# Method for Pythagorean Interval 2-Tuple Linguistic Multiattribute Group Decision Making and Its Application to the Ship Navigation Environment Safety Assessment

Authors: Fuchun Yuan
Journal: Discrete Dynamics in Nature and Society
 (2022)

---

## Abstract

In this paper, some fundamental theories of the Pythagorean interval 2-tuple linguistic numbers (PI2TLNs) combined with the EDAS (evaluation based on distance from average solution) method are proposed. Firstly, we introduced the original concepts of Pythagorean interval 2-tuple linguistic sets (PI2TLSs), and two aggregation operators of PI2TLN were also introduced. Then, the steps of the EDAS method were described briefly. Moreover, the PI2TLN-EDAS model was provided to address the multiattribute group decision-making (MAGDM) problem, and calculation steps were deduced in detail. The effectiveness of the method was illustrated by taking the ship navigation environment safety assessment as an example. Finally, we conducted comparative analysis between PI2TLN-EDAS method and Pythagorean interval 2-tuple linguistic weighted averaging (PI2TLWA) operator/Pythagorean interval 2-tuple linguistic weighted geometric (PI2TLWG) operator.


---

## Body

## 1. Introduction

The EDAS method proposed by Ghorabaee et al. [1] is an evaluation method that calculates the positive and negative distances between each alternative and the average alternative. Different methods are used to deal with the multiattribute decision-making (MADM) problems where the decision information is a certain number [2–7]. Similar to the VIKOR method [8–12], TOPSIS method [13–17], and GRA method [18], the EDAS method is effective when conflicting criteria are present in the MADM issues. Aiming at the case where the decision information is a definite number [19–21], the EDAS method is proposed. Based on this idea, many scholars extended the EDAS method to the fuzzy set environment. For example, Kahraman et al. [22] extended the classical EDAS method to ordinary fuzzy sets. Liang et al. [23] used the EDAS-ELECTRE comprehensive evaluation method to evaluate the clean production performance of the gold mining enterprises. Jiang et al. [24] built the EDAS method based on cumulative prospect theory for MAGDM under picture fuzzy environment. Peng et al. [25] applied the EDAS method to solve the interval value fuzzy soft decision problem. Huang et al. [26] defined an enhancement EDAS method based on prospect theory. Wei et al. [27] built the EDAS method for probabilistic linguistic MAGDM. Lei et al. [28] proposed the PDHL-EDAS method for MAGDM.In the process of decision making, there is a wide range of ambiguity and uncertainty [29–33] such as classical fuzzy set [34], intuitionistic fuzzy sets (IFSs) [35–41], probabilistic hesitant fuzzy set [42–47], probabilistic linguistic sets [48–53], and probabilistic double hierarchy linguistic sets [54–56]. Yager [57] put forward the concept of Pythagorean fuzzy set (PFSs). Zhang [58] defined the accuracy function and new aggregation operator of Pythagorean fuzzy numbers (PFNs) and proposed the similarity measurement method of PFNs based on the method of similarity measure. Zeng et al. [59] combined the distance measure with the PFS and proposed the Pythagorean ordered weighted, average weighted, and average distance operators. Garg [60] introduced the statistical concept of confidence level into the Pythagorean fuzzy set. To better provide decision information in a form that is consistent with the language habits of decision makers, Huang and Wei [61] defined the Pythagorean 2-tuple linguistic fuzzy sets (P2TLSs) and introduced its related properties. The concept of Pythagorean interval 2-tuple linguistic sets (PI2TLSs) first appeared [61] on the basis of PFSs [62] and 2-tuple linguistic information [63].Because the EDAS method is an effective MADM method and Pythagorean interval 2-tuple linguistic fuzzy set has the characteristics of two-stage description, it can relatively contain more decision-making information and describe objective facts more comprehensively and accurately. Based on the advantages of both, this paper attempts to extend the EDAS method to decision environment of Pythagorean interval 2-tuple linguistic fuzzy set (PI2TLS) and proposes a Pythagorean interval 2-tuple linguistic MADM model based on the EDAS method.The rest of this paper is arranged as follows. In Section2, some fundamental definitions of Pythagorean interval 2-tuple linguistic numbers (PI2TLNs) are given. In Section 3, the EDAS method is extended using PI2TLNs. In Section 4, a case study for evaluating the ship navigation environment safety is given. Section 5 gives some conclusions.


