Computer Modeling in Engineering & Sciences

DOI: 10.32604/cmes.2022.019501

ARTICLE

A Personalized Comprehensive Cloud-Based Method for Heterogeneous MAGDM and Application in COVID-19

Xiaobing Mao, Hao Wu and Shuping Wan*

School of Information Management, Jiangxi University of Finance and Economics, Nanchang, 330013, China
*Corresponding Author: Shuping Wan. Email: shupingwan@163.com
Received: 27 September 2021; Accepted: 30 November 2021

1  Introduction

Multi-attribute group decision-making (MAGDM) refers to a decision situation where a group of decision-makers (DMs) provide their own opinions on a given set of alternatives under a set of attributes, and then select the optimal alternative(s) by aggregating their opinions [1–6]. Since the real-life MAGDM problems often involve multiple different types of attributes, it is not easy for DMs to evaluate all attributes in only one form of evaluation information, which results in the appearance of heterogeneous MAGDM. In heterogeneous MAGDM process, the evaluations of different attributes can be expressed by qualitative and quantitative forms. For example, when a customer selects a car, a real number or an interval number can be used to evaluate its price, but a LT (linguistic term) or its extended forms will be preferred than quantitative value to evaluate its safety. Due to the growing uncertainty of actual decision-making environments, it is more convenient and flexible for DMs to employ qualitative forms, e.g., LT, PLTS (probabilistic linguistic term set), LHFS (linguistic hesitant fuzzy set), to characterize the evaluation information of alternatives on attributes. Both PLTS and LHFS are two important extensions of LT. PLTS, proposed by Pang et al. [7], consists of LTs and their corresponding probabilities. LHFS, initiated by Meng et al. [8], contains LTs and their corresponding memberships. For example, a group of DMs are invited to select a site for emergency medical waste disposal during the outbreak of COVID-19. Five attributes, i.e., geographical location, equipment, process technologies, disposal capacity and transport capacity, are chosen to evaluate the alternatives. LTs are suitable to evaluate the geographical location. Since the evaluations for equipment and process technologies are divided into two parts: LTs and corresponding probabilities, PLTSs are suitable to evaluate the equipment and process technologies. Besides, it is easy for DMs to evaluate the disposal capacity and the transportation capacity by using LHFSs. Therefore, the site selection of emergency medical waste disposal is a typical problem of heterogeneous MAGDM with different types of qualitative evaluations. Currently, many scholars have studied heterogeneous MAGDM problems. Yu et al. [1] developed a fusion method based on trust and behavior analysis for heterogeneous MAGDM scenarios. Liu et al. [9] proposed a new axiomatic design-based mathematical programming method for heterogeneous MAGDM with linguistic fuzzy truth degrees. Gao et al. [10] provided a consensus model for heterogeneous MAGDM with several attribute sets. Wan et al. [11] initiated a new prospect theory based method for heterogeneous MAGDM with hybrid fuzzy truth degrees of alternative comparisons. With the in-depth study of previous literature, many heterogeneous MAGDM problems have been effectively solved. However, there is little research on heterogeneous MAGDM with multiple qualitative forms (especially LT, PLTS and LHFS). To fill the gap, this paper intends to use LTs, PLTSs and LHFSs to portray heterogeneous evaluations.

Qualitative evaluations are not easy to be computed directly, especially when DMs use diverse forms of qualitative evaluations. At present, some models have been developed to deal with the calculations of qualitative evaluations, such as linguistic symbolic model [12], two-tuple linguistic model [13], cloud model [14,15]. Linguistic symbolic model and two-tuple linguistic model deal with LTs by converting them into real numbers. Cloud model proposed by Li et al. [14,15] is a more effective tool to describe qualitative concepts since it has strong power in capturing the fuzziness and randomness of LTs, simultaneously. Based on the probability theory and fuzzy set theory, the cloud model utilizes three numerical characteristics, i.e., mathematical expectation Ex, entropy En and hyper entropy He, to realize the nimble and effective inter-transformation between qualitative evaluations and quantitative values. Cloud model has attracted extensive attention from scholars and has been successfully applied to various fields, such as behavioral analysis [16], artificial intelligence [17,18], system assessment [19], data mining [20], knowledge discovery [21] and decision-making [22–34], etc.

Although the above mentioned cloud-based methods [22–32] are efficient in handing various practical decision-making problems, there still exist some defects as follows:

(1)   Some previous studies [22–27,29,31,32] depicted the evaluations only with a single qualitative form, which might limit their applications in practical decision-making problems.

