fuzzy set college

8/8/2019 Fuzzy Set College

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INTRODUCTION

Fuzzy Sets were introduced by Zadeh in 1965 to represent or manipulate dataand information possessing non-statistical uncertainties. It was specificallydesigned to mathematically represent uncertainty and vagueness and to provideformalized tools for dealing with the imprecision intrinsic to many problems.Fuzzy logic provides an inference morphology that enables approximate humanreasoning capabilities to be applied to knowledge -based systems. The Theory of Fuzzy logic provides a mathematical strength to capture the uncertaintiesassociated with human cognitive processes, such as thinking and reasoning.

Some of the essential characteristics of Fuzzy logic relate to the following(Zadeh, 1992)

In Fuzzy logic, exact reasoning is viewed as a limiting case of approximatereasoning.In Fuzzy logic, everything is a matter of degree.In Fuzzy logic, knowledge is interpreted a collection of elastic or

equivalently, fuzzy constraint on a collection of variables.Inference is viewed as a process of propagation of elastic constraints.Any logical system can be fuzzified.There are two main characteristics of fuzzy systems that give them better performance for specific applications.Fuzzy systems are suitable for uncertain or approximate reasoning,especially for the system with a mathematical model that is difficult toderive.

Fuzzy logic allows decision making with estimated values under incomplete or uncertain information.

In Classical set theory, a subset A of a set X can be defined by its characteristicfunction X A as a mapping from the elements of the set {0,1},

X A : X {0, 1}


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This mapping may be represented as a set of ordered pairs, with exactly oneordered pair present for each element of X. The first element of the ordered pair

is an element of the set X, and the second element is an element of the set {0, 1}.The value zero is used to represent non membership, and the value one is usedto represent membership. The truth or falsity of the statement x is in A isdetermined by the ordered pair (x, X A (x)). The statement is true, if the secondelement of the ordered pair is 1, and statement is false if it is 0.

Similarly a Fuzzy subset A of a set X can be defined as a set of ordered pairs,each with the first element from X, and the second element from the interval [0,

1], with exactly one ordered pair present for each element of X. This defined amapping, A, between elements of the set X and values in the interval [0, 1]. Thevalue zero is used to represent complete non-membership, the value one is usedto represent complete membership, and values in between are used to representintermediate degrees of membership.

The set X is referred to as the universe of discourage for the fuzzy subset A.Frequently, the mapping A is described as a function, the membership function

of A.The degree to which the statement x in A is true is determined by findingthe ordered pair (x, A (x)).

The degree of truth of the statement is the second element of the ordered pair. Itshould be noted that the terms membership function and fuzzy subset get usedinterchangeably.

The development of fuzzy set theory to fuzzy technology during the first half of

the 1990s has been very fast. More than 16,000 publications have appearedsince 1965. Most of them have advanced the theory in many areas. Quite anumber of these publications describe, however, applications of fuzzy set theoryto existing methodology or to real problems. In addition, the transition from fuzzyset theory to fuzzy technology has been achieved by providing numeroussoftware and hardware tools that considerably improve the design of fuzzy


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systems and make them more applicable in practice. Since 1994, fuzzy settheory, artificial neural nets, and genetic algorithms have also moved closer together and are now normally called computational intelligence.

In this study we have presented three Chapters. Chapter 1 highlights the basicconcepts of fuzzy set theory. The fuzzy measures and relations are explained inChapter 2. In Chapter 3, the fuzzy set theory has been applied to solve thefollowing problems :

I. Fuzzy Linear Programming Problem.II. Fuzzy Dynamic Programming Problem.

III. Two stage Fuzzy Transportation Problem.

Finally we have given some of the important references.


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Chapter -1

BASIC CONCEPTS OF FUZZY SET THEORY

1.1 Crispness and Vagueness

Most of our traditional tools for formal modeling, reasoning and computing arecrisp, deterministic and precise in character. By crisp we can dichotomous, that isyes-or-no-type rather than more-or-less type. An conventional dual logic, for instance, a statement can be true or false and nothing in between. In set theoryan element can either belong to a set or not; and in optimization, a solution iseither feasible or not. Precision assumes that the parameters of model representexactly either our perception of the phenomenon modeled or the features of thereal system that has been modeled. Generally, precision also implies that themodel is unequivocal, that is, that it contains no ambiguities.

A crisp set can be considered as a container and the elements belong to this setas the objects contained in it. In this sense, an object will be either in thecontainer or not in the container. We can define a membership function A of anelement x for a crisp set A as follows.

A (x) = 1 , if x A0 , otherwise.

Similarly, we can define the membership function for the set operations,like union, intersection and complement as follows:

A B(x) = max ( A (x), B (x)) A B (x) = min ( A (x), B (x)) and

(x) = 1- A (x)respectively. Thus a membership function for a crisp set A defined as

A : X {0, 1}


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Moreover, A is a subset of B if and only if x A x B, for all x.

In terms of membership function A is a subset of B if and only if Ax B (x),

for all x.

Example : (1.1)

Suppose U = {a, b, c, d}, A = {a, b} & B = {b, d}Therefore

A (a) = 1, A (b) = 1, A(c) =0, A (d) =0

B (a) = 0, B (b) = 1, B(c) =0, B (d) =1

and also U (x) = 1 for x=a, b, c (or) d A B = {b}. Thus

A B(a) = min ( A (a), B (a)) =min (1, 0) = 0

A B(b) = min ( A (b), B (b)) =min (1, 1) = 1

It is easy to verify that

A B(c) = 0 and A B(d) = 0

Vagueness

At the heart of the problems arising out of vagueness is the problem regardingsorites paradox. The sorites is a types of arguments which proceed in a chain of deductions, involving predicates like heap, bold, tall, old, red, orange,etc. which lack sharp boundaries between their applications and non-application.Centuries back Eubulides, the Megarian thinker, observed that a chain of apparently valid deductions setting of f with evidently true premises ultimately

leads to apparently false conclusion. The Paradox as originally formulated byEubuildes is known as the paradox of heap. Suppose from a heap of wheats onegrain of wheat is taken away. Obviously it remains a heap. If another grain istaken away, there still is a heap, and so on. In this way at the end one wouldarrive at an assured consequence that zero number of grains makes a heap.Extension of any such predicate can be arranged in a series, each member of


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which is indiscriminable from its immediate successor and the latter involvesinsignificantly small change from its immediate predecessor. Because thesepredicates tolerate insignificantly small changes, one cannot pin point exactly at

which point out the exact location of the cross over. Never the less,indiscriminability is a relation which becomes non -transitive in a long chain. Thisactually is the source of the paradox. One popular form in which sorites argumentis presented in the literatures is conditional form, where the argument proceedsby a series of conditionals and repeated application of modus ponens:

Fan If Fan then Fan-1 If Fan-1 then Fan-2

If Fan-9999 then Fan-10,000 (=Fam)~ Fam

Here F is a vague predicate, say .is a heap; n is an arbitarly large number 10,000 and m is either 0 or is a small number. For each i, m i n, ai is the setcontaining i-nember of grains. Then Fai means the set containing i grains of aheap.

1. 2. Fuzzy Set

The first publication in Fuzzy set theory by Zadeh (1965) and Goguen (1967,1969). Zadeh (1965) writes, The notion of a fuzzy set provides a convenientpoint of departure for the construction of a conceptual frame-work which parallelsin many respects the frame work used in the case of ordinary sets, but is moregeneral than the letter and, potentially, may prove to have a much wider scope of

applicability, particularly in the fields of pattern classification and informationprocessing. Essentially, such a frame work provides a natural way of deals withproblems in which the source of imprecision is the absence of sharply definedcriteria of class membership rather than the presence of random variables.


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Impression here is meant in the sense of vagueness rather than the lack of knowledge about the value of a parameter. Fuzzy set theory provides a strictmathematical frame work in which vague conceptual phenomena can be

precisely and rigorously studied. It can also be considered as a modelinglanguage well suited for situations in which fuzzy relations, criteria, andphenomena exist.

Fuzziness has so far not been defined uniquely semantically, and probably never will be .It will mean different things, depending on the application area and theway it is measured. In the meantime, numerous authors have contributed to thistheory. In 1984, as many as 4,000 publications have already existed and in 2000

there were already more than 30,000.

