A new method for triangular fuzzy compare wise judgment matrix process based on consistency analysis Fanyong Meng, Xiaohong Chen
Abstract: To cope with the uncertainty in the process of decision making, fuzzy preference relations are proposed and commonly applied in many fields. In practical decision-making problems, the decision maker may use triangular fuzzy preference relations to express his/her uncertainty. Based on the row weighted arithmetic mean method, this paper develops an approach for deriving the fuzzy priority vector from triangular fuzzy compare wise judgment matrices. To do this, this paper first analyzes the upper and lower bounds of the triangular fuzzy priority weight of each alternative, which indicate the decision maker’s optimistic and pessimistic attitudes. Based on (acceptably) consistent multiplicative preference relations, the triangular fuzzy priority vector is obtained. Meanwhile, a consistency concept of triangular fuzzy compare wise judgment matrices is defined, and the consistent relationship between triangular fuzzy and crisp preference relations is studied. Different to the existing methods, the new approach calculates the triangular fuzzy priority weights separately. Furthermore, the fuzzy priority vector from trapezoidal fuzzy reciprocal preference relations is considered. Finally, the application of the new method to new product development (NDP) project screening is tested, and comparative analyses are also offered. Keywords: multiple criteria decision making; triangular fuzzy compare wise judgment matrix; fuzzy priority vector; row weighted arithmetic mean method; consistency analysis 1. Introduction Crisp preference relations need the decision maker to give his/her preference relations with precise judgments, which restricts its application. With the socioeconomic development, there usually exist some uncertain and even unknown information in practical decision-making problems. Crisp preference relations [33, 59] seem to be insufficient to cope with this situation. Saaty and Vargas [34] first considered preference relations with uncertain information and introduced the concept of interval multiplicative preference relations. Later, Xu [48] presented interval fuzzy preference relations. Because the interval preference relation can simply indicate the upper and lower boundary of the decision maker’s uncertainty, after the pioneer works of Saaty and Vargas [34] and Xu [48], many researchers devoted themselves into the studying of preference relations with intervals [2, 12, 13, 15, 19-21, 25, 30, 32, 42, 49, 54, 56, 57, 63, 64]. However, this kind of uncertain preference relations considers every element in an interval to be the same importance. Based on fuzzy set theory [65], van Laarhoven and Pedrycz [37] presented preference relations with triangular fuzzy numbers and presented a method to derive the fuzzy priority vector using logarithmic least squares method (LLSM), which was further considered by Boender et al. [4]. Wagenknecht and Hartmann [38] employed the least squares method to calculate the fuzzy priority vector from triangular fuzzy preference relations, which is shown by Saaty [35] that it can be only used for consistent triangular fuzzy preference relations. Buckley [3] adopted the row geometric mean method to derive the fuzzy priority vector from triangular fuzzy preference relations. Chang [6] proposed a method for triangular fuzzy preference relations, which derives crisp weights. However, Wang et al. [41] ________________________ Corresponding Author: Fanyong Meng is with Business School, Central South University, Changsha 410083, China E-mail:
[email protected]. Xiaohong Chen is with Business School, Central South University, Changsha 410083, China, and with School of Accounting, Hunan University of Commerce, Changsha 410205, China E-mail:
[email protected].
1
found that the extent analysis method proposed by Chang [6] cannot derive the true weights from a triangular fuzzy preference relation and has resulted in a huge number of misapplications. Furthermore, Xu [58] gave a fuzzy least squares priority method (FLSPM), and Mikhailov [23] developed a fuzzy preference programming method (PPM) that is based on the α-cut decomposition of the triangular fuzzy judgment. Both of them derive crisp weights from triangular fuzzy preference relations, but the weights determined by these methods cannot represent the relative importance of alternatives at all [41]. Wang and Elhag [43] pointed out that the normalization formula for triangular fuzzy numbers given by Chang [6] is wrong and defined a correct one. Meanwhile, Wang et al. [44] developed an approach to triangular fuzzy compare wise judgment matrices using the modified fuzzy LLSM, which has the narrower support intervals than the fuzzy LLSM [37]. Recently, Wang and Chin [45] constructed a linear goal programming model to derive the normalized fuzzy priority vector from triangular or trapezoidal fuzzy reference relations. Based on the defined similarity degree (SD) and proximity degree (PD), Wu and Chiclana [47] studied group decision making with triangular fuzzy complementary preference relations. Without regard to the advantages and disadvantages of all above-mentioned researches, the leading criticism is their failure to handle inconsistency [18], which may lead to a misleading solution. As for this issue, Saaty [33] defined the concept of consistent multiplicative preference relations and gave a consistency ratio to measure the inconsistent degree of multiplicative preference relations. Meanwhile, Crawford and Williams [7] introduced the geometric consistency index (GCI) to judge the consistency of a multiplicative preference relation, and the thresholds are established by Aguarón and Moreno-Jiménez [1]. Recently, Meng and Chen [26] developed a method to group decision making with incomplete fuzzy preference relations, which is based on the consistent and consensus analysis. Meng and Chen [27] gave the multiplicative geometric consistent index (MGCI) for multiplicative preference relations and studied the average values of the MGCI in simulation method. Then, the authors developed a consistency and consensus based approach to group decision making with incomplete multiplicative preference relations. Furthermore, Buckley [3] presented a concept of consistent trapezoidal fuzzy preference relations, which is an extension of the crisp case [33]. Considering the fuzzy ratios of relative importance as constraints, Salo [36] developed a method for calculating the priority vector from fuzzy preference relations. Following the idea of Salo [36], Leung and Cao [18] introduced a fuzzy consistency definition using the fuzzy ratios of relative importance, which allows a certain tolerance deviation. Then, the authors calculated the fuzzy local and global weights via the extension principle. Ramik and Korviny [31] introduced a new consistency index for triangular fuzzy preference relations using the logarithmic least squares method, which is later shown by Brunelli [5] that this consistency index may fail to fairly capture inconsistency. Furthermore, following the work of Buckley [3], Wang and Chen [46] applied a fuzzy linguistic preference relation (Fuzzy LinPreRa) to construct a consistent triangular fuzzy preference relation, which only needs n−1 pairwise comparisons for a decision-making problem with n alternatives. Recently, Liu et al. [22] pointed out that some conclusions given by Wang and Chen [46] do not true, and they presented another consistency concept of triangular fuzzy preference relations, which is based on the assumption that the decision maker has the same risk preference for all his/her fuzzy preferences. It is worth pointing out that Liu et al. [22] and Wang and Chen [46] adopted the Herrera-Viedma’s method [14] to construct a consistent triangular fuzzy preference relation, and then the fuzzy priority vector is obtained using the row geometric mean method [3]. There are two main issues of the existing methods for triangular fuzzy preference relations. One is the concept of consistency [3], which is a direct extension of the crisp case [33]. As Dubois [10] pointed out that the Buckley’s consistency concept for triangular fuzzy preference relations is not true. The other is the calculation of triangular fuzzy priority vector [3, 22, 46]. Although the authors pointed out that the consistent triangular fuzzy preference
2
relation obtained by Herrera-Viedma et al.’s method is independent of the chosen n−1 pairwise comparisons when the decision maker’s preferences are consistent. However, this assumption is very difficult to achieve. Contrary, in most cases, the decision maker’s preferences are inconsistent. Then, whether the chosen n−1 pairwise comparisons can fully denote the decision maker’s preferences? When the decision maker’s fuzzy preference relation is inconsistent how to select the pairwise comparisons? What are they based on? Furthermore, whether the different ranking orders are obtained by using the different pairwise comparisons? With these issues, this paper develops a new method for deriving the fuzzy priority vector from triangular fuzzy preference relations, which is based on the consistency analysis. The main features of the new method include the following four aspects: (i) The new method calculates the fuzzy priority weights separately, which are derived from acceptably consistent multiplicative preference relations; (ii) A new concept of consistent triangular fuzzy preference relations is defined, which allows the decision maker to have different risk-attitudes for his/her fuzzy judgments; (iii) The upper and lower bounds of the triangular fuzzy priority weight indicate the decision maker’s optimistic and pessimistic preferences, respectively, which are determined by the boundary of triangular fuzzy preferences. Thus, the fuzzy priority weights can be easily derived; (iv) The new method can be easily extended to the other kinds of fuzzy preference relations. This paper is organized as follows: In section 2, some basic concepts are reviewed, such as the consistency ratio (CR), the eigenvector method (EM), the row weighted arithmetic mean method and triangular fuzzy numbers. In section 3, a new method for driving the triangular fuzzy priority vector from triangular fuzzy compare wise judgment matrices is proposed, which is based on consistency analysis. Meanwhile, a new concept of consistent triangular fuzzy compare wise judgment matrices is defined. Then, an algorithm to derive the triangular fuzzy priority vector is developed that can address inconsistent case. Applying the relational index for triangular fuzzy numbers, the ranking order of alternatives is derived. After that, a numerical example is offered to show the application of the new method. Meanwhile, the comparison and analysis are made with respect to several existing methods. In section 4, we further briefly discuss the application of the new method to triangular fuzzy reciprocal preference relations and trapezoidal fuzzy preference relations. In section 5, a practical application of the new method to new product development (NPD) project screening is offered. The conclusions are shown in Section 6. 2. Preliminaries Without loss of generality, throughout the paper, let X ={x1, x2, …, xn} be the set of compared objects. In the traditional AHP, an n order multiplicative preference relation, A, is defined by A (aij )nn such that aij 1 a ji , where 1≤ i, j ≤n [31]. If the elements in A satisfy aij aik akj for all i, j, k = 1, 2, …, n, then A is considered to be consistent; otherwise, A is said to be inconsistent [33]. To judge the consistency of a multiplicative preference relation, Saaty [33] introduced the following consistency ratio (CR) n CI with CI max , (1) CR RI n 1 where max is the largest eigenvalue of A, and RI is the given random index [35]. If CR ≤ 0.1, then A is considered to be acceptably consistent; otherwise, A is unacceptably consistent. For the (acceptably) consistent multiplicative preference relation A, Saaty [33] utilized the right principal eigenvector of A as the priority vector w, which is derived by
3
Aw =λmaxw s.t. eTw = 1. (2) The priority vector w obtained by Eq.(2) is the well-known eigenvector method (EM). For the (acceptably) consistent multiplicative preference relation A, the priority vector can be derived by the row weighted arithmetic mean method [6]
a n
wi
n
j 1
j 1 ij n
,
i 1 , 2 , .n. .. ,
(3)
a
i 1 ij
To cope with the uncertain information, van Laarhoven and Pedrycz [37] presented a fuzzy preference relation with triangular fuzzy numbers. First, let us review the concept of triangular fuzzy numbers and some of their operations. A triangular fuzzy number a is expressed by an ordered triple a (l , m, u) such that l m u with the following membership function mx ll , l x m a ( x) uumx , m x u , 0, otherwise
(4)
where l and u respectively denote the lower and upper bounds of a , and m is the median value. If l > 0, then a is said to be a positive triangular fuzzy number. Let a1 (l1 , m1 , u1 ) and a2 (l2 , m2 , u2 ) be any two positive triangular fuzzy numbers, then some of their operations are defined by [37] (i) Addition: a1 a2 l1 l2 , m1 m2 , u1 u2 ;
(ii) Subtraction:
a2 l1 u2 , m1 m2 , u1 l2 ;
(iii) Multiplication: a1 a2 l1l2 , m1m2 , u1u2 ; (iv) Division: a1 / a2 l1 u2 , m1 m2 l2 u1 ; (v) Exponent: r a1 (r l1 , r m1 , r u1 ) , where r > 0; (vi) Logarithm: logr a1 (logr l1 ,logr m1 ,logr u1 ) , where r > 0; (vii) Reciprocal: a11 1 u1 ,1 m1 ,1 l1 .
