A decision making method based on weighted interval- valued fuzzy reasoning algorithm

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Procedia

Engineering

Procedia Engineering 00 (2011) 000–000 www.elsevier.com/locate/procedia

Advanced in Control Engineering and Information Science

A decision making method based on weighted interval-

valued fuzzy reasoning algorithm

Qiansheng Zhang

a

，

Shihua Luo

b

，

a*

School of Informatics, Guangdong University of Foreign Studies,Guangzhou 510420, P.R. China

a

b_{School of Information Management, Jiangxi University of Finance and Economics, Nanchang 330013, P.R. China}

Abstract

This paper presents a weighted interval-valued fuzzy (IvF) reasoning algorithm for handling decision making problem where interval-valued fuzzy sets (IvFSs) concept and IvF decision rules are used for uncertain knowledge representation. The proposed algorithm can perform IvF matching between the unknown model and the antecedent portion of IvF rule by employing a weighted similarity measure to determine the presence of decision model with a condence interval.

Selection and/or peer-review under responsibility of [CEIS 2011]

Keywords: Interval-valued fuzzy set; Weighted similarity measure; Decision rule; Interval-valued fuzzy reasoning; Decision making; Medical Diagnosis

1. Introduction

It has achieved great success in the application elds such as fuzzy reasoning [2] and fuzzy decision making [7], since fuzzy set theory was proposed by Zadeh. In 1999 [10], Raha studied fuzzy reasoning with fuzzy truth. In [4, 5], Chen investigated fuzzy decision problem and introduced a weighted fuzzy medical diagnosis algorithm. In 2002 [11], Yeung also resented a multilevel weighted fuzzy reasoning algorithm for expert systems. Recently, much knowledge in medical diagnosis, pattern recognition,

* Corresponding author. Tel.: +86-20-26270362; fax: +86-20-36334425. E-mail address: [email protected].

Open access under CC BY-NC-ND license. Open access under CC BY-NC-ND license.

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decision making and expert system, involves interval -valued fuzzy concepts. There is an increasing demand to design a decision algorithm on computer system to handle interval - valued fuzzy decision making problem.

Some applications of interval-valued fuzzy set (IvFS ) theory in approximate reasoning [1] and decision making have been introduced in the literature. For example, Gorzalczany [ 9] proposed some methods of inference in approximate reasoning based on interval - valued fuzzy sets and compatibility measure. And in [3] Chen investigated interval -valued fuzzy reasoning based on vector similarity measure. Also, Dziech in [8] investigated interval-valued fuzzy decision problems.

In this paper, we extend the work of Chen [5] to deal with interval-valued fuzzy decision problem where interval -valued fuzzy production rule [6] are used for decision knowledge representation. And a weighted interval-valued fuzzy reasoning algorithm by employing the new weighted similarity measure of IvF sets and rules is proposed to handle interval-valued fuzzy decision making or medical diagnosis.

2. Similarity Measure between Interval-valued Fuzzy Sets

Deﬁnition 1. Let denote the set of all the closed subintervals of [0, 1], then a preference relation “≤” on [I] can be dened as, [a1, b1 ] ≤[a2, b2] if

[I

]

(

a

₁

+

b

₁

)

/

2

≤

(

a

₂

+

b

₂

)

/

2

.

Deﬁnition 2. Let [a1, b1], [a2 , b2]∈

[I

]

_{, some basic operations are defined as}

[a1 , b1] + [a2 , b2] = [a1 + a2, b1 + b2]; [a1 , b1]

⋅

[

a2, b2] = [a1·a2, b1·b2].

Deﬁnition 3. A mapping

A

:

X →

[I

]

is called interval-valued fuzzy set in universe

X

, if

)]

(

),

(

[

)

(

x

i

A

x

i

A

x

i

A

=

− +

(

IvF

∈

[I

)

]

,

∀

x

i

∈

X

.

Throughout this paper, we let denote the set of all the interval-valued fuzzy sets in universe

X

.