## 2. Preliminaries

### 2.1. Pythagorean Interval 2-Tuple Linguistic Sets

The concept of Pythagorean interval 2-tuple linguistic sets (PI2TLSs) first appeared [61] on the basis of PFSs [62] and 2-tuple linguistic information [63].Definition 1 (see [61]).
In a PI2TLS,A in the theoretical field X is defined as follows:(1)A=ψx,αx,δx,βx,uAx,νAx,x∈X,where ψx,αx,δx,βx is an interval 2-tuple linguistic variable, ψx,δx∈S=S0,S1⋯ST, S is a linguistic term set, and αx,βx∈−0.5,0.5, uAx,νAx∈0,1 represent the degree of membership and degree of non-membership of the element x to an interval 2-tuple linguistic ψx,αx,δx,βx and meet the condition 0≤uAx2+νAx2≤1∈0,1.
For convenience,A=ψA,αA,δA,βA,uA,vA can be called a Pythagorean interval 2-tuple linguistic number (PI2TLN).Definition 2 (see [61]).
For a PI2TLNa˜=ψa,αa,δa,βa,ua,va, its score function is(2)Sfa˜=ΔΔ−1ψa,αa+Δ−1δa,βa2×1+ua2−νa22.Definition 3 (see [61]).
For a PI2TLNa˜=ψa,αa,δa,βa,ua,va, its accuracy function is(3)Afa˜=ΔΔ−1ψa,αa+Δ−1δa,βa2×ua2+νa22.Definition 4 (see [61]).
For two PI2TLNsa˜1=ψa1,αa1,δa1,βa1,ua1,va1 and a˜2=ψa2,αa2,δa2,βa2,ua2,va2, respectively, the scores of a˜1 and a˜2 are Sfa˜1=ΔΔ−1ψa1,αa1+Δ−1δa1,βa1/2×1+ua12−νa12/2 and Sfa˜2=ΔΔ−1ψa2,αa2+Δ−1δa2,βa2/2×1+ua22−νa22/2, and let Afa˜1=ΔΔ−1ψa1,αa1+Δ−1δa1,βa1/2×ua12+νa12/2 and Afa˜2=ΔΔ−1ψa1,αa1+Δ−1δa2,βa2/2×ua22+νa22/2 be the accuracy degrees of a˜1 and a˜2; then, some operational laws of PI2TLNs are defined as follows:(1)
IfSfa˜1<Sfa˜2,a˜1<a˜2.(2)
IfSfa˜1>Sfa˜2,a˜1>a˜2.(3)
IfSfa˜1=Sfa˜2,Afa˜1<Afa˜2,thena˜1<a˜2.(4)
IfSfa˜1=Sfa˜2,Afa˜1>Afa˜2,thena˜1>a˜2.(5)
IfSfa˜1=Sfa˜2,Afa˜1=Afa˜2,thena˜1=a˜2.Definition 5 (see [61]).
For two PI2TLNsa˜1=ψa1,αa1,δa1,βa1,ua1,va1 and a˜2=ψa2,αa2,δa2,βa2,ua2,va2, the normalized Hamming distance (Hd) between a˜1 and a˜2 is defined as(4)Hda˜1,a˜2=14L1+ua12−va12⋅Δ−1ψa1,αa1+Δ−1δa1,βa1−1+ua22−va22⋅Δ−1ψa2,αa2+Δ−1δa2,βa2,where L represents the length of the language scale. It is a numerical value.Definition 6 (see [61]).
For two PI2TLNsa˜1=ψa1,αa1,δa1,βa1,ua1,va1 and a˜2=ψa2,αa2,δa2,βa2,ua2,va2, then(5)a˜1⊕a˜2=ΔΔ−1ψa1,αa1+Δ−1ψa2,αa2,ΔΔ−1δa1,βa1+Δ−1δa2,βa2,ua12+ua22−ua12ua22,νa1νa2;a˜1⊗a˜2=ΔΔ−1ψa1,αa1×Δ−1ψa2,αa2,ΔΔ−1δa1,βa1×Δ−1δa2,βa2,ua1ua2,νa12+νa22−νa12νa22;λa˜1=ΔλΔ−1ψa1,αa1,ΔλΔ−1δa1,βa1,1−1−ua12λ,νa1λ;a˜1λ=ΔΔ−1ψa1,αa1λ,ΔΔ−1δa1,βa1λ,ua1λ,1−1−νa12λ.


