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Algorithms to Extend Crisp Functions andTheir Inverse Functions to Fuzzy NumbersO. G. Duarte,1,* M. Delgado,2,† I. Requena2,‡

1Department of Electrical Engineering, Universidad Nacional de Colombia,Bogota, Colombia2Department of Computer Sciences and A.I./E.T.T.S.I. Informatica,Universidad de Granada, 18071 Granada, Spain

In this article we present an algorithm to extend continuous crisp functions to fuzzy numbersusing the extension principle; the functions must be monotonically increasing in some of thearguments and monotonically decreasing in the others. Then, we present two different solutionsto the problem of extending inverse functions that we have called possible extension andnecessary extension. Finally, using these solutions we have generated a family of intermediateextensions that let us define a parameter to measure the existence of the extended inversefunctions. © 2003 Wiley Periodicals, Inc.

1. INTRODUCTION

Mathematical models are widely used to represent real phenomena in differentsciences and disciplines such as physics, engineering, economics, and sociology.These models use numerical variables connected with numerical functions todescribe their object of study. However, it is not normally possible to know exactlythe numerical value that the variables should have, either because they are esti-mated quantities or because they are measured with instruments of limited preci-sion. Consequently, uncertainty is always present in these mathematical models.Fuzzy numbers are a useful tool to study this uncertainty, but there are other usefulstrategies such as the theory of errors and sensitivity analysis.

The notion of a fuzzy number was introduced by Zadeh in 19751–3 and hasbeen studied and developed by many authors including Dubois and Prade,4–7 Jain,8

Yager,9 Baas and Kwakernaak,10 Mizumoto et al.,11,12 and Sanchez.13,14 Althoughfuzzy numbers can successfully represent the uncertainty of a numerical variablein a mathematical model, there are some difficulties representing the functions that

*Author to whom all correspondence should be addressed: e-mail: [email protected].†e-mail: [email protected].‡e-mail: [email protected].

INTERNATIONAL JOURNAL OF INTELLIGENT SYSTEMS, VOL. 18, 855–876 (2003)© 2003 Wiley Periodicals, Inc. Published online in Wiley InterScience(www.interscience.wiley.com). • DOI 10.1002/int.10121

relate these variables, mainly because with the most usual definition (used in thispaper and shown later), the fuzzy numbers do not have a group algebraic structure.However, there are alternative definitions that give them a ring structure (Tamuraand Horiuchi15) or multiplicative groups and even linear spaces (Mares16), but theyloose some of their ability to represent uncertainty.

Another difficulty is that mathematical models generally are designed to dealwith nonfuzzy (crisp) numerical variables and therefore the functions in the modelsoperate on crisp numbers. If we attempt to use fuzzy numbers to representnumerical variables, it is necessary to modify the functions. In other words, wehave to extend the crisp functions included in the model to fuzzy numbers. Theusual way to extend a function is to use the extension principle postulated byZadeh;1–3 Although generally this is an effective and useful strategy, in some casesit is not the best way to deal with inverse functions, because there are specialdifficulties associated with inverse functions; this has been studied in depth byYager.9

Even an elementary operation such as addition cannot be inverted when fuzzynumbers are used, because if X and Y are fuzzy numbers, (X � Y) � Y � X; inother words, when dealing with fuzzy numbers, subtraction is not the inverseoperation of addition. Bouchon-Meunier et al.17 have shown that if a crisp functionis invertible, then fuzzy numbers are the only fuzzy sets that maintain the invert-ibility of the function.

Furthermore, as Giachetti18 notes, when dealing with functions of more thanone argument, we can have diverse definitions of “invertibility” according to theapplication of the inverse function; if we suppose that X, Y, and Z are fuzzynumbers related by the equation X � Y � Z, then we can distinguish at least twodifferent situations:

● Y and Z are known and we want to know possible values of X● Y is known and we want to know definite values of X to ensure that Z has certain values.

The second approach does not always have a solution, even when the crisp inversefunction is well defined, because of the uncertainty of the fuzzy numbers: if Y hasa lot of uncertainty, and we want Z to only have a little uncertainty, the situationmay be incoherent, because uncertainty is accumulative.

The main aim of this study is to present some algorithms that allow functionsthat relate crisp numerical variables in a mathematical model to be used when thesevariables are represented by fuzzy numbers, without the need to modify theoriginal crisp functions. The algorithms related to inverse functions ensure theexistence of a solution, but sometimes it is necessary to modify the problemconditions. The functions included in this article are multiple argument functions[representing multiple input single output (MISO) models] and monotonicallyincrease in some of the arguments and monotonically decrease in the others.

Section 2 sets out basic definitions and notations; Section 3 shows thealgorithm used to extend crisp functions to fuzzy numbers; in Section 4 we developsome algorithms to extend crisp inverse functions; several examples are given inSection 5; and the conclusions are set out in Section 6.

