Grey Language Hesitant Fuzzy Group Decision Making Method Based on Kernel and Grey Scale

Internationa Journa of Environmenta Research and Pubic Heath Artice Grey Language Hesitant Fuzzy Group Decision Making Method Based on Kerne and Grey Scae Qingsheng Li * ID, Yuzhu Diao, Zaiwu Gong ID and Aqin Hu Schoo of Business, Linyi University, Linyi 276000, China; diaoyuzhu@126.com (Y.D.); zwgong26@163.com (Z.G.); huaqin@yu.edu.cn (A.H.) * Correspondence: iqingshengguani@163.com; Te.: +86-138-5496-9086 Received: 29 January 2018; Accepted: 27 February 2018; Pubished: 2 March 2018 Abstract: Based on grey anguage muti-attribute group decision making, a kerne and grey scae scoring function is put forward according to the definition of grey anguage and the meaning of the kerne and grey scae. The function introduces grey scae into the decision-making method to avoid information distortion. This method is appied to the grey anguage hesitant fuzzy group decision making, and the grey correation degree is used to sort the schemes. The effectiveness and practicabiity of the decision-making method are further verified by the industry chain sustainabe deveopment abiity evauation exampe of a circuar economy. Moreover, its simpicity and feasibiity are verified by comparing it with the traditiona grey anguage decision-making method and the grey anguage hesitant fuzzy weighted arithmetic averaging (GLHWAA) operator integration method after determining the index weight based on the grey correation. Keywords: kerne and grey scae; grey anguage; hesitant fuzzy; grey correation 1. Introduction The deepening interdependence and mutua infuence of the word economy and the appication of the new generation of Internet information technoogy have significanty infuenced the decisions of the state and enterprises. The compexity, uncertainty and coectivization of decision-making information in poitics, the economy and miitary have become the new norma of decision making. A compex decision-making environment and the imited cognitive abiity of decision makers require overcoming the constraints of traditiona sets. In 1965, the American cybernetics expert Zadeh [1] first proposed the concept of the fuzzy set, which expresses the idea of uncertainty through membership degree and has been widey used in various fieds, such as management decisions, medica diagnosis and engineering contro. The Bugarian schoar Atanassov [2] argued that the membership degree of traditiona fuzzy sets can ony be the numbers between 0 and 1, which cannot dig deeper into the fuzzy degree in actua decision making. Based on the above consideration, Atanassov studied three dimensions of membership degree, non-membership degree and hesitation degree, and gave the reationa expression, which was further deveoped into an intuitive fuzzy set. Since Zadeh proposed fuzzy sets, many schoars have studied their extension forms, such as the intuitionistic fuzzy set (IFS) [3], the type 2 fuzzy set [4,5], the fuzzy mutipe set, and [6,7] the hesitant fuzzy set. This incudes group decision making based on information integration and the measurement of hesitant fuzzy sets. Based on Archimedes T-modue and S-modue, Xia Meimei gave the rues of hesitant fuzzy operation and the information integration operator of hesitant fuzzy decision; defined the distance, simiarity, correation, entropy and cross-entropy of hesitant fuzzy sets; and discussed their properties. Moreover, their correation was studied in order to appy this to group decision-making probems [8]. These studies have aid a foundation for the study of hesitant fuzzy group decision making based on the kerne and gray eve. Int. J. Environ. Res. Pubic Heath 2018, 15, 436; doi:10.3390/ijerph15030436 www.mdpi.com/journa/ijerph

Int. J. Environ. Res. Pubic Heath 2018, 15, 436 2 of 14 Increasingy compex decision information requires experts to make group decisions in more cases. Experts need to face the compex environment with both compete and incompete information, both definite information and uncertain information. The compex group decision-making environment has aso attracted the attention of domestic schoars. In 1982, Deng Juong, a Chinese schoar, pubished the first paper about the grey system entited The Contro Probems of Grey Systems in Systems and Contro Letters [9]. In the same year, Deng Juong pubished the first Chinese grey system paper Grey Contro System in the Journa of the Huazhong Institute of Technoogy [10]. These two innovative cassica papers have aroused wide attention and successfuy initiated an emerging trans-discipinary science grey system theory [11]. The grey muti-attribute decision-making method was studied by [12 15]. Youchen et a. studied the inverse order probem of the grey decision mode [16]. Zhang Na et a. estabished the muti-stage grey situation group decision mode based on the Orness measure constraint [17]. Liu Sifeng et a. expored the two-stage grey comprehensive measurement decision mode and the improvement of the trianguar whitening weight function [18]. Luo Dang and Liu Sifeng studied the grey reationa decision method of the grey reationa decision method and an incompete information system [19,20]. Dang Yaoguo et a. studied the grey reationa decision mode based on a dynamic muti-index [21]. Jiang Shiquan et a. put forward a grey reationa decision mode based on area [22]. Xie Ming et a. studied the grey reationa anaysis method of muti-attribute group decision making based on the anaytic hierarchy process (AHP) and its appication [23]. Chen Xiaoxin and Liu Sifeng put forward a grey reationa decision method with partia weight information and a preference to scheme [24]. The rapid deveopment of the grey reationa group decision-making method provides the research foundation for the fuzzy grey reationa decision making in this paper. The differences of knowedge eve, judgment experience and preference of decision experts wi inevitaby ead to the deviation of information interpretation, which further causes uncertain