GRA Method for Probabilistic Linguistic Multiple Attribute Group Decision Making with Incomplete Weight Information and Its Application to Waste Incineration Plants Location Problem

Fan Lei; Guiwu Wei; Jianping Lu; Jiang Wu; Cun Wei

doi:10.2991/ijcis.d.191203.002

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Volume 12, Issue 2, 2019, Pages 1547 - 1556

GRA Method for Probabilistic Linguistic Multiple Attribute Group Decision Making with Incomplete Weight Information and Its Application to Waste Incineration Plants Location Problem

Authors

Fan Lei¹, Guiwu Wei²^{, *}, Jianping Lu², Jiang Wu³, Cun Wei³

¹School of Mathematical Sciences, Sichuan Normal University, Chengdu, 610101, P. R. China

²School of Business, Sichuan Normal University, Chengdu, 610101, P. R. China

³School of Statistics, Southwestern University of Finance and Economics, Chengdu, 611130, P. R. China

^*Corresponding author. Email: weiguiwu@163.com

Corresponding Author

Guiwu Wei

Received 28 August 2019, Accepted 7 November 2019, Available Online 9 December 2019.

DOI: 10.2991/ijcis.d.191203.002 How to use a DOI?
Keywords: Multiple attribute group decision making (MAGDM); Probabilistic linguistic term sets (PLTSs); GRA method; Incomplete weight information; Waste incineration plants
Abstract: In this essay, we investigate the probabilistic linguistic multiple attribute group decision making (PL-MAGDM) with incomplete weight information. In this method, the linguistic representation developed recently is converted into probabilistic linguistic information. For deriving the weight information of the attribute, an optimization model is built on the basis of the fundamental idea of grey relational analysis (GRA), by which the attribute weights can be decided. Then, the optimal alternative is chosen through calculating largest relative relational degree from the probabilistic linguistic positive ideal solution (PLPIS) which considers both the largest grey relational coefficient (GRC) from the PLPIS and the smallest GRC form probabilistic linguistic negative ideal solution (PLNIS). In the end, a case study concerning waste incineration plants location problem is given to demonstrate the merits of the developed methods.
Copyright: © 2019 The Authors. Published by Atlantis Press SARL.
Open Access: This is an open access article distributed under the CC BY-NC 4.0 license (http://creativecommons.org/licenses/by-nc/4.0/).

1. INTRODUCTION

The grey relational analysis (GRA) method was initially designed by Deng [1] for tackling the MADM issue. In addition, GRA method is one of the very popular and useful tools to analyze diverse relationships among the discrete information and make decisions in different situations [2–4]. The major merits of the GRA method are that the analyzed results are depended upon the original data, the calculating process are simple and straightforward and, finally, it is one of the optimal methods to make decisions under diverse business environment [5–7]. Kung and Wen [5] employed GRA to study the grey MADM issue for venture capital enterprises. Tan, Chen and Wu [8] studied the green design alternatives and GRA connected with AHP. Chiang [9] extended GRA for dependent criteria MADM issue. Malek, Ebrahimnejad and Tavakkoli-Moghaddam [10] proposed an improved hybrid GRA method for green resilient supply. Alptekin, Alptekin and Sarac [11] assessed the low carbon development with GRA model in some countries. Zhu, Yuan and Ye [12] aimed at discerning the multi-timescales of carbon market through GRA method and empirical mode decomposition (EMD). Yazdani, Kahraman, Zarate and Onar [13] provided a good platform to ease decision process through the integration of quality function deployment (QFD) and GRA in showing main supply chain drivers under fuzzy setting. Chen [14] connected intuitionistic fuzzy GRA techniques with entropy-based TOPSIS for selecting building materials supplier. Wei [15] tackled dynamic hybrid MADM issue based on the GRA.

