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Note on “Hesitant fuzzy prioritized operators and their application to multiple attribute decision making”
Knowledge-Based Systems 96 (2016) 115–119
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Note on “Hesitant fuzzy prioritized operators and their application to multiple attribute decision making” Feifei Jin a,b,∗, Zhiwei Ni a,b, Huayou Chen c a
School of Management, Hefei University of Technology, Hefei, Anhui 230009, China Key Laboratory of Process Optimization and Intelligent Decision-Making, Ministry of Education, Hefei, Anhui 230009, China c School of Mathematical Sciences, Anhui University, Hefei, Anhui 230601, China b
a r t i c l e
i n f o
Article history: Received 27 November 2014 Revised 3 October 2015 Accepted 27 December 2015 Available online 11 January 2016 Keywords: Decision making Hesitant fuzzy sets Prioritized aggregation operator Idempotency
a b s t r a c t Motivated by the idea of prioritized aggregation (PA) operators, Wei (2012) developed two hesitant fuzzy prioritized aggregation (HFPA) operators, and discussed their desirable properties, but the definitions for the HFPA operators and their properties still need to be improved. In this short note, a numerical example is given to show that the idempotency of the HFPA operators suffers from certain shortcomings. Then, based on some adjusted operations on the hesitant fuzzy elements (HFEs), two improved aggregation operators are investigated to aggregate the collective of attribute values. We further prove that the improved operators have the properties of idempotency and boundedness. Finally, the comparison with the method proposed by Wei (2012) is performed to demonstrate that the proposed information aggregation method is both valid and practical to deal with decision making problems. © 2016 Elsevier B.V. All rights reserved.
1. Introduction Since Zadeh introduced the fuzzy sets (FSs) [2], fuzzy sets have been achieved a great success in various fields. The concept of intuitionistic fuzzy sets (IFSs) [3] put forward by Atanassov is a generalization of the fuzzy set. Atanassov and Gargov further introduced the concept of interval-valued intuitionistic fuzzy sets (IVIFSs) [4], the components of which are intervals rather than exact numbers. Torra [5] proposed the concept of hesitant fuzzy set (HFS) considered as another generalization of FSs, which permits the membership degree having a collection of possible values. Wei [1] extended the prioritized averaging (PA) operator to accommodate the situations where the input arguments are hesitant fuzzy information, and developed two prioritized aggregation operators. He further studied some desirable properties of the proposed operators. However, a close examination demonstrates that some properties suffer from serious drawbacks. The purpose of this note is to point out and correct errors in the properties of HFPA operators. 2. Preliminaries By the relationship between the HFEs and intuitionistic fuzzy values (IFVs), Xia and Xu [6] defined some operations on the HFEs.
∗ Corresponding author at: School of Management, Hefei University of Technology, Hefei, Anhui 230009, China. Tel.: +86 13856942010. E-mail address: [email protected] (F. Jin).
http://dx.doi.org/10.1016/j.knosys.2015.12.023 0950-7051/© 2016 Elsevier B.V. All rights reserved.
Combined with the PA operator, Wei [1] developed two HFPA operators, including the hesitant fuzzy prioritized weighted average (HFPWA) operator and the hesitant fuzzy prioritized weighted geometric (HFPWG) operator. Definition 1. (See Wei [1], Definitions 7 and 8) Let h j , j = 1, 2, . . . , n be a collection of HFEs, then the HFPWA operator and the HFPWG operator are defined as follows, respectively: n
HF PWA(h1 , h2 , . . . , hn ) = ⊕ =
n
j=1
1−
γ1 ∈h1 ,γ2 ∈h2 ,...,γn ∈hn
n
(1 − γ j )
γ1 ∈h1 ,γ2 ∈h2 ,...,γn ∈hn
hj
Tj n T j=1 j
,
(1)
j=1
n
=
Tj
j=1
Tj n T j=1 j
HF PWG(h1 , h2 , . . . , hn ) = ⊗ h j
Tj
j=1
n
(γ j )
Tj n T j=1 j
,
(2)
j=1
j−1 where T1 = 1, T j = k=1 s(hk ), j = 2, . . . , n, and s(hk ) is the score values of hk , k = 1, 2, . . . , n. Then, Wei [1] proved that both the HFPWA operator and the HFPWG operator are idempotent. Theorem 1. (Idempotency, see Wei [1], Theorems 2 and 6) Let h j , j = 1, 2, . . . , n, be a collection of HFEs, where T1 = 1,
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F. Jin et al. / Knowledge-Based Systems 96 (2016) 115–119
j−1 T j = k=1 s(hk ), j = 2, . . . , n, and s(hk ) is the score values of hk , k = 1, 2, . . . , n. If all h j , j = 1, 2, . . . , n, are equal, i.e. h j = h for all j, then
HF PWA(h1 , h2 , . . . , hn ) = h,
(3)
HF PWG(h1 , h2 , . . . , hn ) = h.
(4)
Thus, we calculate the score value of HFPWG(h1 , h2 ) and get that
s(HF PWG(h1 , h2 )) = 0.5274, and then
s(HF PWG(h1 , h2 )) = 0.5274 < 0.54 = s(h ), which indicates that
3. Numerical example
HF PWG(h1 , h2 ) < h.
Now, we furnish the following example to demonstrate that Theorem 1 is technically incorrect.
Example 1 demonstrates that Theorem 1 of the HFPA operators cannot be tenable, which suffer from serious drawbacks. In this case, the operations on the HFEs need to be improved. In the following section, some adjusted operations for HFEs are presented, and two new HFPA operators are developed, which satisfy the properties of idempotency and boundedness.
Example 1. Suppose that h1 and h2 are two HFEs, and h1 = h2 = h = {0.35, 0.50, 0.77}, then the score values of h1 and h2 are s(h1 ) = s(h2 ) = 0.54, and then
T1 = 1, T2 =
2−1
s(hk ) = s(h1 ) = 0.54.
4. The improved operators and their properties
k=1
According to Definition 1, it follows that
HF PWA(h1 , h2 ) = HF PA(h, h ) =
1−
2
γ1 ∈h1 ,γ2 ∈h2
= 1 − (1 − 0.35 ) 1 − (1 − 0.35 )
1 1+0.54
1 − (1 − 0.35 )
1 1+0.54
1 1+0.54
× (1 − 0.35 )
× (1 − 0.50 )
0.54 1+0.54
× (1 − 0.77 )
0.54 1+0.54
1
Tj
(1 − γ j )
2 T j=1 j
j=1
0.54 1+0.54
,
,
1 − (1 − 0.50 ) 1 − (1 − 0.50 )
1 1+0.54
0.54
0.54
× (1 − 0.50 ) 1+0.54 , × (1 − 0.77 )
0.54 1+0.54
1
0.54
1
0.54
1
0.54
(R1) All the elements in each HFE h are arranged in decreasing order, and γ (i) is the ith largest value in h; (R2) If lh = lg , then l = max{lh , lg }. To have a correct comparison, the two HFEs h and g should have the same length. If there are fewer elements in h than in g, an extension of h should be considered optimistically by repeating its maximum element until it has the same length with g; (R3) For convenience, we assume that all the HFEs have the same length l, i.e., h = {γ (1 ) , γ (2 ) , . . . , γ (l ) }.
,
1 − (1 − 0.77 ) 1+0.54 × (1 − 0.35 ) 1+0.54 , 1 − (1 − 0.77 ) 1+0.54 × (1 − 0.50 ) 1+0.54 , 1 − (1 − 0.77 ) 1+0.54 × (1 − 0.77 ) 1+0.54
= {0.3500, 0.4071, 0.5486, 0.4518,0.5000, 0.6192, 0.6689, 0.6980, 0.7700}. Therefore, the score value of HFPWA(h1 , h2 ) can be calculated as
s(HF PWA(h1 , h2 )) = 0.5571 > 0.54 = s(h ), which indicates that
HF PWA(h1 , h2 ) > h.
