Expanding Grey Relational Analysis With the Comparable Degree for Dual Probabilistic Multiplicative Linguistic Term Sets and Its Application on the Cloud Enterprise
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Postgraduate Research and Practice Innovation Program of Jiangsu Province under Grant KYCX18_0199
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Received May 4, 2019, accepted May 23, 2019, date of publication May 28, 2019, date of current version June 20, 2019. Digital Object Identifier 10.1109/ACCESS.2019.2919505 Expanding Grey Relational Analysis With the Comparable Degree for Dual Probabilistic Multiplicative Linguistic Term Sets and Its Application on the Cloud Enterprise WANYING XIE 1,3, ZESHUI XU 1,2, (Fellow, IEEE), ZHILIANG REN 1, AND ENRIQUE HERRERA-VIEDMA 3,4 1School of Economics and Management, Southeast University, Nanjing 211189, China 2State Key Laboratory of Hydraulics and Mountain River Engineering, Business School, Sichuan University, Chengdu 610064, China 3Andalusian Research Institute in Data Science and Computational Intelligence, University of Granada, E-18071 Granada, Spain 4Department of Electrical and Computer Engineering, Faculty of Engineering, King Abdulaziz University, Jeddah 21589, Saudi Arabia Corresponding authors: Zeshui Xu ([email protected]) and Enrique Herrera-Viedma ([email protected].es) This work was supported in part by the Scientific Research Foundation of the Graduate School of Southeast University under Grant YBJJ1832, in part by the Postgraduate Research and Practice Innovation Program of Jiangsu Province under Grant KYCX18_0199, and in part by the FEDER Financial Support under Grant TIN2016-75850-R. ABSTRACT Under the cloud trend of enterprises, how do traditional businesses get on the cloud becomes a worth pondering question. To help those traditional businesses that have no experience to dispel the clouds and see the sun as soon as possible, we are planning to choose one corporation with rich experience to take them into the cloud market. The quintessence of dual probabilistic linguistic term sets (DPLTSs) is that it uses the combination of several linguistic terms and their proportions to reveal decision information by opposite angles. This paper proposes the dual probabilistic multiplicative linguistic preference relations (DPMLPRs) based upon the dual probabilistic multiplicative linguistic term sets (DPMLTSs). Then, it defines the comparable degree between the DPMLPRs and studies the consensus of the group DPMLPR. Moreover, it probes the expanding grey relational analysis (EGRA) under the proposed comparable degree between the DPMLTSs. After that, one example of choosing the experienced cloud cooperative partner is simulated under the dual probabilistic linguistic circumstance. Besides, the comparative analysis is performed by considering the similarity among the EGRA, TODIM, and VIKOR. INDEX TERMS Dual probabilistic multiplicative linguistic preference relations, comparable degree, consensus, expanding grey relational analysis, multi-criteria decision making. I. INTRODUCTION Just like domino effect, since cloud computing [1] was first proposed by Eric Schmidt in 2006, the market for cloud computing is booming. Its research has been gotten a lot of attention from experts in different fields, such as internet of things [2]–[4], cloud storage [5], [6], cloud security [7], [8], cloud education [9], [10] and so on. The essence of cloud computing is to provide services through the network, so its architecture is centered on services, and its objective is to The associate editor coordinating the review of this manuscript and approving it for publication was Muhammad Imran Tariq. offer customers with faster and more convenient information services. The currently acknowledged traits of cloud computing can be summarized as follows: (1) Supersize dimension, such as Amazon, IBM, Microsoft and Yahoo, each has hundreds of thousands of servers, ‘‘Cloud’’ is able to offer consumers unheard-of calculating strength; (2) Virtualization, cloud computing permits consumers to make use of application services from facultative situation utilizing all kinds of terminals. The desired resource is derived from the ‘‘Cloud’’ rather than an established concrete existence. The app operates someplace in the ‘‘Cloud’’. However, as a matter of fact, the consumers are not necessary to learn about or concern VOLUME 7, 2019 2169-3536 2019 IEEE. Translations and content mining are permitted for academic research only. Personal use is also permitted, but republication/redistribution requires IEEE permission. See http://www.ieee.org/publications_standards/publications/rights/index.html for more information. 75041
