# The Service Quality Evaluation of Agricultural E-Commerce Based on Interval-Valued Intuitionistic Fuzzy GRA Method

Authors: Haibin Zhang; Lei Wang
Journal: Journal of Mathematics
 (2022)

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## Abstract

The service quality evaluation of agricultural business-to-customer (B2C) e-commerce is viewed as a multiattribute group decision-making (MAGDM) activity. Thus, a useful MAGDM process is required. Based on the grey relational analysis (GRA) process and the interval-valued intuitionistic fuzzy set (IVIFS), this study defines an interval-valued intuitionistic fuzzy (IVIF) GRA process to depict the service quality of agricultural B2C e-commerce. This is important to agricultural B2C e-commerce because this industry increases rapidly and many new services are innovated. In this article, some necessary definitions related to IVIFSs are reviewed. Additionally, criteria weights are derived using the Criteria Importance Through Intercriteria Correlation method (CRITIC). Subsequently, the GRA method is extended to incorporate IVIFs to obtain a final service alternative. All alternatives can then be ranked, and the best service quality option can be identified and promulgated. Finally, a numerical example and some useful comparative studies are obtained. The analysis results show that the defined algorithm is effective for identifying the service qualities of agricultural B2C e-commerce, which provide a new assessment method for MAGDM.


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## Body

## 1. Introduction

Because the process of making decisions is full of uncertainty and ambiguity [1–3], Zadeh [4] designed fuzzy sets (FSs) to deal with decision-making accuracy issues when binary criteria and alternatives were not available. Atanassov [5] built intuitionistic fuzzy sets (IFSs) to depict uncertain issues. Then, Gupta et al. [6] designed fuzzy mathematical entropy under IFSs.He et al. [7] built power interaction information-fused operators under IFSs. Chen et al. [8] developed the IF-TOPSIS (Technique for Order Preference by Similarity to an Ideal Solution) method using similarity measures. Those research studies extended intuitionistic fuzzy set theory significantly. Interval-valued intuitionistic fuzzy set is very important for fuzzy theory, which not only considers membership degree and non-membership degree but also considers hesitant situation. Jin et al. [9] defined two methods to cope with priority weights under IFSs. Liu et al. [10] devised intuitionistic fuzzy decision-fused operators along with BM and Dombi for real MAGDM. After that, subsequent research studies concentrated on the combination of traditional MAGDM methods and IFSs. The MABAC (multiattribute border approximation area comparison) has a systematic procedure with simple computation process and a sound logic that represents the rationale of human decision making. The IF-MABAC method with distance measures was innovated by Liang et al. [11]. Zhang and He [12] built geometric interaction information-fused operators under IFSs. Xu and Yager [13] defined weighted geometric operators under IVIFs (IVIFWG). This method can improve opinion leader’s weight between experts’ assessment. Krishankumar et al. [14] built IF-PROMETHEE (Preference Ranking Organization Method for Enrichment Evaluation). Li and Wu [15] defined cross-entropy distance under IFSs. Joshi and Kumar [16] defined an extended VIKOR (VlseKriterijumska Optimizacija I Kompromisno Resenje) method using IFSs. Phochanikorn and Tan [17] merged DEMATEL (Decision-Making Trial and Evaluation Laboratory) with ANP (Analytic Network Process) to obtain interdependencies and uncertainties under IFSs. Li et al. [18] defined grey-target real decision making under IFSs. This method employs the thought of gray systems, which provides an innovative idea for this topic. Kumar and Garg [19] defined the TOPSIS method using IVIFSs. Different from the GRA method, TOPSIS did not concentrate on correlation. Based on the extension of IFSs and IVIFSs, He et al. [20] defined the Pythagorean 2-tuple linguistic EDAS method. The CODAS (Combinative Distance-Based Assessment) method uses the Euclidean distance and taxicab distance to measure alternatives according to the negative-ideal point.GRA was initially defined by Deng [21] to cope with a real MAGDM. Compared with other real MAGDM methods, the GRA method can consider the shape similarity of every given alternative from PIS and NIS. Tan et al. [22] defined design alternatives as well as the GRA process combined with AHP. Malek et al. [23] built a revised hybrid GRA for green supply. Yazdani et al. [24] provided the QFD and the GRA methods in dealing with supply chain drivers. Furthermore, some significant research studies are demonstrated as follows. Chen [25] linked the IF-GRA method with the entropy-TOPSIS process to select a material supplier. To discern the carbon market, Zhu et al. [26] took advantage of the GRA process as well as EMD. Chiang [27] used the GRA for dependent MADM. Alptekin et al. [28] solved low-carbon development based on the GRA process. Kung and Wen [29] used the GRA process to solve the grey MADM. Lei et al. [30] defined a probabilistic linguistic GRA method. Wei and Lan [31] solved MADM under IVIFSs using the GRA method. However, that study had the following limitations: it could not solve the MAGDM problem with the GRA method under IVIFSs, and it could solve the MAGDM problem under IVIFSs with GRA with unknown weight information.Unfortunately, in the existing literature, it is very difficult to determine the corresponding works of the GRA method with CRITIC through IVIFS information. The GRA method is often used in grey systems which are partially understood by researchers. The IVIFS method considers both membership and non-membership and can effectively deal with uncertainty. Theoretical ideas behind these two methods were used to tackle unclear problems. Thus, it is imperative to extend the GRA process to IVIFSs. The primary aim of this study is to address some real MAGDM issues efficiently using GRA and IVIFSs. Moreover, this study extends the GRA process to IVIFSs. However, the CRITIC method is employed to derive the given attribute’s weight. Then, a certified application is utilized to certify the defined model, and several useful comparative studies are utilized to certify the advantages of the defined model.The major contribution and novelty of the work can be outlined as follows:(1)
The GRA method is used to cope with the MAGDM issue under IVIFSs.(2)
The weights of attributes are derived objectively through the CRITIC method.(3)
A numerical example study for evaluating the service quality of agricultural B2C e-commerce is given to show the defined approach.(4)
Some useful comparative studies are provided with the existing methods.The remainder of this paper is organized as follows. Some necessary ideas for IVIFSs are given in Section2. The GRA process for real MAGDM is revised with IVIFSs, and the calculation procedures are described in Section 3. A numerical example for evaluating the service quality of agricultural B2C e-commerce is given, and a comparative analysis is presented in Section 4. Finally, we give our conclusions in Section 5.


