A linear goal programming priority method for fuzzy analytic hierarchy process and its applications in new product screening

International Journal of Approximate Reasoning 49 (2008) 451–465

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International Journal of Approximate Reasoning journal homepage: www.elsevier.com/locate/ijar

A linear goal programming priority method for fuzzy analytic hierarchy process and its applications in new product screening q Ying-Ming Wang a,b,*, Kwai-Sang Chin b a b

School of Public Administration, Fuzhou University, Fuzhou 350002, PR China Department of Manufacturing Engineering and Engineering Management, City University of Hong Kong, 83 Tat Chee Avenue, Kowloon Tong, Hong Kong

a r t i c l e

i n f o

Article history: Received 23 November 2007 Received in revised form 12 April 2008 Accepted 20 April 2008 Available online 29 April 2008

Keywords: Fuzzy analytic hierarchy process Multiple criteria decision making Normalized fuzzy weights Linear goal programming New product development Project screening

a b s t r a c t Fuzzy analytic hierarchy process (AHP) has been widely used for a variety of applications such as supplier selection, customer requirements assessment and the like. The vast majority of the applications, however, were found avoiding the use of sophisticated approaches for fuzzy AHP such as fuzzy least squares method while using a simple extent analysis for the sake of simplicity. The extent analysis proves to be incorrect and may lead to a wrong decision being made. This paper proposes a sound yet simple priority method for fuzzy AHP which utilizes a linear goal programming (LGP) model to derive normalized fuzzy weights for fuzzy pairwise comparison matrices. The proposed LGP priority method is tested with three numerical examples including an application of fuzzy AHP to new product development (NDP) project screening decision making. Ó 2008 Elsevier Inc. All rights reserved.

1. Introduction Analytic hierarchy process (AHP) developed by Saaty [35] has been extensively applied in many areas such as selection, evaluation, planning and development, decision making, forecasting, and the like [39]. The traditional AHP requires precise judgments from decision makers (DMs). Precise judgments, however, are not always available and may sometimes be difficult to be elicited from DMs. To overcome this difficulty, fuzzy judgments have been suggested and fuzzy AHP has been applied in a wide variety of applications [1–4,8–11,13–15,19–32,34,36–38]. The key issue of the use of fuzzy AHP for decision making is how to derive priority vectors from fuzzy pairwise comparison matrices. A number of priority methods have been suggested for fuzzy AHP in the literature. For example, Van Laarhoven and Pedrycz [40] utilized triangular fuzzy judgments instead of precise judgments and suggested a logarithmic least squares method (LLSM) for fuzzy AHP. Boender et al. [1] pointed out a mistake of Van Laarhoven and Pedrycz’s LLSM in normalizing fuzzy weights and presented a modified normalization method. The modified normalization method for LLSM was also found incorrect by Wang et al. [43] who therefore proposed a modified fuzzy LLSM for fuzzy AHP. Xu and Zhai [47] also presented an LLSM for fuzzy judgment matrices, but their LLSM was based on a different Euclidean distance metric that was defined as the integral of the distance of every t-level set. As a result, the weights derived by their LLSM were intervals characterized with different t-levels. Xu [46] used the same distance metric to develop a fuzzy least-square priority method for fuzzy judgment matrices.

q The work described in this paper was supported by a grant from the Research Grants Council of the Hong Kong Special Administrative Region, China [Grant No. CityU 111906], and also partially supported by the Natural Science Foundation of Fujian Province of China under the Project No. A0710005. * Corresponding author. Address: School of Public Administration, Fuzhou University, Fuzhou 350002, PR China. Tel.: +86 591 87893307; fax: +86 591 87892545. E-mail address: [email protected] (Y.-M. Wang).

0888-613X/$ - see front matter Ó 2008 Elsevier Inc. All rights reserved. doi:10.1016/j.ijar.2008.04.004

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Buckley [6] employed the geometric mean method to calculate fuzzy weights for each fuzzy matrix and combined them in the usual manner to determine the final fuzzy weights for decision alternatives. Chang [16] proposed an extent analysis method to derive crisp weights from fuzzy comparison matrices. Buckley et al. [7] directly fuzzified Saaty’s original procedure of computing weights in hierarchical analysis to get fuzzy weights in fuzzy hierarchical analysis. Such a method proved to be quite complicated and required an evolutionary algorithm to be designed for obtaining fuzzy weights. Csutora and Buckley [18] proposed a Lambda–Max method to find fuzzy weights, which was also the direct fuzzification of the kmax method. The Lambda– Max method was found only able to generate nonnormalized fuzzy weights, which make no sense because some of them are too wide to be true. To derive a set of normalized fuzzy weights, Wang and Chin [44] came up with an eigenvector method, but found that not every fuzzy pairwise comparison matrix could derive a normalized fuzzy weight vector. Mikhailov [32–34] developed a fuzzy preference programming (FPP) method, which derives crisp weights from fuzzy comparison matrices. Among the above approaches, the extent analysis was found to be the most widely used approach due to its computational simplicity. However, it proves to be incorrect and may result in a wrong decision to be made [44]. Detailed analysis on the extent analysis can be found in Wang et al. [44] and its misapplications are provided in Refs. [2–4,8–11,13–15,19– 31,36–38,48]. The purpose of this paper is to develop a sound yet simple priority method for fuzzy AHP so that the fuzzy weights of fuzzy pairwise comparison matrices can be derived more easily than those sophisticated approaches such as fuzzy LLSM, fuzzy preference programming method and the like. The proposed priority method uses a linear goal programming (LGP) model to derive normalized fuzzy weights for triangular fuzzy pairwise comparison matrices and will be tested with a number of numerical examples including an application of fuzzy AHP to new product development (NDP) project screening decision making. The rest of the paper is organized as follows. Section 2 develops the LGP priority method for fuzzy AHP. Section 3 discusses the aggregation of local fuzzy weights to global fuzzy weights. Numerical examples are tested in Section 4 to provide an application of fuzzy AHP in NPD project screening. The paper concludes in Section 5. 2. The LGP priority method for fuzzy AHP Consider a fuzzy pairwise comparison matrix:

3 1 ðl12 ; m12 ; u12 Þ    ðl1n ; m1n ; u1n Þ 6 ðl ; m ; u Þ 1    ðl2n ; m2n ; u2n Þ 7 7 6 21 21 21 7; ¼6 .. .. .. .. 7 6 5 4 . . . . ðln1 ; mn1 ; un1 Þ ðln2 ; mn2 ; un2 Þ    1 2

e ¼ ða~ij Þ A nn

ð1Þ

where lij = 1/uji, mij = 1/mji and uij = 1/lji for all i, j = 1, . . ., n; j 6¼ i. The above fuzzy comparison matrix can be split into three crisp nonnegative matrices:

2

1 6l 6 21 AL ¼ 6 6 .. 4 . ln1

l12 1 .. . ln2

3    l1n    l2n 7 7 .. 7 7;  . 5  1

2

1 6 m21 6 AM ¼ 6 . 4 .. mn1

m12 1 .. . mn2

3 2    m1n 1 7 6 u21    m2n 7 6 and AU ¼ 6 . .. 7 4 ..  . 5 un1  1

3    u1n    u2n 7 7 ; .. 7  . 5  1

u12 1 .. . un2

ð2Þ

e ¼ ðAL ; AM ; AU Þ. Note that AL and AU are no longer reciprocal matrices. where A e there should exist a normalized fuzzy weight vector, W f ¼ ððwL ; wM ; wU Þ; . . . ; For the fuzzy comparison matrix A, 1 1 1 T e in the sense that a ~ij ¼ ðlij ; mij ; uij Þ  ðwL ; wM ; wU Þ=ðwL ; wM ; wU Þ for all i, j = 1, . . ., n; j 6¼ i. ðwL ; wM ; wU ÞÞ , which is close to A n

n

n

i

i

i

j

j

j

f is normalized if and only if According to Wang and Elhag [42], the fuzzy weight vector W n X i¼1 n X i¼1 n X

wUi  maxðwUj  wLj Þ P 1;

