectives orconstraints. This amounts to a full compensation oflower degrees of membership by the maximum degreeof membership. No membership will result, however,which is larger than the largest degree of member-ship of any of the fuzzy sets involved. Observingmanagerial decisions, one finds that there are hardlyany decisions with no compensation between eitherdifferent degrees of goal achievement or the degreesto which restrictions are limiting the scope of deci-sions. It may be argued that compensatory tendenciesin human aggregation are responsible for the failureof some classical operators (min, product, max) inempirical investigations.38

The following conclusions can probably bedrawn: neither t-norms nor t-conorms can alonecover the scope of human aggregating behavior. Itis very unlikely that a single nonparametric operatorcan model appropriately the meaning of ‘and’ or‘or’ context independently, that is, for all persons,at any time and in each context. There seem to bethree ways to remedy this weakness of t-norms andt-conorms: one can either define parameter-dependentt-norms or t-conorms that cover with their parametersthe scope of some of the nonparametric norms andcan, therefore, be adapted to the context. A secondway is to combine t-norms and their respectivet-conorms and such cover also the range betweent-norms and t-conorms (which may be called the rangeof partial positive compensation). The disadvantageis that generally some of the useful properties oft- and s-norms get lost. The third way is, eventually,to design operators, which are neither t-norms nort-conorms, but which model specific contexts well.Empirical validations of suggested operators are veryscarce. The ‘min’, ‘product’, ‘geometric mean’, andthe γ -operator have been tested empirically and ithas turned out, the γ -operator models the ‘linguisticand’, which lies between36,39 the ‘logical and’ and the‘logical exclusive or’, best and context dependently.It is the convex combination of the product (asa t-norm) and the generalized algebraic sum (asa t-conorm). In addition, it could also be shownthat this operator is pointwise injective, continuous,monotonous, commutative and in accordance with

the truth tables of dual logic. Limited empirical testshave also been executed for shapes of membershipfunctions, linguistic approximation, and hedges.40–45

APPLICATIONSIt shall be stressed that ‘applications’ in this reviewmean applications of fuzzy set theory to other formaltheories or techniques and not to real problems. Thelatter would certainly exceed the scope of this review.The interested reader is referred to Refs 46–54.

Fuzzy Logic, Approximate Reasoning, andPlausible ReasoningLogics as bases for reasoning can be distinguishedessentially by three topic-neutral items: truth values,vocabulary (operators), and reasoning procedures(tautologies, syllogisms). In dual logic, truth valuescan be‘true’ (1) or ‘false’ (0) and operators are definedvia truth tables (e.g., Table 2).

A and B represent two sentences or statementswhich can be true (1) or false (0). These statementscan be combined by operators. The truth values ofthe combined statements are shown in the columnsunder the respective operators ‘and’, ‘inclusive or’,‘exclusive or’, implication, and so on. Hence, the truthvalues in the columns define the respective operators.Considering the modus ponens as one tautology:

(A ∧ (A ⇒ B)) ⇒ B (40)

or:

Premise: A is trueImplication: If A then BConclusion: B is true.

Here four assumptions are being made:

1. A and B are crisp.

2. A in premise is identical to A in implication.

3. True = absolutely true

False = absolutely false.

4. There exist only two quantifiers: ‘All’ and ‘Thereexists at least one case’.

TABLE 2 Truth Table of Dual Logic

A B ∧ ∨ x ∨ ⇒ ⇔ ?

1 1 1 1 0 1 1 1

1 0 0 1 1 0 0 1

0 1 0 1 1 1 0 0

0 0 0 0 0 1 1 0



Expert

Knowledge modules

(acquisition)

Inference engine

Explanatory module

Knowledge engineer

Dialogue modules

Knowledge base (rules, facts, domain knowledge)

Expert system

User

FIGURE 3 | Expert system.

These assumptions are relaxed in fuzzyreasoning.55–61

In fuzzy logic, the truth values are no longerrestricted to the two values ‘true’ and ‘false’ butare expressed by the linguistic variables ‘true’ and‘false’. In approximate reasoning62 additionally thestatements A and/or B can be fuzzy sets. Inplausible reasoning,21 the A in the premise does nothave to be identical (but similar) to the A in theimplication. In all forms of fuzzy reasoning, theimplications can be modeled in many different ways.60

Which one is the most appropriate can be evaluatedeither empirically36 or axiomatically.60 Of course,the models for implications can also be chosen withrespect to their computational efficiency (which doesnot guaranty that the proper model has been chosenfor a certain context).