Definition 1 [37]: A triangular fuzzy compare wise judgment matrix, A , is defined by
A (aij )nn
(l12 , m12 , u12 ) ... (l1n , m1n , u1n ) (1,1,1) (l , m , u ) (1,1,1) ... (l2 n , m2 n , u2 n ) 21 21 21 , (1,1,1) (ln1 , mn1 , un1 ) (ln 2 , mn 2 , un 2 ) ...
(5)
where aij (lij , mij , uij ) is a positive triangular fuzzy number, and a ji 1 / aij for all i, j = 1, 2, …, n. When 4
lij mij uij for each pair (i, j), we obtain a multiplicative preference relation [33].
Definition 2 [37]: The triangular fuzzy compare wise judgment matrix, A , is said to be consistent if and only if aij aik akj for all i, j, k =1, 2, …, n.
Later, Dubois [10] and Ohnishi [28] noted that the above consistency concept is not true, for example, aij aik akj does not hold for all i, j, k =1, 2, …, n. The authors considered the fuzzy entries to be a set of flexible constraints of precise consistent preference relations whose entries are inside the support of the fuzzy entries and derived the priority vector by solving the building programming model. Different to Definition 2, Liu et al. [22] introduced another consistency concept of triangular fuzzy compare wise judgment matrices. With respect to the triangular fuzzy compare wise judgment matrix A (aij )nn as shown in Definition 1, let
AL (aijL ) nn
1 l12 ... l1n 1 u 1 ... l2 n M m 21 , A (aijM )nn 21 un1 un 2 ... 1 mn1
m12 1 mn 2
... m1n 1 u12 ... u1n ... m2 n l 1 ... u2 n , AU (aijU )nn 21 . (6) ... 1 ln1 ln 2 ... 1
Similar to Liu [19], using a convex combination method, Liu et al. [22] defined the multiplicative preference relation D( , , ) dij ( , , )
, where dij ( , , ) aijL
nn
a a M ij
U ij
with , , [0,1] and 1 for
all i, j =1, 2, …, n. From Eq.(6), one can easily see that D( , , ) is a multiplicative preference relation. Since the consistency of D( , , ) can be obtained by that of AL , AM and AU , Liu et al. [22] presented the following consistency concept. Definition 3 [22]: Let A be a triangular fuzzy compare wise judgment matrix. If the multiplicative preference relations AL , AM and AU as shown in Eq.(6) are consistent, then A is said to be a consistent triangular fuzzy compare wise judgment matrix; otherwise, A is said to be inconsistent. The values of , and respectively reflect the decision maker’s pessimistic, neutral and optimistic preferences. Definition 3 is in fact based on the assumption that the decision maker has the same risk preference for all his/her triangular fuzzy preferences. When the decision maker’s risk preferences are different with respect to the different triangular fuzzy preferences, the consistency of A is not true. (1,1,1) (1, 2, 2) (2,6,12) Example 1: Let A ( 12 , 12 ,1) (1,1,1) (2,3,6) . According to Definition 3, it is easy to know that A is a ( 1 , 1 , 1 ) ( 1 , 1 , 1 ) (1,1,1) 6 3 2 12 6 2
consistent triangular fuzzy compare wise judgment matrix. However, the multiplicative preference relation
5
1 A 12 1 12
2 12 1 2 is inconsistent. 1 1 2
3. A new method for triangular fuzzy compare wise judgment matrices Just as the crisp case, the most important issue is to derive the priority vector from fuzzy preference relations, and consistency analysis plays a crucial role in ensuring a reasonable decision. This section focuses on the fuzzy priority vector from a triangular fuzzy compare wise judgment matrix. First, a consistency concept of triangular fuzzy compare wise judgment matrices is defined, and some properties are studied. Then, a new method to derive the fuzzy priority vector is introduced that is based on consistency analysis. A. Triangular fuzzy priority vector With respect to the triangular fuzzy compare wise judgment matrix A (aij )nn , consider the following interval multiplicative preference relation B (bij )nn , which is defined by
[1,1] [al12 , au12 ] ... [al1n , au1n ] [1 au12 ,1 al12 ] [1,1] ... [al2 n , au2 n ] B , [1,1] [1 au1n ,1 al1n ] [1 au 2 n ,1 al 2 n ] ...
(7)
where bij [blij , buij ] [lij (mij lij ) , uij (uij mij ) ] aij for each pair (i, j) with i ≤ j and [0,1] , namely, bij is the α−cut of aij for each pair (i, j) with i ≤ j. Especially, when 1 , B is the multiplicative preference relation
AM . It is worth noting that the interval multiplicative preference relation B (bij )nn is not the α−cut of the triangular fuzzy compare wise judgment matrix A (aij )nn . One can easily check that the α−cut of A (aij )nn is not an interval multiplicative preference relation. Thus, we only require the upper triangular part of B is the α−cut of A . This requirement does not influence the following discussion. With respect to the interval multiplicative preference relation B , [0,1] , let
Bl (blij )nn
1 al12 1a 1 l12 1 al1n 1 al 2 n
1 ... al1n au12 ... al2 n 1 1 au12 B ( b ) and u uij n n ... 1 1 au1n 1 au 2 n
6
... au1n ... au2 n . ... 1
Furthermore, let
B (bij )nn blij
ij
b
1 ij
uij
nn
,
(8)
where ij [0,1] for all i, j =1, 2, …, n such that ij ji . One can easily find that B is a multiplicative preference relation. Definition 4: Let A (aij )nn be a triangular fuzzy compare wise judgment matrix, and B (bij )nn be an interval multiplicative preference relation as shown in Eq.(7). If all multiplicative preference relations B (bij )nn as shown in Eq.(8) are (acceptably) consistent, then B is said to be (acceptably) consistent. Definition 5: Let A (aij )nn be a triangular fuzzy compare wise judgment matrix. If, for any [0,1] , the interval multiplicative preference relation B (bij )nn is (acceptably) consistent, then A is said to be (acceptably) consistent. From Definition 4, we know that the associated interval multiplicative preference relation B (bij )nn is consistent when any multiplicative preference relation B in Eq.(8) is consistent, namely, for any ij [0,1] , B in Eq.(8) is consistent. From Definition 5, we know that a triangular fuzzy compare wise judgment matrix A is consistent when, for any [0,1] , the associated interval reciprocal preference relation B (bij )nn is consistent. Thus, a triangular fuzzy compare wise judgment matrix A is consistent when the multiplicative preference relation B in Eq.(8) is consistent for any [0,1] and any ij [0,1] for all i, j =1, 2, …, n with ij ji .
When B (bij )nn expressed by Eq.(8) is a consistent multiplicative preference relation, by Eq.(3) we obtain
b
n
wi
n
k 1
j 1 ij n
b
j 1 kj
1
i j 1
lij
ij
j i 1 alij n
a a
1 ij
ij
a
1 ij
uij
uij
kj 1 kj k n 1 k 1 j 1 kj 1 kj j k 1 alkj aukj alkj aukj n
,
i 1 , 2 , .n. .. ,
(9)
Theorem 1: Le B (bij )nn be a consistent multiplicative preference relation as shown in Eq.(8) and w w1 , w2 ,..., wn be the priority vector derived by Eq.(9), then, for any i = 1, 2, …, n,
wi
i
i ci i (n 2) 7
,
(10)
where i j 1 i
n i n n 1 1 1 n 1 and i min lkj , ukj . j i 1 uij , ci lki lkj ukj lij k 1, k i j k 1, j i k 1 k i 1 uki
Proof: For a given i, let ij 1 for all j= 1, 2, …, i, and ij 0 for all j=i+1, …, n, by Eq.(9) we derive
wi
i
i
where i j 1
i
,
(11)
i
n k 1 n and a k 1,k i j 1 i j i 1 uij alij alkj
1
ij
a
1 ij
uij
1 kj ukj
kj
aukj
1 kj
1 y1 y2 . From alij x y1 x y2 alij
Since, for all x, y1, y2 such that y1 ≤ y2, we have auij alij
n
a kj
j k 1 alkj
ki
ij
1 ij
and
uij
, we obtain
a
1
a
wi wi . Since alki
.
1 ki
(12)
alki for any ki [0,1] with k =1, 2, …, i−1, and
uki
1
a a
lik
kj
1 kj
1 for any ki [0,1] auik
uik
with k = i+1, i+2, …, n, we have i
i 1 k 1
n k i 1
,
(13)
where
1 k 1 k 1 alki kj 1 kj 1 1 kj nj k 2 alkj kj aukj kj alkj aukj i 1
i 1
and
1 1 n k 2 n kj 1 kj k i 1 j 1 j k alkj aukj 1 kj kj k i 1 auik a a lkj ukj
n
Let c alki
i
i
k 1
n
1
k i 1
uik
a
, we obtain
i 1
where i
k 1
n k i 1
c (n 2) i ,
i
n 1 a kj a 1 kj lkj ukj 1 kj k 1, k i j k 1, j i alkj aukj kj n
Let kji alkj
kj
a
ukj
1 kj
.
1
a a
lkj
kj
1 kj
.
for any kj [0,1] , we have
ukj
8
(14)
1 kj 1 kj min alkj aukj 1 kj alkj aukj kj For any [0,1] , we derive
min a 1 , a 1 . lkj ukj alkj aukj
c
i
(15)
ci .
(16)
Furthermore, for any [0,1] , we have 1 1 1 1 min alkj , aukj min lkj , ukj a a l u lkj ukj kj kj
(17)
for any k =1, 2, …, n with k ≠ i and any j = k+1, k+2, …, n with j ≠ i. Thus, according to Eqs.(13)-(17), we obtain
i ci i (n 2) .
(18)
For any [0,1] , we derive i i . From Eqs.(12) and (18), we have wi
i
c i (n 2) i
i
.
□
Proposition 1: Let A (aij )nn be a consistent triangular fuzzy compare wise judgment matrix, and
w w1 , w2 ,..., wn be the triangular fuzzy priority vector. Then, for any wi wi , i = 1, 2, …, n, we have wi
i
i ci i (n 2)
,
where the notations as shown in Theorem 1. Theorem 2: Let A (aij )nn be a consistent triangular fuzzy compare wise judgment matrix, let B (bij )nn be the consistent multiplicative preference relation as shown in Eq.(8), and let w w1 , w2 ,..., wn be the priority vector derived by Eq.(9), then, for any i = 1, 2, …, n, wi
where i j 1 i
i
i ci i (n 2)
,
(19)
n i n n 1 1 1 n 1 j i 1 lij , ci uki and i max lkj , ukj . lkj ukj uij k 1, k i j k 1, j i k 1 k i 1 lki
Proof: The proof of Theorem 2 is similar to that of Theorem 1.