ð3Þ

wLi þ maxðwUj  wLj Þ 6 1;

ð4Þ

wM i ¼ 1;

ð5Þ

j

j

i¼1

which can be equivalently rewritten as

wLi þ wUi þ

n X j¼1;j6¼i n X

wUj P 1;

i ¼ 1; . . . ; n;

ð6Þ

wLj 6 1;

i ¼ 1; . . . ; n;

ð7Þ

j¼1;j6¼i n X i¼1

wM i ¼ 1:

ð8Þ

Y.-M. Wang, K.-S. Chin / International Journal of Approximate Reasoning 49 (2008) 451–465

453

e defined by Eq. (1) is a precise comparison matrix about the fuzzy weight vector W f , namely, If the fuzzy comparison matrix A U L M U e must be able to be written as ~ij ¼ ðlij ; mij ; uij Þ ¼ ðwLi ; wM a ; w Þ=ðw ; w ; w Þ for all i, j = 1, . . ., n but j ¼ 6 i, then A i i j j j

2

ðwL1 ;wM ;wU Þ 1 1

1 6 6 L M U 6 ðw2 ;w2 ;w2 Þ 6 L M U ðw ;w ;w Þ e¼6 A 6 1 1 1 6 6 ... 6 4 L M U

1

7 7 7: 7 7 7 5

ðwL2 ;wM ;wU Þ7 2 2 U ðwLn ;wM n ;wn Þ 7



U ðwLn ;wM n ;wn Þ UÞ ;w ðwL2 ;wM 2 2

3

U ðwLn ;wM n ;wn Þ 7



.. .

ðwn ;wn ;wn Þ ;wU Þ ðwL1 ;wM 1 1

ðwL1 ;wM ;wU Þ 1 1



ðwL2 ;wM ;wU Þ 2 2

1



ð9Þ

According to the division operation rule of fuzzy arithmetic, i.e. (bL,bM, bU)/(dL, dM, dU) = (bL/dU, bM/dM, bU/dL), where (bL, bM, bU) e defined by Eq. (9) can be further and (dL, dM, dU) are two positive triangular fuzzy numbers, the fuzzy comparison matrix A expressed as

 L M U w1 w1 w1 ; wM ; wL 1 wU 6 2 2 2 6  L M U 6 w2 w2 w2 6 U; M; L 1 w w w e¼6 A 6 1 1 1 6 .. .. 6 . . 6 4  L M U  L M U wn wn wn wn wn wn ; ; ; ; wU wM wL wU wM wL 2

1

1

1

2

2

 



wL1

wU n

wM

wU

; w1M ; w1L

3

7 7 7 2 ; M ; wL 7 wU 7 n wn n 7; 7 .. 7 . 7 5 1

 wL

 

n

wM 2

n

wU 2

ð10Þ

2

which can be split into three crisp nonnegative matrices, as shown below:

2

1

6 6 L 6 w2 6 U 6w AL ¼ 6 1 6. 6. 6. 4 L wn wU 1

wL1

wU 2



wL1

3

wU n

7 7 7    wU 7 n 7 7; .. 7 7  . 7 5  1 wL2

1 .. . wLn wU 2

2

1

6 6 M 6 w2 6 M 6w AM ¼ 6 1 6. 6. 6. 4 M wn wM 1

wM 1 wM 2

1 .. . wM n wM 2



wM 1 wM n

3

2

1

7 6 7 6 U 7 6 w2 6 L    wM 7 6w n 7 7 and AU ¼ 6 1 7 6. .. 7 6.  . 7 6. 5 4 U wn  1 wL wM 2

1

wU 1 wL2

1 .. . wU n wL2



wU 1

3

wLn

7 7 7    wL 7 n 7 7: .. 7 7  . 7 5  1 wU 2

It is easy to verify that

AL W U ¼ W U þ ðn  1ÞW L ;

ð11Þ

AU W L ¼ W L þ ðn  1ÞW U ;

ð12Þ

AM W M ¼ nW M ;

ð13Þ

M T U U T where W L ¼ ðwL1 ; . . . ; wLn ÞT , W M ¼ ðwM 1 ; . . . ; wn Þ and W U ¼ ðw1 ; . . . ; wn Þ are three crisp weight vectors, based on which the f can be expressed as W f ¼ ðW L ; W M ; W U Þ. Eqs. (11) and (12) are important links between the lower fuzzy weight vector W

f. and upper bounds of the fuzzy weight vector W It can also be verified from the above three crisp nonnegative matrices that M M aM ij ¼ aik akj

and aLij aUij ¼ ðaLik aUik ÞðaLkj aUkj Þ for any i; j; k ¼ 1; . . . ; n; i 6¼ j 6¼ k;

ð14Þ

which is the perfectly consistent condition 2 for a triangular fuzzy comparison matrix to3be perfectly consistent. Consider the 1 ð1; 5=3; 3Þ ð4=3; 5=2; 6Þ e ¼ 4 ð1=3; 3=5; 1Þ fuzzy pairwise comparison matrix A 1 ð2=3; 3=2; 4Þ 5, which meets the condition of (14). ð1=6; 2=5; 3=4Þ ð1=4; 2=3; 3=2Þ 1 It is therefore a perfectly consistent fuzzy pairwise comparison matrix. However, due to subjectivity and uncertainty in real judgments, DMs’ subjective judgments cannot always be 100 percent accurate. In other words, Eqs. (11)–(13) cannot always hold. In the case that they do not hold, we introduce the following deviation vectors:

E ¼ ðAL  IÞW U  ðn  1ÞW L ;

ð15Þ

C ¼ ðAU  IÞW L  ðn  1ÞW U ; D ¼ ðAM  nIÞW M ;

ð16Þ ð17Þ

where E = (e1, . . ., en)T, C = (c1, . . ., cn)T, D = (d1, . . ., dn)T, I is an n  n unit matrix, ei, ci and di for i = 1, . . ., n are all deviation variables. It is most desirable that the absolute values of the deviation variables be kept as small as possible, which enables us to f: construct the following nonlinear goal programming (NGP) model for determining the fuzzy weight vector W

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Y.-M. Wang, K.-S. Chin / International Journal of Approximate Reasoning 49 (2008) 451–465

