An improved ranking method for fuzzy numbers with integral values

Applied Soft Computing 14 (2014) 603–608 Contents lists available at ScienceDirect Applied Soft Computing j ourna l h o mepage: www.elsev ier .co An imp rs Vincent F a Department o ection b Faculty of Dev ity, 14 Viet Nam a r t i c l Article history: Received 29 O Received in re Accepted 17 O Available onlin Keywords: Ranking fuzzy Integral value Index of optim ortan een a es are ber b . Des t diffe at hav the in y num 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 num- bers 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 1. Introdu Ranking making, opt numerous r gated [1–14 proposed b fuzzy numb cepts. Liou an integral [2] approac numbers by an approac the centroi introduced Chen and T norm trape and Hajjari fuzzy numb fuzzy numb ∗ Correspon E-mail add 1568-4946/$ – method for fuzzy numbers. © 2013 Elsevier B.V. All rights reserved. ction fuzzy numbers plays a very important role in decision- imization, and other usages. Over the last few decades anking approaches have been proposed and investi- ], with the first method for ranking fuzzy numbers y Jain [1]. Chen [2] offered an approach for ranking ers by using maximizing set and minimizing set con- and Wang [3] developed a ranking approach based on value index to overcome the shortcomings of Chen’s h. Cheng [4] presented an approach for ranking fuzzy using the distance method. Chu and Tsao [5] proposed h for ranking fuzzy numbers with the area between d point and original point. Abbasbandy and Asady [6] an approach to rank fuzzy numbers by sign distance. ang [7] presented an approach to rank non-normal p- zoidal fuzzy numbers with integral value. Abbasbandy [8] showed a new approach for ranking of trapezoidal ers. Wang and Luo [9] proposed an area ranking of ers based on positive and negative ideal points. Kumar ding author. Tel.: +886 2 2737 6333; fax: +886 2 2737 6344. ress: vincent@mail.ntust.edu.tw (V.F. Yu). 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 differenti- ate 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. see front matter © 2013 Elsevier B.V. All rights reserved. rg/10.1016/j.asoc.2013.10.012roved ranking method for fuzzy numbe . Yua,∗, Luu Quoc Datb f Industrial Management, National Taiwan University of Science and Technology, 43, S elopment Economics, University of Economics and Business, Vietnam National Univers e i n f o ctober 2011 vised form 16 February 2012 ctober 2013 e 26 October 2013 numbers ism a b s t r a c t Ranking fuzzy numbers is a very imp cations. The last few decades have s numbers, yet some of these approach posed an approach to rank fuzzy num values through an index of optimism Wang’s approach include: (i) it canno rank effectively the fuzzy numbers th values of the fuzzy numbers are zero, or the right integral value of the fuzz and their images.m/locate /asoc with integral values 4, Keelung Road, Taipei 10607, Taiwan 4 Xuan Thuy Road, Cau Giay Dist., Hanoi, t decision-making procedure in decision analysis and appli- large number of approaches investigated for ranking fuzzy non-intuitive and inconsistent. In 1992, Liou and Wang pro- ased a convex combination of the right and the left integral pite its merits, some shortcomings associated with Liou and rentiate normal and non-normal fuzzy numbers, (ii) it cannot e a compensation of areas, (iii) when the left or right integral dex of optimism has no effect in either the left integral