A new three-way group decision-making model based on geometric heronian mean operators with q-rung orthopair uncertain linguistic information

Abstract

As an effective tool for three-way decisions (3WD) problems, decision-theoretic rough sets (DTRSs) have raised increasing attention recently. In view of the advantages of q-rung orthopair uncertain linguistic variables (q-ROULVs) in depicting uncertain information, a new DTRSs model based on q-ROULVs is proposed to solve three-way group decision-making (3WGDM) problems. Firstly, the loss function of DTRSs is depicted by q-ROULVs and a q-rung orthopair uncertain linguistic DTRSs model is constructed subsequently. Secondly, to aggregate different experts’ evaluation results on loss function in group decision-making (GDM) scenario, the q-rung orthopair uncertain linguistic geometric Heronian mean (q-ROULGHM) operator and the q-rung orthopair uncertain linguistic weighted geometric Heronian mean (q-ROULWGHM) operator are presented. Related properties of the proposed operators are investigated. Thirdly, to compare the expected loss of each alternative, a new score function of q-ROULVs is defined and the corresponding decision rules for 3WGDM are deduced. Finally, an illustrative example of venture capital in high-tech projects is provided to verify the rationality and effectiveness of our method. The influence of different conditional probabilities and parameter values on decision results is comprehensively discussed.

Keywords

q-rung orthopair uncertain linguistic variable decision-theoretic rough set three-way group decision-making geometric Heronian mean operator

1 Introduction

Traditional two-way decisions problems need complete information before the decision-makers (DMs) make a two-way choice of acceptance or rejection. However, with the increasing complexity of decision-making environment, it is often impossible to obtain the complete information for many real-world decision-making issues. To fill in the gap, Yao proposed three-way decisions (3WD) theory [1, 2], which combined with Bayesian decision procedure and decision-theoretic rough sets (DTRSs). As an extension of two-way decisions, 3WD carry out a deferment decision to perform further investigation in the situation of incomplete information. 3WD successfully give semantic explanations to the positive, negative and boundary domains of rough sets with the corresponding acceptance, rejection and deferment decisions, respectively. Many studies on the theory extensions and applications of 3WD have been investigated recently, e.g., conflict analysis [3], multiple criteria decision-making [4], three-way concept analysis [5], three-way clustering [6], granular computing [7, 8], just to name a few.

The research topics on 3WD mainly focus on two significant aspects: conditional probability and loss function. Conditional probability has been investigated from various perspectives. For example, by utilizing the logistic regression, Liu et al. [9] estimated the conditional probability. Combined with the Pythagorean fuzzy information proposed by Yager [10, 11], Liang et al. [12] estimated the conditional probability by computing the relative closeness with ideal solution. Taking an objective set of states with fuzzy variables into consideration, Ye et al. [13] proposed a new determination method of conditional probability. Another crucial problem in 3WD is the representation of loss function, which determine the threshold parameters of the DTRSs model. Combining with utility theory, Zhang et al. [14] proposed an improved utility function to measure the risk. Subsequently, a prospect theory-based 3WD model was constructed by Wang et al. [15] by introducing prospect theory into 3WD theory. Also, some new 3WD models have been developed by utilizing different fuzzy sets to describe the loss function. For example, Zhang and Ma [16] constructed a Pythagorean fuzzy β-covering decision-theoretic rough set model utilizing PF β-covering and PF β-neighborhood. Using q-rung orthopair fuzzy sets (q-ROFSs) to portray the loss function, Liang and Cao [17] constructed a q-rung orthopair fuzzy DTRSs model. Ye et al. [18] proposed interval-valued intuitionistic fuzzy DTRSs by introducing interval-valued intuitionistic fuzzy sets into 3WD. In addition, 3WD have also garnered broad researches from different perspectives, such as multi-granulation three-way decisions [19, 20], three-valued logics [21], three-way decision spaces [22, 23], multi-agent three-way decisions [24], which are all conducive to the theoretical development of 3WD.

With the increasing complexity of decision making problems, it is difficult for the DMs to give accurate evaluation information due to the constraints of knowledge and experience. As an effective tool to deal with uncertain information, Fuzzy sets (FSs) have been extensively introduced to multi-criterion or multi-attribute decision-making problems [25 –27]. As a generalization of Intuitionistic fuzzy sets (IFSs) [28] and PFSs, the q-ROFSs initially proposed by Yager [29 –31] in 2016. In IFSs, the summation of membership degree μ and non-membership degree v has to satisfy the condition μ + v ≤ 1.While in PFSs, the restriction is loosed to that the square sum of μ and v does not exceed 1, i.e., μ² + v² ≤ 1. Similar to IFSs and PFSs, a q-ROFS is characterized by the sum of the q powers of μ and v bounded by 1, i.e., μ^q + v^q ≤ 1 (q ≥ 1). Obviously, as q increases, the space of acceptable orthopairs (μ, v) increases, which has less limitations on value description. Therefore, q-ROFSs provide DMs more freedom in expressing their assessments. Ulteriorly, a q-ROFS will degenerate to an IFS if q = 1 and a PFS if q = 2. Considering the prominent advantages of q-ROFSs, the q-ROFSs have been investigated from different perspectives according to actual demand, including aggregation operators [32 –34], decision-making technologies [35, 36] and some extensions of q-ROFS [37].

However, for many real problems, it is more convenient for DMs to provide qualitative assessments rather than quantitative expressions. Therefore, the linguistic variables (LVs) which were first proposed by Zadeh [38], have emerged as an effective tool for DMs to describe qualitative assessment information. Furthermore, Xu [39] presented the concept of hierarchical linguistic term set. Due to the prominent characteristics, there are so many researches on LVs from different perspectives, such as Hesitant fuzzy linguistic sets [40], 2-tuple linguistic Pythagorean fuzzy sets [41], Pythagorean uncertain linguistic sets [42], and so on. Combined with q-ROFS, Li et al. [43] developed the Heronian Mean (HM) operators of q-rung orthopair linguistic set. And a family of q-rung orthopair linguistic HM operators were proposed further to cope with multi-attribute group decision-making (MAGDM) issues. Combining the q-ROFSs and uncertain linguistic variables (ULVs), Liu et al. [44] defined the q-rung orthopair uncertain linguistic set (q-ROULS) and several q-rung orthopair uncertain linguistic Bonferroni mean (BM) aggregation operators were investigated to settle problems in MAGDM. In the context of probabilistic linguistic q-ROFSs, Liu and Huang [45] developed a consensus reaching process for fuzzy behavioral TOPSIS method according to the correlation measure. In this paper, a new DTRSs model of q-ROULVs is constructed, which is applied to the three-way group decision-making (3WGDM) issues.

Simultaneously, in GDM environment, the integration of information between different experts is an important segment of the decision-making process, therefore choosing a suitable aggregation technique is crucial for obtaining scientific results. Considering the interrelationship of input arguments, q-rung orthopair uncertain linguistic, Bonferroni mean (BM) aggregation operators were investigated by Liu et al. [44]. Xing et al. [46] proposed a family of q-rung orthopair fuzzy uncertain linguistic Choquet integral operators to settle problems in multi-attribute decision-making (MADM) issues. As an effective aggregation technique, Heronian mean (HM) operator is developed to deal with the exact numerical values [47]. Compared with the BM operator featuring repeated interaction operation, HM operator can effectively distinguish the correlation between the two variables in reverse order and reduce the operational amount to half of BM operator. In addition, different from Choquet integral focusing on changing the weight vector of the aggregation operators, the HM operator focuses on the aggregated arguments [48]. Combined with q-ROFS, Li et al. [43] developed the HM operators of q-rung orthopair linguistic set. Yan et al. [49] defined some single-valued neutrosophic numbers Heronian mean operators. Yu et al. [50] presented some HM operators under dual hesitant fuzzy environment. Compared with arithmetic aggregation operators and harmonic aggregation operator, geometric aggregation operators are less affected by extreme values, so they can provide a robust information integration method for MADM problems. Combining the expression of geometric operator and HM operator. Yu [48] developed the intuitionistic fuzzy geometric Heronian mean operator. Wei et al. [51] studied q-rung orthopair fuzzy geometric Heronian mean (q-ROFGHM) operator.In order to calculate the comprehensive loss function in the 3WGDM problem based on in the DTRs model based on q-ROULVs, this paper develops some new aggregation operators for q-ROULVs based HM and GHM operator.

To solve the three-way group decision-making problems in uncertain situation, three aspects need to be considered: (1) In terms of the representation of loss function, a more general information expression approach needs to be designed for three-way decisions case. (2) To fulfill the information integration of virous experts for group decision-making problems, new integration operators should be developed, considering the interrelationship of arguments. (3) In perspective of decision rules in 3WGDM, a score function considering the influence of indeterminacy does good to obtain more rational results. Taking the merits of q-ROULS in depicting uncertainty and GHM operator in integrating information in mind, a new three-way group decision-making model is proposed based on geometric heronian mean operators with q-rung orthopair uncertain linguistic information.

In the whole, the main contributions of this paper are listed as follows:

By using q-ROULVs, a new evaluation form is defined to represent the loss function in DTRS model, which is the generalization of intuitionistic uncertain linguistic DTRSs model.

To integrate the evaluation results of different experts in GDM scenario, two new aggregation operators of q-ROULVs are developed, i.e., the q-rung orthopair uncertain linguistic geometric Heronian mean (q-ROULGHM) operator and the q-rung orthopair uncertain linguistic weighted geometric Heronian mean (q-ROULWGHM) operator. Related properties of the two operators are proved subsequently.

Inspired by herd mentality, a new score function of q-ROULVs is proposed to compare the expect loss, considering the influence of indeterminacy simultaneously. Furtherly, new three-way decision rules are built on the basis of the minimum-score-value principle.