Minimize

J¼

n X ðjei j þ jci j þ jdi jÞ i¼1

Subject to ðAL  IÞW U  ðn  1ÞW L  E ¼ 0; ðAU  IÞW L  ðn  1ÞW U  C ¼ 0; ðAM  nIÞW M  D ¼ 0; n X wUj P 1; i ¼ 1; . . . ; n; wLi þ j¼1;j6¼i

wUi

n X

þ

ð18Þ wLj 6 1;

i ¼ 1; . . . ; n;

j¼1;j6¼i n X

wM i ¼ 1;

i¼1

W U  W M P 0; W M  W L P 0; W L P 0; where the first three constraints are Eqs. (15)–(17), the middle three constraints are the normalization constraints on the f , and the last three constraints are those on the lower and upper bounds of W f and its nonnegativity. fuzzy weight vector W From Saaty’s eigenvector method (EM) [35], it is known that for any crisp pairwise comparison matrix AM, there will exist c M such that AM W c M . So, deviation vector D can always be nonnegative. c M P nW a maximum principal right eigenvector W That is D P 0. There is no guarantee, however, that the deviation vectors E and C can also be nonnegative. Let

ei þ jei j ei þ jei j and ei ¼ ; i ¼ 1; . . . ; n; 2 2 c þ jc i j ci þ jci j and ci ¼ ; i ¼ 1; . . . ; n: ¼ i 2 2

eþi ¼

ð19Þ

cþi

ð20Þ

 þ  þ þ T   T þ T   T Then Eþ ¼ ðeþ 1 ; . . . ; en Þ P 0, E ¼ ðe1 ; . . . ; en Þ P 0, C ¼ ðc1 ; . . . ; cn Þ P 0 and C ¼ ðc1 ; . . . ; cn Þ P 0. Based upon the definiþ  tions of ei and ei , ei and jeij can be expressed as

ei ¼ eþi  ei ; jei j ¼

eþi

þ

i ¼ 1; . . . ; n;

ei ;

ð21Þ

i ¼ 1; . . . ; n;

ð22Þ

 where eþ i  ei ¼ 0 for i = 1 to n. As such, ci and jcij can be expressed as

ci ¼ cþi  ci ; i ¼ 1; . . . ; n; jci j ¼ cþi þ ci ; i ¼ 1; . . . ; n; þ i

ð23Þ ð24Þ

where c  c ¼ 0 for i = 1 to n. As a result, the NGP model (18) can be rewritten as  i

Minimize

J¼

n X ðeþi þ ei þ cþi þ ci þ di Þ ¼ eT ðEþ þ E þ Cþ þ C þ DÞ i¼1

Subject to ðAL  IÞW U  ðn  1ÞW L  Eþ þ E ¼ 0; ðAU  IÞW L  ðn  1ÞW U  Cþ þ C ¼ 0; ðAM  nIÞW M  D ¼ 0; n X wUj P 1; i ¼ 1; . . . ; n; wLi þ j¼1;j6¼i

wUi

þ

n X

ð25Þ wLj

6 1;

i ¼ 1; . . . ; n;

j¼1;j6¼i n X

wM i ¼ 1;

i¼1

W U  W M P 0; W M  W L P 0; W L ; Eþ ; E ; Cþ ; C ; D P 0;

455

Y.-M. Wang, K.-S. Chin / International Journal of Approximate Reasoning 49 (2008) 451–465

þ   which is a linear goal programming (LGP) model, where eT = (1, . . ., 1), eþ i and ei as well as ci and ci for i = 1, . . ., n cannot be selected as basic variables in the simplex method at the same time. This is the LGP model we develop for fuzzy AHP. Such a method using the LGP model (25) for obtaining fuzzy weights from fuzzy pairwise comparison matrices is referred to as the LGP method. For any perfectly consistent triangular fuzzy comparison matrix, the optimal objective function value J* is always zero. In another word, if the optimal objective function value of LGP model (25) turns out to be zero, then it can be concluded that the corresponding triangular fuzzy comparison matrix is perfectly consistent; otherwise, it is inconsistent. The magnitude of the optimal objective function value of LGP model (25) reflects to somewhat extent the degree of inconsistency of a fuzzy pairwise comparison matrix. Despite the fact that LGP model is developed for triangular fuzzy comparison matrices, it can be easily extended to the situation of trapezoidal fuzzy comparison matrices. Consider for example a trapezoidal fuzzy comparison matrix shown below:

3 1 ðl12 ; m12 ; n12 ; u12 Þ    ðl1n ; m1n ; n1n ; u1n Þ 6 ðl ; m ; n ; u Þ 1    ðl2n ; m2n ; n2n ; u2n Þ 7 7 6 21 21 21 21 e¼6 7; B .. .. .. .. 7 6 5 4 . . . . ðln1 ; mn1 ; nn1 ; un1 Þ ðln2 ; mn2 ; nn2 ; un2 Þ    1 2

ð26Þ

e can be broken down where lij = 1/uji and mij = 1/nji for all i, j = 1, . . ., n but j 6¼ i. This trapezoidal fuzzy comparison matrix B into the following four crisp nonnegative matrices:

2

1 6l 6 21 BL ¼ 6 6 .. 4. ln1 2 1 6u 6 21 BU ¼ 6 6 .. 4. un1

   l1n

l12 1 .. .

3

   l2n 7 7 . 7 7; .  . 5

2

6m 6 21 BM ¼ 6 6 .. 4.

ln2    1 3 u12    u1n 1    u2n 7 7 .. .. 7 7 .  . 5 un2

1

mn1

3

m12

   m1n

1 .. .

   m2n 7 7 7; . 7 . 5  .

mn2

 1

2

1

6n 6 21 BN ¼ 6 6 .. 4. nn1

3

n12

   n1n

1 .. .

   n2n 7 7 . 7 7 and .  . 5

nn2

 1

ð27Þ

 1

based on which the following LGP model can be constructed to derive a normalized trapezoidal fuzzy weight vector f ¼ ððwL ; wM ; wN ; wU Þ; . . . ; ðwL ; wM ; wN ; wU ÞÞT for B: e W 1 1 1 1 n n n n

Minimize Subject to

J ¼ eT ðEþ þ E þ Cþ þ C þ Dþ þ D þ Kþ þ K Þ ðBL  IÞW U  ðn  1ÞW L  Eþ þ E ¼ 0; ðBU  IÞW L  ðn  1ÞW U  Cþ þ C ¼ 0; ðBM  IÞW N  ðn  1ÞW M  Dþ þ D ¼ 0; ðBN  IÞW M  ðn  1ÞW N  Kþ þ K ¼ 0; n X wLi þ wUj P 1; i ¼ 1; . . . ; n; j¼1;j6¼i

wUi þ

n X

wLj 6 1;

i ¼ 1; . . . ; n;

ð28Þ

j¼1;j6¼i

wM i þ

n X

wNj P 1;

i ¼ 1; . . . ; n;

wM j 6 1;

i ¼ 1; . . . ; n;

j¼1;j6¼i

wNi þ

n X j¼1;j6¼i

W U  W N P 0; W N  W M P 0; W M  W L P 0; W L ; Eþ ; E ; Cþ ; C ; Dþ ; D ; Kþ ; K P 0; M T N N T U U T where W L ¼ ðwL1 ; . . . ; wLn ÞT , W M ¼ ðwM 1 ; . . . ; wn Þ , W N ¼ ðw1 ; . . . ; wn Þ and W U ¼ ðw1 ; . . . ; wn Þ are the four crisp weight vecþ f ¼ ðW L ; W M ; W N ; W U Þ, E ¼ ðeþ ; . . . ; eþ ÞT , E ¼ ðe ; . . . ; e ÞT , Cþ ¼ tors consisting of the trapezoidal fuzzy weight vector W 1

n

1

n

 þ  þ þ T   T þ  þ þ T   T þ T   T ðcþ 1 ; . . . ; cn Þ , C ¼ ðc1 ; . . . ; cn Þ , D ¼ ðd1 ; . . . ; dn Þ , D ¼ ðd1 ; . . . ; dn Þ , K ¼ ðg1 ; . . . ; gn Þ , and K ¼ ðg1 ; . . . ; gn Þ are all nonnegative deviation vectors. It is noted that the above LGP model (28) is also applicable to interval comparison matrices, which can be seen as a special case of trapezoidal fuzzy comparison matrices with BL  BM and BN  BU. As such, LGP model (25) is also applicable to crisp