In real life, the decision makers (DMs) always employ linguistic terms rather than using the exact numbers because of the complex socioeconomic environment and fuzzy human beings' thinking [16–18]. Under this situations, Zadeh [19] proposed fuzzy linguistic method to depict the qualitative assessment information. In certain environments, DMs may be possible to be uncertain about some diverse linguistic terms when they depict the objects. Thus, Rodriguez, Martinez and Herrera [20] devised the tool of hesitant fuzzy linguistic term set (HFLTS) to snatch the hesitancy degree in the context of the linguistic. Gou, Xu and Liao [21] expanded the entropy and cross-entropy measures to HFLTS for MADM. Wei [22] gave the generalized dice similarity measures for MADM with HFLTS. Liao, Xu and Zeng [23] provided the VIKOR method for qualitative MADM with HFLTS. However, at certain times, the DMs may intend to express the linguistic term “medium” to the linguistic term “bad,” that's to say, these two linguistic terms' probabilities are not same [24]. It can be easily found that the HFLTS can't describe this sort of complicated qualitative information. To overcome this drawback and exactly describe this kind of complicated qualitative information, Pang, Wang and Xu [25] developed the probabilistic linguistic term set (PLTS), which permits DMs to express the preference information of themselves as a collection of several possible linguistic terms combined with probabilities. Lin, Chen, Liao and Xu [26] proposed the ELECTRE II method to deal with PLTSs for edge computing. Feng, Liu and Wei [27] constructed possibility degree comparison of probabilistic linguistic QUALIFLEX method. Bai, Zhang, Qian and Wu [28] gave comparative method and proposed a more efficient mean to tackle PLTSs. Chen, Wang and Wang [29] employed the probabilistic linguistic MULTIMOORA for cloud-based ERP system selection. Lin, Chen, Liao and Xu [26] developed the ELECTRE II method to deal with PLTSs for edge computing. Liu and Teng [30] proposed some Muirhead mean operators for PLTSs. Liu and Teng [31] defined the probabilistic linguistic TODIM method. Liu and Li [32] designed the generalized maclaurin symmetric mean aggregation operators for probabilistic linguistic information. Liu and Li [32] gave the bidirectional projection method for probabilistic linguistic multi-criteria group decision making based on power average operator. Liang, Kobina and Quan [33] designed the probabilistic linguistic grey relational analysis (PL-GRA) for MAGDM based on geometric Bonferroni mean [34–37]. Lu, Wei, Wu and Wei [38] designed the TOPSIS method for probabilistic linguistic MAGDM with entropy weight foro supplier selection of new agricultural machinery products.

But the PL-GRA [33] method have two shortcomings: (1) this method think that the optimal alternative is chosen through calculating largest relative relational degree from the probabilistic linguistic positive ideal solution (PLPIS) which considers both the largest grey relational coefficient (GRC) from the PLPIS which neglects the smallest GRC form probabilistic linguistic negative ideal solution (PLNIS); (2) this method can't tackled the probabilistic linguistic MADM (PL-MADM) or PL-MAGDM issues with incomplete weight information. As a supplement to it, we shall develop a novel GRA-based method for PL-MAGDM in this study and then apply it to select the waste incineration plants sites. The innovativeness of the paper can be summarized as follows: (1) an optimization model is built to derive the weight information of the attribute on the basis of the fundamental idea of conventional GRA method under PLTSs; (2) the PL-GRA method is proposed to solve the probabilistic linguistic MAGDM problems with incomplete weight information; (3) a case study for selecting the waste incineration plants sites is supplied to show the developed approach and (4) some comparative studies are provided with the existing methods.

The remainder of this paper is arranged in the following way: Section 2 gives some fundamental ideas connected to PLTSs. In Section 3, the GRA approach is proposed for PL-MAGDM with incomplete weight information. In Section 4, given a numerical example a case study concerning waste incineration plants location problem to illustrate the developed PL-GRA method and some comparative analysis is provided. In the end, this study makes some conclusions in Section 5.

2. PRELIMINARIES

As an essential and useful tool, The HFLTS [20] is utilized to tackle the hesitancy in the context of the linguistic. Zhao, Xu and Ren [39] designed some distance measures for HFLTS. Wei, Zhao and Tang [40] devised some operations on HFLTSs and possibility degree formulas for comparing HFLTSs. In order to strengthen the modeling capability of HFLTSs, Pang, Wang and Xu [25] proposed the definition of PLTSs to link each linguistic term with a probability value.

Definition 1.