The reminder of this paper is organized as follows. Basic concepts of ULVs, q-ROFSs, and q-ROULFSs are briefly reviewed in Section 2. Section 3 extends the geometric HM to q-ROULVs and proposes the q-ROULGHM and q-ROULWGHM operators. Section 4 constructs a new DTRSs model based on q-ROULVs considering the loss function. A new score function for q-ROULVs is proposed and the corresponding decision rules are also built. Decision-making steps of the proposed three-way group decision-making mehtod are described in Section 5. In Section 6, an illustrative example is provided to demostrate our method. Comparative analysis and sensitivity analysis are conducted subsequently. Section 7 comes to the conclusion of the paper and some future directions are explored.

2 Preliminaries

The definitions of ULVs, q-ROFSs, q-ROULSs are briefly reviewed in this section.

2.1 Uncertain linguistic variables (ULVs)

Suppose S ={ s_i|i = 0, 1, 2, . . . , 2ζ } is a finite and completely ordered discrete linguistic term set (LTS), s_i denotes a possible value of a linguistic variable and 2ζ + 1 is the cardinality of S. A finite LTS should generally satisfy the following properties [52]:

Orderliness: s_i > s_j if and only if i > j;

Negative operator: Neg (s_i) = s_j, wherej = 2ζ - i;

Maximize operator: Max (s_i, s_j) = s_i, if i ≥ j;

Minimize operator: Min (s_i, s_j) = s_i, if i ≤ j.

For example, a LTS S with seven cardinalities can be given as: S ={s₀ = extremely cold, s₁ = very cold,s₂ = cold,s₃ =fair,s₄ =hot,s₅ =very hot,s₇ = extremely hot }. To minimize the information loss, a continuous linguistic term set $\tilde{S} = {{\tilde{s}}_{ψ} | ψ \in R}$ is developed by Xu [39] which also satisfies these properties.

Definition 1. [53] Assume $\hat{s} = [s_{a}, s_{b}]$ , s_a, s_b ∈ S and a ≤ b, $\hat{s}$ is called an ULV, where s_a,s_brepresent the inferior and superior limits of $\hat{s}$ respectively.

Suppose ${\hat{s}}_{1} = [s_{a_{1}}, s_{b_{1}}]$ and ${\hat{s}}_{2} = [s_{a_{2}}, s_{b_{2}}]$ are any two ULVs, λ ≥ 0. Their basic operations are as follows [53]:

${\hat{s}}_{1} \oplus {\hat{s}}_{2} = [s_{a_{1}}, s_{b_{1}}] \oplus [s_{a_{2}}, s_{b_{2}}] = [s_{a_{1} + a_{2} - \frac{a_{1} \times a_{2}}{2 ζ + 1}}, s_{b_{1} + b_{2} - \frac{b_{1} \times b_{2}}{2 ζ + 1}}]$ ;

${\hat{s}}_{1} \otimes {\hat{s}}_{2} = [s_{a_{1}}, s_{b_{1}}] \otimes [s_{a_{2}}, s_{b_{2}}] = [s_{\frac{a_{1} \times a_{2}}{2 ζ + 1}}, s_{\frac{b_{1} \times b_{2}}{2 ζ + 1}}]$ ;

$λ {\hat{s}}_{1} = λ [s_{a_{1}}, s_{b_{1}}] = [s_{(2 ζ + 1) - (2 ζ + 1) {(1 - \frac{a_{1}}{2 ζ + 1})}^{λ}}, s_{(2 ζ + 1) - (2 ζ + 1) {(1 - \frac{b_{1}}{2 ζ + 1})}^{λ}}]$ ;

${({\hat{s}}_{1})}^{λ} = {[s_{a_{1}}, s_{b_{1}}]}^{λ} = [s_{(2 ζ + 1) {(\frac{a_{1}}{(2 ζ + 1)})}^{λ}}, s_{(2 ζ + 1) {(\frac{b_{1}}{(2 ζ + 1)})}^{λ}}]$ .

2.2 q-Rung orthopair fuzzy sets (q-ROFSs)

Definition 2. [29] Suppose X is a nonempty finite set, a q-ROFS F on X is given by; $F = {〈 x, u_{F} (x), v_{F} (x) 〉 x \in X}$ (1) where u_F (x) denotes the membership degree (MD) and v_F (x) ∈ [0, 1]denotes non-membership degree (NMD) ofx ∈ X, respectively, satisfying the restrictive condition 0 ≤ (u_F (x)) ^q + (v_F (x)) ^q ≤ 1 (q ≥ 1). And the indeterminacy degree is expressed asπ_F (x) = (1 - (u_F (x)) ^q - (v_F (x)) ^q) ^¹/_q,0 ≤ π_F (x) ≤ 1 (x ∈ X).

For the sake of simplicity, γ = (u, v) is called a q-rung orthopair fuzzy number (q-ROFN). Suppose γ₁ = (u_{γ
₁}, v_{γ
₁}) and γ₂ = (u_{γ
₂}, v_{γ
₂}) are any two q-ROFNs. To compare γ₁ and γ₂, a score function $f_{s} = \frac{1}{2} (1 + μ^{q} - v^{q})$ and an accuracy function f_h = μ^q + v^q are proposed, respectively. The specific comparison rules are as follows,

If $f_{s}^{1} > f_{s}^{2}$ ,γ₁ > γ₂;

If $f_{s}^{1} = f_{s}^{2}$ , then

If $f_{h}^{1} > f_{h}^{2}$ , γ₁ > γ₂;

If $f_{h}^{1} = f_{h}^{2}$ , γ₁ = γ₂.

The basic operation rules of them are provided as follows (λ ≥ 0) [29],

γ₁ ⊕ γ₂ = ((μ_{γ
₁}^q + μ_{γ
₂}^q - μ_{γ
₁}^qμ_{_γ2}^q) ^¹/_q, v_{_γ1}v_{_γ2});

γ₁ ⊗ γ₂ = (μ_{γ
₁}μ_{γ
₂}, (v_{γ
₁}^q + v_{γ
₂}^q - v_{γ
₁}^qv_{γ
₂}^q) ^¹/_q);

λγ₁ = ((1 - (1 - μ_{γ
₁}^q) ^λ) ^¹/_q, v_{γ
₁}^λ);

(γ₁) ^λ = (μ_{γ
₁}^λ, (1 - (1 - v_{γ
₁}^q) ^λ) ^¹/_q).

2.3 q-Rung orthopair uncertain linguistic sets (q-ROULSs)

Definition 3. [44] LetX ={ x_i|i = 1, 2, . . . , n } be a universe of discourse and $\hat{s}$ be an ULV, a q-ROULS Q defined on X is expressed as $Q = {〈 x_{i} | [s_{A (x_{i})}, s_{B (x_{i})}], μ_{Q} (x_{i}), v_{Q} (x_{i}) 〉 | x_{i} \in X}$ (2) where $[s_{A (x_{i})}, s_{B (x_{i})}] \in \hat{s}$ ,μ_Q (x_i) , v_Q (x_i) ∈ [0, 1] are the MD and NMD of the element x_i to the ULV [s_{A(x_i)}, s_{B(x_i)}] separately, which satisfy 0 ≤ (u_Q (x_i)) ^q + (v_Q (x_i)) ^q ≤ 1 (q ≥ 1), and the indeterminacy degree is given as π_Q (x_i) = (1 - (u_Q (x_i)) ^q - (v_Q (x_i)) ^q) ^¹/_q.

For the sake of convenience, the 4-tuple $〈 [s_{A (x_{i})}, s_{B (x_{i})}], μ_{Q} (x_{i}), v_{Q} (x_{i}) 〉$ is called a q-ROULV, which is expressed by Q =〈 [s_A, s_B] , μ_Q, v_Q 〉.

Suppose x₁ =〈 [s_{A
₁}, s_{B
₁}] , μ_{x
₁}, v_{x
₁} 〉, x₂ =〈 [s_{A
₂}, s_{B
₂}] , μ_{x
₂}, v_{x
₂} 〉 are any two q-ROULVs and λ ≥ 0, then some basic operations are described as follows [54]:

$x_{1} \oplus x_{2} = [s_{A_{1} + A_{2} - \frac{A_{1} \times A_{2}}{(2 ζ + 1)}}, s_{B_{1} + B_{2} - \frac{B_{1} \times B_{2}}{(2 ζ + 1)}}], 〉〈 {(μ_{x_{1}}^{q} + μ_{x_{2}}^{q} - μ_{x_{1}}^{q} μ_{x_{2}}^{q})}^{^{1} /_{q}}, v_{x_{1}} v_{x_{2}}$

$x_{1} \otimes x_{2} = [s_{\frac{A_{1} \times A_{2}}{(2 ζ + 1)}}, s_{\frac{B_{1} \times B_{2}}{(2 ζ + 1)}}], 〉〈 μ_{x_{1}} μ_{x_{2}}, {(v_{x_{1}}^{q} + v_{x_{2}}^{q} - v_{x_{1}}^{q} v_{x_{2}}^{q})}^{^{1} /_{q}}$

$λ x_{1} = [s_{(2 ζ + 1) - (2 ζ + 1) {(1 - \frac{A_{1}}{(2 ζ + 1)})}^{λ}}, s_{(2 ζ + 1) - (2 ζ + 1) {(1 - \frac{B_{1}}{(2 ζ + 1)})}^{λ}}], 〉〈 {(1 - {(1 - μ_{x_{1}}^{q})}^{λ})}^{^{1} /_{q}}, v_{x_{1}}^{λ}$

${(x_{1})}^{λ} = [s_{(2 ζ + 1) {(\frac{A_{1}}{(2 ζ + 1)})}^{λ}}, s_{(2 ζ + 1) {(\frac{B_{1}}{(2 ζ + 1)})}^{λ}}], 〉〈 μ_{x_{1}}^{λ}, {(1 - {(1 - v_{x_{1}}^{q})}^{λ})}^{^{1} /_{q}}$

3 q-Rung orthopair uncertain linguistic Geometric Heronian Mean operator

In this section, we propose two new geometric Heronian mean operator for q-ROULVs, named as q-ROULGHM and q-ROULWGHM operators, respectively. At the same time, the related properties of the q-ROULGHM operator are investigated and proven.

