Group decision making under hesitant fuzzy ... - Semantic Scholar
Knowledge-Based Systems 52 (2013) 1–10
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Group decision making under hesitant fuzzy environment with application to personnel evaluation Dejian Yu a, Wenyu Zhang a,⇑, Yejun Xu b a b
School of Information, Zhejiang University of Finance & Economics, 18 Xueyuan Street, Hangzhou 310018, China Research Institute of Management Science, Business School, Hohai University, Nanjing 211100, China
a r t i c l e
i n f o
Article history: Received 15 October 2012 Received in revised form 13 April 2013 Accepted 15 April 2013 Available online 3 May 2013 Keywords: Hesitant fuzzy set Group decision making Generalized hesitant fuzzy aggregation operator Prioritized aggregation operator
a b s t r a c t In many personnel evaluation scenarios, decision makers are asked to provide their preferences anonymously to both ensure privacy and avoid psychic contagion. The use of hesitant fuzzy sets is a powerful technique for representing this type of information and has been well studied. This paper explores aggregation methods for prioritized hesitant fuzzy elements and their application on personnel evaluation. First, the generalized hesitant fuzzy prioritized weighted average (GHFPWA) and generalized hesitant fuzzy prioritized weighted geometric (GHFPWG) operators are presented. Some desirable properties of the methods are discussed and special cases are investigated in detail. Previous research has indicated that many existing hesitant fuzzy aggregation operators are special cases of the proposed operators. Then, a procedure and algorithm for group decision making is provided using these proposed generalized hesitant fuzzy aggregation operators. Finally, the group decision making method is applied to a representative personnel evaluation problem that involves a prioritization relationship over the evaluation index. Published by Elsevier B.V.
1. Introduction Probation is an important component of the evaluation process. It allows an employee to demonstrate his/her suitability for a position. Probation gives the company an opportunity to observe an employee’s work ethic and training status, assists the employee in adjusting to a new position, and removes an employee whose performance fails to meet expectations. To expand a company’s potential market and maintain growth, it is important to provide sales engineers with professional and systematic trainings. Training is typically provided using a full scale development platform that allows engineers to learn a broad range of topics, from product knowledge to business skills, case studies to simulation, and classroom training to field practice. Effective training methods leverage the participants’ full potential, provide both opportunities and challenges, develop creativity, identify strengths and assist in career planning. After training, employees may receive systematic assessment and evaluation over the course of a probationary period. Let us consider an interesting but realistic problem. Currently, five sales engineers’ probation end dates are approaching, and we need to evaluate their performances to determine if their contracts should be renewed. Since sales engineers are responsible for establishing and maintaining productive working relationships with ⇑ Corresponding author. E-mail addresses: [email protected] (D. Yu), [email protected] (W. Zhang), [email protected] (Y. Xu). 0950-7051/$ - see front matter Published by Elsevier B.V. http://dx.doi.org/10.1016/j.knosys.2013.04.010
target customers in critical business areas, the company is very interested in evaluating their performance. A sales manager (e1), manufacturing manager (e2), engineering manager (e3) and human resource manager (e4) compose the panel of decision makers that will take responsibility for this evaluation. A strict evaluation of each of the five employees xi(i = 1, 2, . . . , 5) is performed from four aspects: work attitude C1, communication skill C2, problem solving skill C3, and learning skill C4. First, work attitude (C1) is considered a must for all professionals. This position also requires the sales engineer to work closely with internal colleagues and customers to develop effective sales proposals, thus communication skills (C2) represent another important competency area. In addition, sales engineers must be able to pair prospective customers with ongoing projects and follow through with appropriate sales activities to get the orders. This means that sales engineers must also possess solid problem solving skills (C3). Finally, learning skills (C4) help us to grow and adapt so that we may achieve more challenging targets. The prioritization of the criteria can be expressed as C1 C2 C3 C4, where ‘‘’’ indicates ‘‘priority to’’. It should be noted that there are two key issues inherent to the above problems. The first key issue is anonymously depicting the decision makers’ preferences. The second key issue is mathematically expressing the criteria prioritization relationships. The focus of this paper is to develop a new decision making method that addresses both of the above problems. To address these issues, the remainder of this paper is organized as follows. In Section 2, a literature review of decision making
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D. Yu et al. / Knowledge-Based Systems 52 (2013) 1–10
methods for personnel evaluation is provided, and the concepts of hesitant fuzzy set and prioritized operators are introduced. Section 3 proposes the generalized hesitant fuzzy prioritized weighted average (GHFPWA) and generalized hesitant fuzzy prioritized weighted geometric (GHFPWG) operators to aggregate the hesitant fuzzy elements (HFEs), whose desirable properties and special cases are also studied in this section. Section 4 develops a multicriteria group decision making method based on the proposed operators under the hesitant fuzzy environment, and applies this decision making method to the illustrative evaluation problem regarding sales engineers presented in this section. Section 5 provides some concluding remarks. 2. Literature and basic concept review Personnel evaluation has been widely studied by a large number of research institutions, and is an important aspect of human resources management. In highly competitive markets, a company’s personnel play a crucial role in the future development of the company. In other words, talent can be a company’s biggest asset. Therefore, staffing is important for any company’s business development. Putting the right people in the right positions can produce many positive impacts on a company, such as lowering employee turnover rate, improving enthusiasm and productivity, and increasing customer satisfaction. Alternatively, waste and other negative impacts can result from poor hiring decisions. Multi-criteria decision making (MCDM) has been identified as an essential technique in the personnel selection process [3,15,11]. Karsak [6] and Gil-Aluja [4] proposed an MCDM algorithm based on the concepts of ideal and anti-ideal solutions for personnel selection. Li [8] combined analytic network process (ANP) with fuzzy data envelopment analysis (DEA) and proposed an integrated method to deal with the personnel selection problem. An example is provided with an electric and machinery company in Taiwan. Chen and Cheng [2] proposed a fuzzy group decision support system based on metric distances to solve the personnel selection problem. Dursun and Karsak [3] developed a MCDM method based on the principles of fusion of fuzzy information and 2-tuple linguistic information. Zhang and Liu [37] used grey relational analysis (GRA) to solve the personnel selection problem under an intuitionistic fuzzy environment. Boran et al. [1] used the Technique for Order Preference by Similarity to Ideal Solution (TOPSIS) to solve the personnel selection problem under an intuitionistic fuzzy environment and illustrated the method with an example involving a sales manager position in a manufacturing company. Merigó and Gil-Lafuente [12,13] studied the human resource selection problem using the ordered weighted average (OWA) operator. The key differences between the above studies are illustrated in Table 1. Table 1 Comparison between existing research results. Author(s)
Published year
Key technology
Karsak, EE
2001
Chen LS, Cheng CH
2005
Lin HT
2010
Dursun M, Karsak EE. Zhang SF, Liu SY
2010 2011
Boran FE, Genc S, Akay D Merigó JM, GilLafuente AM
2011
Fuzzy MCDM, Ideal and Anti-Ideal Solutions Fuzzy Multi-criteria Group Decision Support System Analytic network process (ANP), Fuzzy DEA 2-Tuple linguistic, TOPSIS Grey Relation Analysis (GRA),Intuitionistic fuzzy set TOPSIS, Intuitionistic fuzzy set
2011
OWA operator, Hamming distance; Adequacy coefficient
The above personnel selection methods operate under the assumption that the evaluation criteria are independent and have similar priority levels. This assumption is not always practical, however. Take the problem described in Section 1 as an example. If an alternative (i.e., a sales engineer) has a poor work attitude C1, then the company will likely terminate employment regardless of the employee’s communication skills C2, problem solving skills C3 or learning skills C4. In other words, work attitude C1 has a higher priority than the other criteria. The prioritized average (PA) operator is a useful tool to deal with the above situation. To facilitate later study, the PA operator and hesitant fuzzy sets are introduced here. A hesitant fuzzy set (HFS) is a generalized fuzzy set that is characterized by the membership degree of an element to a set presented as several possible values between 0 and 1. Torra [20,21] first proposed the concept of HFS, which has attracted significant attention from other researchers [23,36,38,27]. Definition 1 ([20,21]). Let X be a fixed set. A HFS on X is defined in terms of a function that when applied to members ofX returns a subset with values in the range of [0, 1], which can be represented as the following mathematical symbol:
E ¼ f< x; f E ðxÞ > jx 2 Xg
ð1Þ
where fE(x) is a set of some values in the range of [0, 1], that denotes the possible membership degrees of the element x 2 X to the set E. For convenience, Xia and Xu [25,26] call fE(x) a hesitant fuzzy element (HFE). Take the problem illustrated in Section 1 as an example. A sales manager (e1), manufacturing manager (e2), engineering manager (e3) and human resource manager (e4) are each asked to estimate the degree to which an alternative (i.e., a sales engineer) presents an excellent work attitude. The sales manager gives the sales engineer a score of 0.6, the manufacturing manager rates the engineer with a 0.7, the engineering manager gives a 0.8 and the human resource manager (e4) determines the score to be 0.9. Thus, the degree to which the alternative presents an excellent work attitude can be represented by a hesitant fuzzy element {0.6, 0.7, 0.8, 0.9}. The HFS is utilized in this work because of its ability to efficiently express the uncertainty presented by this type of realistic situation. HFS therefore can address the first key issue mentioned in Section 1, i.e., how to anonymously describe the decision makers’ preferences. For three HFEs f, f1 and f2, Xia [24] gave the following operational laws for the HFEs as follows: Definition 2. Let f, f1 and f2 be three HFEs, then (1) (2) (3) (4)
kf = [ n2f{l1(kl(n))}, k > 0, fk = [ n2f{k1(kk(n))}, k > 0, 1 f1 f2 ¼ [n1 2f1 ;n2 2f2 fk ðkðn1 Þ þ kðn2 ÞÞg, and 1 f1 f2 ¼ [n1 2f1 ;n2 2f2 fl ðlðn1 Þ þ lðn2 ÞÞg,
where l(t) = k(1 t). Klir et al. [7] pointed out that an additive generator of a continuous Archimedean t-norm is a strictly decreasing function, i.e., k: [0, 1] ? [0, 1], such that k(1) = 0. P 1 Definition 3 [24]. For a HFE f ; Sðf Þ ¼ #f n2f n is called the score function of f, where #f is the number of the elements in f. For two HFEs f1 and f2, if S(f1) > S(f2), then f1 > f2; if S(f1) = S(f2), then f1 = f2. Since n 2 [0, 1], then S(f) 2 [0, 1]. Yager [33] first proposed the prioritized average (PA) operator, which was defined as follows.
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D. Yu et al. / Knowledge-Based Systems 52 (2013) 1–10
Definition 4. [33–35] Let C = {C1, C2, . . . , Cn} be a collection of criteria and a prioritization between the criteria can be expressed by the linear ordering C1 C2 C3 Cn, indicating criteria Cj has a higher priority than Ck if j < k. Then the value Cj(x) is the performance of any alternative x under criteria Cj, and satisfies Cj(x) 2 [0, 1]. If
PAðC i ðxÞÞ ¼
n X
T
j¼1
Tj
; Tj ¼
Qj1
k¼1 C k ðxÞðj
j¼1
!)
Tj Pn
j¼1 T j
lðnj Þ
ð5Þ
can be proven by using mathematical induction on n: (1) For n = 2, since
(
ð2Þ
f1 ¼ [n1 2f1 l
¼ 2; . . . ; nÞ; T 1 ¼ 1, then PA is
T2 Pn
f2 ¼ [n2 2f2 l
wj C j ðxÞ
n X
1
GHFPWA ðf1 ; f2 ; . . . ; fn Þ ¼ [nj 2fj l
T1 Pn
j¼1
where wj ¼ Pnj
(
j¼1 T j
1
!)
T1 Pn
lðn1 Þ
T2 Pn
lðn2 Þ
j¼1 T j
(
j¼1 T j
called the prioritized average (PA) operator.
1
j¼1 T j
ð6Þ !) ð7Þ
then, Example 1 (Paragraph 2 of Section 1). Consider four criteria: work attitude C1, communication skill C2, problem solving skill C3, and learning skill C4. The prioritization relationship for the criteria is C1 C2 C3 C4. Suppose that the satisfaction of the four criteria are C1(x) = 0, C2(x) = 1, C3(x) = 1, and C4(x) = 1. Based on the PA operator, the calculation is performed as follows
T 1 ¼ 1;
T 2 ¼ T 1 S1 ¼ 1 0 ¼ 0; T 3 ¼ T 2 S2 ¼ 0 1 ¼ 0; and T 4
T1 Pn
T2 f 1 Pn
j¼1 T j
j¼1 T j
( 1
¼ [n1 2f1 ;n2 2f2 l (
1
¼ [n1 2f1 ;n2 2f2 l
¼ T 3 S3 ¼ 0 1 ¼ 0
( ¼ [nj 2fj l
Therefore, the normalized weighting vectors are:
f2
1
2 X j¼1
l l
j¼1 T j
T1 Pn
j¼1 T j
Tj Pn
j¼1 T j
!!