The specialization of those publications conceivably increases, making it moreand more difficult for new comers to this area to find a good entry and tounderstand and appreciate the philosophy formalism, and applications potentialof this theory. Roughly speaking, fuzzy set theory in the last two decades hasdeveloped along two lines:

1. As a formal theory that, when maturing become more sophisticated andspecified and was enlarged by original ideas and concepts as well as byembracing classical mathematical areas such as algebra, graph theory,topology, and so on by generalizing them.

2. As an application oriented fuzzy-technology, i.e. as a tool for modelingproblem solving and data mining that has proven superior to existingmethods in many cases and as an attractive add-on to classical

approaches in other cases.

In this context it may be useful to cite and comment the major goals of thistechnology briefly and to correct the still very common view that fuzzy set theoryor fuzzy technology is exclusively or primarily useful to model uncertainly.


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1. 2.1 Modeling of Uncertainly

Uncertainly has been a very important topic for several centuries. There are

numerous methods and theories which claim to be the only proper tool to modeluncertainties. In general, however, they do not even define sufficiently or only ina very specific and limited sense what is meant by uncertainty. It is believed thatuncertainty, if considered as a subjective phenomenon, can and ought to bemodeled by very theories, depending on the causes of uncertain ty, the type andquantity of available information, the requirements of the observer etc. In thissense fuzzy set theory is certainly also one of the theories which can be used tomodel specific types of uncertainty under specific types of circumstances. It might

then complete with other theories, but it might also be the most appropriate wayto model this phenomenon for will-specified situations.

1. 2.2 Relaxation

Classical models and methods are normally based on dual logic. They, therefore ,distinguish between feasible and infeasible belonging to cluster or not, optimal or suboptimal, etc. Often this view does not capture reality adequately. Fuzzy set

theory has been used extensively to relax or generalize classical methods from adichotomous to a gradual character. Examples of this are fuzzy mathematicalprogramming (Zimmermann, 1996) fuzzy clustering (Bezdek and Pal, 1992),fuzzy Petri Nets (Lipp et.al., 1989), fuzzy multi criteria analysis (Zimmermann,1986).

1. 2.3 Compactification

Due to the limited capacity of the human short term memory or technologicalsystems it is often not possible to either store all relavent data , or to presentmasses of data to a human observer in such a way, that he or she can perceivethe information contained in this data. Fuzzy technology has been used to reducethe complexity of data to an acceptable degree usually either via linguisticvariables or via fuzzy data analysis.


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1. 2.4 Meaning Preserving Reasoning

Expert system technology has already been used since two decades and has ledin many cases to disappointment. One of the reasons for this might be, thatexpert systems in their inference engines, when they are based on dual logic,perform symbol processing rather than knowledge processing. In approximatereasoning meanings are attached towords and sentences via linguistic variables.Inference engines then have to be able to process meaningful linguisticexpressions, rather than symbols, and arrive at membership functions of fuzzyset, which can then be retranslated into words and sentences via linguistic

approximation.

1. 2.5 Efficient Determination of Approximate Solutions

Already in the 70s professor. Zadeh expressed his intention to have fuzzy settheory considered as a tool to determine approximate solutions of real problemsin an efficient or affordable way. This goal has never really been achievedsuccessfully. In the recent past, however, cases have become known which are

very good examples for this goal Bardossy (1996), for instance, showed in thecontext of water flow modeling that it can be much more efficient to use fuzzy rulebased systems to solve the problems than systems of differential equations.Comparing the results achieved by these two alternative approaches showed thatthe accuracy of the results was almost the same for all practical purpose. This isparticularly true if one considers the inaccuracies uncertainties contained in theinput data.

1. 3 Basic Definitions

A classical (crisp) set is normally defined as a collection of elements or objectsx X that can be finite, countable or over countable. Each single element can

either belong to or not belong to a set A, A X. In the former case, the statementx belongs to A is true, where as in the latter case this statement is false.


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Such a classical set can be described in different ways. One can either enumerate the elements that belong to the set; describe the set analytically, for

instance, by stating conditions for membership (A={x/x 5}); or define themember elements by using the characteristic function, in which 1 indicatesmembership and 0 non membership For a fuzzy set, the characteristic functionallows various degrees of membership for the elements of a given set.

1. 3.1 Definition

If x is a collection of objects denoted generically by x, then a fuzzy set in X is a

set of ordered pairs: = {(x, (x))|x X}

(x) is called the membership function or grade of membership of x in thatmaps X to the membership space . The range of the membership function is asubset of the non negative real number whose supremum is finite. Elements witha zero degree of membership are normally not listed.

Example : (1. 2)

= real number considerably larger than 10 = {(x, (x)) | x X}

Where (x) = 0 , x 10

(1+(x-10)-2)-1, x>10

Example : (1. 3)

= real numbers close to 10 = {(x, (x)) | (x) = (1+(x-10)-2)-1}

A fuzzy set is represented solely by stating its membership function (for instance,Negoita and Ralesu, 1975).


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bedrooms in a house. Then the fuzzy set comfortable type of house for a four-person family may be described as

= { (1,0.2), (2,0.5), (3,0.8), (4,0.1), (5,0.7), (6,0.3) }

In the literature one finds different ways of denoting fuzzy sets:

A fuzzy set is denoted by an ordered set of pairs, the first element of whichdenotes the elements and the second the degree of membership.

The support of S()={1,2,3,4,5,6}. The elements {7,8,9,10} are not part of thesupport of !

1. 3.3 Definition

The (crisp) set of elements that belong to the fuzzy set at least to the degree is called the - level set.

A = { x X | (x) }

A = {x X| (x) > } is called strong - level set or strong - cut.

Example: (1. 6 )

We refer again to example (1.5) and list possible - level sets. A2 = {1, 2, 3, 4, 5, 6} A5 = {2, 3, 4, 5} A8 = {3, 4} A1 = {4}

The strong - level set = 0.8 is A8 = {4}Convexity also a role in fuzzy set theory. By contrast to classical set theoryhowever, convexity conditions are defined with reference to the membershipfunction rather than the support of the fuzzy set.


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1. 3.4 Definition

A fuzzy set is convex if

( x1+ (1- ) x2) min { (x1), (x2)}, x1, x2 X, [0,1] Alternatively, a fuzzy set is convex if all - level set are convex.

Figure 1.2 Convex fuzzy set.

Figure 1.3 Non convex fuzzy set.Example : (1. 7 )

Figure 1.2 depicts a convex fuzzy set, where as figure 1.3 illustrates a non -convex fuzzy set.

1. 3.5 Definition

For a finite fuzzy set , the cordiality || is defined as

Q

1

x

Q

1

x


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is called the relative cordiality of .

Obviously, the relative cardinality of a fuzzy set depends on the cardinality of theuniverse. So you have to choose the same universe if you want to compare fuzzysets by their relative cardinality.

Example : (1. 8 )

For the fuzzy set comfortable type of house for a four -person family from

example in (Zimmermann, 1985), the cardinality is|| = .2 + .5 + .8 + 1 + .7 + .3 = 3.5Its relative cardinality is

|||| =

The relative cardinality can be interpreted as the fraction of elements of X beingin weighted by their degrees of membership in . For infinite X the cardinality isdefined by

|| = x (x) dx.Of course || does not always exist

1. 4 Operations

let be three fuzzy sets defined on the universe of discourse X. For givenelement x of the universe the following function-theoretic operations of union,intersection and complements are defined as follows:Union:

)x(V)x()x(

~ ~

~ ~ QQ!Q7


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In figure,

Union of Fuzzy setsI ntersection

)x()x()x(

~ ~

~ ~ Q0Q!Q7

Intersection of fuzzy sets.C omplement

(x) = 1- (x)

Complement of fuzzy set.

Any fuzzy set defined on the universe X is a subset of the universe. Also bydefinition null set has a membership 0 and x in X has a membership 1. Note thatthe null set and the whole set are not fuzzy sets.

Q A

B

Q A B

Q A

A


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Example : (1. 9 )

A simple hollow shaft is 1-m radius and has a wall thickness of (1/2T) m. Theshaft is built up stacking a ductile section and a brittle section. A downward force

p and q torque T are simultaneously applied to the shaft. The failure properties of the two section can be described by the following fuzzy sets A and B for theductile and brittle sections as follows:

!

!