decision-making probems accompanied by fuzzy and grey characteristics, namey, the grey fuzzy muti-attribute group decision-making probem. The intuitionistic fuzzy decision making method was studied by [25 29] on the basis of grey correation combined with evidence reasoning and the expert system uncertainty factor. Hu Lifang et a. organicay combined grey with fuzzy, and put forward the grey fuzzy muti-attribute decision making method under the frame of the cosed word [30]. Li Peng et a. put forward the combination method of grey reation and Dempster-Shafer evidence theory, which was appied to mutipe-criteria decision making (MCDM) based on the intuitionistic fuzzy set (IFS) [31]. Some schoars from Nanjing University of Aeronautics and Astronautics combined grey system with fuzzy theory, and put forward reevant decision methods. For exampe, Li Peng, Liu Sifeng and Zhu Jianjun studied the combination of GR, DS and MYCIN, which was appied to IFS decision-making [32,33]. Liu Yong and Liu Sifeng studied the dynamic muti-attribute grey reationa decision making method based on interva intuitionistic fuzzy [26]. In [34], the grey correation method and non-inear programming mode were used to obtain the optima weight and study the reevant decision methods. The study of grey system and fuzzy combination promotes the fusion and promotion of the uncertainty decision method. Research in the fied of grey anguage muti-attribute decision-making is reativey scarce. Wu Jianwen studied the grey anguage muti-criteria decision method and its appication [35]. Wang Jiang et a. put forward a muti-criteria decision method based on interva grey uncertain anguage [36]. Liu Peide et a. proposed a muti-attribute group decision-making method based on a geometric weighted integration operator of interva grey anguage variabes [37]. Wang Hehua studied the information aggregation mode of anguage evauation based on grey decision and its appication [38]. Li Qingsheng et a. [39 41] put forward the grey hesitant fuzzy set and extended it to the grey anguage form, namey, the grey anguage hesitant fuzzy set, which can convenienty use the specia decision method of a grey system and effectivey sove the probem of hesitant fuzzy decision making. How to take the advantages of a grey system to make grey anguage hesitant fuzzy decision

Int. J. Environ. Res. Pubic Heath 2018, 15, 436 3 of 14 expand the appication scope of a grey system is significant for enriching and perfecting grey system theory and the organic combination of uncertain decision making. This paper proposes a kerne and grey scae-scoring function of the grey anguage hesitant fuzzy set which uses the grey correation degree to make decisions. 2. Preiminaries 2.1. Grey Language Set The concepts of anguage vaue and grey scae are generay combined in some practica decision-making processes, so that the decision information is more in ine with the decision habit, which is more ike the description of kerne and grey scae based on a grey system. Wu Jianwen and Wang Jianjiang put forward the grey anguage set by studying the muti-criteria decision and grey system of anguage, and discussed the probem of muti-attribute decision-making based on grey anguage. For exampe, a coege counseor gives the evauation resuts of exceent, good, fair, quaified and faiing. If a coege student is rated as good, the membership degree of the comprehensive evauation vaue of good is 1. However, the anguage evauation vaue of membership degree of 1 is often impractica, as it requires counseors to grasp comprehensive information about the student, incuding earning, interpersona reations, comprehensive abiity in the famiy, society, cassroom and dormitory, etc. Obviousy, this is impossibe in reaity. The decision maker s imited rationaity principe proposed by Simon is coser to reaity. If the counseor can give the membership degree and incompete degree of information (grey scae), it wi be very hepfu for the scientific evauation of coege students. Therefore, evauation vaues in the foowing form can be obtained: A = {x, H i, µ i, v i x X } = {x, good, 0.8, 0.4}, which represents the membership degree to which the coege student is evauated as good at 0.8, and the information uncertainty at 0.4. 2.1.1. Definition of Grey Language Set Definition 1. Set the discourse domain X, and anguage phase set H = {H 0, H 1, H 2, H T 1, H T }. For x X, there is a corresponding anguage vaue H i (H i H). If the membership degree of x to H i is the grey number in [0, 1], its point grey scae is v(x), A = {x, H i, µ i, v i x X } is a grey anguage set (GLS) on X. µ Hi (x) is the membership degree, which can be understood as the cose degree of x to H i [38]. Different from intuitionistic fuzzy sets,v(x) represents the extent to which the decision maker's information is incompete, namey, grey scae, instead of non-membership degree. Athough there is no necessary correation between grey scae v(x) and membership degree µ Hi (x), the smaer the vaue of µ i + v i 1, the better the reiabiity. In genera, the grey anguage set A is abbreviated as A = {H(x), µ(x), v(x) x X }. If a = {H(x), µ(x), v(x)} is GLS, any specific grey anguage eement a in A can be abbreviated as GLE. If the vaues of membership degree and grey scae in a is µ(x) = 1, v(x) = 0, a does not have incompete information degree, and retreats into the genera anguage number. 2.1.2. Agorithm of Grey Language Number (1) Set a i = {H i, µ i, v i } and a j = { } H j, µ j, v j as arbitrary GLE, according to the foowing rues [35]. a i a j = (H i H j, i µ i+j µ j i+j, max(v i, v j )), when H i and H j are not equa to H 0 simutaneousy. a i a j = (H i H j, max(µ i, µ j ), max(v i, v j )), when H i and H j are equa to H 0. In the expression and cacuation process of grey anguage in this paper, if there is no specia expanation, it is operated as H i and H j are not equa to H 0 simutaneousy. (2) ra i = {rh i, µ i, v i } [35]. (3) a i a j = (H i H j, µ i µ j, v i + v j v i v j ) [35]. (4) a λ i = (H λ i, µλ i, 1 (1 v i) λ ) [35].