T1 Pn
1
þl l
lðn1 Þ
!! !)
T2 Pn
1
j¼1 T j
lðn2 Þ
!)
T2 lðn1 Þ þ Pn
j¼1 T j
lðn2 Þ
!) lðnj Þ
x1 ¼ 1; x2 ¼ 0; x3 ¼ 0; and x4 ¼ 0
ð8Þ
The overall satisfaction of the four criteria can be obtained as:
CðxÞ ¼ 1 0 þ 0 1 þ 0 1 þ 0 1 ¼ 0 Thus, the overall satisfaction of the sales engineer is zero, and the alternative (i.e., sales engineer) does not meet the stated needs of the company.
3. Generalized hesitant fuzzy prioritized aggregation operators In this section, the generalized hesitant fuzzy prioritized weighted average (GHFPWA) operator and the generalized hesitant fuzzy prioritized weighted geometric (GHFPWG) operator are proposed to aggregate the HFEs. Some desirable properties and special cases of these operators are also studied in this section. Definition 5. Let fj(j = 1, 2, . . . , n) be a collection of HFEs, and let GHFPWA: Vn ? V. If
T1 GHFPWA ðf1 ; f2 ; . . . ; fn Þ ¼ Pn
j¼1 T j
T2 f1 Pn
j¼1 T j
That is, Eq. (5) holds when n = 2. Suppose that Eq. (5) also holds when n = k:
( 1
GHFPWA ðf1 ; f2 ; . . . ; fk Þ ¼ [nj 2fj l
j¼1
j¼1 T j
k X
T kþ1 f j Pn fkþ1 j¼1 T j j¼1 ( Xk Tj 1 Pn ¼ [n1 2f1 ;n2 2f2 ;...;nk 2fk l j¼1 ¼
1
(
l
þ l
n X j¼1
where T j ¼ fk.
Qj1
k¼1 Sðfk Þðj
Tj
Pn
j¼1 T j
!) lðnj Þ
ð4Þ
¼ 2; . . . ; nÞ; T 1 ¼ 1 and S(ak) is the score of HFE
1
T kþ1 Pn lðnkþ1 Þ j¼1 T j
lðnj Þ
1
T kþ1 Pn lðnkþ1 Þ j¼1 T j
1
l l
1
k X
1
l l
1
k X j¼1
þ l
T kþ1 Pn lðnkþ1 Þ j¼1 T j
Tj
Pn
j¼1 T j
!! lðnj Þ
!!!) (
1
[nkþ1 2fkþ1
!)
Tj
Pn
j¼1 T j
!! lðnj Þ
!!!) (
¼ [n1 2f1 ;n2 2f2 ;...;nk 2fk ;nkþ1 2fkþ1 l
1
( ¼ [n1 2f1 ;n2 2f2 ;...;nk 2fk ;nkþ1 2fkþ1 l
Proof. The first result follows quickly from Definition 2 and Theorem 1. The equation
ð9Þ
!)
j¼1 T j
¼ [n1 2f1 ;n2 2f2 ;...;nk 2fk ;nkþ1 2fkþ1 l
(
:
j¼1 T j
ð3Þ
Theorem 1. Let fj(j = 1, 2, . . . , n) be a collection of HFEs, then their aggregated value by using the GHFPWA operator is also an HFE, and
lðnj Þ
Tj Pn
¼ [n1 2f1 ;n2 2f2 ;...;nk 2fk ;nkþ1 2fkþ1 l
fn
j¼1 T j
GHFPWA ðf1 ; f2 ; . . . ; fkþ1 Þ
then the function GHFPWA is called a GHFPWA operator, where Q T j ¼ j1 k¼1 Sðfk Þ ðj ¼ 2; . . . ; nÞ, T1 = 1 and S(fk) is the score of HFE fk.
GHFPWA ðf1 ; f2 ; . . . ; fn Þ ¼ [nj 2fj l
Pn
j¼1
Tn Pn
!)
Tj
then, when n = k + 1, the operational laws described in Definition 2 state that
(
f2
k X
1
!) T kþ1 P lðnj Þ þ n lðnkþ1 Þ j¼1 T j j¼1 T j j¼1 !) kþ1 X Tj Pn ; ð10Þ lðnj Þ j¼1 T j j¼1 k X
Tj Pn
i.e. Eq. (5) holds for n = k + 1. Thus, Eq. (5) holds for all n. Then
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D. Yu et al. / Knowledge-Based Systems 52 (2013) 1–10
( GHFPWA ðf1 ; f2 ; . . . ; fn Þ ¼ [nj 2fj l
1
n X j¼1
!)
Tj
Pn
j¼1 T j
lðnj Þ
:
ð11Þ
T1 GHFPWA ðf1 ; f2 ; . . . ; fn Þ ¼ Pn
j¼1 T j
GHFPWA ðf1 ; f2 ; . . . ; fn Þ ¼ f :
ð12Þ
Proof. By Theorem 1,
( GHFPWAðf1 ; f2 ; . . . ; fn Þ ¼ [nj 2fj l
1
j¼1
( ¼ [n2f l
1
n X j¼1
( ¼ [n2f l
n X
1
j¼1 T j
Tj Pn
Pn
j¼1 T j
j¼1 T j Pn lðnÞ j¼1 T j
fn
j¼1 T j
¼
f2
1 0 0 f1 f2 fn ¼ f1 ¼ f0g: 1 1 1
ð20Þ
Corollary 2 indicates that when the smallest HFE is associated with the highest priority criteria, the HFE for other criteria will be ignored. h Theorem 3. Let fj(j = 1, 2, . . . , n) be a collection of HFEs, where Q T j ¼ j1 k¼1 Sðfk Þðj ¼ 2; ; nÞ; T 1 ¼ 1 and S(fk) is the score of HFE fk. If h is an HFE, c are elements of hesitant fuzzy set h, and nj are elements of hesitant fuzzy set fj, then
GHFPWAðf1 h; f2 h; ; fn hÞ
!)
Tj Pn
j¼1 T j
Tn Pn
It must be pointed out that the above proving method is based on Xu [29], Zhao et al. [40]. Now, consider some desirable properties of the GHFPWA operator. h Theorem 2. Let fj(j = 1, 2, . . . , n) be a collection of HFEs, where Q T j ¼ j1 k¼1 Sðfk Þðj ¼ 2; . . . ; nÞ; T 1 ¼ 1, and S(fk) is the score of HFE fk. If for all j, nj = n, where nj are elements of hesitant fuzzy set fj, and n is an element of hesitant fuzzy set f, then
T2 f1 Pn
¼ GHFPWAðf1 ; f2 ; ; fn Þ h:
lðnj Þ !)
ð21Þ
Proof. Since for any j,
lðnÞ 1
fj h ¼ [nj 2fj ;c2h fl ðlðnj Þ þ lðcÞÞg;
!)
1
¼ [n2f fl ðlðnÞÞg ¼ [n2f fng ¼ f ;
then Theorem 1 states that
ð13Þ
which completes the proof of Theorem 2. h Corollary 1. If fj(j = 1, 2, . . . , n) is a collection of the largest HFEs, i.e., fj = f⁄ = {1} for all j, then
GHFPWA ðf1 ; f2 ; . . . ; fn Þ ¼ GHFPWA ðf ; f ; . . . ; f Þ ¼ f1g;
ð22Þ
GHFPWAðf1 h; f2 h; ; fn hÞ ( !!) n X Tj 1 1 Pn ¼ [nj 2fj ;c2h l lðl ðlðnj Þ þ lðcÞÞÞ j¼1 T j j¼1 ( ! !) n X Tj 1 Pn : lðnj Þ þ lðcÞ ¼ [nj 2fj ;c2h l j¼1 T j j¼1 According to Definition 2, (
ð14Þ
GHFPWA ðf1 ;f2 ;...;fn Þ h ¼ [nj 2fj l
which is also the largest HFE.
n X
1
j¼1
(
1
Pn
j¼1 T j
n X
1
j¼1
ð15Þ
Tj Pn
!!!
j¼1 T j
j¼1
(
[c2h fcg
lðnj Þ
n X
l l
¼ [nj 2fj ;c2h l
GHFPWA ðf1 ; f2 ; . . . ; fn Þ ¼ GHFPWA ðf ; f2 ; . . . ; fn Þ ¼ f0g;
!)
Tj
1
¼ [nj 2fj ;c2h l
Proof. The proof for Corollary 1 may be obtained similar to the proof of Theorem 2. h Corollary 2 (Non-compensatory). If f1 is the smallest HFE, i.e., f1 = f⁄ = {0}, then
ð23Þ
!
Tj Pn
j¼1 T j
!) þ lðcÞ
lðnj Þ !)
lðnj Þ þ lðcÞ
ð24Þ
:
Thus,
GHFPWA ðf1 h; f2 h; . . . ; fn hÞ ¼ GHFPWA ðf1 ; f2 ; . . . ; fn Þ h;
which is also the smallest HFE.
ð25Þ
which completes the proof of Theorem 3. h Proof. Since f1 = {0}, then by the Definition 3,
Sðf1 Þ ¼ 0:
ð16Þ
Since j1 Y Tj ¼ Sðfk Þðj ¼ 2; ; nÞ and T 1 ¼ 1;
Theorem 4. Let fj(j = 1, 2, . . . , n) be a collection of HFEs, Q T j ¼ j1 k¼1 Sðfk Þðj ¼ 2; . . . ; nÞ; T 1 ¼ 1, and S(fk) is the score of HFE fk. If r > 0, then
GHFPWA ðrf1 ; rf2 ; . . . ; rfn Þ ¼ r GHFPWAðf1 ; f2 ; . . . ; fn Þ: ð17Þ Proof. According to Definition 2,
k¼1
then
1
kfj ¼ [nj 2fj fl ðklðnj ÞÞg; j1 Y Tj ¼ Sðfk Þ ¼ Sðf1 Þ Sðf2 Þ . . . Sðfj1 Þ
n X T j ¼ 1: j¼1
By Definition 5,
k > 0:
Theorem 1 states that
k¼1
¼ 0 Sðf2 Þ . . . Sðfj1 Þ ¼ 0ðj ¼ 2; ; nÞ and
ð26Þ
ð27Þ ( 1
ð18Þ ð19Þ
GHFPWAðrf1 ; rf2 ; . . . ; rfn Þ ¼ [nj 2fj l
n X j¼1
1
¼l
k
n X j¼1
Tj
Pn
j¼1 T j
Tj
Pn
j¼1 T j
!) 1
lðl ðklðnj ÞÞÞ
lðnj Þ
!! : ð28Þ
5
D. Yu et al. / Knowledge-Based Systems 52 (2013) 1–10
On the other hand,
( r GHFPWA ðf1 ;f2 ;...;fn Þ ¼ [nj 2fj l
1
kl l
1
j¼1 T j
j¼1 T j
j¼1
lðnj Þ
!!)
n X Tj Pn k
1
!!!!)
Tj Pn
j¼1
( ¼ [nj 2fj l
n X
lðnj Þ
: ð29Þ
Thus,
GHFPWA ðrf1 ; rf2 ; . . . ; rfn Þ ¼ r GHFPWA ðf1 ; f2 ; . . . ; fn Þ Theorem 5 follows from Theorems 3 and 4.
ð30Þ
h
GHFPWA ðf1 h1 ; f2 h2 ; . . . ; fn hn Þ ¼ GHFPWAðf1 ; f2 ; . . . ; fn Þ GHFPWAðh1 ; h2 ; . . . ; hn Þ:
It should be noted that Tan [19] studied serials of properties of generalized intuitionistic fuzzy geometric aggregation operator, which are the important reference material for the proving methods of Theorems 2, 3, 4, 6. Klir et al. [7] pointed out that an additive generator should satisfy k: [0, 1] ? [0, 1] such that k(1) = 0. If the additive generator k is assigned with different forms, then some specific hesitant fuzzy aggregation operators can be obtained as follows: Case 1. If k(t) = log (t), then the GHFPWA operator is reduced to the following:
Theorem 5. Let fj(j = 1, 2, . . . , n) be a collection of HFEs, Q T j ¼ j1 k¼1 Sðfk Þðj ¼ 2; . . . ; nÞ; T 1 ¼ 1, and S(fk) is the score of HFE fk. If r > 0 and h is an hesitant fuzzy element, then
T1 GHFPWAðf1 ; f2 ; . . . ; fn Þ ¼ Pn
j¼1 T j
¼ r GHFPWA ðf1 ; f2 ; . . . ; fn Þ h:
ð31Þ
Theorem 6. Let fj(j = 1, 2, . . . , n) and hj(j = 1, 2, . . . , n) be two collecQ tions of HFEs, T j ¼ j1 k¼1 Sðfk Þðj ¼ 2; . . . ; nÞ; T 1 ¼ 1, and S(fk) is the score of HFE fk, then
GHFPWA ðf1 h1 ; f2 h2 ; . . . ; fn hn Þ ¼ GHFPWA ðf1 ; f2 ; . . . ; fn Þ GHFPWA ðh1 ; h2 ; . . . ; hn Þ:
ð32Þ
1
ð33Þ
GHFPWA ðf1 h1 ; f2 h2 ; . . . ; fn hn Þ ( !!) n X Tj 1 1 Pn ¼ [nj 2fj ;cj 2hj l lðl ðlðnj Þ þ lðcj ÞÞÞ j¼1 T j j¼1 ( ! !) n X Tj Tj 1 Pn lðnj Þ þ Pn lðcj Þ ¼ [nj 2fj ;cj 2hj l : j¼1 T j j¼1 T j j¼1
n X
1
j¼1
( GHFPWA ðh1 ; h2 ; . . . ; hn Þ ¼ [cj 2hj l
n X
1
j¼1
j¼1 T j
ð34Þ
ð35Þ
!)