value ber, and (iv) it cannot rank consistently the fuzzy numbers 604 V.F. Yu, L.Q. Dat / Applied Soft Computing 14 (2014) 603–608 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 m mistic attitu value ranki numbers th comparativ uation are g proposed ra that the pro shortcomin The rest the prelimi tion 3 briefl 4 uses four and Wang’ ing approac through fiv ranking me evaluation and efficien 2. Prelimin This sect follows. A fuzzy that is both 0}. Normali A in R is 1 Maxx {A(x For conv d ; ], and d ; ] can b fA(x) = ⎧⎪⎪⎪⎨ ⎪⎪⎪⎩ f LA  f RA 0 where f LA : [ f RA (x) are bo inverse func tonical. The by gLA : [0, ω gLA(y) and gA other word site) of a fu number −A 3. A review Combini proposed a of optimism number A a which refle sion maker sm ˛ ˛IR( gLA(y (x). T of a d sm. F of pe oder fuzz = ( 1 2 grea its r rtco s sec roac proa able t, Lio n-no rma ump be n g”. S rresp fB(x) ented IL(B¯ uzzy of no le 1. = (3, is A1 tegr tivel f fuz 1]. Li en fuzzy numbers. ddition, it is also found that when the fuzzy numbers to be red have the compensation of areas, i.e. the fuzzy numbers 1, b1, c1, d1; w) and A2 = (a2, b2, c2, d2; w) with IL(A) = IL(B) (A) = IR(B), they cannot differentiate using Liou and Wang’s ch [16]. le 2. Consider the normal triangular fuzzy numbers ,4,5) and A3 = (2,3,6) as in Fig. 2 which is taken from [16]. Itethod to have flexibility, the decision maker’s opti- de of fuzzy numbers is taken into account. The median ng approach is further applied to differentiate fuzzy at have the compensation of areas. Finally, several e examples and an application for market segment eval- iven to demonstrate the usages and advantages of the nking approach for fuzzy numbers. The results show posed ranking approach can effectively overcome the gs of Liou and Wang’s approach. of the paper is organized as follows. Section 2 states nary concepts and definitions of fuzzy numbers. Sec- y reviews Liou and Wang’s ranking approach. Section numerical examples to show the shortcomings of Liou ranking approach. Section 5 proposes the revised rank- h. Section 6 illustrates the proposed ranking method e numerical examples. Section 7 applies the proposed thod for ranking fuzzy numbers to a market segment and selection problem, demonstrating its applicability cy. Finally, Section 8 draws the conclusions. aries ion briefly reviews the definitions of fuzzy numbers as number is a fuzzy subset in support R (real number) “normal” and “convex”, where supp(A) = { x ∈ R ∣∣A > ty implies that the maximum value of the fuzzy set . Therefore, the non-normal fuzzy number is ∀x ∈ R, )} <1 [4]. enience, the fuzzy number can be denoted by [a, b, c, the membership function of fuzzy number A = [a, b, c, e expressed as: (x), a ≤ x ≤ b, , b ≤ x ≤ c, (x), c ≤ x ≤ d, , otherwise, (1) a, b] → [0, ] and f RA : [c, d] → [0, ]. Since f LA (x) and th strictly monotonical and continuous functions, their tions exist and should be continuous and strictly mono- inverse functions of f LA (x) and f R A (x) can be denoted ] → [a, b] and gRA : [0, ω] → [c, d], respectively. As such, R(y) are then integrable on the closed interval [0, ω]. In s, both ∫ ω 0 gLA(y) and ∫ ω 0 gRA(y) exist. The image (or oppo- zzy number A = (a, b, c, d ; ) can be given by fuzzy = (− d, − c, − b, − a ; ω) as it is given in [21]. on Liou and Wang’s ranking approach ng the left and right integral values, Liou and Wang [3] n approach for ranking fuzzy numbers with an index ˛ ∈ [0, 1]. The left