An integrated approach to green supplier selection based on the interval type-2 fuzzy best-worst and extended VIKOR methods

Information Sciences 502 (2019) 394–417

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An integrated approach to green supplier selection based on the interval type-2 fuzzy best-worst and extended VIKOR methods Qun Wu a, Ligang Zhou a,b,∗, Yu Chen a,c, Huayou Chen a a

School of Mathematical Sciences, Anhui University, Hefei, Anhui, 230601, China China Institute of Manufacturing Development, Nanjing University of Information Science and Technology, Nanjing, Jiangsu, 210044, China c School of Mathematics, University of Manchester, Booth Street West, M156PB Manchester, UK b

a r t i c l e

i n f o

Article history: Received 13 November 2018 Revised 9 June 2019 Accepted 16 June 2019 Available online 21 June 2019 Keywords: Multiple criteria group decision making Green supplier selection VIKOR Interval type-2 fuzzy sets Interval type-2 fuzzy best-worst method

a b s t r a c t Green supply chain management, which considers both the environment and supply chain management, is becoming a hot topic in modern operations management. In a manufacturing enterprise, it is crucial for supply chain managers to select an appropriate supplier for their long-term development prospects to and to pursue the business strategy and remain in a competitive position. Green supplier selection (GSS) can be viewed as a multiple-criteria group decision-making (MCGDM) problem that involves many unmeasurable and conflicting criteria. Considering the advantage of interval type-2 fuzzy sets (IT2FSs) in modeling such complexity and uncertainty, this study provides an integrated methodology to address MCGDM problems based on the best-worst method (BWM) and the VlseKriterijumska Optimizacija I Kompromisno Resenje (VIKOR) technique in an interval type-2 fuzzy environment. First, we extend a new decision-making method called the BWM to IT2FSs. Unlike the BWM and fuzzy BWM, our IT2FSs stem from a questionnaire survey and related interval approaches. Second, we develop an integrated model based on interval type-2 fuzzy BWM and VIKOR to solve MCGDM problems. Finally, we present a GSS example to illustrate on the performance of our approach. We also provide sensitivity and comparative analyses and a discussion of the effectiveness and advantages of the proposed method. © 2019 Elsevier Inc. All rights reserved.

1. Introduction As the world becomes increasingly prosperous, people are becoming richer and the standard of living is improving. However, many manufacturing enterprises blindly pursue their own economic interests while ignoring social responsibility and ecological balance. This approach yields many environmental problems, such as contamination of water supplies and noise pollution. Governments worldwide at all levels are taking effective actions to prevent environmental pollution. Among the many policies, one of the most approbatory measures is green production [38], which has led greenness to become a direction of economic development in many ways.

∗

Corresponding author at: School of Mathematical Sciences, Anhui University, Hefei, Anhui, 230601, China. E-mail address: [email protected] (L. Zhou).

https://doi.org/10.1016/j.ins.2019.06.049 0020-0255/© 2019 Elsevier Inc. All rights reserved.

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With respect to the green economy, manufacturing enterprises must be more critical of their production and operations, and they must place much more importance on their production quality, waste elimination, energy recovery, and so on. Green supply chain management (GSCM) [23], which considers both the environment and supply chain management, is an emerging field. GSCM is a sustainable development pattern in the modern manufacturing industry. In GSCM operations, a green supplier focuses strictly on green product design, green packing, resource recycling, and constant green innovation [29]. For supply chain managers in a manufacturing enterprise, finding and selecting a suitable supplier is crucial for pursuing the company’s business strategy, remaining in a competitive position, and ensuring the future of the enterprise [2]. Accordingly, research on green supplier selection (GSS) has significant theoretical importance and practical implications. GSS is becoming a popular new problem in modern operations management. The multiple-criteria decision-making (MCDM) process, in which decision makers (DMs) evaluate various criteria and select the most appropriate alternative(s), is now widely used in many fields. In real GSCM, supply chain managers are required to consider all suppliers with many conflicting criteria and to consider the tradeoffs to select the optimal supplier(s); therefore, GSS is commonly viewed as a multiple-criteria group decision-making (MCGDM) problem. Kuo et al. [16] developed an integrated GSS model by adopting artificial neural network (ANN), data envelopment analysis (DEA), and the analytic network process (ANP). Yazdani et al. [48] focused on a complex proportional assessment method to select a suitable green supplier. They also identified the relationship between issue parameters using the decision-making trial and evaluation laboratory (DEMATEL) and quality function deployment (QFD) methods. Govindan et al. [11] developed an integrated model to select suitable green suppliers based on a revised Simos procedure and the PROMETHEE methods. Considering the complexity for objective features and disturbances from internal or external uncertainty and ambiguity, it is almost impossible for DMs to access information by exact values: expressing preferences via fuzzy information is more realistic. Therefore, many studies based on fuzzy sets (FSs) aimed to solve GSS problems. By extending the Axiomatic Design principles to a fuzzy environment, Kannan et al. [15] proposed the Fuzzy Axiomatic Design decision-making method and applied it to GSS. Bakeshlou et al. [3] presented an integrated MCDM approach to select an appropriate supplier in a green supply chain by using the fuzzy ANP, DEMATEL, and multiobjective mathematical programming methods. Rostamzadeh et al. [34] developed an extended fuzzy VIKOR (VlseKriterijumska Optimizacija I Kompromisno Resenje) method to evaluate the GSS problem. In their study, they express the preferences of DMs in the form of triangular fuzzy numbers (TFNs). Owing to the convenience and flexibility of FSs in representing uncertainty, studies related to the GSS transitioned from precise numbers to a fuzzy environment. In most existing studies, the method used to address with linguistic words (LWs) is to transform them into type-1 fuzzy sets (T1FSs). T1FSs were initially proposed by Zadeh [49], who described the semantic fuzziness in a quantitative way via degree of membership function (MF) values in [0,1]. Computing with words (CWWs) using T1FSs has been studied widely. However, as Mendel and Wu [21], Li et al. [17] claimed, “words mean different things to different people”, so linguistic uncertainties, both to an individual and across a group of subjects, must be captured by an appropriate type of FSs [14]. As an extension of T1FSs, Zadeh [50] proposed type-2 fuzzy sets (T2FSs), which are characterized by two MFs, the primary and secondary MFs. These two MFs provide more freedom and flexibility to DMs to better express their uncertainties. Therefore, T2FSs are more accurate than T1FSs in modeling uncertainty. Although the general T2FSs have a wide range of advantages, they often lead to high computational complexity. As a special case of T2FSs, interval type-2 fuzzy sets (IT2FSs) [20–22,41] are currently the most widely used T2FSs. Because T1FSs do not have enough degrees of freedom, Liu and Mendel [18] presented a type-2 fuzzistics methodology to obtain IT2FSs models for words. In their method, each subject provides the endpoints of an interval associated with a word on a prescribed scale, as in the following example. Example 1. [21] A private citizen has some capital and wishes to make an investment. The citizen selected possible investment areas and investment criteria by which to judge them. The following six words were chosen according to the weight that he or she assigns to each criteria: unimportant, moderately unimportant, more or less unimportant, moderately important, more or less important, and very important. Mendel and his team conducted a word survey and collected data from 40 adult (male and female) subjects. The subjects were told that each of the 6 words describes an interval falling somewhere between 0 and 10. For example, the range “moderately important” might start at 5.3 and end at 7.6. All the subjects are asked where this range would start and where it would stop for each label. Liu and Mendel [18] collected this data and obtained a mathematical model for the footprint of uncertainty (FOU) for these words using an interval approach (IA). In this way, we can easily create a codebook for a new application of CWWs. With low computational complexity, high efficiency in handling uncertainty, and a relatively simple form, IT2FSs have been used in many applications with satisfactory results. Chen and Lee [5] proposed a multiple-attribute group decisionmaking approach using the TOPSIS method and IT2FSs. Wu et al. [44] presented an integrated approach to large group decision-making problems based on an improved interval type-2 fuzzy TOPSIS method and social network analysis. Qin and Liu [27] focused on the combined ranking value of IT2FSs and applied them to handle MAGDM problems. Chen [7] presented a linear programming technique for multidimensional analysis of preference (LINMAP) methodology to handle multiple criteria decision analysis problems with IT2FSs information. The best-worst method (BWM) was proposed by Rezaei [31]. The BWM provides an innovative and simple approach to determining criteria weights in MCDM problems. Compared with the AHP method [35], the BWM is efficient and flexible and has two key advantages: (1) the BWM constructs the comparison relations such that fewer comparisons are needed than in the pairwise comparison matrix in the AHP method, and (2) the criteria weights obtained by the BWM are more consistent with practical cases. Several studies have utilized the BWM for real-world applications. For example, Aboutorab

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et al. [1] proposed an integrated evaluation model for supplier development problems using the BWM and Z-numbers. Rezaei [33] focused on an innovative three-phase methodology to select the most suitable supplier. In their studies, the BWM is introduced in the selection phase. Gupta and Barua [13] developed a hybrid model for what may be perceived as barriers to green innovation using the BWM and fuzzy TOPSIS. The BWM is also beneficial in other areas, such as assessing the performance of smart bike-sharing programs [40], evaluating firms’ research and development [37], measuring scientific output quality [36], and so on [32]. Up to now, although many studies on the BWM and IT2FSs and their applications have been published, almost no attention has been paid to the BWM in more fuzzy environments. The traditional BWM is limited by its use of only crisp values when making comparisons; thus, extending the BWM to more complex fuzzy environments is of theoretical and instrumental value. This study presents an integrated approach to GSS by combining the interval type-2 fuzzy best-worst method (IT2FBWM) and VIKOR methods. In general, the main contributions of this paper are as follows: •

•

•

First, T1FSs may no longer express linguistic information effectively due to the increasing complexity from the GSS process itself and different DMs’ behavioral characteristics. IT2FSs are powerful tools for representing uncertainty and solving this type of problem. Our IT2FSs model stems from a questionnaire survey and related interval approaches [14,18,43] and thus perfectly coincides with practice and can handle uncertainty more effectively. Second, in MCGDM, the weights of DMs and criteria play vital roles in determining the final result. To obtain the DMs’ weights, we design a mathematical programming model based on the IT2FBWM. We also construct an optimal optimization model to obtain the criteria weights, in which the relative importance of each criterion provided by each DM is in the form of linguistic information. The weights of the DMs and criteria we obtain from these two optimal optimization models are all IT2FSs rather than precise values, thereby retaining the inherent linguistic information. Finally, the VIKOR method developed by Opricovic and Tzeng [25] can be applied to handle MCDM problems with immeasurable and conflicting criteria. This method provides a novel decision mechanism in which DMs can balance the maximum group utility and minimum individual regret; thus, the VIKOR-based MCGDM can obtain sets of compromise results rather than a single result. Therefore, the DMs can easily respond to various situations, even if an emergency or accidents occurs.