Tj
Pn
j¼1 T j
lðcj Þ
;
ð36Þ
l l
1
n X j¼1
( ¼ [nj 2fj ;cj 2hj l
1
n X j¼1
Tj Pn
j¼1 T j
Tj Pn
j¼1 T j
!!! þl l
lðnj Þ
1
n X j¼1
! lðnj Þ þ
n X j¼1
Tj Pn
j¼1 T j
Tj Pn
j¼1 T j
!!!!)
Tj Pn
!
j¼1 T j
n X
logð1 nj Þ
lðnj Þ ¼ log
n Y
ð40Þ T
Pnj T ð1 nj Þ j¼1 j
ð41Þ
j¼1
Tj
Pn
j¼1 T j
T
!! lðnj Þ
¼ 1e
log
Qn j¼1
Pnj ð1nj Þ
T j¼1 j
Tj
¼1
Pn n Y T ð1 nj Þ j¼1 j ; ð42Þ j¼1
Furthermore, when the priority level of the aggregated arguments is reduced to the same level, then Eq. (39) is rewritten as
n o GHFPWAðf1 ; f2 ; . . . ; fn Þ ¼ [nj 2fj 1 Pnj¼1 ð1 nj Þwj ;
ð43Þ
which is the hesitant fuzzy weighted average operator proposed by Xia and Xu [25,26].
9 > > > = j¼1 ð1 nj Þ j¼1 ð1 þ nj Þ GHFPWAðf1 ;f2 ;...;fn Þ ¼ [nj 2fj ; Tj Tj > Pn Pn > > > > ; :Qn ð1 þ n Þ j¼1 T j þ Qn ð1 n Þ j¼1 T j > j j j¼1 j¼1 T
Pnj
T j¼1 j
T
Qn
Pnj
T j¼1 j
which is the hesitant fuzzy Einstein prioritized weighted average operator.
: ð37Þ
Thus,
ð39Þ
ð44Þ
lðcj Þ
!!) lðcj Þ
j¼1 T j
8 > > > 0 and h is an hesitant fuzzy element, then,
j¼1 ð1
nj Þ
j¼1 ð1
nj Þ
¼ GHFPWGðf1 ; f2 ; . . . ; fn Þ GHFPWGðh1 ; h2 ; . . . ; hn Þ:
P
Pnj j¼1
ð47Þ
: Tj
Case 1. If k(t) = log (t), then the GHFPWG operator is reduced to the following:
PTn1
GHFPWGðf1 ; f2 ; . . . ; fn Þ ¼ f1
PTn2 f2
T j¼1 j
PTnn Tj fn j¼1 ;
ð48Þ
then the function GHFPWG is called a GHFPWG operator, where Q T j ¼ j1 k¼1 Sðfk Þðj ¼ 2; . . . ; nÞ; T 1 ¼ 1, and S(fk) is the score of HFE fk. Theorem 7. Let fj(j = 1, 2, . . . , n) be a collection of HFEs, then their aggregated value obtained with the HFPWG operator is also an HFE, and
( GHFPWG ðf1 ; f2 ; ; fn Þ ¼ [nj 2fj k
1
n X j¼1
where T j ¼ fk.
Qj1
k¼1 Sðfk Þðj
Tj
Pn
j¼1 T j
!) kðnj Þ
;
ð49Þ
f2
T j¼1 j
Proof. Since k(t) = log (t), then l(t) = log (1 t), k1(t) = et, l1(t) = 1 et, and
Tj Pn
j¼1 T j
)
Tj kðnj Þ ¼ Pn
n X
j¼1 T j
!
Tj Pn
j¼1 T j
kðnj Þ
logðnj Þ
ð55Þ
n X Tj Pn ¼
j¼1 T j
j¼1
0 1 T Pnj n Tj BY C ¼ log @ nj j¼1 A
! logðnj Þ
ð56Þ
j¼1
)k
1
n X j¼1
Theorem 8. Let fj(j = 1, 2, . . . , n) be a collection of HFEs, where Q T j ¼ j1 k¼1 Sðfk Þðj ¼ 2; . . . ; nÞ; T 1 ¼ 1, and S(fk) is the score of HFE fk. If for all j, nj = n, where nj are elements of hesitant fuzzy set fj, and n is the element of hesitant fuzzy set f, then
ð50Þ
0 log @
¼e
Tj Pn
j¼1 T j
Tj n
n P Y nj
j¼1
T j¼1 j
!! kðnj Þ
0
0 11 T Pnj n Y Tj B CC 1 B ¼ k @ log @ nj j¼1 AA j¼1
1 A
T
j n Pn T Y j¼1 j nj ; ¼
Theorem 9. Let fj(j = 1, 2, . . . , n) be a collection of HFEs, where Q T j ¼ j1 k¼1 Sðfk Þðj ¼ 2; . . . ; nÞ, T1 = 1, and S(fk) is the score of HFE fk. If h is an HFE, c are elements of hesitant fuzzy set h, and nj are elements of hesitant fuzzy set fj, then
ð57Þ
j¼1
which completes the proof of the Case 1.
h
Furthermore, when the priority level of the aggregated arguments is reduced to the same level, the Eq. (39) is transformed to
( ) n Y wj GHFPWGðf1 ; f2 ; . . . ; fn Þ ¼ [nj 2fj nj ;
ð58Þ
j¼1
GHFPWG ðf1 h; f2 h; ; fn hÞ ¼ GHFPWG ðf1 ; f2 ; . . . ; fn Þ h:
PTnn
T j¼1 j
which is the hesitant fuzzy prioritized weighted geometric (HFPWG) operator studied by Wei [23], Yu [36] in detail.
Proof. Theorem 7 can be proven similar to Theorem 1. h
GHFPWG ðf1 ; f2 ; . . . ; fn Þ ¼ f :
PTn2
fn 8 9 Tj P > > n n > : j¼1 ;
j¼1
¼ 2; . . . ; nÞ; T 1 ¼ 1, and S(fk) is the score of HFE
T j¼1 j
GHFPWG ðf1 ; f2 ; . . . ; fn Þ ¼ f1
Definition 6. Let fj(j = 1, 2, . . . , n) be a collection of HFEs and GHFPWG: Vn ? V. If T j¼1 j
ð54Þ
T j¼1 j
Here we define the generalized hesitant fuzzy prioritized weighted geometric (GHFPWG) operator based on the GHFPWA operator and the geometric mean. h
PTn1
ð53Þ
GHFPWGðf1 h1 ; f2 h2 ; . . . ; fn hn Þ
Tj n
T
Tj
ð52Þ
Theorem 12. Let fj(j = 1, 2, . . . , n) and hj(j = 1, 2, . . . , n) be two colQ lections of HFEs, T j ¼ j1 k¼1 Sðfk Þðj ¼ 2; . . . ; nÞ; T 1 ¼ 1, and S(fk) is the score of HFV fk, then
1
T
j Qn 1þnj Pn
T
Pnj
T j¼1 j
j¼1 1nj
þ1 Tj n
r
¼ r GHFPWGðf1 ; f2 ; . . . ; fn Þ h:
T
Pj Qn 1þnj nj¼1 T j
r
GHFPWGðrf1 h; rf2 h; . . . ; rfn hÞ
lðnj Þ
log
r
GHFPWGððf1 Þ ; ðf2 Þ ; . . . ; ðfn Þ Þ ¼ ðGHFPWAðf1 ; f2 ; . . . ; fn ÞÞ :
Tj
¼ log
Theorem 10. Let fj(j = 1, 2, . . . , n) be a collection of HFEs, Q T j ¼ j1 k¼1 Sðfk Þðj ¼ 2; . . . ; nÞ; T 1 ¼ 1, and S(fk) is the score of HFEfk. If r > 0, then
ð51Þ
which was the hesitant fuzzy weighted geometric operator proposed by Xia and Xu [25,26].
7
D. Yu et al. / Knowledge-Based Systems 52 (2013) 1–10
2t
Case 2. If kðtÞ ¼ log t , then the GHFPWG operator is reduced to the following:
GHFPWGðf1 ; f2 ; ; fn Þ ¼ [nj 2fj
8 > > > >
> > > :Qn
T
i¼1 ð2
nj Þ
Pnj j¼1
Tj
þ
Qn
(
n X Tj Pn GHFPWA f1c ; f2c ; . . . ; fnc ¼ [nj 2fj 1 k
9 > > > > = ; Tj Pn > > T > > j¼1 j ; n
j¼1
0
!) 1
j¼1 T j
k ðnj Þ
1c Tj n Pn T Y j B C ¼ @ fj j¼1 A j¼1
c
¼ ðGHFPWGðf1 ; f2 ; . . . ; fn ÞÞ :
i¼1 j
ð59Þ
ð65Þ
which we call the hesitant fuzzy Einstein prioritized weighted geometric operator. 1þt 1 2 Proof. Since kðtÞ ¼ log 2t t , then lðtÞ ¼ log 1t ; k ðtÞ ¼ et þ1, and 1 et 1 l ðtÞ ¼ et þ1, then
(2) According to Definition 2 and the definition of a GHFPWG operator,
(
PTnj 2 nj 2 nj T j¼1 j ¼ log n n T T j j j j j¼1 j¼1 0 1 ! PTnj n n X X 2 n Tj Tj j @log j¼1 A Pn kðnj Þ ¼ ) nj j¼1 T j j¼1 j¼1
Tj Pn
Tj kðnj Þ ¼ Pn
log
Tj Yn 2 nj Pn T j¼1 j ¼ log j¼1 nj !! n X Tj 1 Pn )k kðnj Þ ¼ j¼1 T j j¼1
ð60Þ
j¼1
nj
j¼1
þ1
i¼1 ð2
nj Þ
which completes the proof of Case 2.
;
T
Pnj j¼1
Tj
þ
Qn
Pnj
i¼1 nj
j¼1
h
2
GHFPWG ðf1 ; f2 ; . . . ; fn Þ ¼ [nj 2fj Qn
i¼1 ð2
) wj i¼1 nj Q w nj Þwj þ ni¼1 nj j
Qn
ð63Þ
which is referred to as the hesitant fuzzy Einstein weighted geometric operator. In the following, we study the relationship between GHFPWA operator and GHFPWG operator. Theorem 13. Let fj(j = 1, 2, . . . , n) be a collection of HFEs, Q T j ¼ j1 Sðfk Þðj ¼ 2; . . . ; nÞ; T 1 ¼ 1 and S(fk) is the score of HFE fk, k¼1 then c (1) GHFPWA f1c ; f2c ; . . . ; fnc ¼ ðGHFPWGðf1 ; f2 ; . . . ; fn ÞÞ , and c c c (2) GHFPWG f1 ; f2 ; . . . ; fnc ¼ ðGHFPWAðf1 ; f2 ; . . . ; fn ÞÞ .
j¼1 T j
!c fj
c
¼ ðGHFPWAðf1 ; f2 ; . . . ; fn ÞÞ ;
ð67Þ
This section introduces the methods to rank alternatives based on hesitant fuzzy information. In many group decision problems a set of alternatives must be evaluated on the basis of criteria with prioritized relationships. Consider a group decision making problem [5,28,31,16,22, 18,39,9,10,12,14,17,30,32,41]. Let X = {x1, x2, . . . , xm} be the set of alternatives, C = {C1, C2, . . . , Cn} be the set of criteria, and E = {e1, e2, . . . , ep} be the set of decision makers. It is assumed that there is a prioritization between the criteria expressed by the linear ordering C1 C2 C3 Cn indicating criterion Cj has a higher priority than Ci if j < i. The decision maker ek evaluates the alternative xi under criterion Cj anonymously so as to protect his/her privacy or avoid psychic contagion. The evaluation of alternative xi under criteria Cj is provided by each of the decision makers ek (k = 1, 2, . . . , p) using several values. If two decision makers provide the same value, then that value will emerge only once, and the evaluation can be represented by HFEs. The hesitant fuzzy group decision matrix F = (fij)mn is constructed from these HFEs. Here, fij is the attribute value provided by the decision makers ek(k = 1, 2, . . . , p), which is expressed in a HFE. Based on the above analysis, the main steps of the multi-criteria group decision making method are as follows: Step 1. Calculate the values of Tij (i = 1, 2, . . . , m, j = 1, 2, . . . , n) based on the following equations.
