)
Quasi-fuzzy Computation Model
by
Chien-Chung Wang
Thesis submitted in fulfillment of the requirements for the degree of Doctor of Philosophy in the School of Computer Science and Engineering of the University of New South Wales Sydney, NSW 2052, Australia
January 1997 PLEASE TYPE UNIVERSITY OF NEW SOUTH WALES Thesis/Project Report Sheet
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Abstract 350 words maximum: (PLEASE TYPE)
The thesis presents a new model of computation on fuzzy numbers. It permits for linear complexity of the arithmetic canputations, while retaining a good approximation to their membership functions. For func tion evaluation each argument is approximated by the rrembership curve, whose geometric shape is a family of trapezoids, placed one above the other. The heights and the number of trapezoids are the same for all arguments. The computation is perforrred by the pointwise evaluation separately for the corresponding vertices of the multitrapezoidal curves� The concluding step is to convert resulting multitrapezoidal shape to a fuzzy number of the type rratching that of the arguments. This method is universal, in that it can be applied to any fuzzy num bers. In particular, it serves well in arithmetic on mixed type arguments - a continuous and discrete ones. This method is based on a new concept of fuzzy multisets. They permit any given element of the basic da:nain to occur with several, different membership grades. This is more specific than the older notion of fuzzy bags which constitutes crisp multisets on fuzzy domains. It is used to formalize several constructs of fuzzy theory, both from quasi fuzzy computations and from other applications. Two applications are discussed, showing the usefulness of these con cepts. One is the economic inventory problem on fuzzy quantities, the other deals with multiple possibility assignments. The thesis concludes by considering a few possible lines of future investigations. Quasi-fuzzy arithmetic on nonnumerical domains is con sidered; a very general form of multisets is outlined; uncertainty measures for multipossibility assignments are suggested.
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THISSHEET IS TO BEOLUEDTOTHEINSIDE FRONT COVER OFTHESIS THE Abstract
The thesis presents a new model of computations on fuzzy numbers. It per mits for linear complexity of the arithmetic computations, while retaining a good approximation to their membership functions. For function evaluation each ar gument is approximated by the membership curve, whose geometric shape is a family of trapezoids, placed one above the other. The heights and the number of trapezoids are the same for all arguments. The computation is performed by the pointwise evaluation separately for the corresponding vertices of the multitrape zoidal curves. The concluding step is to convert resulting multitrapezoidal shape to a fuzzy number of the type matching that of the arguments. This method is universal, in that it can be applied to any fuzzy numbers. In particular, it serves well in arithmetic on mixed type arguments - a continuous and discrete ones. This method is based on a new concept of fuzzy multisets. They permit any given element of the basic domain to occur with several, different membership grades. This is more specific than the older notion of fuzzy bags which constitutes crisp multisets on fuzzy domains. It is used to formalize several constructs of fuzzy theory, both from quasi-fuzzy computations and from other applications. Two applications are discussed, showing the usefulness of these concepts. One is the economic inventory problem on fuzzy quantities, the other deals with mul tiple possibility assignments. The thesis concludes by considering a few possible lines of future investi gations. Quasi-fuzzy arithmetic on nonnumerical domains is considered; a very general form of multisets is outlined; uncertainty measures for multipossibility assignments are suggested.
i Acknowledgements
I am much indebted to my PhD Advisor, Professor Arthur Ramer, for his guid ance and support throught the four years of my doctorate research. I express special thanks to my MBA advisor, Professor Shan-Huo Chen, for his valuable discussions and insights. I would like to thank my colleagues and friends, Banchong Harangsri, Hairong Yu, Qi Chen, Jiaming Li, Yuling Chen, Werasak Kurutach, Jinsong Ouyang, Bing Ngu, Ivan Lou, Martin Li, Raymond Liu, and Shinn Sun, for their assistance in these four years. I would like to acknowledge the support of the Ministry of Defense, Republic of China, for offering me the scholarship during my studies 1993-96. I thank the University of New South Wales for providing travel fees for the international conferences in USA, Brazil and Taiwan. Lastly, I would like to express my gratitude to my family for their constant support and encouragement.
11 Contents
1 Overview 1
2 Definitions and notations 4 2.1 Basic terminology . 4 2.2 Fuzzy sets ..... 8 2.3 Trapezoidal and multitrapezoidal numbers 13 2.4 Multisets and fuzzy multisets ...... 17
3 Arithmetic of fuzzy sets 20 3.1 Earlier computational models . 23 3.2 Quasi-fuzzy computation . . . 29 3.3 Multitrapezoidal approximations 30 3.4 Approximating discrete fuzzy numbers . 36 3.5 Conversion of the results ...... 41 3.6 Properties of quasi-fuzzy arithmetic 43
4 Fuzzy Multisets 47 4.1 Fuzzy multisets ...... 48 4.2 Arithmetic on fuzzy multisets . 50 4.3 Coordinate arithmetic on fuzzy multisets . 53
111 5 Two applications 56 5.1 Economic order quantity 57 5.2 Multi-possibility 66
6 Concluding remarks 70
Bibliography 72
iv Chapter 1
Overview
Fuzzy quantities are more informative than the crisp values. When a numeri cal concept, like price, temperature, distance, or many others, is described by a crisp real number there is no room for expressing any imprecission of the values. When the same concept is described by a fuzzy number, several possible val ues are indicated, together with the likehoods that these values are in fact precise [10, 21, 26, 42]. Arithmetic operations on crisp numbers are one step procedures. In the analy sis of computer algorithms each of the four arithmetic operations is considered as a constant time computation. Arithmetic operations on the fuzzy numbers require processing sets of values [9, 11, 17]. Traditional method of extension principle [43, 47] has quadratic complexity. In addition, this method, whether implemented directly or through a-cuts [10, 19], may lead to unexpected, sometimes paradoxi cal results. If the fuzzy numbers are restricted to the simplest possible shape - trapezoids, an alternative method of function principle [6, 7] can be applied. It can be ex pected that combining certain features of these two methods would be beneficial. Our investigations were motivated by combining the advantages of the shapes
1 based on trapezoids with the relative simplicity of the a-cuts. It led to proposing a new model for fuzzy arithmetic, termed quasi-fuzzy computations. In this model all the fuzzy numbers, whether continuous or discrete, are ap proximated by certain piecewise-linear membership functions. These functions have a geometric outline of a family of trapezoids, placed one above the other. The resulting fuzzy numbers and their membership functions are termed multi trapezoidal. Computations on such multitrapezoidal structures are conducted by operations on pairs of the matching vertices. In case of single trapezoids with four vertices, this corresponds to function principle. For multi trapezoids the number of vertices can be any multiple 4n, where the number n corresponds to the number of a cuts on which the computation is based. This method of computing keeps the complexity linear, while allowing for a good approximation of arbitrary fuzzy numbers. We describe the method and its algebraic properties in Chapter 3. In our investigation we found necessary to permit for fuzzy entities whose membership functions have graphs containing 'vertical intervals'. They are mul tivalued functions, which would not be admissible in the basic definition of fuzzy sets. However, their exclusion leads to all definitions and most of the computa tional procedures becoming much more complicated. This is caused by having to consider several special cases, while carefully avoiding multiple membership values. We proposed a simpler method of fuzzy multisets. We define them by permit ting multiple membership values for the same element, but disallowing multiple occurrences of the same element with the same membership. Thus our definition is more specific than fuzzy bags which are crisp multisets based on fuzzy domains [18, 39, 40]. We define fuzzy arithmetic on the multisets. We prove several results on links
2 between fuzzy sets and fuzzy multisets for various computing methods. Fuzzy multisets are analyzed in Chapter 4. Two applications are discussed in Chapter 5. First we discuss a problem of economic optimization, dealing with the question of the optimal reorder quan tity [38]. We demonstrate how quasi-fuzzy computations can be used when the parameters of the problem are fuzzy [12, 29, 32, 33, 37]. The other topic is defining multipossibilities. We use fuzzy multisets to permit constructing assignments where the possibility of any given element need not be unique. We use it to extend the original Zadeh's definitions of possibilities of subsets and cartesian products [44]. We follow by defining continuous fuzzy multisets. We show how they can be used to present formally some concepts, which were in the past introduced not very rigourously. In the last chapter we outline a few possible directions for further investi gations. We believe that quasi-fuzzy operations could be extended to selected nonnumerical domains. We think that fuzzy multisets can be generalized further, to encompass both their present structure and that of crisp multisets. We sug gest that it might be possible to extend uncertainty measures to multi-possibility assignments [20, 30].
3 Chapter 2
Definitions and notations
2.1 Basic terminology
Our terminology includes terms from mathematics and from fuzzy theory. In one section we need a few specialized terms to describe certain economic quantities, which we define in that section. Mathematical concepts we need will refer to sets, functions, and to some extensions of these concepts. They are mostly studied in combinatorics, and we found an introductory chapter of one of several mono graphs to be the best overall reference. We leave the actual definitions to the next section. The terminology we use is kept as simple as possible to make it most descriptive and to make the arguments and proofs readily accessible. For that purpose we relax the use of certain terms as long as it would not cause an ambiguity. If a stricter definition would be required we state it explicitly. We use terms increasing function and decreasing function to denote, in gen eral a nondecreasing function and nonincreasing one respectively. We denote fi nite sets by enumerating their elements, but we do mark the ranges of various subscripts if they are clear from the context. In particular we may refer to a set
4 S = {s 1, ... , sn} the first time as S = {Si : i = 1, ... , n }, but later only as S = { si}. We adopt a similar method for describing multisets. We use the term multiset throughtout; it has been used in mathematics for at least 50 years, replac ing the older term bag. That latter term is used occasionally in fuzzy theory and in computer programming. From topology we need the notion of upper semicontinuous mapping. A real
function is upper semicontinuous at X 0 E X if
lim sup f(xn) ~ f(xo) n
for every sequence Xn --+ x 0 • This definition can be extended for set-valued map pings, defined on arbitrary metric spaces. In such general setting on can define an upper semicontinuous set-valued mapping, or an upper hemicontinuous mapping. Domains we need are either closed intervals or discrete sets of points. For such domains both extensions coincide and, moreover, are equivalent to a very useful closed graph property. If X is the domain of a set valued mapping F, with values being subsets of Y, we write
The mapping F is upper semicontinuous at x 0 E X if for every open set B con taining F(x0 )
F(x0 ) C B C Y
there exists an open set A containing x 0
x 0 E A C X
5 such that for every x E X F(x) c B.
