A q-Rung orthopair fuzzy generalized TODIM method for prioritizing barriers to sustainable food consumption and production

Abstract

Sustainable food consumption and production (SFCP) has become increasingly significant for creating new value, reducing costs, and reducing greenhouse gas emissions. However, there are some challenges and barriers to implementing SFCP in practice. Moreover, current methods for prioritizing barriers to SFCP seldom consider the behavioral preference of experts and interactions among factors, especially with q-Rung orthopair fuzzy set (q-ROFS)-based information. Thus, this study aims to construct a hybrid q-ROFS-based framework for ranking these barriers. First, the q-ROFS is introduced to express the experts’ uncertain information. Then, the q-ROF- CRITIC (CRiteria importance through intercriteria correlation) method is utilized to determine criteria weights considering the interrelations among barriers. Next, the q-ROF generalized TODIM method is built to rank the barriers to SFCP by considering the impact of experts’ behavioral preferences. Finally, a numerical case of barriers analysis for SFCP is organized to display the application procedures of the constructed ranking method. The result indicates that the top-priority set is education and culture (a₄), with the most significant overall dominance value (0.839). Further, a comparison exploration is given to demonstrate the preponderances of the present barriers ranking method. The outcomes demonstrate that the proposed ranking method can provide a synthetic and reliable framework to handle the prioritizing issue for the barriers to SFCP within a complex and uncertain context.

Keywords

Sustainable food consumption and production q-Rung orthopair fuzzy set generalized TODIM method CRITIC approach

1 Introduction

In the past few years, regional conflicts, extreme weather events, the COVID-19 pandemic, and other emerging crises further exacerbated the global situation. The report shows that nearly one-third of the world’s population cannot obtain good grains and nutrition, while almost one-third of all the grains produced are wasted [1]. Further, previous studies showed that annual food waste in Europe varied from 110*10⁶ tonnes to 209*10⁶ tonnes [2], that in China was around 600*10⁶ tonnes [3], and India discarded about 6.24 35*10⁶ tonnes per year [4]. Approximately one-third of all grains harvested yearly, or 1.3 billion tons worth around $1 trillion, ultimately spoil or go to waste in the rubbish due to inefficient transportation, harvesting techniques, etc. [5]. Hence, adopting food waste management and sustainable food consumption and production(SFCP)has become helpful in mitigating food and waste [6]. However, the implementation of SFCP should overcome many barriers in practice. In such cases, some research has been conducted to mitigate food and waste to explore the obstacles of SFCP. For example, Yuan [7] discussed barriers to SFCP and identified the potential challenges involving the inadequate regulatory environment, uncoordinated stakeholder involvement, and insufficient attention to waste management. Ranta, et al. [8] identified cognitive and cultural impediments to increased reuse. However, no research has been done to rank barriers to SFCP, especially considering the expert’s bounded rational behavior. Therefore, it is essential to create a behavior decision-making technique to enhance the performance of previous analysis methods for barriers to SFCP.

The prioritization of barriers to SFCP is indeed a multiple-criteria decision problem. Further, this prioritizing process should consider multiple elements such as cost management, technology, regulations, and education. However, these barriers are qualitative and heavily rely on the experiences and opinions of experts because of the constraints of human cognition and the inherent fuzziness of human thought. Thus, it is insufficient to address the prioritization of barriers to SFCP using accurate data [9, 10]. For this, the fuzzy sets have been incorporated into the prioritization of barriers, such as the intuitionistic fuzzy set (IFS) [11], the Pythagorean fuzzy set (PFS) [12], the hesitant fuzzy set (HFS) [13], and the q-Rung orthopair fuzzy set (q-ROFS) [14]. The q-ROFS is a more flexible and broader range of uncertain information expression means than other fuzzy sets listed above [15]. Moreover, the IFS and PFS are two unique types of q-ROFS [16]. Furthermore, the q-ROFS has been widely employed in current literature to tackle uncertainty expression problems in multi-criteria decision-making (MCDM) issues [17–22]. For example, Alkan and Kahraman [23] introduced a q-ROFS-based decision model for assessing performance. Deveci, et al. [24] reported an integrated q-ROF selection framework mining sites. A combined q-ROFS-based evaluation approach was proposed by Mishra, et al. [25] to determine suitable waste disposal means. Mishra, et al. [26] designed a q-ROF-based decision method for provider selection. Consequently, the q-ROFS is an appropriate means for depicting the experts’ uncertain judgment information in prioritizing barriers to SFCP.

As described above, implementing barriers prioritization to SFCP requires the involvement of MCDM methodology [6, 9, 27]. Besides, the identification and prioritization of barriers to SFCP may be influenced by the behavioral preferences of experts, especially the bounded rational behavior [28, 29]. Hence, behavioral MCDM techniques are the considerable means of resolving the barriers ranking issue. Furthermore, the TODIM approach, as a classical behavioral modeling technique, has been widely applied to tackle diverse ranking issues. The conventional TODIM method has been successfully adopted for various prioritization problems [30–34]; however, it also has some drawbacks, like relative weights calculation [35] and inconsistent ranking results [36]. The generalized TODIM, proposed by Llamazares [37], is a modified version of the classical TODIM method. This method can overcome the limitations of the classical TODIM method. Further, the generalized TODIM method has been successfully applied to tackle options prioritization problems. For example, Wang, et al. [36] established a hybrid generalized TODIM model to identify the most severe failure mode. Alattas and Wu [38] reported an HFS-based generalized TODIM model for ranking the medical industry’s barriers to the Internet of Things. A portfolio-determining model based on generalized TODIM is established by Wu, et al. [39] for investors to choose stocks. Xiao, et al. [40] proposed a Z-fuzzy cloud-based generalized TODIM to rank failures. These extant studies show the practicability of the generalized TODIM model. Thus, we select the generalized TODIM to prioritize the adoption barriers to SFCP.

As discussion listed above, the motivations of this paper are summarized as follows:

Previous studies utilize the triangular fuzzy sets to model the uncertain evaluation information for ranking barriers to SFCP [6, 41], which are insufficient to capture the non-membership degree involved in uncertain information. The q-ROFS can fully overcome these limitations; thus, this study adopts the q-ROFS to transform experts’ uncertain and subjective rating data.

Although the existing method for determining the weights of barriers to SFCP has considered the inter-dependences between each pair, it can not fully capture the inter-correlation relationships among these barriers. The criteria importance through intercriteria correlation (CRITIC) method is a suitable tool to overcome this limitation. Thus, the CRITIC method is integrated with q-ROFS to assign the weights of barriers.