(5)
On the other hand, by the HFPWG operator in Definition 1, it is obtained that
HF PWG(h1 , h2 ) = HF PG(h, h ) =
γ1 ∈h1 ,γ2 ∈h2
= 0.35
0.77
1 1+0.54
1 1+0.54 1 1+0.54
× 0.35
0.54 1+0.54
× 0.35
0.54 1+0.54
× 0.35
0.54 1+0.54
, 0.35
1 1+0.54
, 0.50
1 1+0.54
, 0.77
1 1+0.54
2
(γ j )
Tj 2 T j=1 j
j=1
× 0.50
0.54 1+0.54
× 0.50
0.54 1+0.54
× 0.50
0.54 1+0.54
,0.35
1 1+0.54
, 0.50
1 1+0.54
, 0.77
1 1+0.54
× 0.77
0.54 1+0.54
× 0.77
0.54 1+0.54
× 0.77
0.54 1+0.54
= {0.3500, 0.3966, 0.4615, 0.4412,0.5000, 0.5817, 0.5840, 0.6618, 0.7700}.
l
{γ1(i) + γ2(i) − γ1(i) γ2(i) }; ˙ 2 = i=1 {γ1(i ) γ2(i ) }; (2) h1 ⊗h λ (3) λh = li=1 {1 − (1 − γ (i ) ) }, λ > 0; λ (4) hλ = li=1 {(γ (i ) ) }, λ > 0. ˙ 2= (1) h1 ⊕h
then
0.50
Now, let’s review some adjusted operations on the HFEs as follows: Definition 2. [7] Suppose that h1 , h2 and h are three HFEs, then
s(HF PWA(h1 , h2 )) = 0.5571,
In this section, some adjusted operations on the HFEs are reviewed, and two new improved aggregation operators based on these adjusted operations are investigated to aggregate the collective of attribute values. We further prove that the improved operators have the properties of idempotency and boundedness. In the end, an example shows that our method is easier than that of Wei [1] in some cases. Remark 1. Notice that the number of values in different HFEs may be different. Suppose that lh stands for the number of values in h, then the following assumptions are made:
,
1 − (1 − 0.50 ) 1+0.54 × (1 − 0.35 ) 1+0.54 , 1 1+0.54
(6)
,
,
i=1
l
˙ 2 and According to the Definition 2, we know that h1 , h2 , h1 ⊕h ˙ 2 have the same length l. h1 ⊗h Example 2. Suppose that l = 3, λ = 2, h1 and h2 are two HFEs, and h1 = {0.77, 0.50, 0.35}, h2 = {0.86, 0.59, 0.22}, then we have ˙ 2 = 3i=1 {γ1(i ) + γ2(i ) − γ1(i ) γ2(i ) } = (1) h1 ⊕h {0.9678, 0.7950, 0.4930}; ˙ 2 = 3i=1 {γ1(i ) γ2(i ) } = {0.6622, 0.2950, 0.0770}; (2) h1 ⊗h 2 (3) 2h1 = 3i=1 {1 − (1 − γ (i ) ) } = {0.9471, 0.7500, 0.5775}; 2 3 (4) h2 = i=1 {(γ (i ) ) } = {0.5929, 0.2500, 0.1225}. Based on the adjusted operational principle for HFEs, we investigate the improved HFPA operators as follows:
F. Jin et al. / Knowledge-Based Systems 96 (2016) 115–119
Definition 3. Let h j , j = 1, 2, . . . , n, be a collection of HFEs, then the improved hesitant fuzzy prioritized weighted average (IHFPWA) operator is defined as follows:
n
IHF PWA(h1 , h2 , . . . , hn ) = ⊕˙ =
l
1−
i=1
n
j=1
n
(1 − γ j(i) )
Tj n T j=1 j
Tj
j=1
Tj
Therefore, it follows that
where T1 = 1, T j = k=1 s(hk ), j = 2, . . . , n, and s(hk ) is the score value of hk , k = 1, 2, . . . , n. Definition 4. Let h j , j = 1, 2, . . . , n, be a collection of HFEs, then the improved hesitant fuzzy prioritized weighted geometric (IHFPWG) operator is defined as follows:
=
l n i=1
(γ j(i) )
Tj n T j=1 j
, (8)
j=1
j−1 where T1 = 1, T j = k=1 s(hk ), j = 2, . . . , n, and s(hk ) is the score value of hk , k = 1, 2, . . . , n. According to Definitions 3 and 4, the properties corresponding to Theorems 2 and 6 in Wei [1] can be obtained: Theorem 2. (Idempotency) Let h j , j = 1, 2, . . . , n, be a collection of j−1 HFEs, where T1 = 1, T j = k=1 s(hk ), j = 2, 3, . . . , n, and s(hk ) is the score value of hk , k = 1, 2, . . . , n. If all h j , j = 1, 2, . . . , n, are equal, i.e. h j = h for all j, then
IHF PWA(h1 , h2 , . . . , hn ) = h.
(9)
h1 =h2 = · · · =hn = h = {γ
,γ
then we have
IHF PWA(h1 , h2 , . . . , hn ) =
=
l i=1
=
l
1−
l
(2 )
n
(i )
(1 − γ )
j=1 n
1 − ( 1 − γ (i ) )
,...,γ
1−
Tj n T j=1 j
Tj n T j=1 j j=1
n
(l )
},
=
j=1
1
0.54
1
0.54
1
0.54
1 − (1 − 0.35 ) 1+0.54 × (1 − 0.35 ) 1+0.54
= {0.7700,0.5000, 0.3500} = h, and
3 2
IHF PWG(h1 , h2 ) =
i=1
1
(i )
(γ j )
Tj 2 T j=1 j
j=1 0.54
1
0.54
= 0.77 1+0.54 × 0.77 1+0.54 , 0.50 1+0.54 × 0.50 1+0.54 , 1
0.54
0.35 1+0.54 × 0.35 1+0.54
= {0.7700,0.5000, 0.3500} = h. Inspired by Liao and Xu [8,9], it can be proved that the improved HFPA operators also satisfy the property of boundedness. Theorem 4. (Boundedness) Let h j , j = 1, 2, . . . , n, be a collection j−1 of HFEs, where T1 = 1, T j = k=1 s(hk ), j = 2, 3, . . . , n. And s(hk ) is the score value of hk , k = 1, 2, . . . , n. Let h− = min min γ j(i ) , h+ = 1≤i≤l 1≤ j≤n
(1 − γ j(i) )
h− ≤ IHF PWA(h1 , h2 , . . . , hn ) ≤ h+ . Tj n T j=1 j
Proof. n
Let
(1 − γ j(i) )
j=1
j=1
Tj n T j=1 j
(11)
˜ = IHF PWA(h1 , h2 , . . . , hn ) h
and
γ˜ (i) = 1 −
, i = 1, 2, . . . , l.