and right integral values of fuzzy re defined as IL(A) = 1∫ 0 gLA(y)dy and IR(A) = 1∫ 0 gRA(y)dy, ct the pessimistic and optimistic viewpoints of the deci- , respectively. The total integral value with an index of optimi I˛T (A) = where and f RA mism optimi points For a m of each I0.5T (A) The higher 4. Sho Thi [3] app ing ap reason Firs and no non-no ing ass always rankin the co f¯B(x) = repres IL(B) = gular f values Examp and A2 order total in respec order o ˛ ∈ [0, the giv In a compa A1 = (a and IR approa Examp A1 = (1Fig. 1. Fuzzy numbers A1 and A2 in Example 1. ∈ [0, 1] is defined as: A) + (1 − ˛)IL(A) = ˛ ∫ 1 0 gRA(y)dy + (1 − ˛) ∫ 1 0 gLA(y)dy (2) ) and gRA(y) are respectively the inverse functions of f L A (x) he index of optimism ˛ represents the degree of opti- ecision maker. A larger ˛ indicates a higher degree of or ˛ = 0 and ˛ = 1, the values of I˛T (A) represent the view- ssimistic and optimistic decision makers, respectively. ate decision maker, with ˛ = 0.5, the total integral value y number A becomes:) [IR(A) + IL(A)] ter is I˛T (A), the bigger the fuzzy number Ai is and the anking order. mings of Liou and Wang’s ranking approach tion points out the shortcomings of Liou and Wang’s h. Several examples are chosen to prove that the rank- ch, proposed by Liou and Wang, does not satisfy the properties for the ordering of fuzzy numbers. u and Wang’s method [3] cannot differentiate normal rmal triangular/trapezoidal fuzzy numbers [4]. For the l fuzzy numbers, Liou and Wang [3] made the follow- tion: “When B is a non-normal fuzzy number, fB can ormalized by dividing the maximal value of fB before pecifically, let the normalized fuzzy number of B and onding membership function respectively be B¯ and /ω, where ω = maxxfB(x). The integral value of B is then by the integral value of B when being ranked. Hence, ) and IR(B) = IR(B¯). For the case of trapezoidal and trian- numbers, Liou and Wang proved that the total integral rmal and non-normal fuzzy numbers are the same [7]. Consider the triangular fuzzy numbers A1 = (3, 5, 7 ; 1) 5, 7 ; 0.8) as in Fig. 1, from [3]. Intuitively, the ranking  A2. However, through Liou and Wang’s method, the al values of the triangular fuzzy numbers A1 and A2 are y I˛T (A1) = 4 + 2 ˛ and I˛T (A2) = 4 + 2˛. Thus, the ranking zy numbers A1 and A2 is the same, i.e., A1 ∼ A2. for every ou and Wang’s approach therefore fails to correctly rank V.F. Yu, L.Q. Dat / Applied Soft Computing 14 (2014) 603–608 605 is easy to ve therefore ca Liou and reflect the d left or right optimism h integral val number A = the left and or IR(A) = 0, Example 3. and B = (−3 either the l gral value o for all ˛ ∈ [0 of optimism number A o reasonable. The follo cannot cons Example 4 bers, A = (1, Wang’s [3] B, and C are respectively C are A ≺ B = From this, o of these fuz 1, an s app are espe B = − d W eir im pose rderFig. 2. Fuzzy numbers A1 and A2 in Example 2. Fig. 3. Fuzzy numbers A and B in Example 3. for ˛ = Wang’ and −C 2.5˛, r −A  − Liou an and th 5. Pro In o Fig. 4. Fuzzy numbers A, B and C in Example 4. rify that A1 and A2 have the compensation of areas and nnot be distinguished by Liou and Wang’s [3] approach. Wang’ [3] approach also used an index of optimism to ecision maker’s optimistic attitude. However, when the integral values of fuzzy numbers are zero, the index of as no effect in either the left integral value or the right ue of the fuzzy number. When considering the fuzzy (a, b, c, d ; ω), either a = b or a + b = 0 or c = d or c + d = 0 right integral values of fuzzy number A will be IL(A) = 0 respectively. Consider the triangular fuzzy numbers, A = (−1,1,2), ,−2,2) as in Fig. 3. Using Liou and Wang’s [3] approach, eft integral value of fuzzy number A or the right inte- f fuzzy number B are 0, i.e. either IL(A) = 0 or IR(B) = 0, , 1]. Thus, the decision maker’s viewpoint or the index has no effect in either the left integral value of fuzzy r the right integral value of fuzzy number B. This is not wing example shows that Liou and Wang’s [3] approach istently rank the fuzzy numbers and its images. . Fig. 4 presents three normal triangular fuzzy num- 3, 5), B = (2, 3, 4), and C = (1, 4, 6). Using Liou and approach, the total integral values of fuzzy numbers A, I˛T (A) = 2 + 2˛, I˛T (B) = 2.5 + ˛, and I˛T (C) = 2.5 + 2.5˛, . Thus, the ranking orders of fuzzy numbers A, B, and C for ˛ = 0, B ≺ A ≺ C for ˛ = 1, and A = B ≺ C for ˛ = 1/2. ne can logically infer the ranking order of the images zy numbers as −A  − B = − C for ˛ = 0. −B  − A  − C ranking app based on th bers as follo Definition each with t function f RAi as: SL(Ai) = ωi( SR(Ai) = ωi( where xmin Both SL(Ai) The mea respectively and SR(Ai) a The nov is then defi S˛T (Ai) = ˛S The prop larger S˛T (Ai distinct fuz (1) if S˛T (Au (2) if S˛T (Au (3) if S˛T (AuFig. 5. The novel left integral value SL(Ai) of Ai . Fig. 6. The novel right integral value SR(Ai) of Ai . d −A = − B  − C for ˛ = 1/2. However, using Liou and roach, the total integral values of fuzzy numbers −A, −B, I˛T (−A) = −4 + 2˛, I˛T (−B) = −3.5 + ˛, and I˛T (C) = −5 + ctively. The results show that −B  − A  − C for ˛ = 0, C for ˛ = 1, and −A = − B  − C for ˛ = 1/2. Obviously, ang’s [3] approach inconsistently ranks fuzzy numbers ages. d method to overcome the shortcomings of Liou and Wang’s [3] roach, this section proposes a revised ranking approach e novel integral values and median value of fuzzy num- ws. 3. Suppose there are n fuzzy numbers Ai, i = 1, 2, ..., n, he left membership function f LAi and right membership . The novel left and right integral values of Ai are defined bi − xmin) − ∫ bi ai f LAi (x)dx, (3) ci − xmin) + ∫ di f RAi (x)dx, (4)ci = infP, P = Un i=1Pi, Pi = {x/fAi (x)  0}, wi = supxfAi (x). and SR(Ai) ≥ 0. nings of SL(Ai) and SR(Ai) are expressed in Figs. 1 and 2, . Clearly, the fuzzy number Ai becomes larger if SL(Ai) re larger (see Figs. 5 and 6). el total integral value with index of optimism ˛ ∈ [0, 1] ned as: R(Ai) + (1 − ˛)SL(Ai) (5) osed approach uses S˛T (Ai) to rank fuzzy numbers. The ) is, the larger is the fuzzy number Ai. Therefore, for any zy numbers Au and Av, we have the following properties: ) < S˛T (Av), then Au ≺ Av, ) > S˛T (Av), then Au > Av, and ) = S˛T (Av), then 606 V.F. Yu, L.Q. Dat / Applied Soft Computing 14 (2014) 603–608 (a) if Meu > Mev, then Au > Av, (b) if Meu < Mev, then Au < Av, (c) if Meu = Mev, then Au = Av. where Me i Me are defin Conside median Me d are the low (1) When a Me = a (2) When b Me = a (3) When c Me = d In short, rank Au and for when th compared u In order ranking me fuzzy numb Definition A = (a, b, c, bership fun f RA (x) = (x − SL(A) = b − = (1/2)(a and SR(A) = c − = (1/2)(c Given ˛ fuzzy numb S˛T (A) = ( 1 2 = ˛IR(A) + Notably, Since tri fuzzy numb gular fuzzy S˛T (A) = ( 1 2 Similarly Definition A = (a, b, c, d f LA (x) = ω(x − a)/(b − a) and f RA (x) = ω(x − d)/(c − d), respectively. Thus: SL(A) = ω(b − xmin) − ∫ b ω (x − a) dx, = ( ω ) (c + d − 2xmin) (13) ω(c en ˛( ω 2 ce tri umb lar f( ω 2 tion ; 1), emb = ω( −c − (c + d 2 −b m = − I en ˛ r −A = ( IL(A tion = (a s the of a ch. en th ≤ d2 d as: pres left i essim fuzz The ate d par pres d Ws the median value of fuzzy numbers, and the values of ed as follows [22,23]. ring the trapezoidal fuzzy number A = (a, b, c, d), the of A is derived by the following three conditions (a and er limit value and upper limit value, respectively) [22]. ≤ Me ≤ b, Me is given by + √ (b − a)(c + d − a − b) 2 (6) ≤ Me ≤ c, Me is given by + b + c + d 4 (7) ≤ Me ≤ d, Me is given by − √ (d − c)(c + d − a − b) 2 (8) the total integral values, i.e., S˛T (Ai) values, are used to Av if their total integral values are different. In the case ey are equal, the fuzzy numbers Au and Av are further sing their Me values. to simplify the computational procedures, the proposed thod is further applied for trapezoidal and triangular ers as follows. 4. Considering the normal trapezoidal fuzzy number d ; 1), where a < b ≤ c < d, then the left and right mem- ctions of fuzzy number A are f LA (x) = (x − a)/(b − a) and d)/(c − d), respectively. Thus, xmin − ∫ b a (x − a)/(b − a)dx, + b − 2xmin) = IL(A) − xmin (9) xmin + ∫ d c (x − d) (c − d)dx + d − 2xmin) = IR(A) − xmin (10) ∈ [0, 1], the total integral value of the normal trapezoidal er A = (a, b, c, d ; 1) can be obtained as:) [˛(c + d) + (1 − ˛)(b + a) − 2xmin] (1 − ˛)IL(A) − xmin (11) when xmin = 0, formula (11) is the same as in [3]. angular fuzzy numbers are special cases of trapezoidal ers when b = c, the total integral value of normal trian- number A = (a, b, d ; 1) can be determined by:) [˛d + b + (1 − ˛)a − 2xmin] (12) , when xmin = 0, formula (12) is the same as in [3]. 5. Consider the non-normal trapezoidal fuzzy number ; ). The left and right membership functions of A are and SR(A) = Giv S˛T (A) = Sin fuzzy n triangu S˛T (A) = Defini b, c, d right m f R−A(x) SL(A) = = − and SR(A) = Fro IL(− A) Giv numbe S˛T (−A) = −˛ Defini bers A1 define mism approa Wh a1 ≤ a2 define that re to the for a p of each value. moder 6. Com To Liou ana (b − a) 2 − xmin) + ∫ d c ω (x − d) (c − d)dx = ( ω 2 ) (a + b − 2xmin) (14) ∈ [0, 1], the total integral value of A can be obtained as:) [˛(c + d) + (1 − ˛)(b + a) − 2xmin] (15) angular fuzzy numbers are special cases of trapezoidal ers when b = c, the total integral value of non-normal uzzy number A = (a, b, d ; ω) can be determined by:) [˛d + b + (1 − ˛)a − 2xmin] (16) 6. Suppose there is an opposite fuzzy number of A = (a, denoted by −A = (− d, − c, − b, − a, 1). The left and ership functions of −A are f L−A(x) = ω(x + d)/(d − c) and x + a)/(a − b), respectively. Thus: xmin − ∫ −c −d (x + d) (d − c)dx, ) − xmin = −IR(A) − xmin (17) − xmin + ∫ −a −b (x + a) (a − b)dx = −(a + b) 2 − xmin = −IL(A) − xmin (18) (17) and (18), we have: IR(− A) = − IL(A) and R(A). ∈ [0, 1], the total integral value of the trapezoidal fuzzy = (− d, − c, − b, − a ; 1) can therefore be obtained as: 1 2 ) [−˛(a + b) − (1 − ˛)(c + d)] − xmin ) − (1 − ˛)IR(A) − xmin (19) 7. Suppose there are two trapezoidal fuzzy num- 1, b1, c1, d1 ; ω1) and A2 = (a2, b2, c2, d2; w2). This paper index of optimism (˛) representing the degree of opti- decision maker as in Liou and Wang’s [3] ranking e two fuzzy numbers A1 and A2 are negative and satisfy ≤ d1, the degree of optimism of a decision maker will be when ˛ = 0, the total integral value of each fuzzy