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An improved ranking method for fuzzy numbers with integral values
Applied Soft Computing 14 (2014) 603–608
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Applied Soft Computing journal homepage: www.elsevier.com/locate/asoc
An improved ranking method for fuzzy numbers with integral values Vincent F. Yu a,∗ , Luu Quoc Dat b a
Department of Industrial Management, National Taiwan University of Science and Technology, 43, Section 4, Keelung Road, Taipei 10607, Taiwan Faculty of Development Economics, University of Economics and Business, Vietnam National University, 144 Xuan Thuy Road, Cau Giay Dist., Hanoi, Viet Nam b
a r t i c l e
i n f o
Article history: Received 29 October 2011 Received in revised form 16 February 2012 Accepted 17 October 2013 Available online 26 October 2013 Keywords: Ranking fuzzy numbers Integral value Index of optimism
a b s t r a c t Ranking fuzzy numbers is a very important decision-making procedure in decision analysis and applications. The last few decades have seen a large number of approaches investigated for ranking fuzzy numbers, yet some of these approaches are non-intuitive and inconsistent. In 1992, Liou and Wang proposed an approach to rank fuzzy number based a convex combination of the right and the left integral values through an index of optimism. Despite its merits, some shortcomings associated with Liou and Wang’s approach include: (i) it cannot differentiate normal and non-normal fuzzy numbers, (ii) it cannot rank effectively the fuzzy numbers that have a compensation of areas, (iii) when the left or right integral values of the fuzzy numbers are zero, the index of optimism has no effect in either the left integral value or the right integral value of the fuzzy number, and (iv) it cannot rank consistently the fuzzy numbers and their images. This paper proposes a revised ranking approach to overcome the shortcomings of Liou and Wang’s ranking approach. The proposed ranking approach presents the novel left, right, and total integral values of the fuzzy numbers. The median value ranking approach is further applied to differentiate fuzzy numbers that have the compensation of areas. Finally, several comparative examples and an application for market segment evaluation are given herein to demonstrate the usages and advantages of the proposed ranking method for fuzzy numbers. © 2013 Elsevier B.V. All rights reserved.
1. Introduction Ranking fuzzy numbers plays a very important role in decisionmaking, optimization, and other usages. Over the last few decades numerous ranking approaches have been proposed and investigated [1–14], with the first method for ranking fuzzy numbers proposed by Jain [1]. Chen [2] offered an approach for ranking fuzzy numbers by using maximizing set and minimizing set concepts. Liou and Wang [3] developed a ranking approach based on an integral value index to overcome the shortcomings of Chen’s [2] approach. Cheng [4] presented an approach for ranking fuzzy numbers by using the distance method. Chu and Tsao [5] proposed an approach for ranking fuzzy numbers with the area between the centroid point and original point. Abbasbandy and Asady [6] introduced an approach to rank fuzzy numbers by sign distance. Chen and Tang [7] presented an approach to rank non-normal pnorm trapezoidal fuzzy numbers with integral value. Abbasbandy and Hajjari [8] showed a new approach for ranking of trapezoidal fuzzy numbers. Wang and Luo [9] proposed an area ranking of fuzzy numbers based on positive and negative ideal points. Kumar
∗ Corresponding author. Tel.: +886 2 2737 6333; fax: +886 2 2737 6344. E-mail address: [email protected] (V.F. Yu). 1568-4946/$ – see front matter © 2013 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.asoc.2013.10.012
et al. [10] offered an approach for ranking generalized exponential fuzzy numbers using an integral value approach. Kumar et al. [11] modified Liou and Wang’s [3] approach for the ranking of an L–R type generalized fuzzy number. Chou et al. [12] presented a revised maximizing set and minimizing set ranking approach. Among the ranking approaches, Liou and Wang’s [3] method is a commonly used approach that is highly cited and has wide applications [7,10,11,15–20], but there are some shortcomings associated with their ranking approach. For the triangular and trapezoidal fuzzy numbers, Liou and Wang [3] showed that the integral values of normal and non-normal fuzzy numbers are equal. In other words, the fuzzy numbers A1 = (a, b, c, d; w1 ) and A2 = (a, b, c, d; w2 ) with (ω1 = / ω2 ), are considered the same. Cheng [4] indicated that Liou and Wang’s [3] approach could not differentiate normal and non-normal triangular/trapezoidal fuzzy numbers, because of equivalence between these fuzzy numbers. In addition, Garcia and Lamata [16] showed that if the fuzzy numbers to be compared have the compensation of areas, then they cannot be ranked by the Liou and Wang’s [3] approach. Furthermore, when the left or right integral values of fuzzy numbers are zero, the index of optimism has no effect in either the left integral value or the right integral value of the fuzzy number. Finally, Liou and Wang’s [3] approach may result in inconsistency between the ranking of the fuzzy numbers and the ranking of their images.
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To overcome all the aforementioned shortcomings with Liou and Wang’s [3] ranking approach, this paper proposes a revised approach for ranking fuzzy numbers, presenting the novel left, right, and total integral values of fuzzy numbers. In order for the proposed method to have flexibility, the decision maker’s optimistic attitude of fuzzy numbers is taken into account. The median value ranking approach is further applied to differentiate fuzzy numbers that have the compensation of areas. Finally, several comparative examples and an application for market segment evaluation are given to demonstrate the usages and advantages of the proposed ranking approach for fuzzy numbers. The results show that the proposed ranking approach can effectively overcome the shortcomings of Liou and Wang’s approach. The rest of the paper is organized as follows. Section 2 states the preliminary concepts and definitions of fuzzy numbers. Section 3 briefly reviews Liou and Wang’s ranking approach. Section 4 uses four numerical examples to show the shortcomings of Liou and Wang’ ranking approach. Section 5 proposes the revised ranking approach. Section 6 illustrates the proposed ranking method through five numerical examples. Section 7 applies the proposed ranking method for ranking fuzzy numbers to a market segment evaluation and selection problem, demonstrating its applicability and efficiency. Finally, Section 8 draws the conclusions.
2. Preliminaries This section briefly reviews the definitions of fuzzy numbers as follows. A fuzzy number is a fuzzy subset in support R (real number) that is both “normal” and “convex”, where supp(A) = { x ∈ R A > 0}. Normality implies that the maximum value of the fuzzy set A in R is 1. Therefore, the non-normal fuzzy number is ∀x ∈ R, Maxx {A (x)} <1 [4]. For convenience, the fuzzy number can be denoted by [a, b, c, d ; ], and the membership function of fuzzy number A = [a, b, c, d ; ] can be expressed as:
fA (x) =
⎧ L fA (x), a ≤ x ≤ b, ⎪ ⎪ ⎪ ⎨ ,
b ≤ x ≤ c,
(1)
fAR (x), c ≤ x ≤ d, ⎪ ⎪ ⎪ ⎩ 0,
otherwise,
where fAL : [a, b] → [0, ] and fAR : [c, d] → [0, ]. Since fAL (x) and fAR (x) are both strictly monotonical and continuous functions, their inverse functions exist and should be continuous and strictly monotonical. The inverse functions of fAL (x) and fAR (x) can be denoted by gAL : [0, ω] → [a, b] and gAR : [0, ω] → [c, d], respectively. As such, gAL (y) and gAR (y) are then integrable on the closed interval [0, ω]. In
ω
ω
other words, both 0 gAL (y) and 0 gAR (y) exist. The image (or opposite) of a fuzzy number A = (a, b, c, d ; ) can be given by fuzzy number −A = (− d, − c, − b, − a ; ω) as it is given in [21].