Fuzzy Rule-Based Systems (Fuzzy ExpertSystems and Fuzzy Control)Knowledge-based systems are computer-based sys-tems, normally to support decisions in which mathe-matical algorithms63 are replaced by a knowledge baseand an inference engine. The knowledge base64 con-tains expert knowledge. There are different ways toacquire and store expert knowledge.65,66 The most fre-quently used way to store this knowledge are if–thenrules. These are then considered as ‘logical’ state-ments, which are processed in the inference engineto derive a conclusion or decision. Generally, thesesystems are called ‘expert systems’ (Figure 3). Classi-cal expert systems processed the truth values of thestatements. Hence, they were actually not processingknowledge but symbols.

In the 1970s, crisp rules in the knowledgebase were substituted by fuzzy statements, which

semantically contained the context of the rules.67

Naturally the inference engine had to be substituted bya system that was able to infer from fuzzy statements.It shall be called here ‘computational unit’. The first,very successful, applications of these systems werein control engineering. They were, therefore, called‘fuzzy controllers’.20,68–71 The input to these systemswas numerical (measures of the process output) andhad to be transformed into fuzzy statements (fuzzysets), which was called ‘fuzzification’. This input,together with the fuzzy statements of the rule base,was then processed in the computational unit, whichdelivered again fuzzy sets as output. Because the out-put was to control processes, it had to be transformedagain into real numbers. This process was called‘defuzzification’72 (Figure 4). This is one difference tofuzzy expert systems, which are supposed to replace orsupport human experts. Hence, the output should belinguistic and, rather than having ‘defuzzification’ atthe end, these systems use ‘linguistic approximation’73

to provide user-friendly output.Several methods have been developed for

fuzzification, inference, and defuzzification, and atthe beginning of the 1980s the first commercialsystems were put on the market (control of videocameras, cement kilns, cameras, washing machines,etc.). This development started primarily in Japanand then spread to Europe and the United States. Itboosted the interest in fuzzy set theory tremendously,such that in Europe one was talking of a ‘fuzzyboom’ around 1990. Research and development inthe area of fuzzy control is, however, still ongoingto day.

Fuzzy Data MiningWith the development of electronic data processing,more and more data were available electronically.This led to the situation in which the masses ofdata, for instance in data warehouses, exceeded thehuman capabilities to recognize important structuresin these data. Classical methods to ‘data mine’, suchas cluster techniques, and so on, were available, butoften they did not match the needs. Cluster techniques,for instance, assumed that data could be subdividedcrisply into clusters, which did not fit the structuresthat existed in reality. Fuzzy set theory seemed tooffer good opportunities to improve existing concepts.Bezdek31,74 was one of the first, who developed fuzzycluster methods with the goals, to search for structurein data to reduce complexity and to provide input forcontrol and decision making (Figure 5).

Different approaches have been followed: hierar-chical approaches, semi-formal heuristic approaches,and objective function clustering. Because the area



Fuzzificatione

w

r

Defuzzification

Fuzzy controller

Rule base

Primary detector

Computational unit

Process

z

u x

FIGURE 4 | Fuzzycontroller.

Processdescription and pre-processing

Determ. of memberships Determ. of scale-levels

Classification

discrete

continuous

Pattern recognition Diagnosis Ling. approximation

Defuzzification

Feature analysis

Factor analysis Discriminant analysis Regression analysis

Classifier design

Clustering structured modelling

FIGURE 5 | Fuzzy data mining.

of (intelligent) data mining becomes more and moreimportant, the development of further fuzzy methodsin this area can be expected.46,75–82

Fuzzy DecisionsIn the logic of decision making, a decision is definedas the choice of a (feasible) action by which the utilityfunction is maximized. Hence, it is the search foran optimal and feasible action or strategy. Feasibilityis either defined by enumerating the feasible actionsor by constraints that define the feasibility space.The feasibility space is unordered with respect toutility and the utility or objective function definesan order in this space. Hence, the problem is notsymmetric. The problem becomes more complicatedif several objective functions exist (several orders onthe same space). This leads to the area of ‘multicriteriaanalysis’. This area has grown very much since the1970s. Many approaches have been suggested to solve

problems with several objective functions. In all theseapproaches, objective functions were considered to bereal valued and the actions as crisply defined. For thecase that the objective function or the constraints arenot crisply defined Bellman and Zadeh suggested in1970 the following symmetric model:37

Definition 19 Let µCi, i = 1, . . . , m be membership

functions of constraints on X, defining the decisionspace and µGj

, j = 1, . . . , n the membership functions

of objective (utility) functions or goals on X.A decision is then defined by its membership

function

µD = (µC1∗ · · · ∗ µCm

) × (µG1∗ · · · ∗ µGn

)

= ∗iµCi× ∗jµGj

(41)

where ×, ∗ denote appropriate, possibly contextdependent, ‘aggregators’ (connectives).