## 2. Preliminaries

### 2.1. IVIFSs

Definition 1.
(see [32]). The IVIFS based on X is(1)K=x,μ˜Kx,ν˜Kx|x∈X,where μ˜Kx⊂0,1 is depicted as “membership of K” and ν˜Kx⊂0,1 is depicted as “non-membership of K,” and μ˜Kx, ν˜Kx could meet real condition: 0≤supμ˜Kx+supν˜Kx≤1, ∀x∈X.Definition 2.
(see [33]). Let K1=μ1L,μ1R,ν1L,ν1R and K2=μ2L,μ2R,ν2L,ν2R be two IVIFs; then, the operation rules between them are(2)K1⊕K2=μ1L+μ2L−μ1Lμ2L,μ1R+μ2R−μ1Rμ2R,ν1Lν2L,ν1Rν2R,K1⊗K2=μ1Lμ2L,μ1Rμ2R,ν1L+ν2L−ν1Lν2L,ν1R+ν2R−ν1Rν2R,λK1=1−1−μ1Lλ,1−1−μ1Rλ,ν1Lλ,ν1Rλ,λ>0,K1λ=μ1Lλ,μ1Rλ,1−1−λ1Lλ,1−1−λ1Rλ,λ>0.Definition 3.
(see [34]). Let SK=μL,μR,vL,vR be given IVIFNs, and the score and accuracy values are defined as follows:(3)SK=μL−vL+μR−vR2,HK=μL+νL+μR+νR2.
For two given IVIFNsK1 and K2, from Definition 3,(4)1ifsK1<sK2,thenK1<K2;2ifsK1=sK2,hK1<hK2,thenK1<K2;3ifsK1=sK2,hK1=hK2,thenK1=K2.Definition 4.
(see [35]). Let K1=μ1L,μ1R,ν1L,ν1R and K2=μ2L,μ2R,ν2L,ν2R be given IVIFNs; the Euclidean distance between two given IVIFNs is defined as(5)EDK1,K2=14μ1L−μ2L2+μ1R−μ2R2+ν1L−ν2L2+ν1R−ν2R2.


### 2.2. Two Operators under IVIFSs

Under the setting of the IVIFSs, some operators could be reviewed, including IVIFWA operator and IVIFWG operator [36].Definition 5.
(see [36]). Let Kj=μKjL,μKjR,νKjL,νKjRj=1,2,…,n be a set of given IVIFNs, and the given IVIFWA operator is(6)IVIFWAωK1,K2,…,Kn=⊕j=1nωjKj=1−∏j=1n1−μKjLωj,1−∏j=1n1−μKjRωj,∏j=1nνKjLωj,∏j=1nνKjRωj,where ω=ω1,ω2,…,ωnT is the weight of Kjj=1,2,…,n and ωj>0,∑j=1nωj=1.Definition 6.
(see [37]). Let Kj=μKjL,μKjR,νKjL,νKjRj=1,2,…,n be a set of given IVIFNs, and the given IVIFWG operator is(7)IVIFWGωK1,K2,…,Kn=⊗j=1nKjωj=∏j=1nμKjLωj,∏j=1nμKjRωj,1−∏j=1n1−νKjLωj,1−∏j=1n1−νKjRωj,where ω=ω1,ω2,…,ωnT is the weight of Kjj=1,2,…,n and ωj>0,∑j=1nωj=1.


## 2.1. IVIFSs

Definition 1.
(see [32]). The IVIFS based on X is(1)K=x,μ˜Kx,ν˜Kx|x∈X,where μ˜Kx⊂0,1 is depicted as “membership of K” and ν˜Kx⊂0,1 is depicted as “non-membership of K,” and μ˜Kx, ν˜Kx could meet real condition: 0≤supμ˜Kx+supν˜Kx≤1, ∀x∈X.Definition 2.
(see [33]). Let K1=μ1L,μ1R,ν1L,ν1R and K2=μ2L,μ2R,ν2L,ν2R be two IVIFs; then, the operation rules between them are(2)K1⊕K2=μ1L+μ2L−μ1Lμ2L,μ1R+μ2R−μ1Rμ2R,ν1Lν2L,ν1Rν2R,K1⊗K2=μ1Lμ2L,μ1Rμ2R,ν1L+ν2L−ν1Lν2L,ν1R+ν2R−ν1Rν2R,λK1=1−1−μ1Lλ,1−1−μ1Rλ,ν1Lλ,ν1Rλ,λ>0,K1λ=μ1Lλ,μ1Rλ,1−1−λ1Lλ,1−1−λ1Rλ,λ>0.Definition 3.
(see [34]). Let SK=μL,μR,vL,vR be given IVIFNs, and the score and accuracy values are defined as follows:(3)SK=μL−vL+μR−vR2,HK=μL+νL+μR+νR2.
For two given IVIFNsK1 and K2, from Definition 3,(4)1ifsK1<sK2,thenK1<K2;2ifsK1=sK2,hK1<hK2,thenK1<K2;3ifsK1=sK2,hK1=hK2,thenK1=K2.Definition 4.
(see [35]). Let K1=μ1L,μ1R,ν1L,ν1R and K2=μ2L,μ2R,ν2L,ν2R be given IVIFNs; the Euclidean distance between two given IVIFNs is defined as(5)EDK1,K2=14μ1L−μ2L2+μ1R−μ2R2+ν1L−ν2L2+ν1R−ν2R2.


## 2.2. Two Operators under IVIFSs

Under the setting of the IVIFSs, some operators could be reviewed, including IVIFWA operator and IVIFWG operator [36].Definition 5.
(see [36]). Let Kj=μKjL,μKjR,νKjL,νKjRj=1,2,…,n be a set of given IVIFNs, and the given IVIFWA operator is(6)IVIFWAωK1,K2,…,Kn=⊕j=1nωjKj=1−∏j=1n1−μKjLωj,1−∏j=1n1−μKjRωj,∏j=1nνKjLωj,∏j=1nνKjRωj,where ω=ω1,ω2,…,ωnT is the weight of Kjj=1,2,…,n and ωj>0,∑j=1nωj=1.Definition 6.
(see [37]). Let Kj=μKjL,μKjR,νKjL,νKjRj=1,2,…,n be a set of given IVIFNs, and the given IVIFWG operator is(7)IVIFWGωK1,K2,…,Kn=⊗j=1nKjωj=∏j=1nμKjLωj,∏j=1nμKjRωj,1−∏j=1n1−νKjLωj,1−∏j=1n1−νKjRωj,where ω=ω1,ω2,…,ωnT is the weight of Kjj=1,2,…,n and ωj>0,∑j=1nωj=1.


