A new information fusion method based on interval-valued fuzzy numbers for handling multi-criteria fuzzy decision-making problems

A New Information Fusion Method Based on Interval-Valued Fuzzy Numbers for Handling Multi-Criteria Fuzzy Decision-Making Problems

Shi-Jay Chen and Shyi-Ming Chen

Department of Computer Science and Information Engineering National Taiwan University of Science and Technology

Taipei, Taiwan, R. 0. C.

Abstract - I n this paper, we present a new information fusion method for fusing decision-maker's linguistic opinions to deal with multi-criteria fuzzy decision-making problems. The proposed information fusion algorithm uses interval-valued fuzzy numbers to represent the decision-makers' linguistic opinions. I t uses linguistic quantifiers based on the FN-IOWA operator to flexibly determine the weights of linguistic opinions of each decision-maker for aggregating these linguistic opinions in different linguistic constraints. The proposed method can handle multi-criteria fuzzy decision-making problems in a more intelligent and more flexible manner.

I. INTRODUCTION

From [I], PI, [81-[101, [151-[191, POI, we can see that funy numbers are very useful to represent decision-maker's linguistic opinions in multicriteria fuzzy decisionlnaking problems.

Furthermore, some "hers pointed out that it is more flexile using g e n e m l i i funy numbers [I], [3]-[7], [I I] or interval-valued fuzzy numbers [12], [14], [23] to represent decision-maker's linguistic opinions due to the fact that these two kinds of fuzzy numbers not only can represent decisionmaker's linguistic opinions, but also can represent the degrees of confidence or degrees of unceminty of decisionmaker's linguistic opinions. "E, in this p a p , we present a nem information h i o n methcd based on interval-valued fuzzy numkrs for handling multicriteria f u n y decision-makinp problems. The proposed infomation fusion method can handle the decision-making problems in a more intelligent and more flexible manner due to the fact that the proposed fusion algorithm not only uses interval-valued funy numbers to represent the decision-makers' linguistic opinions, but also uses the linguistic quantifiers to ^aggregate the decision-makers' linguistic opinions in diffemt linguistic constraints.

The rest of this paper is organized as follows. In Section 11.

we briefly review the defmitions of interval-valued fuzzy sets [12], interval-valued fuzzy numbers [I41 and the FN-IOWA operator [SI. In Section 111, we present a new information fusion method based on interval-valued fuzzy numbers to deal with multi-criteria fuzzy decisionmaking problems. In Section IV, we use an example to illustrate the multi-criteria fuzzy decision-making process. The conclusions are discussed in Section V.

11. PRELIMINARIES

In this section, we briefly review the definitions of interval-valued fuzzy sets [12], interval-valued fuzzy numbers [14] and the FN-IOWA operator [SI.

Definition 2.1: An interval-valued fuzzy set C defined in the universe of discourse X is represented by

c ^{== { ( x ,} [Put:

P? (XI]) I x E xi.

where 0 s & ( x ) S & ( X ) S 1, and the membership grade of the element x belonging to the interval-valued fuzzy set C is represented by an interval [P:(x), &(x)] (i.e.,

-

,&Ti,(*) = rP:(d3

Definition 2.2 [14]: If an interval-valued fuzzy _set 2

satisfies the following properties:

( I ) 2 is normal,

(2) 2 is defined in a closed bounded interval, (3) A is a convex set,

then 2 is called an interval-valued fuzzy number in the universe of discourse X.

Assuge that there is a trapezoidal interval- valued fuzzy number 2 ⁼ [ P ~ L , Pzu 1. ^as shown in Fig. 1.

-

-

A

X

CI

L L

0 a y a? a, a2 04 a," 04" a,"

Fig. 1 . Trapezoidal mterval-valued fmq number A

;vP ...

' A 7 '

I!

X

CI

L L

0 a y a? a, a2 04 a," 04" a,"

Fig. 1 . Trapezoidal mterval-valued fmq number A

From Fig. I. we can see that the intenal- valued f u w number

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b: , bp , by , b," and bp are any real values, 0

< ~ L 5 i 9 5 1 a n d O < ~ . L S ~ ( i 5 1 .

( 2 ) Interval-Valued Fuzzy Number Multiplication @:

L L . . L U

U U .