W. Xie et al.: EGRA With the Comparable Degree for DPMLTSs and Its Application on the Cloud Enterprise about where the app is operating. With just one laptop or one mobile phone, you are able to do everything we need through web services, even tasks like supercomputing. (3) Dynamic extendibility, the dimension of the cloud can be vibrantly scaled to fulfill the demands of adhibition and consumers scale growth. (4) High reliability, ‘‘Cloud’’ uses measures such as the fault tolerance for multiple copies of data and computational node isomorphism to ensure high reliability of services. Cloud computing is more responsible than utilizing local computers. (5) Commonality, cloud computing is not targeted at particular applications. With the help of ‘‘Cloud’’, it can structure protean applications. The identical ‘‘Cloud’’ can encourage diverse application operations in the mean time. (6) Service on demand, ‘‘Cloud’’ is a large resource pool that you are able to purchase according to the requirement, and clouds are able to be charged like water, electricity or gas. (7) Low cost and green energy saving. Because the particular fault-tolerant measures of ‘‘Cloud’’ can utilize rare cheap nodes to constitute a cloud, the cloud’s automated centralized management eliminates the need for big business to afford cumulatively advanced data center management costs, and the versatility of ‘‘Cloud’’ enables the exploitation rate of resources much higher than the conventional system. Moreover, consumers are able to thoroughly enjoy the low-cost benefit of ‘‘Cloud’’. Therefore, many traditional businesses begin to transform the cloud computing industry. However, majority of them do not have the relative experience, it is full of hazard for them to join in the cloud market. So it is a good choice for them to look for a good partner that with the rich experience to get twofold results with half the effort. As far as it goes, the world’s four largest cloud computing companies are Amazon Web Services (AWS), Microsoft, Google and Alibaba Cloud. According to their own features, choosing one to collaborate with the four companies is the short cut for those traditional businesses that want to transform in the demand explosion period of cloud industry. How to determine the selected company becomes the question that we will solve in this paper. The DPLTSs [11] enlarges probabilistic linguistic term sets (PLTSs)’ [12] quintessence that uses the combination of several linguistic terms and their proportions to reveal decision information into the membership sentiment and non-membership sentiment. We extend it into the multiplicative linguistic scale [13] and define the dual probabilistic multiplicative linguistic term sets (DPMLTSs). Then we propose the notion of dual probabilistic multiplicative linguistic preference relations (DPMLPRs), and use the DPMLPRs as the implement to do the decision. As most of the studies on the preference relations (PRs) [14]–[20], the consistency [21]–[26] is the common and essential condition for applying the PRs into the material decision. Different from the majority of researchers [27], [28], this paper defines the comparable degree between the DPMLPRs and utilizes it as the measure to judge the consistency of the DPMLPRs. The reason why we use the comparable degree is that the intrinsic quality between the comparable degree [29]–[31] and the distance measure [32], [33] is same. Moreover, because of the structure of the operator itself, the computation of the comparable degree is also separated into two angles: the membership viewpoint and the non-membership viewpoint. After acquiring the consistent DPMLPRs, on account of the defined dual probabilistic linguistic weighted geometric aggregation operator (DPLWGA), we can obtain the group DPMLPR. Then on the foundation of the established comparable degree between the individual DPMLPRs and the group DPMLPR, the group consensus [34]–[38] can be checked directly. Moreover, if the consensus cannot be satisfied in the decision-making procedure, then the decision makers (DMs) need to adjust their PRs, until the consensus is satisfied in the end, and the checking is over. The crucial intention of decision-making is to judge the sort of the alternatives. For the multi-criteria decisionmaking, the research for weights has been done a lot [39]–[42]. Most of them are divided into the following types: partially known [43]–[45], fully known [46], [47], total unknown [48]–[51]. The weights of criteria in this paper is belong to the third type that is total unknown. On the foundation of classic arithmetic averaging method [52], this paper considers the structural characteristics of DPMLTSs and designs the modified arithmetic averaging method to calculate the weights for criteria. After that, the grey relational analysis (GRA) [53] as one of the more common multi-criteria decision-making method, its superiority lies in that it does not require much of the quantity involved in the decision-making. Moreover, it does not require that the quantities to be determined conform to a typical distribution. The amount of calculation is relatively small, and the results agree well with the qualitative analysis. So the GRA has been expanded in this paper by merging with the proposed comparable degree to calculate the relational coefficient. The GRA based upon the comparable degree is named as expanding GRA (EGRA). Together with the weights of the criteria, the final priority of the alternatives is able to be procured at length. Furthermore, we apply the proposed procedure