• 沒有找到結果。

Fuzzy credibility relation method for multiple criteria decision-making problems

N/A
N/A
Protected

Academic year: 2021

Share "Fuzzy credibility relation method for multiple criteria decision-making problems"

Copied!
13
0
0

加載中.... (立即查看全文)

全文

(1)

Intelligent Systems

Fuzzy Credibility Relation Method for Multiple Criteria Decision-Making Problems

HSI-MEI HSU

Institute of Industrial Engineering, National Chaio Tung University, 1001, Ta Shueh Road, Hsin Chu, 30050, Taiwan, Republic of China

and

CHEN-TUNG CHEN

Department of Industrial Engineering and Management, Nan Tai College, 1 Nan Tai Street, Yung Kang City, Tainan County, Taiwan, Republic of China

ABSTRACT

This paper deals with the problem of ranking alternatives under multiple criteria. A fuzzy credibility relation (FCR) method is proposed. Owing to vague concepts repre- sented in decision data, in this study the rating of each alternative and the weight of each criterion are expressed in fuzzy numbers. Then we define the concordance, discordance, and support indices. By aggregating the concordance index and support index, a fuzzy credibility relation is calculated to represent the intensity of the prefer- ences of one alternative over another. Finally, according to the fuzzy credibility relation, the ranking order of all alternatives can be determined. A numerical example is solved to highlight the procedure of the FCR method at the end of this paper. ©Elsevier Science Inc. 1997

INFORMATION SCIENCES 96, 79-91 (1997) © Elsevier Science Inc. 1997

(2)

80 H. M. HSU AND C. T. CHEN 1. I N T R O D U C T I O N

In general, a multiple criteria decision-making (MCDM) problem can be concisely expressed in matrix format as

D = C 1 C 2

I"

A1

Xll X12 A2 x21 x22

Am LXml Xm2

° . . ° . o ° ° ° C n Xln X2n Xmn (A) w = [ w , w2 ... wo],

where A1, A2,..., Am are possible alternatives, C1, C 2 . . . C n are criteria with which performances of alternatives are measured, xij is the rating of alternative A i with respect to criterion Cj, and wj is the weight of criterion Cj. For a crisp MCDM problem, the ratings x u of alternative Ai and the weights wj of the criteria are given as real numbers.

In general, the methods to solve the crisp MCDM problems can be classified into three categories: aggregation into a unique criterion, out- ranking methods, and interactive methods [8, 20]. The outranking method was initially suggested by Benayoun et al. [2] and was improved by Roy [16, 17]. However, this method ignores much vital information present in the concordance and discordance matrices and at times leads to wrong deci- sions [18]. Owing to the fact that an outranking relation is inherently fuzzy in nature, it is undesirable to determine the outranking relation with a crisp relation• Therefore, many authors [4, 13, 15, 19] considered the uncertainty and fuzziness of decision data and proposed different types of preference functions to deal with the strength of the outranking relation.

In the outranking methods mentioned above, the ratings x u and the weights wj are given by crisp numbers, but under many conditions, crisp data are inadequate to model real-life situations. In addition, in order to deal with the uncertainty and fuzziness of decision data, some outranking methods mentioned above defined some threshold values and functions to determine the strength of the outranking relations between alternatives. However, in fact using decision-makers makes it difficult to determine these threshold values and functions, and the final solution will be influ- enced by these threshold values; the resulting preference functions might be unacceptable or unrealistic in some applications [10, 20, 21].

(3)

F U Z Z Y C R E D I B I L I T Y R E L A T I O N M E T H O D 81 To consider the vague concepts expressed in decision data and avoid the final solution influenced by the threshold values, the use of fuzzy numbers is an adequate means to model uncertainty arising from imprecision in human behavior or incomplete knowledge about the external environment [1]. Therefore, in this study the ratings xij and the weights w i are given as trapezoidal fuzzy numbers. Then we retain all the vital information and develop a fuzzy creditability relation (FCR) method to avoid subjective determination of the threshold values.