The rest of this paper proceeds as follows. Section 2 briefly introduces recommended changes to the definitions and notations of IT2FSs and reviews the process of encoding LWs into IT2FSs. Section 3 extends the BWM to IT2FSs and introduces the IT2FBWM. Section 4 integrates the VIKOR and IT2FBWM to address MCGDM problems. Section 5 provides a case study with a GSS example to illustrate the efficiency of the proposed method, and discusses the sensitivity analysis and some comparative analyses to illustrate the advantages of the proposed method. Finally, Section 6 gives some concluding remarks. 2. Preliminaries 2.1. Interval type-2 fuzzy sets Definition 1. [22] Let X be the universe of discourse. A type-2 fuzzy sets (T2FSs) A˜ can be represented by the type-2 membership function μA˜ (x, u ) such that:

A˜ = {(x, u ), μA˜ (x, u )|x ∈ X, u ∈ [0, 1]},

(1)

moreover, T2FSs can be expressed in the following form:

Jx = {(x, u )|u ∈ [0, 1], μA˜ (x, u ) > 0}.

(2)

Definition 2. [22] Let X be the universe of discourse. If all μA˜ (x, u ) = 1, then A˜ is called an interval type-2 fuzzy sets (IT2FSs) and can be expressed as

Ix = {u ∈ [0, 1]|μA˜ (x, u ) = 1}.

(3)

Definition 3. [22] Let X be the universe of discourse. The FOU of IT2FSs A˜ denoted by F OU (A˜ ), can be expressed as

F OU (A˜ ) = {(x, u )|x ∈ X and u ∈ [μA˜ (x ), μ ¯ A˜ (x )]},

(4)

where μA˜ (x ) and μ ¯ A˜ (x ) are defined as

μ¯ A˜ (x ) = sup{u|u ∈ [0, 1], μA˜ (x, u ) > 0},

(5)

μA˜ (x ) = inf{u|u ∈ [0, 1], μA˜ (x, u ) > 0}.

(6)

and

IT2FSs should be called a closed IT2FSs (CIT2FSs) because Ix is a closed interval for every x ∈ X. Throughout this paper, we use IT2FSs to refer to CIT2FSs.

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Fig. 1. An IT2FSs A˜ with geometrical interpretation.

For an IT2FSs A˜ , μA˜ (x ) and μ ¯ A˜ (x ) are the lower MF (LMF) and the upper MF (UMF), respectively. IT2FSs are usually considered in a simplified form, such as a trapezoidal fuzzy set for the UMF and a triangular fuzzy set for the LMF (Fig. 1), such that

A˜ = (A˜U , A˜ L ) = [(aU1 , aU2 , aU3 , aU4 ; hUA˜ ), (aL1 , aL2 , aL3 ; hLA˜ )], where aU1 , aU2 , aU3 , aU4 , hU˜ , aL1 , aL2 , aL3 and hL˜ are all real values, aU1 ≤ aU2 ≤ aU3 ≤ aU4 , aL1 ≤ aL2 ≤ aL3 , aU1 A A hU˜ ≤ 1. In the case of hU˜ = hL˜ = 1, IT2FSs degenerate to normal IT2FSs, and the canonical form A A A

(7) ≤

aL1 , aL3

≤

aU4

and 0 ≤

hL˜ A

≤

of a normal IT2FSs can be

further expressed by

A˜ = (A˜U , A˜ L ) = [(aU1 , aU2 , aU3 , aU4 ), (aL1 , aL2 , aL3 )].

(8)

The corresponding UMF μ ¯ A˜ (x ) and LMF μA˜ (x ) are

μ¯ A˜ (x ) =

⎧ U x−a ⎪ ⎨ aU2 −a1U1 1

⎪ ⎩

aU4 −x aU4 −aU3

 x−aL

1

μA˜ (x ) =

aL2 −aL1 aL3 −x aL3 −aL2

i f aU1 ≤ x ≤ aU2 i f aU2 < x < aU3

(9)

i f aU3 ≤ x ≤ aU4 i f aL1 ≤ x < aL2

(10)

i f aL2 ≤ x ≤ aL3

Mendel and Wu [20] proposed a centroid-based ranking method for IT2FSs that is defined as Definition 4. [20] The centroid CA˜ (x ) of an IT2FSs is the union of the centroids of all its embedded T1FSs c(Ae ), that is

CA˜ = ∪∀ Ae cA˜ (Ae ) = {cl (A˜ ), . . . , cr (A˜ )}, where

cl (A˜ ) = min cA˜ (Ae ) = ∀ Ae

(11)

 N min

∀ θi ∈[μA˜ (xi ),μ¯ A˜ (xi )]

cr (A˜ ) = max cA˜ (Ae ) = ∀ Ae

max

∀ θi ∈[μA˜ (xi ),μ¯ A˜ (xi )]

θ θ

i=1 xi i N i=1 i

 N

θ θ



i=1 xi i N i=1 i

,

(12)

,

(13)

cl (A˜ ) and cr (A˜ ) can be computed by the iterative KM algorithm [24] or EKM algorithm [42]. We can rank A˜ according to the value of C (A˜ ), where

C (A˜ ) =

cl (A˜ ) + cr (A˜ ) . 2

(14)

For any two IT2FSs A˜ and B˜, we have the following order relationships: 1. If C (A˜ ) > C (B˜ ), then A˜  B˜; 2. If C (A˜ ) = C (B˜ ), then A˜ ∼ B˜; 3. If C (A˜ ) < C (B˜ ), then A˜ ≺ B˜. Definition 5. The centroid-based distance measure between any two IT2FSs A˜ and B˜ is

d (A˜ , B˜ ) = |C (A˜ ) − C (B˜ )|. Clearly, the centroid-based distance measure satisfies the following properties:

(15)

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Fig. 2. Data part [43].

P(1). P(2). P(3). P(4).

Reflexivity: d (A˜ , B˜ ) = 0 if and only if A˜ = B˜; Boundedness: 0 ≤ d (A˜ , B˜ ) ≤ max{aU4 , bU4 }; Symmetry: d (A˜ , B˜ ) = d (B˜, A˜ ); Triangle inequality: For any three IT2FSs A˜ , B˜ and C˜, d (A˜ , B˜ ) ≤ d (A˜ , C˜ ) + d (C˜, B˜ ).

2.2. Encoding linguistic words into normal IT2FSs from interval-valued data FSs models for LWs should originate from real life, and the data should be collected from individuals [21,26] because LWs mean different things to different people. We know that an IT2FSs characterized by its FOU can accurately reflect this uncertainty. Thus, the IT2FS is an excellent choice capturing uncertainty. Unexpectedly, few studies have focused on IT2FSs models for words from interval-valued data: we summarize three representative approaches in Table 1. The three abovementioned approaches have advantages and limitations in obtaining effective and reasonable IT2FSs computational models for LWs. We combine the best aspects of each 1. The data part (Fig. 2) we use is exactly the same as in the EIA [43] and the HMA [14]. 2. The FS part (Fig. 3) we use is identical to that in the IA [18] and the EIA [43], except that in the final step, we modify the procedure for computing the LMF of the interior FOUs to handle the non-normal triangle LMF. 3. The location of the apex is determined to be the same as in the EIA, but we set membership grade 1 to the apex to make the LMF a normal triangle. In doing so, we can take less care when addressing the theoretical and computational results and set reasonable FOUs that are not too fat or too thin. We design one questionnaire related to individual preference information, in which we ask a question as in [14,18,43]. For example, on a scale of 1–9 (Table 2), what are the endpoints of an interval that you associate with the importance degree of A over B? Assume that the data intervals [a(i ) , b(i ) ](i = 1, 2, . . . , n ) corresponding to the 9 LWs, confined to [1,9], have been collected. In general, after data processing, only m data intervals are preserved. We use these intervals to generate T1FSs models, as displayed in Fig. 4. Furthermore, we construct an IT2FSs model from the m embedded T1FSs. Table 3 presents the corresponding normal IT2FSs of the 9 LWs. In general, neither the trapezoid nor the triangle is symmetric. Fig. 5 shows the FOUs for all LWs. 2.3. Numerical operations for normal IT2fss Note that we can use the IT2FSs as indicators of uncertainty, and asymmetrical IT2FSs may create significant differences with the BWM. To measure the difference and reflect the multiple relationships among IT2FSs, we provide some numerical operations for normal IT2FSs as follows:

Methology Interval approach (IA) [18]

Enhanced interval approach (EIA) [43]

HMA [14]

Brief description

Advantages

The core idea of the IA is first processing collected data intervals. Second, mapping each subject’s data interval into a type-1 membership function (T1MF). Third, aggregating all the T1FSs to lead to IT2FSs, and finally obtaining the FOU for these words. The trace of the EIA is very similar to the IA. It again consists of a data part and a fuzzy part. More rigorous constraints are added in the data part to obtain surviving intervals. The procedures for computing the LMF in the FS part is improved. The normal IT2FSs models for words are obtained for the first time. The data part is the same as EIA, while in the FS part, the common overlap of data intervals is interpreted as agreement by all of the subjects, and a membership grade of 1 is assigned to the common overlap.

1. Data intervals are collected from subjects, subjects do not need knowledge about FSs; 2. Straightforward mappings from data intervals to FOUs are set, no priori assumption is required; 3. A great deal of data statistics are computed, which increase the accuracy and precision of collected data 1. All of the advantages IA has 2. The data part of the EIA has more strict and reasonable tests than the IA; 3. The FS part of the EIA has an improvement FOUs when computing the LMF 1. Fewer probability assumptions about the intervals are made, which makes the process simpler; 2. Using more information from the data intervals, the common overlaps are addressed seriously here; 3. Membership grade 1 is assigned to the common overlap, which is consistent with our thought.

Limitations 1. The FOUs seem too fat and too wide; 2. The LMFs of the interior FOUs obtained usually have very small height; 3. The non-normal LMF of an interior FOU leads to extra care when dealing with computation. 1. The EIA does not assign a MF value of 1 to the common overlapping data intervals; 2. The non-normal LMF of an interior FOU leads to extra care when dealing with computation. 1. After the data part, no random assumption is made, this might seem too simple for IT2FSs; 2. Data intervals lead to FOUs that are thinner than those from the EIA, which considered less uncertainty in dealing with words.

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Table 1 Summary of existing approaches to encoding words into IT2FSs.

399

400

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Fig. 3. FS part [43].

Fig. 4. Transformation of the data interval [a, b] into the T1FSs [18].

