# A Novel Group Decision-Making Method Based on Generalized Distance Measures of PLTSs on E-Commerce Shopping

Authors: Juxiang Wang; Jian Yuan; Jiajing Zhang; Miao Tang
Journal: Complexity
 (2021)

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## Abstract

In multiattribute group decision-making (MAGDM), due to quantity, fuzziness, and complexity of evaluation linguistic information on commodities, traditional distance measures need to be extended to the integration of evaluation information under a multigranular probabilistic linguistic environment. A more reasonable method is proposed to deal with the missing value in the evaluation information. On the basis of the generalized distance measures and filling in the missing evaluation information, some novel distance measures between two multigranular probabilistic linguistic term sets (PLTSs) are presented in this paper. Based on these distance measures, three extended decision-making (DM) algorithms based on TOPSIS, the extended TOPSIS, and VIKOR are proposed, which are MGPL-TOPSIS, MGPL-ETOPSIS, and MGPL-VIKOR, respectively. The case analyses on purchasing a car are provided to illustrate the application of the extended multiattribute group decision-making (MAGDM) algorithms. Then, sensitivity analyses based on PT are proposed as well. In particular, the extended TOPSIS method is presented. These results demonstrate the novelty, feasibility, and rationality of the distance measures between two multigranular PLTSs proposed in this paper.


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## Body

## 1. Introduction

With the popularity of the Internet, online shopping has become an important way of daily shopping. Shopping on an e-commerce platform, one alternative can be evaluated on different platforms, or one alternative may be described on the same platform by different granular fuzzy linguistic information; for example, we suppose that a consumer wants to buy a new energy car from USD 20,000 to USD 30,000. In this case, they can visit the auto home website to learn about comments of these cars from other consumers and make a reasonable decision to buy a car. Consumers who have purchased these cars can comment on them through many channels. They can score cars on the same platform, post word-of-mouth comments, post through the community, or evaluate through different media. The granularity of evaluation information is different in these evaluation channels. How can we more effectively make purchase decisions of products with different granularity evaluation?In DM, due to the complexity of the real world, Zadeh proposed fuzzy set [1]. Furthermore, Zadeh proposed linguistic variables to represent uncertain and imprecise information intuitively, expressing human thoughts better [2]. In practice, the DMs are always hesitant among some evaluation values, and then, Torra proposed hesitant fuzzy set (HFS) first [3]. The hesitant fuzzy linguistic term set (HFLTS) was proposed to tackle the flexibility of the membership degree of HFS [4]. However, in some cases, the linguistic variables may be uncertain because of the complexity of DM problems, and the DMs may have different preferences with different belief degrees [5], possibility distributions [6], and importance degrees [7]. Then, using several linguistic terms to express evaluation information is more scientific. In this case, the hesitant fuzzy set based on probability was proposed as the probability-based hesitant fuzzy set (PHFS) [8, 9], and probabilistic linguistic term set (PLTS) by Qi Pang et al. is proposed to describe the object more effectively [10]. The PLTSs allow the DMs to give several linguistic terms, serving as the value of a linguistic variable, which enriches the flexibility of the expression of linguistic information. The DMs can express their linguistic evaluations or preference information better. Meanwhile, the PLTSs can provide different importance degrees or weights of all the possible evaluation preferences of one object.The traditional method for group decision-making (GDM) under the same granular linguistic information cannot integrate hybrid evaluation information. Therefore, multigranular linguistic term sets need to be described efﬁciently. Then, how can two PLTSs with multigranular probabilistic linguistic information be measured? Considerable research has been conducted about the distance measures; for example, Zhai et al. presented probabilistic interval-valued intuitionistic hesitant fuzzy sets [11]. Wu et al. proposed a probabilistic linguistic MULTIMOORA method in multicriteria group decision-making (MCGDM) based on the probabilistic linguistic expectation function [12]. However, a few research types on distance measures of PLTSs with multigranular linguistic information remain. Then, distance measures for PLTSs with multigranular linguistic details need to be extended. Some DM models have been used to deal with probabilistic linguistic information. Gou and Xu proposed a new score function of linguistic terms and defined the operations of PLTSs [13]. Pang et al. proposed a probabilistic linguistic representation model based on TOPSIS [10]. Liu and Li presented the PROMTHEE II method [14]. Liao et al. gave the PL-LINMAP method for multiple criteria decision-making (MCDM) with PLTSs [15]. Li and Wang presented an extended QUALIFLEX plan for selecting green suppliers [16]. Wu and Liao proposed the ORESTE method with probabilistic linguistic information [17]. Liao et al. proposed the PL-ELECTRE III method with PLTSs [18]. Abdolhamid et al. extended the VIKOR method for GDM with extended hesitant fuzzy linguistic information [19]. Zhang et al. proposed a probabilistic linguistic method based on VIKOR to evaluate green supply chain initiatives [20]. Zhang and Xu et al. used the probabilistic linguistic VIKOR method to tackle water-human harmony evaluation [21]. Gou et al. improved the VIKOR method for the application of smart healthcare with probabilistic double hierarchy linguistic term set [22].Some researchers have applied the PLTS to solve some practical problems; for example, Hao et al. presented a probabilistic dual-hesitant fuzzy set and its application in risk evaluation [23]. Gao et al. proposed a dynamic reference point method for emergency response [24]. Sharaf extended TOPSIS to similarity measures for MADM and applied it to network selection [25]. Muhammad Sajjad Ali Khan et al. extended TOPSIS for MCDM [26]. Asif Ali and Tabasam Rashid presented a generalized interval-valued trapezoidal fuzzy best-worst MCDM method [27]. Rajkumar Verma presented MAGDM based on aggregation operators for linguistic trapezoidal fuzzy intuitionistic fuzzy sets [28].However, the traditional linguistic information missing is usually filled with the minimum value or ignored. This method is flawed. Linguistic evaluation information is closely related with the psychological activities of decision-makers. On the basis of the discussion above, we present a novel method to deal with the missing value in the evaluation information and generalized distance measures for the PLTSs with multigranular linguistic information. Then, we apply it to solve the problems of MAGDM on the decision-making of purchasing a car.Based on the discussion above, this paper proposes distance measures for the PLTSs with multigranular linguistic information and then applies them in MAGDM.


## 2. Preliminaries

### 2.1. Linguistic Term Sets

The DMs can use LTSs to describe their preferences on the considered alternatives. The additive LTS is used most widely, which is defined as follows [28]:(1)S=Sα|α=0,1,…,g−1,where S is a g-granular fuzzy linguistic set; Sα is a linguistic variable with S0 and Sg, namely, the lower and upper limits of the linguistic terms; and g is a positive integer.Considering the situations where the DMs may hesitate among several possible values in DM, which is similar to the hesitant fuzzy set, the concept of HFLTSs is as follows.Definition 1.
(see [29]). We let S=S0,S1,…Sg−1 be a LTS, and then, HFLTs bs is an ordered finite subset of consecutive linguistic terms S.


### 2.2. Probabilistic Linguistic Term Sets

Definition 2.
(see [10]). We let S=S0,S1,…Sg−1 be an LTS. A PLTS is defined as(2)LP=LkPk|Lk∈S,Pk≥0,k=1,2,…,#LP,∑k=1#LPPk≤1,where LkPk is the linguistic term Lk associated with the probability Pk and #LP is the number of all different linguistic terms in LP.
If∑k=1#LPPk=1, then we obtain the complete information on the probabilistic distribution with all the possible linguistic terms. If ∑k=1#LPPk<1, then partial ignorance exists because of current insufficient evaluation information. Especially, ∑k=1#LPPk=0 means complete ignorance. Therefore, handling ignorance LP is crucial research for the application of PLTSs.Definition 3 (see [10]).
Given a PLTSLP with ∑k=1#LPPk<1, then the associated PLTS L˙P is defined by(3)L˙P=LkP˙k|k=1,2,…,#LP,where P˙k=Pk/∑k=1#LPPk for all k=1,2,…,#LP.


### 2.3. Multigranular Probabilistic Linguistic Term Sets

Definition 4.
(see [30]). We let S=S0,S1,…Sg−1 be g-granular LTS and S′=S0,S1,…Sg′−1 be g′-granular LTS. L1P and L2P are two different granular PLTSs on the attribute set x=x1,x2,…xn. Multigranular PLTSs can be defined as(4)L1P=L1k1P1k1|L1k1∈S,P1k1≥0,k1=1,2,…,#L1P1,∑k1=1#L1PP1k1≤1,L2P=L2k2P2k2|L2k2∈S′,P2k2≥0,k2=1,2,…,#L2P2,∑k2=1#L2PP2k2≤1,where L1k1P1k1 is the linguistic term L1k1 associated with the probability P1k1 and L2k2P2k2 is the linguistic term L2k2 related to the probability P2k2.
The numbers of linguistic terms in PLTSs are usually different for a DM. Therefore, the numbers of linguistic terms need to be added, in which numbers are relatively small. Then, the numbers of linguistic terms are the same.
The numbers ofL1P and L2P are denoted as #L1P and#L2P, respectively. If #L1P<#L2P, then#L1P−#L2P linguistic terms are added to L2P, leading to the numbers of L1P and L2P to be equal. The added linguistic terms are the smallest ones,L2P, and all the linguistic probabilities are zero.Definition 5.
(see [30]). We let L1P and L2P be two multigranular PLTSs. Then, the normalization processes are as follows:(1)
If∑ki=1#LiPPiki<1 by Definition 3, we calculate L˙iP,i=1,2.(2)
If#L1P≠#L2P, then by Definition 4, we add some elements to the one with the smaller number of elements.
The PLTSs obtained by Definition5 are denoted the normalized PLTSs. Conveniently, the normalized PLTSs are marked by L1P and L2P as well.
Given the positions of elements in a PLTS are arbitrary, we need to obtain the ordered PLTSs first, leading to the operational results in PLTSs being determined directly.Definition 6.
(see [30]). We let S=S0,S1,…Sg−1 be g-granular LTS. Given a PLTS, LP=LkPk|Lk∈S, Pk≥0,k=1,2,…#LP, ∑k=1#LPPk≤1, and rkLk is the subscript of the linguistic term LkLP. It is named an ordered multigranular PLTS if the LkPkk=1,2,…,#LP descending order’s values arrangeαLk=rkLk/g×Pkk=1,2,…,#LP linguistic terms.