The organization of this paper is as follows. First, we introduce the basic definitions and notations. Next we define the concordance and support indices to derive the fuzzy credibility relation, and propose a fuzzy credibil- ity relation (FCR) method to solve M C D M problems. Then the F C R method is illustrated with an example. Finally, we give some conclusion at the end of this paper.

2. DEFINITIONS A N D NOTATIONS

The basic definitions and notations that follow will be used through out the paper unless otherwise stated.

DEFINITION 2.1 [11]. If fi is a fuzzy number with membership function p~(x) and whose a-cut is denoted by h a = {x: ~ ( x ) >t a} = [n~', nu ~ ] for

n~2]ctn ~1 n~'] when a l < ~ a 2 Val, a2~[0,1], t~ a is a ~ [ 0 , 1 ] , then [n~':, u t t ,

the set of a-cuts and n 7 and n~ are the lower and upper bounds of the set of a-cuts, respectively.

DEFINITION 2.2 [5]. fi = (ni, n2, n3, n 4) denotes a trapezoidal fuzzy num- ber if its membership function /x~(x) is defined as

x ) =

0,

x < ~ n l , x - - n I l'l I < ~ x <~ n 2 , n 2 - - n 1 , 1, n 2 ~ x ~ n 3 , x - n 4 n 3 < ~ x < ~ n 4 , n 3 - - n 4 , O , x > ~ n 4 .

(1)

DEFINITION 2.3 [11]. If fi is a trapezoidal fuzzy number and n 7 > 0 for a ~ [0,1], then fi is called a positive trapezoidal fuzzy number (PTFN).

(4)

82 H . M . HSU A N D C. T. C H E N Given any two positive trapezoidal fuzzy numbers rh = (rnl, m2, m3, m 4) and h = (nl, n 2, n3, n4) , and a real number r > 0, we know that [11]

th( + )t~ = ( m I + n i , m 2 + n z , m 3 + n 3 , m 4 + n 4 ) , (2) if't( - ) n --- ( m 1 - n 4 , m 2 - n 3 , m 3 - n 2 , m 4 - n l ) , (3) t h ( . ) n = ( m 1.n 1, m 2 . n 2 , m 3 . n 3 , ma.n4) , (4) t h ( : ) h = ml m2 m3 , n 3 , n 2 , , (5) ( 1 1 1 1 ) ( i n ) - 1 _ ~ Brl4 ' m 3 ' m 2 ' m l , (6) r h ( . ) r = ( m 1 .r, m z . r , m 3.r, m a . r ) , (7) r h ( : ) r = ( ml m 2 m 3 m 4 ) r ' r ' r ' r " (8)

DEFINITION 2.4 [14]. If h is a trapezoidal fuzzy number, and n T' > 0 and ct<

n u .~ 1 for a ~ [0,1], then fi is called a normalized positive trapezoidal fuzzy number.

DEFINITION 2.5 [5]. ,~ is called a fuzzy matrix if there exists at least an entry in .i, is a fuzzy number.

3. C O N C O R D A N C E A N D D I S C O R D A N C E ANALYSIS

In this study, we consider the following decision matrix fl by modifying matrix D:

l

J~ll XI1 "'" 3~11

]

~ = "~21 222 "'" 3~2n ° . . X m l "~ra2 "'" fgrnn ~ / = [ W1 I'V2 "'" 14'n ],

where iij = ( a i j , bij , c i j , d i j ) , i = 1, 2 . . . m , j = 1, 2 . . . n , and ffj =

(5)

FUZZY CREDIBILITY RELATION METHOD 83

aij = bjj = cij =dij, then the rating iii is a crisp value. 6’ is a normalized fuzzy criterion weight vector which can be determined by Hsu and Chen’s method [9].