Table 2 Fundamental scale. Linguistic terms

Abbreviations

Explanations

Equal importance Weak importance Moderate importance Moderate plus importance Strong importance Strong plus importance Very strong importance Very, very strong Importance Extreme importance

EI WI MI MP SI SP VS VVS EX

Two activities contribute equally to the objective An activity is favored weakly over another An activity is favored moderately over another An activity is favored moderately plus over another An activity is favored strongly over another An activity is favored strongly plus over another An activity is favored very strongly over another An activity is favored very, very strongly over another An activity is favored extremely over another

Definition 6. Let A˜ = [(aU1 , aU2 , aU3 , aU4 ), (aL1 , aL2 , aL3 )] and B˜ = [(bU1 , bU2 , bU3 , bU4 ), (bL1 , bL2 , bL3 )] be two normal IT2FSs, then some numerical operations for them are (1) Subtraction:

A˜ − B˜ = [(aU1 − bU1 , aU2 − bU2 , aU3 − bU3 , aU4 − bU4 ), (aL1 − bL1 , aL2 − bL2 , aL3 − bL3 )].

Q. Wu, L. Zhou and Y. Chen et al. / Information Sciences 502 (2019) 394–417 Table 3 FOU data for linguistic terms. Words

Normal IT2FSs

Centroids

Mean of centroids

EI WI MI MP SI SP VS VVS EX

[(1.000,1.000,1.000,1.000),(1.000,1.000,1.000)] [(1.00,1.00,1.7184,2.6165),(1.000,1.0734,1.9266)] [(1.4308,2.35,2.80,3.3968),(2.5172,2.6941,3.0828)] [(2.1515,3.00,3.85,4.8107),(3.3550,3.5368,3.8278)] [(3.3101,4.25,5.05,6.0107),(4.4136,4.8900,5.0278)] [(4.6893,5.50,6.20,6.9485),(5.6379,5.8889,6.0621)] [(5.9686,6.750,7.1,8.2314),(6.7172,6.8889,7.1036)] [(7.0136,7.65,8.00,8.7071),(7.5172,7.8125,8.0828)] [(7.0253,8.8624,9.000,9.000),(8.8684,8.9908,9.000)]

[1.000,1.000] [1.3105,1.6489] [2.2339,2.9247] [2.8388,4.1499] [4.0868,5.2602] [5.3207,6.3536] [6.5486,7.4660] [7.5781,8.0816] [7.9099,8.9506]

1.000 1.4797 2.5793 3.4943 4.6735 5.8372 7.0073 7.8299 8.4302

Fig. 5. Normal IT2FSs word models.

(2) Addition:

A˜ + B˜ = [(aU1 + bU1 , aU2 + bU2 , aU3 + bU3 , aU4 + bU4 ), (aL1 + bL1 , aL2 + bL2 , aL3 + bL3 )]. (3) Multiplication:

A˜ × B˜ = [(aU1 bU1 , aU2 bU2 , aU3 bU3 , aU4 bU4 ), (aL1 bL1 , aL2 bL2 , aL3 bL3 )]. (4) Scalar multiplication:

kA˜ = [(kaU1 , kaU2 , kaU3 , kaU4 ), (kaL1 , kaL2 , kaL3 )]. (5) Exponential operation:

(A˜ )k = [( (aU1 )k , (aU2 )k , (aU3 )k , (aU4 )k ), ( (aL1 )k , (aL2 )k , (aL3 )k )]. (6) Division:

A˜ /B˜ = [(aU1 /bU1 , aU2 /bU2 , aU3 /bU3 , aU4 /bU4 ), (aL1 /bL1 , aL2 /bL2 , aL3 /bL3 )].

401

402

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Fig. 6. Reference comparisons [31].

Definition 7. Let A˜ = [(aU1 , aU2 , aU3 , aU4 ), (aL1 , aL2 , aL3 )] and B˜ = [(bU1 , bU2 , bU3 , bU4 ), (bL1 , bL2 , bL3 )] be two normal IT2FSs, then the approximated absolute deviation degree (AADD) between them is

AADD(A˜ , B˜ ) =

3 1 4 |aUi − bUi | + i=1 |aLi − bLi | . i =1 7

(16)

Remark 1. The AADD can be viewed as a tool to reflect the differences among IT2FSs. Remark 2. The AADD is approximated because we consider normal IT2FSs here. Property 1. For any two normal IT2FSs, if AADD(A˜ , B˜ ) → 0, which means there is almost no difference between them, and thus A˜ ≈ B˜, specifically, if AADD(A˜ , B˜ ) = 0, then A˜ = B˜. Property 2. For any two normal IT2FSs, the smaller the AADD(A˜ , B˜ ) is, the smaller the difference between them. Correspondingly, the larger the AADD(A˜ , B˜ ) is, the greater the difference. 3. Interval type-2 fuzzy best-worst method Consider an evaluation problem with n criteria. A general approach to obtain the criteria weights is to construct a preference relation (PR) for these n criteria. The PR is

c1 c2 A= . .. cn

⎛ c1 a11 ⎜a21 ⎜ . ⎝ .. an1

c2 a12 a22 .. . an2

· · · cn⎞ · · · a1n · · · a2n ⎟ , .. ⎟ .. ⎠ . . · · · ann

(17)

where aij represents the preference degree of criterion i over criterion j. The PR A is perfectly consistent if it satisfies ai j = aik × ak j , ∀ i, j, k ∈ N. Unfortunately, inconsistencies often exist in PRs, and expressing the degree of preference is the leading cause of inconsistency. Rezaei [31] noted that when facing a set of criteria, determining the most important and the least important is rather convenient and achievable. Once the best and worst criteria are determined, the linguistic preferences can be viewed in two ways: • •

Best-worst linguistic reference vectors (BWLRVs); Secondary linguistic preference (SLP) elements.

Definition 8. [31] (ai1 , ai2 , . . . , ain ) and (a1 j , a2 j , . . . , an j ) are defined as BWLRVs such that i is the best criteria and j is the worst criteria. Definition 9. [31] A linguistic preference element aij is defined as a SLP if neither i nor j is the best or worst criteria. For a PR with n alternatives, the total number of comparisons is n2 . After removing the diagonal elements, n(n − 1 ) comparisons remain. Considering the reciprocity of a PR, at least n(n − 1 )/2 comparisons are needed. With regard to the BWM, only 2n − 3 comparisons are needed, and the rest are SLPs, as shown in Fig. 6. In fact, we can obtain the SLPs using the BWLRVs. Each SLP aij appears in two relation chains such that aBest,i × ai j = aBest, j , ai j × a j,Worst = ai,Worst . The SLPs act as intermediaries in the comparison chain. Next, we would like to show how to derive the weights based only on the BWLRVs. 3.1. Determine the optimal IT2F criteria weights After obtaining the BWLRVs, they are then transformed into IT2FSs based on Table 3. The obtained IT2F best-to-others (IT2FBO) and IT2F others-to-worst (IT2FOW) vectors are

A˜ B = (A˜ B1 , A˜ B2 , . . . , A˜ Bn ),

(18)

Q. Wu, L. Zhou and Y. Chen et al. / Information Sciences 502 (2019) 394–417

403

A˜W = (A˜ 1W , A˜ 2W , . . . , A˜ nW ),

(19)

clearly, A˜ BB = A˜W W = [(1, 1, 1, 1 ), (1, 1, 1 )]. The definition of a consistent IT2F preference (IT2FP) is as follows: Definition 10. An interval type-2 fuzzy preference A˜ jk is consistent if

A˜ Best, j × A˜ jk = A˜ Best,k , A˜ jk × A˜ k,Worst = A˜ jW , j, k ∈ N.

(20)

˜ = (w ˜ 1, w ˜ 2, . . . , w ˜ n )T . In MCDM, determining the criteria weights is crucial. Assume the optimal IT2F weighting vector is W ˜ B , and that of the worst criteria is w ˜ W . Consider the elements For all criteria weights, the IT2F weight of the best criteria is w ˜ B /w ˜ j = A˜ B j and w ˜ j /w ˜ W = A˜ jW . in the IT2FBO and IT2FOW vectors. If the IT2FP is perfectly consistent, then it should have w In general, perfectly consistent IT2FPs are difficult to obtain. To achieve the highest consistency, a creative solution is to ˜ B /w ˜ j − A˜ B j | and |w ˜ j /w ˜ W − A˜ jW |. After obtaining the IT2F criterion weights, minimize the maximum absolute gaps between |w a normalization process is needed: here, we consider the centroid of the IT2FSs. Based on the above analysis, we construct ˜ ∗ = (w ˜ ∗1 , w ˜ ∗2 , . . . , w ˜ ∗n )T , such that the following constrained optimization model to obtain the optimal IT2F weights W

˜ B /w ˜ j − A˜ B j |, |w ˜ j /w ˜ W − A˜ jW )|} min max{|w j⎧ n ⎪ j=1 C (w˜ j ) = 1

⎪ ⎪ ⎪ U L L U ⎪ ⎨w j1 ≤ w j1 , w j3 ≤ w j4

s.t.

(21)

wL ≤ wL ≤ wL

j1 j2 j3 ⎪ U U U ⎪ w ≤ w ≤ w ≤ wUj4 ⎪ j1 j2 j3 ⎪ ⎪ ⎩ U

w j1 ≥ 0, j = 1, 2, . . . , n

˜ B = [ (w ˜ UB1 , w ˜ UB2 , w ˜ UB3 , w ˜ UB4 ), (w ˜ LB1 , w ˜ LB2 , w ˜ LB3 )], w ˜ j = [ (w ˜ Uj1 , w ˜ Uj2 , w ˜ Uj3 , w ˜ Uj4 ), (w ˜ Lj1 , w ˜ Lj2 , w ˜ Lj3 )], w ), (w˜ L , w˜ L , w˜ L )], A˜ B. j = [(w˜ U , w˜ U , w˜ U , w˜ U ), (w˜ L , w˜ L ,

where ˜U w W4

W1

W2

W3

B, j1

B, j2

B, j3

B, j1

B, j4

˜ W = [ (w ˜U ˜U ˜U w ,w ,w , W1 W2 W3

B, j2

˜ LB, j3 )] and A˜ j,W = [(w ˜ Uj,W 1 , w ˜ Uj,W 2 , w ˜ Uj,W 3 , w ˜ Uj,W 4 ), (w ˜ Lj,W 1 , w ˜ Lj,W 2 , w ˜ Lj,W 3 )]. w To avoid obtaining multiple optimal solutions [32] from model (21), we can minimize the maximum absolute gaps ˜B − w ˜j −w ˜ j × A˜ B j |} and {|w ˜ W × A˜ jW |}. To solve model (21), assume that the maximum absolute gap is δ˜ ∗ = between {|w ∗ ∗ ∗ ∗ ∗ ∗ ∗ [(δ , δ , δ , δ ), (δ , δ , δ )]; then, we can transform Eq. (21) into the following programming model:

min δ ∗ ⎧ U U U ∗ |w˜ B1 − w˜ j1 w˜ B j,1 | ≤ δ , |w˜ UB2 − w˜ Uj2 w˜ UB j,2 | ≤ δ ∗ , |w˜ UB3 − w˜ Uj3 w˜ UB j,3 | ≤ δ ∗ , ⎪ ⎪ ⎪ ⎪ |w˜ UB4 − w˜ Uj4 w˜ UB j,4 | ≤ δ ∗ , |w˜ LB1 − w˜ Lj1 w˜ LB j,1 | ≤ δ ∗ , |w˜ LB2 − w˜ Lj2 w˜ LB j,2 | ≤ δ ∗ , ⎪ ⎪ ⎪ ⎨|w˜ L − w˜ L w˜ L | ≤ δ ∗ , |w˜ U − w˜ U w˜ U | ≤ δ ∗ , |w˜ U − w˜ U w˜ U | ≤ δ ∗ , B3 W 1 jW,1 W 2 jW,2 j3 B j,3 j1 j2 s.t. U U U ∗ U U U ∗ L L ∗ ˜ ˜ ˜W ˜ ˜ ˜ ˜ ˜ ˜L | w − w w | ≤ δ , | w − w w | ≤ δ , | w − w ⎪ 1 w jW,1 | ≤ δ , W 3 jW,3 W 4 jW,4 j1 j3 j4 ⎪ ⎪  n ⎪ L ∗ ∗ ⎪ ˜L ˜L ˜L ˜L ˜ j ) = 1, wUj1 ≤ wLj1 , |w˜ Lj2 − w˜ W j=1 C (w ⎪ 2 w jW,2 | ≤ δ , |w j3 − wW 3 w jW,3 | ≤ δ , ⎪ ⎩wL ≤ wU , wL ≤ wL ≤ wL , wU ≤ wU ≤ wU ≤ wU , wU ≥ 0, j = 1, 2, . . . , n. j3

j4

j1

j2

j3

j1

j2

j3

j4

(22)

j1

The solution space of model (22) is an intersection of linear constraints, one for the sum of the weights and some for the IT2FSs. For a sufficiently a large δ ∗ , the solution space is non-empty; hence, a feasible region must exist. Solving model ˜ 1, w ˜ 2, . . . , w ˜ n )T and δ ∗ . (22), we obtain the optimal weights (w 3.2. Consistency ratio for IT2FBWM The consistency ratio (CR) is a common and effective index to reflect the degree of consistency in PRs. We present a CR to check the reliability of the IT2FBWM. When A˜ B j × A˜ jW = A˜ BW , the IT2FBWM will be inconsistent. To make the equality relation A˜ B j × A˜ jW = A˜ BW true, an IT2FSs δ˜ such that δ˜ = [(δ, δ, δ, δ ), (δ, δ, δ )] is added so that the following equation makes sense:

(A˜ B j − δ˜ ) × (A˜ jW − δ˜ ) = A˜ BW + δ˜ .

(23)

Considering the greatest inequality of (23), where A˜ B j = A˜ jW = A˜ BW , we can rewrite this equation as

(A˜ BW − δ˜ ) × (A˜ BW − δ˜ ) = A˜ BW + δ˜ .

(24)

Eq. (24) can be derived as

δ˜2 − (1∗ + 2A˜ BW )δ˜ + (A˜ 2BW − A˜ BW ) = 0∗ , where

1∗

= [(1, 1, 1, 1 ), (1, 1, 1 )] and

0∗

= [ ( 0 , 0 , 0 , 0 ) , ( 0 , 0 , 0 )] .

(25)

404

Q. Wu, L. Zhou and Y. Chen et al. / Information Sciences 502 (2019) 394–417 Table 4 Consistency index for IT2FBWM. LTs

EI

WI

MI

MP

SI

SP

VS

VVS

EX

Centroids CI

1.0000 0

1.7751 0.1882

3.3551 0.7537

4.3403 1.3038

5.7189 2.0756

6.5797 2.8840

7.3902 3.7304

8.3475 4.3412

8.4302 4.7937

Table 5 Comparison of the best criteria to all the criteria. Best-to-Others

c1

c2

c3

c3

VVS(8)

WI(2)

EI(1)

Table 6 Comparison of all the criteria to the worst criteria. Others-to-Worst

c1

c2

c3

c1

EI(1)

SI(5)

VVS(8)

For A˜ BW , the maximum possible IT2FSs is [(7.0253,8.8624,9,9),(8.8684,8.9908,9)]. Under such circumstances, if we use C (A˜ BW ) to calculate the consistency index, all data affiliated with the IT2FSs A˜ BW can use this CI to maintain the effectiveness and workability of the CR. δ˜ can be also represented by a crisp value δ ; thus, we can change Eq. (25) to

δ 2 − (1 + 2C (A˜ BW ))δ + ( (C (A˜ BW ))2 − C (A˜ BW )) = 0.

(26)

After solving the above equation, we can use different values of C (A˜ BW ) to obtain the smallest consistency and the corresponding maximum possible values δ . These maximum values δ can be considered as the CI (see Table 4). Note that the CI is the maximum deviation value when the IT2FPs have the minimum consistency. Thus, the difference between the optimal solution δ ∗ in (22) and the CI should be as large as possible. At the extreme, when δ ∗ = 0, the deviation in (22) is the minimum value. Hence, the IT2FPs are fully consistent. When δ ∗ = CI, the deviation in (22) is the maximum value; consequently, the IT2FPs have the worst consistency. Therefore, we propose a CR to check the degree of consistency and the reliability of the obtained weights:

CR = δ ∗ /CI,

(27)

where CR ∈ [0, 1], CR → 0 indicates greater consistency, and CR → 1 indicates less consistency. Example 1. Consider a transportation mode selection problem in [31]. The three evaluation criteria are load flexibility, accessibility and cost. First, the best and worst criteria are selected by consensus; then, the BWLRVs are determined on scales of 1–9. Tables 5, 6 show the preference degrees. After completing the LWs comparison among the criteria, we can obtain the IT2FBO vector:



A˜ B =



[(7.0136, 7.6500, 8.0000, 8.7071 ), (7.5172, 7.8125, 8.0828 )], [(1.0 0 0 0, 1.0 0 0 0, 1.7184, 2.6125 ), (1.0 0 0 0, 1.0734, 1.9266 )], , [ ( 1 . 0 0 0 0 , 1 . 0 0 0 0 , 1 . 0 0 0 0 , 1 . 0 0 0 0 ) , ( 1 . 0 0 0 0 , 1 . 0 0 0 0 , 1 . 0 0 0 0 )]

and the IT2FOW vector:



A˜W =



[ ( 1 . 0 0 0 0 , 1 . 0 0 0 0 , 1 . 0 0 0 0 , 1 . 0 0 0 0 ) , ( 1 . 0 0 0 0 , 1 . 0 0 0 0 , 1 . 0 0 0 0 )] , [(3.3101, 4.2500, 5.0500, 6.0107 ), (4.4136, 4.6900, 5.0278 )], . [(7.0136, 7.6500, 8.0000, 8.7071 ), (7.5172, 7.8125, 8.0828 )]

To obtain the optimal IT2F criteria weights, we represent the following constrained optimization model based on the IT2FBWM using concrete numbers, such that

min δ ∗ ⎧ U U |w˜ − 7.0136w˜ 11 | ≤ δ ∗ , |w˜ U32 − 7.6500w˜ U12 | ≤ δ ∗ , |w˜ U33 − 8.0 0 0wU13 | ≤ δ ∗ , ⎪ ⎪ ⎪ U31 ⎪ |w˜ − 8.7071w˜ U14 | ≤ δ ∗ , |w˜ L31 − 7.5172w˜ L11 | ≤ δ ∗ , |w˜ L32 − 7.8125w˜ L12 | ≤ δ ∗ , ⎪ ⎪ ⎪ 34 ⎪ L L ∗ U U ∗ U U ∗ ⎪ ⎪ ⎪|w˜ 33 − 8.0828w˜ 13 | ≤ δ , |w˜ 31 − 1.0 0 0 0w˜ 21 | ≤ δ , |w˜ 32 − 1.0 0 0 0w˜ 22 | ≤ δ , ⎪ U U ∗ U U ∗ L L ∗ ⎪ ⎪ ⎨|w˜ 33 − 1.7184w˜ 23 | ≤ δ , |w˜ 34 − 2.6125w˜ 24 | ≤ δ , |w˜ 31 − 1.0 0 0 0w˜ 21 | ≤ δ , |w˜ L32 − 1.0734w˜ L22 | ≤ δ ∗ , |w˜ L33 − 1.9266w˜ L23 | ≤ δ ∗ , |w˜ U21 − 3.3101w˜ U11 | ≤ δ ∗ , s.t. ⎪ ⎪ |w˜ U22 − 4.2500w˜ U12 | ≤ δ ∗ , |w˜ U23 − 5.0500w˜ U13 | ≤ δ ∗ , |w˜ U24 − 6.0107w˜ U14 | ≤ δ ∗ , ⎪ ⎪ ⎪ ⎪ L L ∗ L L ∗ L L ∗ ⎪ ⎪|w˜ 21 − 4.4136w˜ 11 | ≤ δ , |w˜ 22 − 4.6900w˜ 12 | ≤ δ , |w˜ 23 − 5.0287w˜ 13 | ≤ δ , ⎪  ⎪ ⎪ ⎪ 3j=1 C (w˜ j ) = 1, wUj1 ≤ wLj1 , wLj3 ≤ wUj4 , wLj1 ≤ wLj2 ≤ wLj3 , ⎪ ⎪ ⎩wU ≤ wU ≤ wU ≤ wU , wU ≥ 0, j = 1, 2, . . . , 3. j1

j2

j3

j4

j1

Q. Wu, L. Zhou and Y. Chen et al. / Information Sciences 502 (2019) 394–417

405

The optimal IT2F weights of the three criteria can be obtained, where

˜ 1 = [(0.0750, 0.0750, 0.0762, 0.0762 ), (0.0758, 0.0762, 0.0762 )], w ˜ 2 = [(0.3426, 0.3426, 0.3426, 0.3426 ), (0.3426, 0.3426, 0.3426 )], w ˜ 3 = [(0.4582, 0.4582, 0.7043, 0.7794 ), (0.4582, 0.4833, 0.7318 )]. w The crisp weights of the three criteria are calculated as

˜ 1 ) = 0.0758, C (w ˜ 2 ) = 0.3426, C (w ˜ 3 ) = 0.5816. C (w The optimal δ ∗ is 0.1156, so CR = 0.0266. The CR is very close to 0, indicating high consistency in our IT2FBWM model. In the same case, the CR for the BWM in [31] is 0.058, which is almost 2 times that of IT2FBWM. This difference yields significant advantages because the IT2FBWM not only considers the vague ambiguity of DMs but also leads to a higher CI than the BWM does. The obtained criteria weights of the BWM in [31] are 0.0714, 0.3387, and 0.5899, respectively, while the results of FBWM in [12] are 0.1431, 0.3496 and 0.5073. Although the importance rankings of the three criteria are almost the same for the three methods, a difference exists between the FBWM and IT2FBWM, mainly in the constraints of their optimization models and operational laws for TFNs. The difference is explained as follows: •

•

Guo and Zhao [12], Tian et al. [39] extended the BWM to the fuzzy environment and expressed the LWs in TFNs. In fact, the expression of LWs using symmetrical TFNs is unreasonable because the type-1 triangular fuzzy MFs are obtained by simple symmetrical fuzzification. This process cannot handle uncertainty and fuzziness very well, while our IT2FSs model stems from a questionnaire survey and related interval approaches. Thus, our method is in line with practice and can effectively handle uncertainty. Guo and Zhao [12], Tian et al. [39] constructed similar constrained optimization models, while their programming models of the operations among TFNs collide with their defined operational laws. This practice may change the feasible region. For instance, if the fuzzy comparison itself is fully consistent, then it should satisfy a˜B j × a˜ jW = a˜BW , ∀ j ∈ N, where a˜B j , a˜ jW , a˜BW are all TFNs. Their defined operational laws do not satisfy the property

a˜BW a˜ jW

= a˜B j , ∀ j ∈ N in most cases.

