optimal fuzzy reasoning and its robustness analysis

Optimal Fuzzy Reasoning andIts Robustness AnalysisLei Zhang,* Kai-Yuan Cai†

Department of Automatic Control, Beijing University of Aeronautics andAstronautics, Beijing 100083, China

Fuzzy reasoning methods are extensively used in intelligent systems and fuzzy control. Mostexisting fuzzy reasoning methods follow rules of logical inference. In this article, fuzzy reason-ing is treated as an optimization problem. The idea of optimal fuzzy reasoning is reviewed andthree new optimal fuzzy reasoning methods are given by using new optimization objective func-tions. The robustness of fuzzy reasoning, that is, how errors in premises affect conclusions infuzzy reasoning, is evaluated in a probabilistic or statistical context by using the Monte Carlosimulation method. Six optimal fuzzy reasoning methods are evaluated in comparison with theCRI method in terms of probabilistic robustness. © 2004 Wiley Periodicals, Inc.

1. INTRODUCTION

Fuzzy reasoning has been a research topic in the fuzzy community since theinception of Zadeh’s pioneering work.1 Various methods have been proposed forfuzzy reasoning, including the compositional rule of inference,1 evidence reason-ing,2,3 an approximate analogical reasoning approach based on similarity mea-sures,4 the triple implication method,5,6 and so on. The idea of the CRI method isthat a fuzzy relation R is first determined by an implicational operation from theexperience of experts or existing data, and the consequence B is then determinedby a compositional operation from R and the given fuzzy premise A. Roughlyspeaking, most existing fuzzy reasoning methods view fuzzy reasoning as a logi-cal inference process and have a strong relation to conventional logical inference.A natural question is whether fuzzy reasoning can be formulated on a basis otherthan a logical one, because a solid logical basis for existing fuzzy reasoning is notvery clear yet. On the other hand, there are a few methods about how to optimallydetermine R7; however, few papers are concerned with how to optimally determineB for the given R and A. The triple implication method incorporates the idea ofoptimally determining B based on logic foundation, but does not treat it as a basis

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

INTERNATIONAL JOURNAL OF INTELLIGENT SYSTEMS, VOL. 19, 1033–1049 (2004)© 2004 Wiley Periodicals, Inc. Published online in Wiley InterScience(www.interscience.wiley.com). • DOI 10.1002/int.20035

of fuzzy reasoning. In our previous paper,8 we treat optimization as a basis offuzzy reasoning and introduce three methods of optimal fuzzy reasoning. In thisway, fuzzy reasoning is no longer logical inference. Rather, it is a process of com-putation and optimization. In the present article we introduce three more methodsof optimal fuzzy reasoning.

The robustness problem of fuzzy reasoning is concerned with how errors inpremises affect conclusions in fuzzy reasoning. It is analyzed in Ref. 9 in terms ofd-equalities of fuzzy sets.10 It is shown that the robustness of the CRI methods isnot very satisfactory. An interesting question is whether optimal fuzzy reasoningmethods can behave better in terms of robustness than the CRI method. To answerthis question, in this article we propose to compare the robustness of fuzzy reason-ing methods in a probabilistic or statistical context by using the popular MonteCarlo simulation method because the analytic robustness analysis of various fuzzyreasoning methods is often very complicated.

The rest of the article is organized as follows. Section 2 reviews the mainresults of our previous paper.8 Section 3 gives three new optimal fuzzy reasoningmethods. Section 4 gives the notion of robustness measure. Sections 5, 6, and 7compare the probabilistic robustness of the six optimal fuzzy reasoning methodsand the CRI method under three different scenarios. Concluding remarks are con-tained in Section 8.

2. PREVIOUS WORK ON OPTIMAL FUZZY REASONING

The basic idea of optimal fuzzy reasoning was first introduced in Ref. 8 asfollows:

(1) Suppose a fuzzy relation matrix R is determined from the experience of experts (fuzzyrules) or data. Then we should trust R and treat it as a basis for evaluating the qualityof a fuzzy reasoning method.

(2) If a fuzzy premise A is given, then a fuzzy reasoning method will generate the corre-sponding consequence B. A fuzzy reasoning method is optimal if the matrix R ' is theclosest to R, where R ' � A r B.