3. A review on Liou and Wang’s ranking approach Combining the left and right integral values, Liou and Wang [3] proposed an approach for ranking fuzzy numbers with an index of optimism ˛ ∈ [0, 1]. The left and right integral values of fuzzy
1
1
gAL (y)dy
number A are defined as IL (A) = 0
gAR (y)dy,
and IR (A) =
Fig. 1. Fuzzy numbers A1 and A2 in Example 1.
optimism ˛ ∈ [0, 1] is defined as:
IT˛ (A) = ˛IR (A) + (1 − ˛)IL (A) = ˛
1
1
gAR (y)dy + (1 − ˛) 0
gAL (y)dy 0
(2)
where gAL (y) and gAR (y) are respectively the inverse functions of fAL (x) and fAR (x). The index of optimism ˛ represents the degree of opti-
mism of a decision maker. A larger ˛ indicates a higher degree of optimism. For ˛ = 0 and ˛ = 1, the values of IT˛ (A) represent the viewpoints of pessimistic and optimistic decision makers, respectively. For a moderate decision maker, with ˛ = 0.5, the total integral value of each fuzzy number A becomes: IT0.5 (A) =
1
2
[IR (A) + IL (A)]
The greater is IT˛ (A), the bigger the fuzzy number Ai is and the higher its ranking order. 4. Shortcomings of Liou and Wang’s ranking approach This section points out the shortcomings of Liou and Wang’s [3] approach. Several examples are chosen to prove that the ranking approach, proposed by Liou and Wang, does not satisfy the reasonable properties for the ordering of fuzzy numbers. First, Liou and Wang’s method [3] cannot differentiate normal and non-normal triangular/trapezoidal fuzzy numbers [4]. For the non-normal fuzzy numbers, Liou and Wang [3] made the following assumption: “When B is a non-normal fuzzy number, fB can always be normalized by dividing the maximal value of fB before ranking”. Specifically, let the normalized fuzzy number of B and the corresponding membership function respectively be B¯ and f¯B (x) = fB (x)/ω, where ω = maxx fB (x). The integral value of B is then represented by the integral value of B when being ranked. Hence, ¯ and IR (B) = IR (B). ¯ For the case of trapezoidal and trianIL (B) = IL (B) gular fuzzy numbers, Liou and Wang proved that the total integral values of normal and non-normal fuzzy numbers are the same [7]. Example 1. Consider the triangular fuzzy numbers A1 = (3, 5, 7 ; 1) and A2 = (3, 5, 7 ; 0.8) as in Fig. 1, from [3]. Intuitively, the ranking order is A1 A2 . However, through Liou and Wang’s method, the total integral values of the triangular fuzzy numbers A1 and A2 are respectively IT˛ (A1 ) = 4 + 2˛ and IT˛ (A2 ) = 4 + 2˛. Thus, the ranking order of fuzzy numbers A1 and A2 is the same, i.e., A1 ∼ A2 . for every ˛ ∈ [0, 1]. Liou and Wang’s approach therefore fails to correctly rank the given fuzzy numbers. In addition, it is also found that when the fuzzy numbers to be compared have the compensation of areas, i.e. the fuzzy numbers A1 = (a1 , b1 , c1 , d1 ; w) and A2 = (a2 , b2 , c2 , d2 ; w) with IL (A) = IL (B) and IR (A) = IR (B), they cannot differentiate using Liou and Wang’s approach [16].
0
which reflect the pessimistic and optimistic viewpoints of the decision maker, respectively. The total integral value with an index of
Example 2. Consider the normal triangular fuzzy numbers A1 = (1,4,5) and A3 = (2,3,6) as in Fig. 2 which is taken from [16]. It
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Fig. 5. The novel left integral value SL (Ai ) of Ai . Fig. 2. Fuzzy numbers A1 and A2 in Example 2.
Fig. 6. The novel right integral value SR (Ai ) of Ai . Fig. 3. Fuzzy numbers A and B in Example 3.
for ˛ = 1, and −A = − B − C for ˛ = 1/2. However, using Liou and Wang’s approach, the total integral values of fuzzy numbers −A, −B, and −C are IT˛ (−A) = −4 + 2˛, IT˛ (−B) = −3.5 + ˛, and IT˛ (C) = −5 + 2.5˛, respectively. The results show that −B − A − C for ˛ = 0, −A − B = − C for ˛ = 1, and −A = − B − C for ˛ = 1/2. Obviously, Liou and Wang’s [3] approach inconsistently ranks fuzzy numbers and their images. 5. Proposed method
Fig. 4. Fuzzy numbers A, B and C in Example 4.
is easy to verify that A1 and A2 have the compensation of areas and therefore cannot be distinguished by Liou and Wang’s [3] approach. Liou and Wang’ [3] approach also used an index of optimism to reflect the decision maker’s optimistic attitude. However, when the left or right integral values of fuzzy numbers are zero, the index of optimism has no effect in either the left integral value or the right integral value of the fuzzy number. When considering the fuzzy number A = (a, b, c, d ; ω), either a = b or a + b = 0 or c = d or c + d = 0 the left and right integral values of fuzzy number A will be IL (A) = 0 or IR (A) = 0, respectively. Example 3. Consider the triangular fuzzy numbers, A = (−1,1,2), and B = (−3,−2,2) as in Fig. 3. Using Liou and Wang’s [3] approach, either the left integral value of fuzzy number A or the right integral value of fuzzy number B are 0, i.e. either IL (A) = 0 or IR (B) = 0, for all ˛ ∈ [0, 1]. Thus, the decision maker’s viewpoint or the index of optimism has no effect in either the left integral value of fuzzy number A or the right integral value of fuzzy number B. This is not reasonable. The following example shows that Liou and Wang’s [3] approach cannot consistently rank the fuzzy numbers and its images. Example 4. Fig. 4 presents three normal triangular fuzzy numbers, A = (1, 3, 5), B = (2, 3, 4), and C = (1, 4, 6). Using Liou and Wang’s [3] approach, the total integral values of fuzzy numbers A, B, and C are IT˛ (A) = 2 + 2˛, IT˛ (B) = 2.5 + ˛, and IT˛ (C) = 2.5 + 2.5˛, respectively. Thus, the ranking orders of fuzzy numbers A, B, and C are A ≺ B = C for ˛ = 0, B ≺ A ≺ C for ˛ = 1, and A = B ≺ C for ˛ = 1/2. From this, one can logically infer the ranking order of the images of these fuzzy numbers as −A − B = − C for ˛ = 0. −B − A − C
In order to overcome the shortcomings of Liou and Wang’s [3] ranking approach, this section proposes a revised ranking approach based on the novel integral values and median value of fuzzy numbers as follows. Definition 3. Suppose there are n fuzzy numbers Ai , i = 1, 2, ..., n, each with the left membership function fAL and right membership i
function fAR . The novel left and right integral values of Ai are defined i as:
bi
SL (Ai ) = ωi (bi − xmin ) − ai
SR (Ai ) = ωi (c i − xmin ) +
ci
di
fAL (x)dx,
(3)
fAR (x)dx,
(4)
i
i
n P , P = {x/f (x) 0}, w = sup f (x). where xmin = infP, P = Ui=1 Ai i i i x Ai Both SL (Ai ) and SR (Ai ) ≥ 0. The meanings of SL (Ai ) and SR (Ai ) are expressed in Figs. 1 and 2, respectively. Clearly, the fuzzy number Ai becomes larger if SL (Ai ) and SR (Ai ) are larger (see Figs. 5 and 6). The novel total integral value with index of optimism ˛ ∈ [0, 1] is then defined as:
ST˛ (Ai ) = ˛SR (Ai ) + (1 − ˛)SL (Ai )
(5)
The proposed approach uses ST˛ (Ai ) to rank fuzzy numbers. The larger ST˛ (Ai ) is, the larger is the fuzzy number Ai . Therefore, for any distinct fuzzy numbers Au and Av , we have the following properties: (1) if ST˛ (Au ) < ST˛ (Av ), then Au ≺ Av , (2) if ST˛ (Au ) > ST˛ (Av ), then Au > Av , and (3) if ST˛ (Au ) = ST˛ (Av ), then
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fAL (x) = ω(x − a)/(b − a) and fAR (x) = ω(x − d)/(c − d), respectively. Thus:
(a) if Meu > Mev , then Au > Av , (b) if Meu < Mev , then Au < Av , (c) if Meu = Mev , then Au = Av .
where Me is the median value of fuzzy numbers, and the values of Me are defined as follows [22,23]. Considering the trapezoidal fuzzy number A = (a, b, c, d), the median Me of A is derived by the following three conditions (a and d are the lower limit value and upper limit value, respectively) [22].