Therefore, their approaches are limited when extending the BWM to fuzzy environments. In summary, our developed IT2FBMW can efficiently handle indeterminate fuzzy information, the IT2F weights obtained by the IT2FBWM are more consistent with practical needs, and the CR for the IT2FBMW to confirm the consistency of preferences is more reliable.

4. Extended VIKOR method for MCGDM problems based on IT2FBWM 4.1. Problem description and weights determination for DMs and criteria Consider the MCGDM problem with m alternatives xi (i = 1, 2, . . . , m ) for selection and k DMs dt (t = 1, 2, . . . , k ) expressing their preferences for n criteria c j ( j = 1, 2, . . . , n ). First, as Fig. 7 illustrates, we should determine the weights of the DMs and criteria. This procedure includes two parts. The first part is determined by the managing director, who empowers the DMs in terms of their knowledge, proficiency, and talent. The steps are as follows: Step 1. Specify the best (i.e., the most authoritative) and the worst (i.e., the least authoritative) DMs according to the managing director. Step 2. Specify the linguistic preference degree of the best DM dB (with the highest expertise) over each DM dt . The linguistic best-to-others vector is

EB = (eB1 , eB2 , . . . ., eBk ).

(28)

Step 3. Specify the linguistic preference degree of the worst DM dW (with the lowest expertise) over each DM dt . The linguistic others-to-worst vector is

EW = (eW 1 , eW 2 , . . . ., eW k ).

(29)

Step 4. Transform the obtained linguistic preferences into IT2FSs based on Table 3. Then obtain two IT2F vectors:

E˜B = (e˜B1 , e˜B2 , . . . ., e˜Bk ), E˜W = (e˜W 1 , e˜W 2 , . . . ., e˜W k ). ˜ = (λ ˜ 1, λ ˜ 2, . . . , λ ˜ )T for DMs by the following model: Step 5. Compute the optimal weights λ k

(30)

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Q. Wu, L. Zhou and Y. Chen et al. / Information Sciences 502 (2019) 394–417

Fig. 7. Procedure of weights determination for DMs and criteria.

˜ t e˜Bt |, |λ ˜ W e˜tW |} ˜B −λ ˜t −λ min max{|λ t⎧ k ˜ ⎪ ⎪ t=1 C (λt ) = 1

⎪ ⎪ L L ⎪ , λt3 ≤ λUt4 ⎨λUt1 ≤ λt1 L L L s.t. λt1 ≤ λt2 ≤ λt3 ⎪ ⎪ ⎪ λU ≤ λUt2 ≤ λUt3 ≤ λUt4 ⎪ ⎪ ⎩ Ut1 λt1 ≥ 0, t = 1, 2, . . . , k

(31)

Eq. (31) can be converted into a crisp equivalently. We ignore the detailed explanations here. The second part is determined by the DMs. In this part, the DMs evaluate the importance degrees of all criteria by LWs (Table 2). The managing director collects comments from each DM, and the individual evaluations are transformed into IT2FSs. The obtained IT2F vectors are

˜ (t ) = (w ˜ 1(t ) , w ˜ 2(t ) , . . . , w ˜ n(t ) ), t = 1, 2, . . . , k. W

(32)

To obtain objective weights for different criteria, an ideal reference point (IRP) is required to achieve more rational and discrepant results. Inspired by the ideal solution in the classical TOPSIS method and the averaging operator, for criteria cj , we ˜ j = [(wUj1 , wUj2 , wUj3 , wUj4 ), (wLj1 , wLj2 , wLj3 )], construct the following single-objective optimization model to determine the IRP w

Q. Wu, L. Zhou and Y. Chen et al. / Information Sciences 502 (2019) 394–417

407

where k 

γ j = 1k AADD(w˜ (jt ) , w˜ j ) ⎧ U t=1L L w j1 ≤ w j1 , w j3 ≤ wUj4 ⎪ ⎪ ⎨ L L L

min

(33)

w j1 ≤ w j2 ≤ w j3

s.t.

⎪ ⎪ ⎩

wUj1 wUj1

≤

wUj2

≤

wUj3

≤

wUj4

≥ 0, j = 1, 2, . . . , n

The objective function of the optimization model aims to minimize the deviation between individual opinion and group opinion in the IT2F environment. In fact, the optimization model above is based on the principle of the least absolute value method; hence, an optimal solution exists. By solving the optimization model (33), we can obtain the unique optimal IT2F ˜ = (w ˜ 1, w ˜ 2, . . . , w ˜ n )T . Then, we can calculate the normalized criteria weights w∗j ( j = 1, 2, . . . , n ) as weights w

˜ j) C (w w∗j = n . ˜ j) C j=1 (w

(34)

4.2. The extended VIKOR method for interval type-2 fuzzy information Opricovic and Tzeng [25] introduced the VIKOR method. Let (ω1 , ω2 , . . . , ωn )T be the weighting vector of the criteria. The VIKOR method uses the following Lp -metric:



n 

L p,i =

j=1

 p 1/p f j∗ − fi j , ωj ∗ − fj − fj

1 ≤ p ≤ ∞, i = 1, 2, . . . , m,

(35)

where f j∗ = max fi j and f j− = min fi j are the positive and negative points, respectively. i

i

Definition 11. The IT2F group utility measure (IT2F-GUM) can be measured based on the following formulation:

Si =

n  j=1

ωj

d ( f j∗ , fi j ) d ( f j∗ , f j− )

.

(36)

Definition 12. The IT2F individual regret measure (IT2F-IRM) can be expressed by the following formulation:

 d ( f j∗ , fi j ) Ri = max ω j , d ( f j∗ , f j− ) j

(37)

The general VIKOR index Qi of alternative xi is

Qi = v

Si − S − Ri − R− + (1 − v ) ∗ , S∗ − S− R − R−

(38)

where S∗ = max{Si }, S− = min{Si }, R∗ = max{Ri }, R− = min{Ri }, and v ∈ [0, 1] denotes the strategy of the maximum group i

i

i

i

utility. In fact, we can treat Ri and Si as “cost” criteria, and the general VIKOR index Qi is based on the following two simple linear normalization functions such that

Qi1 =

Si − min{Si } Si − S − i = , ∗ − S −S max{Si } − min{Si }

(39)

Ri − min{Ri } Ri − R− i = . R∗ − R− max{Ri } − min{Ri }

(40)

i

i

and

Qi2 =

i

i

To better reflect both linear and nonlinear normalization conditions, we consider a power function such that y = xα (α > 0 ). We can use the power function specifically when the preferences are monotonically increasing, linear, nonlinear, convex and concave distributed, and to reflect the following special cases: 1. If α = 1, then the normalization method is linear, which means that the two preference measures are subject to uniform distributions; 2. If α > 1, then the normalization method is convex, which means that the two preference measures located in a rather broad range are always welcome;

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Q. Wu, L. Zhou and Y. Chen et al. / Information Sciences 502 (2019) 394–417

3. If α < 1, then the normalization method is concave, which means that only when the two preference measures are sufficiently small can they approach the ideal points. The two extended indexes QiS and QiR use the following Lα -metric:

QiS = (Qi1 )α and QiR = (Qi2 )α , α > 0.

(41)

Then, we define an extended VIKOR index of alternative xi as

Qi = v · QiS + (1 − v ) · QiR ,

v ∈ [0, 1],

(42)

where v denotes the strategy of the maximum group utility. Combining the derived DMs’ weights based on the IT2FBWM, we propose an extended IT2F-VIKOR method for MCGDM, including the following steps: Step 1. Construct the individual linguistic decision matrix A(t ) = (ai(jt ) )m×n .

Step 2. Transform the individual linguistic decision matrix A(t ) = (ai(jt ) )m×n into IT2F decision matrix A˜ (t ) = (a˜i(jt ) )m×n . Step 3. Determine the weights of the DMs and criteria with models (31) and (33). ˜ t (t = 1, 2, . . . , k ) and criteria w ˜ j ( j = 1, 2, . . . , n ) using the EKM algorithm, Step 4. Calculate the crisp weights of DMs λ denoted as λt∗ (t = 1, 2, . . . , k ) and w∗j ( j = 1, 2, . . . , n ), respectively. Step 5. Aggregate the individual decision matrix A˜ (t ) = (a˜i(jt ) )m×n into a collective decision matrix A˜ = (a˜i j )m×n using the interval type-2 fuzzy weighted average (IT2FWA) operator, where

a˜i j = IT 2F W Aλ∗ (a˜i(j1) , a˜i(j2) , . . . , a˜i(jk ) )  k ∗ (t ) k ∗ (t ) k ∗ (t ) k ∗ (t )  ( t=1 λt a¯ i j1 , t=1 λt a¯ i j2 , t=1 λt a¯ i j3 , t=1 λt a¯ i j4 ; min(h¯ a˜(t ) )), ij = k . t ) k t ) k t ) k t) ( t=1 λt∗ ai(j1 , t=1 λt∗ ai(j2 , t=1 λt∗ ai(j3 , t=1 λt∗ ai(j4 ; min(h (t ) ))

(43)

a˜i j

Step 6. Identify the IT2F positive ideal solution (IT2F-PIS) a˜+ = (a˜+ , a˜+ , . . . , a˜+ n ) and the IT2F negative ideal solution (IT2F1 2 − − ˜ ˜ NIS) a˜− = (a˜− , a , . . . , a ) , where n 1 2



a˜+j

=

max{a˜i j } for benefit criteria c j j

min{a˜i j } j

(44)

for cost criteria c j

 a˜−j

=

min{a˜i j } for benefit criteria c j j

max{a˜i j } j

(45)

for cost criteria c j

Step 7. Calculate the IT2F-GUM and IT2F-IRM of the alternatives. Eqs. (36) and (37) define the IT2F-GUM and IT2F-IRM for alternative xi , respectively. Step 8. Compute IT2F compromise measure Qi for alternative xi . Step 9. Rank the alternatives according to the Q values, which sorts based on the smallest Qi and satisfies the following two conditions. Condition 1: Acceptable advantage:

Q ( x (2 ) ) − Q ( x (1 ) ) ≥

1 , m−1

(46)

where x(2) is with the second-smallest Q value and m is the number of alternatives. Condition 2: Acceptable stability in decision making: The alternatives x(1) should also be the best ranked by Si and/or Ri , which indicates that this compromise solution is stable within a decision-making process. If one of these two conditions is not satisfied, compromise solutions could be obtained: •

If Condition 1 is not satisfied, x(N) is determined by the relationship Q (x(N ) ) − Q (x(1 ) ) < all alternatives x(1 ) , x(2 ) , . . . , x(N ) are compromise solutions.