5

4.0

4

2.0

3

7.0

2

5.0~

5

2.0

4

3.0

3

5.0

2

1~

B

and A

We can see the following:1. The set of loading for which either material B or material D will be s afe can

be obtained by getting B~~ 7

2. The set of loading for which one excepts that both material B and material D

safe can be obtained by forming B~~ +

3. The complements ~ and B~ represents the set of loading for material D andB are unsafe.

4. B~~ gives the set of loadings for which the ductile material is safe but the

brittle is not.

5. ~B~ gives the set of loading for which the brittle material safe but the ductile

not.

6. De Morgans laws can be used to find B~~B~~ 7+ ! which asserts that the

loadings that are not safe with respect to both materials are the union of thethose that are unsafe with respect to the brittle material with those that are unsafefor with respect to the ductile material.

7. De Morgans law B AB A + ! asserts that the loads that are safe for

neither material D nor material B are the intersection of those that are unsa fe for material D with those that are unsafe for material B.


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Chapter : 2

FUZZY MEASURES AND RELATIONS

2.1 Fuzzy Measures

We shall first briefly describe the meaning and features of fuzzy measures. In thelate 1970s, sugeno (1977) defined a fuzzy measures as follows : Fuzzy measuretheory considered a number of special classes of measure each of which ischaracterized by a special property. Some of the measures used in this theoryare plausibility and belief measures, fuzzy set membership function and theclassical probability measures. In the fuzzy measure theory the condition areprecise, but the information about an element alone is insufficient to determinewhich special classes of measure should be used. The central concept of fuzzymeasure theory is the fuzzy measure which was introduced by Choquet in (1953)and independently defined by Sugeno (1974) in the context of fuzzy integrals.

2.1.1 Definition

A set function g defined onB a boreal field B of the arbitrary set X that has thefollowing properties is called a fuzzy measure.1) g (0) = 0, g (X) = 1

2) If A, B B and A B, then g (A) g(B).

3) If An B , A1 A2 then

-

gp

!gp n

) A(Limg) A(gLim nnn

Sugenos measure differs from the classical measure essentially by re laxing theadditively property Murofushi and Sugeno (1989). A different approach, however is used by Klement and Schwyhla (1982).


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2.1. 2 Definition

Let p(X) is the power set of X. A possibility measure is a function

: p(X) [0, 1] with the properties1. (0) = 0, (X) =1.2. A B = (A) (B)

3. ( iIi AU ) = IiSup (Ai) with an index set I.

It can be uniquely determined by a po ssibility distribution function f : X [0, 1] by (A) =

AxSup f (x), A X. It follows directly that f is defined by f(x) = ({x}),

for all x X, Klir and Folger, (1988).

Example : ( 2.1)

Let X = {0, 1 10} ({x}): = Possibility that x is close to 8.

x 0 1 2 3 4 5 6 7 8 9 10

({x}) 0.0 0.0 0.0 0.0 0.0 0.1 0.5 0.8 1 0.8 0.8

(A) : = Possibility that A contains an integer close to 8. A X (A) =

AxSup

(

({x})

For A = {2, 5, 9} we compu te: (A) =

AxSup

(

({x})

= sup { ({1}), ({5}), ({9})}= sup {0.0, 0.1, 0.8}= 0.8

2.1. 3 Axioms

Fuzzy measure can be considered as generalization of the classical probabilitymeasure. A Fuzzy measure g over a set X (the universe of discourse with thesubsets E, F, ,) satisfies the following conditions.1. When E is the empty set then g(E) = 0.


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2. When E is a subset of F, then g(E) g(F). A fuzzy measure g is called normalized if g(X) =1.

2.1. 4 Properties of fuzzy measures

For any E, F X, a fuzzy measure is:

Additive, if g (E F) = g (E) + g(F), for all E F=J ;

Supermodular, if g (E F)+g(E F) g(E) + g(F);

Submodular, if g (E F) + g(E F) e g(E) + g(F);

Superadditive, if g (E F)+g(E F) g(E) + g(F) for all E F = J ;

Subadditive, if g (E F)+g(E F) g(E) + g(F) for all EF= J ;

Symmetric, if |E| = |F| implies g(E) = g(F);Boolean, if g(E) = 0 or g (F) = 1.

Understanding the properties of fuzzy measures is useful in application. When afuzzy measure is used to define a function such as the Sugeno integral or Choquet integral, these properties will be crucial in understanding the function sbehavior. For instance, the Choquet integral with respect to an additive fuzzymeasure reduces to the Lebesgue integral. In discrete cases, a symmetric fuzzymeasures will result in the Ordered Weighted Averaging (OWA) operator submodular fuzzy measures result in convex functions, while supermodelar fuzzymeasures result in concave functions when used to define a Choquet integral.

2.1. 5 Mobius representation

Let g be a fuzzy measure the Mobius representation of g is given by the setfunction M, where for every E, F X

)()1()( ) 0

0 )

! The equivalent axioms in obius representation are:

1. (J ) =0;


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2. 1

2

3 i4 3 M( ) 0, for all 5 X and all i6 ,

A fuzzy measure in Mobius representation M is called normalized, if

7 X8

M( ) = 1

Mobius representation can be used to give an indication of which subsets of Xinteract with one another. or instance, an additive fuzzy measure has Mobiusvalues all equal to zero except for singletons. The fuzzy measure g in standardrepresentation can be recovered from the Mobius from using the Zeta transform.

g( ) = 7

8 9

M( ), for all 5 X.

2.1. 6 Simplification Assumptions for Fuzzy Measures

Since fuzzy measures are defined on the power set, even in disc rete cases thenumber of variables can be quite high (2X). For this reason, in the context of Multi-Criteria decision analysis and other disciplines, simplification assumptionson the fuzzy measure have been introduced so that it is less computationallyexpensive to determine and use. For instance, when it is assumed the fuzzy

measure is additive, it will hold that g (E) =Ei

g ({i}) and the values of the fuzzy

measure can be evaluated from the values on X. Similarly, a symmetric fuzzymeasure is defined uniquely by |X| values. Two important fuzzy measures thatcan be used are the Sugeno-or -fuzzy measure and k-additive measuresintroduced by Sugeno and Grabisch respectively.

2.1. 7 Sugeno- -measure

The sugeno -measure is a special case of fuzzy measures defined iteratively. It

has the following definition.

2.1. 7.1 Definition

Let X = {x1, , xn} be a finite set and let (-1,+ ). A sugeno -measure is afunction g from 2X to [0,1] with properties:1. g (X) = 1


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2. if A, B X (or A, B 2X) with A+ B = J then

g(A7 B) = g(A) + g(B) + g(A) g(B).

As a convention, the value of g at a singleton set {xi} is called a density and is

denoted by g i=g({xi}). In addition, we have that satisfies the property +1 = T

!

n

1i(1+ gi)

Taha n i an d Keller as well as Wan g an d Klir have showed that on ce the de n sitiesare kn own , it is possible to use the previous polyn omial to obtain the values of un iquely.

2.1. 8 K- additive fuzzy measure

The K-additive fuzzy measure limits the interaction between the subsets E X tosize |E| = K. This drastically reduces the number of variables needed to definethe fuzzy measure, and as K can be anything from 1 to |X|, it allows for acompromise between modeling ability and simplicity.

2.1. 8 .1 Definition

A discrete fuzzy measure g on a set X is called k-additive (1 k |X|) if itsMobius representation verifies M(E)=0, whenever |E| >k for any EX, and thereexists a subset F with K elements such that M(F) 0.

2.1. 9 Shapley and interaction indices

In game theory, the shapely value or shapely index is used to indicate the weightof a game. Shapely values can be calculated for fuzzy measures in order to givesome indication of the importance of each singleton . In the case of additive fuzzymeasures, the shapely value will be the same as each singleton. For a givenfuzzy measure g, and |X| = n the shapely index for every i, ,n X is :

(i) = )](})i{([!n

!!)1En(}i{XE

7


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The shapely value is the vector as(g) = (] (1), , ] (n)).

2.2. Measures of FuzzinessMeasures of fuzziness, in contrast to fuzzy measures, try to indicate the degreeof fuzziness of a fuzzy set. A number of approaches to this end have becomeknown. Some authors, strongly influenced by the Shannon entropy as a measureof information and following de Luca and Termini (1972), consider a measure of fuzziness as a mapping d from the power set p(X) to [0,+ ] that satisfies anumber of conditions. Others (Kaufmann, 1979) suggested an index of fuzzinessas a normalized distance, and others (Yager, 1979; Higashi and Klir, 1982) base

their concept of a measure of fuzziness on the degree of distinction between thefuzzy set and its complement.