Proof. (1) According to Definition 2 and the Definition of a GHFPWA operator,
( GHFPWAðf1 ; f2 ; . . . ; fn Þ ¼ [nj 2fj l
1
n X j¼1
and
Tj Pn
!) 1
l ðnj Þ
ð62Þ
Tj
Furthermore, when the priority level of the aggregated arguments is reduced to the same level, then the Eq. (59) is transformed to
(
n X
j¼1 T j
; and ð66Þ
4. Personnel evaluation based on prioritized operators under hesitant fuzzy environment
T
Qn
kðnj Þ
þ1
Pnj Tj Qn 2 i¼1 nj j¼1
¼ Tj
j¼1 T j
which completes the proof of Theorem 13. h
T
T
Qn 2nj
¼
j¼1
nj
Pn
j¼1
2 j¼1
!)
Tj
( n X Tj Pn GHFPWG f1c ; f2c ; . . . ; fnc ¼ [nj 2fj 1 l
Pj Qn 2nj nj¼1 T j
e
¼
GHFPWGðf1 ; f2 ; . . . ; fn Þ ¼ [nj 2fj k
n X j¼1
ð61Þ
log
2 T Pnj
1
Tj Pn
j¼1 T j
!) lðnj Þ
T ij ¼
j1 Y
Sðr ik Þ ði ¼ 1; 2; . . . ; m; j ¼ 2; . . . ; nÞ
ð68Þ
k¼1
ð64Þ
T i1 ¼ 1 i ¼ 1; 2; . . . ; m
ð69Þ
Step 2. Aggregate the hesitant fuzzy values fij for each alternative xi by the GHFPWA (or GHFPWG) operator.
8
D. Yu et al. / Knowledge-Based Systems 52 (2013) 1–10
( GHFPWAðf1 ; f2 ; . . . ; fn Þ ¼ [nj 2fj l
n X
1
j¼1
!)
Tj
Pn
j¼1 T j
i
lðnj Þ
¼ 1; 2; . . . ; m ð70Þ or
(
n X
1
GHFPWGðf1 ; f2 ; . . . ; fn Þ ¼ [nj 2fj k
j¼1
Tj Pn
j¼1 T j
!)
Table 2 Hesitant fuzzy group decision matrix.
x1 x2 x3 x4 x5
ð71Þ
i
C3
C4
{0.6, 0.7} {0.2, 0.4, 0.7} {0.3, 0.7, 0.8} {0.3, 0.5, 0.6} {0.2, 0.3, 0.5}
{0.2, 0.4, 0.5, 0.6} {0.6, 0.8} {0.3, 0.6, 0.7} {0.2, 0.4} {0.3, 0.5}
{0.5, 0.6, 0.9} {0.6, 0.7, 0.9} {0.7} {0.5, 0.6, 0.7} {0.4}
{0.5, 0.7} {0.3, 0.4, 0.5, 0.8} {0.5, 0.6, 0.8, 0.9} {0.8, 0.9} {0.2, 0.6, 0.7}
Step 3. Calculate the scores of fi(i = 1, 2, 3, 4, 5).
Step 3. Rank all the alternatives by the score function defined by Definition 3.
1 X n; #fi n 2f i
C2
kðnj Þ
i ¼ 1; 2; . . . ; m
Sðfi Þ ¼
C1
i ¼ 1; 2; . . . ; m
S1 ¼ 0:6040;
S2 ¼ 0:5923;
S3 ¼ 0:6360;
S4 ¼ 0:4797;
S5 ¼ 0:3671
ð72Þ Since
i
S3 > S1 > S2 > S4 > S5 ; Using the above process, the bigger the value of S(fi), the larger is the overall HFE fi. So is the alternative xi (i = 1, 2, . . . , m).
the following results are obtained
x3 x1 x2 x4 x5 Example 2 (Paragraph 2 of Section 1). Section 1 introduced a personnel evaluation problem that included a prioritization relationship between multiple criteria. To draw a scientific conclusion that is free of psychic contagion, this evaluation will be made anonymously. If two decision makers provide the same value, then the value will emerge only once, and the evaluation can be represented by HFEs. The hesitant fuzzy group decision matrix F = (fij)mn is constructed as follows (Table 2). Approach 1: The ranking of alternatives can be obtained as follows using the GHFPWA operator and letting k(t) = log (t): Step 1. Calculate the values of Tij(i = 1, 2, . . . , 5, j = 1, 2, . . . , 4) based on Eqs. (68) and (69).
0
1:0000 0:6500 B B 1:0000 0:4333 B T ij ¼ B B 1:0000 0:6000 B @ 1:0000 0:4667
0:2763 0:1842
1
C 0:3033 0:2224 C C 0:3200 0:2240 C C C 0:1400 0:0840 A 1:0000 0:3333 0:1333 0:0533
( 1
n X j¼1
Tj Pn
j¼1 T j
!) lðnj Þ
Step 10 . Same as the above Step 1. Step 20 . Since kðtÞ ¼ log 2t , then t
( GHFPWAðf1 ; f2 ; . . . ; fn Þ ¼ [nj 2fj l
1
n X j¼1
Tj Pn
!)
j¼1 T j
lðnj Þ
;
ð75Þ
which is reduced to the
GHFPWAðf1 ; f2 ; . . . ; fn Þ 8 9 T T Pnj Pnj > > > > Q Q > T T < n ð1 þ n Þ j¼1 j n ð1 n Þ j¼1 j > = j j j¼1 j¼1 : ¼ [nj 2fj Tj Tj > Pn Pn > > > > > Q Q T T : n ð1 þ n Þ j¼1 j þ n ð1 n Þ j¼1 j ; j j j¼1 j¼1
ð76Þ
The overall preference values fi are obtained using the GHFPWA operator in Eq. (76) to aggregate all of the preference values fij (i = 1, 2, 3, 4, 5) in the ith line of F. Due to the space limit, f1 is provided as a representative example.
Step 2. Since k(t) = log (t), then
GHFPWAðf1 ; f2 ; . . . ; fn Þ ¼ [nj 2fj l
Clearly, the best option is alternative x3. Approach 2: The ranking of alternatives can be obtained as fol lows using the GHFPWA operator and letting kðtÞ ¼ log 2t : t
f1 ¼ f0:4341;0:4506;0:4455;0:4616;0:5064;0:5208;0:4748;0:4900; ð73Þ
0:4854;0:5003;0:5419;0:5552;0:4976;0:5122;0:5078;0:5221; 0:5618;0:5745;0:5234;0:5373;0:5330;0:5466;0:5843;0:5964; 0:4939;0:5086;0:5041;0:5185;0:5586;0:5714;0:5303;0:5439;
which is reduced to the
8
S1 > S2 > S4 > S5 ;
0:5463;0:5661;0:5594;0:5786;0:6325;0:6485;0:5848;0:6029; 0:5968;0:6144;0:6637;0:6784;0:6075;0:6246;0:6188;0:6354; 0:6820;0:6959;0:6336;0:6495;0:6441;0:6596;0:7032;0:7161g
S3 ¼ 0:5666;
¼ 0:3391
The final ranking can be obtained as
x3 x1 x2 x4 x5 :
S4 ¼ 0:4315;
S5
9
D. Yu et al. / Knowledge-Based Systems 52 (2013) 1–10
The best option with this approach is also alternative x3. Approach 3: The ranking of alternatives can be obtained as follows, by using the GHFPWG operator and letting k(t) = log (t):
GHFPWGðf1 ; f2 ; . . . ; fn Þ ¼ [nj 2fj k
1
n X j¼1
!)
Tj
Pn
j¼1 T j
kðnj Þ
ð77Þ Step 3000 . Calculate the scores of fi (i = 1, 2, 3, 4, 5).
8 9 T Pnj > > n > : j¼1 ;
S1 ¼ 0:6015; ð78Þ
f10 ¼ f0:4111;0:4233;0:4210;0:4336;0:4440;0:4572;0:5089;0:5241; 0:5212;0:5367;0:5496;0:5660;0:5451;0:5614;0:5583;0:5749; 0:5887;0:6063;0:5766;0:5938;0:5905;0:6081;0:6227;0:6413; 0:4422;0:4554;0:4529;0:4664;0:4776;0:4918;0:5475;0:5638; 0:5607;0:5774;0:5913;0:6089;0:5864;0:6039;0:6006;0:6185; 0:6333;0:6522;0:6203;0:6388;0:6353;0:6542;0:6699;0:6899g
Step 300 . Calculate the scores of fi (i = 1, 2, 3, 4, 5) respectively.
S3 ¼ 0:5834;
S4 ¼ 0:4251;
S5
¼ 0:3485 Since
S3 > S1 > S2 > S4 > S5 ; the final ranking is obtained as
x3 x1 x2 x4 x5 ; Which again indicates the best option is alternative x3. Approach 4: The ranking of alternatives can be obtained as fol lows using the GHFPWG operator and letting kðtÞ ¼ log 2t : t
GHFPWGðf1 ; f2 ; . . . ; fn Þ ¼ [nj 2fj k
1
n X j¼1
Tj
GHFPWGðf1 ; f2 ; . . . ; fn Þ ¼ [nj 2fj
S4 ¼ 0:4763;
S5
¼ 0:3938
S3 > S1 > S2 > S4 > S5 ; We can easily get
x3 x1 x2 x4 x5 : The best option is therefore alternative x3. The above examples show that the same conclusion can be obtained with all four different approaches. This consistency demonstrates the stability of the proposed method. 5. Conclusions In this age of increasingly competitive markets, talent has become essential for business success. Talent can be a critical factor in a company’s ability to achieve operational excellence. To address the disadvantages of traditional personnel evaluation methods, this paper proposes the use of a hesitant fuzzy group decision making method. The method is based on the GHFPWA and GHFPWG operators. The use of hesitant fuzzy sets is a powerful technique that captures the decision makers’ preferences. The ability to prioritize criteria not only addresses the actual needs of companies from a human resources perspective, but also opens the approach up to a wide range of applications. A realistic example is used to illustrate the effectiveness of the proposed method for personnel evaluation. It is worth noting that in addition to human resource issues, the proposed method is equally applicable to factory location, supplier selection and many other management decision problems. Acknowledgement
Pn
j¼1 T j
!) kðnj Þ
ð79Þ
;
which is reduced to
8 > > > >
> Pnj > > :Qn ð2 n Þ j¼1 T j þ Qn j i¼1 i¼1
9 > > > > = : Tj Pn > > Tj > > n j¼1 ; j
ð80Þ The overall preference values fi are obtained using the GHFPWG operator in Eq. (80) to aggregate the preference values fij (i = 1, 2, 3, 4, 5) in the ith line of F. Due to the space limit, f1 is provided as a representative example.