Another very useful notion is that of a graph ofF. We define
G(F) = {(x,y): x EX, y E F(x)}.
For the two types of domains we need (intervals and discrete sets) upper semicon tinuity is equivalent to the graph G(F) being closed. The domains of fuzzy sets we use are either finite sets or intervals of the real axis. The first type is often referred to as being discrete, the second type as con tinuous. Membership grades (membership function) of the continuous type often have an uoutline of a trapezoid or a family of trapezoids, placed one above the other. Such trapezoids will be described by specifying the coordinates of their ver tices. In the diagrams we usually indicate all the more important coordinates
6 1 w
a b c d
We refer to the second case as multitrapezoidal; however, the term is a little cumbersome and we often simplify it to trapezoidal, even though a multitrape zoidal shape is meant. As we discuss in the next session, we shall consider a general multitrapezoid to have 4n vertices. In the specific cases, some of these coordinates, as well some vertices, may collapse into one. Such trapezoids rep resent mathematical, single-valued functions only if there are no vertical intervals present in their outline. Thus the diagram below presents not a standard function but a multi valued function.
7 a 4n
To make our discussion in the later chapters free from the conflicting terminol ogy we may prefer to describe such geometric features of the membership grades by using the terms trapezoidal curve or trapezoidal shape. A similar terminology, motivated by geometry, would be used for triangular fuzzy numbers or related objects.
2.2 Fuzzy sets
The most fundamental concept for our work is the fuzzy set. The notion itself was first proposed in 1965 and led to the development of a very large body of theoret ical and applied research. The basic definitions were proposed by Zadeh [43]and used ever since by all researchers in the field. A few excellent overview mono graphs were published in the last 10 year, and we adopt some of their notation. Although fuzzy theory originated from a single source only about 30 years ago, there is a fair variation of terminology and notation. We found those used in two monographs, by Dubois and Prade [10] and by Klir and Yuan [21] particularly clear and useful. A fuzzy set F is a pair consisting of the basic domain X and a membership function Jl : X -t [0, 1]. Occasionally, such F is referred to as a fuzzy subset of X.
8 The basic domain X is a standard set; in our thesis it will take one of two forms, either a finite set { a 1 , ... , an} or a closed interval of the real axis [a, b], a, b E R. In the latter case the domain is infinite, hence the membership function JL must be given by a formula. In the former case of a finite domain one can enumerate explicitly the membership values JL(ai), i = 1, ... , n. If done so, it is customary to use the sum notation proposed by Zadeh and write
p = JL(ai) + JL(a2) + ... + JL(an) a1 a2 an or even
We adopt one feature of notation from the publications of J. Buckley [ ]. Whenever using a letter symbol for a fuzzy set, he always places a tilde '""'' above it. It does not, in general indicate any transformation; it is there to remind us that the object is fuzzy. Therefore we always write A, B, F, etc. for fuzzy sets, sub sets, and multisets. We write X, Y, ... for ordinary sets, usually the basic domains of fuzzy sets.
The value of JL(x), x E X is often called the membership grade of x in F. It is always a number between 0 and 1, interpreted as signifying to what extent x should be counted as the element of F. Several authors require that the membership functions are normalized by the condition sup JL(x) = 1. xEX This property is often justified by saying that there must be at least one element in X which is 'fully possible'. For the analysis of the computational model this condition is not germane; our definition and computations work equally well for arbitrary w = supxEX JL(x) :::; 1. However, in the examples and diagrams we
9 adhere to the convention to keeping the membership grades normalized. The value JL(x) = 1 signifies a full membership; the collection of such ele ments is called the core C of F
C={xEX:JL(x)=1}.
A large part of our thesis deals with evaluating real functions f(x 1, •.• , xn), when applied to fuzzy arguments. Often, at the same time we need to speak of the membership functions which define these fuzzy arguments. An individual value of the membership function JL( x) is often called the membership grade of x. We adopt the word grade to denote the entire membership function. For example, we may write about a membership grade v(x) being a continuous one. We find that this makes clear the distinction between the real functions, which are evaluated, and the membership function, which describe objects. Our thesis deals with fuzzy arithmetic and fuzzy sets we consider are always defined on a numerical domain X. In particular, all a 1, ... , an would be real numbers. Although arbitrary finite sets X could be considered, there is no loss generality in using integers, and usually consecutive integers. Such finite sets X and their fuzzy subsets will be used in the examples. Finite fuzzy numbers are defined in analogy with the continuous ones. Their base domain is a finite range of integers [k, k + 1, ... , l]. Unimodality means that for some m and n, such that k ::; m :S n :S l the following hold for the membership grade v(x)
v(m) v(m + 1) = · · · = v(n) = argmaxv(x) X v(k) < v(k + 1) :S .. · :S v(m) v(n) > v(n + 1) 2: .. · 2: v(l)
10 w ------~ ...... / . .
k k+l m n 1
We use the lower case Greek letters J.L, v, ry, ... to denote the membership grades. They represent the fuzzy arguments on which we evaluate real func tions. The results of such evaluation will be always a fuzzy number on a real basic domain. The membership grades based on other domains do not appear in the applications of fuzzy theory; if needed, they could be evaluated using formu lae similar to those used for the real domains. We often need to have a symbol to denote the elements of the basic domain of either fuzzy arguments of a func tion or of the fuzzy number which is the value of the function. For the evaluation
B = f(Ar, ... , An) we typically write x 1 , ... , Xn for the elements of domains of
A 1 , ... , An, and z for the elements of B. When we need fuzzy variables, denoting entire fuzzy numbers, we use X, Y, Z or similar. A fuzzy set F is, in the strict terminology, a fuzzy subset of its basic domain X where every element x E X has a certain membership grade in F. Accordingly, we can refer to any element of X as also being an element of F. We adopt the same convention for fuzzy multisets. Describing fuzzy sets by their membership functions can be view as a descrip tion by 'vertical cuts' - for each x E X, we state the vertical placement of J.L( x). They could also be described by 'horizontal cuts'; this leads to one of the most important concepts linking fuzzy and crisp sets.
11 Given a fuzzy set F = (X, Jl(x)), for every a, 0 < a::::; 1 we define an a-cut
C1AF) = {x EX: Jl(X) ~a}.
It is immediate from this definition that Ca (F) C Cf3 (F) for a ::::; j3. Given any family of crisp sets {Ia}, indexed by a, 0 It may fail in general, but it is possible in the most important case when all Ia are closed intervals. In particular, it holds for the fuzzy numbers with the upper sernicontinuous membership functions. We discuss now some more important cases of the continuous membership function. In majority of applications of fuzzy theory the fuzzy sets based on a numerical domain have unimodal membership functions. For F with membership grade M( x) it means that the values of Jl( x) first increase with the increasing value of x, then remain at the maximum value within the core, and then decrease with x continuing to increase. In other words, the membership grade comprises three parts - monotonically increasing, core and monotonically decreasing. Such fuzzy sets are called fuzzy numbers. In all applications of fuzzy theory they serve to replace real numbers of the crisp computations. 12 A special case of this pattern is when the membership function assumes the shape of a trapezoid. It means that the increasing and decreasing parts are linear. We discuss this case and its generalization in the next section. 2.3 Trapezoidal and multitrapezoidal numbers In applications, the most common fuzzy numbers are those represented as geo metric trapezoids. They are called trapezoidal and give a special short notation. If the monotonically increasing region is [a, b], the core is [b, c), and the decreas ing region is [c, d] we denote the trapezoidal number (a, b, c, d; w), where w is its maximum membership value. 1 w a b c d An even more specialized case is when the membership function has the shape of a triangle, the number itself called triangular. As this is special case of a trapezoid with the upper base of length 0 we do not introduce new notation and simply write (a, b, b, c; w) to denote a triangular number. 13 w a b c For any fuzzy set F, based on the domain X, those elements x E X for which Jl( x) = 0 are considered as not belonging to F. Therefore, when the sum notation is used for the discrete numbers, it is customary to exclude such elements from the list. This creates a problem of distinguishing between the discrete fuzzy sets which originated from different continuous trapezoidal membership functions. Let us illustrate the situation with an example. 14 1 5 8 10 Number T 1 3 5 8 12 Number T 2 In some applications we may want to represent these trapezoidal numbers by the discrete numbers. The simplest approach would seem to be to list the four vertices of the trapezoid and hope that this would permit for reconstruction of the entire trapezoid. However, if applied directly, it does not work. In the examples above, the vertices ofT1 lie at the x values 1, 5, 8, and 10, leading to a number ~ = {0/1,1/5,1/8,0/10}. For the trapezoid T2 the corresponding discrete number becomes i2 = {0/3, 1/5, 1/8, 0/12}. 15 If we disregard the entries with 0 membership grade, we get il = i2 = {1/5, 1/8}. This makes impossible to reconstruct the original trapezoids. To permit such reconstruction we somehow need to indicate the base of each trapezoid. Our so lution is to consider the end-point of the lower base to have the membership grade not 0, but an infinitesimal, positive value E. In actual application such E can be set to some small positive value like 0.00001. Now we can make the distinction between t1 and t2 t1 {c/1,1/5,1/8,c/10} t 2 {c/3,1/5,1/8,c/12}. We will use the same method later, when dealing with more complicated shapes. Multi-trapezoidal numbers consist, as geometric objects, of several trapezoids, placed one above the other. To describe then we need to indicate the coordinate values of all vertices and their levels, that is their membership grades. Let us consider a general multitrapezoid T consisting of n simple trapezoids. Each simple trapezoid has 4 vertices, hence we expect T to have 4n vertices. We can verify it by counting the vertices at each level. Let us denote by w1 , w 2 , ... , Wn the height of the individual trapezoids which form T. Then the levels (membership grades) of the vertices will be 16 For convenience, we often use symbols There are 2 vertices at level 0, forming the lower base of the lowest trapezoid. There are 4 vertices at each of the levels h, ... , ln-I· For each levelli, two vertices form the top base of the i'th trapezoid, two other vertices form the bottom base of the i + 1 'st trapezoid. Finally there are two vertices at the levelln, forming the top base of the n 'th trapezoid. The total number of vertices, counted by level, is 2 + 4(n- 1) + 2 = 4n in agreement with the earlier number. To denote such multitrapezoidal structures we list all the coordinates first (separated by commas), then, after a semicolon, we list all the heights w 1, ... , Wn· A typical representation is We observe that the suport of the fuzzy number Tis the interval [a1 , a4n], while its core is [a2n, a2n+ll· 2.4 Multisets and fuzzy multisets The basic concept was introduced on many occasions in combinatorics and in computer science. Both these areas of research deal primarily with finite objects, in particular finite multisets. A finite multiset is defined as a finite collection of elements, each entering with a finite, positive multiplicity. 