Since h− = min min γ j(i ) , h+ = max max γ j(i ) , then we have 1≤i≤l 1≤ j≤n
1≤i≤l 1≤ j≤n
h− ≤ γ j(i) ≤ h+ , i = 1, 2, . . . , l, j = 1, 2, . . . , n, and then
i=1 l
1≤i≤l 1≤ j≤n
i=1
(1 − γ j(i) )
Tj 2 T j=1 j
max max γ j(i ) , then
Proof. Since h j = h for all j, i.e., (1 )
2
1 − (1 − 0.50 ) 1+0.54 × (1 − 0.50 ) 1+0.54 ,
j−1
j=1
1−
i=1
j=1
IHF PWG(h1 , h2 , . . . , hn ) = ⊗˙ h j
= 1 − (1 − 0.77 ) 1+0.54 × (1 − 0.77 ) 1+0.54 , (7)
Tj n T j=1 j
3
IHF PWA(h1 , h2 ) =
hj
,
n
117
1 − ( 1 − γ (i ) ) =
i=1
l
γ (i )
n
i=1
≤
n
(1 − γ j(i) )
Tj n T j=1 j
≤
j=1
n
(1 − h− )
Tj n T j=1 j
(12)
j=1
it follows that
which completes the proof of Theorem 2. Theorem 3. (Idempotency) Let h j , j = 1, 2, . . . , n, be a collection of j−1 HFEs, where T1 = 1, T j = k=1 s(hk ), j = 2, 3, . . . , n. And s(hk ) is the score value of hk , k = 1, 2, . . . , n. If all h j , j = 1, 2, . . . , n, are equal, i.e. h j = h for all j, then
1−
n
(1 − h− )
Tj n T j=1 j
≤1−
j=1
≤1−
n
(1 − γ j(i) )
Tj n T j=1 j
j=1 n
(1 − h+ )
Tj n T j=1 j
, i = 1, 2, . . . , l,
(13)
j=1
(10)
Proof. The proof of Theorem 3 is similar to that of Theorem 2. So it is omitted here. Example 3. (Adapted from Example 1) Suppose that h1 and h2 are two HFEs, and h1 = h2 = h = {0.77, 0.50, 0.35}, then l = 3. According to the Example 1, we have
T1 = 1, T2 = 0.54.
Tj n T j=1 j
j=1
= {γ (1) , γ (2) , . . . , γ (l ) } = h,
IHF PWG(h1 , h2 , . . . , hn ) = h.
(1 − h+ )
which is equivalent to
1 − (1 − h− )
n T j=1 j n T j=1 j
≤ γ˜ (i) ≤ 1 − (1 − h+ )
n T j=1 j n T j=1 j
, i = 1, 2, . . . , l,
i.e.,
h− ≤ γ˜ (i) ≤ h+ , i = 1, 2, . . . , l.
(14)
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F. Jin et al. / Knowledge-Based Systems 96 (2016) 115–119
˜) = As s(h
h− =
1 l
l i=1
γ˜ (i) , according to Eq. (14), we have
Table 1 Hesitant fuzzy decision matrix H = (hi j )5×4
l l l 1 − 1 (i ) 1 + h ≤ γ˜ ≤ h = h+ , l l l i=1
i=1
X1 X2 X3 X4 X5
i=1
i.e.,
˜ ) ≤ s(h+ ). s(h− ) ≤ s(h
G1
G2
G3
G4
{0.4,0.5,0.7} {0.6,0.7,0.8} {0.6,0.8} {0.5,0.6,0.7} {0.6,0.7}
{0.5,0.8} {0.5,0.6} {0.2,0.3,0.5} {0.4,0.5} {0.5,0.7}
{0.6,0.7,0.9} {0.4,0.6,0.7} {0.4,0.6} {0.8,0.9} {0.7,0.8}
{0.5,0.6} {0.4,0.5} {0.5,0.7} {0.3,0.4,0.5} {0.2,0.3,0.4}
Therefore,
h ≤ IHF PWA(h1 , h2 , . . . , hn ) ≤ h+ , −
which completes the proof of Theorem 4. Theorem 5. (Boundedness) Suppose that h j , j = 1, 2, . . . , n, is a j−1 collection of HFEs, where T1 = 1, T j = k=1 s(hk ), j = 2, 3, . . . , n, and s(hk ) is the score values of hk , k = 1, 2, . . . , n. Let h− = min min γ j(i ) , h+ = max max γ j(i ) , then 1≤i≤l 1≤ j≤n
1≤i≤l 1≤ j≤n
h ≤ IHF PWG(h1 , h2 , . . . , hn ) ≤ h+ . −
(15)
Proof. The proof of Theorem 5 is similar to that of Theorem 4. So it is omitted here. In order to analyze the relationship among the IHFPWA operator and IHFPWG operator, we introduce the following lemma: Lemma 1. [10] Let xj > 0, λ j > 0, j = 1, 2, . . . , n, and nj=1 λ j = 1, n n λj then j=1 λ j x j ≥ j=1 x j , with equality if and only if x1 = x2 = · · · = xn . Theorem 6. Let h j , j = 1, 2, . . . , n, be a collection of HFEs, where j−1 T1 = 1, T j = k=1 s(hk ), j = 2, 3, . . . , n. And s(hk ) is the score value of hk , k = 1, 2, . . . , n, then
IHF PWA(h1 , h2 , . . . , hn ) ≥ IHF PWG(h1 , h2 , . . . , hn ). (i )
(i )
Proof. For any γ1 ∈ h1 , γ2 using the Lemma 1, we have
1−
n j=1
(1 − γ j(i) )
=1− 1−
n
Tj n T j=1 j
j=1
≥
n
(γ j(i) )
Tj n T j=1 j
≥1−
n j=1
Tj
n
j=1
(i )
Tj
(i )
∈ h2 , . . . , γn
· γj
j=1
=
∈ hn , i = 1, 2, . . . , l,
⎛
{0.4, 0.5, 0.7} ⎜{0.6, 0.7, 0.8} ⎜ H˜ = ⎜ ⎜{0.6, 0.8, 0.8} ⎝{0.5, 0.6, 0.7} {0.6, 0.7, 0.7}
{0.5, 0.8, 0.8} {0.5, 0.6, 0.6} {0.2, 0.3, 0.5} {0.4, 0.5, 0.5} {0.5, 0.7, 0.7}
{0.6, 0.7, 0.9} {0.4, 0.6, 0.7} {0.4, 0.6, 0.6} {0.8, 0.9, 0.9} {0.7, 0.8, 0.8}
⎞ {0.5, 0.6, 0.6} {0.4, 0.5, 0.5}⎟ ⎟ {0.5, 0.7, 0.7}⎟ ⎟ {0.3, 0.4, 0.5}⎠ {0.2, 0.3, 0.4}
Step 2: Utilizing Definition 3 to calculate Ti j , i = 1, 2, . . . , 5, j = 1, 2, 3, 4, as follows:
⎛1.000
⎜1.000 T = (Ti j )5×4 = ⎜1.000 ⎝ 1.000 1.000
0.533 0.700 0.733 0.600 0.667
0.373 0.397 0.244 0.280 0.422
⎞
0.274 0.225⎟ 0.130 ⎟ ⎠ 0.243 0.324
Step 3: Without loss of generality, we use the IHFPWA operator ˜ i (i = ˜ i j , j = 1, 2, 3, 4, into a collective HFEs h to aggregate all HFEs h 1, 2, . . . , 5 ) of candidates Xi (i = 1, 2, . . . , 5 ), then we have
˜ 1 = {0.4768, 0.6440, 0.7666}, h ˜ 2 = {0.5231, 0.6389, 0.7113}, h ˜ 4 = {0.5152, 0.6282, 0.6821}, ˜h3 = {0.4590, 0.6564, 0.6943}, h
Tj
n
(16)
With this information, we utilize the proposed information aggregation method to get the ranking of these candidates. The following steps are involved: Step 1: Using Remark 1 to obtain a normalized decision matrix ˜ i j )5×4 from H = (hi j )5×4 as follows: H˜ = (h
Tj
n j=1
· (1 − γ j(i) )
˜ i ), i = 1, 2, . . . , 5, i.e., Step 4: Calculating the score functions s(h
Tj
n
j=1
˜ 5 = {0.5560, 0.6869, 0.6933}. h
(i )
Tj
· γj
, i = 1, 2, . . . , l,
˜ 1 ) = 0.6291, s(h ˜ 2 ) = 0.6244, s(h ˜ 3 ) = 0.6032, s (h (17)
j=1
it follows that
s(IHF PWA(h1 , h2 , . . . , hn )) ≥ s(IHF PWG(h1 , h2 , . . . , hn )),
˜ 4 ) = 0.6085, s(h ˜ 5 ) = 0.6454. s (h ˜ i ), Step 5: According to the ordering of the score functions s(h ˜ i , i = 1, 2, . . . , 5, in dei = 1, 2, . . . , 5,we rank the overall HFEs h scending order:
i.e.,
˜1 > h ˜2 > h ˜4 > h ˜ 3, ˜5 > h h
IHF PWA(h1 , h2 , . . . , hn ) ≥ IHF PWG(h1 , h2 , . . . , hn ),
then the ranking of five candidates is that
which completes the proof of Theorem 6. In what follows, Example 4 will show that our method is valid and practical in some cases.