### 2.2. Pythagorean Interval 2-Tuple Linguistic Arithmetic Aggregation Operators

In this section, the PI2TLWA operator and PI2TLWG operator are introduced.Definition 7 (see [64]).
Fora˜j=ψaj,αaj,δaj,βaj,uaj,vajj=1,2,…,n is a collection of PI2TLNs. PI2TLWA operator is defined as(6)PI2TLWAωa˜1,a˜2,…,a˜n=⊕j=1nωja˜j=Δ∑j=1nωjΔ−1ψaj,αaj,Δ∑j=1nωjΔ−1δaj,βaj,1−∏j=1n1−uaj2ωj,∏j=1nνajωj,where ω=ω1,ω2,…,ωnT is the weight vector of a˜jj=1,2,…,n and ωj>0,∑j=1nωj=1.Definition 8 (see [31]).
Fora˜j=ψaj,αaj,δaj,βaj,uaj,vajj=1,2,…,n is a collection of PI2TLNs. PI2TLWG operator is defined as(7)PI2TLWGωa˜1,a˜2,…,a˜n=⊗j=1nωja˜j=Δ∏j=1nΔ−1ψaj,αajωj,Δ∏j=1nΔ−1δaj,βajωj,,∏j=1nuajωj1−∏j=1n1−νaj2ωj,where ω=ω1,ω2,...,ωnT is the weight vector of a˜jj=1,2,...,n and ωj>0,∑j=1nωj=1.