Suppose R � ~rij !m�n and A � ~ai !1�m are given, and B � ~bj !1�n is theconsequence. Let R ' � ~rij

' !m�n � ~ai ∧ bj !m�n , where ai ∧ bj � min~ai , bj !.Reference 8 presents three optimal fuzzy reasoning methods according to three

objective functions as follows:

Objective function 1: J1 � (i�1

m

(j�1

n

6rij � rij' 6

Objective function 2: J2 � �(i�1

m

(j�1

n

~rij � rij' !2

Objective function 3: J3 � max1�i�m,1�j�n

6rij � rij' 6

1034 ZHANG AND CAI

The three objective functions measure the absolute distance from R ' to R. Inthis section we review the basic results presented in Ref. 8.

2.1. Optimal Fuzzy Reasoning Method Based on J1

Objective function J1 can be expressed as

J1 � (j�1

n

J1j

where

J1j � (i�1

m

6rij � ai ∧ bj 6

We can easily see that if J1j achieves its minimum at bj* ~ j � 1,2, . . . , n!, J1

achieves its minimum at B* � ~b1* , . . . , bn

* !. So B* can be obtained as follows:

(1) Rearrange A � ~ai !1�m into A' � ~ai' !1�m , where ai

' � ai�1' ~i � 1,2, . . . , m � 1!.

And rearrange the row ordering of R � ~rij !m�n into R ' � ~rij' !m�n according to the

change of the row ordering of A. Thus the consequence cannot be changed.(2) Attain bj

* ~ j � 1,2, . . . , n! as follows: Let

0 � a0' � a1

' � {{{� am' � am�1

' � 1

J1jp � (

i�1

p

6rij' � a ' 6� (

i�p�1

m

6rij' � bj 6 ap

' � bj � ap�1' ~ p � 0,1, . . . , m!

Rearrange ~rp�1, j' , . . . , rm, j

' ! into ~rp�1, j'' , . . . , rm, j

'' !, where ri, j'' � ri�1, j

'' ~i � p � 1,p � 2, . . . , m � 1!. Let

mid � �r ~m�p�1!

2, j

'' ~m � p � 1! is an even number

r ~m�p!

2, j

'' � r ~m�p!

2�1, j

''

2~m � p � 1! is an odd number

If mid � {ap' , ap�1

' } , bjp* � mid, else bj

p* should be ap' or ap�1

' .

J1jp* � (

i�1

p

6rij' � a ' 6� (

i�p�1

m

6rij' � bj

p* 6 ap' � bj

p*� ap�1

' ~ p � 0,1, . . . , m!

So

J1j* � min

0�p�mJ1j

p* , q* � arg min0�p�m

J1jp* and bj

*� bjq* .

(3) J1* � (

j�1

n

J1j* , B* � ~b1

* , . . . , bn* !

OPTIMAL FUZZY REASONING AND ROBUSTNESS 1035



J22 � (

j�1

n

J2j

where

J2j � (i�1

m

~rij � ai ∧ bj !2



*!. So B* can be obtained as follows:


' � ai�1' ~i � 1,2, . . . , m � 1!.



* ~ j � 1,2, . . . , n! as follows: Let

0 � a0' � a1

' � {{{� am' � am�1

' � 1

J2jp � (

i�1

p

~rij' � ai

' !2 � (i�p�1

m

~rij' � bj !

2 ap' � bj � ap�1

' ~ p � 0,1, . . . , m!

If

mid �1

m � p (i�p�1

m

rij' � @ap

' , ap�1' #

bjp* � mid, else bj


' :

J2jp* � (

i�1

p

~rij' � ai

' !2 � (i�p�1

m

~rij' � bj

p* !2 ap' � bj

p*� ap�1

' ~ p � 0,1, . . . , m!

So

J2j* � min

0�p�mJ2j


J2jp* and bj

*� bjq*

(3) J2* � �(

j�1

n

J2j* , B* � ~b1

* , . . . , bn* !

1036 ZHANG AND CAI



J3 � max1�j�n

J3j ,

where

J3j � max1�i�m

6rij � ai ∧ bj 6





' � ai�1' ~i � 1,2, . . . , m � 1!.



