An integrated decision-making methodology for green supplier selection based on the improved IVIF-CPT-MABAC method

Wang, Jing; Cai, Qiang; Wang, Hongjun; Wei, Guiwu; Liao, Ningna

doi:10.3233/JIFS-224206

An integrated decision-making methodology for green supplier selection based on the improved IVIF-CPT-MABAC method

Article type: Research Article

Authors: Wang, Jing^a | Cai, Qiang^b | Wang, Hongjun^{c; *} | Wei, Guiwu^b | Liao, Ningna^b

Affiliations: [a] School of Mathematical Sciences, Sichuan Normal University, Chengdu, P.R. China | [b] School of Business, Sichuan Normal University, Chengdu, P.R. China | [c] School of Economics and Management, Chongqing University of Arts and Sciences, Chongqing, China

Correspondence: [*] Corresponding author. Hongjun Wang, School of Economics and Management, Chongqing University of Arts and Sciences, Chongqing 402160, China. E-mail: [email protected]; [email protected].

Keywords: Multiple attribute group decision making (MAGDM), interval-valued intuitionistic fuzzy sets (IVIFSs), MABAC, cumulative prospect theory (CPT), CRITIC, green supplier selection

DOI: 10.3233/JIFS-224206

Journal: Journal of Intelligent & Fuzzy Systems, vol. 44, no. 5, pp. 8535-8560, 2023

Published: 04 May 2023

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Abstract

Green supply chain management attaches great importance to the coordinated development of social economy and ecological environment, and requires enterprises to consider environmental protection factors in product design, packaging, procurement, production, sales, logistics, waste and recycling. Suppliers are the “source” of the entire supply chain, and the choice of green suppliers is the basis of green supply chain management, and their quality will directly affect the environmental performance of enterprises. The green supplier selection is a classical multiple attribute group decision making (MAGDM) problems. Interval-valued intuitionistic fuzzy sets (IVIFSs) are the extension of intuitionistic fuzzy sets (IFSs), and are utilized to depict the complex and changeable circumstance. To better adapt to complex environment, the purpose of this paper is to construct a new method to solve the MAGDM problems for green supplier selection. Taking the fuzzy and uncertain character of the IVIFSs and the psychological preference into consideration, the original MABAC method based on the cumulative prospect theory (CPT) is extended into IVIFSs (IVIF-CPT-MABAC) method for MAGDM issues. Meanwhile, the method to evaluate the attribute weighting vector is calculated by CRITIC method. Finally, a numerical example for green supplier selection has been given and some comparisons is used to illustrate advantages of IVIF-CPT-MABAC method and some comparison analysis and sensitivity analysis are applied to prove this new method’s effectiveness and stability.

1Introduction

The concept of fuzzy sets (FSs) proposed by Zadeh [1] descripts the complex decision-making environment, and it has been extended into more complicated fuzzy sets such as IFSs [2, 3] and the hesitant fuzzy set (HFS) [4] which are the two main branches of FSs. The IFSs is the classical fuzzy set and it has been applied to many fields, such as business management, decision -making and so on [5–7]. Constrained by the fact that the IFSs is in the range of real number, the IVIFSs is the extension of IFSs with the interval numbers to express the membership and non-membership degree which could satisfy the increasingly complex decision environments. Thus, Atanassov [8] proposed the conception of IVIFSs, and the corresponding operators of IVIFSs are the introduced [9]. With the basic research set up, the corresponding innovation of IVIFSs has been proposed [10–14]. Mondal and Samanta [15] defined the topology and studied the characteristic of IVIFSs which could prove the IVIFSs could construct the topological category. Grzegorzewski [16] defined new distance method based on Hausdorff metric which is the well-known Hamming distance. Xu [17] based on some operational laws of IVIFSs, such as the score function and accuracy, gave the introduction of geometric aggregation operator. Zhang, Dong, Zhang and Song [18] introduced the inclusion and similarity measure in IVIFSs. Senapati, Chen, Mesiar and Yager [19] proposed novel Aczel-Alsina operations and applied it in the MADM process. Besides, the latest research on IFSs and IVIFSs in MADM is enough to attract everyone’s attention. For instance, Hayat et al. proposed the new aggregation operators on group-based generalized intuitionistic fuzzy soft sets [20]; Karaaslan explored new aggregation operators on group-based generalized intuitionistic fuzzy soft sets from a new perspective [21]; Hayat et al. applied the new group-based generalized interval-valued q-rung orthopair fuzzy soft aggregation operator to sports decision-making problems, and extended the generalized intuitionistic fuzzy soft set [22–24]; Yang et al. explored the application of aggregation and interactive aggregation soft operators under the interval-valued q-order orthomorphic fuzzy soft environment in the evaluation of automation companies [25] and so on. Meanwhile, many MADM methods have been developed into IVIFSs to solve the practical MADM problems. TOPSIS method [26], grey relational analysis (GRA) method [27, 28], maximizing deviation method [29], VIKOR method [30, 31], MABAC method [13, 32], and so on.

MADM contains multiple criteria, conflict among criteria, incommensurable units, alternatives and preference decision [33, 34]. Group decision making (GDM) means one more decision-makers (DMs) to take part in the decision-making progress [35], and means the results do not determined by one DM. The MAGDM is the integration of MADM and GDM which means the multi-attribute decision making is made by one more DM. Thus, MAGDM problems are too scientific and complex to evaluate by simple real number and operation. The FSs and its extension provide to contain more information [36–39] and MADM methods provide different theories to evaluate the attribute weight and the alternatives [40–43]. Different from other methods, MABAC [44] evaluates the alternative by defining the distance between possible solution and border approximation area, and is a classical method utilized in different fuzzy sets which include Pythagorean Fuzzy sets (PFSs) Peng and Yang [45], interval neutrosophic sets (INSs) [46], interval type-2 fuzzy [47]. The above studies all consider the assumption that DMs are all rational. However, in the practical situation, the psychological preference should be fully considered for the influence on decision outcomes. According to the related theory, such as CPT [48], DMs are always risk-averse.

Selecting long-term stable suppliers is the most important problem in building a green supply chain under the green supply chain management mode, which directly affects the competitiveness of enterprises in the market [49–51]. How to grasp the changing internal and external environment, determine reasonable evaluation indicators according to their own needs, and choose green suppliers that suit them has become an urgent problem to be solved [52–54]. The selection of green suppliers is a process of coordinating with the economic benefit objective on the basis of considering environmental factors and giving them certain weights. The selection process of green suppliers can neither overemphasize environmental protection factors, nor overemphasize economic interests. Only by coordinating the development of the two, can the long-term cooperation between enterprises and suppliers be promoted. Green supplier selection is a very complex decision-making problem [55–58]. Due to the complexity of the green supply chain system and the uncertainty, ambiguity and hesitation of human thinking, the evaluation of green suppliers will be affected by many uncertain factors, which makes it difficult to describe the evaluation indicators with exact values [59–62]. In order to be more in line with the thinking and cognitive mode of decision makers, this paper uses IVIFSs to describe attribute values. With the rapid development of science and technology, many decision-making problems in life have become more and more complex, and it is difficult to better solve these problems by relying solely on a certain decision-maker. The MAGDM together not only to reduce mistakes, but also to improve the accuracy of decision-making. It can be seen that it is of great significance to apply the MAGDM method under IVIFS to the selection of green suppliers. Therefore, it is necessary to integrate the CPT which takes the DMs’ risk preference into consideration and classical MAGDM methods. In this study, the concepts of IVIFS, MABAC and some distance in interval-value intuitionistic fuzzy environment are reviewed, however, some problems are also found in the review process. The motivations of this study are put forward as follows: (1) In the face of increasingly complex decision-making environment and complicated reality, it is particularly important to obtain evaluation information efficiently. IVIFS can more flexibly and completely express the preference evaluation information of DMs. (2) Since the existing CRITIC method is only applicable to real numbers, this paper uses precise functions to extend CRITIC algorithm to IVIFS environment to improve the rationality of attribute weight information. (3) Selecting good green suppliers has a far-reaching positive impact on promoting the improvement of green supply chain management and the development of green economy. (4) The method proposed in this paper can effectively solve the MAGDM problems which the decision-making information is expressed by IVIFSs and the attributes have local advantages.

The aim of this paper is to extend MABAC method based on CPT for MAGDM under IVIFSs and apply it into green supplier selection. The contributions of this paper are shown as follows: (1) The integration of CPT and MABAC method in IVIFSs which not only considers relatively simple and reasonable classical method, but also considers the psychological state of DMs which is more realistic has been constructed. This new method has clear decision-making logic and stable results, which makes the new theory more practical. (2) The CRITIC method is extended into IVIFSs to calculate the weighting vector of attributes, making the method proposed in this paper more practical. (3) the IVIF-CPT-MABAC method is not only proposed for MAGDM issues, but also broadens the practical application range of IVIF theory. (4) Finally, a numerical example of green supplier selection is built to illustrate the practicality of the proposed and some comparisons is used to illustrate advantages of IVIF-CPT-MABAC method and some comparison analysis and sensitivity analysis are applied to prove this new method’s effectiveness and stability.

The following is the structure of the article. Section 2 gave the basic introduction and calculation operators of IVIFSs and the classical CPT. In section 3, the IVIF-CPT-MABAC method is constructed for MAGDM and is applied into green supplier selection in Section 4. Meanwhile, Section 4 gave a numerical example for green supplier selection and gave the sensitivity and comparison analysis to prove this method’s stability and effectiveness. In the conclusion, we prospect the application of this new method.

2Preliminary knowledge

In this section, the basic elements related to decision-making such as IVIFS and CPT are described below.

2.1IVIFSs

IVIFS, as an extension of the IFS, breaks the limitation of IFS on the representation of real numbers and makes information description more accurate and scientific. In IVIFS, the membership, non-membership and hesitancy degree are depicted with interval value for the complexity and uncertainty of objective things. The specific definition is depicted as follows.

Definition 1. [63]. Let Y be a finitely nonempty set, and the IFS is described as Equation (2.1):

(2.1)

IFS={<y,φIFS(y),ψIFS(y)>|y∈Y},

where φ_IFS (y), ψ_IFS (y) and π_IFS (y) denote the membership, non-membership and hesitancy degree in IFSs which are defined as follows:

(2.2)

φIFS(y):y∈Y→φIFS(y)∈[0,1];

(2.3)

ψIFS(y):y∈Y→ψIFS(y)∈[0,1];

(2.4)

0⩽φIFS(y)+ψIFS(y)⩽1;

(2.5)

πIFS(y)=1-φIFS(y)-ψIFS(y).

Definition 2 [8]. Let Y be a finitely nonempty set, then the IVIFS is described as follows:

(2.6)

IVIF˜={<y,φ˜IVIF˜(y),ψ˜IVIF˜(y)>|y∈Y},

where φ˜IVIF˜(y) and ψ˜IVIF˜(y) represent the membership and non-membership degree of IVIFS, and are defined as follows:

(2.7)

φ˜IVIF˜(y):y∈Y→φ˜IVIF˜(y)=[LM(y),RM(y)]⊆[0,1];

(2.8)

ψ˜IVIF˜(y):y∈Y→ψ˜IVIF˜(y)=[LN(y),RN(y)]⊆[0,1];

(2.9)

0⩽RM(y)+RN(y)⩽1.