X

Deﬁnition 4. Let

A

,

B

∈

IvF

(

X

)

, some elementary notations are given as follows: A ∩ B = {( , [

x

_i

A

−

(

x

_i

)

∧

B

−

(

x

_i

),

A

+

(

x

_i

)

∧

B

+

(

x

_i

)

]) / }, + + − −

X

x

i

∈

A∪B = {( , [

x

_i

A

(

x

_i

)

∨

B

(

x

_i

),

A

(

x

_i

)

∨

B

(

x

_i

)

]) /

x

_i

∈

X

}, Supp

A

= {

x

_i

∈

X

/

A

−

(

x

_i

)

≥ 0,

A

+

(

x

_i

)

> 0}.

Deﬁnition 5. Function

S

:

IvF

(

X

)

× → [0, 1] is called the similarity measure between A, B , if it satises the following properties:

)

(

X

IvF

(P1) 0 ≤ S(A, B) ≤ 1 ; (P2) S(A, B) = 1, iff A = B ;

(P3) S(A, B) = S(B, A); (P4) If A ⊆_B⊆_{C, then S(A, C) ≤ S(A, B), S(A, C) ≤ S(B, C).}

Deﬁnition 6. The weighted similarity measure between interval-valued fuzzy sets A and B is dened as

(

,

)

1

₂1

{

(

)

(

)

(

)

(

)

},

i i i i i i W

A

B

A

x

B

x

A

x

B

x

S

=

−

∑

ω

−

+

−

+

where

ω

iis the weight of

x

i

∈

X

, and it also fulfils the properties (P1)-(P4) of similarity measure.

3. A Weighted interval-valued fuzzy reasoning technique

It is well known that much knowledge in the real -life world is interval valued fuzzy expression rather than precise concept. For example, “ If one has a very strong headache and has a fairly serious stomach, more or less high temperature, less medium cough, then he might have caught a Viral”, where

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“very strong ”, “ fairly serious”, “ more or less high”, “less medium” can be viewed as IvF concepts, which are not clear and also context dependent. In order to make such real-life world knowledge suitable for being processed by computer, we need to use some valued fuzzy production rules for interval-valued fuzzy knowledge representations which contain many interval-interval-valued fuzzy terms.

Let _j ₁_≤_j_≤_p be a set of IvF production rules. The general formulation of the interval-valued fuzzy decision model are described as follows.

R

=

{

R

}

R

1

: T

1

→ d

1

, CF

1

=[

t

₁−

,

t

₁+

]

;

R

2

: T

2

→ d

2

, CF

2

= [

t

₂−

,t

₂+

]

; ……….

R

p

: T

p

→ d

p

, CF

p

= [

t

−_p

,

t

_p+

]

; +

Fact: T

∗

,

c

₁

=

[

c

₁−

,

c

₁

]

,

₂ ₂ ₂

, .. ,

_m _m ;

———————————————————

[

,

]

+ −

=

c

=

[

−

,

+

]

m

c

Consequence: ,

d

*

CF

*

=

[

t

*−

,

t

*+

]

;

where = is

T

_j and is and and is , (j = 1, 2, · · · , p),

T

*

u

1

A

1j

u

2

A

2j

u

m

A

mj

,

A

= is

∈

1

IvF

1* and

)

2is 2* and· · and m is m*,

and _ij

A

_i_*

(

X

_i , (i = 1, 2, · · · , m; j = 1, 2, ·· · , p).

u u

1

u

,

d

j ,

d

*2, · ,

D

mare linguistic variables,

X

iis the range of variable ;

u

i

d

∈_D,

=

{

d

₁

,

d

₂

,

L

,

d

_p

}

_{is the the decision model set containing all decision alterna-}

tives, _j is the consequent portion of IvF decision rule

R

_j.

CF

j

= [

]

is the interval-valued certainty level which indicates the certainty that the IvF

decision rule can be condent;

+ − j j

t

,

j

R

+ −

,

]

[

=

_i _i i

c

is the condence interval of the fact “ is

u

_i

A

_i_*” can be condent.

In what follows, a weighted interval-valued fuzzy reasoning approach will be presented based on the proposed weighted similarity function

S

_W.