* ~ j � 1,2, . . . , n! as follows: Let

0 � a0' � a1

' � {{{� am' � am�1

' � 1

J3jp � max� max

1�i�p6rij' � ai

' 6, maxp�1�i�m

6rij' � bj 6� ap

' � bj � ap�1' ~ p � 0,1, . . . , m!

If

mid �

maxp�1�i�m

rij' � min

p�1�i�mrij'

2� @ap

' , ap�1' #



' :

J3jp* � max� max

1�i�p6rij' � ai

' 6, maxp�1�i�m

6rij' � bj

p* 6�ap' � bj

p*� ap�1

' ~ p � 0,1, . . . , m!

So

J3j* � min

0�p�mJ3j


J3jp* and bj

*� bjq*

(3) J3* � max

1�j�nJ3j* , B* � ~b1

* , . . . , bn* !.

3. NEW OPTIMAL FUZZY REASONING METHODS

In this section, we propose another three objective functions that measure therelative distance from R ' to R and introduce three new optimal fuzzy reasoningmethods based on the three objective functions.


Objective function 4: J4 � (i�1

m

(j�1

n

� rij � rij'

rij�

Objective function 5: J5 � �(i�1

m

(j�1

n � rij � rij'

rij�2

Objective function 6: J6 � max1�i�m,1�j�n � rij � rij

'

rij�



J4 � (j�1

n

J4j

where

J4j � (i�1

m

� rij � ai ∧ bj

rij�





' � ai�1' ~i � 1,2, . . . , m � 1!.



* ~ j � 1,2, . . . , n! as follows: Let

0 � a0' � a1

' � {{{� am' � am�1

' � 1

J4jp � (

i�1

p

� rij' � ai

'

rij' �� (

i�p�1

m

� rij' � bj

rij' � ap

' � bj � ap�1' ~ p � 0,1, . . . , m!

Rearrange ~rp�1, j' , . . . , rm, j

' ! into ~rp�1, j'' , . . . , rm, j

'' !, where ri, j'' � ri�1, j

'' ~i � p � 1,p � 2, . . . , m � 1!. Because

(i�p�1

m

� rij''� x

rij'' � ~x � {ri, j

'' , ri�1, j'' }; i � p � 1, p � 2, . . . , m � 1!

is a monotony function, bjp* should be ap

' , ap�1' , or ri, j

'' ~ri, j'' � {ap

' , ap�1' }; i �

p � 1, p � 2, . . . , m � 1!:

1038 ZHANG AND CAI

J4jp* � (

i�1

p � rij' � ai

'

rij' �� (

i�p�1

m � rij' � bj

p*

rij' � ap

' � bjp*

� ap�1' ~ p � 0,1, . . . , m!

So

J4j* � min

0�p�mJ4j


J4jp* and bj

*� bjq*

(3) J4* � (

j�1

n

J4j* , B* � ~b1

* , . . . , bn* !



J52 � (

j�1

n

J5j

where

J5j � (i�1

m � rij � ai ∧ bj

rij�2





' � ai�1' ~i � 1,2, . . . , m � 1!.



* ~ j � 1,2, . . . , n! as follows: Let

0 � a0' � a1

' � {{{� am' � am�1

' � 1

J5jp � (

i�1

p � rij' � ai

'

rij' �2

� (i�p�1

m � rij' � bj

rij' �2

ap' � bj � ap�1

' ~ p � 0,1, . . . , m!

If

mid �

(i�p�1

n 1

rij'

(i�p�1

n 1

rij'2

� @ap' , ap�1

' # ,




' :

J5jp* � (

i�1

p � rij' � ai

'

rij' �2

� (i�p�1

� rij' � bj

p*

rij' �2

ap' � bj

p*� ap�1

' ~ p � 0,1, . . . , m!

So

J5j* � min

0�p�mJ5j


J5jp* and bj

*� bjq*

(3) J5* � �(

j�1

n

J5j* , B* � ~b1

*, . . . , bn* !



J6 � maxl�j�n

J6j

where

J6j � maxl�i�m � rij � ai ∧ bj

rij�





' � ai�1' ~i � 1,2, . . . , m � 1!.



* ~ j � 1,2, . . . , n! as follows: Let

0 � a0' � a1

' � {{{� am' � am�1

' � 1

J6jp � max� max

1�i�p� rij' � ai

'

rij' �, max

p�1�i�m� rij' � bj

rij' �� ap

' � bj � ap�1' ~ p � 0,1, . . . , m!