•

1 m−1

for the maximum N, and

If Condition 2 is not satisfied, then alternatives x(1) and x(2) are both compromise solutions. Step 10. End.

5. Illustrative example To explain some application areas of the proposed method, we consider a typical decision-making problem: the GSS in a GSCM.

Q. Wu, L. Zhou and Y. Chen et al. / Information Sciences 502 (2019) 394–417

409

Table 7 Expertise degrees of the three DMs. Managing director

Best DM

Wost DM

D

d3

d1

Evaluation of expertise degrees of three DMs

EB EW

d1

d2

MP(4) EI(1)

WI(2) MI(2)

d3 EI(1) MI(3)

Table 8 The weight preferences of the criteria by the DMs.

d1 d2 d3

c1

c2

c3

c4

c5

c6

c7

WI VS SI

MP MI MP

VS SI VS

SI SP SI

SP MP MI

MP WI WI

MI SI MP

5.1. Problem description and the assessment criteria for GSS With the continuous development of agriculture and industry, the world’s natural environment is worsening. The green economy provides a new mode for simultaneous economic development and environmental protection. The GSCM concept emerged as a sustainable development pattern in the modern manufacturing industry to reflect the development potential and competitiveness of an enterprise in the long run. GSS is one of the most important activities in GSCM. Consider an electronic enterprise seeking a long-term green supplier partner to purchase electronic assemblies for its new electronic products. The enterprise will assess four potential suppliers xi (i = 1, 2, 3, 4 ). A managing director D forms a group of three DMs d1 , d2 , d3 with different professional skills for the evaluation. During the selection process, seven major perspectives are considered according to green economic aspects: •

•

•

•

•

•

•

Green product innovation (c1 ): Green product innovation aims to address the whole product manufacturing process to reduce material and energy depletion, increase material recovery and recycling, and minimize hazardous materials usage. Environmental regime (c2 ): An environmental regime seeks to apply management techniques to establish an effective environmental production management system. The two most widely used and authoritative standards are ISO 140 0 0 and ISO 14001. Use of green technology (c3 ): The use of green technology (UGT) aims to protect human health and the environment and to promote sustainable economic development. UGT integrates closely with the use of green energy, green manufacturing, green management, and so on. Product quality management (c4 ): This criterion represents the supplier’s ability to control service and product quality. Advanced management ideas, perfect product quality, and good post sale service contribute to product quality and cost reduction, while increasing benefits. Currently, the most widely used standard in product quality management systems is ISO9001. Total green product cost (c5 ): The total green product cost is a measure of the total cost paid by suppliers. Total green product cost accounts for the design, manufacture, packaging, stockpiling, transportation, repair, recycle, and so on throughout the product life cycle. Resource consumption (c6 ): Resource consumption is a measure of the total amount of resources consumed in the production process, including raw and processed materials, and the expenditure of energy, water, and so on. Green process planning has great importance in the process of reducing resource consumption and decreasing waster emission. Environmental pollution of production (c7 ): This feature represents the supplier’s pollution per time unit. Much of the environmental pollution is linked to harmful materials, random discharge of sewage, disorderly emissions, and so on during the production phase.

The managing director formulates the degrees of relative importance of the DMs based on LWs. Table 7 illustrates the LBO and LOW vectors of the DMs’ relative importance degrees. Table 8 shows the estimates of the DMs for criteria c1 to c7 . The DMs are provided with LWs (Table 9) on the seven criteria ci (i = 1, 2, . . . , 7 ) for each alternative xi (i = 1, 2, 3, 4 ). The decision matrices are listed inTables 10–12. To obtain more objective and reasonable evaluation results, we investigate an effective MCGDM approach to evaluating suppliers.

5.2. The evaluation steps Step 1. Derive the weights of the DMs.

410

Q. Wu, L. Zhou and Y. Chen et al. / Information Sciences 502 (2019) 394–417 Table 9 Linguistic terms and their corresponding IT2FSs [21]. Linguistic terms

Corresponding interval type-2 fuzzy sets

Very bad (VB) Bad (B) Somewhat bad (SB) Fair (F) Somewhat Good (SG) Good (G) Very Good (VG)

[(0,0,0.59,3.95), (0,0.09,1.32;1)] [(0.28,2.00,3.00,5.22), (1.79,2.37,2.71;0.48)] [(0.98,2.75,4.00,5.41), (2.79,3.30,3.71;0.42)] [(2.38,4.50,6.00,8.18), (4.79,5.12,5.35;0.27)] [(4.02,5.65,7.00,8.41), (5.89,6.34,6.81;0.40)] [(4.38,6.50,7.75,9.62),(6.79,7.25,7.91;0.47)] [(5.21,8.27,10,10), (7.66,9.82,10;1)]

Table 10 Decision matrix A(1) .

x1 x2 x3 x4

c1

c2

c3

c4

c5

c6

c7

SB SB B G

VB SB B F

F F SG SG

G G SG F

G SB SG SB

F B F B

VB SG VB B

Table 11 Decision matrix A(2) .

x1 x2 x3 x4

c1

c2

c3

c4

c5

c6

c7

SB VB VB B

SG SB B B

SG VG G SG

SG F SG SG

B G F G

VB SB SG F

VG SB F SB

Table 12 Decision matrix A(3) .

x1 x2 x3 x4

c1

c2

c3

c4

c5

c6

c7

VB SB B F

SB B SB F

SG F G SG

SG G SG G

F G F G

G SG SG SB

SG VB SB B

Transform the linguistic importance degrees of the DMs into IT2FSs. The IT2FBO and IT2FOW vectors are



E˜B = and



[(2.1515, 3.0 0 0 0, 3.8500, 4.8107 ), (3.3550, 3.5368, 3.8278 )] [(1.0 0 0 0, 1.0 0 0 0, 1.7184, 2.6165 ), (1.0 0 0 0, 1.0734, 1.9266 )] [ ( 1 . 0 0 0 0 , 1 . 0 0 0 0 , 1 . 0 0 0 0 , 1 . 0 0 0 0 ) , ( 1 . 0 0 0 0 , 1 . 0 0 0 0 , 1 . 0 0 0 0 )]

 E˜W =



[ ( 1 . 0 0 0 0 , 1 . 0 0 0 0 , 1 . 0 0 0 0 , 1 . 0 0 0 0 ) , ( 1 . 0 0 0 0 , 1 . 0 0 0 0 , 1 . 0 0 0 0 )] [(1.0 0 0 0, 1.0 0 0 0, 1.7184, 2.6165 ), (1.0 0 0 0, 1.0734, 1.9266 )] , [(1.4308, 2.3500, 2.80 0 0, 3.3968 ), (2.5172, 2.6941, 3.0828 )]

respectively. Apply the IT2FBWM to construct the mathematical programming model:

δ∗ ⎧|λU − 2.1515λU | ≤ δ ∗ , |λU − 3.0 0 0 0λU | ≤ δ ∗ , |λU − 3.8500λU | ≤ δ ∗ , 31 11 32 12 33 13 ⎪ ⎪ ⎪|λU − 4.8107λU | ≤ δ ∗ , |λL − 3.3550λL | ≤ δ ∗ , |λL − 3.5368λL | ≤ δ ∗ , ⎪ ⎪ 31 11 32 12 34 14 ⎪ ⎪|λL − 3.8278λL | ≤ δ ∗ , |λU − 1.0 0 0 0λU | ≤ δ ∗ , |λU − 1.0 0 0 0λU | ≤ δ ∗ , ⎪ ⎪ 13 31 21 32 22 ⎪ ⎪ U33 ⎪ U ∗ U U ∗ L L ∗ ⎪ ⎨|λ33 − 1.7184λ23 | ≤ δ , |λ34 − 2.6125λ24 | ≤ δ , |λ31 − 1.0 0 0 0λ21 | ≤ δ , |λL32 − 1.0734λL22 | ≤ δ ∗ , |λL33 − 1.9266λL23 | ≤ δ ∗ , |λU21 − 1.4308λU11 | ≤ δ ∗ , s.t. ⎪ ⎪ |λU22 − 2.3500λU12 | ≤ δ ∗ , |λU23 − 2.80 0 0λU13 | ≤ δ ∗ , |λU24 − 3.3968λU14 | ≤ δ ∗ , ⎪ ⎪ ⎪ ⎪ ⎪ |λL21 − 2.5172λL11 | ≤ δ ∗ , |λL22 − 2.6941λL12 | ≤ δ ∗ , |λL23 − 3.0828λL13 | ≤ δ ∗ , ⎪ ⎪ ⎪ 3 ⎪ U L L U L L L ⎪ ⎪ k=1 C (λ˜ k ) = 1, λk1 ≤ λk1 , λk3 ≤ λk4 , λk1 ≤ λk2 ≤ λk3 , ⎩ λUk1 ≤ λUk2 ≤ λUk3 ≤ λUk4 , λUk1 ≥ 0, k = 1, 2, 3. min

Q. Wu, L. Zhou and Y. Chen et al. / Information Sciences 502 (2019) 394–417

411

The optimal normal IT2F weights can be solved by using MATLAB software, where

λ˜ 1 = [(0.1330, 0.1330, 0.1330, 0.1330 ), (0.1330, 0.1330, 0.1330 )]; λ˜ 2 = [(0.3074, 0.3345, 0.3346, 0.3347 ), (0.3290, 0.3341, 0.3346 )]; λ˜ 3 = [(0.4032, 0.4516, 0.6292, 0.7571 ), (0.4446, 0.4545, 0.6263 )], and the optimal CI is 0.1172. Given that CR = 0.0899, the obtained result is acceptable. The centroids of the relative DMs’ weights λt∗ (t = 1, 2, 3 ) are