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[24] M.M. Xia, Research on Fuzzy Decision Information Aggregation Techniques and Measures, Doctoral Dissertation, Southeast University, 2012. [25] M.M. Xia, Z.S. Xu, Hesitant fuzzy information aggregation in decision making, International Journal of Approximate Reasoning 52 (2011) 395–407. [26] M.M. Xia, Z.S. Xu, On distance and correlation measures of hesitant fuzzy information, International Journal of Intelligent Systems 26 (2011) 410–425. [27] M.M. Xia, Z.S. Xu, N. Chen, Some hesitant fuzzy aggregation operators with their application in group decision making, Group Decision and Negotiation 22 (2013) 259–279. [28] Z.S. Xu, Approaches to multi-stage multi-attribute group decision making, International Journal of Information Technology and Decision Making 10 (2011) 121–146. [29] Z.S. Xu, Intuitionistic fuzzy aggregation operators, IEEE Transactions on Fuzzy Systems 15 (2007) 1179–1187. [30] Z.S. Xu, X.Q. Cai, Minimizing group discordance optimization model for deriving expert weights, Group Decision and Negotiation 21 (2012) 863–875. [31] Z.S. Xu, M.M. Xia, Distance and similarity measures for hesitant fuzzy sets, Information Sciences 181 (2011) 2128–2138. [32] Z.S. Xu, M.M. Xia, Hesitant fuzzy entropy and cross-entropy and their use in multi-attribute decision making, International Journal of Intelligent Systems 27 (2012) 799–822. [33] R.R. Yager, Prioritized aggregation operators, International Journal of Approximate Reasoning 48 (2008) 263–274. [34] R.R. Yager, Prioritized OWA aggregation, Fuzzy Optimization Decision Making 8 (2009) 245–262. [35] R.R. Yager, G. Giray, Z. Marek, Using a web personnel evaluation Tool-PET for lexicographic multi-criteria service selection, Knowledge-Based Systems 24 (2011) 929–942. [36] J.D. Yu, Hesitant fuzzy prioritized operators and their application in multicriteria group decision making, African Journal of Business and Management 6 (2012) 9370–9380. [37] S.F. Zhang, S.Y. Liu, A GRA-based intuitionistic fuzzy multi-criteria group decision making method for personnel selection, Expert Systems with Applications 38 (2011) 11401–11405. [38] B. Zhu, Z.S. Xu, M.M. Xia, Hesitant fuzzy geometric Bonferroni means, Information Sciences 205 (2012) 72–85. [39] G. Zhang, J. Lu, An integrated group decision-making method dealing with fuzzy preferences for alternatives and individual judgments for selection criteria, Group Decision and Negotiation 12 (2003) 501–515. [40] H. Zhao, Z.S. Xu, M.F. Ni, S.S. Liu, Generalized aggregation operators for intuitionistic fuzzy sets, International Journal of Intelligent Systems 25 (2010) 1–30. [41] L.G. Zhou, H.Y. Chen, A generalization of the power aggregation operators for linguistic environment and its application in group decision making, Knowledge-Based Systems 26 (2012) 216–224.
Contents lists available at ScienceDirect
Knowledge-Based Systems journal homepage: www.elsevier.com/locate/knosys
Group decision making under hesitant fuzzy environment with application to personnel evaluation Dejian Yu a, Wenyu Zhang a,⇑, Yejun Xu b a b
School of Information, Zhejiang University of Finance & Economics, 18 Xueyuan Street, Hangzhou 310018, China Research Institute of Management Science, Business School, Hohai University, Nanjing 211100, China
a r t i c l e
i n f o
Article history: Received 15 October 2012 Received in revised form 13 April 2013 Accepted 15 April 2013 Available online 3 May 2013 Keywords: Hesitant fuzzy set Group decision making Generalized hesitant fuzzy aggregation operator Prioritized aggregation operator
a b s t r a c t In many personnel evaluation scenarios, decision makers are asked to provide their preferences anonymously to both ensure privacy and avoid psychic contagion. The use of hesitant fuzzy sets is a powerful technique for representing this type of information and has been well studied. This paper explores aggregation methods for prioritized hesitant fuzzy elements and their application on personnel evaluation. First, the generalized hesitant fuzzy prioritized weighted average (GHFPWA) and generalized hesitant fuzzy prioritized weighted geometric (GHFPWG) operators are presented. Some desirable properties of the methods are discussed and special cases are investigated in detail. Previous research has indicated that many existing hesitant fuzzy aggregation operators are special cases of the proposed operators. Then, a procedure and algorithm for group decision making is provided using these proposed generalized hesitant fuzzy aggregation operators. Finally, the group decision making method is applied to a representative personnel evaluation problem that involves a prioritization relationship over the evaluation index. Published by Elsevier B.V.
1. Introduction Probation is an important component of the evaluation process. It allows an employee to demonstrate his/her suitability for a position. Probation gives the company an opportunity to observe an employee’s work ethic and training status, assists the employee in adjusting to a new position, and removes an employee whose performance fails to meet expectations. To expand a company’s potential market and maintain growth, it is important to provide sales engineers with professional and systematic trainings. Training is typically provided using a full scale development platform that allows engineers to learn a broad range of topics, from product knowledge to business skills, case studies to simulation, and classroom training to field practice. Effective training methods leverage the participants’ full potential, provide both opportunities and challenges, develop creativity, identify strengths and assist in career planning. After training, employees may receive systematic assessment and evaluation over the course of a probationary period. Let us consider an interesting but realistic problem. Currently, five sales engineers’ probation end dates are approaching, and we need to evaluate their performances to determine if their contracts should be renewed. Since sales engineers are responsible for establishing and maintaining productive working relationships with ⇑ Corresponding author. E-mail addresses: [email protected] (D. Yu), [email protected] (W. Zhang), [email protected] (Y. Xu). 0950-7051/$ - see front matter Published by Elsevier B.V. http://dx.doi.org/10.1016/j.knosys.2013.04.010
target customers in critical business areas, the company is very interested in evaluating their performance. A sales manager (e1), manufacturing manager (e2), engineering manager (e3) and human resource manager (e4) compose the panel of decision makers that will take responsibility for this evaluation. A strict evaluation of each of the five employees xi(i = 1, 2, . . . , 5) is performed from four aspects: work attitude C1, communication skill C2, problem solving skill C3, and learning skill C4. First, work attitude (C1) is considered a must for all professionals. This position also requires the sales engineer to work closely with internal colleagues and customers to develop effective sales proposals, thus communication skills (C2) represent another important competency area. In addition, sales engineers must be able to pair prospective customers with ongoing projects and follow through with appropriate sales activities to get the orders. This means that sales engineers must also possess solid problem solving skills (C3). Finally, learning skills (C4) help us to grow and adapt so that we may achieve more challenging targets. The prioritization of the criteria can be expressed as C1 C2 C3 C4, where ‘‘’’ indicates ‘‘priority to’’. It should be noted that there are two key issues inherent to the above problems. The first key issue is anonymously depicting the decision makers’ preferences. The second key issue is mathematically expressing the criteria prioritization relationships. The focus of this paper is to develop a new decision making method that addresses both of the above problems. To address these issues, the remainder of this paper is organized as follows. In Section 2, a literature review of decision making
2
D. Yu et al. / Knowledge-Based Systems 52 (2013) 1–10
methods for personnel evaluation is provided, and the concepts of hesitant fuzzy set and prioritized operators are introduced. Section 3 proposes the generalized hesitant fuzzy prioritized weighted average (GHFPWA) and generalized hesitant fuzzy prioritized weighted geometric (GHFPWG) operators to aggregate the hesitant fuzzy elements (HFEs), whose desirable properties and special cases are also studied in this section. Section 4 develops a multicriteria group decision making method based on the proposed operators under the hesitant fuzzy environment, and applies this decision making method to the illustrative evaluation problem regarding sales engineers presented in this section. Section 5 provides some concluding remarks. 2. Literature and basic concept review Personnel evaluation has been widely studied by a large number of research institutions, and is an important aspect of human resources management. In highly competitive markets, a company’s personnel play a crucial role in the future development of the company. In other words, talent can be a company’s biggest asset. Therefore, staffing is important for any company’s business development. Putting the right people in the right positions can produce many positive impacts on a company, such as lowering employee turnover rate, improving enthusiasm and productivity, and increasing customer satisfaction. Alternatively, waste and other negative impacts can result from poor hiring decisions. Multi-criteria decision making (MCDM) has been identified as an essential technique in the personnel selection process [3,15,11]. Karsak [6] and Gil-Aluja [4] proposed an MCDM algorithm based on the concepts of ideal and anti-ideal solutions for personnel selection. Li [8] combined analytic network process (ANP) with fuzzy data envelopment analysis (DEA) and proposed an integrated method to deal with the personnel selection problem. An example is provided with an electric and machinery company in Taiwan. Chen and Cheng [2] proposed a fuzzy group decision support system based on metric distances to solve the personnel selection problem. Dursun and Karsak [3] developed a MCDM method based on the principles of fusion of fuzzy information and 2-tuple linguistic information. Zhang and Liu [37] used grey relational analysis (GRA) to solve the personnel selection problem under an intuitionistic fuzzy environment. Boran et al. [1] used the Technique for Order Preference by Similarity to Ideal Solution (TOPSIS) to solve the personnel selection problem under an intuitionistic fuzzy environment and illustrated the method with an example involving a sales manager position in a manufacturing company. Merigó and Gil-Lafuente [12,13] studied the human resource selection problem using the ordered weighted average (OWA) operator. The key differences between the above studies are illustrated in Table 1. Table 1 Comparison between existing research results. Author(s)
Published year
Key technology
Karsak, EE
2001
Chen LS, Cheng CH
2005
Lin HT
2010
Dursun M, Karsak EE. Zhang SF, Liu SY
2010 2011
Boran FE, Genc S, Akay D Merigó JM, GilLafuente AM
2011
Fuzzy MCDM, Ideal and Anti-Ideal Solutions Fuzzy Multi-criteria Group Decision Support System Analytic network process (ANP), Fuzzy DEA 2-Tuple linguistic, TOPSIS Grey Relation Analysis (GRA),Intuitionistic fuzzy set TOPSIS, Intuitionistic fuzzy set
2011
OWA operator, Hamming distance; Adequacy coefficient
The above personnel selection methods operate under the assumption that the evaluation criteria are independent and have similar priority levels. This assumption is not always practical, however. Take the problem described in Section 1 as an example. If an alternative (i.e., a sales engineer) has a poor work attitude C1, then the company will likely terminate employment regardless of the employee’s communication skills C2, problem solving skills C3 or learning skills C4. In other words, work attitude C1 has a higher priority than the other criteria. The prioritized average (PA) operator is a useful tool to deal with the above situation. To facilitate later study, the PA operator and hesitant fuzzy sets are introduced here. A hesitant fuzzy set (HFS) is a generalized fuzzy set that is characterized by the membership degree of an element to a set presented as several possible values between 0 and 1. Torra [20,21] first proposed the concept of HFS, which has attracted significant attention from other researchers [23,36,38,27]. Definition 1 ([20,21]). Let X be a fixed set. A HFS on X is defined in terms of a function that when applied to members ofX returns a subset with values in the range of [0, 1], which can be represented as the following mathematical symbol:
E ¼ f< x; f E ðxÞ > jx 2 Xg
ð1Þ
where fE(x) is a set of some values in the range of [0, 1], that denotes the possible membership degrees of the element x 2 X to the set E. For convenience, Xia and Xu [25,26] call fE(x) a hesitant fuzzy element (HFE). Take the problem illustrated in Section 1 as an example. A sales manager (e1), manufacturing manager (e2), engineering manager (e3) and human resource manager (e4) are each asked to estimate the degree to which an alternative (i.e., a sales engineer) presents an excellent work attitude. The sales manager gives the sales engineer a score of 0.6, the manufacturing manager rates the engineer with a 0.7, the engineering manager gives a 0.8 and the human resource manager (e4) determines the score to be 0.9. Thus, the degree to which the alternative presents an excellent work attitude can be represented by a hesitant fuzzy element {0.6, 0.7, 0.8, 0.9}. The HFS is utilized in this work because of its ability to efficiently express the uncertainty presented by this type of realistic situation. HFS therefore can address the first key issue mentioned in Section 1, i.e., how to anonymously describe the decision makers’ preferences. For three HFEs f, f1 and f2, Xia [24] gave the following operational laws for the HFEs as follows: Definition 2. Let f, f1 and f2 be three HFEs, then (1) (2) (3) (4)
kf = [ n2f{l1(kl(n))}, k > 0, fk = [ n2f{k1(kk(n))}, k > 0, 1 f1 f2 ¼ [n1 2f1 ;n2 2f2 fk ðkðn1 Þ þ kðn2 ÞÞg, and 1 f1 f2 ¼ [n1 2f1 ;n2 2f2 fl ðlðn1 Þ þ lðn2 ÞÞg,
where l(t) = k(1 t). Klir et al. [7] pointed out that an additive generator of a continuous Archimedean t-norm is a strictly decreasing function, i.e., k: [0, 1] ? [0, 1], such that k(1) = 0. P 1 Definition 3 [24]. For a HFE f ; Sðf Þ ¼ #f n2f n is called the score function of f, where #f is the number of the elements in f. For two HFEs f1 and f2, if S(f1) > S(f2), then f1 > f2; if S(f1) = S(f2), then f1 = f2. Since n 2 [0, 1], then S(f) 2 [0, 1]. Yager [33] first proposed the prioritized average (PA) operator, which was defined as follows.
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D. Yu et al. / Knowledge-Based Systems 52 (2013) 1–10
Definition 4. [33–35] Let C = {C1, C2, . . . , Cn} be a collection of criteria and a prioritization between the criteria can be expressed by the linear ordering C1 C2 C3 Cn, indicating criteria Cj has a higher priority than Ck if j < k. Then the value Cj(x) is the performance of any alternative x under criteria Cj, and satisfies Cj(x) 2 [0, 1]. If
PAðC i ðxÞÞ ¼
n X
T
j¼1
Tj
; Tj ¼
Qj1
k¼1 C k ðxÞðj
j¼1
!)