[25] Let L=lα|α=−θ,⋯,−2,−1,0,1,2,⋯θ be an LTS, the linguistic terms lα can express the equivalent information to β is derived by the transformation function g:

(1)g:l−θ,lθ→0,1,glα=α+θ2θ=β

At the same time, β can be expressed the equivalent information to the linguistic terms lα, β is derived by the transformation function g−1:

(2)g−1:0,1→l−θ,lθ,g−1β=l2β−1θ=lα

Definition 2.

Given an LTS L={lj|j=−θ,⋯,−2,−1,0,1,2,⋯θ}, a PLTS L is devised as

(3)L(p)={∑ϕ=1#L(p)l(ϕ)(p(ϕ))|l(ϕ)∈L,p(ϕ)≥0,ϕ=1,2,⋯,#L(p),∑ϕ=1#L(p)p(ϕ)≤1}

where lϕpϕ is the ϕth linguistic term lϕ linked with the probability pϕ, and #Lp is the number of all different linguistic terms in Lp. The linguistic term lϕ in Lp is listed in ascending rank.

In order to facile computation, Pang, Wang and Xu [25] normalized the PLTS Lp as L~p=lϕp~ϕ|lϕ∈L,p~ϕ≥0,ϕ=1,2,⋯,#Lp~,∑ϕ=1#Lp~p~ϕ=1, where p~(ϕ)=p(ϕ)/∑ϕ=1#L(p)p(ϕ) for all ϕ=1,2,⋯,#Lp~.

Definition 3.

Let L=lα|α=−θ,⋯,−1,0,1,⋯θ be an LTS, L~1p~=l1ϕp~1ϕ|ϕ=1,2,⋯,#L~1p~ and L~2p~=l2ϕp~2ϕ|ϕ=1,2,⋯,#L~2p~ be two PLTSs, where #L~1p~ and #L~2p~ are the numbers of LTS in L~1p~ and L~2p~, respectively. If #L~1p~>#L~2p~, then add #L~1p~−#L~2p~ linguistic terms to L~2p~. What's more, the newly added linguistic terms should be the smallest linguistic term in L~2p~ and the corresponding probabilities of newly added linguistic terms should be zero.

Definition 4.

For a PLTS L~p~=lϕp~ϕ|ϕ=1,2,⋯,#L~ p~, the score sL~p~ and deviation degree σL~p~ of L~p~ is illustrated as follows:

(4)sL~p~=∑ϕ=1#L~p~glϕp~ϕ/∑ϕ=1#L~p~p~ϕ

(5)σL~p~=∑ϕ=1#L~ p~glϕp~ϕ−sL~p~2/∑ϕ=1#L~ p~p~ϕ

By using the Eqs. (4) and (5), the order of two PLTSs is defined in the following: (1) if sL~1p~>sL~2p~, then L~1(p~)>L~2(p~); (2) if sL~1p~=sL~2p~, then if σL~1p~=σL~2p~, then L~1p~=L~2p~; if σL~1p~<σL~2p~, then, L~1p~>L~2p~.

Definition 5.

[41] Let L=lα|α=θ,⋯,−1,0,1,⋯θ be an LTS. And let L~1p~=l1ϕp~1ϕ|ϕ=1,2,⋯,#L~1p~ and L~2p~=l2ϕp~2ϕ|ϕ=1,2,⋯,#L~2p~ be two PLTSs with #L~1p~=#L~2p~, then the Hamming distance dL~1p~,L~2p~ between L~1p~ and L~2p~ is given as follows:

(6)dL~1p~,L~2p~=∑ϕ=1#L~1p~(p~1ϕgl1ϕ−p~2ϕgl2ϕ)/#L~1p~

3. GRA METHOD FOR PROBABILISTIC LINGUISTIC MAGDM WITH INCOMPLETE WEIGHT INFORMATION

In such section, we propose a novel PL-GRA method to tackle the MAGDM issues with incomplete weight information. Assume that A=A1,A2,⋯,Am is a collection of potential alternatives, G=G1,G2,⋯,Gn is the collection of chosen attributes, and E=E1,E2,⋯,Eq is the collection of qualified experts. Suppose that Lk=lijkm×n is the group decision making matrix, where lijki=1,2,⋯,m,j=1,2,⋯,n,k=1,2,⋯,q is linguistic variables, given by the DM Ek∈E, for the alternative Ai∈A with regard to the attribute Gj∈G, w=w1,w2,⋯,wn is the weight vector of the attributes Gjj=1,2,⋯,n, where wj∈0,1, ∑j=1nwj=1. H is a sort of partially known weight information, which can be listed by the following forms, for i≠j [42–44]: Case 1. A weak ranking: wi≥wj; Case 2. A strict ranking: wi−wj≥αi, αi>0; Case 3. A ranking of differences: wi−wj≥wk−wl, for j≠k≠l; Case 4. A ranking with multiples: wi≥βiwj, 0≤βi≤1; Case 5. An interval form: αi≤wi≤αi+εi,0≤αi<αi+εi≤1.

Subsequently, we shall use PL-GRA to tackle MAGDM issues with incomplete weight information. The detailed calculating steps of the proposed method are presented in the following:

Step 1. Shift cost attributes into beneficial attributes. If L=lα|α=−θ,⋯,−1,0,1,⋯θ is an LTS, then the cost attribute value is lα, then the corresponding beneficial attribute value is l−α.

Step 2. Switch the linguistic variables lijki=1,2,⋯,m,j=1,2,⋯,n,k=1,2,⋯,q into probabilistic linguistic information Lij p, and build the probabilistic linguistic decision matrix L=Lij pm×n, Lij p~=lij ϕpij ϕ|ϕ=1,2,⋯,#Lij p i=1,2,⋯,m,j=1,2,⋯,n. Thus, probabilistic linguistic information for the alternative Ai∈A with regard to the all the attribute G can be expressed as PLAi=li1 ϕpi1 ϕ,li2 ϕpi2 ϕ,⋯,lin ϕpin ϕ, ϕ=1,2,⋯,#Lij p.

Step 3. Derive the normalized assessing matrix L~=L~ij p~m×n with PLTSs, L~ij p~=lij ϕp~ij ϕ|ϕ=1,2,⋯,#Lij p~ i=1,2,⋯,m,j=1,2,⋯,n.

Step 4. Giving the definition of PLPIS and PLNIS as follows:

(7)PLPIS+=(PLPIS1,PLPIS2,⋯,PLPISn)

(8)PLNIS+=(PLNIS1,PLNIS2,⋯,PLNISn)

where

(9)PLPISj={plj(ϕ)(ppj(ϕ))|ϕ=1,2,⋯,#Lij (p~)}={maxis(Lij (p~))}

(10)PLNISj={nlj(ϕ)(npj(ϕ))|ϕ=1,2,⋯,#Lij (p~)}={minis(Lij (p~))}

Step 5. Computing the corresponding GRC of each alternative from PLPIS and PLNIS by utilizing the following equation, respectively:

(11)PLPISξij=min1≤i≤mmin1≤j≤ndPLAij,PLPISj+ρmax1≤i≤mmax1≤j≤ndPLAij,PLPISjdPLAij,PLPISj+ρmax1≤i≤mmax1≤j≤ndPLAij,PLPISj i=1,2,⋯,m,j=1,2,⋯,n,

(12)PLNISξij=min1≤i≤mmin1≤j≤ndPLAij,PLNISj+ρmax1≤i≤mmax1≤j≤ndPLAij,PLNISjdPLAij,PLNISj+ρmax1≤i≤mmax1≤j≤ndPLAij,PLNISj i=1,2,⋯,m,j=1,2,⋯,n,

(13)dPLAij,PLPISj=∑ϕ=1#Lijp~|pij ϕglij ϕ−p~jϕgpljϕ|/#Lijp~

(14)dPLAij,PLNISj=∑ϕ=1#Lijp~|pij ϕglij ϕ−p~jϕgnljϕ|/#Lijp~

Step 6. Calculating the degree of GRC of all possible alternatives from PLPIS and PLNIS, respectively:

(15)PLPISξi=∑j=1nwjPLPISξij,i=1,2,⋯,m

(16)PLNISξi=∑j=1nwjPLNISξij,i=1,2,⋯,m

The fundamental idea of GRA method is that the optimal alternative is supposed to possess the “largest degree of GRC” from PLPIS and “smallest degree of GRC” from PLNIS. Evidently, the larger PLPISξi along with smaller PLNISξi, the better alternative ALi is. But the attribute weights' information is incompletely known. So, to derive the PLPISξi and PLNISξi, we can build the multiple objective optimization models MOOM−1 in the following:

maxPLPISξi=∑j=1nwjPLPISξijminPLNISξi=∑j=1nwjPLNISξijsubject to:w∈H,i=1,2,⋯,m.

