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ARTICLE IN PRESS Computers & Industrial Engineering xxx (2009) xxx–xxx

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Centroid defuzzification and the maximizing set and minimizing set ranking based on alpha level sets q Ying-Ming Wang * School of Public Administration, Fuzhou University, Fuzhou 350002, PR China

a r t i c l e

i n f o

Article history: Received 27 April 2007 Accepted 17 November 2008 Available online xxxx Keywords: Centroid defuzzification Maximizing set and minimizing set Alpha level sets Risk assessment

a b s t r a c t Centroid defuzzification and the maximizing set and minimizing set methods are two commonly used approaches to ranking fuzzy numbers and often require membership functions to be known. In this paper, the two methods are reinvestigated when explicit membership functions are not known but alpha level sets are available. Two analytical formulas are derived under the assumption that the exact membership functions can be approximated by using piecewise linear functions based on alpha level sets. The derived analytical formulas are of significant importance and provide very useful decision supports for a wide variety of applications of the two methods in industrial engineering and other areas. Numerical examples are offered to test the derived formulas and illustrate their computational processes and applications in risk assessment of a software development. Ó 2008 Elsevier Ltd. All rights reserved.

1. Introduction In many industrial engineering applications such as fuzzy weighted average (FWA) (Detyniecki & Yager, 2000; Dong & Wong, 1987; Guh, Hon, & Lee, 2001; Guh, Hon, Wang, & Lee, 1996; Guu, 2002; Kao & Liu, 1999, 2001; Lee & Park, 1997; Vanegas & Labib, 2001), fuzzy data envelopment analysis (FDEA) (Kao & Liu, 2000, 2003; León, Liern, Ruiz, & Sirvent, 2003; Wang, Greatbanks, & Yang, 2005), fuzzy multiple criteria decision making (FMCDM) (Chen & Klein, 1997a; Hon, Guh, Wang, & Lee, 1996; Tseng & Klein, 1992), and so on, explicit membership functions are often not available. The only information available is that their a-level sets (or called acuts). In order to compare or rank the fuzzy numbers without explicit membership functions but with a-level sets available, ranking approaches based on a-level sets are necessary for the fuzzy numbers. Several approaches have been suggested in the literature for ranking fuzzy numbers based on their a-level sets. Chen and Klein (1997a, 1997b) suggested an index of difference based on a-level sets, fuzzy subtraction operation and area measurement. Dubios and Parade suggested a defuzzification procedure by averaging the a-cuts, which is called averaging level cuts (ALC) (Oussalah, 2002). Yager (1981) suggested a valuation method, which ranks fuzzy numbers using valuations (see also Yager and Filev (1999) and Detyniecki and Yager (2000). Filev and Yager (1993) also suggested a generalized level set defuzzification (LSD) method. The

q

This work described in this paper was supported by the Natural Science Foundation of Fujian Province of China (No. A0710005) and the National Natural Science Foundation of China (NSFC) (No. 70771027). * Tel.: +86 591 87893307; fax: +86 591 22866677. E-mail address: [email protected].

purpose of this paper is not to suggest a new approach for defuzzification, but to derive analytical formulas for the most widely used centroid defuzzification method and the maximizing set and minimizing set ranking approach when only a-level sets are available and provide decision supports for their applications in industrial engineering and other areas. This is of significant importance because the two methods are often utilized to convert a FMCDM problem into a crisp MCDM. However, FMCDM problems can also be solved using a-level sets. If a comparison analysis needs to be conducted to find the difference between the two solution processes, the defuzzification methods employed by the two processes have to be the same; otherwise, their results will be incomparable. The rest of the paper is organized as follows. In Section 2, we briefly review the centroid defuzzification method and the maximizing set and minimizing set approach and give their formulas for known fuzzy membership functions. In Section 3, we assume that exact membership functions can be approximated by using piecewise linear functions based on a-level sets and derive analytical formulas for the two methods. Numerical examples are provided and examined in Section 4. The paper is concluded in Section 5. 2. Centroid and utilities for known membership functions ~ is a normal fuzzy number, whose membership function l ~ Let A A is defined by

8 fA~L ðxÞ; > > > > < 1; lA~ ðxÞ ¼ R > fA~ ðxÞ; > > > : 0;

a  x  b; b  x  c; c 6 x 6 d;

ð1Þ

otherwise;

0360-8352/$ - see front matter Ó 2008 Elsevier Ltd. All rights reserved. doi:10.1016/j.cie.2008.11.014

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where fA~L : ½a; b ! ½0; 1 and fA~R : ½c; d ! ½0; 1 are two continuous mappings from the real line R to the closed interval [0, 1]. The former is a strictly increasing function called left membership function and the latter is a monotonically decreasing function called right ~ is referred to as a fuzzy interval membership function. If b – c, A (Dubois & Prade, 1983), or a flat fuzzy number (Matarazzo & Munda, ~ is referred to as a trapezoi2001). If fA~L and fA~R are both linear, then A ~ ¼ ða; b; c; dÞ, which is dal fuzzy number and is usually denoted by A plotted in Fig. 1. In particular, when b = c, the trapezoidal fuzzy number is reduced to a triangular fuzzy number, denoted by ~ ¼ ða; b; dÞ. So, triangular fuzzy numbers are special cases of trapA ezoidal fuzzy numbers. The centroid defuzzification method defines the centroid coor~ in horizontal axis as its defuzzified value, which can dinate of A be expressed as (Uehara & Hirota, 1998; Wang & Luoh, 2000)

di  xMi xMi  xmin ¼ ; di  ci xmax  xmin xGi  ai xmax  xGi ¼ ; bi  ai xmax  xmin di  xMi di  xmin uMi ¼ ¼ ; di  ci ðdi  ci Þ þ ðxmax  xmin Þ xG  ai xmax  ai ¼ uGi ¼ i ; bi  ai ðbi  ai Þ þ ðxmax  xmin Þ

Rc Rd R Rd Rb L ~ Þdx þ b ðxÞdx þ c ðxf A ~ Þdx A ~ ¼ Ra xlA~ ðxÞdx ¼ a ðxf x0 ðAÞ : Rc Rd R Rb L d l ðxÞdx ðf Þdx þ dx þ ðf Þdx ~ ~ ~ A a a A b c A

uT ðiÞ ¼ ½uMi þ 1  uGi =2;

ð2Þ

~ ¼ ða; b; c; dÞ, its centroid turns out For a trapezoidal fuzzy number A to be

  dc  ab ~ ¼1 aþbþcþd x0 ðAÞ : 3 ðd þ cÞ  ða þ bÞ

ð3Þ

Especially when b = c, the above formula is simplified as

~ ¼ x0 ðAÞ

aþbþd ; 3

ð4Þ

~ ¼ ða; b; dÞ . which is the centroid of triangular fuzzy number A The maximizing set and minimizing set method ranks fuzzy ~ N be N fuzzy ~1; . . . ; A numbers according to their total utilities. Let A numbers to be compared or ranked, whose membership functions are denoted by lA~ i ðxÞ; i ¼ 1; . . . ; N. The method first defines a max~ whose membership func~ and a minimizing set G, imizing set M tions are, respectively, defined by (Chen, 1985)