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Y.-M. Wang, K.-S. Chin / International Journal of Approximate Reasoning 49 (2008) 451–465

pairwise comparison matrices which can be viewed as a special case of triangular fuzzy comparison matrices with AL  AM  AU. 3. Global fuzzy weights and defuzzifications In hierarchical structures, local fuzzy weights derived from fuzzy pairwise comparison matrices need to be aggregated into global fuzzy weights. Consider a simple three-level hierarchical structure shown in Fig. 1. Suppose the local triangular fuzzy weights for upper-level criteria and lower-level alternatives have all been obtained using the LGP method, as shown in U L M U Table 1, where ðwLj ; wM j ; wj Þ is the normalized triangular fuzzy weight for criterion j(j = 1, . . ., m) and ðwij ; wij ; wij Þ is the normalized triangular fuzzy weight of alternative Ai with respect to the criterion j (i = 1, . . ., n; j = 1, . . ., m). Table 1 forms a multiple criteria decision matrix and the global fuzzy weights can therefore be obtained by using the simple additive weighting method in multiple attribute decision making. That is

~ Ai ¼ w

m X U L M U ðwLj ; wM j ; wj Þðwij ; wij ; wij Þ;

ð29Þ

j¼1 U ~ Ai ¼ ðwLA ; wM where w Ai ; wAi Þ. By fuzzy arithmetic, (29) can be approximately unfolded as i

wLAi ¼

m X

wLij wLj ;

m X

wM Ai ¼

j¼1

M wM ij wj

and wUAi ¼

j¼1

m X

wUij wUj ;

i ¼ 1; . . . ; n:

ð30Þ

j¼1

It is often found that the support intervals ½wLAi ; wUAi  ði ¼ 1; . . . ; nÞ determined by fuzzy arithmetic are too wide to be reached. Therefore, the more precise global fuzzy weights with narrower supports can be obtained by the following equation and linear programming (LP) models [5]:

wM Ai ¼

m X

M wM ij wj ;

ð31Þ

j¼1

Minimize

wLAi ¼

m X

wLij wj ;

ð32Þ

j¼1

Subject to

wLj 6 wj 6 wUj ; m X wj ¼ 1:

j ¼ 1; . . . ; m;

j¼1

Maximize

wUAi ¼

m X

wUij wj ;

ð33Þ

j¼1

Subject to

wLj 6 wj 6 wUj ; m X wj ¼ 1:

j ¼ 1; . . . ; m;

j¼1

Decision Goal

Criterion 1 (C1)

…

Criterion j (Cj)

Alternative 1 (A1 )

…

Alternative i (Ai)

Criterion m (Cm)

…

…

Alternative n (An)

Fig. 1. A three-level hierarchical structure for fuzzy AHP.

Table 1 Aggregation of local triangular fuzzy weights Alternatives

U Criterion 1 ðwL1 ; wM 1 ; w1 Þ

A1 .. . Ai .. . An

U ðwL11 ; wM 11 ; w11 Þ .

.. U ðwLi1 ; wM i1 ; wi1 Þ .. . U ðwLn1 ; wM n1 ; wn1 Þ

...

U Criterion j ðwLj ; wM j ; wj Þ

...

U Criterion m ðwLm ; wM m ; wm Þ

Global fuzzy weights

... ... ... ... ...

U ðwL1j ; wM 1j ; w1j Þ .

... ... ... ... ...

U ðwL1m ; wM 1m ; w1m Þ .. . U ðwLim ; wM im ; wim Þ .. . U ðwLnm ; wM nm ; wnm Þ

ðwL ; wM ; wU Þ .. A1 A1 A1 . ðwL ; wM ; wU Þ .. Ai Ai Ai . U ðwLAn ; wM An ; w An Þ

.. U ðwLij ; wM ij ; wij Þ .. . U ðwLnj ; wM nj ; wnj Þ

Y.-M. Wang, K.-S. Chin / International Journal of Approximate Reasoning 49 (2008) 451–465

457

The following theorem shows that the global fuzzy weights obtained in this way are always normalized. U Theorem 1. Let ðwLAi ; wM Ai ; wAi Þði ¼ 1; . . . ; nÞ be the global fuzzy weights determined by (31)–(33). Then there exist n X i¼1 n X i¼1 n X

wUAi  maxðwUAj  wLAj Þ P 1;

ð34Þ

wLAi þ maxðwUAj  wLAj Þ 6 1;

ð35Þ

wM Ai ¼ 1:

ð36Þ

j

j

i¼1

P Pn Pm M M Pm Pn Pm M M M   Proof. From Eq. (31) it can be derived that ni¼1 wM Ai ¼ i¼1 j¼1 wij wj ¼ j¼1 ð i¼1 wij Þwj ¼ j¼1 wj ¼ 1. Let ðw1 ; . . . ; wm Þ     ^ ^ and ðw1 ; . . . ; wm Þ be the optimal solutions to LP model (32) and LP model (33), respectively. Then, ðw1 ; . . . ; wm Þ is usually P Pm L  L  ^; . . . ; w ^ m Þ is not optimal to wLA . As a result, we have wLA ¼ m ^ not optimal to wUAi and ðw j¼1 wij wj 6 j¼1 wij wj and i i Pm U  Pm U 1 U ^ j P j¼1 wij wj , based on which it can be derived that wAi ¼ j¼1 wij w n X

wLAk

6

k¼1;k6¼i n X

wUAk P

k¼1;k6¼i

n X

m X

k¼1;k6¼i

j¼1

n X

m X

wLAi þ

m X

wUAk P

n X

wUAi þ

! wUkj wj

wLAk 6

m X

k¼1;k6¼i

¼

m X

n X

j¼1

k¼1;k6¼i

! wLkj

m X

n X

j¼1

k¼1;k6¼i

^ j 6 w

m X ^ j ; ð1  wUij Þw

! wUkj wj P

j¼1 m X ð1  wLij Þwj ; j¼1

m m X X wLij wj þ ð1  wLij Þwj ¼ wj ¼ 1;

j¼1

k¼1;k6¼i

¼

j¼1

k¼1;k6¼i n X

! ^ j wLkj w

j¼1

j¼1

m m X X ^ j ¼ ^ j þ ^ j ¼ 1: wUij w ð1  wUij Þw w

j¼1

j¼1

j¼1

Since the above inequalities hold for all i = 1, . . ., n, it can therefore be concluded that Pn L U L h i¼1 wAi þ maxj ðwAj  wAj Þ 6 1. This completes the proof.