Int. J. Environ. Res. Pubic Heath 2018, 15, 436 4 of 14 2.1.3. Properties of Grey Language Number Agorithm Assume that a i = {H i, µ i, v i }, a j = { } H j, µ j, v j and ak = {H k, µ k, v k } are any three grey anguage numbers, and the arithmetic operation has the foowing properties: (1) a i a j = a j a i. (2) (a i a j )a k = a i (a j a k ). (3) a i a j = a j a i. (4) (a i a j )a k = a i (a j a k ). (5) λa i λa j = λ(a i a j ), λ > 0. (6) λ 1 a i λ 2 a i = (λ 1 + λ 2 )a i, λ 1 > 0, λ 2 > 0. (7) (a i a j ) λ = a λ i a λ j, λ 0. (8) a i λ 1 a i λ 2 = a i (λ 1 +λ 2 ), λ 1 0, λ 2 0. 2.1.4. Size of Grey Language Numbers Definition 2. Assume a i = {H i, µ i, v i } and a j = { H j, µ j, v j } are any two grey anguage numbers [38]. If (1 v i ) µ i IH i < (1 v j ) µ j IH j, a i is smaer than a j, denoted as a i < a j ; If (1 v i ) µ i IH i > (1 v j ) µ j IH j, a i is greater than a j, denoted as a i > a j ; If (1 v i ) µ i IH i = (1 v j ) µ j IH j, and H i = H j, µ i = µ j, v i = v j, a i is strongy equa to a j, denoted as a i = a j ; If (1 v i ) µ i IH i = (1 v j ) µ j IH j, and at east one of H i = H j, µ i = µ j, v i = v j, is tenabe, a i is weaky equa to a j, denoted as a i a j. Where, I is subscript operator. 2.2. Hesitant Fuzzy Set When the attribute of a given scheme beongs to the membership degree of a certain set, it is not necessariy a random interva nor does it obey a certain distribution. The performance wi be more prominent in group decision making. The membership degree given by different groups may be uncoordinated, and sometimes it is not necessary to reach consistent possibe vaues. To this end, Torra [42,43] proposed the hesitant fuzzy set which can better meet the needs of group decision making. 2.2.1. Definitions Reated to Hesitant Fuzzy Sets Definition 3. Assume X is a given set, and hesitant fuzzy set is the projection from X [0,1], abbreviated as A = { x, h(x) x X }. Where, h(x) is the set of severa hesitant fuzzy numbers (HFN) in [0,1]. HFN is the membership degree vaue of x X beonging to set A. Obviousy, there is 0 HFN 1 [42,43]. The hesitant fuzzy eement contains the possibe hesitant fuzzy number in the set. The ength of data in each hesitant fuzzy eement is ikey to vary. The foowing method is used to determine the size of the hesitant fuzzy eement. Definition 4. Any hesitation fuzzy eement h, s(h) = 1/ 1 γ hγ is score function, where is the number of eements in h. There are two hesitant fuzzy eements h 1, h 2. If s(h 1 ) > s(h 2 ), h 1 > h 2. If s(h 1 ) = s(h 2 ), h 1 h 2. Torra beieved that membership degree at both ends of hesitant fuzzy can be understood as an intuitionistic fuzzy number. Definition 5. The enveope of any hesitant fuzzy number h can be converted to intuitionistic fuzzy number through the foowing formua: A env (h) = (h, (1 h + )), where h = minγ, h + = maxγ γ h [42,43]. Based on the connotation of hesitant fuzzy enveope reation [8] studied the enveop reation set operation. (1) A env (h c ) = (A env (h)) c.