Me = a +
(b − a)(c + d − a − b) 2
(6)
a+b+c+d 4
Me = d −
(d − c)(c + d − a − b) 2
(8)
Definition 4. Considering the normal trapezoidal fuzzy number A = (a, b, c, d ; 1), where a < b ≤ c < d, then the left and right membership functions of fuzzy number A are fAL (x) = (x − a)/(b − a) and fAR (x) = (x − d)/(c − d), respectively. Thus, (x − a)/(b − a)dx, a
= (1/2)(a + b − 2xmin ) = IL (A) − xmin
SR (A) = c − xmin + c
(9)
d
(x − d) dx (c − d)
2
(10)
[˛(c + d) + (1 − ˛)(b + a) − 2xmin ]
= ˛IR (A) + (1 − ˛)IL (A) − xmin
(11)
Notably, when xmin = 0, formula (11) is the same as in [3]. Since triangular fuzzy numbers are special cases of trapezoidal fuzzy numbers when b = c, the total integral value of normal triangular fuzzy number A = (a, b, d ; 1) can be determined by: ST˛ (A) =
1
2
ST˛ (A) =
ω
2
2
ω
2
(c + d − 2xmin ) (13)
(a + b − 2xmin )
[˛(c + d) + (1 − ˛)(b + a) − 2xmin ]
ω
2
[˛d + b + (1 − ˛)a − 2xmin ]
(14)
(15)
(16)
Definition 6. Suppose there is an opposite fuzzy number of A = (a, b, c, d ; 1), denoted by −A = (− d, − c, − b, − a, 1). The left and L (x) = ω(x + d)/(d − c) and right membership functions of −A are f−A R (x) = ω(x + a)/(a − b), respectively. Thus: f−A
−c
SL (A) = −c − xmin − −d
=
(x + d) dx, (d − c)
−(c + d) − xmin = −IR (A) − xmin 2
and
−a
SR (A) = −b − xmin + −b
(17)
−(a + b) (x + a) = −IL (A) − xmin dx = 2 − xmin (a − b) (18)
From (17) and (18), we have: IR (− A) = − IL (A) and IL (− A) = − IR (A). Given ˛ ∈ [0, 1], the total integral value of the trapezoidal fuzzy number −A = (− d, − c, − b, − a ; 1) can therefore be obtained as:
[˛d + b + (1 − ˛)a − 2xmin ]
1
2
[−˛(a + b) − (1 − ˛)(c + d)] − xmin
= −˛IL (A) − (1 − ˛)IR (A) − xmin
Given ˛ ∈ [0, 1], the total integral value of the normal trapezoidal fuzzy number A = (a, b, c, d ; 1) can be obtained as:
1
(x − d) dx = (c − d)
d
c
ST˛ (−A) =
= (1/2)(c + d − 2xmin ) = IR (A) − xmin
ST˛ (A) =
SR (A) = ω(c − xmin ) +
ω
b
SL (A) = b − xmin −
and
and
ST˛ (A) =
In short, the total integral values, i.e., ST˛ (Ai ) values, are used to rank Au and Av if their total integral values are different. In the case for when they are equal, the fuzzy numbers Au and Av are further compared using their Me values. In order to simplify the computational procedures, the proposed ranking method is further applied for trapezoidal and triangular fuzzy numbers as follows.
ω
a
(7)
(3) When c ≤ Me ≤ d, Me is given by
(x − a) dx, = (b − a)
Since triangular fuzzy numbers are special cases of trapezoidal fuzzy numbers when b = c, the total integral value of non-normal triangular fuzzy number A = (a, b, d ; ω) can be determined by:
(2) When b ≤ Me ≤ c, Me is given by Me =
ω
Given ˛ ∈ [0, 1], the total integral value of A can be obtained as:
(1) When a ≤ Me ≤ b, Me is given by
b
SL (A) = ω(b − xmin ) −
(12)
Similarly, when xmin = 0, formula (12) is the same as in [3]. Definition 5. Consider the non-normal trapezoidal fuzzy number A = (a, b, c, d ; ). The left and right membership functions of A are
(19)
Definition 7. Suppose there are two trapezoidal fuzzy numbers A1 = (a1 , b1 , c1 , d1 ; ω1 ) and A2 = (a2 , b2 , c2 , d2 ; w2 ). This paper defines the index of optimism (˛) representing the degree of optimism of a decision maker as in Liou and Wang’s [3] ranking approach. When the two fuzzy numbers A1 and A2 are negative and satisfy a1 ≤ a2 ≤ d2 ≤ d1 , the degree of optimism of a decision maker will be defined as: when ˛ = 0, the total integral value of each fuzzy number that represents an optimistic decision maker’s viewpoint is equal to the left integral values of fuzzy numbers A1 and A2 . Conversely, for a pessimistic decision maker, i.e. ˛ = 1, the total integral value of each fuzzy number is equal to the fuzzy number’s right integral value. The total integral value of each fuzzy number represents a moderate decision maker when ˛ = 0.5. 6. Comparative examples To present the rationality and necessity for the revision of Liou and Wang’s [3] ranking approach, the following examples are
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employed to compare our proposed approach with their original one and other methods in the literature. Example 5. Re-consider the two normal triangular fuzzy numbers A1 = (3, 5, 7 ; 1) and A2 = (3, 5, 7 ; 0.8) in Example 1. Using the revised ranking approach, the total integral values of A1 and A2 are ST˛ (A1 ) = 1 + 2˛ and ST˛ (A2 ) = 0.8 + 1.6˛, respectively. Since ˛ = 0 for a pessimistic decision maker, the total integral values of A1 and A2 are respectively ST0 (A1 ) = 1 and ST0 (A1 ) = 0.8, and thus A1 A2 . Conversely, ˛ = 1 for an optimistic decision maker, whereby the total integral values of A1 and A2 are respectively ST0 (A1 ) = 3 and ST0 (A1 ) = 2.4, and thus A1 A2 . For a moderate decision maker, ˛ = 0.5, the total integral values of A1 and A2 are respectively ST0 (A1 ) = 2 and ST0 (A1 ) = 1.6, and thus A1 A2 . Therefore, the ranking order is A1 A2 for every ˛ ∈ [0, 1]. However, the results from Example 1 show that using Liou and Wang’ [3] ranking approach, the ranking order is A1 ∼ A2 , for every ˛ ∈ [0, 1]. Clearly, the proposed ranking approach can overcome the shortcomings of the inconsistency of Liou and Wang’s method in ranking fuzzy numbers. Example 6. The two normal triangular fuzzy numbers A1 = (1, 4, 5) and A2 = (2, 3, 6) in Example 2 are reconsidered by using the proposed ranking approach. According to Eq. (12), the total integral values of A1 and A2 are the same, i.e., ST˛ (A1 ) = ST˛ (A2 ) = 1.5 + 2˛. Since ST˛ (A1 ) = ST˛ (A2 ) for every ˛ ∈ [0, 1], the fuzzy numbers A1 and A2 are further compared using their median values, i.e., Me(A1 ) and Me(A2 ). Since Me(A1 ) = 3.45 < 3.55 = Me(A2 ), the ranking order is A2 > A1 . However, the results from Example 2 show that the ranking order of A1 and A2 is always the same, i.e., A1 ∼ A2 . Obviously, the ranking order obtained by the proposed ranking approach is more reasonable than the outcome obtained by Liou and Wang’s approach. Example 7. Re-consider the triangular fuzzy numbers, A = (−1, 1, 2) and B = (−3, − 2, 2) in Example 3. Using the proposed approach, the left, right, and total integral values of A and B are SL (A) = 3, SR (A) = 4.5, ST˛ (A) = 1.5˛ + 3, SL (B) = 0.5, SR (B) = 3 and ST˛ (B) = 2.5˛ + 0.5 respectively. Therefore, the ranking order is A B for every ˛ ∈ [0, 1]. Clearly, this example shows that by using the proposed approach, the decision maker’s viewpoint or the index of optimism is represented by both the left and right integral values of fuzzy numbers A and B. Example 8. Re-consider the three normal triangular fuzzy numbers, A = (1, 3, 5), B = (2, 3, 4), and C = (1, 4, 6) in Example 4. Using formula (12), the total integral values of fuzzy numbers A, B, and C are IT˛ (A) = 1 + 2˛, IT˛ (B) = 1.5 + ˛, and IT˛ (C) = 1.5 + 2.5˛, respectively. Thus, A ≺ B = C for ˛ = 0, B ≺ A ≺ C for ˛ = 1, and A = B ≺ C for ˛ = 1/2. The opposite of the three fuzzy number are −A = (−5, − 3, − 1), −B = (−4, − 3, − 2), and −C = (−6, − 4, − 1), respectively. Since −5 ≺ −4 ≺ −2 ≺ −1, −6 ≺ −5 ≺ −3 ≺ −1 and −6 ≺ −4 ≺ −2 ≺ −1, one can logically infer the ranking order of the images of these fuzzy numbers as −B − A − C for ˛ = 0, −A − B = − C for ˛ = 1, and −A = − B − C for ˛ = 1/2. Using the proposed method, the total integral values of fuzzy numbers −A −B, and −C are IT˛ (−A) = 2 + 2˛, IT˛ (−B) = 2.5 + ˛, and IT˛ (−C) = 1 + 2.5˛, respectively. Thus, it can be concluded that −B − A − C for ˛ = 0, −A − B = − C for ˛ = 1, and −A = − B − C for ˛ = 1/2. Again, this example shows that the proposed approach overcomes the shortcomings of the inconsistency of Liou and Wang’s approach in ranking fuzzy numbers and its images. Example 9. Consider the data used in [12], i.e., the three normal fuzzy numbers A1 = (5, 6, 7), A2 = (5.9, 6, 7) and A3 = (6, 6, 7) as shown in Fig. 7. According to formula (12), the total integral values are obtained as IT˛ (A1 ) = 0.5 + ˛, IT˛ (A2 ) = 0.95 + 0.55˛, and
Fig. 7. Fuzzy numbers A1 , A2 and A3 in Example 9.