number ents an optimistic decision maker’s viewpoint is equal ntegral values of fuzzy numbers A1 and A2. Conversely, istic decision maker, i.e. ˛ = 1, the total integral value y number is equal to the fuzzy number’s right integral total integral value of each fuzzy number represents a ecision maker when ˛ = 0.5. ative examples ent the rationality and necessity for the revision of ang’s [3] ranking approach, the following examples are V.F. Yu, L.Q. Dat / Applied Soft Computing 14 (2014) 603–608 607 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 num- bers A1 = (3 revised ran S˛T (A1) = 1 + for a pessim A2 are resp Conversely, total integr S0T (A1) = 2.4 ˛ = 0.5, the S0T (A1) = 2 a ing order is Example 1 the ranking posed rank inconsisten bers. Example 6. 5) and A2 = ( posed rank values of A1 Since S˛T (A1 A2 are furt and Me(A2) is A2 > A1. H ing order o the ranking more reaso approach. Example 7 1, 2) and approach, t are SL(A) = 3 S˛T (B) = 2.5 A  B for ev the propose of optimism of fuzzy num Example 8. bers, A = (1, formula (12 are I˛T (A) = tively. Thus ˛ = 1/2. The opp − 1), −B = (− −5 ≺ −4 ≺ − can logicall numbers as −A = − B  − gral values I˛T (−B) = 2. be conclude and −A = − proposed a tency of Lio its images. Example 9 mal fuzzy n as shown i values are o Fig. 7. Fuzzy numbers A1, A2 and A3 in Example 9. ative results of Example 9. g approach A1 A2 A3 Ranking et al. [14] 0.25 0.5339 0.5625 A1 ≺ A2 ≺ A3 and Luo [9] 0.5 0.571 0.583 A1 ≺ A2 ≺ A3 [13] 0.66667 0.81818 1 A1 ≺ A2 ≺ A3 2] 0.5 0.5714 0.5833 A1 ≺ A2 ≺ A3 istance (p = 1) [6] 6.12 12.45 12.5 A1 ≺ A2 ≺ A3 istance (p = 2) [6] 8.52 8.82 8.85 A1 ≺ A2 ≺ A3 [4] bandy = 1 + , i.e. ˛ ode aker nd t ppro ed by ed by d no the ch a lyin ers to s sect t seg ting d cust pons A3. [24]: the bargaining power of customers (C1), the bargaining of suppliers (C2), the threat of new entrants (C3), the threat of alternatives versus criteria. a Alternatives D1 D2 D3 rij A1 G G VG (0.733, 0.833, 1.0) A2 F F F (0.3, 0.5, 0.7) A3 G F G (0.567, 0.7, 0.9) A1 VG G G (0.733, 0.833, 1.0) A2 G G F (0.567, 0.7, 0.9) A3 G G G (0.7, 0.8, 1.0) A1 G VG G (0.733, 0.833, 1.0) A2 F G F (0.433, 0.6, 0.8) A3 G F G (0.567, 0.7, 0.9) A1 VG G VG (0.767, 0.867, 1.0) A2 F P F (0.267, 0.433, 0.6) A3 G G G (0.7, 0.8, 1.0) A1 G G G (0.7, 0.8, 1.0) A2 P F F (0.267, 0.433, 0.6) A3 F F G (0.433, 0.6, 0.8), 5, 7 ; 1) and A2 = (3, 5, 7 ; 0.8) in Example 1. Using the king approach, the total integral values of A1 and A2 are 2 ˛ and S˛T (A2) = 0.8 + 1.6˛, respectively. Since ˛ = 0 istic decision maker, the total integral values of A1 and ectively S0T (A1) = 1 and S0T (A1) = 0.8, and thus A1  A2. ˛ = 1 for an optimistic decision maker, whereby the al values of A1 and A2 are respectively S0T (A1) = 3 and , and thus A1  A2. For a moderate decision maker, total integral values of A1 and A2 are respectively nd S0T (A1) = 1.6, and thus A1  A2. Therefore, the rank- A1  A2 for every ˛ ∈ [0, 1]. However, the results from show that using Liou and Wang’ [3] ranking approach, order is A1 ∼ A2, for every ˛ ∈ [0, 1]. Clearly, the pro- ing approach can overcome the shortcomings of the cy of Liou and Wang’s method in ranking fuzzy num- The two normal triangular fuzzy numbers A1 = (1, 4, 2, 3, 6) in Example 2 are reconsidered by using the pro- ing approach. According to Eq. (12), the total integral and A2 are the same, i.e., S˛T (A1) = S˛T (A2) = 1.5 + 2˛. ) = S˛T (A2) for every ˛ ∈ [0, 1], the fuzzy numbers A1 and her