□
Proposition 2: Let A (aij )nn be a consistent triangular fuzzy compare wise judgment matrix, and
w w1 , w2 ,..., wn be the triangular fuzzy priority vector. Then, for any wi wi , i = 1, 2, …, n, we have 9
wi
i
i ci i (n 2)
,
where the notations as shown in Theorem 2.
According to Propositions 1 and 2, we know that
i c (n 2) i
i
i
and
i c i (n 2) i
i
can be
considered as the lower and upper bounds of the triangular fuzzy priority weight wi , i = 1, 2, …, n, respectively. The remaining issue is how to confirm the median value of wi . When A (aij )nn is a consistent triangular fuzzy compare wise judgment matrix, A1 AM (mij )nn is a consistent multiplicative preference relation. Thus, by Eq.(3) the median value of wi , i = 1, 2, …, n, is determined by
m m n
m i
w
j 1
ij
n
n
j 1
i 1
.
(20)
ij
Definition 6: Let A (aij )nn be a consistent triangular fuzzy compare wise judgment matrix. Then, the triangular fuzzy priority vector w w1 , w2 ,..., wn with wi (wil , wim , wiu ) , i = 1, 2, …, n, is defined by n m i i j 1 ij , wi , n , n i ci i (n 2) c ( n 2) m i i i j 1 i 1 ij
(21)
where the notations as shown in Theorems 1 and 2. Proposition 3: Let x1, x2 such that 1≤ x1 ≤ x2, then 1 1 1 1 1 1 min x1 , x2 x1 , max x1 , x2 x2 . x1 x2 x1 x1 x2 x2
As we know, in most situations, it is difficult or even impossible to require a decision maker to give a consistent triangular fuzzy compare wise judgment matrix. Thus, we need to consider the consistent degree, which is very important to ensure a reasonable decision. With respect to AM , one can apply the CR [33] to judge its consistent degree. If AM is acceptably consistent, then the priority vector wm w1m , w2m ,..., wnm can be approximately calculated by Eq.(3). For wil
i
i ci i (n 2)
and wiu
i
i ci i (n 2)
, i = 1, 2, …, n, by Proposition 3 one
can easily obtain the associated crisp multiplicative preference relations Bi (bij )nn and Bi (bij )nn .
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When Bi and Bi are both acceptably consistent, then the priority weights wil and wiu can be approximately derived by Eq.(3). From Proposition 3, we derive bij lij and bij uij , or bij uij and bij lij for each pair (i, j). Define the multiplicative preference relation Ki (kij )nn , where kij bij bij
1
and k ji 1 kij with [0,1] and 1≤ i ≤ j ≤ n.
One can easily obtain that 1 (l ) (u ) , b l , b u kij ij 1 ij ij ij ij ij (lij ) (uij ) , bij uij , bij lij
(22)
for 1≤ i ≤ j ≤ n. Especially, when 1 , we have K Bi , and K Bi for 0 .
Theorem 3: Let Bi and Bi be both (acceptably) consistent, then Ki (kij )nn defined by Eq.(22), for any [0,1] , is (acceptably) consistent.
Definition 7: Let Ki kij
nn
be an interval fuzzy multiplicative preference relation, where kij [kij , kij ]
min{bij , bij }, max{bij , bij } [lij , uij ] and k ji 1 kij 1 kij ,1 kij for all 1≤ i ≤ j ≤ n. If, for any [0,1] , Ki (kij )nn is (acceptably) consistent, then K i is said to be (acceptably) consistent.
Proposition 4: Let Bi and Bi be both (acceptably) consistent. Then, the interval fuzzy multiplicative preference
relation Ki kij
nn
is (acceptably) consistent.
From Proposition 4, we derive that the triangular fuzzy priority vector w w1 , w2 ,..., wn can be obtained from the (acceptably) consistent multiplicative preference relations AM , Bi and Bi , i = 1, 2, …, n. However, in most practical decision-making problems, this requirement usually does not hold. To cope with the unacceptably consistent case and retain the original preference information as much as possible, we adopt the following method to adjust the associated multiplicative preference relation. Without loss of generality, suppose that Bi (i = 1, 2, …, n) is unacceptably consistent. Using Eq.(3), we derive
b
j 1 ij n
n
w
Bi p
n
j 1
b i 1 ij
,
p 1 , 2 , .n. .. ,
Then, construct the consistent multiplicative preference relation Bi bij
11
nn
, denoted by
(23)
ij
b
Let Bi bij
nn
wiBi
wBji
,
p, q = 1, 2, …, n.
(24)
with
b b ij b ij bij ij
1
CRij max np, q 1, p q bpq bpq
(25)
otherwise
for each pair (i, j) such that i < j and bij 1 bji for j < i, where [0,1] . Repeat this process until CR Bi 0.1 . Then, use Eq.(3) to calculate the priority vector. Now, let us consider an algorithm to derive the triangular fuzzy priority vector from triangular fuzzy compare wise judgment matrices. Let A be a triangular fuzzy compare wise judgment matrix, and w w1 , w2 ,..., wn be the triangular fuzzy priority vector. According to the above analysis, the main procedure can be described as follows: Step 1: According to A (aij )nn (lij , mij , uij ) , we derive the multiplicative preference relation AM (mij )nn , nn and judge its consistency. If AM is (acceptably) consistent, then use Eq.(3) to calculate the priority vector wm w1m , w2m ,..., wnm from AM with wim being the mean value of wi , i = 1, 2, …, n. When AM is unacceptably
consistent, we apply Eq.(24) to construct the consistent multiplicative preference relation AM mij
nn
with
mij wim wmj for each pair (i, j). Then, we utilize Eq.(25) to adjust the consistency of AM until it is
acceptably consistent according to Eq.(1). Suppose that AM mij
nn
is the associated acceptably consistent
multiplicative preference relation, where 1 mij mij mij mij
CRij max np, q 1, p q mij mij otherwise
for each pair (i, j) such that i < j and mij 1 m ji for j < i. Solve the priority vector wm w1m , w2m ,..., wnm from
AM by using Eq.(3), then wim is the mean value of wi , i = 1, 2, …, n. Step 2: For the given i, suppose that Bi and Bi are the multiplicative preference relations with respect to the lower and upper bounds of wi . If Bi and Bi are both (acceptably) consistent, then wil and wiu are determined according to Eq.(21). Otherwise, without loss of generality, suppose that Bi is unacceptably consistent. Then, according to Eqs.(23)-(25), adjust the consistency of Bi until the associated multiplicative preference
relation Bi bij
nn
is acceptably consistent. Finally, we use Eq.(3) to calculate the priority weight wil , 12
b
j 1 ij n
n
l i
where w
n
. Similarly, one can obtain the priority weight wiu .
i 1 ij
j 1
b
Step 3: Repeat Step 2, until each triangular fuzzy priority weight wi (wil , wim , wiu ) , i = 1, 2, …, n, is obtained. Step 4: Suppose that w w1 , w2 ,..., wn is the derived triangular fuzzy priority vector, where wi (wil , wim , wiu ) , i = 1, 2, …, n. Rank alternatives according to w . Applying this procedure, one can obtain the triangular fuzzy priority vector from triangular fuzzy compare wise judgment matrices. The rest issue is how to rank triangular fuzzy priority weights. At present, many ranking methods for triangular fuzzy numbers are presented [17, 24, 59-62]. Dubois and Prade [11] and Wang and Kerre [39, 40] reviewed ranking methods for fuzzy sets and classified them, respectively. It is worth pointing out that their classified standards are different. The reader is referred to [11, 39, 40] for details. In the fuzzy AHP, the authors [22, 31, 45, 46] mainly adopted the fuzzy mean method [17] to rank triangular fuzzy weights, which is based on the assumption that there is a uniform probability distribution on triangular fuzzy numbers. Let a (l , m, u) be a positive triangular fuzzy number, its expectation value is l mu . 3
ELL (a)
(26)
However, this method only gives the ranking order of triangular fuzzy numbers. Their preferred degrees cannot be reflected. To both consider their ranking orders and preferred degrees; Zeng and Meng [66] introduced a relational index based on computing areas limited by the membership function of triangular fuzzy numbers. According to the lower and upper bounds of two triangular fuzzy numbers, by symmetry the authors classified six kinds of position relationships. This method is based on the assumption that the right enclosed area composed by the x axis and the membership function is better than the left enclosed area. For the composed common area, one of which is better than the other with 1/2 possibility. Furthermore, it considers that there is a uniform distribution on the enclosed area. Let a1 (l1 , m1 , u1 ) and a2 (l2 , m2 , u2 ) be any two positive triangular fuzzy numbers. By symmetry, there are only five possible relationships between their membership functions. The relational index P(a1 a2 ) is defined as follows [66]: 1
1
1
S
0
l2 m2
u2 l1 m1
Fig 1.
0
u1
0
l1
0
P ( a1 a 2 )
4 4
S 45
m2 m1 u1 u2
Fig 5. m2 m1 , l2 l1 u1 u2
S (S S ) S S 1 (S ) S a1 S a 2 S a1 S a 2 2 S a1 S a 2 4 2
l2 l1
4 2
2
P(a1 a2 )
S15 ( S 25 S35 ) S 25 S35 1 ( S 25 ) 2 S a1 S a2 S a1 S a2 2 S a1 S a2
13
l2
P(a1 a2 )
S15 S 25
l1
3 3
m2
S13
m1 u2 u1
Fig 3. m2 m1, l1 l2 u2 u1
S35
Fig 4. m2 m1 , l1 l2 u1 u2 4 4
S 0
S12 1 ( S22 ) 2 S22 S32 Sa1 2 Sa1 Sa2 Sa1 Sa2
4 2
m2 m1 u1 u2 4 2
S 23
u2 m1 u1
S54
l2 4 1
l2 m2 l1
S 22
1
S14 S
S34
S
P(a1 a2 )
S 44
S 43
2 1
Fig 2. m2 m1 , l2 l1 u2 u1
u 2 l1
P(a1 a2 ) 1
1
2 3
S13 S 3S 3 1 ( S 23) 2 2 4 S a1 S a 1 S a 2 2 S a 1S a 2
u1
u2
l1
l2
In the listed relational index, Sa1 a1 ( x)dx and Sa2 a2 ( x)dx with a1 ( x) and a2 ( x) being the membership functions of a1 and a2 . In Fig.3, when m1 m2 , we derive the sixth case:
1
S 26 S36 0
l1
l2
m2(m1 )
S16
u2 u1
Fig 6 m2 m1 , l1 l2 u2 u1
P(a1 a2 )
S16 1 S 26 S a1 2 S a1
1 when u1 u2 l2 l1 . In this case, the relational index is defined by 2 l1u1 P(a1 a2 ) . (27) l1u1 l2u2
From Fig.6, one can derive that P(a1 a2 )
Definition 8: Let a1 (l1 , m1 , u1 ) and a2 (l2 , m2 , u2 ) be any two positive triangular fuzzy numbers. Their relational index is defined by 1 2 S1 1 ( S22 )2 S22 S32 Sa1 2 Sa1 Sa2 Sa1 Sa2 S13 S23 S43 1 ( S23 )2 Sa1 Sa1 Sa2 2 Sa1 Sa2 P(a1 a2 ) 4 4 4 4 4 4 2 S1 ( S2 S4 ) S2 S4 1 ( S2 ) Sa1 Sa2 Sa1 Sa2 2 Sa1 Sa2 5 5 5 S1 ( S2 S3 ) S25 S35 1 ( S25 )2 S S Sa1 Sa2 2 Sa1 Sa2 a1 a2
When P(a1 a2 )
u2 l1 m2 m1 , l2 l1 u2 u1 m2 m1 , l1 l2 u2 u1
,
(28)
m2 m1 , l1 l2 u1 u2 m2 m1 , l2 l1 u1 u2
1 , we use Eq.(27) to calculate the relational index between a1 and a2 . 2
Proposition 4: Let a1 (l1 , m1 , u1 ) and a2 (l2 , m2 , u2 ) be any two positive triangular fuzzy numbers. Then, the relational index P as shown in Definition 8 satisfies (i) P(a1 a2 ) P(a2 a1 ) 1 ; (ii) 0 P(a1 a2 ) 1 .