Definition 4. [55] Suppose a_i (i = 1, 2, . . . , n) is a set of non-negative numbers, the HM operator is defined as $HM (a_{1}, a_{2}, . . ., a_{n}) = \frac{2}{n (n + 1)} \sum_{i = 1}^{n} \sum_{j = i}^{n} \sqrt{a_{i} a_{j}}$ (3)

Definition 5. [48] Suppose α, β ≥ 0 and α, βdo not take the value 0 simultaneously, a_i (i = 1, 2, . . . , n) is a set of non-negative numbers. The geometric Heronian mean (GHM) operator is expressed as $G H M^{α, β} (a_{1}, a_{2}, ..., a_{n}) = \frac{1}{α + β} {(\prod_{i = 1, j = i}^{n} (α a_{i} + β a_{j}))}^{^{2} /_{n (n + 1)}}$ (4)

GHM satisfies the following three properties.

GHM^α,β (0, 0, . . . , 0) = 0, GHM^α,β (a, a, . . . , a) = a;

GHM^α,β (a₁, a₂, . . . , a_n) ≥ GHM^α,β (b₁, b₂, . . . , b_n), if a_i ≥ b_i, for all i;

$min_{i} {a_{i}} \leq GH M^{α, β} (a_{1}, a_{2}, \dots, a_{n}) \leq max_{i} {a_{i}}$ .

Definition 6. Let $\tilde{a_{i}} = 〈 [s_{A_{i}}, s_{B_{i}}], μ_{a_{i}}, v_{a_{i}} 〉$ (i = 1, 2, . . . , n) be a q-ROULV, the geometric Heronian mean operator of a group of q-ROULVs (q-ROULGHM) is defined as follows. $\begin{matrix} q - {ROULGHM}^{α, β} ({\tilde{a}}_{1}, {\tilde{a}}_{2}, \dots, {\tilde{a}}_{n}) = \\ \frac{1}{α + β} {(\underset{i = 1, j = i}{\overset{n}{\otimes}} (α {\tilde{a}}_{i} \oplus β {\tilde{a}}_{j}))}^{2 / n (n + 1)} \end{matrix}$ (5) where α, β ≥ 0 and α, β do not take the value 0 simultaneously.

With the GHM operator of q-ROULVs in mind, we can further derive that the aggregated result of q-ROULVs is still a q-ROULV, which is described as Theorem 3.

Theorem 1. Suppose $\tilde{a_{i}} = 〈 [s_{A_{i}}, s_{B_{i}}], μ_{a_{i}}, v_{a_{i}} 〉 (i = 1, 2, . . ., n)$ is a group of q-ROULVs, the aggregated result of q-ROULVs with the q-ROULGHM operator is still a q-ROULV, as shown in Eq. (6). $\begin{array}{l} q - R O U L W G H M^{α, β} (\tilde{a_{1}}, \tilde{a_{2}}, ..., \tilde{a_{n}}) = 〈 [s_{(2 ζ + 1) - (2 ζ + 1) {(1 - {(\prod_{i = 1, j = i}^{n} (1 - {(1 - \frac{A_{i}}{2 ζ + 1})}^{α} {(1 - \frac{A_{j}}{2 ζ + 1})}^{β}))}^{^{2} /_{n (n + 1)}})}^{^{1} /_{α + β}}} 〉, s_{(2 ζ + 1) - (2 ζ + 1) {(1 - {(\prod_{i = 1, j = i}^{n} (1 - {(1 - \frac{B_{i}}{2 ζ + 1})}^{α} {(1 - \frac{B_{j}}{2 ζ + 1})}^{β}))}^{^{2} /_{n (n + 1)}})}^{^{1} /_{α + β}}}], \\ {(1 - {(1 - {(\prod_{i = 1, j = i}^{n} (1 - {(1 - μ_{a_{i}}^{q})}^{α} {(1 - μ_{a_{j}}^{q})}^{β}))}^{^{2} /_{n (n + 1)}})}^{^{1} /_{α + β}})}^{^{1} /_{q}}, {(1 - {(\prod_{i = 1, j = i}^{n} (1 - v_{a_{i}}^{α q} v_{a_{j}}^{β q}))}^{^{2} /_{n (n + 1)}})}^{^{1} /_{q (α + β)}} 〉 \end{array}$ (6) where α, β ≥ 0 and α, β do not take the value 0 simultaneously.

See Appendix A for the proof of Theorem 1.

What’s more, the following three properties are satisfied by q-ROULGHM operator and the corresponding proof process is given.

Property 1. (Idempotency) Suppose $\tilde{a_{i}} = 〈 [s_{A_{i}}, s_{B_{i}}], μ_{a_{i}}, v_{a_{i}} 〉 (i = 1, 2, . . ., n)$ are a group of q-ROULVs, if all $\tilde{a_{i}} (i = 1, 2, . . ., n)$ are identical, that is, $\tilde{a_{i}} = \tilde{a} = 〈 [s_{A_{(a)}}, s_{B_{(a)}}], μ_{\tilde{a}}, v_{\tilde{a}} 〉$ ,then $\begin{matrix} q - ROULGH M^{α, β} (\tilde{a_{1}}, \tilde{a_{2}}, . . ., \tilde{a_{n}}) \\ = q - ROULGH M^{α, β} (\tilde{a}, \tilde{a}, . . ., \tilde{a}) \\ = \tilde{a} \end{matrix}$ (7) where α, β ≥ 0, α, β do not take the value 0 simultaneously.

Appendix B gives the proof of Property 1 in detail.

Property 2. (Monotonicity) Suppose $\tilde{a_{i}} = 〈 [s_{A_{i}}, s_{B_{i}}], μ_{a_{i}}, v_{a_{i}} 〉$ (i = 1, 2, . . . , n) and ${\tilde{a_{i}}}^{'} = 〈 [s_{{A_{i}}^{'}}, s_{{B_{i}}^{'}}], {μ_{a_{i}}}^{'}, {v_{a_{i}}}^{'} 〉 (i = 1, 2, . . ., n)$ are two collections of q-ROULVs, if the conditions s_{A
_i} ≤ s_{A_i′},s_{B
_i} ≤ s_{B_i′},μ_{a
_i} ≤ μ_{a
_i}′,v_{a
_i} ≥ v_{a
_i}′hold for all i = 1, 2, . . . , n, then $\begin{matrix} q - ROULGH M^{α, β} (\tilde{a_{1}}, \tilde{a_{2}}, . . ., \tilde{a_{n}}) \\ \leq q - ROULGH M^{α, β} ({\tilde{a_{1}}}^{'}, {\tilde{a_{2}}}^{'}, . . ., {\tilde{a_{n}}}^{'}) \end{matrix}$ (8) where α, β ≥ 0, α, βdo not take the value 0 simultaneously.

The proof of Property 2 is given in Appendix C.

Property 3. (Boundedness) Suppose $\tilde{a_{i}} = 〈 [s_{A_{i}}, s_{B_{i}}], μ_{a_{i}}, v_{a_{i}} 〉 (i = 1, 2, . . ., n)$ are a group of q-ROULVs, α, β ≥ 0, α, β do not take the value 0 simultaneously. Then $\begin{matrix} \min (\tilde{a_{1}}, \tilde{a_{2}}, . . ., \tilde{a_{n}}) & \leq q - ROULGH M^{α, β} (\tilde{a_{1}}, \tilde{a_{2}}, . . ., \tilde{a_{n}}) \\ \leq \max (\tilde{a_{1}}, \tilde{a_{2}}, . . ., \tilde{a_{n}}) \end{matrix}$ (9)

Assume ${\tilde{a}}^{-} = \min (\tilde{a_{1}}, \tilde{a_{2}}, . . ., \tilde{a_{n}})$ , ${\tilde{a}}^{+} = \max$ $(\tilde{a_{1}}, \tilde{a_{2}}, . . ., \tilde{a_{n}})$ , according to Properties 1 and 2, we have ${\tilde{a}}^{-} = q - ROULGH M^{α, β} ({\tilde{a}}^{-}, {\tilde{a}}^{-}, . . ., {\tilde{a}}^{-}) \leq q - ROULGH M^{α, β} (\tilde{a_{1}}, \tilde{a_{2}}, . . ., \tilde{a_{n}}) \leq q - ROULGH M^{α, β} ({\tilde{a}}^{+}, {\tilde{a}}^{+}, . . ., {\tilde{a}}^{+}) = {\tilde{a}}^{+}$ .

In many practical applications, especially in GDM environment, caused by different knowledge and experience, the evaluation information of each expert is unique. However, from the expression form of q-ROULGHM operator in Definition 6, it can be noticed that the importance of aggregated arguments is not be taken into consideration. Therefore, to overcome the limitation of q-ROULGHM, we develop the q-ROULWGHM operator by introducing a weight vector.

Definition 7. Let $\tilde{a_{i}} = 〈 [s_{A_{i}}, s_{B_{i}}], μ_{a_{i}}, v_{a_{i}} 〉 (i = 1, 2, . . ., n)$ be a collection of q-ROULVs, the weight vector of $\tilde{a_{i}} (i = 1, 2, . . ., n)$ is represented by W = (w₁, w₂, . . . , w_n) ^T where w_i ∈ [0, 1] is the weight of $\tilde{a_{i}}$ , $\sum_{i = 1}^{n} w_{i} = 1$ . Then q-ROULWGHM operator is denoted as the following expression $q - R O U L W G H M^{α, β} (\tilde{a_{1}}, \tilde{a_{2}}, ..., \tilde{a_{n}}) = \frac{1}{α + β} {(\underset{i = 1, j = i}{\overset{n}{\otimes}} (α {(\tilde{a_{i}})}^{n w_{i}} \oplus β {(\tilde{a_{j}})}^{n w_{j}}))}^{^{2} /_{n (n + 1)}}$ (10) where α, β ≥ 0, α, β do not take the value 0 simultaneously.