(2)   Few studies took DMs’ personalities into account during the transformation process. Wang et al. [24] introduced overlap parameter into the transformation process to reflect the DMs’ personality and preference. But the determination of overlap parameter is a little subjective, which may lead to unreasonable decision results.

(3)   The comprehensive clouds in existing approaches [23,32] may cause the loss and distortion of evaluation information.

(4)   Methods in [22,24,26,29,32] used the expected score values of clouds to rank the alternatives, while methods in [23,27,30,33,34] utilized the closeness coefficient and priority vector to rank the alternatives. However, the expected score values of clouds sometimes are unstable since the expected score values are generated randomly. The closeness coefficient and priority vector depend on the distances between clouds, but different definitions of distance between clouds usually generate different ranking results.

To overcome the above limitations, this paper develops a personalized comprehensive cloud-based method for heterogeneous MAGDM, in which the evaluations of alternatives on attributes are represented as LTs, PLTSs and LHFSs. Regulation parameters of entropy and hyper entropy are proposed to reflect the DMs’ personalities. Two approaches are put forward to transform PLTS and LHFS into comprehensive cloud of PLTS (C-PLTS) and comprehensive cloud of LHFS (C-LHFS), respectively. The cloud almost stochastic dominance (CASD) relationship and CASD degree are initiated to compare clouds and further rank the alternatives. In addition, a novel approach is presented to obtain DMs’ weights and a comprehensive tri-objective programing model is constructed to determine the attribute weights. The proposed method is employed to the site selection of emergency medical waste disposal in COVID-19. Compared with existing studies, the major contributions of this paper are highlighted in the following four aspects:

(1)   Regulation parameters of entropy and hyper entropy are defined objectively. By incorporating regulation parameters into the transformation process, DMs’ personalities are reflected well. Moreover, DMs’ weights are objectively determined based on the proposed regulation parameters.

(2)   From the perspectives of probability and membership degree, two approaches are put forward to transform PLTS and LHFS into C-PLTS and C-LHFS, respectively. The modified ratios of LTs decrease the loss and distortion of evaluation information.

(3)   CASD relationship and CASD degree are defined and used to compare clouds. Based on the proposed comparison approach for clouds, the alternatives are ranked and the ranking results are stable and effective.

(4)   A comprehensive tri-objective programing model is constructed to determine the attribute weights. In this model, three perspectives are considered, including differentiation between evaluation values, relationship between attributes and the amount of information contained in evaluation values. The setting of balance coefficients enables DMs to make a tradeoff in the three perspectives, which can improve the flexibility of the proposed method.

The remainder of this paper is organized as follows: Section 2 briefly introduces some concepts related to LTs and reviews cloud model as well as almost first-degree stochastic dominance (AFSD). Section 3 describes the heterogeneous MAGDM problem and develops two novel transformation approaches from PLTS and LHFS to comprehensive clouds. In Section 4, a personalized comprehensive cloud-based method is proposed for heterogeneous MAGDM problem. A numerical example and sensitivity analyses are conducted to illustrate the proposed method in Section 5. Section 6 performs some comparison analyses to explain the superiorities of the proposed method. Some conclusions are summarized in Section 7.

2  Preliminaries

This section briefly introduces some concepts related to LTs and reviews cloud model as well as AFSD.

2.1 LT and Some Related Concepts

Let S={si|i=1,2,…,2τ+1} be a finite and completely ordered discrete term set with odd cardinality [35], where τ is a nonnegative integer, and si represents a possible value for a LT. The set S is a linguistic term set (LTS) if si,sj∈S satisfy the following properties:

(i)  Ordered set: si≤sj if and only if i≤j;

(ii)  Negation operation: neg(si)=sj, if i+j=2τ+1.

In linguistic evaluation scales, the absolute deviation of semantics between any two adjacent LTs may increase, decrease or remain unchanged with increasing linguistic subscripts. To reflect various semantics deviation, linguistic scale functions (LSFs) [22] are used to flexibly portray evaluation scales according to specific semantic situations.

Definition 1. [22,36] Let si∈S be a LT. When θi∈[0,1] is a numerical value, the LSF is mapped from si to θi(i=1,2,⋯,2τ+1) as follows:

F:si→θi,(i=1,2,⋯,2τ+1)

where 0≤θ1<θ2<⋯<θ2τ+1≤1. θi represents the evaluation of DM when he/she chooses the LT si. As a result, the function F describes the semantics of si(i=1,2,⋯,2τ+1). LSFs are strictly monotonously increasing with respect to the subscript i.