Int. J. Environ. Res. Pubic Heath 2018, 15, 436 5 of 14 (2) A env (h 1 h 2 ) = A env (h 1 ) A env (h 2 ). (3) A env (h 1 h 2 ) = A env (h 1 ) A env (h 2 ). (4) A env (h λ ) = (A env (h)) λ. (5) A env (λh) = λa env (h). (6) A env (h 1 h 2 ) = A env (h 1 ) A env (h 2 ). (7) A env (h 1 + h 2 ) = A env (h 1 ) + A env (h 2 ). 2.2.2. Agorithms of Hesitant Fuzzy Set Assume any three hesitant fuzzy numbers h, h 1, h 2, and Torra [42,43] studied their agorithms: (1) h c = γ h {1 γ}. (2) h 1 h 2 = γ1 h 1,γ 2 h 2 max{γ 1, γ 2 }. (3) h 1 h 2 = γ1 h 1,γ 2 h 2 min{γ 1, γ 2 }. On the basis of studying the operation of Archimedes T-modue and S-modue [8] studied the agorithm of hesitant fuzzy operation. There are arbitrary hesitant fuzzy eements h, h 1, h 2. (1) h λ = γ h { k 1 (λk(γ)) }. (2) λh = γ h { 1 (λ(γ)) }. (3) h 1 h 2 = γ1 h 1,γ 2 h 2 { k 1 (k(γ 1 ) + k(γ 2 )) }. (4) h 1 + h 2 = γ1 h 1,γ 2 h 2 { 1 ((γ 1 ) + (γ 2 )) }. where (t) = k(1 t), and k : [0, 1] [0, + ) is a stricty decreasing function. The hesitant fuzzy operation rue can be defined as beow: Definition 6. There are three hesitant fuzzy numbers h, h 1, h 2. (1) (h c ) λ = (λh) c. (2) λ(h c ) = (h λ ) c. (3) h c 1 hc 2 = (h 1 + h 2 )c. (4) h c 1 + hc 2 = (h 1 h 2 )c. where, h c = γ h {1 γ} [8]. Definition 7. Assume h i (i = 1, 2,, n) as a set of hesitant fuzzy numbers, ω = (ω 1, ω 2,, ω n ) T satisfie, n 1 ω i = 1, ω i 0, (i = 1, 2,, n), ATS HFWA(h 1, h 2,, h n ) = n ω i h i. i=1 ATS HFWA is a hesitant fuzzy weighted arithmetic averaging operator based on Archimedes T-Modue and S-Modue. ATS HFWG(h 1, h 2,, h n ) = ATS HFWG is a hesitant fuzzy weighted geometric averaging operator based on Archimedes T-Modue and S-Modue [8]. 2.3. Kerne and Grey Scae 2.3.1. Definitions of Kerne and Grey Scae The study of grey number operation in grey system theory has attracted attention for a ong period of time, but no satisfactory resut has been achieved. Definition 8. Let the grey number [a, a], a < a in the absence of grey number vaue distribution information [44]: If is a continuous grey number, ˆ = 1 2 (a + a) is the kerne of grey number. n i=1 h i ω i.

Int. J. Environ. Res. Pubic Heath 2018, 15, 436 6 of 14 If is discrete grey number, a i [a, a], (i = 1, 2,, n) is the possibe vaue of grey number. Its average vaue is the kerne of grey number, namey, ˆ = 1 n n i=1 a i is the kerne of grey number. Definition 9. If is a random grey number, and the mathematica expectation E( ) can be cacuated, E( ) is the kerne of random and can expressed as ˆ = E( ) [44]. Definition 10. Set the background or a of the grey number is Ω, µ( ) as the measure of interva grey number taking the number fied. g o ( ) = µ( )/µ(ω) is the grey eve of grey number, which can be understood as the ratio of a possibe membership degree to background, abbreviated as g o [44]. The properties of Ω and the measure show that the grey scae satisfies the foowing properties: grey scae satisfies the criterion, namey, 0 g o ( ) 1. When g o = 0, the uncertainty is 0, which is the white number for compete certainty; When g o = 1, the uncertainty is 1, which is the back number for compete uncertainty; When 0 < g o ( ) < 1, it is an uncertain grey number. The grey scae refects the uncertainty degree of the things described by the grey number. The white number is a competey certain number, and the back number is competey unknown. When g o is coser to 0, the uncertainty of the grey number is smaer. When g o is coser 1, the uncertainty of the vaue range is greater. Definition 11. ˆ is the kerne of grey number, and g o is the grey scae. ˆ g o is the simpified form of grey numbers. ˆ g o incudes a information of vaue, and has one-to-one correspondence [44]. On the one hand, ˆ g o can be cacuated according to the definition of the kerne and grey scae. On the other hand, the range of can aso be cacuated according to ˆ g o [45] proposed the agorithm based on ˆ and g o and the grey scae non-decreasing axiom. 2.3.2. Agorithms of Kerne and Grey Scae Axiom 1. Grey scae non-decreasing axiom: grey scae is equa to the maximum grey eve of a grey numbers invoved in the operation, and it wi not decrease [44]. According to Axiom 1, the simpified form k (gk o ) based on the grey number can obtain the grey data agebraic agorithm [45]. (1) k (gk o) + m (gm) o = ( k + m )(gk o( go m). (2) k (gk o) m (gm) o = ( k m )(gk o go m). (3) k (gk o) m (gm) o = ( k m )(gk o( go m). (4) k (gk o) m (gm) o = ( k m )(gk o( go m) ( m = 0). (5) m k (gk o) = (m k )(gk o). (6) ( k (gk o))r = ( k ) r (gk o). Grey heterogeneous data may be incuded by the interva grey number, discrete grey number, rea number or other grey information. A the different data structures and grey information characteristics beong to the same grey category, have a common attribute kerne and grey scae (rea is a specia grey number whose grey scae is 0 ). So the agebraic agorithm between grey heterogeneous data can be studied through the kerne and grey scae. Definition 12. Assume F( ) is a grey heterogeneous data set. For arbitrary i and j, i + j, i j, i j and i j ( j = 0) beong to F( ), and is grey heterogeneous data fied [46].