X5 X1 X2 X4 X3 .
Example 4. In order to promote the academic education, a Chinese university wants to introduce oversea outstanding teachers [11]. The panel of decision makers have strict evaluation for five candidates X1 , X2 , X3 , X4 and X5 , considering the following four attributes, G1 : morality, G2 : research capability, G3 : teaching skills, and G4 : education background, where the prioritization relationship for the criteria is that G1 G2 G3 G4 . The evaluation values of the five candidates with respect to four attributes are given by using the HFEs listed in Table 1, which is constructed by a hesitant fuzzy decision matrix H = (hi j )5×4 .
And the most desirable candidate is X5 . Different from our method, that of Wei [1] is described as following: Step 1’: Based on the hesitant fuzzy decision matrix H = (hij )5 × 4 and Definition 1, we can get Ti j , i = 1, 2, . . . , 5, j = 1, 2, 3, 4, as follows:
⎛1.000
⎜1.000 T = (Ti j )5×4 = ⎜1.000 ⎝ 1.000 1.000
0.533 0.700 0.700 0.600 0.650
0.347 0.385 0.233 0.270 0.390
⎞
0.254 0.218 ⎟ 0.117 ⎟. ⎠ 0.230 0.293
F. Jin et al. / Knowledge-Based Systems 96 (2016) 115–119
119
Step 2’: Using HFPWA operator [1] to aggregate all HFEs hi j , j = 1, 2, 3, 4 into the over HFEs hi , i = 1, 2, . . . , 5, as follows:
Step 4’: According to the score functions s(hi ), i = 1, 2, . . . , 5, we rank the overall HFEs hi , i = 1, 2, . . . , 5, in descending order:
h1 = {0.4747, 0.4885, 0.4986, 0.5118, 0.5178, 0.5304, 0.5397,
h5 > h1 > h2 > h4 > h3 .
0.5518, 0.5804, 0.5822, 0.5914, 0.5932, 0.6013, 0.6117, 0.6148, 0.6165, 0.6205, 0.6249, 0.6265, 0.6304, 0.6339, 0.6378, 0.6435, 0.6473, 0.6663, 0.6750, 0.6936, 0.6968, 0.6982, 0.7017, 0.7048, 0.7061, 0.7119, 0.7195, 0.7589, 0.7652}, h2 = {0.4752, 0.4878, 0.4999, 0.5124, 0.5158, 0.5311, 0.5398, 0.5523, 0.5814, 0.5816, 0.5909, 0.5930, 0.6012, 0.6121, 0.6144, 0.6172, 0.6197, 0.6250, 0.6265, 0.6312, 0.6340, 0.6368, 0.6433, 0.6480, 0.6667, 0.6753, 0.6940, 0.6967, 0.6981, 0.7021, 0.7043, 0.7064, 0.7118, 0.7192, 0.7595, 0.7653}, h3 = {0.4351, 0.4499, 0.4588, 0.4734, 0.4789, 0.4911, 0.5004, 0.5129, 0.5409, 0.5435, 0.5523, 0.5544, 0.5622, 0.5723,
Then the ranking of five candidates is that
X5 X1 X2 X4 X3 , and the most desirable candidate is X5 . From the above analysis, we can see that the proposed method in this article has the same ranking result with that of Wei [1]. However, compared with the method developed by Wei [1], we find that the proposed method has some advantages, which are that our proposed operators meet idempotency, and the decision making process of the information aggregation proposed by our method is more simplified than that of Wei [1]. Example 4 demonstrates that the proposed information aggregation operators are both valid and practical to deal with group decision making problems. This note develops two new operators to aggregate the collective of attribute values to overcome the drawbacks of Wei [1], and some properties, such as the idempotency and boundedness, of these operators are investigated. Acknowledgments
0.5754, 0.5777, 0.5815, 0.5857, 0.5876, 0.5911, 0.5945, 0.5990, 0.6042, 0.6084, 0.6269, 0.6462, 0.6549, 0.6573, 0.6587, 0.6628, 0.6659, 0.6668, 0.6729, 0.6806, 0.7198, 0.7259}, h4 = {0.4429, 0.4561, 0.4660, 0.4802, 0.4857, 0.4985, 0.5074, 0.5199, 0.5481, 0.5491, 0.5598, 0.5613, 0.5693, 0.5798, 0.5829, 0.5842, 0.5881, 0.5931, 0.5940, 0.5985, 0.6020, 0.6061, 0.6117, 0.6153, 0.6344, 0.6525, 0.6619, 0.6648, 0.6666, 0.6689, 0.6729, 0.6730, 0.6797, 0.6875, 0.7267, 0.7331}, h5 = {0.4820, 0.4936, 0.5056, 0.5181, 0.5215, 0.5374, 0.5458, 0.5580, 0.5875, 0.5878, 0.5961, 0.5994, 0.6076, 0.6175, 0.6210, 0.6235, 0.6255, 0.6309, 0.6329, 0.6371, 0.6397, 0.6421, 0.6494, 0.6541, 0.6729, 0.6816, 0.7002, 0.7025, 0.7041, 0.7078, 0.7100, 0.7122, 0.7179, 0.7253, 0.7655, 0.7719}. Step 3’: Calculating the score functions s(hi ), i = 1, 2, . . . , 5:
s(h1 ) = 0.6241, s(h2 ) = 0.6242, s(h3 ) = 0.5850, s(h4 ) = 0.6020, s(h5 ) = 0.6302.
The work was supported by National Natural Science Foundation of China (Nos. 71271071, 71371011, 71301041, 71490725, 91546108). The authors are thankful to the anonymous reviewers and the editor for their valuable comments and constructive suggestions that have led to an improved version of this paper. References [1] G.W. Wei, Hesitant fuzzy prioritized operators and their application to multiple attribute decision making, Knowl. Based Syst. 31 (2012) 176–182. [2] L.A. Zadeh, Fuzzy Sets, Inf. Control 8 (1965) 338–353. [3] K. Atanassov, Intuitionistic fuzzy sets, Fuzzy Sets Syst. 20 (1986) 87–96. [4] K. Atanassov, G. Gargov, Interval-valued intuitionistic fuzzy sets, Fuzzy Sets Syst. 31 (1989) 343–349. [5] V. Torra, Hesitant fuzzy sets, Int. J. Intell. Syst. 25 (2010) 529–539. [6] M.M. Xia, Z.S. Xu, Hesitant fuzzy information aggregation in decision making, Int. J. Approx. Reason. 52 (2011) 395–407. [7] H.C. Liao, Z.S. Xu, M.M. Xia, Multiplicative consistency of hesitant fuzzy preference relation and its application in group decision making, Int. J. Inf. Technol. Decis. Mak. 13 (1) (2014) 47–76. [8] H.C. Liao, Z.S. Xu, Some new hybrid weighted aggregation operators under hesitant fuzzy multi-criteria decision making environment, J Intell Fuzzy Syst. 26 (4) (2014) 1601–1617. [9] H.C. Liao, Z.S. Xu, Extended hesitant fuzzy hybrid weighted aggregation operators and their application in decision making, Soft Comput 19 (2015) 2551– 2564. [10] Z.S. Xu, On consistency of the weighted geometric mean complex judgment matrix in AHP, Eur. J. Oper. Res. 126 (2000) 683–687. [11] D.J. Yu, Y.Y. Wu, T. Lu, Interval-valued intuitionistic fuzzy prioritized operators and their application in group decision making, Knowl. Based Syst. 30 (2012) 57–66.