The generalized TODIM method has been extended to the different fuzzy environments [35, 42]; however, these versions are incapable of handling the barriers ranking problem with q-ROF information. Therefore, it is necessary to integrate the generalized TODIM method with q-ROFS for prioritizing the barriers to SFCP.

Based on the above challenges, the current research proposes an integrated q-ROF environment-based MCDM approach that ranks barriers to SFCP. This hybrid approach combines the q-ROFS, CRITIC method, and generalized TODIM method for prioritizing barriers of SFCP under complex uncertainty. The contributions and innovativeness of this paper are summarized as follows:

The q-ROFS-based ranking method provides a new way to address the complex uncertainty expression problem in the current literature addressing the barriers to SFCP. In addition, this method gives an approach to generating inaccurate data in the evaluation using q-ROFNs.

In the q-ROFS-based ranking method, a new weights calculation method can reflect the inter-correlation relationships among barriers to SFCP within the q-ROF environment. The q-ROF-CRITIC method is the first time integrated with the barriers ranking problem.

In the q-ROF-based ranking method, the q-ROF-generalized TODIM method is the first time constructed to analyze and prioritize the barriers to SFCP. This method can demonstrate the impact of an expert’s preference on the ranking result of barriers to SFCP.

A case study of barriers analysis for SFCP is organized to demonstrate the practicability of the proposed q-ROF-CRITIC-generalized TODIM method.

The structure of this study is arranged as follows: the preliminaries are provided in Section 2. Section 3 presents the calculation steps of the extended q-ROF-CRITIC-generalized TODIM method. Section 4 deliberates a case study of prioritizing barriers to SFCP, including the comparative analysis. Finally, section 5 displays the conclusion and future directions.

2 Preliminaries

This section briefly reviews the conceptions of q-ROFS and its basic algorithms.

Definition 1. [36] Let Y be a universe of discourse, then the q-ROFS Z is displayed in the following form: $Z (y) = {〈 α_{z} (y), β_{z} (y) 〉 | y \in Y}$ (1)

Where q ⩾ 1 the parameters α (y) and β (y) mean the degree of membership and non-membership of Y in Z, and satisfy (α (y)) ^q + (β (y)) ^q ⩽ 1, ∀ y ∈ Y.

Definition 2. [37] Assume Z₁ =〈 α₁, β₁ 〉 and Z₂ =〈 α₂, β₂ 〉 are any two q-ROFNS, and q ⩾ 1, t ⩾ 0, then the operational rules are defined as follows:

$Z_{1} \oplus Z_{2} = 〈 ((α_{1})^{q} + (α_{2})^{q} - (α_{1})^{q} \cdot (α_{2})^{q}), β_{1}^{q} β_{2}^{q} 〉$

Z₁⊗ Z₂ = 〈 (α₁) ^q (α₂) ^q, (β₁) ^q + (β₂) ^q - (β₁) ^q (β₂) ^q 〉

$t \cdot Z_{1} = 〈 {(1 - (1 - (α_{1})^{q})^{t})}^{\frac{1}{q}}, {(β_{1})}^{t} 〉$

${(Z_{1})}^{t} = 〈 {(α_{1})}^{t}, {(1 - {(1 - (β_{1})^{q})}^{t})}^{\frac{1}{q}} 〉$

Definition 3. [38] The normalized Hamming distance d (Z₁, Z₂) between the q-ROFNS Z₁ and Z₂ is denoted as follows:

$\begin{matrix} d (Z_{1}, Z_{2}) = \\ \frac{| {(α_{1})}^{q} - {(α_{2})}^{q} | + | {(β_{1})}^{q} - {(β_{2})}^{q} | + | {(π_{1})}^{q} - {(π_{2})}^{q} |}{2} \end{matrix}$ (2)

Where d (Z₁, Z₂) ∈ [0, 1], $π_{1} = (1 - (α_{1})^{q} - {{(β_{1})}^{q})}^{\frac{1}{q}}$ , $π_{2} = {(1 - (α_{2})^{q} - (β_{2})^{q})}^{\frac{1}{q}}$ , and q ⩾ 1.

Definition 4. [38] Let Z =〈 α, β 〉 be a q-ROFN, where 0 ⩽ α ⩽ 1, 0 ⩽ α + β ⩽ 1 and q ⩾ 1. Then, the score value S(Z) of the q-ROFN Z =〈 α, β 〉 is defined as: $S (Z) = α^{q} - β^{q}$ (3)

Where S (Z) ∈ [- 1, 1].

3 An extended generalized TODIM method

Inspired by the discussion above, in this phrase, we develop an integrated generalized TODIM framework to rank barriers to SFCP. This framework adopts the q-ROFNs-based scales to express experts’ opinions. Then, the q-ROF-CRITIC approach is presented to determine criteria weights that can depict the inter-dependencies among barriers. Next, an extended generalized TODIM based on q-ROFSs is presented to prioritize the barriers which reflect the impact of the expert’s bounded rational behavior characteristics. Figure 1 shows the prioritizing method based on the q-ROF generalized TODIM method

Fig. 1

A prioritizing framework with q-Rung orthopair fuzzy generalized TODIM method.

3.1 Problem statement

This subsection mainly describes the definition of the MCDM problem.

Step1.1: Acquire the information from experts

Consider a barriers ranking issue as an MCDM problem, including m alternatives a_i ={ a₁, a₂, … a_m } and n criteria B_j ={ B₁, B₂, ⋯ , B_n }. A group of experts E ={ E₁, E₂, …, E_t } is invited to judge ranking barriers using the linguistic terms in Table 1. The vectors $w_{τ}^{'} = {(w_{1}^{'}, w_{2}^{'}, \dots, w_{t})}^{T}$ and w_j = (w₁, w₂, …, w_n) ^T are the weights of experts and criteria that satisfy the conditions $\sum_{j = 1}^{n} w_{j} = 1$ , w_j ∈ [0, 1], $\sum_{τ = 1}^{t} w_{τ}^{'} = 1$ , and $w_{j}^{'} \in [0, 1]$ . Let the matrix $X^{τ} = {(x_{ij}^{τ})}_{m \times n}$ be the individual decision matrix, which is expressed as.