## 2.1. Pythagorean Interval 2-Tuple Linguistic Sets

The concept of Pythagorean interval 2-tuple linguistic sets (PI2TLSs) first appeared [61] on the basis of PFSs [62] and 2-tuple linguistic information [63].Definition 1 (see [61]).
In a PI2TLS,A in the theoretical field X is defined as follows:(1)A=ψx,αx,δx,βx,uAx,νAx,x∈X,where ψx,αx,δx,βx is an interval 2-tuple linguistic variable, ψx,δx∈S=S0,S1⋯ST, S is a linguistic term set, and αx,βx∈−0.5,0.5, uAx,νAx∈0,1 represent the degree of membership and degree of non-membership of the element x to an interval 2-tuple linguistic ψx,αx,δx,βx and meet the condition 0≤uAx2+νAx2≤1∈0,1.
For convenience,A=ψA,αA,δA,βA,uA,vA can be called a Pythagorean interval 2-tuple linguistic number (PI2TLN).Definition 2 (see [61]).
For a PI2TLNa˜=ψa,αa,δa,βa,ua,va, its score function is(2)Sfa˜=ΔΔ−1ψa,αa+Δ−1δa,βa2×1+ua2−νa22.Definition 3 (see [61]).
For a PI2TLNa˜=ψa,αa,δa,βa,ua,va, its accuracy function is(3)Afa˜=ΔΔ−1ψa,αa+Δ−1δa,βa2×ua2+νa22.Definition 4 (see [61]).
For two PI2TLNsa˜1=ψa1,αa1,δa1,βa1,ua1,va1 and a˜2=ψa2,αa2,δa2,βa2,ua2,va2, respectively, the scores of a˜1 and a˜2 are Sfa˜1=ΔΔ−1ψa1,αa1+Δ−1δa1,βa1/2×1+ua12−νa12/2 and Sfa˜2=ΔΔ−1ψa2,αa2+Δ−1δa2,βa2/2×1+ua22−νa22/2, and let Afa˜1=ΔΔ−1ψa1,αa1+Δ−1δa1,βa1/2×ua12+νa12/2 and Afa˜2=ΔΔ−1ψa1,αa1+Δ−1δa2,βa2/2×ua22+νa22/2 be the accuracy degrees of a˜1 and a˜2; then, some operational laws of PI2TLNs are defined as follows:(1)
IfSfa˜1<Sfa˜2,a˜1<a˜2.(2)
IfSfa˜1>Sfa˜2,a˜1>a˜2.(3)
IfSfa˜1=Sfa˜2,Afa˜1<Afa˜2,thena˜1<a˜2.(4)
IfSfa˜1=Sfa˜2,Afa˜1>Afa˜2,thena˜1>a˜2.(5)
IfSfa˜1=Sfa˜2,Afa˜1=Afa˜2,thena˜1=a˜2.Definition 5 (see [61]).
For two PI2TLNsa˜1=ψa1,αa1,δa1,βa1,ua1,va1 and a˜2=ψa2,αa2,δa2,βa2,ua2,va2, the normalized Hamming distance (Hd) between a˜1 and a˜2 is defined as(4)Hda˜1,a˜2=14L1+ua12−va12⋅Δ−1ψa1,αa1+Δ−1δa1,βa1−1+ua22−va22⋅Δ−1ψa2,αa2+Δ−1δa2,βa2,where L represents the length of the language scale. It is a numerical value.Definition 6 (see [61]).
For two PI2TLNsa˜1=ψa1,αa1,δa1,βa1,ua1,va1 and a˜2=ψa2,αa2,δa2,βa2,ua2,va2, then(5)a˜1⊕a˜2=ΔΔ−1ψa1,αa1+Δ−1ψa2,αa2,ΔΔ−1δa1,βa1+Δ−1δa2,βa2,ua12+ua22−ua12ua22,νa1νa2;a˜1⊗a˜2=ΔΔ−1ψa1,αa1×Δ−1ψa2,αa2,ΔΔ−1δa1,βa1×Δ−1δa2,βa2,ua1ua2,νa12+νa22−νa12νa22;λa˜1=ΔλΔ−1ψa1,αa1,ΔλΔ−1δa1,βa1,1−1−ua12λ,νa1λ;a˜1λ=ΔΔ−1ψa1,αa1λ,ΔΔ−1δa1,βa1λ,ua1λ,1−1−νa12λ.


## 2.2. Pythagorean Interval 2-Tuple Linguistic Arithmetic Aggregation Operators

In this section, the PI2TLWA operator and PI2TLWG operator are introduced.Definition 7 (see [64]).
Fora˜j=ψaj,αaj,δaj,βaj,uaj,vajj=1,2,…,n is a collection of PI2TLNs. PI2TLWA operator is defined as(6)PI2TLWAωa˜1,a˜2,…,a˜n=⊕j=1nωja˜j=Δ∑j=1nωjΔ−1ψaj,αaj,Δ∑j=1nωjΔ−1δaj,βaj,1−∏j=1n1−uaj2ωj,∏j=1nνajωj,where ω=ω1,ω2,…,ωnT is the weight vector of a˜jj=1,2,…,n and ωj>0,∑j=1nωj=1.Definition 8 (see [31]).
Fora˜j=ψaj,αaj,δaj,βaj,uaj,vajj=1,2,…,n is a collection of PI2TLNs. PI2TLWG operator is defined as(7)PI2TLWGωa˜1,a˜2,…,a˜n=⊗j=1nωja˜j=Δ∏j=1nΔ−1ψaj,αajωj,Δ∏j=1nΔ−1δaj,βajωj,,∏j=1nuajωj1−∏j=1n1−νaj2ωj,where ω=ω1,ω2,...,ωnT is the weight vector of a˜jj=1,2,...,n and ωj>0,∑j=1nωj=1.