(

lM~ ðxÞ ¼ (

½ðx  xmin Þ=ðxmax  xmin Þk ; xmin  x  xmax ; 0;

otherwise;

ð5Þ

ð7Þ ð8Þ ð9Þ ð10Þ

where uMi ¼ supx ðlM~ ðxÞ ^ lA~ i ðxÞÞ is referred to as the right utility ~ i and uG ¼ sup ðl ~ ðxÞ ^ l ~ ðxÞÞ as its left utility value. It value of A x G Ai i ~ i away from the minimizing set G, ~ is obvious that the farther the A ~ i to the maximizing set M, ~ the smaller the uGi and the closer the A ~ i is defined the larger the uMi . So, the final total utility value of A as (Chen, 1985)

i ¼ 1; . . . ; N:

ð11Þ

~ i and the higher its order. The N The greater the uT(i), the bigger the A ~ ~ fuzzy numbers A1 ; . . . ; AN can all be ranked according to their total utilities. 3. Centroid and utilities by a-level sets In the situations that only a-level sets are available, exact membership functions are usually not known. In this study, we assume that exact membership functions can be approximated by using piecewise linear functions based on a-level sets. ~ be a fuzzy number. Its a-level sets Aa or a-cuts Definition 1. Let A are defined as

Aa ¼ fx 2 XjlA~ ðxÞ  ag ¼ ½minfx 2 XjlA~ ðxÞ  ag; maxfx 2 XjlA~ ðxÞ  ag ¼ ½ðxÞLa ; ðxÞUa ;

0 < a  1:

ð12Þ

According to Zadeh’s extension principle (Dubois & Prade, 1980), ~ can also be expressed as the fuzzy number A

~ ¼ [ a a  Aa ; A

0 < a  1:

ð13Þ

k

lG~ ðxÞ ¼ ½ðxmax  xÞ=ðxmax  xmin Þ ; xmin  x  xmax ; 0;

otherwise;

ð6Þ

S where xmin = inf X, xmax = sup X, X ¼ Ni¼1 X i ; X i ¼ fxjlA~ i ðxÞ > 0g, and k is a constant reflecting decision maker (DM)’s attitude towards risk with k > 1 representing risk-seeking, k < 1 corresponding to risk-averse and k = 1 standing for risk-neutral (Raj & Kumar, 1999; Raja & Kumar, 1998). Usually, k is set as one. Fig. 2 shows the graphical representations of the maximizing set and minimizing set. ~ inter~ and the minimizing set G In Fig. 2, the maximizing set M sect the right and left membership functions of trapezoidal fuzzy ~ i ¼ ðai ; bi ; ci ; di Þ, respectively, at the points Mi and Gi, number A whose coordinates are determined by the following equations:

~ ¼ [a a  Aa ¼ [a a  ½ðxÞL ; ðxÞU  ð0 < a  1Þ be a Definition 2. Let A a a fuzzy number represented by a-level sets. Its membership function fA~ ðxÞ is approximately defined as

lA~ ðxÞ ¼

8 0; > > > > > > > > < ai þ > > 1; > > > > > > : ai þ

x < ðxÞLa0 or x > ðxÞUan ; Dai ðxðxÞLa Þ ðxÞLa

iþ1

i

ðxÞLa

i

; ðxÞLai  x  ðxÞLaiþ1 ; i ¼ 0; 1; . . . ; n  1; ðxÞLan  x  ðxÞUan ;

Dai ððxÞU a xÞ i

U ðxÞU a ðxÞa i

iþ1

; ðxÞUaiþ1  x  ðxÞUai ; i ¼ 0; 1; . . . ; n  1; ð14Þ

μ ~A( x)

where Dai = ai+1  ai, i = 0, 1,. . . , n  1 and 0 = a0 < a1 <  < an  1 < an = 1. Fig. 3 shows the graphical representation of the above piecewise linear membership function. Under the assumption of piecewise linearity, we have the following theorem.

~

A

1

f ~AL

0

a

~ ¼ [a a  Aa ¼ [a a  ½ðxÞL ; ðxÞU ð0 < a  1Þ be a Theorem 1. Let A a a fuzzy number represented by a-level sets, whose membership function ~ can be is defined by Eq. (14). Then the defuzzified centroid of A determined by

f A~R

b

c

d

x

Fig. 1. Membership functions of trapezoidal fuzzy numbers.

Rd ~ ¼ Ra xlA~ ðxÞdx ; x0 ðAÞ d lA~ ðxÞdx a

ð15Þ

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μ (x)

~

1

μ M~ (x)

Ai

μ G~ (x)

Gi

uG i

Mi

u Mi

0 xmin

ai

xG i bi

ci

di

xMi

xmax

x

Fig. 2. Graphical representations of maximizing set and minimizing set.

μ A~ (x) αn =1 α i+1

f

L ~ A

f

R ~ A

αi

x 0

(x) αL0 (x) αLi

(x) αLi+1

(x) αLn

(x) U αn

(x) U αi+1

(x) U αi

(x) U α0

Fig. 3. Piecewise linear membership function represented by a-level sets.

where

Z

lA~ ðxÞdx ¼

a

Remark 1. Let n = 1. Then Eqs. (18), (19), and (15) become

"

d

1 ðxÞUan  ðxÞLan  2 þ

n1 X

n1 X i¼1

ai ððxÞUaiþ1  ðxÞLaiþ1 Þ

Z

#

aiþ1 ððxÞUai  ðxÞLai Þ ;

ð16Þ

i¼0

Z

"

d

xlA~ ðxÞdx ¼

a

ð18Þ d

a

" # n1 X 1 2U 2L 2U 2L 2U 2L ððxÞa0  ðxÞa0 Þ þ ððxÞan  ðxÞan Þ þ 2 xlA~ ðxÞdx ¼ ððxÞai  ðxÞai Þ 6n i¼1 þ

1 6n

n1  X



ðxÞUai  ðxÞUaiþ1  ðxÞLai  ðxÞLaiþ1 :

d

xlA~ ðxÞdx ¼

1h 2L 2U 2L ððxÞ2U a0  ðxÞa0 Þ þ ððxÞan  ðxÞan Þ 6 i þððxÞUa0  ðxÞUan  ðxÞLa0  ðxÞLan Þ ;

ð19Þ

ð20Þ

ð21Þ

" # ðxÞUa0  ðxÞUan  ðxÞLa0  ðxÞLan 1 ; ðxÞLa0 þ ðxÞLan þ ðxÞUan þ ðxÞUa0  3 ððxÞUa0  ðxÞLa0 Þ þ ððxÞUan  ðxÞLan Þ ð22Þ

which is exactly the centroid defuzzication formula of trapezoidal fuzzy numbers (see Eq. (3)). Remark 2. If ðxÞLan ¼ ðxÞUan , then Eqs. (18), (19), and (15) can be simplified as

Z

d

lA~ ðxÞdx ¼

a

Z a

i¼0

The proof of the theorem is provided in Appendix A.