If

mid �

2 minp�1�i�m

rij' max

p�1�i�mrij'

minp�1�i�m

rij' � max

p�1�i�mrij'

� @ap' , ap�1

' #

1040 ZHANG AND CAI



' :

J6jp* � max� max

1�i�p� rij' � ai

'

rij' �, max

p�1�i�m� rij' � bj

p*

rij' ��

ap' � bj

p*� ap�1

' ~ p � 0,1, . . . , m!

So

J6j* � min

0�p�mJ6j


J6jp* and bj

*� bjq*

(3) J6* � max

1�j�nJ6j* , B* � ~b1

* , . . . , bn* !

4. ROBUSTNESS MEASURE

We hope that a fuzzy reasoning method can have two desirable propertieswhen the fuzzy rule base (or fuzzy relation matrix R! is determined. First, theconsequences B and B ' should be similar when the premise A deviates slightlyfrom A' owing to system noises or parameter perturbations, where A � R � B andA' � R � B ' ; second, B and B ' should not be similar if A deviates significantlyfrom A' . The first property is concerned with the robustness problem of fuzzyreasoning, which is discussed in this article. And optimal fuzzy reasoning meth-ods should normally satisfy the second property already. Suppose there is big dif-ference between A and A' , whereas B is similar to B ' ; then the matrix R '� A'r B '

will not be closest to R. This violates the basic idea of optimal fuzzy reasoning.In the present article, we compare the robustness of the seven methods as

follows:

Methods 1–3: Optimal fuzzy reasoning methods based on J1–J3, respectively;Method 4: The max-min CRI method, that is,

B � ~bj !1�n � ~max1�i�m~ai ∧ rij !!1�n

for the given R � ~rij !m�n and A � ~ai !1�m ;Methods 5–7: Optimal fuzzy reasoning methods based on J4–J6, respectively.

A fuzzy reasoning method determines the consequence of a given fuzzy prem-ise based on given fuzzy rules or inference rules. The given rules are usually com-bined to make up a fuzzy relation matrix. So the robustness measure RM of afuzzy reasoning method should be a function of the corresponding fuzzy relationmatrix and the given fuzzy premise:

RM � M~R, A!

where R is a fuzzy relation matrix and A is a fuzzy premise.A natural method to evaluate the function M is to define and derive M ana-

lytically, as shown in Ref. 9 by using the so-called d-equalities of fuzzy sets.


However analytical evaluation of M is often complicated, and this is particularlytrue for the optimal fuzzy reasoning methods. Alternatively, in this article, we pro-pose to evaluate the function M in a statistical context by using the popular MonteCarlo simulation method.

In the simulations, we assume the following:

(1) For R � ~rij !m�n and A � ~ai !1�m , where rij , ai � @0,1# ~i � 1,2, . . . , m; j � 1,2, . . . , n! and max1�i�m ai � 1, the corresponding consequence is B � ~bj !1�n .

(2) For A � ~ai !1�m and DA � ~Dai !1�m , where DA is the perturbation of A and Dai ;U~� 1

40_ , 1

40_ !, that is, Dai is a random variable that takes values in the interval (� 1

40_ , 1

40_)

in accordance with a uniform distribution, let A' � A � DA � ~ai' !1�m , where

ai' � �

1 ai � Dai � @1,�`!

ai � Dai ai � Dai � ~0,1!

0 ai � Dai � ~�`,0#

i � 1,2, . . . , m

(3) For R � ~rij !m�n and A' � ~ai' !1�m , the corresponding consequence is B ' � ~bj

' !1�n .(4) For Methods 5–7, let rij � « when rij � 0, where 0 � « �� 1. We assume that« � 0.001 in the simulation.

For the given R � ~rij !m�n and A � ~ai !1�m , the Monte Carlo simulationcan be done as follows:

(1) Randomly generate $DAs %s�1p , where DAs � ~Dasi !1�m and Dasi ; U~� 1

40_ , 1

40_ !.

(2) For the given R and A, B ~i ! � ~b1~i ! , . . . , bn

~i ! ! is the consequence of the ith method~i � 1,2, . . . ,7!.