Due to each alternative is non-inferior, for all the alternatives, there is no preference relation. Besides, the above multiple objective optimization models MOOM−1 might be aggregated with equal weights into the following single objective optimization model SOOM−1:

minPLPISξ=∑i=1mwjPLNISξi−PLPISξi=∑i=1m∑j=1nwjPLNISξij−PLPISξij=∑i=1m∑j=1nwjmin1≤i≤mmin1≤j≤ndPLAij,PLPISj+ρmax1≤i≤mmax1≤j≤ndPLAij,PLPISjdPLAij,PLPISj+ρmax1≤i≤mmax1≤j≤ndPLAij,PLPISj−min1≤i≤mmin1≤j≤ndPLAij,PLNISj+ρmax1≤i≤mmax1≤j≤ndPLAij,PLNISjdPLAij,PLNISj+ρmax1≤i≤mmax1≤j≤ndPLAij,PLNISjsubject to:w∈H

By solving the model SOOM−1, we get the optimal solution w=w1,w2,⋯,wn, which can be regarded as the weight vector of attributes. Then, we can get PLPISξi and PLNISξi by Eqs. (15) and (16).

Step 7. Derive the probabilistic linguistic relative relational degree (PLRRD) of all possible alternatives from PLPIS.

(17)PLRRDξi=PLPISξiPLPISξi+PLNISξi,i=1,2,⋯,m

Step 8. According to PLRRDξi, the sorting order of all possible alternatives can be obtained. If any alternative has the largest PLRRDξi, then, it is optimal choice.

4. NUMERICAL CASE AND COMPARATIVE ANALYSIS

4.1. Numerical Case

Along with the acceleration of urbanization and the rapid growth of urban population, the output of municipal solid waste also increases rapidly. More and more city or region will create more and more waste incineration plants in accordance with “The Twelfth Five-Year Plan.” However, the waste incinerator is a NIMBY facility, and if the site selection is not scientific and reasonable, it is likely to cause NIMBY conflicts. Therefore, how to reasonably carry on the waste incineration plant scientific location is particularly important. Waste incineration plants location problem should be regarded as the corresponding MAGDM [45–51]. Thus, in this chapter we developed a case study concerning waste incineration plants location problem to demonstrate the approach presented in this essay. There are five potential waste incineration plants sites Aii=1,2,3,4,5 proposed to select. The experts select the following four beneficial attributes to assess the five potential waste incineration plants sites: ① G₁ is waste incineration plants area; ② G₂ is transportation cost of waste; ③ G₃ is site far from the residential area; ④ G₄ is seepage control rainwater and sewage diversion. The transportation cost (G₂) is not beneficial attribute, others are beneficial attributes. The five potential waste incineration plants sites Aii=1,2,3,4,5 are to be assessed by utilizing the linguistic term set

L=l−3=extremely poor(EP),l−2=very poor(VP), l−1=poor(P),l0=medium(M),l1=good(G), l2=very good(VG),l3=extremely good(EG)

by the five DMs within the above four attributes, as listed in the Tables 1–5. In the light of the three experts' judgment, the attribute weights are partly known in the following:

H=0.15≤ω1≤0.21,0.19≤ω2≤0.29, 0.22≤ω3≤0.35,ω3−ω4≥0.11

Alternatives	G₁	G₂	G₃	G₄
A₁	G	VG	M	VG
A₂	G	P	VG	P
A₃	VG	VG	P	VG
A₄	VG	VP	VP	P
A₅	EG	VP	G	EG

DM, decision maker.

Table 1

Linguistic decision matrix by the first DM.

Alternatives	G₁	G₂	G₃	G₄
A₁	EG	VP	VG	VG
A₂	G	EP	EG	M
A₃	VG	M	G	EG
A₄	VG	VP	P	P
A₅	VG	P	G	VG

DM, decision maker.

