Fuzzy mathematics in economics and engineering

This book is basically about solving fuzzy equations linear, eigenvalue, differential, etc.. Any reader familiar with fuzzy sets, fuzzy numbers, the extension principle, a-cuts, interval

Fuzzy Mathematics

in Economics and Engineering

Editor-in-chief

Prof Janusz Kacprzyk

Systems Research Institute

Polish Academy of Sciences

ul Newelska 6

01-447 Warsaw, Poland

E-mail: kacprzyk@ibspan.waw.pl

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Further volumes of this series can

be found at our homepage

Vol 71 K Leiviska (Ed.)

Industrial Applications of Soft Computing, 2001

ISBN 3-7908-1388-5

Vol 72 M Mares

Fuzzy Cooperative Games, 2001

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Vol 73 Y Yoshida (Ed.)

Dynamical Aspects in Fuzzy Decision, 2001

Vol 75 V Loia and S Sessa (Eds.)

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Soft Computingfor Risk Evaluation and Management,

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Vol 79 A Osyczka

Evolutionary Algorithms for Single and

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Fuzzy Reasoning in Decision Making and Optimization, 2002

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Belief Functions in Business Decisions, 2002

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University of Alabama at Birmingham

Mathematics Department

Birmingham, AL 35294

USA

buckley@math.uab.edu

Professor Esfandiar Eslami 1

Shahid Bahonar University

ISSN 1434-9922

ISBN 978-3-7908-2505-3 ISBN 978-3-7908-1795-9 (eBook)

10.1007/978-3-7908-1795-9

Cataloging-in-Publication Data applied for

Die Deutsche Bibliothek - CIP-Einheitsaufnahme

Buckley, James J.: Fuzzy mathematics in economics and engineering: with 27 tables / James J Buckley; Esfandiar Eslami; Thomas Feuring - Heidelberg; New York: Physica-VerI., 2002

(Studies in fuzziness and soft computing; Vol 91)

This work is subject to copyright All rights are reserved, whether the whole or part of the material is concerned specifically the rights of translation, reprinting, reuse of illustrations, recitation broadcasting, reproduction on microfilm or in any other way, and storage in data banks Duplication of this publication

or parts thereof is permitted only under the provisions of the German Copyright Law of September 9,

1965, in its current version, and permission for use must always be obtained from Physica-Verlag tions are liable for prosecution under the German Copyright Law

Viola-© Springer-Verlag Berlin Heidelberg 2002

Originally published by Physica-Verlag Heidelberg in 2002

Softcover reprint of the hardcover I st edition 2002

The use of general descriptive names, registered names, trademarks, etc in this publication does not imply even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use

Hardcover Design: Erich Kirchner, Heidelberg

DOI

To Julianne, Birgit and Mehra

Helen, Lioba, Jason, Pooya, Peyman and Payam

3.4.1 Fuzzy Linear Equation

3.4.2 Fuzzy Quadratic Equation

3.4.3 System of Linear Equations

3.5 Fuzzy Input-Output Analysis

3.5.1 The Open Model

5 Fuzzy Non-Linear Regression

5.1 Univariate Non-Linear Fuzzy Regression

6.2.1 Job Times Fuzzy Numbers

6.2.2 Job Times Discrete Fuzzy Sets

6.2.3 Summary

6.3 Fuzzy Inventory Control

6.3.1 Demand Not Fuzzy

6.4.3 Finite or Infinite System Capacity

6.4.4 Machine Servicing Problem

6.4.5 Fuzzy Queuing Decision Problem

6.4.6 Summary and Conclusions

6.5 Fuzzy Network Analysis

6.5.1 Fuzzy Shortest Route

6.5.2 Fuzzy Min-Cost Capacitated Network

6.5.3 Evolutionary Algorithm

6.5.4 Summary and Conclusions

6.6 Summary and Conclusions

Bibliography

7 Fuzzy Differential Equations

7.1 Fuzzy Initial Conditions

7.1.1 Electrical Circuit

7.1.2 Vibrating Mass

7.1.3 Dynamic Supply and Demand

7.2 Other Fuzzy Parameters

7.3 Summary and Conclusions

8.2.2 Extension Principle Solution

8.2.3 Interval Arithmetic Solution

8.4.3 Fuzzy Fibonacci Numbers

8.5 Summary and Conclusions

Bibliography

9 Fuzzy Partial Differential Equations

9.1 Elementary Partial Differential Equations

9.2 Classical Solution

9.3 Extension Principle Solution

9.4 Summary and Conclusions

Bibliography

10 Fuzzy Eigenvalues

10.1 Fuzzy Eigenvalue Problem

10.1.1 Algorithm

10.2 Fuzzy Input-Output Analysis

10.3 Fuzzy Hierarchical Analysis

x

10.3.2 Fuzzy Amax-Method

10.3.3 Fuzzy Geometric Row Mean Method

10.4 Summary and Conclusions

11 Fuzzy Integral Equations

11.1 Resolvent Kernel Method

11.1.2 Second Solution Method

11.2 Symmetric Kernel Method

11.2.1 Classical Solution

11.2.2 Second Solution Method

11.3 Summary and Conclusions

Bibliography

12.1 Summary 243 12.1.1 Chapter 3: Solving Fuzzy Equations 243

12.2.7 Chapter 9: Fuzzy Partial Differential Equations 252

List of Figures

List of Tables

267

271

applica-No previous knowledge of fuzzy sets is needed because in Chapter 2 we survey the basic ideas needed for the rest of the book The basic prerequisite

is elementary differential calculus because we use derivatives, and partial derivatives, from time to time The chapters on differential equations are all elementary and should be understandable from basic calculus