17 Therefore, given any multiset M we can explicitly enumerate its elements M ={a, a, b, a, c, c, b, .. .} or, more compactly In the first representation we simply list all the element of M. Each element is listed as many times as its multiplicity; the order in which the elements are listed does not matter. In the second representation we list only the distinct elements of M (again, the order does not matter), but we state their multiplicity explicitly. All the standard set operation can be defined on multisets. They are obtained by a direct extension of the corresponding definitions for sets, but may require a certain caution on occasion. For example, a powerset of a finite set is finite, but a power multiset is infinite. A function from set X to set Y is defined as a subset of X x Y. A function from multiset S to multiset Tis still a subset (not a submultiset) ofthe productS x T. A typical formal definition of a multiset M is a (base) set S and a function r : S--+ N, such that LsES r(8) < oo. The integer LsES r(8) is called the cardinality 1 or the number of elements of M. If M = {8k. : 8 E S} and P = {3 • : 8 E S} then the boolean operations are the natural definitions. Multiset M is contained in P whenever ks ::; ls for all 8 E S; intersection and union are MnP { 8min(k.,l.) : 8 E S} MUP { 8max(k.,l.) : 8 E S}. We discuss the detailed arguments leading to fuzzy multisets in Chapter 3. Here we only present the basic definition. This definition is presented indepen- 18 dently of the definition of fuzzy sets, which we introduce in the next section. We first remark that in the few articles of the fuzzy theory literature, the term 'fuzzy multiset' or 'fuzzy bag' are used to denote ordinary multiset, based on a fuzzy domain. We define it in a more restricted way, designed to capture certain specific features of fuzzy membership grades. For a finite basic domain X, we define a fuzzy multiset G as where any given xi can repeat a finite number of times, but if xi = xi then mi =f mi. This definition differs from [ ] in that we do not permit repeating the same x E X with the same membership grade. For a continuous X = [a, b] we consider only the multisets whose membership grade appear geometrically as a unimodal, continuous curve. It means that for any given x E [a, b], its membership grade r(x) is a unique real number in between 0 and 1, or r (x) = [o:, ,8], 0 .:::; o: .:::; ,8 .:::; 1. In the terminology of set-valued mappings, fuzzy multisets are upper-semicontinuous 2 and upper-hemicontinuous. In particular, their graphs are closed subsets of R . 19 Chapter 3 Arithmetic of fuzzy sets Fuzzy sets were introduced, at first, to give a quantitative expression to soft clas sification of concepts and their properties. The notion of the membership rage permitted qualifying the nuances of membership of an object in one of several categories, without the mandatory 0-1 commitment. One of the most important classifications is the one effected by the assignment of a numerical value to a selected parameter. Use of fuzzy sets permits such as signment become a soft one - the parameter in question can have one of several numerical values, each with a certain likelihood. Our knowledge about the actual value of the parameter is imprecise, and consists of the assignment of the likeli hoods. To make the consideration concrete, let us suppose that we are interested in the price of a pen, which we might expect to be about $5. Let us further assume that prices can be listed in steps of 5 cents (the least coin currently in circulation in Australia). Although we cannot be sure (without further inquiry) what pre cise price the pen may fetch on a given day in a given shop, we can reasonably summarize our knowledge of the expected prices. Our initial understanding of the price as 'about $5' implies that the membership grade of 5.00 is 1; we may also postulate that the prices very near $5 will be also at the maximum possibility. 20 For concreteness, let us suppose that any price in the range $4.50- $5.25 has the membership grade 1. We may also assume that the pen in question is costing no less than $3 (inflation!) and no more than $5.95 (competition!). While in some analyses the precise membership grade for each price in between $3.00-- $5.95 could be important, in our example we can assume the simplest pattern of mem bership grades. Remembering that the prices of less than $3.00 or more than $5.95 are excluded, we assign such values membership grade 0. We now assume a linear increase of membership grade from $2.95 to $4.50, and then a linear decrease of membership grade for price from $5.30 to $6.00. 1 I 3: 4.50 5.25 I 3.05 5.95 This simple model has all the main characteristics of modeling numerical quantities in the fuzzy context. The domain of the permissible values is dis crete, but corresponds to a continuous numerical range. In our example we have a set of integer multiples of $0.05; if we ignore the dollar sign and remember that the prices cannot be negative, the domain would consist of the numbers {0.0, 0.05, 0.1, 0.15, ...}. If we further assume that the price we may contem plate whether for a single pen, or even a collection of pens, cannot be above $100 (our personal limit!) the domain of prices becomes a finite, discrete set of num bers from 0.0 to 100.0, in increments of 0.05. Such domains are indeed standard in all the applications of fuzzy techniques to numerical analyses. 21 Our selection of the membership function, for the range of relevant prices, is also fairly typical. In our example there is a certain core range, within which each element (each price) has a full possibility value 1. There is a certain, reasonably wide, range of prices with non-zero, positive possibility. Finally, the change of the possibility value is linear with the increase (resp. decrease) of the prices. When the membership function is plotted as a function in the two-dimensional coordinate system, the shape it assumes is trapezoidal. Most of our assumptions about the likely price are necessary. There must be a contiguous support interval where the membership grade is greater than zero. There must be a smaller core interval, where the membership grade is maximal. It is unreasonable to assume that the 'maximal membership' would be reached in two or more separate segments of the support interval. The reason is that our membership function expresses our imprecise knowledge about the price, but the price itself concretely determined by the store. Permitting disjoint core segments would imply that the price itself is not definite but subject to some chance mech anism. This would require a probabilistic model and not a fuzzy one. A similar argument shows that the price change outside the core must imply a monotonic change of the membership grade. The intuition is obvious - we know the price imprecisely, but being closer to the core is more likely. The formalization of this intuition is best done using the a-cuts. If the price change was nonmono tonic, then there would exist a value a 0 , 0 < a 0 < 1, such that the cut Ca0 would not be contiguous. As the cut Ca0 represents the segment of prices of which values our certainty is at the level a 0 or higher, such segment must be contiguous. Oth erwise we would have two separate ranges, both of likehood at least a0 . Again, it would suggest presence of a chance mechanism, hence a probabilistic model. We conclude that the membership function representing imprecise knowledge of a numerical parameter should be unimodal. It should have a finite interval 22 as the domain of support, a smaller finite interval as its core, be monotonically increasing on the left of the core and monotonically decreasing on the right of the core. The only simplifying assumption we made was selecting a linear pattern to model the change of the membership grade. The very same assumptions hold true in almost all practical applications. Again, the use of trapezoidal membership functions is most common due to its conve nience in the numerical computations. However, this may sacrifice too much of the expressive power of the fuzzy grade. Moreover, the gain in simplicity is of ten only temporal. As we will see in a moment, trapezoidal shapes are preserved under addition and substraction, but not under other most basic operations, like multiplication or division. If maintaining the trapezoidal shape is a key objective, it can be achieved through use of the function principle. It permits a great simpli fication of all the fuzzy computations, but at a cost of significant departure of the results from those obtained through extension principle. A compromise method is the subject of this chapter. In essence, it approxi mates an arbitrary fuzzy number by a family of trapezoids, rather than a simple one, and then preforms all the operations by computations on the vertices alone. The form and quality of approximation can be varied depending on a specific ap plication. Another advantage of this compromise method is a uniform treatment of both the discrete and continuous fuzzy numbers. Lastly, it leads to the plau sible results for all the standard operations, avoiding certain patological cases of extension principle. 