According to Theorem 1, we can derive the following theorem easily.

Theorem 2. Let $\tilde{a_{i}} = 〈 [s_{A_{i}}, s_{B_{i}}], μ_{a_{i}}, v_{a_{i}} 〉 (i = 1, 2, . . ., n)$ be a collection of q-ROULVs, and the weight vector of $\tilde{a_{i}} (i = 1, 2, . . ., n)$ is represented by W = (w₁, w₂, . . . , w_n) ^T where ω_i ∈ [0, 1] is the weight of $\tilde{a_{i}}$ , $\sum_{i = 1}^{n} w_{i} = 1$ . Then, the result aggregated by q-ROULWGHM is still an q-ROULV which is expressed by Eq. (11) $\begin{array}{l} q - R O U L W G H M^{α, β} (\tilde{a_{1}}, \tilde{a_{2}}, ..., \tilde{a_{n}}) = 〈 [s_{(2 ζ + 1) - (2 ζ + 1) {(1 - {(\prod_{i = 1, j = i}^{n} (1 - {(1 - {(\frac{A_{i}}{2 ζ + 1})}^{n w_{i}})}^{α} {(1 - {(\frac{A_{j}}{2 ζ + 1})}^{n w_{j}})}^{β}))}^{^{2} /_{n (n + 1)}})}^{^{1} /_{α + β}}}, 〉 s_{(2 ζ + 1) - (2 ζ + 1) {(1 - {(\prod_{i = 1, j = i}^{n} (1 - {(1 - {(\frac{B_{i}}{2 ζ + 1})}^{n w_{i}})}^{α} {(1 - {(\frac{B_{j}}{2 ζ + 1})}^{n w_{j}})}^{β}))}^{^{2} /_{n (n + 1)}})}^{^{1} /_{α + β}}}], \\ {(1 - {(1 - {(\prod_{i = 1, j = i}^{n} (1 - {(1 - μ_{a_{i}}^{n w_{i} q})}^{α} {(1 - μ_{a_{j}}^{n w_{j} q})}^{β}))}^{^{2} /_{n (n + 1)}})}^{^{1} /_{α + β}})}^{^{1} /_{q}} \\ {(1 - {(\prod_{i = 1, j = i}^{n} (1 - {(1 - {(1 - v_{a_{i}}^{q})}^{n w_{i}})}^{α} {(1 - {(1 - v_{a_{j}}^{q})}^{n w_{j}})}^{β}))}^{^{2} /_{n (n + 1)}})}^{^{1} /_{q (α + β)}} 〉 \end{array}$ (11) where α, β ≥ 0, α, β do not take the value 0 simultaneously.

Taking the proof of Theorem 1 in mind, the proof of Theorem 2 can be similarly obtained.

4 A New DTRSs Model Based on q-ROULVs

Utilizing the q-ROULVs to portray the loss function, a new basic model for 3WD is constructed in this section. In addition, in light of the minimum-risk decision rules suggested by the Bayesian decision procedure, the decision rules of 3WD by a new score function are also derived here.

4.1 The loss function matrix

Bayesian decision procedure assumed that the state set Ω ={ g₁, g₂, . . . , g_m }is a non-empty finite set containing m states, A ={ a₁, a₂, . . . , a_n }is a non-empty finite set containing n actions. P (g_i ∣ X) (i = 1, 2, . . . , m) is the conditional probability that objectx belongs to stateg_i when object x is described by concept g_i, λ (a_j|g_i)(j = 1, 2, . . . , n) indicates the risk or loss caused by taking action a_j when the object is in state g_i. In the framework of Bayesian decision procedure [56], a new DTRSs model of q-ROULVs can be constructed, which consists of two states and three actions uniformly. The states set is given by Ω ={ G, G^c }, denoting that an object is in the state G or not. For example, traditional channels or e-commerce channels in commodity sales channels; profit and non-profit status of investment projects; good presentation and bad presentation in manuscript reviewing. Suppose P (G|x) is the conditional probability of an object x belonging to G, P (G^c|x) is the conditional probability of an object x not belonging to G, respectively, P (G|x) + P (G^c|x) = 1. The set of actions is expressed as D ={ d_P, d_B, d_N }, where d_P, d_B and d_N represent x ∈ Pos (G), x ∈ Bnd (G) and x ∈ Neg (G), separately. That is, the object x can be partitioned into three regions according to the loss function: positive, boundary, and negative regions, which correspond to the final decision result of 3WD: accept, reject and non-commitment.

In the context of the q-ROULVs, the loss function matrix is constructed based of the states and actions as shown in the Table 1. When the object x belongs to state G, λ_P1, λ_B1 and λ_N1 which take the form of q-ROULVs indicate the costs or risk caused by taking actions d_P, d_B and d_N, respectively. Oppositely, λ_P2, λ_B2 and λ_N2 indicate the costs or risk for the identical actions when the object is not in the G. On basis of the characteristics of q-ROULVs and the semantics of DTRSs, a logical relationship can be derived as Eqs. (12)-(13): $λ_{P 1} \leq λ_{B 1} < λ_{N 1}$ (12) $λ_{P 2} > λ_{B 2} \geq λ_{N 2}$ (13)

Table 1

The loss function matrix with q-ROULVs

	G	G ^c
d _P	λ_P1 =〈 [s_{A _P1}, s_{B _P1}] , μ_{λ _P1}, v_{λ _{_P1}} 〉	λ_P2 =〈 [s_{A _P2}, s_{B _P2}] , μ_{λ _P2}, v_{λ ₂} 〉
d _B	λ_B1 =〈 [s_{A _B1}, s_{B _B1}] , μ_{λ _B1}, v_{λ _{_B1}} 〉	λ_B2 =〈 [s_{A _B2}, s_{B _B2}] , μ_{λ _B2}, v_{λ _{_B2}} 〉
d _N	λ_N1 =〈 [s_{A _N1}, s_{B _N1}] , μ_{λ _N1}, v_{λ _{_N1}} 〉	λ_N2 =〈 [s_{A _N2}, s_{B _N2}] , μ_{λ _N2}, v_{λ _{_N2}} 〉

For Eq.(12), when the object x is in the state G, the loss of classifying it into the positive region (i.e., taking actions d_P) will not exceed the loss of dividing it into the boundary region. Both of them are strictly less than the loss of distinguishing x into the negative region. Similarly, the Eq.(13) for the object x not belonging to G can be interpreted in the same way. In addition, to ensure that x is partitioned into only one region, another assumption should be satisfied as: (λ_P2 - λ_B2) (λ_N1 - λ_B1) > (λ_B1 - λ_P1) (λ_B2 - λ_N2).

According to the Bayesian decision procedure, the expected loss L (d_i|x)(i = P, B, N) corresponding individual actions can be expressed as Eqs. (14)-(16):

$L (d_{P} | x) = P (G | x) λ_{P 1} \oplus P (G^{c} | x) λ_{P 2}$ (14) $L (d_{P} | x) = P (G | x) λ_{P 1} \oplus P (G^{c} | x) λ_{P 2}$ (15) $L (d_{P} | x) = P (G | x) λ_{P 1} \oplus P (G^{c} | x) λ_{P 2}$ (16)

For the sake of convenience, let ω and ϖ represent P (G|x) and P (G^c|x), separately. Based on the operation rules of q-ROULVs, L (d_i|x) (i = P, B, N) can be calculated by Eqs. (17)-(19):

$\begin{array}{l} L (d_{P} | x) = 〈 [s_{(2 ζ + 1) - (2 ζ + 1) {(1 - \frac{A_{P 1}}{2 ζ + 1})}^{ω}}, s_{(2 ζ + 1) - (2 ζ + 1) {(1 - \frac{B_{P 1}}{2 ζ + 1})}^{ω}}] 〉, \\ {(1 - {(1 - μ_{λ_{P 1}}^{q})}^{ω})}^{^{1} /_{q}}, v_{λ_{P 1}}^{ω} 〉 \oplus \\ 〈 [s_{(2 ζ + 1) - (2 ζ + 1) {(1 - \frac{A_{P 2}}{2 ζ + 1})}^{ϖ}}, s_{(2 ζ + 1) - (2 ζ + 1) {(1 - \frac{B_{P 2}}{2 ζ + 1})}^{ϖ}}] 〉, {(1 - {(1 - μ_{λ_{P 2}}^{q})}^{ϖ})}^{^{1} /_{q}}, v_{λ_{P 2}}^{ϖ} 〉 \\ = 〈 [s_{(2 ζ + 1) - (2 ζ + 1) {(1 - \frac{A_{P 1}}{2 ζ + 1})}^{ω} {(1 - \frac{A_{P 2}}{2 ζ + 1})}^{ϖ}}, s_{(2 ζ + 1) - (2 ζ + 1) {(1 - \frac{B_{P 1}}{2 ζ + 1})}^{ω} {(1 - \frac{B_{P 2}}{2 ζ + 1})}^{ϖ}}], \\ {(1 - {(1 - μ_{λ_{P 1}}^{q})}^{ω} {(1 - μ_{λ_{P 2}}^{q})}^{ϖ})}^{^{1} /_{q}}, v_{λ_{P 1}}^{ω} v_{λ_{P 2}}^{ϖ} 〉 \end{array}$ (17) $\begin{array}{l} L (d_{B} | x) = 〈 [s_{(2 ζ + 1) - (2 ζ + 1) {(1 - \frac{A_{P 1}}{2 ζ + 1})}^{ω} {(1 - \frac{A_{B 2}}{2 ζ + 1})}^{ϖ}}, s_{(2 ζ + 1) - (2 ζ + 1) {(1 - \frac{B_{B 1}}{2 ζ + 1})}^{ω} {(1 - \frac{B_{B 2}}{2 ζ + 1})}^{ϖ}}] \\ , {(1 - {(1 - μ_{λ_{B 1}}^{q})}^{ω} {(1 - μ_{λ_{B 2}}^{q})}^{ϖ})}^{^{1} /_{q}}, v_{λ_{B 1}}^{ω} v_{λ_{B 2}}^{ϖ} 〉 \end{array}$ (18) $\begin{array}{l} L (d_{N} | x) = 〈 [s_{(2 ζ + 1) - (2 ζ + 1) {(1 - \frac{A_{P 1}}{2 ζ + 1})}^{ω} {(1 - \frac{A_{B 2}}{2 ζ + 1})}^{ϖ}}, s_{(2 ζ + 1) - (2 ζ + 1) {(1 - \frac{B_{B 1}}{2 ζ + 1})}^{ω} {(1 - \frac{B_{B 2}}{2 ζ + 1})}^{ϖ}}], \\ {(1 - {(1 - μ_{λ_{N 1}}^{q})}^{ω} {(1 - μ_{λ_{N 2}}^{q})}^{ϖ})}^{1 / q}, v_{λ_{N 1}}^{ω} v_{λ_{N 2}}^{ϖ} 〉 \end{array}$ (19)