Z

 i 1 h U ðxÞa0  ðxÞLa0 þ ððxÞUan  ðxÞLan Þ ; 2

ð17Þ

Especially when Dai  1n and ai ¼ ni ; i ¼ 0; . . . ; n; the equations are simplified as " # Z d n1 X 1 lA~ ðxÞdx ¼ ððxÞUai  ðxÞLai Þ ; ððxÞUa0  ðxÞLa0 Þ þ ððxÞUan  ðxÞLan Þ þ 2 2n a i¼1

Z

a

~ ¼ x0 ðAÞ

i¼0 n1 1X Dai ððxÞUai  ðxÞUaiþ1  ðxÞLai  ðxÞLaiþ1 Þ: 6 i¼0

lA~ ðxÞdx ¼

a

n1 X 1 2L 2L ai ððxÞ2U ðxÞ2U an  ðxÞan  aiþ1  ðxÞaiþ1 Þ 6 i¼1 # n1 X 2U 2L þ aiþ1 ððxÞai  ðxÞai Þ

þ

d

d

" # n1 X 1 ððxÞUai  ðxÞLai Þ ; ððxÞUa0  ðxÞLa0 Þ þ 2 2n i¼1

" n1 X 1 2L 2L ððxÞ2U xlA~ ðxÞdx ¼ ððxÞ2U a0  ðxÞa0 Þ þ 2 ai  ðxÞai Þ 6n i¼1 # n1 X U U L L þ ððxÞai  ðxÞaiþ1  ðxÞai  ðxÞaiþ1 Þ ;

ð23Þ

ð24Þ

i¼0

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2L ððxÞ2U a0  ðxÞa0 Þ þ 2

~ ¼1 x0 ðAÞ 3

n1 P i¼1

2L ððxÞ2U ai  ðxÞai Þ þ

ððxÞUa0  ðxÞLa0 Þ þ 2

nP 1

i¼0 n1 P i¼1

ððxÞUai  ðxÞUaiþ1  ðxÞLai  ðxÞLaiþ1 Þ ð25Þ

:

ððxÞUai  ðxÞLai Þ

Remark 3. Let n ¼ 1: Then Eqs. (23)–(25) become

uM ¼

aj0 þ1  ððxÞUaj  xmin Þ  aj0  ððxÞUaj þ1  xmin Þ 0

0 þ1

0

Z

d

a

1 2

lA~ ðxÞdx ¼ ððxÞUa0  ðxÞLa0 Þ;

ð26Þ

uG ¼

d

a

xlA~ ðxÞdx ¼

1 2L U U L ½ððxÞ2U a0  ðxÞa0 Þ þ ððxÞa0  ðxÞan  ðxÞa0 6  ðxÞLan Þ;

0

ðxÞLai

2U 2L U U L L ~ ¼ 1  ððxÞa0  ðxÞa0 Þ þ ððxÞa0  ðxÞan  ðxÞa0  ðxÞan Þ x0 ðAÞ 3 ððxÞUa  ðxÞLa Þ

¼

ð35Þ

:

ð36Þ

0

Remark 5. Let n = 1. Then Eqs. (35) and (36) become

uM ¼

0

1 ððxÞUa0 þ ðxÞUan þ ðxÞLa0 Þ; 3

0

 ðxÞLai þ ðai0 þ1  ai0 Þðxmax  xmin Þ

;

The proof of the theorem is provided in Appendix B.

ð27Þ

0

þ ðaj0 þ1  aj0 Þðxmax  xmin Þ

ai0 þ1  ðxmax  ðxÞLai Þ  ai0  ðxmax  ðxÞLai þ1 Þ 0 þ1

Z

0

ðxÞUaj  ðxÞUaj

ð28Þ

which is exactly the centroid defuzzication formula of triangular fuzzy numbers (see Eq. (4)). Remark 4. The equations below developed in Oussalah (2002), Uehara and Hirota (1998), Yager (1981), Yager and Filev (1999), respectively, are not the centroid defuzzification formulas:

uG ¼

ðxÞUa0  xmin U

;

ð37Þ

;

ð38Þ

ðxÞa0  ðxÞUan þ ðxmax  xmin Þ xmax  ðxÞLa0 ðxÞLan  ðxÞLa0 þ ðxmax  xmin Þ

which are exactly the same as Eqs. (9) and (10). In order to determine which intervals the intersection points G and M lie in, we introduce the following sign functions:

S1 ðaÞ ¼ lG~ ððxÞLa Þ  a;

ð39Þ

U

L

U

S2 ðaÞ ¼ lM~ ððxÞa Þ  a:

!

n ðxÞai þ ðxÞai 1X ; n i¼1 2 P ½ððxÞUai  ðxÞLai ÞððxÞUai þ ðxÞLai Þ

xALC ¼

x ¼

i

~ ¼ ValðAÞ

2 Z

P i

1

Av eðAa Þda ¼

0

0

1

L

U

S1 ðaÞ > 0; a  ai0

and

ðxÞa þ ðxÞa da: 2

0

ð31Þ

ð32Þ

~ ¼ ða; b; cÞ, (31) is simplified as For triangular fuzzy numbers A



S2 ðaÞ > 0; a  aj0 S2 ðaÞ < 0; a  aj0 þ1

0 þ1

 and ½ðxÞUaj

0 þ1

; ðxÞUaj  can be readily 0

determined. Tables 2 and 3 show an illustrative example. 4. Numerical examples In this section, we examine two numerical examples. One is a test example, in which exact membership function is known. The purpose of the test is to verify whether the results based on a-level sets are identical with those obtained from known membership function. The other is a risk assessment example, which is taken from Chen (2001).

ð33Þ

The computation of defuzzified centroid can be easily implemented on a table. Table 1 shows an illustrative example.

Table 1 Alpha-level sets and the computation of defuzzified centroid.

~ ¼ [a a  ðAÞ ¼ [a a  ½ðxÞL ; ðxÞU  ð0 < a  1Þ be a Theorem 2. Let A a a a fuzzy number represented by a-level sets. Its membership function is ~ defined by (5) with defined by (14). Suppose the maximizing set M at k = 1 intersects the right membership function fA~R ~ defined by (6) with xM 2 ½ðxÞUaj þ1 ; ðxÞUaj  and the minimizing set G 0 0 k = 1 intersects the left membership function fA~L at xG 2 ½ðxÞLai ; ðxÞLai þ1 , 0 0 ~ can be determined as shown in Fig. 4. Then the total utility value of A by

ai

ðxÞLai

ðxÞUai

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 Sum

34.78 38.18 41.64 45.15 48.72 52.33 55.98 59.66 63.36 67.08 70.81

99.11 96.38 93.62 90.86 88.08 85.28 82.46 79.61 76.72 73.79 70.81

uT ¼ ðuM þ 1  uG Þ=2; where

:

So, by checking the signs of the above two functions and observing their changes, ½ðxÞLai ; ðxÞLai

!