3.1 Earlier computational models The earlier method of calculating a fuzzy quantities was proposed by the founder of the theory. It is known as Extension Principle [13, 28, 43], a computational 23 rule which might be termed a fuzzy convolution [28]. We describe it here for the functions of two arguments, the modification to arbitrary many arguments immediate. Let f(x, y) be an arbitrary real function of two real arguments x and y. We wish to compute a fuzzy quantity j(A, B), for two fuzzy arguments A and B. For every z E Z, a possible value in the range of j(x, y) we consider all the pairs (x, y) such that f (x, y) = z. We then define 1-1 (z) as the highest possibility of the pair (x, y) which would lead to that value z. Possibility of the pair of values (x, y) is defined as the minimum of the possibilities of x andy separately. We can summarize it in a formal definition of 1-1(z) as the membership function of f(A, B). Let 17(x) be the membership function of A and v(y) the membership function of B. Then tL(z) = sup (rJ(x) A v(y)). x,y:z=f(x,y) If there are no pairs (x, y) such that j(x, y) = z then the supremum is over the empty set and it is thus considered to have value 0. This definition is equivalent to another method, based on a-cuts, for isotonic, upper-semicontinuous functions j(x, y) [10, 13]. Isotonicity means monotonicity (whether increasing or decreasing) of f(x, y) separately in each variable. Upper semicontinuity is a mild technical condition from general topology, satisfied both for the continuous membership functions and for the membership functions on the discrete domains. As these are the only types of functions we study in our work, we assume this condition throughout and consider the two methods equivalent for all properly monotonic functions of fuzzy arguments. In particular all arithmetic functions, all boolean functions, and all their compositions fall into their class. This extension principle and a-cuts is widely used in applications, generally with the trapezoidal fuzzy numbers as arguments [24, 25]. However, certain sig- 24 nificant compromises must be made to make the method work successfully in practice. These compromises are related to two theoretical difficulties. The first one is the matter of nontrapezoidal results. Assuming for a moment that A, B, 6, and so on, are all trapezoidal, we find that A+ B, A- B, min( A, B) and max( A, B) are also trapezoidal. But A x Band A/Bare not at all trapezoidal; rather, they are bound by the curves of higher degree. It is of degree 2 for a sin gle product or division, but of higher degree for more complex expressions. This means that in practice all such computations require numerical approximations. Such approximations require computational resources which grow quadratically with the complexity of the problem. This in turn implies that only fairly simple formulae, involving a handful of operations of multiplication, division, or higher algebraic function can be evaluated. Indeed, even a cursory scan of the articles presenting fuzzy applications, shows that almost all the formulae consist almost exclusively of addition, subtraction, and multiplication by real, crisp constants. The other difficulty lies in applying the extension principle to discrete ar guments. We illustrate it on two extremely simple examples. First, let E = {c/O, 0.5/1, 1/2, 0.5/3, c/4} and let us compute F = E +E. A simple ver ification shows that F = c/0, c/1, 0.5/2, 0.5/3, 1/4, 0.5/5, 0.5/6, c/7, c/8. E 1 • 0.5 • • '2. • 0 1 2 3 4 25 1 F 0.5 • • • • 0 1 2 3 4 5 6 7 8 - - - Two facts become apparent. Firstly, E + E is not the same as 2 x E. Sec- - - ondly, F has a somewhat unexpected shape. Although strictly speaking, both E and F are discrete, finite fuzzy numbers for which geometric shapes are not de fined, these numbers are ordinarily considered as representing some continuous membership function. If we assume, as it is usual in practice, that such contin uous functions are piecewise linear, we can find their shapes by connecting the dots. And this has an element of surprise. The argument E leads to a simple tri angular membership function on the interval [0, 2]. We could expect that E + E be also triangular or trapezoidal over the interval [0, 4]. This would be the case of - - 2 x E. However, F gives a plateau shape with a peak in the middle. This effect leads to a significant practical difficulty. The point is that in applications, such fuzzy number E would be viewed as representing a continuous triangular number - - - - - Z. However, E + E would not agree with Z + Z. The second example deal with boolean operations. Let us take two symmetric fuzzy numbers, both defined on the integers ranging from 0 to 4. We take A c/0, 0.5/1, 1/2, 0.5/3, c/4 B c/0,0.7/1,1/2,0.7/3,c/4 26 A straightforward verfication shows that c/0, 0.7 /1,1/2, 0.5/3, c/4 c/0,0.5/1,1/2,0.7/3,c/4 The result is somewhat puzzling. Neither A 1\ B nor A V B is symmetric. Also, the membership grades of A 1\ B lie above those of A, while the grades of A V B lie below those of B. Another method of defining arithmetic operations was proposed in 1986. It is called a function principle, and is defined only for the trapezoidal fuzzy numbers. Let us assume first that are two fuzzy numbers of the same height. Then where it is understood that the list of values on the right is put in the increasing order. For example, let A (1, 4, 6, 10; 1) B (0, 2, 5, 9; 1) Then A- B = (1, 2, 1, 1; 1) = (1, 1, 1, 2; 1). For the trapezoids of differing heights we first trim the trapezoids to the same height 27 w I _ _:______+_ w I I I I I I I I a b c d e f g h bl c I Tl T2 We replace the number (a, b, c, d; w) with (a, b', c', d; w'). Then the arithmetic proceeds as before. This trimming agrees with the extension principle computa tions, and all the cuts of T2 above the level w' are empty, the result of any oper ation will also have empty cuts at the levels above w'. This means that we can simply ignore that part of T1 which lies above the level w', which justifies the trimming. This method has the advantage of great computational efficiency. Only four operations are required to evaluate a function of two fuzzy arguments. The number of operations required to evaluate a more complex formula increases only linearly with the number of arguments in the formula. However, the method cannot be applied to the discrete fuzzy numbers or two fuzzy numbers of shape different than trapezoidal. In the next session we present our proposed method, which combines the lin ear complexity of the function principle with the universality of the extension principle. 28 3.2 Quasi-fuzzy computation We shall start by describing the intuitive model of the method proposed. Again we illustrate it with the ease of evaluating the function of f(A, B) of two fuzzy arguments. We begin by approximating both A and B by a multi-trapezoidal membership functions. We recall from Chapter 2 that a multi-trapezoidal function is defined as the boundary of family of trapezoids stacked one on top of the other. However, we do not permit completely arbitrary multi-trapezoids. We require that both A and B have the same number of heights for all the trapezoids. This means, in particular, that both A and B are approximated by the same number of trapezoids. Let us suppose that we require the same (w1 , ... , wn) for both A and B, but place no particular re- strictions on the values a 1 , ... , a4n, b1 ... , b4n. We note that the number of entries (ai) and (bi) is a multiple of 4; it is to present the multi-trapezoids is a general position. We remarked in Chapter 2 that often some of these entries may coincide, but they all must be included during the computations. The formula is now a direct coordinate-wise application of the function f (x, y). (We think of a1, ... , a4n as coordinates of A, b1 , ... , b4n as coordinates of B.) We define where it is understood that the entries f (ai, bi) on the right hand side are listed in the increasing order. 29 The formula, as presented, can be applied only to two multi-trapezoidal num bers of the same structure. We recall that being of identical structure means, for A and B, to have the same number of coordinates - 4n, and the same sequence of heights- w1, ... Wn. We also reiterate, as discussed in the Introduction, that we assume function f(x, y) to be defined for the entire ranges of x from a 1 to a4n andy from b1 to b4n. As a simple example, if f (x, y) = ~, we require that either the coordinates b1, ... b4n are all positive or all negative. In the next session we discuss how to apply the outlined method to arbitrary fuzzy numbers, whether discrete or continuous. 3.3 Multitrapezoidal approximations The basic premise is simple- if the arguments of a function are not all in the same structure trapezoidal form, we find a suitable approximation and perform all the arithmetic, algebraic, boolean and so on, operations on these approximate values. This apparent simplicity hides several problems which must be resolved. Some are technical, to which we give definite answers. Some are judgemental and de pend on the specific values of arguments as well as the specific application. An example of such problem is the selection of the sequence of heights for the trape zoidal approximations. As we wish to work with general fuzzy numbers, we denote them by X, Y, Z, ... , while retaining A, B, ... for the trapezoidal ones, usually the approximation of the general numbers. Once an aproximation was selected, say A :::: X, B :::: Y, and the function f(A, B) was evaluated, there remains a final step, It consists of interpreting the computed result. The value computed, say that of f(A, B), rep resents an approximation of the (hypothetical) f (X, Y). The latter should be de- 30 - - fined on a domain similar to those of X andY (for example, discrete) and should - - have a shape similar to those of X andY (for example, a smooth membership grade). We discuss the issues of such interpretation separately in Section 3.4. The remainder of this section is devoted to questions arising from the approximating process of the continuous fuzzy numbers. There is no generally applicable rule for selecting the sequence of heights wi, ... , Wn, or even for deciding how many heights should be selected. All we say is that the sum of the heights WI + · · · + Wn should be equal to the height of the arguments. If the arguments have different heights initially, the lowest height becomes the common height of all the approximations. This re quirement is technical and not judgemental; we discussed its necessity in Chapter 2. If the fuzzy numbers are normalized- common height 1 -the problem does not arise. This is an almost universal case in the applications of fuzzy arithmetic, and we assume in the following that WI + · · · + Wn = 1. If there is a need to discuss an exception, it will stated explicitly. Selecting specific WI, ... Wn is a task which resembles knowledge engineering. We want the trapezoidal shapes to approximate the original fuzzy numbers closely. This could suggest selecting several WI, ... Wn. At the same time we want the computations efficient, hence we prefer to keep the number n of heights rather small. 31 Finally, we will need to convert back the result f(A, B) to the continuous - - shape of the same type as the shape of X and Y. Having a large value of n makes the trapezoidal numbers appear smoother, but having a smaller n makes the actual conversion easier. We assume that the domain expert is the person who will decide on the specific w1, ... wn. Such decision need - - not be fixed for the duration. If the specific values of A and B change, the use of different sequences of heights might be indicated. Once selected, the heights determine the levels of the top edges of the trape zoids. In notation of Chapter 2: 32 ln WI + · · · + Wn = 1. We start by forming a-cuts at these levels, and an additional cut at level c. Each cut will be a closed interval, higher level cuts nested inside those of lower levels. We shall use the endpoints of these intervals as indications of the locations of the location of the vertices of trapezoids. Except for the lowest, extra cut at level c and the highest cut (at level1), the remaining cuts should indicate the location of four coordinates. 10=1 ------~~------~ 1~1 ------ 1 2 ----- 11 ----- We select these coordinates in two pairs, each pair surrounding the endpoint of the a-cut. The precise selection of the coordinates is a matter of judgement, and we usually refer to the domain expert. He may suggest that the two surrounding coordinates coincide, thus giving a flat region of length 0. We shall list both coordinates, simply repeating the values. Two external cut, at 0 and at 1, should produce the support for the trapezoidal number and the core, respectively. 