to the case mentioned above and to help to determine the selected cooperative partner. Besides, given that the similar principle among the GRA, TODIM [54] and VIKOR [55] that studies the comparable degree between the alternative and ideal alternative, we also expand the TODIM, VIKOR into the expanding TODIM (ETODIM), expanding VIKOR (EVIKOR). Then we compare the EGRA, ETODIM and EVIKOR in the comparative analysis section, and show their several advantages and disadvantages. In a word, the innovation points of the whole paper can be listed as follows: (1) Define the DPMLPRs; (2) Denote the comparable degree between the individual DPMLPRs; (3) Study the consistency of the individual DPMLPRs; (4) Research the consensus of group DPMLPR; (5) Propose the EGRA method based on the defined comparable degree 75042 VOLUME 7, 2019
W. Xie et al.: EGRA With the Comparable Degree for DPMLTSs and Its Application on the Cloud Enterprise between the DPMLTSs; (6) Expand the TODIM and VIKOR methods. The remaining of this paper is structured as follows: Section II lists some necessary notions. Section III defines the DPMLTSs, the basic operations among the DPMLTSs, the comparable degree between the individual DPMLPRs, and study the consistency, consensus of the DPMLPRs. Section IV computes the weights of criteria, introduces the EGRA method, and the integrated multi-criteria decisionmaking procedure. Section V utilizes a simulation case relevant to the cloud computing industry to clarify the potential and reality of the dual probabilistic multiplicative linguistic multi-criteria group decision-making procedure. Section VI ends with some conclusions. II. PRELIMINARIES In this section, we will briefly recall some essential concepts, such as the linguistic terms, the dual probabilistic linguistic term set (DPLTS) and the normalized dual probabilistic linguistic term element (NDPLTE). A. THE LINGUISTIC TERM SETS Let S=sαα∈1q,qbe a continuous multiplicative linguistic label set, and qis a adequately large positive integer [13]. Moreover, if α > β, then sα>sβ; if rec (sα)= sβ, then αβ =1; peculiarly, rec (s1)=s1. Based on the multiplicative linguistic label set S, Xu [13] introduced some basic operational laws for them as follows: (sα)µ=sαµ, µ ∈[0,1]; sα⊗sβ=max s1/q,min sαβ ,sq; sα⊕sβ=max s1/q,min sα+β,sq. B. THE DPLTS Let Xbe a fixed set, a DPLTS on Xcan be signified into the coming type [11]: D={hx, ℘ (p), ϒ (p)i,x∈X}(1) where ℘(p)= ℘(i)p(i)℘(i)∈S1,p(i)≥0, #℘(p) X i=1 p(i)≤1 , ϒ(p)= ϒ(j)p(j)ϒ(j)∈S1,p(j)≥0, #ϒ(p) X j=1 p(j)≤1 . ℘(p)and ϒ(p)stand for the conceivable membership and non-membership degrees to the element x∈Xfor the set Dwith the conditions that s−q≤℘+⊕ϒ+≤sq,s−q≤ ℘−⊕ϒ−≤sq,S1={sα|α∈[−q,q]}. In addition to that, we call the pair D=h℘(p), ϒ (p)ithe dual probabilistic linguistic element (DPLTE). Moreover, in the cause of reducing the trouble of the computation, Xie et al. [11] further designed the coming procedure to normalize the DPLTEs (NDPLTEs) as follows: Assume that D1=h℘1(p), ϒ1(p)iand D2= h℘2(p), ϒ2(p)iare two unlike DPLTEs. For the first step, similar to earn the NPLTSs, there is a need to avoid the deviations in the cardinalities of the two PLTSs ℘1(p)and ℘2(p), and to score the PLTSs ℘1(p)and ℘2(p)with the identical cardinal numbers: #℘1(p)=#℘2(p). For the second step, we need to replume the PLTSs ℘1(p)and ℘2(p) separately in the downward sort. Likewise, the PLTSs ϒ1(p) and ϒ2(p)also need to be treated with the same way. Then we can obtain two new DPLTEs D0 1=℘0 1(p), ϒ0 1(p), D0 2=℘0 2(p), ϒ0 2(p), where ℘0l(p)= ℘0(i) lp0 l(i)℘0(i) l∈S,p0(i) l≥0, #℘0l(p) X i=1 p0(i) l≤1 and ϒ0l(p)= ϒ0(j)p0(j) l lϒ0(j) l∈S,p0(j) l≥0, #ϒ0l(p) X j=1 p0(j) l≤1 are revealed in falling sort, #℘0 1(p)=#℘0 2(p), #ϒ0 1(p)= #ϒ0 2(p),l=1,2. Moreover, we offer the definition of score function and accuracy function [11] to compare the different DPLTEs as follows: For a DPLTE D=h℘(p), ϒ (p)i, it’s score function is: S(D)=s¯α−¯ β(2) where ¯α=P#℘(p) i=1I℘(i)p(i)/P#℘(p) i=1p(i),¯ β= P#ϒ(p) j=1Iϒ(j)r(j)p(j)/P#ϒ(p) j=1p(j)and I(·)is the function that can obtain the subscript of the corresponding linguistic term. With regard to two DPLTEs Dl(l=1,2), if S(D1)> S(D2), then D1is superior to D2, denoted by D1D2; if S(D1)<S(D2), then D1is inferior to D2, denoted by D1≺D2. If S(D1)=S(D2), it is tight to tell from two DPLTEs. Thus, we state the accuracy function for the DPLTE as follows: For a DPLTE D=h℘(p), ϒ (p)i, it’s accuracy function can be ruled as: A(D) =X#℘(p) i=1p(i)I℘(i)− ¯α21/2 /X#℘(p) i=1p(i) +X#ϒ(p) j=1p(j)Iϒ(j)−¯ β21/2 /X#ϒ(p) j=1p(j) (3) Hence, with regard to two DPLTEs Dl(l=1,2)with S(D1)=S(D2), if A(D1)<A(D2), then D1D2; if A(D1)>A(D2), then D1≺D2; if A(D1)=A(D2), then D1∼D2. VOLUME 7, 2019 75043