## 2.1. Linguistic Term Sets

The DMs can use LTSs to describe their preferences on the considered alternatives. The additive LTS is used most widely, which is defined as follows [28]:(1)S=Sα|α=0,1,…,g−1,where S is a g-granular fuzzy linguistic set; Sα is a linguistic variable with S0 and Sg, namely, the lower and upper limits of the linguistic terms; and g is a positive integer.Considering the situations where the DMs may hesitate among several possible values in DM, which is similar to the hesitant fuzzy set, the concept of HFLTSs is as follows.Definition 1.
(see [29]). We let S=S0,S1,…Sg−1 be a LTS, and then, HFLTs bs is an ordered finite subset of consecutive linguistic terms S.


## 2.2. Probabilistic Linguistic Term Sets

Definition 2.
(see [10]). We let S=S0,S1,…Sg−1 be an LTS. A PLTS is defined as(2)LP=LkPk|Lk∈S,Pk≥0,k=1,2,…,#LP,∑k=1#LPPk≤1,where LkPk is the linguistic term Lk associated with the probability Pk and #LP is the number of all different linguistic terms in LP.
If∑k=1#LPPk=1, then we obtain the complete information on the probabilistic distribution with all the possible linguistic terms. If ∑k=1#LPPk<1, then partial ignorance exists because of current insufficient evaluation information. Especially, ∑k=1#LPPk=0 means complete ignorance. Therefore, handling ignorance LP is crucial research for the application of PLTSs.Definition 3 (see [10]).
Given a PLTSLP with ∑k=1#LPPk<1, then the associated PLTS L˙P is defined by(3)L˙P=LkP˙k|k=1,2,…,#LP,where P˙k=Pk/∑k=1#LPPk for all k=1,2,…,#LP.


## 2.3. Multigranular Probabilistic Linguistic Term Sets

Definition 4.
(see [30]). We let S=S0,S1,…Sg−1 be g-granular LTS and S′=S0,S1,…Sg′−1 be g′-granular LTS. L1P and L2P are two different granular PLTSs on the attribute set x=x1,x2,…xn. Multigranular PLTSs can be defined as(4)L1P=L1k1P1k1|L1k1∈S,P1k1≥0,k1=1,2,…,#L1P1,∑k1=1#L1PP1k1≤1,L2P=L2k2P2k2|L2k2∈S′,P2k2≥0,k2=1,2,…,#L2P2,∑k2=1#L2PP2k2≤1,where L1k1P1k1 is the linguistic term L1k1 associated with the probability P1k1 and L2k2P2k2 is the linguistic term L2k2 related to the probability P2k2.
The numbers of linguistic terms in PLTSs are usually different for a DM. Therefore, the numbers of linguistic terms need to be added, in which numbers are relatively small. Then, the numbers of linguistic terms are the same.
The numbers ofL1P and L2P are denoted as #L1P and#L2P, respectively. If #L1P<#L2P, then#L1P−#L2P linguistic terms are added to L2P, leading to the numbers of L1P and L2P to be equal. The added linguistic terms are the smallest ones,L2P, and all the linguistic probabilities are zero.Definition 5.
(see [30]). We let L1P and L2P be two multigranular PLTSs. Then, the normalization processes are as follows:(1)
If∑ki=1#LiPPiki<1 by Definition 3, we calculate L˙iP,i=1,2.(2)
If#L1P≠#L2P, then by Definition 4, we add some elements to the one with the smaller number of elements.
The PLTSs obtained by Definition5 are denoted the normalized PLTSs. Conveniently, the normalized PLTSs are marked by L1P and L2P as well.
Given the positions of elements in a PLTS are arbitrary, we need to obtain the ordered PLTSs first, leading to the operational results in PLTSs being determined directly.Definition 6.
(see [30]). We let S=S0,S1,…Sg−1 be g-granular LTS. Given a PLTS, LP=LkPk|Lk∈S, Pk≥0,k=1,2,…#LP, ∑k=1#LPPk≤1, and rkLk is the subscript of the linguistic term LkLP. It is named an ordered multigranular PLTS if the LkPkk=1,2,…,#LP descending order’s values arrangeαLk=rkLk/g×Pkk=1,2,…,#LP linguistic terms.


## 3. Main Results in Discrete Case

### 3.1. Generalized Distance Measures between Multigranular PLTSs

The traditional method of handling ignorance is not very scientific. Then, we extend the method and present the novel method to calculate the missing values. Inspired by [31], we present Definition 7 as follows.Definition 7.
We letL1P and L2P be two multigranular PLTSs. Then, the extended normalization processes are as follows:(1)
If∑ki=1#LiPPiki<1, then by Definition 3, we calculate L˙1P,i=1,2.(2)
If#L1P≠#L2P, then we add some elements to the one with(5)αLk=t×maxrkLkg+1−tminrkLkg,0≤t≤1,k=1,2,…,#LP,where t represents the risk preferences of the DMs. If t>0.5, it means the DMs are optimistic. If t>0.5, it means they are pessimistic. The value t should be given by the DMs previously.
Conveniently, we suppose that the PLTEs are the extended normalized and ordered multigranular PLTEs as Definitions5 and 7 in all the following sections in this paper.
The normalized distance measures are extended, and the generalized distance measures between two multigranular PLTSs in discrete cases are presented as follows.Example 1.
LetS=S0,S1,…,S6, L1P=S30.4,S20.2,S10.2 and L2P=S30.1,S20.3 be two PLTSs, then (1) according to Definition 7, L˙1P=S30.5,S20.25,S10.25, L˙2P=S20.75,S30.25. (2) Since #L2P<#L1P, then we add the linguistic term t×1.5/5+1−t×0.75/5=0.15t+0.15. When t=0.2, then after normalization, αL˙1k=0.3,0.1,0.05,αL˙2k=0.3,0.18,0.15.
Conveniently, suppose the PLTEs are the extended normalized and ordered multigranular PLTEs as Definition5 and Definition 7 in all the following sections in this paper.
The normalized distance measures are extended, and the generalized distance measures between two multigranular PLTSs in discrete cases are proposed as follows.Definition 8.
We letL1kP1k∈L1P and L2kP2k∈L2P be two PLTEs as in Definition 4. Then, the distance measured between them is defined as(6)dL1kP1k,L2kP2k=αL1kP1k−αL2kP2k.Example 2.
LetαL˙1k=0.1, αL˙2k=0.18, then dL1kP1k,L2kP2k=0.1−0.18=0.08.Definition 9.
We letL1kP1k∈L1P and L2kP2k∈L2P be two PLTEs on the attribute set, denoted by x=x1,x2,…,xn, where xj is the jth attribute of the alternatives and j=1,2,…,n. Then, the generalized Hamming distance between L1P and L2P is defined as follows:(7)dhdL1P,L2P=1L∑k=1LdL1kP1kL2kP2k,where #L1P=#L2P=L.Example 3.
LetαL˙1k=0.3,0.1,0.05 and αL˙2k=0.3,0.18,0.15, then dhdL1P,L2P=1/30.3−0.3+0.1−0.18+0.05−0.15=0.06.
The generalized Euclidean distance betweenL1(P) and L2(P) is as follows:(8)dedL1P,L2P=1L∑k=1LdL1kP1k,L2kP2k21/2.Example 4.
LetαL˙1k=0.3,0.1,0.05 and αL˙2k=0.3,0.18,0.15, then dedL1P,L2P=1/30.3−0.32+0.1−0.182+0.05−0.152=0.0739.
The generalized distance betweenL1P and L2P is as follows:(9)dgdL1P,L2P=1L∑k=1LdL1kP1k,L2kP2kλ1/λ,λ>0
Significantly, ifg=g′, the generalized distance reduces to the generalized Hamming distance. If n=1,λ=1, the generalized distance reduces to the normalized Hamming distance. If n=1,λ=2, it reduces to the normalized Euclidean distance. Definition 9 extends the normalized Hamming distance and Euclidean distance.


### 3.2. Generalized Weighted Distance Measures between Multigranular PLTSs

We letS=S0,S1,…Sg−1 be g-granular LTS and S′=S0,S1,…Sg′−1 be g′-granular LTS. L1P and L2P are two different granular PLTSs on the attribute set x=x1,x2,…,xn with the weight vector w=w1,w2,…,wnT, where xj is the jth attribute of the alternatives, j=1,2,…,n,0≤wj≤1,∑j=1nwj=1. Then, the normalized weighted distance measures are extended similar to Section 3.1. The generalized weighted distance measures between L1P and L2P are defined as follows.Definition 10.
A generalized weighted distance betweenL1P and L2P is defined as(10)dgwdL1P,L2P=∑j=1nwjL∑k=1LdL1kP1k,L2kP2kλ1/λ,λ>0.
Primarily, two exceptional cases of the generalized weighted distance are as follows:(1)
Ifλ=1, then generalized weighted distance reduces to the generalized weighted Hamming distance as follows:(11)dgwhdL1P,L2P=∑j=1nwjL∑k=1LdL1kP1k,L2kP2k.(2)
Ifλ=2, then generalized weighted distance reduces to the generalized weighted Euclidean distance as follows:(12)dgwedL1P,L2P=∑j=1nwjL∑k=1LdL1kP1k,L2kP2k21/2.