We sometimes define new concepts in the chapters Usually, when we do this the concept is useful only for that chapter and therefore was not included

in Chapter 2

Our policy on "theorems" is always to state the theorem and present the proof whenever the proof is short and elementary Longer, and more complicated proofs, are relegated to the references

We do not give a complete list of references We, of course, give all our references since the book is based on these papers For other references we: (1) give the recent (last couple of years) references; and (2) for older references we only give a few "key" citations from which the reader can find other relevant papers

An overview of the book can be seen from the table of contents A more detailed overview is in Chapter 12 So, if you want a quick reading about what is in the book, please turn to the summary section of Chapter 12

To achieve a uniform notation in a book with lots of mathematics is always difficult What we have done is introduce the basic notation, to be uniform throughout the book, in Chapter 2 Other notation is chapter dependent By chapter dependent we mean some symbols may change their meaning from chapter to chapter For example, the letters "a" and "b" may be used as [a, b]

to represent a closed interval in one chapter but they can be parameters in a

J J Buckley et al., Fuzzy Mathematics in Economics and Engineering

© Springer-Verlag Berlin Heidelberg 2002

differential equation in another chapter

Our recommendation reading for applications to economics is

I-t 2 -t {3,4,5,6,8, 10} -t 12, where "1" stands for Chapter 1, etc For engineering we suggest

I-t 2 -t {3, 7,8,9,10, 11} -t 12 This book is basically about solving fuzzy equations (linear, eigenvalue, differential, etc.) and fuzzy optimization What is new is that we introduce three solution concepts for fuzzy equations: (1) the classical solution; (2) the extension principle solution; and (3) the a-cut and interval arithmetic solution Therefore, it is most important that the reader understand the ideas involved in these three methods of solving fuzzy equations before proceeding

to the rest of the book The easiest introduction to the solution methods is

in Sections 3.1 and 3.2 of Chapter 3 where we look at solving a simple fuzzy linear equation

Also what is new is that we use an evolutionary algorithm to solve fuzzy problems, especially the fuzzy optimization problems Usually the classical methods (calculus, etc.) do not apply to solving fuzzy optimization prob-lems so we employ a direct search algorithm to generate good (approximate) solutions to these fuzzy problems

Our general purpose evolutionary algorithm (abbreviated EA ) is scribed in Chapter 13 at the end of this book It has to be adopted to the different fuzzy problems and this is usually discussed in the chapters where it is used We have not included our evolutionary algorithm in the book since you can download this software (genetic, evolutionary) from the internet Use the terms "genetic", or "evolutionary" algorithm in your search engine and then you can solve your own fuzzy optimization problems Some of the figures in the book are difficult to obtain so they were created using different methods First, many were made using the graphics package

de-in LaTeX2, For example, Figures 3.6,3.11, many de-in Chapters 6 and 10, and

Figures 7.1, 7.3, were done this way Some others, impossible to do within

the graphics package in LaTeX2" were first drawn in Maple [1] and then exported to LaTeX2, We did those in Chapter 7 ( not Figures 7.1 and

7.3) and some in Chapter 3 this way There are some other figures that we

considered easier to do in Maple and then export to LaTeX2, ( all those in

Chapter 2 plus some in Chapter 3) These figures are defined as x = f(y),

for a ~ y ~ 1, which is backwards from the usual y a function of x, and the

"implicitplot" command in Maple made it easy to graph x a function of y ( y

axis vertical and the x axis horizontal) Finally, there were figures done using

the graphics package in LaTeX2" but the data for the graph was obtained

from our evolutionary algorithm or from Maple The graphs in Chapter 4 plus some in Chapter 3, 6 and Figure 10.9 were done this way

Bibliography

[1] Maple 6, Waterloo Maple Inc., Waterloo, Canada

Chapter 2

Fuzzy Sets

In this chapter we have collected together the basic ideas from fuzzy sets and fuzzy functions needed for the book Any reader familiar with fuzzy sets, fuzzy numbers, the extension principle, a-cuts, interval arithmetic, possibility theory and fuzzy functions may go on to the rest of the book A good general reference for fuzzy sets and fuzzy logic is [1], [6]

Our notation specifying a fuzzy set is to place a "bar" over a letter SO

A, B, , X, Y, , Ct, /3, , will all denote fuzzy sets

If 0 is some set, then a fuzzy subset A of 0 is defined by its membership function, written A(x), which produces values in [0,1] for all x in O So, A(x)

is a function mapping 0 into [0,1] If A(xo) = 1, then we say Xo belongs to

A, if A(xt} = 0 we say Xl does not belong to A, and if A(X2) = 0.6 we say the membership value of X2 in A is 0.6 When A(x) is always equal to one or zero we obtain a crisp (non-fuzzy) subset of O For all fuzzy sets B, C,

we use B(x), C(x), for the value of their membership function at x Most

of the fuzzy sets we will be using will be fuzzy numbers

The term "crisp" will mean not fuzzy A crisp set is a regular set A crisp number is just a real number A crisp matrix (vector) has real numbers

as its elements A crisp function maps real numbers (or real vectors) into real numbers A crisp solution to a problem is a solution involving crisp sets, crisp numbers, crisp functions, etc

2.1.1 Fuzzy Numbers

A general definition of fuzzy number may be found in [1],[6], however our fuzzy numbers will be almost always triangular (shaped), or trapezoidal (shaped), fuzzy numbers A triangular fuzzy number N is defined by three

J J Buckley et al., Fuzzy Mathematics in Economics and Engineering

© Springer-Verlag Berlin Heidelberg 2002

6 CHAPTER 2 FUZZY SETS

numbers a < b < c where the base of the triangle is the interval [a, c] and its

vertex is at x = b Triangular fuzzy numbers will be written as N = (a/b/c)