W. Xie et al.: EGRA With the Comparable Degree for DPMLTSs and Its Application on the Cloud Enterprise III. THE DUAL PROBABILISTIC MULTIPLICATIVE LINGUISTIC TERM SETS Considering the multiplicative linguistic label set [13] and the defined DPLTS together, next we extend the DPLTS into the environment of multiplicative linguistic label set, and study the basic operations in the following section. A. THE DPMLTS Let Xbe a fixed set, a DPMLTS on Xcan be shown as the following style: D={hx, ℘ (p), ϒ (p)i,x∈X}(4) where ℘(p)= ℘(i)p(i)℘(i)∈S,p(i)≥0, #℘(p) X i=1 p(i)≤1 , ϒ(p)= ϒ(j)p(j)ϒ(j)∈S,p(j)≥0, #ϒ(p) X j=1 p(j)≤1 . ℘(p)and ϒ(p)stand for the conceivable membership and non-membership degrees to the element x∈Xfor the set Dwith the situations that s1/q≤℘+⊗ϒ+≤sq, s1/q≤℘−⊗ϒ−≤sq. Additionally, we call the pair D= h℘(p), ϒ (p)ithe dual multiplicative probabilistic linguistic element (DPMLTE). Then on behalf of better applying the DPMLTEs in to the practical case, we regulate the essential operation for the DPMLTEs as follows: For two DPMLTEs D1=h℘1(p), ϒ1(p)iand D2= h℘2(p), ϒ2(p)i, then the multiplicative operation is D1⊗D2=h℘1(p), ϒ1(p)i⊗h℘2(p), ϒ2(p)i =h℘1(p)⊗℘2(p), ϒ1(p)⊗ϒ2(p)i(5) Based on the Ref. [3], where ℘1(p)⊗℘2(p)=[℘(i1) 1∈℘1(p),℘(i2) 2∈℘2(p)n℘(i1) 1⊗℘(i2) 2 p(i1) 1p(i2) 2|i1=1,2,...,#℘1(p),i2=1,2,...,#℘2(p)} ϒ1(p)⊗ϒ2(p)=[ϒ(j1) 1∈ϒ1(p),ϒ(j2) 2∈ϒ2(p)nϒ(j1) 1⊗ϒ(j2) 2 p(j1) 1p(j2) 2|j1=1,2,...,#ϒ1(p),j2=1,2,...,#ϒ2(p)} The power operation is (D1)λ=h℘1(p), ϒ1(p)iλ=(℘1(p))λ,(ϒ1(p))λ(6) where (℘1(p))λ= ∪℘(i1) 1∈℘1(p) ×℘(i1) 1λp(i1) 1λ|i1=1,2,...,#℘1(p). Then let D1,D2,...,Dnbe a set of DPMLTEs, then the dual probabilistic multiplicative linguistic weighted geometric aggregated (DPMLWGA) operator can be expressed as: DPMLWGA (D1,D2,...,Dn) =n ⊗ i=1 (℘i(p))ωi,n ⊗ i=1ϒj(p)ωi(7) where ω=(ω1, ω2, . . . , ωn)Tis the weight vector with respect to the DPMLTEs, and fulfills ωi∈[0,1]and n P i=1 ωi=1. B. THE DPMLPR In the cause of applying the DPMLTSs to the decisionmaking procedure, in the following, we define the dual probabilistic multiplicative linguistic preference relation (DPMLPR) as follows: A DPMLPR on the mentioned set S=sαα∈1q,q is defined as the matrix D=dijn×n,dij =℘ij (p), ϒij (p), which meets ℘ij (p)=ϒji (p), ϒij (p)=℘ji (p)(8) i6= j, for i,j=1,2,...,n. Moreover, if i=j, then ℘ii (p)=ϒii (p)=h{s1(1)},{s1(1)}i . It is common knowledge that the consistency of PRs is the essential requirement for logical decision-making. So it is no exception to study the consistency of the defined DPMLPRs. For a DPMLPR D=℘ij (p), ϒij (p), if Dis consistent, then it should satisfy the following conditions: for ∀i,k,j= 1,2,...,n, (℘ij (p)=℘ik (p)⊗℘kj (p) ϒij (p)=ϒik (p)⊗ϒkj (p)(9) which means that the DPMLPR Dis consistent if and only if the membership PR ℘=℘ij (p)and the non-membership ϒ=ϒij (p)are consistent at the same time. For the single membership PR ℘=℘ij (p), its consistent PR C℘=C℘ij (p), where C℘ij (p)=n rn ⊗ k=1℘ik (p)⊗℘kj (p). By learning from the Ref. [56], its consistency index can be calculated as follows: CI℘=1− n X i,j=1,i<j 2 (n−1) (n−2)log e℘ij −log eC℘ij (10) where for a DPMLTE D=h℘(p), ϒ (p)i, the expected value of the DPMLTE is ED=*#℘(p) X i=1 p(i)I℘(i), #ϒ(p) X j=1 p(j)Iϒ(j)+=e℘(p),eϒ(p) (11) 75044 VOLUME 7, 2019
W. Xie et al.: EGRA With the Comparable Degree for DPMLTSs and Its Application on the Cloud Enterprise Example 1: For one DPMLTE D=s1/2(0.4),s3(0.6), {s2(0.3),s1(0.5)}i on the certain linguistic term set S= sαα∈19,9, then expected value of the DPMLTE is ED=*#℘(p) X i=1 p(i)I℘(i), #ϒ(p) X j=1 p(j)Iϒ(j)+=h2.0,1.1i. Then for the DPMLPR D, its consistency index can be computed as below: CI =1− n X i,j=1,i<j 1 (n−1) (n−2) ×log e℘ij −log eC℘ij +log eϒij −log eCϒij (12) Moreover, the consistency procedure can be expressed as the Algorithm 1: Algorithm 1 The Procedure to Adjust the Consistency Step 1. Set the threshold value for the consistency index 4, and calculate the respective consistency index CIi(i=1,2,...,n)for the DPMLPRs; Step 2. Judge the consistency of the DPMLPRs, if CIi> 4, then go to Step 4; Otherwise, go to the next step. Step 3. Modify the elements of the DPMLPRs according to the following method: (℘0 ij (p)=℘ij (p)θ⊗C℘ij (p)1−θ ϒ0 ij (p)=ϒij (p)θ⊗Cϒij (p)1−θ(13) where θ∈[0,1]is a regulation parameter. Step 4. Let D0=D, then go back to Step 1. C. THE COMPATIBILITY DEGREE FOR DPMLPRS For the obtained consistent DPMLPRs, we are devote to study the consensus of the group DPMLPRs. Usually, people like to choose the distance measure [32], [33] or the similarity measure [57]–[59] as the foundation to analyze the consensus of the group PR. In this paper, with an eye to the similar practical meaning among the distance measure, similarity measure and comparable degree, we utilize the comparable degree between the DPMLPRs as the foundation to research the consensus of group PR in the following subsection: Before introducing the comparable degree between the DPMLPRs, we first give the notion of comparable degree for two DPMLTEs. For any two DPMLTEs D1= h℘1(p), ϒ1(p)iand D2=h℘2(p), ϒ2(p)i, the comparable degree between two DPMLTEs can be calculated as follows: C(D1,D2)=1 2log e℘ 1−log e℘ 2+log eϒ 1−log eϒ 2(14) Furthermore, for two different DPMLPRs D1= d1 ijn×n=D℘1 ij (p), ϒ1 ij (p)En×nand D2=d2 ijn×n= D℘2 ij (p), ϒ2 ij (p)En×n, the comparable degree of D1and D2 can be defined as: C(D1,D2) =1 n(n−1) 1 2 n X i=1 n X j=1 log e℘ ij1−log e℘ ij2+ log eϒ ij1−log eϒ ij2 (15) where ED1=De℘ ij1,eϒ ij1Eand ED2=De℘ ij2,eϒ ij2Eare the homologous expected value of the different