33 a a a 2n 2n+l 4n It is immediate that the total number of coordinates is 4n. The inside n - 1 cuts, at the levels l1, ... , ln-l produce four coordinates each. The cut atE produces two vertices of the core. The only condition on the coordinates, which needs to be satisfied, is that they form an increasing sequence. As we deal with fuzzy numbers which are represented by unimodal membership grades, this condition is satisfied in a natural fashion. Once the selection of the heights w1, ... , Wn and the following determination of the coordinates a 1 , ... , a4n, b1 , ... , b4n is completed, the evaluation of f(A, B) can be completed. The first stage is the evaluation of f(ai, bi) for all i = 1, ... , 4n. This involves no furthur concepts of fuzzy theory; it is simply a sequence of evaluations of a crisp real crisp function on crisp real arguments. Computationally it is very efficient, especially when compared to the complexities of the extension principle. If we assume that the complexity of single function evaluation f (x, y) as constant- a standard assumption for the four arithmetic operations and the boolean comparisions - then the computation of f(A, B) can be completed in O(n) time (in the sense of algorithmic complexity). If parallel processing is available, the parallel time can be reduced further. With an unlimited number of processors the complexity is constant 0(1)- we simple evaluate all f(ai, bi) simultaneously. If the number of processors is bound by some functionp(n), the parallel time becomes O(njp(n)). 34 The second stage consists of sorting f (ai, bi) in the increasing order. This stage can be omitted if the function f(x, y) is monotonically increasing in both arguments. Such functions are addition, multiplication, and all the boolean com putations. The computed list of the values f(a1 , b1), ... , f(a4n, b4n), combined with the heights w1, ... , Wn, gives a full description of the result as a trapezoidal number. On some rare occasion it might be desirable to write the explicit lin ear equations for each of the sides of the trapezoidal figure. It is a constant time operation for each side separately, hence complexity 0 (n) for the entire trapezoid. If the function is monotonically decreasing in all its arguments, sorting of the values f (ai, bi) can be replaced by reversal of the order- simply list them as Examples of such functions are 1/x, yi1 - x2 - y2, and many other common ex pressions. We conclude that both of these monotonic cases can be handled in O(n) sequential time complexity, or 0(1) (constant) parallel time complexity. Cases of arbitrary isotonic function (for example, monotonically increasing in its first argument and monotonically decreasing in the second) and of a gen eral function are similar. We cannot predict the order of the values of f(ai, bi), hence we need to sort them. This can be done in 0 (n log n) sequential time or 0 (log n) parallel time. This sorting step dominates the complexity of the entire computation, hence represents its combined complexity. There remains one more issue that will be addressed the next chapter. It may happen that two or more consecutive values of f(ai, bi) are the same, although their heights (membership grades) are different. Graphically, it corresponds to several coordinate points (vertices of the trapezoid) lying one above another. The trapezoidal number would then have vertical sides. This is admissible as a ge ometric representation, but it does not represent an ordinary function. Namely, 35 the membership function must assign a unique membership grade to each ele ment of the domain; such vertical line would suggest multiple membership grades for the same element of the domain. The solution we propose is to view all the trapezoidal numbers as fuzzy multisets rather than simple fuzzy sets. We discuss the concept and its uses in the next chapter. Here we only observe that once the fuzzy multisets are permitted, there is no further difficulty in performing arbitrary quasi-fuzzy computations. 3.4 Approximating discrete fuzzy numbers It could be argued, as the first impression, that there is little to approximate and that the discrete values are themselves the coordinates for quassi-fuzzy compu tations. However, this approach would work well only in certain special cases. In practice it would apply to the standard trapezoidal numbers (four vertices) or triangular numbers (three vertices). For general discrete numbers there could be several difficulties. The most obvious is that different arguments could have dif ferent sequences of heights. Even for a single argument it might not be clear how to define all the coordi nates. A number of elements lying at the same height could vary. It would often be either on or two; then, on some rare occasions could be more than four. It 36 would not be the regular pattern of four coordinates to a level, characteristic of a trapezoidal number. ------·---·----·------·------·------ -----~------4----·-·-·------·-- ---·------ Quite clearly, we need a systematic method of constructing trapezoidal num bers with prescribed heights, based on the arbitrary discrete fuzzy numbers. We propose a two step approach. In the first step we construct a continuous fuzzy number which has the original discrete number as its set of vertices. It can be done in one of the two ways. Either we connect the vertices using straight line intervals, or we build a staircase like structure with all the edges horizontal or vertical, but not slanted. 37 ------D 0 0 ------o 0 0 The first construction is unique - we simply 'connect the dots'. The second construction is less clear as there are usually different choices for the placement of the vertical edges. They could be placed to make the horizontal bases as small as possible, as large as possible, or of some intermediate length. Also, such con struction should be interpreted as a fuzzy multiset and not just a simple set. In a majority of applications the first construction is the most natural; still, on some occasions the second construction may be preferred. Once the contnuous fuzzy number is constructed, we proceed as in the previ ous section. We find the family of a-cuts corresponding to the selected sequence 38 of heights WI, ... Wn and construct a multi-trapezoidal number. We carry out the same construction for all the arguments of the function to be evaluated. If the height sequence consisted of only one element WI = 1, our construction would amount to approximating each number by simple trapezoid or a triangle. The greatest advantage of our method of approximating all the arguments by the multitrapezoidal number lies in the possibility of mixing discrete and contin uous values. To the best of our knowledge such computation are not discussed anywhere in the literature, except perhaps at the level of basic definitions. The reason is very obvious, once a few examples are examined. In fact, even a simple addition of two very regular numbers may lead to the most unsatisfactory results. Let A be a triangular (continuous) number on vertices at locations (0, 1, 2); let B and 6 be discrete fuzzy numbers, each of 5 elements B = {c-/0, 0.5/1, 1/2, 0.5/3, c-/4} 6 = { c-/0, 0.5/3, 1/6, 0.5/9, c-/12} We first examine the effect of computation of A+ B and A+ 6 using the extension principle (hence also the a-cut method). Easy verification produces the figure below: It is continuous and unimodal, although its shape with several 'steps' is rather 39 unintuitive, not at all like the essentially triangular shape of the arguments. How ever, the second example of A + 6 produces an almost disastrous result: 1 0.5 0 2 3 5 6 8 9 11 12 14 It is obviously useless in practice, consisting of essentially five disjoint pieces. Slightly less regular or less symmetric input values can produce even more grosteque results. The use of quasi-fuzzy computations offers a dramatic contrast. Both num bers, A and 6, lead naturally to triangular shapes. Selecting as heights w1 = w2 = 0.5 (a natural choice in light of tthe structure of C) gives as the sum also a triangular number 1 0.5 0 3.5 7 10.5 14 40 It has (geometrically) five vertices, some occuring twice in the full listing. Its formal representation is (0, 3.5, 3.5, 7, 7, 10.5, 10.5, 14; E, 0.5, 0.5, 1, 1, 0.5, 0.5, c) or in Zadeh's sum notation { c/0, 0.5/3.5, 0.5/3.5, 1/7, 1/7, 0.5/10.5, 0.5/10.5, c/14}. In the next section we discuss the passage back from the multitrapezoidal re sult to the domain of our choosing. In this example such conversion is straightfor ward. If the domain is selected as continuous, there is nothing more to be done - the resulting triangular shape is the answer. If the domain should be discrete then an obvoius choice is selection at the levels E, 0.5 and 1. The result is simply { c/0, 0.5/3.5, 1/7, 0.5/10.5, c/14} Either one of these answers is useful and also agrees well with our intuition. 3.5 Conversion of the results The computation of f(X, Y) is performed by evaluating f(A, B), where A is a multi trapezoidal approximation of X and similarly B that of Y. Although f (A, B) could be given as the final result, it is often convenient to convert it back to a shape similar to that of its arguments. However, they could themselves have different shapes, which means that the target shape should be stated explicitly. We consider now several main cases. First, let the target domain be continuous, but with no particular conditions on 41 the shape of membership grade. Then often f(A, B) itself is a satisfactory answer. It gives a continuous, unimodal, piecewise-linear membership function. If the target function must be particularly simple, say a triangle or a simple trapezoid, it is enough to approximate f(A, B) by the desired shape. Sometimes the arguments - - X and Y were originably given as smooth, differentiable functions. Then it is likely that the result off (X, Y) should have the same smoothness properties. The method consists of fitting a smooth function to the vertices of the multitrapezoid f(A, B). This can be effected by one of the algorithm of numerical analysis. The process itself has little relationship to fuzzy theory and can be handled by any of several standard numerical packages. Another type of conversion is required if the target domain should be discrete. Such conversion can be performed either on the basis of the specified heights of the elements. If we specify the heights, we must determine the values of the corresponding elements. If the values of the elements are specified, the heights can be computed using f(A, B). Let us first consider the specified heights, which we call ( u 1, ... um). In prac tice they usually would come as the heights of one of the arguments. We simply compute the a-cuts of f(A, B) for each ofthe corresponding levels r 1 = u1, r 2 = u +u + ··· +u =r 1 2 m m u+u =r 1 2 2 u =r 1 1 42 If the cut contains no flat region, that is if the cut line at level r i intersects the trapezoid f(A, B) in two points, we are done. We simply take these points as part of the answer. If the cut line contains a flat region, say to left of the maximum, this region is a contiguous interval which is a part of the basis of one of the trapezoids. We can then pick any point of such interval; the center would be usually a natural choice. ------~.