(3) For the given R and As' , Bs

'~i ! � ~bs1'~i ! , . . . , bsn

'~i ! ! is the consequence of the ith method~s � 1,2, . . . , p; i � 1,2, . . . ,7!, where As

' � A � DAs .(4) Compute Js

~i ! � (j�1n ~bj

~i ! � bsj'~i ! !2 ~s � 1,2, . . . , p; i � 1,2, . . . ,7!.

(5) For Js~i ! ~i � 1,2, . . . ,7!, rearrange the sequence $1,2, . . . ,7% into $ks

1 , ks2 , . . . , ks

7 % tosatisfy Js

~ksd ! � Js

~ksd�1 ! ~d � 1,2, . . . ,6!.

(6) Let Q � ~qsi !p�7, where qsi � d and ksd � i ~d � 1,2, . . . ,7!. So qsi expresses the

order of the ith method in the s experiment ~s � 1,2, . . . , p!.(7) Let T ~s! � ~tij~s!!7�7, where

tij ~s! �1

s (z�1

s

eij ~z!

and

eij ~z! � �1 qzi � j

0 qzi � j~s � 1,2, . . . , p!

Example 1. Let m � n � 7, p � 10000,

A � @0.2717 0.2772 0.9775 0.4338 0.7761 0.2162 1.0000#

1042 ZHANG AND CAI

and

R �

0.9501 0.2311 0.6068 0.4860 0.8913 0.7621 0.4565

0.0185 0.8214 0.4447 0.6154 0.7919 0.9218 0.7382

0.1763 0.4057 0.9355 0.9169 0.4103 0.8936 0.0579

0.3529 0.8132 0.0099 0.1389 0.2028 0.1987 0.6038

0.2722 0.1988 0.0153 0.7468 0.4451 0.9318 0.4660

0.4186 0.8462 0.5252 0.2026 0.6721 0.8381 0.0196

0.6813 0.3795 0.8318 0.5028 0.7095 0.4289 0.3046

.

Figure 1 shows that $tij~s!; s � 1,2, . . . ,10000% converges as s increases forgiven i, j. And by comparing Js

~g! ~g � 1,2, . . . ,7; s � 1,2, . . . ,10000! and Fig-ure 1, we can arrive at the notion that the order of the robustness of the sevenmethods is

Method 5 � Method 3 � Method 1 � Method 7 � Method 4

� Method 2 � Method 6

where “�” is interpreted as “is less robust than.” For example, Method 5 is lessrobust than Method 3 against errors in the premises.

Figure 1. Behavior of $tij~s!; s � 1,2, . . . ,10000% is represented in a matrix-like form: Ele-ment ~i, j ! depicts the behavior of tij~s! with the horizontal axis representing s; i, j � 1,2, . . . ,7.


The next three sections, which are based on the robustness measure definedin this section, discuss the statistical or probabilistic robustness of the seven meth-ods under three different scenarios.

5. ROBUSTNESS ANALYSIS I

In this section, we assume that R is randomly generated and A is given. TheMonte Carlo simulation can be done as follows:

(1) Randomly generate $Rl %l�1p and $DAls %s�1

w ~l � 1, . . . , p!, where Rl � ~rlij !m�n , rlij ;U~0,1!, DAls � ~Dalsi !1�m and Dalsi ; U~� 1

40_ , 1

40_ !.

(2) For the given Rl and A, Bl~i ! � ~b1l

~i ! , . . . , bln~i ! ! is the consequence of the ith method

~i � 1,2, . . . ,7).(3) For the given Rl and Als

' , Bls'~i ! � ~bls1

'~i ! , . . . , blsn'~i ! ! is the consequence of the ith method

~s � 1,2, . . . ,w; l � 1,2, . . . , p; i � 1,2, . . . ,7!, where Als' � A � DAls .

(4) Compute Jls~i ! � (j�1

n ~blj~i ! � blsj

'~i ! !2 ~s � 1,2, . . . ,w; l � 1,2, . . . , p; i � 1,2, . . . ,7!.(5) For Jls

~i ! ~i � 1,2, . . . ,7!, rearrange the sequence $1,2, . . . ,7% into $k~l�1!w�s1 ,

k~l�1!w�s2 , . . . , k~l�1!w�s

7 % to satisfy Jls~k~l�1!w�s

d !� Js

~k~l�1!w�sd�1 ! ~d � 1,2, . . . ,6!.