## 3. EDAS Method for MAGDM with PI2TLNs

LetP=P1,P2,…,Pm be a solution set, σ=σ1,σ2,…,σn be an evaluation attribute set, and D=D1,D2,…,Dk be an expert set. ϖ1,ϖ2,…,ϖk and ω1,ω2,…,ωn are, respectively, the expert’s weighting vector and attribute’s weighting vector which satisfy ϖt∈0,1,ωj∈0,1 and ∑t=1kϖt=1,∑j=1nωj=1. The calculation steps of this method are as follows:Step 1. A DM group is constructed to select the best attribute for the performance measurement of the alternative, and the evaluation matrix Tk=tijkm×n of each decision maker with PI2TLNs is finally formed:(8)Tk=tijkm×n=t11kt12k…t1nkt21kt22k…t2nk⋮⋮⋮⋮tm1ktm2k…tmnk.Step 2. Use PI2TLWA operator or PI2TLWG operator to obtain the group evaluation matrix T=tijm×n:(9)T=tEm×n=t11t12…t1nt21t22…t2n⋮⋮⋮⋮tm1tm2…tmn,tij=⊕t=1ktijk=PI2TLWAtij1,tij2,…,tijk=Δ∑t=1kϖtΔ−1ψgijt,αgijt,Δ∑t=1kϖtΔ−1δgijt,βgijt,1−∏t=1k1−ugijt2ϖt,∏t=1kνgijtϖt,or(10)tij=⊗t=1ktijk=P2TLWGtij1,tij2,…,tijk=Δ∑t=1kΔ−1ψgijt,αgijtϖt,Δ∑t=1kΔ−1δgijt,βgijtϖt,,∏t=1kugijtϖt1−∏t=1k1−νgijt2ϖt.Step 3. Calculate AV:(11)AV=AVj1×n=∑i=1mtijm1×n.According to Definition6,(12)∑i=1mtij=Δ∑i=1mΔ−1ψgij,αgij,Δ∑i=1mΔ−1δgij,βgij,1−∏i=1m1−ugij2,∏i=1mνgij.Thus,(13)AV=AVj1×n=∑i=1mtijm1×n=Δ∑i=1m1mΔ−1ψgij,αgij,Δ∑i=1m1mΔ−1δgij,βgij,1−∏i=1m1−ugij21/m,∏i=1mνgij1/m.Step 4. According to AV, calculate the values of PDA and NDA depending on the type of attribute.(14)PDAij=PDAijm×n=max0,tij−AVjAVj,(15)NDAij=NDAijm×n=max0,AVj−tijAVj.To facilitate calculation, the PI2TLNs’ scoring function proposed in Definition2 is used to calculate the PDA and NDA values as follows:(16)PDAij=PDAijm×n=max0,stij−sAVjsAVj,NDAij=NDAijm×n=max0,sAVj−stijsAVj.Step 5. Calculate weighting distance SPi and SNi.(17)SPi=∑j=1nωjPDAij,SNi=∑j=1nωjNDAij.Step 6. Normalize the calculating results of the following equation:(18)NSPi=SPimaxSPi,NSNi=1−SNimaxSNi.Step 7. Calculate the values of appraisal score (AS).(19)ASi=12NSPi+NSNi.Step 8. All alternatives are ranked according to their final evaluation value.


