The determination and elimination of hidden inherent preference using q-ROFNs multicriteria group decision making problem

Abstract

The group decision-making problem usually involves decision makers (DMs) from different professional backgrounds, which leads to a considerable point, that it is the fact that there will be a certain difference in the professional cognition, risk preference and other hidden inherent factors of these DMs to the objective things that need to be evaluated. To improve the reasonability of decision-making, these hidden inherent preference (HIP) of DMs should be determined and eliminated prior to decision making. As a special form of fuzzy set, q-rung orthopair fuzzy numbers (q-ROFNs) is a useful tool to process uncertain information in decision making problems. Hence, under the environment of q-ROFNs, the determination of HIP based on distance from average score is proposed and a risk model is established to eliminate the HIP by analyzing the possible impact. Meanwhile, a dominant function is proposed, which extends the comparison method between q-ROFNs and an integrated decision-making method is provided. Finally, considering the application background of double carbon economy, an example by selecting the best design of electric vehicles charging station (EVCS) is conducted to illustrate the proposed method, and the feasibility and efficiency are verified.

Keywords

group decision-making q-ROFNs hidden inherent preference risk model double carbon economy

1 Introduction

Multicriteria group decision making problem (MCG DM), which means a group of decision makers evaluate a set of alternatives according to their own preferences, in order to seek the optimal alternative or the ordering of alternatives. Over the past few decades, the MCGDM has attracted considerable attention and has been extensively studied by researchers. Various methods have been developed in different situations, such as the preference ranking organization method for enrichment evaluations (PROMETHEE) method [3,4, 3,4], the evaluation based on distance from average solution (EDAS) method [5] and the novel integrated improved fuzzy stepwise weight assessment ratio analysis (IMF SWARA) method [35].

With the increasing complexity of the situation needed to process in MCGDM, most information becomes imprecise and fuzzy, which leads to many uncertain issues. Therefore, Zadeh [6] proposed fuzzy set (FS) theory in 1965. By establishing appropriate membership degree functions, FS takes the object to be investigated and its membership degree(MD) as a certain fuzzy set, and analyzes the fuzzy object through the relevant operations and transformations. As an effective tool, FS have been widely studied [1 , 32]. However, when the information we faced is in the form of MD and non-membership degree (NMD), then there will be some limitations in the application of FS in real life. Thus, it has become popular to present these information by multi-valued sets. Atanssov proposed the concept of intuitionistic fuzzy sets (IFSs) [7] in 1986, which describes the fuzziness using MD and NMD, where the sum of MD and NMD is between 0 and 1. Further, to deal with the more general case, Yager [8] changed the constraints of IFSs to that the sum of their squares of MD and NMD is between 0 and 1 and proposed a new model called Pythagorean fuzzy sets (PFSs). More recently years, Yager [9] introduced a general class of IFSs and PFSs called q-rung orthopair fuzzy sets (q-ROFSs) in which the sum of the qth power of MD and the qth power of NMD is limited to [0,1], which provides a wider range of MD and NMD. Obviously, IFSs, and PFSs are special issues of q-ROFSs.

As an effective tool, q-ROFSs have outstanding capability to solve MCGDM problems, which have attracted considerable attention of scholars [10–12 , 19] and have been widely applied in recent years. At present, the research on q-ROFSs mainly focuses on decision making [13 , 24–28]; the extended definition of q-ROFNs [14–18 , 33] and aggregation operator-based methods [20–22 , 34].

As mentioned at the beginning, the same decision-making problem involves DMs from different professional backgrounds. This leads to a considerable point, that it is the fact that there will be a certain difference in the professional cognition, risk preference and other hidden inherent factors of these DMs to the objective things that need to be evaluated. For example, in the given decision-making information, the evaluation of decision maker A is obviously higher than the average evaluation information, while that of decision maker B is obviously lower than the average evaluation information. There is no doubt that the optimistic attitude of A and the pessimistic attitude of B will lead to evaluations being better or worse than their actual level, which may have a certain influence on the rationality of the decision.