Table 2

Linguistic decision matrix by the second DM.

Alternatives	G₁	G₂	G₃	G₄
A₁	EG	G	VG	VG
A₂	G	EP	EG	G
A₃	G	M	G	VG
A₄	G	EP	P	P
A₅	VG	VP	M	G

DM, decision maker.

Table 3

Linguistic decision matrix by the third DM.

Alternatives	G₁	G₂	G₃	G₄
A₁	VG	G	VG	VG
A₂	G	P	VG	M
A₃	G	EG	G	VG
A₄	G	EP	VP	EP
A₅	EG	P	VG	VG

DM, decision maker.

Table 4

Linguistic decision matrix by the fourth DM.

Alternatives	G₁	G₂	G₃	G₄
A₁	EG	VP	M	G
A₂	P	EP	VG	M
A₃	VG	VG	G	VG
A₄	VP	VP	P	P
A₅	EG	VP	VG	VG

DM, decision maker.

Table 5

Linguistic decision matrix by the fifth DM.

Following that, the PL-GRA method is utilized to select the optimal waste incineration plants sites.

Step 1. Shift cost attribute G₂ into beneficial attribute. If the cost attribute value is lτ, then the corresponding beneficial attribute value is l−ττ=−3,−2,−1,0,1,2,3 (See Tables 6–10).

Alternatives	G₁	G₂	G₃	G₄
A₁	G	VP	M	VG
A₂	G	G	VG	P
A₃	VG	VP	P	VG
A₄	VG	VG	VP	P
A₅	EG	VG	G	EG

DM, decision maker.

Table 6

Linguistic decision matrix by the first DM.

Alternatives	G₁	G₂	G₃	G₄
A₁	EG	VG	VG	VG
A₂	G	EG	EG	M
A₃	VG	M	G	EG
A₄	VG	VG	P	P
A₅	VG	G	G	VG

DM, decision maker.

Table 7

Linguistic decision matrix by the second DM.

Alternatives	G₁	G₂	G₃	G₄
A₁	EG	P	VG	VG
A₂	G	EG	EG	G
A₃	G	M	G	VG
A₄	G	EG	P	P
A₅	VG	VG	M	G

DM, decision maker.

Table 8

Linguistic decision matrix by the third DM.

Alternatives	G₁	G₂	G₃	G₄
A₁	VG	P	VG	VG
A₂	G	G	VG	M
A₃	G	EP	G	VG
A₄	G	EG	VP	EP
A₅	EG	G	VG	VG

DM, decision maker.

Table 9

Linguistic decision matrix by the fourth DM.

Alternatives	G₁	G₂	G₃	G₄
A₁	EG	VG	M	G
A₂	P	EG	VG	M
A₃	VG	VP	G	VG
A₄	VP	VG	P	P
A₅	EG	VG	VG	VG

DM, decision maker.

Table 10

Linguistic decision matrix by the fifth DM.

Step 2. Transform the linguistic variables into PLTSs (Table 11).

Alternatives	G₁	G₂
A₁	l10.2,l20.2,l30.6	l−20.2,l−10.4,l20.4
A₂	l−10.2,l10.8	l10.4,l30.6
A₃	l10.4,l20.6	l−30.2,l−20.4,l00.4
A₄	l−20.2,l10.4,l20.4	l20.6,l30.4
A₅	l20.4,l30.6	l10.4,l20.6

Alternatives	G₃	G₄

A₁	l00.4,l20.6	l10.2,l20.8
A₂	l20.6,l30.4	l−10.2,l00.6,l10.2
A₃	l−10.2,l10.8	l20.8,l30.2
A₄	l−20.4,l−10.6	l−30.2,l−10.8
A₅	l00.2,l10.4,l20.4	l10.2,l20.6,l30.2

PLTS, probabilistic linguistic term set.

Table 11

Decision matrix with PLTSs.

Step 3. Calculate the decision matrix with normalized PLTSs (Table 12).