We shall, as an illustration, discusses two of those measures, suppose for bothcases that the support of A is finite. The first is as follows : Let (x) be themembership function of the fuzzy set for x X, X is finite. It seems plausiblethat the measure of fuzziness d () should then have the following properties (deLuca and Termini, 1972).

1. d () =0, if is a crisp set in x.2. d () assumes a unique maximum if (x) = , for all x X.3. d() d() if is crisper than , ie., if (x) (x) for (x) and

(x) (x) for (x) 4. d ( ) = d () where is the complement of .

De Luca and Termini suggested as a measure of fuzziness the entropy of afuzzy set (de Luca and Termini, 1972) which they defined as follows.

2.2.1 Definition

The entropy as a measure of a fuzzy set = {(x, (x))} is defined as


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d () = H() + H( ), x X.

H () = -k!

n

1i (xi) In ( (xi))

Where n is the number of elements in the support of and k is a positiveconstant. Using Shannons function,

S(x) = -xln x-(1-x) ln (1-x), De Luca and Termini simplify the expression indefinition 2.2.1 to arrive at the following definition.

2.2.2 Definition

The entropy d as a measure of fuzziness of fuzzy set. = {x, (x)} is defined as

d() = k!

n

1iS ( (xi))

Example : ( 2.2 )

Let = integers close to 10 = {(7,0.1), (8,0.5), (9,0.8), (10,1), (11,0.8), (12,0.5), (13,0.1)}Let k=1, sod () = 0.325 + 0.693 + 0.593 + 0 + .501 + 0.693 + 0.611 + 0.325 = 3.038

Furthermore, let ~ = integers quite close to 10~ = {(6,0.1),(7,0.3),(8,0.4),(9,0.7),(10,1),(11,0.8),(12,0.5),(13,0.3), (14,0.1)}

d (~ )= 0.325+0.611+0.673+0.611+0+0.501+0.693+0.611+0.325 = 4.35

The second measure is as follows :

Knopfmacher (1975), Loo (1977), Gottwald (1979b) and others based their contributions on the luca and Terminais suggestion in some respects.If is a fuzzy set in X and is its complement, then in contrast to crisp sets, itis not necessarily true that.


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=X

=J

This means that fuzzy sets do not always satisfy the law of the excluded middle,which is one of their major distinctions from traditional crisp sets. Some authors(Yager, 1979; Higashi and Klir, 1982) consider the relationship between and to be the essence of fuzziness.

Yager (1979) notes that the requirement of distinction between and is notsatisfied by fuzzy sets. He therefore suggests that any measure of fuzzinessshould be a measure of the lack of distinction between and or (x) and

(x).

2.2.3 Definition

Dp (, ) =p

1p

i A~i A~

n

1i)x()x(

-

QQ

!

, p = 1, 2, 3

Let S = supp () : Dp (S, S) = ||s|| 1/p.

2.2.4 Definition (Yager, 1 979 )

The measure of the fuzziness of can be defined as

f p () =) A(psup) A, A(D1 P@

So f p() [0, 1]. This measure also satisfies properties 1 to 4 required by deLuca and Termini, (1972).For p = 1, Dp (, ) yields the Humming metric.

D1 (, ) = !

n

1i| (xi) - (xi)|

Because (x) =1- (x), this becomes


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D1 (, ) = !

n

1i|2 (xi) - 1|

For p=2, we arrive at the Euclidean metric.

D2, (, ) = (!

n

1i( (xi) - (xi))2)1/2, an d for (x) =1- (x), we have

D2 (, ) = (!

n

1i(2 (xi) -1)2)1/2

Example : ( 2.3)

Let = integers close to 10 and

B = integers quite close to 10 be defined as in above example. Applying the above derived formula, we compute for p=1;

D1 (, ) = 0.8 + 0 + 0.6 + 1 + 0.6 + 0 + 0.8 = 3.8||supp () || = 7.

So f 1 () = 1 - 78.3 = 0.457

Analogously,

D1 (B , B ) = 4.6

||supp ( B)|| = 9

So f 1 (B) = 1 - 96.4

= 0.489

Similarly, for P = 2, we obtainD2 (, ) = 1.73||supp () || = 2.65

So f 2 () = 1 - 65.273.1 = 0.347, and

D2 (B, B) = 1.78

||supp ( B) || = 1So f 2 (B) = 1- 3

78.1 = 0.407

2.3 Fuzzy Relations


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Fuzzy relations are fuzzy subsets of XxY, that is, mapping from X Y. They havebeen studied by a number of authors, in particular by Zadeh (1965, 1971),Kaufmann (1975), and Rosen feld (1975). Applications of fuzzy relations are

widespread and important. We shall consider some of them and point to morepossible use at the end of this chapter. We shall exemplarily consider only binaryrelations. A generalization to n any relations is straight forward.

2.3.1 Definition

Let X, Y R be universal sets; then

R = {((x, y), RA Q (x, y) | (x, y) XxY} is called a fuzzy relation on XxY.

Example : ( 2.4 )

Let X =Y=R and R : = Considerably larger than. The membership function of

the fuzzy relation, which is, of course, a fuzzy set on XxY, can then be

"

e

e

!Q

y11xfor 1

y11xyfor y10 yx

yxfor 0

)y,x(RB

A different membership function for this relation could be

"e

!Qyxfor ))xy(1(yxfor 0

)y,x( 12RB

For discrete supports, fuzzy relations can also be defined by matrixes.

2.3.2 Definition


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Let R and Z be two fuzzy relations in the same product space. The

union/intersection of R with Z is then defined by

ZC

RC

7Q (x, y) = max {D

~Q (x, y), E ~Q (x, y)}, (x, y) X x Y.

F ~G

~+

Q (x, y) = min { RH Q (x, y), ZH Q (x, y)}, (x, y) X x Y.

Example : ( 2.5 )

Let R and Z be the two fuzzy relations defined as follows.

R = x considerably larger than y

and

Z - y very close to x:

The union of R and Z , which can be interpreted as x considerably larger or very

close to y, is then given by

y1 y2 y3 y4

x1 0.8 1 0.1 0.7

x2 0 0.8 0 0

x3 0.9 1 0.7 0.8

y1 y2 y3 y4

x1 0.4 0 0.9 0.6

x2 0.9 0.4 0.5 0.7

x3 0.3 0 0.8 0.5


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:ZR7

The intersection of R and Z is represented by

:ZR+

So far min and max have been used to define intersection and union. Since

fuzzy relations are fuzzy sets, operations can also be defined using thealternative definitions (Zimmermann, 1985). Some additional concepts, such asthe projection and the cylindrical extension of fuzzy relations, have been shownto be useful.

A classical relation can be considered as a set of tuples, where a tuple is anordered pair. A binary tuple is denoted by (u, v), an example of a ternary tuple is(u, v, w) and an example of n- ary tuple is (x1, ....., x n).

Example: ( 2.6 )

Let X be the domain of man {John, Charles, James} and Y the domain of women{Diana, Rita, Eva}, then the relation married to on X x Y is for example{(Charles, Diana), (John, Eva), (James, Rita)}

y1 y2 y3 y4

x1 0.8 1 0.9 0.7

x2 0.9 0.8 0.5 0.7

x3 0.9 1 0.8 0.8

y1 y2 y3 y4

x1 0.4 0 0.1 0.6

x2 0 0.4 0 0

x3 0.3 0 0.7 0.5


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2.3.3 Definition

Let X1,., X n be classical sets. The subsets of the Cartesian productX1 x ..x X n are called n-ary relations. If X1 =. = X n and R Xn then R iscalled an n-ary relation in X.

Let R be a binary relation inR . Then the characteristic function of R is defined as

XR (u, v) =

I

o t r is),u(if 1

Example : ( 2.7)

Consider the following relation (u, v) R u [a, b] and v [o, c].

XR (u, v) =

P

ot r is ]c,o[x]b,a[)v,u(if 1

L t R b binary r lation in a classical s t X.

2.3 .4 Definition

R is reflexive if u U: (u, v) R.