Similarity, the hesitancy degree of IVIFS is shown as follows:

(2.10)

π˜IFS(y)=[1-RM(y)+RN(y),1-LM(y)-LN(y)].

The interval-valued intuitionistic fuzzy numbers (IVIFN) in Equation (2.6) are shown as follows:

(2.11)

IVIFN˜=(φ˜IVIF˜(y),ψ˜IVIF˜(y)) =([LM(y),RM(y)],[LN(y),RN(y)]),

Which could be abbreviated to the form in Equation (2.12)

(2.12)

IVIFN˜=([LM,RM],[LN,RN]).

Specially, when LM = RM, LN = RN, IVIFS degenerate into IFS. The maximum IVIFN is IVIFN˜max=([1,1],[0,0]) and the minimum IVIFN is IVIFN˜min=([0,0],[1,1]) .

Definition 3 [17]. Suppose any three IVIFNs, IVIFN˜1=([LM1,RM1],[LN1,RN1]) , IVIFN˜2=([LM2,RM2],[LN2,RN2]) and IVIFN˜3=([LM3,RM3],[LN3,RN3]) , and the calculation rules of IVIFNs are defined as follows:

(1) (IVIFN˜1)c=([LN1,RN1],[LM1,RM1]);
(2) IVIFN˜1⊕IVIFN˜2=([LM1+LM2-LM1LM2,RM1+RM2-RM1RM2] , [LN₁LN₂, RN₁RN₂]) ;
(3) IVIFN˜1⊗IVIFN˜2=([LM1LM2,RM1RM2] , [LN₁+ LN₂ - LN₁LN₂, RN₁ + RN₂ - RN₁RN₂]) ;
(4) λIVIFN˜1=([1-(1-LM1)λ,1-(1-RM1)λ],[LN1λ,RN1λ]),λ>0 .
(5) IVIFN˜1λ=([LM1λ,RM1λ],[1-(1-LN1)λ,1-(1-RN1)λ]),λ>0 .

Besides, the calculation formulas also meet the following rules [9]:

(i) Commutative law
- a) IVIFN˜1⊕IVIFN˜2=IVIFN˜2⊕IVIFN˜1;
- b) IVIFN˜1⊗IVIFN˜2=IVIFN˜2⊗IVIFN˜1 .
(ii) Associative law
- a) (IVIFN˜1⊕IVIFN˜2)⊕IVIFN˜3=IVIFN˜1⊕(IVIFN˜2⊕IVIFN˜3);
- b) (IVIFN˜1⊗IVIFN˜2)⊗IVIFN˜3=IVIFN˜1⊗(IVIFN˜2⊗IVIFN˜3) .
(iii) Distributive law
- a) λ(IVIFN˜1⊕IVIFN˜2)=λIVIFN˜1⊕λIVIFN˜2;
- b) λ1IVIFN˜1⊕λ2IVIFN˜1=(λ1+λ2)IVIFN˜1 .
(iv) Exponential operation law
- a) (IVIFN˜1⊗IVIFN˜2)λ=IVIFN˜1λ⊗IVIFN˜2λ;
- b) IVIFN˜1λ1⊗IVIFN˜1λ2=IVIFN˜1(λ1+λ2) .

where λ, λ₁, λ₂, ⩾ 0.

Definition 4 [64]. Let IVIFN˜=([LM,RM],[LN,RN]) be IVIFN, and the score functions of IVIFN ( SF(IVIFN˜) and AF(IVIFN˜) ) are defined as follows:

(2.13)

SF(IVIFN˜)=(LM+RM)(LM+LN)-(LN+RN)(RM+RN)2,

where SF(IVIFN˜)∈[-1,1] , if the value of SF(IVIFN˜) is greater, the corresponding IVIFN is larger.

(2.14)

AF(IVIFN˜)=(1-LM+RM)(1-LM-LN)+(1-LN+RN)(1-RM-RN)2,

where AF(IVIFN˜)∈[0,1] , if the value of AF(IVIFN˜) is greater, the corresponding IVIFN is larger.

Definition 5 [64]. According to Definition 4, suppose any two IVIFNs, IVIFN˜1=([LM1,RM1],[LN1,RN1]) and IVIFN˜2=([LM2,RM2],[LN2,RN2]) , and the comparison between these two IVIFNs could be done as follows:

If SF(IVIFN˜1)>SF(IVIFN2˜) , then IVIFN˜1>IVIFN˜2 ;

If SF(IVIFN˜1)<SF(IVIFN2˜) , then IVIFN˜1<IVIFN˜2 ;

If SF(IVIFN˜1)=SF(IVIFN2˜) , and it can be divided into the following three cases:

a) If AF(IVIFN˜1)<AF(IVIFN˜2) , then IVIFN˜1<IVIFN˜2 ;
b) If AF(IVIFN˜1)>AF(IVIFN˜2) , then IVIFN˜1>IVIFN˜2 ;
c) If AF(IVIFN˜1)=AF(IVIFN˜2) , then IVIFN˜1=IVIFN˜2 .

Definition 6 [17]. Let a set of m-dimensional fuzzy numbers be set IVIFN˜k=([LMk,RMk],[LNk,RNk]) , (k = 1, 2, ⋯ , m). According to Definition 3, the interval-valued intuitionistic fuzzy weighted average operator is defined as follows:

(2.15)

IVIFWAω(IVIFN˜1,IVIFN˜2,⋯,IVIFN˜m)=∑k=1mωkIVIFN˜k=([1-∏k=1m(1-LMk)ωk,1-∏k=1m(1-RMk)ωk],[∏k=1mLNkωk,∏k=1mRNkωk]),

where ω = (ω₁, ω₂, …, ω_m) ^T is the weighting vector of IVIFN˜k(k=1,2,⋯,m) , and ∀ω_k ∈ [0, 1], ∑k=1mωk=1 .

Specially, if ω=(1m,1m,…,1m)T , IVIFWA operators would degenerate to interval-valued fuzzy average operator (IVIFA).

Definition 7 [17]. Set IVIFN˜k=([LMk,RMk],[LNk,RNk]) , (k = 1, 2, ⋯ , m), and according to Definition 3, the IVIFWG operators are defined as follows:

(2.16)

IVIFWGω(IVIFN˜1,IVIFN˜2,⋯,IVIFN˜m)=∏k=1m(IVIFN˜k)ωk=([∏k=1mLMkωk,∏k=1mRMkωk],[1-∏k=1m(1-LNk)ωk,1-∏k=1m(1-RNk)ωk]),

where ω = (ω₁, ω₂, …, ω_m) ^T is the weighting vector of IVIFN˜k(k=1,2,⋯,m) , and ∀ω_k ∈ [0, 1], ∑k=1mωk=1 .

Specially, if ω=(1m,1m,…,1m)T , IVIFWG operator would be degenerated to interval-valued intuitionistic operator (IVIFG).

Definition 8 [16, 65]. Set two IVIFNs IVIFN˜1=([LM1,RM1],[LN1,RN1]) and IVIFN˜2=([LM2,RM2],[LN2,RN2]) , the Hamming distance and Hausdorff distance between two IVIFNs are defined as follows:

(2.17)

dHM(IVIFN˜1,IVIFN˜2)=14(|LM1-LM2|+|RM1-RM2|+|LN1-LN2|+|RN1-RN2|);

(2.18)

dHA(IVIFN˜1,IVIFN˜2)=12max{|LM1-LM2|,|RM1-RM2|,|LN1-LN2|,|RN1-RN2|};

(2.19)

dM(IVIFN˜1,IVIFN˜2)=14(|LM1-LM2|+|RM1-RM2|+|LN1-LN2|+|RN1-RN2|)+12max{|LM1-LM2|,|RM1-RM2|,|LN1-LN2|,|RN1-RN2|}.

2.2Cumulative prospect theory

There are mainly two kinds of risk decision-making theories in uncertain environment. One is expected utility theory, which assumes that people are in a completely rational state when making risk decisions, and this is also the common cognition of many scholars before the prospect theory is put forward. The other is prospect theory (PT) developed by Tversky and Kahneman [66] on the basis of bounded rationality theory in 1979. It realizes the influence of DMs’ risk preference on decision-making by comparing reference points, that is, when decision makers face risks, The sensitivity to loss is greater than the sensitivity to gain There are mainly two kinds of risk decision making theories in uncertain environment. One is expected utility theory, which assumes that people are in a completely rational state when making risk decisions, and this is also the common cognition of many scholars before the prospect theory is put forward. The other is PT on the basis of bounded rationality theory in 1979 [66]. It realizes the influence of decision maker’s risk preference on decision making by comparing reference points, that is, when decision makers face risks. The sensitivity to “loss” is greater than the sensitivity to “gain”. Such measures, which add to the psychology of policymakers, are more realistic. In 1992, Tversky and Kahneman [48] proposed the cumulative prospect theory (CPT) on the basis of prospect theory through a large number of experiments, which mainly improved and extended the scope of application to decision-making situations with arbitrary number of results and random dominance problems. Such measures, which add to the psychology of policymakers, are more realistic.

The prospect function P (z) combined with value function V (z_ɛ) and weighting function I (r_ɛ), and its expression formula is as Equation (2.20) [66].

(2.20)

P(z)=∑ɛ=1kV(zɛ)I(rɛ).

A value function described by a piecewise function can be presented as Equation (2.21) [66].

(2.21)

V(zɛ)={(zɛ-z0)α,ifzɛ⩾z0;-ρ·(y0-yɛ)β,ifzɛ<z0;

where α (0 < α ⩽ 1) and β (0 < β ⩽ 1) are the value function curvature parameters, ρ (ρ > 1) means the sensitivity parameters for DMs to avoid loss, the larger the value of ρ, the more DMs are concerned about the extent of the loss.

When facing with gains and losses, the weighting function are shown as follows [66]:

(2.22)

I(rɛ)={I+(rɛ)=rɛη/(rɛη+(1-rɛ)η)1η,rɛ-r0⩾0;I-(rɛ)=rɛτ/(rɛτ+(1-rɛ)τ)1τ,rɛ-r0<0;

where η and τ is the curve parameter of weighting function, and 0 < η ⩽ 1, 0 < τ ⩽ 1.

3Extended MABAC method for MAGDM based on CPT under IVIFSs

In 2015, Pamučar and Ćirović [67] firstly proposed multi-attributive border approximation area comparison (MABAC). MABAC method defines the distance between alternative and the border approximation are (BAA), and divides the BAA into positive and negative BAA according to the value of distance. In the end, the final ranking results are determined by the equations. MABAC method has the character of simple, logic and feasible.

3.1Classical MABAC method

Suppose that in some multi-attribute decision-making problem, there are the set of alternatives ML = {ML₁, ML₂, ⋯ , ML_n}, the set of attributes MT = {MT₁, MT₂, ⋯ , MT_s} and the weighting vector of attributes ω = (ω₁, ω₂, ⋯ , ω_s) ^T, and ∀ω_j ∈ [0, 1], ∑j=1sωj=1 . Thus, the evaluation value of i - th alternative ML_i and the j - th attribute MT_j is x_ij (i = 1, 2, ⋯ n ; j = 1, 2, ⋯ , s), then the MADM matrix (3.1)is constructed according to the evaluation value.