Table 1 Comparative results of Example 9. Ranking approach
A1
A2
A3
Ranking
Wang et al. [14] Wang and Luo [9] Asady [13] Chen [2] Sign distance (p = 1) [6] Sign distance (p = 2) [6] Cheng [4] Abbasbandy and Hajjari [8]
0.25 0.5 0.66667 0.5 6.12 8.52 6.021 6
0.5339 0.571 0.81818 0.5714 12.45 8.82 6.349 6.075
0.5625 0.583 1 0.5833 12.5 8.85 6.7519 6.0834
A1 ≺ A2 ≺ A3 A1 ≺ A2 ≺ A3 A1 ≺ A2 ≺ A3 A1 ≺ A2 ≺ A3 A1 ≺ A2 ≺ A3 A1 ≺ A2 ≺ A3 A3 ≺ A2 ≺ A1 A1 ≺ A2 ≺ A3
IT˛ (A3 ) = 1 + 0.5˛. It is observed that for an optimistic decision maker, i.e. ˛ = 1, the three fuzzy numbers are the same. Conversely, for a moderate decision maker, i.e. ˛ = 0.5, or a pessimistic decision maker, i.e. ˛ = 0, A3 is the most preferred alternative, followed by A2 and then A1 This is consistent with the ranking obtained by other approaches [2,6,8,9,13,14]. Table 1 summarizes the results obtained by different methods. Note that the ranking A1 A2 A3 obtained by the CV index of Cheng [4] is thought of as unreasonable and not consistent with human intuition [9,12]. This example shows the strong discrimination power of the proposed ranking approach and its advantages.
7. Applying the proposed method for ranking fuzzy numbers to solve a market segment selection problem This section applies the proposed ranking approach to deal with market segment evaluation and selection problem. Assume that a marketing department is looking for a suitable market segment to satisfy customer needs. Three decision makers, i.e., D1 , D2 , and D3 , are responsible for the evaluation of three market segments, i.e., A1 , A2 , and A3 . Five criteria are chosen for evaluating the market segments [24]: the bargaining power of customers (C1 ), the bargaining power of suppliers (C2 ), the threat of new entrants (C3 ), the threat Table 2 Ratings of alternatives versus criteria. Criteria
Alternatives
D1
D2
D3
rij
C1
A1 A2 A3 A1 A2 A3 A1 A2 A3 A1 A2 A3 A1 A2 A3
G F G VG G G G F G VG F G G P F
G F F G G G VG G F G P G G F F
VG F G G F G G F G VG F G G F G
(0.733, 0.833, 1.0) (0.3, 0.5, 0.7) (0.567, 0.7, 0.9) (0.733, 0.833, 1.0) (0.567, 0.7, 0.9) (0.7, 0.8, 1.0) (0.733, 0.833, 1.0) (0.433, 0.6, 0.8) (0.567, 0.7, 0.9) (0.767, 0.867, 1.0) (0.267, 0.433, 0.6) (0.7, 0.8, 1.0) (0.7, 0.8, 1.0) (0.267, 0.433, 0.6) (0.433, 0.6, 0.8)
C2
C3
C4
C5
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Table 3 The importance weights of the criteria and the aggregated weights. Criteria
D1
D2
D3
rij
C1 C2 C3 C4 C5
VH H M H H
VH VH H M H
H VH H H H
(0.733, 0.933, 0.967) (0.733, 0.933, 0.967) (0.533, 0.7, 0.833) (0.533, 0.7, 0.833) (0.6, 0.8, 0.9)
Table 4 The left, right, and total integral values of each alternative. Alternative
SL (Ai )
SR (Ai )
ST0.5 (Ai )
Ranking
A1 A2 A3
0.321 0.099 0.251
0.539 0.315 0.479
0.430 0.207 0.365
1 3 2
of substitute products (C4 ), and the intensity of competitive rivalry (C5 ). Step 1. Aggregate ratings of alternatives versus criteria makers use the linguistic rating Assume that the decision
set S = VP, P, F, G, VG , where VP = Very Poor = (0.0, 0.1, 0.2), P = Poor = (0.2, 0.3, 0.4), F = Fair = (0.3, 0.5, 0.7), G = Good = (0.7, 0.8, 1.0), and VG = Very Good = (0.8, 0.9, 1.0), to evaluate the suitability of the alternative market segments under each criteria. Table 2 presents the suitability ratings of alternatives versus the five criteria. According to Chu and Lin’s method (2003), the aggregated suitability ratings of three alternatives, A1 , A2 and A3 , versus five criteria C1 , C2 , C3 , C4 and C5 from three decision makers can be obtained as shown in Table 2. Step 2. Aggregate the importance weights Assume that the decision
makers employ a linguistic weighting set Q = VL, L, M, H, VH , where VL = Very Low = (0.1, 0.2, 0.3), L = Low = (0.2, 0.3, 0.4), M = Medium = (0.4, 0.5, 0.7), H = High = (0.6, 0.8, 0.9), and VH = Very High = (0.8, 1.0, 1.0), to assess the importance of all the criteria. Table 3 displays the importance weights of five criteria from the three decision-makers. By Chu and Lin’s [25] approach, the aggregated weights of the criteria from the decisionmaking committee can be obtained as presented in Table 3. Step 3. Normalize the performance of alternatives versus objective criteria To put forth an easier and practical procedure, this paper defines all of the fuzzy numbers in [0,1]. The calculation of normalization is thus no longer needed. Step 4. Develop the membership function of each normalized weighted rating The final fuzzy evaluation values can be developed via arithmetic operation of the fuzzy numbers as in [25]. Step 5. Defuzzification Using Eqs. (3)–(5), the left, right, and total integral values of each alternative with ˛ = 1/2 can be obtained, as shown in Table 4. According to Table 4, the ranking order of the three market segments is A1 A3 A2 . Thus, the best selection is market segment A1 having the largest total integral value. 8. Conclusions This paper has proposed a revised ranking approach to overcome the shortcomings of Liou and Wang’s ranking approach. The
proposed ranking approach presents the novel left, right, and total integral values of fuzzy numbers. To differentiate fuzzy numbers that have the compensation of areas, the median value ranking approach is further applied. Several comparative examples and an application for market segment evaluation have been given to illustrate the usage, applicability, and advantages of the proposed ranking approach. The results indicate that the ranking order obtained by the proposed approach is more consistent with human intuitions than Liou and Wang’s original method. Furthermore, the revised ranking approach is easy to apply and can effectively rank a mix of various types of fuzzy numbers, including normal, non-normal, triangular, trapezoidal, and other types of fuzzy numbers. References [1] R. Jain, Decision-making in the presence of fuzzy variables, IEEE Trans. Systems Man Cybernet. 6 (1976) 698–703. [2] S.H. Chen, Ranking fuzzy numbers with maximizing set and minimizing set, Fuzzy Sets Syst. 17 (1985) 113–129. [3] T.S. Liou, M.J. Wang, Ranking fuzzy numbers with integral value, Fuzzy Sets Syst. 50 (1992) 247–255. [4] C.H. Cheng, A new approach for ranking fuzzy numbers by distance method, Fuzzy Sets Syst. 95 (1998) 307–317. [5] T.C. Chu, C.T. Tsao, Ranking fuzzy numbers with an area between the centroid point and original point, Comput. Math. Appl. 43 (2002) 111–117. [6] S. Abbasbandy, B. Asady, Ranking of fuzzy numbers by sign distance, Inform. Sci. 176 (2006) 2405–2416. [7] C.C. Chen, H.C. Tang, Ranking nonnormal p-norm trapezoidal fuzzy numbers with integral value, Comput. Math. Appl. 56 (2008) 2340–2346. [8] S. Abbasbandy, T. Hajjari, A new approach for ranking of trapezoidal fuzzy numbers, Comput. Math. Appl. 57 (3) (2009) 413–419. [9] Y.M. Wang, Y. Luo, Area ranking of fuzzy numbers based on positive and negative ideal points, Comput. Math. Appl. 58 (2009) 1769–1779. [10] A. Kumar, P. Singh, A. Kaur, Ranking of generalized exponential fuzzy numbers using integral value approach, Int. J. Adv. Soft. Comput. Appl. 2 (2) (2010) 221–230. [11] A. Kumar, P. Singh, P. Kaur, A. Kaur, A new approach for ranking of L–R type generalized fuzzy numbers, Expert Syst. Appl. 38 (2011) 10906–10910. [12] S.Y. Chou, L.Q. Dat, F.Y. Vincent, A revised method for ranking fuzzy numbers using maximizing set and minimizing set, Comput. Ind. Eng. 61 (2011) 1342–1348. [13] B. Asady, The revised method of ranking LR fuzzy number based on deviation degree, Expert Syst. Appl. 37 (7) (2010) 5056–5060. [14] Z.X. Wang, Y.J. Liu, Z.P. Fan, B. Feng, Ranking L–R fuzzy number based on deviation degree, Inform. Sci. 179 (13) (2009) 2070–2077. [15] K. Deep, M.L. Kansal, K.P. Singh, Ranking of alternatives in fuzzy environment using integral value, J. Math. Stat. Allied Fields 1 (2) (2007) 2007. [16] M.S. Garcia, M.T. Lamata, A modification of the index of Liou and Wang for ranking fuzzy numbers, Int. J. Uncer. Fuzz. Know. Based Syst. 14 (4) (2007) 411–424. [17] T.W. Liao, A fuzzy multicriteria decision-making method for material selection, J. Manuf. Syst. 15 (1) (1996) 1–12. [18] C.Y. Tsai, C.C. Lo, A.C. Chang, Using fuzzy QFD to enhance manufacturing strategic planning, J. Chin. Inst. Ind. Eng. 18 (3) (2003) 33–41. [19] C.T. Tsao, T.C. Chu, An improved fuzzy MCDM model based on ideal and antiideal concepts, J. Oper. Res. Soc. Jpn. 45 (2) (2002) 185–197. [20] C.C. Wei, G.S. Liang, M.J. Wang, A comprehensive supply chain management project selection framework under fuzzy environment, Int. J. Project Manage. 25 (2007) 627–636. [21] A. Kaufmann, M.M. Gupta, Introduction to fuzzy arithmetic: theory and application, Van Nostrand Reinhold, New York, 1991. [22] M. Yamashiro, The median for a trapezoidal fuzzy number, Microelectron. Reliab. 34 (9) (1994) 1509–1511. [23] M. Yamashiro, The median for a L–R fuzzy number, Microelectron. Reliab. 35 (2) (1995) 269–271. [24] C.W. Ou, S.Y. Chou, Y.H. Chang, Using a strategy-aligned fuzzy competitive analysis approach for market segments evaluation and selection, Expert Syst. Appl. 36 (2009) 527–541. [25] T.C. Chu, Y.C. Lin, A fuzzy TOPSIS method for robot selection, Int. J. Adv. Manuf. Technol. 21 (2003) 284–290.