~ ¼ a þ b þ c þ d: ValðAÞ 4

a þ 2b þ c : 4



ð30Þ

~ ¼ ða; b; c; dÞ, (31) turns Especially for trapezoidal fuzzy numbers A out to be

~ ¼ ValðAÞ

It can be seen very clearly from Fig. 4 that

S1 ðaÞ < 0; a  ai0 þ1 ;

ððxÞUai  ðxÞLai Þ Z

ð29Þ

ð40Þ

ð34Þ

ðxÞUai  ðxÞLai 64.33 58.20 51.98 45.71 39.36 32.95 26.48 19.95 13.36 6.71 359.03

2L ðxÞ2U ai  ðxÞai

ðxÞLai  ðxÞLaiþ1

ðxÞUai  ðxÞUaiþ1

8613.144 7831.392 7030.815 6217.017 5384.448 4534.250 3665.891 2778.437 1871.469 945.238

1327.900 1589.815 1880.046 2199.708 2549.518 2929.433 3339.767 3780.058 4250.189 4749.935

9552.222 9023.096 8506.313 8002.949 7511.462 7032.189 6564.641 6107.679 5661.169 5225.070

48872.1

28596.37

73186.79

~ ¼ 1 8613:144þ2 ð48872:18613:144Þþð73186:7928596:37Þ ¼ 68:18  x0 ðAÞ 64:33þ2 ð359:0364:33Þ 3

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μ M~

~ α n = 1 μG α i +1

0

α i0

uG

αj

0+1

uM

αj

0

x

(x) αLi

xmin

0

(x) αLi

(x) Uα j (x) Uα

0 +1

0+1

xmax

j 0

Fig. 4. Maximizing and minimizing sets for fuzzy numbers represented by a-level sets.

Table 2 Alpha-level sets for the overall risk scores of three new products development (NPD) and the changes of their sign functions.

a

Alpha-level sets

Changes of sign functions

NPD1 ðxÞLa 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 xmin ¼ 5:19;

NPD2 ðxÞUa

ðxÞLa

NPD3 ðxÞUa

NPD1

ðxÞLa

34.78 99.11 25.71 89.21 5.19 38.18 96.38 28.58 86.04 6.29 41.64 93.62 31.55 82.89 7.52 45.15 90.86 34.61 79.77 8.91 48.72 88.08 37.77 76.67 10.46 52.33 85.28 41.01 73.59 12.2 55.98 82.46 44.33 70.53 14.16 59.66 79.61 47.74 67.49 16.36 63.36 76.72 51.23 64.46 18.84 67.08 73.79 54.78 61.43 21.63 70.81 70.81 58.41 58.41 24.79 L U xmax ¼ 99:11; S1 ¼ ðxmax  ðxÞa Þ=ðxmax  xmin Þ  a and S2 ¼ ððxÞa  xmax Þ=ðxmax

Table 3 Utilities of the overall risk scores of the three new products development and their rankings. New product development

uM

uG

uT

Rank

NPD1 NPD2 NPD3

0.7706 0.6723 0.4345

0.4986 0.5876 0.8421

0.6360 0.5423 0.2962

1 2 3

~ ¼ ð3; 5; 7; 10Þ Example 1. Given the trapezoidal fuzzy number A and its a-level sets at a = 0, 0.1,. . . , 1.0, respectively, together with ~ ¼ ð0; 15; 15Þ and the minimizing set the maximizing set M ~ ¼ ð0; 0; 15Þ. G By Eqs. (3), and (9), (10), (11), we have its defuzzified centroid and utilities based on known membership function as follows:

  dc  ab ~ ¼1 aþbþcþd x0 ðAÞ 3 ðd þ cÞ  ða þ bÞ   1 10 7  3 5 ¼ 3 þ 5 þ 7 þ 10  ¼ 6:2963; 3 ð10 þ 7Þ  ð3 þ 5Þ uM ¼

d  xmin 10  0 ¼ 0:5556; ¼ ðd  cÞ þ ðxmax  xmin Þ ð10  7Þ þ ð15  0Þ

uG ¼

xmax  a 15  3 ¼ 0:7059; ¼ ðb  aÞ þ ðxmax  xmin Þ ð5  3Þ þ ð15  0Þ

NPD2

NPD3

ðxÞUa

S1

S2

S1

S2

S1

S2

61.85 58.23 54.6 50.94 47.27 43.58 39.87 36.13 32.37 28.59 24.79  xmin Þ  a

+ + + + +      

+ + + + + + + +   

+ + + + + +     

+ + + + + + +    

+ + + + + + + + +  

+ + + + +      

uT ¼

uM þ 1  uG 0:5556 þ 1  0:7059 ¼ 0:4248: ¼ 2 2

The defuzzified centroid and utilities based on a-level sets are computed in Tables 4 and 5, form which it can be seen quite clearly that the results based on a-level sets are identical with those obtained from known membership function. This verifies the validity of the two analytical formulas developed in this study.

Table 4 ~ ¼ ð3; 5; 7; 10Þ and the computation Alpha-level sets of the trapezoidal fuzzy number A of its defuzzified centroid.

ai

ðxÞLai

ðxÞUai

ðxÞUai  ðxÞLai

2L ðxÞ2U ai  ðxÞai

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 Sum

3.0 3.2 3.4 3.6 3.8 4.0 4.2 4.4 4.6 4.8 5.0

10.0 9.7 9.4 9.1 8.8 8.5 8.2 7.9 7.6 7.3 7.0

7.0 6.5 6.0 5.5 5.0 4.5 4.0 3.5 3.0 2.5 2.0 49.5

91.00 83.85 76.80 69.85 63.00 56.25 49.60 43.05 36.60 30.25 24.00 624.25

ðxÞLai  ðxÞLaiþ1

ðxÞUai  ðxÞUaiþ1

9.60 10.88 12.24 13.68 15.20 16.80 18.48 20.24 22.08 24.00

97.00 91.18 85.54 80.08 74.80 69.70 64.78 60.04 55.48 51.10

163.20

729.70

~ ¼ 1 91þ24þ2 ð624:259124Þþð729:7163:2Þ ¼ 6:2963  x0 ðAÞ 7þ2þ2 ð4972Þ 3

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Despite the fact that the problem has been investigated many times, all the investigations focus on transforming the problem into a crisp multiple attribute decision analysis. There has been no attempt to solve the problem using a-level sets and the extension principle, which are supposed to provide exact solution to the problem. So, in this study, the problem is regarded as a fuzzy weighted average (FWA) and solved of using a-level sets and the extension principle. Assessments of risk items under each attribute are first aggregated using FWA to generate an aggregated assessment for the attribute and the aggregated assessments of the six attributes are then aggregated again to generate an overall risk assessment for each software project. Eleven a-levels, a = 0, 0.1, . . . , 1.0, are set for computation. The overall risk assessments for the three projects are presented in Table 7, together with their defuzzified centroids, utilities and rankings. The rankings turn out to be the same as those obtained by Chen (2001), but our approach provides much more information on the overall risk of each project and also offers a full picture rather than only a point estimate for the overall risk assessment of each project, as shown in Fig. 6.