Tj Pn
j¼1 T j
lðnj Þ
ð5Þ
can be proven by using mathematical induction on n: (1) For n = 2, since
(
ð2Þ
f1 ¼ [n1 2f1 l
¼ 2; . . . ; nÞ; T 1 ¼ 1, then PA is
T2 Pn
f2 ¼ [n2 2f2 l
wj C j ðxÞ
n X
1
GHFPWA ðf1 ; f2 ; . . . ; fn Þ ¼ [nj 2fj l
T1 Pn
j¼1
where wj ¼ Pnj
(
j¼1 T j
1
!)
T1 Pn
lðn1 Þ
T2 Pn
lðn2 Þ
j¼1 T j
(
j¼1 T j
called the prioritized average (PA) operator.
1
j¼1 T j
ð6Þ !) ð7Þ
then, Example 1 (Paragraph 2 of Section 1). Consider four criteria: work attitude C1, communication skill C2, problem solving skill C3, and learning skill C4. The prioritization relationship for the criteria is C1 C2 C3 C4. Suppose that the satisfaction of the four criteria are C1(x) = 0, C2(x) = 1, C3(x) = 1, and C4(x) = 1. Based on the PA operator, the calculation is performed as follows
T 1 ¼ 1;
T 2 ¼ T 1 S1 ¼ 1 0 ¼ 0; T 3 ¼ T 2 S2 ¼ 0 1 ¼ 0; and T 4
T1 Pn
T2 f 1 Pn
j¼1 T j
j¼1 T j
( 1
¼ [n1 2f1 ;n2 2f2 l (
1
¼ [n1 2f1 ;n2 2f2 l
¼ T 3 S3 ¼ 0 1 ¼ 0
( ¼ [nj 2fj l
Therefore, the normalized weighting vectors are:
f2
1
2 X j¼1
l l
j¼1 T j
T1 Pn
j¼1 T j
Tj Pn
j¼1 T j
!!
T1 Pn
1
þl l
lðn1 Þ
!! !)
T2 Pn
1
j¼1 T j
lðn2 Þ
!)
T2 lðn1 Þ þ Pn
j¼1 T j
lðn2 Þ
!) lðnj Þ
x1 ¼ 1; x2 ¼ 0; x3 ¼ 0; and x4 ¼ 0
ð8Þ
The overall satisfaction of the four criteria can be obtained as:
CðxÞ ¼ 1 0 þ 0 1 þ 0 1 þ 0 1 ¼ 0 Thus, the overall satisfaction of the sales engineer is zero, and the alternative (i.e., sales engineer) does not meet the stated needs of the company.
3. Generalized hesitant fuzzy prioritized aggregation operators In this section, the generalized hesitant fuzzy prioritized weighted average (GHFPWA) operator and the generalized hesitant fuzzy prioritized weighted geometric (GHFPWG) operator are proposed to aggregate the HFEs. Some desirable properties and special cases of these operators are also studied in this section. Definition 5. Let fj(j = 1, 2, . . . , n) be a collection of HFEs, and let GHFPWA: Vn ? V. If
T1 GHFPWA ðf1 ; f2 ; . . . ; fn Þ ¼ Pn
j¼1 T j
T2 f1 Pn
j¼1 T j
That is, Eq. (5) holds when n = 2. Suppose that Eq. (5) also holds when n = k:
( 1
GHFPWA ðf1 ; f2 ; . . . ; fk Þ ¼ [nj 2fj l
j¼1
j¼1 T j
k X
T kþ1 f j Pn fkþ1 j¼1 T j j¼1 ( Xk Tj 1 Pn ¼ [n1 2f1 ;n2 2f2 ;...;nk 2fk l j¼1 ¼
1
(
l
þ l
n X j¼1
where T j ¼ fk.
Qj1
k¼1 Sðfk Þðj
Tj
Pn
j¼1 T j
!) lðnj Þ
ð4Þ
¼ 2; . . . ; nÞ; T 1 ¼ 1 and S(ak) is the score of HFE
1
T kþ1 Pn lðnkþ1 Þ j¼1 T j
lðnj Þ
1
T kþ1 Pn lðnkþ1 Þ j¼1 T j
1
l l
1
k X
1
l l
1
k X j¼1
þ l
T kþ1 Pn lðnkþ1 Þ j¼1 T j
Tj
Pn
j¼1 T j
!! lðnj Þ
!!!) (
1
[nkþ1 2fkþ1
!)
Tj
Pn
j¼1 T j
!! lðnj Þ
!!!) (
¼ [n1 2f1 ;n2 2f2 ;...;nk 2fk ;nkþ1 2fkþ1 l
1
( ¼ [n1 2f1 ;n2 2f2 ;...;nk 2fk ;nkþ1 2fkþ1 l
Proof. The first result follows quickly from Definition 2 and Theorem 1. The equation
ð9Þ
!)
j¼1 T j
¼ [n1 2f1 ;n2 2f2 ;...;nk 2fk ;nkþ1 2fkþ1 l
(
:
j¼1 T j
ð3Þ
Theorem 1. Let fj(j = 1, 2, . . . , n) be a collection of HFEs, then their aggregated value by using the GHFPWA operator is also an HFE, and
lðnj Þ
Tj Pn
¼ [n1 2f1 ;n2 2f2 ;...;nk 2fk ;nkþ1 2fkþ1 l
fn
j¼1 T j
GHFPWA ðf1 ; f2 ; . . . ; fkþ1 Þ
then the function GHFPWA is called a GHFPWA operator, where Q T j ¼ j1 k¼1 Sðfk Þ ðj ¼ 2; . . . ; nÞ, T1 = 1 and S(fk) is the score of HFE fk.
GHFPWA ðf1 ; f2 ; . . . ; fn Þ ¼ [nj 2fj l
Pn
j¼1
Tn Pn
!)
Tj
then, when n = k + 1, the operational laws described in Definition 2 state that
(
f2
k X
1
!) T kþ1 P lðnj Þ þ n lðnkþ1 Þ j¼1 T j j¼1 T j j¼1 !) kþ1 X Tj Pn ; ð10Þ lðnj Þ j¼1 T j j¼1 k X
Tj Pn
i.e. Eq. (5) holds for n = k + 1. Thus, Eq. (5) holds for all n. Then
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D. Yu et al. / Knowledge-Based Systems 52 (2013) 1–10
( GHFPWA ðf1 ; f2 ; . . . ; fn Þ ¼ [nj 2fj l
1
n X j¼1
!)
Tj
Pn
j¼1 T j
lðnj Þ
:
ð11Þ
T1 GHFPWA ðf1 ; f2 ; . . . ; fn Þ ¼ Pn
j¼1 T j
GHFPWA ðf1 ; f2 ; . . . ; fn Þ ¼ f :
ð12Þ
Proof. By Theorem 1,
( GHFPWAðf1 ; f2 ; . . . ; fn Þ ¼ [nj 2fj l
1
j¼1
( ¼ [n2f l
1
n X j¼1
( ¼ [n2f l
n X
1
j¼1 T j
Tj Pn
Pn
j¼1 T j
j¼1 T j Pn lðnÞ j¼1 T j
fn
j¼1 T j
¼
f2
1 0 0 f1 f2 fn ¼ f1 ¼ f0g: 1 1 1
ð20Þ
Corollary 2 indicates that when the smallest HFE is associated with the highest priority criteria, the HFE for other criteria will be ignored. h Theorem 3. Let fj(j = 1, 2, . . . , n) be a collection of HFEs, where Q T j ¼ j1 k¼1 Sðfk Þðj ¼ 2; ; nÞ; T 1 ¼ 1 and S(fk) is the score of HFE fk. If h is an HFE, c are elements of hesitant fuzzy set h, and nj are elements of hesitant fuzzy set fj, then
GHFPWAðf1 h; f2 h; ; fn hÞ
!)
Tj Pn
j¼1 T j
Tn Pn
It must be pointed out that the above proving method is based on Xu [29], Zhao et al. [40]. Now, consider some desirable properties of the GHFPWA operator. h Theorem 2. Let fj(j = 1, 2, . . . , n) be a collection of HFEs, where Q T j ¼ j1 k¼1 Sðfk Þðj ¼ 2; . . . ; nÞ; T 1 ¼ 1, and S(fk) is the score of HFE fk. If for all j, nj = n, where nj are elements of hesitant fuzzy set fj, and n is an element of hesitant fuzzy set f, then
T2 f1 Pn
¼ GHFPWAðf1 ; f2 ; ; fn Þ h:
lðnj Þ !)
ð21Þ
Proof. Since for any j,
lðnÞ 1
fj h ¼ [nj 2fj ;c2h fl ðlðnj Þ þ lðcÞÞg;
!)
1
¼ [n2f fl ðlðnÞÞg ¼ [n2f fng ¼ f ;
then Theorem 1 states that
ð13Þ
which completes the proof of Theorem 2. h Corollary 1. If fj(j = 1, 2, . . . , n) is a collection of the largest HFEs, i.e., fj = f⁄ = {1} for all j, then
GHFPWA ðf1 ; f2 ; . . . ; fn Þ ¼ GHFPWA ðf ; f ; . . . ; f Þ ¼ f1g;
ð22Þ
GHFPWAðf1 h; f2 h; ; fn hÞ ( !!) n X Tj 1 1 Pn ¼ [nj 2fj ;c2h l lðl ðlðnj Þ þ lðcÞÞÞ j¼1 T j j¼1 ( ! !) n X Tj 1 Pn : lðnj Þ þ lðcÞ ¼ [nj 2fj ;c2h l j¼1 T j j¼1 According to Definition 2, (
ð14Þ
GHFPWA ðf1 ;f2 ;...;fn Þ h ¼ [nj 2fj l
which is also the largest HFE.
n X
1
j¼1
(
1
Pn
j¼1 T j
n X
1
j¼1
ð15Þ
Tj Pn
!!!
j¼1 T j
j¼1
(
[c2h fcg
lðnj Þ
n X
l l
¼ [nj 2fj ;c2h l
GHFPWA ðf1 ; f2 ; . . . ; fn Þ ¼ GHFPWA ðf ; f2 ; . . . ; fn Þ ¼ f0g;
!)
Tj
1
¼ [nj 2fj ;c2h l
Proof. The proof for Corollary 1 may be obtained similar to the proof of Theorem 2. h Corollary 2 (Non-compensatory). If f1 is the smallest HFE, i.e., f1 = f⁄ = {0}, then
ð23Þ
!
Tj Pn
j¼1 T j
!) þ lðcÞ
lðnj Þ !)
lðnj Þ þ lðcÞ
ð24Þ
:
Thus,
GHFPWA ðf1 h; f2 h; . . . ; fn hÞ ¼ GHFPWA ðf1 ; f2 ; . . . ; fn Þ h;
which is also the smallest HFE.
ð25Þ
which completes the proof of Theorem 3. h Proof. Since f1 = {0}, then by the Definition 3,
Sðf1 Þ ¼ 0:
ð16Þ
Since j1 Y Tj ¼ Sðfk Þðj ¼ 2; ; nÞ and T 1 ¼ 1;
Theorem 4. Let fj(j = 1, 2, . . . , n) be a collection of HFEs, Q T j ¼ j1 k¼1 Sðfk Þðj ¼ 2; . . . ; nÞ; T 1 ¼ 1, and S(fk) is the score of HFE fk. If r > 0, then
GHFPWA ðrf1 ; rf2 ; . . . ; rfn Þ ¼ r GHFPWAðf1 ; f2 ; . . . ; fn Þ: ð17Þ Proof. According to Definition 2,
k¼1
then
1
kfj ¼ [nj 2fj fl ðklðnj ÞÞg; j1 Y Tj ¼ Sðfk Þ ¼ Sðf1 Þ Sðf2 Þ . . . Sðfj1 Þ
n X T j ¼ 1: j¼1
By Definition 5,
k > 0:
Theorem 1 states that
k¼1
¼ 0 Sðf2 Þ . . . Sðfj1 Þ ¼ 0ðj ¼ 2; ; nÞ and
ð26Þ
ð27Þ ( 1
ð18Þ ð19Þ
GHFPWAðrf1 ; rf2 ; . . . ; rfn Þ ¼ [nj 2fj l
n X j¼1
1
¼l
k
n X j¼1
Tj
Pn
j¼1 T j
Tj
Pn
j¼1 T j
!) 1
lðl ðklðnj ÞÞÞ
lðnj Þ
!! : ð28Þ
5
D. Yu et al. / Knowledge-Based Systems 52 (2013) 1–10
On the other hand,
( r GHFPWA ðf1 ;f2 ;...;fn Þ ¼ [nj 2fj l
1
kl l
1
j¼1 T j
j¼1 T j
j¼1
lðnj Þ
!!)
n X Tj Pn k
1
!!!!)
Tj Pn
j¼1
( ¼ [nj 2fj l
n X
lðnj Þ
: ð29Þ
Thus,
GHFPWA ðrf1 ; rf2 ; . . . ; rfn Þ ¼ r GHFPWA ðf1 ; f2 ; . . . ; fn Þ Theorem 5 follows from Theorems 3 and 4.