856 DUARTE, DELGADO, AND REQUENA

2. BASIC DEFINITIONS AND NOTATIONS

DEFINITION 1. Fuzzy numbers. A fuzzy number usually is defined as a normal,convex, and upper semicontinuous fuzzy set on R. Let A be one of such fuzzynumbers with membership function A(x); the �-cuts of A are

A� � � �x�A�x� � �� 0, 1 lim

�30�

�inf�A��, lim�30�

�sup�A�� 0

A� is always a closed interval, even for � � 0 (A0 is the support of the fuzzynumber). The fuzzy number A is defined completely using two functions LA(�) andRA(�)

LA�� LA� � inf�A��

RA�� RA� � sup�A��

Using these functions, the �-cuts of A can be written in an interval format:

A� � LA��, RA��

A is a fuzzy number if and only if LA(�) and RA(�) satisfy the following conditions:

● LA(�) must be monotonically increasing and right continuous● RA(�) must be monotonically decreasing and left continuous (1)● LA(1) � RA(1)

DEFINITION 2. The fuzzy number A also may be represented using the functionDA(�, d) : [0, 1] � {�1, 1} 3 R as follows:

DA��, d� � �LA�� if d � 1RA�� if d � �1

The �-cuts of A may now be written as follows:

A� � DA��, 1�, DA��, �1�

Note that the representation of a fuzzy number using the function DA(�, d) isconceptually the same as if we use �-cuts. In this study we will use the functionDA(�, d) so that the algorithms are more compact.

2.1. Discrete Representation of a Fuzzy Number

Although the parameter � used in Definitions 1 and 2 may take any valuebetween 0 and 1, a practical implementation in digital computers entails assigning� a discrete set of values. The algorithms presented in this article suppose that wewant to know a discrete number of �-cuts, associated with an ordered set Alfa �{�1, �2, . . . , �p} where �i � �i�1, �p � 1, and �1 � 0. It should be noted thatif we restrict the possible values of � to the set Alfa � {0, 1} and we assume a

ALGORITHMS TO EXTEND CRISP FUNCTIONS 857

linear interpolation for the other values of �, then we are representing trapezoidalfuzzy numbers.

DEFINITION 3. In the rest of the article we assume that y � f(X), and f : U1 � U2

� . . . � Un 3 V is a continuous function, strictly monotonically increasing withsome of the n variables and strictly monotonically decreasing with the others. Theterms Ui and V are convex subsets of R. We also assume that xk � fk

�1(y, Xk), andXk � [x1 x2

. . . xk�1 xk�1. . . xn] is the function that calculates the

value of xk when we know the output y and the other input variables xi; further-more, we assume that f(X) and fk

�1(y, Xk) are well defined in all the possiblecombinations of their respective arguments.

We define the function df : I 3 {1, �1} (I is the index set {1, 2, . . . , n}) asfollows:

df�i� � �1 if f�X� increases with xi

�1 if f�X� decreases with xi

We can assert that if the variables such that df (i) � 1 increase and/or thevariables such that df (i) � �1 decrease, then y � f(X) increases. On the otherhand, if the variables such that df (i) � 1 decrease and/or the variables such thatdf (i) � �1 increase, then y � f(X) decreases.

DEFINITION 4. Let A1, A2, . . . , An by n fuzzy numbers defined on U1, U2, . . . , Un,respectively, using the functions DAi(�, d) (Definition 2), X � [A1 A2

. . . An]and f(X) as in the previous definitions. We define

DX��, df� � DA1��, df�1�� DA1��, df�2�� · · · DA1��, df�n��

DX��, �df� � DA1��, �df�1�� DA1��, �df�2�� · · · DA1��, �df�n��

(2)

Each of the elements DAi(�, df (i)) and DAi(�, �df (i)) belong to the �-cut of Ai.Furthermore, they are the lowest or highest value of the �-cut of Ai depending onthe increasing-decreasing condition of the variable xi. So, DX(�, df) is the n-upleof values of X belonging to the respective �-cuts of X1, X2, . . . , Xn that gives y thelowest value. Moreover, DX(�, �df) is the n-uple of values of X belonging to therespective �-cuts of X1, X2, . . . , Xn that gives y the highest value. Note that DAi(�,df (i)) is an increasing function of �, whereas DAi(�, �df (i)) is a decreasing one.

3. EXTENDING CRISP FUNCTIONS TO FUZZY NUMBERS

Let y � f(X) be a crisp function with the characteristics indicated inDefinition 3. Suppose we want to extend it to fuzzy numbers; i.e., we want to finda function that allows us to calculate y (or more exactly, a fuzzy number repre-senting y) when we represent the n arguments of f(X) using the fuzzy numbers A1,A2, . . . , An. Because of the continuity of f(X), we can extend the function y �f(X) to fuzzy numbers using the extension principle with the �-cuts as follows:


y� � f�A1�, A2�, . . . , An��

The foregoing expression implies that an �-cut of y is the set of all the valuesobtained when all the possible values of xi belonging to the respective �-cuts of theinput variables are used as arguments of f(X). As f(X) is monotonically increasingor decreasing with all its arguments, and because the �-cuts of the input variablesare closed intervals, then y� also is a closed interval in which its extremes can becalculated using Equation 3, according to Definition 4.

y� � Ly�, Ry�

Ly� � f�DX��, df��

Ry� � f�DX��, �df�� (3)

As stated in Definition 4, Ly� is increasing and Ry� is decreasing; furthermore,Ly1 � Ry1. Because of the continuity of f( x), the conditions in Equation 1 aresatisfied, and the y�’s generate a fuzzy number.

A possible interpretation of Equation 3 could be that the extremes of the �-cuty� are calculated using the extremes of the �-cuts of the arguments A1, A2, . . . ,An. Furthermore, df (i) is a variable that tells us whether f( x) increases ordecreases with xi and is used in Equation 3 in order to establish which of the twoextremes of the �-cut of Ai must be used in the calculus. Figure 1 helps us toillustrate the idea; we have supposed n � 2, so y � f( x1, x2); we have alsosupposed that y increases with x1 and decreases with x2. In Figure 1 we show thatin order to calculate the lowest value of a certain �-cut of y (Ly�), we have to usethe lowest value of the corresponding �-cut of A1 [because df (1) � 1] and thehighest value of the corresponding �-cut of A2 [because df (2) � �1]; to calculateRy�, we have to use the other extrems of the �-cut of A1 and A2.