2.3.5 Definition

R is anti-reflexive if for all u U: (u, v) R.

2.3.6 Definition

R is Symmetric if from (u, v) R p (v, u) R, for all u, v U.

c

a b


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2.3 .7 Definition

R is anti-symmetric if (u, v) R and (v, u) R then u=v, for all u, v U.

2.3.8 DefinitionR is transitivity if (u, v) R and (v, w) R then (u, w) R, for all u, v, w U.

Example : ( 2.8)

Consider the classical inequality relations on the real line R . It is clear that is

reflexive, anti-symmetrical and transitive < is anti-reflexive, anti-symmetric andtransitive.

Other Important Properties of Binary Relations

2.3 .9 Property (Equivalence) R is an equivalence relations if, R is reflexive, symmetric and transitive.

2.3.10 Property (Partial order)R is a partial order relation if it is reflexive, anti -symmetric and transitive.

2.3.11 Property (Total order)

R is a total order relation if it is partial order and for all u, v R, (u, v) R or

(v,u) R.

Example: ( 2.9 )

Let us consider the binary relation subset of . It is clear that we have a partialorder relation. The relation on natural numbers is a total order relation.Consider the relation mod 3 on natural numbers.

{(m, n)| (n-m) mod 3| 0}This is an equivalence relation.

2.3.1 2 Definition Let X and Y be non empty sets. A fuzzy relation R is a fuzzy subset of XxY. In

other words, R F (XxY). If X = Y then we say that R is a binary fuzzy relation in


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X. Let R be a binary fuzzy relation onR . Then R (u, v) is interpreted as the

degree of membership of the ordered pair (u, v) in R.

Example : ( 2.10) A simple example of a binary fuzzy relation on

U= {1, 2, 3}, called approximately equal can be defined asR (1, 1) = R (2, 2) = R (3, 3) =1.R (1, 2) = R (2, 1) = R (2, 3) = R (3, 2) =0.8R (1, 3) = R (3, 1) = 0.3The membership function R is given by

Q

!

!

!

!

2vuif .

1vuif 8.

vuif 1

)v,u(

In matrix notation it can be represented as

18..8.18.23.8.11

321

2.4 Fuzzy Graphs

It was already mentioned that definitions 2.3.1 of a fuzzy relation can also beinterpreted as defining a fuzzy graph. In order to stay in line with the terminology

of traditional graph we shall use the following definition of a fuzzy graph.

2.4.1 Definition

Let E be the (crisp) set of nodes. A fuzzy graph is then defined by


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G (xi, x j) = {(xi, x j), GR Q (xi, x j) | (xi, x j) ExE}

If E is a fuzzy set, a fuzzy graph would have to be defined in analogy to definition(Zimmermann, 1985).

Example : ( 2.11)

a) Let E = {A, B, C}Considering only three possible degrees of membership, graphs could bedescribed as shown in figure (2.4)b) Let E = {x1, x2, x3, x4}, then a fuzzy graph could be described as~ (xi, x j) = {[(x1, x2), 0.3], [(x1, x3), 0.6], [(x1, x1), 1], [(x2, x1), 0.4], [(x3, x1), 0.2],[(x

3, x

2), 0.5], [(x

4, x

3), 0.8]}

Figure 2.4Example (2.11) (a) shows directed fuzzy binary graphs.

2.4.2 Definition

H~ (xi, x j) is a fuzzy sub graph of ~ (xi, x j) if H~Q (xi, x j) GS Q (xi, x j),

for all (xi, x j) E x E

H (xi, x j) spans graph G (xi, x j) if the node sets of H (xi, x j) and G (xi, x j) areequal, that is if they differ only in there ar c weights.

A B C

A

B

C

A

B C


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Fig 2.4 (b) Graphs are not forests

2.4.4 Definition

A fuzzy graph G is a pair of functions G: ( , ) where is a fuzzy subset of a nonempty set and is a symmetric fuzzy relation on . The underlying crisp graphof G: ( , ) is denoted by G* ( , E) where E x . A fuzzy graph G is complete

if (uv) = (u) (v) for all u, v where uv denotes the edge between u and v.

2.4.5 Definition

Let G: ( , ) be a fuzzy graph. The degree of a vertex u is dG (u) = { vu

(uv).

Since (uv) > 0 for uv E and (uv) = 0 for uv E, this is equivalent to

dG (u) = QEuv

(u v).

The minimum degree of G is

H(G) = {d(v) / v }The maximum degree of G is

(G) = {d (v) / v }.

1

1

1

1

1


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2.4.6 Definition

The strength of connectedness between two vertices u and v is (u, v) = sup {Qk (u, v) / k = 1, 2}

Where Qk

(u, v) = sup {Q(uu1) Q(u1 u2) .. Q(uk-1,v) / u1, u k-1 }

2.4.7 Definition An edge uv is a fuzzy bridge G: ( , ), if deletion of uv reduces the strength of connectedness between pair of vertices.

2.4.8 Definition A vertex u is a fuzzy cutvertex of G: ( , ) if deletion of u reduces the strength

of connectedness between some other pair of vertices.

2.4.9 Definition Let G: ( , ), be a fuzzy graph such that G*: ( , E) is a cycle. Then G is a fuzzycycle if and only if there does not exist a unique edge xy such thatQ(xy) = {Q(uv) / (uv)>0}

2.4.10 Definition

The order of a fuzzy graph G isO (G) =

vu

(u)

The size of a f uzzy graph G is

S (G) = Euv

Q(uv)

2.4.11 Definition

Let G: ( , ) be a fuzzy graph on G*: ( , E). if dG (v) = k for all v , (i.e) if eachvertex has same degree k, then G is said to be a regular fuzzy graph of degree K(or) a K-regular fuzzy graph.


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Example: ( 2.1 4)

Consider G*: ( , E) where = {u, v, w} and E = {uv, vw, wu}. Define G: ( , ) by (u) =0.5, (v) =0.7, (w) =0.6 and (uv) =0.5, (vw) =0.6, (wu) =0.5. Then

G is a complete fuzzy graph. But d (u) = (uv) + (vw) = 0.5 + 0.5 = 1 and d (v)= d (w) = 1.1 so G is not regular.

2.4.1 2 Definition

Let G: ( , ) be a fuzzy graph on G*. The total degree of a vertex u v is definedby

dG (u) = {

Qvu

(uv) + (u)

= Euv

Q(uv) + (u) = dG (u) + (u)

If each vertex of G has the same total degree K, then G is said to be a totallyregular fuzzy graph of total degree K or a K totally regular fuzzy graph.

Example: ( 2.1 5)

Consider G*: ( , E) where = {v1, v2, v3, v4} and E = {v1v2, v2v3, v3v4, v4v1}. DefineG: ( , ) by (v1) = 0.5 and (v2) = 0.4, (v3) = 0.7, (v4) = 0.5 and Q (v1v2) =

0.2, Q(v2v3) = 0.4, Q(v3v4) = 0.2, Q( v4v1) = 0.4 then d(vi) = 0.6 for all i = 1, 2, 3, 4.

So G is a regular fuzzy graph. But td (v1) = 1.1 { 1 = td (v2). So G is not totallyregular.

Example: ( 2.1 6 )

Consider G*: ( , E) where = {v1, v2, v3} and E = {v1v2, v1v3}. Define G: ( , ) by (v1) = 0.4 and (v2) = 0.8, (v3) = 0.7, and Q( v1v2) = 0.3 Q( v1v3) = 0.4 thentd(vi) = 1.1 for all i = 1, 2, 3. So G is a totally regular fuzzy graph. But d (v1) = 0.7{ 0.3 = d (v2). So G is not regular.


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Example: ( 2.1 7)

Consider G*: ( , E) where = {v1, v2, v3} and E = {v1 v2, v2 v3, v3 v1}. Define G: ( ,

) by (v1) = (v2) = (v3) = 0.4 and Q(v1 v2) = Q(v2 v3) = Q(v3 v1) = 0.3.Thend(vi) = 0.6 for all i = 1, 2, 3. So G is a regular fuzzy graph. Also td (vi) = 1 for all i =1, 2, 3. Hence G is also a totally regular fuzzy graph.