A triangular fuzzy number N = (1.2/2/2.4) is shown in Figure 2.1 We see

Figure 2.1: Triangular Fuzzy Number N

A trapezoidal fuzzy number M is defined by four numbers a < b < c < d

where the base of the trapezoid is the interval [a,d] and its top (where the

membership equal one) is over [b,c] We write M = (a/b,c/d) for trapezoidal

fuzzy numbers Figure 2.2 shows M = (1.2/2,2.4/2.7)

A triangular shaped fuzzy number P is given in Figure 2.3 P is only partially specified by the three numbers 1.2, 2, 2.4 since the graph on [1.2,2], and [2,2.4], is not a straight line segment To be a triangular shaped fuzzy number we require the graph to be continuous and: (1) monotonically in-creasing on [1.2,2]; and (2) monotonically decreasing on [2,2.4] For tri-

angular shaped fuzzy number P we use the notation P ~ (1.2/2/2.4) to show that it is partially defined by the three numbers 1.2, 2, and 2.4 If

P ~ (1.2/2/2.4) we know its base is on the interval [1.2,2.4] with vertex (membership value one) at x = 2 Similarly we define trapezoidal shaped fuzzy number Q ~ (1.2/2,2.4/2.7) whose base is [1.2,2.7] and top is over the interval [2,2.4] The graph of Q is similar to M in Figure 2.2 but it has

continuous curves for its sides

Although we will be using triangular (shaped) and trapezoidal (shaped) fuzzy numbers throughout the book, many results can be extended to more

Figure 2.2: Trapezoidal Fuzzy Number M

general fuzzy numbers, but we shall be content to work with only these special fuzzy numbers

We will be using fuzzy numbers in this book to describe uncertainty For example, in Chapter 4 if a future interest rate is uncertain but be-lieved to be between 7 and 8%, we could model it using a trapezoidal fuzzy number (0.065/0.07,0.08/0.085) Also, in Chapter 7, if the initial value of the derivative is uncertain but known to be around 5, we may model it as

y'(O) = (4/5/6)

2.1.2 Alpha-Cuts

Alpha -cuts are slices through a fuzzy set producing regular (non-fuzzy) sets

If A is a fuzzy subset of some set n, then an a-cut of A, written A[a] is

defined as

A[a] = {x E nIA(x) ~ a} , (2.1) for all a, 0 < a ~ 1 The a = 0 cut, or A[O], must be defined separately Let N be the fuzzy number in Figure 2.1 Then N[O] = [1.2,2.4] Using equation (2.1) to define N[O] would give N[O] = all the real numbers Sim-ilarly, M[O] = [1.2,2.7] from Figure 2.2 and in Figure 2.3 prO] = [1.2,2.4] For any fuzzy set A, A[O] is called the support, or base, of A Many authors call the support of a fuzzy number the open interval (a, b) like the support

8 CHAPTER 2 FUZZY SETS

Figure 2.3: Triangular Shaped Fuzzy Number P

of N in Figure 2.1 would then be (1.2,2.4) However in this book we use the closed interval [a, b] for the support (base) of the fuzzy number

The core of a fuzzy number is the set of values where the membership value equals one If N = (a/b/c), or N ~ (a/b/c), then the core of N is the single point b However, if M = (a/b, c/d), or M ~ (a/b, c/d), then the core

of M = [b,cJ

For any fuzzy number Q we know that Q[a] is a closed, bounded, interval for 0 ::; a ::; 1 We will write this as

(2.2) where ql (a) (q2(a)) will be an increasing (decreasing) function of a with ql(l) ::; q2(1) If Q is a triangular shaped or a trapezoidal shaped fuzzy number then: (1) ql (a) will be a continuous, monotonically increasing func-

tion of a in [0,1]; (2) q2(a) will be a continuous, monotonically decreasing function of a, 0 ::; a ::; 1; and (3) ql(l) = q2(1) (ql (1) < q2(1) for trape-zoids) We sometimes check monotone increasing (decreasing) by showing that dql (a)/da > 0 (dq2(a)/da < 0) holds

For the N in Figure 2.1 we obtain N[a] = [nl(a),n2(a)J, nl(a) = 1.2 +

0.8a and n2(a) = 2.4 - O.4a, 0 S; a ::; 1 Similarly, M in Figure 2.2 has

M[a] = [ml(a), m2(a)], miCa) = 1.2 + 0.8a and m2(a) = 2.7 - 0.3a, 0 S;

a ::; 1 The equations for ni(a) and mi(a) are backwards With the y-axis vertical and the x-axis horizontal the equation nl (a) = 1.2 + 0.8a means

x = 1.2 + O.By, 0 :::; y :::; 1 That is, the straight line segment from (1.2,0) to (2,1) in Figure 2.1 is given as x a function of y whereas it is usually stated as

y a function of x This is how it will be done for all a-cuts of fuzzy numbers

2.1.3 Inequalities

Let N = (a/b/c) We write N ~ d, d some real number, if a ~ d, N > d when

a> d, N :::; d for c :::; d and N < d if c < d We use the same notation for triangular shaped and trapezoidal (shaped) fuzzy numbers whose support is the interval [a, c]

If A and B are two fuzzy subsets of a set 0, then A :::; B means A(x) :::;

B(x) for all x in 0, or A is a fuzzy subset of B A < B holds when A(x) <

B(x), for all x There is a potential problem with the symbol :::; In some places in the book (Chapters 4, 6 and 10) M :::; N, for fuzzy numbers M and

N, means that M is less than or equal to N It should be clear on how we use ":::;" as to which meaning is correct