DPMLPRs D1 and D2, respectively. Moreover, if e℘ ij1=e℘ ij2,eϒ ij1=eϒ ij2,i,j=1,2,...,n, than we call DPMLPRs D1and D2are perfectly compatible. Theorem 1: For two DPMLPRs D1and D2, then (a)C(D1,D2)≥0; (b)C(D1,D2)=C(D2,D1); (c)C(D1,D2)=0, if D1and D2are perfectly compatible. It is easy to see that Eqs. (a-c) are apparent. Therefore, the proof is omitted. Theorem 2: For three different DPMLPRs D1,D2and D3, we have C(D1,D3)≤C(D1,D2)+C(D2,D3) Proof: C (D1,D3), as shown at the top of the next page. Definition 1: For two DPMLPRs D1and D2, if C(D1,D2)≤δ(16) then we call that D1and D2are of acceptable compatibility, where δis the threshold value of acceptable compatibility. For a set of DPMLPRs D1,D2,...,Dn, the group DPMLPR Dcan be expressed as the following form, D, as shown at the top of the next page. Theorem 3: For a set of DPMLPRs Di=di st n×n= ℘i st (p), ϒi st (p)n×n, one DPMLPR D∗=d∗ st n×n= ℘∗ st (p), ϒ∗ st (p)n×n,s=1,2,...,n,t=1,2,...,n and D=(dst )n×n=(h℘st (p), ϒst (p)i)n×nis the group DPMLPR of the set of DPMLPRs Di(i=1,2,...,n)by utilizing the weight vector ω=(ω1, ω2, . . . , ωn)T, then if C(Di,D∗)≤δ,i=1,2,...,n, then C(D,D∗)≤δ, where δis the threshold value of acceptable compatibility. Proof: D =(dst )n×n=(hLst (p),Ust (p)i)n×n, where ℘st (p)=n ⊗ i=1 (℘sti (p))ωi =n ⊗ i=1℘ωi sti pωi sti=I−1 n Y i=1 (I(℘sti))ωi! n Y i=1 pωi sti!, ϒst (p)=n ⊗ i=1 (ϒsti (p))ωi =n ⊗ i=1ϒωi sti pωi sti=I−1 n Y i=1 (I(ϒi))ωi! n Y i=1 pωi sti!. Then EDi=e℘ sti,eϒ sti(i=1,2,...,n),ED=e℘ st ,eϒ st , ED∗=e℘∗ st ,eϒ∗ st ,e℘ st =n Q i=1 (I(℘sti))ωin Q i=1 pωi sti,eϒ st = n Q i=1 (I(ϒi))ωin Q i=1 pωi sti. VOLUME 7, 2019 75045
W. Xie et al.: EGRA With the Comparable Degree for DPMLTSs and Its Application on the Cloud Enterprise C(D1,D3)=1 n(n−1) 1 2 n X i=1 n X j=1 log e℘ ij1−log e℘ ij3+ log eϒ ij1−log eϒ ij3 =1 n(n−1) 1 2 n X i=1 n X j=1 log e℘ ij1−log e℘ ij2+log e℘ ij2−log e℘ ij3+ log eϒ ij1−log eϒ ij2+log eϒ ij2−log eϒ ij3 ≤1 n(n−1) 1 2 n X i=1 n X j=1 log e℘ ij1−log e℘ ij2+log e℘ ij2−log e℘ ij3+ log eϒ ij1−log eϒ ij2+log eϒ ij2−log eϒ ij3 ≤1 n(n−1) 1 2 n X i=1 n X j=1log e℘ ij1−log e℘ ij2+log eϒ ij1−log eϒ ij2 +1 n(n−1) 1 2 n X i=1 n X j=1log e℘ ij2−log e℘ ij3+log eϒ ij2−log eϒ ij3 =C(D1,D2)+C(D2,D3) D= D11 D12 · · · D1n D21 D22 · · · D2n . . .. . ..... . . Dm1Dm2· · · Dmn = n ⊗ i=1℘i 11 (p)ωi,n ⊗ i=1ϒi 11 (p)ωi n ⊗ i=1℘i 12 (p)ωi,n ⊗ i=1ϒi 12 (p)ωi· · · n ⊗ i=1℘i 1n(p)ωi,n ⊗ i=1ϒi 1n(p)ωi n ⊗ i=1℘i 21 (p)ωi,n ⊗ i=1ϒi 21 (p)ωi n ⊗ i=1℘i 22 (p)ωi,n ⊗ i=1ϒi 22 (p)ωi· · · n ⊗ i=1℘i 2n(p)ωi,n ⊗ i=1ϒi 2n(p)ωi . . .. . ..... . . n ⊗ i=1℘i n1(p)ωi,n ⊗ i=1ϒi n1(p)ωi n ⊗ i=1℘i n2(p)ωi,n ⊗ i=1ϒi n2(p)ωi· · · n ⊗ i=1℘i nn (p)ωi,n ⊗ i=1ϒi nn (p)ωi Since CDi,D∗ =1 n(n−1) "1 2 n X s=1 n X t=1log e℘ sti −log e℘∗ st +log eϒ sti −log eϒ∗ st #, ≤δ then, C(D,D∗), as shown at the top of the next page. Thus the proof is completed. Theorem 4: For two sets of DPMLPRs Di(i=1,2,...,n), Di(i=1,2,...,n),Dis the group DPMLPR of the set of DPMLPRs Di(i=1,2,...,n)and Dis the group DPMLPR of the set of DPMLPRs Di(i=1,2,...,n)by utilizing the same weight vector ω=(ω1, ω2, . . . , ωn)T, respectively, then if C(Di,Di)≤δ,i=1,2,...,n, then C(D,D)≤δ, where δis the threshold value of acceptable compatibility. The proof is similar to that of Theorem 4, so the specific proof process is omitted. Then the group consensus procedure can be listed in Algorithm 2. IV. METHODOLOGY In this section, the determination of weights for criteria based on the group DPMLPR and the patulous GRA are presented in detail. A. THE WEIGHTS FOR CRITERIA Based on the algorithm in Section III, we can get a group DPMLPR D=Dijn×n=℘ij (p), ϒij (p)n×nwith the acceptable consensus degree. Then for the DPMLPR D=Dijn×n, with a view to the construction features of the elements in DPMLPR, the classic arithmetic averaging method [52] cannot be used directly. So we give the following equation to calculate the weights for criteria: $i= n X j=1 ISDijn X i=i n X j=1 ISDij (17) where S(·)is the score function of Dij. 75046 VOLUME 7, 2019
W. Xie et al.: EGRA With the Comparable Degree for DPMLTSs and Its Application on the Cloud Enterprise CD,D∗=1 n(n−1) "1 2 n X s=1 n X t=1 log e℘ st −log e℘∗ st +log eϒ st −log eϒ∗ st !# =1 n(n−1) 1 2 n X s=1 n X t=1 log ( n Y i=1 (I(℘sti))ωi! n Y i=1 (psti)ωi!)−log e℘∗ st + log ( n Y i=1 (I(ϒsti))ωi! n Y i=1 (psti)ωi!)−log eϒ∗ st =1 n(n−1) 1 2 n X s=1 n X t=1 log ( n Y i=1 I(℘sti)!ωi n Y i=1 psti!ωi)−log e℘∗ st + log ( n Y i=1 I(ϒsti)!ωi n Y i=1 psti!ωi)−log eϒ∗ st =1 n(n−1) 1 2 n X s=1 n X t=1 n X i=1 ωilog {I(℘sti)psti}−log e℘∗ st + n X i=1 ωilog {I(ϒsti)psti}−log eϒ∗ st ≤1 n(n−1) 1 2 n X s=1 n X t=1 n X i=1 ωilog {I(℘sti)psti}−log e℘∗ st + n X i=1 ωilog {I(ϒsti)psti}−log eϒ∗ st = n X i=1 ωi(1 n(n−1) "1 2 n X s=1 n X t=1 log {I(℘sti)psti}−log e℘∗ st + log {I(ϒsti)psti}−log eϒ∗ st !#) = n X i=1 ωi(1 n(n−1) "1 2 n X s=1 n X t=1 log e℘ sti −log e℘∗ st + log eϒ sti −log eϒ∗ st !