The most frequent interpretation of ×, ∗ is themin-operator.

It should be clear that the intersection ofthe objective functions and the constraints (called‘decision’) is again a fuzzy set. If a crisp decision isneeded, one could, for instance, determine the solutionthat has the highest degree of membership in this fuzzyset. Let us call this solution the ‘maximizing solution’.

This definition complicated and facilitated theclassical model at the same time: it facilitatedthe model by suggesting a symmetric rather thanan asymmetric model. It complicated the problembecause now to determine the optimal action, notreal numbers (of the real-valued utility function)but functions (the membership functions of fuzzynumbers) had to be compared and ranked. The laterproblem led to many publications that focused onthe ranking of fuzzy numbers.83,84 The symmetry ofthe model led to very efficient approaches in thearea of ‘multiobjective decision making’, that is, inmultiobjective mathematical programming, in whichseveral continuous objective functions have to beoptimized subject to constraints. In fact, mathematicalprogramming can be considered as a special case ofdecisions in the logic of decision making. This willbe considered in more detail in the next section. Forfuzzy multistage decisions, see Refs 85,86.

Fuzzy OptimizationFuzzy optimization87 was already discussed briefly inSection Fuzzy Analysis. Here, we shall concentrate on‘constrained optimization’, which is generally called‘mathematical programming’.88–95 It will show thepotential of the application of fuzzy set theory toclassical techniques very clearly. We shall look at theeasiest, furthest developed, and most frequently usedversion, the linear programming. It can be defined asfollows:

Definition 20 Linear programming model:

max f (x) = z = cTxs.t. Ax ≤ b

x ≥ 0

with c, x ∈ Rn, b ∈ R

m, A ∈ Rm×n.

In this model it is normally assumed that allcoefficients of A, b, and c are real (crisp) numbers,that ‘≤’ is meant in a crisp sense, and that ‘maximize’ isa strict imperative. This also implies that the violationof any single constraint renders the solution unfeasibleand that all constraints are of equal importance(weight). Strictly speaking, these are rather unrealisticassumptions, which are partly relaxed in fuzzy linearprogramming.96–100

If we assume that the LP-decision has to bemade in fuzzy environments, quite a number ofpossible modifications exist. First of all, the decisionmaker might really not want to actually maximizeor minimize the objective function. Rather he mightwant to reach some aspiration levels which mightnot even be definable crisply. Thus, he might wantto ‘improve the present cost situation considerably’,and so on. Second, the constraints might be vaguein one of the following ways: the ‘≤’ sign mightnot be meant in the strictly mathematical sense butsmaller violations might well be acceptable. This canhappen if the constraints represent aspiration levelsas mentioned above or if, for instance, the constraintsrepresent sensory requirements (taste, color, smell,etc.) which cannot adequately be approximated bya crisp constraint. Of course, the coefficients of thevectors b or c or of the matrix A itself can have a fuzzycharacter either because they are fuzzy in nature orbecause perception of them is fuzzy. Finally, the roleof the constraints can be different from that in classicallinear programming where all constraints are of equalweight. For the decision maker, constraints might be ofdifferent importance or possible violations of differentconstraints may be acceptable to him to differentdegrees. Fuzzy linear programming offers a numberof ways to allow for all those types of vagueness andwe shall discuss some of them below. If we assumethat the decision maker can establish an aspirationlevel, z, of the objective function, which he wants toachieve as far as possible and if the constraints ofthis model can be slightly violated—without causingunfeasibility of the solution—then the model can bewritten as follows:

Find xs.t. cTx≥z

Ax≤bx≥0.

(42)

Here, ≤ denotes the fuzzified version of ≤ andhas the linguistic interpretation ‘essentially smallerthan or equal’. ≥ denotes the fuzzified version of≥ and has the linguistic interpretation ‘essentiallygreater than or equal’. The objective function mighthave to be written as a minimizing goal to consider zas an upper bound. Obviously the asymmetric linearprogramming model has now been transformed intoa symmetric model. To make this even more visible,we shall rewrite the model as

Find xs.t. Bx≤d

x ≥ 0.