If we have a discrete representation as proposed in Section 2.1, then thefollowing algorithm is used to extend y � f(X) to fuzzy numbers:

Figure 1. Extension of crisp functions.


Algorithm 1 Direct extension of crisp functions

Inputs • A crisp function y � f(X), which will be extended to fuzzy numbers. The f(X) mustbe a continuous function, strictly monotonically increasing with some of the n vari-ables and strictly monotonically decreasing with the others (see Definition 3).

• The function df (i) associated to y � f(X) (see Definition 3)• A1, A2, . . . , An fuzzy numbers that will be used as the arguments of the extension

of f(X). Every fuzzy number Ai is represented by its function DAi(a, d) for a setAlfa of values of � (see Definition 2). Alfa � {�1, �2, . . . , �p} for a fixed p.

Outputs • A set of p �-cuts of the fuzzy number Y � f(A1, A2, . . . , An). Every �-cut will bedenoted by y(�i) � [Ly�, Ry�], j � 1, 2, . . . , p

Procedure 1. j � 12. calculate y(�j) � [Ly�, Ry�] using Equation 33. j � j � 14. if j p and then stop, else go to 2

4. EXTENDING CRISP INVERSE FUNCTIONSTO FUZZY NUMBERS

Taking the same conditions as in Section 3, suppose that we now want toextend the crisp inverse function xk � fk

�1( y, Xk) and Xk �[ x1

. . . xk�1 xk�1. . . xn]�, which calculates the value of the input

variable xk when we know the output y and the other input variables xi. Thesolution to this problem may have at least two different approaches when eachvariable is represented by a fuzzy number:

● We want to know which values the variable xk could take if we know the values of y,x1, x2, . . . , xk�1, xk�1, . . . , xn. In this paper, this approach will be called the possibleextension of the inverse function, and will be denoted by xk

pos.● We want to know which values the variable xk must take in order to ensure that the

output variable y takes certain predetermined values when we know the values of theother input variables x1, x2, . . . , xk�1, xk�1, . . . , xn. In this article, this approach willbe called the necessary extension of the inverse function, and will be denoted by xk

nec.

The difference between the two approaches may be better visualized and under-stood by using an interval example: let us suppose that the function y � f( x1,x2) � x1 � x2 relates the physical variables x1, x2, and y. We now have at leasttwo different situations where we need to use the inverse function x1 � f1

�1( y, x2):

● We have measured the physical variables y and x2, and we want to estimate the possiblevalues of x1. For example, if y � [5, 8] and x2 � [2, 4], then we have x1

pos � [1, 6]because [5, 8] � [2, 4] � [1, 6].

● We have measured the value of x2 and we want to establish which values x1 must takein order to ensure that the output variable v belongs to a certain interval. For example,if x2 � [2, 4] and we want y to belong to [5, 8], then we have x1

nec � [3, 4] because[3, 4] � [2, 4] � [5, 8]. If we want y to belong to [5, 6], then we cannot find any interval[a, b] such that [a, b] � [2, 4] � [5, 6].

If the variables x1, x2, and y were crisp, then the two foregoing situations wouldbe solved using the same crisp inverse function x1 � f1

�1( y, x2) � y � x2. But


because they are intervals, the uncertainty included in these intervals implies adifferent mathematical manipulation in each situation. The first situation corre-sponds to our possible extension and the second corresponds to the necessaryextension.

4.1. Possible Extension

The procedure used to obtain the possible extension consists of consideringthe crisp inverse function xk � fk

�( y, Xk) as a direct function and extending itusing the procedure described in Section 3. However, we must take into accounthow xk varies with regard to the other variables:

If f(X) increases with xk, thenfk

�1( y, Xk) is increasing with yfk

�1( y, Xk) is decreasing with xi if f(X) increases with xi

fk�1( y, Xk) is increasing with xi if f(X) decreases with xi

If f(X) decreases with xk, thenfk

�1( y, Xk) is decreasing with yfk

�1( y, Xk) is increasing with xi if f(X) increases with con xi

fk�1( y, Xk) is decreasing with xi if f(X) decreases with xi

Using the foregoing definitions of df (i) and DA(�, d), we can write Equation 4 tocalculate xk

pos(�) as follows:

xkpos(�) � Lxk

pos��, Rxkpos��

Lxkpos�� fk

�1�D��, df�k��, DXk��, �dfk��

Rxkpos�� fk

�1�DY��, �df�k��, DXk��, dfk��

DXk��, dfk� � · · · DAi��, df�k�df�i�� · · · i � 1, 2 · · · k � 1, k

� 1, . . . n

DXk��, �dfk� � · · · DAi��, �df�k�df�i�� · · · i � 1, 2 · · · k � 1, k � 1, . . . n

(4)

On account of having extended the crisp function fk�1 to fuzzy numbers using the

procedure of Section 3, xkpos(�) represents a fuzzy number; Dy(�, df (k)), Dxk(�,

dfk), and Dxk(�, �dfk) allow us to write in shorthand if xk decreases or increaseswhen the other variables increase. Figure 2 helps to illustrate this idea; we havesupposed, as in Figure 1, that n � 2 and x1 � fk

�1( y, x2) and that y � f(X)increases with x1 and decreases with x2.