## 3.1. Generalized Distance Measures between Multigranular PLTSs

The traditional method of handling ignorance is not very scientific. Then, we extend the method and present the novel method to calculate the missing values. Inspired by [31], we present Definition 7 as follows.Definition 7.
We letL1P and L2P be two multigranular PLTSs. Then, the extended normalization processes are as follows:(1)
If∑ki=1#LiPPiki<1, then by Definition 3, we calculate L˙1P,i=1,2.(2)
If#L1P≠#L2P, then we add some elements to the one with(5)αLk=t×maxrkLkg+1−tminrkLkg,0≤t≤1,k=1,2,…,#LP,where t represents the risk preferences of the DMs. If t>0.5, it means the DMs are optimistic. If t>0.5, it means they are pessimistic. The value t should be given by the DMs previously.
Conveniently, we suppose that the PLTEs are the extended normalized and ordered multigranular PLTEs as Definitions5 and 7 in all the following sections in this paper.
The normalized distance measures are extended, and the generalized distance measures between two multigranular PLTSs in discrete cases are presented as follows.Example 1.
LetS=S0,S1,…,S6, L1P=S30.4,S20.2,S10.2 and L2P=S30.1,S20.3 be two PLTSs, then (1) according to Definition 7, L˙1P=S30.5,S20.25,S10.25, L˙2P=S20.75,S30.25. (2) Since #L2P<#L1P, then we add the linguistic term t×1.5/5+1−t×0.75/5=0.15t+0.15. When t=0.2, then after normalization, αL˙1k=0.3,0.1,0.05,αL˙2k=0.3,0.18,0.15.
Conveniently, suppose the PLTEs are the extended normalized and ordered multigranular PLTEs as Definition5 and Definition 7 in all the following sections in this paper.
The normalized distance measures are extended, and the generalized distance measures between two multigranular PLTSs in discrete cases are proposed as follows.Definition 8.
We letL1kP1k∈L1P and L2kP2k∈L2P be two PLTEs as in Definition 4. Then, the distance measured between them is defined as(6)dL1kP1k,L2kP2k=αL1kP1k−αL2kP2k.Example 2.
LetαL˙1k=0.1, αL˙2k=0.18, then dL1kP1k,L2kP2k=0.1−0.18=0.08.Definition 9.
We letL1kP1k∈L1P and L2kP2k∈L2P be two PLTEs on the attribute set, denoted by x=x1,x2,…,xn, where xj is the jth attribute of the alternatives and j=1,2,…,n. Then, the generalized Hamming distance between L1P and L2P is defined as follows:(7)dhdL1P,L2P=1L∑k=1LdL1kP1kL2kP2k,where #L1P=#L2P=L.Example 3.
LetαL˙1k=0.3,0.1,0.05 and αL˙2k=0.3,0.18,0.15, then dhdL1P,L2P=1/30.3−0.3+0.1−0.18+0.05−0.15=0.06.
The generalized Euclidean distance betweenL1(P) and L2(P) is as follows:(8)dedL1P,L2P=1L∑k=1LdL1kP1k,L2kP2k21/2.Example 4.
LetαL˙1k=0.3,0.1,0.05 and αL˙2k=0.3,0.18,0.15, then dedL1P,L2P=1/30.3−0.32+0.1−0.182+0.05−0.152=0.0739.
The generalized distance betweenL1P and L2P is as follows:(9)dgdL1P,L2P=1L∑k=1LdL1kP1k,L2kP2kλ1/λ,λ>0
Significantly, ifg=g′, the generalized distance reduces to the generalized Hamming distance. If n=1,λ=1, the generalized distance reduces to the normalized Hamming distance. If n=1,λ=2, it reduces to the normalized Euclidean distance. Definition 9 extends the normalized Hamming distance and Euclidean distance.


## 3.2. Generalized Weighted Distance Measures between Multigranular PLTSs

We letS=S0,S1,…Sg−1 be g-granular LTS and S′=S0,S1,…Sg′−1 be g′-granular LTS. L1P and L2P are two different granular PLTSs on the attribute set x=x1,x2,…,xn with the weight vector w=w1,w2,…,wnT, where xj is the jth attribute of the alternatives, j=1,2,…,n,0≤wj≤1,∑j=1nwj=1. Then, the normalized weighted distance measures are extended similar to Section 3.1. The generalized weighted distance measures between L1P and L2P are defined as follows.Definition 10.
A generalized weighted distance betweenL1P and L2P is defined as(10)dgwdL1P,L2P=∑j=1nwjL∑k=1LdL1kP1k,L2kP2kλ1/λ,λ>0.
Primarily, two exceptional cases of the generalized weighted distance are as follows:(1)
Ifλ=1, then generalized weighted distance reduces to the generalized weighted Hamming distance as follows:(11)dgwhdL1P,L2P=∑j=1nwjL∑k=1LdL1kP1k,L2kP2k.(2)
Ifλ=2, then generalized weighted distance reduces to the generalized weighted Euclidean distance as follows:(12)dgwedL1P,L2P=∑j=1nwjL∑k=1LdL1kP1k,L2kP2k21/2.


## 4. Applications of Generalized Distance Measures in MAGDM

### 4.1. Description of the Problem

A set of alternativesA=A1,A2,…,Am is presented, the attribute vector is x=x1,x2,…,xnT, w=w1,w2,…,wnT is the weight vector, and xj is the jth attribute of the alternatives, j=1,2.…,n,0≤wj≤1, ∑j=1nwj=1. The DMs assess m alternatives on n attributes by utilizing a linguistic term set to get a set of linguistic decision matrices.Then, the evaluation of linguistic information is used to make up a multigranular probabilistic linguistic decision matrix as follows:(13)R=LijPm×n=L11PL12P⋯L1nPL21PL22P⋯L2nP⋮⋮⋱⋮Lm1PLm2P⋯LmnP,where LijP=LijkijPijkij|Lijkij∈Si,Pijkij≤0,kij=1,2,⋯,#LijP,∑kij=1#LijPPijkij≤1 is a multigranular PLTS denoting the degree of the alternative Ai on the attribute xj, Si=S0,S1,…,Sgi−1 is a gi-granular fuzzy linguistic set, and rijkij is the subscript of the linguistic term LijkijPijkij, which is associated with the probability Pijkij, i=1,2,…,m,j=1,2,…,n.In MAGDM problems, the attributes can be classified into two types: benefits and costs. The higher the benefit attribute, the better the situation, whereas the opposite it applies to the cost attribute. In this paper, we suppose that the attributes are benefits.On the basis of the generalized distance measures of Section3, the extended TOPSIS is presented as follows.


### 4.2. MGPL-TOPSIS Algorithm

MGPL-TOPSIS algorithm is a MAGDM approach based on TOPSIS under multigranular probabilistic fuzzy linguistic environment proposed as follows.Step 1.
Individual preferences over the alternatives on different attributes provided by experts are gathered asR=LijPm×n.Step 2.
(see [30]). The weight vector w=w1,w2,…,wnT of n attributes x=x1,x2,…,xnT is computed as follows:(14)Ej=1m∑i=1m1-2LjT∑i=1Lj1+qrijkij⋅ln1+qrijkij+1+q1-rLj-i+1jkij⋅ln1+q1-rLj-i+1jkij/2-2+qrijkij+q1-rLj-i+1jkij/2⋅ln2+qrijkij+q1-rLj-i+1jkij/2,wj=1-Ejn-∑j=1nEj,i=1,2,…,m,j=1,2,…,n,Lj=#LijP,where T=1+qIn1+q−2+qIn2+q−In2,q>0, 0≤wj≤1, and ∑j=1nwj=1. In this paper, we let q=2 [30].Step 3.
The positive ideal solution and the negative ideal solution, respectively, are calculated.
The probabilistic linguistic positive ideal solution (PLPIS) and the probabilistic linguistic negative ideal solution (PLNIS) are defined, respectively.
The PLNIS of the alternatives is(15)L+=L1P+,L2P+,…,LnP+.
The PLNIS of the alternatives is(16)L−=L1P−,L2P−,…,LnP−,where LjP+=LΔ, in which △=maxi,j,kαLijkij, and LjP−=L∇, in which ∇=mini,j,kαLijkij.(17)αLijkij=rijkijLijkijgi×Pijkij,i=1,2,…,m,j=1,2,…,n,kij=1,2,…,#LijP.Step 4.
The distance betweenAi and L+, denoted by dAi,L+, and the distance between Ai and L−, denoted by dAi,L−, are computed.Step 5.
The closeness degree of each alternative is computed as follows:(18)CDi=1−δdAi,L−δdAi,L++1−δdAi,L−,where the parameter δ∈0,1 represents the risk preferences of the decision-makers. If δ<0.5, then the DMs are optimistic. If δ>0.5, then they are pessimistic. The value δ should be given by the DMs previously.Step 6.
The alternatives are ranked according to the valuesCDi of Ai.
The larger the closeness degree, the better the alternative.


### 4.3. MGPL-ETOPSIS Algorithm

MGPL-ETOPSIS algorithm is based on the extended TOPSIS by Qi Pang et al. [10], which is a MAGDM approach under multigranular probabilistic fuzzy linguistic environment proposed as follows:Step 1: individual preferences over the alternatives on different attributes provided by experts are gathered asR=LijPm×n.Step 2: the weight vector (see [30]) w=w1,w2,…,wnT of n attributes x=x1,x2,…,xnT is computed as follows, as seen in equation (14).Step 3: the positive ideal solution and the negative ideal solution, respectively, are calculated.Step 4: the distance betweenAi and L+, denoted by dAi,L+, and the distance between Ai and L−, denoted by dAi,L−, are computed.Step 5: compute the closeness coefficientCIi of each alternative Ai as follows:(19)CIi=dAi,L−dmaxAi,L−−dAi,L+dminAi,L+.Rank the alternatives byCIi. Obviously, the bigger the closeness coefficient, the better the alternative.