Alternatives	G₁	G₂
A₁	l10.2,l20.2,l30.6	l−20.2,l−10.4,l20.4
A₂	l−10,l−10.2,l10.8	l10,l10.4,l30.6
A₃	l10,l10.4,l20.6	l−30.2,l−20.4,l00.4
A₄	l−20.2,l10.4,l20.4	l20,l20.6,l30.4
A₅	l20,l20.4,l30.6	l10,l10.4,l20.6

Alternatives	G₃	G₄

A₁	l00,l00.4,l20.6	l10,l10.2,l20.8
A₂	l20,l20.6,l30.4	l−10.2,l00.6,l10.2
A₃	l−10,l−10.2,l10.8	l20,l20.8,l30.2
A₄	l−20,l−20.4,l−10.6	l−30,l−30.2,l−10.8
A₅	l00.2,l10.4,l20.4	l10.2,l20.6,l30.2

PLTS, probabilistic linguistic term set.

Table 12

Decision matrix with Nnormalized PLTSs.

Step 4. Defining the PLPIS and PLNIS by Eqs. (7–9) (Table 13):

	G₁	G₂
PLPIS	l20,l20.4,l30.6	l20,l20.6,l30.4
PLNIS	l−10,l−10.2,l10.8	l−30.2,l−20.4,l00.4

	G₃	G₄

PLPIS	l20,l20.6,l30.4	l20,l20.8,l30.2
PLNIS	l−20,l−20.4,l−10.6	l−30,l−30.2,l−10.8

PLPIS, probabilistic linguistic positive ideal solution; PLNIS, probabilistic linguistic negative ideal solution.

Table 13

PLPIS and PLNIS.

Step 5. Computing the corresponding GRC of each alternative from PLPIS and PLNIS (Tables 14 and 15):

Alternatives	G₁	G₂	G₃	G₄
A₁	0.6241	0.4864	0.5284	0.3333
A₂	0.5631	0.5355	1.0000	0.4835
A₃	0.7467	0.4261	0.4387	1.0000
A₄	0.5613	1.0000	0.4261	0.3459
A₅	1.0000	0.5826	0.5745	0.6241

GRC, grey relational coefficient; PLPIS, probabilistic linguistic positive ideal solution.

Table 14

GRC of each alternative from PLPIS.

Alternatives	G₁	G₂	G₃	G₄
A₁	0.6511	0.6868	0.5051	0.4427
A₂	1.0000	0.4283	0.4124	0.5000
A₃	0.6230	1.0000	0.5012	0.3333
A₄	0.5405	0.4124	1.0000	1.0000
A₅	0.5493	0.4816	0.5627	0.3910

GRC, grey relational coefficient; PLNIS, probabilistic linguistic negative ideal solution.

Table 15

GRC of each alternative from PLNIS.

Step 6. The model SOOM−1 is utilized to set up the single-objective programming model:

mind(w)=−0.1315w1−0.0215w2+0.0137w3−0.1197w4subject:w∈H

Solve this model, the weight vector of attributes can be got: w=0.2100,0.2000,0.3500,0.2400T.

Step 7. Calculating the degree of GRC of all possible alternatives from PLPIS and PLNIS, respectively (Table 16):

Alternatives	PLPISξi	PLNISξi
A₁	0.4933	0.5571
A₂	0.6914	0.5600
A₃	0.6356	0.5863
A₄	0.5500	0.7860
A₅	0.6774	0.5025

PLPIS, probabilistic linguistic positive ideal solution; PLNIS, probabilistic linguistic negative ideal solution.

Table 16

PLPIS(ξi) and PLNIS(ξi) of each alternative.

Step 8. Calculating the PLRRDξi of each alternative from PLPIS by Eq. (14) (Table 17).

Alternatives	A₁	A₂	A₃	A₄	A₅
PLRRDξi	0.4696	0.5525	0.5202	0.4117	0.5741

PLRRD, probabilistic linguistic relative relational degree; PLPIS, probabilistic linguistic positive ideal solution.

Table 17

PLRRD of each alternative from PLPIS.

Step 9. According to the PLRRDξii=1,2,3,4,5, all the waste incineration plants sites can be ranked. Evidently, the order is A5>A2>A3>A1>A4 and the most desirable waste incineration plants site among five alternatives is A5.

