Soft Comput (2006) 10: 94–103 DOI 10.1007/s00500-004-0429-9
O R I GI N A L P A P E R
Jing-Rong Chang Æ Ching-Hsue Cheng Æ Chen-Yi Kuo
Conceptual procedure for ranking fuzzy numbers based on adaptive two-dimensions dominance
Published online: 7 April 2005 Springer-Verlag 2005
Abstract Many methods for ranking of fuzzy numbers have been proposed. However, these methods just can apply to rank some types of fuzzy numbers (i.e. normal, non-normal, positive, and negative fuzzy numbers), and many ranking cases can just rank by their graphs intuitively. So, it is important to use proper methods in the right condition. In this paper, a conceptual procedure is proposed to describe how to use intuitive ranking and some technical ranking methods properly. We also introduce a new ranking fuzzy numbers approach that can adjust experts’ confidence and optimistic index of decision maker using two parameters (a and b) to handle the problems and find the best solutions. After illustrate many numerical examples following our conceptual procedure the ranking results are validity. Keywords Fuzzy number Æ Ranking procedure Æ Ranking fuzzy number method Æ Intuitively method Æ Decision making
1 Introduction The concept of fuzzy numbers was presented by Jain (1978), and Dubois and Prade (1978). In many multiple criteria decision making situations, the values for the qualitative criteria are often imprecisely defined for the
J.-R. Chang Æ C.-H. Cheng Æ C.-Y. Kuo Department of Information Management, National Yunlin University of Science and Technology, 123 University Road, Section 3, Touliu, Yunlin, 640, Taiwan J.-R. Chang (&) Graduate School of Management, National Yunlin University of Science and Technology, 123 University Road, Section 3, Touliu, Yunlin, 640, Taiwan E-mail:
[email protected] Tel.: +886-5-5342601 Fax: +886-5-5312077
decision-makers (DMs) and the final scores of alternatives are represented in terms of fuzzy numbers. In order to choose a best alternative, we need to build a crisp total ordering from fuzzy numbers. In recent years, many methods are proposed for ranking different types of fuzzy numbers (Chen and Lu 2001; Chen and Cheng 2004; Chu and Tsao 2002; Lee et al. 2004; Modarres and Soheil 2001; Tang 2003; Tran and Duckstein 2002; Wang and Kerre 2001a, 2001b), and can be classified into four major classes: preference relation, fuzzy mean and spread, fuzzy scoring, and linguistic expression, but each method appears to have advantages as well as disadvantages (Chen and Hwang 1992). However, some ranking methods assume that the membership function is normal, but in many cases, limitation to the normal membership function is not adequate. Besides, some are counterintuitive, not discriminating or complex; and most of them consider only on point of view on comparing fuzzy quantities. Further, many of them produce different rankings for the same problem. Yager’s x index (Yager 1980b) measures the general mean of fuzzy numbers, but the x index alone provides very poor discrimination ability. Murakami et al.’s (1983) x and y can be derived only when g(x)=x and gðxÞ ¼ 1=2 uA~ ðxÞ; so Murakami et al.’s method is not logically sound either. For overcoming those problems, Cheng (1998) proposed a distance method to rank fuzzy numbers based on calculating both x and y values. However, Cheng’s method can not rank the negative fuzzy number. Recently, Chen and Cheng (2004) proposed a more easy formula by metric distance to deal with positive and negative, symmetry and nonsymmetry, triangular/trapezoidal, general and normal fuzzy numbers at the same time. Lee and Li’s (1998) method uses the generalized mean and standard deviation to rank fuzzy numbers based on two kinds of probability distributions (uniform distribution and proportional distribution). This method ranks fuzzy numbers based on two different criteria, namely, the fuzzy mean and the fuzzy spread of the