First, we use the linear scale transformation to transform the various criteria scales into a compcrable scale. We obtain the normalized fuzzy decision matrix denoted by R:

(9)

where B and C are the set of benefit criteria and COST criteria, respectively, and

dy = maxdij, if jEB,

i

a,: = min uij, if jEC.

i

The normalization method mentioned above is to preserve the property that the ranges of normalized fuzzy numbers belong to [0, 11.

Then we calculate the weighted normalized fuzzy decision matrix as

where ~ij=fij(‘)~j, i=l,Z ,..., m, j=1,2 ,.,., n.

After the construction of weighted normalized fuzzy decision matrix ?‘, the pairwise comparison of the preference relationships between the alternatives A, and A,, can be established as stated in the following section.

3.1. THE CONCORDANCE SET AND DISCORDANCE SET

The weighted normalized fuzzy ratings of A, and A, (g, h = 1,2,. . . , m

and g#h) in V are denoted as

tg=[6g,,,fig2,...,&l

and ij,,= K&h*,..., zThn], respectively. Then we compare the fuzzy number ~~j to

(6)

84 H . M . H S U A N D C. T. C H E N

(:hi"

If 17gj is larger than (or equal to) t3h/, we say alternative

Ag

is at least

as good a s Z h with respect to the jth criterion. In this way, we partition

the criteria J =

{jlj

= 1,2 .... , n} into concordance set

Cg h

and discordance

set

Dg h

designated as

Cgh={jlF;gi>_.gah/, j =

1 , 2 , . . . , n } (11)

and

Dgh={jlg;g j

< t3hj, j = 1,2 ... n}. (12)

Many authors [3, 6, 7, 12] have been devoted to the investigation with regard to the comparison of fuzzy numbers. One of the useful methods to compare fuzzy numbers was proposed by Lee and Li [12]. It is probably the most logical ranking method [10], which ranks fuzzy numbers based on the fuzzy mean and the fuzzy spread of the fuzzy numbers. In this paper, we use Lee and Li's method to compare the fuzzy numbers to determine the concordance set and the discordance set.

If the discordance set is empty, i.e.,

Dg h

= Q, then it indicates that Ag fully outranks

Ah;

otherwise, if the concordance set is empty, i.e.,

Cg h = Q,

then it indicates that Ag does not outrank Z h absolutely. Practically, the discordance set and the concordance set are usually not empty. In order to determine the degree of "Ag outranks

Zh,"

we must consider the concor- dance set and the discordance set simultaneously. With respect to criterion j in the concordance set, a larger distance between ~Tgj and t3hj indicates a higher concordance degree to say "Ag dominates Ah." In other words, with respect to criterion k in the discordance set, a larger distance between t3g k and Vhk indicates a higher discordance degree to say "Ag dominates Ah."

In order to determine the difference between fuzzy numbers, we use the dissemblance index for fuzzy numbers to calculate the distance between fuzzy numbers [11, 22]. The dissemblance index of ~g/and Vhj is expressed

a s

d(~gj,~;hj )

1 1

-,~-,~

= fo=oa(V,j,vh )

(13)

where A(t3g~, vh~) = IVg3t-

Vh~,l +

IVg~ --

Vh~,[, Vg3 = [Vg~" Vg~u]'

and Vh~ =

[V~/t, VT, j,,].

Referring to the E L E C T R E method [2, 10], we define the

(7)

F U Z Z Y C R E D I B I L I T Y R E L A T I O N M E T H O D criteria to represent the strength of Ag dominates A h as

85

Ej c,h a(0 j, 0hi)

Clgh

Ej ~ j a(0gj, 0hi)

(14)

Similarly, we also define the discordance index as

maxj ~ Og h d( ugj,

Ohj )

DIg h = . (15)

maxj ~ j d(Ogj, Ohj )