ð30Þ
h
GHFPWA ðf1 h1 ; f2 h2 ; . . . ; fn hn Þ ¼ GHFPWAðf1 ; f2 ; . . . ; fn Þ GHFPWAðh1 ; h2 ; . . . ; hn Þ:
It should be noted that Tan [19] studied serials of properties of generalized intuitionistic fuzzy geometric aggregation operator, which are the important reference material for the proving methods of Theorems 2, 3, 4, 6. Klir et al. [7] pointed out that an additive generator should satisfy k: [0, 1] ? [0, 1] such that k(1) = 0. If the additive generator k is assigned with different forms, then some specific hesitant fuzzy aggregation operators can be obtained as follows: Case 1. If k(t) = log (t), then the GHFPWA operator is reduced to the following:
Theorem 5. Let fj(j = 1, 2, . . . , n) be a collection of HFEs, Q T j ¼ j1 k¼1 Sðfk Þðj ¼ 2; . . . ; nÞ; T 1 ¼ 1, and S(fk) is the score of HFE fk. If r > 0 and h is an hesitant fuzzy element, then
T1 GHFPWAðf1 ; f2 ; . . . ; fn Þ ¼ Pn
j¼1 T j
¼ r GHFPWA ðf1 ; f2 ; . . . ; fn Þ h:
ð31Þ
Theorem 6. Let fj(j = 1, 2, . . . , n) and hj(j = 1, 2, . . . , n) be two collecQ tions of HFEs, T j ¼ j1 k¼1 Sðfk Þðj ¼ 2; . . . ; nÞ; T 1 ¼ 1, and S(fk) is the score of HFE fk, then
GHFPWA ðf1 h1 ; f2 h2 ; . . . ; fn hn Þ ¼ GHFPWA ðf1 ; f2 ; . . . ; fn Þ GHFPWA ðh1 ; h2 ; . . . ; hn Þ:
ð32Þ
1
ð33Þ
GHFPWA ðf1 h1 ; f2 h2 ; . . . ; fn hn Þ ( !!) n X Tj 1 1 Pn ¼ [nj 2fj ;cj 2hj l lðl ðlðnj Þ þ lðcj ÞÞÞ j¼1 T j j¼1 ( ! !) n X Tj Tj 1 Pn lðnj Þ þ Pn lðcj Þ ¼ [nj 2fj ;cj 2hj l : j¼1 T j j¼1 T j j¼1
n X
1
j¼1
( GHFPWA ðh1 ; h2 ; . . . ; hn Þ ¼ [cj 2hj l
n X
1
j¼1
j¼1 T j
ð34Þ
ð35Þ
!)
Tj
Pn
j¼1 T j
lðcj Þ
;
ð36Þ
l l
1
n X j¼1
( ¼ [nj 2fj ;cj 2hj l
1
n X j¼1
Tj Pn
j¼1 T j
Tj Pn
j¼1 T j
!!! þl l
lðnj Þ
1
n X j¼1
! lðnj Þ þ
n X j¼1
Tj Pn
j¼1 T j
Tj Pn
j¼1 T j
!!!!)
Tj Pn
!
j¼1 T j
n X
logð1 nj Þ
lðnj Þ ¼ log
n Y
ð40Þ T
Pnj T ð1 nj Þ j¼1 j
ð41Þ
j¼1
Tj
Pn
j¼1 T j
T
!! lðnj Þ
¼ 1e
log
Qn j¼1
Pnj ð1nj Þ
T j¼1 j
Tj
¼1
Pn n Y T ð1 nj Þ j¼1 j ; ð42Þ j¼1
Furthermore, when the priority level of the aggregated arguments is reduced to the same level, then Eq. (39) is rewritten as
n o GHFPWAðf1 ; f2 ; . . . ; fn Þ ¼ [nj 2fj 1 Pnj¼1 ð1 nj Þwj ;
ð43Þ
which is the hesitant fuzzy weighted average operator proposed by Xia and Xu [25,26].
9 > > > = j¼1 ð1 nj Þ j¼1 ð1 þ nj Þ GHFPWAðf1 ;f2 ;...;fn Þ ¼ [nj 2fj ; Tj Tj > Pn Pn > > > > ; :Qn ð1 þ n Þ j¼1 T j þ Qn ð1 n Þ j¼1 T j > j j j¼1 j¼1 T
Pnj
T j¼1 j
T
Qn
Pnj
T j¼1 j
which is the hesitant fuzzy Einstein prioritized weighted average operator.
: ð37Þ
Thus,
ð39Þ
ð44Þ
lðcj Þ
!!) lðcj Þ
j¼1 T j
8 > > > 0 and h is an hesitant fuzzy element, then,
j¼1 ð1
nj Þ
j¼1 ð1
nj Þ
¼ GHFPWGðf1 ; f2 ; . . . ; fn Þ GHFPWGðh1 ; h2 ; . . . ; hn Þ:
P
Pnj j¼1
ð47Þ
: Tj
Case 1. If k(t) = log (t), then the GHFPWG operator is reduced to the following:
PTn1
GHFPWGðf1 ; f2 ; . . . ; fn Þ ¼ f1
PTn2 f2
T j¼1 j
PTnn Tj fn j¼1 ;
ð48Þ
then the function GHFPWG is called a GHFPWG operator, where Q T j ¼ j1 k¼1 Sðfk Þðj ¼ 2; . . . ; nÞ; T 1 ¼ 1, and S(fk) is the score of HFE fk. Theorem 7. Let fj(j = 1, 2, . . . , n) be a collection of HFEs, then their aggregated value obtained with the HFPWG operator is also an HFE, and
( GHFPWG ðf1 ; f2 ; ; fn Þ ¼ [nj 2fj k
1
n X j¼1
where T j ¼ fk.
Qj1
k¼1 Sðfk Þðj
Tj
Pn
j¼1 T j
!) kðnj Þ
;
ð49Þ
f2
T j¼1 j
Proof. Since k(t) = log (t), then l(t) = log (1 t), k1(t) = et, l1(t) = 1 et, and
Tj Pn
j¼1 T j
)
Tj kðnj Þ ¼ Pn
n X
j¼1 T j
!
Tj Pn
j¼1 T j
kðnj Þ
logðnj Þ
ð55Þ
n X Tj Pn ¼
j¼1 T j
j¼1
0 1 T Pnj n Tj BY C ¼ log @ nj j¼1 A
! logðnj Þ
ð56Þ
j¼1
)k
1
n X j¼1
Theorem 8. Let fj(j = 1, 2, . . . , n) be a collection of HFEs, where Q T j ¼ j1 k¼1 Sðfk Þðj ¼ 2; . . . ; nÞ; T 1 ¼ 1, and S(fk) is the score of HFE fk. If for all j, nj = n, where nj are elements of hesitant fuzzy set fj, and n is the element of hesitant fuzzy set f, then
ð50Þ
0 log @
¼e
Tj Pn
j¼1 T j
Tj n
n P Y nj
j¼1
T j¼1 j
!! kðnj Þ
0
0 11 T Pnj n Y Tj B CC 1 B ¼ k @ log @ nj j¼1 AA j¼1
1 A
T
j n Pn T Y j¼1 j nj ; ¼
Theorem 9. Let fj(j = 1, 2, . . . , n) be a collection of HFEs, where Q T j ¼ j1 k¼1 Sðfk Þðj ¼ 2; . . . ; nÞ, T1 = 1, and S(fk) is the score of HFE fk. If h is an HFE, c are elements of hesitant fuzzy set h, and nj are elements of hesitant fuzzy set fj, then
ð57Þ
j¼1
which completes the proof of the Case 1.
h
Furthermore, when the priority level of the aggregated arguments is reduced to the same level, the Eq. (39) is transformed to
( ) n Y wj GHFPWGðf1 ; f2 ; . . . ; fn Þ ¼ [nj 2fj nj ;
ð58Þ
j¼1
GHFPWG ðf1 h; f2 h; ; fn hÞ ¼ GHFPWG ðf1 ; f2 ; . . . ; fn Þ h:
PTnn
T j¼1 j
which is the hesitant fuzzy prioritized weighted geometric (HFPWG) operator studied by Wei [23], Yu [36] in detail.
Proof. Theorem 7 can be proven similar to Theorem 1. h
GHFPWG ðf1 ; f2 ; . . . ; fn Þ ¼ f :
PTn2
fn 8 9 Tj P > > n n > : j¼1 ;
j¼1
¼ 2; . . . ; nÞ; T 1 ¼ 1, and S(fk) is the score of HFE
T j¼1 j
GHFPWG ðf1 ; f2 ; . . . ; fn Þ ¼ f1
Definition 6. Let fj(j = 1, 2, . . . , n) be a collection of HFEs and GHFPWG: Vn ? V. If T j¼1 j
ð54Þ
T j¼1 j
Here we define the generalized hesitant fuzzy prioritized weighted geometric (GHFPWG) operator based on the GHFPWA operator and the geometric mean. h
PTn1
ð53Þ
GHFPWGðf1 h1 ; f2 h2 ; . . . ; fn hn Þ
Tj n
T
Tj
ð52Þ
Theorem 12. Let fj(j = 1, 2, . . . , n) and hj(j = 1, 2, . . . , n) be two colQ lections of HFEs, T j ¼ j1 k¼1 Sðfk Þðj ¼ 2; . . . ; nÞ; T 1 ¼ 1, and S(fk) is the score of HFV fk, then
1
T
j Qn 1þnj Pn
T
Pnj
T j¼1 j
j¼1 1nj
þ1 Tj n
r
¼ r GHFPWGðf1 ; f2 ; . . . ; fn Þ h:
T
Pj Qn 1þnj nj¼1 T j
r
GHFPWGðrf1 h; rf2 h; . . . ; rfn hÞ
lðnj Þ
log
r
GHFPWGððf1 Þ ; ðf2 Þ ; . . . ; ðfn Þ Þ ¼ ðGHFPWAðf1 ; f2 ; . . . ; fn ÞÞ :
Tj
¼ log
Theorem 10. Let fj(j = 1, 2, . . . , n) be a collection of HFEs, Q T j ¼ j1 k¼1 Sðfk Þðj ¼ 2; . . . ; nÞ; T 1 ¼ 1, and S(fk) is the score of HFEfk. If r > 0, then
ð51Þ
which was the hesitant fuzzy weighted geometric operator proposed by Xia and Xu [25,26].
7
D. Yu et al. / Knowledge-Based Systems 52 (2013) 1–10
2t
Case 2. If kðtÞ ¼ log t , then the GHFPWG operator is reduced to the following:
GHFPWGðf1 ; f2 ; ; fn Þ ¼ [nj 2fj
8 > > > >
> > > :Qn
T
i¼1 ð2
nj Þ
Pnj j¼1
Tj
þ
Qn
(
n X Tj Pn GHFPWA f1c ; f2c ; . . . ; fnc ¼ [nj 2fj 1 k
9 > > > > = ; Tj Pn > > T > > j¼1 j ; n
j¼1
0
!) 1
j¼1 T j
k ðnj Þ
1c Tj n Pn T Y j B C ¼ @ fj j¼1 A j¼1
c
¼ ðGHFPWGðf1 ; f2 ; . . . ; fn ÞÞ :
i¼1 j
ð59Þ
ð65Þ
which we call the hesitant fuzzy Einstein prioritized weighted geometric operator. 1þt 1 2 Proof. Since kðtÞ ¼ log 2t t , then lðtÞ ¼ log 1t ; k ðtÞ ¼ et þ1, and 1 et 1 l ðtÞ ¼ et þ1, then
(2) According to Definition 2 and the definition of a GHFPWG operator,
(
PTnj 2 nj 2 nj T j¼1 j ¼ log n n T T j j j j j¼1 j¼1 0 1 ! PTnj n n X X 2 n Tj Tj j @log j¼1 A Pn kðnj Þ ¼ ) nj j¼1 T j j¼1 j¼1
Tj Pn
Tj kðnj Þ ¼ Pn
log
Tj Yn 2 nj Pn T j¼1 j ¼ log j¼1 nj !! n X Tj 1 Pn )k kðnj Þ ¼ j¼1 T j j¼1
ð60Þ
j¼1
nj
j¼1
þ1
i¼1 ð2
nj Þ
which completes the proof of Case 2.