4.2 Decision rules derived from a new ranking method for q-ROULVs

The Bayesian decision process suggests the minimum-risk decision rules (P), (N) and (B) in the following,

(P) If L (d_P|x) ≤ L (d_B|x) and L (d_P|x) ≤ L (d_N|x), then decide x ∈ Pos (G);

(B) If L (d_B|x) ≤ L (d_P|x) and L (d_B|x) ≤ L (d_N|x), then decide x ∈ Bnd (G);

(N) If L (d_N|x) ≤ L (d_P|x) and L (d_N|x) ≤ L (d_B|x), then decide x ∈ Neg (G).

As can be seen from Section 4.1, in the q-rung orthopair uncertain linguistic environment, the expected loss calculated by Eqs. (17)-(19) is still a q-ROULV. Therefore, the decision rules cannot be utilized directly. To compare the expected loss as q-ROULVs accurately, a new score function is presented considering the membership degree, non-membership degree, and indeterminacy degree simultaneously.

For a q-ROFN δ = (μ_δ, v_δ), μ_δ,v_δ and π_δ represent the choice of support, rejection and indeterminacy in decision-making process, respectively. According to herd mentality, in GDM scenario, DMs who are hesitated are more likely to be influenced by the others and thus change their own judgment. Inspired by this, we consider the influence of indeterminacy on score function by comparing the membership degree and non-membership degree. When μ_δ > v_δ, people who hesitate have a stronger disposition to support, in other words, the π_δ has a positive effect on score value. Conversely, people who hesitate have a stronger disposition to reject when μ_δ < v_δ, which means the π_δ has a negative effect on score value [55]. If μ_δ = v_δ, which means the desire of DMs to support is equivalent to reject, the influence of hesitation on score value is not considered. That is, the influence value of indeterminacy on score is regarded as 0. In addition, the influence of indeterminacy on score value is assumed to be symmetrical.

As a common S-type function, Sigmoid function is a monotone increasing continuous function which expression is $Y (x) = \frac{1}{1 + e^{- x}} (x \in R)$ . The image of it passes and is centrosymmetric with respect to (0,0.5). Based on Sigmoid function, let x = μ^q - v^q, the value of x ranges from -1 to 1 because of u, v ∈ [0, 1]. Then suppose the function $f (x) = \frac{1}{1 + e^{- x}} - \frac{1}{2} (x \in R)$ , according to the Sigmoid function, the image of f (x) passes and is symmetric with respect to the origin. That is to say, when μ = v, the function value is 0. Therefore, a new score function of q-ROFNs can be defined as Definition 8.

Definition 8. For a q-ROFN δ = (μ_δ, v_δ), where π_δ = (1 - μ_δ^q - v_δ^q) ^¹/_q, the new score function is denoted as Eq. (20): $S_{q} (δ) = \frac{1}{2} (1 + {μ_{δ}}^{q} - {v_{δ}}^{q}) + (\frac{1}{e^{{v_{δ}}^{q} - {μ_{δ}}^{q}} + 1} - \frac{1}{2}) {π_{δ}}^{q}$ (20)

According to the characteristics of q-ROFNs, two properties of S_q (δ) are deduced as Property 4 and Property 5.

Property 4. (Monotonicity) For a q-ROFN δ = (μ_δ, v_δ), S_q (δ) monotonically increases with respect to μ_δ, and monotonically decreases with respect to v_δ.

Proof. From π_δ = (1 - μ_δ^q - v_δ^q) ^¹/_q, the first order partial derivative of S_q (δ) with respect to μ_δ is obtained.

$\begin{matrix} \frac{\partial s_{q} (δ)}{\partial μ_{δ}} & = q {μ_{δ}}^{q - 1} (1 - \frac{1}{e^{{v_{δ}}^{q} - {μ_{δ}}^{q}} + 1}) \\ - \frac{q {μ_{δ}}^{q - 1} e^{{v_{δ}}^{q} - {μ_{δ}}^{q}} ({μ_{δ}}^{q} + {v_{δ}}^{q} - 1)}{{(e^{{v_{δ}}^{q} - {μ_{δ}}^{q}} + 1)}^{2}} \end{matrix}$ (21)

Obviously, μ_δ^q + v_δ^q - 1 ≤0, and e^{v_δ^q-μ_δ^q} ≥ 0, $1 - \frac{1}{e^{{v_{δ}}^{q} - {μ_{δ}}^{q}} + 1} \geq 0$ . Then, we can know that $\frac{\partial s_{q} (δ)}{\partial μ_{δ}} \geq 0$ , thus S_q (δ) is a monotonous increasing function of the membership degree.

Similarly, the first order partial derivative of S_q (δ) with respect to v_δ is calculated as

$\frac{\partial s_{q} (δ)}{\partial v_{δ}} = \frac{q {v_{δ}}^{q - 1} e^{{v_{δ}}^{q} - {μ_{δ}}^{q}} ({μ_{δ}}^{q} + {v_{δ}}^{q} - 1)}{{(e^{{v_{δ}}^{q} - {μ_{δ}}^{q}} + 1)}^{2}} - \frac{q {v_{δ}}^{q - 1}}{e^{{v_{δ}}^{q} - {μ_{δ}}^{q}} + 1}$ (22)

We can have $\frac{\partial s_{q} (δ)}{\partial v_{δ}} \leq 0$ easily, thus S_q (δ) monotonically decreases with respect to the non-membership degree. □

Property 5. (Boundedness) For a q-ROFN δ = (μ_δ, v_δ), the value of S_q (δ) satisfies the following conditions:

0 ≤ S_q (δ) ≤ 1;

If and only if δ = (0, 1), S_q (δ) = 0; if and only if δ = (1, 0), S_q (δ) = 1.

Proof.

According the properties of the function $f (x) = \frac{1}{1 + e^{- x}} (x \in R)$ , we can have $- \frac{1}{2} \leq \frac{1}{e^{v^{q} - μ^{q}} + 1} - \frac{1}{2} \leq \frac{1}{2}$ .

Furthermore, it is obvious that $- \frac{1}{2} {π_{δ}}^{q} \leq (\frac{1}{e^{{v_{δ}}^{q} - {μ_{δ}}^{q}} + 1} - \frac{1}{2}) {π_{δ}}^{q} \leq \frac{1}{2} {π_{δ}}^{q} \Rightarrow \frac{1}{2} (1 + {μ_{δ}}^{q} - {v_{δ}}^{q}) - \frac{1}{2} {π_{δ}}^{q} \leq \frac{1}{2} (1 + {μ_{δ}}^{q} - {v_{δ}}^{q}) + (\frac{1}{e^{{v_{δ}}^{q} - {μ_{δ}}^{q}} + 1} - \frac{1}{2}) {π_{δ}}^{q} \leq \frac{1}{2} (1 + {μ_{δ}}^{q} - {v_{δ}}^{q}) + \frac{1}{2} {π_{δ}}^{q}$ .

Since $0 = \frac{1}{2} (1 - {μ_{δ}}^{q} - {v_{δ}}^{q} - {π_{δ}}^{q}) \leq \frac{1}{2} (1 + {μ_{δ}}^{q} - {v_{δ}}^{q} - {π_{δ}}^{q})$ and $\frac{1}{2} (1 + {μ_{δ}}^{q} - {v_{δ}}^{q} - {π_{δ}}^{q}) \leq \frac{1}{2} (1 + {μ_{δ}}^{q} + {v_{δ}}^{q} + {π_{δ}}^{q}) = 1$ , we can obtain 0 ≤ S_q (δ) ≤1, in other words, S_q (δ) ∈ [0, 1].

According to the monotonicity of S_q (δ) described in Theorem 3, S_q (δ) monotonically increases with respect to μ_δ, and monotonically decreases with respect to v_δ. Therefore, S_q (δ) will take the maximum value of 1 only when μ = 1, v = 0; similarly, S_q (δ) will take the minimum value of 0 only when the conditions μ = 0, v = 1 are met simultaneously.