A @ B = [ ( a : , a 2 , a:, a d , w ~ ) , ( a : , a ? , a , , a , , G ; ) ] @ [ ( b f , b f , b:, b i ; $), ( b y , b,", b y , b," ; Gel1

= [ ( a t b:, a i

bf , ai ^{x b:, a: x} b," ; Min ( ~ t ,

A A B B

- _ - -

B

G & ) ) , ( a F by , a y b t ,a," b y , a," b, 8 . ,

Min ( G ; , G;NL ( 2 )

where a:, a t , a t , a t , a,", a y , a y , a,", b:, b i , b:, b: , b: , by , by and b p are any real values, 0

<i,L;5,&'5 1 andO<;L;SGgS 1.

A A n B

Definition 2.3: The FN-IOWA operator for aggregating the OWA pair is shown as follows:

F ( < u , , ii,>,<u,, &>, ...,< ^U,, iin>) = W ' B , where < U , , Zi > denotes an OWA pair, ^U, denotes an order inducing value, and ZZ denotes an argument value; B is an ordered argument vector,

- -

B =[bl bl ... bn]',

6; is an argument value of the OWA pair which is the jth largest order inducing value U , , where I 5 i 5 n; W is a weighting vector,

w'=[w1w2... 4

where the weight w, of the weighting vector W is associated with the jth position of the ordering sequence in the

weighting vector FV and 1 5 j 5 n. The weighting vector W h,as the following properties: (1) w, E [0, 11 and ( 2 )

ZW, ^{= I ; FV'} denotes the transpose of the weighting vector

111. A NEW INFORhMTION FUSION METHOD BASED ON INTERVAL-VALUED FUZZY NWERS FOR HANDLING MULTI-CRITERIA FUZZY DECISION-MAKING PROBLEMS

In the following, we present a new information fusion method based on interval-valued fuzzy numbers to deal with fuzzy decisionmaking problems. Let U be the universe of discourse, U ⁼ [0, k]. Assume that there are some linguistic opinions Er given by a decision-maker. These opinions are represented by interval-valued fuzzy numbers c, ^{= [(a:, b: ,}

c :

, d : ; i f ) , ( a " , b!", CY, d,"; $,')I, where 0 5 a, 5 b, 5 c; 5 d, 5 k, 0 <G,! 5G;5 1 , l 5 i 5 n, and the two values GtLand 1;" represent the interval-degrees of confidence [ Gf , G''] of the linguistic opinions 5, . The proposed information fusion method is now presented as follows:

Step 1: Translate each interval-valued fuzzy number 4 ,

where I 5 i 5 n, into a standardized interval- valued fuzzy number E;, shown as follows:

-

-

- - - _

c,? ^{= [G, , G, ]}

b;" c: d f . .

a;" bu cu dy k ' k ' k ' k k ' k ' k ' k

L

1

w, 1, (- -; w, 11

=[(% - - -

=[(a:', b y , e:', d y ; G,!), (ay, ^{b y ,} cy, dy; +;")I, ⁽⁵⁾

where 0 S a y 5 b y 5 c ~ d,L'S S 1, 0 5 ay*< b y 5 $'5 d,u'5 1, and 0 <G,!5;V" 5 1.

Step 2: Decide the order of the pairs E,'>, - where 1 5 i 5 n.

Fmm [7], we can use the ordered inducing value U , to decide the ordw of the pairs

c2'>. Furthermore, we can see that there are two different situations for deciding the order of the pairs shown as follows:

Situation 1: We use the relative weight between 0 and 1 to Wresent the ordeed inducing value u Z , where 1 S i S n. The larger the value of the ordered inducing value ^U, , the more the importance ofthe linguistic opinions E,*.

Situation 2: We can use the ordered sequence I, 2, _ _ _ , k, . . ., n to repraent the ordered inducing value U , , where I 5 i 5 n. If there is a linguistic opinion 4 and its ordered sequence is 1 (i.e., ut =I), then we can see that the opinion G; is the most important one among the other opinions E,*, where 1 5 i 5 n andifk.