However, if we aggregate the discordance index and the concordance index directly with the same procedures as the E L E C T R E [10, 17] method, the E L E C T R E III [15] method, or Singh's m e t h o d [18], we will not distinguish the difference between two alternatives effectively in many cases. F o r example, in two cases we obtain (a) Clij = 0.8, Dlij = 1.0 and (b) Clji = 0.2, Dlji = 1.0. Intuitively, Aj is dominated by A i. However, if we follow the same procedure as in the E L E C T R E m e t h o d with concordance threshold ? = 0.8 and discordance threshold d = 0.2, then the outranking degree eij of A i over Aj is equal to zero and the outranking degree eji of A/ over A i is also equal to zero. Because the discordance indices are 1 in situations (a) and (b), the outranking degrees between alternative A i and Aj are also equal to zero when we adopt the E L E C T R E III method. According to the m e t h o d of Singh et al. [18] we obtain the following results:

(i) Eli. / = 0.8, d'ij = 1 - Dlij = 0, ely = min{Clij, d'il} = min(0.8,0} = 0. (ii) CIji = 0.2, d)t = 1 - DIj~ = 0, eji = min{CIji, d ) i / = min{0.2, 0} = 0. These methods mentioned above neglect the information provided by the values of concordance indices when the values of discordance indices are equal to 1. Therefore, the contribution of the difference values of concor- dance indices must be considered. Meanwhile, a higher discordance index indicates a lower outranking degree for one alternative over another. Thus, according to the discordance indices of each pair of alternatives, we transform the discordance index into a support index CIg h as

Dlhg

cI h -- . (16)

(8)

86 H. M. HSU AND C. T. CHEN With the transformation of the discordance index into the support index, the larger value o f f i g h represents the higher degree of "Ag outranks

h h . "

Combining the concordance and support indices, the credibility degree of "Ag outranks a h " c a n then be expressed as eg h ---min{Clsh, f I g h } . W e call E = [egh]mx m the fuzzy credibility relation matrix.

In the example mentioned above, we obtain the following results: (i) Cliy = 0.8, CI~ = 0.5, ei. i = min{0.8,0.5} = 0.5.

(ii) Clji = 0.2, CIj* = 0.5, e)i = min{0.2,0.5} = 0.2.

It means that the credibility value of "A i outranks A / ' is higher than the credibility value of "Aj outranks

hi."

After constructing the fuzzy credibility relation matrix, a ranking proce- dure is developed to determine the ranking order of each alternative.

3.2. RANKING PROCEDURE

According to the fuzzy credibility relation matrix E, the fuzzy strict credibility relation matrix can be defined as

where

ES=[eiSj]mxm, (17)

eiy--eji , when eij ~eji ,

t Z E s ( A i ' A j ) = e i S j = O, otherwise. (18)

The value of eTj indicates the degree of strict dominance of alternative A i over alternative Aj. Then, using the fuzzy strict credibility relation matrix

ES=[eiSj]m×m, the nondominated degree of each alternative A i ( i = 1,2,..., m) can be defined as

tLND(Ai) = min { 1 - 1 x E , ( A ~ , A i ) } = I - max lzE,(Aj,Ai), (19)

Aje£t A jell

A]~Ai Aj:~A i

where 1) = {A1, A 2 ... Am}.

A large value of I~ND(Ai) indicates that the alternative A i has a higher nondominated degree than others. Then we can use the /xND(A/) values to

(9)

F U Z Z Y CREDIBILITY RELATION METHOD 87 rank a set of alternatives. The ranking procedure is described as follows:

Step 1. Set K = 0 and f~ = {A1, A 2 ... Am}.

Step 2. Select the alternatives which have the highest nondominated degree, say h h , a h = max/{/~ND(Ai)}. The ranking for A h is r ( h h) = K + 1. Step 3. Delete the alternatives A h from ~'~, that is, f ~ = ~ \ A h. The corresponding row and column of A h are deleted from the fuzzy strict credibility relation matrix.

Step 4. Recalculate the nondominated degree for each alternative Ai, h i E ~ . If l~ = O, then stop. Otherwise, set K = K + 1 and return to step 2.