## 4. Numerical Example

In recent years, due to the increase in ship types, larger ships, faster speeds, congested routes, and more bridges and port facilities, the navigation environment has undergone great changes and become more and more complex, but operators must operate in a certain environment. Based on the severe changes around sailing environment, the difficulty for operation becomes harder and harder for operators to finish tasks safely and efficiently under this environment. More mistakes will occur because the pressure of the operators becomes heavier and heavier. To prevent and reduce occurrence of accidents and to make safe and efficient transportation on the sea, not only the quality of operators should be improved and the performance and management of the ship should be improved but also the sailing environment should be improved. The improvement of environment is based on systematic analysis and evaluation of the environment, so we have to understand what the danger elements are in the sailing and what level the danger has reached, so as to take pointed measure to improve the environment. According to the relevant conclusions, ship operation simulation results, and domestic and foreign expert consultation, this paper sets the risk evaluation level for each evaluation target and uses the PI2TLN-EDAS method to evaluate the safety of the ship's navigation environment and then to evaluate the safety of the ship's navigation environment. Obviously, evaluation of the ship navigation environment safety is a classical MAGDM problem. In this section, a numerical example is provided to evaluate the ship navigation environment safety by using EDAS method with PI2TLNs. Assume that there are five possible ship navigationsPii=1,2,3,4,5 to be assessed by four evaluation attributes σjj=1,2,3,4: (1) σ1 is the hydrometeorological factor; (2) σ2 is the channel condition factor; (3) σ3 is the channel traffic factor; and (4) σ4 is the port condition factor. Five possible ship navigations Pii=1,2,3,4,5 are assessed with PI2TLNs under four attributes by three experts Dk (expert’s weight ϖ=0.25,0.39,0.36 and attributes’ weight ω=0.22,0.32,0.26,0.20).In order to get the most desirable ship navigations, the relevant calculating steps are listed below:Step 1. Construct the evaluation matrix Tk=tijk5×4i=1,2,…,5,j=1,2,…,4 of each decision maker as in Tables 1–3.Step 2. Use PI2TLWA operator to integrate evaluation information from decision makers (expert’s weight ϖ=0.25,0.39,0.36). The results are presented in Table 4.Step 3. Calculate the value of (AVj):(20)AV1=Δ15×1.5000+15×4.5800+15×4.2500+15×3.8300+15×3.6100,Δ15×3.6100+15×4.2500+15×5.6400+15×2.2500+15×2.0800,1−1−0.607221/5×1−0.635021/5×1−0.496521/5×1−0.404121/5×1−0.417421/5,0.61521/5×0.58911/5×0.76271/5×0.35771/5×0.62561/5=s4,−0.4460,s4,−0.4340,0.5273,0.5732,AV2=Δ15×3.4400+15×3.3600+15×1.7500+15×4.2000+15×4.0000,Δ15×4.2500+15×3.7500+15×3.4500+15×1.9700+15×3.5800,1−1−0.610421/5×1−0.438521/5×1−0.478821/5×1−0.485321/5×1−0.485921/5,0.56411/5×0.44421/5×