## 3. GRA Process for MAGDM with IVIFNs

This section builds the IVIF-GRA process for defined real MAGDM. The designed calculating steps of defined method could be always depicted. LetTA=TA1,TA2,…TAn be the set of attributes and ta=ta1,ta2,…tan be the given attribute weight TAj, where taj∈0,1,j=1,2,…,n,∑j=1ntaj=1. Let HA=HA1,HA2,…HAl be a group of DMs that have remarkable degree of ha=ha1,ha2,…hal, where hak∈0,1, k=1,2,…,l. ∑k=1lhak=1. Let PA=PA1,PA2,…PAm be a group of given alternatives. QA=qaijm×n is the decision matrix, and qaij depicts the value of PAi for TAj. Afterwards, the given calculating steps are listed.Step 1. Build the matrix QAk=qaijkm×n; then, derive the overall matrix QA=qaijm×n.(8)QAk=qaijkm×n=qa11kqa12k⋯qa1nkqa21kqa22k⋯qa2nk⋮⋮⋮⋮qam1kqam2k⋯qamnk,kk=1,2,…,k,(9)QA=qaijm×n=qa11qa12⋯qa1nqa21qa22⋯qa2n⋮⋮⋮⋮qam1qam2⋯qamn,(10)qaij=1−∏k=1l1−μqaijkLhak,1−∏k=1l1−μqaijkRhak,∏k=1lνqaijkLhak,∏k=1lνqaijkRhak,whereqaijk is the assessment information of PAii=1,2,…,m for TAjj=1,2,…,n and HAkk=1,2,…,l.Step 2. Normalize the mathematical matrix QA=qaijm×n to QAN=qaijNm×n with IVIFNs.(11)qaijN=μijL,μijR,νijL,νijR,TAjis a benefit criterion,νijL,νijR,μijL,μijR,TAjis a cost criterion.Step 3. Take advantage of the CRITIC to obtain the attributes’ weight.The CRITIC process was defined to decide attributes’ weights. This method was defined by Diakoulaki et al. [38] which took the correlations relationship between given attributes into consideration, whereafter the computing procedures of such method are designed.Depending on the normalized matrix,QN=qaijnm×n, the correlation coefficient between attributes could be obtained.(12)CCjr=∑i=1mSqaijN−SqajNSqairN−SqarN∑i=1mSqaijN−SqajN2∑i=1mSqairN−SqarN2,j,r=1,2,…,n,whereSqajN=1/m∑i=1mSqaijN and SqarN=1/m∑i=1mSqairN.Calculate attributes’ standard deviation.(13)SDj=1m−1∑i=1mSqaijN−SqajN2,j=1,2,…,n,whereSqjN=1/m∑i=1mSqijN.Calculate the attributes’ weights.(14)taj=SDj∑t=1n1−CCjr∑j=1nSDj∑t=1n1−CCjr,j=1,2…,n,wheretaj∈0,1 and ∑j=1ntaj=1.Step 4. Build positive ideal solutions (PISs) IVIFPISj and the corresponding negative ideal solutions (NISs) IVIFNISj through equations (15) and (16):(15)IVIFPISj=μjL+,μjR+,νjL+,νjR+,(16)IVIFNISj=μjL+,μjR+,νjL+,νjR+,whereIVIFPISj=maxjμijL,maxjμijR,minjνijL,minjνijR and IVIFNISj=minjμijL,minjμijR,maxjνijL,maxjνijR.Step 5. The important point of computing grey rational coefficients emphasizes shape similarity degree of every given attribute of every given alternative from the IVIFPIS and IVIFNIS. Thus, compute the grey relational coefficient (GRC) of each given alternative between each given alternative and IVIF-IVPIS and the GRC between each given alternative and IVIF-PIS as(17)IVIFPISξij=min1≤i≤mmin1≤j≤nEDqaijN,IVIFPISj+ρmax1≤i≤mmax1≤j≤nEDqaijN,IVIFPISjEDqaijN,IVIFPISj+ρmax1≤i≤mmax1≤j≤nEDqaijN,IVIFPISjIVIFNISξij=min1≤i≤mmin1≤j≤nEDqaijN,IVIFPISj+ρmax1≤i≤mmax1≤j≤nEDqaijN,IVIFPISjEDqaijN,IVIFNISj+ρmax1≤i≤mmax1≤j≤nEDqaijN,IVIFNISj,i=1,2,…,m,j=1,2,…,n.Step 6. Figure out the degree of GRC of all given alternatives from IVIFPIS as well as IVIFNIS:(18)IVIFPISξi=∑j=1ntajIVIFPISξij,i=1,2,…,m,IVIFNISξi=∑j=1ntajIVIFNISξij,i=1,2,…,m.The basic idea of the GRA process is that the given alternative could have the “largest degree of grey relation” from the IVIFPIS and the “smallest degree of grey relation” from the IVIFNIS.Step 7. Compute each alternative’s interval-valued intuitionistic fuzzy relative relational degree (IVIFRRD) of all given alternatives from IVIFPIS:(19)IVIFRRDi=IVIFNISξiIVIFNISξi+IVIFPISξi,i=1,2,…,m.Step 8. The optimal choice can be determined by the highest value of IVIFRRD.