4. NUMERICAL EXAMPLE

A hypothetical example is designed to demonstrate the computational process of this fuzzy credibility relation (FCR) method. Suppose that a manufacturing company desires to select a suitable city for establishing a new factory. After preliminary screening, three candidates A1, A2, and A 3 remain for further evaluation. The company considers five criteria to select the most suitable candidate:

(1) land cost (C 1)

(2) transportation distance (C 2) (3) numbers of satellite factory (C 3) (4) human resource (C 4)

(5) the flexibility of government policy (C 5)

The benefit and cost criteria sets are B={3,4,5} and C={1,2}, respec- tively.

Now we apply the fuzzy credibility relation (FCR) method to solve this problem. The computational procedure is summarized as follows:

Step 1. The fuzzy decision matrix and the normalized fuzzy weight of each criterion are given as

b___

4.5 55 100 (3,5,6,7) (4,5,6,8) } 5.5 70 60 (6,7,8,9) (4,4,5,5,7) , 6.0 60 120 (4,5,6,7) (6,7,8,9) ~, = (0.5,0.6,0.8,1.0), ~'2 = (0.35,0.5,0.6,0.75), if'3 = (0.35,0.4, 0.55,0.7),

(10)

88 H . M . HSU AND C. T. CHEN

1~ 4 = (0.4,0.55,0.7,0.8)

if5 = (0.4, 0.5,0.6, 0.7).

Step

2. The normalized fuzzy decision matrix is calculated as

R =

1.0 1.0 0.83 (0.33,0.56,0.67,0.78) (0.44,0.56,0.67,0.89)]

0.82 0.79 0.5 (0.67,0.78,0.89,1.0) (0.44,0.5,0.56,0.78).

[0.75 0.92 1.0 (0.44,0.56,0.67,0.78) (0.67,0.78,0.89,1.0)

Step

3. The weighted normalized fuzzy decision matrix is calculated as

= (0.5,0.6,0.8,1.0) (0.41,0.49,0.66,0.82) (0.38,0.45,0.60,0.75) (0.35,0.5,0.6,0.75) (0.28,0.4,0.47,0.59) (0.32,0.46,0.55,0.69) (0.29,0.33,0.46,0.58) (0.12,0.2,0.28,0.56) (0.35,0.4,0.55,0.7) (0.13, 0.31,0.47, 0.62) (0.27,0.43,0.62,0.8) (0.18,0.31,0.47,0.62) (0.18,0.28,0.4,0.62) } (0.18,0.25,0.34,0.55) . (0.27,0.39,0.53,0.7)

Step

4. The concordance and discordance sets are determined, respec- tively, as

C12 = {1,2,3,5}, C13 = {1,2}, C21 = {4,5},

Cz3

= {1,4}, C31 = {3,4,5}, C32 = {2,3,5}, DIE = {4}, D13 = {3,4,5},

D21 = {1,2,3}, 023 = {2,3,5}, D31 = {1,2}, D32 = {1,4}.

Step

5. The distances of each pair of alternatives with respect to each criterion are computed as shown in Table 1. According to the results of Table 1, three matrices CI, DI, and CI* defined in (14), (15), and (16), respectively, can be shown as

C I =

- - 0.74 0.53]

0.26 - - 0.31 ,

(11)

F U Z Z Y C R E D I B I L I T Y R E L A T I O N M E T H O D TABLE 1

The Distance Measurements

89

Distance C 1 C 2 C 3 C 4 C 5 Max Sum

(A1, A2) 0.13 0.12 0.13 0.15 0.04 0.15 0.57 (A1, A 2) 0.18 0.05 0.09 0.01 0.10 0.18 0.43 (A2, A 3) 0.05 0.07 0.21 0.14 0.14 0.21 0.61 D I = - - 1.0 0.56 ] 0.87 - - 1.0 , [ 1.0 0.67 - - - - 0.47 0.64] C I * = 0.53 - - 0.40 • [0.36 0.60 - -

In this case, using the method of Singh et al. [18], we obtain the outranking degrees e12 - 0 and e23 = 0. However, the concordance degree for alterna- tive A1 over A 2 (CI~2 =0.74) is larger than the concordance degree for alternative A 2 over A 3 (CI23 = 0.31). Intuitively considering concordance and discordance indices simultaneously, the credibility degree el2 should be larger than e23. Thus, this method shows an unacceptable result.