Int. J. Environ. Res. Pubic Heath 2018, 15, 436 7 of 14 Assume F( ) as the grey heterogeneous data set. i, j and k F( ). The kerne and grey scae are i (gi o), j (g o j ) and k (gk o ). Grey heterogeneous data agebraic agorithms have the foowing properties [45]: (1) i (gi o) + j (g o j ) = j (g o j ) + i (gi o). (2) ( i (g o i ) + j (g o j )) + k (g o k ) = i (g o i ) + ( j (g o j ) + k (g o k )). (3) 0 F( ), Make i (gi o) + 0 = i (gi o). (4) For s F( ), s F( ) and make s + s = 0. (5) i (gi o) ( j (g o j ) k (gk o)) = ( i (gi o) j (g o j )) k (gk o). (6) There is unit eement 1 F( ) which makes 1 i (gi o) = i (gi o) 1 = i (gi o). (7) ( i (gi o) + j (g o j )) k = i (gi o) k (gk o) + j (g o j ) k (gk o). (8) i (g o i ) ( j (g o j ) + k (g o k )) = i (g o i ) j (g o j ) + i (g o i ) k (g o k ). 3. Grey Language Hesitant Fuzzy Information Aggregation and Kerne and Grey Scae Scoring Function 3.1. Information Aggregation of Grey Language Hesitant Fuzzy Decision Since the evauation criteria are given in the form of grey anguage, the number of decision experts is the ength of the data, which is denoted as r ij = {(H σ1, µ 1, v 1 ), (H σ2, µ 2, v 2 ), (H σ, µ, v )}, where σk is the IH vaue of the kth item. The data are aggregated by the agorithm of the grey anguage fuzzy number to obtain z ij. z ij = (H 1 (, σ1 µ 1 + σ2 µ 2 + σ µ, max(v σk) 1, v 2 v )) (1) σk Definition 13. Assume A = {a k k = 1, 2, 3 } as the grey anguage set, a k = {H k, µ k, v k } and g(a 1, a 2, a 3, a ) = w k a k. Where, w k is the weight vector corresponding to the grey anguage array a k.w k [0, 1] and w k = 1, g is grey anguage weighted average operator, abbreviated as the GLHFWAA operator. GLHFWAA(a 1, a 2, a ) = w k a k = (w k H k, µ k, v k ) (2) If w = ( 1, 1,, 1 ), GLHFWAA degenerates to grey anguage averaging operator GLAA, GLHFAA(a 1, a 2, a ) = 3.2. Grey Language Hesitant Fuzzy Score Function w k a k = ( 1 H k, µ k, v k ). Definition 14. Assume a i = {H i, µ i, v i } as any grey anguage number, and S(a k ) is the score function of a k. S(a k ) = (1 v k ) µ k IH k (3) Assume gh GLH, GLH = {(H 1, µ 1, v 1 ), (H 2, µ 2, v 2 ), (H, µ, v )} are grey anguage decision data given by experts.