Example: ( 2.1 8 )

Consider G*: ( , E) where = {v1, v2, v3} and E = {v1v2, v3v1}. Define G: ( , ) by (v1) = 0.3, (v2) = (v3)= 0.4 and Q(v1v2) = 0.1, Q(v1v3) = 0.2 then d(v1) = 0.3 {

0.1 = d (v2). So G is not a regular fuzzy graph. Also td (v1) = 0.6 { 0.5 = td (v2). SoG is not totally regular fuzzy graph.


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Chapter - 3

APPLICATIONS OF FUZZY SET THEORY

3.1 Introduction

Applications of Fuzzy set theory can already be found in many different areas.These are classified as follows :-

1) Applications to mathematics, that is, generalizations of traditionalmathematics, such as topology, graph theory, algebra logic a nd so on.

2) Applications to algorithms such as clustering methods, control algorithms,mathematical programming, and so on.

3) Applications to standard models such as the transportation model,inventory control models, maintenance models, and so on.

4) Finally, applications to real -world problems of different kinds

3.2 Fuzzy Linear Programming

Linear programming models shall be considered as a special kind of decision

model. The decision space is defined by constraints; the goal (utility function) isdefined by the objective function; and the type of decision is decision makingunder certainty. The classical model of linear programming can be stated as,

maximize f(x) = cTxsuch that Ax b

x 0 ----- (1)

with c, x R n, b R m , A R mxn .

Let us now depart from the classical assumptions that all coefficients of A, b andc are crisp numbers, that e is meant in a crisp sense and that maximize is astrict imperative !

If we assume that the LP-decision has to be made in fuzzy environments, quite anumber of possible modifications model Fuzzy decisions exist. First of all, the


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decision maker might not really want to actually maximize or minimize theobjective function. Rather, he or she might want to reach some aspiration levelsthat might not even be definable crisply. Thus he or she might want to improve

the present cost situation considerably, and so on .

Secondly, the constraints might be vague in one of the following ways : The sign might not be meant in the strictly mathematical sense, but smaller violationsmight well be acceptable. This can happen if the constraints represent aspirationlevels as mentioned above or it, for instance, the constraints represent sensoryrequirements (taste, colour, smell. etc) that cannot adequately be approximatedby a crisp constraint. Of course, the coefficients of the vectors b or c or the matrix

A itself can have fuzzy character either because they are fuzzy in nature or because perception of them is fuzzy.

Finally, the role of the constraints can be different from that in classical linear programming, where the violation of any single constraint by any amount rendersthe solution infeasible. The decision maker might accept small violat ions of constraints but might also attach different (crisp or fuzzy) degrees of importanceto violations of different constraints. Fuzzy linear programming offers a number of

ways to allow for these types of agueness and we shall discuss some of thembelow.

First of all, one can either accept Bellman Zadehs concept of a symmetricaldecision model or develop specific models on the basis of a nonsymmetricalbasic model of a fuzzy decision (Orlovsky, 1980; Asia et.al., 1975). Here weshall adopt the former, more common, approach. Secondly, one has to decidehow a fuzzy maximize is to be interpreted, or whether to stick to a crisp

maximize. In the latter case, complications arise on how to connect a crispobjective function with a fuzzy solution space. We will discuss one approach for afuzzy goal and one approach for a crisp objective function.

Finally, one has to decide where and how fuzziness enters the constraints. Someauthors (Tanaka and Asai 1984) consider the coefficients of A, b, c as fuzzy


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numbers and the constraints as fuzzy functions. We shall here adapt another approach that seems to be more efficient computationally and more closelyresembles Bellman Zedehs model. We shall represent the goal and the

constraints by fuzzy sets and then aggregate them in order to derive amaximizing decision.

In both approaches, one also has to decide on the type of membership functioncharacterizing either the fuzzy members or the fuzzy sets representing goal andconstraints. In classical LP, the iolation of any constraint in model (1) rendersthe solution infeasible. Hence all constraints are considered to be of equal weightor importance. When departing from classical LP, this conclusion is no longer

true, and one also has to worry about the relative weights attached to theconstraints.

Before we develop a specific model of linear programming in a fuzzyenvironment, it should have become clear that in contrast to classical linear programming, fuzzy linear programming is not a uniquely defined type of model;many variations are possible, depending on the assumption or features of thereal situation to be modeled.

3.1.1 Symmetric Fuzzy Linear Programming

Let us now turn to a first basic model for fuzzy linear programming. Weshall assume that the decision maker can establish as aspiration level Z, for thevalue of the objective function he or she wants to achieve and that each of theconstraints is modeled as a fuzzy set. Our fuzzy LP then becomes :

Find x such thatcTx u Z

Ax ~e b

x 0 ------(2)


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are very well satisfied (i.e satisfied in the crisp sense); andQi (x) should increasemonotonously from 0 to 1, that is,

Qi (x) =

V

W

X

W !e

e

iii

iiii

ii

pdxif 1m,.......,1i

pdxdif ]1,[dxif 1

------(7)

Using the simp lest type of membership function, we assume them to be linearlyincreasing over the tolerance intervalp i :

Qi (x) =

Y

"

!e

e

iii

iiii

i

ii

ii

pdxif

1m......,,2,1i,pdxdif dx1

dxif 1

------(8)

The p i are subjectively chosen constants of admissible violations of theconstraints and the objective function substituting equation (8) into problem (6)yields, after some rearrangements [Zimmermann 1976] and with some additionalassumptions,

Qu

i

ii dxB1i0x

minmax ------(9)

Introducing one new variableP, which corresponds essentially to equation (4)we arrive at

maximizeP such that Ppi + Bix di + pi , i = 1, ., m+1

x 0 ------(10)If the optimal solution to problem (10) is the vector (P, x0), then x0 is themaximizing solution (6) of model (2), assuming membership function as specified

in (8).

We should realize that this maximizing solution can be found by solving onestandard (crisp) LP with only one more variable and one more constraint than inmodel (3). Consequently, this approach is computationally very efficient.


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A slightly modified version of models (9) and (10) respectively, results if themembership functions are defined as follows : A variable ti, i=1, ., m+1, 0 t i pi is defined that measures the degree of violation of the ith constraint. The

membership function of the ith

row is then

Qi (x) = 1 -i

i

pt ------(11)

The crisp equivalent model is thenmaximizeP

such that Ppi + ti pi, i=1, .., m+1Bix ti di

ti pi x, t 0 ------(12)

This model is larger than model (10) even though the set of constraints ti pi isactually redundant. Model (12) has some advantages, however, in particular when performing sensitivity analysis, which will be discussed in the secondvolume on decisions in fuzzy environments.

Example : ( 3.1)

A company wanted to decide on the size and structure of its truck fleet. Four differently sized trucks (x1 through x4) were considered. The object was tominimize cost, and the constraints were to supply all customers. This meantcertain qualities had to be moved and a minimum number of customers per dayhad to be contacted. For other reasons, it was required that at least six of thesmallest trucks be included in the fleet. The management wanted to usequantitative analysis and agreed to the following sugg ested linear programmingapproach.

minimize41,400 x1 + 44,300 x2 + 48,100 x3 + 49,100 x4

subject to constraints0.84 x1 + 1.44 x2 + 2.16 x3 + 2.4 x4 170


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16 x1 + 16 x2 + 16 x3 + 16 x4 1,300x1 6

x2, x3, x4 0

The solution was x1 = 6, x2 = 16.29, x3 = 0, x4 = 58.96, and min cost = 3,864,975.When the results were presented to management, it turned out that the findingswere considered acceptable but that the management would rather have someleeway in the constraints. Managements felt that because demand forecastshad been used to formulate the constraints, there was a danger of not being ableto meet higher demands by their customers. When they were asked whether or not they really wanted to minimize transportation cost, they answered : Now you

are joking. A few months ago you told us that we have to minimize cost;otherwise, you could not model our problem. Nobody knows minimum costanyway. The budget shows a cost figure of $4.2 million, a figure that must not beexecuted. If you want to keep your contract, you better stay considerably belowthis figure.