#) ≤ n X i=1 ωiδ=δ B. THE EXPANDING GREY RELATIVE ANALYSIS METHOD With regard to the individual dual probabilistic linguistic decision-making matrices given by the DMs for the alternatives (a1,a2,...,am)with respect to the criteria (c1,c2,...,cn), the group dual linguistic decision-making matrix M=Mijm×ncan be acquired by Eq. (7) as follows: M= c1c2· · · cn a1 a2 . . . am M11 M12 · · · M1n M21 M22 · · · M2n . . .. . ..... . . Mm1Mm2· · · Mmn (18) Let M+=M+ 1,M+ 1,...,M+ nTand M−= M− 1,M− 1,...,M− nTbe the positive ideal element (PIE) and the negative ideal element (PIE) in M, respectively, where M+ j=max iMij,M− j=min iMij,M+ jand M− jare determined through Eq. (2) or Eq. (3). Due to the reality that the comparable degree is similar to the distance measure in physical significance, in the light of the proposed comparable degree between the DPMLTSs, the grey relative coefficient matrices based on the PIE and the NIE are extended as follows: µ+ ij = min 1≤i≤mmin 1≤j≤nC(Mij,M+ j)+ξmax 1≤i≤mmax 1≤j≤nC(Mij,M+ j) C(Mij,M+ j)+ξmax 1≤i≤mmax 1≤j≤nC(Mij,M+ j) (19) µ− ij = min 1≤i≤mmin 1≤j≤nC(Mij,M− j)+ξmax 1≤i≤mmax 1≤j≤nC(Mij,M− j) C(Mij,M− j)+ξmax 1≤i≤mmax 1≤j≤nC(Mij,M− j) (20) Combined with the weights of criteria, the opposite closeness coefficient to the PIE can be determined by the VOLUME 7, 2019 75047
W. Xie et al.: EGRA With the Comparable Degree for DPMLTSs and Its Application on the Cloud Enterprise Algorithm 2 The Procedure to Adjust the Consensus Step 1: For the set of consistent DPMLPRs Di(i=1,2,...,n), with the Eq. (7) and the subjective weight vector of the DMs, it is easy to obtain the group DPMLPR D. Step 2: Let δbe the threshold value of acceptable compatibility, then calculate the compatibility degree between the individual DPMLPRs Dι(ι=1,2,...,n)and the group DPMLPR D=Dijn×n=℘ij (p), ϒij (p)n×n, if C(Dιo,D)≤σ, then group DMLPR is of the acceptable consensus, go to Step 4; Otherwise, go to the next step. Step 3: Let ιo=ιo+1, Dιo+1=Dιo, where Dιo+1 ij =D℘ιo+1 ij (p), ϒιo+1 ij (p)En×n, ℘ιo+1 ij (p)=η℘ιo ij (p)⊗(1−η)℘ij (p), ϒιo+1 ij (p)=ηϒιo ij (p)⊗(1−η)ϒij (p). Then go back to Step 2 until C(Dιo,D)≤σ. Step 4: Let ιo=ιo+1, Dιo+1=Dιo, then go back to Step 1. coming equation: OCi= n P j=1 µ+ ij $j n P j=1 µ+ ij $j+ n P j=1 µ− ij $j (21) The bigger the opposite closeness coefficient OCi, the better the alternative. Then the EGRA method can be illustrated as Algorithm 3: Step 1: Identify the PIE and the NIE of the group dual probabilistic decision-making matrix; Step 2: Calculate the respective grey relative coefficients on the foundation of PIE and NIE; Step 3: Obtain the opposite closeness coefficient for the alternative. C. THE INTEGRATED PROCESS FOR SOLVING MULTI-CRITERIA GROUP DECISION-MAKING PROBLEM On the foundation of Section III and the remaining subsection of Section IV, the integrated decision-making procedure can be concluded as follows: V. SIMULATION EXPERIMENT So as to make the decision-making procedure more detailed, this section performs a concrete simulation experiment relevant to the assessment for the manifestation of cloud enterprise mentioned above. Moreover, this section has four subsections: the first subsection is the practical experimental procedure to make Section II, III and IV particular; the second and third subsections are the comparative analysis; the four subsection is the sensitivity analysis. A. EXPERIMENTAL PROCESS Cloud computing [1] is a type of computing in which vibrantly scalable and always virtualized resources are supplied as a service over the internet. It was first proposed by the CEO Eric Schmidt of Google at the search engine conference in 2006. According to service types, cloud computing is able to be divided into three types: IaaS (Infrastructure-as-a-Service), consumers can get services from a complete computer infrastructure over the internet; PaaS (Platform-as-aService), it uses the software development environment, application environment, etc. as a service to directly provide users with the application platform required by the software; SaaS (Software-as-aService), it is a model for providing software over the internet. Instead of purchasing software, users rent web-based software from providers to manage business operations. The emergence of cloud computing will reshape the IT industry landscape. There will be two clear investment opportunities: one is the new market capacity brought about by the rapid development of the cloud computing industry, and the other is to reshape the emerging industry opportunities brought about by the IT landscape. Considering the broader trend, many corporations are going to the ‘‘Cloud’’. For those IT corporations, they already have own IT costs and IT technology. It is much easier for them to the ‘‘Cloud’’. While for those traditional corporations that lack of network experience want to the ‘‘Cloud’’, they need to bear the cost of trial and error and the risk of failure. It is good choice for those traditional corporations to choose a good partner. Obviously, the so-called good partner shall have rich experience and enough funds to support the traditional industries in need of assistance. Globally, the four giants of the cloud industry are AWS, Microsoft, Google and Alibaba Cloud. As mentioned in Ref. [11], one good partner corporations shall equip with the four features: Corporate value, Independent research and development