## 4. Illustrative Example and Some Comparative Analyses

### 4.1. Numerical Example

In the last few years, Internet shopping has been developing rapidly in China, exerting a great influence on traditional pure offline formats such as department stores, supermarkets, and physical fresh stores. According to the data released by China’s agricultural fresh e-commerce development BBS in 2016, more than 80% of traditional fresh e-commerce enterprises failed to make profits. The pure online Internet shopping model can no longer satisfy the needs of Chinese consumers. The new retail model of “online + offline + strong logistics + big data” has become the favorite of the new era. As an emerging business model, the new retail model is based on the Internet, big data technology, and intelligent logistics to effectively organize and integrate online and offline data and resources. Gronroos [39] defined service quality as “a kind of perception, which is determined by the comparison between customer's service expectation and actual service experience.” There are two classical theory frameworks for service quality measurement. Parasuraman et al. [40] proposed the SERVQUAL (service quality) model for service quality evaluation, which is also named the PZB model. The other one is SERVPERF (service performance) which was proposed by Cronin and Taylor [41]. Compared with SERVQUAL, SERVPERF is easier for investigation because it supports one-shot measurement. However, this study did not aim to develop or extend this famous theory; this framework was used to develop a new methodology. Thus, a few service research papers were reviewed. Three attributes were borrowed directly from SERVQUAL. Furthermore, “customer complaint” was the fourth attribute to represent cost attribute in the research. In this part, an empirical application is given to evaluate to the service quality of agricultural B2C e-commerce, which could be solved through IVIF-GRA model. Since the local government hopes to choose the urban road through the best environmental behaviors, there are five latent enterprises EPii=1,2,3,4,5. For evaluating the service quality evaluation of agricultural B2C e-commerce fairly, five experts EH=EH1,EH2,EH3,EH4,EH5 (expert’s weight ew = (0.2, 0.2, 0.2, 0.2) are asked. All invited experts express their assessed information through four attributes: ① ET1 is assurance of agriculture e-commerce; ② ET2 is customer complaint of agriculture e-commerce; ③ ET3 is reliability of e-commerce; and ④ ET4 is responsiveness of e-commerce. Evidently, ET2 is cost attribute, while ET1, ET3, and ET4 are given benefit attributes.(i)
Step 1. Build each given IVIF matrix QAk=qaijkm×n as in Tables 1–5. From the tables and equations (8)–(10), the overall matrix shall be obtained. The calculating results are defined in Tables 1–6.(ii)
Step 2. Normalize the IVIF information matrix EQ=qaijm×n to EQN=qaijNm×n (see Table 7).(iii)
Step 3. The weights can be calculated tajj=1,2,…,n through CRITIC (see Table 8).(iv)
Step 4. Determine the IVIFPISj and IVIFNISj by using equations (15) and (16).(20)IVIFPIS1=0.5876,0.6829,0.2512,0.3172,IVIFPIS2=0.4522,0.4195,0.5082,0.5805,IVIFPIS3=0.4703,0.5187,0.3848,0.4723,IVIFPIS4=0.5627,0.6405,0.2908,0.3171593,IVIFNIS1=0.3011,0.3678,0.5678,0.6323,IVIFNIS2=0.1661,0.2502,0.6574,0.7498,IVIFNIS3=0.3036,0.3589,0.5872,0.6411,IVIFNIS4=0.3162,0.4103,0.5152,0.5892.(v)
Step 5. Figure out the GRC of every alternative from IVIFPIS as well as IVIFNIS (Tables 9 and 10).(vi)
Step 6. Figure out the degree of GRC of all alternatives from IVIFPIS as well as IVIFNIS (Table 11).(vii)
Step 7. Calculate the IVIFRRDξi of each given alternative from IVIFPIS (Table 12).Table 1
IVIF matrix byH1.
ET1ET2ET3ET4EP1([0.55, 0.62], [0.27, 0.38])([0.29, 0.35], [0.60, 0.65])([0.32, 0.40], [0.55, 0.60])([0.35, 0.40], [0.51, 0.60])EP2([0.28, 0.46], [0.50, 0.54])([0.53, 0.60], [0.35, 0.40])([0.60, 0.65], [0.30, 0.35])([0.62, 0.70], [0.25, 0.30])EP3([0.52, 0.60], [0.35, 0.40])([0.46, 0.52], [0.40, 0.48])([0.47, 0.55], [0.40, 0.45])([0.25, 0.31], [0.66, 0.71])EP4([0.36, 0.41], [0.56, 0.59])([0.41, 0.45], [0.50, 0.55])([0.60, 0.70], [0.25, 0.30])([0.73, 0.81], [0.14, 0.22])EP5([0.70, 0.80], [0.15, 0.20])([0.36, 0.40], [0.57, 0.60])([0.29, 0.36], [0.58, 0.64])([0.57, 0.62], [0.30, 0.38])Table 2
IVIF matrix byH2.