(3.1)

X=[xij]n×s=[x11x12⋯x1j⋯x1sx21x22⋯x2j⋯x2s⋮⋮⋱⋮⋱⋮xi1xi2⋯xij⋯xis⋮⋮⋱⋮⋱⋮xn1xn2⋯xnj⋯xns]

The specific steps of classical MABAC method are as follows:

Step 1. Normalized decision.

Normalize the decision matrix X = [x_ij] _n×s using Equation (3.2) and get normalized decision matrix X′=[xij′]n×s .

(3.2)

xij′={x′-mini{x′}maxi{x′}-mini{x′},MTj is positiveattribute;x′-maxi{x′}mini{x′}-maxi{x′},MTj is negativeattribute;i=1,2,⋯,n;j=1,2,⋯,s.

Step 2. Weighted normalized decision matrix.

The normalized decision matrix is determined by step 1, and with the weighting vector ω = (ω₁, ω₂, ⋯ , ω_s) ^T, and the weighted normalized decision matrix X″=[xij″]n×s is defined as follows:

xij″=ωj(1+xij′),i=1,2,⋯,n;j=1,2,⋯,s.

Step 3. The boundary approximates area value matrix.

The 1 × s boundary approximates area value matrix BAA = [g_j] _1×s is obtained using Equation (3.3)

(3.3)

gj=(∏i=1sxij′′)1/n,j=1,2,⋯,s.

Step 4. Calculate the distance between BAA.

The distance Dq_ij between alternative ML_i under each attribute and BAA are calculated as follows:

(3.4)

Dqij=xij″-gj,i=1,2,⋯,n;j=1,2,⋯,s.

Step 5. Obtain the overall distance between alternative ML_i and BAA.

(3.5)

Si=∑j=1sDqij,i=1,2,⋯,n.

Step 6. Rank the alternatives.

Rank all the alternatives ML_i (i = 1, 2, ⋯ , n) according to the overall distance S_i (i = 1, 2, ⋯ , n). If the value of overall distance S_i is larger, the corresponding alternative is better. Otherwise, it indicates that the corresponding alternative is far from the standard and should be abandoned.

3.2The extended MABAC method for MAGDM based on CPT under IVIFSs

The integration of CPT and MABAC method takes the DMs’ psychological factor into consideration. The weighting vector is calculated by CRITIC method [68] calculates and the CPT-MABAC method is then extended in the IVIFSs. Thus, the extended CPT-MABAC applies IVIFNs to express information and extends the application range of MABAC method which is more in line with the uncertainty of the real environment. By the previously mentioned, the specific steps of IVIF-CPT-MABAC model for MAGDM are shown in Fig. 3.1.

Fig. 3.1

The framework of IVIF-CPT-MABAC method.

According to the IVIF correlation theory, the decision matrix is reconstructed based on PT. Suppose that there are a set of experts MD = {MD₁, MD₂, ⋯ , MD_e}, a set of alternatives ML = {ML₁, ML₂, ⋯ , ML_n}, and a set of attributes MT = {MT₁, MT₂, ⋯ , MT_s}. The weighting vector of attributes ω = (ω₁, ω₂, ⋯ , ω_s) ^T, and ∀ω_j ∈ [0, 1], ∑j=1sωj=1 . Besides, the weighting vector od experts is ϑ = (ϑ₁, ϑ₂, ⋯ , ϑ_e) ^T, ∑c=1eϑc=1 . Therefore, the i - th alternative under j - th attributes evaluation value evaluated by the c - th experts can be expressed by IVIFNs E˜ij(c)=([LMij(c),RMij(c)],[LNij(c),RNij(c)]) , where [LMij(c),RMij(c)] and [LNij(c),RNij(c)] , (i = 1, 2, ⋯ n ; j = 1, 2, ⋯ , s ; c = 1, 2, ⋯ , e) represents the membership and non-membership interval respectively. Thus, the IVIF MAGDM matrices are constructed as follows:

(3.6)

Q(c)=[E˜ij(c)]n×s=[E˜11(c)E˜12(c)⋯E˜1j(c)⋯E˜1s(c)E˜12(c)E˜12(c)⋯E˜2j(c)⋯E˜2s(c)⋮⋮⋱⋮⋱⋮E˜i1(c)E˜i2(c)⋯E˜ij(c)⋯E˜is(c)⋮⋮⋱⋮⋱⋮E˜n1(c)E˜n2(c)⋯E˜nj(c)⋯E˜ns(c)]

Step 1. IVIF MAGDM matrices Q.

Equation (3.7) is determined by the IVIFWA operators to aggerate the IVIF MADM matrices Q(c)=[E˜ij(c)]n×s,(i=1,2,⋯n;j=1,2,⋯,s;c=1,2,⋯,e) , and obtain the overall decision matrix Q=[E˜ij]n×s=[([LMij,RMij],[LNij,RNij])]n×s using Equation (3.7)

E˜ij=([LMij,RMij],[LNij,RNij])=IVIFWAϑ(E˜ij(1),E˜ij(2),⋯,E˜ij(e))

(3.7)

=([1-∏c=1e(1-LMij(c))ϑc,1-∏c=1e(1-RMij(c))ϑc],[∏c=1e(LNij(c))ϑc,∏c=1e(RNij(c))ϑc]).

Step 2. Obtain the normalized IVIF MADM decision matrix Q′.

The normalized IVIF decision matrix Q′=[E˜ij′]n×s=[([LMij′,RMij′],[LNij′,RNij′])]n×s is obtained by Equation (3.8)

(3.8)

E˜ij′={([LMij,RMij],[LNij,RNij]),MTjispositiveattribute;([LNij,RNij],[LMij,RMij]),MTjisnegativeattribute.

Step 3. Use the extended CRITIC method to get the original attribute weight.

The IVIF-CRITIC method is mainly divided into the following steps:

(i) Determine the correlation coefficient matrix R = [RC_jp] _s×s of attribute MT_j and MT_p using Equation (3.9)

(3.9)

RCjp=∑i=1n(AF(E˜ij′)-AF(E˜j′))·(AF(E˜ip′)-AF(E˜p′))∑i=1n(AF(E˜ij′)-AF(E˜j′))2∑i=1n(AF(E˜ip′)-AF(E˜p′))2,

(3.10)

AF(E˜j′)=1n∑i=1nAF(E˜ij′),j=1,2,⋯,s.

(3.11)

AF(E˜p′)=1n∑i=1nAF(E˜ip′),p=1,2,⋯,s.

(ii) Determine volatility index DX_j.

a. Calculate the standard deviation σ_j of each attribute MT_j.
(3.12)
σj=1n∑i=1n(AF(E˜ij′)-AF(E˜j′))2,j=1,2,⋯s.
b. Calculate the volatility index DX_j of each attribute.
(3.13)
DXj=σj∑p=1s(1-RCjp),j=1,2,⋯,s.
c. Determine the initial attributes weight ω_j.
(3.14)
ωj=DXj∑jsDXj,j=1,2,⋯s.

Step 4. Determine BAA.

According to the information in normalized IVIF decision matrix Q′, the BAA=[gj*]1×s is calculated using Equation (3.15).

(3.15)

[gj*]1×s=[([LMij*,RMij*],[LNij*,RNij*])]1×s=[([∏i=1nLMij′n,∏i=1nRMij′n],[1-∏i=1n(1-LNij′)n,1-∏i=1n(1-RNij′)n])]1×s.

Step 5. Calculate the relative attribute weight matrix I*=[Iij*(ωj)]n×s .

Compare the magnitude of IVIFNs E˜ij′=([LMij′,RMij′],[LNij′,RNij′]) and gj*=([LMij*,RMij*],[LNij*,RNij*]) , the relative attribute weight Iij*(ωj) is determined by Equation (3.16)

(3.16)

Iij*(ωj)={ωjη(ωjη+(1-ωj)η)1η,E˜ij′⩾gj*;ωjτ(ωjτ+(1-ωj)τ)1τ,E˜ij′<gj*;,i=1,2,⋯n;j=1,2,⋯s.

Step 6. Obtain the weighted value function V(dMij(E˜ij′,gj*)) in IVIFs.

(3.17)

V(dMij(E˜ij′,gj*))={Iij*(ωj)·(dMij(E˜ij′,gj*))α,ifE˜ij′>gj*;0, ifE˜ij′=gj*;-ρ·Iij*(ωj)·(dMij(E˜ij′,gj*))β,ifE˜ij′<gj*;i=1,2,⋯n;j=1,2,⋯s.

where α, β and ρ are the relative parameters which show the DMs’ risk preference, and the distance dMij(E˜ij′,gj*) between possible solution under each attribute and BAA is calculated as Equation (3.18).

(3.18)

dMij(E˜ij′,gj*)=14(|LMij′-LMij*|+|RMij′-RMij*|+|LNij′-LNij*|+|RNij′-RNij*|)+12max{|LMij′-LMij*|,|RMij′-RMij*|,|LNij′-LNij*|,|RNij′-RNij*|}.

Step 7. Obtain the overall distance Si*(MLi),(i=1,2,⋯n) between each alternative ML_i and BAA.

(3.19)

Si*(MLi)=∑j=1sV(dMij(E˜ij′,gj*)).

Step 8. Rank all the alternatives according to the overall distance Si* in descending order, If the value of overall distance Si* is larger, the corresponding alternative is better. Otherwise, it indicates that the corresponding alternative is far from the standard and should be abandoned.

4Application of IVIF-CPT-MABAC method in GSS

4.1Background description

The concept of green supply chain was first proposed by the Manufacturing Research Association (MRC) of Michigan State University in 1996. It is a modern management mode that has the least negative impact on the environment and the most efficient use of resources [69–71]. In recent years, with the continuous occurrence of extreme weather, development and environmental protection should always keep pace, especially Chinese government has put forward the strategic requirement of sustainable development. Therefore, implementing green supply chain management is a sustainable development approach combining green consciousness and economic development [72, 73]. Again because of the green supplier located in the upstream of the supply chain, for the green supply chain had a significant influence on environmental protection and resource conservation, is an important part of the green supply chain management, so in this section, we based on IVIFSs environment, constructs to the third quarter of the model is applied to green supplier selection. As we all know, the GSS is a classical MAGDM issue [74–77]. To select the most appropriate green supplier, we invite five experts MD_c (c = 1, 2, ⋯ , 5) to evaluate alternatives ML_i (i = 1, 2, ⋯ , 5) under six attributes MT_j (j = 1, 2, ⋯ , 6). By summarizing the relevant literature, the main six attributes are shown as follows: MT₁: green manufacturing process; MT₂: waste management; MT₃: green Product design; MT₄: green image; MT₅: resource consumption; MT₆: green environmental management system. MT₂ and MT₅ are negative attribute, and the weighting vector of experts is ϑ = (ϑ₁, ϑ₂, ⋯ , ϑ₅) = (0.27, 0.16, 0.23, 0.16, 0.18). In order to be more in line with the realistic evaluation environment, we ask the experts to express their evaluation views clearly in language, and collect the evaluation of the experts to form the initial language evaluation matrix in Tables 4.2–4.6. Then according to the IVIF linguistic evaluation scale in Table 4.1, the expert’s literal language is transformed into an IVIFN adapted to our model, and the results are shown in Tables 4.7–4.11.