To solve the above problem, Li [23] proposed the concept personalized individual semantics(PIS) to reduce the impact of this risk on language terms in 2017. A few years later, these factors that could lead to the deterioration of information or lead to the improvement of information are considered a specific risk and these risk are further modeled by analyzing the possible sources of bias [30, 31]. In addition, in 2022, Chen [36] proposed a novel confidence-consensus model to manage these possible risk preference arising from different backgrounds. Nevertheless, the determination and quantification of such risks are not proposed. Therefore, determining and eliminating the risks that may be brought about by the inherent preferences of DMs is the focus of this paper. Based on the above introduction, the primary motivations of this paper are as follows:

With the increase of social demands, more and more individuals can participate in the q-ROFNs MCGDM process. However, due to different experience and psychological cognition of decision-makers, the hidden inherent preferences (HIP) of them are obviously different. Therefore, it is necessary to determine HIP before making decisions. For this reason, the determination of HIP based on distance from average score is proposed. Compared with the subjective given preference, the proposed method extracts the preference from initial objective evaluation information of DMs, which is more reasonable.

Different HIP of DMs corresponds to the different evaluation attitude of them. More deeply, optimistic or pessimistic evaluation attitude will lead the evaluations being better or worse than their actual level. Thus, MD and NMD of q-ROFNs are modeled under different types of criteria by analyzing the possible impact.

Real-life information is full of uncertainty, which makes decision making problem difficult, while q-ROFNs can solve this problem. However, at present, the comparison methods of q-ROFNs is mainly through distance and score function, while there are few studies on the dominance relationship of q-ROFNs. Hence, the dominant relation between q-ROFNs is proposed to extend the comparison method of q-ROFNs and the PROMETHEE method with q-ROFSs based on dominant relation is provided.

The remainder of the paper is organized as follows: In Section 2, some definitions are briefly reviewed. In Section 3, a novel score function is introduced and some basic properties are proofed. In Section 4, the methods for determining and eliminating inherent perference is introduced. In Section 5, an extended decision-making method is introduced. In Section 6, the method introduced in Section 5 is applied to multi-criteria group decision making problems and in Section 7 the feasibility of the method are discussed. In Section 8, some remarkable conclusions are drawn.

2 Preliminaries

In this section, some related concepts are briefly reviewed.

Definition 1 (Q-rung orthopair fuzzy sets [9]) Let X be a fixed set, then q-rung orthopair fuzzy sets (q-ROFSs) on X can be defined as follows: $A = {< x, μ_{A} (x), v_{A} (x) > ∣ x \in X}$ (1) where μ_A(x) , v_A(x) represent the membership degree and non-membership degree of x belongs to X respectively. Moreover, (μ_A(x)) ^q +(v_A(x)) ^q ≤ 1, q ≥ 1 is valid. In order to facilitate calculation, denoting α =(μ_α, v_α) be a q-rung orthopair number (q-ROFN), the hesitation degree of α is calculated by: $π = \sqrt[q]{1 - μ^{q} - v^{q}}$ .

Definition2(Scorefunctionofq - ROFN [19, 20]) : Let α =(μ_α, v_α) be a q-ROFN, then the score function of α can be defined as follows: $S_{α} = (μ^{q} - v^{q})$ (2) In order to limit the range of the score function to [0, 1], Wei [20] proposed a new definition of the score function: $S_{α}^{'} = \frac{1}{2} (1 + μ^{q} - v^{q})$ (3)

Definition3(Accuracyfunctionofq - ROFN [19]) : Let α =(μ_α, v_α) be a q-ROFN, then the accuracy function of α can be defined as follows: $H_{α} = μ^{q} + v^{q}$ (4)

Definition4(Comparisonofq - ROFNs [19]) Let α₁ =(μ₁, v₁) , α₂ =(μ₂, v₂) be two q-ROFNs, based on the the score function and accuracy function, the following comparative scheme are given: (i) if S_α₁ < S_α₂,then α₁ ≺ α₂ (ii) if S_α₁ = S_α₂,then