(6) Let Q � ~qui !pw�7, where qui � d and kud � i ~d � 1,2, . . . ,7!. So qui expresses the

order of the ith method in the u experiment ~u � 1,2, . . . , pw!.(7) Let T ~u! � ~tij~u!!7�7, where

tij ~u! �1

u (z�1

u

eij ~z!

and


0 qzi � j~u � 1,2, . . . , pw!

Example 2. Let m � n � 7, p � 1000, w � 100 and

A � @1.0000 0.2433 0.6387 0.5115 0.9381 0.8021 0.4804#

Figure 2 shows that $tij~u!;u � 1,2, . . . ,100000% converges as u increases forgiven i, j. And by comparing Jls

~g! ~g � 1,2, . . . ,7; s � 1,2, . . . ,100; l �1,2, . . . ,1000! and Figure 2, we can arrive at the notion that the order of the robust-ness of the seven methods is



6. ROBUSTNESS ANALYSIS II

In this section, we assume that R is given and A is randomly generated. TheMonte Carlo simulation can be done as follows:

1044 ZHANG AND CAI

(1) Randomly generate $Al %l�1p and $DAls %s�1

w ~l � 1, . . . , p!, where NAl � ~ Sali !1�m ,Sali ; U~0,1!, Al � ~ali !1�m � ~ Sali/max1�i�m Sali!1�m, DAls � ~Dalsi !1�m , andDalsi ; U~� 1

40_ , 1

40_ !.

(2) For the given R and Al , Bl~i ! � ~bl1


~i � 1,2, . . . ,7!.(3) For the given R and Als

' , Bls'~i ! � ~bls1


~s � 1,2, . . . ,w; l � 1,2, . . . , p; i � 1,2, . . . ,7!, where Als' � Al � DAls .


n ~blj~i ! � blsj

'~i ! !2 ~s � 1,2, . . . ,w; l � 1,2, . . . , p; i � 1,2, . . . ,7!.(5) For Jls


k~l�1!w�s , . . . , k~l�1!w�s7 % to satisfy Jls

~k~l�1!w�sd !

� Js~k~l�1!w�s

d�1 ! ~d � 1,2, . . . ,6!.(6) Let Q � ~qui !pw�7, where qui � d and ku

d � i ~d � 1,2, . . . ,7!. So qui expresses theorder of the ith method in the u experiment ~u � 1,2, . . . , pw!.

(7) Let T ~u! � ~tij~u!!7�7, where

tij ~u! �1

u (z�1

u

eij ~z!

and


0 qzi � j~u � 1,2, . . . , pw!

Figure 2. Behavior of $tij~u!;u � 1,2, . . . ,100000% is represented in a matrix-like form: Ele-ment ~i, j ! depicts the behavior of tij~u! with the horizontal axis representing u; i, j � 1,2, . . . ,7.


Example 3. Let m � n � 7, p � 1000, w � 100 and

R �

0.9501 0.2311 0.6068 0.4860 0.8913 0.7621 0.4565

0.0185 0.8214 0.4447 0.6154 0.7919 0.9218 0.7382

0.1763 0.4057 0.9355 0.9169 0.4103 0.8936 0.0579

0.3529 0.8132 0.0099 0.1389 0.2028 0.1987 0.6038

0.2722 0.1988 0.0153 0.7468 0.4451 0.9318 0.4660

0.4186 0.8462 0.5252 0.2026 0.6721 0.8381 0.0196

0.6813 0.3795 0.8318 0.5028 0.7095 0.4289 0.3046

Figure 3 shows that $tij~u!; u � 1,2, . . . ,100000% converges as u increases forgiven i, j. And by comparing Jls

~g! ~g � 1,2, . . . ,7; s � 1,2, . . . ,100; l �1,2, . . . ,1000! and Figure 3, we can arrive at the notion that the order of the robust-ness of the seven methods is




1046 ZHANG AND CAI

7. ROBUSTNESS ANALYSIS III

In this section, we assume that R and A are randomly generated. The MonteCarlo simulation can be done as follows:

(1) Randomly generate $Rl %l�1p , $Al %l�1

p , and $DAls %s�12 ~l � 1, . . . , p!, where Rl �

~rlij !m�n , rlij ; U~0,1!, NAl � ~ Sali !1�m , Sali ; U~0,1!, Al � ~ali !1�m � ~ Sali/max1�i�m Sali!1�m , DAls � ~Dalsi !1�m , and Dalsi ; U~� 1

40_ , 1

40_ !.