Table 4.2

The linguistic evaluation matrix given by the first expert

	MT₁	MT₂	MT₃	MT₄	MT₅	MT₆
ML₁	G	VG	M	MG	MT	MG
ML₂	G	M	G	M	M	G
ML₃	EG	MG	PG	VG	M	VG
ML₄	MG	G	VG	B	G	VB
ML₅	VG	B	EG	PB	MG	VB

Table 4.3

The linguistic evaluation matrix given by the second expert

	MT₁	MT₂	MT₃	MT₄	MT₅	MT₆
ML₁	MG	VG	M	G	M	G
ML₂	G	M	G	MG	M	G
ML₃	PG	G	EG	EG	M	VG
ML₄	MG	G	VG	MB	MG	VB
ML₅	G	B	VG	B	MG	B

Table 4.4

Linguistic evaluation matrix given by the third expert

	MT₁	MT₂	MT₃	MT₄	MT₅	MT₆
ML₁	G	G	M	G	M	MG
ML₂	MG	M	G	M	B	MG
ML₃	EG	G	VG	VG	MG	PG
ML₄	MG	MG	G	M	G	B
ML₅	MG	MB	VG	MB	M	MB

Table 4.5

Linguistic evaluation matrix given by the fourth expert

	MT₁	MT₂	MT₃	MT₄	MT₅	MT₆
ML₁	MG	G	M	MG	MB	MG
ML₂	MG	M	MG	G	M	G
ML₃	EG	G	PG	VG	MG	EG
ML₄	G	G	G	M	G	B
ML₅	G	M	G	M	G	B

Table 4.6

Linguistic evaluation matrix given by the fifth expert

	MT₁	MT₂	MT₃	MT₄	MT₅	MT₆
ML₁	G	VG	M	MG	MB	MG
ML₂	VG	M	G	M	MG	G
ML₃	EG	M	PG	VG	M	VG
ML₄	G	G	VG	M	G	VB
ML₅	VG	B	EG	B	MG	VB

Table 4.1

4.1 IVIF linguistic evaluation scale

Linguistic scale	IVIFN
Perfectly bad (PB)	〈 [0.00, 0.10] , [0.85, 0.90]〉
Very bad (VB)	〈 [0.00, 0.10] , [0.70, 0.75]〉
Bad (B)	〈 [0.15, 0.25] , [0.55, 0.60]〉
Middle bad (MB)	〈 [0.30, 0.40] , [0.45, 0.50]〉
Middle (M)	〈 [0.40, 0.50] , [0.35, 0.40]〉
Middle good (MG)	〈 [0.50, 0.60] , [0.25, 0.30]〉
Good (G)	〈 [0.60, 0.70] , [0.15, 0.20]〉
Very good (VG)	〈 [0.70, 0.80] , [0.05, 0.10]〉
Extremely good (EG)	〈 [0.80, 0.90] , [0.05, 0.10]〉
Perfectly good (PG)	〈 [1.00, 1.00] , [0.00, 0.00]〉

Table 4.7

The IVIF evaluation matrix Q⁽¹⁾ given by the first expert

	MT₁	MT₂	MT₃
ML₁	〈 [0.60, 0.70] , [0.15, 0.20]〉	〈 [0.70, 0.80] , [0.05, 0.10]〉	〈 [0.40, 0.50] , [0.35, 0.40]〉
ML₂	〈 [0.60, 0.70] , [0.15, 0.20]〉	〈 [0.40, 0.50] , [0.35, 0.40]〉	〈 [0.60, 0.70] , [0.15, 0.20]〉
ML₃	〈 [0.80, 0.90] , [0.05, 0.10]〉	〈 [0.40, 0.50] , [0.35, 0.40]〉	〈 [1.00, 1.00] , [0.00, 0.00]〉
ML₄	〈 [0.50, 0.60] , [0.25, 0.30]〉	〈 [0.60, 0.70] , [0.15, 0.20]〉	〈 [0.70, 0.80] , [0.05, 0.10]〉
ML₅	〈 [0.70, 0.80] , [0.05, 0.10]〉	〈 [0.15, 0.25] , [0.55, 0.60]〉	〈 [0.80, 0.90] , [0.05, 0.10]〉
	MT₄	MT₅	MT₆
ML₁	〈 [0.50, 0.60] , [0.25, 0.30]〉	〈 [0.30, 0.40] , [0.45, 0.50]〉	〈 [0.50, 0.60] , [0.25, 0.30]〉
ML₂	〈 [0.40, 0.50] , [0.35, 0.40]〉	〈 [0.40, 0.50] , [0.35, 0.40]〉	〈 [0.60, 0.70] , [0.15, 0.20]〉
ML₃	〈 [0.70, 0.80] , [0.05, 0.10]〉	〈 [0.40, 0.50] , [0.35, 0.40]〉	〈 [0.70, 0.80] , [0.05, 0.10]〉
ML₄	〈 [0.15, 0.25] , [0.55, 0.60]〉	〈 [0.60, 0.70] , [0.15, 0.20]〉	〈 [0.00, 0.10] , [0.70, 0.75]〉
ML₅	〈 [0.00, 0.10] , [0.85, 0.90]〉	〈 [0.50, 0.60] , [0.25, 0.30]〉	〈 [0.00, 0.10] , [0.70, 0.75]〉

Table 4.8

The IVIF evaluation matrix Q⁽²⁾ given by the second expert

	MT₁	MT₂	MT₃
ML₁	〈 [0.50, 0.60] , [0.25, 0.30]〉	〈 [0.70, 0.80] , [0.05, 0.10]〉	〈 [0.40, 0.50] , [0.35, 0.40]〉
ML₂	〈 [0.60, 0.70] , [0.15, 0.20]〉	〈 [0.40, 0.50] , [0.35, 0.40]〉	〈 [0.60, 0.70] , [0.15, 0.20]〉
ML₃	〈 [1.00, 1.00] , [0.00, 0.00]〉	〈 [0.60, 0.70] , [0.15, 0.20]〉	〈 [0.80, 0.90] , [0.05, 0.10]〉
ML₄	〈 [0.50, 0.60] , [0.25, 0.30]〉	〈 [0.60, 0.70] , [0.15, 0.20]〉	〈 [0.70, 0.80] , [0.05, 0.10]〉
ML₅	〈 [0.60, 0.70] , [0.15, 0.20]〉	〈 [0.15, 0.25] , [0.55, 0.60]〉	〈 [0.70, 0.80] , [0.05, 0.10]〉
	MT₄	MT₅	MT₆
ML₁	〈 [0.60, 0.70] , [0.15, 0.20]〉	〈 [0.40, 0.50] , [0.35, 0.40]〉	〈 [0.60, 0.70] , [0.15, 0.20]〉
ML₂	〈 [0.50, 0.60] , [0.25, 0.30]〉	〈 [0.40, 0.50] , [0.35, 0.40]〉	〈 [0.60, 0.70] , [0.15, 0.20]〉
ML₃	〈 [0.80, 0.90] , [0.05, 0.10]〉	〈 [0.40, 0.50] , [0.35, 0.40]〉	〈 [0.70, 0.80] , [0.05, 0.10]〉
ML₄	〈 [0.30, 0.40] , [0.45, 0.50]〉	〈 [0.50, 0.60] , [0.25, 0.30]〉	〈 [0.00, 0.10] , [0.70, 0.75]〉
ML₅	〈 [0.15, 0.25] , [0.55, 0.60]〉	〈 [0.50, 0.60] , [0.25, 0.30]〉	〈 [0.15, 0.25] , [0.55, 0.60]〉

Table 4.9

The IVIF evaluation matrix Q⁽³⁾ given by the third expert

	MT₁	MT₂	MT₃
ML₁	〈 [0.60, 0.70] , [0.15, 0.20]〉	〈 [0.60, 0.70] , [0.15, 0.20]〉	〈 [0.40, 0.50] , [0.35, 0.40]〉
ML₂	〈 [0.50, 0.60] , [0.25, 0.30]〉	〈 [0.40, 0.50] , [0.35, 0.40]〉	〈 [0.60, 0.70] , [0.15, 0.20]〉
ML₃	〈 [0.80, 0.90] , [0.05, 0.10]〉	〈 [0.60, 0.70] , [0.15, 0.20]〉	〈 [0.70, 0.80] , [0.05, 0.10]〉
ML₄	〈 [0.50, 0.60] , [0.25, 0.30]〉	〈 [0.50, 0.60] , [0.25, 0.30]〉	〈 [0.60, 0.70] , [0.15, 0.20]〉
ML₅	〈 [0.50, 0.60] , [0.25, 0.30]〉	〈 [0.30, 0.40] , [0.45, 0.50]〉	〈 [0.70, 0.80] , [0.05, 0.10]〉
	MT₄	MT₅	MT₆
ML₁	〈 [0.60, 0.70] , [0.15, 0.20]〉	〈 [0.40, 0.50] , [0.35, 0.40]〉	〈 [0.50, 0.60] , [0.25, 0.30]〉
ML₂	〈 [0.40, 0.50] , [0.35, 0.40]〉	〈 [0.15, 0.25] , [0.55, 0.60]〉	〈 [0.50, 0.60] , [0.25, 0.30]〉
ML₃	〈 [0.70, 0.80] , [0.05, 0.10]〉	〈 [0.50, 0.60] , [0.25, 0.30]〉	〈 [1.00, 1.00] , [0.00, 0.00]〉
ML₄	〈 [0.40, 0.50] , [0.35, 0.40]〉	〈 [0.60, 0.70] , [0.15, 0.20]〉	〈 [0.15, 0.25] , [0.55, 0.60]〉
ML₅	〈 [0.30, 0.40] , [0.45, 0.50]〉	〈 [0.40, 0.50] , [0.35, 0.40]〉	〈 [0.30, 0.40] , [0.45, 0.50]〉

Table 4.10

The IVIF evaluation matrix Q⁽⁴⁾ given by the fourth expert

	MT₁	MT₂	MT₃
ML₁	〈 [0.50, 0.60] , [0.25, 0.30]〉	〈 [0.60, 0.70] , [0.15, 0.20]〉	〈 [0.40, 0.50] , [0.35, 0.40]〉
ML₂	〈 [0.50, 0.60] , [0.25, 0.30]〉	〈 [0.40, 0.50] , [0.35, 0.40]〉	〈 [0.50, 0.60] , [0.25, 0.30]〉
ML₃	〈 [0.80, 0.90] , [0.05, 0.10]〉	〈 [0.60, 0.70] , [0.15, 0.20]〉	〈 [1.00, 1.00] , [0.00, 0.00]〉
ML₄	〈 [0.60, 0.70] , [0.15, 0.20]〉	〈 [0.60, 0.70] , [0.15, 0.20]〉	〈 [0.60, 0.70] , [0.15, 0.20]〉
ML₅	〈 [0.60, 0.70] , [0.15, 0.20]〉	〈 [0.40, 0.50] , [0.35, 0.40]〉	〈 [0.60, 0.70] , [0.15, 0.20]〉
	MT₄	MT₅	MT₆
ML₁	〈 [0.50, 0.60] , [0.25, 0.30]〉	〈 [0.30, 0.40] , [0.45, 0.50]〉	〈 [0.50, 0.60] , [0.25, 0.30]〉
ML₂	〈 [0.60, 0.70] , [0.15, 0.20]〉	〈 [0.40, 0.50] , [0.35, 0.40]〉	〈 [0.60, 0.70] , [0.15, 0.20]〉
ML₃	〈 [0.70, 0.80] , [0.05, 0.10]〉	〈 [0.50, 0.60] , [0.25, 0.30]〉	〈 [0.80, 0.90] , [0.05, 0.10]〉
ML₄	〈 [0.40, 0.50] , [0.35, 0.40]〉	〈 [0.60, 0.70] , [0.15, 0.20]〉	〈 [0.15, 0.25] , [0.55, 0.60]〉
ML₅	〈 [0.40, 0.50] , [0.35, 0.40]〉	〈 [0.60, 0.70] , [0.15, 0.20]〉	〈 [0.15, 0.25] , [0.55, 0.60]〉