Contents lists available at ScienceDirect
Applied Soft Computing journal homepage: www.elsevier.com/locate/asoc
An improved ranking method for fuzzy numbers with integral values Vincent F. Yu a,∗ , Luu Quoc Dat b a
Department of Industrial Management, National Taiwan University of Science and Technology, 43, Section 4, Keelung Road, Taipei 10607, Taiwan Faculty of Development Economics, University of Economics and Business, Vietnam National University, 144 Xuan Thuy Road, Cau Giay Dist., Hanoi, Viet Nam b
a r t i c l e
i n f o
Article history: Received 29 October 2011 Received in revised form 16 February 2012 Accepted 17 October 2013 Available online 26 October 2013 Keywords: Ranking fuzzy numbers Integral value Index of optimism
a b s t r a c t Ranking fuzzy numbers is a very important decision-making procedure in decision analysis and applications. The last few decades have seen a large number of approaches investigated for ranking fuzzy numbers, yet some of these approaches are non-intuitive and inconsistent. In 1992, Liou and Wang proposed an approach to rank fuzzy number based a convex combination of the right and the left integral values through an index of optimism. Despite its merits, some shortcomings associated with Liou and Wang’s approach include: (i) it cannot differentiate normal and non-normal fuzzy numbers, (ii) it cannot rank effectively the fuzzy numbers that have a compensation of areas, (iii) when the left or right integral values of the fuzzy numbers are zero, the index of optimism has no effect in either the left integral value or the right integral value of the fuzzy number, and (iv) it cannot rank consistently the fuzzy numbers and their images. This paper proposes a revised ranking approach to overcome the shortcomings of Liou and Wang’s ranking approach. The proposed ranking approach presents the novel left, right, and total integral values of the fuzzy numbers. The median value ranking approach is further applied to differentiate fuzzy numbers that have the compensation of areas. Finally, several comparative examples and an application for market segment evaluation are given herein to demonstrate the usages and advantages of the proposed ranking method for fuzzy numbers. © 2013 Elsevier B.V. All rights reserved.
1. Introduction Ranking fuzzy numbers plays a very important role in decisionmaking, optimization, and other usages. Over the last few decades numerous ranking approaches have been proposed and investigated [1–14], with the first method for ranking fuzzy numbers proposed by Jain [1]. Chen [2] offered an approach for ranking fuzzy numbers by using maximizing set and minimizing set concepts. Liou and Wang [3] developed a ranking approach based on an integral value index to overcome the shortcomings of Chen’s [2] approach. Cheng [4] presented an approach for ranking fuzzy numbers by using the distance method. Chu and Tsao [5] proposed an approach for ranking fuzzy numbers with the area between the centroid point and original point. Abbasbandy and Asady [6] introduced an approach to rank fuzzy numbers by sign distance. Chen and Tang [7] presented an approach to rank non-normal pnorm trapezoidal fuzzy numbers with integral value. Abbasbandy and Hajjari [8] showed a new approach for ranking of trapezoidal fuzzy numbers. Wang and Luo [9] proposed an area ranking of fuzzy numbers based on positive and negative ideal points. Kumar
∗ Corresponding author. Tel.: +886 2 2737 6333; fax: +886 2 2737 6344. E-mail address: [email protected] (V.F. Yu). 1568-4946/$ – see front matter © 2013 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.asoc.2013.10.012
et al. [10] offered an approach for ranking generalized exponential fuzzy numbers using an integral value approach. Kumar et al. [11] modified Liou and Wang’s [3] approach for the ranking of an L–R type generalized fuzzy number. Chou et al. [12] presented a revised maximizing set and minimizing set ranking approach. Among the ranking approaches, Liou and Wang’s [3] method is a commonly used approach that is highly cited and has wide applications [7,10,11,15–20], but there are some shortcomings associated with their ranking approach. For the triangular and trapezoidal fuzzy numbers, Liou and Wang [3] showed that the integral values of normal and non-normal fuzzy numbers are equal. In other words, the fuzzy numbers A1 = (a, b, c, d; w1 ) and A2 = (a, b, c, d; w2 ) with (ω1 = / ω2 ), are considered the same. Cheng [4] indicated that Liou and Wang’s [3] approach could not differentiate normal and non-normal triangular/trapezoidal fuzzy numbers, because of equivalence between these fuzzy numbers. In addition, Garcia and Lamata [16] showed that if the fuzzy numbers to be compared have the compensation of areas, then they cannot be ranked by the Liou and Wang’s [3] approach. Furthermore, when the left or right integral values of fuzzy numbers are zero, the index of optimism has no effect in either the left integral value or the right integral value of the fuzzy number. Finally, Liou and Wang’s [3] approach may result in inconsistency between the ranking of the fuzzy numbers and the ranking of their images.
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To overcome all the aforementioned shortcomings with Liou and Wang’s [3] ranking approach, this paper proposes a revised approach for ranking fuzzy numbers, presenting the novel left, right, and total integral values of fuzzy numbers. In order for the proposed method to have flexibility, the decision maker’s optimistic attitude of fuzzy numbers is taken into account. The median value ranking approach is further applied to differentiate fuzzy numbers that have the compensation of areas. Finally, several comparative examples and an application for market segment evaluation are given to demonstrate the usages and advantages of the proposed ranking approach for fuzzy numbers. The results show that the proposed ranking approach can effectively overcome the shortcomings of Liou and Wang’s approach. The rest of the paper is organized as follows. Section 2 states the preliminary concepts and definitions of fuzzy numbers. Section 3 briefly reviews Liou and Wang’s ranking approach. Section 4 uses four numerical examples to show the shortcomings of Liou and Wang’ ranking approach. Section 5 proposes the revised ranking approach. Section 6 illustrates the proposed ranking method through five numerical examples. Section 7 applies the proposed ranking method for ranking fuzzy numbers to a market segment evaluation and selection problem, demonstrating its applicability and efficiency. Finally, Section 8 draws the conclusions.