If we have a discrete representation as proposed in Section 2.1, then thefollowing algorithm is used to make a possible extension of xk � fk

�1( y, Xk) tofuzzy numbers.


Algorithm 2 Possible extension of inverse crisp functions

Inputs • A crisp function y � f(X) that must be a continuous function, strictly monotoni-cally increasing with some of the n variables and strictly monotonically decreasingwith the others (see Definition 3)

• The function df (i) associated to y � f(X) (see Definition 3)• An inverse crisp function associated with y � f(X), xk � fk

�1(y, Xk), Xk �[x1 x2

. . . xk�1 xk�1. . . xn] that calculates the value of xk when we

know the output y and the other input variables xi (see Definition 3). xk � fk�1(y,

Xk) is the function that will be extended• A1, . . . , Ak�1, Ak�1, . . . , An fuzzy numbers that will be used as the arguments

of the extension of f(X). Every fuzzy number Ai is represented by its functionDAi(a, d) for a set Alfa of values of �. (See Definition 2). Alfa � {�1, �2, . . . ,�p} for a fixed p

• The fuzzy number Y that is the output of the direct function Y � f(X), Y isrepresented by its function DY(a, d) for the same set Alfa

Outputs • A set of p �-cuts of the unknown fuzzy number xkpos. Every �-cut will be denoted

by xkpos(�i) � [Lxk�

pos, Rxk�pos], j � 1, 2, . . . , p

Procedure 1. j � 12. Calculate xk

pos(�j) � [Lxk�pos, Rxk�

pos] using Equation 43. j � j � 14. if j p and then stop else go to 2

4.2. Necessary Extension

The procedure used to obtain the necessary extension consists of calculatingthe extremes of every �-cut of xk using xk � fk

�1( y, Xk) with the most restrictiveextremes of the �-cuts of y and Xk, as can be seen in Figure 3. Again, we havesupposed that n � 2, x1 � fk

�1( y, x2), y � f(X), increasing with x1 anddecreasing with x2. The following expression is used to calculate xk

nec(�):

Figure 2. Possible extension of crisp inverse functions.


xknec�� Lxk

nec��, Rxknec��

Lxknec�� fk

�1�DY��, df�k��, DXk��, dfk��

Rxknec�� fk

�1�DY��, �df�k��, DXk��, �dfk�

DXk��, dfk� � · · · DAi��, df�k�df�i�� · · · i � 1, 2 . . . k � 1, k � 1, . . . n

DXk��, �dfk� � · · · DAi��, �df�k�df�i�� · · · i � 1, 2 . . . k � 1, k � 1, . . . n

(5)

However, we cannot ensure that Lxknec(�) always will be monotonically increasing, or

that Rxknec(�) always will be monotonically decreasing, or that Lxk

nec(�) � Rxknec(�) for

every �, and as a result, Equation 5 will not generally represent a fuzzy number. Onlyfor certain fuzzy numbers Y will we have a fuzzy number represented by xk

nec(�) inEquation 5. The reason for this is explained in the following example.

Let us suppose that the fuzzy numbers A1, . . . Ak�1, Ak�1, . . . An represent thenumerical values of the variables x1, x2, . . . , xk�1, xk�1, . . . , xn and that each has atrapezoidal shape, and also that the fuzzy number Y represents the output variable y andhas a singleton shape. We want to obtain xk

nec(�); in other words, we want to obtain afuzzy number Ak [in which its �-cuts are xk

nec(�)] so that it ensures that the output willbe a singleton when the other input variables are trapezoidal ones. Obviously, such afuzzy number cannot be found, because the uncertainty included in the trapezoidalnumbers cannot “just disappear.” Because the accumulative characteristic of uncer-tainty, whatever fuzzy number Ak we use, the output will not be a singleton.

To solve this problem, we propose an algorithm that verifies whether thefuzzy number Y is a coherent value with the uncertainties of the input variables x1,x2, . . . , xk�1, and xk�1, . . . , xn. If it is not, then the algorithm modifies the fuzzynumber Y adding uncertainty (by widening its �-cuts) in a convenient way, so thatwe can ensure that Y and xk

nec(�) will be fuzzy numbers. Conditions in Equation 1are satisfied because of the procedure we use to calculate them (see Steps 1.2, 2.1,2.2.2, and 2.3.2 in Algorithm 3).

Figure 3. Necessary extension of crisp inverse functions.


The algorithm supposes that we have the discrete representation proposed inSection 2.1. The algorithm is as follows.

Algorithm 3 Necessary extension of inverse crisp functions

Inputs • A crisp function y � f(X) that must be a continuous function, strictly monotonicallyincreasing with some of the n variables and strictly monotonically decreasing with theothers (see Definition 3)

• The function df (i) associated with y � f(X) (see Definition 3)• An inverse crisp function associated with y � f(X), xk � fk

�1(y, Xk), Xk �[x1 x2

. . . xk�1 xk�1. . . xn] that calculates the value of xk when we know

the output y and the other input variables xi (see Definition 3); xk � fk�1(y, Xk) is the

function that will be extendedA1, . . . , Ak�1 and Ak�1, . . . , An fuzzy numbers that will be used as the arguments ofthe extension of f(X). Every Ai is represented by its function DAi(a, d) for a set Alfaof values of � (see Definition 2). Alfa � {�1, �2, . . . , �p} for a fixed p

• The fuzzy number Y that is the desired output of the direct function Y � f(X); Y isrepresented by its function DY(a, d) for the same set Alfa