Table 4.11

The IVIF evaluation matrix Q⁽⁵⁾ given by the fifth expert

	MT₁	MT₂	MT₃
ML₁	〈 [0.60, 0.70] , [0.15, 0.20]〉	〈 [0.70, 0.80] , [0.05, 0.10]〉	〈 [0.40, 0.50] , [0.35, 0.40]〉
ML₂	〈 [0.70, 0.80] , [0.05, 0.10]〉	〈 [0.40, 0.50] , [0.35, 0.40]〉	〈 [0.60, 0.70] , [0.15, 0.20]〉
ML₃	〈 [0.80, 0.90] , [0.05, 0.10]〉	〈 [0.40, 0.50] , [0.35, 0.40]〉	〈 [1.00, 1.00] , [0.00, 0.00]〉
ML₄	〈 [0.60, 0.70] , [0.15, 0.20]〉	〈 [0.60, 0.70] , [0.15, 0.20]〉	〈 [0.70, 0.80] , [0.05, 0.10]〉
ML₅	〈 [0.70, 0.80] , [0.05, 0.10]〉	〈 [0.15, 0.25] , [0.55, 0.60]〉	〈 [0.80, 0.90] , [0.05, 0.10]〉
	MT₄	MT₅	MT₆
ML₁	〈 [0.50, 0.60] , [0.25, 0.30]〉	〈 [0.30, 0.40] , [0.45, 0.50]〉	〈 [0.50, 0.60] , [0.25, 0.30]〉
ML₂	〈 [0.40, 0.50] , [0.35, 0.40]〉	〈 [0.40, 0.50] , [0.35, 0.40]〉	〈 [0.60, 0.70] , [0.15, 0.20]〉
ML₃	〈 [0.70, 0.80] , [0.05, 0.10]〉	〈 [0.40, 0.50] , [0.35, 0.40]〉	〈 [0.70, 0.80] , [0.05, 0.10]〉
ML₄	〈 [0.40, 0.50] , [0.35, 0.40]〉	〈 [0.60, 0.70] , [0.15, 0.20]〉	〈 [0.00, 0.10] , [0.70, 0.75]〉
ML₅	〈 [0.15, 0.25] , [0.55, 0.60]〉	〈 [0.50, 0.60] , [0.25, 0.30]〉	〈 [0.15, 0.25] , [0.55, 0.60]〉

We use IVIF-CPT-MABAC method to describe the specific process of the GSS.

Table 4.13

Normalized IVIF decision matrix Q′

	MT₁	MT₂	MT₃
ML₁	〈 [0.570, 0.671] , [0.177, 0.228]〉	〈 [0.077, 0.131] , [0.664, 0.766]〉	〈 [0.400, 0.500] , [0.350, 0.400]〉
ML₂	〈 [0.586, 0.688] , [0.150, 0.207]〉	〈 [0.350, 0.400] , [0.400, 0.500]〉	〈 [0.585, 0.686] , [0.163, 0.213]〉
ML₃	〈 [1.000, 1.000] , [0.000, 0.000]〉	〈 [0.220, 0.273] , [0.520, 0.622]〉	〈 [1.000, 1.000] , [0.000, 0.000]〉
ML₄	〈 [0.537, 0.637] , [0.210, 0.261]〉	〈 [0.169, 0.220] , [0.579, 0.679]〉	〈 [0.664, 0.766] , [0.077, 0.131]〉
ML₅	〈 [0.630, 0.733] , [0.103, 0.161]〉	〈 [0.489, 0.539] , [0.231, 0.332]〉	〈 [0.738, 0.844] , [0.060, 0.112]〉
	MT₄	MT₅	MT₆
ML₁	〈 [0.542, 0.642] , [0.205, 0.256]〉	〈 [0.408, 0.458] , [0.341, 0.441]〉	〈 [0.518, 0.618] , [0.230, 0.281]〉
ML₂	〈 [0.454, 0.555] , [0.290, 0.342]〉	〈 [0.388, 0.439] , [0.350, 0.451]〉	〈 [0.579, 0.679] , [0.169, 0.220]〉
ML₃	〈 [0.719, 0.821] , [0.050, 0.100]〉	〈 [0.307, 0.358] , [0.441, 0.542]〉	〈 [1.000, 1.000] , [0.000, 0.000]〉
ML₄	〈 [0.324, 0.426] , [0.412, 0.462]〉	〈 [0.163, 0.213] , [0.585, 0.686]〉	〈 [0.061, 0.162] , [0.637, 0.687]〉
ML₅	〈 [0.197, 0.299] , [0.549, 0.602]〉	〈 [0.249, 0.300] , [0.497, 0.598]〉	〈 [0.151, 0.252] , [0.561, 0.611]〉

Table 4.18

Relative attribute weight matrixI^*

Relative attribute weight	MT₁	MT₂	MT₃	MT₄	MT₅	MT₆
I1j*(ωj)	0.289	0.183	0.316	0.122	0.100	0.312
I2j*(ωj)	0.289	0.198	0.316	0.122	0.100	0.312
I3j*(ωj)	0.287	0.183	0.309	0.122	0.100	0.312
I4j*(ωj)	0.289	0.183	0.309	0.102	0.080	0.319
I5j*(ωj)	0.289	0.198	0.309	0.102	0.080	0.319

Table 4.20

Weighted value function V

Weighted value function	MT₁	MT₂	MT₃	MT₄	MT₅	MT₆
ML₁	–0.084	–0.115	–0.285	0.028	0.022	0.098
ML₂	–0.062	0.044	–0.072	0.010	0.019	0.127
ML₃	0.130	–0.014	0.141	0.060	0.004	0.271
ML₄	–0.121	–0.057	0.030	–0.039	–0.043	–0.327
ML₅	–0.029	0.089	0.054	–0.087	–0.016	–0.236

Table 4.21

Overall distance matrix S^*

Overall distance	ML₁	ML₂	ML₃	ML₄	ML₅
Si*(MLi)	–0.336	0.065	0.592	–0.556	–0.225

Step 1. Aggerate the IVIF decision matrices Q^(c), (c = 1, 2, ⋯ , 5) using Equation (3.7) to get the IVIF decision matrix Q in Table 4.12.

Table 4.12

IVIF decision matrix Q

	MT₁	MT₂	MT₃
ML₁	〈 [0.570, 0.671] , [0.177, 0.228]〉	〈 [0.664, 0.766] , [0.077, 0.131]〉	〈 [0.400, 0.500] , [0.350, 0.400]〉
ML₂	〈 [0.586, 0.688] , [0.150, 0.207]〉	〈 [0.400, 0.500] , [0.350, 0.400]〉	〈 [0.585, 0.686] , [0.163, 0.213]〉
ML₃	〈 [1.000, 1.000] , [0.000, 0.000]〉	〈 [0.520, 0.622] , [0.220, 0.273]〉	〈 [1.000, 1.000] , [0.000, 0.000]〉
ML₄	〈 [0.537, 0.637] , [0.210, 0.261]〉	〈 [0.579, 0.679] , [0.169, 0.220]〉	〈 [0.664, 0.766] , [0.077, 0.131]〉
ML₅	〈 [0.630, 0.733] , [0.103, 0.161]〉	〈 [0.231, 0.332] , [0.489, 0.539]〉	〈 [0.738, 0.844] , [0.060, 0.112]〉
	MT₄	MT₅	MT₆
ML₁	〈 [0.542, 0.642] , [0.205, 0.256]〉	〈 [0.341, 0.441] , [0.408, 0.458]〉	〈 [0.518, 0.618] , [0.230, 0.281]〉
ML₂	〈 [0.454, 0.555] , [0.290, 0.342]〉	〈 [0.350, 0.451] , [0.388, 0.439]〉	〈 [0.579, 0.679] , [0.169, 0.220]〉
ML₃	〈 [0.719, 0.821] , [0.050, 0.100]〉	〈 [0.441, 0.542] , [0.307, 0.358]〉	〈 [1.000, 1.000] , [0.000, 0.000]〉
ML₄	〈 [0.324, 0.426] , [0.412, 0.462]〉	〈 [0.585, 0.686] , [0.163, 0.213]〉	〈 [0.061, 0.162] , [0.637, 0.687]〉
ML₅	〈 [0.197, 0.299] , [0.549, 0.602]〉	〈 [0.497, 0.598] , [0.249, 0.300]〉	〈 [0.151, 0.252] , [0.561, 0.611]〉

Step 2. Transform the IVIF decision matrix Q into normalized IVIF decision matrix Q′ using Equation (3.8)

Step 3. Calculate the initial attribute weight using extended CRITIC method, and the result is shown in Table 4.16

Table 4.16

Original attribute weight ω_j

Attribute weight	MT₁	MT₂	MT₃	MT₄	MT₅	MT₆
ω_j	0.243	0.113	0.283	0.043	0.029	0.288

(i) The correlation coefficient matrix R=[RCjp]s×s,i=1,2,⋯,6 calculated by Equation (3.9) to (3.11) is shown in Table 4.14.

Table 4.14

The correlation coefficient matrix R

Correlation coefficient	MT₁	MT₂	MT₃	MT₄	MT₅	MT₆
R1	1.000	0.074	0.941	0.919	0.394	0.969
R2	0.074	1.000	–0.255	–0.159	–0.219	0.117
R3	0.941	–0.255	1.000	0.950	0.368	0.906
R4	0.919	–0.159	0.950	1.000	0.274	0.958
R5	0.394	–0.219	0.368	0.274	1.000	0.255
R6	0.969	0.117	0.906	0.958	0.255	1.000

(ii) Calculate the volatility index of each attribute using Equation (3.12) to (3.13), and the result is shown in Table 4.15.

Table 4.15

The volatility index

Volatility index	MT₁	MT₂	MT₃	MT₄	MT₅	MT₆
DX_j	0.135	0.062	0.157	0.024	0.016	0.159

(iii) Calculate the original attribute weight using Equation (3.14).

Step 4. Obtain the BAA matrix BAA = [g * _j] _1×s using normalized IVIF matrix Q′ and Equation (3.15), and the result is shown in Table 4.17.