}

Deﬁnition 7. Let , the weighted similarity measure between fact and the antece-dent of the jth rule can be dened by

,

{

T

1

T

2

T

p

T

=

L

j

R

m *

T

j

T

)

,

(

)

,

(

T

*

T

j _i ₁ i

S

W

A

i*

A

ij

S

η

=

∑

₌

η

_i ，

1

≤

j

≤

p

； where

{

(

)

(

)

(

)

(

)

}

2

1

)

,

(

* 1 * * i it ij it n t it i it ij it ij i W

A

x

A

x

A

x

A

x

S

i + + = − −

−

+

−

=

∑

ω

is the weighted similarity measure between IvFSs

A

_i_*

,

A

_ij.

i

W

= {

ω

_i₁,

ω

_i₂,· · ,

ω

_in_i} is the weight vector of the elements in universe

X

_i,

i

n

is the number of elements in the universe

X

_i= {

x

_i₁,

x

_i₂,· · · , }， i

in

x

i

η

is the weight of linguistic variable in production rule, which indicates the importance degree of contributing to the consequence of IvF decision rule. i

u

i

The larger value of

S

(

T

_*

,

T

_j

,

η

)

T

, the higher the weighted similarity matching

d

egree between the fact

and the antecedent of the jth rule .

*

T

j j

In the sequel, we need to propose a reasonable method to evaluate the weight vector

R

η

= (

η

₁

,

η

₂

,

m

η

,

L

) of all the variables in the antecedent portion of IvF rule by using information entropy theory.

u

_i

≅

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l

T

≅

T

_j_{iff → , →}

T

_l

d

_l

T

_j

d

_j_,

d

_l

=

d

_j_{, (1 ≤}

l,

j

_{≤ p).}

If there are only different output states in all the consequent portions of this system, then the output set is partitioned into

s

,

L

d

}

,

{

d

1

d

2

d

p

D

=

≅

[

],

[

d

Θ/ ={ ₁ ₂

],

L

[,

d

_s

]}

(1≤s≤p), And T / = {[ ], [ ], · · · , [ ]}.

≅

T

₁

T

₂

T

_s

Note that each class

[

T

_r

]

also corresponds to the IvF decision class

[

d

_r

]

.

Therefore,

η

= (

η

₁

,

η

₂

,

L

,

η

_m) can be evaluated from the statistical information of each variable in

this decision system as follows: i

u

The information gain-loss of variable in the IvF decision system can be computed by the following mutual information entropy

u

i

=

Θ

/

)

(

u

i

I

H(Θ)−H(Θ/ ) ,

u

_i

which indicates the information of variable about the decision output portion of this IvF decision system; where

u

i

)

(

Θ

H

= −

∑

₌ ( )

is the uncertainty information of all the decision outputs in this IvF system,

s

r 1

P

[

d

r

]

log

2

P

([

d

r

])

P ( ) = denotes the distribution probability of in space Θ, is the number o elements contained in the decision class

[

d

r

]

[

d

u

p

N

_r

/

∑

Ii

P

]

[

d

r

N

r f .

]

r

A

)

/

And H(Θ/ ) =i _k₌₁ ,

which shows the uncertain effect degree of variable on the decision output of the IvF system,

(

A

ik

)

H

(

Θ

ik

u

_i

X

i

I

denotes the number of IvFSs dened on universe _i，

A

where P ( ) = denotes the frequency of occurring in all the decision rules, is the number of rules containing IvF term

A

ik

N

ik

/

p

A

_ik, ik

p

N

ik

H(Θ/

A

_ik) = −

∑

_{r 1}s₌

P

([

d

_r

]

/

A

_ik)

log

₂

P

([

d

_r

]

/

A

_ik

)

,

P ( / ) =P( and [dr])/ P( )) is the ratio of the number of rules containing both antecedent and consequent to the number of rules containing .

]

[

d

r ik

A

ik

A

ik

A

ik r

d

A

_ik

Thus, the weight

η

_iof variable can be computed by

u

_i

∑

=

Θ

=

_i _i _i i

I

u

)

/

₁

(

/

u

)

η

,

(

Θ

/

(

m

I

.