### 4.4. MGPL-VIKOR Algorithm

MGPL-VIKOR algorithm is a MAGDM approach based on VIKOR under multigranular probabilistic fuzzy linguistic environment proposed as follows:Step 1: individual preferences over the alternatives on different attributes provided by experts are gathered asR=LijPm×n.Step 2: the weight vector (see [30]) w=w1,w2,…,wnT of n attributes x=x1,x2,…,xnT is computed as follows, as seen in equation (14).Step 3: compute the distancedLijP,LjP+ and dLjP+,LjP−.Step 4: compute the whole benefitMUi and individual regret MRi, MU+, MU−, MR+, and MR−, respectively, i = 1, 2, …, m, j = 1, 2, …, n.Step 5: compute the compromise indexMCi of Ai. Rank the alternatives, according to MCi. Obviously, the bigger the compromise index, the better the alternative [31].The definitions of whole benefit,MUi, individual regret MRi, and the compromise index MCi, are as follows:(20)MUi=∑j=1nwjdLijP,LjP+dLjP+,LjP−,MRi=maxwjdLijP,LjP+dLjP+,LjP−,MCi=vMUi−MU+MU−−MU++1−vMRi−MR+MR−−MR+,where MU+=maxMUi, MU−=minMUi, MR+=maxMRi, and MR−=minMRi, i=1,2,…,m. The parameter v denotes the weight of the strategy of the maximum whole benefits, whereas 1−v is the weight of the individual regret strategy.Rank the alternatives byMCi. The higher the MCi, the more preferred the alternative.


## 4.1. Description of the Problem

A set of alternativesA=A1,A2,…,Am is presented, the attribute vector is x=x1,x2,…,xnT, w=w1,w2,…,wnT is the weight vector, and xj is the jth attribute of the alternatives, j=1,2.…,n,0≤wj≤1, ∑j=1nwj=1. The DMs assess m alternatives on n attributes by utilizing a linguistic term set to get a set of linguistic decision matrices.Then, the evaluation of linguistic information is used to make up a multigranular probabilistic linguistic decision matrix as follows:(13)R=LijPm×n=L11PL12P⋯L1nPL21PL22P⋯L2nP⋮⋮⋱⋮Lm1PLm2P⋯LmnP,where LijP=LijkijPijkij|Lijkij∈Si,Pijkij≤0,kij=1,2,⋯,#LijP,∑kij=1#LijPPijkij≤1 is a multigranular PLTS denoting the degree of the alternative Ai on the attribute xj, Si=S0,S1,…,Sgi−1 is a gi-granular fuzzy linguistic set, and rijkij is the subscript of the linguistic term LijkijPijkij, which is associated with the probability Pijkij, i=1,2,…,m,j=1,2,…,n.In MAGDM problems, the attributes can be classified into two types: benefits and costs. The higher the benefit attribute, the better the situation, whereas the opposite it applies to the cost attribute. In this paper, we suppose that the attributes are benefits.On the basis of the generalized distance measures of Section3, the extended TOPSIS is presented as follows.


## 4.2. MGPL-TOPSIS Algorithm

MGPL-TOPSIS algorithm is a MAGDM approach based on TOPSIS under multigranular probabilistic fuzzy linguistic environment proposed as follows.Step 1.
Individual preferences over the alternatives on different attributes provided by experts are gathered asR=LijPm×n.Step 2.
(see [30]). The weight vector w=w1,w2,…,wnT of n attributes x=x1,x2,…,xnT is computed as follows:(14)Ej=1m∑i=1m1-2LjT∑i=1Lj1+qrijkij⋅ln1+qrijkij+1+q1-rLj-i+1jkij⋅ln1+q1-rLj-i+1jkij/2-2+qrijkij+q1-rLj-i+1jkij/2⋅ln2+qrijkij+q1-rLj-i+1jkij/2,wj=1-Ejn-∑j=1nEj,i=1,2,…,m,j=1,2,…,n,Lj=#LijP,where T=1+qIn1+q−2+qIn2+q−In2,q>0, 0≤wj≤1, and ∑j=1nwj=1. In this paper, we let q=2 [30].Step 3.
The positive ideal solution and the negative ideal solution, respectively, are calculated.
The probabilistic linguistic positive ideal solution (PLPIS) and the probabilistic linguistic negative ideal solution (PLNIS) are defined, respectively.
The PLNIS of the alternatives is(15)L+=L1P+,L2P+,…,LnP+.
The PLNIS of the alternatives is(16)L−=L1P−,L2P−,…,LnP−,where LjP+=LΔ, in which △=maxi,j,kαLijkij, and LjP−=L∇, in which ∇=mini,j,kαLijkij.(17)αLijkij=rijkijLijkijgi×Pijkij,i=1,2,…,m,j=1,2,…,n,kij=1,2,…,#LijP.Step 4.
The distance betweenAi and L+, denoted by dAi,L+, and the distance between Ai and L−, denoted by dAi,L−, are computed.Step 5.
The closeness degree of each alternative is computed as follows:(18)CDi=1−δdAi,L−δdAi,L++1−δdAi,L−,where the parameter δ∈0,1 represents the risk preferences of the decision-makers. If δ<0.5, then the DMs are optimistic. If δ>0.5, then they are pessimistic. The value δ should be given by the DMs previously.Step 6.
The alternatives are ranked according to the valuesCDi of Ai.
The larger the closeness degree, the better the alternative.


## 4.3. MGPL-ETOPSIS Algorithm

MGPL-ETOPSIS algorithm is based on the extended TOPSIS by Qi Pang et al. [10], which is a MAGDM approach under multigranular probabilistic fuzzy linguistic environment proposed as follows:Step 1: individual preferences over the alternatives on different attributes provided by experts are gathered asR=LijPm×n.Step 2: the weight vector (see [30]) w=w1,w2,…,wnT of n attributes x=x1,x2,…,xnT is computed as follows, as seen in equation (14).Step 3: the positive ideal solution and the negative ideal solution, respectively, are calculated.Step 4: the distance betweenAi and L+, denoted by dAi,L+, and the distance between Ai and L−, denoted by dAi,L−, are computed.Step 5: compute the closeness coefficientCIi of each alternative Ai as follows:(19)CIi=dAi,L−dmaxAi,L−−dAi,L+dminAi,L+.Rank the alternatives byCIi. Obviously, the bigger the closeness coefficient, the better the alternative.


## 4.4. MGPL-VIKOR Algorithm

MGPL-VIKOR algorithm is a MAGDM approach based on VIKOR under multigranular probabilistic fuzzy linguistic environment proposed as follows:Step 1: individual preferences over the alternatives on different attributes provided by experts are gathered asR=LijPm×n.Step 2: the weight vector (see [30]) w=w1,w2,…,wnT of n attributes x=x1,x2,…,xnT is computed as follows, as seen in equation (14).Step 3: compute the distancedLijP,LjP+ and dLjP+,LjP−.Step 4: compute the whole benefitMUi and individual regret MRi, MU+, MU−, MR+, and MR−, respectively, i = 1, 2, …, m, j = 1, 2, …, n.Step 5: compute the compromise indexMCi of Ai. Rank the alternatives, according to MCi. Obviously, the bigger the compromise index, the better the alternative [31].The definitions of whole benefit,MUi, individual regret MRi, and the compromise index MCi, are as follows:(20)MUi=∑j=1nwjdLijP,LjP+dLjP+,LjP−,MRi=maxwjdLijP,LjP+dLjP+,LjP−,MCi=vMUi−MU+MU−−MU++1−vMRi−MR+MR−−MR+,where MU+=maxMUi, MU−=minMUi, MR+=maxMRi, and MR−=minMRi, i=1,2,…,m. The parameter v denotes the weight of the strategy of the maximum whole benefits, whereas 1−v is the weight of the individual regret strategy.Rank the alternatives byMCi. The higher the MCi, the more preferred the alternative.


## 5. Illustrative Example

In the real world, people usually encounter the DM problems, such as healthcare management, project evaluation, education assessment, emergency management, and smart city construction, especially COVID-19 prevention and control; for example, someone will purchase one of the five new energy cars, who can find all kinds of evaluation information of these five cars through the network for the development of Internet information. The more professional and popular website about auto information is the “Auto Home” website. Some evaluation information of these cars is presented on the “Auto Home” website in three ways: scoring data, word-of-mouth data, and forum reviews on eight attributes. The eight attributes are spacex1, power x2, manipulate (x3), power consumption (x4), comfort x5, appearance x6, interior decoration x7, and cost performance x8, respectively. The five cars are Tiggo3Xe A1, ZhongTaiE200 A2, Yuan New Energy A3, Song New Energy A4, and Qin Pro New Energy A5. Given that scoring data online is a five-point system, and the scoring data can be mapped to 5-granular linguistic term sets. The average word-of-mouth data can be mapped to 7-granular linguistic term sets. Because of the complexity of forum reviews, this information can be mapped to 9-granular linguistic term sets.Then, we use the generalized distance measure formula (equation (10)) as an example to apply the algorithm (Section 4.2) as follows.