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Knowledge-Based Systems journal homepage: www.elsevier.com/locate/knosys
Note on “Hesitant fuzzy prioritized operators and their application to multiple attribute decision making” Feifei Jin a,b,∗, Zhiwei Ni a,b, Huayou Chen c a
School of Management, Hefei University of Technology, Hefei, Anhui 230009, China Key Laboratory of Process Optimization and Intelligent Decision-Making, Ministry of Education, Hefei, Anhui 230009, China c School of Mathematical Sciences, Anhui University, Hefei, Anhui 230601, China b
a r t i c l e
i n f o
Article history: Received 27 November 2014 Revised 3 October 2015 Accepted 27 December 2015 Available online 11 January 2016 Keywords: Decision making Hesitant fuzzy sets Prioritized aggregation operator Idempotency
a b s t r a c t Motivated by the idea of prioritized aggregation (PA) operators, Wei (2012) developed two hesitant fuzzy prioritized aggregation (HFPA) operators, and discussed their desirable properties, but the definitions for the HFPA operators and their properties still need to be improved. In this short note, a numerical example is given to show that the idempotency of the HFPA operators suffers from certain shortcomings. Then, based on some adjusted operations on the hesitant fuzzy elements (HFEs), two improved aggregation operators are investigated to aggregate the collective of attribute values. We further prove that the improved operators have the properties of idempotency and boundedness. Finally, the comparison with the method proposed by Wei (2012) is performed to demonstrate that the proposed information aggregation method is both valid and practical to deal with decision making problems. © 2016 Elsevier B.V. All rights reserved.
1. Introduction Since Zadeh introduced the fuzzy sets (FSs) [2], fuzzy sets have been achieved a great success in various fields. The concept of intuitionistic fuzzy sets (IFSs) [3] put forward by Atanassov is a generalization of the fuzzy set. Atanassov and Gargov further introduced the concept of interval-valued intuitionistic fuzzy sets (IVIFSs) [4], the components of which are intervals rather than exact numbers. Torra [5] proposed the concept of hesitant fuzzy set (HFS) considered as another generalization of FSs, which permits the membership degree having a collection of possible values. Wei [1] extended the prioritized averaging (PA) operator to accommodate the situations where the input arguments are hesitant fuzzy information, and developed two prioritized aggregation operators. He further studied some desirable properties of the proposed operators. However, a close examination demonstrates that some properties suffer from serious drawbacks. The purpose of this note is to point out and correct errors in the properties of HFPA operators. 2. Preliminaries By the relationship between the HFEs and intuitionistic fuzzy values (IFVs), Xia and Xu [6] defined some operations on the HFEs.
∗ Corresponding author at: School of Management, Hefei University of Technology, Hefei, Anhui 230009, China. Tel.: +86 13856942010. E-mail address: [email protected] (F. Jin).
http://dx.doi.org/10.1016/j.knosys.2015.12.023 0950-7051/© 2016 Elsevier B.V. All rights reserved.
Combined with the PA operator, Wei [1] developed two HFPA operators, including the hesitant fuzzy prioritized weighted average (HFPWA) operator and the hesitant fuzzy prioritized weighted geometric (HFPWG) operator. Definition 1. (See Wei [1], Definitions 7 and 8) Let h j , j = 1, 2, . . . , n be a collection of HFEs, then the HFPWA operator and the HFPWG operator are defined as follows, respectively: n
HF PWA(h1 , h2 , . . . , hn ) = ⊕ =
n
j=1
1−
γ1 ∈h1 ,γ2 ∈h2 ,...,γn ∈hn
n
(1 − γ j )
γ1 ∈h1 ,γ2 ∈h2 ,...,γn ∈hn
hj
Tj n T j=1 j
,
(1)
j=1
n
=
Tj
j=1
Tj n T j=1 j
HF PWG(h1 , h2 , . . . , hn ) = ⊗ h j
Tj
j=1
n
(γ j )
Tj n T j=1 j
,
(2)
j=1
j−1 where T1 = 1, T j = k=1 s(hk ), j = 2, . . . , n, and s(hk ) is the score values of hk , k = 1, 2, . . . , n. Then, Wei [1] proved that both the HFPWA operator and the HFPWG operator are idempotent. Theorem 1. (Idempotency, see Wei [1], Theorems 2 and 6) Let h j , j = 1, 2, . . . , n, be a collection of HFEs, where T1 = 1,
116
F. Jin et al. / Knowledge-Based Systems 96 (2016) 115–119
j−1 T j = k=1 s(hk ), j = 2, . . . , n, and s(hk ) is the score values of hk , k = 1, 2, . . . , n. If all h j , j = 1, 2, . . . , n, are equal, i.e. h j = h for all j, then
HF PWA(h1 , h2 , . . . , hn ) = h,
(3)
HF PWG(h1 , h2 , . . . , hn ) = h.
(4)
Thus, we calculate the score value of HFPWG(h1 , h2 ) and get that
s(HF PWG(h1 , h2 )) = 0.5274, and then
s(HF PWG(h1 , h2 )) = 0.5274 < 0.54 = s(h ), which indicates that
3. Numerical example
HF PWG(h1 , h2 ) < h.
Now, we furnish the following example to demonstrate that Theorem 1 is technically incorrect.
Example 1 demonstrates that Theorem 1 of the HFPA operators cannot be tenable, which suffer from serious drawbacks. In this case, the operations on the HFEs need to be improved. In the following section, some adjusted operations for HFEs are presented, and two new HFPA operators are developed, which satisfy the properties of idempotency and boundedness.
Example 1. Suppose that h1 and h2 are two HFEs, and h1 = h2 = h = {0.35, 0.50, 0.77}, then the score values of h1 and h2 are s(h1 ) = s(h2 ) = 0.54, and then
T1 = 1, T2 =
2−1
s(hk ) = s(h1 ) = 0.54.
4. The improved operators and their properties
k=1
According to Definition 1, it follows that
HF PWA(h1 , h2 ) = HF PA(h, h ) =
1−
2
γ1 ∈h1 ,γ2 ∈h2
= 1 − (1 − 0.35 ) 1 − (1 − 0.35 )
1 1+0.54
1 − (1 − 0.35 )
1 1+0.54
1 1+0.54
× (1 − 0.35 )
× (1 − 0.50 )
0.54 1+0.54
× (1 − 0.77 )
0.54 1+0.54
1
Tj
(1 − γ j )
2 T j=1 j
j=1
0.54 1+0.54
,
,
1 − (1 − 0.50 ) 1 − (1 − 0.50 )
1 1+0.54
0.54
0.54
× (1 − 0.50 ) 1+0.54 , × (1 − 0.77 )
0.54 1+0.54
1
0.54
1
0.54
1
0.54
(R1) All the elements in each HFE h are arranged in decreasing order, and γ (i) is the ith largest value in h; (R2) If lh = lg , then l = max{lh , lg }. To have a correct comparison, the two HFEs h and g should have the same length. If there are fewer elements in h than in g, an extension of h should be considered optimistically by repeating its maximum element until it has the same length with g; (R3) For convenience, we assume that all the HFEs have the same length l, i.e., h = {γ (1 ) , γ (2 ) , . . . , γ (l ) }.
,
1 − (1 − 0.77 ) 1+0.54 × (1 − 0.35 ) 1+0.54 , 1 − (1 − 0.77 ) 1+0.54 × (1 − 0.50 ) 1+0.54 , 1 − (1 − 0.77 ) 1+0.54 × (1 − 0.77 ) 1+0.54
= {0.3500, 0.4071, 0.5486, 0.4518,0.5000, 0.6192, 0.6689, 0.6980, 0.7700}. Therefore, the score value of HFPWA(h1 , h2 ) can be calculated as
s(HF PWA(h1 , h2 )) = 0.5571 > 0.54 = s(h ), which indicates that
HF PWA(h1 , h2 ) > h.