λ∗1 = 0.1330, λ∗2 = 0.3281, λ∗3 = 0.5389. Step 2. Determine the optimal criteria weights. The optimal criteria weights are computed based on the linguistic preferences provided by the DMs (Table 8) and (Eq. (33). The optimal IT2F criteria weights are

˜ 1 = [(3.4262, 4.0 0 0 0, 4.6228, 5.6182 ), (4.0436, 4.2841, 4.686 )]; w ˜ 2 = [(1.9113, 2.7833, 3.50 0 0, 4.3394 ), (3.0757, 3.2559, 3.5795 )]; w ˜ 3 = [(5.0824, 5.9167, 6.4167, 7.4912 ), (5.9493, 6.2226, 6.4117 )]; w ˜ 4 = [(3.7698, 4.6667, 5.4333, 6.3233 ), (4.8217, 5.2230, 5.3726 )]; w ˜ 5 = [(2.7572, 3.6167, 4.2833, 5.0520 ), (3.8367, 4.0399, 4.3242 )]; w ˜ 6 = [(1.3838, 1.6667, 2.4289, 3.3452 ), (1.7850, 1.8945, 2.5603 )]; w ˜ 7 = [(2.2975, 3.20 0 0, 3.90 0 0, 4.7394 ), (3.4286, 3.7070, 3.9795 )]. w The centroids of the relative criteria weights w∗j ( j = 1, 2, . . . , 7 ) are

w∗1 = 0.1536, w∗2 = 0.1116, w∗3 = 0.2174, w∗4 = 0.1774, w∗5 = 0.1388, w∗6 = 0.0756, w∗7 = 0.1256. Step 3. Aggregate the individual decision matrix. The aggregated IT2F decision matrix A˜ = (a˜i j )m×n is obtained based on the DMs’ weights and the IT2FWA aggregation operator. Alternative x1 is taken as a representative example, with the results determined as

a˜11 = [(0.4519, 1.2680, 2.1624, 4.6232], (1.2865, 1.5216, 1.5701, 2.4220; 0.42 )]; a˜12 = [(1.8471, 3.3357, 4.5308, 6.2001], (3.4360, 3.8585, 3.8705, 4.4092; 0.40 )]; a˜13 = [(3.8019, 5.4971, 6.8670, 8.3794], (5.7437, 6.1777, 6.1777, 6.6158; 0.27 )]; a˜14 = [(4.0679, 5.6500, 7.0998, 8.5709], (6.0097, 6.4610, 6.4610, 6.9563; 0.40 )]; a˜15 = [(1.9570, 3.8327, 5.2485, 7.4003], (4.0717, 4.5010, 4.5010, 4.8243; 0.27 )]; a˜16 = [(2.6769, 3.6433, 5.1681, 7.5682], (4.2962, 4.5880, 4.6175, 5.4073; 0.27 )]; a˜17 = [(3.8758, 5.7582, 7.1318, 8.3385], (5.6874, 6.6386, 6.7096, 7.1265; 0.40 )]. Step 4. Employ Eqs. (44) and (45), the centroids based IT2F-PIS a˜+ = (a˜+ , a˜+ , . . . , a˜+ ), and the IT2F-PIS a˜− = 1 2 7

(a˜− , a˜− , . . . , a˜− ) are 1 2 7

C (a˜+ ) = (4.6116, 4.3765, 6.9233, 6.5331, 4.6116, 3.8090, 2.5335 ), C (a˜− ) = (2.1043, 2.8834, 6.1275, 6.2729, 6.5229, 6.1275, 6.3052 ). Step 5. Calculate Si and Ri (i = 1, 2, 3, 4 ) by Eqs. (36) and (37). The results are computed as

S1 = 0.6695, S2 = 0.7351, S3 = 0.6064, S4 = 0.3252; R1 = 0.2174, R2 = 0.1901, R3 = 0.1774, R4 = 0.1777. Step 6. Compute the Qi (v = 0.5, α = 1 ) of alternative xi (i = 1, 2, 3, 4 ) by Eq. (42). According to the results in Step 5, obtain the Qi of the alternatives as follows:

Q1 = 0.9200, Q2 = 0.6584, Q3 = 0.3431, Q4 = 0.0035. Step 7. Rank the alternatives. We find S4 < S2 < S1 < S3 , R3 ≈ R4 < R2 < R1 and Q4 < Q3 < Q2 < Q1 . Therefore, x4 ranks first in terms of all three values. 1 According to Condition (1), because Q3 − Q4 = 0.3396 ≥ 4−1 , alternative x4 can satisfy the acceptable conditions. Thus, the most desirable alternative is x4 .

412

Q. Wu, L. Zhou and Y. Chen et al. / Information Sciences 502 (2019) 394–417 Table 13 Sensitivity analysis of parameters α and v for the compromise measure.

α

v

Q1

Q2

Q3

Q4

Final ordering

0.5

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

1 0.9917 0.9833 0.9750 0.9666 0.9583 0.9499 0.9416 0.9332 0.9249 0.9165 1 0.9840 0.9680 0.9520 0.9360 0.9200 0.9040 0.8880 0.8720 0.8560 0.8400 1 0.9770 0.9540 0.9310 0.9080 0.8850 0.8620 0.8389 0.8159 0.7929 0.7699

0.5629 0.6066 0.6503 0.6940 0.7377 0.7814 0.8252 0.8689 0.9126 0.9563 1 0.3168 0.3852 0.4535 0.5218 0.5901 0.6584 0.7267 0.7951 0.8634 0.9317 1 0.1784 0.2605 0.3427 0.4248 0.5070 0.5892 0.6713 0.7535 0.8357 0.9178 1.0000

0 0.0828 0.1657 0.2485 0.3313 0.4142 0.4970 0.5798 0.6627 0.7455 0.8283 0 0.0686 0.1372 0.2058 0.2745 0.3431 0.4117 0.4803 0.5489 0.6175 0.6861 0 0.0568 0.1137 0.1705 0.2273 0.2842 0.3410 0.3978 0.4547 0.5115 0.5683

0.0836 0.0752 0.0668 0.0585 0.0501 0.0418 0.0334 0.0251 0.0167 0.0084 0 0.0070 0.0063 0.0056 0.0049 0.0042 0.0035 0.0028 0.0021 0.0014 0.0007 0 0.0006 0.0005 0.0005 0.0004 0.0004 0.0003 0.0002 0.0002 0.0001 0.0001 0.0000

x3 x4 x2 x1 x4 x3 x2 x1 x4 x3 x2 x1 x4 x3 x2 x1 x4 x3 x2 x1 x4 x3 x2 x1 x4 x3 x2 x1 x4 x3 x2 x1 x4 x3 x2 x1 x4 x3 x1 x2 x4 x3 x1 x2 x3 x4 x2 x1 x4 x3 x2 x1 x4 x3 x2 x1 x4 x3 x2 x1 x4 x3 x2 x1 x4 x3 x2 x1 x4 x3 x2 x1 x4 x3 x2 x1 x4 x3 x2 x1 x4 x3 x1 x2 x4 x3 x1 x2 x3 x4 x2 x1 x4 x3 x2 x1 x4 x3 x2 x1 x4 x3 x2 x1 x4 x3 x2 x1 x4 x3 x2 x1 x4 x3 x2 x1 x4 x3 x2 x1 x4 x3 x1 x2 x4 x3 x1 x2 x4 x3 x1 x2

1

1.5

5.3. Sensitivity analysis In this subsection, we discuss how the parameters α and v in the proposed VIKOR index affect the calculation results and final orderings. The value of α in Eq. (41) represents different types of normalization methods, and v in Eq. (42) represents a decision mechanism. In practice, the parameter α is determined by the environment, and v is based on the DMs’ preferences. Here, α is chosen from 0 to 3 in increments of 0.25 to reflect the three normalization methods. We use a value of v from 0 to 1 in increment of 0.1 to analyze the sensitivity. We present three typical results in Table 13 along with their corresponding radar diagrams in Fig. 8. Furthermore, we show that two parameters we change continuously explain the stability of our proposed method, and the 3D graphics are shown in Fig. 9. In Fig. 9, the images with different settings of α and v have a little overlap. This overlap would have only a limited effect on the final ranking orders of the alternatives. However, the best supplier is almost always x4 ; therefore, our proposed method is robust and reliable. Additionally, the rank of x1 improves as v increases, while the rank of x2 degrades as v increases, which reveals that the decision mechanisms affect the ranking. In general, Fig. 9 illustrates the stability of our method in a simple and direct manner.

5.4. Comparative analysis and discussion To illustrate the viability and applicability of our MCGDM method, we provide some comparative analysis with respect to interval type-2 fuzzy TOPSIS (IT2F-TOPSIS) [5] and interval type-2 fuzzy TODIM (IT2F-TODIM) [29]. First, we compare our method to the IT2F-TOPSIS method [5]. In that method, for an IT2FSs A˜ j = [(aUj1 , aUj2 , aUj3 , aUj4 ; hU˜ ), (aLj1 , aLj2 , aLj3 , aLj4 ; hL˜ )], the ranking value Rank(A˜ j ) is Aj

Aj

Rank(A˜ j ) = M1 (A˜Uj ) + M1 (A˜ Lj ) + M2 (A˜Uj ) + M2 (A˜ Lj ) + M3 (A˜Uj ) + M3 (A˜ Lj ) 1 − (S1 (A˜Uj ) + S1 (A˜ Lj ) + S2 (A˜Uj ) + S2 (A˜ Lj ) + S3 (A˜Uj ) + S3 (A˜ Lj ) 4 +S4 (A˜U ) + S4 (A˜ L )) + H1 (A˜U ) + H1 (A˜ L ) + H2 (A˜U ) + H2 (A˜ L ), j

j

j

j

j

j

(47)

Q. Wu, L. Zhou and Y. Chen et al. / Information Sciences 502 (2019) 394–417

Fig. 8. Qi values for discrete settings of parameters α and v.

Fig. 9. Qi values for continuous settings of parameters α and v.