Three kinds of LSFs are shown below:

LSF1: F(si)=θi=i−12τ,(i=1,2,⋯,2τ+1)(1)

In LFS1, the absolute deviation between adjacent LTs remains unchanged with increasing linguistic subscripts. Take τ=3 as an example, and the LTs are graphically shown in Fig. 1.

Figure 1: LSF1 (τ=3)

LSF2: F(si)=θi={aτ−aτ−i+12(aτ−1),(i=1,2,…,τ+1)aτ+ai−τ−1−22(aτ−1),(i=τ+2,τ+3,⋯,2τ+1).(2)

Lots of experimental studies [37] have illustrated that a generally lies in the interval [1.36,1.4]. Moreover, a also can be determined by a subjective approach [22]. In LFS2, the absolute deviation between adjacent LTs gradually increases from the middle of the given LTs to both ends. If we take τ=3 and set a=1.36, the LTs are graphically shown in Fig. 2.

Figure 2: LSF2 (τ=3, a=1.36)

LSF3: F(si)=θi={τα−(τ−i+1)α2τα,(i=1,2,…,τ+1)τβ+(i−τ−1)β2τβ,(i=τ+2,τ+3,⋯,2τ+1).(3)

LSF3 is defined based on prospect theory's value function and the DMs’ different sensation for the absolute deviation between adjacent linguistic subscripts. α and β(α,β∈[0,1]) represent the curvature of the subjective value function for gain and loss, respectively [38]. LSF3 reduces to LSF1 when α=β=1. The absolute deviation between adjacent LTs gradually decreases from the middle of the given LTs to both ends. If we take τ=3 and set α=β=0.8, the LTs are graphically shown in Fig. 3.

Figure 3: LSF3 (τ=3, α=β=0.8)

In order to save all of the given information and facilitate calculation, the aforesaid functions can be extended into F∗:S~→R+, where F∗(si)=θi is a continuous and strictly monotonously increasing function.

Definition 2. [7] Let S={si|i=1,2,…,2τ+1} be a LTS. A PLTS L(p) can be defined as L(p)={s(l)(p(l))|s(l)∈S,p(l)≥0,i=1,2,…,# L(p),∑l=1# L(p)p(l)≤1}, where s(l)(p(l)) is the LT s(l) associated with the probability p(l), and # L(p) denotes the number of all different LTs in L(p). If ∑l=1# L(p)p(l)<1, then p~(l)=p(l)/∑l=1# L(p)p(l) is used to normalize the PLTS.

In this paper, it is assumed that all PLTSs have already been normalized.

Definition 3. [8] Let S={si|i=1,2,…,2τ+1} be a LTS. A LHFS LH in S is defined as LH={(s(l),lh(s(l)))|s(l)∈S}, where lh(s(l))={r1,r2,…,r# lh(s(l))} is a set with # lh(s(l)) values in (0, 1] and denotes the possible membership degrees of the element s(l)∈S to the set LH. #LH denotes the number of all different LTs in LH, and # lh(s(l)) represents the count of real numbers in lh(s(l)).

2.2 Cloud Model

Definition 4. [14] Let U be the universe of discourse and T be a qualitative concept in U. If x∈U is a random instantiation of concept T that satisfies x∼N(Ex,En′2) and En′∼N(En,He2), and y∈[0,1] is the certainty degree of x belonging to T that satisfies y=exp⁡(−(x−Ex)22(En′)2), then the distribution of x in the universe U is defined as a normal cloud, and (x,y) represents a cloud drop.

For simplicity, normal cloud is called as cloud hereafter. The degree of certainty of x belonging to concept T is a probability distribution rather than a fixed number. Hence, ∀x∈U, y=exp⁡(−(x−Ex)22(En′)2) is a one-to-many mapping.