-

-

- -

I

826

According to the ordered inducing value

of each linguistic opinion c:, ^{where 1 5 i 5 n,} we can obtain the ordered pair <

u p ,

G;>, where I S p 2 n. If uI < ^u2 < ... <

< ... <

I l n ,

then the ordering sequence ofthe pairs is shown as follows:

< u I , G,*>,

G 2 > .... ,

(?;>, ... , < U ” , Z;>.

Furthermore, we can get the ordered opinion vector B,

- - -

- - -

-.

-

- - - -

B = [ T b;’ ._. ; ; ... ;“’I,

where ; ; is the opinion G; of the pair < Z.j, > having the pth largest ordered inducing value u p , 1 5 p 2 n, and it is an

interval-valued fuzzy number.

Step 3: Use the linguistic quantifiers and the inducing value

up

of the ordered pair $>, where 1 5 p S n, to determine the weight wp of each opinion z.‘ where 1 S p 2 n. Assume that there are six linguistic constraints, shown as follows:

“The relutive importunt opinions of all evaluating criteria are

“Tlte most important opinions of all evaluating criteria are

“The leusf importunr opinions of all evaluating criteria are

“The relative imporfunf opinions of the useful evaluating

“The most important opinions of the useful evaluating

“The leusf imporfunt opinions of the useful evaluating

- - -

U P ,

-

P ’ P

-

P ’

considered”, considered”, considered”,

criteria are considered”, criteria are considered”, criteria are considered”.

’The above linguistic constraints have the following form:

“Q opinions of N evaluating criteria are considered, (4) where Q is a linguistic quantifier, Q €{The relative important, the most important, the least important}, N is a linguistic term, N €{all, the useful}. The membership function of a linguistic quantifier Q is represented as follows [7]:

0, if x < a

+’ ^{, if a 5 I < h , where r E (112. I. 2 ) ( 5 )}

where a E [0, I], b E [0, I] andx

[0, I]. Ifthe linguistic quantifier Q

“nre relative imporfunf”, then r

1; if h e lin,&ic quantifier Q = “The most impurfunt”, then ^r

2; ifthe linguistic quantifier Q

=

“nze le& importunf”, then r

ID. If the linguistic term N

“alP’, then the parmeters a and b of the linguistic quantifier Q are 0 and I, respectively; if the linguistic term N

“the useful”, then the parameters ^a and b of the linguistic quantifier Q are 0.4 and 1,

1, if

> b

( I ) If we use the relative weight between 0 and 1 (i.e., Situation 1) to represent the ordered inducing value u p , where I S p 5 n, then we can obtain the weighted value w,,

where c w , = I .

(2) If we use the ordered sequence I, 2, ..., i, ..., n (i.e., Situation 2) to represent the ordered inducing value u p ,

where 1 5 p S n, then we can obtain the weighted value w,,

/=1

(7)

where i is the sequence and c w , = I .

Step 4: Calculate the interval-degree confidence [ $&, $7’ 1,

and obtain the fusing result F((l, ,E;))=

/=I

I], where 1 5 p ⁵ n, as follows:

[+fl

L

= ( a L , bL,cL,dL;G&),

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(11) Based on formulas ( I O ) and (1 I), we can get the fusion result

R( ^{,, G} =:) ⁾ of the evaluating opinions En*, where 1 5 i 5 n.

IV. AN EXAMPLE

= ( a " , P ,

d " , "y-,.

-

In this section, we apply the proposed information fusion algorithm to deal with fuzzy decisionmaking problems.

Assume that there are three evaluating criteria C,, C, and C,, and assume that we can use intervakvalwd fuzzy Embers to represent the linguistic opinions E;, a; and of the evaluating criteria C,, C, and Cj, respectively. Furthermore, we consider the following two situations:

[Step 11 Because the-universe of disc_urse of the three linguistic opinions 2., , C, and 5, represented by intervahalued fuzzy numbers are between _O and 10, respectively, we use formula ( 3 ) to translate G I , cl ^{E d}

into th_e standardized interval-valued fuzzy numbers El*,

and c;, respectively, shown as follows:

G; - - ⁼ [(o. i,0.2,0.3,0.4; o.s),(o. i,o.2,0.3,0.4; 1 .n)].

Gi ⁼ [(0.4,0.5,0.6,0.7;0.7),(0.4,0.5,0.6,0.7;0.8)],

9 -

[(0.6.0.7,0.8.0.90.7),(0.6.0.7.0.8,0.9;0.9)].