Proof: To prove (i), without loss of generality, take the fourth case for example, from Fig.4 we have P(a2 a1 )
S54 S34 (S24 S44 ) 1 ( S24 )2 . Sa2 Sa1 Sa2 2 Sa1 Sa2
14
Thus, P(a1 a2 ) P(a2 a1 )
S54 S34 (S24 S44 ) ( S24 )2 S14 ( S24 S44 ) S24 S44 Sa2 Sa1 Sa2 Sa1 Sa2 Sa1 Sa2 Sa1 Sa2
S54 S14 S24 S14 S44 S24 S44 (S24 )2 S24 S34 S34 S44 Sa2 Sa1 Sa2
S54 S24 ( S14 S24 S34 ) S44 (S14 S24 S34 ) Sa2 Sa1 Sa2
4 4 S54 S2 Sa1 S4 Sa1 Sa2 Sa1 Sa2
S24 S44 S54 Sa2
1. For (ii): From the expression of P and (i), we can easily obtain the conclusion.
□
B. A numerical example and comparison analysis Example 2: Consider the following triangular fuzzy compare wise judgment matrix (1,1,1) (2,3, 4) ( 13 , 12 ,1) (1, 2,3) 1 1 1 ( , , ) (1,1,1) (4,5,6) (1,1, 2) A 4 3 2 . (1, 2,3) ( 16 , 15 , 14 ) (1,1,1) (2,3, 4) 1 1 1 1 1 1 ( 3 , 2 ,1) ( 2 ,1,1) ( 4 , 3 , 2 ) (1,1,1) 1 3 12 2 1 1 5 1 From A , we derive AM 3 1 . With respect to AM , we obtain max 5.4024 and CR( AM ) 0.5252 , 2 5 1 3 1 1 2 1 3 1
namely, AM is unacceptably consistent. According to Eqs.(23)-(25), the adjusted multiplicative preference relation is listed as follows: 2.1485 0.5 2 1 0.4654 1 1.1285 1 . AM 2 0.8861 1 3 1 0.3333 1 0.5
By AM , we derive max 4.2647 and CR( AM ) 0.0991 . Applying Eq.(3), we derive wm 0.2979,0.1895,0.3632,0.1494 . 15
(29)
With respect to each object xi, the multiplicative preference relation, the adjusted multiplicative preference relation, the CR and the lower bound of the triangular fuzzy weight are listed as shown in Table 1. Table 1 The lower bound of the triangular fuzzy weight Multiplicative preference relation and the CR 1 2 13 1 1 1 6 2 B 2 1 3 6 1 4 1 1 1 1 2 4
Adjusted multiplicative preference relation and the CR
max 5.5530
1.3271 0.3333 1 1 0.7535 1 1.1551 2 B 3 0.8657 1 3.817 1 0.5 0.2619 1
1
1 4 13 3 1 1 4 1 B2 4 1 3 4 1 4 1 1 1 1 4 3 1 4 1 3 1 1 6 2 B3 4 1 1 6 1 2 1 1 1 1 2 2 1 4 13 3 1 1 6 2 B4 4 1 3 6 1 4 1 1 1 1 3 2 4
max 4.2562 w1l 0.1829
1
1
CR( B ) 0.5816
max 5.7862
2.5318 0.3604 3 1 0.3949 1 0.6914 1 B 2.7746 1.4463 1 4 0.3333 1 0.25 1
1
CR( B ) 0.0960
max 4.2658
2
2
CR( B ) 0.6690
max 5.0458
2.2842 1 3 1 0.4378 1 2.3864 2 B 1 0.419 1 2 0.3333 0.5 0.5 1
2
CR( B ) 0.3917
max 6.0998
1.8297 0.3333 3 1 0.5465 1 0.9618 2 B4 3 1.0397 1 4 0.3333 0.5 0.25 1
4
CR( B ) 0.7864
w2l 0.1417
CR( B ) 0.0996
max 4.2599
3
3
The lower bound of the triangular fuzzy weight
3
w3l 0.2225
CR( B ) 0.0973
max 4.2606 w4l 0.0956 4
CR( B ) 0.0976
Furthermore, the multiplicative preference relation, the adjusted multiplicative preference relation, the CR and the upper bound of the triangular fuzzy weight are listed as shown in Table 2. Table 2 The upper bound of the associated triangular fuzzy weight Multiplicative preference relation and the CR 1 4 1 3 1 1 4 1 B1 4 1 1 4 1 2 1 3 1 1 1 1 2 1 1 1 1 6 2 B 2 1 1 6 1 2 1 1 1 2 2 1
Adjusted multiplicative preference relation and the CR
max 4.8569
2.4 1 3 1 0.4167 1 1.815 1 B1 1 0.5509 1 2 1 0.5 1 0.3333
1
CR( B ) 0.3209
max 4.8000
1.6789 1 1 1 0.5956 1 2.1047 2 B2 1 0.4751 1 2 0.5 0.5 1 1
2
1 2 13 1 1 1 4 1 B 2 1 3 4 1 4 1 1 1 1 4
2
CR( B ) 0.2996
max 5.3201
1.7047 0.3333 1 1 0.5866 1 0.9858 1 B3 3 1.0144 1 2.9389 1 0.3403 1 1
3
1 2 1 1 1 1 4 1 B 2 1 1 4 1 2 1 1 1 2 1
3
CR( B ) 0.4944
max 4.6447 4
CR( B ) 0.2415
max 4.2618 w1u 0.3891 1
CR( B ) 0.0981
max 4.2626 2
w2u 0.3193
CR( B ) 0.0984
max 4.2625 3
w3u 0.4207
CR( B ) 0.0983
max 4.2652
2 1 1 1 0.5 1 1.7613 1 B4 1 0.5678 1 2 1 0.5 1 1
4
The upper bound of the triangular fuzzy weight
w4u 0.202 4
CR( B ) 0.0993
From Eq.(29), Tables 1 and 2, we derive
w w1 , w2 , w3 , w4 (0.1829,0.2979,0.3891),(0.1417,0.1895,0.3193),(0.2225,0.3632,0.4207),(0.0956,0.1494,0.202) . Using Eq.(28), we obtain w3
0.8247
w1
0.9211
w2
0.9653
w4 , and x3 > x1 > x2 > x4.
16
In this example, if Liu et al.’s method [22] is used to make decision, and the triangular fuzzy pairwise comparisons a12 , a23 and a34 are chosen to construct the consistent triangular fuzzy compare wise judgment matrix, we derive
w ' w '1 , w '2 , w '3 , w '4 (0.2951,0.7031,1.5991),(0.1241,0.2344,0.4754),(0.0253,0.0469,0.097),(0.0075,0.0156,0408) , and the ranking order is w1 w2 w3 w4 . It shows that x1 > x2 > x3 > x4, which is different to that obtained by the new method. Furthermore, if a13 , a23 and a24 are chosen to construct the consistent triangular fuzzy compare wise judgment matrix, we have
w '' w"1 , w"2 , w"3 , w"4 (0.0123,0.0279,0.079),(0.1475,0.2787,0.6257),(0.0301,0.0557,0.1277),(0.0877,0.2787,0.5261) , and the ranking order is w2 w4 w3 w1 , which shows that x2 > x4 > x3 > x1. Another ranking order is obtained, which is obviously different to that derived by w ' . The alternative x1 is from the best to the worst. Thus, when a decision maker cannot give his/her consistent preference relation, this method to construct a consistent preference relation needs to be further studied. The same issue also exists in [46]. In this example, if the fuzzy arithmetic mean method [6] is adopted to calculate the priority vector, we derive w 0.3264,0.3626,0.3086.0.0024 . Thus, the ranking order is x2 > x1 > x3 > x4, which is different to the
above-ranking orders. When the fuzzy geometric mean method [3] is applied to calculate the fuzzy priority vector, we derive w (0.1618,0.3181,0.5975),(0.1791,0.2746,0.5024),(0.1361,0.2579,0.4225),(0.0809,0.1544,0.2699) .
Thus, the ranking order is x1 > x2 > x3 > x4, which is the same as the first case of the Liu’s method [43]. It is worth noting that neither the fuzzy arithmetic mean method nor the fuzzy geometric mean method is based on acceptably consistent fuzzy preference relations. Furthermore, when the Wang and Chin’s method [45] is applied in this example, by using the linear goal programming (LGP) model (25) [45] we have J * 3.1516 and
w w1, w2 , w3 , w4 (0.3124,0.4138,0.4138),(0.3794,0.3794,0.3794),(0.2068,0.2068,0.2335),(0,0,0.1005) , and the ranking order is w1 w2 w3 w4 , which shows that x1 > x2 > x3 > x4. It is the same as that obtained by the fuzzy geometric mean method [3] and the first case of the Liu’s method [43]. According to the ranking method adopted in [45], we obtain r ( x1 ) r ( x2 ) 0.0006 . One can see that the difference between x1 and x2 is very small.
17
Furthermore, it is worth noting that the variables and constraints in the LGP model (25) [45] will be greatly increased with respect to the increasing order of a triangular fuzzy compare wise judgment matrix. For example, in a 4 order triangular fuzzy compare wise judgment matrix, there are 32 variables and 29 constraints, and there are 40 variables and 36 constraints for a 5 order triangular fuzzy compare wise judgment matrix. Furthermore, in some cases, the crisp priority weight is obtained from a triangular fuzzy compare wise judgment matrix, which seems to be unreasonable. From the priority vector in this example, one can see that the priority weight of the alternative x2 is crisp. Meanwhile, the mean and upper values of the alternatives x1 and x3 are respectively equal; and the mean and lower values of the alternative x4 are equal. Moreover, this method does not consider the consistency of triangular fuzzy compare wise judgment matrices. With respect to different methods, the ranking order is listed as shown in Table 3. Table 3 Ranking order with respect to different methods
Methods
The ranking order
Liu et al.’s method [22]
x1 > x2 > x3 > x4 or x2 > x4 > x3 > x1
The fuzzy arithmetic mean method [6]
x2 > x1 > x3 > x4
The fuzzy geometric mean method [3]
x1 > x2 > x3 > x4
Wang and Chin’s method [45]
x1 > x2 > x3 > x4
The new method
x3 > x1 > x2 > x4
Remark 1: From this example, one can see that it is necessary to analyze consistency. When the consistency is considered in this example, one can find that the triangular fuzzy preference value a23 (4,5,6) is abnormal. Thus, when we calculate the fuzzy priority vector based on acceptably consistency, the triangular fuzzy preference value a23 (4,5,6) will be adjusted, and the object x3 is the best. However, when the priority vector is directly calculated
according to A , we cannot get this conclusion. 4. A further discussion Besides triangular fuzzy compare wise judgment matrices, triangular fuzzy reciprocal preference relations are another important kind of fuzzy preference relations. First, let us consider its definition. Definition 9: A triangular fuzzy reciprocal preference relation, P , is defined by
P ( pij )nn
(0.5,0.5,0.5) ( p12l , p12m , p12u ) ... ( p1ln , p1mn , p1un ) l m u ( p , p , p ) (0.5,0.5,0.5) ... ( p2l n , p2mn , p2un ) 21 21 21 , l m u l m u ( pn1 , pn1 , pn1 ) ( pn 2 , pn 2 , pn 2 ) ... (0.5,0.5,0.5)
(30)
where pij ( pijl , pijm , piju ) is a positive triangular fuzzy number with pij p ji pijl puji , pijm p mji , piju plji (1,1,1) for all i, j = 1, 2, …, n. When pijl ijm piju for each pair (i, j), we derive a fuzzy preference relation [29].