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fuzzy numbers, and the fuzzy number ranked higher when it both with higher mean value and at the same time lower spread. However, when higher mean value and higher spread or lower mean value at the same time with lower spread, it is not easy to compare the ordering clearly by this method. In spite of these many proposed methods, the results may not satisfy decision makers. In daily life, we will choose alternatives according to the experts; however, different people have different weights of experts’ opinion. For instance, some people consider the expert’s confidence is very important to them, others do not. Furthermore, we must take care of people who are optimistic or pessimistic, which will lead to complete different results. For above reasons, a conceptual procedure is proposed in this paper to describe those methods’ using condition to ensure those methods can rank fuzzy numbers effectively. We also introduce a new ranking fuzzy numbers approach that can adjust experts’ confidence and optimistic index of decision maker using two parameters (a and b) to handle the problems and find the best solutions. This new approach for ranking fuzzy numbers cannot only compute more quickly and correctly but also identify the normal and non-normal fuzzy numbers. According to weights of experts’ confidence and optimistic index, DMs can determine which alternative to be adopted. We also compare with other methods using many examples to validate and verify our method. This paper is organized in six sections. Following this introduction, Sect. 2 presents a brief overview of fuzzy number and some of the ranking fuzzy numbers methods. Section 3 proposes a new ranking fuzzy numbers method based on adaptive two-dimensions dominance. The conceptual procedure for ranking fuzzy numbers was presented in Sect. 4. In Sect. 5, we illustrate many numerical examples to validate this conceptual procedure. Finally, we give some conclusion at the end of this paper.
2 Preliminary 2.1 Fuzzy numbers A fuzzy number is a fuzzy subset in support R (real number) which is both ‘‘normal’’ and ‘‘convex’’, where ~ ¼ x 2 R u ~ > 0 (Cheng 1998). supp supp A A Normality implies that: 9x 2 R; [x uA~ ðxÞ ¼ 1; that ~ in R is 1. is, the maximum value of the fuzzy set A Therefore, the non-normal fuzzy number is: 8x 2 R; Maxx uA~ ðxÞ \1: For convenience, the trapezoid fuzzy number can be denoted by [a, b, c, d; 1], if b=c mean this fuzzy ~ is a trinumber is a degenerative trapezoid means A angular fuzzy number, the membership function fA~ of ~ ¼ ½a; b; c; d; 1 can be the trapezoid fuzzy number A expressed as
8 f L ðxÞ; > > < A~ 1; fA~ ¼ f R ðxÞ; > > : A~ 0;
a x b; b x c; c x d; otherwise;
ð1Þ
where fA~L : ½a; b ! ½0; 1 and fA~R : ½c; d ! ½0; 1: Since fA~L : ½a; b ! ½0; 1 is continuous and strictly increasing, the inverse function of fA~L exists. Similarly, fA~R : ½c; d ! ½0; 1 is continuous and strictly decreasing, the inverse function of fA~R also exists. The inverse functions of fA~L and fA~R are denoted by gLA~ L and gR ~ ; respectively. Since fA ~ : ½a; b ! ½0; 1 is conA tinuous and strictly increasing, gLA~ : ½0; 1 ! ½a; b is also continuous and strictly increasing. Similarly, if fA~R : ½c; d ! ½0; 1 is continuous and strictly decreasing, then gR ~ : ½0; 1 ! ½c; d is continuous and strictly A decreasing; gLA~ and gR ~ are continuous on a closed A interval R 1 are integrable on [0, 1], that is, R 1 [0, 1] and they both 0 gLA~ ðyÞ dy and 0 gR ~ ðyÞ dy exist. A 2.2 Ranking fuzzy number methods In practical use, ranking fuzzy numbers is very important (Hong and Lee 2002; Iskader 2002; Lin 2002). For example, the concept of optimum or best choice to come true is completely based on ranking or comparison. Therefore, how to set the rank of fuzzy numbers has been one of the main problems. To resolve the task of comparing fuzzy numbers, many authors have proposed fuzzy ranking methods which yield a totally ordered set or ranking. These methods range from the trivial to the complex, from including one fuzzy number attribute to including many fuzzy number attributes. A review and comparison of these existing methods can be found in (Cheng and Hwang (1989), Lee and Li (1988) and Zimmermann (1987)). Those ranking methods are classified into four major classes according to Chen and Hwang (1992), which are listed as follows: 1. Preference relation a. Degree of optimality (as in (Bass and Kwakernann (1977), Baldwin and Guild (1979) and Watson et al. (1979))). b. Hamming distance (as in (Kerre (1982), Kolodziejezyk (1986) and Yager (1980a))). c. a-cut (as in (Adamo (1980), Buckely and Channas (1989) and Mabuchi (1988))). d. Comparison function (as in (Delgado et al. (1988), Dubois and Parde (1983) and Tsukamato et al. (1983))). 2. Fuzzy mean and spread a. Probability distribution (as in (Delgoda et al. (1988), Dubais and Prade (1983) and Tsukamoto et al. (1983))).