Table 1
The linguistic terms for the relative importance evaluation [39]

Linguistic terms for barriers q-ROFNs

Very very Preferable (VVP) (0.95,0.15)

Very Preferable (VP) (0.85,0.25)

Preferable (P) (0.75,0.35)

Moderate Preferable (MP) (0.65,0.45)

Moderate (M) (0.55,0.55)

Moderate Undesirable (MU) (0.45,0.65)

Undesirable (U) (0.35, 0.75)

Low (L) (0.25, 0.85)

Very low (VL) (0.15, 0.95)

Linguistic terms for barriers	q-ROFNs
Very very Preferable (VVP)	(0.95,0.15)
Very Preferable (VP)	(0.85,0.25)
Preferable (P)	(0.75,0.35)
Moderate Preferable (MP)	(0.65,0.45)
Moderate (M)	(0.55,0.55)
Moderate Undesirable (MU)	(0.45,0.65)
Undesirable (U)	(0.35, 0.75)
Low (L)	(0.25, 0.85)
Very low (VL)	(0.15, 0.95)

$\begin{matrix} \begin{matrix} \end{matrix} \begin{matrix} B_{1} & B_{2} & \dots & B_{n} \end{matrix} \\ X^{τ} = \begin{matrix} a_{1} \\ a_{2} \\ ⋮ \\ a_{m} \end{matrix} (\begin{matrix} x_{11}^{τ} & x_{12}^{τ} & \dots & x_{1 n}^{τ} \\ x_{21}^{τ} & x_{22}^{τ} & \dots & x_{2 n}^{τ} \\ ⋮ & ⋮ & ⋱ & ⋮ \\ x_{m 1}^{τ} & x_{m 2}^{τ} & \dots & x_{mn}^{τ} \end{matrix}) \end{matrix}$ (4)

Where $x_{ij}^{τ}$ denotes the preference assessment of alternatives a_i under criteria B_j by expert E_τ.

3.2 The collective matrix construction

Aggregating individual experts’ rating information is crucial for prioritizing barriers to SFCP. Therefore, the weighted averaging operator is applied in this subsection to establish the group matrix. Further, the distance-based weighting method is presented to determine the weights of the decision experts. Then, the collective matrix is computed using the weighted average operator. The steps are listed as follows.

Step 2.1: Obtain the average rating information of experts. The average rating is computed as follows: $\overset{x_{ij ˜}}{¯} = 〈 {\bar{α}}_{{\tilde{x}}_{ij}}, {\bar{β}}_{{\tilde{x}}_{ij}} 〉 = \frac{1}{t} \sum_{τ = 1}^{t} ({\tilde{x}}_{ij}^{τ})$ (5)

In which, ${\tilde{x}}_{ij}^{τ}$ is the element of the rating matrix from the τth expert.

Step 2.2: Compute the similarity measure between ${\tilde{x}}_{ij}^{τ}$ and $\overset{x_{ij ˜}}{¯}$ as: $s ({\tilde{x}}_{ij}^{τ}, \overset{x_{ij ˜}}{¯}) = 1 - \frac{d ({\tilde{x}}_{ij}^{τ}, \overset{x_{ij ˜}}{¯})}{\sum_{τ = 1}^{t} d ({\tilde{x}}_{ij}^{τ}, \overset{x_{ij ˜}}{¯})}$ (6)

In which, the function $d ({\tilde{x}}_{ij}^{τ}, \overset{x_{ij ˜}}{¯})$ is defined as a distance measure.

Step 2.3: Calculate the weights of experts as: $w_{τ}^{'} = \frac{\sum_{i = 1}^{m} \sum_{j = 1}^{n} s ({\tilde{x}}_{ij}^{τ}, \overset{x_{ij ˜}}{¯})}{\sum_{τ = 1}^{t} (\sum_{i = 1}^{m} \sum_{j = 1}^{n} s ({\tilde{x}}_{ij}^{τ}, \overset{x_{ij ˜}}{¯}))}$ (7)

Step 2.4: Construct the collective matrix $x = [{\tilde{x}}_{ij}]$ as: ${\tilde{x}}_{ij} = \sum_{τ = 1}^{t} w_{τ}^{'} {\tilde{x}}_{ij}^{τ} = \oplus_{τ = 1}^{t} (w_{τ}^{'} \cdot 〈 α_{{\tilde{x}}_{ij}^{τ}}, β_{{\tilde{x}}_{ij}^{τ}} 〉)$ (8)

3.3 Determine the weight of the barrier

Based on previous literature [43, 44], there are inter-dependences among the barriers to SFCP. Thus, determining the weights for these barriers should consider this interaction. Therefore, we adopt the q-ROF-CRITIC method to assess the importance of barriers. The calculation procedures of the approach are as follows:

Step 3.1: Construct the score matrix. $k_{ij} = α_{{\tilde{x}}_{ij}}^{q} - β_{{\tilde{x}}_{ij}}^{q}$ (9)

Step 3.2: Convert the score matrix into a standard q-ROF matrix $G (Z) = {({\tilde{k}}_{ij})}_{m \times n}$ as: ${\tilde{k}}_{ij} = {\begin{matrix} \frac{k_{ij} - k_{j}^{-}}{k_{j}^{+} - k_{j}^{-}}, j \in N_{b} \\ \frac{k_{j}^{+} - k_{ij}}{k_{j}^{+} - k_{j}^{-}}, j \in N_{n} \end{matrix}$ (10)

Where $k_{j}^{-} = min_{i} k_{ij}$ and $k_{j}^{+} = max_{i} k_{ij}$ .

Step 3.3: Calculate the barrier standard deviations using the formula below: $σ_{j} = \sqrt{\frac{\sum_{i = 1}^{m} {({\tilde{k}}_{ij} - \bar{k_{j}})}^{2}}{m}}$ (11)

Where ${\bar{k}}_{j} = \frac{1}{m} \sum_{i = 1}^{m} {\tilde{k}}_{ij}$ .

Step 3.4: Estimate the correlation among barriers by using the formula below: $r_{jt} = \frac{\sum_{i = 1}^{m} ({\tilde{k}}_{ij} - {\bar{k}}_{j}) ({\tilde{k}}_{it} - {\bar{k}}_{t})}{\sqrt{\sum_{i = 1}^{m} {({\tilde{k}}_{ij} - {\bar{k}}_{j})}^{2} \sum_{i = 1}^{m} {({\tilde{k}}_{it} - {\bar{k}}_{t})}^{2}}}$ (12)

Step 3.5: Determine the quantity of information on each barrier as follows: $v_{j} = σ_{j} \sum_{t = 1}^{n} (1 - r_{jt})$ (13)

Step 3.6: Determine the weight of each barrier using the formula below: $w_{j} = \frac{v_{j}}{\sum_{j = 1}^{n} v_{j}}$ (14)

3.4 The q-ROF- generalized TODIM for the barriers prioritizing

Step 4.1: Evaluate the dominance degree between barriers.