2. Preliminaries This section briefly reviews the definitions of fuzzy numbers as follows. A fuzzy number is a fuzzy subset in support R (real number) that is both “normal” and “convex”, where supp(A) = { x ∈ R A > 0}. Normality implies that the maximum value of the fuzzy set A in R is 1. Therefore, the non-normal fuzzy number is ∀x ∈ R, Maxx {A (x)} <1 [4]. For convenience, the fuzzy number can be denoted by [a, b, c, d ; ], and the membership function of fuzzy number A = [a, b, c, d ; ] can be expressed as:
fA (x) =
⎧ L fA (x), a ≤ x ≤ b, ⎪ ⎪ ⎪ ⎨ ,
b ≤ x ≤ c,
(1)
fAR (x), c ≤ x ≤ d, ⎪ ⎪ ⎪ ⎩ 0,
otherwise,
where fAL : [a, b] → [0, ] and fAR : [c, d] → [0, ]. Since fAL (x) and fAR (x) are both strictly monotonical and continuous functions, their inverse functions exist and should be continuous and strictly monotonical. The inverse functions of fAL (x) and fAR (x) can be denoted by gAL : [0, ω] → [a, b] and gAR : [0, ω] → [c, d], respectively. As such, gAL (y) and gAR (y) are then integrable on the closed interval [0, ω]. In
ω
ω
other words, both 0 gAL (y) and 0 gAR (y) exist. The image (or opposite) of a fuzzy number A = (a, b, c, d ; ) can be given by fuzzy number −A = (− d, − c, − b, − a ; ω) as it is given in [21].
3. A review on Liou and Wang’s ranking approach Combining the left and right integral values, Liou and Wang [3] proposed an approach for ranking fuzzy numbers with an index of optimism ˛ ∈ [0, 1]. The left and right integral values of fuzzy
1
1
gAL (y)dy
number A are defined as IL (A) = 0
gAR (y)dy,
and IR (A) =
Fig. 1. Fuzzy numbers A1 and A2 in Example 1.
optimism ˛ ∈ [0, 1] is defined as:
IT˛ (A) = ˛IR (A) + (1 − ˛)IL (A) = ˛
1
1
gAR (y)dy + (1 − ˛) 0
gAL (y)dy 0
(2)
where gAL (y) and gAR (y) are respectively the inverse functions of fAL (x) and fAR (x). The index of optimism ˛ represents the degree of opti-
mism of a decision maker. A larger ˛ indicates a higher degree of optimism. For ˛ = 0 and ˛ = 1, the values of IT˛ (A) represent the viewpoints of pessimistic and optimistic decision makers, respectively. For a moderate decision maker, with ˛ = 0.5, the total integral value of each fuzzy number A becomes: IT0.5 (A) =
1
2
[IR (A) + IL (A)]
The greater is IT˛ (A), the bigger the fuzzy number Ai is and the higher its ranking order. 4. Shortcomings of Liou and Wang’s ranking approach This section points out the shortcomings of Liou and Wang’s [3] approach. Several examples are chosen to prove that the ranking approach, proposed by Liou and Wang, does not satisfy the reasonable properties for the ordering of fuzzy numbers. First, Liou and Wang’s method [3] cannot differentiate normal and non-normal triangular/trapezoidal fuzzy numbers [4]. For the non-normal fuzzy numbers, Liou and Wang [3] made the following assumption: “When B is a non-normal fuzzy number, fB can always be normalized by dividing the maximal value of fB before ranking”. Specifically, let the normalized fuzzy number of B and the corresponding membership function respectively be B¯ and f¯B (x) = fB (x)/ω, where ω = maxx fB (x). The integral value of B is then represented by the integral value of B when being ranked. Hence, ¯ and IR (B) = IR (B). ¯ For the case of trapezoidal and trianIL (B) = IL (B) gular fuzzy numbers, Liou and Wang proved that the total integral values of normal and non-normal fuzzy numbers are the same [7]. Example 1. Consider the triangular fuzzy numbers A1 = (3, 5, 7 ; 1) and A2 = (3, 5, 7 ; 0.8) as in Fig. 1, from [3]. Intuitively, the ranking order is A1 A2 . However, through Liou and Wang’s method, the total integral values of the triangular fuzzy numbers A1 and A2 are respectively IT˛ (A1 ) = 4 + 2˛ and IT˛ (A2 ) = 4 + 2˛. Thus, the ranking order of fuzzy numbers A1 and A2 is the same, i.e., A1 ∼ A2 . for every ˛ ∈ [0, 1]. Liou and Wang’s approach therefore fails to correctly rank the given fuzzy numbers. In addition, it is also found that when the fuzzy numbers to be compared have the compensation of areas, i.e. the fuzzy numbers A1 = (a1 , b1 , c1 , d1 ; w) and A2 = (a2 , b2 , c2 , d2 ; w) with IL (A) = IL (B) and IR (A) = IR (B), they cannot differentiate using Liou and Wang’s approach [16].
0
which reflect the pessimistic and optimistic viewpoints of the decision maker, respectively. The total integral value with an index of
Example 2. Consider the normal triangular fuzzy numbers A1 = (1,4,5) and A3 = (2,3,6) as in Fig. 2 which is taken from [16]. It
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605
Fig. 5. The novel left integral value SL (Ai ) of Ai . Fig. 2. Fuzzy numbers A1 and A2 in Example 2.
Fig. 6. The novel right integral value SR (Ai ) of Ai . Fig. 3. Fuzzy numbers A and B in Example 3.
for ˛ = 1, and −A = − B − C for ˛ = 1/2. However, using Liou and Wang’s approach, the total integral values of fuzzy numbers −A, −B, and −C are IT˛ (−A) = −4 + 2˛, IT˛ (−B) = −3.5 + ˛, and IT˛ (C) = −5 + 2.5˛, respectively. The results show that −B − A − C for ˛ = 0, −A − B = − C for ˛ = 1, and −A = − B − C for ˛ = 1/2. Obviously, Liou and Wang’s [3] approach inconsistently ranks fuzzy numbers and their images. 5. Proposed method
Fig. 4. Fuzzy numbers A, B and C in Example 4.
is easy to verify that A1 and A2 have the compensation of areas and therefore cannot be distinguished by Liou and Wang’s [3] approach. Liou and Wang’ [3] approach also used an index of optimism to reflect the decision maker’s optimistic attitude. However, when the left or right integral values of fuzzy numbers are zero, the index of optimism has no effect in either the left integral value or the right integral value of the fuzzy number. When considering the fuzzy number A = (a, b, c, d ; ω), either a = b or a + b = 0 or c = d or c + d = 0 the left and right integral values of fuzzy number A will be IL (A) = 0 or IR (A) = 0, respectively. Example 3. Consider the triangular fuzzy numbers, A = (−1,1,2), and B = (−3,−2,2) as in Fig. 3. Using Liou and Wang’s [3] approach, either the left integral value of fuzzy number A or the right integral value of fuzzy number B are 0, i.e. either IL (A) = 0 or IR (B) = 0, for all ˛ ∈ [0, 1]. Thus, the decision maker’s viewpoint or the index of optimism has no effect in either the left integral value of fuzzy number A or the right integral value of fuzzy number B. This is not reasonable. The following example shows that Liou and Wang’s [3] approach cannot consistently rank the fuzzy numbers and its images. Example 4. Fig. 4 presents three normal triangular fuzzy numbers, A = (1, 3, 5), B = (2, 3, 4), and C = (1, 4, 6). Using Liou and Wang’s [3] approach, the total integral values of fuzzy numbers A, B, and C are IT˛ (A) = 2 + 2˛, IT˛ (B) = 2.5 + ˛, and IT˛ (C) = 2.5 + 2.5˛, respectively. Thus, the ranking orders of fuzzy numbers A, B, and C are A ≺ B = C for ˛ = 0, B ≺ A ≺ C for ˛ = 1, and A = B ≺ C for ˛ = 1/2. From this, one can logically infer the ranking order of the images of these fuzzy numbers as −A − B = − C for ˛ = 0. −B − A − C
In order to overcome the shortcomings of Liou and Wang’s [3] ranking approach, this section proposes a revised ranking approach based on the novel integral values and median value of fuzzy numbers as follows. Definition 3. Suppose there are n fuzzy numbers Ai , i = 1, 2, ..., n, each with the left membership function fAL and right membership i
function fAR . The novel left and right integral values of Ai are defined i as:
bi
SL (Ai ) = ωi (bi − xmin ) − ai
SR (Ai ) = ωi (c i − xmin ) +
ci
di
fAL (x)dx,
(3)
fAR (x)dx,
(4)
i
i
n P , P = {x/f (x) 0}, w = sup f (x). where xmin = infP, P = Ui=1 Ai i i i x Ai Both SL (Ai ) and SR (Ai ) ≥ 0. The meanings of SL (Ai ) and SR (Ai ) are expressed in Figs. 1 and 2, respectively. Clearly, the fuzzy number Ai becomes larger if SL (Ai ) and SR (Ai ) are larger (see Figs. 5 and 6). The novel total integral value with index of optimism ˛ ∈ [0, 1] is then defined as:
ST˛ (Ai ) = ˛SR (Ai ) + (1 − ˛)SL (Ai )
(5)
The proposed approach uses ST˛ (Ai ) to rank fuzzy numbers. The larger ST˛ (Ai ) is, the larger is the fuzzy number Ai . Therefore, for any distinct fuzzy numbers Au and Av , we have the following properties: (1) if ST˛ (Au ) < ST˛ (Av ), then Au ≺ Av , (2) if ST˛ (Au ) > ST˛ (Av ), then Au > Av , and (3) if ST˛ (Au ) = ST˛ (Av ), then
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fAL (x) = ω(x − a)/(b − a) and fAR (x) = ω(x − d)/(c − d), respectively. Thus:
(a) if Meu > Mev , then Au > Av , (b) if Meu < Mev , then Au < Av , (c) if Meu = Mev , then Au = Av .