### 5.1. Application of MGPL-TOPSIS Algorithm

The application of the algorithm based on MGPL-TOPSIS is shown as follows (Tables1–13):Step 1: the users’ evaluation information on the “Auto Home” website until Feb 17 in 2019 is collected (Tables1–3).Here, the scoring data are the five cars’ final average values on eight attributes from the scoring data (Table1). The evaluation information is the general impression of the word-of-mouth data (Table 2). The evaluation information is from the forum review data (Table 3). These data are obtained on the “Auto Home” website.Then, we obtain the users’ overall evaluation linguistic term sets (Table4).Then, we obtain the users’ overall evaluation probabilistic linguistic term sets by Definition7. We suppose that the DMs are the most pessimistic t=0. The probability of evaluation information is calculated by probability definition, and we obtain the probabilistic linguistic evaluation matrix as follows (Table 5).Then, we obtain the extended normalized DM matrix by Definition7 (Table 6).Step 2: the weight vector is calculated on the eight attributes by equation (14) (Table 7).Step 3: the PLPIS and PLNIS are calculated, respectively (Tables8 and 9). To calculate conveniently, we only denote αi instead of Sαi.Step 4:dAi,L− and dAi,L+,i=1,2,…,5 are calculated. The results are as follows (Tables 10 and 11).Step 5: the closeness coefficientCDi of Aii=1,2,…,5 by equation (18) is calculated. The results are as follows (Table 12).Step 6: The alternatives are ranked byCDi. Here, we let δ=0.5 (Table 13 and Figure 1).Table 1
Evaluation information bys5 (scoring data).
Alternativex1x2x3x4x5x6x7x8A1S4.875S55S55S55S4.475S4.635S4.35S4.835A2S4.795S4.815S4.755S4.655S4.025S4.755S4.645S4.755A3S4.335S4.715S4.675S4.455S4.395S4.855S4.245S4.755A4S4.485S4.775S4.555S4.305S4.395S4.645S4.475S4.555A5S4.405S4.785S4.615S4.455S4.165S4.815S4.275S4.455Table 2
Evaluation information bys7 (word-of-mouth data).
Alternativex1x2x3x4x5x6x7x8A1S77S77S77—S77S67—S77A2S57S77S67—S47S77S67S67A3S37S57S67S17S17S77S47S47A4S77S37S77S17S67S67S77S57A5S77—S77—S77S77S67S77Table 3
Evaluation information bys9 (forum reviews data).
Alternativex1x2x3x4x5x6x7x8A1S99S89S79S39S79S89S59S99A2S79S89S79—S59S29S19S39A3S59S69S59S29S39S99S59S49A4S89S49S89S39S49S59S99S79A5S99S19S99—S99S89S79S99Table 4
The linguistic evaluation matrix.
Alternativex1x2x3x4A1S4.875,S77,S99S55,S77,S89S55,S77,S79S55,S39A2S4.795,S57,S79S4.815,S77,S89S4.755,S67,S79S4.655A3S4.335,S37,S59S4.715,S57,S69S4.675,S67,S59S4.455,S17,S29A4S4.485,S77,S89S4.775,S37,S49S4.555,S77,S89S4.35,S17,S39A5S4.45,S77,S99S4.785,S19S4.615,S77,S99S4.455Alternativex5x6x7x8A1S4.475,S77,S79S4.635,S67,S89S4.35,S47,S59S4.835,S77,S99A2S4.025,S47,S59S4.755,S77,S29S5.645,S67,S19S4.755,S67,S39A3S4.395,S17,S39S4.855,S77,S99S4.245,S47,S59S4.755,S47,S49A4S4.395,S67,S49S4.645,S67,S59S4.475,S77,S99S4.555,S57,S79A5S4.165,S77,S99S4.815,S77,S89S4.275,S67,S79S4.455,S77,S99Table 5
The probabilistic linguistic evaluation matrix.
Alternativex1x2x3x4A1S4.8751/3,S771/3,S991/3S551/3,S771/3,S891/3S551/3,S771/3,S791/3S551/3,S171/3,S391/3A2S4.7951/3,S571/3,S791/3S4.8151/3,S771/3,S891/3S4.7551/3,S671/3,S791/3S4.6551/3,S171/3,S291/3A3S4.3351/3,S371/3,S591/3S4.7151/3,S571/3,S691/3S4.6751/3,S671/3,S591/3S4.4551/3,S171/3,S291/3A4S4.4851/3,S771/3,S891/3S4.7751/3,S371/3,S491/3S4.5551/3,S771/3,S891/3S4.351/3,S171/3,S391/3A5S4.451/3,S771/3,S991/3S4.7851/3,S371/3,S191/3S4.6151/3,S771/3,S991/3S4.4551/3,S171/3,S291/3Alternativex5x6x7x8A1S4.4751/3,S771/3,S791/3S4.6351/3,S671/3,S891/3S4.351/3,S471/3,S591/3S4.8351/3,S771/3,S991/3A2S4.0251/3,S471/3,S591/3S4.7551/3,S771/3,S291/3S5.6451/3,S671/3,S191/3S4.7551/3,S671/3,S391/3A3S4.3951/3,S171/3,S391/3S4.8551/3,S771/3,S991/3S4.2451/3,S471/3,S591/3S4.7551/3,S471/3,S491/3A4S4.3951/3,S671/3,S491/3S4.6451/3,S671/3,S591/3S4.4751/3,S771/3,S991/3S4.5551/3,S571/3,S791/3A5S4.1651/3,S771/3,S991/3S4.8151/3,S771/3,S891/3S4.2751/3,S671/3,S791/3S4.4551/3,S771/3,S991/3Table 6
The normalized DM matrix.
Alternativex1x2x3x4x5x6x7x8A1S2/3,S4.87/15,S0S2/3,S8/27,S0S2/3,S7/27,S0S1/3,S1/9,S1/21S1/3,S4.47/15,S7/27S4.63/15,S8/27,S2/7S4.3/15,S4/21,S5/27S2/3,S4.83/15,S0A2S4.79/15,S7/27,S5/21S1/3,S4.81/15,S8/27S4.75/15,S2/7,S7/27S4.65/15,S2/27,S1/21S4.02/15,S4/21,S5/27S1/3,S4.75/15,S2/27S4.64/15,S2/7,S1/27S4.75/15,S2/7,S1/9A3S4.33/15,S5/27,S1/7S4.71/15,S5/21,S2/9S4.67/15,S2/7,S5/27S4.45/15,S2/27,S1/21S4.39/15,S1/9,S1/21S2/3,S4.85/15,S0S4.24/15,S4/21,S5/27S4.75/15,S4/21,S4/27A4S1/3,S4.48/15,S8/27S4.77/15,S4/27,S1/7S1/3,S4.55/15,S8/27S4.3/15,S1/9,S1/21S4.39/15,S2/7,S4/27S4.64/15,S2/7,S5/27S2/3,S4.47/15,S0S4.55/15,S7/27,S5/21A5S2/3,S4.4/15,S0S4.78/15,S1/7,S1/27S2/3,S4.61/15,S0S4.45/15,S2/27,S1/21S2/3,S4.16/15,S0S1/3,S4.81/15,S8/27S2/7,S4.27/15,S7/27S2/3,S4.45/15,S0Table 7
The weights of the attributes.
Attributex1x2x3x4x5x6x7x8Weight0.09210.12650.08070.24660.13500.08950.12620.1034Table 8
The normalized PLPIS.
Attributex1x2x3x4PLPIS(0.6667, 0.3247, 0.2963)(0.6667, 0.3207, 0.2963)(0.6667, 0.3073, 0.2963)(0.3333, 0.1111, 0.0476)Attributex5x6x7x8PLPIS(0.6667, 0.2980, 0.2593)(0.6667, 0.3233, 0.2963)(0.6667, 0.2980, 0.2593)(0.6667, 0.3220, 0.2381)Table 9
The normalized PLNIS.
Attributex1x2x3x4PLNIS(0.2887, 0.1852, 0.2963)(0.3140, 0.1429, 0.2963)(0.3113, 0.2593, 0.2963)(0.2867, 0.0741, 0.0476)Attributex5x6x7x8PLNIS(0.2680, 0.1111, 0.2593)(0.3087, 0.2857, 0.2963)(0.2827, 0.1905, 0.2593)(0.3033, 0.1905, 0.2381)Table 10
The results ofdAi,L−.
AlternativeA1A2A3A4A5λ=10.12380.05490.05490.05490.0549λ=20.18100.08460.10360.11680.1867λ=30.21600.10710.14120.15510.2236λ=40.23950.12410.17020.18590.2481λ=50.25640.13720.19280.21060.2658λ=60.26920.14730.21070.23040.2795λ=70.27930.15540.22520.24660.2904λ=80.28740.16190.23720.25990.2993λ=90.29410.16730.24720.27090.3068λ=100.29980.17190.25570.28030.3132Table 11
The results ofdAi,L+.
AlternativeA1A2A3A4A5λ=10.09050.12990.14840.11550.1055λ=20.16260.19260.19990.18050.1704λ=30.20390.23090.23440.21910.2075λ=40.23020.25570.25830.24440.2313λ=50.24860.27290.27540.26210.2483λ=60.26220.28560.28820.27510.2612λ=70.27290.29540.29810.28520.2715λ=80.28150.30330.30590.29310.2801λ=90.28880.30980.31220.29960.2873λ=100.29490.31530.31750.30500.2936Table 12
The closeness coefficientCDi.
AlternativeA1A2A3A4A5λ=10.57760.29690.26800.36990.5366λ=20.52680.30530.34140.39280.5228λ=30.51440.31680.37600.41450.5186λ=40.50990.32680.39720.43200.5175λ=50.50780.33450.41180.44550.5171λ=60.50660.34030.42230.45580.5169λ=70.50580.34470.43040.46370.5168λ=80.50520.34810.43670.46990.5166λ=90.50460.35070.44190.47490.5164λ=100.50410.35280.44610.47890.5162Table 13
The ranking ofAi.
Distance parameterRankλ=1A1≻A5≻A4≻A2≻A3λ=2A1≻A5≻A4≻A3≻A2λ=3A5≻A1≻A4≻A3≻A2λ=4A5≻A1≻A4≻A3≻A2λ=5A5≻A1≻A4≻A3≻A2λ=6A5≻A1≻A4≻A3≻A2λ=7A5≻A1≻A4≻A3≻A2λ=8A5≻A1≻A4≻A3≻A2λ=9A5≻A1≻A4≻A3≻A2λ=10A5≻A1≻A4≻A3≻A2Table13 and Figure 1 show that whenλ=1, the ranking ofAi(i=1,2,…,5) is “A1≻A5≻A4≻A2≻A3,” whenλ=2, the ranking is “A1≻A5≻A4≻A3≻A2,” and when λ=3,4,…,10, the ranking is “A5≻A1≻A4≻A3≻A2,” illustrating that the ranking results are changed with λ. When λ is different, the ranking result is different. When λ=3, the ranking is stable, demonstrating that the algorithm based on TOPSIS is available.Figure 1
CDi based on MGPL-TOPSIS.In real life, we can use the MGPL-TOPSIS algorithm to deal with some GDM problems, such as environmental pollution management, urban traffic planning, treatment options, project evaluation, education assessment, emergency management, and smart city construction, especially COVID-19 prevention and control. The calculation method of the MGPL-TOPSIS algorithm is more convenient and effective, and the scientific method to make up for the DMs’ missing information reduces the loss of effective information, so as to make the DM results more objective and effective.