□

In general, the value of q is the minimum integer value satisfying the sum of the qth power of membership μ and non-membership v not exceeds 1. Suppose δ₁ = (0.9, 0.1),δ₂ = (0.8, 0)(q = 1), according to f_s and f_h, $f_{s}^{1} = 0.9$ , $f_{h}^{1} = 1$ , $f_{s}^{2} = 0.9$ , $f_{h}^{2} = 0.8$ ,we get δ₁ > δ₂. However, in reality, due to the high membership degree and the small non-membership degree of both, people tend to choose the object that everyone supports. That is, δ₂ > δ₁ is more in line with the human intuition. According to the new score function proposed, S (δ₁) = 0.9,S (δ₂) = 0.938,we get δ₁ < δ₂. Therefore, due to the consideration of indeterminacy degree, the new score function not only makes the score value more accurate, but also makes the result closer to actual human judgment.

Based on the analysis above, we can further define a new score function to compare q-ROULVs. For a q-ROULV Q =〈 [s_A, s_B] , μ_Q, v_Q 〉, its score function takes the form as follows Eq. (23), $\begin{array}{l} S (Q) = I_{(^{A + B} /_{2})} \times [\frac{1}{2} (1 + μ_{Q}^{q} - v_{Q}^{q}) g \\ + (\frac{1}{e^{v_{Q}^{q} - μ_{Q}^{q}} + 1} - \frac{1}{2}) π_{Q}^{q}] \end{array}$ (23) where $I_{(^{A + B} /_{2})} = \frac{A + B}{2}$ . Apparently, S (Q) also satisfies the properties of monotonically increasing with respect to μ_Q and monotonically decreasing with v_Q.

Therefore, the expected loss can be calculated and compared according to the new score function. The new decision rules are on the basis of the following minimum-score-value principle:

(P) If S (d_P|x) = min (S (d_P|x) , S (d_B|x) , S (d_N|x)), then x ∈ Pos (G);

(B) If S (d_B|x) = min (S (d_P|x) , S (d_B|x) , S (d_N|x)), then x ∈ Bnd (G);

(N) If S (d_N|x) = min (S (d_P|x) , S (d_B|x) , S (d_N|x)), then x ∈ Neg (G).

5 The Decision-making Process of 3WGDM

Based on the GHM operator developed in section 2 and the DTRSs model proposed in section 3, we propose a new three-way group decision-making method under the q-ROULVs environment. Suppose X ={ x₁, x₂, . . . , x_m } is a finite set of alternatives and E ={ e₁, e₂, . . . , e_n } is an expert set whose weight vector is W = (w₁, w₂, . . . , w_n) ^T satisfying the constraints of w_i ∈ [0, 1], $\sum_{i = 1}^{n} w_{i} = 1$ . According to three-way group decisions, the set of states is expressed as Omega ={ G, G^c } and the actions set is expressed as D ={ d_P, d_B, d_N }. Then let P (G|x) and P (G^c|x) be the conditional probability of an object x belonging to G and G^c, respectively, P (G|x) + P (G^c|x) = 1.

The three-way group decision-making process can be described in detail as follows.

Step 1. Construct the evaluation matrix.

For each alternative, the experts will provide the evaluation information of the loss functions, which takes the form of q-ROULVs described in Table 1.

Step 2. Integrate the value of the loss function.

Considering the weights of experts, the individual information is aggregated into collective information by the proposed q-ROULWGHM operator.

Step 3. Determine conditional probability.

We focus on the decision-making solution. Thus, the condition probability is supposed known here.

Step 4. Calculate the expected loss L (d_i|x) (i = P, B, N) according to Eqs. (17)-(19).

Step 5. Calculate the score values S (d_i|x) (i = P, B, N) of each alternative by Eq. (23).

Step 6. Obtain the final decision result.

Based on the new minimum-score-value decision rules, the final decision result can be derived by comparing the score value.

6 An Illustrative Example

In this section, to verify the feasibility and effectiveness of the proposed method, a numerical example is provided.

When an investment institution has the intention of venture capital, it should first make a comprehensive evaluation of the primary investment project, and the accuracy of the evaluation will directly affect the investment effect. Therefore, for venture capital institutions, having a set of more scientific, perfect, rigorous and operational risk investment project decision-making method is related to the final investment selection and investment success or failure. Science and technology project investment is one of the important forces to promote the development of science and technology innovation. As more and more attention has been paid to the role of venture capital in promoting high-tech industry, venture capital of scientific and technological innovation projects is facing more development opportunities.

Z company is a venture capital company whose business scope is to provide equity investment, asset restructuring and merger, management consulting services to high-tech enterprises with market potential, high growth and relatively mature. After conducting market research, Z selects four alternative projects for cooperation defined as X ={ x₁, x₂, x₃, x₄ }. At this time, if the company refuse a promising project, it may lead to loss. On the contrary, if it chooses to accept an immature one, it will also result in significant financial losses. Therefore, to evade risks better, it is also an option for DMs to take further consideration of a project. In order to make investment decision more efficient and scientific, the company invites three experts in venture capital E ={ e₁, e₂, e₃ } to evaluate the risk of each project. And the weight vector of experts involved in different fields has been determined which is W = (0.3, 0.2, 0.5) ^T. The set of states is expressed as Ω ={ G, G^c } denoting that the alternative projects are promising and unpromising for investment. The actions set is expressed as D ={ d_P, d_B, d_N } denotes the acceptance, rejection and non-commitment decisions.

6.1 Cooperation platforms selections

According to three-way group decision-making, the following is the specific decision-making process by applying the newly proposed method.

Step 1. Based on the comprehensive situation and development orientation of each alternative project, experts are invited to provide their evaluation (q = 3) about the loss function, which are shown in Table 2, Table 3 and Table 4, respectively.

Table 2
The evaluation results of the expert e₁

G G ^c G G ^c

x ₁ x ₂

d _P 〈 [s₂, s₄] , 0.7, 0.4〉〈 [s₄, s₅] , 0.5, 0.3〉〈 [s₁, s₃] , 0.5, 0.3〉〈 [s₄, s₆] , 0.7, 0.6〉

d _B 〈 [s₃, s₆] , 0.6, 0.5〉〈 [s₂, s₄] , 0.7, 0.2〉〈 [s₂, s₄] , 0.9, 0.5〉〈 [s₃, s₄] , 0.4, 0.7〉

d _N 〈 [s₄, s₆] , 0.8, 0.1〉〈 [s₁, s₂] , 0.7, 0.3〉〈 [s₃, s₅] , 0.6, 0.2〉〈 [s₁, s₂] , 0.7, 0.2〉

x ₃ x ₄

d _P 〈 [s₂, s₃] , 0.6, 0.1〉〈 [s₅, s₅] , 0.8, 0.2〉〈 [s₁, s₂] , 0.9, 0.3〉〈 [s₃, s₅] , 0.7, 0.5〉

d _B 〈 [s₃, s₄] , 0.8, 0.4〉〈 [s₃, s₄] , 0.6, 0.7〉〈 [s₄, s₅] , 0.7, 0.6〉〈 [s₃, s₃] , 0.6, 0.3〉

d _N 〈 [s₅, s₆] , 0.6, 0.3〉〈 [s₂, s₂] , 0.7, 0.4〉〈 [s₅, s₆] , 0.8, 0.1〉〈 [s₁, s₂] , 0.7, 0.2〉

	G	G ^c
d _P	〈 [s₂, s₄] , 0.7, 0.4〉	〈 [s₄, s₅] , 0.5, 0.3〉	〈 [s₁, s₃] , 0.5, 0.3〉	〈 [s₄, s₆] , 0.7, 0.6〉
d _B	〈 [s₃, s₆] , 0.6, 0.5〉	〈 [s₂, s₄] , 0.7, 0.2〉	〈 [s₂, s₄] , 0.9, 0.5〉	〈 [s₃, s₄] , 0.4, 0.7〉
d _N	〈 [s₄, s₆] , 0.8, 0.1〉	〈 [s₁, s₂] , 0.7, 0.3〉	〈 [s₃, s₅] , 0.6, 0.2〉	〈 [s₁, s₂] , 0.7, 0.2〉
	x ₃	x ₄
d _P	〈 [s₂, s₃] , 0.6, 0.1〉	〈 [s₅, s₅] , 0.8, 0.2〉	〈 [s₁, s₂] , 0.9, 0.3〉	〈 [s₃, s₅] , 0.7, 0.5〉
d _B	〈 [s₃, s₄] , 0.8, 0.4〉	〈 [s₃, s₄] , 0.6, 0.7〉	〈 [s₄, s₅] , 0.7, 0.6〉	〈 [s₃, s₃] , 0.6, 0.3〉
d _N	〈 [s₅, s₆] , 0.6, 0.3〉	〈 [s₂, s₂] , 0.7, 0.4〉	〈 [s₅, s₆] , 0.8, 0.1〉	〈 [s₁, s₂] , 0.7, 0.2〉

Table 3

The evaluation results of the expert e₂

	G	G ^c	G		G ^c
	x ₁		x ₂
d _P	〈 [s₂, s₃] , 0.6, 0.3〉	〈 [s₅, s₆] , 0.7, 0.2〉		〈 [s₁, s₂] , 0.8, 0.4〉		〈 [s₅, s₆] , 0.9, 0.1〉
d _B	〈 [s₃, s₄] , 0.5, 0.2〉	〈 [s₂, s₃] , 0.6, 0.1〉		〈 [s₃, s₅] , 0.7, 0.1〉		〈 [s₄, s₅] , 0.6, 0.2〉
d _N	〈 [s₅, s₅] , 0.7, 0.1〉	〈 [s₁, s₂] , 0.8, 0.1〉		〈 [s₅, s₆] , 0.9, 0.3〉		〈 [s₂, s₃] , 0.6, 0.3〉
	x ₃		x ₄
d _P	〈 [s₁, s₁] , 0.7, 0.2〉	〈 [s₅, s₆] , 0.6, 0.4〉		〈 [s₁, s₃] , 0.6, 0.2〉		〈 [s₂, s₅] , 0.9, 0.3〉
d _B	〈 [s₂, s₄] , 0.8, 0.5〉	〈 [s₂, s₄] , 0.7, 0.2〉		〈 [s₃, s₃] , 0.6, 0.4〉		〈 [s₂, s₄] , 0.8, 0.3〉
d _N	〈 [s₄, s₅] , 0.7, 0.1〉	〈 [s₁, s₃] , 0.6, 0.3〉		〈 [s₄, s₅] , 0.7, 0.1〉		〈 [s₁, s₂] , 0.7, 0.1〉