where Me is the median value of fuzzy numbers, and the values of Me are defined as follows [22,23]. Considering the trapezoidal fuzzy number A = (a, b, c, d), the median Me of A is derived by the following three conditions (a and d are the lower limit value and upper limit value, respectively) [22].
Me = a +
(b − a)(c + d − a − b) 2
(6)
a+b+c+d 4
Me = d −
(d − c)(c + d − a − b) 2
(8)
Definition 4. Considering the normal trapezoidal fuzzy number A = (a, b, c, d ; 1), where a < b ≤ c < d, then the left and right membership functions of fuzzy number A are fAL (x) = (x − a)/(b − a) and fAR (x) = (x − d)/(c − d), respectively. Thus, (x − a)/(b − a)dx, a
= (1/2)(a + b − 2xmin ) = IL (A) − xmin
SR (A) = c − xmin + c
(9)
d
(x − d) dx (c − d)
2
(10)
[˛(c + d) + (1 − ˛)(b + a) − 2xmin ]
= ˛IR (A) + (1 − ˛)IL (A) − xmin
(11)
Notably, when xmin = 0, formula (11) is the same as in [3]. Since triangular fuzzy numbers are special cases of trapezoidal fuzzy numbers when b = c, the total integral value of normal triangular fuzzy number A = (a, b, d ; 1) can be determined by: ST˛ (A) =
1
2
ST˛ (A) =
ω
2
2
ω
2
(c + d − 2xmin ) (13)
(a + b − 2xmin )
[˛(c + d) + (1 − ˛)(b + a) − 2xmin ]
ω
2
[˛d + b + (1 − ˛)a − 2xmin ]
(14)
(15)
(16)
Definition 6. Suppose there is an opposite fuzzy number of A = (a, b, c, d ; 1), denoted by −A = (− d, − c, − b, − a, 1). The left and L (x) = ω(x + d)/(d − c) and right membership functions of −A are f−A R (x) = ω(x + a)/(a − b), respectively. Thus: f−A
−c
SL (A) = −c − xmin − −d
=
(x + d) dx, (d − c)
−(c + d) − xmin = −IR (A) − xmin 2
and
−a
SR (A) = −b − xmin + −b
(17)
−(a + b) (x + a) = −IL (A) − xmin dx = 2 − xmin (a − b) (18)
From (17) and (18), we have: IR (− A) = − IL (A) and IL (− A) = − IR (A). Given ˛ ∈ [0, 1], the total integral value of the trapezoidal fuzzy number −A = (− d, − c, − b, − a ; 1) can therefore be obtained as:
[˛d + b + (1 − ˛)a − 2xmin ]
1
2
[−˛(a + b) − (1 − ˛)(c + d)] − xmin
= −˛IL (A) − (1 − ˛)IR (A) − xmin
Given ˛ ∈ [0, 1], the total integral value of the normal trapezoidal fuzzy number A = (a, b, c, d ; 1) can be obtained as:
1
(x − d) dx = (c − d)
d
c
ST˛ (−A) =
= (1/2)(c + d − 2xmin ) = IR (A) − xmin
ST˛ (A) =
SR (A) = ω(c − xmin ) +
ω
b
SL (A) = b − xmin −
and
and
ST˛ (A) =
In short, the total integral values, i.e., ST˛ (Ai ) values, are used to rank Au and Av if their total integral values are different. In the case for when they are equal, the fuzzy numbers Au and Av are further compared using their Me values. In order to simplify the computational procedures, the proposed ranking method is further applied for trapezoidal and triangular fuzzy numbers as follows.
ω
a
(7)
(3) When c ≤ Me ≤ d, Me is given by
(x − a) dx, = (b − a)
Since triangular fuzzy numbers are special cases of trapezoidal fuzzy numbers when b = c, the total integral value of non-normal triangular fuzzy number A = (a, b, d ; ω) can be determined by:
(2) When b ≤ Me ≤ c, Me is given by Me =
ω
Given ˛ ∈ [0, 1], the total integral value of A can be obtained as:
(1) When a ≤ Me ≤ b, Me is given by
b
SL (A) = ω(b − xmin ) −
(12)
Similarly, when xmin = 0, formula (12) is the same as in [3]. Definition 5. Consider the non-normal trapezoidal fuzzy number A = (a, b, c, d ; ). The left and right membership functions of A are
(19)
Definition 7. Suppose there are two trapezoidal fuzzy numbers A1 = (a1 , b1 , c1 , d1 ; ω1 ) and A2 = (a2 , b2 , c2 , d2 ; w2 ). This paper defines the index of optimism (˛) representing the degree of optimism of a decision maker as in Liou and Wang’s [3] ranking approach. When the two fuzzy numbers A1 and A2 are negative and satisfy a1 ≤ a2 ≤ d2 ≤ d1 , the degree of optimism of a decision maker will be defined as: when ˛ = 0, the total integral value of each fuzzy number that represents an optimistic decision maker’s viewpoint is equal to the left integral values of fuzzy numbers A1 and A2 . Conversely, for a pessimistic decision maker, i.e. ˛ = 1, the total integral value of each fuzzy number is equal to the fuzzy number’s right integral value. The total integral value of each fuzzy number represents a moderate decision maker when ˛ = 0.5. 6. Comparative examples To present the rationality and necessity for the revision of Liou and Wang’s [3] ranking approach, the following examples are
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607
employed to compare our proposed approach with their original one and other methods in the literature. Example 5. Re-consider the two normal triangular fuzzy numbers A1 = (3, 5, 7 ; 1) and A2 = (3, 5, 7 ; 0.8) in Example 1. Using the revised ranking approach, the total integral values of A1 and A2 are ST˛ (A1 ) = 1 + 2˛ and ST˛ (A2 ) = 0.8 + 1.6˛, respectively. Since ˛ = 0 for a pessimistic decision maker, the total integral values of A1 and A2 are respectively ST0 (A1 ) = 1 and ST0 (A1 ) = 0.8, and thus A1 A2 . Conversely, ˛ = 1 for an optimistic decision maker, whereby the total integral values of A1 and A2 are respectively ST0 (A1 ) = 3 and ST0 (A1 ) = 2.4, and thus A1 A2 . For a moderate decision maker, ˛ = 0.5, the total integral values of A1 and A2 are respectively ST0 (A1 ) = 2 and ST0 (A1 ) = 1.6, and thus A1 A2 . Therefore, the ranking order is A1 A2 for every ˛ ∈ [0, 1]. However, the results from Example 1 show that using Liou and Wang’ [3] ranking approach, the ranking order is A1 ∼ A2 , for every ˛ ∈ [0, 1]. Clearly, the proposed ranking approach can overcome the shortcomings of the inconsistency of Liou and Wang’s method in ranking fuzzy numbers. Example 6. The two normal triangular fuzzy numbers A1 = (1, 4, 5) and A2 = (2, 3, 6) in Example 2 are reconsidered by using the proposed ranking approach. According to Eq. (12), the total integral values of A1 and A2 are the same, i.e., ST˛ (A1 ) = ST˛ (A2 ) = 1.5 + 2˛. Since ST˛ (A1 ) = ST˛ (A2 ) for every ˛ ∈ [0, 1], the fuzzy numbers A1 and A2 are further compared using their median values, i.e., Me(A1 ) and Me(A2 ). Since Me(A1 ) = 3.45 < 3.55 = Me(A2 ), the ranking order is A2 > A1 . However, the results from Example 2 show that the ranking order of A1 and A2 is always the same, i.e., A1 ∼ A2 . Obviously, the ranking order obtained by the proposed ranking approach is more reasonable than the outcome obtained by Liou and Wang’s approach. Example 7. Re-consider the triangular fuzzy numbers, A = (−1, 1, 2) and B = (−3, − 2, 2) in Example 3. Using the proposed approach, the left, right, and total integral values of A and B are SL (A) = 3, SR (A) = 4.5, ST˛ (A) = 1.5˛ + 3, SL (B) = 0.5, SR (B) = 3 and ST˛ (B) = 2.5˛ + 0.5 respectively. Therefore, the ranking order is A B for every ˛ ∈ [0, 1]. Clearly, this example shows that by using the proposed