2.1.4 Discrete Fuzzy Sets

Let A be a fuzzy subset of O If A(x) is not zero only at a finite number of

x values in 0, then A is called a discrete fuzzy set Suppose A(x) is not zero

only at Xl, X2, X3 and X4 in O Then we write the fuzzy set as

(2.3)

where the /li are the membership values That is, A(Xi) = /li, 1 :::; i :::; 4, and A(x) = 0 otherwise We can have discrete fuzzy subsets of any space O Notice that a-cuts of discrete fuzzy sets of R, the set of real numbers, do not produce closed, bounded, intervals

2.2 Fuzzy Arithmetic

If A and B are two fuzzy numbers we will need to add, subtract, multiply and divide them There are two basic methods of computing A + B, A - B, etc which are: (1) extension principle; and (2) a-cuts and interval arithmetic

10 CHAPTER 2 FUZZY SETS

In all cases C is also a fuzzy number [6] We assume that zero does not belong

to the support of B in C = A/B If A and B are triangular (trapezoidal)

fuzzy numbers then so are A + B and A - B, but A B and A/ B will be

triangular (trapezoidal) shaped fuzzy numbers

We should mention something about the operator "sup" in equations (2.4)

- (2.7) If 0 is a set of real numbers bounded above (there is a M so that

x :::; M, for all x in 0), then sup(O) = the least upper bound for O If 0 has a maximum member, then sup(O) = max(O) For example, if 0 = [0,1), sup(O) = 1 but if 0 = [0,1], then sup(O) = max(O) = 1 The dual operator

to "sup" is "inf' If 0 is bounded below (there is a M so that M :::; x for all

x EO), then infCO) = the greatest lower bound For example, for 0 = (0,1] inf(O) = 0 but if 0 = [0,1], then inf(O) = minCO) = O

Obviously, given A and B, equations (2.4) - (2.7) appear quite

compli-cated to compute A + B, A - B, etc So, we now present an equivalent

procedure based on a-cuts and interval arithmetic First, we present the basics of interval arithmetic

2.2.2 Interval Arithmetic

We only give a brief introduction to interval arithmetic For more tion the reader is referred to ([7],[8]) Let [al,bl ] and [a2,b2] be two closed, bounded, intervals of real numbers If * denotes addition, subtraction, mul-tiplication, or division, then [al, bd * [a2, b2] = [a,,8] where

[al,b 1 ]· [b~' a~] ,

(2.9) (2.10) (2.11)

al > ° and b2 < 0, or b1 > 0 and b2 < 0, etc For example, if al ~ ° and

a2 ~ 0, then

(2.15) and if b1 < 0 but a2 ~ 0, we see that

(2.16) Also, assuming b1 < 0 and b 2 < 0 we get

(2.17) but al ~ 0, b 2 < 0 produces

C[a] = A[a] + B[a] (2.19)

We add the intervals using equation (2.9) Setting C = A - B we get

C[a] = A[a] - B[a] , (2.20) for all a in [0,1] Also

C[a] = A[a] B[a] , (2.21) for C = A· Band

C[a] = A[a]/ B[a] , (2.22) when C = A/ B This method is equivalent to the extension principle method

of fuzzy arithmetic [6] Obviously, this procedure, of a-cuts plus interval arithmetic, is more user (and computer) friendly

12 CHAPTER 2 FUZZY SETS

[4+a,6-a) So, if C = A·B we obtain C[a) = [(a-3)(6-a), (-1-a)(4+a)),

o :::; a :::; 1 The graph of C is shown in Figure 2.4

2.3 Fuzzy Functions

In this book a fuzzy function is a mapping from fuzzy numbers into fuzzy

numbers We write H(X) = Z for a fuzzy function with one independent

variable X Usually X will be a triangular (trapezoidal) fuzzy number and then we usually obtain Z as a triangular (trapezoidal) shaped fuzzy number

For two independent variables we have H(X,Y) = z

Where do these fuzzy functions come from? They are usually extensions

of real-valued functions Let h: [a,b) -t R This notation means z = hex)

for x in [a, b) and z a real number One extends h: [a, b) -t R to H(X) = Z

in two ways: (1) the extension principle; or (2) using a-cuts and interval arithmetic

If h is continuous, then we have a way to find a-cuts of Z Let Z[a] =

[Zl (a), z2(a)] Then [3]

for 0 ~ a ~ 1

mini hex) I x E X[a] } ,

maxi hex) I x E X[a] } ,

(2.24) (2.25)

If we have two independent variables, then let z = hex, y) for x in [aI, b l ],

mini h(x,y) I x E X[a], y E Y[a] } ,

maxi h(x,y) I x E X[a], y E Y[a] } ,

(2.27) (2.28)

o ~ a ~ 1 We use equations (2.24) ~ (2.25) and (2.27) ~ (2.28) throughout this book

2.3.2 Alpha-Cuts and Interval Arithmetic

All the functions we usually use in engineering and science have a computer algorithm which, using a finite number of additions, subtractions, multipli-cations and divisions, can evaluate the function to required accuracy ([2]) Such functions can be extended, using a-cuts and interval arithmetic, to fuzzy functions Let h : [a, b] -t R be such a function Then its exten-sion H(X) = Z, X in [a, b] is done, via interval arithmetic, in computing

h(X[a]) = Z[a], a in [0,1] We input the interval X[a], perform the

arith-metic operations needed to evaluate h on this interval, and obtain the interval

Z[a] Then put these a-cuts together to obtain the value Z The extension

to more independent variables is straightforward

For example, consider the fuzzy function

Z = H(X) = A X + 13

CX+D' (2.29)

14 CHAPTER 2 FUZZY SETS

for triangular fuzzy numbers A, B, C, D and triangular fuzzy number X in

[0,10] We assume that C ~ 0, D> 0 so that C X + D > O This would be the extension of