(5)
On the other hand, by the HFPWG operator in Definition 1, it is obtained that
HF PWG(h1 , h2 ) = HF PG(h, h ) =
γ1 ∈h1 ,γ2 ∈h2
= 0.35
0.77
1 1+0.54
1 1+0.54 1 1+0.54
× 0.35
0.54 1+0.54
× 0.35
0.54 1+0.54
× 0.35
0.54 1+0.54
, 0.35
1 1+0.54
, 0.50
1 1+0.54
, 0.77
1 1+0.54
2
(γ j )
Tj 2 T j=1 j
j=1
× 0.50
0.54 1+0.54
× 0.50
0.54 1+0.54
× 0.50
0.54 1+0.54
,0.35
1 1+0.54
, 0.50
1 1+0.54
, 0.77
1 1+0.54
× 0.77
0.54 1+0.54
× 0.77
0.54 1+0.54
× 0.77
0.54 1+0.54
= {0.3500, 0.3966, 0.4615, 0.4412,0.5000, 0.5817, 0.5840, 0.6618, 0.7700}.
l
{γ1(i) + γ2(i) − γ1(i) γ2(i) }; ˙ 2 = i=1 {γ1(i ) γ2(i ) }; (2) h1 ⊗h λ (3) λh = li=1 {1 − (1 − γ (i ) ) }, λ > 0; λ (4) hλ = li=1 {(γ (i ) ) }, λ > 0. ˙ 2= (1) h1 ⊕h
then
0.50
Now, let’s review some adjusted operations on the HFEs as follows: Definition 2. [7] Suppose that h1 , h2 and h are three HFEs, then
s(HF PWA(h1 , h2 )) = 0.5571,
In this section, some adjusted operations on the HFEs are reviewed, and two new improved aggregation operators based on these adjusted operations are investigated to aggregate the collective of attribute values. We further prove that the improved operators have the properties of idempotency and boundedness. In the end, an example shows that our method is easier than that of Wei [1] in some cases. Remark 1. Notice that the number of values in different HFEs may be different. Suppose that lh stands for the number of values in h, then the following assumptions are made:
,
1 − (1 − 0.50 ) 1+0.54 × (1 − 0.35 ) 1+0.54 , 1 1+0.54
(6)
,
,
i=1
l
˙ 2 and According to the Definition 2, we know that h1 , h2 , h1 ⊕h ˙ 2 have the same length l. h1 ⊗h Example 2. Suppose that l = 3, λ = 2, h1 and h2 are two HFEs, and h1 = {0.77, 0.50, 0.35}, h2 = {0.86, 0.59, 0.22}, then we have ˙ 2 = 3i=1 {γ1(i ) + γ2(i ) − γ1(i ) γ2(i ) } = (1) h1 ⊕h {0.9678, 0.7950, 0.4930}; ˙ 2 = 3i=1 {γ1(i ) γ2(i ) } = {0.6622, 0.2950, 0.0770}; (2) h1 ⊗h 2 (3) 2h1 = 3i=1 {1 − (1 − γ (i ) ) } = {0.9471, 0.7500, 0.5775}; 2 3 (4) h2 = i=1 {(γ (i ) ) } = {0.5929, 0.2500, 0.1225}. Based on the adjusted operational principle for HFEs, we investigate the improved HFPA operators as follows:
F. Jin et al. / Knowledge-Based Systems 96 (2016) 115–119
Definition 3. Let h j , j = 1, 2, . . . , n, be a collection of HFEs, then the improved hesitant fuzzy prioritized weighted average (IHFPWA) operator is defined as follows:
n
IHF PWA(h1 , h2 , . . . , hn ) = ⊕˙ =
l
1−
i=1
n
j=1
n
(1 − γ j(i) )
Tj n T j=1 j
Tj
j=1
Tj
Therefore, it follows that
where T1 = 1, T j = k=1 s(hk ), j = 2, . . . , n, and s(hk ) is the score value of hk , k = 1, 2, . . . , n. Definition 4. Let h j , j = 1, 2, . . . , n, be a collection of HFEs, then the improved hesitant fuzzy prioritized weighted geometric (IHFPWG) operator is defined as follows:
=
l n i=1
(γ j(i) )
Tj n T j=1 j
, (8)
j=1
j−1 where T1 = 1, T j = k=1 s(hk ), j = 2, . . . , n, and s(hk ) is the score value of hk , k = 1, 2, . . . , n. According to Definitions 3 and 4, the properties corresponding to Theorems 2 and 6 in Wei [1] can be obtained: Theorem 2. (Idempotency) Let h j , j = 1, 2, . . . , n, be a collection of j−1 HFEs, where T1 = 1, T j = k=1 s(hk ), j = 2, 3, . . . , n, and s(hk ) is the score value of hk , k = 1, 2, . . . , n. If all h j , j = 1, 2, . . . , n, are equal, i.e. h j = h for all j, then
IHF PWA(h1 , h2 , . . . , hn ) = h.
(9)
h1 =h2 = · · · =hn = h = {γ
,γ
then we have
IHF PWA(h1 , h2 , . . . , hn ) =
=
l i=1
=
l
1−
l
(2 )
n
(i )
(1 − γ )
j=1 n
1 − ( 1 − γ (i ) )
,...,γ
1−
Tj n T j=1 j
Tj n T j=1 j j=1
n
(l )
},
=
j=1
1
0.54
1
0.54
1
0.54
1 − (1 − 0.35 ) 1+0.54 × (1 − 0.35 ) 1+0.54
= {0.7700,0.5000, 0.3500} = h, and
3 2
IHF PWG(h1 , h2 ) =
i=1
1
(i )
(γ j )
Tj 2 T j=1 j
j=1 0.54
1
0.54
= 0.77 1+0.54 × 0.77 1+0.54 , 0.50 1+0.54 × 0.50 1+0.54 , 1
0.54
0.35 1+0.54 × 0.35 1+0.54
= {0.7700,0.5000, 0.3500} = h. Inspired by Liao and Xu [8,9], it can be proved that the improved HFPA operators also satisfy the property of boundedness. Theorem 4. (Boundedness) Let h j , j = 1, 2, . . . , n, be a collection j−1 of HFEs, where T1 = 1, T j = k=1 s(hk ), j = 2, 3, . . . , n. And s(hk ) is the score value of hk , k = 1, 2, . . . , n. Let h− = min min γ j(i ) , h+ = 1≤i≤l 1≤ j≤n
(1 − γ j(i) )
h− ≤ IHF PWA(h1 , h2 , . . . , hn ) ≤ h+ . Tj n T j=1 j
Proof. n
Let
(1 − γ j(i) )
j=1
j=1
Tj n T j=1 j
(11)
˜ = IHF PWA(h1 , h2 , . . . , hn ) h
and
γ˜ (i) = 1 −
, i = 1, 2, . . . , l.