Based on the definition given in reference [37], the dominance degree is determined as follows: ${\tilde{Φ}}_{j} (a_{i}, a_{l}) = {\begin{matrix} w_{j} {(d ({\tilde{x}}_{ij}, {\tilde{x}}_{lj}))}^{α}, if {\tilde{x}}_{ij} ⩾ {\tilde{x}}_{lj} \\ - θ w_{j} {(d ({\tilde{x}}_{lj}, {\tilde{x}}_{ij}))}^{β}, if {\tilde{x}}_{ij} < {\tilde{x}}_{lj} \end{matrix}$ (15)

Where w_j is the weight of the jth barrier.

Step 4.2: Calculate the global value of each alternative.

The global value of each alternative a_i is obtained as follows: $Φ (a_{i}) = \sum_{l = 1}^{m} \sum_{j = 1}^{n} {\tilde{Φ}}_{j} (a_{i}, a_{l})$ (16)

Based on the global value Φ (a_i) of each alternative, we can rank the barrier priority of each option.

4 An illustrative example

This section introduces the application process of the current approach, the sensitivity, and comparative analyses.

4.1 The problem description

This sub-section describes a case analysis for prioritizing barriers to SFCP. Accordingly, the barriers to SFCP are chosen to display the illustrative example analysis. Food production and consumption hugely impact maintaining natural ecosystems and surroundings, and SFCP is a crucial component of the agriculture and food system’s sustainable development. According to previous research, the main criteria and sub-criteria affecting the implementation of the SFCP are displayed in Table 2. In such cases, we can find that this case includes four alternatives (a₁, a₂, a₃, a₄) and ten criteria (B₁, B₂, B₃, B₄, B₅, B₆, B₇, B₈, B₉, B₁₀), as shown in Fig. 2. Then, we organize a committee of three skilled decision-makers (E₁, E₂, E₃) to process the current decision-making issue.

Fig. 2

The hierarchical tree of factors affecting SFCP implementation.

Table 2

Main criteria and sub-criteria affecting the implementation of SFCP

Criteria	Sub-criteria	References
Cost management	Inadequate infrastructure	(Zhang and Wen 2014), (K. Nyo 2016)
	Cost barrier	(Weinrich 2019), (Werning and Spinler 2020)
	Lack of economies of scale	(Borrello et al. 2016a), (Sauvé, Bernard and Sloan 2016)
Technology	Lack of investment in advanced equipment/technologies	(Garland et al. 2009), (Kaur et al. 2018)
	Lack of expertise	(Tay et al. 2015), (Kaur et al. 2018), (Borrello et al. 2016b)
	Lack of cross-sector collaboration	(Mangla et al. 2018), (Kelly and Lange 2019)
Regulations and policies	Lack of benchmarking and standards	(Galli et al. 2020)
	Weak legal enforcement	(Junior and Soares 2016), (García-Quevedo, Jové-Llopis and Martínez-Ros 2020), (Farooque, Zhang and Liu 2019)
Education and culture	Behavioral barrier	(Hoek et al. 2021), (Weinrich 2019)
	Lack of environmental education and accountability	(Liu et al. 2021a), (Farooque, Zhang and Liu 2019)

4.2 Application of the proposed method

In this subsection, the proposed framework is applied to address the prioritization issue for adoption barriers to SFCP under uncertain and complex circumstances. The implementation procedures of the q-ROF- generalized TODIM method for prioritizing barriers to SFCP are given as follows:

Based on Step 1.1, the individual evaluation information provided by experts E_τ (τ = 1, 2, 3) is displayed in Table 3.

Table 3
The personal decision opinions given by experts

Experts Alternatives Criteria

B1 B2 B3 B4 B5 B6 B7 B8 B9 B10

E1 a ₁ P VVP VP U U L L L L L

a ₂ VP VVP VP VVP P P P P L L

a ₃ L L L L L L VVP VP L L

a ₄ VP VP P P VVP U VP VVP VVP VVP

E2 a ₁ VP VVP P L U U U L L L

a ₂ VP VP VVP VVP VP MP P P L U

a ₃ U L L U L L V.P. L L L

a ₄ P VP P VP VP U VP VP VVP VVP

E3 a ₁ P VP VP U L L L U L U

a ₂ VVP VP VP VP P VP MP P U L

a ₃ L U U L L L VP VVP L L

a ₄ VP P VP P VVP U VP VP VVP VVP

Experts	Alternatives	Criteria
E1	a ₁	P	VVP	VP	U	U	L	L	L	L	L
	a ₂	VP	VVP	VP	VVP	P	P	P	P	L	L
	a ₃	L	L	L	L	L	L	VVP	VP	L	L
	a ₄	VP	VP	P	P	VVP	U	VP	VVP	VVP	VVP
E2	a ₁	VP	VVP	P	L	U	U	U	L	L	L
	a ₂	VP	VP	VVP	VVP	VP	MP	P	P	L	U
	a ₃	U	L	L	U	L	L	V.P.	L	L	L
	a ₄	P	VP	P	VP	VP	U	VP	VP	VVP	VVP
E3	a ₁	P	VP	VP	U	L	L	L	U	L	U
	a ₂	VVP	VP	VP	VP	P	VP	MP	P	U	L
	a ₃	L	U	U	L	L	L	VP	VVP	L	L
	a ₄	VP	P	VP	P	VVP	U	VP	VP	VVP	VVP

Then, the experts’ linguistic decision opinions can be transformed into the q-ROF decision matrix, shown in Table 4.