Table 5 ~ ¼ ð3; 5; 7; 10Þ and the computation Alpha-level sets of the trapezoidal fuzzy number A of its utilities.

ai

ðxÞLai

ðxÞUai

S1(ai)

S2(ai)

Utilities

0.0 0.1 0.2 0.3

3.0 3.2 3.4 3.6

10.0 9.7 9.4 9.1

+ + + +

+ + + 

¼ 0:7059 uG ¼ 1:0 ð154:8Þ0:9 ð155:0Þ ð5:04:8Þþ0:1 ð150Þ

0.4 0.5

3.8 4.0

8.8 8.5

+ +

 

uM ¼ 0:3 ð9:40Þ0:2 ð9:10Þ ð9:49:1Þþ0:1 ð150Þ ¼ 0:5556

0.6 0.7 0.8 0.9 1.0

4.2 4.4 4.6 4.8 5.0

8.2 7.9 7.6 7.3 7.0

+ + + + 

    

G ¼ 0:4248 uT ¼ uM þ1u 2

Example 2. Consider a software development risk assessment problem which was investigated by Chen (2001), Lee (1996a, 1996b, 1999) and Lee, Lee, Lee, and Chen (2003, 2004). The hierarchical structure for the assessment problem is shown in Fig. 5 and consists of six attributes and 14 risk items (or called risk factors). The risk of each item is defined as the product of grade of risk and grade of importance that are described by linguistic variables and characterized by triangular fuzzy numbers. Table 6 shows the assessments of each risk item and the weights of the attributes for three software projects (A1, A2 and A3) provided by two experts (decision makers). For brevity, the original assessments and weights presented in Chen (2001) by two experts have been averaged as a whole in Table 6.

5. Concluding remarks Many applications of fuzzy set theory require defuzzification and ranking approaches based on alpha level sets because exact membership functions may not always be available. In this paper, we have assumed that exact membership functions can be approximated using piecewise linear functions based on alpha level sets and derived two analytical formulas to meet such a requirement. The two formulas are, respectively, extensions of the most widely used centroid defuzzification approach and the maximizing set and minimizing set method to alpha level sets. One is to capture the

Attribute W1

X1: Personnel

Risk item W11

X11: Personnel shortfalls, key person(s) quit W21 W22

W2

X2: System requirement

W23 W24

X21: Requirement ambiguity X22: Developing the wrong software function X23: Developing the wrong user interface X24: Continuing stream requirement changes

W31 W3

X3: Schedules and budgets

X31: Schedule not accurate W32

X32: Budget not sufficient

W41

Aggregative risk assessment

X41: Gold-piating W42 W4

W5

W6

X4: Developing technology

X5: External resource

X6: Performance

W43

X42: Skill levels inadequate X43: Straining hardware

W44

X44: Straining software

W51

X51: Shortfalls in externally furished components

W52

X52: Shortfalls in externally performed tasks

W61

X61: Real-time performance shortfalls

Fig. 5. Hierarchical structure for software development risk assessment (Chen, 2001).

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Y.-M. Wang / Computers & Industrial Engineering xxx (2009) xxx–xxx Table 6 Weights and risk assessment data for three software development projects. Risk item

X1 X11

X2 X21

X22

Weight

Grade of risk

(0.125, 0.275, 0.425) (0.75, 0.875, 1) A1: (0.45, 0.55, 0.65) A2: (0.65, 0.75, 0.85) A3: (0, 0.05, 0.15) (0.275, 0.45, 0.6) (0.2, 0.325, A1: (0.2, 0.3, 0.4) 0.425) A2: (0.7, 0.8, 0.9) A3: (0.15, 0.25, 0.35) (0.2, 0.35, 0.5)

Grade of importance (0.5, 0.675, 0.8) (0.7, 0.8, 0.9) (0, 0, 0.1) (0.4, 0.5, 0.6) (0.7, 0.8, 0.9) (0.1, 0.2, 0.3)

A1: (0.25, 0.4, 0.55) A2: (0.5, 0.6, 0.7) A3: (0, 0.05, 0.15)

(0.3, 0.4, 0.5) (0.6, 0.7, 0.8) (0, 0.1, 0.2)

X23

(0.175, 0.3, 0.4)

A1: (0.25, 0.45, 0.55) A2: (0.7, 0.8, 0.9) A3: (0.05, 0.15, 0.25)

(0.25, 0.35, 0.45) (0.6, 0.7, 0.8) (0.05, 0.15, 0.25)

X24

(0.15, 0.325, 0.5)

A1: (0.3, 0.4, 0.5)

(0.25, 0.375, 0.5)

A2: (0.7, 0.8, 0.9) A3: (0.15, 0.25, 0.35)

(0.7, 0.8, 0.9) (0.1, 0.2, 0.3)

X3 X31

X32

X4 X41

(0.15, 0.25, 0.35) (0.225, 0.35, A1: (0.3, 0.4, 0.5) 0.55) A2: (0.7, 0.8, 0.9) A3: (0.1, 0.2, 0.3) (0.3, 0.525, 0.625)

A1: (0.3, 0.5, 0.7)

(0.2, 0.3, 0.4)

A2: (0.5, 0.6, 0.7) A3: (0.15, 0.25, 0.35)

(0.6, 0.7, 0.8) (0.15, 0.25, 0.35)

(0.2, 0.325, 0.45) (0.25, 0.35, A1: (0.225, 0.35, 0.45) 0.475) A2: (0.65, 0.75, 0.85) A3: (0.2, 0.3, 0.4)

ðR2 ÞUa

ðR3 ÞLa

ðR3 ÞUa

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

0.077 0.086 0.096 0.106 0.116 0.127 0.138 0.150 0.161 0.174 0.186

0.345 0.325 0.306 0.288 0.271 0.255 0.240 0.226 0.212 0.199 0.186

0.327 0.345 0.363 0.380 0.398 0.417 0.435 0.454 0.473 0.492 0.511

0.719 0.696 0.674 0.653 0.632 0.611 0.591 0.571 0.551 0.531 0.511

0.009 0.011 0.014 0.017 0.020 0.024 0.028 0.032 0.037 0.041 0.046

0.116 0.108 0.100 0.092 0.085 0.077 0.071 0.064 0.058 0.052 0.046

Centroid Total utility Rank

0.198 0.294 2

0.517 0.665 1

Project 1

0.055 0.092 3

Project 2

Project 3

1 0.8 0.6

(0.2, 0.3, 0.4)