18
Wang and Chen [46] presented a method for the fuzzy linguistic AHP using triangular fuzzy reciprocal preference relations, which is based on the consistency concept. Later, Liu et al. [22] pointed out that there are some drawbacks of this consistency concept. With respect to multiplicative and reciprocal preference relations, Kacprzyk and Fedrizzi [16] researched their relationship and introduced the following proposition. Proposition 5 [16]: Let A (aij )nn be a multiplicative preference relation with aij [ 19 ,9] . Then, P ( pij )nn is a 1 reciprocal preference relation with pij [0,1] , where pij (1 log9 aij ) . 2
Let P ( pij )nn be a reciprocal preference relation with pij [0,1] . According to Proposition 5, one can easily check that A (aij )nn is a multiplicative preference relation with aij [ 19 ,9] , where aij 9
2 pij 1
. Similar to Proposition 5, we
can have the similar relationship between triangular fuzzy compare wise judgment matrices and triangular fuzzy reciprocal preference relations. Proposition 6: Let P ( pij )nn be a triangular fuzzy reciprocal preference relation as shown in Eq.(30). Then, A (aij )nn is a triangular fuzzy compare wise judgment matrix with aij (lij , mij , uij ) and lij , mij , uij [ 19 ,9] , where
aij 9
2 pij 1
for all i, j 1, 2,..., n .
Similar to Proposition 6, for a triangular fuzzy compare wise judgment matrix A (aij )nn with aij (lij , mij , uij ) and lij , mij , uij [ 19 ,9] , P ( pij )nn is a triangular fuzzy reciprocal preference relation as shown in Eq.(30), where 1 pij (1 log9 aij ) . 2
According to Proposition 6, one can convert a triangular fuzzy reciprocal preference relation into a triangular fuzzy compare wise judgment matrix and then apply the method introduced in subsection 3.1 to derive the triangular fuzzy priority vector. It is worth pointing out that Xu [50, 51] also introduced two methods to calculate the fuzzy priority weights from triangular fuzzy reciprocal preference relations. There are two main differences between the new method and the methods in [50, 51]. One is that the new method is based on the consistency analysis, while the methods in [50, 51] do not. The other is that the new method calculates the lower and upper bounds of the fuzzy priority weights with respect to difference multiplicative preference relations, while Xu [50, 51] calculated the lower and upper bounds of the fuzzy priority weights with respect to the same multiplicative preference relation. Similar to triangular fuzzy preference relations, one can apply the new method to trapezoidal fuzzy preference relations. Definition 10 [3]: A trapezoidal fuzzy compare wise judgment matrix, B , is defined by 19
T (tij )nn
where tij (lij , ij , ij , uij ) is
(1,1,1,1) (l12 , 12 , 12 , u12 ) ... (l1n , 1n , 1n , u1n ) (l , , , u ) (1,1,1,1) ... (l2 n , 2 n , 2 n , u2 n ) , 21 21 21 21 (1,1,1,1) (ln1 , n1 , n1 , un1 ) (ln 2 , n 2 , n 2 , un 2 ) ...
a
positive
trapezoidal
fuzzy
number,
(31)
namely, 0 lij ij ij uij
and
t ji 1/ tij (1 uij ,1 ij ,1 ij ,1 lij ) for all i, j = 1, 2, …, n with i ≤ j.
Let w w1 , w2 ,..., wn be the trapezoidal fuzzy priority vector for the trapezoidal fuzzy compare wise judgment
l u matrix B , where wi wi , wi , wi , wi , i = 1, 2, …, n. Similar to triangular fuzzy compare wise judgment matrices,
one can see that the lower and upper bounds wil and wiu (i = 1, 2, …, n) are determined by the boundary of tij (i, j = 1, 2, …, n). To solve the left and right mean values wi and wi (i = 1, 2, …, n), it needs to consider the multiplicative
1ij
preference relation R (rij )nn , where rij ( ij ) ij ( ij )
and rji
1 for all i, j = 1, 2, …, n with i ≤ j. From rij
Theorems 1 and 2, one can have the following theorem. Theorem 5: Let T (tij )nn be a trapezoidal fuzzy compare wise judgment matrix as shown in Eq.(31), and
1ij
R (rij )nn be a multiplicative preference relation, where rij ( ij ) ij ( ij )
and rji
1 for all i, j = 1, 2, …, n such rij
that i ≤ j. Let w w1 , w2 ,..., wn be the priority vector from R by using Eq.(3), then, for any i = 1, 2, …, n, we have
i
i i i (n 2) where i j 1 i
i
1
ij
j i 1 ij , i j 1 n
i
1
ij
wi
i
i i i (n 2) i
j i 1 ij , i ki n
k 1
n
,
1
k i 1
(32)
i
, i ki
ki
k 1
n
1
k i 1
,
ki
n n n 1 1 1 1 max kj , kj and i min kj , kj . j k 1, j i k 1, k i j k 1, j i k 1, k i kj kj kj kj n
Similar to triangular fuzzy compare wise judgment matrices, we introduce the following consistency concept of trapezoidal fuzzy compare wise judgment matrices Definition 11: Let T (tij )nn be a trapezoidal fuzzy compare wise judgment matrix, let D (dij )nn be an interval 20
,1 tlij for multiplicative preference relation, where dij [dlij , duij ] [lij ( ij lij ) , uij (uij ij ) ] and t ji 1 tuij
all i, j = 1, 2, …, n such that i≤j , [0,1] . If all multiplicative preference relations D (dij )nn with ij [0,1] , dij dlij
ij
d
uij
1 ij
and t ji 1 tij for all i, j =1, 2, …, n such that i ≤ j are (acceptably) consistent, then D is said to
be (acceptably) consistent. Definition 12: Let T (tij )nn be a trapezoidal fuzzy compare wise judgment matrix. If, for any [0,1] , the interval multiplicative preference relation D (dij )nn is (acceptably) consistent, then T is said to be (acceptably) consistent. Definition 13: Let T (tij )nn be a consistent trapezoidal fuzzy compare wise judgment matrix. Then, the
l u trapezoidal fuzzy priority vector w w1 , w2 ,..., wn with wi wi , wi , wi , wi , i = 1, 2, …, n, is defined by
i i i i wi , , , , i ci i (n 2) i i i (n 2) i i i (n 2) i ci i (n 2)
(33)
where the notations as shown in Theorems 1, 2 and 5. When a trapezoidal fuzzy compare wise judgment matrix is inconsistent, one can use the introduced method for triangular fuzzy compare wise judgment matrices to adjust the associated multiplicative preference relation and then use Eq.(3) to approximately calculate the trapezoidal fuzzy priority vector. Example 3: Consider the following trapezoidal fuzzy compare wise judgment matrix (1,1,1,1) ( 15 , 13 , 12 ,1) (2,3,5,6) (1,3,3,5) (1, 2,3,5) (1,1,1,1) (3,5,6,7) ( 14 , 13 , 12 , 12 ) T 1 1 1 1 . ( 6 , 5 , 3 , 2 ) ( 71 , 16 , 15 , 13 ) (1,1,1,1) (2,3,5,5) 1 1 1 1 1 1 1 ( 5 , 3 , 3 ,1) (2, 2,3, 4) ( 5 , 5 , 3 , 2 ) (1,1,1,1)
Similar to Example 2, we derive
w w1 , w2 , w3 , w4 (0.19116,0.3016,0.3289,0.3832),(0.2034,0.3557,0.408,0.5684), (0.1283,0.1462,0.1789,0.2502),(0.0484,0.1008,0.2076,0.3959) .
Using the relational index for triangular fuzzy numbers, we obtain w2
0.8663
w1
0.9001
w4
0.5540
w3 , and x2 > x1 > x4 > x3.
According to Definition 4 in [22], one can easily check that the trapezoidal fuzzy compare wise judgment matrix T is inconsistent. Thus, the different ranking orders will be obtained using the method in [22]. Furthermore, when the Wang and Chin’s method is adopted in this example, by the LGP model (28) in [46] we derive J * 4.2452 and
21
w* w *1 , w *2 , w *3 , w *4 (0.1727,0.2519,0.3238,0.3238),(0.2053,0.4361,0.5080,0.5080), (0.0913,0.1244,0.1244,0.1244),(0.0437,0.0437,0.1156,0.3465) .
From w * , we have x2 > x1 > x4 > x3, which is the same as that obtained by the new method. To obtain the trapezoidal fuzzy priority vector w * , we need to solve a LGP model with 48 variables and 44 constraints. As above pointed out, this method is not based on acceptably consistent trapezoidal fuzzy compare wise judgment matrices. When the fuzzy arithmetic mean method [6] and the fuzzy geometric mean method [3] are used in this example, we obtain x2 > x1 > x3 > x4 for the former and x1 > x2 > x3 > x4 for the latter. The different ranking orders are obtained using these two methods. 5. Application to new product development (NDP) project screening In this section, we pay our attention to the application of the new method to NPD project screening [8, 9, 46], which can be modeled by a hierarchical structure, as shown in Fig. 7. MKFIT, MANUFIT, CUSTFIT, FINRISK and UNCERT are five screening criteria, each with some sub-criteria. The definitions for these criteria and sub-criteria are listed in Table 4. Fuzzy scales are defined in Table 5, and the decision maker’s subjective judgments on the pairwise comparison matrices for the five screening criteria, 13 subcriteria and three NPD projects are listed in Tables 6-16. Since 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 31 1 4,1 3,1 2 rather than 3 2,3, 4 .