### 5.2. Comparative Analysis and Discussion

To demonstrate the feasibility and efficiency of the algorithm based on generalized distance measures of the PLTSs, we calculate the other results by the two different algorithms based on MGPL-ETOPSIS and MGPL-VIKOR, respectively. Here, we letδ=0.5 (Tables 14 and 15).Table 14
The compromise indexCIi based on MGPL-ETOPSIS.
AlternativeA1A2A3A4A5λ=1−0.0005−0.9926−1.2005−0.7282−0.1789λ=2−0.0304−0.7314−0.6742−0.4844−0.0481λ=3−0.0340−0.6535−0.5179−0.3809−0.0178λ=4−0.0347−0.6103−0.4359−0.3123−0.0050λ=5−0.0363−0.5830−0.3839−0.26330.0003λ=6−0.0407−0.5663−0.3495−0.22890.0000λ=7−0.0434−0.5531−0.3223−0.2013−0.0003λ=8−0.0449−0.5418−0.2996−0.17830.0000λ=9−0.0463−0.5329−0.2810−0.1597−0.0001λ=10−0.0471−0.5251−0.2649−0.14380.0001Table 15
The compromise indexMCi based on MGPL-VIKOR.
AlternativeA1A2A3A4A5λ=10.92430.45340.00000.78230.5470λ=20.84730.29730.00000.67970.4548λ=30.92380.16040.00000.62620.5279λ=41.00000.04590.20720.57430.6898λ=51.00000.04270.32410.52300.7447λ=61.00000.04070.37170.50210.7564λ=71.00000.03900.39310.49220.7599λ=81.00000.03740.40320.48670.7625λ=91.00000.03600.40810.48340.7644λ=101.00000.03470.41050.48110.7659In Figures1 and 2, we can find that the results based on MGPL-TOPSIS and MGPL-ETOPSIS are the same as follows. When λ=1, the ranking of Ai(i=1,2,…,5) is “A1≻A5≻A4≻A2≻A3.” When λ=2, the ranking is “A1≻A5≻A4≻A3≻A2,” and when λ=3,4,…,10, the ranking is “A5≻A1≻A4≻A3≻A2.” The results based on MGPL-VIKOR are as follows (Figure 3). When λ=1, λ=2, andλ=3, the ranking is “A1≻A4≻A5≻A2≻A3,” and whenλ=4, 5,…, 10, the ranking is “A1≻A5≻A4≻A3≻A2.” The results based on PT are as follows. When λ=1 and 2, the ranking is “A5≻A1≻A2≻A4≻A3,” and when λ=3,4,…,10, the ranking is “A5≻A1≻A2≻A3≻A4.”Figure 2
CIi based on MGPL-ETOPSIS.Figure 3
TheMCi based on MGPL-VIKOR.


### 5.3. Sensitivity Analysis

To calculate the sensitivity parameterλ, we take the algorithm’s different parameters based on PT [29] and rank Aii=1,2,…,5. Then, the results are shown in Tables 16–19.Table 16
The rationCPi based on PT (α=0.85,θ=4.1,andβ=0.85).
AlternativeA1A2A3A4A5λ=10.31300.11450.10760.13820.2853λ=20.24090.11090.11950.12830.2355λ=30.22030.10940.12090.12470.2222λ=40.21150.10850.12050.12300.2168λ=50.20710.10800.11990.12210.2141λ=60.20450.10770.11940.12170.2127λ=70.20290.10740.11900.12140.2118λ=80.20190.10720.11880.12130.2112λ=90.20120.10710.11860.12120.2109λ=100.20060.10700.11840.12110.2106Table 17
The rationCPi based on PT (α=0.88,θ=2.25,andβ=0.88).
AlternativeA1A2A3A4A5λ=10.57830.20360.19210.24870.5267λ=20.44070.19700.21360.22980.4314λ=30.40150.19430.21610.22300.4060λ=40.38500.19280.21520.21980.3958λ=50.37660.19180.21410.21820.3908λ=60.37170.19120.21310.21730.3880λ=70.36870.19070.21240.21680.3863λ=80.36670.19040.21190.21650.3852λ=90.36540.19010.21160.21640.3845λ=100.36440.18990.21130.21630.3841Table 18
The rationCPi based on PT (α=0.89,θ=2.25,andβ=0.92).
AlternativeA1A2A3A4A5λ=10.61890.21340.20080.26250.5625λ=20.46420.20470.22230.24000.4549λ=30.42020.20100.22400.23160.4259λ=40.40170.19900.22260.22770.4140λ=50.39210.19770.22110.22570.4081λ=60.38660.19680.21980.22450.4047λ=70.38310.19620.21890.22380.4026λ=80.38080.19580.21830.22340.4013λ=90.37920.19540.21780.22310.4004λ=100.37800.19520.21740.22290.3997Table 19
The rationCi (α=0.725,θ=2.04,andβ=0.717).
AlternativeA1A2A3A4A5λ=10.58610.25170.23420.28940.5353λ=20.47210.24520.25810.27450.4575λ=30.43850.24240.26140.26880.4365λ=40.42420.24080.26100.26620.4281λ=50.41690.23980.26020.26480.4240λ=60.41270.23920.25950.26410.4217λ=70.41010.23870.25890.26370.4204λ=80.40830.23840.25850.26350.4742λ=90.40720.23810.25820.26340.4190λ=100.40630.23790.25800.26330.4187Tables16–20 show that the results are as follows. When λ=1, the ranking of Ai (i=1,2,…,5) is “A1 ≻ A5 ≻ A4 ≻ A2 ≻ A3.” When λ=2, the ranking is “A1 ≻ A5 ≻ A4 ≻ A3 ≻ A2,” and when λ=3,4,…,10, the ranking is “A5 ≻ A1 ≻ A4 ≻ A3 ≻ A2” although the parameters θ, α, and β are changed. Therefore, the larger the parameter λ, the ranking Ai tends to be stable, which is not affected by the parameters’ subjective psychological factors θ, α, and β. The definitions of distance measures between two PLTSs under a multigranular linguistic environment are scientifically presented in Section 3.Table 20
The ranking ofAi.
AlgorithmParameterRankPTα=0.85,θ=4.1,β=0.85α=0.88,θ=2.25,β=0.88α=0.89,θ=2.25,β=0.92α=0.725,θ=2.04,β=0.717λ=1A1≻A5≻A4≻A2≻A3λ=2A1≻A5≻A4≻A3≻A2λ=3A5≻A1≻A4≻A3≻A2λ=4A5≻A1≻A4≻A3≻A2λ=5A5≻A1≻A4≻A3≻A2λ=6A5≻A1≻A4≻A3≻A2λ=7A5≻A1≻A4≻A3≻A2λ=8A5≻A1≻A4≻A3≻A2λ=9A5≻A1≻A4≻A3≻A2λ=10A5≻A1≻A4≻A3≻A2