[Step 21 We can see that the ordered pairs of the three linguistic opinions c;, ^{5; and} 4 ^are asfollows:

<0.9, E;>,<O.6, c:>,<O.76, E;>.

- I

-

- -

We can use the ordered inducing value U , to decide the ordered sequence of the pairs, shown as follows:

- -

<0.6, 2 > , < 0 . 7 6 , ~ ; > , i n . 9 , G;>,

- I _ _

- -

where b,'=c;, &'=a; and F,*=a;. Furthermore, we can get theorderedopinionvectorB=[~' - - - C b;'].

Situation 1: We use the relative weight between 0 and I to represent the ordered inducing value U , , where I 5 i 5 3 . The larger the value of the ordered kducing value U , , the more the importance of the opinions G:, where I 5 i 5 3 .

,Step 3l Based on ( 5 ) and (6), we can get the absolute weight w, of the ordered opinion q , ^{shown as}

follows:

I

Situation 2: We use the ordered sequence 1, 2, 3 to represent the order@ inducing value ui , where I C i 5 3. If there is an opinion 2; and its sequence is 1 (is., U * = I), then we can see that the opinion G; is the most important one among the other opinions E:, where 1 5 i C 3 and i # k.

In the following, we use the following limguistic constraint to illustrate how to use the linguistic quantifier for dealing with the fimy decisionmaking problem:

"The relative important opinions of all evaluating criteria are considered".

- -

Assume that there are three linguistic opinions a, ^, C, and

c, represented by interval-valued fuzzy numbers, shown as follows:

-

E

[(I, 2,3,4; 0.8), ( 1 , 2 , 3 , 4 ; 1.0)],

6

^[(4, 5 , 6, 7; 0.7), (4, 5 , 6, 7; 0.8)]:

a,= [(6,7, 8,9; 0.7), ( 6 , 7 , 8 , 9 ; 0.9)].

-2

0.6 n )

'(0.9+ 0.76 + 0 . 6 ) - ~ ( 0 . 9 + 0.7610.6 -

-

= 0.265.

In the same way, we can get the absolute weights w2 and wjof the ordered linguistic opinions &* and b;: , respectively, shown as follows:

-

-

0.76+0.6 0.6 )

1v

-

2

Q(0.9+0.76 +o.6)-Q(o.9+n.7,+o,6

= 0.337,

= 0.398, Situation I: Let us consider Situation 1 to deal with the

fuzzy decisionmaking problem. Assume that the relative weight of the linguistic opinion Cl is 0.9, the relative weight of the linguistic opinion Z2 is 0.6 and the relative weight of the linguistic opinion E3 is 0.76, then

-

where C1Va ^{= 1.} Thus, the weighted vector W

wj]

=

[0.265 0.337 0.3981.

I = ,

828

[Step 41 By applying formulas (8) and (9), we can obtain the aggregated interval-degree confidence [$;a, $& 1, shown as follows:

0.7

= [O.Z6S 0.337 0.398]x~J~]

= 0.7398,

" I/

WAm

= [ w , wi wJr

=[0.265 0.337 0.3981~

=0.9133.

Furthermore, by applying formulas ( I O ) and ( I I), we can get the fusion result

= [ f ' ( ( u , . z : ) ) ,

T u ( ( , . ?))I of the three linguistic opinions e,', ^{2.; and} a;, shown as follows:

- -

-

I

=([0.265 0.337 0.398]@1;]; 0.7398)

L J

=(0.348,0.448,0.548,0.648; 0.7398),

= (10.265 0.337 O.398lr '8 E:]

b;"'

; 0.9133)

=(0.348.0.448,0.548, 0.648;0.9133)

Situation 2: Let us consider Situation 2 to deal with the fuzzy decision-making problem. Assume that the ordered sequence of the opinion 5, is 1, the ordered sequence of the opinion

c2 is 3, and the ordered sequence of the opinion 5, is 2.