ET1ET2ET3ET4EP1([0.38, 0.42], [0.52, 0.58])([0.61, 0.65], [0.30, 0.35])([0.37, 0.45], [0.50, 0.55])([0.55, 0.60], [0.32, 0.40])EP2([0.61, 0.65], [0.30, 0.35])([0.35, 0.45], [0.50, 0.55])([0.70, 0.80], [0.10, 0.20])([0.52, 0.62], [0.30, 0.38])EP3([0.63, 0.70], [0.25, 0.30])([0.59, 0.62], [0.26, 0.38])([0.19, 0.25], [0.70, 0.75])([0.59, 0.65], [0.30, 0.35])EP4([0.35, 0.40], [0.55, 0.60])([0.65, 0.75], [0.20, 0.25])([0.62, 0.70], [0.25, 0.30])([0.37, 0.45], [0.55, 0.60])EP5([0.42, 0.48], [0.50, 0.52])([0.37, 0.40], [0.53, 0.60])([0.36, 0.40], [0.55, 0.60])([0.55, 0.62], [0.28, 0.38])Table 3
IVIF matrix byH3.
ET1ET2ET3ET4EP1([0.28, 0.36], [0.57, 0.64])([0.32, 0.39], [0.41, 0.61])([0.35, 0.40], [0.55, 0.60])([0.68, 0.75], [0.20, 0.25])EP2([0.44, 0.48], [0.50, 0.52])([0.25, 0.30], [0.55, 0.70])([0.58, 0.63], [0.30, 0.37])([0.41, 0.52], [0.40, 0.48])EP3([0.34, 0.42], [0.50, 0.58])([0.17, 0.22], [0.65, 0.78])([0.52, 0.62], [0.32, 0.38])([0.74, 0.80], [0.15, 0.21])EP4([0.18, 0.25], [0.70, 0.75])([0.32, 0.40], [0.55, 0.60])([0.66, 0.75], [0.20, 0.25])([0.59, 0.66], [0.30, 0.35])EP5([0.32, 0.40], [0.55, 0.60])([0.43, 0.47], [0.50, 0.53])([0.60, 0.65], [0.30, 0.35])([0.57, 0.62], [0.32, 0.38])Table 4
IVIF matrix byH4.
ET1ET2ET3ET4EP1([0.36, 0.42], [0.52, 0.59])([0.56, 0.62], [0.30, 0.38])([0.31, 0.42], [0.50, 0.58])([0.71, 0.78], [0.17, 0.22])EP2([0.59, 0.65], [0.30, 0.35])([0.52, 0.60], [0.35, 0.41])([0.19, 0.30], [0.65, 0.70])([0.23, 0.34], [0.58, 0.66])EP3([0.38, 0.45], [0.50, 0.55])([0.72, 0.80], [0.15, 0.20])([0.73, 0.80], [0.10, 0.20])([0.60, 0.66], [0.30, 0.34])EP4([0.63, 0.75], [0.15, 0.25])([0.65, 0.72], [0.21, 0.28])([0.55, 0.60], [0.30, 0.40])([0.50, 0.50], [0.50, 0.50])EP5([0.55, 0.60], [0.30, 0.40])([0.65, 0.70], [0.25, 0.30])([0.24, 0.35], [0.58, 0.65])([0.43, 0.46], [0.51, 0.54])Table 5
IVIF matrix byH5.
ET1ET2ET3ET4EP1([0.59, 0.65], [0.27, 0.35])([0.33, 0.43], [0.51, 0.57])([0.26, 0.32], [0.60, 0.68])([0.19, 0.25], [0.19, 0.25])EP2([0.40, 0.45], [0.50, 0.55])([0.41, 0.45], [0.50, 0.55])([0.34, 0.41], [0.52, 0.59])([0.33, 0.42], [0.50, 0.58])EP3([0.46, 0.55], [0.30, 0.45])([0.35, 0.40], [0.50, 0.60])([0.57, 0.67], [0.26, 0.33])([0.28, 0.35], [0.58, 0.65])EP4([0.34, 0.41], [0.53, 0.59])([0.28, 0.32], [0.60, 0.68])([0.69, 0.75], [0.18, 0.25])([0.41, 0.55], [0.37, 0.45])EP5([0.27, 0.34], [0.60, 0.66])([0.49, 0.55], [0.40, 0.45])([0.39, 0.45], [0.48, 0.55])([0.62, 0.72], [0.20, 0.28])Table 6
Overall IVIF evaluation matrix.
ET1ET2ET3ET4EP1([0.432, 0.494], [0.432, 0.508])([0.422, 0.468], [0.424, 0.512])([0.322, 0.398], [0.540, 0.602])([0.496, 0.556], [0.278, 0.344])EP2([0.464, 0.538], [0.420, 0.462])([0.412, 0.480], [0.450, 0.522])([0.482, 0.558], [0.374, 0.442])([0.422, 0.520], [0.406, 0.480])EP3([0.466, 0.544], [0.380, 0.456])([0.458, 0.512], [0.392, 0.488])([0.402, 0.578], [0.364, 0.422])([0.492, 0.554], [0.398, 0.452])EP4([0.372, 0.444], [0.498, 0.616])([0.462, 0.528], [0.412, 0.472])([0.624, 0.700], [0.236, 0.300])([0.520, 0.594], [0.372, 0.424])EP5([0.452, 0.474], [0.420, 0.476])([0.460, 0.504], [0.450, 0.496])([0.376, 0.442], [0.498, 0.558])([0.548, 0.608], [0.322, 0.392])According to theIVIFRRDξi, all given alternatives are ranked, the higher IVIFRRDξi, the better the alternative selected. Evidently, the order is EP3>EP1>EP5>EP4>EP2, and EP3 is the best one.Table 7
The normalized IVIF matrix.
ET1ET2ET3ET4EP1([0.432, 0.494], [0.432, 0.508])([0.424, 0.512], [0.422, 0.468])([0.322, 0.398], [0.540, 0.602])([0.496, 0.556], [0.278, 0.344])EP2([0.464, 0.538], [0.420, 0.462])([0.450, 0.522], [0.412, 0.480])([0.482, 0.558], [0.374, 0.442])([0.422, 0.520], [0.406, 0.480])EP3([0.466, 0.544], [0.380, 0.456])([0.392, 0.488], [0.458, 0.512])([0.402, 0.578], [0.364, 0.422])([0.492, 0.554], [0.398, 0.452])EP4([0.372, 0.444], [0.498, 0.616])([0.412, 0.472], [0.462, 0.528])([0.624, 0.700], [0.236, 0.300])([0.520, 0.594], [0.372, 0.424])EP5([0.452, 0.474], [0.420, 0.476])([0.450, 0.496], [0.460, 0.504])([0.376, 0.442], [0.498, 0.558])([0.548, 0.608], [0.322, 0.392])Table 8
The weightstaj.
ET1ET2ET3ET4taj0.33680.17330.12470.3562Table 9
GRC of each alternative from IVIFPIS.
AlternativesET1ET2ET3ET4EP10.68251.00000.33330.4095EP20.54430.43430.41750.4943EP30.70490.68251.00001.0000EP40.51810.49430.43430.4831EP51.00000.43430.41750.5181Table 10
GRC of each alternative from IVIFNIS.
AlternativesET1ET2ET3ET4EP10.51720.34880.41670.4545EP20.78951.00001.00000.8333EP30.44120.36590.33330.3947EP41.00000.51720.71431.0000EP50.42860.45450.48390.4769Table 11
IVIFPISξi and IVIFNISξi of every alternative.
AlternativesIVIFPISξiIVIFNISξiEP10.56530.4146EP20.47890.9398EP30.85240.3746EP40.48560.7449EP50.62750.4866Table 12
IVIFRRD of each alternative from IFPIS.
AlternativesEP1EP2EP3EP4EP5IVIFRRDξi0.57700.33750.69470.39460.5632