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3. Fuzzy scoring
2.2.2 Lee and Li’s method
a. Proportion to optimal (as in (Lee and Li (1988))). b. Left/right scores (as in (Chen (1985), Chen and Hwang (unpublished paper) and Jain (1976, 1978))). c. Centroid index (as in (Murakami et al. (1983) and Yager (1980b))). d. Area measurement (as in (Liau and Wang (1992) and Yager (1981))). 4. Linguistic expression a. Intuition (as in (Efstathiou and Tong (1980))). b. Linguistic approximation (as in (Tong and Bonissone (1984))). 2.2.1 Ranking fuzzy number by distance method Method for ranking fuzzy number by distance is based on calculating both center ~x and ~y values. For example, Cheng (1998) proposed a distance method to rank fuzzy numbers based on calculating centriod-index values. For ~ ¼ ½a; b; c; d; 1; its mema trapezoidal fuzzy number A bership function fA~ is given by 8 xa ; a x b; > > < ba 1; b x c; fA~ ¼ xd ð2Þ ; c x d; > > : cd 0; otherwise; ~ is The centroid point ð~x0 ; ~y0 Þ for a fuzzy number A defined as follows: Rc R d R R b L dx þ xf x dx þ ~ ~ dx b c xfA ~ ¼ aR A ~x0 ðAÞ ; ð3Þ R R b L dx þ c dx þ d f R dx f ~ ~ a b c A A R 1 R L dy þ yg ~ ~ dy 0 ygA ~ ¼ R0 A ~y0 ðAÞ : R 1 L dy þ 1 gR dy ~ ~ 0 gA 0 A
Lee and Li (1988) proposed the use of generalized mean and standard deviation based on the probability measures of fuzzy events to rank fuzzy numbers. The method ranks fuzzy numbers based on two different criteria, namely the fuzzy mean and the fuzzy spread of the fuzzy numbers. They assume two kinds of probability distributions for fuzzy events and derive corresponding indices as follows: ~ ¼ 1=jAj ~ and A ~ 2 U : Its 1. Uniform distribution: f A ~ and standard deviation rU A ~ are mean ~xU A defined as R ~ ðxÞdx ~ xlA ~ ¼ RSðAÞ ~xU A ; ð8Þ ~ ðxÞdx ~ lA SðAÞ " R ~ ¼ rU ðAÞ
~ SðAÞ
R
x2 lA~ ðxÞdx
~ SðAÞ
#1=2
!
lA~ ðxÞdx
~ ð~xU ðAÞÞ
2
;
ð9Þ
~ is the support of fuzzy number A: ~ where S A ~ ~ 2 P; 2. Proportional distribution: f A ¼ klA~ ðxÞ; A where k is the proportional constant. Then, we have R 2 ~ ðxÞdx ~ Þ x lA S ðA ~ ~xP A ¼ R ð10Þ 2 ~ ðxÞ dx ~ Þ lA S ðA 20R 31=2 2 1 2 x l ðxÞ dx ~ ~ A ~ ¼ 4@ RS ðAÞ ~ 25 : rP A 2 A ~xP A ~ ðxÞ dx ~ Þ lA S ðA
ð11Þ
the higher value of Eqs. 8 and 10 means the higher fuzzy number.