Table 4.17

BAA matrix

BAA	MT₁	MT₂	MT₃
g * _j	〈 [0.647, 0.736] , [0.131, 0.176]〉	〈 [0.218, 0.279] , [0.500, 0.606]〉	〈 [0.648, 0.740] , [0.139, 0.183]〉
BAA	MT₄	MT₅	MT₆
g * _j	〈 [0.408, 0.518] , [0.323, 0.376]〉	〈 [0.288, 0.341] , [0.451, 0.554]〉	〈 [0.308, 0.443] , [0.367, 0.415]〉

Step 5. Transform the original weight ω_j into relative weight Iij*(ωj) using Equation (2.16) and parameter [48]: η = 0.61, τ = 0.69.

Step 6. Obtain the weighted value function V(dM1j(E˜1j′,gj*)) using Equation (3.17), where the distance dM1j(E˜1j′,gj*) between alternative under each attribute and BAA are shown in Table 4.19.

Table 4.19

The distance between alternative and BAA

Distance	MT₁	MT₂	MT₃	MT₄	MT₅	MT₆
dM1j(E˜1j′,gj*)	0.098	0.235	0.353	0.191	0.175	0.269
dM1j(E˜2j′,gj*)	0.070	0.181	0.075	0.061	0.152	0.361
dM1j(E˜3j′,gj*)	0.408	0.021	0.409	0.446	0.024	0.854
dM1j(E˜4j′,gj*)	0.148	0.105	0.070	0.134	0.197	0.408
dM1j(E˜5j′,gj*)	0.030	0.406	0.138	0.333	0.066	0.282

Step 7. Calculate the overall distance Si*(MLi) between alternative and BAA using Equation (3.19)

It follows from the above: S1*(ML1)=-0.336 , S2*(ML2)=0.065 , S3*(ML3)=0.592 , S4*(ML4)=-0.556 , S5*(ML5)=-0.225 .

Step 8. Based on the overall distance, the optimal alternative is selected, and the five alternatives are ranked as follows:

ML3>ML2>ML5>ML1>ML4

4.2Sensitivity analysis

In section 4.1, the influence of CPT has been taken into consideration, and the parameters (η = 0.61, τ = 0.69, α = 0.88, β = 0.88, ρ = 2.25) are referenced to obtain the last decision results. At this point, it is also observed that a large number of parameters are involved in Equation (3.16) and (3.17). Naturally, we consider whether the parameter values will affect the decision results. In order to explore the robustness and effectiveness of IVIF-CPT-MABAC method, then we will discuss the impact of individual parameter changes on our decision results.

4.2.1The sensitivity analysis of weighting parameter η

When τ = 0.69 and the parameter of value functions are constant, and the parameter η in different weighting function get different overall distance and ranking results. The results are shown in Table 4.22

Table 4.22

The corresponding results of different value of parameters η

η	ML₁	ML₂	ML₃	ML₄	ML₅	Ranking results	Optimal alternative
0.61	–0.336	0.065	0.592	–0.556	–0.225	ML₃ > ML₂ > ML₅ > ML₁ > ML₄	ML₃
0.05	–0.484	–0.135	–0.014	–0.586	–0.369	ML₃ > ML₂ > ML₅ > ML₁ > ML₄	ML₃
0.15	–0.467	–0.115	0.031	–0.584	–0.356	ML₃ > ML₂ > ML₅ > ML₁ > ML₄	ML₃
0.25	–0.406	–0.043	0.217	–0.577	–0.306	ML₃ > ML₂ > ML₅ > ML₁ > ML₄	ML₃
0.35	–0.358	0.020	0.402	–0.568	–0.259	ML₃ > ML₂ > ML₅ > ML₁ > ML₄	ML₃
0.45	–0.337	0.055	0.521	–0.561	–0.233	ML₃ > ML₂ > ML₅ > ML₁ > ML₄	ML₃
0.55	–0.334	0.066	0.579	–0.557	–0.225	ML₃ > ML₂ > ML₅ > ML₁ > ML₄	ML₃
0.65	–0.339	0.063	0.595	–0.556	–0.228	ML₃ > ML₂ > ML₅ > ML₁ > ML₄	ML₃
0.75	–0.349	0.053	0.583	–0.556	–0.237	ML₃ > ML₂ > ML₅ > ML₁ > ML₄	ML₃
0.85	–0.360	0.039	0.555	–0.556	–0.249	ML₃ > ML₂ > ML₅ > ML₁ > ML₄	ML₃
0.95	–0.372	0.024	0.518	–0.558	–0.262	ML₃ > ML₂ > ML₅ > ML₁ > ML₄	ML₃

From the results of Table 4.22, Figs. 4.1 and 4.2, the results of IVIF-CPT-MABAC method will fluctuate to some extent with different value of η. With the value of η increase, the change trend of the total distance of all alternatives is to increase first and then decrease, where ML₃ changes the most and ML₄ changes the least. The order of the options remains the same, and the best and worst option is ML₃ and ML₄ respectively.

Fig. 4.1

The overall distance between different parameters η.

Fig. 4.2

The alternative ranking results of different parameters η.

4.2.2The sensitivity analysis of parameter τ in value function

When η = 0.61 and the parameter in value function are constant, then the overall distance and ranking results of alternatives can be obtained by different value of parameter τ in weighting function in Table 4.23.

Table 4.23

The corresponding results of different value of parameter τ

τ	ML₁	ML₂	ML₃	ML₄	ML₅	Ranking results	Optimal alternative
0.69	–0.336	0.065	0.592	–0.556	–0.225	ML₃ > ML₂ > ML₅ > ML₁ > ML₄	ML₃
0.05	0.148	0.200	0.606	0.030	0.143	ML₃ > ML₂ > ML₅ > ML₁ > ML₄	ML₃
0.15	0.114	0.192	0.605	–0.027	0.101	ML₃ > ML₂ > ML₅ > ML₁ > ML₄	ML₃
0.25	–0.032	0.156	0.599	–0.240	–0.051	ML₃ > ML₂ > ML₅ > ML₁ > ML₄	ML₃
0.35	–0.183	0.117	0.593	–0.430	–0.175	ML₃ > ML₂ > ML₅ > ML₁ > ML₄	ML₃
0.45	–0.282	0.088	0.591	–0.534	–0.234	ML₃ > ML₂ > ML₅ > ML₁ > ML₄	ML₃
0.55	–0.329	0.072	0.591	–0.570	–0.246	ML₃ > ML₂ > ML₅ > ML₁ > ML₄	ML₃
0.65	–0.339	0.066	0.592	–0.565	–0.234	ML₃ > ML₂ > ML₅ > ML₁ > ML₄	ML₃
0.75	–0.326	0.066	0.593	–0.538	–0.210	ML₃ > ML₂ > ML₅ > ML₁ > ML₄	ML₃
0.85	–0.300	0.071	0.595	–0.499	–0.181	ML₃ > ML₂ > ML₅ > ML₁ > ML₄	ML₃
0.95	–0.267	0.079	0.597	–0.456	–0.152	ML₃ > ML₂ > ML₅ > ML₁ > ML₄	ML₃

From the results of Table 4.23, Figs. 4.3 and 4.4, the results of IVIF-CPT-MABAC method will fluctuate to some extent with different value of τ. With the value of τ increase, the change trend of the total distance of all alternatives is to decrease first and then increase, where ML₃ changes the most and ML₄ changes the least. Though the order of the options has a little difference when τ = 0.05, 0.15, 0.25, where ML₃ changes the most and ML₄ changes the least. The order of the options has little difference, and the best and worst option is ML₃ and ML₄ respectively.

Fig. 4.3

The overall distance of different parameter τ.

Fig. 4.4

The alternative ranking results of different parameters τ.

4.2.3The sensitivity analysis of parameter α in value function

When β = 0.88, ρ = 2.25 and the parameter in weighting function are constant, then the overall distance and ranking results of alternatives are shown in Table 4.24.

Table 4.24

The corresponding results of different value of parameters α

α	ML₁	ML₂	ML₃	ML₄	ML₅	Ranking results	Optimal alternative
0.88	–0.336	0.065	0.592	–0.556	–0.225	ML₃ > ML₂ > ML₅ > ML₁ > ML₄	ML₃
0.05	0.011	0.540	1.066	–0.315	0.100	ML₃ > ML₂ > ML₅ > ML₁ > ML₄	ML₃
0.15	–0.056	0.441	0.977	–0.378	0.034	ML₃ > ML₂ > ML₅ > ML₁ > ML₄	ML₃
0.25	–0.115	0.359	0.902	–0.427	–0.023	ML₃ > ML₂ > ML₅ > ML₁ > ML₄	ML₃
0.35	–0.165	0.289	0.836	–0.464	–0.070	ML₃ > ML₂ > ML₅ > ML₁ > ML₄	ML₃
0.45	–0.208	0.231	0.779	–0.492	–0.110	ML₃ > ML₂ > ML₅ > ML₁ > ML₄	ML₃
0.55	–0.246	0.182	0.728	–0.514	–0.145	ML₃ > ML₂ > ML₅ > ML₁ > ML₄	ML₃
0.65	–0.278	0.140	0.682	–0.531	–0.174	ML₃ > ML₂ > ML₅ > ML₁ > ML₄	ML₃
0.75	–0.306	0.104	0.640	–0.544	–0.198	ML₃ > ML₂ > ML₅ > ML₁ > ML₄	ML₃
0.85	–0.330	0.074	0.603	–0.554	–0.220	ML₃ > ML₂ > ML₅ > ML₁ > ML₄	ML₃
0.95	–0.350	0.047	0.569	–0.561	–0.238	ML₃ > ML₂ > ML₅ > ML₁ > ML₄	ML₃

From the results of Table 4.24, Figs. 4.5 and 4.6, the results of IVIF-CPT-MABAC method will fluctuate to some extent with different value of α. With the value of α increase, the change trend of the total distance of all alternatives is decrease. The order of the options remains constantly, and the best and worst option is ML₃ and ML₄ respectively.

Fig. 4.5

The overall distance of different parameters α.

Fig. 4.6

The alternative ranking results of different parameters α.

4.2.4The sensitivity analysis of parameter β in value function

When α = 0.88, ρ = 2.25 and the parameter in weighting function are constant, and the overall distance and ranking results of alternatives with different value of parameter β are shown in Table 4.25.

Table 4.25

The corresponding results of different value of parameter β

β	ML₁	ML₂	ML₃	ML₄	ML₅	Ranking results	Optimal alternative
0.88	–0.336	0.065	0.592	–0.556	–0.225	ML₃ > ML₂ > ML₅ > ML₁ > ML₄	ML₃
0.05	–1.489	–0.994	0.267	–1.989	–1.450	ML₃ > ML₂ > ML₅ > ML₁ > ML₄	ML₃
0.15	–1.251	–0.718	0.375	–1.690	-1.148	ML₃ > ML₂ > ML₅ > ML₁ > ML₄	ML₃
0.25	–1.051	–0.506	0.449	–1.440	–0.915	ML₃ > ML₂ > ML₅ > ML₁ > ML₄	ML₃
0.35	–0.883	–0.343	0.499	–1.230	–0.733	ML₃ > ML₂ > ML₅ > ML₁ > ML₄	ML₃
0.45	–0.741	–0.217	0.534	–1.054	–0.589	ML₃ > ML₂ > ML₅ > ML₁ > ML₄	ML₃
0.55	–0.621	–0.121	0.557	–0.905	–0.474	ML₃ > ML₂ > ML₅ > ML₁ > ML₄	ML₃
0.65	–0.518	–0.047	0.573	–0.779	–0.381	ML₃ > ML₂ > ML₅ > ML₁ > ML₄	ML₃
0.75	–0.431	0.010	0.583	–0.672	–0.305	ML₃ > ML₂ > ML₅ > ML₁ > ML₄	ML₃
0.85	–0.357	0.054	0.591	–0.581	–0.242	ML₃ > ML₂ > ML₅ > ML₁ > ML₄	ML₃
0.95	–0.292	0.088	0.596	–0.503	–0.190	ML₃ > ML₂ > ML₅ > ML₁ > ML₄	ML₃

From the results of Table 4.25, Figs. 4.7 and 4.8, the results of IVIF-CPT-MABAC method will fluctuate to some extent with different value of β. With the value of β increase, the change trend of the total distance of all alternatives is increase. The order of the options remains constantly, and the best and worst option is ML₃ and ML₄ respectively.