Pn

U i¼1 wAi

 maxj ðwUAj  wLAj Þ P 1 and

In the case that a hierarchical structure has more than three levels, the obtained global fuzzy weights need to be further aggregated with the local fuzzy weights in the higher level to produce final global fuzzy weights for final decision making. To make the final decision making easier, the final global fuzzy weights can be defuzzified as crisp numbers for comparison by the following centroid defuzzification formula [45]:

1 U  0 ðAi Þ ¼ ðwLA þ wM w Ai þ wAi Þ; i 3

i ¼ 1; . . . ; n;

ð37Þ

 0 ðAi Þ is the defuzzified centroid weight of decision alternative Ai, based on which decision alternatives can be comwhere w pared and ranked. 4. Numerical examples In this section, three numerical examples are examined using the proposed LGP method to show its applications. Example 1 is a perfectly consistent fuzzy pairwise comparison matrix and the LGP method can produce a precise fuzzy weight vector to perfectly match it. Example 2 is an inconsistent triangular fuzzy comparison matrix and the LGP method produces a normalized fuzzy weight vector to match it in the sense of optimality. Example 3 is a hierarchical structure for new product development (NDP) project screening decision making and is resolved using fuzzy AHP. Example 1. Consider the following 4  4 triangular fuzzy comparison matrix

3 1 ð1=5; 1=2; 1Þ ð1=6; 2=5; 3=4Þ ð1=7; 1=3; 3=5Þ 6 ð1; 2; 5Þ 1 ð1=2; 4=5; 5=4Þ ð3=7; 2=3; 1Þ 7 7 e¼6 A 7: 6 4 ð4=3; 5=2; 6Þ ð4=5; 5=4; 2Þ 1 ð4=7; 5=6; 6=5Þ 5 ð5=3; 3; 7Þ ð1; 3=2; 7=3Þ ð5=6; 6=5; 7=4Þ 1 2

This triangular fuzzy comparison matrix can be decomposed into three crisp nonnegative matrices, as shown below:

3 1 1=5 1=6 1=7 6 1 1 1=2 3=7 7 7 6 AL ¼ 6 7; 4 4=3 4=5 1 4=7 5 5=3 1 5=6 1 2

3 1 1=2 2=5 1=3 6 2 1 4=5 2=3 7 7 6 AM ¼ 6 7; 4 5=2 5=4 1 5=6 5 3 3=2 6=5 1 2

3 1 1 3=4 3=5 6 5 1 5=4 1 7 7 6 AU ¼ 6 7: 46 2 1 6=5 5 7 7=3 7=4 1 2

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By solving LGP model (25) for the above three crisp matrices, we obtain the following results: T

W L ¼ ð0:0526; 0:1579; 0:2105; 0:2632Þ ; W M W U

¼ ð0:1176; 0:2353; 0:2941; 0:3529ÞT ; ¼ ð0:1579; 0:2632; 0:3158; 0:3684Þ;

and J ¼ 0:

e is a perfectly consistent triangular fuzzy comparison matrix because of J* = 0 and its normalized triangular It is obvious that A fuzzy weight vector is obtained as

2

ð0:0526; 0:1176; 0:1579Þ

3

7 6 6 ð0:1579; 0:2353; 0:2632Þ 7 f  ¼ ðW  ; W  ; W  Þ ¼ 6 7 W L M U 6 ð0:2105; 0:2941; 0:3158Þ 7: 5 4 ð0:2632; 0:3529; 0:3684Þ e perIt can be verified that this normalized triangular fuzzy weight vector matches the triangular fuzzy comparison matrix A fectly well. Example 2. Consider the 5  5 triangular fuzzy comparison matrix given below:

2

1

ð3=2; 2; 5=2Þ

6 ð2=5; 1=2; 2=3Þ 1 6 6 e A¼6 ð2=7; 1=3; 2=5Þ ð2=5; 1=2; 2=3Þ 6 6 4 ð2=7; 1=3; 2=5Þ ð2=5; 1=2; 2=3Þ

3

ð5=2; 3; 7=2Þ

ð5=2; 3; 7=2Þ

ð7=2; 4; 9=2Þ

ð3=2; 2; 5=2Þ

ð3=2; 2; 5=2Þ

1

ð2=3; 1; 3=2Þ

ð2=3; 1; 3=2Þ

1

ð5=2; 3; 7=2Þ 7 7 7 ð3=2; 2; 5=2Þ 7 7; 7 ð3=2; 2; 5=2Þ 5

ð2=9; 1=4; 2=7Þ ð2=7; 1=3; 2=5Þ ð2=5; 1=2; 2=3Þ ð2=5; 1=2; 2=3Þ

1

which is taken from Chan et al. [15]. The above triangular fuzzy comparison matrix can be broken down into three crisp nonnegative matrices, which are shown as follows:

2

1

3=2 5=2 5=2 7=2

3

6 2=5 1 3=2 3=2 5=2 7 7 6 7 6 7; AL ¼ 6 2=7 2=5 1 2=3 3=2 7 6 7 6 4 2=7 2=5 2=3 1 3=2 5 2=9 2=7 2=5 2=5

1

2

1

2

6 1=2 1 6 6 AM ¼ 6 6 1=3 1=2 6 4 1=3 1=2

3

3

3

4

2

2

1

1

1

1

37 7 7 27 7; 7 25

2

1

5=2 7=2 7=2 9=2

3

6 2=3 1 5=2 5=2 7=2 7 7 6 7 6 7: and AU ¼ 6 2=5 2=3 1 3=2 5=2 7 6 7 6 4 2=5 2=3 3=2 1 5=2 5

1=4 1=3 1=2 1=2 1

2=7 2=5 2=3 2=3

1

By solving LGP model (25) for the above three crisp matrices, we get the results below:

W L ¼ ð0:3672; 0:2106; 0:1128; 0:1128; 0:0671ÞT ; W M ¼ ð0:4045; 0:2450; 0:1369; 0:1369; 0:0767ÞT ; W U ¼ ð0:4045; 0:2609; 0:1561; 0:1561; 0:0849Þ;

and J ¼ 0:033902:

e is therefore an inconsistent triangular fuzzy comparison matrix and its normalized triangular fuzzy Since J* = 0.033902 6¼ 0, A weight vector can be generated as

3 ð0:3672; 0:4045; 0:4045Þ 7 6 6 ð0:2106; 0:2450; 0:2609Þ 7 7 6 f  ¼ ðW  ; W  ; W  Þ ¼ 6 ð0:1128; 0:1369; 0:1561Þ 7: W L M U 7 6 7 6 4 ð0:1128; 0:1369; 0:1561Þ 5 ð0:0671; 0:0767; 0:0849Þ 2

e in the sense of minimum This normalized triangular fuzzy weight vector matches the triangular fuzzy comparison matrix A deviation or inconsistency. By using the extent analysis method, Chan et al. [15] got a crisp weight vector which is W0 = (0.46, 0.19, 0.12, 0.12, 0.11)T. It is observed that 0.46, 0.19 and 0.11 are all beyond their intervals and therefore not believable. It is also found that Chan et al. did not get the results correctly. By the extent analysis [16,48], the weights for the above triangular fuzzy comparison matrix should be W0 = (0.71, 0.29, 0, 0, 0)T, which is completely unacceptable. Wang et al. [44] have revealed that the weights determined by the extent analysis method do not represent the relative importance of decision criteria or alternatives and therefore cannot be used as their priorities. This is the reason why the crisp weight vector obtained by the extent analysis are beyond the fuzzy intervals determined by the LGP method. Example 3. NPD is highly risky because of tough competition and rapid technological and market changes. It demands advanced decision-making tools that can help manufacturing companies screen NPD projects in early product development stages to enhance the success of NPD projects. As a widely used decision-making tool, AHP has been illustrated by Calantone