R1
ð4Þ
Clearly, the inverse functions of fA~L and fA~R are L gA~ ¼ a þ ðb aÞy and gR ~ ¼ d þ ðc dÞy: Then, A R b xa R d xd Rc w x x dx þ w dx þ w dx a b c x ba ~ ¼ ~x0 ðAÞ Rc R d xd cd R b xa w a ba dx þ w b dx þ w c cd dx R d R R b L Rc ~ dx þ b x dx þ c xfA ~ dx a xfA ; ð5Þ ¼ Rb R R c d L dx þ R dx f dx þ f ~ ~ a b c A A hR
i
R1 ðyÞgLA~ ðyÞdy þ 0 ðyÞgR ~ ðyÞdy A ~ ¼ ~y0 ðAÞ : ð6Þ R1 L R1 R ~ ðyÞdy þ 0 gA ~ ðyÞdy 0 gA ~ of fuzzy number A ~ is The ranking index R A qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ~ ¼ ð~x0 Þ2 þð~y0 Þ2 ; R A ð7Þ ~ is, the better the ranking of A: ~ The larger the value of R A w
1 0
2.2.3 Chen and Lu’s method Chen and Lu (2001) proposed an approximate approach for ranking fuzzy numbers based on the left and right dominance. It only requires a few left and right spreads at some a-levels of fuzzy numbers to determine the respective dominance. However, this method cannot apply in ranking the non-normal fuzzy numbers. Chen and Lu define the lower and upper limit of the ~ i as kth a-cut for the fuzzy number A n o ð12Þ li;k ¼ inf xjlA~ i ðxÞ ak ; x2R
n o ri;k ¼ sup xjlA~ i ðxÞ ak ;
ð13Þ
x2R
where li,k and ri,k are left and right spreads, respectively. L ~ ~ The left (right) dominance Di,j (DR i,j) of Ai over Aj is defined as: DLi;j ¼
n 1 X li;k lj;k ; n þ 1 k¼0
ð14Þ
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DR i;j ¼
n 1 X ri;k rj;k ; n þ 1 k¼0
ð15Þ
where n+1 a-cut are used to calculate the dominance. ~ j with the index of ~ i over A Then, the total dominance of A optimism b 2 [0, 1] can be defined as the convex combination of DLi,j and DR i,j by " # n 1 X R L Di;j ðbÞ ¼ bDi;j þ ð1 bÞDi;j ¼ b ri;k rj;k n þ 1 k¼0 " # n 1 X þ ð1 bÞ li;k lj;k n þ 1 k¼0 (" # n n X X 1 b ri;k þ ð1 bÞ li;k ¼ nþ1 k¼0 k¼0 " #) n n X X ð16Þ rj;k þ ð1 bÞ lj;k : b k¼0
k¼0
maker can rank two fuzzy numbers A decision ~ j based on the following rules: ~ i and A A ~ i\A ~j; 1. if Di,j (b)0, then Ai > A
3 A new ranking fuzzy numbers method From Chen and Lu’s method (see Eq. 16), ranking fuzzy numbers must take the left and right spread (x-axis) into consideration (i.e. LS(i) and RS(i) in Eq. 17). However, their method cannot rank non-normal fuzzy number. For overcoming this problem, the proposed method also considers DMs’ confidence level (i.e. LH(i) and RH(i) in Eq. 17). This concept is consistent with Murakami et al.’s method (1983) (the vertical value (y-axis) was used an important index for ranking fuzzy numbers in Murakami et al.’s method). For above concepts, we will introduce a new method, which is named ‘‘new ranking fuzzy numbers method based on adaptive two-dimensions dominance’’. ‘‘Adaptive’’ of this method’s title means our method can be applied by DMs’ preference and perception. DMs can decide by their thought using two parameters, a and b. ~ i can be defined as A trapezoid fuzzy number A ~ Ai ¼ ða; b; c; d : LH ; RH Þ: When ranking triangular fuzzy number (i.e. b=c), it still can be computed by our proposed approach. The proposed method is summarized in Eq. 17 Di ¼