Fig. 7. Hierarchical structure for NPD project screening. Table 4 Definitions of screening criteria and sub-criteria for NPD projects [8, 9, 46] Criteria or sub-criteria
Definition
22
Fit with company’s core marketing competencies
MKTFIT TIMING
Project matches the target launch timing needed by target segments
PRICE
Project matches the target price of target segments
LOGISTICS
Project fits with company’s logistics and distribution strengths
SALES
Project fits with company’s salesforce coverage and strengths Fit with company’s core manufacturing competencies
MANUFIT MFGTECH
Project fits with company’s manufacturing technology
MFGCAP
Manufacturing capacity matches demands
SUPPLY
Project allows companies to use their good suppliers Fit with customers’ requirements
CUSTFIT DESIGN
Project is designed to meet the requirements of customers
RELIA
Project meets the target reliability level
FINRISK
Financial risks of projects
REVENUE
Expected revenue
EXPEND
Expected expenditure Uncertainties about projects’ outcomes
UNCERT R&DUNC
Technological uncertainties in research and project design
NONR&D
Uncertainties which are not related to research and project design
Table 5 Fuzzy scales for pairwise comparisons in fuzzy AHP Importance intensity
Definition
1 1,1, 2
Equal importance
3 2,3, 4
Moderate importance of one over another
5 4,5,6
Strong importance of one over another
7 6,7,8
Very strong importance of one over another
9 8,9,9
Extreme importance of one over another
2 1, 2,3 , 4 3, 4,5 ,6 5,6,7 ,8 7,8,9
Intermediate values
Reciprocals
Reciprocals for inverse comparison
Table 6 Triangular fuzzy compare wise judgment matrix of five screening criteria with respect to NPD project screening and its triangular fuzzy weights Criteria
MKTFIT
MANUFIT
CUSTFIT
FINRISK
UNCERT
Fuzzy weights
MKTFIT
(1, 1, 1)
(2, 3, 4)
(1, 1, 2)
(1, 2, 3)
(2, 3, 4)
(0.2105, 0.3175, 0.4398)
MANUFIT
(1/4, 1/3, 1/2)
(1, 1, 1)
(1/4, 1/3, 1/2)
(1/3, 1/2, 1)
(1/3, 1/2, 1)
(0.0350, 0.0847, 0.1509)
CUSTFIT
(1/2, 1, 1)
(2, 3, 4)
(1, 1, 1)
(1, 2, 3)
(1, 2, 3)
(0.1571, 0.2857, 0.3881)
FINRISK
(1/3, 1/2, 1)
(1, 2, 3)
(1/3, 1/2, 1)
(1, 1, 1)
(1, 1, 2)
(0.0976, 0.1587, 0.2824)
UNCERT
(1/4, 1/3, 1/2)
(1, 2, 3)
(1/3, 1/2, 1)
(1/2, 1, 1)
(1, 1, 1)
(0.0872, 0.1534, 0.2335)
Table 7 Triangular fuzzy compare wise judgment matrix of four sub-criteria with respect to MKTFIT and its triangular fuzzy weights Sub-criteria
TIMING
PRICE
LOGISTICS
SALES
Fuzzy weights
TIMING
(1, 1, 1)
(1, 1, 2)
(1, 2, 3)
(2, 3, 4)
(0.2429, 0.3559, 0.4858)
PRICE
(1/2, 1, 1)
(1, 1, 1)
(1, 1, 2)
(2, 3, 4)
(0.2053, 0.3051, 0.4156)
LOGISTICS
(1/3, 1/2, 1)
(1/2, 1, 1)
(1, 1, 1)
(1, 2, 3)
(0.1241, 0.2288, 0.3273)
SALES
(1/4, 1/3, 1/2)
(1/4, 1/3, 1/2)
(1/3, 1/2, 1)
(1, 1, 1)
(0.0759, 0.1102, 0.1765)
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Table 8 Triangular fuzzy compare wise judgment matrix of three sub-criteria with respect to MANUFIT and its triangular fuzzy weights Sub-criteria
MFGTECH
MFGCAP
SUPPLY
Fuzzy weights
MFGTECH
(1, 1, 1)
(1, 2, 3)
(2, 3, 4)
(0.3692, 0.5294, 0.6358)
MFGCAP
(1/3, 1/2, 1)
(1, 1, 1)
(1, 2, 3)
(0.1854, 0.3088, 0.4615)
SUPPLY
(1/4, 1/3, 1/2)
(1/3, 1/2, 1)
(1, 1, 1)
(0.1138, 0.1618, 0.2632)
Table 9 Triangular fuzzy compare wise judgment matrix of two sub-criteria with respect to CUSTFIT and its triangular fuzzy weights Sub-criteria
DESIGN
RELIA
Fuzzy weights
DESIGN
(1, 1, 1)
(1, 2, 3)
(0.5000, 0.6667, 0.7500)
RELIA
(1/3, 1/2, 1)
(1, 1, 1)
(0.2500, 0.3333, 0.5000)
Table 10 Triangular fuzzy compare wise judgment matrix of two sub-criteria with respect to FINRISK and its triangular fuzzy weights Sub-criteria
REVENUE
EXPEND
Fuzzy weights
REVENUE
(1, 1, 1)
(2, 3, 4)
(0.6667, 0.7500, 0.8000)
EXPEND
(1/4, 1/3, 1/2)
(1, 1, 1)
(0.2000, 0.2500, 0.3333)
Table 11 Triangular fuzzy compare wise judgment matrix of two sub-criteria with respect to UNCERT and its triangular fuzzy weights Sub-criteria
R&DUNC
NONR&D
Fuzzy weights
R&DUNC
(1, 1, 1)
(3, 4, 5)
(0.7500, 0.8000, 0.8333)
NONR&D
(1/5, 1/4, 1/3)
(1, 1, 1)
(0.1667, 0.2000, 0.2500)
Table 12 Triangular fuzzy compare wise judgment matrix of three NPD projects with respect to each sub-criterion of MKTFIT and its triangular fuzzy weights Projects
Project A
Project B
Project C
Fuzzy weights
Triangular fuzzy compare wise judgment matrix of three projects with respect to TIMING and its triangular fuzzy weights Project A
(1, 1, 1)
(1, 1, 2)
(1, 2, 3)
(0.3053, 0.4000, 0.5538)
Project B
(1/2, 1, 1)
(1, 1, 1)
(1, 2, 3)
(0.2308, 0.4000, 0.4491)
Project C
(1/3, 1/2, 1)
(1/3, 1/2, 1)
(1, 1, 1)
(0.1370, 0.2000, 0.3333)
Triangular fuzzy compare wise judgment matrix of three projects with respect to PRICE and its triangular fuzzy weights Project A
(1, 1, 1)
(1/3, 1/2, 1)
(1, 2, 3)
(0.1854, 0.3088, 0.4615)
Project B
(1, 2, 3)
(1, 1, 1)
(2, 3, 4)
(0.3692, 0.5298, 0.5538)
Project C
(1/3, 1/2, 1)
(1/4, 1/3, 1/2)
(1, 1, 1)
(0.1132, 0.1618, 0.2632)
Triangular fuzzy compare wise judgment matrix of three projects with respect to LOGISTICS and its triangular fuzzy weights Project A
(1, 1, 1)
(1, 2, 3)
(1/3, 1/2, 1)
(0.1854, 0.3088, 0.4615)
Project B
(1/3, 1/2, 1)
(1, 1, 1)
(1/4, 1/3, 1/2)
(0.1138, 0.1618, 0.2632)
Project C
(1, 2, 3)
(2, 3, 4)
(1, 1, 1)
(0.3692, 0.5294, 0.6358)
Triangular fuzzy compare wise judgment matrix of three projects with respect to SALES and its triangular fuzzy weights Project A
(1, 1, 1)
(1/3, 1/2, 1)
(1/6, 1/5, 1/4)
(0.0896, 0.1211, 0.1915)
Project B
(1, 2, 3)
(1, 1, 1)
(1/4, 1/3, 1/2)
(0.1459, 0.2375, 0.3439)
Project C
(4, 5, 6)
(2, 3, 4)
(1, 1, 1)
(0.5350, 0.6413, 0.7135)
Table 13 Triangular fuzzy compare wise judgment matrix of three NPD projects with respect to each sub-criterion of MANUFIT and its triangular fuzzy weights Projects
Project A
Project B
Project C
Triangular fuzzy compare wise judgment matrix of three projects with respect to MFGTECH and its triangular fuzzy weights
24
Fuzzy weights
Project A
(1, 1, 1)
(1, 2, 3)
(3, 4, 5)
(0.3974, 0.5350, 0.6413)
Project B
(1/3, 1/2, 1)
(1, 1, 1)
(2, 3, 4)
(0.2375, 0.3439, 0.4768)
Project C
(1/5, 1/4, 1/3)
(1/4, 1/3, 1/2)
(1, 1, 1)
(0.0919, 0.1210, 0.1692)
Triangular fuzzy compare wise judgment matrix of three projects with respect to MFGCAP and its triangular fuzzy weights Project A
(1, 1, 1)
(1/6, 1/5, 1/4)
(1/3, 1/2, 1)
(0.0847, 0.1137, 0.1788)
Project B
(4, 5, 6)
(1, 1, 1)
(3, 4, 5)
(0.5749, 0.6689, 0.7332)
Project C
(1, 2, 3)
(1/5, 1/4, 1/3)
(1, 1, 1)
(0.1344, 0.2174, 0.3114)
Triangular fuzzy compare wise judgment matrix of three projects with respect to SUPPLY and its triangular fuzzy weights Project A
(1, 1, 1)
(1, 2, 3)
(1/3, 1/2, 1)
(0.1854, 0.3088, 0.4615)
Project B
(1/3, 1/2, 1)
(1, 1, 1)
(1/4, 1/3, 1/2)
(0.1138, 0.1618, 0.2632)
Project C
(1, 2, 3)
(2, 3, 4)
(1, 1, 1)
(0.3692, 0.5294, 0.6358)
Table 14 Triangular fuzzy compare wise judgment matrix of three NPD projects with respect to each sub-criterion of CUSTFIT and its triangular fuzzy weights Projects
Project A
Project B
Project C
Fuzzy weights
Triangular fuzzy compare wise judgment matrix of three projects with respect to DESIGN and its triangular fuzzy weights Project A
(1, 1, 1)
(2, 3, 4)
(3, 4, 5)
(0.4932, 0.6115, 0.6920)
Project B
(1/4, 1/3, 1/2)
(1, 1, 1)
(1, 2, 3)
(0.1557, 0.2548, 0.3699)
Project C
(1/5, 1/4, 1/3)
(1/3, 1/2, 1)
(1, 1, 1)
(0.0971, 0.1338, 0.2154)
Triangular fuzzy compare wise judgment matrix of three projects with respect to RELIA and its triangular fuzzy weights Project A
(1, 1, 1)
(1/3, 1/2, 1)
(1, 2, 3)
(0.1724, 0.2857, 0.4286)
Project B
(1, 2, 3)
(1, 1, 1)
(3, 4, 5)
(0.4286, 0.5714, 0.6650)
Project C
(1/3, 1/2, 1)
(1/5, 1/4, 1/3)
(1, 1, 1)
(0.1031, 0.1429, 0.2457)
Table 15 Triangular fuzzy compare wise judgment matrix of three NPD projects with respect to each sub-criterion of FINRISK and its triangular fuzzy weights Projects
Project A
Project B
Project C
Fuzzy weights
Triangular fuzzy compare wise judgment matrix of three projects with respect to REVENUE and its triangular fuzzy weights Project A