[Step 11 Because the universe of discourse of the three opinions 6, , e, ^{and 4} represented by interval-valued fuzzy numbers are between 0 and IO, respectively, we use formula (3) to translate G", , G"* and d ^{into the}

standardized interval-valued fuzzy numbers GI , 5; and

G;, respectively, shown as follows:

- - -

- -

- -

2 . -

I

cl* - ⁼ [(0.1,0.2,0.3,0.4; 0.8),(0.1,0.2,0.3,0.4; I.O)],

cl =[(0.4,0.5,0.6,0.7;0.7),(0.4,0.5,0.6,0.7;0.8)],

5; =[(0.6,0.7,0.8,0.9;0.7),(0.6,0.7,0.8,0.9;0.9)~.

- -

[Step 21 We can see that the three ordered pairs of the three linguistic opinions G : , - - - 6: and a; ^{are as} follows:

- - -

< I , e,*>, ^<3, e;>, ^0, e;>.

We can use the ordered inducing value U, to decide the ordered sequence ofthe pairs, shown in follows:

- -

4, p, <2, b;'>, ^<3, g;>.

- - - _ ^{- -}

where &'=a;, &+=Cl and &'=E;. Furthermore, we can get the ordered opinion vector B ⁼ [c ^{b;' 6 I.}

[Step 31 Based on formulas (5) and (7), we can get the absolute weight w, of the ordered opinion &*, shown as follows:

- - _ -

- 1 - -

3'

h the Same way, we can get the absolute weights - I and - 1 of the order$ linguistic opinions gi ^{and ,}

W ,

= 1. Thus, the weighted vector W =

-

3 - A

respeaively, and where

[Step 4) By applying formulas (8) and (9), we can obtain the aggregated interval-degree of confidence [ $& ,

1,

shown as follows:

= 0.7333,

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- - -

the three opinions a,’ ^, a; ^and 4, shown as follows:

I

TL(((a,. g:))= (0.3667, 0.4667, 0.5667, 0.6667; 0.7333),

+ ( ( U > ,

?))= (0.3667,0.4667, 0.5667,0.6667; 0.9)

Therefore, we can obtain the fusion result F( u,.G, 1 - [(0.3667, 0.4667, 0.5667, 0.6667; 0.7333), (0.3667, 0.4667, 0.5667, 0.6667; 0.9)] of the three linguistic opinions e,, e,

and a, of the three evaluating criteria C1, C, and C,.

= ( =*) ^-

- - -

V. CONCLUSIONS

Io this paper, we have presented a new fuzzy information fusion method based on interval-valued fuuy numbers for handling the multi-criteria fuzzy decision-making problems. ?he proposed information h i o n method can handle the multi-criteria fuzzy decision-making problems in a more intelligent and more flexible manner due to the fact that the proposed method not only uses interval-valued fuzzy numbers to represent decision-makers’ linguistic opinions, but also uses the linguistic quantifiers based on the FN-IOWA operator to aggregate the decision-makers’ linguistic opinions in different linguistic constraints.

ACKNOWLEDGEMENTS

This work was supported in part by the National Science Council, Republic of China, under Grant NSC 92-22 13-E-01 1-074.

D l

PI

131

[41

15 I

[61

I71

830

REFERENCES

G I. Adamapoulos and C. P. Pappis, “A fuzzy-linguistic approach

a mullisriteria sequencing problem” Eumpem J o u m l of Operaham1 Research, vol. 92, no. 8, pp. 628-636, 1996.

F. T. S . C h , M H. C h q and N. K. H. Tang ‘Evaluating altemative praduaion cycles wing the extended limy AHP ndmd,“ Eumperm J ~ - l ~ O p e ~ m i ~ m l R , ~ ~ l . 100,no. 2,pp. 351-365.1997.

S . H. Chen, “Operations on fuzzy numbers with function principal,”

Tamkong J o u m l of Manogrment Sciences. vol. 6, no. 1. pp. 13-25.

1985.

S. H. Chen, “Ranking generalized fuzzy number with graded mean integration,” Proceedings of the Eighth lntemotionnl Fuzzy Systems A S S O C ~ U I ~ O ~ World Congress. vol. 2, Taipei. Taiwan, Republic of China. 1999, pp. 899-902.

S . J. Chen and S . M. Chen, “A new method for handling multi-criteria fuzzy decision making problems using M-IOWA operators,”

Cybernetics and Systems: An Inremotional J o u m l , vol. 34. no. _{2, pp.}

109-137,2003.