---~ ,-+-~------ . . -----~------~------. . . . Last case consist of being given a set of specified values of the elements of the answer. Let us call these values z1, ... , zk. Then the answer should be of the form where all the question marks are replaced by suitable membership grades. This is simply done by evaluating the multitrapezoidal membership function f(A, B) at those values z1 , ... , Zk· 3.6 Properties of quasi-fuzzy arithmetic Arithmetic, in the traditional meaning of the term, concerns itself with four ba sic operations - addition, substraction, multiplication, and division. These also are the elementary operations in computer implementations. All other numerical 43 functions, given through analytical formulae, are either finite combinations of the arithmetic operations, or are approximated by such combinations. The four basic operations have algebraic properties which can be briefly sum marized as saying that the real numbers R form a field. It means that both addition and multiplication are associative and commutative, have ventral elements and in verses, and lastly, multiplication distributes over addition. Quasi-fuzzy operations utilize coordinatewise crisp operations; therefore, it may be hoped that many of the algebraic properties of R will apply to fuzzy numbers. This is indeed the case, and we state the main properties in the next few theorems. Theorem 1. Addition and multiplication are associative and commutative. Proof. Associativity (of an abstract operation '*') means a * (b * c) = (a * b) *c. For fuzzy numbers A, B, 6 with coordinates (ai), (bi), (ci) associativity follows from the associativity of the corresponding crisp operations and the fact that addition and multiplication are monotonic (have greater values for greater elements). Taking as an example addition, we have This defines uniquely the sum ai + bi + ci. Now and the coordinate of the sum A + B + 6 are simply the ordered sequence (ai + bi + ci)· Theorem 2. Multiplication over addition 44 Proof. It is a direct consequence of the distributivity for crisp operations and the monotonicity of these operations. As a simple corollary we get Lemma 1. For any A (-1)A =-A. At the same time A-B ::j=. A+ (-B) can occur even if all the coordinates of A, B, and both expressions above, are positive. For example, two trapezoidal numbers provide inequality. Let A= (10, 20, 30, 40; 1), B = (o, 1, 2, 3; 1). Then A-B (10, 19, 28, 37; 1) A+ (-B) (7, 18, 29, 40; 1). However, certain more specialized algebraic relationships, involving substrac tion and division, may hold. We have Lemma 2. For arbitrary A, B for positive A and B 45 Similar relationship will hold for more complex operations and functions, as long as they are monotonic, that is, whenever we can be assured that the order of the coordinates is preserved. Quasifuzzy operations are performed on the discrete fuzzy numbers, repre senting multitrapezoidal structures. For general numbers X, Y, .. ., the definition of f(X, Y) involves a reconstruction of the continuous number from the set of the discrete coordinates. As we discussed ib the previous section, this process involoves case-by-case expert analysis. Hence, no algebraic identity can hold for general continuous number. However, the most common case of trapezoidal or multitrapezoidal numbers permits for formulating such algebraic rules. In fact we have a general theorem. Theorem 3. In quasi-fuzzy arithmetic, all algebraic formulae which hold for the discrete numbers hold as well for the multitrapezoidal numbers. Proof. All the trapezoidal numbers have a unique conversion to the discrete num bers - we take their vertices as coordinates. In the opposite direction, all the dis crete numbers, which result from the quasi-fuzzy operations, have unique trape zoidal representations - we 'connect the dots'. The two conversion processes preserve the order of the coordinates. Hence, if there is equality for two algebraic expressions on discrete arguments, the same equality holds for the multitrape zoidal arguments. 46 Chapter 4 Fuzzy Multisets The motivation for our investigation was given by certain problems arising in com puting with multitrapezoidal number. The proposed solution led to a new concept -fuzzy multiset, generalizing the notion of a fuzzy set. The concept is of impor tance in its own right and is introduced here independently of any computational model. In the later part of the chapter we discuss the interaction of multisets and two computing methods- extension principle and quasi-fuzzy arithmetic. The first part of the chapter is devoted to basic definitions, introducing the concept of multiset. We do not attempt to define it in the greatest possible generality; in stead, we are guided by the pragmatical considerations and the usefulness in fuzzy arithmetic. The idea of multiple entries for the same value elements was studied before in the literature, in the work of Yager on fuzzy bags [39, 40]. He defined that concept in a manner fairly similar to our fuzzy multiset. In fact, his defintion was in many cases more general than our definition. However, he was motivated by the analysis of cardinality of fuzzy sets and did not investigate any deeper links with fuzzy arithmetic. Recently, certain corrections and extensions were proposed Kim and Miyamoto [18]. 47 4.1 Fuzzy multisets We define fuzzy multisets only in a restricted fashion. We introduce them for one purpose only - to model conveniently certain types of membership grade which would not be permitted as ordinary fuzzy sets. We do not attempt to generalize simultaneously crisp sets and fuzzy sets [31]. Given the basic domain X, we define a finite fuzzy multiset M as a collection {mdai}, where mi E [0, 1) are the membership grades, ai EX are the elements, and where (ai) can be repeated, but only with different membership grades. In other words, our definition is a somewhat restricted form of a multiset comprising fuzzy elements. At the same time they are not simply multisets which happened to have a fuzzy set as their domain. The purpose of the definition is to model conveniently the membership grades which would correspond, in the graphical representation to the trapezoids with with some sides vertical [35, 36]. 0.5 ------....------, 0.3 ------.------4 a b In the example above, element a of the domain has all the membership grades from 0.3 to 0.5 while element b has the membership ranging from c to 0.3. To rep resent such situation directly as a fuzzy multiset would require allowing infinite 48 multiplicities of the elements. Our main interest lies in defining arithmetic oper ations, and use of quasi-fuzzy model requires only a few, discrete points of the membership grade. To return to the example above, we would represent the mem bership grade for the element by just two entries{ ... 0.3/a, 0.5/a, .. . } selecting the lowest and the highest membership values. 0.5 I------.-----.. 0.4 r------0.3 I------.....----... a If in the course of some computation there would be used an a-cut at level 0.4, we would add an entry 0.4/ a. This approach permits using finite multisets in all applications in which ordinary fuzzy sets would be used. Moreover, as we show shortly, fuzzy multisets will appear only as the input values (in function evaluation) or as intermediate results. The final results will be invariably standard fuzzy sets. Before we analyze the behavior of multisets in fuzzy arithmetic, we need to introduce a very basic operator. We denote it S and use it to convert a fuzzy mulitset to a fuzzy set. If M is a multiset we construct S ( M) by taking each element in M with its highest membership grades. For exmaple, if M = {0.3/a, 0.5/a, 0.2/b, 1.0/b, .. .} then S(M) = {0.5/a, 1.0/b ... .} 49 1.0 ------~ 0.5 ------~ 0.3 ------· 0.2 ------~------·. . a b Obviously, ifF is already a fuzzy set then S (F) = F. From this we conclude that applying operator S several times is equivalent to applying it just once; for any multiset S(S(M)) = S(M). 4.2 Arithmetic on fuzzy multisets We consider here the use of extension principle and the o:-cut arithmetic. First we observe that any function evaluation using the extension principle leads always to a fuzzy set. Theorem 4. For arbitrary f(x1 , ... Xn) and arbitrary multisets M1, ... Mn is a fuzzy set. In particular fext(MI, · · .) = S(fext(MI, · · .)). Proof. The expression on the left fext(M1, .. . ) has as its membership grade Jt(z) 50 defined using the rule of supremum f.l(z) = sup (vi(xi) 1\ ···I\ vn(xn)) Xl,···Xn XI, ... Xn. , ... over the suitable selection of Here supx 1 xn is an ordinary function, returning the maximum value, hence returning a unique value. We conclude that every z E Z has a unique membership grade and that !ext (MI, ... ) is a fuzzy set. From this follows an observation that we could convert all the arguments - - MI, ... , Mn to fuzzy sets before begining the computations. Theorem 5. With notation as above , ... Proof. To compute f.l(z) we take the supremum supx1 xn (vi (xi) 1\ ···I\ vn(xn)). If any xi has more than one membership grade, that is if vi (Xi) is a set of values, the supremum can be computed using only the greatest value. And taking such greatest value is the same as converting Mi to S (Mi). Another observation of interest is that an identity function actually performs a conversion to fuzzy sets. Theorem 6. If I (x) = x then for any multi set M. Proof. If f.L(z) is the membership function of Iext(M) and v(z) is the membership function of M then f.L(z) =sup v(z), supremum taken over all the values of v(z). This matches precisely the definition 51 of the operator S. We can now discuss the interaction of a-cut arithmetic and fuzzy multisets. For standard fuzzy sets, in most situations, extension principle gives the same result as a-cuts. This is no longer true for arbitrary multisets. The reason is that a-cuts operate separately at each level, hence they do not remove duplicate elements. A simple example isM= N = {0.3/1, 0.5/1, 1/2}. Now computing the sum gives (M + N)ext {0.5/2, 0.5/3, 1/4} (M + N)a {0.3/2, 0.5/2, 0.3/3, 0.5/3, 1/4}. However, application of the operator S restores the equivalence. We state the result only for the discrete multisets. Theorem 7. For any f(x1 , ... xn) and M1 , ... Mn, finite multisets Proof. It is sufficient to reflect that the presence of elements with multiple mem bership grades is caused by performing the same evaluation of f(x1 , ... xn) at several a-cut levels. There is nothing in the formal rules of a-cut arithmetic that , ... forces removing such duplicates. This is exactly equivalent to using supx1 xn in evaluating fext(XI, ... Xn)· This proof actually shows more. It shows that if we would remove the du- - - plicates from the arguments M1 , ... Mn the result would be a fuzzy set. We can summarize it in separate theorem. Theorem 8. S(fa(Ml, ... )) = !a(S(Ml, .. .) The results above show fuzzy multisets can only occur as input values in ex tension principle computations. They could occur in a-cut computations, but their 52 removal is straightforward. 4.3 Coordinate arithmetic on fuzzy multisets There are two models that perform the operations by direct matching of the co ordinates. They are function principle and quasi-fuzzy computations. Because they perform computations separately at each selected level they can be applied directly to the multisets. Compared with the operations on fuzzy sets it introduce no computational difficulties. Fuzzy multisets occur naturally in quasi-fuzzy computations. Approximating continuous membership grades by piecewise linear functions can lead to the trape zoidal shapes which have some vertical edges. Then the cuts at the consecutive heights may intersect the same edge. This would lead to the same domain element being included twice, with two different membership values. ----- On some occasion the membership function may be specified initially as a set of rectangles, one above the other. 