A1: (0.3, 0.4, 0.5) A2: (0.7, 0.8, 0.9) A3: (0.15, 0.25, 0.35)

(0.2, 0.3, 0.4) (0.65, 0.75, 0.85) (0.2, 0.3, 0.4)

X44

(0.2, 0.3, 0.4)

A1: (0.2, 0.3, 0.4) A2: (0.6, 0.7, 0.8) A3: (0.15, 0.25, 0.35)

(0.2, 0.3, 0.4) (0.65, 0.75, 0.85) (0.15, 0.25, 0.35

X51

(0.1, 0.275, 0.4) (0.45, 0.55, 0.65)

A1: (0.2, 0.3, 0.4)

(0.45, 0.55, 0.65)

(0.5, 0.6, 0.7)

A2: (0.55, 0.65, 0.75) A3: (0.15, 0.25, 0.35) A1: (0.3, 0.4, 0.5) A2: (0.7, 0.8,0.9) A3: (0.15, 0.25, 0.35) A1: (0.4, 0.5, 0.6) A2: (0.5, 0.6, 0.7) A3: (0.15, 0.25, 0.35)

0.2 0

0

0.1

0.2

0.3

(0.55, 0.65, 0.75) (0.1, 0.2, 0.3)

X43

X61

ðR2 ÞLa

(0.275, 0.4, 0.525)

(0.1, 0.25, 0.4) (0.65, 0.75, 0.85) (0.1, 0.2, 0.3)

(0.125, 0.3, 0.5) (0.85, 0.95, 1)

Project 3

ðR1 ÞUa

0.4

A1: (0.15, 0.25, 0.35) A2: (0.65, 0.75, 0.85) A3: (0.05, 0.15, 0.25)

X6

Project 2

ðR1 ÞLa

(0.6, 0.7, 0.8) (0.05, 0.15, 0.25)

(0.1, 0.2, 0.3)

X52

Project 1

(0.225, 0.35, 0.45)

X42

X5

a

Alpha

Attribute

Table 7 Alpha-level sets for the overall risks of the three software projects at different alevels.

(0.55, 0.65, 0.75) (0.2, 0.3, 0.4) (0.45, 0.55, 0.65) (0.7, 0.8, 0.9) (0.15, 0.25, 0.35)

0.5

0.6

0.7

Fig. 6. The overall risks of the three software development projects.

also been illustrated in detail. A software development risk assessment example has been reinvestigated as a fuzzy weighted average and has demonstrated the applications of alpha level sets and the two formulas. It can be concluded that the two formulas provide very helpful support to the applications of fuzzy logic. Appendix A. The proof of Theorem 1 From Definition 2 and Fig. 3, it is known that

Z

d

lA~ ðxÞdx ¼

a

Z

þ

a

1

lA~ ðxÞdx þ    þ

ðxÞLa

þ

Z

ðxÞLa

Z

d

xlA~ ðxÞdx ¼

Z

ðxÞU an

lA~ ðxÞdx þ

ðxÞU a

0

ðxÞLa

1

ðxÞLa

Z

ðxÞU a

n1

ð41Þ

xlA~ ðxÞdx þ    þ

0

þ þ

Z

Z

ðxÞLan

ðxÞLa

xlA~ ðxÞdx

n1

ðxÞU an

ðxÞLan

Z

lA~ ðxÞdx þ   

ðxÞU an

lA~ ðxÞdx;

ðxÞU a1

Z

lA~ ðxÞdx

n1

ðxÞLan

Z

ðxÞLan

ðxÞLa

0

(0.25, 0.35, 0.45) (0.45, 0.55, 0.65) (0.2, 0.3, 0.4)

centroid of a fuzzy number from its alpha level sets. The other is to calculate the utilities of a fuzzy number using its alpha level sets and a predetermined maximizing set and minimizing set. The validity of the two formulas has been examined and verified through a test example and their computational processes have

0.4

Overall risk

ðxÞU a

ðxÞU a1

0

xlA~ ðxÞdx þ xlA~ ðxÞdx:

Z

ðxÞU a

ðxÞU an

n1

xlA~ ðxÞdx þ    ð42Þ

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Let

Z

Q iL ¼

iþ1

lA~ ðxÞdx; i ¼ 0; 1; . . . ; n  1;

ðxÞLa

ðxÞU an

lA~ ðxÞdx;

ðxÞLan

Q iU ¼

Z

i

ð44Þ

xlA~ ðxÞdx;

ð46Þ

¼

iþ1 iþ1

i ¼ 0; 1; . . . ; n  1;

Z

xlA~ ðxÞdx;

ðxÞU a

ð47Þ

Z

xlA~ ðxÞdx;

ðxÞU a

# dx

ai

2U ððxÞ2U ai  ðxÞaiþ1 Þ   Dai 1 U 1 2U 3U 3U þ U ðxÞai ððxÞ2U ai  ðxÞaiþ1 Þ  ððxÞai  ðxÞaiþ1 Þ U 2 3 ðxÞa  ðxÞa

2

ai

iþ1

Dai U U ððxÞai  ðxÞ2U ððxÞ2U aiþ1 Þ þ ai þ ðxÞai  ðxÞaiþ1 2 6  2ðxÞ2U aiþ1 Þ; i ¼ 0; 1; . . . ; n  1: 2U

lA~ ðxÞdx ¼ Q m þ

ð54Þ

i ¼ 0; 1; . . . ; n  1:

ð48Þ

"

iþ1

ai þ

ðxÞLa

# Dai ðx  ðxÞLai Þ ðxÞLaiþ1  ðxÞLai

i

dx

¼ ai ððxÞLaiþ1  ðxÞLai Þ   Dai 1 2L 2L L L L  ðxÞ Þ  ðxÞ ððxÞ  ðxÞ Þ þ L ððxÞ aiþ1 ai ai aiþ1 ai ðxÞaiþ1  ðxÞLai 2 1 ¼ ðai þ Dai ÞððxÞLaiþ1  ðxÞLai Þ; 2

ðQ iL þ Q iU Þ

¼ ðxÞUan  ðxÞLan n1  X 1 þ ðai þ Dai ÞððxÞLaiþ1  ðxÞLai Þ 2 i¼0  1 þðai þ Dai ÞððxÞUai  ðxÞUaiþ1 Þ 2 n1 X 1 ¼ ðxÞUan  ðxÞLan þ ðai þ Dai Þ½ððxÞUai  ðxÞLai Þ 2 i¼0

Then we have ðxÞLa

n1 X i¼0

iþ1

Q iL ¼

ðxÞUai  ðxÞUaiþ1

d

a

i

Z

x ai þ

ðxÞU a

Dai ððxÞUai  xÞ

Further, we have

i

ðxÞU an

ðxÞLan

RiU ¼

¼

ð45Þ

ðxÞLa

Rm ¼

ð43Þ

lA~ ðxÞdx; i ¼ 0; 1; . . . ; n  1;