(43)



The fuzzy set ‘decision’ D is then

µD(x) = mini=1,...,m+1

{µi(x)}. (44)

µi(x) can be interpreted as the degree to which xfulfills (satisfies) the fuzzy inequality Bix ≤ di (whereBi denotes the ith row of B). Assuming that thedecision maker is interested not in a fuzzy set but ina crisp ‘maximizing’ solution, we could suggest the‘maximizing solution’, which is the solution to thepossibly nonlinear programming problem

maxx≥0

mini=1,...,m+1

{µi(x)} = maxx≥0

µD(x). (45)

As membership functions seem suitable.

µi(x) =

1 if Bix ≤ di

1 − Bix−dipi

if di < Bix ≤ di + pi,i = 1, . . . , m + 1

0 if Bix > di + pi.

(46)

The pi are subjectively chosen constants ofadmissible violations of the constraints and the objec-tive function. Substituting these membership functionsinto the model yields, after some rearrangements andwith some additional assumptions

maxx≥0

mini=1,...,m+1

{1 − Bix − di

pi

}. (47)

Introducing one new variable, λ, which corre-sponds essentially to the degree of membership of x inthe fuzzy set decision, we arrive at

max λ

s.t. λpi + Bi(x) ≤ di + pi, i = 1, . . . , m + 1x ≥ 0.

(48)

A slightly modified version of this model resultsif the membership functions are defined as follows:a variable ti, i = 1, . . . , m + 1, 0 ≤ ti ≤ pi is definedwhich measures the degree of violation of the ithconstraint. The membership function of the ith row isthen

µi(x) = 1 − ti

pi. (49)

The crisp equivalent model is then

max λ

s.t. λpi + ti ≤ piBix − ti ≤ di, i = 1, . . . , m + 1

ti ≤ pix, t ≥ 0.

(50)

This is a normal crisp linear programming modelfor which very efficient solution methods exist.

The main advantage, compared with the unfuzzyproblem formulation, is the fact that the decisionmaker is not forced into a precise formulation becauseof mathematical reasons although he might only beable or willing to describe his problem in fuzzy terms.Linear membership functions are obviously only a veryrough approximation. Membership functions whichmonotonically increase or decrease, respectively, in theinterval of [di, di + pi] can also be handled quite easily.It should also be observed that the classical assumptionof equal importance of constraints has been relaxed:the slope of the membership functions determines the‘weight’ or importance of the constraint. The slopes,however, are determined by the pis. The smaller the pithe higher the importance of the constraint. For pi = 0,the constraint becomes crisp, that is, no violation isallowed. Because of the symmetry of the model, itis very easy to add additional objective functions,which solves the problem of multiobjective decisionmaking.101–107 Because this is a crisp model, existingcrisp constraints can also be added easily. So far,two major assumptions have been made to arrive at‘equivalent models’ which can be solved efficiently bystandard LP-methods:

1. Linear membership functions were assumed forall fuzzy sets involved.

2. The use of the minimum-operator for theaggregation of fuzzy sets was considered to beadequate.

re1 The linear membership functions used so farcould all be defined by fixing two points, theupper and lower aspiration levels or the twobounds of the tolerance interval. The obviousway to handle nonlinear membership functionsis probably to approximate them piecewise bylinear functions.108,109

re2 As already mentioned earlier, the min-operatormodels not always the real meaning of the‘linguistic and’. Hence, the modeler might wantto use other operators. The computationalefficiency, by which the problem can then besolved, depends only on the type of crispequivalent model, for example, whether it islinear or nonlinear, and that depends on thecombination of the type of membership functionand the operator used in the fuzzy model. Table 3shows these possible combinations and the typeof resulting crisp equivalent model.



TABLE 3 Resulting Equivalent Models

Membership function Operator Model

Linear min LP

Logistic min LP

Hyperbolic min LP

Linear γ min +(1 − γ ) max MILP

Linear/nonlinear Product nonconvex NLP

Linear/nonlinear γ -operator nonconvex NLP

Linear ˜and LP

Linear or MILP

Logistic ˜and NLP with lin. const.

With and = γ min(x, y) + (1 − γ ) 12 (x + y) and

or = γ max(x, y) + (1 − γ ) 12 (x + y).

CONCLUSION

Since its inception in 1965 as a generalization of duallogic and/or classical set theory, fuzzy set theory hasbeen advanced to a powerful mathematical theory. Inmore than 30,000 publications, it has been applied

to many mathematical areas, such as algebra, analy-sis, clustering, control theory, graph theory, measuretheory, optimization, operations research, topology,and so on. In addition, alone or in combination withclassical approaches it has been applied in practice invarious disciplines, such as control, data processing,decision support, engineering, management, logistics,medicine, and others. It is particularly well suited as a‘bridge’ between natural language and formal modelsand for the modeling of nonstochastic uncertainties.

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FURTHER READING

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