ability, Corporate size and Product market share. Apparently, the four features are benefit, which means the four features are positively related to the direction of growth. Considering the future development potential of cloud computing, the enterprise who wants to get twofold results with half the effort chooses to collaborate with one of the four giants of the cloud industries: AWS, Microsoft, Google and Alibaba Cloud. Supposed that the four giants of the cloud industries are four evaluated alternatives xi(i=1,2,3,4). To evaluate the four enterprises, they entrust one questionnaire enterprise to investigate the impact of four cloud enterprises under the four previously mentioned aspects. The questionnaire enterprise regards the four mentioned-above aspects as four criteria: Corporate value (c1), Independent research and development ability (c2), Corporate size (c3) Product market share (c4). Obviously, all of the four criteria are benefit. In order to make the evaluation as objective as possible, and consider the DPMLTSs can from the two 75048 VOLUME 7, 2019
W. Xie et al.: EGRA With the Comparable Degree for DPMLTSs and Its Application on the Cloud Enterprise opposite aspects display the decision-making information, the questionnaire enterprise choose the DPMLTSs as the decision-making tool for evaluation. To some extent, not only reflect the membership degree of the decision-making information, but also the non-membership degree. Assume that the DPMLPRs that are given by four DMs for the four alternatives with respect to four criteria are as D1, D2,D3, and D4, as shown at the bottom of this page. Step 1: Let 4=0.9, then we check and improve the consistency of individual DPMLPRs ˜ Di(i=1,2,3,4) by Algorithm 1 as follows: TABLE 1. The consistent degree of individual DPMLPRS. Obviously, based on Table 1, all of the four individual DPMLPRs are not consistent. On the foundation of Algorithm 1, they can be adjusted as D1,D2,D3, and D4, as shown at the next page. The consistent degree of four adjusted individual DPMLPRs are listed as follows: TABLE 2. The consistent degree of adjusted individual DPMLPRs. Step 2: Let the subjective weight of the DMs ˆω= (0.3,0.2,0.15,0.35)T, then we utilize the aggregation operator (7) to figure out the group DPMLPR Das shown at the bottom of the page 11. D1= h{s1(1)},{s1(1)}i h{s6(0.1),s7(0.7),s8(0.2)},{s1/4(0.4),s1/3(0.4)}i h{s1/4(0.4),s1/3(0.4)},{s6(0.1),s7(0.7),s8(0.2)}i h{s1(1)},{s1(1)}i h{s1(0.2),s2(0.2),s3(0.2)},{s4(0.3),s5(0.4),s6(0.3)}i h{s1/4(0.5),s1/3(0.3),s1/2(0.1)},{s1/3(0.2),s1/2(0.6),s1(0.2)}i h{s1/8(0.2),s1/7(0.2)},{s4(0.3),s5(0.2),s6(0.2)}i h{s4(0.4),s6(0.6)},{s1/5(0.3),s1/4(0.2),s1/3(0.5)}i h{s4(0.3),s5(0.4),s6(0.3)},{s1(0.2),s2(0.2),s3(0.2)}i h{s4(0.3),s5(0.2),s6(0.2)},{s1/8(0.2),s1/7(0.2)}i h{s1/3(0.2),s1/2(0.6),s1(0.2)},{s1/4(0.5),s1/3(0.3),s1/2(0.1)}i h{s1/5(0.3),s1/4(0.2),s1/3(0.5)},{s4(0.4),s6(0.6)}i h{s1(1)},{s1(1)}i h{s3(0.6),s4(0.4)},{s1/3(0.7),s1(0.3)}i h{s1/3(0.7),s1(0.3)},{s3(0.6),s4(0.4)}i h{s1(1)},{s1(1)}i D2= h{s1(1)},{s1(1)}i h{s4(0.4),s5(0.4),s6(0.2)},{s1/5(0.3),s1/3(0.3)}i h{s1/5(0.3),s1/3(0.3)},{s4(0.4),s5(0.4),s6(0.2)}i h{s1(1)},{s1(1)}i h{s3(0.3),s4(0.2),s5(0.3)},{s1/8(0.3),s1/6(0.3)}i h{s3(0.3),s4(0.2)},{s3(0.2),s4(0.6)}i h{s1/6(0.8),s1/4(0.1)},{s3(0.1),s4(0.7),s5(0.2)}i h{s1/2(0.2),s1(0.3),s2(0.5)},{s1/7(0.9),s1/6(0.1)}i h{s1/8(0.3),s1/6(0.3)},{s3(0.3),s4(0.2),s5(0.3)}i h{s3(0.1),s4(0.7),s5(0.2)},{s1/6(0.8),s1/4(0.1)}i h{s3(0.2),s4(0.6)},{s3(0.3),s4(0.2)}i h{s1/7(0.9),s1/6(0.1)},{s1/2(0.2),s1(0.3),s2(0.5)}i h{s1(1)},{s1(1)}i h{s1/4(0.5),s1/3(0.3),s1/2(0.1)},{s6(0.4),s7(0.1),s8(0.4)}i h{s6(0.4),s7(0.1),s8(0.4)},{s1/4(0.5),s1/3(0.3),s1/2(0.1)}i h{s1(1)},{s1(1)}i D3= h{s1(1)},{s1(1)}i h{s1/2(0.8),s2(0.1)},{s1/6(0.3),s1/5(0.2),s1/4(0.4)}i h{s1/6(0.3),s1/5(0.2),s1/4(0.4)},{s1/2(0.8),s2(0.1)}i h{s1(1)},{s1(1)}i h{s1/5(0.5),s1/4(0.3),s1/3(0.1)},{s1/9(0.3),s1/7(0.2)}i h{s1/2(0.1),s1(0.3),s2(0.3)},{s1/8(0.3),s1/7(0.2),s1/6(0.4)}i h{s1/7(0.5),s1/6(0.2)},{s1/4(0.3),s1/3(0.1),s1/2(0.6)}i h{s1/2(0.2),s1(0.2),s2(0.5)},{s1/8(0.6),s1/7(0.1),s1/6(0.3)}i h{s1/9(0.3),s1/7(0.2)},{s1/5(0.5),s1/4(0.3),s1/3(0.1)}i h{s1/4(0.3),s1/3(0.1),s1/2(0.6)},{s1/7(0.5),s1/6(0.2)}i h{s1/8(0.3),s1/7(0.2),s1/6(0.4)},{s1/2(0.1),s1(0.3),s2(0.3)}i h{s1/8(0.6),s1/7(0.1),s1/6(0.3)}, {s1/2(0.2),s1(0.2),s2(0.5)}i h{s1(1)},{s1(1)}i h{s1/2(0.6),s2(0.1)},{s1/5(0.6),s1/4(0.3),s1/3(0.1)}i h{s1/5(0.6),s1/4(0.3),s1/3(0.1)},{s1/2(0.6),s2(0.1)}i h{s1(1)},{s1(1)}i D4= h{s1(1)},{s1(1)}i h{s4(0.2),s5(0.3),s6(0.2)},{s1(0.5),s2(0.1),s3(0.3)}i h{s1(0.5),s2(0.1),s3(0.3)},{s4(0.2),s5(0.3),s6(0.2)}i h{s1(1)},{s1(1)}i h{s1/3(0.2),s1/2(0.4),s1(0.2)},{s1(0.1),s3(0.1)}i h{s1/7(0.5),s1/5(0.5)},{s1(0.2),s2(0.3),s3(0.3)}i h{s2(0.2),s3(0.2),s4(0.2)},{s1/5(0.3),s1/3(0.6)}i h{s1/4(0.3),s1/3(0.3),s1/2(0.3)},{s1/8(0.1),s1/7(0.3),s1/6(0.6)}i h{s1(0.1),s3(0.1)},{s1/3(0.2),s1/2(0.4),s1(0.2)}i h{s1/5(0.3),s1/3(0.6)},{s2(0.2),s3(0.2),s4(0.2)}i h{s1(0.2),s2(0.3),s3(0.3)},{s1/7(0.5),s1/5(0.5)}i h{s1/8(0.1),s1/7(0.3),s1/6(0.6)},{s1/4(0.3),s1/3(0.3),s1/2(0.3)}i h{s1(1)},{s1(1)}i h{s6(0.4),s7(0.1),s8(0.3)},{s1/2(0.3),s1(0.3),s2(0.1)}i h{s1/2(0.3),s1(0.3),s2(0.1)},{s6(0.4),s7(0.1),s8(0.3)}i h{s1(1)},{s1(1)}i VOLUME 7, 2019 75049