ðxÞLa

Z

i

i

ðxÞU a

ðxÞU a

Z

"

ðxÞU a

iþ1

i

Z

Qm ¼

RiL ¼

RiU ¼

ðxÞLa

Z

 ððxÞUaiþ1  ðxÞLaiþ1 Þ " n1 X 1 ðxÞUan  ðxÞLan  ai ððxÞUaiþ1  ðxÞLaiþ1 Þ ¼ 2 i¼1 # n1 X U L þ aiþ1 ððxÞai  ðxÞai Þ ;

i ¼ 0; 1; . . . ; n  1; ð49Þ

ð55Þ

i¼0

Z

Qm ¼

ðxÞU an

lA~ ðxÞdx ¼

ðxÞLan

Q iU ¼

Z

"

ðxÞU a

i

ai þ

ðxÞU a

iþ1

Z

ðxÞU an

ðxÞLan

dx ¼ ðxÞUan  ðxÞLan ;

Z a

Dai ððxÞUai  xÞ U

ð50Þ

#

U

ðxÞai  ðxÞaiþ1

d

xlA~ ðxÞdx ¼ Rm þ

1 2L ½ðxÞ2U an  ðxÞan  2 n1  X ai Dai 2L þ ððxÞ2L ð2ðxÞ2L aiþ1  ðxÞai Þ þ aiþ1 2 6 i¼0 i L L ðxÞ2L ai  ðxÞai  ðxÞaiþ1 Þ n1  X ai Dai 2U þ ððxÞ2U ððxÞ2U ai  ðxÞaiþ1 Þ þ ai 2 6 i¼0 i þðxÞUai  ðxÞUaiþ1  2ðxÞ2U aiþ1 Þ

¼

dx

¼ ai ððxÞai  ðxÞUaiþ1 Þ   Dai 1 2U ðxÞUai ððxÞUai  ðxÞUaiþ1 Þ  ððxÞ2U þ U ai  ðxÞaiþ1 Þ U 2 ðxÞai  ðxÞaiþ1    1 ¼ ai þ Dai ðxÞUai  ðxÞUaiþ1 ; i ¼ 0; 1; . . . ; n  1; 2 ð51Þ RiL ¼

ðxÞLa

iþ1

ðxÞLa

" x ai þ

# Dai ðx  ðxÞLai Þ ðxÞLaiþ1  ðxÞLai

i

¼

ai

2L

dx

Z

n1 1X 2L Dai ½ððxÞ2U ai  ðxÞai Þ 6 i¼0

2L þððxÞUai  ðxÞUaiþ1  ðxÞLai  ðxÞLaiþ1 Þ  2ððxÞ2U aiþ1  ðxÞaiþ1 Þ

i

Dai 2L L ¼ ððxÞaiþ1  ðxÞai Þ þ ð2ðxÞ2L aiþ1  ðxÞai  ðxÞai 2 6  ðxÞLaiþ1 Þ; i ¼ 0; 1; . . . ; n  1; 2L

n1 1 1X 2L 2L ai ½ððxÞ2U ½ðxÞ2U an  ðxÞan  þ ai  ðxÞai Þ 2 2 i¼0 2L ððxÞ2U aiþ1  ðxÞaiþ1 Þ þ

2L

ððxÞaiþ1  ðxÞai Þ   Da i 1 1 L 3L 3L 2L 2L  ðxÞ Þ  ððxÞ  ðxÞ Þ ððxÞ ðxÞ þ L aiþ1 ai aiþ1 ai 2 ai ðxÞa  ðxÞLa 3 iþ1

Rm ¼

¼

2

ai

ðRiL þ RiU Þ

i¼0

U

Z

n1 X

¼

2L

ð52Þ

n1 X 1 2L 2L ai ððxÞ2U ½ðxÞ2U an  ðxÞan  aiþ1  ðxÞaiþ1 Þ 6 i¼1

þ

n1 X

2L aiþ1 ððxÞ2U ai  ðxÞai Þ

i¼0 ðxÞU an

ðxÞLan

xdx ¼

1 2L ½ðxÞ2U an  ðxÞan ; 2

ð53Þ

þ

n1 1X Dai ððxÞUai  ðxÞUaiþ1  ðxÞLai  ðxÞLaiþ1 Þ; 6 i¼0

ð56Þ

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Rd

2L ðxÞ2U an  ðxÞan 

xlA~ ðxÞdx

1 ~ ¼ Ra x0 ðAÞ ¼  d lA~ ðxÞdx 3 a

n1 P i¼1

2L ai ððxÞ2U aiþ1  ðxÞaiþ1 Þ þ

ðxÞUan  ðxÞLan 

n1 P i¼1

n1 P i¼0

2L 1 aiþ1 ððxÞ2U ai  ðxÞai Þ þ 6

ai ððxÞUaiþ1  ðxÞLaiþ1 Þ þ

Appendix B. The proof of Theorem 2 Let

Dai0 ðxG  ðxÞLai Þ xmax  xG 0 ¼ ai0 þ : xmax  xmin ðxÞLai þ1  ðxÞLai 0

ð58Þ

0

From Eq. (58), we get

xG ¼

xmax  ððxÞLai

0 þ1

 ðxÞLai Þ þ ðai0 þ1  ðxÞLai  ai0  ðxÞLai 0

L

0

0 þ1

Þðxmax  xmin Þ :

L

ðxÞai

0 þ1

 ðxÞai þ Dai0  ðxmax  xmin Þ 0

ð59Þ Accordingly, we have

xmax  Dai0  ai0 þ1  ðxÞLai þ ai0  ðxÞLai þ1 xmax  xG 0 0 uG ¼ ¼ xmax  xmin ðxÞLai þ1  ðxÞLai þ Dai0  ðxmax  xmin Þ 0

¼

0

ai0 þ1  ðxmax  ðxÞLai Þ  ai0  ðxmax  ðxÞLai þ1 Þ 0

ðxÞLai

0 þ1

0

 ðxÞLai þ ðai0 þ1  ai0 Þðxmax  xmin Þ

ð60Þ

:

0

Let

Daj0 ððxÞUaj  xM Þ xM  xmin 0 ¼ aj0 þ : xmax  xmin ðxÞUaj  ðxÞUaj þ1 0

ð61Þ

0

Accordingly, we obtain

xM ¼

xmin  ððxÞUaj  ðxÞUaj

0 þ1

0

U

Þ þ ðaj0 þ1  ðxÞUaj  aj0  ðxÞUaj

ðxÞaj  ðxÞaj 0

0 þ1

0

U

0 þ1

Þðxmax  xmin Þ

þ Daj0  ðxmax  xmin Þ

; ð62Þ

uM ¼

aj0 þ1  ðxÞUaj  aj0  ðxÞUaj þ1  xmin  Daj0 xM  xmin 0 0 ¼ xmax  xmin ðxÞUaj  ðxÞUaj þ1 þ Daj0  ðxmax  xmin Þ 0