(2.30)

We would substitute the intervals A[a] for Xl, B[a] for X2, O[a] for X3, D[a]

for X4 and X[a] for X, do interval arithmetic, to obtain interval Z[a] for Z

Alternatively, the fuzzy function

would be the extension of

Z = H(X) for the a-cut and interval arithmetic extension of h

We know that Z can be different from Z* But for basic fuzzy arithmetic

in Section 2.2 the two methods give the same results In the example below

we show that for h(x) = x(1 - x), X in [0,1], we can get Z* i= Z for some

X in [0,1] What is known ([3],[7]) is that for usual functions in science and engineering Z* ~ Z Otherwise, there is no known necessary and sufficient

conditions on h so that Z* = Z for all X in [a, b]

There is nothing wrong in using a-cuts and interval arithmetic to evaluate fuzzy functions Surely, it is user, and computer, friendly However, we should

be aware that whenever we use a-cuts plus interval arithmetic to compute

Z = H(X) we may be getting something larger than that obtained from the extension principle The same results hold for functions of two or more independent variables

(2.34) (2.35)

for Z[o:] = [Zl (0:), Z2(0:)], 0: in [0,1]

The extension principle extends the regular equation Z = (1 - x)x, 0 :::;

x :::; 1, to fuzzy numbers as follows

0.50 - 0.250: Equations (2.34) and (2.35) give Z[0.50] = [5/64,21/64] but equations (2.37) and (2.38) produce Z* [0.50] = [7/64, 15/64] Therefore,

Z* -I- Z We do know that if each fuzzy number appears only once in the fuzzy expression, the two methods produce the same results ([3],[7]) However,

if a fuzzy number is used more than once, as in equation (2.33), the two procedures can give different results

2.4 Possibility Theory

We will be using some of possibility theory only in Section 6.2 (Fuzzy PERT) and Section 6.4 (Fuzzy Queuing Theory) For a general introduction to possi-bility theory we suggest ([4],[5]), and a brief review of the parts of possibility theory needed in Chapter 6 is in this section

Let X be a fuzzy variable whose values are restricted by a possibility distribution A, where A is a fuzzy subset of R Possibility distributions need

to be normalized which means that A(x) = 1 for some x in It If E is any subset of R, not a fuzzy set, we compute the possibility that X takes its

values in E as follows

Poss[X E E] = sup {A(x) I x E E} (2.39) This is analogous to probability theory where we use in fuzzy set theory sup (or max) in place of summing (addition) in probability and min in place of multiplication If X is a random variable with probability density f(x), then the probability that X takes its values in E is

Prob[X E E] = f f(x)dx (2.40)

E

In place of integration (summing) we use "sup" in fuzzy set theory

Now consider Xl, , Xn fuzzy variables with associated possibility tributions AI' ' An, respectively Let X = (Xl' ' Xn) and x =

dis-16 CHAPTER 2 FUZZY SETS

(Xl, , Xn) ERn Assuming that the fuzzy variables are non-interactive (analogous to independent in probability theory) we form their joint 7r pos-sibility distribution as

Poss[X E EI U~] = max {Poss[X E EIJ,Poss[X E E 2 ]} • (2.43)

The possibility of a union is the maximum of the individual possibilities

Bibliography

[1 J J.J Buckley and E.Eslami: Introduction to Fuzzy Logic and Fuzzy Sets, Physica-Verlag, Heidelberg, Germany, 2001

[2J J.J Buckley and Y Hayashi: Can Neural Nets

be Universal Approximators for Fuzzy Functions? Fuzzy Sets and Systems, 101 (1999), pp.323-330 [3J J.J Buckley and Y Qu: On Using a-cuts to Eval-uate Fuzzy Equations, Fuzzy Sets and Systems, 38 (1990), pp 309-312

[4J D Dubois and H Prade: Possibility Theory: An proach to Computerized Processing of Uncertainty, Plenum Press, N.Y., 1988

Ap-[5J D Dubois and H Prade (eds.): Fundamentals of Fuzzy Sets, Kluwer, The Netherlands, 2000

[6J G.J Klir and B Yuan: Fuzzy Sets and Fuzzy Logic: Theory and Applications, Prentice Hall, Upper Sad-dle River, N.J., 1995

[7J R.E Moore: Methods and Applications of val Analysis, SIAM Studies in Applied Mathematics, Philadelphia, 1979

Inter-[8J A Neumaier: Interval Methods for Systems of tions, Cambridge University Press, Cambridge, U.K.,

Equa-1990

Chapter 3

Solving Fuzzy Equations

In this chapter we first look at different types of solutions to the simple fuzzy linear equation A X + B = C and then systems of fuzzy linear equations Solving fuzzy differential equations, fuzzy difference equations and fuzzy in-tegral equations, come later on in the book In the applications section we

-also look at solving A X + B X = C, the fuzzy quadratic At the end

of the chapter we discuss fuzzy input-output analysis Solutions to more complicated fuzzy equations are discussed in ([2], [3], [6]) Throughout this chapter, except in Section 3.5, we use triangular, and triangular shaped, fuzzy numbers In Section 3.5 we use trapezoidal (shaped) fuzzy numbers

3.1 AX +B = C

A, Band C will be triangular fuzzy numbers so let A = (al/a2/a3)' B =

(bl/b 2 /b 3 ) and C = (Cl/C2/C3)' X, if it exists, will be a triangular shaped fuzzy number so let X ~ (Xl/X2/X3)' In the crisp equation

we immediately obtain X = (c - b) / a, if a "I-O We used the important facts

b - b = 0 and (l/a)a = 1 from real numbers to get the solution

We try this same approach with the fuzzy equation

(3.2)

we get

(l/A)(A X + (B - B)) = (l/A)(C - B) (3.3) But the left side of the equation (3.3) does not equal X since B - B "I- 0 and (l/A)(A) "I- 1 For example, if B = (1/2/3), then B - B = (-2/0/2) not zero Also, if A = (1/2/3), (l/A)(A) ~ ((1/3)/1/3), a triangular shaped fuzzy number, not one