Table 4

The q-ROFs decision matrix given by experts in the form of

Experts	Alternatives	Criteria
		B1	B2	B3	B4	B5	B6	B7	B8	B9	B10
E1	a ₁	(0.75,0.35)	(0.95,0.15)	(0.85,0.25)	(0.35,0.75)	(0.35,0.75)	(0.25,0.85)	(0.25,0.85)	(0.25,0.85)	(0.25,0.85)	(0.25,0.85)
	a ₂	(0.85,0.25)	(0.95,0.15)	(0.85,0.25)	(0.95,0.15)	(0.75,0.35)	(0.75,0.35)	(0.75,0.35)	(0.75,0.35)	(0.25,0.85)	(0.25,0.85)
	a ₃	(0.25,0.85)	(0.25,0.85)	(0.25,0.85)	(0.25,0.85)	(0.25,0.85)	(0.25,0.85)	(0.95,0.15)	(0.85,0.25)	(0.25,0.85)	(0.25,0.85)
	a ₄	(0.85,0.25)	(0.85,0.25)	(0.75,0.35)	(0.75,0.35)	(0.95,0.15)	(0.35,0.75)	(0.85,0.25)	(0.95,0.15)	(0.95,0.15)	(0.95,0.15)
E2	a ₁	(0.85,0.25)	(0.95,0.15)	(0.75,0.35)	(0.25,0.85)	(0.35,0.75)	(0.35,0.75)	(0.35,0.75)	(0.25,0.85)	(0.25,0.85)	(0.25,0.85)
	a ₂	(0.85,0.25)	(0.85,0.25)	(0.95,0.15)	(0.95,0.15)	(0.85,0.25)	(0.65,0.45)	(0.75,0.35)	(0.75,0.35)	(0.25,0.85)	(0.35,0.75)
	a ₃	(0.35,0.75)	(0.25,0.85)	(0.25,0.85)	(0.35,0.75)	(0.25,0.85)	(0.25,0.85)	(0.85,0.25)	(0.25,0.85)	(0.25,0.85)	(0.25,0.85)
	a ₄	(0.75,0.35)	(0.85,0.25)	(0.75,0.35)	(0.85,0.25)	(0.85,0.25)	(0.35,0.75)	(0.85,0.25)	(0.85,0.25)	(0.95,0.15)	(0.95,0.15)
E3	a ₁	(0.75,0.35)	(0.85,0.25)	(0.85,0.25)	(0.35,0.75)	(0.25,0.85)	(0.25,0.85)	(0.25,0.85)	(0.35,0.75)	(0.25,0.85)	(0.35,0.75)
	a ₂	(0.95,0.15)	(0.85,0.25)	(0.85,0.25)	(0.85,0.25)	(0.75,0.35)	(0.85,0.25)	(0.65,0.45)	(0.75,0.35)	(0.35,0.75)	(0.25,0.85)
	a ₃	(0.25,0.85)	(0.35,0.75)	(0.35,0.75)	(0.25,0.85)	(0.25,0.85)	(0.25,0.85)	(0.85,0.25)	(0.95,0.15)	(0.25,0.85)	(0.25,0.85)
	a ₄	(0.85,0.25)	(0.75,0.35)	(0.85,0.25)	(0.75,0.35)	(0.95,0.15)	(0.35,0.75)	(0.85,0.25)	(0.85,0.25)	(0.95,0.15)	(0.95,0.15)

According to Steps 2.1–2.3, the importance of the experts is determined using Equation (7), denoted as ${w_{1}^{'} = 0.358, w_{2}^{'} = 0.300, w_{3}^{'} = 0.342}$ .

Considering Step 2.4, the collective matrix is calculated using Equation (8), as shown in Table 5.

Table 5

The collective matrix

	a ₁	a ₂	a ₃	a ₄
B1	(0.787,0.000)	(0.898,0.000)	(0.288,0.230)	(0.826,0.000)
B2	(0.928,0.000)	(0.900,0.000)	(0.292,0.284)	(0.823,0.000)
B3	(0.826,0.001)	(0.894,0.001)	(0.292,0.381)	(0.792,0.003)
B4	(0.326,0.190)	(0.928,0.000)	(0.288,0.230)	(0.787,0.000)
B5	(0.323,0.154)	(0.787,0.000)	(0.250,0.323)	(0.931,0.000)
B6	(0.288,0.230)	(0.772,0.000)	(0.250,0.323)	(0.350,0.135)
B7	(0.288,0.230)	(0.721,0.000)	(0.900,0.000)	(0.850,0.000)
B8	(0.292,0.284)	(0.750,0.001)	(0.860,0.001)	(0.900,0.000)
B9	(0.250,0.323)	(0.292,0.284)	(0.250,0.323)	(0.950,0.00)
B10	(0.292,0.284)	(0.288,0.230)	(0.250,0.323)	(0.950,0.000)

Considering the second phase, the q-ROF-CRITIC approach is used to determine the criteria weights, which are described as follows.

Considering Step 3.1, the score values for the collective matrix are derived using Equation (9), as shown in Table 6.

Table 6

The calculation result of the CRITIC approach

	a ₁	a ₂	a ₃	a ₄	σ_j	v _j	w _j
B1	0.488	0.725	0.012	0.564	0.481	5.965	0.168
B2	0.800	0.730	0.002	0.557	0.392	3.561	0.100
B3	0.564	0.714	–0.030	0.496	0.377	3.087	0.087
B4	0.028	0.800	0.012	0.488	0.420	2.732	0.077
B5	0.030	0.488	–0.018	0.808	0.413	2.672	0.075
B6	0.012	0.461	–0.018	0.040	0.408	2.683	0.075
B7	0.012	0.375	0.730	0.614	0.382	3.292	0.093
B8	0.002	0.422	0.636	0.730	0.385	3.005	0.084
B9	–0.018	0.002	–0.018	0.857	0.430	5.282	0.149
B10	0.002	0.012	–0.018	0.857	0.425	3.282	0.092

Based on Step 3.2, we convert the score matrix into a standard Fermatean fuzzy matrix using Equation (10). Then, considering Steps 3.2–3.6, the final weights for barriers are obtained using Equations (11)–(14), and the results are given in Table 6.

Considering the third phase, the developed q-ROF-generalized TODIM method is adopted to rank the barriers. First, based on Step 4.1, we can obtain the dominance values between any two alternatives under each criterion using Equation (15) with α = β = 0.88 and θ = 2.25, and the result is given in Table 7.