53 I I Lastly, function principle computations use trapezoids. If one of the inside angles of such trapezoid is a right angle, we have a multiset with a certain element repeated twice. For example, the trapezoid T = (1, 1, 2, 3; 1) is most naturally represented by the multiset of its vertices T= {c/1,1/1,1/2,c/3;1} 1 2 3 We can inquire about the interaction of the operator S and function evaluation using quasi-fuzzy computations. If we denote by fq(M1 , ... Mn) such evaluation, we have a theorem similar to the results on extension principle and a-cut compu tations. 54 Theorem 9. S(Jq(MI, ... Mn)) = S(Jq(S(MI), ... S(Mn))) Proof. Evaluation of f(M1 , ... Mn), using quasi-fuzzy model, is performed sep arately at each selected height. If the evaluation is performed twice at two different heights, the S operator on the left and the internal S operator on the right will remove the duplicate values of f(x1 , ... xn) and retain only the one with the higher membership grade. On the other hand, if for some collection of input values (x1 , ... xn) and a different (y1 , ... Yn) the equality holds, then the external S operator on the right will remove the duplication. A simpler equality formula could be expected, but need not hold. The follow ing is not necessarily true The example of subtraction makes it clear. Let A1 {c/2,1/3,1/5,c/7} lV {c/1,1/2,1/3,c/4} Then S(M) = M and S(N) = N, as both M and N are fuzzy sets. The difference M- N = {c/1, 1/1, 1/2,c/3} and is a fuzzy multiset. This is the result of S(M) - S(N), while the result of S(M- N) is {1/1, 1/2, c/3}. 55 Chapter 5 Two applications In this chapter we study two very different applications of the framework we de veloped. The first one is a solution of the standard economic optimization problem -finding the reorder quantity. We solve the problems when some of the constraints are fuzzy. This is a very common case in decision making in problems arising from control systems or intelligent systems [3, 46]. It leads to a fuzzy number as the answer, which has then to be defuzzified. We perform the latter using the cen troid method, which is the very common in fuzzy decision problems. We further study the example numerically, using several thousands of randomly generated examples. We use those to compare the effect of two approaches to defuzzifica tion: (1) compute with fuzzy inputs, then defuzzify the output, (2) defuzzify the inputs, then compute with the crisp quantities. The other application is of somewhat theoretical nature. It deals with assign ing a rigorous meaning to the vertical segments in some membership grades. Such grades have appeared in published papers, usually dealing with some decision making problems. In the chapter on fuzzy multisets we mentioned such mem bership grades, but also immediately concluded that the use of quasi-fuzzy com putations will reduce such cases to finite fuzzy multisets. The question of their 56 interpretation remains, and we discuss it in the second section of this chapter. 5.1 Economic order quantity A key inventory control problem is establishing the EOQ-Economic Order Quan tity [34, 38]. Its basic form assumes constant annual demand and constant lead time. The former means the the total amount consumed (utilized) by an enterprise during the one-year period ; the latter means that the time elapsed between the placement of the order and the receipt order is known and constant. As the total demand in a year is constant, so is the cost of the commodity acquired. The cost of ordering (sometimes called a set-up cost) is constant per order, hence proportional to the number of orders in a year. The other variable component is the holding cost (cost of warehousing). This cost is assumed to be proportional to the amount of the commodity held. It has discontinuous jumps when orders are delivered (the actual delivery step is essentially instantaneous) and decreases linearly afterwards, due to the steady consumption in the produc tion process. The pattern of the stock held is ideally (for the optimal amount Q ordered) Q ------ time If the quantity ordered were larger than Q the pattern would become 57 c+Q Q c time which would imply storing of unwanted amounts; such amounts could not be utilized as the consumption rate is steady. If the quantity ordered were below Q we would have production interrupts Q Q-c ,I ' I ' I time -c ------~ 'I 'I ,I As the enterprise is deemed profitable, any idle periods are undesirable, even without considering the cost of resuming the production when the commodity becomes available through a reorder. We conclude that the optimal design is the one when we allow, in uniform time intervals, for the amount of the commodity stored decrease to 0 and, at that precise moment, to accept a new delivery. This leads to the equation which we first state using linguistic terms and then in a symbolic form Total Cost =Production Cost + Ordering Cost + Warehousing Cost. The same problem can be considered when some, or all of the input parame ters are fuzzy [8, 29, 32, 33]. This model well the real applications- in practice 58 such parameters as costs of ordering or warehousing, rates of demand or produc tion, are known imprecisely. The quantity resulting from the computation will be also fuzzy. The ensuing computations are best conducted using quasi-fuzzy computations. We now introduce several variables to express various components of the above equation. We state them in two forms - an upper case letter to denote a crisp quantity and the same letter with tilde to denote corresponding fuzzy quan tity. The following variables will be used: Q - quantity ordered per cycle R - demand rate D - annual demand P - production rate E - ordering cost U - unit production cost A - unit warehousing cost. The variable that can be controlled is Q - quantity ordered. More frequent orders (shorter cycle) mean lower storage costs, but more cycles per year, hence higher annual ordering costs. Standard formulae, used in economic theory of inventory management are Production cost -U·D Reordering cost Warehousing cost- PPR ·A·~- The production cost is independent of Q. The reordering cost is proportional to number of supply periods in a given year, hence inversely proportional to Q. The warehousing cost has the term PPR as the relative rate at which the commodity is depleted, A as the unit cost which becomes a normalizing constant, and Q/2 59 as the average amount of the commodity stored. The last term reflects a linear decrease of the amount stored within any given reordering cycle. Our objective is to minimize the total cost as a function of Q. With the pro duction cost constant, the minimization applies to C(Q) = E ~D + (P ;PR)A. Q. The two terms of the sum have a constant product; therefore, their sum is mini mum when they are equal. The unique minimum corresponds to the positive Q satisfying E·D (P- R)A. Q Q 2P j2P · E · D Q V(P- R)A The same formula can be applied to fuzzy computations, each variable re placed with the corresponding fuzzy variable. The results of a fuzzy computation may differ a little in the final value of Q, but they always will have to be defuzzi fied to produce a useful optimalizing quantity. We now address this last problem. A recent monograph by Yager and Filev [41] presents as two most often used methods- the Center ofArea (COA) and the Mean of Maxima (MOM). The COA, also known as the fuzzy centroid, consists of finding a value c such that the vertical line at c would divide the area under the membership grade into halves. 60 In general it requires solving a numerical analysis problem of evaluating fyF(y)dy f F(y) dy for the membership function F(y), with integrations taken over the entire basic domain [ ]. In the case of multitrapezoids the problem could be solved in an elementary fashion by case analysis. The method would consist of several steps. First, we would compute the complete area under the multitrapezoidal curve. In our case it is 1 2w1(a4n- a1 + a4n-l- a2) + 1 2w2(a4n-l- a2 + a4n-2- a3) + (5.1) 1 2wn(a2n+2- a2n-l + a2n+l - a2n) The next would consist of partitioning the area under the curve by a vertical line at one of the ai coordinates 61 _I I I a. c a. 1 I+ 1 Using a formula like ( 5.1) we would compute the area to the left of the vertical line and using the value from (5.1) compute the area to the right of that line. By progressively advancing to the index of the coordinate ai and repeating the computations we could find two consecutive aj and aj+l such that the area to the left of aj is less than one half of the sum (5.1), while the area to the left of aJ+1 is greater than that half. Then by solving a linear equation we could determine an exact position c in between aj and aj+l such that the vertical line at x partitions the area (5.1) into equal parts. The full implementation of the method outlined is quite cumbersome, as no closed form formula is available for computing value c. A significant simplifi cation obtains if we can compute that value as the centroid of a discrete fuzzy set. This approach by Yager and Filev [41] works for more general cases, and can certainly be applied to multitrapezoids. For a general discrete fuzzy set { F(yi)/yi} the centroid is L F(yi)Yi c = 'LF(yi) . In our case, F(yi) correspond to li 62 fori = 1, ... , 2n, and [i/2] denoting the integer rounded down. Fori = 2n + 1, ... , 4n we need to recall thatF(ai) = F(4n+1-i), thus using the same li values, but in the reverse order. A further simplification results from selecting the heights wi equal, that is by arranging the defining cuts of the multitrapezoids uniformly spaced. Then the simple arithmetic average gives the defuzzified value .L: ai C=--. 4n An interesting special case occurs when this value of c lies in between a2n and a2n+ 1. Then the vertical line at c intersects all the bases of the trapezoids that are used in forming the multitrapezoidal membership curve. It is immediate that in that case the same vertical line partitions the area under the curve into equal parts. In other words the centroid of the vertices becomes also the centroid of the entire membership function. This comparison brings us to the second most popular defuzzification method. Termed Mean of Maxima (MOM), it reduces in our case to a simple mean of two numbers For many applications these two methods- COA and MOM will produce very close results. Membership functions are often symmetric, when an outright equal ity results. We designed the quasifuzzy method to approximate arbitrary fuzzy numbers; we feel that ignoring all but the two central coordinates a2n and a2n+l is somewhat extreme. At the same time the property of the defuzzified answer fall in a central location among all the coordinates would be quite satisfying. We propose a method which combines the features of two methods described above. We take as the defuzzified value such d which lies within the interval [a 2n, a2n+ 1] 63 and is closest to the centroid The formula becomes d= We term this method the Central Centroid (CC). We used the quasi-fuzzy computations and the CC defuzzification in comparing the fuzzy economic order quantity computations with their crisp counterparts. Using a symbolic algebra system MAPLE [23] we generated 10,000 fuzzy input sets for the problem. Each set consisted of the following fuzzy entities: E, iJ, P, R, and A. We performed first computation on the fuzzy inputs, and then defuzzified the resulting fuzzy quantity obtaining the answer Q 1. For com parisian, we defuzzified all the inputs first, and then performed the computation, obtaining the answer Qm. A part of our simulation, when the inputs were simple trapezoids is summarized in the table below. 