Since management felt forced into giving precise constraints in spite of the factthat it would rather have given some intervals, model (3) was selected to model

the managements perception of the problem satisfactorily. The followingparameters were estimated :Lower bounds of the tolerance interval :

d1 = 3,700,000, d2 = 170, d3 = 1,300, d4 = 6Spreads of tolerance intervals,

p1 = 500,000, p2 = 10, p3 = 100, p4 = 6 After dividing all rows by their respective pis and rearranging in such a way thatonlyP remains on the left-hand side, our problem in the form of (10) became

MaximizeP subject to constraints0.083 x1 + 0.089 x2 + 0.096 x3 + 0.098 x4 + P 8.4

0.084 x1 + 0.144 x2 + 0.216 x3 + 0.24 x4 - P 17

0.16 x1 + 0.16 x2 + 0.16 x3 + 0.16 x4 - P 13

0.167 x1 - P 1


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Zimmermann (1983, 1991, 2000) reviewed development and applications in thefield of fuzzy dynamic programming.

Traditional economic criterion (maximization of profit and minimization of cost)models are indeed successful for many real inventory problems. However, thereare some inventory problems for which the economic criterion models are notapplicable including reservoir operation problems as well as s ome retail inventoryproblems. But, the fuzzy criterion models are just able to incorporate the expertknowledge via fuzzy membership functions. So, fuzzy criterion models are closer to the spirit of modern decision-making Turban (1998), than the existing inventorymodels. Consider a multistage decision making inventory control system in which

recorder quantities di (i = 1, 2, ) are the decision variables and x i (1,2,) arethe different states of the system. A recorder for di items is to be done at thebeginning of each state. The decision maker should be able to evaluate the finalstate. We assume that back logging is not allowed, since items in the inventoryare perishable. The following questions should be able to be answered by thedecision maker, which states are the best? Which states are qualified and whichstates are too bad? Another important issue in perishable inventory controlsystem is due to the nature of the stock. The goal of the problem is to minimize

the stock at the end of the planning horizon. In this model, we focus our attentionon inventory systems having perishable items. The main purpose of this modelfor these types of inventory systems. We also show that the optimal inventorycontrol for this kind of inventory the system is obtained as a natural extension of ordinary inventory control system. Optimal schedules, the final inventory with lowor zero level are obtained for different values of perishable parameter.

3.3.2 Model Description (Fuzzy)

The following notations and assumptions are used in developing themodel. Consider an inventory control system with N periods. The order quantity ineach period i is assumed to be a fuzzy variable, di D where D = {0,H, 2H, ..} is

the set of values permitted for the decision. Let xi X, i = 1, 2, .., N+1 be thestate variable representing the inventory level at the beginning of period I, whereX = {0,X, 2X, .}. Most of the cases X= H, where Xand Hare fundamental units of


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Fig 3.1 illustrates the basic fuzzy dynamic programming structure. Theformulation of the problem is as follows.

max Ri (xi, di) = max { r i (xi, di) * Ri+1 (xi + 1) }di diSuch thatxi+1 = ti (xi, di)

= xi pi + di ai, i = 1, 2, ., N-1

where pi = ,xi X

-

Xe

0 e E e 1

(or)

max Ri (xi, di) = max { r i (xi, di) * Ri+1 (ti (xi, di)) }di di

3.3.4 Theorem

The fuzzy dynamic programming problemmax (Ri (xi, di)) = max { r i (xi, di) * Ri+1 (xi+1)}di di

Wherexi+1 = ti (xi, di)

= xi pi + di ai, i = 1, 2, , N-1, 0 e E e 1 subject to the fuzzy constraints

),d( if ~iQ ),x( g h ~ iQ has the optimal maximizingdecision.

.....,,0i},d{D~ i0 !! for a givenx0

x0 x1 x2 xN xN+ 1i 0

d 0 0 d 1 1 d N P N

Z0 ZN

0 0 0 000 0 0


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wherever K is a constant and g is an arbitrary function of dN-1, the abovemembership can be rewritten as

)d....,,d( 0 1N00D~ 0Q = 1N0 d....,,dminmax

QQ

Q

)x(....,),d(

....,),d(

1NG~

2NC~

0C~

1N

2N

0

where,

!Q )x( 1NG~ 1N 1Ndminmax

QQ

))d,x(t(

),d(

1N1NN

G~1NC~ N1N

The optimal decision set 0D~ can be obtained by the above recursive function.

3.3.5 Solution Procedure

The general solution procedure of solving inventory control problem by fuzzydynamic program (Kacprzyk and Esogbue, 1996) approach is described.

Step 1 : We calculate the lower 'lix and upper 'uix , bounds using the forward

calculation

t

X

-

X

E! 1i'l 1i'l

1i'l1i

'li d

xx,0maxx

1iu

1i

'u1i'u

1i'u

i adx

xx X

-

X

E!

Step 2 : Then, the bounds 'uix and'lix from backward calculation are computed.

u1i1i

"li"l

i"l

1i daxxx X

-

X

E!


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l1i1i

"ui"u

i"u1i da

xxx X

-

X

E!

Step 3 : Then the final bounds are computed as follows.}x,x{minX "ui'uiui !

}x,x{maxX "li'lili !

Step 4 : From the above, compute

i

iiG~iC~iD~

d})}d,x(),d({min{max)x( QQ!Q

i

1iiiG~iC~iu ~

d})adpx(),d((m in{max)x( QQ!Q

Step 5 : For the specificE, observing the table for )x( 1D~Q , we get the optimal pairs (xi, di) with positive values. For each pair (xi, di)

select the corresponding pairs (xi, d j) from the table for )x( 2D~Q and continuing

this process, till D~Q (xN-1) to get all possible optimal schedule.

Example : ( 3.2 )

The general procedure explained in the theorem 3. 3.4 is applied for the periodicreview perishable inventory problem with number of periods N = 4. Assume thatthe demand occurs in each period t = 1,2,3,4 be a 1=45, a2=50, a3=45, a4=60. Theinventory on hand at the beginning of each period in nature is of perishable innature. The number of items perished pi in each period i may be directly

proportional to the inventory on hand in that period. So, we assume pi = X-

X

E ix

where E [0,1] the perishability factor is, andX is the fundamental unit of

inventory. Here, we take X= H= 5 and E = 0 for non perishable or usual inventory

system and E = 1, case is a perfect perishable inventory system which is non-

existance. So we assume the value E strictly lies between 0 and 1 (0e E 1).


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Case 1 :

In our problem, if we assume that E = 0.25, let the membership function of the

fuzzy constraints iC~ on the decision variables di be of the form,

)d( iC~ iQ = 0 if 0 di 50 10i

-3 + 0.5i + di / 20 50 10i < di 70-10i5 0.5i di / 20 70 10i < di 90 10i0 90 10i < di, where i = 1,2,3,4

The membership function of fuzzy goal iG~ representing the decision to have a

low stock at the end of the planning horizon )0x( 5 $ is given by

v

ee!Q

020x0if 20/x1

)x( 555G~5

Let x0 be the initial stock or level at the beginning. The inventory level which issupposed to be zero and the permitted state values for the recorder quantitiesdi D, be given by ak {0, 5, 10, } and that of the possible inventory levels xi x

be given by X j {0, 5, 10, }. We are only concerned _ a0)d(/d iC~i i "Q thesupport of fuzzy constraint set tC~ . Now the bounded decision variables are

obtained as in the following table ( lid - lower bound, uid - upper bound).

Here 0 x5 20Table 1

i lid uid

1 55 85

2 45 75

3 35 65

4 25 55


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Using the transformation function, we can also find upper and lower bounds for the state variables on the different intermediate stages by the following three

steps. By using step 1, calculate the lower 'lix and upper 'uix bounds state

variables xi as follows :

Table 2

i 'lix'u

ix 1 - -

2 10 40

3 0 55

4 0 60

5 - -

By step -2, starting with x5 and assumingIIl5x ,

IIu5x = 20, we obtain recursively

the following upper and lower bounds.