0.44961/5×0.65911/5×0.74781/5=s3,−0.3500,s3,0.4000,0.5059,0.5609,AV3=Δ15×2.0300+15×4.0000+15×2.7200+15×4.3900+15×3.6900,Δ15×1.6100+15×3.8600+15×4.5000+15×4.4700+15×3.2800,1−1−0.442621/5×1−0.500121/5×1−0.643621/5×1−0.311821/5×1−0.567821/5,0.67801/5×0.84591/5×0.67121/5×0.50301/5×0.68171/5=s3,0.3660,s4,−0.4560,0.5137,0.6670,AV4=Δ15×2.6100+15×4.0000+15×3.0600+15×1.8600+15×4.0600,Δ15×2.4700+15×2.0300+15×3.5300+15×4.7500+15×1.2500,1−1−0.328721/5×1−0.494521/5×1−0.297921/5×1−0.491021/5×1−0.537021/5,0.56411/5×0.41591/5×0.63491/5×0.76531/5×0.79851/5=s3,0.1180,s3,−0.1940,0.4451,0.6192.So, the values ofAVj are determined as follows:(21)AV1×4=s4,−0.4460,s4,−0.4340,0.5273,0.5732,s3,−0.3500,s3,0.4000,0.5059,0.5609,s3,0.3660,s4,−0.4560,0.5137,0.6670,s3,0.1180,s3,−0.1940,0.4451,0.6192.Step 4. Based on the results of average solution (AVj), equations (14) and (15) are used to calculate PDAij and NDAij (see Tables 5–7).Step 5. Compute the values of SPi and SNi by equation (17) (weighting vector ω=0.22,0.32,0.26,0.20), and the results are depicted in Table 8.Step 6. Normalize the results of Step 6 by equation (18), and the results are presented in Table 9.Step 7. Compute ASii=1,2,3,4,5:(22)AS1=0.2356,AS2=0.8518,AS3=0.5046,AS4=0.4868,AS5=0.1109.Step 8. According to the calculation results, the greater the value of ASi, the better the alternative, so the order of all alternatives is P2>P3>P4>P1>P5; P2 is the best alternative.Table 1
Rating alternatives on each criterion byD1.
σ1σ2P1<[(S3, 0), (S4, 0)], (0.4, 0.7)><[(S5, 0), (S2, 0)], (0.6, 0.1)>P2<[(S5, 0), (S2, 0)], (0.6, 0.4)><[(S3, 0), (S6, 0)], (0.6, 0.5)>P3<[(S5, 0), (S6, 0)], (0.2, 0.6)><[(S1, 0), (S3, 0)], (0.3, 0.8)>P4<[(S2, 0), (S3, 0)], (0.6, 0.8)><[(S6, 0), (S2, 0)], (0.2, 0.9)>P5<[(S1, 0), (S1, 0)], (0.5, 0.6)><[(S4, 0), (S1, 0)], (0.5, 0.7)>σ3σ4P1<[(S2, 0), (S2, 0)], (0.5, 0.5)><[(S3, 0), (S1, 0)], (0.4, 0.5)>P2<[(S4, 0), (S5, 0)], (0.2, 0.5)><[(S4, 0), (S2, 0)], (0.4, 0.7)>P3<[(S5, 0), (S3, 0)], (0.6, 0.2)><[(S3, 0), (S5, 0)], (0.2, 0.4)>P4<[(S1, 0), (S6, 0)], (0.3, 0.3)><[(S3, 0), (S4, 0)], (0.7, 0.5)>P5<[(S3, 0), (S4, 0)], (0.6, 0.1)><[(S4, 0), (S2, 0)], (0.6, 0.4)>Table 2
Rating alternatives on each criterion byD2.
σ1σ2P1<[(S1, 0), (S3, 0)], (0.6, 0.4)><[(S1, 0), (S5, 0)], (0.5, 0.7)>P2<[(S3, 0), (S5, 0)], (0.3, 0.6)><[(S3, 0), (S3, 0)], (0.4, 0.2)>P3<[(S4, 0), (S6, 0)], (0.5, 0.8)><[(S2, 0), (S6, 0)], (0.6, 0.3)>P4<[(S3, 0), (S2, 0)], (0.1, 0.6)><[(S6, 0), (S1, 0)], (0.2, 0.8)>P5<[(S4, 0), (S1, 0)], (0.2, 0.7)><[(S4, 0), (S3, 0)], (0.3, 0.9)>σ3σ4P1<[(S3, 0), (S1, 0)], (0.5, 0.7)><[(S2, 0), (S2, 0)], (0.3, 0.7)>P2<[(S4, 0), (S3, 0)], (0.6, 0.8)><[(S4, 0), (S3, 0)], (0.6, 0.2)>P3<[(S1, 0), (S5, 0)], (0.7, 0.5)><[(S5, 0), (S4, 0)], (0.4, 0.5)>P4<[(S6, 0), (S3, 0)], (0.2, 0.4)><[(S1, 0), (S5, 0)], (0.1, 0.7)>P5<[(S2, 0), (S4, 0)], (0.6, 0.6)><[(S6, 0), (S1, 0)], (0.4, 0.9)>P5<[(S3, 0), (S4, 0)], (0.6, 0.1)><[(S4, 0), (S2, 0)], (0.6, 0.4)>Table 3
Rating alternatives on each criterion byD3.