Outputs • A set of p �-cuts of the unknown fuzzy number xknec. Every �-cut will be denoted by

xkpos(�j) � [Lxk�

nec, Rxk�nec], j � 1, 2, . . . , p

• A set of p �-cuts of the fuzzy number Y, that may be different than those of the input,because the algorithm can modify Y in order to ensure that xk

nec existsProcedure 1. Calculate for the upper �-cut

1.1. Calculate Lxknec(�p), Rxk

nec(�p) using Equation 51.2. If Lxk

nec(�p) Rxknec(�p), then widen the �-cut Y�p � [Ly(�p), Ry(�p)] on the

left and right sides until we get a new �-cut Y��p � [L�y(�p), R�y(�p)] such thatLxk

nec(�p) � Rxknec(�p)

2. Calculate for the remaining �-cuts: for �j, j � p � 1, p � 2, . . . , 12.1. Widen the �-cut of y if the previous is wider than the actual as follows:

2.1.1. If Ly(�i) L�y(�j�1), then let L�y(�i) � L�y(�j�1)2.1.2. If Ry(�i) � R�y(�j�1), then let R�y(�i) � R�y(�j�1)

2.2. Calculate Lxknec(�p) as follows:

2.2.1. calculate Lxknec(�p) using Equation 5

2.2.2. If Lxknec(�i) Lxk

nec(�j�1), then calculate y*� as follows:

y*� � f�DX��*j, df��

DX��*j, dfk� � · · · DX��*ji, df�i�df�k�� · · · i � 1, 2 . . . n

�*ji � ��j if i � k�j�1 if i � k

2.2.1.1. If df (k) � 1, then widen the �-cut Y�j � [Ly(�j), Ry(�j)] on the leftside transforming it into the new �-cut Y��p � [y*

�, Ry(�j)]2.2.1.2. If df (k) � �1, then widen the �-cut Y�j � [Ly(�j), Ry(�j)] on the

right side transforming it into the new �-cut Y��p � [Ly(�j), y*�]

2.3. Calculate Rxknec(�p) as follows:

2.3.1. Calculate Rxknec(�p), using Equation 5

2.3.2. If Rxknec(�i) � Rxk

nec(�j�1) then calculate y*� as follows:

y*� � f�DX��*j, �df��

DX��*j, �djk� � · · · DX��*ji, �df�i�df�k�� · · · i � 1, 2 . . . n

�*ji � ��j if i � k�j�1 if i � k

2.3.2.1. If df (k) � 1, then widen the �-cut Y�j � [Ly(�j), Ry(�j)] on the rightside transforming it into the new �-cut Y��p � [Ly(�j), y*

�]2.3.2.2. If df (k) � �1, then widen the �-cut Y�j � [Ly(�j), Ry(�j)] on the left

side transforming it into the new �-cut Y��p � [y*�, Ry(�j)]


A useful strategy to widen the �-cut in Step 1.2 is the following:

● L�y(�p) � Ly(�p) � (Ry(�p) � Ly(�p) � ZERO) F● R�y(�p) � Ry(�p) � (Ry(�p) � Ly(�p) � ZERO) F

ZERO is a small positive number that enables the algorithm to work even if Ry(�p) �Ly(�p). The term F is a positive number that serves as a step-widening factor.

4.3. Family of Intermediate Extensions

By comparing Figures 2 and 3, it can be observed that the main differencebetween the possible and necessary extensions lies in the selection of the �-cutextreme of every variable in Xk used to calculate xk

pos(�) or xknec(�). This allows us

to define intermediate extensions using intermediate values of the �-cuts.To organize these intermediate extensions, we define a parameter r that can

take any value in the interval [0, 1]. The possible extension will be associated witha value of r � 0 and the necessary solution with r � 1. When r changes from 0to 1, we obtain different intermediate extensions denoted by xk

int(�, r), which varyslowly from the possible to the necessary extension. The aim of r is to choosewhich value of the �-cuts must be used to calculate the extension, according toEquation 6. Figure 4 shows the use of parameter r.

xkint��, r� � Lxk

int��, 4�, Rxkint��, r�

Lxkint��, r� � fk

�1�DY��, df�k��, DXk��, dfk, r��

Rxkint��, r� � fk

�1�DY��, �df�k��, DXk��, �dfk, r�

DXk��, dfk, r� � · · · DAi��, df�k�df�i�, r� · · · i � 1, 2 . . . k � 1,

k � 1, . . . n

Figure 4. Intermediate family of extension of crisp inverse functions.


DXk��, �dfk, r� � · · · DAi��, �df�k�df�i�, r� · · · i � 1, 2 . . . k � 1, k

� 1, . . . n

DXk��, df�k�df�i�, r� � DAi��, df�k�df�i�� r�DAi��, �df�k�df�i��

� DAi��, df�k�df�i��

DXk��, �dfk, r� � DAi��, �df�k�df�i�� r�DAi��, df�k�df�i�� DAi��, �df�k�df�i��

(6)

To obtain the intermediate family of extensions, we propose the same algorithmused to obtain the necessary solution (Algorithm 3), but using Equation 6 insteadof Equation 5 in Steps 1, 3, and 4. As in the case of the necessary extension, thealgorithm will verify if the fuzzy number Y is a coherent one with the uncertaintiesof the input variables x1, x2, . . . , xk�1, and xk�1, . . . , xn. If it is not, then thealgorithm will conveniently modify Y to ensure that Y and xk

nec(�) are fuzzynumbers. The algorithm supposes that we have the discrete representation pro-posed in Step 2.3. The algorithm is as follows.