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WANYING XIE received the master’s degree from Xiangtan University, Xiangtan, China, in 2016. She is currently pursuing the Ph.D. degree with the School of Economics and Management, Southeast University. Her research results have been published in the International Journal of Information Technology and Decision Making,KnowledgeBased Systems, and the International Journal of Fuzzy Systems. Her current research interests include decision analysis, decision evaluation, information fusion, and computing with words. 75056 VOLUME 7, 2019
W. Xie et al.: EGRA With the Comparable Degree for DPMLTSs and Its Application on the Cloud Enterprise ZESHUI XU received the Ph.D. degree in management science and engineering from Southeast University, Nanjing, China, in 2003. From 2003 to 2005, he was a Postdoctoral Researcher with the School of Economics and Management, Southeast University. From 2005 to 2007, he was a Postdoctoral Researcher with the School of Economics and Management, Tsinghua University, Beijing, China. He is a Distinguished Young Scholar with the National Natural Science Foundation of China and the Chang Jiang Scholars of the Ministry of Education of China. He is currently a Professor with the Business School, Sichuan University, Chengdu, China. He is an IEEE Fellow, an IFSA Fellow, an IET Fellow, a BCS Fellow, and an RSA Fellow. He has been selected as a 2014 Thomson Reuters Highly Cited Researcher (in the fields of computer science and engineering) and also included in The World’s Most Influential Scientific Minds 2014. His h-index is 86 and has authored the following books: Uncertain Multi-Attribute Decision Making: Methods and Applications (Springer, 2015), Intuitionistic Fuzzy Information Aggregation: Theory and Applications (Science Press and Springer, 2012), Linguistic Decision Making: Theory and Methods (Science Press and Springer, 2012), Intuitionistic Fuzzy Preference Modeling and Interactive Decision Making (Springer, 2013), Intuitionistic Fuzzy Aggregation and Clustering (Springer, 2013), and Hesitant Fuzzy Sets Theory (Springer, 2014). He has contributed more than 400 journal articles to professional journals. He is an Advisory Member of the journal Granular Computing, an Associate Editor of Fuzzy Optimization and Decision Making and the Journal of Intelligence Systems, a Section Editor of the Asian Journal of Social and Economic Sciences, and the Chief Editor of Scholars Journal of Economics, Business and Management. He is also a member of the Editorial Boards of Information Fusion,Information: An International Journal, the International Journal of Applied Management Science, the International Journal of Data Analysis Techniques and Strategies, the Journal of Applied and Computational Mathematics, the International Journal of Research in Industrial Engineering,System Engineering—Theory and Practice,Fuzzy Systems and Mathematics, the Journal of Systems Engineering, and the Chinese Journal of Management Science. His current research interests include information fusion, group decision making, computing with words, and aggregation operators. ZHILIANG REN received the master’s degree from Qufu Normal University, Qufu, China, in 2015. He is currently pursuing the Ph.D. degree with the School of Economics and Management, Southeast University. His research results have been published in Information Sciences, Knowledge-Based Systems, and Applied Soft Computing. His current research interests include decision analysis, decision evaluation, information fusion, and computing with words. ENRIQUE HERRERA-VIEDMA received the M.Sc. and Ph.D. degrees in computer science from the University of Granada, Granada, Spain, in 1993 and 1996, respectively. He is currently a Professor of computer science and the Vice-President for Research and Knowledge Transfer with the University of Granada. His h-index is 69 with more than 18 000 citations received in Web of Science and 85 in Google Scholar with more than 29 000 cites received. He has been identified as one of the world’s most influential researchers by the Shanghai Center and Thomson Reuters/Clarivate Analytics in both computer science and engineering in the years 2014–2018. His current research interests include group decision making, consensus models, linguistic modeling, aggregation of information, information retrieval, bibliometric, digital libraries, web quality evaluation, recommender systems, and social media. Dr. Herrera-Viedma is the Vice-President for Publications in SMC Society and an Associate Editor in several journals, such as the IEEE TRANSACTIONS ON FUZZY SYSTEMS, the IEEE TRANSACTIONS ON SYSTEMS, MAN, AND CYBERNETICS: SYSTEMS,Information Sciences,Applied Soft Computing, Soft Computing,Fuzzy Optimization and Decision Making,International Journal of Fuzzy Systems,Journal of Intelligent & Fuzzy Systems,Engineering Applications of Artificial Intelligence,Journal of Ambient Intelligence and Humanized Computing,International Journal of Machine Learning and Cybernetics, and Knowledge-Based Systems. VOLUME 7, 2019 75057