Table 7

The dominance values among alternatives concerning the criteria

criteria	B1				B2
Alternatives	a ₁	a ₂	a ₃	a ₄	a ₁	a ₂	a ₃	a ₄
a ₁	0	–0.1064	0.0854	–0.0392	0	0.0096	0.08	0.0288
a ₂	0.0473	0	0.1228	0.03361	–0.0216	0	0.0736	0.02137
a ₃	–0.19208	–0.2762	0	–0.2196	–0.18	–0.166	0	–0.1293
a ₄	0.01743	–0.0756	0.0976	0	–0.0648	–0.048	0.0575	0
criteria	B3				B4
Alternatives	a ₁	a ₂	a ₃	a ₄	a ₁	a ₂	a ₃	a ₄
a ₁	0	–0.0367	0.0504	0.00814	0	–0.137	0.0014	–0.0861
a ₂	0.01631	0	0.0625	0.02267	0.06068	0	0.0614	0.02754
a ₃	–0.11343	–0.1407	0	–0.1008	–0.0032	–0.138	0	–0.088
a ₄	–0.00362	–0.051	0.0448	0	0.03828	–0.062	0.0391	0
criteria	B5				B6
Alternatives	a ₁	a ₂	a ₃	a ₄	a ₁	a ₂	a ₃	a ₄
a ₁	0	–0.0844	0.0034	–0.1349	0	–0.082	0.0026	–0.0052
a ₂	0.03752	0	0.0388	–0.062	0.03641	0	0.037	0.03502
a ₃	–0.00774	–0.0874	0	–0.1377	–0.0058	–0.083	0	–0.008
a ₄	0.05996	0.02754	0.0612	0	0.0023	–0.079	0.0036	0
Criteria	B7				B8
Alternatives	a ₁	a ₂	a ₃	a ₄	a ₁	a ₂	a ₃	a ₄
a ₁	0	–0.083	–0.1533	–0.131	0	–0.084	–0.1232	–0.1398
a ₂	0.03689	0	–0.0836	–0.0591	0.03747	0	–0.049	–0.0675
a ₃	0.06815	0.03717	0	0.01388	0.05477	0.0218	0	–0.0237
a ₄	0.05821	0.02625	–0.0312	0	0.06211	0.03	0.0105	0
criteria	B9				B10
Alternatives	a ₁	a ₂	a ₃	a ₄	a ₁	a ₂	a ₃	a ₄
a ₁	0	–0.0062	0	–0.2872	0	–0.004	0.0017	–0.1767
a ₂	0.00275	0	0.0028	–0.2844	0.00186	0	0.0031	–0.1769
a ₃	0	–0.0062	0	–0.2872	–0.0038	–0.007	0	–0.1785
a ₄	0.12764	0.12638	0.1276	0	0.07853	0.0786	0.0793	0

Then, according to Step 4.2, the dominance matrices of all criteria are obtained using Equation (16). The result is expressed as: Φ (a₁) = -1.629, Φ (a₂) = 0.016, Φ (a₃) = -2.388, Φ (a₄) = 0.839. Thus, we can find that the alternative a₄ is the best based on the overall dominance value of each option.

4.3 Comparative analysis

To illustrate the effectiveness and advantage of the approach, we compare this model with several popular q-ROF-MCDM models, including the q-ROF-VIKOR [45], the q-ROF-TOPSIS [23], and the q-ROF-TODIM [46]. The detailed rating indices and final rankings determined by various techniques are given in Table 8.

Table 8
The comparisons of different methods

Methods Alternatives Ranking indices Final ranking orders

$d_{i}^{-}$ $d_{i}^{+}$ CCI _i

q-ROF–TOPSIS a ₁ 0.639691 2.22306 0.223453 a₄ ≻ a₂ ≻ a₁ ≻ a₃

a ₂ 1.598151 0.990255 0.617427

a ₃ 0.334822 2.652372 0.112086

a ₄ 2.253572 0.240949 0.903409

q-ROF–VIKOR S _i R _i Q _i

a ₁ 0.3965 0.148529 0.5 a₂ ≻ a₃ ≻ a₁ ≻ a₄

a ₂ 0.566793 0.146874 0.985279

a ₃ 0.472052 0.148529 0.721828

a ₄ 0.519104 0.092296 0.35998

∑_l θ (H_i, H_l) φ (H_i)

q-ROF–TODIM a ₁ –15.7963 0.261232 a₃ ≻ a₂ ≻ a₁ ≻ a₄

a ₂ –10.8262 0.749837

a ₃ –8.28158 1

a ₄ –18.4536 0

Q (a_i)

The proposed method a ₁ –1.62881 a₄ ≻ a₂ ≻ a₁ ≻ a₃

a ₂ 0.0155

a ₃ –2.38779

a ₄ 0.839328

Methods	Alternatives	Ranking indices	Final ranking orders
q-ROF–TOPSIS	a ₁	0.639691	2.22306	0.223453	a₄ ≻ a₂ ≻ a₁ ≻ a₃
	a ₂	1.598151	0.990255	0.617427
	a ₃	0.334822	2.652372	0.112086
	a ₄	2.253572	0.240949	0.903409
q-ROF–VIKOR	S _i	R _i	Q _i
	a ₁	0.3965	0.148529	0.5	a₂ ≻ a₃ ≻ a₁ ≻ a₄
	a ₂	0.566793	0.146874	0.985279
	a ₃	0.472052	0.148529	0.721828
	a ₄	0.519104	0.092296	0.35998
		∑_l θ (H_i, H_l)	φ (H_i)
q-ROF–TODIM	a ₁	–15.7963	0.261232		a₃ ≻ a₂ ≻ a₁ ≻ a₄
	a ₂	–10.8262	0.749837
	a ₃	–8.28158	1
	a ₄	–18.4536	0
		Q (a_i)
The proposed method	a ₁	–1.62881			a₄ ≻ a₂ ≻ a₁ ≻ a₃
	a ₂	0.0155
	a ₃	–2.38779
	a ₄	0.839328

As seen from Table 8, we can find that the final order derived by the suggested model is consistent with that obtained by the q-ROF-TOPSIS. Further, alternatives a₁ and a₄ have the same ranking orders in the q-ROF-VIKOR and the q-ROF-TODIM methods. The alternative a₃ has the same ranking order in all four decision models. This result shows that different weights calculation methods and ranking arithmetics may impact the barriers identification and prioritization result.

Moreover, from Table 8, we also can find that the prioritization result of barriers gained by the suggested method is distinct from the other frameworks. The main reasons for these different ranking results are summarized as follows. Firstly, there are quantitative and qualitative barriers in the decision-making process, and the experts’ decision behavior may impact the accurate judgment of each barrier. The VIKOR and TOPSIS methods can not capture this feature because the two methods rank option just based on a distance measure. Second, compared with the q-ROF-TODIM method, the suggested generalized TODIM approach can eliminate the reverse ordering issue in ranking alternatives. Moreover, the suggested way can consider the interactions among barriers. Therefore, the suggested q-ROF-generalized TODIM method can provide a more accurate and practical barrier analysis result for SFCP application.