## 5.1. Application of MGPL-TOPSIS Algorithm

The application of the algorithm based on MGPL-TOPSIS is shown as follows (Tables1–13):Step 1: the users’ evaluation information on the “Auto Home” website until Feb 17 in 2019 is collected (Tables1–3).Here, the scoring data are the five cars’ final average values on eight attributes from the scoring data (Table1). The evaluation information is the general impression of the word-of-mouth data (Table 2). The evaluation information is from the forum review data (Table 3). These data are obtained on the “Auto Home” website.Then, we obtain the users’ overall evaluation linguistic term sets (Table4).Then, we obtain the users’ overall evaluation probabilistic linguistic term sets by Definition7. We suppose that the DMs are the most pessimistic t=0. The probability of evaluation information is calculated by probability definition, and we obtain the probabilistic linguistic evaluation matrix as follows (Table 5).Then, we obtain the extended normalized DM matrix by Definition7 (Table 6).Step 2: the weight vector is calculated on the eight attributes by equation (14) (Table 7).Step 3: the PLPIS and PLNIS are calculated, respectively (Tables8 and 9). To calculate conveniently, we only denote αi instead of Sαi.Step 4:dAi,L− and dAi,L+,i=1,2,…,5 are calculated. The results are as follows (Tables 10 and 11).Step 5: the closeness coefficientCDi of Aii=1,2,…,5 by equation (18) is calculated. The results are as follows (Table 12).Step 6: The alternatives are ranked byCDi. Here, we let δ=0.5 (Table 13 and Figure 1).Table 1
Evaluation information bys5 (scoring data).
Alternativex1x2x3x4x5x6x7x8A1S4.875S55S55S55S4.475S4.635S4.35S4.835A2S4.795S4.815S4.755S4.655S4.025S4.755S4.645S4.755A3S4.335S4.715S4.675S4.455S4.395S4.855S4.245S4.755A4S4.485S4.775S4.555S4.305S4.395S4.645S4.475S4.555A5S4.405S4.785S4.615S4.455S4.165S4.815S4.275S4.455Table 2
Evaluation information bys7 (word-of-mouth data).
Alternativex1x2x3x4x5x6x7x8A1S77S77S77—S77S67—S77A2S57S77S67—S47S77S67S67A3S37S57S67S17S17S77S47S47A4S77S37S77S17S67S67S77S57A5S77—S77—S77S77S67S77Table 3
Evaluation information bys9 (forum reviews data).
Alternativex1x2x3x4x5x6x7x8A1S99S89S79S39S79S89S59S99A2S79S89S79—S59S29S19S39A3S59S69S59S29S39S99S59S49A4S89S49S89S39S49S59S99S79A5S99S19S99—S99S89S79S99Table 4
The linguistic evaluation matrix.
Alternativex1x2x3x4A1S4.875,S77,S99S55,S77,S89S55,S77,S79S55,S39A2S4.795,S57,S79S4.815,S77,S89S4.755,S67,S79S4.655A3S4.335,S37,S59S4.715,S57,S69S4.675,S67,S59S4.455,S17,S29A4S4.485,S77,S89S4.775,S37,S49S4.555,S77,S89S4.35,S17,S39A5S4.45,S77,S99S4.785,S19S4.615,S77,S99S4.455Alternativex5x6x7x8A1S4.475,S77,S79S4.635,S67,S89S4.35,S47,S59S4.835,S77,S99A2S4.025,S47,S59S4.755,S77,S29S5.645,S67,S19S4.755,S67,S39A3S4.395,S17,S39S4.855,S77,S99S4.245,S47,S59S4.755,S47,S49A4S4.395,S67,S49S4.645,S67,S59S4.475,S77,S99S4.555,S57,S79A5S4.165,S77,S99S4.815,S77,S89S4.275,S67,S79S4.455,S77,S99Table 5
The probabilistic linguistic evaluation matrix.
Alternativex1x2x3x4A1S4.8751/3,S771/3,S991/3S551/3,S771/3,S891/3S551/3,S771/3,S791/3S551/3,S171/3,S391/3A2S4.7951/3,S571/3,S791/3S4.8151/3,S771/3,S891/3S4.7551/3,S671/3,S791/3S4.6551/3,S171/3,S291/3A3S4.3351/3,S371/3,S591/3S4.7151/3,S571/3,S691/3S4.6751/3,S671/3,S591/3S4.4551/3,S171/3,S291/3A4S4.4851/3,S771/3,S891/3S4.7751/3,S371/3,S491/3S4.5551/3,S771/3,S891/3S4.351/3,S171/3,S391/3A5S4.451/3,S771/3,S991/3S4.7851/3,S371/3,S191/3S4.6151/3,S771/3,S991/3S4.4551/3,S171/3,S291/3Alternativex5x6x7x8A1S4.4751/3,S771/3,S791/3S4.6351/3,S671/3,S891/3S4.351/3,S471/3,S591/3S4.8351/3,S771/3,S991/3A2S4.0251/3,S471/3,S591/3S4.7551/3,S771/3,S291/3S5.6451/3,S671/3,S191/3S4.7551/3,S671/3,S391/3A3S4.3951/3,S171/3,S391/3S4.8551/3,S771/3,S991/3S4.2451/3,S471/3,S591/3S4.7551/3,S471/3,S491/3A4S4.3951/3,S671/3,S491/3S4.6451/3,S671/3,S591/3S4.4751/3,S771/3,S991/3S4.5551/3,S571/3,S791/3A5S4.1651/3,S771/3,S991/3S4.8151/3,S771/3,S891/3S4.2751/3,S671/3,S791/3S4.4551/3,S771/3,S991/3Table 6
The normalized DM matrix.
Alternativex1x2x3x4x5x6x7x8A1S2/3,S4.87/15,S0S2/3,S8/27,S0S2/3,S7/27,S0S1/3,S1/9,S1/21S1/3,S4.47/15,S7/27S4.63/15,S8/27,S2/7S4.3/15,S4/21,S5/27S2/3,S4.83/15,S0A2S4.79/15,S7/27,S5/21S1/3,S4.81/15,S8/27S4.75/15,S2/7,S7/27S4.65/15,S2/27,S1/21S4.02/15,S4/21,S5/27S1/3,S4.75/15,S2/27S4.64/15,S2/7,S1/27S4.75/15,S2/7,S1/9A3S4.33/15,S5/27,S1/7S4.71/15,S5/21,S2/9S4.67/15,S2/7,S5/27S4.45/15,S2/27,S1/21S4.39/15,S1/9,S1/21S2/3,S4.85/15,S0S4.24/15,S4/21,S5/27S4.75/15,S4/21,S4/27A4S1/3,S4.48/15,S8/27S4.77/15,S4/27,S1/7S1/3,S4.55/15,S8/27S4.3/15,S1/9,S1/21S4.39/15,S2/7,S4/27S4.64/15,S2/7,S5/27S2/3,S4.47/15,S0S4.55/15,S7/27,S5/21A5S2/3,S4.4/15,S0S4.78/15,S1/7,S1/27S2/3,S4.61/15,S0S4.45/15,S2/27,S1/21S2/3,S4.16/15,S0S1/3,S4.81/15,S8/27S2/7,S4.27/15,S7/27S2/3,S4.45/15,S0Table 7
The weights of the attributes.
Attributex1x2x3x4x5x6x7x8Weight0.09210.12650.08070.24660.13500.08950.12620.1034Table 8
The normalized PLPIS.
Attributex1x2x3x4PLPIS(0.6667, 0.3247, 0.2963)(0.6667, 0.3207, 0.2963)(0.6667, 0.3073, 0.2963)(0.3333, 0.1111, 0.0476)Attributex5x6x7x8PLPIS(0.6667, 0.2980, 0.2593)(0.6667, 0.3233, 0.2963)(0.6667, 0.2980, 0.2593)(0.6667, 0.3220, 0.2381)Table 9
The normalized PLNIS.
Attributex1x2x3x4PLNIS(0.2887, 0.1852, 0.2963)(0.3140, 0.1429, 0.2963)(0.3113, 0.2593, 0.2963)(0.2867, 0.0741, 0.0476)Attributex5x6x7x8PLNIS(0.2680, 0.1111, 0.2593)(0.3087, 0.2857, 0.2963)(0.2827, 0.1905, 0.2593)(0.3033, 0.1905, 0.2381)Table 10
The results ofdAi,L−.
AlternativeA1A2A3A4A5λ=10.12380.05490.05490.05490.0549λ=20.18100.08460.10360.11680.1867λ=30.21600.10710.14120.15510.2236λ=40.23950.12410.17020.18590.2481λ=50.25640.13720.19280.21060.2658λ=60.26920.14730.21070.23040.2795λ=70.27930.15540.22520.24660.2904λ=80.28740.16190.23720.25990.2993λ=90.29410.16730.24720.27090.3068λ=100.29980.17190.25570.28030.3132Table 11
The results ofdAi,L+.
AlternativeA1A2A3A4A5λ=10.09050.12990.14840.11550.1055λ=20.16260.19260.19990.18050.1704λ=30.20390.23090.23440.21910.2075λ=40.23020.25570.25830.24440.2313λ=50.24860.27290.27540.26210.2483λ=60.26220.28560.28820.27510.2612λ=70.27290.29540.29810.28520.2715λ=80.28150.30330.30590.29310.2801λ=90.28880.30980.31220.29960.2873λ=100.29490.31530.31750.30500.2936Table 12
The closeness coefficientCDi.
AlternativeA1A2A3A4A5λ=10.57760.29690.26800.36990.5366λ=20.52680.30530.34140.39280.5228λ=30.51440.31680.37600.41450.5186λ=40.50990.32680.39720.43200.5175λ=50.50780.33450.41180.44550.5171λ=60.50660.34030.42230.45580.5169λ=70.50580.34470.43040.46370.5168λ=80.50520.34810.43670.46990.5166λ=90.50460.35070.44190.47490.5164λ=100.50410.35280.44610.47890.5162Table 13
The ranking ofAi.
Distance parameterRankλ=1A1≻A5≻A4≻A2≻A3λ=2A1≻A5≻A4≻A3≻A2λ=3A5≻A1≻A4≻A3≻A2λ=4A5≻A1≻A4≻A3≻A2λ=5A5≻A1≻A4≻A3≻A2λ=6A5≻A1≻A4≻A3≻A2λ=7A5≻A1≻A4≻A3≻A2λ=8A5≻A1≻A4≻A3≻A2λ=9A5≻A1≻A4≻A3≻A2λ=10A5≻A1≻A4≻A3≻A2Table13 and Figure 1 show that whenλ=1, the ranking ofAi(i=1,2,…,5) is “A1≻A5≻A4≻A2≻A3,” whenλ=2, the ranking is “A1≻A5≻A4≻A3≻A2,” and when λ=3,4,…,10, the ranking is “A5≻A1≻A4≻A3≻A2,” illustrating that the ranking results are changed with λ. When λ is different, the ranking result is different. When λ=3, the ranking is stable, demonstrating that the algorithm based on TOPSIS is available.Figure 1
CDi based on MGPL-TOPSIS.In real life, we can use the MGPL-TOPSIS algorithm to deal with some GDM problems, such as environmental pollution management, urban traffic planning, treatment options, project evaluation, education assessment, emergency management, and smart city construction, especially COVID-19 prevention and control. The calculation method of the MGPL-TOPSIS algorithm is more convenient and effective, and the scientific method to make up for the DMs’ missing information reduces the loss of effective information, so as to make the DM results more objective and effective.


## 5.2. Comparative Analysis and Discussion

To demonstrate the feasibility and efficiency of the algorithm based on generalized distance measures of the PLTSs, we calculate the other results by the two different algorithms based on MGPL-ETOPSIS and MGPL-VIKOR, respectively. Here, we letδ=0.5 (Tables 14 and 15).Table 14
The compromise indexCIi based on MGPL-ETOPSIS.
AlternativeA1A2A3A4A5λ=1−0.0005−0.9926−1.2005−0.7282−0.1789λ=2−0.0304−0.7314−0.6742−0.4844−0.0481λ=3−0.0340−0.6535−0.5179−0.3809−0.0178λ=4−0.0347−0.6103−0.4359−0.3123−0.0050λ=5−0.0363−0.5830−0.3839−0.26330.0003λ=6−0.0407−0.5663−0.3495−0.22890.0000λ=7−0.0434−0.5531−0.3223−0.2013−0.0003λ=8−0.0449−0.5418−0.2996−0.17830.0000λ=9−0.0463−0.5329−0.2810−0.1597−0.0001λ=10−0.0471−0.5251−0.2649−0.14380.0001Table 15
The compromise indexMCi based on MGPL-VIKOR.
AlternativeA1A2A3A4A5λ=10.92430.45340.00000.78230.5470λ=20.84730.29730.00000.67970.4548λ=30.92380.16040.00000.62620.5279λ=41.00000.04590.20720.57430.6898λ=51.00000.04270.32410.52300.7447λ=61.00000.04070.37170.50210.7564λ=71.00000.03900.39310.49220.7599λ=81.00000.03740.40320.48670.7625λ=91.00000.03600.40810.48340.7644λ=101.00000.03470.41050.48110.7659In Figures1 and 2, we can find that the results based on MGPL-TOPSIS and MGPL-ETOPSIS are the same as follows. When λ=1, the ranking of Ai(i=1,2,…,5) is “A1≻A5≻A4≻A2≻A3.” When λ=2, the ranking is “A1≻A5≻A4≻A3≻A2,” and when λ=3,4,…,10, the ranking is “A5≻A1≻A4≻A3≻A2.” The results based on MGPL-VIKOR are as follows (Figure 3). When λ=1, λ=2, andλ=3, the ranking is “A1≻A4≻A5≻A2≻A3,” and whenλ=4, 5,…, 10, the ranking is “A1≻A5≻A4≻A3≻A2.” The results based on PT are as follows. When λ=1 and 2, the ranking is “A5≻A1≻A2≻A4≻A3,” and when λ=3,4,…,10, the ranking is “A5≻A1≻A2≻A3≻A4.”Figure 2
CIi based on MGPL-ETOPSIS.Figure 3
TheMCi based on MGPL-VIKOR.