Since h− = min min γ j(i ) , h+ = max max γ j(i ) , then we have 1≤i≤l 1≤ j≤n
1≤i≤l 1≤ j≤n
h− ≤ γ j(i) ≤ h+ , i = 1, 2, . . . , l, j = 1, 2, . . . , n, and then
i=1 l
1≤i≤l 1≤ j≤n
i=1
(1 − γ j(i) )
Tj 2 T j=1 j
max max γ j(i ) , then
Proof. Since h j = h for all j, i.e., (1 )
2
1 − (1 − 0.50 ) 1+0.54 × (1 − 0.50 ) 1+0.54 ,
j−1
j=1
1−
i=1
j=1
IHF PWG(h1 , h2 , . . . , hn ) = ⊗˙ h j
= 1 − (1 − 0.77 ) 1+0.54 × (1 − 0.77 ) 1+0.54 , (7)
Tj n T j=1 j
3
IHF PWA(h1 , h2 ) =
hj
,
n
117
1 − ( 1 − γ (i ) ) =
i=1
l
γ (i )
n
i=1
≤
n
(1 − γ j(i) )
Tj n T j=1 j
≤
j=1
n
(1 − h− )
Tj n T j=1 j
(12)
j=1
it follows that
which completes the proof of Theorem 2. Theorem 3. (Idempotency) Let h j , j = 1, 2, . . . , n, be a collection of j−1 HFEs, where T1 = 1, T j = k=1 s(hk ), j = 2, 3, . . . , n. And s(hk ) is the score value of hk , k = 1, 2, . . . , n. If all h j , j = 1, 2, . . . , n, are equal, i.e. h j = h for all j, then
1−
n
(1 − h− )
Tj n T j=1 j
≤1−
j=1
≤1−
n
(1 − γ j(i) )
Tj n T j=1 j
j=1 n
(1 − h+ )
Tj n T j=1 j
, i = 1, 2, . . . , l,
(13)
j=1
(10)
Proof. The proof of Theorem 3 is similar to that of Theorem 2. So it is omitted here. Example 3. (Adapted from Example 1) Suppose that h1 and h2 are two HFEs, and h1 = h2 = h = {0.77, 0.50, 0.35}, then l = 3. According to the Example 1, we have
T1 = 1, T2 = 0.54.
Tj n T j=1 j
j=1
= {γ (1) , γ (2) , . . . , γ (l ) } = h,
IHF PWG(h1 , h2 , . . . , hn ) = h.
(1 − h+ )
which is equivalent to
1 − (1 − h− )
n T j=1 j n T j=1 j
≤ γ˜ (i) ≤ 1 − (1 − h+ )
n T j=1 j n T j=1 j
, i = 1, 2, . . . , l,
i.e.,
h− ≤ γ˜ (i) ≤ h+ , i = 1, 2, . . . , l.
(14)
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F. Jin et al. / Knowledge-Based Systems 96 (2016) 115–119
˜) = As s(h
h− =
1 l
l i=1
γ˜ (i) , according to Eq. (14), we have
Table 1 Hesitant fuzzy decision matrix H = (hi j )5×4
l l l 1 − 1 (i ) 1 + h ≤ γ˜ ≤ h = h+ , l l l i=1
i=1
X1 X2 X3 X4 X5
i=1
i.e.,
˜ ) ≤ s(h+ ). s(h− ) ≤ s(h
G1
G2
G3
G4
{0.4,0.5,0.7} {0.6,0.7,0.8} {0.6,0.8} {0.5,0.6,0.7} {0.6,0.7}
{0.5,0.8} {0.5,0.6} {0.2,0.3,0.5} {0.4,0.5} {0.5,0.7}
{0.6,0.7,0.9} {0.4,0.6,0.7} {0.4,0.6} {0.8,0.9} {0.7,0.8}
{0.5,0.6} {0.4,0.5} {0.5,0.7} {0.3,0.4,0.5} {0.2,0.3,0.4}
Therefore,
h ≤ IHF PWA(h1 , h2 , . . . , hn ) ≤ h+ , −
which completes the proof of Theorem 4. Theorem 5. (Boundedness) Suppose that h j , j = 1, 2, . . . , n, is a j−1 collection of HFEs, where T1 = 1, T j = k=1 s(hk ), j = 2, 3, . . . , n, and s(hk ) is the score values of hk , k = 1, 2, . . . , n. Let h− = min min γ j(i ) , h+ = max max γ j(i ) , then 1≤i≤l 1≤ j≤n
1≤i≤l 1≤ j≤n
h ≤ IHF PWG(h1 , h2 , . . . , hn ) ≤ h+ . −
(15)
Proof. The proof of Theorem 5 is similar to that of Theorem 4. So it is omitted here. In order to analyze the relationship among the IHFPWA operator and IHFPWG operator, we introduce the following lemma: Lemma 1. [10] Let xj > 0, λ j > 0, j = 1, 2, . . . , n, and nj=1 λ j = 1, n n λj then j=1 λ j x j ≥ j=1 x j , with equality if and only if x1 = x2 = · · · = xn . Theorem 6. Let h j , j = 1, 2, . . . , n, be a collection of HFEs, where j−1 T1 = 1, T j = k=1 s(hk ), j = 2, 3, . . . , n. And s(hk ) is the score value of hk , k = 1, 2, . . . , n, then
IHF PWA(h1 , h2 , . . . , hn ) ≥ IHF PWG(h1 , h2 , . . . , hn ). (i )
(i )
Proof. For any γ1 ∈ h1 , γ2 using the Lemma 1, we have
1−
n j=1
(1 − γ j(i) )
=1− 1−
n
Tj n T j=1 j
j=1
≥
n
(γ j(i) )
Tj n T j=1 j
≥1−
n j=1
Tj
n
j=1
(i )
Tj
(i )
∈ h2 , . . . , γn
· γj
j=1
=
∈ hn , i = 1, 2, . . . , l,
⎛
{0.4, 0.5, 0.7} ⎜{0.6, 0.7, 0.8} ⎜ H˜ = ⎜ ⎜{0.6, 0.8, 0.8} ⎝{0.5, 0.6, 0.7} {0.6, 0.7, 0.7}
{0.5, 0.8, 0.8} {0.5, 0.6, 0.6} {0.2, 0.3, 0.5} {0.4, 0.5, 0.5} {0.5, 0.7, 0.7}
{0.6, 0.7, 0.9} {0.4, 0.6, 0.7} {0.4, 0.6, 0.6} {0.8, 0.9, 0.9} {0.7, 0.8, 0.8}
⎞ {0.5, 0.6, 0.6} {0.4, 0.5, 0.5}⎟ ⎟ {0.5, 0.7, 0.7}⎟ ⎟ {0.3, 0.4, 0.5}⎠ {0.2, 0.3, 0.4}
Step 2: Utilizing Definition 3 to calculate Ti j , i = 1, 2, . . . , 5, j = 1, 2, 3, 4, as follows:
⎛1.000
⎜1.000 T = (Ti j )5×4 = ⎜1.000 ⎝ 1.000 1.000
0.533 0.700 0.733 0.600 0.667
0.373 0.397 0.244 0.280 0.422
⎞
0.274 0.225⎟ 0.130 ⎟ ⎠ 0.243 0.324
Step 3: Without loss of generality, we use the IHFPWA operator ˜ i (i = ˜ i j , j = 1, 2, 3, 4, into a collective HFEs h to aggregate all HFEs h 1, 2, . . . , 5 ) of candidates Xi (i = 1, 2, . . . , 5 ), then we have
˜ 1 = {0.4768, 0.6440, 0.7666}, h ˜ 2 = {0.5231, 0.6389, 0.7113}, h ˜ 4 = {0.5152, 0.6282, 0.6821}, ˜h3 = {0.4590, 0.6564, 0.6943}, h
Tj
n
(16)
With this information, we utilize the proposed information aggregation method to get the ranking of these candidates. The following steps are involved: Step 1: Using Remark 1 to obtain a normalized decision matrix ˜ i j )5×4 from H = (hi j )5×4 as follows: H˜ = (h
Tj
n j=1
· (1 − γ j(i) )
˜ i ), i = 1, 2, . . . , 5, i.e., Step 4: Calculating the score functions s(h
Tj
n
j=1
˜ 5 = {0.5560, 0.6869, 0.6933}. h
(i )
Tj
· γj
, i = 1, 2, . . . , l,
˜ 1 ) = 0.6291, s(h ˜ 2 ) = 0.6244, s(h ˜ 3 ) = 0.6032, s (h (17)
j=1
it follows that
s(IHF PWA(h1 , h2 , . . . , hn )) ≥ s(IHF PWG(h1 , h2 , . . . , hn )),
˜ 4 ) = 0.6085, s(h ˜ 5 ) = 0.6454. s (h ˜ i ), Step 5: According to the ordering of the score functions s(h ˜ i , i = 1, 2, . . . , 5, in dei = 1, 2, . . . , 5,we rank the overall HFEs h scending order:
i.e.,
˜1 > h ˜2 > h ˜4 > h ˜ 3, ˜5 > h h
IHF PWA(h1 , h2 , . . . , hn ) ≥ IHF PWG(h1 , h2 , . . . , hn ),
then the ranking of five candidates is that
which completes the proof of Theorem 6. In what follows, Example 4 will show that our method is valid and practical in some cases.