Int. J. Environ. Res. Pubic Heath 2018, 15, 436 8 of 14 S(GLH) = 1 S(GLH) is the score function of GLH. S(a k ) k = 1, 2 (4) Exampe. Let GLH = {(3, 0.6, 0.2), (4, 0.6, 0.3), (4, 0.7, 0.2)}, S(GLH) = 1 S(a k ) = 1 3 [(1 0.2) 0.6 3 + (1 0.3) 0.6 4 + (1 0.2) 0.7 4] = 1.7867. The above definition transforms uncertain grey anguage information into a score function and becomes a definite vaue, which is theoreticay inadequate. In this paper, it is considered that the score function shoud be a grey number [45] beieved that the kerne ˆ of the grey number is an important representative of the vaue of the grey scae. Kerne is used in a kinds of synthesis operations, and has an important vaue. Aso, the grey agebra operation properties of ˆ g o are discussed, and the reevant agorithm is proved. According to the gradation function of grey anguage and the definition of the kerne and grey scae, the grey anguage scoring function based on the kerne and grey scae is studied in this paper. Definition 15. Assume gh GLH, GLH = {(H 1, µ 1, v 1 ), (H 2, µ 2, v 2 ), (H, µ, v )} S( ˆ g o) = ( 1 ˆ k, max(g 0 k )) = ( 1 µ i IH k, max(v k )) k = 1, 2 (5) S( ˆ g o) is the score function of the grey anguage based on the kerne and grey scae. 4. Kerne and Grey Scae Group Decision Making Method Based on Grey Language Hesitant Fuzzy Information 4.1. Mode Construction Assume that there are m feasibe schemes A 1, A 2,, A m and n evauation criteria of a hesitant fuzzy group decision making probem based on grey anguage. w = (w 1, w 2,, w n ) T is weight vector corresponding to the evauation criteria, where w i represents the weight of P i. Assume the ordered anguage evauation set H = (H 0, H 1,, HT ). According to the evauation set, experts evauate various scheme indexes. In the grey anguage hesitant fuzzy set, beongs to the discrete grey numbers. gh 11 gh 12... gh 1n gh GLH = 21 gh 22... gh 2n............ gh m1 gh m2... gh mn 4.2. Decision-Making Steps Step 1. In this step, the decision matrix P m n = (p ij ) m n is normaized as R m n = (r ij ) m n. It is necessary to eiminate the infuence of different physica quantities before making a decision and to normaize different types of index vaues. Generay speaking, the cost index vaue shoud be converted to the benefit index. In specia cases, the benefit index vaue aso needs to be converted to the cost index vaue. Cost index vaue (H i, µ, v) is converted to benefit index vaue. Benefit index vaue (H i, µ, v) is converted to cost index vaue: r ij = (H T i, µ, v) (6) r ij = (H T i, µ, v) (7)

Int. J. Environ. Res. Pubic Heath 2018, 15, 436 9 of 14 Step 2. Data suppement. Torra studied the hesitancy faced by experts in group decision making and proposed a hesitant fuzzy set suitabe for group decision making. Add the maximum membership scae to the hesitant fuzzy eement when the decision maker has risk preference. Add the minimum membership scae to the hesitant fuzzy eement when the decision maker is risk-averse and pessimistic. Step 3. Transform grey anguage data into a score function of kerne and grey scae through the Formua (4). Step 4. Cacuate grey correation degree and grey scae. Definition 16. ˆ is the kerne part of score function of grey anguage eement based on kerne and grey scae. The grey correation coefficient ζ i (j) and grey correation degree γ(j) between the empirica judgment vaue and the reference weight vaue of each evauation index are obtained by the foowing formua: Make D = ˆ i (j) ˆ 0 (j), and ξ i (j) = min min i j D + ρ max max D i j D + ρ max max D i j (8) γ(i) = 1 n n j=1 ξ i (j), 1 i m (9) where, ρ is the resoution ratio, ρ (0, 1), generay, ρ = 0.5. The correation degree of each sequence directy refects the degree to which each evauation is cose to the optima idea scheme. Definition 17. Assume g 0 (j max ) as the maximum grey scae of kerne vaue in j coumn and g 0 (i max ) as the maximum grey scae of kerne vaue in i row. g 0 (i) = max(g 0 (j max ), g 0 (i max )) 1 j n (10) γ g o(i) = (γ (g1 o )(1), γ (g2 o )(2),, γ (gn o ) (m))t Where, 1 i m (11) Step 5. The schemes are sorted according to the grey correation degree and grey scae. The foowing principes shoud be foowed: (1) When γ(i) > γ(m), i m; (2) When γ(i) = γ(m), and g 0 (i) > g 0 (m), i m; (3) When γ(i) = γ(m), and g 0 (i) = g 0 (m), i m. 5. Exampe Anaysis As an economic deveopment mode harmonious with the environment, the circuar economy