(a): H_α₁ < H_α₂, α₁ ≺ α₂

(b): H_α₁ = H_α₂, α₁ = α₂

Definition5(Operationallawsofq - ROFNs [19]) Let α =(μ_α, v_α) , β =(μ_β, v_β) be two q-ROFNs, then the following operational laws are defined: $\begin{matrix} α \oplus β = ({({μ_{α}}^{q} + {μ_{β}}^{q} - {μ_{α}}^{q} {μ_{β}}^{q})}^{1 / q}, v_{α} v_{β}) \\ α \otimes β = (μ_{α} μ_{β}, {({v_{α}}^{q} + {v_{β}}^{q} - {v_{α}}^{q} {v_{β}}^{q})}^{1 / q}) \\ λ α = (1 - {({(1 - {μ_{α}}^{q})}^{λ})}^{1 / q}, {v_{α}}^{λ}) \\ α^{λ} = ({μ_{α}}^{λ}, 1 - {({(1 - {v_{α}}^{q})}^{λ})}^{1 / q}) \end{matrix}$

3 Rigorous score function of q-ROFN

As an important tool to measure the superiority and inferiority of q-ROFN information, the traditional score functions have limitations in dealing with some special cases. Therefore, in this section, a novel score function is introduced.

Before putting forward the definition of rigorous score function, the limitations of previous score function are briefly reviewed.

Example3.1: Considering the following three q-ROFNs (Assume q = 1): $α_{1} = (0.5, 0.49), α_{2} = (0.3, 0.29), α_{3} = (0.02, 0.01)$ Even if the hesitation degree of three q-ROFNs are significantly different, the following results can be obtained by calculating the scores using score function in definition 2. $\begin{matrix} S_{α_{1}} = S_{α_{2}} = S_{α_{3}} = 0.01; S_{α_{1}}^{'} = S_{α_{2}}^{'} = S_{α_{3}}^{'} = 0.505 \end{matrix}$

Example3.2: For the special case where the membership degree is equal to the non-membership degree: $α_{4} = {(0.5, 0.5)}_{2}, α_{5} = {(0.5, 0.5)}_{3}, α_{6} = {(0.2, 0.2)}_{2}$ where (μ, v) _p, p = 2, 3 stands for different q value in q-ROFN,then the following results can also be obtained by score function in definition 2: $\begin{matrix} S_{α_{4}} = S_{α_{5}} = S_{α_{6}} = 0; S_{α_{4}}^{'} = S_{α_{5}}^{'} = S_{α_{6}}^{'} = 0.5 \end{matrix}$ It can be easily found that as long as the membership degree is equal to the non-membership degree,the score function has the same value.

As an effective tool to represent uncertainty information, one of the concerns of q-ROFN is the degree of hesitation, which represents the unknown degree of membership and non-membership; another point of concern is the q value, different q values reflect different domain ranges. However, the above score functions do not reflect the variation tendency with the changing of hesitation degree. Based on the above analysis, the definition of q-ROFN rigorous score function is given below.

Definition6(Rigorousscorefunctionofq - ROFN) Let α =(μ_α, v_α) , β =(μ_β, v_β) be two q-ROFNs, S_q(α) is called a rigorous score function if and only if the following properties are satisfied.

(P1): 0 ≤ S_q(α) ≤ 1

(P2): S_q(α) = 0 if and only if α =(0, 1)

(P3): S_q(α) = 1 if and only if α =(1, 0)

(P4): S_q(α) increases with respect to μ_α and decreases with respect to v_α.

(P5): S_q(α) = S_q(β) if and only if μ_α = μ_β, v_α = v_β

(P6): S_q(α) = S_p(α) ⇔ p = q

In the above definition, P5 ensures that under the same q value, only the membership degree and the non-membership degree are all the same, the value of the score function is equal; P6 guarantees that under the same numerical expression, only the q value is the same, the value of score function is equal. In fact, this is a strengthened restriction on hesitation degree, which better embodies the characteristics of q-ROFN information

According to definition 6, a rigorous score function of q-ROFN is given below. Let α =(μ, v) be a q-ROFN, a novel score function of α is defined as follows: $S_{q} (α) = \frac{ln (1 + μ^{q}) + ln (2 - v^{q})}{2 ln 2}$ (5)