### 4.2. Comparative Analysis

In this part, the designed method is compared with four methods to show the superiority. Firstly, the research makes a comparison between our designed methods, IVIFWA and IVIFWG. For given IVIFWA operator, the calculating value isSEP1=0.5821, SEP2=0.4709, SEP3=0.6324, SEP4=0.5013, and SEP5=0.5632. Thus, the ranking order is EP3>EP1>EP5>EP4>EP2. For the given IVIFWG information operator, the order value is SEP1=0.5633, SEP2=0.4543, SEP3=0.6288, SEP4=0.5087, and SEP5=0.5431. So, the order is EP3>EP1>EP5>EP4>EP2.Furthermore, the designed method is compared with IVIF-MABAC [42]. The given overall value of each given alternative is as follows: I1=2.6576, I2=0.5324, I3=4.0232, I4=0.9855, and I5=1.2577. Therefore, the order of all given alternatives is EP3>EP1>EP5>EP4>EP2. Finally, our defined method is compared with IVIF-CODAS [43]. The total assessment score (AS) of each given alternative is as follows: AS1=0.9253, AS2=−1.6588, AS3=1.6754, AS4=−1.1277, and AS5=0.6577. Therefore, the order of alternatives is EP3>EP1>EP5>EP4>EP2.Ultimately, the results of these given methods are shown in Table13.Table 13
The obtained results of other methods.
ModelsOrderThe best choiceThe worst choiceIVIFWA operator [44]EP3>EP1>EP5>EP4>EP2EP3EP2IVIFWG operator [13]EP3>EP1>EP5>EP4>EP2EP3EP2IVIF-MABAC method [42]EP3>EP1>EP5>EP4>EP2EP3EP2IVIF-CODAS method [43]EP3>EP1>EP5>EP4>EP2EP3EP2The designed methodEP3>EP1>EP5>EP4>EP2EP3EP2From Table13, it is evident that the most optimal choice is EP3, whereas the worst one is EP2. These methods could also cope with MAGDM from different angles. The IVIFWA and IVIFWG operator directly fuse given decision values. The IVIF-MABAC process emphasizes the risk preferences and interactive relationship among given criteria. The IVIF-GRA emphasizes shape similarity degree between two given sequences. The IVIF-CODAS model often emphasizes the connection of practical Euclidean and concrete Hamming distance. However, compared with these previous four methods, the designed model may be more accurate, since it not only considers each alternative’s closest degree to IVIFPIS but also considers each alternative’s farthest degree from IVIFNIS.