(2) For the given Rl and Al , Bl~i ! � ~bl1


~i � 1,2, . . . ,7).(3) For the given Rl and Als

' , Bls'~i ! � ~bls1


~s � 1,2, . . . ,w; l � 1,2, . . . , p; i � 1,2, . . . ,7!, where Als' � Al � DAls .


n ~blj~i ! � blsj

'~i ! !2 ~s � 1,2, . . . ,w; l � 1,2, . . . , p; i � 1,2, . . . ,7!.(5) For Jls


k~l�1!w�s2 , . . . , k~l�1!w�s

7 % to satisfy Jls~k~l�1!w�s

d !� Js

~k~l�1!w�sd�1 ! ~d � 1,2, . . . ,6!.

(6) Let Q � ~qui !pw�7, where qui � d and kud � i ~d � 1,2, . . . ,7!. So qui expresses the

order of the ith method in the u experiment ~u � 1,2, . . . , pw!.(7) Let T ~u! � ~tij~u!!7�7, where

tij ~u! �1

u (z�1

u

eij ~z!

and


0 qzi � j~u � 1,2, . . . , pw!

Example 4. Let m � n � 7, p � 1000, and w � 100. Figure 4 shows that$tij~u!;u � 1,2, . . . ,100000% converges as u increases for given i, j. And by com-paring Jls

~g! ~g � 1,2, . . . ,7; s � 1,2, . . . ,100; l � 1,2, . . . ,1000! and Figure 4, wecan arrive at the notion that the order of the robustness of the seven methods is



8. CONCLUSION

In a previous paper,8 we introduced three optimal fuzzy reasoning methodsbased on three different objective functions concerning the absolute distancebetween two fuzzy relations. In this article, we introduced another three optimalfuzzy reasoning methods based on three objective functions concerning the rela-tive distance between fuzzy relations. Because analytical evaluation of the robust-ness of optimal fuzzy reasoning methods is rather complicated, in this article weevaluated the robustness in a probabilistic or statistical context by using the MonteCarlo simulation method. Among the six optimal fuzzy reasoning methods and theconventional CRI method, the Monte Carlo simulation shows that the optimal fuzzyreasoning method based on J5 is the best or the most robust against errors in thepremises. Also, the CRI method is usually not the best one in comparison with the


optimal fuzzy reasoning methods in terms of robustness. However the CRI methodis not the worst one in terms of robustness. In other words, the robustness of theCRI method can be improved by properly treating fuzzy reasoning as an optimi-zation problem. The CRI method is acceptable is the sense that it is simple interms of computation, and improper use of optimization objective function in opti-mal fuzzy reasoning may impair the robustness of fuzzy reasoning.

In summary, the contribution of the article is as follows. First, we showedthat fuzzy reasoning can be viewed as a process of computation and optimizationinstead of a process of logical inference by reviewing the idea of optimal fuzzyreasoning and introducing three new optimal fuzzy reasoning methods. Second,we proposed to evaluate the robustness of a fuzzy reasoning method in a probabi-listic or statistical context by using the Monte Carlo simulation method in the casesin which analytic robustness evaluation is complicated. Finally, we compared thesix optimal fuzzy reasoning methods with the CRI method in terms of probabilis-tic robustness.

There is much work that deserves investigation in the future. First, we shouldexamine in what context and when optimal fuzzy reasoning methods should beadopted. A possible application area of optimal fuzzy reasoning methods is fuzzydecision making, because fuzzy decision making involves a process of fuzzy opti-mization. Second, a theoretical foundation should be developed for probabilisticor statistical evaluation of robustness of fuzzy reasoning. Third, the advantagesand limitations of optimal fuzzy reasoning in particular, and fuzzy reasoning in


1048 ZHANG AND CAI

general, should be examined. Treating fuzzy reasoning as an optimization prob-lem offers a new angle for dealing with fuzzy reasoning problems.

Acknowledgments

The work was supported by the National Natural Science Foundation of China under Grants60274057 and 60204011.

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optimal fuzzy reasoning and its robustness analysis

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