4.2. Comparative Analysis

Then, our proposed method is compared with probabilistic linguistic weighted average (PLWA) operator [25] and PL-TOPSIS method [25] as in Table 18.

Methods	Computing Results	Ordering
PL-TOPSIS [25]	d1=−1.2035,d2=−0.3420,d3=−1.4208,d4=−2.1670,d5=0.0000.	A5≻A2≻A1≻A3≻A4
PLWA operator [25]	EZ1w=s0.4520,EZ2w=s0.4687,EZ3w=s0.2647,EZ4w=s−0.0593,EZ5w=s0.5887.	A5≻A2≻A1≻A3≻A4
PL-GRA method	PLRRDξ1=0.4696,PLRRDξ2=0.5525PLRRDξ3=0.5202,PLRRDξ4=0.4117PLRRDξ5=0.5741	A5>A2>A3>A1>A4

PLWA, probabilistic linguistic weighted average; PL_GRA, probabilistic linguistic grey relational analysis.

Table 18

Ordering of the waste incineration plants sites by using diverse methods.

In terms of the above analysis, it can be found that these abovementioned methods have the same best waste incineration plants site A5 and the bad waste incineration plants site A4, and there are slightly different in three methods' ranking results, which can confirm the presented method is feasible and effective in this essay. All these methods have their good advantages: (1) PL-TOPSIS method emphasis the distance similarity degree from the positive and negative ideal solution; (2) PLWA operator emphasis group influences; (3) our proposed PL-GRA method emphasis emphasizes he shape similarity degree from PIS and NIS simultaneously with incomplete weight information.

5. CONCLUSION

In this essay, the GRA method is expanded to the PL-MAGDM with incomplete weight information. First and foremost, the definition, comparative method and distance of PLTs are simply reviewed. Additionally, the extended GRA method is employed to tackle PL-MAGDM issues with incomplete weight information. We construct the multiple objective optimization models on the basis of the conventional GRA method. Besides, the multiple objective optimization models can be converted into a single-objective programming model by making use of the linear equal weighted method. By calculating the single-objective programming model, the weight information can be acquired. In the light of the conventional GRA, the optimal choice is derived by obtaining “largest degree of GRC” from PLPIS and “smallest degree of GRC” from PLNIS. Finally, a practical case study concerning waste incineration plants location problem is designed to validate the proposed algorithms and some comparative studies are also designed to verify the applicability. In our future research, the proposed methods and algorithm will be needful and meaningful for other real decision making problems [52–60] and the developed approaches can also be extended to other fuzzy [48,61,62] and uncertain information [63–73].

CONFLICT OF INTEREST

The authors declare that they have no conflict of interest.

AUTHORS' CONTRIBUTIONS

Fan Lei, Guiwu Wei, Jianping Lu, Jiang Wu and Cun Wei conceived and worked together to achieve this work, Fan Lei compiled the computing program by Excel and analyzed the data, Fan Lei and Guiwu Wei wrote the paper. Finally, all the authors have read and approved the final manuscript.

Funding Statement

The work was supported by the National Natural Science Foundation of China under Grant No. 71571128 and the Humanities and Social Sciences Foundation of Ministry of Education of the People's Republic of China (14YJCZH082).

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Journal: International Journal of Computational Intelligence Systems
Volume-Issue: 12 - 2
Pages: 1547 - 1556
Publication Date: 2019/12/09
ISSN (Online): 1875-6883
ISSN (Print): 1875-6891
DOI: 10.2991/ijcis.d.191203.002 How to use a DOI?
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ris enw bib

TY  - JOUR
AU  - Fan Lei
AU  - Guiwu Wei
AU  - Jianping Lu
AU  - Jiang Wu
AU  - Cun Wei
PY  - 2019
DA  - 2019/12/09
TI  - GRA Method for Probabilistic Linguistic Multiple Attribute Group Decision Making with Incomplete Weight Information and Its Application to Waste Incineration Plants Location Problem
JO  - International Journal of Computational Intelligence Systems
SP  - 1547
EP  - 1556
VL  - 12
IS  - 2
SN  - 1875-6883
UR  - https://doi.org/10.2991/ijcis.d.191203.002
DO  - 10.2991/ijcis.d.191203.002
ID  - Lei2019
ER  -

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