Fig. 4.7

The overall distance of different value of parameter β.

Fig. 4.8

The alternative ranking results of different parameters β.

4.2.5The sensitivity analysis of parameter ρ in value function

When α = 0.88, β = 0.88 and the parameter in weighting function are constant, and the overall distance and ranking results of alternatives with different value of parameter ρ are shown in Table 4.26.

Table 4.26

The corresponding results of different value of parameter ρ

ρ	ML₁	ML₂	ML₃	ML₄	ML₅	Ranking results	Optimal alternative
2.25	–0.336	0.065	0.592	–0.556	–0.225	ML₃ > ML₂ > ML₅ > ML₁ > ML₄	ML₃
1.55	–0.186	0.107	0.597	–0.374	–0.111	ML₃ > ML₂ > ML₅ > ML₁ > ML₄	ML₃
2.55	–0.401	0.047	0.591	–0.634	–0.275	ML₃ > ML₂ > ML₅ > ML₁ > ML₄	ML₃
3.55	-0.616	–0.013	0.584	–0.895	–0.439	ML₃ > ML₂ > ML₅ > ML₁ > ML₄	ML₃
4.55	–0.831	–0.073	0.578	–1.155	–0.603	ML₃ > ML₂ > ML₅ > ML₁ > ML₄	ML₃
5.55	–1.047	–0.132	0.572	–1.416	–0.767	ML₃ > ML₂ > ML₅ > ML₁ > ML₄	ML₃
6.55	–1.262	–0.192	0.566	–1.676	–0.931	ML₃ > ML₂ > ML₅ > ML₁ > ML₄	ML₃
7.55	–1.477	–0.252	0.560	–1.937	–1.094	ML₃ > ML₂ > ML₅ > ML₁ > ML₄	ML₃
8.55	–1.693	–0.312	0.553	–2.197	–1.258	ML₃ > ML₂ > ML₅ > ML₁ > ML₄	ML₃
9.55	–1.908	–0.372	0.547	–2.457	–1.422	ML₃ > ML₂ > ML₅ > ML₁ > ML₄	ML₃
10.00	–2.005	–0.399	0.545	–2.575	–1.496	ML₃ > ML₂ > ML₅ > ML₁ > ML₄	ML₃

From the results of Table 4.26, Figs. 4.9 and 4.10, the results of IVIF-CPT-MABAC method will fluctuate to some extent with different value of ρ. With the value of ρ increase, the change trend of the total distance of all alternatives is decrease, and alternative ML₄ decreases the most specially. The order of the options remains constantly, and the best and worst option is ML₃ and ML₄ respectively.

Fig. 4.9

The overall distance of different value of parameter ρ.

Fig. 4.10

The alternative ranking results of different parameters ρ.

According to the above observation and analysis results, although the change of parameters in the weight function and value function will cause certain fluctuations to the calculation results of the model, it will not change the optimal solution and the worst solution in the decision results. Therefore, the IVIF-CPT-MABAC method is robust and effective. Meanwhile, when DMs’ psychological preference is not obvious, the difference between the superior and inferior alternatives does not change much; when DM’s psychological preference is obvious, the difference between the superior and inferior alternatives change much. Thus, the IVIF-CPT-MABAC method in this paper is more in line with the realistic decision environment.

4.3Comparison analysis

In this section, in order to prove the effectiveness of the IVIF-CPT-MABAC method, we compare the IVIF-CPT-MABAC method with several classical MADM method based on the same initial matrix. The classical method is listed as follows: IVIFWA operator [17], IVIFWG operator [17], IVIF-TAXONOMY method [78], IVIF-WASPAS method [79], IVIF-TOPSIS method [80] and IVIF-CPT-TODIM method [81]. The specific calculation results are shown in Table 4.27 to 4.33.

Table 4.27

The calculation results of IVIFWA operator

Alternative	IVIFWA	SF (ML_r)	AF (ML_r)	Ranking results
ML₁	〈 [0.461, 0.560] , [0.276, 0.334]〉	0.205	0.816	4
ML₂	〈 [0.551, 0.649] , [0.188, 0.246]〉	0.384	0.817	2
ML₃	〈 [1.000, 1.000] , [0.000, 0.000]〉	1.000	1.000	1
ML₄	〈 [0.427, 0.536] , [0.259, 0.334]〉	0.185	0.778	5
ML₅	〈 [0.533, 0.648] , [0.178, 0.255]〉	0.374	0.807	3

Table 4.28

The calculation results of IVIFWG operator

Alternative	IVIFWG	SF (ML_r)	AF (ML_r)	Ranking results
ML₁	〈 [0.401, 0 . 497] , [0 . 324, 0.391]〉	0.092	0.807	3
ML₂	〈 [0.534, 0.626] , [0 . 212, 0.272]〉	0.338	0.822	2
ML₃	〈 [0 . 699, 0 . 771] , [0 . 117, 0 . 171]〉	0.591	0.879	1
ML₄	〈 [0 . 000, 0 . 367] , [0 . 414, 0 . 480]〉	–0.263	0.630	5
ML₅	〈 [0 . 000, 0 . 479] , [0 . 323, 0 . 386]〉	–0.115	0.594	4

Table 4.29

The calculation results of IVIF-Taxonomy

Alternative	F_i (ML_i)	Ranking results
ML₁	0.472	3
ML₂	0.538	2
ML₃	0.947	1
ML₄	0.468	4
ML₅	0.178	5

Table 4.30

The corresponding calculation results of IVIF-WASPAS

Alternative	SF (ML_i)	AF (ML_i)	Ranking
ML₁	0.204	0.816	4
ML₂	0.346	0.812	2
ML₃	1	1	1
ML₄	0.077	0.764	5
ML₅	0.217	0.793	3

Table 4.31

The corresponding calculation results of IVIF-TOPSIS

Alternative	di+	di-	C_i	Ranking
ML₁	0.071	0.031	0.303	4
ML₂	0.051	0.051	0.502	2
ML₃	0.007	0.112	0.942	1
ML₄	0.084	0.035	0.294	5
ML₅	0.064	0.054	0.456	3

Table 4.32

The original attribute weight ϖ_j

Original	MT₁	MT₂	MT₃	MT₄	MT₅	MT₆
attribute
weight
ϖ_j	0.172	0.162	0.174	0.163	0.159	0.171

Table 4.33

The corresponding calculation results of IVIF-CPT-TODIM

Alternative	Ω (ML_r)	ℏ (ML_r)	Ranking
ML₁	–49.850	0.358	3
ML₂	–28.390	0.696	2
ML₃	–9.075	1	1
ML₄	–72.635	0	5
ML₅	–62.689	0.156	4

4.3.1Compare with IVIFWA operator [17]

Substitute the Table 4.7 to Table 4.11, weighting vector of experts ϑ = (ϑ₁, ϑ₂, ⋯ , ϑ₅) = (0.27, 0.16, 0.23, 0.16, 0.18) and original attribute weight ω = (ω₁, ω₂, ⋯ , ω₆) = (0.243, 0.113, 0.283, 0 . 043, 0 . 0290 . 288) into Equation (2.15), and get the calculation results in Table 4.27.

4.3.2Compare with IVIFWG operator [17]

The IVIFWA and IVIFWG operator are the classical MADM methods. Tables 4.27 and 4.28 show the ranking decision results of IVIFWA operator is ML₃ > ML₂ > ML₅ > ML₁ > ML₄. The decision result is slightly different from the results of improved MABAC method, but it does not affect the selection of the optimal scheme and the worst scheme, so the IVIF-CPT-MABAC method in this paper is effective.

4.3.3Compare with IVIF-Taxonomy method [78]

The IVIF-Taxonomy method mainly classifies and compares alternatives and uses mean and standard deviation to determine the development value of alternatives. The larger the development value, the better the alternative. Substitute the Tables 4.7 to 4.11, weighting vector of experts ϑ = (ϑ₁, ϑ₂, ⋯ , ϑ₅) = (0.27, 0.16, 0.23, 0.16, 0.18) and original attribute weight ω = (ω₁, ω₂, ⋯ , ω₆) = (0.243, 0.113, 0.283, 0 . 043, 0 . 0290 . 288) into Equation (2.16), and get the calculation results in Table 4.29.

The main steps are as follows:

Step 1. Obtain the composite matrix of distance m_pq using Equation (4.1)

(4.1)

mpq=14∑j=1sυj(|μLM(Epj′)-μLM(Eqj′)|+|μRM(Eij′)-μRM(Eqj′)|+|vLN(Eij′)-vLN(Eqj′)|+|vRN(Eij′)-vRN(Eqj′)|),

Step 2. The alternatives are homogenized by formula (4.2)

(4.2)

G=g¯±2Hg

where g¯=1n∑i=1ngi , Hg=1n∑i=1n(gi-g¯) .

Step 3. Determine the alternative and development model using Equation (4.3), and select the optimal alternative in Table 4.29 using Equation (4.4) and Equation (4.5).

(4.3)

Kio=14∑i=1nυj(|μLM(Eij′)-μLM(Ej′∧+)|+|μRM(Eij′)-μRM(Ej′∧+)|+|vLN(Eij′)-vLN(Ej′∧+)|+|vRN(Eij′)-vRN(Ej′∧+)|),

where Ej′∧+=([maxiLNj,maxiLNj], [miniLMj,miniRMj]) means the positive ideal point

(4.4)

K=K¯io+2SKio,i=1,2,⋯,n.

(4.5)

Fi=KioK,i=1,2,⋯,n.

where K¯io and S_{K_io} is the mean and variance of K_io.

Table 4.29 shows that the decision result of IVIF-Taxonomy method is ML₃ > ML₂ > ML₁ > ML₄ > ML₅. The result shows that the optimal alternative is still ML₃, and the IVIF-CPT-MABAC method is effective.

4.3.4Compare with IVIF-WASPAS [79]

The IVIF-WASPAS method, combining with the weighted sum model (WSM) and weighted product model (WPM), ranks the alternative by determining the relative importance of each attribute. the larger the relative importance, the better the alternative. The calculation results obtained by following the main steps are shown in Table 4.30

The specific steps are as follows:

Step 1. Determine the normalize decision matrix Q″(E˜¯ij) by Table (4.12), Equation (4.6) and Equation (4.7).

(4.6)

E˜¯ij=E˜ijmaxiE˜ij=([min(LMij,maxiLMij),min(RMij,maxiRMij)],).

where maxiE˜ij=([maxiLMij, maxiRMij],[miniLNijminiRNij]) .