;
T
Pnj j¼1
Tj
þ
Qn
Pnj
i¼1 nj
j¼1
h
2
GHFPWG ðf1 ; f2 ; . . . ; fn Þ ¼ [nj 2fj Qn
i¼1 ð2
) wj i¼1 nj Q w nj Þwj þ ni¼1 nj j
Qn
ð63Þ
which is referred to as the hesitant fuzzy Einstein weighted geometric operator. In the following, we study the relationship between GHFPWA operator and GHFPWG operator. Theorem 13. Let fj(j = 1, 2, . . . , n) be a collection of HFEs, Q T j ¼ j1 Sðfk Þðj ¼ 2; . . . ; nÞ; T 1 ¼ 1 and S(fk) is the score of HFE fk, k¼1 then c (1) GHFPWA f1c ; f2c ; . . . ; fnc ¼ ðGHFPWGðf1 ; f2 ; . . . ; fn ÞÞ , and c c c (2) GHFPWG f1 ; f2 ; . . . ; fnc ¼ ðGHFPWAðf1 ; f2 ; . . . ; fn ÞÞ .
j¼1 T j
!c fj
c
¼ ðGHFPWAðf1 ; f2 ; . . . ; fn ÞÞ ;
ð67Þ
This section introduces the methods to rank alternatives based on hesitant fuzzy information. In many group decision problems a set of alternatives must be evaluated on the basis of criteria with prioritized relationships. Consider a group decision making problem [5,28,31,16,22, 18,39,9,10,12,14,17,30,32,41]. Let X = {x1, x2, . . . , xm} be the set of alternatives, C = {C1, C2, . . . , Cn} be the set of criteria, and E = {e1, e2, . . . , ep} be the set of decision makers. It is assumed that there is a prioritization between the criteria expressed by the linear ordering C1 C2 C3 Cn indicating criterion Cj has a higher priority than Ci if j < i. The decision maker ek evaluates the alternative xi under criterion Cj anonymously so as to protect his/her privacy or avoid psychic contagion. The evaluation of alternative xi under criteria Cj is provided by each of the decision makers ek (k = 1, 2, . . . , p) using several values. If two decision makers provide the same value, then that value will emerge only once, and the evaluation can be represented by HFEs. The hesitant fuzzy group decision matrix F = (fij)mn is constructed from these HFEs. Here, fij is the attribute value provided by the decision makers ek(k = 1, 2, . . . , p), which is expressed in a HFE. Based on the above analysis, the main steps of the multi-criteria group decision making method are as follows: Step 1. Calculate the values of Tij (i = 1, 2, . . . , m, j = 1, 2, . . . , n) based on the following equations.
Proof. (1) According to Definition 2 and the Definition of a GHFPWA operator,
( GHFPWAðf1 ; f2 ; . . . ; fn Þ ¼ [nj 2fj l
1
n X j¼1
and
Tj Pn
!) 1
l ðnj Þ
ð62Þ
Tj
Furthermore, when the priority level of the aggregated arguments is reduced to the same level, then the Eq. (59) is transformed to
(
n X
j¼1 T j
; and ð66Þ
4. Personnel evaluation based on prioritized operators under hesitant fuzzy environment
T
Qn
kðnj Þ
þ1
Pnj Tj Qn 2 i¼1 nj j¼1
¼ Tj
j¼1 T j
which completes the proof of Theorem 13. h
T
T
Qn 2nj
¼
j¼1
nj
Pn
j¼1
2 j¼1
!)
Tj
( n X Tj Pn GHFPWG f1c ; f2c ; . . . ; fnc ¼ [nj 2fj 1 l
Pj Qn 2nj nj¼1 T j
e
¼
GHFPWGðf1 ; f2 ; . . . ; fn Þ ¼ [nj 2fj k
n X j¼1
ð61Þ
log
2 T Pnj
1
Tj Pn
j¼1 T j
!) lðnj Þ
T ij ¼
j1 Y
Sðr ik Þ ði ¼ 1; 2; . . . ; m; j ¼ 2; . . . ; nÞ
ð68Þ
k¼1
ð64Þ
T i1 ¼ 1 i ¼ 1; 2; . . . ; m
ð69Þ
Step 2. Aggregate the hesitant fuzzy values fij for each alternative xi by the GHFPWA (or GHFPWG) operator.
8
D. Yu et al. / Knowledge-Based Systems 52 (2013) 1–10
( GHFPWAðf1 ; f2 ; . . . ; fn Þ ¼ [nj 2fj l
n X
1
j¼1
!)
Tj
Pn
j¼1 T j
i
lðnj Þ
¼ 1; 2; . . . ; m ð70Þ or
(
n X
1
GHFPWGðf1 ; f2 ; . . . ; fn Þ ¼ [nj 2fj k
j¼1
Tj Pn
j¼1 T j
!)
Table 2 Hesitant fuzzy group decision matrix.
x1 x2 x3 x4 x5
ð71Þ
i
C3
C4
{0.6, 0.7} {0.2, 0.4, 0.7} {0.3, 0.7, 0.8} {0.3, 0.5, 0.6} {0.2, 0.3, 0.5}
{0.2, 0.4, 0.5, 0.6} {0.6, 0.8} {0.3, 0.6, 0.7} {0.2, 0.4} {0.3, 0.5}
{0.5, 0.6, 0.9} {0.6, 0.7, 0.9} {0.7} {0.5, 0.6, 0.7} {0.4}
{0.5, 0.7} {0.3, 0.4, 0.5, 0.8} {0.5, 0.6, 0.8, 0.9} {0.8, 0.9} {0.2, 0.6, 0.7}
Step 3. Calculate the scores of fi(i = 1, 2, 3, 4, 5).
Step 3. Rank all the alternatives by the score function defined by Definition 3.
1 X n; #fi n 2f i
C2
kðnj Þ
i ¼ 1; 2; . . . ; m
Sðfi Þ ¼
C1
i ¼ 1; 2; . . . ; m
S1 ¼ 0:6040;
S2 ¼ 0:5923;
S3 ¼ 0:6360;
S4 ¼ 0:4797;
S5 ¼ 0:3671
ð72Þ Since
i
S3 > S1 > S2 > S4 > S5 ; Using the above process, the bigger the value of S(fi), the larger is the overall HFE fi. So is the alternative xi (i = 1, 2, . . . , m).
the following results are obtained
x3 x1 x2 x4 x5 Example 2 (Paragraph 2 of Section 1). Section 1 introduced a personnel evaluation problem that included a prioritization relationship between multiple criteria. To draw a scientific conclusion that is free of psychic contagion, this evaluation will be made anonymously. If two decision makers provide the same value, then the value will emerge only once, and the evaluation can be represented by HFEs. The hesitant fuzzy group decision matrix F = (fij)mn is constructed as follows (Table 2). Approach 1: The ranking of alternatives can be obtained as follows using the GHFPWA operator and letting k(t) = log (t): Step 1. Calculate the values of Tij(i = 1, 2, . . . , 5, j = 1, 2, . . . , 4) based on Eqs. (68) and (69).
0
1:0000 0:6500 B B 1:0000 0:4333 B T ij ¼ B B 1:0000 0:6000 B @ 1:0000 0:4667
0:2763 0:1842
1
C 0:3033 0:2224 C C 0:3200 0:2240 C C C 0:1400 0:0840 A 1:0000 0:3333 0:1333 0:0533
( 1
n X j¼1
Tj Pn
j¼1 T j
!) lðnj Þ
Step 10 . Same as the above Step 1. Step 20 . Since kðtÞ ¼ log 2t , then t
( GHFPWAðf1 ; f2 ; . . . ; fn Þ ¼ [nj 2fj l
1
n X j¼1
Tj Pn
!)
j¼1 T j
lðnj Þ
;
ð75Þ
which is reduced to the
GHFPWAðf1 ; f2 ; . . . ; fn Þ 8 9 T T Pnj Pnj > > > > Q Q > T T < n ð1 þ n Þ j¼1 j n ð1 n Þ j¼1 j > = j j j¼1 j¼1 : ¼ [nj 2fj Tj Tj > Pn Pn > > > > > Q Q T T : n ð1 þ n Þ j¼1 j þ n ð1 n Þ j¼1 j ; j j j¼1 j¼1
ð76Þ
The overall preference values fi are obtained using the GHFPWA operator in Eq. (76) to aggregate all of the preference values fij (i = 1, 2, 3, 4, 5) in the ith line of F. Due to the space limit, f1 is provided as a representative example.
Step 2. Since k(t) = log (t), then
GHFPWAðf1 ; f2 ; . . . ; fn Þ ¼ [nj 2fj l
Clearly, the best option is alternative x3. Approach 2: The ranking of alternatives can be obtained as fol lows using the GHFPWA operator and letting kðtÞ ¼ log 2t : t
f1 ¼ f0:4341;0:4506;0:4455;0:4616;0:5064;0:5208;0:4748;0:4900; ð73Þ
0:4854;0:5003;0:5419;0:5552;0:4976;0:5122;0:5078;0:5221; 0:5618;0:5745;0:5234;0:5373;0:5330;0:5466;0:5843;0:5964; 0:4939;0:5086;0:5041;0:5185;0:5586;0:5714;0:5303;0:5439;
which is reduced to the
8
S1 > S2 > S4 > S5 ;
0:5463;0:5661;0:5594;0:5786;0:6325;0:6485;0:5848;0:6029; 0:5968;0:6144;0:6637;0:6784;0:6075;0:6246;0:6188;0:6354; 0:6820;0:6959;0:6336;0:6495;0:6441;0:6596;0:7032;0:7161g
S3 ¼ 0:5666;
¼ 0:3391
The final ranking can be obtained as
x3 x1 x2 x4 x5 :
S4 ¼ 0:4315;
S5
9
D. Yu et al. / Knowledge-Based Systems 52 (2013) 1–10
The best option with this approach is also alternative x3. Approach 3: The ranking of alternatives can be obtained as follows, by using the GHFPWG operator and letting k(t) = log (t):
GHFPWGðf1 ; f2 ; . . . ; fn Þ ¼ [nj 2fj k
1
n X j¼1
!)
Tj
Pn
j¼1 T j
kðnj Þ
ð77Þ Step 3000 . Calculate the scores of fi (i = 1, 2, 3, 4, 5).
8 9 T Pnj > > n > : j¼1 ;
S1 ¼ 0:6015; ð78Þ
f10 ¼ f0:4111;0:4233;0:4210;0:4336;0:4440;0:4572;0:5089;0:5241; 0:5212;0:5367;0:5496;0:5660;0:5451;0:5614;0:5583;0:5749; 0:5887;0:6063;0:5766;0:5938;0:5905;0:6081;0:6227;0:6413; 0:4422;0:4554;0:4529;0:4664;0:4776;0:4918;0:5475;0:5638; 0:5607;0:5774;0:5913;0:6089;0:5864;0:6039;0:6006;0:6185; 0:6333;0:6522;0:6203;0:6388;0:6353;0:6542;0:6699;0:6899g
Step 300 . Calculate the scores of fi (i = 1, 2, 3, 4, 5) respectively.
S3 ¼ 0:5834;
S4 ¼ 0:4251;
S5
¼ 0:3485 Since
S3 > S1 > S2 > S4 > S5 ; the final ranking is obtained as
x3 x1 x2 x4 x5 ; Which again indicates the best option is alternative x3. Approach 4: The ranking of alternatives can be obtained as fol lows using the GHFPWG operator and letting kðtÞ ¼ log 2t : t
GHFPWGðf1 ; f2 ; . . . ; fn Þ ¼ [nj 2fj k
1
n X j¼1
Tj
GHFPWGðf1 ; f2 ; . . . ; fn Þ ¼ [nj 2fj
S4 ¼ 0:4763;
S5
¼ 0:3938
S3 > S1 > S2 > S4 > S5 ; We can easily get
x3 x1 x2 x4 x5 : The best option is therefore alternative x3. The above examples show that the same conclusion can be obtained with all four different approaches. This consistency demonstrates the stability of the proposed method. 5. Conclusions In this age of increasingly competitive markets, talent has become essential for business success. Talent can be a critical factor in a company’s ability to achieve operational excellence. To address the disadvantages of traditional personnel evaluation methods, this paper proposes the use of a hesitant fuzzy group decision making method. The method is based on the GHFPWA and GHFPWG operators. The use of hesitant fuzzy sets is a powerful technique that captures the decision makers’ preferences. The ability to prioritize criteria not only addresses the actual needs of companies from a human resources perspective, but also opens the approach up to a wide range of applications. A realistic example is used to illustrate the effectiveness of the proposed method for personnel evaluation. It is worth noting that in addition to human resource issues, the proposed method is equally applicable to factory location, supplier selection and many other management decision problems. Acknowledgement
Pn
j¼1 T j
!) kðnj Þ
ð79Þ
;
which is reduced to
8 > > > >
> Pnj > > :Qn ð2 n Þ j¼1 T j þ Qn j i¼1 i¼1
9 > > > > = : Tj Pn > > Tj > > n j¼1 ; j
ð80Þ The overall preference values fi are obtained using the GHFPWG operator in Eq. (80) to aggregate the preference values fij (i = 1, 2, 3, 4, 5) in the ith line of F. Due to the space limit, f1 is provided as a representative example.
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