Proof: It is not difficult to see that S_q(α) satisfies P(2) , P(3) , P(4) in definition 6, therefore, it is only necessary to prove P(1) , P(5) , P(6)

(P1) : According to the definition of q-ROFN, it can be easily found that μ^q, μ^q ∈ [0, 1], then $ln (1 + μ^{q}), ln (2 - v^{q}) \in [0, 2 ln 2]$ Hence, 0 ≤ S_q(α) ≤ 1

(P5): Denote β =(s, t) be a q-ROFN, then S_q(α) = S_q(β) is equivalent to $\begin{matrix} ln (1 + μ^{q}) + ln (2 - v^{q}) = ln (1 + s^{q}) + ln (2 - t^{q}) \\ \Leftrightarrow ln (\frac{1 + μ^{q}}{1 + s^{q}}) + ln (\frac{2 - v^{q}}{2 - t^{q}}) = 0 \\ \Leftrightarrow ln (\frac{1 + μ^{q}}{1 + s^{q}} \cdot \frac{2 - v^{q}}{2 - t^{q}}) = 0 \\ \Leftrightarrow \frac{1 + μ^{q}}{1 + s^{q}} \cdot \frac{2 - v^{q}}{2 - t^{q}} = 1 (*) \end{matrix}$ it can be easily to get $\frac{1 + μ^{q}}{1 + s^{q}}, \frac{2 - v^{q}}{2 - t^{q}} \in [\frac{1}{2}, 2]$ Thus, there are only two cases where equation (*) holds.

(i): The two numerical results are equal, in this case $\frac{1 + μ^{q}}{1 + s^{q}} = \frac{2 - v^{q}}{2 - t^{q}} = 1$ therefore $μ = s, v = t$

(ii): The two numerical results are not equal, in this case, there must be one range in $[\frac{1}{2}, 1)$ and the other range in (1, 2]. Assume: $\frac{1 + μ^{q}}{1 + s^{q}} \in [\frac{1}{2}, 1), \frac{2 - v^{q}}{2 - t^{q}} \in (1, 2]$ then $\begin{matrix} 1 + μ^{q} < 1 + s^{q}, 2 - v^{q} < 2 - t^{q} \\ \Leftrightarrow μ < s, v > t \end{matrix}$ At this time, S_q(α) < S_q(β), which is contrary to the condition. Thus, this case does not exist, P(5) holds.

(P6):S_q(α) = S_p(α) is equivalent to $\begin{matrix} ln (1 + μ^{q}) + ln (2 - v^{q}) = ln (1 + μ^{p}) + ln (2 - v^{p}) \\ \Leftrightarrow ln (\frac{1 + μ^{q}}{1 + μ^{p}}) + ln (\frac{2 - v^{q}}{2 - v^{p}}) = 0 \\ \Leftrightarrow ln (\frac{1 + μ^{q}}{1 + μ^{p}} \cdot \frac{2 - v^{q}}{2 - v^{p}}) = 0 \\ \Leftrightarrow \frac{1 + μ^{q}}{1 + μ^{p}} \cdot \frac{2 - v^{q}}{2 - v^{p}} = 1 (*) \end{matrix}$

Likewise, there are two cases are considered:

(i): The two numerical results are equal,in this case $\frac{1 + μ^{q}}{1 + μ^{p}} = \frac{2 - v^{q}}{2 - v^{p}} = 1 \Leftrightarrow p = q$

(ii): One range in $[\frac{1}{2}, 1)$ and the other range in (1, 2]. Assume: $\begin{matrix} \frac{1 + μ^{q}}{1 + μ^{p}} \in [\frac{1}{2}, 1), \frac{2 - v^{q}}{2 - v^{p}} \in (1, 2] \\ \Leftrightarrow 1 + μ^{q} < 1 + μ^{p}, 2 - v^{q} < 2 - v^{p} \\ \Leftrightarrow μ^{q} < μ^{p}, v^{q} > v^{p} \end{matrix}$ Since μ^q < μ^p and v^q > v^p cannot be established at the same time, this case does not exist, P(6) holds.