There are two kinds of uncertainty: randomness and fuzziness. Randomness refers to the uncertainty contained in an event that has a clear definition but do not necessarily occur. Fuzziness refers to the uncertainty contained in an event that has appeared but it is difficult to define it accurately [39]. There is a practical demand to describe fuzziness and randomness inherent in LTs simultaneously. Cloud can perfectly depict the overall quantitative properties of a concept through three numerical characteristics: mathematical expectation Ex, entropy En and hyper entropy He, where Ex is the mathematical expectation of cloud drops belonging to a concept in the universe, and En reflects the uncertainty measurement of a qualitative concept, including randomness and fuzziness. From the perspective of probability theory, En is similar to standard variance of random variables. From the point of fuzzy set theory, En represents the scope in which cloud drops are accepted by the concept, and it indicates the support set of the concept with membership degrees larger than 0. As a result, En reflects randomness and fuzziness of a qualitative concept and their correlation, simultaneously. He represents the degree of uncertainty of En, i.e., the second-order entropy of the entropy [15,40]. A cloud can be described by Ex, En, He, and denoted by C=(Ex,En,He).

Definition 5. [15] Given two clouds C1=(Ex1,En1,He1) and C2=(Ex2,En2,He2), some operations of clouds are defined as follows:

(1)   C1+C2=(Ex1+Ex2,En12+En22,He12+He22);

(2)   C1−C2=(Ex1−Ex2,En12+En22,He12+He22);

(3)   γC1=(γEx1,γEn1,γHe1), (γ≥0).

2.3 AFSD

The AFSD is used to compare two stochastic variables. It was proposed by Leshno and Levy [41]. Let X1 and X2 be two stochastic variables, where G1(x) and G2(x) denote two cumulative distribution functions, respectively. Let Ω={x|G1(x)>G2(x)}, Θ={x|G2(x)>G1(x)} and ||G1(x)−G2(x)||=∫ΩG1(x)−G2(x)dx+∫ΘG2(x)−G1(x)dx. Then, AFSD is defined below:

Definition 6. [41,42] For 0<δ<0.5, X1 dominates X2 by δ−AFSD if and only if ∫ΩG1(x)−G2(x)dx≤δ||G1(x)−G2(x)||, where ||G1(x)−G2(x)|| corresponds to the area between G1 and G2, ∫ΩG1(x)−G2(x)dx corresponds to the area that G1 is greater than G2, and δ denotes the degree of first-degree stochastic dominance violation allowed.

3  Heterogeneous MAGDM Problem and Comprehensive Cloud

This section describes the heterogeneous MAGDM problem and introduces the improved transformation approach between LT and cloud in detail. Particularly, we developed two novel transformation approaches from PLTS and LHFS to comprehensive clouds.

3.1 Description for Heterogeneous MAGDM Problem

A heterogeneous MAGDM problem is to find the best solution from all feasible alternatives assessed on multiple attributes by a group of DMs. The evaluation attributes in heterogeneous MAGDM can be classed into several subsets which are expressed by different kinds of forms.

For a heterogeneous MAGDM problem, suppose that DMs de(e=1,2,⋯,k) have to select the optimal alternative(s) from a group of alternatives xu(u=1,2⋯,m) or rank these alternatives based on attributes yv(v=1,2⋯,n). Denote an alternative set by X={x1,x2,⋯,xm}, an attribute set by Y={y1,y2,⋯,yn}, and a DM set by D={d1,d2,⋯,dk}. Denote Y1={y1,y2,⋯,yv1},Y2={yv1+1,yv1+2,⋯,yv2}, Y3={yv2+1,yv2+2,⋯,yv3}, respectively, where 1≤v1≤v2≤v3≤n. Namely, Y is divided into three subsets Yt(t=1,2,3), where Yt(t=1,2,3) are attribute subsets in which attribute values are expressed with LTs, PLTSs and LHFSs respectively. Yt∩Yl=∅(t,l=1,2,3;t≠l), ∪t=13Yt=Y, where ∅ is an empty set. Denote M={1,2,⋯,m}, N1={1,2,⋯,v1},N2={v1+1,v1+2,⋯,v2}, N3={v2+1,v2+2,⋯,n}, N={1,2,⋯,n} and K={1,2,⋯,k}. Denote the DM weight vector by v=(ϖ1,ϖ2,⋯,ϖk)T, where ϖe is the weight of DM de, satisfying that 0≤ϖe≤1(e=1,2,⋯,k) and ∑e=1kϖe=1. Denote the attribute weight vector by w=(w1,w2,⋯,wn)T, where wv is the weight of attribute yv, satisfying that 0≤wv≤1(v=1,2,⋯,n) and ∑v=1nwv=1 [43].