(1, 1, 1)
(1, 2, 3)
(3, 4, 5)
(0.4286, 0.5714, 0.6650)
Project B
(1/3, 1/2, 1)
(1, 1, 1)
(1, 2, 3)
(0.1724, 0.2857, 0.4286)
Project C
(1/5, 1/4, 1/3)
(1/3, 1/2, 1)
(1, 1, 1)
(0.1031, 0.1429, 0.2411)
Triangular fuzzy compare wise judgment matrix of three projects with respect to EXPEND and its triangular fuzzy weights Project A
(1, 1, 1)
(4, 5, 6)
(3, 4, 5)
(0.5749, 0.6689, 0.7332)
Project B
(1/6, 1/5, 1/4)
(1, 1, 1)
(1/3, 1/2, 1)
(0.0847, 0.1137, 0.1788)
Project C
(1/5, 1/4, 1/3)
(1, 2, 3)
(1, 1, 1)
(0.1344, 0.2174, 0.3114)
Table 16 Triangular fuzzy compare wise judgment matrix of three NPD projects with respect to each sub-criterion of UNCERT and its triangular fuzzy weights Projects
Project A
Project B
Project C
Fuzzy weights
Triangular fuzzy compare wise judgment matrix of three projects with respect to R&DUNC and its triangular fuzzy weights Project A
(1, 1, 1)
(2, 3, 4)
(1/4, 1/3, 1/2)
(0.2174, 0.3114, 0.4165)
Project B
(1/4, 1/3, 1/2)
(1, 1, 1)
(1/5, 1/4, 1/3)
(0.0868, 0.1138, 0.1618)
Project C
(2, 3, 4)
(3, 4, 5)
(1, 1, 1)
(0.4614, 0.5749, 0.6689)
Triangular fuzzy compare wise judgment matrix of three projects with respect to NONR&D and its triangular fuzzy weights Project A
(1, 1, 1)
(1/3, 1/2, 1)
(1/5, 1/4, 1/3)
(0.1031, 0.1429, 0.2250)
Project B
(1, 2, 3)
(1, 1, 1)
(1/3, 1/2, 1)
(0.1724, 0.2857, 0.4286)
Project C
(3, 4, 5)
(1, 2, 3)
(1, 1, 1)
(0.4286, 0.5714, 0.6650)
25
The triangular fuzzy priority vectors in the last columns of Tables 6-16 are obtained using Eq.(21). To obtain the goal triangular fuzzy priority vector, we need to aggregate the triangular fuzzy priority vector of the lower-level criteria to that of the upper-level criteria. Suppose that there are m criteria cj (j = 1, 2, …, m) with the triangular fuzzy priority weight w j wlj , wmj , wuj . For the alternative ai, i = 1, 2, …, n, the triangular fuzzy priority weight with respect to the criterion cj is denoted by wij wijl , wijm , wiju . Then, the goal triangular fuzzy priority weight is calculated by wai j 1 w j wij j 1 wlj wijl , wmj wijm , wuj wiju . m
m
(34)
Using Eq.(34), the local and goal triangular fuzzy priority vectors are obtained as shown in Table 17. Table 17 Global triangular fuzzy priority vector of the three NPD projects with respect to NPD project screening Criteria
Fuzzy weights
Project A
Project B
Project C
MKTFIT
(0.2105, 0.3175, 0.4398)
(0.1420, 0.3218, 0.6457)
(0.1571, 0.3685, 0.5952)
(0.1429, 0.3130, 0.6053)
MANUFIT
(0.0350, 0.0847, 0.1509)
(0.1835, 0.3683, 0.6117)
(0.2072, 0.4148, 0.7108)
(0.1009, 0.2169, 0.4186)
CUSTFIT
(0.1571, 0.2857, 0.3881)
(0.2897, 0.5029, 0.7333)
(0.1850, 0.3603, 0.6099)
(0.0743, 0.1368, 0.3344)
FINRISK
(0.0976, 0.1587, 0.2824)
(0.4007, 0.5958, 0.7764)
(0.1319, 0.2427, 0.4025)
(0.0956, 0.1615, 0.2967)
UNCERT
(0.0872, 0.1534, 0.2335)
(0.1802, 0.2777, 0.4033)
(0.0938, 0.1482, 0.2420)
(0.4175, 0.5742, 0.7236)
(0.1367, 0.4142, 0.9743)
(0.0904, 0.3163, 0.7759)
(0.0910, 0.2705, 0.7119)
Global triangular fuzzy weights Relational indices Ranking
w1
0.7324
w2
0.6064
w3
A>B>C
It is interesting that the same ranking order is obtained by Wang and Chin [46]. In this example, if the fuzzy arithmetic mean method [6] is adopted to calculate the global fuzzy priority vector, we have wG 0.5006,0.3021,0.1972 . Thus, the ranking order is A > B > C, which is same as the above-ranking order.
When the fuzzy geometric mean method [3] is applied to calculate the global fuzzy priority vector, we derive wG (0.0734,0.4128,2.3764),(0.0516,0.3153,1.8947),(0.0519,0.2719,1.55) and A > B > C. which is still the
same as the above-ranking order. Furthermore, when the Liu’s method is used to calculate the global triangular fuzzy priority vector, we derive wG (0.0471,0.2624,1.7597),(0.0525,0.2605,1.6815),(0.0559,0.3639,2.6038) .
From wG , we have w3 w1 w2 and C > A > B, which is different to the above-ranking order.
Remark 2: Since all most of triangular fuzzy compare wise judgment matrices given in this example are acceptably consistent, the same ranking order is obtained except for the Liu’s method. So, when the fuzzy compare wise judgment matrix is (acceptably) consistent, one can choose one of them; otherwise, we suggest the decision maker to adopt the new method, which is based on consistency analysis.
26
6. Conclusions To cope with the uncertainty in the process of decision making, some researchers proposed fuzzy preference relations applying fuzzy set theory [65]. In this situation, it needs to obtain the fuzzy priority vector from fuzzy preference relations. This paper discusses fuzzy preference relations with triangular fuzzy numbers. According to the extension principle and the row weighted arithmetic mean method, a new method for obtaining the triangular fuzzy priority vector from triangular fuzzy compare wise judgment matrices is developed. To derive the triangular fuzzy priority vector, we first prove that the boundary of triangular fuzzy weights can be determined by that of triangular fuzzy judgments. Then, each alternative’s triangular fuzzy weight is calculated from acceptably consistent multiplicative preference relations. Meanwhile, a new definition of consistent triangular fuzzy compare wise judgment matrices is presented, and some properties are studied. Based on the relational index for triangular fuzzy numbers, we introduce an algorithm to derive the triangular fuzzy priority vector from triangular fuzzy compare wise judgment matrices that can cope with the inconsistent case. Furthermore, we extend the new method to trapezoidal fuzzy compare wise judgment matrices. It is worth pointing out that the new method can also be used to deal with triangular and trapezoidal fuzzy reciprocal preference relations. There are two main advantages of the new method: (i) it considers triangular fuzzy priority weights separately; (ii) it is based on consistency analysis. To illustrate the application of the new method and to show the difference between the new method and some other approaches, the associated examples are offered, which show the feasibility and efficiency of the new method. Moreover, we discuss its application to NPD projects screening. However, we here only present one approach to triangular fuzzy compare wise judgment matrices, and we will continue to study the other methods for deriving the fuzzy priority vector from fuzzy preference relations. Furthermore, this paper only discusses the complete case, and we shall also study the incomplete case in a similar way to crisp preference relations [52, 55]. Acknowledgment The authors first gratefully thank the Area Editor and three anonymous referees for their valuable and constructive comments which have much improved the paper. This work was supported by the State Key Program of National Natural Science of China (No. 71431006), the National Natural Science Foundation of China (Nos. 71571192, 71501189), the China Postdoctoral Science special Foundation (2015T80901), the China Postdoctoral Science Foundation (2014M560655), and the Innovation-Driven Planning Foundation of Central South University (2015CX010). References [1] J. Aguarón and J. M. Moreno-Jiménez, “The geometric consistency index: Approximated thresholds,” Eur. J. Oper. Res. vol. 147, no. 1, pp. 137–145, 2003. [2] A. Arbel and L. G. Vargas, “The Analytic Hierarchy Process with interval judgements,” Proc. IXth Int. Sympo. on Multi-criteria Decis. Ma. Fair fax, 1990. [3] J. J. Buckley, “Fuzzy hierarchical analysis,” Fuzzy Sets Syst. vol. 17, no. 3, pp. 233–247, 1985. [4] C. G. E. Boender, J. G. de Graan and F. A. Lootsma, “Multi-criteria decision analysis with fuzzy pairwise comparisons,” Fuzzy Sets Syst. vol. 29, no. 1, pp. 133–143, 1989. [5] M. Brunelli, “A note on the article “Inconsistency of pair-wise comparison matrix with fuzzy elements based on geometric mean” [Fuzzy Sets Syst., 161 (2010) 1604–1613],” Fuzzy Sets Syst. vol. 176, no. 1, pp. 76–78, 2011.