J J Buckley et al., Fuzzy Mathematics in Economics and Engineering

© Springer-Verlag Berlin Heidelberg 2002

This shows a major problem in solving fuzzy equations: some basic ations we used to solve crisp equations do not hold for fuzzy equations Actu-ally, this comes as no great surprises because this also happens in probability theory If X is a random variable with positive variance, then X - X # 0 and X/X # 1 since both X - X and X/X will have positive variance

oper-We now introduce our first solution method, called the classical method, producing solution Xc (when it exists) This procedure employs a-cuts and interval arithmetic to solve for Xc Let A[a] = [ada), a2(a)], B[a] =

[b l (a),b2(a)], C[a] = [cI(a),c2(a)] and Xc[a] = [xI(a),x2(a)], 0 ~ a ~ 1

Substitute these into equation (3.2) producing

We now use interval arithmetic to solve equation (3.4) for Xl (a) and x2(a)

We say that this method defines solution Xc when [Xl (a), X2 (a)] defines the a-cuts of a fuzzy number For the Xl (a), X2(a) to specify a fuzzy number

we need:

1 xI(a) monotonically increasing, 0 ~ a ~ 1;

2 x2(a) monotonically decreasing, 0 ~ a ~ 1; and

We did not mention anything about the Xi (a) being continuous because throughout this book xI(a), x2(a) will be continuous If xI(I) < x2(1)

we obtain Xc a trapezoidal shaped fuzzy number

3.1 AX +B = C 21

Example 3.1.2

Now set A = (8/9/10), B = (-3/ - 2/ - 1) and C = (3/5/7) So A[a] =

[8 + a, 10 - a], B[a] = [-3 + a, -1-a], C[a] = [3 + 2a, 7 - 2a] Again we must have Xc> 0 so we obtain

6+a 8+a' 8-a

10-a·

(3.8) (3.9)

We see that xl(a) is increasing (its derivative is positive), x2(a) is decreasing (derivative is negative) and xl(l) = 7/9 = x2(1) The solution Xc exists, with a-cuts [xl(a),x2(a)], shown in Figure 3.1

Figure 3.1: Solution to Example 3.1.2

Working more examples, like Examples 3.1.1 and 3.1.2 above, we conclude that too often fuzzy equations have no solution (X c) This motivated the

authors in ([2], [3], [6], [7]) to propose new solutions for fuzzy equations These new solutions will be introduced in the next section The classical solution, plus the new solutions, will be used throughout this book

3.2 New Solutions

We continue working with the fuzzy equation A X + B = C The new solutions simply fuzzify the crisp solution (c - b)ja, a i- O The fuzzified crisp solution is

(C-B)jA, (3.10) where we assume zero does not belong to the support of A There are two ways to evaluate equations (3.10) The first method is the extension principle

If Xe is the value of equation (3.10) using the extension principle, then

Xe(X) = max{7f(a,b,c) I (c- b)ja = x} , (3.11) where

7f(a, b, c) = min {A(a), B(b), C(c)} (3.12)

Since the expression (c - b) j a, a i- 0, is continuous in a, b, c we know how

to find a-cuts of X e [5]

Xel(a) = min{(c-b)jalaEA[a],bEB[a],cEC[a]}, (3.13) Xe2(a) max {(c - b)ja I a E A[a], bE B[a], c E C[a]) (3.14)

where

(3.15)

o :::; a :::; 1 X e will be a triangular shaped fuzzy number when A, B, Care triangular fuzzy numbers

Theorem 3.1 If Xc exists, Xc:::; Xe'

Proof 3.1 It suffices to show Xc[a] ~ Xe[a], 0:::; a :::; 1 Choose a E [0,1]

and x E Xc[a] From equation (3.,,/.) we see there is an a E A[a], bE B[a],

c E C[a] so thatax+b = c Or, x = (c-b)jafora E A[a], bE B[a], c E C[a] This implies, from equations (3.13) and (3.14), thatxel(a):::; x:::; xe2(a), or

3.2 NEW SOLUTIONS 23

Proof 3.2 Theorem 3.1 in [14J

X I mayor may not satisfy the fuzzy equation X I will be a triangular shaped fuzzy number when A, B, C are all triangular fuzzy numbers We summarize these results as:

2 Xc always satisfies the fuzzy equation;

Our general strategy for solving fuzzy equations will be:

1 the solution is Xc when it exists;

2 if Xc fails to exist, the solution is X e; and

3 if Xc fails to exist and X e is difficult to construct, use X I as the

(approximate) solution

For more complicated fuzzy equations X e will be difficult to compute However, X I is usually easily constructed, since it uses only max, min and the arithmetic of real numbers For this reason we suggest approximating

Xe by XI when we do not have Xe'

Example 3.2.1

This continues Example 3.1.1 where Xc does not exist To calculate Xe we

need to evaluate equations (3.13) and (3.14) But this is easily done since (c - b) / a is increasing in c and decreasing in both band a So

which is the same as Xe[a] because intervals in equation (3.19) are positive

In this example, we get Xe = XI

This continues Example 3.1.2 We notice that, since A > 0 and C - B > 0,

.!L[c-b] oc a = 1 a > 0 'ob a !L[c-b] = _1 a < 0 and .!L[c-b] oa a = b-c a 2 < O This means that the expression c-;;b is increasing in c but decreasing in a and b Then

equations (3.13) and (3.14) become

Let us next look at solutions to systems of fuzzy linear equations

3.3 Systems of Fuzzy Linear Equations

This section is based on [7], see also [15], [16] and [17] Let A = [£iij] be

- t -

-a n x n matrix of triangular fuzzy numbers £iij, B = (bl , , b n ) a n x 1 vector of triangular fuzzy numbers bi and xt = (Xl, ,X n ) a n x 1 vector