Algorithm 4 Intermediate extension of inverse crisp functions

Inputs • A crisp function y � f(X) that must be a continuous function, strictly monotonicallyincreasing with some of the n variables and strictly monotonically decreasing withthe others (see Definition 3)

• The function df (i) associated with y � f(X) (see Definition 3)• An inverse crisp function associated with y � f(X), xk � fk

�1(y, Xk), and Xk �[x1 x2

. . . xk�1 xk�1. . . xn] that calculates the value of xk when we

know the output y and the other input variables xi (see Definition 3). xk � fk�1(y,

Xk) is the function that will be extended• A1, . . . , Ak�1 and Ak�1, . . . , An fuzzy numbers that will be used as the arguments

of the extension of f(X). Every fuzzy number Ai is represented by its functionDAi(a, d) for a set Alfa of values of � (see Definition 2). Alfa � {�1, �2, . . . �p}for a fixed p

• The fuzzy number Y that is the desired output of the direct function Y � f(X), Y isrepresented by its function DY(a, d) for the same set Alfa

• A parameter r � [0, 1]Outputs • A set of p �-cuts of the unknown fuzzy number xk

int(r). Every �-cut will be denotedby xk

int(r)(�j) � [Lxk�int(r), Rxk�

int(r)], j � 1, 2, . . . , p• A set of p �-cuts of the fuzzy number Y, that may be different than those of the

input, because algorithm can modify Y in order to ensure that xkint(r) exists.

Procedure The same as in Algorithm 3, but use Equation 6 instead of Equation 5

It is important to emphasize that possible extension always exists but thatnecessary extension does not (unless we use Algorithm 3 because it can conve-niently modify the problem). When necessary extension does not exist, there mustbe a threshold value of r, e.g., r0, so that if r � r0, then extension exists; r0 canthen be used to measure the existence of a solution to the problem of extending theinverse function. If r0 � 1, then all the extensions exist (possible, necessary, andintermediate), but if r0 � 1, then only the possible extension and some of theintermediate extensions exist. In other words, r0 measures the existence of the


extended inverse function. We can use the following algorithm in order to calculatethe threshold r0.

Algorithm 5 Existence of the inverse extension

Inputs • The same as in Algorithms 3Outputs • The threshold r0 that can be interpreted as a measure of the existence of a solution to

the problem of extending the inverse functionProcedure 1. dr � 1.0/(p � 1), a step factor calculated according to the number of elements of

Alfa (see Definition 2)2. r � 0, flag � 03. Calculate the intermediate extension corresponding to r, using Algorithm 4

3.1. If the fuzzy number Y is modified in the steps 1.2, 2.2.2, or 2.3.2 of Algorithm 4,for any � then flag � 1

4. If r � 1.0, then flag � 15. If flag � 1, then r0 � r and exit else r � r � dr and go to 2

5. EXAMPLES

In the following examples we will consider the crisp function

y � f�x1, x2� �x1

2

x2

defined for strictly positive real numbers x1, x2, and y and their inverse functions

x1 � f1�1�y, x2� � �yx2

x2 � f2�1�y, x1� �

x12

y

These functions can appear in several mathematical models, i.e., relating thevoltage across a resistor, the resistance of the resistor, and the power dissipated ina simple electric circuit ( y would represent the power, x1 would represent thevoltage, and x2 would represent the resistance). In all of the following examples,we have used a discrete representation with Alfa � {0, 0.01, 0.02, . . . , 0.99,1.00}.

Example 1. If x1 and x2 are represented by trapezoidal fuzzy numbers x1 �T(1.0, 1.8, 2.2, 3.0) and x2 � T(0.5, 0.9, 1.1, 1.5), then y can be obtainedusing Algorithm 1. The result is shown in Figure 5.

Figure 5. The y in Example 1.


Example 2.

(a) If y and x2 are represented by trapezoidal fuzzy numbers y � T(1, 1, 1, 1) andx2 � T(0.5, 0.9, 1.1, 1.5), then the possible values of x1 can be obtainedusing a possible extension of f1

�1 (Algorithm 2). The result is shown inFigure 6.

(b) If y and x1 are represented by trapezoidal fuzzy numbers y � T(1, 1, 1, 1) andx1 � T(1.0, 1.8, 2.2, 3.0), then the possible values of x2 can be obtainedusing a possible extension of f2

�1 (Algorithm 2). The result is shown inFigure 7.

Example 3.

(a) If x2 is represented by the trapezoidal fuzzy number x2 � T(0.5, 0.9, 1.1,1.5) and we want to ensure that y can be represented by the trapezoidal fuzzynumber y � T(1, 1, 1, 1), then we can establish what x1 must be using anecessary extension of f1

�1 (Algorithm 3). The result is shown in Figure 8, buty has been modified as is shown in Figure 9.

(b) If x1 is represented by the trapezoidal fuzzy number x1 � T(1.0, 1.8, 2.2,3.0) and we want to ensure that y can be represented by the trapezoidal fuzzynumber y � T(1, 1, 1, 1), then we can establish what x2 must be using anecessary extension of f2


Figure 6. The x1 in Example 2(a).

Figure 7. The x2 in example 2(b).


Figure 8. The x1 in example 3(a).

Figure 9. The y in example 3(a).


Figure 11. The y in example 3(b).


Example 4.