413

414

Q. Wu, L. Zhou and Y. Chen et al. / Information Sciences 502 (2019) 394–417 Table 14 The ranking-based distances for each alternative. Ranking-based distance

x1

x2

x3

x4

D ( xi , a ) D ( xi , a− )

4.1559 1.9882

2.8541 3.2943

3.0211 2.6616

1.9163 3.9809

+

where

  4 1 4

k=1

M p (A˜ ij ) = (aij p + aij ( p+1 ) )/2, 1 ≤ p ≤ 3,

(aijk −

1 4

4

k=1

Sq (A˜ ij ) =

  q+1 1 2

k=q

(aijk −

1 2

q+1 k=q

aijk )2 , 1 ≤ q ≤ 3,

S4 (A˜ ij ) =

aijk )2 , H p (A˜ ij ) denotes the membership value of element aij ( p+1 ) in the trapezoidal MF A˜ ij , 1 ≤ p ≤ 2,

i ∈ {U, L}. Based on the ranking value and related criteria weights, the ranking-based weighted decision matrix is

⎛

1.9342 ⎜2.4691 ˜ Rank(A ) = ⎝ 1.9545 4.3639

2.8113 2.0867 2.1315 3.0052

8.3371 8.6685 9.3718 8.5968

7.1122 6.9532 7.0151 7.1964

3.9406 5.6507 4.6035 5.6507

2.2117 2.3096 2.9030 1.8264

⎞

5.0576 1.8789⎟ . 2.8416 ⎠ 2.2623

Then, the ranking-based IT2F-PIS a+ and IT2F-NIS a− are

a+ = (4.3639, 3.0052, 9.3718, 7.1964, 3.9406, 1.8264, 1.8789 ), a− = (1.9342, 2.0867, 8.3371, 6.9532, 5.6507, 2.9030, 5.0576 ). The geometric distances D(xi , a+ ) between each alternative xi and IT2F-PIS and D(xi , a− ) between each alternative xi and IT2F-NIS for alternative xi (i = 1, 2, 3, 4 ) are shown in Table 14. Finally, the relative closeness coefficient of alternative xi is calculated with respect to the PIS a+ as

RCC (xi ) =

D ( xi , a− ) , i = 1, 2, 3, 4. D ( xi , a+ ) + D ( xi , a− )

(48)

The RCC(xi ) values for alternative xi are

RCC (x1 ) = 0.3236, RCC (x2 ) = 0.5358, RCC (x3 ) = 0.4684, RCC (x4 ) = 0.6750. In light of the values of RCC(xi ), the final ranking is x4 x2 x3 x1 . Observe from Table 13 that the ranking results obtained by the IT2F-TOPSIS method and our method are slightly different, although the optimal choice is the same. Essentially, these two approaches differ in the ranking value for IT2FSs and the decision mechanisms. Compared with the IT2F-TOPSIS method, our method has three advantages: •

•

•

The MCGDM based on the VIKOR method can provide sets of compromise results. Therefore, the DMs can easily respond to various situations, such as emergencies and accidents. While the IT2F-TOPSIS obtains a complete ranking order, our ranking results are flexible. The ranking value for IT2FSs proposed by Chen and Lee [5] increases the real value of an IT2FSs by almost one order of magnitude. For instance, for IT2FSs A˜ = [(1, 1, 1, 1; 1 ), (1, 1, 1, 1; 1 )], using Eq. (47), we have Rank(A˜ ) = 10; if we use the centroid-based ranking value, then C (A˜ ) = 1, which is consistent with our belief. Thus, the centroid-based ranking value in the IT2F-VIKOR method is more reasonable and appropriate. The method proposed in [5] gives DMs the same weight, while in real cases, DMs should be allocated with different weights according to their expertise. Our proposed IT2FBWM enables the managing director to evaluate the expertise of DMs by linguistic judgments, and we construct an optimization model to obtain the individual weights. Furthermore, this optimization model is easy to understand and feasible, which makes the weight allocation to DMs more scientific and reasonable.

Next, we compare our method with the IT2F-TODIM proposed by Qin et al. [29], who proposed a new distance measure for IT2FSs such that

d (A˜ , B˜ ) = |Rd (A˜ , 1˜ ) − Rd (B˜, 1˜ )|,

(49)

where

Rd (A˜ , 1˜ ) = 1 − aL4 − λ(aL4 − aU1 + aU4 − aL4 ) − −(aL4 − aL3 − aL2 + aL1 )) − hLA˜ (aU4 −

1 [hU˜ 2hL˜ hU˜ A A A aU3 − aL4 +

(λ(aL2 − aL1 − aU2 + aU1 ) aL3 )],

(50)

where the parameter λ reflects the attitude of a DM. Notably, the proposed distance function applies only to the universe of discourse X positioned in [0,1]. To overcome this drawback, we use the centroid-based distance measure for IT2FSs in the following discussion. The decision steps are as follows:

Q. Wu, L. Zhou and Y. Chen et al. / Information Sciences 502 (2019) 394–417

415

Table 15 The overall dominance of each alternative.

π (xi )

x1

x2

x3

x4

0.5504

0

0.8593

1

Step 1. Calculate the relative weight wjr of the criteria cj to the reference criteria cr according to the following formula:

w jr =

wj , wr = max{w j }, j = 1, 2, . . . , 7. wr j

(51)

Step 2. Calculate the degree of dominance of each alternative xi over each alternative xk for DM dt under criteria cj using the following expression:

φ (j t ) (xi , xk ) =

⎧ w jr d (a˜i(jt ) ,a˜k(tj) ) ⎪ ⎪ 7 ⎪ ⎪ j=1 w jr ⎨

if C (a˜i(jt ) ) > C (a˜k(tj) ) if C (a˜i(jt ) ) = C (a˜k(tj) )

0

 ⎪ ⎪ (t ) (t ) 7 ⎪ ⎪ ⎩− 1 d(a˜i j ,a˜kwj ) j=1 w jr θ

(52)

if C (a˜i(jt ) ) < C (a˜k(tj) )

jr

where θ > 0 is an attenuation factor of the losses, here we take θ = 1 Step 3. Calculate the global dominance of alternative xi over the other alternatives xk by

δ ( xi , xk ) =

3

t=1

λt∗

7

j=1

φ (j t ) (xi , xk ) .

(53)

Step 4. Calculate the overall value of alternative xi by normalizing the overall dominance:

4

π ( xi ) =

k=1

max{ i

4

δ (xi , xk ) − min{

k=1

4

i

k=1

δ (xi , xk ) } − min{ i

δ (xi , xk )}

4

k=1

δ (xi , xk )}

, i = 1, 2, 3, 4.

(54)

The final results are shown in Table 15. Step 5. Rank all the alternatives by the overall value π (xi ) of alternative xi . The larger π (xi ) is, the better alternative xi is. From Table 15, we have x4 x3 x1 x2 . As shown in Table 13, the final ranking order has some similarities between the IT2-TODIM and our method. Compared to the IT2F-TODIM, our method has the following advantages: •

•

•

The two approaches differ in the ranking value for IT2FSs. The ranking-based distance function proposed by Qin et al. [29] applies only to the universe of discourse X positioned in [0,1], which may lead to an indistinguishable problem when used to measure the distance among IT2FSs. Using the centroid to rank IT2FSs and the centroid-based ranking value to determine the distances between IT2FSs is always effective. In [29], although linguistic preferences are also expressed by IT2FSs, the IT2FSs are in the form of symmetrical TFNs. The same transformation method is used in [5]. This practice is improper to some extent. By contrast, our IT2FSs model is derived from a questionnaire survey and related interval approaches, and neither the UMF nor the LMF is symmetrical; thus, they accord well with practice and can effectively handle uncertainty. The TODIM method itself is vulnerable to two paradoxes affecting the weight of the model [19]. As Llamazares [19] noted, if one criteria weight approaches zero, the paradox appears. Also, the TODIM method is based on pairwise comparisons and thus may suffer the rank-reversal phenomenon when some alternatives are added or deleted [19]. Thus, much room for improvement remains.

5.5. Further discussion of complex and dynamic decision making models In recent years, with the incoming wave of big data, greater uncertainty and complexity are emerging in decision-making problems. Decision-making problems with large-scale DMs, large-scale attributes [45], individual sets of attributes, individual sets of alternatives [9], strategic weight manipulation [10], and so on, make the solutions more complex and difficult. As a comparison with existing complex and dynamic decision-making models, we discuss some desirable research directions and highlight the features of our proposed IT2FSs model. •

Because linguistic words mean different things to different people, a codebook to model LWs with IT2FSs can more accurately represent the subjective preference of an individual and group in an uncertain environment. As decisionmaking processes become more complex and dynamic, different codebooks, such as 15-word, 11-word and 32-word codebooks [21], can make the transformation between complex qualitative linguistic information and quantitative IT2FSs more efficient and flexible.

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In real life, a DM often strategically sets attribute weights to obtain her/his desired ranking of alternatives [10]. This phenomenon probably does not help the company in the long run. Clearly, a consensus reaching process is often necessary to achieve a general consensus on the selected alternatives [9]. Compared with the pairwise comparison matrix in PRs [46,47], the BWM first identifies the best and worst criteria, and then conducts pairwise comparisons between each of these two criteria (best and worst) and the other criteria, thereby simplifying the computations among alternatives. The final weights derived from BWM are also relatively reliable compared to those of these methods.

6. Conclusion With fierce market competition, an increasing number of manufacturing enterprises are willing to collaborate with green suppliers; therefore, the selection of an appropriate green supplier is of utmost importance. The application of fuzzy tools in the process of GSS is developing rapidly. Because DMs tend to express their uncertain opinions in linguistic terms, a suitable linguistic codebook in the form of IT2FSs is constructed through a questionnaire survey [21]. Because of their low computational complexity and high efficiency, the IT2FSs are more accurate than T1FSs in modeling uncertainty. Although many fuzzy MCDM methods have been applied to the GSS problem, those methods cannot obtain compromise solutions. In this work, we focused on the MCGDM in an IT2FSs environment for GSS using the BWM and VIKOR methods. The BWM is a recent, effective MCDM method that constructs a comparison system in a structured manner and reduces inconsistency. The structured comparison is more consistent with rational human thinking. We extend the BWM to IT2FSs and develop the IT2FBWM to compute the DMs’ weights. The proposed IT2BWM requires fewer pairwise comparisons than does fuzzy AHP [4] and obtains more reliable weights than do fuzzy BWM [12] and BWM [31]. Although the mathematical programming solution process is tedious, MATLAB can be used to efficiently complete this complex task. Furthermore, in combination with a well-known technique called VIKOR, we introduced a hybrid MCGDM for GSS. One practical advantage of the proposed method is that it provides a new perspective of research on linguistic decision-making by using IT2FSs rather than T1FSs. Additionally, the proposed VIKOR-based method can provide sets of compromise results rather than a single result, which also improves the flexibility and convenience of the decision-making process. Future research could extend the application scope of the BWM to several fuzzy environments. Additionally, the IT2FBWM can be combined with other MCDM methods, such as TOPSIS [44] and LINMAP [30], to solve GSS problems. Finally, our proposed model can perform very well in many other fields, such as hotel location selection [8], E-commerce service [45], preference relations [46,47], and so on [6,28]. Declaration of Competing Interest The authors declare that they have no competing financial interest statement. Acknowledgements The authors are thankful to the Editor in Chief, Professor Witold Pedrycz, associate editor and reviewers for their valuable comments and constructive suggestions with regard to this paper. 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