5 Conclusion

SFCP has gained more and more attention in recent years, caused by regional conflicts, extreme weather events, the COVID-19 pandemic, and other emerging crises. However, the extant barriers evaluation models are limited to resolving the situations in which the expert’s bounded rational behavior and the interactive risk factors are considered. To this end, this paper organizes a barriers evaluation framework based on the q-ROF-generalized TODIM method. The proposed barriers evaluation framework includes three different phases. This first phase is obtaining experts’ weights using the distance-based weighting technique. Then, the weighted averaging operator is used to build the group assessment matrix. The second phase introduces the q-ROF-CRITIC method to determine the weights of barriers. The last stage is organized to determine the final ranking order of each barrier using the hybrid q-ROF-generalized TODIM model. This method can reflect the influence of interaction relationships between barrier factors and the bounded rational behavior of experts. After that, a case analysis for the evaluation of adoption barriers to SFCP is performed to illustrate the suggested framework.

However, the suggested method has some limitations. First, the consensus between experts is not considered in the barriers analysis procedure. Thus, it is a considerable future direction to incorporate the consensus-reaching methods into the barriers analysis method proposed in the references [47, 48]. Additionally, the ordered prioritizing framework might help address barrier evaluation issues in other disciplines. Consequently, one potential future work path is to use the presented framework to examine and evaluate barriers in other fields.

Footnotes

Acknowledgments

This work was supported in part by the Social Science Planning Project of Anhui province (AHSKQ2021D20).

References

Xue

, Liu

, Lu

, Cheng

, Hu

, Liu

, Dou

, Cheng

and Liu

, China’s food loss and waste embodies increasing environmental impacts, Nature Food 2 (2021), 519–528.

Corrado

and Sala

, Food waste accounting along global and European food supply chains: State of the art and outlook, Waste Management 79 (2018), 120–131.

, Hong

, Yang

, Yu

, Fang

, Bai

, Liu

, Gao

, Yan

, Wang

and Wang

, Effect of hydraulic retention time on anaerobic co-digestion of cattle manure and food waste, Renewable Energy 150 (2020), 213–220.

Hafid

H.S.

, Rahman

N.A.A.

, Shah

U.K.M.

, Baharuddin

A.S.

and Ariff

A.B.

, Feasibility of using kitchen waste as future substrate for bioethanol production: A review, Renewable and Sustainable Energy Reviews 74 (2017), 671–686.

Ardra

and Barua

M.K.

, Halving food waste generation by: The challenges and strategies of monitoring UN sustainable development goal target 12.3, Journal of Cleaner Production 380 (2022), 135042.

Liu

, Wood

L.C.

, Venkatesh

V.G.

, Zhang

and Farooque

, Barriers to sustainable food consumption and production in China: A fuzzy DEMATEL analysis from a circular economy perspective, Sustainable Production and Consumption 28 (2021), 1114–1129.

Yuan

, Barriers and countermeasures for managing construction and demolition waste: A case of Shenzhen in China, Journal of Cleaner Production 157 (2017), 84–93.

Ranta

, Aarikka-Stenroos

, Ritala

and Mäkinen

S.J.

, Exploring institutional drivers and barriers of the circular economy: A cross-regional comparison of China, the US, and Europe, Resources, Conservation and Recycling 135 (2018), 70–82.

Yang

, Gai

, Cao

, Zhang

and Wu

, Application of group decision making in shipping industry 4.0: Bibliometric analysis, trends, and future directions, Systems 11 (2023), 69.

10.

Wang

, Han

, Ding

, Wu

, Chen

and Deveci

, A Fermatean fuzzy Fine–Kinney for occupational risk evaluation using extensible MARCOS with prospect theory, Engineering Applications of Artificial Intelligence 117 (2023), 105518.

11.

Kaswan

M.S.

, Rathi

, Reyes

J.A.G.

and Antony

, Exploration and investigation of green lean six sigma adoption barriers for manufacturing sustainability, IEEE Transactions on Engineering Management (2021). http://doi.org/10.1109/TEM.2021.3108171

12.

Sumrit

, Prioritization of policy initiatives to overcome Industry 4.0 transformation barriers based on a Pythagorean fuzzy multi-criteria decision making approach, Cogent Engineering 8 (2021), 1979712.

13.

Erol

, Murat

, Ar, I. Peker and Searcy

, Alleviating the impact of the barriers to circular economy adoption through blockchain: An investigation using an integrated MCDM-based QFD with hesitant fuzzy linguistic term sets, Computers & Industrial Engineering 165 (2022), 107962.

14.

Yang

, Duan

, Feng

and Mishra

A.R.

, Internet of things challenges of sustainable supply chain management in the manufacturing sector using an integrated q-Rung Orthopair Fuzzy-CRITIC-VIKOR method, Journal of Enterprise Information Management 35 (2022), 1011–1039.

15.

Güneri

and Deveci

, Evaluation of supplier selection in the defense industry using q-rung orthopair fuzzy set based EDAS approach, Expert Systems with Applications 222 (2023), 119846.

16.

Deveci

, Gokasar

, Pamucar

, Coffman

D.M.

and Papadonikolaki

, Safe E-scooter operation alternative prioritization using a q-rung orthopair Fuzzy Einstein based WASPAS approach, Journal of Cleaner Production 347 (2022), 131239.

17.

Krishankumar

, Nimmagadda

S.S.

, Rani

, Mishra

A.R.

, Ravichandran

K.S.

and Gandomi

A.H.

, Solving renewable energy source selection problems using a q-rung orthopair fuzzy-based integrated decision-making approach, Journal of Cleaner Production 279 (2021), 123329.

18.

Albahri

A.S.

, Albahri

O.S.

, Zaidan

A.A.

, Alnoor

, Alsattar

H.A.

, Mohammed

, Alamoodi

A.H.

, Zaidan

B.B.

, Aickelin

, Alazab

, Garfan

, Ahmaro

I.Y.Y.

and Ahmed

M.A.

, Integration of fuzzy-weighted zero-inconsistency and fuzzy decision by opinion score methods under a q-rung orthopair environment: A distribution case study of COVID-19 vaccine doses, Computer Standards & Interfaces 80 (2022), 103572.

19.

Krishankumar

and Ecer

, Selection of IoT service provider for sustainable transport using q-rung orthopair fuzzy CRADIS and unknown weights, Applied Soft Computing 132 (2023), 109870.

20.

Ecer

, Ögel

İ.Y.

, Krishankumar

and Tirkolaee

E.B.

, The q-rung fuzzy LOPCOW-VIKOR model to assess the role of unmanned aerial vehicles for precision agriculture realization in the Agri-Food 4.0 era, Artificial Intelligence Review (2023). https://doi.org/10.1007/s10462-10023-10476-10466

21.