27
[6] D. Y. Chang, “Application of the extent analysis method on fuzzy AHP,” Eur. J. Oper. Res. vol. 95, no. 3, pp. 649–655, 1996. [7] G. B. Crawford and C. Williams, “A note on the analysis of subjective judgment matrices,” J. Math. Psy. vol. 29, no. 4, pp. 387–405, 1985. [8] R. J. Calantone, C. A. Di Benedetto and J. B. Schmidt, “Using the analytic hierarchy process in new product screening,” J. Prod. Innov. Manag. vol.16, no. 1, pp. 65–76, 1999. [9] K. S. Chin, D. L. Xu, J. B. Yang and J. P. K. Lam, “Group-based ER–AHP system for product project screening,” Expert Syst. Appl. vol. 35, no. 4, pp. 1909–1929, 2008. [10] D. Dubois, “The role of fuzzy sets in decision sciences: Old techniques and new directions,” Fuzzy Sets Syst. vol. 184, no. 1, pp. 3–28, 2011. [11] D. Dubois and H. Prade, “A unified view of ranking techniques for fuzzy numbers,” Proc 1999 IEEE Int. Conf. Fuzzy Syst. Seoul, 1999, pp.1328–1333. [12] S. Gen, F. E. Boran, D. Akay and Z. S. Xu, “Interval multiplicative transitivity for consistency, missing values and priority weights of interval fuzzy preference relations,” Inform. Sci. vol. 180, no. 24, pp. 4877–4891, 2010. [13] L. M. Haines, “A statistical approach to the analytic hierarchy process with interval judgments .(I). Distribution on feasible regions,” Eur. J. Oper. Res. vol. 110, no. 1, pp. 112–125, 1998. [14] E. Herrera-Viedma, F. Herrera, F. Chiclana and M. Luque, “Some issues on consistency of fuzzy preference relations,” Eur. J. Oper. Res. vol. 154, no. 2, pp. 98–109, 2004. [15] R. Islam, M. P. Biswal and S. S. Alam, “Preference programming and inconsistent interval judgments,” Eur. J. Oper. Res. vol. 97, no. 1, pp. 53–62, 1997. [16] J. Kacprzyk and M. Fedrizzi, “Multiperson decision making models using fuzzy sets and possibility theory,” Kluwer Academic Publishers, Dordrecht, 1990. [17] E. S. Lee and R. L. Li, “Comparison of fuzzy numbers based on the probability measure of fuzzy events,” Comput. Math. Appl. vol. 15, no. 10, pp. 887–896, 1988. [18] L. C. Leung and D. Cao, “On consistency and ranking of alternatives in fuzzy AHP,” Eur. J. Oper. Res. vol. 124, no. 1, pp. 102–113, 2000. [19] F. Liu, “Acceptable consistency analysis of interval reciprocal comparison matrices,” Fuzzy Sets Syst. vol. 160, no. 18, pp. 2686–2700, 2009. [20] F. Liu, W. G. Zhang and Z. X. Wang, “A goal programming model for incomplete interval multiplicative preference relations and its application in group decision-making,” Eur. J. Oper. Res. vol. 218, no. 3, pp. 747–754, 2012. [21] F. Liu, W. G. Zhang and L. H. Zhang, “A group decision-making model based on a generalized ordered weighted geometric average operator with interval preference matrices,” Fuzzy Sets Syst. vol. 246, no. 1, pp. 1–18, 2014. [22] F. Liu, W. G. Zhang and L. H. Zhang, “Consistency analysis of triangular fuzzy reciprocal preference relations,” Eur. J. Oper. Res. vol. 235, no. 3, pp. 718–726, 2014. [23] L. Mikhailov, “Deriving priorities from fuzzy pairwise comparison judgments,” Fuzzy Sets Syst. vol. 134, no. 3, pp. 365–385, 2003. [24] B. Matarazzo and G. Munda, “New approaches for the comparison of L-R fuzzy numbers: a theoretical and operational analysis,” Fuzzy Sets Syst. vol. 118, no. 3, pp. 407–418, 2001. [25] F. Y. Meng, X. H. Chen, Q. Zhang and J. Lin, “Two new methods for deriving the priority vector from interval comparison matrices,” Inform. Fusion. vol. 26, pp. 122–135, 2015. [26] F. Y. Meng and X. H. Chen, “A new method for group decision making with incomplete fuzzy preference relations,” Knowl-Based Syst. vol. 73, pp. 111–123, 2015. [27] F. Y. Meng and X. H. Chen, “An approach to incomplete multiplicative preference relations and its application in group decision making,” Inform. Sci. vol. 309, pp. 119–1375, 2015. [28] S. Ohnishi, D. Dubois, H. Prade and T. Yamanoi, “A fuzzy constraint-based approach to the Analytic Hierarchy,” Process, in :
28
Uncertainty and intelligent information systems, edited by B. Bouchon-Meunier, C. Marsala, M. Rifqi, & R. R Yager, World Scientific , Singapore, 2008, pp. 217–228. [29] S. A. Orlovsky, “Decision-making with a fuzzy preference relation,” Fuzzy Sets Syst. vol. 1, no. 3, pp. 155–167, 1978. [30] V. V. Podinovski, “Interval articulation of superiority and precise elicitation of priorities,” Eur. J. Oper. Res. vol. 180, no. 1, pp. 406–417, 2007. [31] J. Ramik and P. Korviny, “Inconsistency of pair-wise comparison matrix with fuzzy elements based on geometric mean,” Fuzzy Sets Syst. vol. 161, no. 11, pp. 1604–1613, 2010. [32] J. Rezaei and R. Ortt, “Multi-criteria supplier segmentation using a fuzzy preference relation based AHP,” Eur. J. Oper. Res. vol. 225, no. 1, pp. 75–84, 2013. [33] T. L. Saaty, The Analytic Hierarchy Process, McGraw-Hill, New York, 1980. [34] T. L. Saaty and L. G. Vargas, “Uncertainty and rank order in the analytic hierarchy process,” Eur. J. Oper. Res. vol. 32. No. 1, pp. 107–117, 1987. [35] T. L. Saaty, Multicriteria Decision Making: The Analytic Hierarchy Process, RWS Publications, Pittsburgh, PA, 1988. [36] A. A. Salo, “On fuzzy ratio comparisons in hierarchical decision models,” Fuzzy Sets Syst. vol. 84, no. 1, pp. 21–32, 1996. [37] P. J. M. van Laarhoven and W. Pedrycz, “A fuzzy extension of Saaty’s priority theory,” Fuzzy Sets Syst. vol. 11, no. 1-3, pp. 229–241, 1983. [38] M. Wagenknecht and K. Hartmann, “On fuzzy rank ordering in polyoptimisation,” Fuzzy Sets Syst. vol. 11, pp. 253–264, 1983. [39] X. Z. Wang and E. E. Kerre, “Reasonable properties for the ordering of fuzzy quantities (I),” Fuzzy Sets Syst. vol. 118, no. 3, pp. 375–385, 2001. [40] X. Z. Wang and E. E. Kerre, “Reasonable properties for the ordering of fuzzy quantities (II),” Fuzzy Sets Syst. vol. 118, no. 3, pp. 387–405, 2001. [41] Y. M. Wang, Y. Luo and Z. Hua, “On the extent analysis method for fuzzy AHP and its applications,” Eur. J. Oper. Res. vol. 186, no. 2, pp. 735–747, 2008. [42] Y. M. Wang, J. B. Yang and D. L. Xu, “A two-stage logarithmic goal programming method for generating weights from interval comparison matrices,” Fuzzy Sets Syst. vol. 152, no. 3, pp. 475–498, 2005. [43] Y. M. Wang and T. M. S. Elhag, “On the normalization of interval and fuzzy weights,” Fuzzy Sets Syst. vol. 157, no. 18, pp. 2456–2471, 2006. [44] Y. M. Wang, T. M. S. Elhag and Z. S. Hua, “A modified fuzzy logarithmic least squares method for fuzzy analytic hierarchy process,” Fuzzy Sets Syst. vol. 157, no. 18, pp. 3055–3071, 2006. [45] Y. M. Wang and K. S. Chin, “A linear goal programming priority method for fuzzy analytic hierarchy process and its applications in new product screening,” Int. J. Approx. Reason. vol. 49, no. 2, pp. 451–465, 2008. [46] T. C. Wang and Y. H. Chen, “Applying fuzzy linguistic preference relations to the improvement of consistency of fuzzy AHP,” Inform. Sci. vol. 178, no. 19, pp. 3755–3765, 2008. [47] J. Wu and F. Chiclana, “Visual information feedback mechanism and attitudinal prioritisation method for group decision making with triangular fuzzy complementary preference relations,” Inform. Sci. vol. 279, pp. 716–734, 2014. [48] Z. S. Xu, “A practical method for priority of interval number complementary judgment matrix,” Oper. Res. Manag. Sci. vol. 10, no. 1, pp. 16–19, 2001. [49] Z. S. Xu, “Consistency of interval fuzzy preference relations in group decision making,” Appl. Soft Comput. vol. 11, no. 5, pp. 3898–3909, 2011. [50] Z. S. Xu, “A method for priorities of triangular fuzzy number complementary judgement matrices,” Fuzzy Syst. Math. vol.16, no. 1, pp. 47-50, 2002. [51] Z. S. Xu, “A priority method for triangular fuzzy number complementary judgement matrix,” Syst. Eng. The. Pra. vol. 23, no. 10, pp. 86–89, 2003. [52] Y.J. Xu and H.M. Wang, “Eigenvector method, consistency test and inconsistency repairing for an incomplete fuzzy preference
29
relation,” Appl. Math. Modell. vol. 37, no. 7, pp. 5171–5183, 2013. [53] Y.J. Xu, R. Patnayakuni and H.M. Wang, “The ordinal consistency of a fuzzy preference relation,” Inf. Sci. vol. 224, no. 1, pp. 152–164, 2013. [54] Y.J. Xu, K.W. Li and H.M. Wang, “Incomplete interval fuzzy preference relations and their applications,” Comput. Ind. Eng. vol. 67, no. 1, pp. 93–103, 2014. [55] Y.J. Xu, J.N.D. Gupta and H.M. Wang, “The ordinal consistency of an incomplete reciprocal preference relation,” Fuzzy Sets Syst.vol. 246, no. 1, pp. 62–77, 2014. [56] Y.J. Xu, K.W. Li and H.M. Wang, “Consistency test and weight generation for additive interval fuzzy preference relations,” Soft Comput. vol. 18 no. 8, pp. 1499–1513, 2014. [57] M. M. Xia and Z. S. Xu, “Interval weight generation approaches for reciprocal relations,” Appl. Math. Modell. vol. 38, no. 3, pp. 828–838, 2014. [58] R. Xu, “Fuzzy least-squares priority method in the analytic hierarchy process,” Fuzzy Sets Syst. vol. 112, no. 3, pp. 359–404, 2000. [59] J. S. Yao and F. T. Lin, “Fuzzy critical path method based on signed distance ranking of fuzzy numbers,” IEEE Trans. Syst. Man Cy. vol. 30, no. 1, pp. 76–82, 2000. [60] R. R. Yager, “In a general class of fuzzy connectives,” Fuzzy Sets Syst. vol. 4, no. 3, pp. 235–242, 1980. [61] R. R. Yager and F. Dimitar, “On ranking fuzzy numbers using valuations,” Int. J. Intell. Syst. vol. 14, no. 12, pp. 1249–1268, 1999. [62] R. R. Yager, M. Detyniecki and B. B. Meunier, “A context-dependent method for ordering fuzzy numbers using probabilities,” Inform. Sci. vol. 138, no. 1-4, pp. 237–255, 2001. [63] D.J. Yu, J.M. Merigo and L.G. Zhou, “Interval-valued multiplicative intuitionistic fuzzy preference relations,” Int. J. Fuzzy Syst. vol. 15, no. 4, pp. 412-422, 2013. [64] Z. Zhang and C.H. Guo, “Consistency-based algorithms to estimate missing elements for uncertain 2-tuple linguistic preference relations,” Int. J. Comput. Intell. Syst., vol. 7, no. 5, pp. 924–936, 2014. [65] L. A. Zadeh, “Fuzzy sets,” Inform. Cont. vol. 8, no. 3, pp. 338–353, 1965. [66] X. L. Zeng and F. Y. Meng, “On ranking methods for fuzzy numbers based on possibility degree,” Oper. Res. Manag. Sci. Fuzz. vol. 7, pp. 66–72, 2007.
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Fanyong Meng is an associate Professor in School of Management, Qingdao Technological University and carrying out his post doctoral work in Central South University. He received his master degree in School of Mathematics and Information Science, Guangxi University, China in 2008, and earned his Ph.D. degree in School of Management and Economics, Beijing Institute of Technology, China in 2011. Currently, he has contributed over 70 journal articles to professional journals. His research interests include fuzzy mathematics, decision making and games theory.
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Xiao-hong Chen is a professor at School of Business, Central South University. She received the Ph.D. degree in management from Tokyo Institute of Technology in 1999. She is the director of some organizations, such as the national level key disciplines "management science and Engineering", the national innovative research groups, ministry of education, "the Yangtze River scholars and innovation team", and the national level teaching team. She is a chief professor and one of the first Chinese distinguished social scientists. Since 1998, she enjoys special government allowances of the State Council. Furthermore, she is the first "Lotus Scholars Program" Professors
and won the "Lotus Scholars Program" distinguished professor award. Currently, she has contributed over 230 journal articles to professional journals. Her current research interests include decision analysis and financing and innovation of
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SMEs.
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