## 4.1. Numerical Example

In the last few years, Internet shopping has been developing rapidly in China, exerting a great influence on traditional pure offline formats such as department stores, supermarkets, and physical fresh stores. According to the data released by China’s agricultural fresh e-commerce development BBS in 2016, more than 80% of traditional fresh e-commerce enterprises failed to make profits. The pure online Internet shopping model can no longer satisfy the needs of Chinese consumers. The new retail model of “online + offline + strong logistics + big data” has become the favorite of the new era. As an emerging business model, the new retail model is based on the Internet, big data technology, and intelligent logistics to effectively organize and integrate online and offline data and resources. Gronroos [39] defined service quality as “a kind of perception, which is determined by the comparison between customer's service expectation and actual service experience.” There are two classical theory frameworks for service quality measurement. Parasuraman et al. [40] proposed the SERVQUAL (service quality) model for service quality evaluation, which is also named the PZB model. The other one is SERVPERF (service performance) which was proposed by Cronin and Taylor [41]. Compared with SERVQUAL, SERVPERF is easier for investigation because it supports one-shot measurement. However, this study did not aim to develop or extend this famous theory; this framework was used to develop a new methodology. Thus, a few service research papers were reviewed. Three attributes were borrowed directly from SERVQUAL. Furthermore, “customer complaint” was the fourth attribute to represent cost attribute in the research. In this part, an empirical application is given to evaluate to the service quality of agricultural B2C e-commerce, which could be solved through IVIF-GRA model. Since the local government hopes to choose the urban road through the best environmental behaviors, there are five latent enterprises EPii=1,2,3,4,5. For evaluating the service quality evaluation of agricultural B2C e-commerce fairly, five experts EH=EH1,EH2,EH3,EH4,EH5 (expert’s weight ew = (0.2, 0.2, 0.2, 0.2) are asked. All invited experts express their assessed information through four attributes: ① ET1 is assurance of agriculture e-commerce; ② ET2 is customer complaint of agriculture e-commerce; ③ ET3 is reliability of e-commerce; and ④ ET4 is responsiveness of e-commerce. Evidently, ET2 is cost attribute, while ET1, ET3, and ET4 are given benefit attributes.(i)
Step 1. Build each given IVIF matrix QAk=qaijkm×n as in Tables 1–5. From the tables and equations (8)–(10), the overall matrix shall be obtained. The calculating results are defined in Tables 1–6.(ii)
Step 2. Normalize the IVIF information matrix EQ=qaijm×n to EQN=qaijNm×n (see Table 7).(iii)
Step 3. The weights can be calculated tajj=1,2,…,n through CRITIC (see Table 8).(iv)
Step 4. Determine the IVIFPISj and IVIFNISj by using equations (15) and (16).(20)IVIFPIS1=0.5876,0.6829,0.2512,0.3172,IVIFPIS2=0.4522,0.4195,0.5082,0.5805,IVIFPIS3=0.4703,0.5187,0.3848,0.4723,IVIFPIS4=0.5627,0.6405,0.2908,0.3171593,IVIFNIS1=0.3011,0.3678,0.5678,0.6323,IVIFNIS2=0.1661,0.2502,0.6574,0.7498,IVIFNIS3=0.3036,0.3589,0.5872,0.6411,IVIFNIS4=0.3162,0.4103,0.5152,0.5892.(v)
Step 5. Figure out the GRC of every alternative from IVIFPIS as well as IVIFNIS (Tables 9 and 10).(vi)
Step 6. Figure out the degree of GRC of all alternatives from IVIFPIS as well as IVIFNIS (Table 11).(vii)
Step 7. Calculate the IVIFRRDξi of each given alternative from IVIFPIS (Table 12).Table 1
IVIF matrix byH1.
ET1ET2ET3ET4EP1([0.55, 0.62], [0.27, 0.38])([0.29, 0.35], [0.60, 0.65])([0.32, 0.40], [0.55, 0.60])([0.35, 0.40], [0.51, 0.60])EP2([0.28, 0.46], [0.50, 0.54])([0.53, 0.60], [0.35, 0.40])([0.60, 0.65], [0.30, 0.35])([0.62, 0.70], [0.25, 0.30])EP3([0.52, 0.60], [0.35, 0.40])([0.46, 0.52], [0.40, 0.48])([0.47, 0.55], [0.40, 0.45])([0.25, 0.31], [0.66, 0.71])EP4([0.36, 0.41], [0.56, 0.59])([0.41, 0.45], [0.50, 0.55])([0.60, 0.70], [0.25, 0.30])([0.73, 0.81], [0.14, 0.22])EP5([0.70, 0.80], [0.15, 0.20])([0.36, 0.40], [0.57, 0.60])([0.29, 0.36], [0.58, 0.64])([0.57, 0.62], [0.30, 0.38])Table 2
IVIF matrix byH2.
ET1ET2ET3ET4EP1([0.38, 0.42], [0.52, 0.58])([0.61, 0.65], [0.30, 0.35])([0.37, 0.45], [0.50, 0.55])([0.55, 0.60], [0.32, 0.40])EP2([0.61, 0.65], [0.30, 0.35])([0.35, 0.45], [0.50, 0.55])([0.70, 0.80], [0.10, 0.20])([0.52, 0.62], [0.30, 0.38])EP3([0.63, 0.70], [0.25, 0.30])([0.59, 0.62], [0.26, 0.38])([0.19, 0.25], [0.70, 0.75])([0.59, 0.65], [0.30, 0.35])EP4([0.35, 0.40], [0.55, 0.60])([0.65, 0.75], [0.20, 0.25])([0.62, 0.70], [0.25, 0.30])([0.37, 0.45], [0.55, 0.60])EP5([0.42, 0.48], [0.50, 0.52])([0.37, 0.40], [0.53, 0.60])([0.36, 0.40], [0.55, 0.60])([0.55, 0.62], [0.28, 0.38])Table 3
IVIF matrix byH3.
ET1ET2ET3ET4EP1([0.28, 0.36], [0.57, 0.64])([0.32, 0.39], [0.41, 0.61])([0.35, 0.40], [0.55, 0.60])([0.68, 0.75], [0.20, 0.25])EP2([0.44, 0.48], [0.50, 0.52])([0.25, 0.30], [0.55, 0.70])([0.58, 0.63], [0.30, 0.37])([0.41, 0.52], [0.40, 0.48])EP3([0.34, 0.42], [0.50, 0.58])([0.17, 0.22], [0.65, 0.78])([0.52, 0.62], [0.32, 0.38])([0.74, 0.80], [0.15, 0.21])EP4([0.18, 0.25], [0.70, 0.75])([0.32, 0.40], [0.55, 0.60])([0.66, 0.75], [0.20, 0.25])([0.59, 0.66], [0.30, 0.35])EP5([0.32, 0.40], [0.55, 0.60])([0.43, 0.47], [0.50, 0.53])([0.60, 0.65], [0.30, 0.35])([0.57, 0.62], [0.32, 0.38])Table 4
IVIF matrix byH4.
ET1ET2ET3ET4EP1([0.36, 0.42], [0.52, 0.59])([0.56, 0.62], [0.30, 0.38])([0.31, 0.42], [0.50, 0.58])([0.71, 0.78], [0.17, 0.22])EP2([0.59, 0.65], [0.30, 0.35])([0.52, 0.60], [0.35, 0.41])([0.19, 0.30], [0.65, 0.70])([0.23, 0.34], [0.58, 0.66])EP3([0.38, 0.45], [0.50, 0.55])([0.72, 0.80], [0.15, 0.20])([0.73, 0.80], [0.10, 0.20])([0.60, 0.66], [0.30, 0.34])EP4([0.63, 0.75], [0.15, 0.25])([0.65, 0.72], [0.21, 0.28])([0.55, 0.60], [0.30, 0.40])([0.50, 0.50], [0.50, 0.50])EP5([0.55, 0.60], [0.30, 0.40])([0.65, 0.70], [0.25, 0.30])([0.24, 0.35], [0.58, 0.65])([0.43, 0.46], [0.51, 0.54])Table 5
IVIF matrix byH5.
ET1ET2ET3ET4EP1([0.59, 0.65], [0.27, 0.35])([0.33, 0.43], [0.51, 0.57])([0.26, 0.32], [0.60, 0.68])([0.19, 0.25], [0.19, 0.25])EP2([0.40, 0.45], [0.50, 0.55])([0.41, 0.45], [0.50, 0.55])([0.34, 0.41], [0.52, 0.59])([0.33, 0.42], [0.50, 0.58])EP3([0.46, 0.55], [0.30, 0.45])([0.35, 0.40], [0.50, 0.60])([0.57, 0.67], [0.26, 0.33])([0.28, 0.35], [0.58, 0.65])EP4([0.34, 0.41], [0.53, 0.59])([0.28, 0.32], [0.60, 0.68])([0.69, 0.75], [0.18, 0.25])([0.41, 0.55], [0.37, 0.45])EP5([0.27, 0.34], [0.60, 0.66])([0.49, 0.55], [0.40, 0.45])([0.39, 0.45], [0.48, 0.55])([0.62, 0.72], [0.20, 0.28])Table 6
Overall IVIF evaluation matrix.
ET1ET2ET3ET4EP1([0.432, 0.494], [0.432, 0.508])([0.422, 0.468], [0.424, 0.512])([0.322, 0.398], [0.540, 0.602])([0.496, 0.556], [0.278, 0.344])EP2([0.464, 0.538], [0.420, 0.462])([0.412, 0.480], [0.450, 0.522])([0.482, 0.558], [0.374, 0.442])([0.422, 0.520], [0.406, 0.480])EP3([0.466, 0.544], [0.380, 0.456])([0.458, 0.512], [0.392, 0.488])([0.402, 0.578], [0.364, 0.422])([0.492, 0.554], [0.398, 0.452])EP4([0.372, 0.444], [0.498, 0.616])([0.462, 0.528], [0.412, 0.472])([0.624, 0.700], [0.236, 0.300])([0.520, 0.594], [0.372, 0.424])EP5([0.452, 0.474], [0.420, 0.476])([0.460, 0.504], [0.450, 0.496])([0.376, 0.442], [0.498, 0.558])([0.548, 0.608], [0.322, 0.392])According to theIVIFRRDξi, all given alternatives are ranked, the higher IVIFRRDξi, the better the alternative selected. Evidently, the order is EP3>EP1>EP5>EP4>EP2, and EP3 is the best one.Table 7
The normalized IVIF matrix.
ET1ET2ET3ET4EP1([0.432, 0.494], [0.432, 0.508])([0.424, 0.512], [0.422, 0.468])([0.322, 0.398], [0.540, 0.602])([0.496, 0.556], [0.278, 0.344])EP2([0.464, 0.538], [0.420, 0.462])([0.450, 0.522], [0.412, 0.480])([0.482, 0.558], [0.374, 0.442])([0.422, 0.520], [0.406, 0.480])EP3([0.466, 0.544], [0.380, 0.456])([0.392, 0.488], [0.458, 0.512])([0.402, 0.578], [0.364, 0.422])([0.492, 0.554], [0.398, 0.452])EP4([0.372, 0.444], [0.498, 0.616])([0.412, 0.472], [0.462, 0.528])([0.624, 0.700], [0.236, 0.300])([0.520, 0.594], [0.372, 0.424])EP5([0.452, 0.474], [0.420, 0.476])([0.450, 0.496], [0.460, 0.504])([0.376, 0.442], [0.498, 0.558])([0.548, 0.608], [0.322, 0.392])Table 8
The weightstaj.
ET1ET2ET3ET4taj0.33680.17330.12470.3562Table 9
GRC of each alternative from IVIFPIS.
AlternativesET1ET2ET3ET4EP10.68251.00000.33330.4095EP20.54430.43430.41750.4943EP30.70490.68251.00001.0000EP40.51810.49430.43430.4831EP51.00000.43430.41750.5181Table 10
GRC of each alternative from IVIFNIS.
AlternativesET1ET2ET3ET4EP10.51720.34880.41670.4545EP20.78951.00001.00000.8333EP30.44120.36590.33330.3947EP41.00000.51720.71431.0000EP50.42860.45450.48390.4769Table 11
IVIFPISξi and IVIFNISξi of every alternative.
AlternativesIVIFPISξiIVIFNISξiEP10.56530.4146EP20.47890.9398EP30.85240.3746EP40.48560.7449EP50.62750.4866Table 12
IVIFRRD of each alternative from IFPIS.
AlternativesEP1EP2EP3EP4EP5IVIFRRDξi0.57700.33750.69470.39460.5632