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et al. [12] as a useful decision support tool for NPD project screening. Chin et al. [17] proposes a group-based ER-AHP system for product project screening by integrating the evidential reasoning (ER) approach and the AHP, which can help manufacturers in handling uncertainties and group-based decisions in the early NPD project screening stage. Fuzzy AHP, however, has never been used for NPD projects screening before. In this example, we illustrate that fuzzy AHP can be well used to support NPD project screening and deal with uncertainties and fuzziness in NPD project screening decision making. According to Chin et al. [17], NPD project screening can be modeled as a hierarchical structure, as shown in Fig. 2, where MKFIT, MANUFIT, CUSTFIT, FINRISK and UNCERT are five screening criteria, each with some sub-criteria. The definitions for these criteria and sub-criteria are documented in Table 2. Since precise judgments are not easy to make, fuzzy scales are therefore defined in Table 3 to better capture DM’s subjective judgments. In the case that there are multiple DMs, each DM’s subjective judgments may be weighted and geometrically averaged. Alternatively, LGP models (25) and (28) can be easily extended to the situation of group decision making. In Tables 4–14, we show DM’s subjective judgments on the pairwise comparison matrices for the five screening criteria, 13 subcriteria and three NPD projects. Note that EXPEND, R&DUNC and NONR&D are all cost-type sub-criteria. The cost-type criteria refer to those criteria for minimization rather than for maximization. In making pairwise comparisons for NPD projects, special attention has to be paid to the three cost-type sub-criteria because they are the smaller the better. If the EXPEND of one NPD project is about three times of that of another NPD project, then the ratio of the former to the latter should be ~ ¼ ð1=4; 1=3; 1=2Þ rather than 3. ~ This has never been mentioned or stressed in the AHP literature before. 1=3

NPD Project Screening (Selection of the best NPD Project)

MKTFIT

MANUFIT

UNCERT NONR&D

R&DUNC

EXPEND

Project B

FINRISK REVENUE

RELIA

DESIGN

SUPPLY

MFGCAP

MFGTECH

SALES

LOGISTICS

PRICE

TIMING

Project A

CUSTFIT

Project C

Fig. 2. Hierarchical structure for NPD project screening.

Table 2 Definitions of screening criteria and sub-criteria for NPD projects [12,17] Criteria or sub-criteria

Definition

MKTFIT TIMING PRICE LOGISTICS SALES MANUFIT MFGTECH MFGCAP SUPPLY CUSTFIT DESIGN RELIA FINRISK REVENUE EXPEND UNCERT R&DUNC NONR&D

Fit with company’s core marketing competencies Project matches the target launch timing needed by target segments Project matches the target price of target segments Project fits with company’s logistics and distribution strengths Project fits with company’s salesforce coverage and strengths Fit with company’s core manufacturing competencies Project fits with company’s manufacturing technology Manufacturing capacity matches demands Project allows companies to use their good suppliers Fit with customers’ requirements Project is designed to meet the requirements of customers Project meets the target reliability level Financial risks of projects Expected revenue Expected expenditure Uncertainties about projects’ outcomes Technological uncertainties in research and project design Uncertainties which are not related to research and project design

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Table 3 Fuzzy scales for pairwise comparisons in fuzzy AHP Importance intensity

Definition

~ ¼ ð1; 1; 2Þ 1 ~ ¼ ð2; 3; 4Þ 3 ~ ¼ ð4; 5; 6Þ 5 ~ ¼ ð6; 7; 8Þ 7 ~ ¼ ð8; 9; 9Þ 9 ~ ¼ ð1; 2; 3Þ, 4 ~ ¼ ð3; 4; 5Þ, 6 ~ ¼ ð5; 6; 7Þ, 8 ~ ¼ ð7; 8; 9Þ 2 Reciprocals

Equal importance Moderate importance of one over another Strong importance of one over another Very strong importance of one over another Extreme importance of one over another Intermediate values Reciprocals for inverse comparison

Table 4 Fuzzy comparison matrix of five screening criteria with respect to NPD project screening and its fuzzy weights Criteria

MKTFIT

MANUFIT

CUSTFIT

FINRISK

UNCERT

Fuzzy weights

MKTFIT MANUFIT CUSTFIT FINRISK UNCERT

1 (1/4, 1/3, 1/2) (1/2, 1, 1) (1/3, 1/2, 1) (1/4, 1/3, 1/2)

(2, 3, 4) 1 (2, 3, 4) (1, 2, 3) (1, 2, 3)

(1, 1, 2) (1/4, 1/3, 1/2) 1 (1/3, 1/2, 1) (1/3, 1/2, 1)

(1, 2, 3) (1/3, 1/2, 1) (1, 2, 3) 1 (1/2, 1, 1)

(2, 3, 4) (1/3, 1/2, 1) (1, 2, 3) (1, 1, 2) 1

(0.2606, 0.3225, 0.3493) (0.0624, 0.0842, 0.1119) (0.1915, 0.2936, 0.2936) (0.1209, 0.1552, 0.2101) (0.1005, 0.1445, 0.1575)

Table 5 Fuzzy comparison matrix of four sub-criteria with respect to MKTFIT and its fuzzy weights Sub-criteria

TIMING

PRICE

LOGISTICS

SALES

Fuzzy weights

TIMING PRICE LOGISTICS SALES

1 (1/2, 1, 1) (1/3, 1/2, 1) (1/4, 1/3, 1/2)

(1, 1, 2) 1 (1/2, 1, 1) (1/4, 1/3, 1/2)

(1, 2, 3) (1, 1, 2) 1 (1/3, 1/2, 1)

(2, 3, 4) (2, 3, 4) (1, 2, 3) 1

(0.2846, 0.3605, 0.4323) (0.2534, 0.3027, 0.3097) (0.1457, 0.2313, 0.2676) (0.0882, 0.1054, 0.1382)

Table 6 Fuzzy comparison matrix of three sub-criteria with respect to MANUFIT and its fuzzy weights Sub-criteria

MFGTECH

MFGCAP

SUPPLY

Fuzzy weights

MFGTECH MFGCAP SUPPLY

1 (1/3, 1/2, 1) (1/4, 1/3, 1/2)

(1, 2, 3) 1 (1/3, 1/2, 1)

(2, 3, 4) (1, 2, 3) 1

(0.4194, 0.5405, 0.5927) (0.2016, 0.2973, 0.4274) (0.1452, 0.1622, 0.2056)

Table 7 Fuzzy comparison matrix of two sub-criteria with respect to CUSTFIT and its fuzzy weights Sub-criteria

DESIGN

RELIA

Fuzzy weights

DESIGN RELIA

1 (1/3, 1/2, 1)

(1, 2, 3) 1

(0.5, 0.6667, 0.75) (0.25, 0.3333, 0.5)

Table 8 Fuzzy comparison matrix of two sub-criteria with respect to FINRISK and its fuzzy weights Sub-criteria

REVENUE

EXPEND

Fuzzy weights

REVENUE EXPEND

1 (1/4, 1/3, 1/2)

(2, 3, 4) 1

(0.6667, 0.75, 0.8) (0.2, 0.25, 0.3333)

Table 9 Fuzzy comparison matrix of two sub-criteria with respect to UNCERT and its fuzzy weights Sub-criteria

R&DUNC

NONR&D

Fuzzy weights

R&DUNC NONR&D

1 (1/5, 1/4, 1/3)

(3, 4, 5) 1

(0.75, 0.8, 0.8333) (0.1667, 0.2, 0.25)