1 fa½bRH ðiÞ þ ð1 bÞLH ðiÞ 2 þð1 aÞ½bRS ðiÞ þ ð1 bÞLS ðiÞg;
where Di: the index of ith fuzzy number,
the weighting of experts’ opinion by DM assigned, b: the optimistic index of decision maker, RH (i): the right height of ith fuzzy number, LH (i): the left height of ith fuzzy number, RS (i): the spread of ith fuzzy number right RS A~ ¼ c þ d ; spread of ith fuzzy number LS (i): the left LS A~ ¼ a þ b : ~i: The larger the value of Di is, the better the ranking of A a is the weighting of experts’ opinion by DM assigned, that means, the larger a is, the heavier weigh of experts’ opinion is. In other words, the smaller a is, the less y-axis influences the results. b is the optimistic index of DM, which determines the weightings of left and right spread on the x-axis. That is to say, the larger b is, the more right spread influence the results; the smaller b is, the more left spread influence the results. The algorithm for ranking fuzzy numbers is as follows: a:
Step 1: From given fuzzy numbers, we get the values of RH, LH, RS and LS of the fuzzy numbers, and identify the fuzzy numbers that are normal or nonnormal. Step 2: If a fuzzy number is normal, DMs just consider the index of the optimistic (i.e. a=0); otherwise, DMs must consider the weighting of experts’ confidence level in addition (i.e. a „ 0 and given b’s value). Step 3: Substitute a, b to Eq. 17, then, we will get the Di of each fuzzy number. Consequently, the fuzzy numbers can be ranked by their indices (Dis). Example 3.1 Assume the five fuzzy numbers are to be ranked; they are two triangle fuzzy ~ 1 ¼ ð3; 5; 7; 1Þ; A ~ 2 ¼ ð3; 5; 7; 0:8Þ and three numbers A ~ 1 ¼ ð5; 7; 9; 10; 1Þ; B ~2 ¼ trapezoid fuzzy numbers B ~ 3 ¼ ð7; 8; 9; 10; 0:4Þ (see Fig. 1, ð6; 7; 9; 10; 0:6Þ; B Cheng 1998). By our proposed method, if the weighting of expert’s opinion is very low and DM’s optimistic index is very high, i.e. a=0.1 and b=0.9.
ð17Þ Fig. 1 Five normal/non-normal and triangular/trapezoidal fuzzy ~1; A ~2; B ~1; B ~ 2 and B ~3: numbers A
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According Eq. 17, the indices (Dis) of the fuzzy numbers are 5.27, 5.26, 8.285, 8.31, and 8.39. ~3 > B ~2 > B ~1 The order of these fuzzy numbers is B ~1 > A ~2: >A Example 3.2 Similarly, as Example 3.1, let a=0.5 and b=0.5. By computing the values of the fuzzy numbers, the indices (Dis) of the fuzzy numbers are 2.75, 2.7, 4.125, 4.15, and 4.35. Therefore, we get the order ~1 > A ~ 2 . The results of above two ~2 > B ~1 > A ~3 > B as B examples are same with Cheng (1998).