has become an important way to promote the comprehensive deveopment abiity of an industria chain. Many industria chains have carried out ecoogica transformations under the guidance of circuar economy idea. The sustainabe deveopment of an industria chain based on the circuar economy is a compicated evauation process, with uncertainty and mutipe objectives. Evauation of the sustainabe deveopment abiity of an industria chain based on the circuar economy aims to study the optimization function of the circuar economy on the design and reconstruction of the industria chain, and to anayze the comprehensive deveopment eve and potentia of the industria chain from the perspective of the circuar economy. The evauation system of the sustainabe deveopment abiity of an industria chain based on the characteristics of the circuar economy can provide scientific and effective means for the operation, management and improvement of the industria chain. Therefore, it is necessary to estabish an industria chain sustainabe deveopment capacity evauation index system covering

Int. J. Environ. Res. Pubic Heath 2018, 15, 436 10 of 14 four aspects of economic deveopment: (P 1 ), resource utiization (P 2 ), environmenta protection (P 3 ) and green management (P 4 ), so as to scientificay and objectivey refect the sustainabe deveopment capacity of an industria chain [46]. Four industries A 1, A 2, A 3, A 4 are assumed to be compared in the sustainabe deveopment { of the circuar economy industry chain in a certain area. Three } experts H evauate and seect H = 0 = very poor, H 1 = poor, H 2 = reativey poor, H 3 = genera as the H 4 = good, H 5 = reativey good, H 6 = very good anguage evauation term set. The evauation vaue is represented by grey anguage vaue, forming the grey anguage hesitant fuzzy decision matrix in Tabe 1. Tabe 1. The vaues of the grey anguage hesitant fuzzy decision matrix. Industries Attribute (P 1 ) Attribute (P 2 ) A 1 {(H 3, 0.6, 0.3), (H 4, 0.6, 0.4), (H 4, 0.7, 0.4)} {(H 3, 0.7, 0.4), (H 3, 0.8, 0.4), (H 3, 0.9, 0.3),} A 2 {(H 3, 0.7, 0.4), (H 3, 0.8, 0.4), (H 4, 0.9, 0.1)} {(H 3, 0.9, 0.1), (H 4, 0.7, 0.4), (H 4, 0.8, 0.2)} A 3 {(H 4, 0.7, 0.2), (H 5, 0.6, 0.2), (H 5, 0.8, 0.1)} {(H 4, 0.7, 0.2), (H 5, 0.6, 0.2), (H 6, 0.9, 0.2)} A 4 {(H 3, 0.6, 0.2), (H 4, 0.8, 0.4), (H 5, 0.9, 0.3)} {(H 3, 0.7, 0.4), (H 4, 0.6, 0.4), (H 5, 0.7, 0.4)} Industries Attribute (P 3 ) Attribute (P 4 ) A 1 {(H 2, 0.8, 0.5), (H 3, 0.6, 0.4), (H 4, 0.7, 0.4)} {(H 2, 0.8, 0.3), (H 2, 0.6, 0.4), (H 2, 0.7, 0.4)} A 2 {(H 3, 0.6, 0.2), (H 3, 0.7, 0.4), (H 3, 0.8, 0.3)} {(H 3, 0.7, 0.3), (H 3, 0.9, 0.4), (H 4, 0.8, 0.3)} A 3 {(H 4, 0.8, 0.3), (H 4, 0.8, 0.2), (H 5, 0.7, 0.2)} {(H 4, 0.9, 0.3), (H 5, 0.8, 0.2), (H 6, 0.8, 0.1)} A 4 {(H 3, 0.8, 0.1), (H 3, 0.6, 0.2), (H 4, 0.8, 0.1),} {(H 3, 0.9, 0.2), (H 4, 0.9, 0.3), (H 5, 0.8, 0.2)} 5.1. Method 1 Step 1. Normaize decision making matrix P m n = (p ij ) m n as R m n = (r ij ) m n. A indexes in this exampe are benefit indexes, so they are omitted. Step 2. Suppement data. Since there are three experts in this exampe, none of whom has abstained or given the same score, it is not necessary to suppement the data, and this step is omitted. Step 3. Transform grey anguage data into the score function of the kerne and grey scae through Formua (5), and Tabe 2 is obtained. Tabe 2. ˆ g o form of grey hesitant fuzzy decision matrix. Industries Attribute (P 1 ) Attribute (P 2 ) A 1 2.3333 (0.40) 2.4000 (0.40) A 2 2.7000 (0.40) 2.9000 (0.40) A 3 3.2667 (0.20) 3.7333 (0.20) A 4 3.1667 (0.40) 2.6667 (0.40) Industries Attribute (P 3 ) Attribute (P 4 ) A 1 2.0667 (0.50) 1.4000 (0.40) A 2 2.1000 (0.30) 2.6667 (0.40) A 3 3.3000 (0.30) 4.1333 (0.30) A 4 2.4667 (0.20) 3.4333 (0.30) Step 4. Cacuate grey correation degree and grey scae. ξ i (j) = 0.5942 0.5062 0.5256 0.3333 0.7069 0.6212 0.5325 0.4824 1.0000 1.0000 1.0000 1.0000 0.9318 0.5616 0.6212 0.6613 γ(i) = (0.4898, 0.5857, 1.0000, 0.6940) T