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Project C

Fuzzy weights

Fuzzy comparison matrix of three projects with respect to TIMING and its fuzzy weights Project A 1 (1, 1, 2) Project B (1/2, 1, 1) 1 Project C (1/3, 1/2, 1) (1/3, 1/2, 1)

Project A

Project B

(1, 2, 3) (1, 2, 3) 1

(0.3636, 0.4000, 0.5091) (0.2909, 0.4000, 0.4000) (0.1455, 0.2000, 0.3273)

Fuzzy comparison matrix of three projects with respect to PRICE and its fuzzy weights Project A 1 (1/3, 1/2, 1) Project B (1, 2, 3) 1 Project C (1/3, 1/2, 1) (1/4, 1/3, 1/2)

(1, 2, 3) (2, 3, 4) 1

(0.2016, 0.2973, 0.4274) (0.4194, 0.5405, 0.5927) (0.1452, 0.1622, 0.2056)

Fuzzy comparison matrix of three projects with respect to LOGISTICS and its fuzzy weights Project A 1 (1, 2, 3) Project B (1/3, 1/2, 1) 1 Project C (1, 2, 3) (2, 3, 4)

(1/3, 1/2, 1) (1/4, 1/3, 1/2) 1

(0.2016, 0.2973, 0.4274) (0.1452, 0.1622, 0.2056) (0.4194, 0.5405, 0.5927)

Fuzzy comparison matrix of three projects with respect to SALES and its fuzzy weights Project A 1 (1/3, 1/2, 1) Project B (1, 2, 3) 1 Project C (4, 5, 6) (2, 3, 4)

(1/6, 1/5, 1/4) (1/4, 1/3, 1/2) 1

(0.1023, 0.1216, 0.1648) (0.1761, 0.2297, 0.2841) (0.6136, 0.6486, 0.6591)

Table 11 Fuzzy comparison matrices of three NPD projects with respect to each sub-criterion of MANUFIT and their fuzzy weights Projects

Project C

Fuzzy weights

Fuzzy comparison matrix of three projects with respect to MFGTECH and its fuzzy weights Project A 1 (1, 2, 3) Project B (1/3, 1/2, 1) 1 Project C (1/5, 1/4, 1/3) (1/4, 1/3, 1/2)

Project A

Project B

(3, 4, 5) (2, 3, 4) 1

(0.4212, 0.5600, 0.6480) (0.2320, 0.3200, 0.4506) (0.1200, 0.1200, 0.1282)

Fuzzy comparison matrix of three projects with respect to MFGCAP and its fuzzy weights Project A 1 (1/6, 1/5, 1/4) Project B (4, 5, 6) 1 Project C (1, 2, 3) (1/5, 1/4, 1/3)

(1/3, 1/2, 1) (3, 4, 5) 1

(0.0956, 0.1143, 0.1640) (0.6739, 0.6857, 0.6857) (0.1596, 0.2000, 0.2305)

Fuzzy comparison matrix of three projects with respect to SUPPLY and its fuzzy weights Project A 1 (1, 2, 3) Project B (1/3, 1/2, 1) 1 Project C (1, 2, 3) (2, 3, 4)

(1/3, 1/2, 1) (1/4, 1/3, 1/2) 1

(0.2016, 0.2973, 0.4274) (0.1452, 0.1622, 0.2056) (0.4194, 0.5405, 0.5927)

Table 12 Fuzzy comparison matrices of three NPD projects with respect to each sub-criterion of CUSTFIT and their fuzzy weights Projects

Project C

Fuzzy weights

Fuzzy comparison matrix of three projects with respect to DESIGN and its fuzzy weights Project A 1 (2, 3, 4) Project B (1/4, 1/3, 1/2) 1 Project C (1/5, 1/4, 1/3) (1/3, 1/2, 1)

Project A

Project B

(3, 4, 5) (1, 2, 3) 1

(0.5815, 0.6269, 0.6388) (0.1762, 0.2388, 0.3040) (0.1145, 0.1343, 0.1850)

Fuzzy comparison matrix of three projects with respect to RELIA and its fuzzy weights Project A 1 (1/3, 1/2, 1) Project B (1, 2, 3) 1 Project C (1/3, 1/2, 1) (1/5, 1/4, 1/3)

(1, 2, 3) (3, 4, 5) 1

(0.1975, 0.2857, 0.4222) (0.4444, 0.5714, 0.6296) (0.1333, 0.1429, 0.1728)

Table 13 Fuzzy comparison matrices of three NPD projects with respect to each sub-criterion of FINRISK and their fuzzy weights Projects

Project C

Fuzzy weights

Fuzzy comparison matrix of three projects with respect to REVENUE and its fuzzy weights Project A 1 (1, 2, 3) Project B (1/3, 1/2, 1) 1 Project C (1/5, 1/4, 1/3) (1/3, 1/2, 1)

Project A

Project B

(3, 4, 5) (1, 2, 3) 1

(0.4444, 0.5714, 0.6296) (0.1975, 0.2857, 0.4222) (0.1333, 0.1429, 0.1728)

Fuzzy comparison matrix of three projects with respect to EXPEND and its fuzzy weights Project A 1 (4, 5, 6) Project B (1/6, 1/5, 1/4) 1 Project C (1/5, 1/4, 1/3) (1, 2, 3)

(3, 4, 5) (1/3, 1/2, 1) 1

(0.6739, 0.6857, 0.6857) (0.0956, 0.1143, 0.1640) (0.1596, 0.2000, 0.2305)

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Table 14 Fuzzy comparison matrices of three NPD projects with respect to each sub-criterion of UNCERT and their fuzzy weights Projects

Project C

Fuzzy weights

Fuzzy comparison matrix of three projects with respect to R&DUNC and its fuzzy weights Project A 1 (2, 3, 4) Project B (1/4, 1/3, 1/2) 1 Project C (2, 3, 4) (3, 4, 5)

Project A

Project B

(1/4, 1/3, 1/2) (1/5, 1/4, 1/3) 1

(0.2264, 0.2683, 0.3175) (0.0917, 0.1098, 0.1460) (0.5365, 0.6220, 0.6819)

Fuzzy comparison matrix of three projects with respect to NONR&D and its fuzzy weights Project A 1 (1/3, 1/2, 1) Project B (1, 2, 3) 1 Project C (3, 4, 5) (1, 2, 3)

(1/5, 1/4, 1/3) (1/3, 1/2, 1) 1

(0.1333, 0.1429, 0.1728) (0.1975, 0.2857, 0.4222) (0.4444, 0.5714, 0.6296)

By solving LGP model (25) for each fuzzy pairwise comparison matrix in Tables 4–14, a normalized triangular fuzzy weight vector can be generated for each of them. The results are shown in the last columns of the tables, from which it is easy to find that the DM considers MKTFIT and CUSTFIT more important than the other three screening criteria. It is also observed that NPD project A performs better than projects B and C for screening sub-criteria TIMING, MFGTECH, DESIGN, REVENUE and EXPEND, NPD project B is better than projects A and C for screening sub-criteria PRICE, MFGCAP and RELIA, and NPD project C is superior to projects A and B under screening sub-criteria LOGISTICS, SALES, SUPPLY, R&DUNC and NONR&D. Apparently, the three NPD projects are all non-inferior alternatives and none of them can be judged as the best NPD project without further analysis. Tables 15–19 show the global fuzzy weights of the three NPD projects with respect to each screening criterion, which are obtained by solving Eq. (31) and the pair of LP models (32) and (33), respectively. These global fuzzy weights are then aggre-