Step

6. Construct the fuzzy credibility relation matrix as

E = 0.26 - - [0.36 0.60

Step

7. Construct the fuzzy strict credibility relation matrix as 0 0.21 0.171

E ' = - - 0 .

0.29 - -

Step

8. Compute the nondominated degree of each alternative Ai ( i = 1,2,3) as

~ N D ( A 1 ) = 1.0, /zND(A2) =0.71, /.t, ND ( A 3) =0.83.

Step

9. The alternative A 1 has the highest nondominated degree and set r(A 1) = 1.

(12)

90 H. M. HSU AND C. T. C H E N

Step 10. Delete the alternative A 1 from the fuzzy strict credibility

relation matrix.

Step 11. After deleting the alternative A1, the new fuzzy strict credibil-

ity relation matrix is

E s = [

0 9- 0]

The nondominated degrees of alternatives A 2 and A 3 are 0.71 and 1.0, respectively. Therefore, r(A 3) = 2 and r(A 2) = 3.

5. CONCLUSION

In general, multicriteria problems adhere to uncertain and imprecise data, and fuzzy set theory is adequate to deal with it. In this paper, a fuzzy credibility relation (FCR) method based on the fuzzy ratings and fuzzy weights is proposed to solve fuzzy M C D M problems. Decision-makers are difficult to determine the threshold values. Meantime, the final solution is often influenced by the threshold values. Therefore, the F C R method considers fuzzy assessment data instead of threshold values to model the uncertainty arising from imprecision in human behavior or incomplete knowledge about the external environment.

In this paper, a support index is determined by transformation of the discordance index, which indicates the outranking degree for one alterna- tive over another from the viewpoint of the discordance set. By aggregating the concordance and support indices, denoted by credibility degree, the intensity of the preferences of one alternative over another is effectively represented. Through constructing the fuzzy credibility relation matrix, a systematic and objective procedure is proposed to rank a finite set of alternatives in the F C R method. This method provides a stepwise way to produce the ranking order of each alternative.

The framework of F C R provided in this paper can be easily extended to the analysis of other problems such as project management, selection of a site for an industry, and many other areas of management decision problems.

R E F E R E N C E S

1. R. E. Bellman and L. A. Zadeh, Decision-making in a fuzzy environment. Manage.

Sci. 17(4):141-164 (1970).

2. R. Benayoun, B. Roy, and B. Sussman, ELECTRE: Une methode pour diuder le choix en presence de points rue multiples, Direction Scientifique, Note de Travail 49, Sema (Metra International), Paris, 1966.

(13)

F U Z Z Y C R E D I B I L I T Y R E L A T I O N M E T H O D 91 3. G. Bortolan and R. Degani, A review of some methods for ranking fuzzy subsets.

Fuzzy Sets and Syst. 15:1-19 (1985).

4. J. P. Brans, B. Mareshal, and Ph. Vincke, Promethee: A new family of outranking methods in multicriteria analysis. In J. P. Brans (Ed.), Operational Research '84,

North-Holland, Amsterdam, 1984, pp. 477-490.

5. J. J. Buckley, Fuzzy hierarchical analysis. Fuzzy Sets and Syst. 17:233-247 (1985).

6. S. J. Chen and C. L. Hwang, Fuzzy Multiple Attribute Decision Making: Methods and Applications, Springer-Verlag, Berlin, 1992.

7. D. Dubois and H. Prade, Ranking fuzzy numbers in the setting of possibility theory,

Inform. Sci. 30:183-224 (1983).

8. J. S. Dyer, P. C. Fishburn, R. E. Steuer, J. Wallenius, and S. Zionts, Multiple criteria decision making, Multiattribute utility theory: The next ten years, Manage. Sci.