Let ruve be the evaluation of an alternative xu on attribute yv given by DM de. If v∈N1, ruve is a LT, denoted by si(si∈S); If v∈N2, ruve is a PLTS, denoted by L(p)={s(l)(p(l))|s(l)∈S}; If v∈N3, ruve is a LHFS, denoted by LH={(s(l),lh(s(l)))|s(l)∈S}. After normalizing, the individual original normalized evaluation matrix Re=(ruve)m×n can be obtained as

Re=(ruve)m×n=(ruve)m×n=y1y2⋯ynx1x2⋮xm[r11er12e⋯r1ner21er22e⋯r2ne⋮⋮⋮⋮rm1erm2e⋯rmne](e=1,2,⋯,k)(4)

3.2 Transformation between LT and Cloud

Generally, two kinds of approaches have been proposed for transformation from LTs to clouds so far. One is based on the golden radio [44], and the other is based on the LSF [22]. Wang et al. [24] introduced a parameter named overlapping degree into the transformation approach [22] to determining the degree of overlap between two adjacent clouds. With the parameter ε∈[εmin,εmax], DMs could express their preference for the degree of overlap between two adjacent clouds. However, after processing the calculation formulae for three numerical characteristics, a problem emerges. That is, once the LSF and some related parameters are fixed in [24], D(Exi,i+1)=Exi+1−Exi3, D(Exi−1,i+1)=Exi+1−Exi−16 and D(Exi−1,i)=Exi−Exi−13 are three fixed values. It is easy to see that the values of Eni and Hei are linearly dependent on εmax+εmin2 and εmax−εmin6, respectively. However, the determination of overlapping degree is entirely based on the subjective preference of DMs, which means the determination of Eni and Hei is also subjective. To overcome the above defects, two regulation parameters ς and ζ for entropy and hyper entropy are proposed in this paper. We improve the transformation approach in [24] by replacing εmax+εmin2 (εmax−εmin6) with ς (ζ) during the transformation process. Significantly, the determination of ς and ζ is totally objective and logical. The specific approaches to determining ς and ζ are stated in Sections 3.2.1 and 3.2.2.

Let S={si|i=1,2,⋯,2τ+1} be a LTS, L(p)={s(l)(p(l))|s(l)∈S} be a normalized PLTS, and LH={(s(l),lh(s(l)))|s(l)∈S} be a LHFS. L(p)uve={suv(l)e(puv(l)e)} denotes the evaluation of an alternative xu on attribute yv(v∈N2) given by the DM de, and # L(p)uve∈[1,2τ+1] denotes the number of all different LTs in L(p)uve. LHuve={(s(l),lh(s(l))uve)|s(l)∈S} indicates the evaluation of an alternative xu on attribute yv(v∈N3) given by the DM de. #LHuve∈[1,2τ+1] signifies the number of all different LTs in LHuve. #lh(s(l))uve represents the count of real numbers in lh(s(l))uve={ruv,1e,ruv,2e,…,ruv,#lh(s(l))uvee}. lh(s(l))uve is a set with #lh(s(l))uve values in (0, 1] and denotes the possible membership degrees of the element s(l)∈S to the set LHuve.

3.2.1 Determination of the Regulation Parameter of Entropy

Entropy En reflects the uncertainty measurement of a qualitative concept, specifically randomness and fuzziness. From the perspective of fuzzy set theory, it represents the scope in which the cloud drops are accepted by the concept. It is common that the more elements are used in evaluations, the more hesitant the DM is. Hence, for a hesitant DM, larger En should to be assigned to its LTs. Therefore, the regulation parameter of En should be determined according to DMs’ hesitant degree.

Definition 7. For a heterogeneous MAGDM, the average number of LTs used by DM de at an evaluation in the form of PLTSs or LHFSs, is defined as

ηe=1m(n−v1)∑u=1m[∑v=v1+1v2#L(p)uve+∑v=v2+1n#LHuve].(5)

It is obvious that ηe∈[1,2τ+1].

Definition 8. The hesitant degree of DM de is defined as

HDe=ηe/(2τ+1).(6)

Obviously, HDe∈[12τ+1,1].

The determination of DM's hesitant degree is based on the average number of LTs that DM uses at a single evaluation in the form of PLTSs or LHFSs.