4 The conceptual procedure for ranking fuzzy number In fuzzy multiple criteria decision making (MCDM) problem, many fuzzy numbers can intuitively rank their order by drawing graphs. If their ordering cannot be ranked by graphs, we can use many other methods to rank fuzzy numbers. Generally, we can calculate the x and y value of each fuzzy number then use the x value as the most important index, the y value on the vertical axis as an aid index for ranking fuzzy numbers (as Yager’s x index (1980b)). But, in special cases (such as all x values are equal or left and right spreads are same for all fuzzy numbers), the y value is an important index for ranking fuzzy numbers (as Murakami et al’s method (1983)). However, those two methods were either poor in its discrimination ability or in lacking of logical. So, Cheng (1998) proposed a distance method to rank fuzzy Fig. 2 The conceptual procedure for ranking fuzzy number
numbers based on calculating both x and y values to overcome Yager’s and Murakami et al’s problems. Lee and Li’s (1988) method can rank fuzzy numbers by their generalized mean and standard deviation based on two kinds of probability distributions, but it is not easy to compare in certain condition (the fuzzy number in the same time with higher mean and higher spread). For above reasons, a conceptual procedure is proposed in this paper to describe those methods’ using condition to ensure those methods can rank fuzzy numbers effectively. We also introduce a new ranking fuzzy numbers approach that can adjust experts’ confidence and optimistic index of decision maker using two parameters (a and b) to handle the problems and find the best solutions. This new approach for ranking fuzzy numbers not only can compute more quickly and correctly but also ranking the normal, non-normal, positive, and negative fuzzy numbers. The conceptual procedure is shown as Fig. 2. We describe the steps of Fig. 2’s ranking procedure as following: Step 1. Drawing their graphs and ranking: To draw fuzzy numbers’ graphs, if we can rank their orderings intuitively by those graphs, the ranking results can be easily obtained. Step 2. If the orderings of fuzzy numbers cannot rank by graphs, we have to determine the types of fuzzy numbers. There are four types of fuzzy numbers: (1)
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Normal and positive; (2) Non-normal and positive; (3) Normal and negative; (4) Non-normal and negative. Step 3. Rank different types of fuzzy numbers: 1. Using Lee and Li’s method (or other methods) to rank normal and positive fuzzy numbers a. Ranking their order by both with higher mean value and at the same time lower spread. b. If it is difficult to rank by this method then we can calculate their CV coefficient to rank their order. 2. Using the distance method (or other methods, such as Liou and Wang (1992)) to rank positive and normal/non-normal fuzzy numbers. Table 1 The results of comparison using Bortolan and Degani (1985) examples
3. Using the new ranking fuzzy numbers method based on adaptive two-dimensions dominance to rank normal, non-normal, positive, and negative fuzzy numbers.
5 Validating and comparison First of all, we validate our proposed method with representative examples of Bortolan and Degani (1985), and Tseng and Klein (1989) (see Tables 3 and 4 in Appendix). According to the steps of the ranking procedure in previous section,
Examples Methods Proposed method
Kerre [21] Lee and Li (1988) U
a=0.1, b=0.9 a=0.5, b=0.5
B–K Chen and Lu (2001)
P
b= 1
U Lee and Li’s Uniform; P Lee and Li’s Proportional; B–K Baas and Kwakernaak (1977)