Table 15 Global fuzzy weights of the three NPD projects with respect to MANUFIT Sub-criteria

Local fuzzy weights

Project A

Project B

Project C

TIMING PRICE LOGISTICS SALES

(0.2846, 0.3605, 0.4323) (0.2534, 0.3027, 0.3097) (0.1457, 0.2313, 0.2676) (0.0882, 0.1054, 0.1382)

(0.3636, 0.4000, 0.5091) (0.2016, 0.2973, 0.4274) (0.2016, 0.2973, 0.4274) (0.1023, 0.1216, 0.1648)

(0.2909, 0.4000, 0.4000) (0.4194, 0.5405, 0.5927) (0.1452, 0.1622, 0.2056) (0.1761, 0.2297, 0.2841)

(0.1455, 0.2000, 0.3273) (0.1452, 0.1622, 0.2056) (0.4194, 0.5405, 0.5927) (0.6136, 0.6486, 0.6591)

(0.2340, 0.3158, 0.4395)

(0.2686, 0.3696, 0.4183)

(0.2332, 0.3146, 0.4133)

Global fuzzy weights

Table 16 Global fuzzy weights of the three NPD projects with respect to MANUFIT Projects

MFGTECH (0.4194, 0.5405, 0.5927)

MFGCAP (0.2016, 0.2973, 0.4274)

SUPPLY (0.1452, 0.1622, 0.2056)

Global fuzzy weights

Project A Project B Project C

(0.4212, 0.5600, 0.6480) (0.2320, 0.3200, 0.4506) (0.1200, 0.1200, 0.1282)

(0.0956, 0.1143, 0.1640) (0.6739, 0.6857, 0.6857) (0.1596, 0.2000, 0.2305)

(0.2016, 0.2973, 0.4274) (0.1452, 0.1622, 0.2056) (0.4194, 0.5405, 0.5927)

(0.2484, 0.3849, 0.5051) (0.3032, 0.4031, 0.5155) (0.1738, 0.2120, 0.2621)

Table 17 Global fuzzy weights of the three NPD projects with respect to CUSTFIT Projects

DESIGN (0.5000, 0.6667, 0.7500)

RELIA (0.2500, 0.3333, 0.5000)

Global fuzzy weights

Project A Project B Project C

(0.5815, 0.6269, 0.6388) (0.1762, 0.2388, 0.3040) (0.1145, 0.1343, 0.1850)

(0.1975, 0.2857, 0.4222) (0.4444, 0.5714, 0.6296) (0.1333, 0.1429, 0.1728)

(0.3895, 0.5131, 0.5846) (0.2433, 0.3497, 0.4668) (0.1192, 0.1372, 0.1820)

Table 18 Global fuzzy weights of the three NPD projects with respect to FINRISK Projects

REVENUE (0.6667, 0.7500, 0.8000)

EXPEND (0.2000, 0.2500, 0.3333)

Global fuzzy weights

Project A Project B Project C

(0.4444, 0.5714, 0.6296) (0.1975, 0.2857, 0.4222) (0.1333, 0.1429, 0.1728)

(0.6739, 0.6857, 0.6857) (0.0956, 0.1143, 0.1640) (0.1596, 0.2000, 0.2305)

(0.4903, 0.6000, 0.6483) (0.1635, 0.2429, 0.3706) (0.1386, 0.1571, 0.1921)

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Y.-M. Wang, K.-S. Chin / International Journal of Approximate Reasoning 49 (2008) 451–465 Table 19 Global fuzzy weights of the three NPD projects with respect to UNCERT Projects

R&DUNC (0.7500, 0.8000, 0.8333)

NONR&D (0.1667, 0.2000, 0.2500)

Global fuzzy weights

Project A Project B Project C

(0.2264, 0.2683, 0.3175) (0.0917, 0.1098, 0.1460) (0.5365, 0.6220, 0.6819)

(0.1333, 0.1429, 0.1728) (0.1975, 0.2857, 0.4222) (0.4444, 0.5714, 0.6296)

(0.2031, 0.2432, 0.2934) (0.1093, 0.1449, 0.2151) (0.5135, 0.6118, 0.6732)

Table 20 Global fuzzy weights of the three NPD projects with respect to NPD project screening and their rankings Criteria

Local fuzzy weights

Project A

Project B

Project C

MKTFIT MANUFIT CUSTFIT FINRISK UNCERT

(0.2606, 0.3225, 0.3493) (0.0624, 0.0842, 0.1119) (0.1915, 0.2936, 0.2936) (0.1209, 0.1552, 0.2101) (0.1005, 0.1445, 0.1575)

(0.2340, 0.3158, 0.4395) (0.2484, 0.3849, 0.5051) (0.3895, 0.5131, 0.5846) (0.4903, 0.6000, 0.6483) (0.2031, 0.2432, 0.2934)

(0.2686, 0.3696, 0.4183) (0.3032, 0.4031, 0.5155) (0.2433, 0.3497, 0.4668) (0.1635, 0.2429, 0.3706) (0.1093, 0.1449, 0.2151)

(0.2332, 0.3146, 0.4133) (0.1738, 0.2120, 0.2621) (0.1192, 0.1372, 0.1820) (0.1386, 0.1571, 0.1921) (0.5135, 0.6118, 0.6732)

(0.3022, 0.4132, 0.5186) 0.4113 1

(0.2162, 0.3144, 0.4161) 0.3156 2

(0.2014, 0.2724, 0.3510) 0.2749 3

Membership degree

Global fuzzy weights Centroid defuzzification Ranking

1

0 0.0

C

0.1

0.2

B

0.3

A

0.4

0.5

0.6

Fuzzy weights Fig. 3. The global fuzzy weights of three NPD projects with respect to decision goal.

gated further with the local fuzzy weights of the five screening criteria into the final global fuzzy weights, which are shown in Table 20 and depicted in Fig. 3. It is very clear that NPD project A has the biggest global fuzzy weight with respect to the decision goal ‘‘NPD Project Screening”, followed by project B. Project C has the smallest global fuzzy weight with respect to NPD project screening. Therefore, project A is the best project for new product development. This example successfully shows that fuzzy AHP can provide very good decision supports to NPD project screening decision making. 5. Conclusions Fuzzy AHP has found extensive applications in a variety of areas such as supplier selection, customer requirements assessment and the like. However, the huge majority of the applications just use a simple extent analysis for the sake of simplicity, which has been shown to be incorrect and may result in wrong decision being made. To correctly and successfully apply fuzzy AHP for decision making, we have proposed in this paper a sound yet simple linear goal programming priority method which we refer to as the LGP method for fuzzy AHP. It can be used as a major tool to tackle fuzzy AHP decision making problems due to its simplicity of computation and rationality of weights. We have also tested the proposed LGP method with three numerical examples including an application of fuzzy AHP to NPD project screening decision making. The testing results show that the LGP method can derive precise fuzzy weights for perfectly consistent fuzzy comparison matrices and normalized optimal fuzzy weights for inconsistent fuzzy comparison matrices on the basis of minimum deviation. It is also shown that fuzzy AHP can be used as a very useful decision support tool for NPD project screening. It is also noticed that in a significant number of fuzzy multiple criteria decision making applications, fuzzy numbers such as fuzzy ratings, fuzzy weights etc. are defuzzified as crisp numbers at the very beginning of decision analysis. This simplifies decision making process, but definitely suffers from information loss. In our proposed LGP method, whether or not normal-

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