compared using their median values, i.e., Me(A1) . Since Me(A1) = 3.45 < 3.55 = Me(A2), the ranking order owever, the results from Example 2 show that the rank- f A1 and A2 is always the same, i.e., A1 ∼ A2. Obviously, order obtained by the proposed ranking approach is nable than the outcome obtained by Liou and Wang’s . Re-consider the triangular fuzzy numbers, A = (−1, B = (−3, − 2, 2) in Example 3. Using the proposed he left, right, and total integral values of A and B , SR(A) = 4.5, S˛T (A) = 1.5 ˛ + 3, SL(B) = 0.5, SR(B) = 3 and ˛ + 0.5 respectively. Therefore, the ranking order is ery ˛ ∈ [0, 1]. Clearly, this example shows that by using d approach, the decision maker’s viewpoint or the index is represented by both the left and right integral values bers A and B. Re-consider the three normal triangular fuzzy num- 3, 5), B = (2, 3, 4), and C = (1, 4, 6) in Example 4. Using ), the total integral values of fuzzy numbers A, B, and C 1 + 2˛, I˛T (B) = 1.5 + ˛, and I˛T (C) = 1.5 + 2.5˛, respec- , A ≺ B = C for ˛ = 0, B ≺ A ≺ C for ˛ = 1, and A = B ≺ C for osite of the three fuzzy number are −A = (−5, − 3, 4, − 3, − 2), and −C = (−6, − 4, − 1), respectively. Since 2 ≺ −1, −6 ≺ −5 ≺ −3 ≺ −1 and −6 ≺ −4 ≺ −2 ≺ −1, one y infer the ranking order of the images of these fuzzy −B  − A  − C for ˛ = 0, −A  − B = − C for ˛ = 1, and C for ˛ = 1/2. Using the proposed method, the total inte- of fuzzy numbers −A −B, and −C are I˛T (−A) = 2 + 2˛, 5 + ˛, and I˛T (−C) = 1 + 2.5˛, respectively. Thus, it can d that −B  − A  − C for ˛ = 0, −A  − B = − C for ˛ = 1, B  − C for ˛ = 1/2. Again, this example shows that the pproach overcomes the shortcomings of the inconsis- u and Wang’s approach in ranking fuzzy numbers and . Consider the data used in [12], i.e., the three nor- umbers A1 = (5, 6, 7), A2 = (5.9, 6, 7) and A3 = (6, 6, 7) n Fig. 7. According to formula (12), the total integral btained as I˛T (A1) = 0.5 + ˛, I˛T (A2) = 0.95 + 0.55˛, and Table 1 Compar Rankin Wang Wang Asady Chen [ Sign d Sign d Cheng Abbas I˛T (A3) maker for a m sion m by A2 a other a obtain obtain able an shows approa 7. App numb Thi marke marke satisfy are res A2, and ments power Table 2 Ratings Criteri C1 C2 C3 C4 C56.021 6.349 6.7519 A3 ≺ A2 ≺ A1 and Hajjari [8] 6 6.075 6.0834 A1 ≺ A2 ≺ A3 0.5˛. It is observed that for an optimistic decision = 1, the three fuzzy numbers are the same. Conversely, rate decision maker, i.e. ˛ = 0.5, or a pessimistic deci- , i.e. ˛ = 0, A3 is the most preferred alternative, followed hen A1 This is consistent with the ranking obtained by aches [2,6,8,9,13,14]. Table 1 summarizes the results different methods. Note that the ranking A1  A2  A3 the CV index of Cheng [4] is thought of as unreason- t consistent with human intuition [9,12]. This example strong discrimination power of the proposed ranking nd its advantages. g the proposed method for ranking fuzzy solve a market segment selection problem ion applies the proposed ranking approach to deal with ment evaluation and selection problem. Assume that a epartment is looking for a suitable market segment to omer needs. Three decision makers, i.e., D1, D2, and D3, ible for the evaluation of three market segments, i.e., A1, Five criteria are chosen for evaluating the market seg- 608 V.F. Yu, L.Q. Dat / Applied Soft Computing 14 (2014) 603–608 Table 3 The importance weights of the criteria and the aggregated weights. Criteria D1 D2 D3 rij C1 VH VH H (0.733, 0.933, 0.967) C2 H VH