## 4.2. Comparative Analysis

In this part, the designed method is compared with four methods to show the superiority. Firstly, the research makes a comparison between our designed methods, IVIFWA and IVIFWG. For given IVIFWA operator, the calculating value isSEP1=0.5821, SEP2=0.4709, SEP3=0.6324, SEP4=0.5013, and SEP5=0.5632. Thus, the ranking order is EP3>EP1>EP5>EP4>EP2. For the given IVIFWG information operator, the order value is SEP1=0.5633, SEP2=0.4543, SEP3=0.6288, SEP4=0.5087, and SEP5=0.5431. So, the order is EP3>EP1>EP5>EP4>EP2.Furthermore, the designed method is compared with IVIF-MABAC [42]. The given overall value of each given alternative is as follows: I1=2.6576, I2=0.5324, I3=4.0232, I4=0.9855, and I5=1.2577. Therefore, the order of all given alternatives is EP3>EP1>EP5>EP4>EP2. Finally, our defined method is compared with IVIF-CODAS [43]. The total assessment score (AS) of each given alternative is as follows: AS1=0.9253, AS2=−1.6588, AS3=1.6754, AS4=−1.1277, and AS5=0.6577. Therefore, the order of alternatives is EP3>EP1>EP5>EP4>EP2.Ultimately, the results of these given methods are shown in Table13.Table 13
The obtained results of other methods.
ModelsOrderThe best choiceThe worst choiceIVIFWA operator [44]EP3>EP1>EP5>EP4>EP2EP3EP2IVIFWG operator [13]EP3>EP1>EP5>EP4>EP2EP3EP2IVIF-MABAC method [42]EP3>EP1>EP5>EP4>EP2EP3EP2IVIF-CODAS method [43]EP3>EP1>EP5>EP4>EP2EP3EP2The designed methodEP3>EP1>EP5>EP4>EP2EP3EP2From Table13, it is evident that the most optimal choice is EP3, whereas the worst one is EP2. These methods could also cope with MAGDM from different angles. The IVIFWA and IVIFWG operator directly fuse given decision values. The IVIF-MABAC process emphasizes the risk preferences and interactive relationship among given criteria. The IVIF-GRA emphasizes shape similarity degree between two given sequences. The IVIF-CODAS model often emphasizes the connection of practical Euclidean and concrete Hamming distance. However, compared with these previous four methods, the designed model may be more accurate, since it not only considers each alternative’s closest degree to IVIFPIS but also considers each alternative’s farthest degree from IVIFNIS.


## 5. Discussion and Conclusion

Although the final comparison revealed that the four methods achieved the same ranking result, illustrating the effectiveness of the proposed method, it is difficult to conclude that the results are always the same under all conditions. Their efficiencies may be different in practice. The proposed method calculates the IVIFPIS and IVIFNIS in replaced upper- and lower-approximation areas in the MABAC process [42] and is advantageous to e-commerce service evaluation. However, it does not focus on risk assessment, unlike MABAC, which may be vital for other management areas.The simulation of the real world using models and algorithms has limitations. The proposed method estimated the average weight of all five experts in this research, and IVIFWG could increase the weight of one expert through the weighted geometric mean [13]. Democratic and opinion-leader models are considered by those two methods. In other situations, the weights of opinion leaders should not be increased individually. The assessment process should focus on similar or moderate dispersion comments. Further studies should consider more of such situations.For agriculture e-commerce services, the invertedU model is another knotty problem. High evaluation data of one attribute do not imply that the service is good in terms of that attribute (e.g., excessive enthusiasm). Algorithm improvement and research design avoiding may be two possible solutions for this problem that will be explored in the future.This paper defines a useful method for this kind of issue, even the interaction between membership and non-membership is considered in previous methods, such as Choquet-IVIF and interactive aggregation operators under IVIF [45–47], it builds the IVIF-GRA method for service quality evaluation of agricultural B2C e-commerce. After that, a numerical example gives evaluation to the service quality of agricultural B2C e-commerce. Furthermore, to check on the feasibility as well as availability of the new proposed method, some useful comparative analyses are also designed. In the near future, the defined models could also be extended to many more uncertain cases [48–51] and the designed algorithms could be employed to cope with many other real MAGDM-like evaluation issues and site selection [33,52]. A new possibility degree measure for interval-valued q-rung orthopair fuzzy sets in decision making can be explored for methodology innovation.



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*Source: 3931136-2022-01-07.xml*