, Al-Barakati

and Rani

, Investigating the Internet-of-Things (Iot) risks for supply chain management using Q-rung orthopair fuzzy-SWARA-ARAS Framework, Technological and Economic Development of Economy (2022). https://doi.org/10.3846/tede.2022.16583

22.

Tang

, Yang

, Gu

, Chiclana

, Liu

and Wang

, A new integrated multi-attribute decision-making approach for mobile medical app evaluation under q-rung orthopair fuzzy environment, Expert Systems with Applications 200 (2022), 117034.

23.

Alkan

and Kahraman

, Evaluation of government strategies against COVID-19 pandemic using q-rung orthopair fuzzy TOPSIS method, Applied Soft Computing 110 (2021), 107653.

24.

Deveci

, Gokasar

and Brito-Parada

P.R.

, A comprehensive model for socially responsible rehabilitation of mining sites using Q-rung orthopair fuzzy sets and combinative distance-based assessment, Expert Systems with Applications 200 (2022), 117155.

25.

Mishra

A.R.

, Rani

, Pamucar

, Hezam

I.M.

and Saha

, Entropy and discrimination measures based q-rung orthopair fuzzy MULTIMOORA framework for selecting solid waste disposal method, Environmental Science and Pollution Research 30 (2023), 12988–13011.

26.

Mishra

A.R.

, Rani

, Saha

, Pamucar

and Hezam

I.M.

, A q-rung orthopair fuzzy combined compromise solution approach for selecting sustainable third-party reverse logistics provider, Management Decision 61 (2023), 1816–1853.

27.

Farooque

, Zhang

and Liu

, Barriers to circular food supply chains in China, Supply Chain Management: An International Journal 24 (2019), 677–696.

28.

Feil

A.A.

, Cyrne

C.C.d.S.

, Sindelar

F.C.W.

, Barden

J.E.

and Dalmoro

, Profiles of sustainable food consumption: Consumer behavior toward organic food in southern region of Brazil, Journal of Cleaner Production 258 (2020), 120690.

29.

Hoek

A.C.

, Malekpour

, Raven

, Court

and Byrne

, Towards environmentally sustainable food systems: decision-making factors in sustainable food production and consumption, Sustainable Production and Consumption 26 (2021), 610–626.

30.

Wang

, Liu

, Chen

and Qin

, Risk assessment based on hybrid FMEA framework by considering decision maker’s psychological behavior character, Computers & Industrial Engineering 136 (2019), 516–527.

31.

, Zhao

, Wei

and Chen

, PT-TODIM method for probabilistic linguistic MAGDM and application to industrial control system security supplier selection, International Journal of Fuzzy Systems 24 (2022), 202–215.

32.

Divsalar

, Ahmadi

, Ebrahimi

and Ishizaka

, A probabilistic hesitant fuzzy Choquet integral-based TODIM method for multi-attribute group decision-making, Expert Systems with Applications 191 (2022), 116266.

33.

Ding

, Yu

S.-H.

, Chu

J.-J.

, Chen

and Shu

X.-Y.

, A decision framework for cultural and creative products based on IF-TODIM method and group consensus reaching model, Advanced Engineering Informatics 55 (2023), 101891.

34.

Nie

R.-X.

, Tian

Z.-P.

, Sang

C. Kwai

and Wang

J.-Q.

, Implementing healthcare service quality enhancement using a cloud-support QFD model integrated with TODIM method and linguistic distribution assessments, Journal of the Operational Research Society 73 (2022), 207–229.

35.

, Liu

, Qin

, Wang

and Zhou

, A linguistic distribution behavioral multi-criteria group decision making model integrating extended generalized TODIM and quantum decision theory, Applied Soft Computing 98 (2021).

36.

Wang

, Liu

, Qin

and Liu

, An extended generalized TODIM for risk evaluation and prioritization of failure modes considering risk indicators interaction, IISE Transactions 51 (2019), 1236–1250.

37.

Llamazares

, An analysis of the generalized TODIM method, European Journal of Operational Research 269 (2018), 1041–1049.

38.

Alattas

and Wu

, A framework to evaluate the barriers for adopting the internet of medical things using the extended generalized TODIM method under the hesitant fuzzy environment, Applied Intelligence 52 (2022), 13345–13363.

39.

, Liu

, Qin

, Zhou

, Mardani

and Deveci

, An integrated generalized TODIM model for portfolio selection based on financial performance of firms, Knowledge-Based Systems 249 (2022), 108794.

40.

Xiao

, Huang

and Zhang

, An integrated risk assessment method using Z-fuzzy clouds and generalized TODIM, Quality and Reliability Engineering International 38 (2022), 1909–1943.

41.

Mangla

S.K.

, Govindan

and Luthra

, Prioritizing the barriers to achieve sustainable consumption and production trends in supply chains using fuzzy Analytical Hierarchy Process, Journal of Cleaner Production 151 (2017), 509–525.

42.

Tang

, Liu

and Wang

, A hybrid risk prioritization method based on generalized TODIM and BWM for Fine-Kinney under interval type-2 fuzzy environment, Human and Ecological Risk Assessment 27 (2021), 954–979.

43.

Wang

, Liu

, Qin

and Fu

, A risk evaluation and prioritization method for FMEA with prospect theory and Choquet integral, Safety Science 110 (2018), 152–163.

44.

Wang

, Wang

, Fan

, Han

, Wu

and Pamucar

, A complex spherical fuzzy CRADIS method based Fine-Kinney framework for occupational risk evaluation in natural gas pipeline construction, Journal of Petroleum Science and Engineering 220 (2023), 111246.

45.

Cheng

, Jianfu

, Alrasheedi

, Saeidi

, Mishra

A.R.

and Rani

, A new extended VIKOR approach using q-Rung Orthopair fuzzy sets for sustainable enterprise risk management assessment in manufacturing small and medium-sized enterprises, International Journal of Fuzzy Systems 23 (2021), 1347–1369.

46.

Chen

, Luo

and Gou

, A novel q-rung orthopair fuzzy TODIM approach for multi-criteria group decision making based on Shapley value and relative entropy, Journal of Intelligent & Fuzzy Systems 40 (2021), 235–250.

47.

Zhang

and Li

, Consensus-based TOPSIS-Sort-B for multi-criteria sorting in the context of group decision-making, Annals of Operations Research (2022). https://doi.org/10.1007/s10479-10022-04985-w

48.

, Zhang

and Yu

, Consensus reaching for ordinal classification-based group decision making with heterogeneous preference information, Journal of the Operational Research Society (2023). https://doi.org/10.1080/01605682.01602023.02186806