(a) If x2 is represented by the trapezoidal fuzzy number x2 � T(0.5, 0.9, 1.1,1.5) and we want to ensure that y can be represented by the trapezoidal fuzzynumber y � T(0.5, 1.1, 2.5), then we can establish what x1 must be using anecessary extension of f1


(b) If x1 is represented by the trapezoidal fuzzy number x1 � T(1.0, 1.8, 2.2,3.0) and we want to ensure that y can be represented by the trapezoidal fuzzynumber y � T(0.5, 1, 1, 2.5), then we can establish what x2 must be usinga necessary extension of f2

�1 (Algorithm 3). The result is shown in Figure 14,but y has been modified as is shown in Figure 15.

Example 5.

(a) If y and x2 are represented by trapezoidal fuzzy numbers y � T(1, 1, 1, 1) andx2 � T(0.5, 0.9, 1.1, 1.5), then the values of x1 can be obtained using thefamily of intermediate extensions of f1

�1. The result is shown in Figure 16, buty has been modified as is shown in Figure 17.

(b) If y and x1 are represented by trapezoidal fuzzy numbers y � T(1, 1, 1, 1) andx1 � T(1.0, 1.8, 2.2, 3.0), then the values of x2 can be obtained using thefamily of intermediate extensions of f2

�1. The result is shown in Figure 18, buty has been modified as is shown in Figure 19.




Note in these examples how the uncertainty of x1 and x2, changes when rchanges. Note also that in these examples the threshold of existence of an extensionis r0 � 0.5, because if r takes a value 0.5 the algorithm has to modify y in orderto obtain a solution.

6. AN APPLICATION

In the following paragraphs we present an application of some of the previousalgorithms. The application is a fuzzy methodology to make the environmentalimpact analysis that is used to evaluate the environmental implications of a certainhuman activity. There are some traditional methodologies (see Refs. 19 and 20),one of them is the interactive matrix. There are some versions of this methodology;here, we use the importance matrix also known as the qualitative environmentalImpact Analysis. In short, the procedure is the following:

1. To identify the environmental factors that can be affected by the human activity. It must beassigned a weight (wi) to every factor, such that the total sum of weights is 1000.

2. To identify the actions related with the human activity3. To identify the effects that every action can make on every factor4. To evaluate the importance of every effect5. To summarize all the effects in an importance matrix: row i represents the factor i,

column j represents the action j, and in cell i, j we place the importance of the effecti � j (the effect of the action j on the factor i)

6. To study the importance matrix making additions, averages, and weighted averages ofthe effects, by rows, columns, and/or the whole matrix

7. To decide how the whole environmental impact is




The evaluation of the importance of every effect in Step 4 and the finaldecision in Step 7 are word-based, so in our fuzzy methodology we propose acomputing with words strategy.21 In the traditional methodology, importance (IM)is calculated in Step 4 as

IM � �3I � 2EX � MO � PE � RV � SI � AC � EF � PR � MC (7)

where

I: reflects the intensity of the effectEX reflects the area of the effectMO reflects the moment of the effectPE reflects the persistence of the effectRI reflects the reversibility of the effectSI reflects the sinergy of the effectAC reflects the accumulation of the effectEF reflects the cause-effect relation of the effectPR reflects the periodicity of the effectMC reflects the recoverability of the effect

Importance is positive if the effect is beneficial, otherwise it is negative.All the variables defined previously are evaluated based on linguistic terms, as

we summarize in Table I. The effect is classified according to its importance, ascompatible (IM � 25), moderate (25 � IM � 50), severe (50 � IM � 75), orcritical (75 � IM). Some indexes are calculated in Step 6 such as the followingIMij is the importance of the effect of the action j on the factor i; there are n factorsand m actions):

absolute importance of an action IAj � �i�1

n

Iij (8)

mean importance of an action IAMj �¥i�1

n Iij

n(9)

Table I. Evaluation of the importance: Values of the variables.

Intensity Area Moment Persistence ReversibilityLow � 1 Local � 1 Long term � 1 Brief � 1 Short term � 1Medium � 2 Partial � 2 Medium term � 2 Temporal � 2 Medium term � 2High � 4 Vast � 4 Immediate � 4 Permanent � 4 Nonreversible � 4Very High � 8 Total � 8 Critical � �4Total � 12 Critical � �4

Sinergy Accumulation Cause-effect Periodicity RecoverabilitySimple � 1 Simple � 1 relation Irregular � 1 Immediately � 1

Sinergic � 2Accumulative

� 4 Indirect � 1 Periodical � 2 Medium term � 2Very sinergic

� 4 Direct � 4 Continuous � 4 Minimizable � 4Irrecoverable � 8


relative importance of an action IARj �¥i�1

n wiIij

¥i�1n wi

(10)

absolute importance of a factor IFi � �j�1

m

Iij (11)

mean importance of a factor IFMi �¥j�1

m Iij

m(12)

relative importance of a factor IFRi �¥j�1

m wiIij

¥i�1n wi

� � wi

¥i�1n wi

�IFi (13)

absolute total importance IT � �j�1

m �i�i

n

Iij � �i�1

n

IFi � �j�1

m

IAj (14)

mean total importance ITM �¥j�1

m¥i�1

n Iij

mn�

¥i�1n IFMi

n�

¥j�1m IAMj

m(15)

relative total importance ITR � �j�1

m

IARj � �i�1

n

IFRi (16)

The previous indexes are classified in a similar way to the importance, in order tomake the final decision in Step 7. Because this procedure is clearly word-based, wecan suggest an alternative fuzzy methodology like this:

1F. Equal than Step 12F. Equal than Step 23F. Equal than Step 34F. To Evaluate the importance of every effect replacing Table I by a family

of linguistic variables based on fuzzy sets and using approximate rea-soning

5F. Equal than Step 56F. To study the importance matrix using approximate reasoning based on

fuzzy sets7F. To make the final decision using linguistic approximation

If we use classical approximate reasoning in Steps 4F and 6F, i.e., if we use arule-based inference engine, then the solution is very computationally expensive,because the number of variables and labels is very high. In fact, to calculate theimportance in Step 4F, we should use an inference engine with 10 input variables(see Equation 7), and if we use the same labels as in Table I, the rule base couldhave up to 129.600 rules!. In Step 6F the computational cost is even higher,because the number of factors usually is 20, and the number of actions is usually 10, so we could expect at least 200 inputs for the inference engine.