¼

0

aj0 þ1  ððxÞUaj  xmin Þ  aj0  ððxÞUaj þ1  xmin Þ 0

ðxÞUaj  ðxÞUaj 0

0

0 þ1

þ ðaj0 þ1  aj0 Þðxmax  xmin Þ

:

ð63Þ

References Chen, S. H. (1985). Ranking fuzzy numbers with maximizing set and minimizing set. Fuzzy Sets and Systems, 17, 113–129. Chen, S. M. (2001). Fuzzy group decision making for evaluating the rate of aggregative risk in software development. Fuzzy Sets and Systems, 118, 75–88. Chen, C. B., & Klein, C. M. (1997a). An efficient approach to solving fuzzy MADM problems. Fuzzy Sets and Systems, 88, 51–67. Chen, C. B., & Klein, C. M. (1997b). A simple approach to ranking a group of aggregated fuzzy utilities. IEEE Transactions on Systems, Man, and Cybernetics – Part B: Cybernetics, 27, 26–35. Detyniecki, D., & Yager, R. R. (2000). Ranking fuzzy numbers using a-weighted valuations. International Journal of Uncertainty, Fuzziness and Knowledge-Based Systems, 8, 573–591.

n1 P i¼0

n1 P i¼0

Dai ððxÞUai  ðxÞUaiþ1  ðxÞLai  ðxÞLaiþ1 Þ :

ð57Þ

aiþ1 ððxÞUai  ðxÞLai Þ

Dong, W. M., & Wong, F. S. (1987). Fuzzy weighted averages and implementation of the extension principle. Fuzzy Sets and Systems, 21, 183–199. Dubois, D., & Prade, H. (1980). Fuzzy sets and systems: Theory and application. New York: Academic Press. Dubois, D., & Prade, H. (1983). Ranking fuzzy numbers in the setting of possibility theory. Information Sciences, 30, 183–224. Filev, D. P., & Yager, R. R. (1993). An adaptive approach to defuzzification based on level sets. Fuzzy Sets and Systems, 54, 355–360. Guh, Y. Y., Hon, C. C., & Lee, E. S. (2001). Fuzzy weighted average: The linear programming approach via Charnes and Cooper’s rule. Fuzzy Sets and Systems, 117, 157–160. Guh, Y. Y., Hon, C. C., Wang, K. M., & Lee, E. S. (1996). Fuzzy weighted average: A max–min paired elimination method. Computers & Mathematics with Applications, 32, 115–123. Guu, S. M. (2002). Fuzzy weighted averages revisited. Fuzzy Sets and Systems, 126, 411–414. Hon, C. C., Guh, Y. Y., Wang, K. M., & Lee, E. S. (1996). Fuzzy multiple attributes and multiple hierarchical decision making. Computers & Mathematics with Applications, 32, 109–119. Kao, C., & Liu, S. T. (1999). Competitiveness of manufacturing firms: An application of fuzzy-weighted average. IEEE Transactions on Systems, Man and Cybernetics – Part A: Systems and Humans, 29, 661–667. Kao, C., & Liu, S. T. (2000). Fuzzy efficiency measures in data envelopment analysis. Fuzzy Sets and Systems, 113, 427–437. Kao, C., & Liu, S. T. (2001). Fractional programming approach to fuzzy weighted average. Fuzzy Sets and Systems, 120, 435–444. Kao, C., & Liu, S. T. (2003). A mathematical programming approach to fuzzy efficiency ranking. International Journal of Production Economics, 86, 145–154. Lee, H. M. (1996a). Applying fuzzy set theory to evaluate the rate of aggregative risk in software development. Fuzzy Sets and Systems, 79, 323–336. Lee, H. M. (1996b). Group decision making using fuzzy sets theory for evaluating the rate of aggregative risk in software development. Fuzzy Sets and Systems, 80, 261–271. Lee, H. M. (1999). Generalization of the group decision making using fuzzy sets theory for evaluating the rate of aggregative risk in software development. Information Sciences, 113, 301–311. Lee, H. M., Lee, S. Y., Lee, T. Y., & Chen, J. J. (2003). A new algorithm for applying fuzzy set theory to evaluate the rate of aggregative risk in software development. Information Sciences, 153, 177–197. Lee, H. M., Lee, T. Y., Lee, S. Y., & Chen, J. J. (2004). A fuzzy group assessment model for evaluating the rate of aggregative risk in software development. International Journal of Reliability, Quality and Safety Engineering, 11, 17–33. Lee, D. H., & Park, D. (1997). An efficient algorithm for fuzzy weighted average. Fuzzy Sets and Systems, 87, 39–45. León, T., Liern, V., Ruiz, J. L., & Sirvent, I. (2003). A fuzzy mathematical programming approach to the assessment of efficiency with DEA models. Fuzzy Sets and Systems, 139, 407–419. Matarazzo, B., & Munda, G. (2001). New approaches for the comparison of L-R fuzzy numbers: A theoretical and operational analysis. Fuzzy Sets and Systems, 118, 407–418. Oussalah, M. (2002). On the compatibility between defuzzification and fuzzy arithmetic operations. Fuzzy Sets and Systems, 128, 247–260. Raj, P. A., & Kumar, D. N. (1999). Ranking alternatives with fuzzy weights using maximizing set and minimizing set. Fuzzy Sets and Systems, 105, 365–375. Raja, P. A., & Kumar, D. N. (1998). Ranking multi-criterion river basin planning alternatives using fuzzy numbers. Fuzzy Sets and Systems, 100, 89–99. Tseng, T. Y., & Klein, C. M. (1992). A new algorithm for fuzzy multicriteria decision making. International Journal of Approximate Reasoning, 6, 45–66. Uehara, K., & Hirota, K. (1998). Parallel and multistage fuzzy inference based on families of a-level sets. Information Sciences, 106, 159–195. Vanegas, L. V., & Labib, A. W. (2001). Application of new fuzzy-weighted average (NFWA) method to engineering design evaluation. International Journal of Production Research, 39, 1147–1162. Wang, Y. M., Greatbanks, R., & Yang, J. B. (2005). Interval efficiency assessment using data envelopment analysis. Fuzzy Sets and Systems, 153, 347–370. Wang, W. J., & Luoh, L. (2000). Simple computation for the defuzzifications of center of sum and center of gravity. Journal of Intelligent and Fuzzy Systems, 9, 53–59. Yager, R. R. (1981). A procedure for ordering fuzzy subsets of the unit interval. Information Sciences, 24, 143–161. Yager, R. R., & Filev, D. (1999). On ranking fuzzy numbers using valuations. International Journal of Intelligent Systems, 14, 1249–1268.

Please cite this article in press as: Wang, Y.-M. Centroid defuzzification and the maximizing set and minimizing set ranking based ... Computers & Industrial Engineering (2009), doi:10.1016/j.cie.2008.11.014