Let λ be a threshold value. If _* _j

η

)

j

d

T

S

≥ λ, then the rule can be red, which indicates that the unknown fact has the decision class with condence interval

j

R

j j m i i i j

c

S

T

CF

*

=

(

∑

₌₁

η

)

(

*

,

η

)

*

CF

.

The larger the value _j , the higher condence that the unknown pattern might have decision

T

_*

d

_j. If

S

(

T

_*

,

T

_j

,

η

)

j

d

< λ, then the rule can not be red, then the unknown fact should not be assigned

to decision .

R

j

T

*

In the subsequent, let λ be a given threshold value, assume η = (

η

₁

,

η

₂

,

L

,

η

_m) is the weight vector of linguistic variables , ,…, . In order to get a concluded decision from the fact and the decision rules {

R

_j} 1≤j≤p, we give a weighted interval-valued fuzzy reasoning algorithm for deal-

u

1

u

2

u

m

T

* ing with this decision making system or medical diagnosis problem.

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for j ← 1 to do

p

begin

for i ← 1 to

m

do begin

Let

B

_ij be the result of the intersection of and , i.e.,

B

* i

A

(

A

−

)

A

ij ij

Q

=

A

i*m∩

A

ij= {(

x

it, [

A

i*

(

x

it

)

∧ , ∧ ])/ }. −

B

ij

B

x

it i*

(

x

it

)

+

B

A

ij

(

x

it

)

+

φ

≠

x

it

∈

X

i If _j=

U

Supp = Supp ∪ Supp · · ∪ Supp ,

i=1

B

ij 1j 2j mj then j

ρ

← S(

T

_*

,

T

_j,

η

); begin if

ρ

_j≥ λ, then begin

]

)

(

,

)

[(

)

(

* + = + − = − =

∑

=

←

m_i _i _i _j _j _im _i _i _j _j _im _i _i _j _j j

c

CF

c

t

c

t

CF

₁

η

ρ

₁

η

ρ

₁

η

ρ

}

{

max

* * j j k

CF

=

* ; If ,

then the unknown fact is assigned to the decision

T

_*

d

_k with condence

CF

_k . end

end end

Acknowledgements

This work is supported by the National Natural Science Foundation of China (Nos. 61070061, 60964005), and the Guangdong Province Planning Project of Philosophy and Social Sciences (09O-19).

References

[1] Bustince H, Burillo P. Mathematical analysis of interval-valued fuzzy relations: application to approximate reasoning. Fuzzy Sets and Systems 2000, 113, 205-219

[2] Baldwin J. F, Guild N. Feasible algorithms for approximate reasoning using fuzzy logic. Fuzzy Sets and Systems 1980, 3, 225-251.

[3] Chen S. M. Bidirectional approximate reasoning based on interval-valued fuzzy sets. Fuzzy Sets and Systems 1997, 91, 339-353.

[4] Chen S. M. A new approach to handling fuzzy decision making problems. IEEE Trans. Systems Man Cybernet , 1988, 18, 1012-1016.

[5] Chen S. M. A weighted fuzzy reasoning algorithm for medical diagnosis. Decision Support Systems 1994, 11, 37-43. [6] Chen S.M. An improved algorithm for inexact reasoning based on extended fuzzy production rules. Cybernetics and Systems

1992, 23, 463-481.

[7] Chun M G. A similarity- based bidirectional approximate reasoning method for decision maing systems. Fuzzy Sets and Systems 2001, 117, 269-278.

[8] Dziech A. Decision making in signal transmission problems with Interval-valued fuzzy sets. Fuzzy Sets and Systems 1987, 23,

191-203.

[9] Gorzalczany M B. A method of inference in approximate reasoning based on interval valued fuzzy sets. Fuzzy Sets and Systems 1987, 21, 1-17.

[10] Raha S, Ray K S. Reasoning with vague truth. Fuzzy Sets and Systems 1999, 105, 385-399.

[11] Yeung D S. A multilevel weighted fuzzy reasoning algorithm for expert systems. IEEE Trans. on Systems, Man and Cybernet ics, Part A. 2002, 28, 149-158.

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