## 5.3. Sensitivity Analysis

To calculate the sensitivity parameterλ, we take the algorithm’s different parameters based on PT [29] and rank Aii=1,2,…,5. Then, the results are shown in Tables 16–19.Table 16
The rationCPi based on PT (α=0.85,θ=4.1,andβ=0.85).
AlternativeA1A2A3A4A5λ=10.31300.11450.10760.13820.2853λ=20.24090.11090.11950.12830.2355λ=30.22030.10940.12090.12470.2222λ=40.21150.10850.12050.12300.2168λ=50.20710.10800.11990.12210.2141λ=60.20450.10770.11940.12170.2127λ=70.20290.10740.11900.12140.2118λ=80.20190.10720.11880.12130.2112λ=90.20120.10710.11860.12120.2109λ=100.20060.10700.11840.12110.2106Table 17
The rationCPi based on PT (α=0.88,θ=2.25,andβ=0.88).
AlternativeA1A2A3A4A5λ=10.57830.20360.19210.24870.5267λ=20.44070.19700.21360.22980.4314λ=30.40150.19430.21610.22300.4060λ=40.38500.19280.21520.21980.3958λ=50.37660.19180.21410.21820.3908λ=60.37170.19120.21310.21730.3880λ=70.36870.19070.21240.21680.3863λ=80.36670.19040.21190.21650.3852λ=90.36540.19010.21160.21640.3845λ=100.36440.18990.21130.21630.3841Table 18
The rationCPi based on PT (α=0.89,θ=2.25,andβ=0.92).
AlternativeA1A2A3A4A5λ=10.61890.21340.20080.26250.5625λ=20.46420.20470.22230.24000.4549λ=30.42020.20100.22400.23160.4259λ=40.40170.19900.22260.22770.4140λ=50.39210.19770.22110.22570.4081λ=60.38660.19680.21980.22450.4047λ=70.38310.19620.21890.22380.4026λ=80.38080.19580.21830.22340.4013λ=90.37920.19540.21780.22310.4004λ=100.37800.19520.21740.22290.3997Table 19
The rationCi (α=0.725,θ=2.04,andβ=0.717).
AlternativeA1A2A3A4A5λ=10.58610.25170.23420.28940.5353λ=20.47210.24520.25810.27450.4575λ=30.43850.24240.26140.26880.4365λ=40.42420.24080.26100.26620.4281λ=50.41690.23980.26020.26480.4240λ=60.41270.23920.25950.26410.4217λ=70.41010.23870.25890.26370.4204λ=80.40830.23840.25850.26350.4742λ=90.40720.23810.25820.26340.4190λ=100.40630.23790.25800.26330.4187Tables16–20 show that the results are as follows. When λ=1, the ranking of Ai (i=1,2,…,5) is “A1 ≻ A5 ≻ A4 ≻ A2 ≻ A3.” When λ=2, the ranking is “A1 ≻ A5 ≻ A4 ≻ A3 ≻ A2,” and when λ=3,4,…,10, the ranking is “A5 ≻ A1 ≻ A4 ≻ A3 ≻ A2” although the parameters θ, α, and β are changed. Therefore, the larger the parameter λ, the ranking Ai tends to be stable, which is not affected by the parameters’ subjective psychological factors θ, α, and β. The definitions of distance measures between two PLTSs under a multigranular linguistic environment are scientifically presented in Section 3.Table 20
The ranking ofAi.
AlgorithmParameterRankPTα=0.85,θ=4.1,β=0.85α=0.88,θ=2.25,β=0.88α=0.89,θ=2.25,β=0.92α=0.725,θ=2.04,β=0.717λ=1A1≻A5≻A4≻A2≻A3λ=2A1≻A5≻A4≻A3≻A2λ=3A5≻A1≻A4≻A3≻A2λ=4A5≻A1≻A4≻A3≻A2λ=5A5≻A1≻A4≻A3≻A2λ=6A5≻A1≻A4≻A3≻A2λ=7A5≻A1≻A4≻A3≻A2λ=8A5≻A1≻A4≻A3≻A2λ=9A5≻A1≻A4≻A3≻A2λ=10A5≻A1≻A4≻A3≻A2


## 6. The Extended TOPSIS Method

There is a set of three alternativesA=A1,A2,A3 and the weight vector w=w1,w2,w3,w4T of attribute vector x=x1,x2,x3,x4T, where 0≤wj≤1, ∑j=1nwj=1 [10]. The five DMs assess the three alternatives on four attributes by the multigranular linguistic set, which is S=S0=none,S1=very low,S2=low,S3=medium,S4=high,S5=very high,S6=perfect (Tables 21–25).Step 1: collect the five DMs’ evaluation information:Their original decision matrices are shown in Tables21–25. The probabilistic linguistic decision matrix and the normalized probabilistic linguistic decision matrix of the group are shown in Tables 26 and 27.Step 2: calculate the weight vector of the attributesxjj=1,2,3,4:(21)w=0.2396,0.2340,0.1332,0.3931T.Step 3: determine the PLPISL+ and the PLNIS L−, respectively (Table 28).Step 4: calculate the deviation degrees between each alternative and the PLPIS (PLNIS), respectivelyi=1,2,3:(22)dA1,L+=0.4257,dA2,L+=0.8076,dA3,L+=0.4055,dA1,L−=0.6244,dA2,L−=0.1223,dA3,L−=0.6692,dminAi,L+=0.4055,dmaxAi,L−=0.6692.Step 5: calculate the closeness coefficientCIAii=1,2,3:(23)CIA1=−0.1168,CIA2=−1.8089,CIA3=0.Step 6: rank the alternativesAi according to CIAii=1,2,3:(24)A3>A1>A2Table 21
The linguistic decision matrix provided by the first DM.
Alternativex1x2x3x4A1S3S4S5S4A2S2S3S1S2A3S4S3—S5Table 22
The linguistic decision matrix provided by the second DM.
Alternativex1x2x3x4A1S4S2S4S5A2S3S1—S3A3S5S3S4S5Table 23
The linguistic decision matrix provided by the third DM.
Alternativex1x2x3x4A1S4—S4S4A2S3S2S5S3A3S4S3S1S5Table 24
The linguistic decision matrix provided by the fourth DM.
Alternativex1x2x3x4A1S4S4S5S3A2S5S2S4—A3S3S2S4S4Table 25
The linguistic decision matrix provided by the fifth DM.
Alternativex1x2x3x4A1S3S4S2S5A2S3S3S4S2A3S3—S5S4Table 26
The probabilistic linguistic decision matrix of the group.
Attributesx1x2x3x4A1S30.4,S40.6S40.6,S20.2S50.4,S20.2,S40.4S40.4,S30.2,S50.4A2S20.2,S30.6,S50.2S30.4,S10.2,S20.4S10.2,S50.2,S40.4S30.4,S20.4A3S40.4,S30.4,S50.2S30.6,S20.2S10.2,S50.2,S40.4S40.4,S50.6Table 27
The normalized probabilistic linguistic decision matrix of the group.
Attributesx1x2x3x4A1S40.6,S30.4,S30S40.6,S20.2,S20S50.4,S40.4,S20.2S50.4,S40.4,S30.2A2S30.6,S50.2,S20.2S30.4,S20.4,S10.2S40.4,S50.2,S10.2S30.4,S20.4,S20A3S40.4,S30.4,S50.2S30.6,S20.2,S20S40.4,S50.2,S10.2S50.6,S40.4,S40Table 28
The positive ideal solution and the negative ideal solution.
Attributesx1x2x3x4PLPIS(2.4, 1.2, 1)(2.4, 0.8, 0.2)(2, 1.6, 0.4)(3, 1.6, 0.6)Attributesx1x2x3x4PLNIS(1.6, 1, 0)(1.2, 0.4, 0)(1.6, 1, 0.2)(1.2, 0.8, 0)By comparing the proposed classical algorithm above, we can find it is only a special example of the proposed algorithms; in this paper, whenλ=2, that is, the proposed algorithms extended the classical algorithm.


## 7. Conclusions

This paper presents generalized distance measures between two PLTSs with multigranular probabilistic linguistic information and the method of filling with the missing evaluation information, which can be used to deal with MAGDM problems. The main advantages of this paper can be given as follows: (1) the generalized distance measures improve the accuracy of multigranular linguistic information in the MAGDM issues, and even some evaluation information is null, a more reasonable method is presented to address the missing value of the evaluation information; (2) the parameterλ of the generalized distance measures is a variable that can be used to obtain different distance measures formula according to the DMs’ needs; since then, the rankings of the alternatives are stable; (3) under these distance measures, the presented three algorithms MGPL-TOPSIS, MGPL-ETOPSIS, and MGPL-VIKOR are more available; (4) the proposed calculation method of PLTSs is more effective; (5) the entropy weight method can save more information and become more objective and accurate.The generalized distance measures presented in this paper also have some limitations: (1) whether more appropriate ways exist to measure the distances between two PLTSs with multigranular linguistic information is a worthwhile question; (2) how to select theλ proper to measure the distance between two PLTSs and the algorithm according to the practical problem have potential to be studied further. Some directions for further research are identified. First, the applications of these extended distance measures are interesting to study in cluster analysis and other MAGDM problems. Second, linguistic information can be expressed by interval fuzzy numbers.



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*Source: 5553343-2021-06-09.xml*