F ~ u1 ~ u2 G ~ u1 ~ u2 H ~ u1 ~ u2 I ~ u1 ~ u2 ~ u3 J ~ u1 ~ u2 ~ u3 K ~ u1 ~ u2 ~ u3 L ~ u1 ~ u2 M ~ u1 ~ u2 ~ u3 N ~ u1 ~ u2 O ~ u1 ~ u2 P ~ u1 ~ u2 Q ~ u1 ~ u2 R ~ u1 ~ u2
0.5
0
0.417 0.462
0.519 0.544
0.96 0.89
0.58 0.54 0.84 0.55 0.59 1
0.000 0.000
0.158 0.554
0.45 0.55
0.51 0.89
0.41 0.38 0.82 0.60 0.60 1
0.100 0.200 0.500
0.158 0.644
0.45 0.6
0.42 0.95
0.41 0.38 0.66 0.70 0.70 1
0.000 0.300 0.600
0.878 0.788 0.698
0.65 0.6 0.55
1 0.86 0.76
0.77 0.80 1 0.70 0.70 0.74 0.63 0.60 0.6
0.100 0.100
0.100
0.100 0.100
0.100
0.752 0.743 0.73
0.6 0.575 0.538
1 0.91 0.75
0.70 0.70 1 0.63 0.65 1 0.58 0.57 1
0.000 0.050
0.100
0.000 0.075
0.150
0.775 0.653 0.572
0.563 0.525 0.5
1 0.85 0.75
0.62 0.63 1 0.57 0.55 1 0.50 0.50 1
0.150 0.075
0.000
0.100 0.050
0.000
0.608 0.536
0.5 0.5
0.91 0.91
0.50 0.50 1 0.50 0.50 1
0.100 0.000
0.100
0.635 0.649 0.694
0.475 0.513 0.538
0.76 0.92 0.96
0.44 0.46 1 0.53 0.53 0.88 0.56 0.58 1
0.000 0.075 0.150
0.158 0.688
0.35 0.6
0.64 1
0.20 0.20 0 0.80 0.80 0.8
X
X
X
0.518 0.784
0.55 0.5
0.78 1
0.60 0.60 0 0.90 0.90 0.2
X
X
X
0.118 0.698
0.15 0.65
0.89 0.88
0.20 0.20 0 0.80 0.80 0.2
X
X
X
0.446 0.406
0.55 0.35
0.72 0.97
0.60 0.60 0.2 0.60 0.60 0.2
X
X
X
0.788 0.847
0.7 0.525
0.82 1
0.87 0.90 0.2 0.95 0.95 0.2
X
X
X
0.000
0.050 0.025 0.000
100 Table 2 The results of comparison using Tseng and Klein (1989) examples
Examples Methods Proposed method
T–K Kerre [21] Lee and B–K Chen and Lu (2001) Li (1988) U
a=0.1, b=0.9 a=0.5, b=0.5
P
b= 1
U Lee and Li’s Uniform; P Lee and Li’s Proportional; T–K Tseng and Klein (1989), B–K Baas and Kwakernaak (1977)
B ~ u1 ~ u2 C ~ u1 ~ u2 D ~ u1 ~ u2 E ~ u1 ~ u2 F ~ u1 ~ u2 G ~ u1 ~ u2 H ~ u1 ~ u2 I ~ u1 ~ u2 J ~ u1 ~ u2 K ~ u1 ~ u2 L ~ u1 ~ u2 M ~ u1 ~ u2
0.5
0
0.806 0.68
0.65 0.5
0.87 0.99 0.13 0.54
0.80 0.80 0.56 0.100 0.50 0.50 0.19
0.300
0.500
0.788 0.518
0.6 0.45
0.87 1.0 0.13 0.55
0.70 0.70 0.56 0.300 0.40 0.40 0.19
0.300
0.300
0.68 0.649
0.5 0.513
0.47 0.89 0.53 0.95
0.50 0.50 0.44 0.050 0.57 0.53 0.48
0.030 0.100
0.608 0.608
0.5 0.5
0.49 0.95 0.51 0.96
0.50 0.50 0.36 0.000 0.53 0.50 0.39
0.000
0.000
0.653 0.563
0.525 0.475
0.56 0.93 0.44 0.87
0.50 0.55 0.40 0.100 0.50 0.45 0.36
0.100
0.100
0.572 0.644
0.5 0.5
0.50 0.90 0.50 0.90
0.50 0.50 0.38 0.100 0.000 0.50 0.50 0.38
0.100
0.482 0.473
0.45 0.445
0.52 1.0 0.48 0.98
0.40 0.40 0.29 0.020 0.39 0.39 0.28
0.020
0.020
0.662 0.658
0.55 0.538
0.56 1.0 0.44 0.95
0.60 0.60 0.33 0.000 0.57 0.58 0.29
0.025
0.050
0.662 0.649
0.55 0.513
0.64 1.0 0.36 0.85
0.60 0.60 0.38 0.000 0.53 0.52 0.29
0.075
0.150
0.658 0.649
0.538 0.513
0.58 1.0 0.42 0.90
0.57 0.58 0.38 0.000 0.53 0.52 0.33
0.050
0.100
0.806 0.797
0.55 0.525
0.52 1.0 0.48 0.96
0.60 0.60 0.57 0.070 0.60 0.60 0.44
0.050
0.000
0.811 0.806
0.563 0.55
0.50 0.95 0.50 0.95
0.62 0.63 0.57 0.040 0.005 0.60 0.60 0.53
0.050
1. Drawing their graphs and ranking: We first judge if the fuzzy numbers can rank by graphs. The examples A, B, C, D, E of the Bortolan and Degani (1985) and example A of Tseng and Klein (1989) can easily obtain the ranking fuzzy numbers’ ordering of each example. 2. Determining the types of fuzzy numbers and then ranking by proper method. The other examples in Tables 3 and 4 are all positive fuzzy numbers, so they can rank by other methods. In this paper, Examples I, J, K, L and M of Bortolan and Degani (1985), and Tseng and Klein (1989) are chosen to explain the results. In Appendix, we use other methods to explain the results of these methods, and Tables 1 and 2 show the outcomes. We can easily see that most experimental results are consistent with other methods (Table 1’s example G, H, I,