VH (0.733, 0.933, 0.967) C3 M H H (0.533, 0.7, 0.833) C4 H M H (0.533, 0.7, 0.833) C5 H H H (0.6, 0.8, 0.9) Table 4 The left, right, and total integral values of each alternative. Alternative A1 A2 A3 of substitut (C5). Step 1. A Assume set S = { VP P = Poor = (0 1.0), and V ity of the al presents th ria. Accord suitability r criteria C1, obtained as Step 2. A Assume ing set Q = L = Low = (0 0.8, 0.9), an tance of all five criteria approach, t making com Step 3. N tive criteria To put fo all of the fu is thus no lo Step 4. D weighted ra The fina metic opera Step 5. D Using Eq alternative Accordin ments is A1 having the 8. Conclus This pap come the sh 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 pro- posed 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 nor- on-n rs. nces ain, De n Cyb . Chen zy Set . Liou, t. 50 ( . Chen zy Set . Chu, nt and bbasb . 176 ( . Che bers 0–23 bbasb bers . Wan e idea umar s usin –230 umar eraliz . Chou s usin 2–13 sady, ree, E . Wan n deg eep, ng int . Garc king f –424 . Liao anuf. . Tsai, plann . Tsao al con . Wei, ject se (2007 aufm ion, Va Yama iab. 34SL(Ai) SR(Ai) S0.5T (Ai) Ranking 0.321 0.539 0.430 1 0.099 0.315 0.207 3 0.251 0.479 0.365 2 e products (C4), and the intensity of competitive rivalry ggregate ratings of alternatives versus criteria that the decision makers use the linguistic rating , P, F, G, VG } , where VP = Very Poor = (0.0, 0.1, 0.2), .2, 0.3, 0.4), F = Fair = (0.3, 0.5, 0.7), G = Good = (0.7, 0.8, G = Very Good = (0.8, 0.9, 1.0), to evaluate the suitabil- ternative market segments under each criteria. Table 2 e suitability ratings of alternatives versus the five crite- ing to Chu and Lin’s method (2003), the aggregated atings of three alternatives, A1, A2 and A3, versus five C2, C3, C4 and C5 from three decision makers can be shown in Table 2. ggregate the importance weights that the decision makers employ a linguistic weight-{ VL, L, M, H, VH } , where VL = Very Low = (0.1, 0.2, 0.3), .2, 0.3, 0.4), M = Medium = (0.4, 0.5, 0.7), H = High = (0.6, d VH = Very High = (0.8, 1.0, 1.0), to assess the impor- the criteria. Table 3 displays the importance weights of from the three decision-makers. By Chu and Lin’s [25] he aggregated weights of the criteria from the decision- mittee can be obtained as presented in Table 3. ormalize the performance of alternatives versus objec- rth an easier and practical procedure, this paper defines zzy numbers in [0,1]. The calculation of normalization nger needed. evelop the membership function of each normalized ting l fuzzy evaluation values can be developed via arith- tion of the fuzzy numbers as in [25]. efuzzification s. (3)–(5), the left, right, and total integral values of each with ˛ = 1/2 can be obtained, as shown in Table 4. g to Table 4, the ranking order of the three market seg-  A3  A2. Thus, the best selection is market segment A1 mal, n numbe Refere [1] R. J Ma [2] S.H Fuz [3] T.S Sys [4] C.H Fuz [5] T.C poi [6] S. A Sci [7] C.C num 234 [8] S. A num [9] Y.M ativ [10] A. K ber 221 [11] A. K gen [12] S.Y ber 134 [13] B. A deg [14] Z.X atio [15] K. D usi [16] M.S ran 411 [17] T.W J. M [18] C.Y gic [19] C.T ide [20] C.C pro 25 [21] A. K cat [22] M. Rellargest total integral value. ions er has proposed a revised ranking approach to over- ortcomings of Liou and Wang’s ranking approach. The [23] M. Yama (2) (1995 [24] C.W. Ou, S ysis appr 36 (2009 [25] T.C. 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