A great deal of experimental research has demonstrated that regulation parameter ς generally lies in [1,2]. In fact, if the hesitant degree HDe is 12τ+1, it means that only one LT is used by DM de at each evaluation in the form of PLTSs or LHFSs. In this situation, DM de is regarded as a decisive and confident person. Accordingly, it is appropriate for DM de to take 1 as the value of ς. The more the LTs used by DM de, the bigger the value of ς is. Based on this premise, the regulation parameter ς of entropy is defined as follows:

Definition 9. Let ρ1=1+2τ2τ+1. The regulation parameter of entropy for DM de is defined as

ςe=logρ1(ηe2τ+1+2τ2τ+1)+1=logρ1(HDe+2τ2τ+1)+1,(7)

where ηe∈[1,2τ+1], HDe∈[12τ+1,1]. Clearly, it holds that ςe∈[1,2].

In the following, an example is given to illustrate how to determine the value of ς.

Example 1. Let S={si|i=1,2,⋯,7} be a LTS. There are three alternatives x1, x2 and x3. In order to select the optimal alternative, DM de gives evaluations for x1, x2, x3 on three attributes y1, y2, y3. The evaluations for y1, y2 and y3 are expressed in the forms of LT, PLTS and LHFS, respectively. DM de gives an evaluation matrix as follows:

Re=(ruve)3×3=y1y2y3x1x2x3[s4s6(0.1),s7(0.9)(s3,0.8,0.9),(s4,0.6)s2s3(0.1),s4(0.2),s5(0.7)(s5,0.7)s6s4(1)(s1,0.8) ]

Calculate the average number of LTs used by DM de at an evaluation in the form of PLTSs or LHFSs by Eq. (5):

ηe=13×2×[(2+2)+(3+1)+(1+1)]=1.6667

Calculate the hesitant degree of DM de by Eq. (6):

HDe=1.66677=0.2381

Then, the value of ςe is calculated by Eq. (7):

ςe=log1+0.8571(0.2381+0.8571)+1=1.1470

3.2.2 Determination of the Regulation Parameter of Hyper Entropy

Hyper entropy He represents the degree of uncertainty of En, i.e., the second-order entropy of the entropy. The larger He is, the thicker the cloud is, and the wider the distribution of membership is. Thus, on the one hand, the information entropy as a very important concept to measure the uncertainty in evaluation information provided by DMs, the larger it is, the larger He should be. On the other hand, membership degree as an index to measure the degree that an element belongs to a certain concept, the lower it is, the larger the indeterminacy degree is. Furthermore, the larger indeterminacy degree in the decision matrix provided by DM, the larger He should to be. With the above analysis, the regulation parameter of He should be determined according to indeterminacy degree and information entropy.

Definition 10. [45] The information entropy of L(p) is defined as follows:

H(L(p))=−log2z∑l=1#L(p)(p(l))log2(p(l)),(8)

where z is a constant that is set to 1.28 in this paper as [45] sets. It is easily seen that H(L(p))∈[0,log2z⋅log2(2τ+1)].

Definition 11. The information entropy of DM de is defined as follows:

He=−log2zm(v2−v1)∑u=1m∑v=v1+1v2∑l=1# L(p)uve(p(l))log2(p(l)),(9)

where z is a constant that is set to 1.28 in this paper as [45] sets. Obviously, He∈[0,log2z⋅log2(2τ+1)].

The closer the memberships for corresponding LTs are to 0, the larger indeterminacy degree DMs have for corresponding LT.

Definition 12. The indeterminacy degree of LH={(s(l),lh(s(l)))|s(l)∈S} is defined as follows:

ID(LH)=1#LH∑l=1#LH[1#lh(s(l))∑r∈lh(s(l))(1−r)].(10)

Clearly, it holds that ID(LH)∈[0,1).

Definition 13. The indeterminacy degree of DM de is defined as follows:

IDe=1m(n−v2)∑u=1m∑v=v2+1n{1#LHuve∑l=1#LHuve[1#lh(s(l))uve∑ruve∈lh(s(l))uve(1−ruve)]}.(11)

It is easily seen that IDe∈[0,1).

Definition 14. Let ρ2=1+log2z⋅log2(2τ+1). The regulation parameter of hyper entropy for DM de is defined as

ζe=12[logρ2(He+1)+log2(IDe+1)],(12)

where He∈[0,log2z⋅log2(2τ+1)], IDe∈[0,1). Obviously, ζe∈[0,1).

Take τ=3 as an example. The graphical representation for the regulation parameter of hyper entropy is shown in Fig. 4.