S . I. Chen and S . M. Chen. “A new method for handling the fuzzy ranking and the defuvification problems.” Proceedings ofthe Ezghth Narronal Conference on F c y Theory and Its Applications. Taipei, Taiwan, Republic of China, 2000.

S . J. Chen and S . M. Chen, “Aggregating fuuy opinions in the

heterogeneous group decision- making environment,” Accepted and to appear in Cybernetics andSvstems: An Internoliono1 Journal, vol.

35,2004.

N. Chiadamrang, “An integrated fuzzy multi-criteria decision making method for manufachlring strategies selection.” Computers and Industrial Engmeering, vol. 37, no. 1-2, pp. 433436, 1999.

H. D a g , ‘hulti-criteria analysis with furzy paiwise comparison,”

lnremalionol Joumal of Approximate Reasoning, vol. 21, no. 3, pp.

215-231. 1999.

G. B. Devedzic and E. Pap, “multi-criteria- multistages linguistic evaluation and ranking of machine tools.“ Fuzzy Sets and Systems, vol. 102, no. 3,pp. 451461. 1999.

i\. Gonraler, 0. Pons, and M. A. Vila, “Dealing with uncertainty and imprecision by means of fuzzy numbers,” Inlemotional Journal of 4pproximate Reosonig, vol. 21, no. 2. pp. 233-256, 1999.

M. B. Gomlcrany, “A method of inference in approximate reasoning bared on interval-valued fuzzy sets,” Fuzzy Sels ondSysrems, vol. 21, no. 1,pp. 1-17, 1987.

E Herren, E. H. Viedma, and J. L. Verdegay, ”A linguistic decision process in group decision making,” Group Decision and Negotmtion.

vol. 5 , no. 2, pp. 165.176, 1996.

D. H. Hang and S . Lee, “Some algebraic properties and a distance . . measure far interval-valued f u u y numbers.” Informorion Scienter, . . vol. 148, no. I . pp. 1-10,2002.

E. E. Karsak and E. Tolga. “Fuuy multi-criteria decision-making procedure for evaluating advanced manufacturing system investments,” Internalional Journal oft‘roduction Economics, vol. 69.

no. I, pp. 49-64.2001.

H. S . Lee, “An optimal aggregation method for fuzzy opinions of group decision,” Proceedings of the 1999 IEEE Intermtiom1 Conference on Systems, Mon. and Cvbemetics, vol. 3, Tokyo. Japan, pp. 314-319, 1999.

S. J. Lee, S . 1. L i m and B. S . Ahn, “Service restoration of primary distribution systems based on fuzzy evaluation of multkriteria.” IEEE rranroctionsonPowerSvslems. vol. 13,no. 3, pp. IlS6-1163. 1998.

G S . Liang, “Fuzzy MCDM based on ideal and anti-ideal concepts.”

European J o u m l of Operalimd Research, vol. 112. no. 8. pp.

682491,1999.

T. W. Liao and B. Rouge, “A fuzzy mullisriteria decision making methad for material selection,” Joumnl of ~~mfocrurmg Systems, vol. 15. no. 1.

pp. 1-12. 1996.

R. Petmvic and D. Petrovic, “multi-criteria ranking of inventory replenishment policies in the presence of unceminly in customer demand,” Intemarionol Joumal ofProduclion Economics, vol. 7 1, no.

1-3, pp. 439446,2001.

R. R. Yager. “On ordered weighted averaging aggregation operaton in multi-criteria decision making,” IEEE Transactions on Systems, Man. andCvbernetics,vol. 18,no. I , pp. 183-190, 1988.

R. R. Yager , “A note on the representation ofquantified statements in terms of the implementation operation,” Annals of Operotiom Research. vol. 12, no. 3, pp. 297-314. 1988.

I. S . Yao and F. T. Lin, “Constmcting a funy flow-shop sequencing model based on statistical data,’’ Internotzonal Joumol of Approximate Reasoning, vol. 29, no. 3, pp. 215-234.2002.

L. A. Zadeh, “Acomputation approach

fuzzy quantifiers in nahlral languages,” Computers and Mathematics with Appltcalions. vol. 9, no.

2,pp. 149-184, 1983.