Based on the above analysis, S_q(α) is a rigorous score function and the comparison of counterexamples in Example 3.1 and Example 3.2 under different score functions can be obtained in the following table:

As can be seen from the results in Table 1, considering the potential differences that may be caused by hesitation, the rigorous score function proposed in this paper overcomes the limitation that the hesitation degree changes while the score function does not change. More intuitively, suppose that q = 3, then the variation tendency of S_q(α) with the changing of μ and v is shown in Fig

Table 1

The comparison of different score functions

q-ROFN	S_α in [19]	$S_{α}^{'}$ in [20]	S_q(α) in this paper
α₁	0.01	0.505	0.5898
α₂	0.01	0.505	0.5763
α₃	0.01	0.505	0.5107
α₄	0	0.5	0.5646
α₅	0	0.5	0.5384
α₆	0	0.5	0.5137

Fig. 1

Variation tendency of the proposed score function with the changing of μ and v when q = 3.

4 The determination and elimination of hidden inherent preference under q-ROFS environment

In multi-criteria group decision making (MCGDM) problem, different decision makers (DMs) have subjective evaluations for the same objective result, which may lead to evaluations being better or worse than their actual level. Hence,to improve the reasonability and validity of decision making, it is necessary to identify and address this hidden inherent preferences (HIP) of decision makers. In this section, the composition of HIP is introduced and then the determination and elimination of HIP under q-ROFS environment are proposed.

4.1 Problem description

Suppose X ={ x₁, x₂, …, x_n } is the set of alternatives, DM ={ d₁, d₂, …, d_p } is the set of decision makers and C ={ c₁, c₂, …, c_n } are the criteria set to be evaluated. DMs give their own evaluations based on their own perceptions and the decision matrix is written as: $A^{k} = [\begin{matrix} (μ_{11}^{k}, v_{11}^{k}) & (μ_{12}^{k}, v_{12}^{k}) & \dots & (μ_{1 m}^{k}, v_{1 m}^{k}) \\ (μ_{21}^{k}, v_{21}^{k}) & (μ_{22}^{k}, v_{22}^{k}) & \dots & (μ_{2 m}^{k}, v_{2 m}^{k}) \\ \cdot & \cdot & \cdot & \cdot \\ \cdot & \cdot & \cdot & \cdot \\ \cdot & \cdot & \cdot & \cdot \\ (μ_{n 1}^{k}, v_{n 1}^{k}) & (μ_{n 2}^{k}, v_{n 2}^{k}) & \dots & (μ_{nm}^{k}, v_{nm}^{k}) \end{matrix}]$ where $(μ_{ij}^{k}, v_{ij}^{k})$ represents decision maker d_k′s evaluation of alternatives x_i under criterion c_j. In this section, the above MCGDM problem is represented by an information table S =(X, DM, C).

Definition7(ThecompositionofHIP) : Given a q-ROFS information table S =(X, DM, C), for any d_k ∈ DM, the HIP should consist of two parts: PA^k, PC^k, where $P A^{k} = {P {A_{1}}^{k}, P {A_{2}}^{k}, . . ., P {A_{n}}^{k}}$ (6) $P C^{k} = {P {C_{1}}^{k}, P {C_{2}}^{k}, . . ., P {C_{m}}^{k}}$ (7) represent d_k′s hidden inherent preference for different criteria and different alternatives respectively.

In fact, HIP reflects the positive or negative attitudes of decision makers towards different alternatives and criteria, which may be caused by different behavioral preferences, psychological cognition and other uncertain hidden factors. That’s why it’s called hidden preference. This special preference is reflected in the initial evaluation information, therefore, the determination and elimination of HIP need to be a priority.

4.2 The determination of hidden inherent preference

Inspired by EDAS method, the HIP should be corresponded to the average score matrix (AVS) of all DMs, the degree that was above the AVS was regarded as the relatively positive attitude while the degree that was below the AVS was regarded as the relatively pessimistic attitude. Then the following concepts are introduced.