J, K, M, N, and O, and Table 2’s examples B, C, D, E, H, I, J, and K). In Table 1, most ranking methods cannot distinguish fuzzy numbers in Example L and M, some even cannot tell Example J and K. For example, Kerre (1982), Lee and Li (1988) and Baas and Kwakernaak (1977) cannot separate fuzzy numbers in Example L. Baas and Kwakernaak (1977) also cannot distinguish the example M. For that reason, it is not appropriate for DMs to rank the fuzzy numbers. The results of Chen and Lu (2001) approach will change according to the value of b, and most results correspond with our method. However, it cannot rank the non-normal fuzzy numbers. Besides, in Table 1’s example F, Chen and Lu’s method cannot rank the order of these two fuzzy numbers under different b’s values, and our proposed method obtain the same result with Baas and Kwakernaak’s
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(1977) methods, which follows Adamo’s opinions (1980). The characteristic is that our method and theirs are applied for DMs’ opinions. But our method use two parameters to reflect experts’ confidence and optimistic index of DMs. Furthermore, it can be computed more quickly and easily. In light of the outcome in Table 2, the results of our proposed method are harmonized with those of Table 3 Examples given by Bortolan and Degani (1985)
other methods except for Example M (the same with Table 1’s example L). In Example M, Tseng and Klein (1989) and Kerre (1982) consider the two fuzzy numbers are the same, but our method and Baldwin and Guild (1979) do not think so, both of us agree that the M1 is larger than M2 and their difference is very small. Due to the different b of the hen and Lu (2001) approach, we may get the
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results of ranking reversal. Roughly, there is not much difference in our method and theirs. Approximately, from above verifying and the experiments, the results of our method represent correctly and more briefly in those examples when a=0.5 and b=0.5 or a=0.1 and b=0.9 and. After some advance experiments, we find there are many other pairs of acceptable values of these two parameters. When DMs find it difficult to assign the values of parameter a and b, we suggest DMs that a and b=0.5 are appropriate value for distinguishing the fuzzy numTable 4 Examples given by Tseng and Klein (1989)
bers. That is because this pair of values is more fitting the thinking of the moderate DMs (in average case).
6 Conclusion The existing ranking fuzzy numbers methods all have their advantages and some shortcomings. They may valuable in solving some types of fuzzy numbers (i.e. normal, non-normal, positive, and negative fuzzy numbers). For above reasons, this approach proposes a new
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ranking fuzzy numbers method based on adaptive twodimensions dominance, which has the ability to rank each type of fuzzy numbers. We also propose a conceptual procedure to describe those methods’ using condition and to ensure some existing methods can rank fuzzy numbers effectively. Finally, comparative examples are also presented to illustrate the advantages of the new ranking method and proposed conceptual procedure. The proposed conceptual procedure can be used in some other applications, such as fuzzy multiple criteria or multiple attribute decision making, fuzzy goal programming, fuzzy scheduling, and fuzzy control. Acknowledgements The authors would like to thank the anonymous referees for providing very helpful comments and suggestions. Their insight and comments led to a better presentation of the ideas expressed in this paper.
Appendix Comparison examples
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