Table 4

The evaluation results of the expert e₃

	G	G ^c	G		G ^c
	x ₁		x ₂
d _P	〈 [s₁, s₂] , 0.7, 0.1〉	〈 [s₅, s₆] , 0.7, 0.2〉		〈 [s₂, s₃] , 0.7, 0.1〉		〈 [s₅, s₆] , 0.7, 0.2〉
d _B	〈 [s₂, s₃] , 0.8, 0.1〉	〈 [s₄, s₅] , 0.6, 0.3〉		〈 [s₄, s₅] , 0.7, 0.1〉		〈 [s₃, s₅] , 0.5, 0.1〉
d _N	〈 [s₅, s₆] , 0.6, 0.4〉	〈 [s₂, s₃] , 0.5, 0.1〉		〈 [s₄, s₆] , 0.9, 0.2〉		〈 [s₁, s₂] , 0.7, 0.3〉
	x ₃		x ₄
d _P	〈 [s₁, s₂] , 0.7, 0.1〉	〈 [s₄, s₆] , 0.7, 0.3〉		〈 [s₁, s₂] , 0.6, 0.7〉		〈 [s₃, s₄] , 0.8, 0.4〉
d _B	〈 [s₃, s₅] , 0.7, 0.3〉	〈 [s₃, s₅] , 0.6, 0.4〉		〈 [s₃, s₄] , 0.5, 0.1〉		〈 [s₂, s₃] , 0.8, 0.1〉
d _N	〈 [s₅, s₆] , 0.6, 0.2〉	〈 [s₁, s₂] , 0.8, 0.2〉		〈 [s₅, s₆] , 0.8, 0.2〉		〈 [s₁, s₃] , 0.7, 0.4〉

Step 2. Considering the weights of experts, the collective evaluation results are obtained by utilizing the q-ROULWGHM operator (α = 1, β = 1) shown in Table 6.

Step 3. The conditional probabilities of the states G and G^c are supposed to be known and given in Table 5.

Table 5

The condition probability information

	x ₁	x ₂	x ₃	x ₄
G	0.3	0.2	0.4	0.5
G ^c	0.7	0.8	0.6	0.5

Table 6

The collective evaluation results

	G	G ^c	G		G ^c
	x ₁		x ₂
d _P	〈 [s_1.62, s_2.85] , 0.683, 0.303〉	〈 [s_4.68, s_5.67] , 0.643, 0.243〉		〈 [s_1.44, s_2.80] , 0.664, 0.284〉		〈 [s_4.68, s_5.99] , 0.748, 0.437〉
d _B	〈 [s_2.59, s_4.02] , 0 . 669, 0 . 363〉	〈 [s_2.84, s_4.22] , 0 . 643, 0 . 263〉		〈 [s_3.10, s_4.68] , 0 . 767, 0 . 360〉		〈 [s_3.25, s_4.68] , 0 . 507, 0 . 510〉
d _N	〈 [s_4.68, s_5.78] , 0 . 697, 0 . 339〉	〈 [s_1.44, s_2.47] , 0 . 647, 0 . 218〉		〈 [s_3.88, s_5.67] , 0 . 806, 0 . 229〉		〈 [s_1.30, s_2.29] , 0 . 683, 0 . 282〉
	x ₃		x ₄
d _P	〈 [s_1.38, s_2.04] , 0 . 675, 0 . 137〉	〈 [s_3.98, s_5.18] , 0.714, 0 . 310〉		〈 [s_1.09, s_2.29] , 0 . 701, 0 . 591〉		〈 [s_2.80, s_4.51] , 0 . 792, 0 . 427〉
d _B	〈 [s_2.80, s_4.48] , 0.759, 0 . 389〉	〈 [s_2.37, s_3.55] , 0 . 635, 0 . 537〉		〈 [s_3.34, s_4.06] , 0 . 605, 0 . 448〉		〈 [s_2.37, s_3.25] , 0 . 740, 0 . 246〉
d _N	〈 [s_4.78, s_5.99] , 0 . 635, 0 . 236〉	〈 [s_1.26, s_2.29] , 0 . 726, 0 . 312〉		〈 [s_4.78, s_5.78] , 0 . 780, 0 . 173〉		〈 [s_1.09, s_2.47] , 0.707, 0 . 343〉

Step 4. The expected loss results of each alternative are calculated and shown in Table 7.

Table 7

The expected loss results

	L (d_P\|x)	L (d_B\|x)	L (d_N\|x)
x ₁	〈 [s_4.02, s_5.13] , 0.656, 0.259〉	〈 [s_2.77, s_4.16] , 0.651, 0.290〉	〈 [s_2.72, s_3.95] , 0.663, 0.249〉
x ₂	〈 [s_4.24, s_5.66] , 0.734, 0.401〉	〈 [s_3.22, s_4.68] , 0.591, 0.476〉	〈 [s_1.95, s_3.34] , 0.715, 0.270〉
x ₃	〈 [s_3.36, s_4.55] , 0.703, 0.242〉	〈 [s_2.50, s_3.86] , 0.681, 0.487〉	〈 [s_2.69, s_4.03] , 0.702, 0.287〉
x ₄	〈 [s_2.02, s_3.57] , 0.752, 0.502〉	〈 [s_2.88, s_3.68] , 0.684, 0.332〉	〈 [s_3.38, s_4.65] , 0.747, 0.244〉

Step 5. Using the new score function, the score values of the relevant expected loss results for each alternative are calculated and listed in Table 9.

Table 8

The score values

	S (d_P\|x)	S (d_B\|x)	S (d_N\|x)
x ₁	3.103	2.322	2.287
x ₂	3.514	2.240	1.919
x ₃	2.845	1.998	2.392
x ₄	1.908	2.255	3.045

Table 9

Results comparison with different operators

		S (d_P\|x)	S (d_B\|x)	S (d_N\|x)	Results
q-ROULWAA	x ₁	3.212	2.554	2.394	Pos (G) ={ x₄ }
	x ₂	3.813	2.576	1.966	Bnd (G) ={ x₃ }
	x ₃	2.772	2.194	2.735	Neg (G) ={ x₁, x₂ }
	x ₄	2.088	2.366	3.102
q-ROULWGA	x ₁	3.058	2.275	2.199	Pos (G) ={ x₄ }
	x ₂	2.204	1.838	3.481	Bnd (G) ={ x₃ }
	x ₃	2.593	2.033	2.634	Neg (G) ={ x₁, x₂ }
	x ₄	1.845	2.175	3.025
q-ROULWBM	x ₁	2.483	1.816	1.800	Pos (G) ={ x₄ }
	x ₂	2.849	2.020	1.511	Bnd (G) ={ x₃ }
	x ₃	2.200	1.630	2.059	Neg (G) ={ x₁, x₂ }
	x ₄	1.557	1.789	2.268
q-ROULPWA	x ₁	3.183	2.425	2.360	Pos (G) ={ x₄ }
	x ₂	3.543	2.666	2.302	Bnd (G) ={ x₃ }
	x ₃	2.697	2.177	2.714	Neg (G) ={ x₁, x₂ }
	x ₄	2.079	2.333	3.061
q-ROULPWG	x ₁	3.085	2.281	1.972	Pos (G) ={ x₄ }
	x ₂	3.442	2.551	1.783	Bnd (G) ={ x₃ }
	x ₃	2.578	2.129	2.377	Neg (G) ={ x₁, x₂ }
	x ₄	1.953	2.233	2.300
our method	x ₁	3.103	2.322	2.287	Pos (G) ={ x₄ }
	x ₂	3.514	2.240	1.919	Bnd (G) ={ x₃ }
	x ₃	2.845	1.998	2.392	Neg (G) ={ x₁, x₂ }
	x ₄	1.908	1.255	3.045

Step 6. According to the new minimum-score-value decision rules, we can obtain the final decision result from Table 9:

Pos (G) ={ x₄ };Bnd (G) ={ x₃ };Neg (G) ={ x₁, x₂ }.

That means X company will choose the project x₄ for cooperation and reject x₁,x₂. As for x₃, further consideration and investigation are required.

6.2 Comparative analysis

In this section, to verify the validity and rationality of our method, the comparisons in two aspects are performed. On the one hand, five different operators are used for the same illustrative example in Section 6.1. On the other hand, the proposed method is utilized to solve the issue in ref [57] in intuitionistic uncertain linguistic information environment.

In Ref [54] and [44], Liu et al. not only gave the definition of q-ROULFSs, but also explored related operators conclude q-ROULWAA, q-ROULWGA and q-ROULWBM operators. By adopting these operators to integrate information, corresponding score values and decision results are caculated and shown in Table 8. In addition, Liang et al. [17] constructed a q-rung orthopair fuzzy decision-theoretic rough sets to tackle 3WGDM problems in q-rung orthopair fuzzy environment. To integrate the opinions of DMs, the paper developed q-ROFPWA and q-ROFPWG operators. To conduct the comparation, we developed the q-rung orthopair uncertain linguistic power weighted average (q-ROULPWA) and q-rung orthopair uncertain linguistic power weighted geometric (q-ROULPWG) to tackle the example in Subsection 6.1. The results are also listed in Table 8.