64 Qm Cost(m) Qf Cost(t) (before defuzzification) Cost(f) 3284.8 12798.0 3290.6 (10305.0, 12296.1,13435.1,15277 .1) 12828.3 3485.2 13359.0 3492.8 (11898.4,12729.0,13956.0,14942.1) 13381.4 3570.3 13200.1 3586.2 (10326.8,11896.3,14410.3,16269.6) 13225.8 3648.7 14189.9 3675.7 (11190.9,14610.6,14888.3,16190.8) 14220.1 3693.8 13206.1 3697.0 (1231 0.1' 12585.5,14038.8,13631.1) 13141.4 3815.8 12560.9 3842.0 (10796.9,12914.5,13195.3,13788.2) 12673.7 3833.0 13038.3 3859.1 (1 0012.5,12512.4,13860.5, 15936.8) 13080.6 3867.5 12729.8 3893.5 (10913.7,12025.0,13564.7,14364.9) 12717.1 3873.9 13490.7 3885.2 ( 9733.2,12749.5,15337.2,16419.0) 13559.7 3965.1 12749.2 3993.5 (10194.2,11236.4,14389.9,15317 .3) 12784.4 4035.9 12130.8 4065.2 (10330.3,10549.6,12694.6,15234.1) 12202.2 4079.8 12526.5 4093.8 (10405.2, 11935.5,12760.2, 14912.2) 12503.3 4174.3 12217.1 4198.3 (10877.6,10993.1,13226.0,13757.1) 12213.4 4201.0 12782.4 4216.6 (12271.1,11812.0,12840.5,14261.6) 12796.3 4348.7 12528.7 4368.4 ( 9842.8,11753.5,13401.2,15465.9) 12615.9 4417.2 11552.9 4440.3 (10821. 7' 10293.0,12009.7' 13115.6) 11560.0 4451.3 12719.4 4463.5 (10584.8,12500.4,13446.7,14460.5) 12748.1 4556.1 11870.5 4570.5 (11008.6,11248.1,12427.3,12583.0) 11816.7 4583.7 12274.7 4606.8 (10950.8,12562.4,13076.8,11803.8) 12098.5 4667.9 11829.1 4686.9 (10131.8,11719.5,12388.8,13247.4) 11871.9 For simple trapezoids quasi-fuzzy computations reduce to the function principle. We observe that in about 80% of the cases the total cost C(m), corresponding to the value Qm is less than the cost C(f) corresponding to the value Q1. The absolute difference between these two costs is quite small, permitting in practice for selecting the order of operations on the basis of computational 65 convenience: either defuzzification before computation, or computation followed by defuzzification seem both acceptable. If a greater precision is required, the decision maker may have to conduct a preliminary study on sample data and then decide on sequencing of the computation and defuzzification steps. 5.2 Multi-possibility Multisets can be applied to reasoning about possibilities. The initial formulation by Zadeh [44] introduced a concept of possibility of an 'event' x E X as its membership grade p(x). The fundamental idea was to treat each x E X as the possible 'state of the world', and use its membership to describe numerically how possible is x. Zadeh extended his definition to subsets and to cartesian products. In some applied situations we may wish to start from the original assignment of possibilities which is multivalues. For certain x E X we may allow that its possibility 1r(x) is not a single number but a set of numbers (between 0 and 1). It would seem to represent a common indecission of human reasoners. We can call assignments multipossibilities. When such possibilities are considered for all x E X, we find that 1r (x) becomes a fuzzy multiset. Possibility of a subset is defined by selecting the maximum value 1r(A) =sup 1r(x). xEA When we apply this definition to multisets we must write 1r(A) = sup{y: y E 1r(x)}. xEA The value computed would remain the same if for every x E X we only take its greatest possibility max{ y : y E 1r (x)}. This proves the theorem. 66 Theorem 10. n(A) = S(n)(A) when Sis the operator converting multisets to sets. The term on the right means the membership function S (n) applied to the subset A. A Cartesian product of two possibility assignments, n on X and p on Y, is defined n@ p: (x, y) H min(n(x), p(y)). For multipossibilities there are two reasonable ways of applying the above for mula. One is to use, in computing n@ p, only the greatest values from n(x) and p(y). We recall that now both n(x) and p(y) can be sets. The other method would be to combine all the values from n(x) with all the values from p(y). This latter method would require much more extensive computations. It would also be of little value in applications. As we found in Chapter 3, most computations lead to the results which are fuzzy sets and not multisets. This suggests definition n@p: (x,y) H min(sup{a: a E n(x)},sup{b: bE p(y)}). a b A theorem, similar to the previous one, holds. Theorem 11. Membership functions defining n@ p and S (n)@ S (p) are identical. Introduction of cartesian product led to investigation of the feasibility of defin ing conditional possibilities. There were few different models proposed; the earli est one led to an interesting application of fuzzy multisets Risdal [ ] obtained as a result of conditioning a fuzzy set whose elements had as possibility values either single numbers or intervals [r, 1], for some 0 < r < 1. She did not define such constructs formally. We can do it now by introducing continuous fuzzy multisets. We assume a continuous basic domain and continuous membership functions in the remainder this section. Fuzzy numbers are defined as fuzzy sets with uni modal membership function. Represented geometrically, such membership func- 67 tion is a unimodal curve, with no vertical segments. A slight generalization, which includes many step functions discussed in the previous chapters, is to permit such vertical segments. Now the membership grade of a given element of the domain could be not only a single value, but also an interval of such values. This concept can be formalized as a continuous fuzzy multiset. We define such a multiset as a pair consisting of the interval of definition (the base domain X) and a membership grade represented by a unimodal curve. The grade of a given element x E X is either a single value J.L(x) E [0, 1] or a closed interval [J-C(x), J.L+(x)] c [0, 1]. We define, as before, the operator S which converts such a multiset into a fuzzy set. If M =(X, J.L(x)) then In other words we take as the membership grade in S (M) the upper bound of the membership in S (M). We note S (M) is not a continuous number. For example, if X= [0, 100], and M is trapezoidal (with aright angle) M = (1, 1, 2, 3; 1) 1 2 3 M then the membership function of S(M) is discontinuous at 1. Its left limit at 1 is 68 0, while the right limit is 1. Our results on fuzzy arithmetic carry without change from the discrete to the continuous case. The actual computations would be performed by selecting appro priate cut levels. This is exactly the process that led to designing the quasi-fuzzy computations. 69 Chapter 6 Concluding remarks The model we developed consists of two parts- quasi-fuzzy computations and fuzzy multisets. Both were designed to serve specific practical needs. We il lustrated such applications by two examples taken from the articles on decision making using fuzzy quantities. We did not strive for maximum generality of the definitions. The formulations we give are sufficient for all typical applications found in the literature. Theoret ical investigations of more general models can be pursued and are likely to bring new questions and new results. We list a few directions of a continuing research. We expect that it is possible to define quasi-fuzzy operations for nonnumer ical domains. Lack of numerical values makes difficult defining monotonicity or unimodality of the membership functions. However, some conditions on the geometric shape of the membership function could be imposed. Fuzzy multisets could be generalized. The first step would be to permit multi ple, completely identical entries, where both the domain elements and their mem bership grades coincide. This would bring the model to one very close to the models proposed by Yager [39, 40]. A further step step would be to permit as membership grades arbitrary positive real values, not necessarily less or equal to 70 1. We could then view 0.5/ a as a 'true fuzzy' element with membership 0.5, 2.0/b as the repeated twice element b, typical of multisets, and a new meaning would be assigned to 2.5/c and similar. In some applications requiring that membership functions are continuous functions of the basic domain could be too restrictive. 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Kluwer Academic, Boston, 1992. [43] L.A. Zadeh. Fuzzy sets. Information and Control, 8:338-353, 1965. [44] L.A. Zadeh. Fuzzy sets as a basis for a theory of possibility. Fuzzy Sets and Systems, 1:3-28, 1978. [45] L.A. Zadeh. Possibility theory and soft data analysis. In L. Cobb and R. M. Thrall (editors), Mathematical Frontiers of the Social and Policy Sciences. Westview Press, Boulder, COL, 69-129, 1981. [46] H. J. Zimmermann. Fuzzy Sets, Decision Making and Expert Systems. Kluwer Academic, Boston MA 1987. [47] H. J. Zimmermann. Fuzzy Set Theory and its Applications. Kluwer Aca demic, Boston MA 1991. 76 Erratum 1. Page 6 - Reference number. The reference at the end of the first paragraph of Section 2.2 should read: [13, 19, 41]. 2. Page 26 - Notation to be included after the second paragraph (end of line 8). Trapezoidal fuzzy numbers are conveniently denoted by listing their vertex coordinates. As the base vertices have the y-coordinate 0 and the top two vertices have the same coordinate w, the notation can be simplified to a 5-tuple consiting of the four x-coordinates and the height of the number (value w). For the normalized fuzzy numbers the latter is 1; in such cases the notation can be further simplified to listing a 4-tuple of the x-coordinates. 3. Page 29- Addendum at the end of Section 3.2. First, however, we outline some general reasons for effectiveness of the proposed method. The foremost are its unversality and avoidance of para doxical results. Quasi-fuzzy computations permit mixing arbitrary types of fuzzy numbers as arguments to essentially any function evaluations. This much more than the typical applications of extension priciple would allow. The latter does not, as a rule deal with mixed inputs to the arithmetic operations. It also has difficulty handling exclusively discrete arguments for such simple functions as maximum or minimum. Due to the coordinate-wise application of the function evaluated, the re sults of quasi-fuzzy arithmetic tend to assume the shape that would cor respond to the informal expectations. For example, if the inpouts can be viewed as having a triangular shape, the results of the arthmetic operations will also have that shape. As we shown on several examples, the extension principle may easily lead to quite unusal results of no practical value. Last advantage we should point point out is the facility of fast computing. 2 In the general case, the extension principle requires O(n ) elementary steps to perform the basic arithmetic operations. However, the proposeed quasi fuzzy arithmetic handles these operations in a linear number of elementary steps. 4. Page 64 - Addendum at the bottom of the page. The comparison of the values Q f and Qm shows that the latter are typically greater than the former, albeit by a very small amount. Although our initial expectation would be for these values to fall rather randomly' on either side of each other, there appears to be a very slightly skew towards greater values resulting from the fuzzy computations. However, the difference is statistically insignificant, and we would attribute it to the vagaries of the defuzzifying process.