approach, the decision maker’s viewpoint or the index of optimism is represented by both the left and right integral values of fuzzy numbers A and B. Example 8. Re-consider the three normal triangular fuzzy numbers, A = (1, 3, 5), B = (2, 3, 4), and C = (1, 4, 6) in Example 4. Using formula (12), the total integral values of fuzzy numbers A, B, and C are IT˛ (A) = 1 + 2˛, IT˛ (B) = 1.5 + ˛, and IT˛ (C) = 1.5 + 2.5˛, respectively. Thus, A ≺ B = C for ˛ = 0, B ≺ A ≺ C for ˛ = 1, and A = B ≺ C for ˛ = 1/2. The opposite of the three fuzzy number are −A = (−5, − 3, − 1), −B = (−4, − 3, − 2), and −C = (−6, − 4, − 1), respectively. Since −5 ≺ −4 ≺ −2 ≺ −1, −6 ≺ −5 ≺ −3 ≺ −1 and −6 ≺ −4 ≺ −2 ≺ −1, one can logically infer the ranking order of the images of these fuzzy numbers as −B − A − C for ˛ = 0, −A − B = − C for ˛ = 1, and −A = − B − C for ˛ = 1/2. Using the proposed method, the total integral values of fuzzy numbers −A −B, and −C are IT˛ (−A) = 2 + 2˛, IT˛ (−B) = 2.5 + ˛, and IT˛ (−C) = 1 + 2.5˛, respectively. Thus, it can be concluded that −B − A − C for ˛ = 0, −A − B = − C for ˛ = 1, and −A = − B − C for ˛ = 1/2. Again, this example shows that the proposed approach overcomes the shortcomings of the inconsistency of Liou and Wang’s approach in ranking fuzzy numbers and its images. Example 9. Consider the data used in [12], i.e., the three normal fuzzy numbers A1 = (5, 6, 7), A2 = (5.9, 6, 7) and A3 = (6, 6, 7) as shown in Fig. 7. According to formula (12), the total integral values are obtained as IT˛ (A1 ) = 0.5 + ˛, IT˛ (A2 ) = 0.95 + 0.55˛, and
Fig. 7. Fuzzy numbers A1 , A2 and A3 in Example 9.
Table 1 Comparative results of Example 9. Ranking approach
A1
A2
A3
Ranking
Wang et al. [14] Wang and Luo [9] Asady [13] Chen [2] Sign distance (p = 1) [6] Sign distance (p = 2) [6] Cheng [4] Abbasbandy and Hajjari [8]
0.25 0.5 0.66667 0.5 6.12 8.52 6.021 6
0.5339 0.571 0.81818 0.5714 12.45 8.82 6.349 6.075
0.5625 0.583 1 0.5833 12.5 8.85 6.7519 6.0834
A1 ≺ A2 ≺ A3 A1 ≺ A2 ≺ A3 A1 ≺ A2 ≺ A3 A1 ≺ A2 ≺ A3 A1 ≺ A2 ≺ A3 A1 ≺ A2 ≺ A3 A3 ≺ A2 ≺ A1 A1 ≺ A2 ≺ A3
IT˛ (A3 ) = 1 + 0.5˛. It is observed that for an optimistic decision maker, i.e. ˛ = 1, the three fuzzy numbers are the same. Conversely, for a moderate decision maker, i.e. ˛ = 0.5, or a pessimistic decision maker, i.e. ˛ = 0, A3 is the most preferred alternative, followed by A2 and then A1 This is consistent with the ranking obtained by other approaches [2,6,8,9,13,14]. Table 1 summarizes the results obtained by different methods. Note that the ranking A1 A2 A3 obtained by the CV index of Cheng [4] is thought of as unreasonable and not consistent with human intuition [9,12]. This example shows the strong discrimination power of the proposed ranking approach and its advantages.
7. Applying the proposed method for ranking fuzzy numbers to solve a market segment selection problem This section applies the proposed ranking approach to deal with market segment evaluation and selection problem. Assume that a marketing department is looking for a suitable market segment to satisfy customer needs. Three decision makers, i.e., D1 , D2 , and D3 , are responsible for the evaluation of three market segments, i.e., A1 , A2 , and A3 . Five criteria are chosen for evaluating the market segments [24]: the bargaining power of customers (C1 ), the bargaining power of suppliers (C2 ), the threat of new entrants (C3 ), the threat Table 2 Ratings of alternatives versus criteria. Criteria
Alternatives
D1
D2
D3
rij
C1
A1 A2 A3 A1 A2 A3 A1 A2 A3 A1 A2 A3 A1 A2 A3
G F G VG G G G F G VG F G G P F
G F F G G G VG G F G P G G F F
VG F G G F G G F G VG F G G F G
(0.733, 0.833, 1.0) (0.3, 0.5, 0.7) (0.567, 0.7, 0.9) (0.733, 0.833, 1.0) (0.567, 0.7, 0.9) (0.7, 0.8, 1.0) (0.733, 0.833, 1.0) (0.433, 0.6, 0.8) (0.567, 0.7, 0.9) (0.767, 0.867, 1.0) (0.267, 0.433, 0.6) (0.7, 0.8, 1.0) (0.7, 0.8, 1.0) (0.267, 0.433, 0.6) (0.433, 0.6, 0.8)
C2
C3
C4
C5
608
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Table 3 The importance weights of the criteria and the aggregated weights. Criteria
D1
D2
D3
rij
C1 C2 C3 C4 C5
VH H M H H
VH VH H M H
H VH H H H
(0.733, 0.933, 0.967) (0.733, 0.933, 0.967) (0.533, 0.7, 0.833) (0.533, 0.7, 0.833) (0.6, 0.8, 0.9)
Table 4 The left, right, and total integral values of each alternative. Alternative
SL (Ai )
SR (Ai )
ST0.5 (Ai )
Ranking
A1 A2 A3
0.321 0.099 0.251
0.539 0.315 0.479
0.430 0.207 0.365
1 3 2
of substitute products (C4 ), and the intensity of competitive rivalry (C5 ). Step 1. Aggregate ratings of alternatives versus criteria makers use the linguistic rating Assume that the decision
set S = VP, P, F, G, VG , where VP = Very Poor = (0.0, 0.1, 0.2), P = Poor = (0.2, 0.3, 0.4), F = Fair = (0.3, 0.5, 0.7), G = Good = (0.7, 0.8, 1.0), and VG = Very Good = (0.8, 0.9, 1.0), to evaluate the suitability of the alternative market segments under each criteria. Table 2 presents the suitability ratings of alternatives versus the five criteria. According to Chu and Lin’s method (2003), the aggregated suitability ratings of three alternatives, A1 , A2 and A3 , versus five criteria C1 , C2 , C3 , C4 and C5 from three decision makers can be obtained as shown in Table 2. Step 2. Aggregate the importance weights Assume that the decision
makers employ a linguistic weighting set Q = VL, L, M, H, VH , where VL = Very Low = (0.1, 0.2, 0.3), L = Low = (0.2, 0.3, 0.4), M = Medium = (0.4, 0.5, 0.7), H = High = (0.6, 0.8, 0.9), and VH = Very High = (0.8, 1.0, 1.0), to assess the importance of all the criteria. Table 3 displays the importance weights of five criteria from the three decision-makers. By Chu and Lin’s [25] approach, the aggregated weights of the criteria from the decisionmaking committee can be obtained as presented in Table 3. Step 3. Normalize the performance of alternatives versus objective criteria To put forth an easier and practical procedure, this paper defines all of the fuzzy numbers in [0,1]. The calculation of normalization is thus no longer needed. Step 4. Develop the membership function of each normalized weighted rating The final fuzzy evaluation values can be developed via arithmetic operation of the fuzzy numbers as in [25]. Step 5. Defuzzification Using Eqs. (3)–(5), the left, right, and total integral values of each alternative with ˛ = 1/2 can be obtained, as shown in Table 4. According to Table 4, the ranking order of the three market segments is A1 A3 A2 . Thus, the best selection is market segment A1 having the largest total integral value. 8. Conclusions This paper has proposed a revised ranking approach to overcome the shortcomings of Liou and Wang’s ranking approach. The
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