Table 3

i "li

x "Iuix 1 0 120

2 0 105

3 0 80

4 5 55

5 - 15

From the above 2 tables, the final upper and lower bounds can be determined by

using step 3. Hence we get,


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Stage 2 :

Table - 6

d 3 x 3

35 40 45 50 55 60 65 )( 3G~ xQ 0 0 0 0

5 0 0 0 0

10 0 0 0 0

15 0 0 0 0

20 0 0 0 0

25 0 0 0 0 0

30 0 0 0 0 0 35 0 0 0 0 0 0

40 0 0 0 0 0 0 0 0

45 0 0 0 0 0 0 0 0

50 0 0 0 0 0 0 0 0

55 0 0 0 0 0 0 0 0

Stage 3 :

2

4222G~1C~2D~

d

})adpx(),d((m in{max)x( QQ!Q

Table - 7

d 2

x 2 45 50 55 60 65 70 75 )( 2G~ xQ

10 0 0 0 0

15 0 0 0 0

20 0 0 0 0 0 25 0 0 0 0 0 0

30 0 0 0 0 0 0

35 0 0 0 0 0 0 0 0

40 0 0 0 0 0 0 0 0


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Stage 4 :

1

1111G~1C~1D~

d

})adpx(),d((m in{max)x( QQ!Q

Table 8

d 1

x 1 55 60 65 70 75 80 85 )( 1G~ xQ

0 0 0 0 0 0

Case 2 : If the perishable factor E = 0. 5 , then we have

Table 9

i lix uix 1 0 0

2 10 40

3 0 45

4 5 20

5 - 15

Stage 1 :Table 10

d 4

x 4 25 30 35 40 45 50 55 )( 4G~ xQ

5 0 0 0 0 0 0 0 0

10 0 0 0 0 0 0

15 0 0 0 0 0 0 1 1

20 0 0 0 0 0


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Stage 2 :Table - 11

D3

x 3 35 40 45 50 55 60 65 )(

3G~ xQ

0 0 0 1 0 1

5 0 0 1 0 1

10 0 0 0

15 0 0 0

20 0 0 0 0

25 0 0 0 0

30 0 0 0 0

35 0 0 0 0

40 0 0 0 0 0

45 0 0 0 0 0

Stage 3 :

Table - 1 2

D2

x 2

45 50 55 60 65 70 75 )( 2xD~Q 10 0 0 0

15 0 0 0

20 0 0 0 0

25 0 0 0 0

30 0 0 0 0 0

35 0 0 0 0 0

40 0 0 0 0 0 0 0 0

Stage 4 :

Table - 1 3

d 1

x 1 55 60 65 70 75 80 85 )( 1G~ xQ

0 0 0 0 0 0


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Optimal Schedule : From the above, we form an optimal schedule for variousvalues of perishable parameter E.

Case 1 : when E = 0.25

Case 2 :

When E = 0.5Case 1 :

Case 2 :

It is observed that, the cases with inventory level low or zero gives the optimalinventory. (i.e.) we could get the optimum schedule for perishable parameter. The

y

(0,55 )

045

(10 ,6 0)

5

5 0

(15 ,2 0)

5

45

(15 ,55 )

5

6 0

5

(0,6 0)

045

(15 ,55 )

5

5 0

(15 ,5 0)

5

45

(15 ,55 )

5

6 0

5

(0,6 0)

045

(15 ,6 0)

105 0

(15 ,55 )

1045

1(15 ,55 )

106 0

0

(0,6 0)

045

(15 ,6 0)

105 0

(15 ,5 0)

1045

(10 ,55 )

5

6 0

0


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optimal schedules for various perishable rates E have been compared. We

observe that when E 0.5 the final inventory position becomes exactly zero,

which is the objective of our criteria. Thus asE increases, the optimal schedule

becomes more accurate and crisp.

3.4 Tow stage Fuzzy Transportation Problem

3.4.1. Introduction

The transportation problem refers to a special class of linear programmingproblems. In a typical problem a product is to be transport from m source s to ndesignations and their capacities are a 1, a2, ., a m and b1, b2, ., b n

respectively. In addition there is a penalty cij associated with transporting unit of product from source i to designation j. This penalty may be cost or delivery timeor safety of delivery etc. A variable xij represents the unknown quantity to beshipped from source i to destination j.

In some circumstances due to storage constraints, designations are unable toreceive the quantity in excess of their minimum demand. After consuming part of whole of this initial shipment, they are prepared to receive the excess quantity in

the second stage. According to Sonia and Rita Malhotra (2003), in suchsituations the product transported to the destinations has two stages. Justenough of the product is shipped in Stage I so that the minimum requirementsof the destinations are satisfied and having done this the surplus quan tities at thesources are shipped to the destinations according to cost consideration. In boththe stages the transportation of the product from sources to the destinations isdone in parallel. The aim is to minimize the sum of the transportation costs in thetwo stages.

3.4.2 Definition

A real fuzzy number a is a fuzzy subset of the real number R with

membership function aQ satisfying the following conditions.

1. aQ is continuous from R to the closed interval [0,1]


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2. aQ is strictly increa sing a nd continuous on [a 1, a 2]

3. aQ is strictly decrea sing a nd continuous on [a 3, a 4]where a 1, a 2, a 3 &a 4 a re rea l numbers a nd the fuzzy number denoted bya = [a 1, a 2, a 3, a 4] is ca lled fuzzy tr a pezoida l number.

3.4.3 Definition

The fuzzy number a = [a1, a2, a3, a 4] is trapezoidal number, denoted by [a1, a 2,

a3, a4] its membership function aQ is given by figure (3.1)

Fig (3.1) Membership function of a fuzzy number a~

3.4.4 Definition

The E-level set the fuzzy number ba nda is defined a s the or dina ry set LE (b,a ) for which the degree of their membership function exceeds the level

E [0,1].

}n......,2,1,m......,2,1i,)b,a(/b,a{)b,a(L iam !!EuQ!E

3.4.5. Theoretical Development

Let ~ e the minimum fuzzy requirement of a homogenous product at the

destination and ia~ the fuzzy availa ility of the same at source i. The Two-stage

Fuzzy Cost Minimization Transportation Pro lem (FCMTP) deals with supplyingand destinations their minimum requirements in stage I and quantity

j

ji

i ba is supplied to the destinations in stage II, from the source which

Qa(x)

x0 a1 a2 a3 a4

.


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3.4.6 Definition ( E - optimal solution)

A point x* X ( ia , jb ) is said to be E-optimal solution (E-Two stage FCMTP), if

and only if there does not exist another x, y X (a,b), a, b LE ( ia , jb ), such that

ci j, xi j, ci j xi j, with strict inequality holding for at least one ci j where for

corresponding values of parameters ( ia , jb ) are called E-level optimal

parameters. The problem (E-Two stage FCMTP) can be rewritten in the following

equivalent form (E - Two stage FCMTP)

!!

!!

!

!

!

m

1i ji j

n

1 jii j

n.....,2,1 jbx

m.....,2,1iax

S

o j j

0 j

0i j

0i bhah eeee

xi j 0 , for all i, j

It should be noted that the constraint (a i, b j LE ( ia , jb ) has been replaced by the

constraint. o j j0 j

0ii

0i bhandah eeee where 0ih and 0i and 0 jh and 0 jH are

lower and upper bounds and a i, b j are constraints. The parametric study of theproblem (E Two stage FCMTP) where 0ih , 0iH and 0 jh ,

0 jH are assumed to be

parameters rather than constraints and (renamed h i, Hi and h j, H j) can beunderstood as follows. Let X (h, H) denotes the decision space of pro blem(E - Two stage FCMTP), definedby

X (h,H) = (xi j, ai, b j) R(n(n+1)) | ai - j

xij 0

b j -i

xij 0, Hi ai 0, H j b j 0,

a i h j 0, b j h j 0, xij 0, i I, j J


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2 3 5 11 4 2 3

4 7 9 5 10 4 4

12 25 9 6 26 12 38 7 9 24 10 8 3

1 2 3 3 2 2

After applying the AM method, we get x12 = 1, x15 = 2, x21 = 1, x22 = 1, x26 = 2,x34 = 3, x34 = 3 and minimum z = 75.

Stage II

We take a1 = 3, a2 = 4, a3 = 4, a4 = 4b1 = 2, b2 = 3, b3 = 2, b4 = 3, b5 = 2, b6 = 3

2 3 5 11 4 2 3

4 7 9 5 10 4 4

12 25 9 6 26 12 3

8 7 9 24 10 8 32 3 2 3 2 3

After applying the AM method, we get x12 = 1, x15 = 2, x21 = 2, x22 = 2, x33 = 1,x34 = 3, x43 = 3 and minimum z = 93.

There fore the optimal value of the objective function of the problem (16) is givenby

Minimum (75 + 93) = 168


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Conclusion

In this study, we presented the basic concept of fuzzy set theory, fuzzy

measures and relations and applications of fuzzy set theory to solve linear programming problems, dynamic programming problems and two stagetransportation problems.


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