Figure 4: Graphical representation for the regulation parameter of hyper entropy (τ=3)

Example 2. Following Example 1, the value of ζe can be calculated as follows:

Calculate the information entropy of each evaluation in the form of PLTSs by Eq. (8):

H({s6(0.1),s7(0.9)})=−log21.28×(0.1×log20.1+0.9×log20.9)=0.1670,

H({s3(0.1),s4(0.2),s5(0.7)})=−log21.28×(0.1×log20.1+0.2×log20.2+0.7×log20.7)=0.4120,

H({s4(1)})=−log21.28×(1×log21)=0

Calculate the information entropy of DM de by Eq. (9):

He=13×(0.1670+0.4120+0)=0.1930

Calculate the indeterminacy degree of each evaluation in the form of LHFSs by Eq. (10):

ID({s3(0.8,0.9),s4(0.6)})=12×{12×[(1−0.8)+(1−0.9)]+(1−0.6)}=0.275,

ID({s5(0.7)})=1−0.7=0.3,

ID({s1(0.8)})=1−0.8=0.2

Calculate the indeterminacy degree of DM de by Eq. (11):

IDe=13×(0.275+0.3+0.2)=0.2583

Then, the value of ζe is calculated by Eq. (12):

ζe=12×[log1.9998(0.1930+1)+log2(0.2583+1)]=0.2931

3.2.3 Specific Procedures for Transformation between LT and Cloud

Definition 15. [46] Let S={si|i=1,2,…,2τ+1} be a LTS, where τ is a positive integer. A valid universe [Xmin,Xmax] is provided by DMs. Then, a LT si(i=1,2,…,2τ+1) can be represented by the normal cloud Ci=(Exi,Eni,Hei).

Then, the specific transformation procedures are shown as follows:

(1)   Calculate ς and ζ.

Determination approaches for ς and ζ are shown in Sections 3.2.1 and 3.2.2.

(2)   Calculate θi.

Map si to θi using LSFs.

LSF2 Eq. (2) is adopted in this paper, where a=1.36.

(3)   Calculate Exi.

Exi=Xmin+θi(Xmax−Xmin).(13)

(4)   Calculate Eni.

Let (x,y) be a cloud drop, where x∼N(Ex,En′2). According to 3σ principle of the normal distribution, 99.7% cloud drops of Ci should be located in the interval [Exi−1,Exi+1]. However, since the distances between Exi and Exi−1 are different from the distances between Exi and Exi+1, the entropy of Ci(i=2,3,⋯,2τ) are different for the right side and the left side. For simplicity, Eni(i=2,3,⋯,2τ) take the mean value of ςEni_ and ςEni¯.

Eni={ςEni¯=ςExi+1−Exi3,i=1ςEni¯+Eni_2=ςExi+1−Exi−16,2≤i≤2τςEni_=ςExi−Exi−13,i=2τ+1,(14)

where ςEni_=ςExi−Exi−13, ςEni¯=ςExi+1−Exi3, ςEni_ denotes Eni for the left side, and ςEni¯ denotes Eni for the right side.

(5)   Calculate Hei

Hei={ζExi+1−Exi3,i=1ζExi+1−Exi−16,2≤i≤2τζExi−Exi−13,i=2τ+1.(15)

Based on the above analyses, the corresponding cloud for LT si can be generated by Algorithm 1.

To illustrate the advantages of regulation parameters, an example is given below.

Example 3. Given a universe [Xmin,Xmax] and a LTS S={si|i=1,2,⋯,7}. For alternatives x1, x2, x3 regarding three attributes y1, y2, y3, DMs d1, d2 and d3 give their evaluations. The evaluations for y1, y2 and y3 are expressed in the forms of LT, PLTS and LHFS, respectively. DMs d1, d2 and d3 give their evaluation matrices as follows:

R1=(ruv1)3×3=y1y2y3x1x2x3[ s4s6(1)(s3,0.9)s2s3(1)(s5,1)s6s4(1)(s1,0.95) ]

R2=(ruv2)3×3=y1y2y3x1x2x3[s4s6(0.1),s7(0.9)(s3,0.8,0.9),(s4,0.6)s2s3(0.1),s4(0.2),s5(0.7)(s5,0.7)s6s4(1)(s1,0.8) ]

R3=(ruv3)3×3=y1y2y3x1x2x3[s4s6(0.5),s7(0.5)(s3,0.2),(s4,0.1)s2s3(0.3),s4(0.3),s5(0.4)(s5,0.2)s6s4(1)(s1,0.1) ]

Based on Eqs. (5)–(12), the regulation parameters ς and ζ for each DM are calculated as follows:

ς1=1,ζ1=0.0352,ς2=1.1470,ζ2=0.2931,ς3=1.1470,ζ3=0.6359;

Based on Eqs. (2) and (13)–(15), θi, Exi, Eni, Hei(i=1,2,⋯,7) for DMs d1, d2, d3 are calculated and the results are shown in Tables 1–3, respectively.