Int. J. Environ. Res. Pubic Heath 2018, 15, 436 11 of 14. γ g o(i) = (0.4898 (0.50), 0.5857 (0.40), 1.0000 (0.30), 0.6940 (0.40) ) T. Step 5. Rank the schemes according to the grey correation and grey scae of each industry. Obviousy, A 3 A 4 A 2 A 1. 5.2. Comparison Method 2 The foowing steps can be obtained according to the traditiona decision making method in grey anguage. Step 1. Normaize decision making matrix P m n = (p ij ) m n as R m n = (r ij ) m n. A indexes in this exampe are benefit indexes, so they are omitted. Step 2. Aggregate data r ij through Formua (1) to obtain z ij, as shown in Tabe 3. Tabe 3. z ij of grey anguage decision matrix. Industries Attribute (P 1 ) Attribute (P 2 ) A 1 {(H 3.6667, 0.6364, 0.4)} {(H 3, 0.8, 0.4)} A 2 {(H 3.3333, 0.89, 0.4)} {(H 3.6666, 0.7909, 0.4)} A 3 {(H 4.6667, 0.7, 0.2)} {(H 5, 0.7615, 0.2)} A 4 {(H 4, 0.7917, 0.4)} {(H 4, 0.6667, 0.4)} Industries Attribute (P 3 ) Attribute (P 4 ) A 1 {(H 3, 0.6888, 0.5)} {(H 2, 0.7, 0.4)} A 2 {(H 3, 0.7, 0.4)} {(H 3.6667, 0.8, 0.4)} A 3 {(H 5, 0.8267, 0.3)} {(H 2, 0.7, 0.3)} A 4 {(H 3.3333, 0.74, 0.2)} {(H 4, 0.8583, 0.3)} Step 3. Cacuate the score function S(a ij ) of z ij through Formua (2), as shown in Tabe 4. Tabe 4. S(a ij ) decision matrix of grey anguage hesitation fuzzy. Industries Attribute (P 1 ) Attribute (P 2 ) Attribute (P 3 ) Attribute (P 4 ) A 1 1.4000 1.4400 1.0333 0.8400 A 2 1.7800 1.7400 1.2600 1.7600 A 3 2.6133 3.0462 2.8933 0.9800 A 4 1.9000 1.6000 1.9733 2.4033 Step 4. Cacuate grey correation coefficient based on S(a i ) to obtain index weight w(j). ξ i (j) = 0.4339 0.3667 0.3333 0.3730 0.5274 0.4159 0.3628 0.5911 1.0000 1.0000 1.0000 0.3952 0.5659 0.3914 0.5027 1.0000 w = (0.2729, 0.2348, 0.2375, 0.2548 ) T Step 5. Cacuate the comprehensive criterion vaues of each scheme through the GLHFWAA operator. Z 1 = 0.2729 (H 3.6667, 0.6364, 0.4) + 0.2348 (H 3, 0.8, 0.4) + 0.2375 (H 3, 0.6889, 0.5) +0.2548 (H 2, 0.7, 0.4) = (H 2.9272, 0.6996, 0.5)

Int. J. Environ. Res. Pubic Heath 2018, 15, 436 12 of 14 Simiary, Z 2 = (H 3.4174, 0.8008, 0.4), Z 3 = (H 4.1446, 0.7537, 0.3), Z 4 = (H 3.8417, 0.7681, 0.4). Step 6. Compare Z i (i = 1, 2, 3, 4) and sort the schemes. IH 2.9272 0.6697 (1 0.5) < IH 3.4174 0.8008 (1 0.4) < IH 3.8417 0.7681 (1 0.4) < IH 4.1446 0.7535 (1 0.3) Z 1 < Z 2 < Z 4 < Z 3. Therefore, industry A 3 is the optima, foowed by industry A 4 and industry A 1 is the worst. 5.3. Comparison Method 3 Step 5 can aso be used to continue the decision making after decision step 4 in method 2. Step 5. Seect the grey anguage S(a ij ) decision matrix and grey correation weight w(j) by using the mutipe inear weighting method. The comprehensive criterion vaue of each scheme is cacuated through the foowing formua: n Z i = S(a ij ) w(j) j=1 1.4000 1.4400 1.0333 1.7800 1.7400 1.2600 = 2.6133 3.0462 2.8933 1.9000 1.6000 1.9733 = [1.1796 1.6420 2.3653 1.9752 ] T 0.8400 1.7600 0.9800 2.4033 (0.2729, 0.2348, 0.2375, 0.2548 )T Z 1 < Z 2 < Z 4 < Z 3, which is consistent with the sorting obtained through methods 1 and 2. By comparison, the grey scoring function used in method 1 is more reasonabe and simpe. Method 3 is simper than method 2. On the whoe, it can be concuded that method 1 method 3 method 2. 6. Concusions This paper studied the grey anguage decision making method, and defined the scoring function of grey anguage based on the kerne and grey scae which is more suitabe for uncertain decision making. Moreover, a group decision-making method based on the grey correation degree of the kerne and grey scae was proposed. In comparison, the method is more reasonabe than traditiona methods by introducing the kerne and grey scae to avoid adding subjective data, thereby reducing information distortion; in addition, the grey reationa degree decision method does not need to cacuate the index weight, and hence to some extent this method is more convenient. Of course, it is hard to avoid the ack of depth ony through a comparison of three different decision methods of the same exampe, so we wi conduct systematic comparative research on data simuation in future research so as to improve the credibiity of the comparison. Acknowedgments: Nationa Science Foundation of China under Grants (71671090); Manufacturing Deveopment Research Institute of China (SK20140090-15, SK20150090-7); Aeronautica Science Fund of China (2016ZG52068). Author Contributions: Qingsheng Li designed the structure and improved the manuscript; Zaiwu Gong proposed the idea; Yuzhu Diao performed the cacuations; Aqin Hu wrote the paper. Conficts of Interest: The authors decare no confict of interest. References 1. Zadeh, L.A. Fuzzy sets. Inf. Contro 1965, 8, 338 353. [CrossRef] 2. Nassov, K. Intuitionistic fuzzy sets. Fuzzy Sets Syst. 1986, 20, 87 96.

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