3.3 SYSTEMS OF FUZZY LINEAR EQUATIONS 25

y

1

Figure 3.3: Solution to Example 3.2.2

of unknown triangular shaped fuzzy numbers Xj Set aij = (aijr!aij2/aij3),

bi = (biI/bi2 /bi3), and Xj ~ (Xjr!Xj2/Xj3)' We wish to solve

for 0 :::; a :::; 1 Let v = (an,aI2, ,ann) E Rk, k = n 2, be a vector

in a[O] Each v E a[O] determines a crisp n x n matrix A = [aij] Also,

b t = (bl, ,bn) ERn is a vector in b[O] As in ([3], [7]) we assume A-I

exists for all v in a[O] The existence of A-lover a[O] simplifies the discussion

of the joint solution to be introduced below

The joint solution XJ, a fuzzy subset of Rn, is based on the extension principle

(3.25)

where

(3.26)

The vertex of XJ(x), where the membership value is equal to one, is at

x = A-Ib for v = (a112, aI22, , ann2), b t = (bI2 , , b n2 ) In the crisp case

the solution to Ax = b is a vector x = A-lb in Rn , so for the fuzzy case

A X = B, the (joint) solution is a fuzzy vector about the crisp solution A-lb,

for v and b at the vertex values of all the triangular fuzzy numbers

In the crisp case the marginals, the Xi, are just the components of the

vector x = A-I b In the fuzzy case we obtain the marginals X Ji by projecting

X J onto the coordinate axes Then

(3.27)

for 1 :::; i :::; n Obviously, it will be difficult to compute X J and X Ji,

1 :::; i :::; n, for n ~ 4 We will determine the joint solution, and its marginals,

in two examples at the end of this section for n = 2

Since X J is difficult to determine we now turn to methods of finding the marginals directly without first computing the joint solution As in the previous section there will be three solutions Xci, X ei and X Ii, 1 :::; i :::; n

The classical solution is determined by substituting the intervals aij[a], bi[a] and Xi[a] = [xi1(a),xi2(a)] into A X = B and solving for the xi1(a), xi2(a), 1 :::; i :::; n The resulting equations are evaluated using interval

arithmetic If the intervals [xiI(a),xi2(a)] define a triangular shaped fuzzy

number Xi for 0:::; a :::; 1, 1 :::; i :::; n, then this solution is called the classical

solution and we write Xci = Xi, 1 :::; i :::; n The conditions for [XiI (a), Xi2 (a)]

to define Xci are given in the previous section The equations to solve for xi1(a) and xi2(a) are

n

~)aijl (a),aij2(a)][Xjl (a), Xj2(a)] = [biI (a), bi2(a)] ,

j=l

(3.28)

for 1 :::; i :::; n, where aij[a] = [aijl (a), aij2 (a)], bi[a] = [bil (a), bi2(a)] After

using interval arithmetic we obtain a (2n) x (2n) system to solve for XiI (a), xda), 0:::; a :::; l

As in the previous section too often the Xci fail to exist We only need

X ci, for one value of i, to fail to exist for the classical solution to not exist When the classical solution does not exist we turn to X ei, 1 :::; i :::; n

We will use Cramer's rule ([12, p 218]) on Ax = b to solve for each Xi

A comes from v E a[O] and let b E b[O] Let Aj be A with its j-th column

3.3 SYSTEMS OF FUZZY LINEAR EQUATIONS 27

Xej2 [a] = max {I~II I v E ala], bE bra] } , (3.32)

To get the X Ij we evaluate equation (3.29) using a-cuts and interval arithmetic Substitute intervals aij[a] and bi[a] for aij and bi in IAjl/IAI,

evaluate using interval arithmetic, and the result is XIj[a], 0 :::::; a :::::; 1,

1:::::; j:::::; n

Theorem 3.3 If the Xci exist, 1 :::::; i :::::; n, then Xci :::::; X Ji :::::; Xei :::::; XIi,

1:::::; i :::::; n

Proof 3.3 1 We show Xci :::::; X Ji for all i by showing Xci [a] C

XJi[a] Choose a E [0,1] and Xi E Xci [a], 1 :::::; i :::::; n Then

Xi E [xcil(a),xci2(a)], the a-cuts of Xci, 1 :::::; i :::::; n From equation

(3.28) there are aij E aij[a] and bi E bi[a] so that

n

LaijXj = bi ,

1 :::::; i :::::; n Let xt = (Xl, ,Xn) Let v = (au, ,ann ) in ala],

b t = (b l , , bn) E bra] For this v and b we have x = A-I b This implies, from the definition of X J, thatXJ(x) 2': a HenceXJi(xi) 2': a

and Xi E XJi[a], all i

2 We argue that XJi :::::; Xei by showing XJi[a] C Xei[a], 1 :::::; i :::::; n Choose a E [0,1] and Xi E XJi[a], 1:::::; i:::::; n Let xt = (Xl, ,Xn)