To avoid the computational cost, we propose extending to fuzzy numbersEquation 7 to make the approximate reasoning in Step 4F, and Equations 8–16 tomake it in Step 6F. All the equations involved are strictly monotonic, so we can usethe algorithms presented in this article.

An important additional advantage is that we can make inverse reasoningusing the extensions of the inverse functions; suppose we evaluate a certain project(such as the construction of a highway) and, consequently, we obtain that theproject is not compatible with the environment, so it cannot be executed. Now, wehave to find the corrective activities that would reduce the environmental impact;i.e., we have to calculate the importance of a new effect (a beneficial one) on theenvironment, such that the total importance can take just some predefined values.Clearly, we can use the necessary extension of the inverse function in order tocalculate it.

7. CONCLUSIONS AND REMARKS

In this article we have presented some algorithms to extend crisp functionsand their inverse functions to fuzzy numbers. The type of functions dealt with hereare continuous functions monotonically increasing with some of their argumentsand monotonically decreasing with the others. We have also proposed somedifferent solutions to the problem of extending crisp inverse functions: first, weproposed the possible and the necessary extensions, and then we proposed a familyof intermediate extensions that varies slowly between the first two. Although someof the extensions may not exist, the algorithms proposed ensure that a solution isfound by conveniently modifying the condition of the problem.

We believe that the algorithms proposed in this article can be used in a widerange of problems in several areas of knowledge, because uncertainty is an inherentcondition of many mathematical models. As an example, we have shown anapplication of the Algorithms 1 and 3 in a fuzzy methodology to make anenvironmental impact evaluation.

AcknowledgmentsThis work has been supported in part by the project TIC-99-0563. DGICYT. MECD.

Madrid.

References

1. Zadeh LA. The concept of linguistic variable and its applications to approximate reasoning,Part I, Inform Sci 1975;8:199–249.

2. Zadeh LA. The concept of linguistic variable and its applications to approximate reasoning,Part II, Inform Sci 1976;8:301–357.

3. Zadeh LA. The concept of linguistic variable and its applications to approximate reasoning,Part III, Inform Sci 1976;9:43–80.

4. Dubois D, Prade H. Operations on fuzzy numbers. Int J Syst Sci 1978;9:613–626.5. Dubois D, Prade H. Fuzzy real algebra: Some results. Fuzzy Sets Syst 1979;2:327–348.


6. Dubois D, Prade H. Fuzzy sets and systems, theory, and applications. New York: Aca-demic Press; 1980.

7. Dubois D, Prade H. Addition of interactive fuzzy numbers. IEEE Trans Autom Control1981;26:926–936.

8. Jain R. Tolerance analysis using fuzzy sets. J Syst Sci 1976;12:1393–1401.9. Yager RR. On the lack of inverses in fuzzy arithmetic. Fuzzy Sets Syst 1980;4:73–82.

10. Baas SM, Kwakernaak H. Rating and ranking of multiple aspect alternatives using fuzzysets. Automatica 1977;3:47–58.

11. Mizumoto M, Tanaka H. Algebraic properties of fuzzy numbers. Int Conf Cybernetic Soc,Washington DC, 1976. pp 559–563.

12. Mizumoto M, Tanaka H. Some properties of fuzzy numbers. In: Gupta MM, Ragade RK,Yager RR, editors. Advances in fuzzy sets and theory and applications. Amsterdam:North-Holland; 1979. pp 153–164.

13. Sanchez E. Solution of fuzzy equations with extended operations. Fuzzy Sets Syst 1983;11:163–184.

14. Sanchez E. Nonstandard fuzzy arithmetics. Technical Report, University of California,Berkeley, 1985.

15. Tamura N, Horiuchi K. VSOP fuzzy numbers and fuzzy comparison relations. In: SecondIEEE Int Conf Fuzzy Systems, San Francisco, CA, 1993.

16. Mares M. Weak arithmetics if fuzzy numbers. Fuzzy Sets Syst 1997;2:143–154.17. Bouchon-Meunier B, Kosheleva O, Kreinovich V, Nguyen HT. Fuzzy numbers are the

only fuzzy sets that keep invertible operations invertible. Fuzzy Sets Syst 1997;2:155–164.18. Giachetti R. Evaluating engineering functions with imprecise quantities. Seventh IFSA

World Congress Prague, Vol II; 1997. pp 150–155.19. Canter LW. Environmental impact assessment. New York: McGrawHill; 1997.20. Conesa Fdez V. Guıa metodologica para la eveluacion del impacto ambiental. Madrid:

Ediciones Mundi-Prensa; 1997.21. Zadeh LA. What is computing with words? In: Zadeh LA, Kacprzyk J, editors. Computing

with words in information/intelligent systems 1. New York: Physica Verlag; 1999. pp3–12.


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