Three sets of cloud generated by DMs d1, d2, d3 are graphically shown in Figs. 5–7, respectively.

It can be seen from Figs. 5–7 that the cloud drops distribution varies in width and thickness for different DMs. In previous studies [22,23,25–31,46], clouds for the corresponding LTS are usually the same for all DMs. However, since different DMs have different personalities, knowledge and experience, the width and thickness of clouds for the corresponding LTS should be different for different DMs. The more sure, confident and decisive the DM is for its evaluation matrix, the more concentrated and thinner the cloud distribution should be, vice versa. Unfortunately, most previous studies failed to notice this characteristic, whereas this paper sufficiently takes this characteristic into account. Although the overlap degree is considered in [24] to obtain personalized cloud sets for DMs, the determination of overlap degree depends on DMs’ subjective intuition. This defect is overcome in this paper. The regulation parameters are determined according to the evaluation matrix given by DM, which means the determination of regulation parameters is objective and logical.

Figure 5: Clouds for the LTs used by d1

Figure 6: Clouds for the LTs used by d2

Figure 7: Clouds for the LTs used by d3

3.3 Transformation from PLTS and LHFS to Comprehensive Clouds

In this sub-section, two approaches are brought forward to transform PLTS and LHFS into comprehensive clouds, respectively.

3.3.1 Transformation from PLTS to Comprehensive Cloud

Definition 16. Let S={si|i=1,2,…,2τ+1} be a LTS. The valid universe is [Xmin,Xmax]. Let L(p)={s(l)(p(l))|s(l)∈S} be a normalized PLTS. The cloud Cs(l)(Exs(l),Ens(l),Hes(l)) represents LT s(l)∈S. Then, CL(p)(ExL(p),EnL(p),HeL(p)) can be defined as C-PLTS, which is characterized by three numerical characteristics ExL(p), EnL(p) and HeL(p).

Definition 17. [24] Given a cloud C=(Ex,En,He), if (x,y) is a cloud drop of C, x satisfies x∼N(Ex,En′2) and En′∼N(En,He2). Then, the normal curve (NC) of all cloud drops can be defined as f=exp⁡(−(x−Ex)22En2).

In this paper, g=12πEnexp⁡(−(x−Ex)22En2) is utilized to represent the probability density function curve (PDFC) of C.

The specific procedures to determine ExL(p), EnL(p), HeL(p) of C-PLTS are as follows:

(1)   Let Cs(l) be the corresponding clouds for s(l)(l=1,2,…,# L(p)), Cs(l)(p(l)) be Cs(l) with corresponding probability p(l) and χl,l+1∈[Exs(l),Exs(l+1)] be the abscissa value of intersection point between the PDFCs of Cs(l)(p(l)) and Cs(l+1)(p(l+1)). If Exs(l)+3Ens(l)>Exs(l+1)−3Ens(l+1)(l∈{1,2,…,# L(p)−1}), then Eq. (16) is used to calculate the value of χl,l+1.

p(l)12πEns(l)exp⁡(−(χl,l+1−Exs(l))22(Ens(l))2)=p(l+1)12πEns(l+1)exp⁡(−(χl,l+1−Exs(l+1))22(Ens(l+1))2),(χl,l+1∈[Exs(l),Exs(l+1)]).(16)

(2)   Use Eq. (17) to calculate the area for the PDFC of Cs(l)(p(l)) from lower limit to upper limit, which can be denoted by As(l).

As(l)={p(l)12πExs(l)∫Exs(l)−3Ens(l)χl,l+1exp⁡(−(x−Exs(l))22(Ens(l))2)dx,l=1p(l)12πExs(l)∫χl−1,lχl,l+1exp⁡(−(x−Exs(l))22(Ens(l))2)dx,l∈{2,3,…,# L(p)−1}p(l)12πExs(l)∫χl−1,lExs(l)+3Ens(l)exp⁡(−(x−Exs(l))22(Ens(l))2)dx,l=# L(p).(17)