38(5):645-654 (1992).

9. H. M. Hsu and C. T. Chen, Fuzzy hierarchical weight analysis model for multicrite- ria decision problem, J. Chinese Inst. Industrial Engrg. 11(3):129-136 (1994).

10. C. L. Hwang and K. Yoon, Multiple Attributes Decision Making Methods and Applications, Springer-Verlag, Berlin, (1981).

11. A. Kauffman and M. M. Gupta, Introduction to Fuzzy Arithmetic: Theory and Applications, Van Nostrand Reinhold, New York, 1985.

12. E. S. Lee and R. J. Li, Comparison of fuzzy numbers based on the probability measure of fuzzy events, Comput. Math. Appl. 15(10):887-896 (1988).

13. J. M. Martel and G. R. D'Avignon, A fuzzy outranking relation in multicriteria decision making, Eur. J. Oper. Res. 25:258-271 (1986).

14. D. S. Negri, Fuzzy analysis and optimization, Ph.D. Thesis, Dept. Industrial Engi- neering, Kansas State University, 1989.

15. A. Ostanello, Outranking methods, in: G. Fandel and J. Spron (Eds.), Multiple Criteria Decision Methods and Applications, Springer-Verlag, New York, 1985, pp.

41-60.

16. B. Roy, Partial preference analysis and decision-aid: The fuzzy outranking relation concept, in: D. E. Bell, R. L. Keeney, and H. Raiffa (Eds.), Conflicting Objectiues in Decision, Wiley, New York, 1977, pp. 40-75.

17. B. Roy, Problems and methods with multiple objective function, Math. Program.

1(2):239-266 (1971).

18. D. Singh, J. R. Rao, and S. S. Alam, Partial preference structure with fuzzy relations for MCDM problems, Int. J. Syst. Sci. 20(12):2387-2394 (1989).

19. J. L. Siskos, J. Lodrard, and J. Lombard, A multicriteria decision making methodol- ogy under fuzziness: Application to the evaluation of radiological protection in nuclear power plants, in: H. J. Zimmermann (Ed.), TIMS/Studies in the Management Sciences, Elsevier Science Publishers, New York, 1984, Vol. 20, pp. 261-283.

20. J. Teghem, Jr., C. Delhaye, and P. L. Kunsch, An interactive decision support systems (IDSS) for multicriteria decision aid, Math. Comput. Modelling

12(101):1311-1320 (1989).

21. R. Vetschera, An interactive outranking system for multi-attribute decision making,

Comput. Oper. Res. 15(4):311-322 (1988).

22. R. Zwick, E. Carlstein, and D. V. Budescu, Measures of similarity among fuzzy concepts: A comparative analysis, Int. J. Approximate Reasoning 1:221-242 (1987). Received 28 November 1994; reeised 12 February 1996

參考文獻

相關文件

Teacher starts the lesson with above question and explains to students that making business decision is one of the basic functions of a

For the data sets used in this thesis we find that F-score performs well when the number of features is large, and for small data the two methods using the gradient of the

Biases in Pricing Continuously Monitored Options with Monte Carlo (continued).. • If all of the sampled prices are below the barrier, this sample path pays max(S(t n ) −

Secondly then propose a Fuzzy ISM method to taking account the Fuzzy linguistic consideration to fit in with real complicated situation, and then compare difference of the order of

In order to improve the aforementioned problems, this research proposes a conceptual cost estimation method that integrates a neuro-fuzzy system with the Principal Items

Theory of Project Advancement(TOPA) is one of those theories that consider the above-mentioned decision making processes and is new and continued to develop. For this reason,

【Keywords】Life-City; Multiple attribute decision analysis (MADA); Fuzzy Delphi method (FDM); Fuzzy extented analytic hierarchy process

[2] Baba N., Inoue N., Asakawa H., Utilization of neural networks and GAs for constructing reliable decision support systems to deal stocks, IJCNN 2000 Proceedings of