We first show XJ[a] C rr~=l Xei[a] Let wE XJ[a] From the tion of X J there are aij E aij [a], bi E bi[a] so that if v = (au, , ann),

defini-bt = (bl, ,bn), W = A-lb This means Wj = IAjl/IAI for v E ala],

bE bra] and Wj E Xej[a], 1:::::; j :::::; n

Now we argue that X E rr~=l Xei[a] If X E XJ[a], then this follows from above So assume X ¢ XJ[a] But X must then be in rr~=l Xei[a]

or else there is a value of i, say i = k, so that X Jk(Xk) < a, a diction

contra-But X E rr~=l Xei[a] means that Xi E Xeda] all i

3 Xei :::::; XIi follows from Theorem 3.1 of [14J

Our solution strategy is: (1) use Xci, 1 :::::; i :::::; n, if it exists; (2) if the

classical solution does not exist use X Ji, 1 :::::; i :::::; n However, if the joint solution is too difficult to compute use X ei, 1 :::::; i :::::; n Equations (3.31) and

(3.32) may be hard to evaluate to get the X ei We have used evolutionary algorithms in the past ([8] - [11]) to do this One can always use the X Ii

because they are the easiest to calculate Notice how the fuzziness grows (the supports do not decrease) as we go from Xci to X Ii The only solution guaranteed to satisfy the fuzzy equations in the classical solution

In the following two examples we only consider 2 x 2 fuzzy matrices since then we can easily see pictures of o:-cuts of the joint solution It is known that, in general, o:-cuts of the joint solution need not be convex [13] For example, in two dimensions XJ[O:] need not be a rectangle

[4 + 0:, 7 - 20:] [Xc11 (0:), Xc12 (0:)]

[6 + 20:, 12 - 40:]· [XC21(0:), Xc22 (0:)]

[1 + 0:,3 - 0:] , [2 + 30:,8 - 30:]

which define triangular shaped fuzzy numbers

It was shown in [7] that a way to find o:-cuts of X J is

XJ[O:] = {A-1b I v E a[o:], bE b[o:] }

X J[O:] = b1 [0:] X b2 [0:] ,

a11[O:] a22[O:]

(3.35) (3.36)

(3.37) (3.38)

(3.39)

(3.40)

which is a rectangle in R2 for all 0::; 0:::; 1 The first part of XJ[O:] is XJ1[O:]

and XJ2[O:] is the second part Then

0::;0:::;1

[ 1 + 0: 3 - 0:]

7-20:'4+0: ' [ 2 + 30: 8 - 30:]

12 - 40:' 6 + 20:

(3.41) (3.42)

1

Next we find that IAII/IAI = bI/au and IA21/IAI = b2/a22' From tions (3.31) and (3.32) we obtain Xei = X Ji , i = 1,2

equa-Finally, we substitute the intervals aula], a22[a], bl[a] and b2[a] into

Xl = bI/au and X2 = b2/a22 and we see that XIi = X ei , i = 1,2

For this 2 x 2 fuzzy diagonal matrix A we get

-and b = (bl , ~), where au = (1/2/3), a22 = (2/5/8), bl = (4/5/7) and

b2 = (6/8/12) Then aural = [1 + a,3 - a], a22[a] = [2 + 3a,8 - 3a],

bl[a] = [4 + a, 7 - 2a] and b2[a] = [6 + 2a, 12 - 4a]

As in Example 3.3.1 we solve for XcI> 0 and obtain

[4+ a 7 - 2a]

l+a' 3-a ' (3.45)

which does not define a fuzzy number since 8/8a[(4 + a)/(1 + a)] < 0, or (4 + a)/(1 + a) is a decreasing function of a in [0,1] The classical solution does not exist

Figure 3.5: X c2 and XJ2 in Example 3.3.1

We find a-cuts of XJ using equation (3.39) We only go through the details for a = 0 and a = 1 An equivalent expression to equation (3.39) is

x J[a] = { x E Rn I Ax = b, v E a[a], bE bra]} (3.46)

For a = 1 we get x = (2.5,1.1) For a = 0 first assume Xl ~ 0, X2 ~ O Then

we want all solutions for Xl and X2 so that

([1,3]XI + [0, 0]X2) n [4,7] i- 0 ,

([1, I]XI + [2, 8]X2) n [6,12] i- 0

We have used the a = 0 cuts of au, a22, hI and h2 This means

Xl < 7, 3XI > 4 ,

Xl + 2X2 ~ 12,

Xl + 8X2 > 6 ,

(3.47) (3.48)

(3.49) (3.50) (3.51) (3.52)

for Xl ~ 0, X2 ~ 0 in the first quadrant Now Xl must be non-negative so we can now only consider the fourth quadrant

Assume Xl ~ 0 and X2 ~ O Then the equations become

Xl < 7, (3.53)

Xl + 8X2 < 12, (3.55)

Xl + 2X2 > 6, (3.56)

XJ2[a] It turns out, for this example, that XJ1 = Xel and X J2 = X e2

Using equations (3.31) and (3.32) we find a-cuts of X ej , j = 1,2 It is easy to see that

X el a [] = [4 3 _ a' 1 + a 7 - + 2a] a ' (3.57)

o ::; a ::; 1 However, X e2 [ a] is a little more difficult since we need to find

the max and min of

(3.58)

for au E au [a], a22 E ~2[a], b l E bl[a] and b2 E b2[a] We did this and X e2

is shown in Figure 3.8

X

8

(3.59) with xI21(a) = N1(a)/D2(a), 0 ~ a ~ 0.0981 = (J108 -10)/4, X121(a) = N1(a)/D1(a) for 0.0981 ~ a ~ 1 and XI22 (a) = N2(a)/D2(a) for all a and

Nl(a) = (1 + a)(6 + 2a) - (7 - 2a), N2(a) = (3 - a)(12 - 4a) - (4 + a),

Dl (a) = (3 - a)(8 - 3a),

D2(a) = (1 + a)(2 + 3a)

(3.60) (3.61) (3.62) (3.63)