X5 X1 X2 X4 X3 .
Example 4. In order to promote the academic education, a Chinese university wants to introduce oversea outstanding teachers [11]. The panel of decision makers have strict evaluation for five candidates X1 , X2 , X3 , X4 and X5 , considering the following four attributes, G1 : morality, G2 : research capability, G3 : teaching skills, and G4 : education background, where the prioritization relationship for the criteria is that G1 G2 G3 G4 . The evaluation values of the five candidates with respect to four attributes are given by using the HFEs listed in Table 1, which is constructed by a hesitant fuzzy decision matrix H = (hi j )5×4 .
And the most desirable candidate is X5 . Different from our method, that of Wei [1] is described as following: Step 1’: Based on the hesitant fuzzy decision matrix H = (hij )5 × 4 and Definition 1, we can get Ti j , i = 1, 2, . . . , 5, j = 1, 2, 3, 4, as follows:
⎛1.000
⎜1.000 T = (Ti j )5×4 = ⎜1.000 ⎝ 1.000 1.000
0.533 0.700 0.700 0.600 0.650
0.347 0.385 0.233 0.270 0.390
⎞
0.254 0.218 ⎟ 0.117 ⎟. ⎠ 0.230 0.293
F. Jin et al. / Knowledge-Based Systems 96 (2016) 115–119
119
Step 2’: Using HFPWA operator [1] to aggregate all HFEs hi j , j = 1, 2, 3, 4 into the over HFEs hi , i = 1, 2, . . . , 5, as follows:
Step 4’: According to the score functions s(hi ), i = 1, 2, . . . , 5, we rank the overall HFEs hi , i = 1, 2, . . . , 5, in descending order:
h1 = {0.4747, 0.4885, 0.4986, 0.5118, 0.5178, 0.5304, 0.5397,
h5 > h1 > h2 > h4 > h3 .
0.5518, 0.5804, 0.5822, 0.5914, 0.5932, 0.6013, 0.6117, 0.6148, 0.6165, 0.6205, 0.6249, 0.6265, 0.6304, 0.6339, 0.6378, 0.6435, 0.6473, 0.6663, 0.6750, 0.6936, 0.6968, 0.6982, 0.7017, 0.7048, 0.7061, 0.7119, 0.7195, 0.7589, 0.7652}, h2 = {0.4752, 0.4878, 0.4999, 0.5124, 0.5158, 0.5311, 0.5398, 0.5523, 0.5814, 0.5816, 0.5909, 0.5930, 0.6012, 0.6121, 0.6144, 0.6172, 0.6197, 0.6250, 0.6265, 0.6312, 0.6340, 0.6368, 0.6433, 0.6480, 0.6667, 0.6753, 0.6940, 0.6967, 0.6981, 0.7021, 0.7043, 0.7064, 0.7118, 0.7192, 0.7595, 0.7653}, h3 = {0.4351, 0.4499, 0.4588, 0.4734, 0.4789, 0.4911, 0.5004, 0.5129, 0.5409, 0.5435, 0.5523, 0.5544, 0.5622, 0.5723,
Then the ranking of five candidates is that
X5 X1 X2 X4 X3 , and the most desirable candidate is X5 . From the above analysis, we can see that the proposed method in this article has the same ranking result with that of Wei [1]. However, compared with the method developed by Wei [1], we find that the proposed method has some advantages, which are that our proposed operators meet idempotency, and the decision making process of the information aggregation proposed by our method is more simplified than that of Wei [1]. Example 4 demonstrates that the proposed information aggregation operators are both valid and practical to deal with group decision making problems. This note develops two new operators to aggregate the collective of attribute values to overcome the drawbacks of Wei [1], and some properties, such as the idempotency and boundedness, of these operators are investigated. Acknowledgments
0.5754, 0.5777, 0.5815, 0.5857, 0.5876, 0.5911, 0.5945, 0.5990, 0.6042, 0.6084, 0.6269, 0.6462, 0.6549, 0.6573, 0.6587, 0.6628, 0.6659, 0.6668, 0.6729, 0.6806, 0.7198, 0.7259}, h4 = {0.4429, 0.4561, 0.4660, 0.4802, 0.4857, 0.4985, 0.5074, 0.5199, 0.5481, 0.5491, 0.5598, 0.5613, 0.5693, 0.5798, 0.5829, 0.5842, 0.5881, 0.5931, 0.5940, 0.5985, 0.6020, 0.6061, 0.6117, 0.6153, 0.6344, 0.6525, 0.6619, 0.6648, 0.6666, 0.6689, 0.6729, 0.6730, 0.6797, 0.6875, 0.7267, 0.7331}, h5 = {0.4820, 0.4936, 0.5056, 0.5181, 0.5215, 0.5374, 0.5458, 0.5580, 0.5875, 0.5878, 0.5961, 0.5994, 0.6076, 0.6175, 0.6210, 0.6235, 0.6255, 0.6309, 0.6329, 0.6371, 0.6397, 0.6421, 0.6494, 0.6541, 0.6729, 0.6816, 0.7002, 0.7025, 0.7041, 0.7078, 0.7100, 0.7122, 0.7179, 0.7253, 0.7655, 0.7719}. Step 3’: Calculating the score functions s(hi ), i = 1, 2, . . . , 5:
s(h1 ) = 0.6241, s(h2 ) = 0.6242, s(h3 ) = 0.5850, s(h4 ) = 0.6020, s(h5 ) = 0.6302.
The work was supported by National Natural Science Foundation of China (Nos. 71271071, 71371011, 71301041, 71490725, 91546108). The authors are thankful to the anonymous reviewers and the editor for their valuable comments and constructive suggestions that have led to an improved version of this paper. References [1] G.W. Wei, Hesitant fuzzy prioritized operators and their application to multiple attribute decision making, Knowl. Based Syst. 31 (2012) 176–182. [2] L.A. Zadeh, Fuzzy Sets, Inf. Control 8 (1965) 338–353. [3] K. Atanassov, Intuitionistic fuzzy sets, Fuzzy Sets Syst. 20 (1986) 87–96. [4] K. Atanassov, G. Gargov, Interval-valued intuitionistic fuzzy sets, Fuzzy Sets Syst. 31 (1989) 343–349. [5] V. Torra, Hesitant fuzzy sets, Int. J. Intell. Syst. 25 (2010) 529–539. [6] M.M. Xia, Z.S. Xu, Hesitant fuzzy information aggregation in decision making, Int. J. Approx. Reason. 52 (2011) 395–407. [7] H.C. Liao, Z.S. Xu, M.M. Xia, Multiplicative consistency of hesitant fuzzy preference relation and its application in group decision making, Int. J. Inf. Technol. Decis. Mak. 13 (1) (2014) 47–76. [8] H.C. Liao, Z.S. Xu, Some new hybrid weighted aggregation operators under hesitant fuzzy multi-criteria decision making environment, J Intell Fuzzy Syst. 26 (4) (2014) 1601–1617. [9] H.C. Liao, Z.S. Xu, Extended hesitant fuzzy hybrid weighted aggregation operators and their application in decision making, Soft Comput 19 (2015) 2551– 2564. [10] Z.S. Xu, On consistency of the weighted geometric mean complex judgment matrix in AHP, Eur. J. Oper. Res. 126 (2000) 683–687. [11] D.J. Yu, Y.Y. Wu, T. Lu, Interval-valued intuitionistic fuzzy prioritized operators and their application in group decision making, Knowl. Based Syst. 30 (2012) 57–66.