(4.7)

E˜¯ij=miniE˜ijE˜ij=([min(miniLMij,LMij),min(miniRMij,RMij)],[max(maxiLNij,LNij,max(maxiRNij,RNij)])

where miniE˜ij=([miniLMij,miniRMij], [maxiLNijmaxiRNij]) .

Step 2. Obtain the sum and product of relative importance using Equation (4.8) and Equation (4.9).

(4.8)

ℓ˜j(1)=∑j=1sωj·E˜¯ij,i=1,2,⋯,n.

(4.9)

ℓ˜i(2)=∑j=1s(E˜¯ij)ωj,i=1,2,⋯,n.

Step 3. Calculate the Joint generalized criterion values ℓ˜i=([LM″ij′,RM″′],[LN″ij′,RN″′]) of each alternative using Equation (4.10) and sort the alternatives with score function and accuracy function using Equation (4.11) and Equation (4.12) respectively.

(4.10)

ℓ˜i=λℓ˜j(1)+(1-λ)ℓ˜i(2)

(4.11)

SF(ℓ˜i)=12(LM″ij′-LN″ij′+RM″ij′-RN″ij′)

(4.12)

AF(ℓ˜i)=12(LM″ij′+LN″ij′+RM″ij′+RN″ij′)

Table 4.30 shows that the decision result of IVIF-WASPAS method is ML₃ > ML₂ > ML₅ > ML₁ > ML₄. The result shows that the optimal alternative is still ML₃, and the IVIF-CPT-MABAC method is effective.

4.3.5Compare with IVIF-TOPSIS method [80]

IVIF-TOPSIS method mainly calculates the distance between alternative solutions and positive and negative ideal solutions (PIS and NIS). The smaller the distance between alternative solutions and positive ideal solutions, the larger the distance between alternative solutions and negative ideal solutions, the better the solution. The sorting results in Table 4.31 are obtained by the following steps.

The main steps are as follows:

Step 1. Determine the IVIF PIS E˜ij+=([LMj+,RMj+],[LNj+RNj+]) and IVIF NIS E˜ij-=([LMj-,RMj-],[LNj-RNj-]) using Table (4.12), Equation (4.13) and Equation (4.14).

(4.13)

E˜j+={([maxiLMij,maxiRMij],[miniLNijminiRNij]),MTjisnegativeattribute;([miniLMij,miniRMij],[maxiLNijmaxiRNij]),MTjispositiveattribute.

(4.14)

E˜j-={([miniLMij,miniRMij],[maxiLNijmaxiRNij]),MTjispositiveattribute;([maxiLMij,maxiRMij],[miniLNijminiRNij]),MTjisnegativeattribute.

Step 2. The distance between each alternative between IVIF PIS and IVIF NIS using Equation (4.15) and Equation (4.16).

(4.15)

di+=14n∑j=1sωj{|LMij-LMj+|+|RMij-RMj+|+|LNij-LNj+|+|RNij-RNj+|},

(4.16)

di-=14n∑j=1sωj{|LMij-LMj-|+|RMij-RMj-|+|LNij-LNj-|+|RNij-RNj-|}.

Step 3. Calculate the relative proximity degree between each alternative between each alternative and IVIF PIS. The larger the relative proximity degree, the better the alternative.

(4.17)

Ci=di-di++di-,0⩽Ci⩽1

Table 4.30 shows that the decision result of IVIF-TOPSIS method is ML₃ > ML₂ > ML₅ > ML₁ > ML₄. The result shows that the optimal alternative is still ML₃, and the IVIF-CPT-MABAC method is effective.

4.3.4Compare with IVIF-CPT-TODIM method [81]

IVIF-CPT-TODIM method mainly refers to each other through pairwise alternatives, and uses the overall value to measure the superiority of each alternative compared with other alternatives. The greater the superiority, the better the solution. The main steps are listed as follows:

Step 1. Obtain the original attribute weight using entropy method in Table 4.32.

(4.18)

ej(E˜ij′)=1n∑i=1ncos|LMij′-LNij′|+|RMij′-RNij′|2[2+(1-LMij′-LN′)+(1-RMij′-RN′)]·π,

(4.19)

ϖj=1-ɛj∑j=1s(1-ɛj),

(4.20)

ɛj=ej∑j=1sej,i=1,2,⋯,n,j=1,2,⋯,s.

Step 2. Calculate the percentage weight between ML_r and ML_i using Equation (4.21) and (4.22). (η = 0.61, τ = 0.69)

(4.21)

Θrij(ɛj)={ɛjη(ɛjη+(1-ɛj)η)1η,E˜rj′⩾E˜rj′;ɛjτ(ɛjτ+(1-ɛj)τ)1τ,E˜rj′<E˜rj′.

(4.22)

Θrij*(ɛj)=Θrij(ɛj)max{Θrio(ɛO)|o∈s}.

Step 3. Calculate the Proportional prospect advantage of between ML_r and ML_i under attribute MT_j (α = 0.88, β = 0.88, ρ = 2.25).

(4.23)

℘j(MLr,MLi)= {Θrij*(εj)·(Drij)α∑j=1sΘrij*(εj) , E˜rj′>E˜rj′;0 ,E˜rj′= E˜rj′;−ρ(∑j=1sΘrij*(εj))·(Drij)αΘrij*(εj) ,E˜rj′< E˜rj′.

(4.24)

Drij={14[(LMrj-LMij)2+(RMrj-RMij)2+(LNrj-LNij)2+(RNrj-RNij)2]}12.

Step 4. Calculate the prospect value of ML_r (r = 1, 2, ⋯ , n) using Equation (4.26)

(4.25)

ℏ(MLr)=Ω(MLr)-minr{Ω(MLr)}maxr{Ω(MLr)}-minr{Ω(MLr)};

(4.26)

Ω(MLr)=∑i=1n∑j=1s℘j(MLr,MLi).

Step 5. Rank the alternatives with the calculation results in Step 4. The larger the value of ℏ (ML_r), the better the alternative.

According to Table 4.33, we observe that the decision results of IVIF-CPT-TODIM method and the decision results of improved MABAC method in this paper exchange the order of ML₁ and ML₅, but it does not affect the selection of the optimal scheme and the worst scheme. Therefore, the IVIF-CPT-MABAC method in this manuscript is effective.

Fig. 4.12

The ranking results of different methods.

4.3.7Comprehensive analysis

According to the ranking of the five schemes calculated by the six decision-making methods shows in Table 4.34, Figs. 4.10 and 4.11, we see that although the ranking of the five options is slightly different, the best scheme is always ML₃. Therefore, we can conclude that based on the same initial matrix of this paper, the extended method in this paper is effective. In the same fuzzy environment, different MAGDM methods have their own advantages. IVIFWA operator pays more attention to the overall balance while IVIFWG operator focus on unit, and they have a common shortcoming, that is, they can’t evaluate extreme values. IVIF-TAXONOMY method has less possibility of information distortion in the process of information aggregation. IVIF-WASPAS method is proposed base on IVIFWA and IVIFWG, so this method has the same weakness as them. The IVIF-TOPSIS method and IVIF-CPT-TODIM method have simple decision-making principles and use distance matrix to make decisions, but they are only suitable for conservative decision makers. However, compared with the above decision-making methods, the IVIF-CPT-MABAC method constructed in this paper not only considers the influence of the decision-maker’s psychological factors, but also improves the decision-making method through the cumulative prospect function, so the algorithm logic in this paper is closer to the real decision-making environment. In addition, the improved method proposed in the study better handles the MAGDM problems with IVIF information and attribute indicators with local advantages in the decision-making process, and has strong practicability.

Table 4.34

The ranking results of different methods

Decision-making methods	Alternative order	The optimum	The worst
solution	solution
IVIFWA	ML₃ > ML₂ > ML₅ > ML₁ > ML₄	ML₃	ML₄
IVIFWG	ML₃ > ML₂ > ML₁ > ML₅ > ML₄	ML₃	ML₄
IVIF-TOXONOMY	ML₃ > ML₂ > ML₁ > ML₄ > ML₅	ML₃	ML₅
IVIF-WASPAS	ML₃ > ML₂ > ML₅ > ML₁ > ML₄	ML₃	ML₄
IVIF-TOPSIS	ML₃ > ML₂ > ML₅ > ML₁ > ML₄	ML₃	ML₄
IVIF-CPT-TODIM	ML₃ > ML₂ > ML₁ > ML₅ > ML₄	ML₃	ML₄
IVIF-CPT-MABAC	ML₃ > ML₂ > ML₅ > ML₁ > ML₄	ML₃	ML₄

Fig. 4.11

The score value of different methods.

5Conclusion

In this paper, we study the MABAC method combined with CPT to solve MAGDM problem in the interval intuitionistic fuzzy environment. Firstly, we introduce the basic definition of IVIFS, the comparison formula, the related aggregation operator, and the mixing distance of Hamming and Hausdorff. We know that the calculation formula of MABAC method is simple, the result is stable, the hidden gains and losses are considered, and it is easy to combine with other methods for decision analysis. This method evaluates the alternatives by defining the closeness between the possible solution and the BAA. if the closeness degree is larger, then the alternative is better. However, the calculation of the BAA is greatly affected by the extreme value, so it is more suitable for the situation where the attribute index has local advantages. Therefore, CRITIC method, which comprehensively uses correlation coefficient and standard deviation principle in statistics to determine attribute weight, plays an important role in the improved MABAC method, that is, the larger the standard deviation of indicators with local advantages, the greater the impact of volatility on the overall evaluation value, the greater the weight given to the index, which is just suitable for the decision-making characteristics of MABAC method. Then, we not only integrate the CPT that represents the decision-maker’s risk preference into the classic MABAC decision method, but also use IVIFNs to represent the evaluation value of attributes to jointly construct the IVIF-CPT-MABAC method, so as to retain the collected expert evaluation opinions more clearly and completely, making the decision results more accurate and efficient. Next, we apply the IVIF-CPT-MABAC method to the selection of green suppliers to verify the feasibility of the improved method in this paper. Besides, the sensitivity analysis conducted by controlling the individual changes of each parameter in the model can be seen as simulating the impact of the psychological changes of the decision-maker on the decision results to verify the stability of the construction method in this paper. Finally, compared with other six classical MAGDM methods to verify the effectiveness of the method created in this paper in solving uncertain decision problems.

In future research, we will focus on building new models and functions to determine attribute weights so that attribute weights can dynamically change with data. At the same time, we will explore more applications of the new method in this paper and other more effective solutions of MAGDM. At the same time, with the development of network technology [82], a large number of decision-making methods in network form have emerged, such as microblog, WeChat, QQ and other online voting forms, so DMs are involved in different decision-making complexes [83]. In theory, facing such complex decision-making, we can also expand the research results of this paper to the research direction of complex decision-making with the times.

Acknowledgments

The work was supported by the National Social Science Foundation of China under Grant No. 21XJY003 and Sichuan Province Social Development Key R&D Projects under Grant No. 2023YFS0375.

Compliance with ethical standards

Ethical approval

This article does not contain any studies with human participants or animals performed by any of the authors.

Conflict of interest

The authors declare that they have no conflict of interest.

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