It can be concluded that though scores are different, the final decision results based on different operators are consistent with the results of our method. And, from the comparation with q-ROULWHM, the scores calculated by our method are reasonable because of little difference compared with other operators. Moreover, in some practical applications, due to the difference of knowledge and background, the judgements of experts may be influenced by others. It is noteworthy that q-ROULWAA and q-ROULWGA neglect the correlation between elements. Additionally, from the expression form of q-ROULWBM, the repeated interactive operation results in a certain extent of the complexity in the calculation process. However, q-ROULWGHM overcomes such limitations, which not only captures the interrelationship between any two input arguments, but also improves the computational efficiency compared with q-ROULWBM operator-based method. Compared with q-ROULPWA and q-ROULPWG, the q-ROULWGHM focuses on the aggregated arguments which can set the parameters according to the decision maker’s requirements, reflecting more flexibility in the decision-making process.

In the context of intuitionistic uncertain linguistic, Liu et al. [57] proposed WIULGMSM operator (k=2) to solve an expansion issue of business scope according to three-way decision. The final decision is Pos (G) ={ x₁ };Bnd (G) ={ x₂ };Neg (G) ={ x₃, x₄ }. From the property of q-ROFSs, when q=1, a q-ROFS will degenerate to an IFS. Therefore, we can deal with the same problem by setting q=1 in our method. Slightly different from above decision, the result of our method is presented in Table 10, which suggests we choose to conduct further investigation for x₁. The most important reason for this difference is that, when comparing the expected loss, Liu’s method only considers the influence of membership and non-membership, but does not consider the degree of indeterminacy. In this paper, taking the influence of indeterminacy degree into account, the proposed new score function makes the decision result more accurate and objective. What’s more, the interrelationship between any two input arguments is captured by WIULGMSM, which is the same as our method (α = 1, β = 1). Remarkably, the degree of the interactions of arguments can be controlled by adjusting the values of parameters α, β. And, compared with WIULGMSM which only can be applied in IFS environment, q-ROULWGHM operator is more inclusive, which provides DMs more free space to express their assessments.

Table 10
Results comparsion in the context of IULVs

Q (d_P|x) Q (d_B|x) Q (d_N|x) Results

WIULGMSM x ₁ S _0.314 S _0.317 S _0.316 Pos (G) ={ x₁ }

x ₂ S _0.321 S _0.286 S _0.290 Bnd (G) ={ x₂ }

x ₃ S _0.388 S _0.316 S _0.303 Neg (G) ={ x₃, x₄ }

x ₄ S _0.400 S _0.328 S _0.318

S (d_P|x) S (d_B|x) S (d_N|x) Results

Our method x ₁ 2.529 2.214 2.225 Pos (G) ={ ∅ }

x ₂ 2.552 1.968 2.313 Bnd (G) ={ x₁, x₂ }

x ₃ 2.946 2.244 2.005 Neg (G) ={ x₃, x₄ }

x ₄ 2.779 2.188 2.151

		Q (d_P\|x)	Q (d_B\|x)	Q (d_N\|x)	Results
WIULGMSM	x ₁	S _0.314	S _0.317	S _0.316	Pos (G) ={ x₁ }
	x ₂	S _0.321	S _0.286	S _0.290	Bnd (G) ={ x₂ }
	x ₃	S _0.388	S _0.316	S _0.303	Neg (G) ={ x₃, x₄ }
	x ₄	S _0.400	S _0.328	S _0.318
		S (d_P\|x)	S (d_B\|x)	S (d_N\|x)	Results
Our method	x ₁	2.529	2.214	2.225	Pos (G) ={ ∅ }
	x ₂	2.552	1.968	2.313	Bnd (G) ={ x₁, x₂ }
	x ₃	2.946	2.244	2.005	Neg (G) ={ x₃, x₄ }
	x ₄	2.779	2.188	2.151

6.3 The influence of P (G|x) and the parameter α, β on decision results

Under the background of the example, this part is to further investigate the influence of conditional probability and the parameter values of α, βon results of 3WGDM. Assuming that the conditional probability information of each alternative is same, and the P (G|x) increases from 0.10 to 0.90, we can obtain the mutative decision results shown in Table 11. Utilizing MATLAB, Fig. 1 shows the classification results and change process corresponding to different conditional probabilities ranging from 0 to 1. For example, when 0 ≤ P (G|x₁) < 0.31, x₁ is classificated into negative region, which means to accept it. When 0.31 ≤ P (G|x₁) < 0.75, x₁ is classificated into boundary region, that is further investigation is required. When o . 75 ≤ P (G|x₁) ≤ 1, x₁ is classificated into positive region, which means to reject it. As can be seen from Table 11, when P (G|x) is relatively low, the final decision results investigate that all alternatives should be classified into negative region, that is, to reject all alternatives. Conversely, when the P (G|x) is relatively high, we accept all alternatives. With the increase of conditional probability, the corresponding action will gradually change from rejection to acceptance. Therefore, the reasonable determination considering each expert’s opinions comprehensively of conditional probability is very important and significant to the decision results.

Fig. 1

Classification results with P (G|x).

Table 11

Decision results with different P (G|x)

P (G\|x)	Pos (G)	Bnd (G)	Neg (G)
0.10	∅	∅	{x₁, x₂, x₃, x₄}
0.15	∅	{x₃}	{x₁, x₂, x₄}
0.30	∅	{x₃, x₄}	{x₁, x₂, x₄}
0.45	{x₄}	{x₁, x₂, x₃}	∅
0.60	{x₃, x₄}	{x₁, x₂}	∅
0.75	{x₁, x₂, x₃, x₄}	∅	∅
0.90	{x₁, x₂, x₃, x₄}	∅	∅

From the expressions of GHM operator, different decision results may be produced by choosing different parameter values of α, β. Table 12 lists the decision results for different α, β. We can notice that, with further increasing parameter value, the decision results remain unchanged, which shows the good robustness of the method. Therefore, the decision results are objective and credible. However, when the parameter value of α, β is too large, x₁ is classified to the boundary region which produces little difference. Therefore, considering the correlation of expert opinions, the parameters should be determined according to the actual problems and the computational complexity.

Table 12

Decision results with (α, β)

(α, β)	Pos (G)	Bnd (G)	Neg (G)
(0, 1)	{x₄}	{x₃}	{x₁, x₂}
(1, 0)	{x₄}	{x₃}	{x₁, x₂}
(1, 1)	{x₄}	{x₃}	{x₁, x₂}
(3, 3)	{x₄}	{x₃}	{x₁, x₂}
(5, 5)	{x₄}	{x₃}	{x₁, x₂}
(2, 3)	{x₄}	{x₃}	{x₁, x₂}
(5, 2)	{x₄}	{x₃}	{x₁, x₂}
(13, 13)	{x₄}	{x₁, x₃}	{x₂}
(15, 9)	{x₄}	{x₁, x₃}	{x₂}

6.4 Discussions

With the context of q-rung orthopair uncertain linguistic fuzzy environment, a new DTRS model is constructed this paper to help address problems in uncertainty decision-making situation. Compared with the methods of others, the method proposed in this paper has the following advantages:

From the perspective of information expression of loss function, compared with Intuitionistic fuzzy DTRSs model [58], Pythagorean fuzzy DTRSs model [12], the q-rung orthopair uncertain linguistic DTRSs model constructed in this paper provides DMs more freedom to express their views in membership and non-membership degrees like q-rung orthopair fuzzy DTRSs model [17]. Considering qualitative information which expressed by words, q-rung orthopair uncertain linguistic DTRSs model is the generalization of intuitionistic uncertain linguistic DTRSs model [57].

Compared with the score function and comparison rules in [57], the score function proposed in this paper can not only consider the influence of indeterminacy by comparing the membership degree and non-membership degree, but also simplify comparison process with more reasonable score values. Therefore, the 3WD decision rules based on score-value minimization is more objective and accurate of than rules in ref [12] and ref [57].

From the perspective of information integration, compared with q-ROULWAA, q-ROULWGA [54], q-ROULBM [44], the q-ROULGHM is the gernalization of the Intuitionistic fuzzy geometric Heronian mean [48] and Intuitionistic uncertain linguistic Heronian mean operators [59], therefore providing more applicability and operability. It not only considers the correlation between any input arguments, but also avoids repeated operations compared with BM operator to reducing the computational complexity. In addition, the degree of the interactions of arguments can be controlled by adjusting the values of parameters, making the integration process more flexible and practical. Considering the importance of arguments, the weighting form q-ROULWGHM is developed by introducing the weight vector, which is more in line with the real decision-making situation.

7 Conclusions

With the prominent advantages of q-ROULSs, we provide a new perspective in dealing with 3WGDM problems. Considering the loss function is a significant element, we construct a new DTRS model utilizing the q-ROULVs to measure the loss function. To aggregate different opinions of experts, the q-ROULGHM and q-ROULWGHM operators are designed and some related properties are investigated simultaneously. Then, a new method considering the influence of indeterminacy on score value is proposed to compare the expect loss. Furthermore, in light of the minimum-risk decision rules, the new decision rules are deduced. Finally, a descriptive example is provided to demonstrate the practicability and rationality of our proposed approach.

This paper is based on the decision-making situation with known conditional probability, but it is difficult for decision-makers to directly give conditional probability information in most situations. How to deduce objective conditional probability information from initial decision-making information is a topic worthy of research. In the near future, further investigations will be conducted to deal with the three-way decisions issues when the conditional probability is unknown. Besides, it is also necessary for researchers to find out the three-way decision-making model suitable for each discipline, since the thinking of three-way decisions exists in all fields of life. Therefore, it would be beneficial to extend our research results to applications in other fields.

Conflicts of interest

The authors declare that they have no conflict of interest.

Footnotes

Appendix

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