σ1σ2P1<[(S1, 0), (S4, 0)], (0.7, 0.7)><[(S5, 0), (S5, 0)], (0.7, 0.3)>P2<[(S6, 0), (S5, 0)], (0.8, 0.4)><[(S4, 0), (S3, 0)], (0.3, 0.6)>P3<[(S2, 0), (S1, 0)], (0.4, 0.4)><[(S3, 0), (S5, 0)], (0.6, 0.7)>P4<[(S1, 0), (S3, 0)], (0.7, 0.4)><[(S5, 0), (S5, 0)], (0.4, 0.4)>P5<[(S5, 0), (S4, 0)], (0.5, 0.4)><[(S4, 0), (S6, 0)], (0.6, 0.5)>σ3σ4P1<[(S1, 0), (S2, 0)], (0.3, 0.5)><[(S3, 0), (S4, 0)], (0.3, 0.3)>P2<[(S4, 0), (S4, 0)], (0.5, 0.8)><[(S4, 0), (S1, 0)], (0.4, 0.5)>P3<[(S3, 0), (S5, 0)], (0.6, 0.7)><[(S1, 0), (S2, 0)], (0.2, 0.6)>P4<[(S5, 0), (S5, 0)], (0.4, 0.4)><[(S2, 0), (S5, 0)], (0.5, 0.7)>P5<[(S6, 0), (S2, 0)], (0.5, 0.6)><[(S2, 0), (S1, 0)], (0.6, 0.6)>Table 4
The group Pythagorean interval 2-tuple linguistic decision matrixR.
σ1σ2P1<[(S2, −0.5), (S4, −0.39)], (0.6072, 0.6152)><[(S3, 0.44), (S4, 0.25)], (0.6104, 0.5641)>P2<[(S5,−0.42), (S4, 0.25)], (0.635, 0.5891)><[(S3, 0.36), (S4, −0.25)], (0.4385, 0.4442)>P3<[(S4, 0.25), (S6, −0.36)], (0.4965, 0.7627)><[(S2, −0.25), (S3, 0.45)], (0.4788, 0.4496)>P4<[(S4, −0.17), (S2, 0.25)], (0.4041, 0.3577)><[(S4, 0.2), (S2, −0.03)], (0.4853, 0.6591)>P5<[(S4, −0.39), (S2, 0.08)], (0.4174, 0.6256)><[(S4, 0), (S4, −0.42)], (0.4859, 0.7478)>σ3σ4P1<[(S2, 0.03), (S2, −0.39)], (0.4426, 0.678)><[(S3,−0.39), (S2, 0.47)], (0.3287, 0.5641)>P2<[(S4, 0), (S4, −0.14)], (0.5001, 0.8459)><[(S4, 0), (S2, 0.03)], (0.4945, 0.4159)>P3<[(S3, −0.28), (S5, −0.5)], (0.6436, 0.6712)><[(S3, 0.06), (S4,−0.47)], (0.2979, 0.6349)>P4<[(S4, 0.39), (S4, 0.47)], (0.3118, 0.503)><[(S2, −0.14), (S5, −0.25)], (0.491, 0.7653)>P5<[(S4, −0.31), (S3, 0.28)], (0.5678, 0.6817)><[(S4, 0.06), (S1, 0.25)], (0.537, 0.7985)>Table 5
The score values ofgij and AVj.
σ1σ2σ3σ4P12.53004.05421.34002.0062P24.66273.53722.10083.2306P33.28772.67053.47922.2589P43.14752.47143.73992.1661P52.22702.56562.98891.7277SAVj3.38033.17692.82992.4130Table 6
The results ofPDAij.
σ1σ2σ3σ4P10.00000.27610.00000.0000P20.37940.11340.00000.3389P30.00000.00000.22940.0000P40.00000.00000.32160.0000P50.00000.00000.05620.0000Table 7
The results ofNDAij.
σ1σ2σ3σ4P10.25150.00000.52650.1686P20.00000.00000.25760.0000P30.02740.15940.00000.0638P40.06890.22210.00000.1023P50.34120.19240.00000.2840Table 8
The values ofSPi and SNi.
P1P2P3P4P5SPi0.08840.18750.05960.08360.0146SNi0.22590.06700.06980.10670.1934Table 9
The values ofNSPi and NSNi.
P1P2P3P4P5NSPi0.47121.00000.31810.44580.0779NSNi0.00000.70350.69110.52780.1439


## 5. Conclusion

In MAGDM methods, uncertainty and hesitation are usually unavoidable. PI2TLSs are more likely to reflect uncertain or ambiguous information. At the same time, the EDAS method has the characteristics of small computation and few steps. Therefore, this paper proposes the PI2TLN-EDAS method to solve the MAGDM problem of PI2TLNs. Then, taking ship navigation environment safety evaluation as an example, the effectiveness and practicability of the method are verified through comparative analysis. Further research will carry out other applications of this method in other uncertain MAGDM problems [65–68] and other fields [69–75].



---

*Source: 6881900-2022-02-25.xml*