Definition8(Averagescorematrix) : Given a q-ROFS information table S =(X, DM, C), the average score matrix (AVS) is defined as follows: $AVS = {[S_{q} (A V_{ij})]}_{n \times m}, i = 1, 2, . . n; j = 1, 2, . . ., m$ where $A V_{i j} = (\frac{1}{p} \sum_{k = 1}^{p} μ_{i j}^{k}, \frac{1}{p} \sum_{k = 1}^{p} v_{i j}^{k})$ (8)S_q is the novel score function in definition 6.

Definition9(Positiveandnegativescorematrix) : Given a q-ROFS information table S =(X, DM, C), for every expert, the positive score matrix (PSA) and the negative score matrix (NSA) are defined as follows: $PS A^{k} = {[PS \underset{ij}{A^{k}}]}_{n \times m}; NS A^{k} = {[NS \underset{ij}{A^{k}}]}_{n \times m}$ if criterion c_j is beneficial criterion ${PSA}_{ij}^{k} = max (0, (S_{q} (α_{ij}^{(k)}) - S_{q} (A V_{ij})))$ ${NSA}_{ij}^{k} = max (0, (S_{q} (A V_{ij}) - S_{q} (α_{ij}^{(k)})))$ if criterion c_j is non-beneficial criterion ${PSA}_{ij}^{k} = max (0, (S_{q} (A V_{ij}) - S_{q} (α_{ij}^{(k)})))$ ${NSA}_{ij}^{k} = max (0, (S_{q} (α_{ij}^{(k)}) - S_{q} (A V_{ij})))$

Definition10(ThedeterminationofHIP) : Given a q-ROFS information table S =(X, DM, C), the determination of hidden preference is defined as follows: $\begin{matrix} {PC}_{j}^{k} = \frac{\sum_{i = 1}^{n} ({PSA}_{ij}^{k} - {NSA}_{ij}^{k})}{n}, j = 1, 2, . . ., m \\ {PA}_{i}^{k} = \frac{\sum_{j = 1}^{m} ({PSA}_{ij}^{k} - {NSA}_{ij}^{k})}{m}, i = 1, 2, . . ., n \end{matrix}$ where PC_j^k represents the preference for on criterion c_j,PA_i^k represents the preference for on alternative x_i.

The above definition 8-10 is progressive, in which definition 8 merges the evaluations of all DMs into a reference average score matrix, and definition 9 actually obtains the degree of exceeding or falling below this average score matrix, the definition 10 obtains HIP of different alternatives and different criteria by horizontal and vertical processing of the matrix in definition 9.

4.3 The elimination of hidden inherent preference

The positive and negative values of HIP indicate that the current evaluation information is better or worse than the real information. In other words, the degree of preference can be defined as the deviation between the precise value and the obtained value, which can be described as follows [37]: $r = \frac{| Approximate value - Exact value |}{Exact value}$ (9)

More deeply, for a q-ROFN (μ, v), assume (μ^*, v^*) is the exact value (without preference). Then r is expressed as the deviation degree between μ and μ^*, v and v^*. More intuitively, the deviation caused by different preference is shown in Figure 2

Fig. 2

The deviation caused by diffenent pererence.

Therefore, how to measure this deviation is a problem worth considering. In addition, as an important information of q-ROFN, hesitation degree should be potentially correlated with hesitation about membership and non-membership. For example: Let α =(0.3, 0.5) , β =(0.5, 0.3), even if π_α = π_β, the information in them are different. In other words, if α and β are converted to a crisp set, π_α would allocate more membership to α while π_β would allocate more non-membership to β. Based on the above analysis, the following definition is given:

Definition11: Let α =(μ_α, v_α) be a q-ROFN, π_α is the hesitation degree of α. Assume (μ^*, v^*) is the q-ROFN after preference elimination, then (μ^*, v^*) should meet the following conditions: $0 \leq μ^{*} \leq \sqrt[q]{μ^{q} + \frac{μ^{q}}{μ^{q} + v^{q}} (1 - μ^{q} - v^{q})}$ $0 \leq v^{*} \leq \sqrt[q]{v^{q} + \frac{v^{q}}{μ^{q} + v^{q}} (1 - μ^{q} - v^{q})}$

where $\sqrt[q]{μ^{q} + \frac{μ^{q}}{μ^{q} + v^{q}} (1 - μ^{q} - v^{q})}$ and $\sqrt[q]{v^{q} + \frac{v^{q}}{μ^{q} + v^{q}} (1 - μ^{q} - v^{q})}$ are the maximum possible membership degree and non-membership degree in α. For the convenience of calculation, for a q-ROFN (μ, v), the maximum possible membership degree and the maximum possible non-membership degree are denoted as μ_max and v_max.

Let (μ, v) represents decision maker DM_k′s evaluation of alternative x_i under criterion c_j,PC_j^kand PA_i^k represent DM_k′s preference on criteria c_j and alternative x_i respectively, if we model PA and PC of the beneficial criterion, the following cases can be considered.

CaseA:PA is considered as error factor in the assessment process, for the beneficial criterion, the following formula can be obtained: $\begin{matrix} {(μ, v)}_{A - b} = \\ (min (\frac{μ}{1 + {PA}_{i}^{k}}, μ_{max}), min (\frac{v}{1 - {PA}_{i}^{k}}, v_{max})) \end{matrix}$

Proof: (AP - b): If PA_i^k > 0, for the beneficial criterion, PA will make the current evaluation information be better than the real information. Thus, the current membership degree will be better than the real: μ^A-Pb and the current non-membership degree will be worse than the real: v^A-Pb, from Eq. (9),we obtain: $\begin{matrix} P {A_{i}}^{k} = \frac{μ - μ^{A - Pb}}{μ^{A - Pb}} \to μ^{A - Pb} = \frac{μ}{1 + P {A_{i}}^{k}} \\ P {A_{i}}^{k} = \frac{v^{A - Pb} - v}{v^{A - Pb}} \to v^{A - Pb} = \frac{v}{1 - P {A_{i}}^{k}} \end{matrix}$

(AN - b): If PA_i^k < 0, for the beneficial criterion, PA will make the current evaluation information be worse than the real information. the current membership degree will be worse than the real: μ^A-Nb and the current non-membership degree will be better than the real: v^A-Nb, from Eq. (9), we can get $\begin{matrix} - P {A_{i}}^{k} = \frac{μ^{A - Pb} - μ}{μ^{A - Pb}} \to μ^{A - Nb} = \frac{μ}{1 + P {A_{i}}^{k}} \\ - P {A_{i}}^{k} = \frac{v - v^{A - Pb}}{v^{A - Nb}} \to v^{A - Pb} = \frac{v}{1 - P {A_{i}}^{k}} \end{matrix}$ Meanwhile, μ, v is limited to [0, μ_max] and [0, v_max], obviously 0 < μ^A-Pb, 0 < μ^A-Nb, 0 < v^A-Pb, 0 < v^A-Nb,at this point,the result of case A has been proved.

Now, if we model PA of non-beneficial criterion, then the following formula can be obtained: $\begin{matrix} {(μ_{ij}^{k}, v_{ij}^{k})}_{A - c} = \\ (min (\frac{μ}{1 - {PA}_{i}^{k}}, μ_{\max}), min (\frac{v}{1 + {PA}_{i}^{k}}, v_{\max})) \end{matrix}$

(AP - c):If PA_i^k > 0, for the non-beneficial criterion, PA will make the current membership degree will be worse than the real: μ^A-Pc and the current non-membership degree will be better than the real: v^A-Pc, from Eq. (9),we obtain: $\begin{matrix} P {A_{i}}^{k} = \frac{μ^{A - Pc} - μ}{μ^{A - Pc}} \to μ^{A - Pc} = \frac{μ}{1 - P {A_{i}}^{k}} \\ P {A_{i}}^{k} = \frac{v - v^{A - Pc}}{v^{A - Pc}} \to v^{A - Pc} = \frac{v}{1 + P {A_{i}}^{k}} \end{matrix}$

(AN - c):If PA_i^k < 0, for the non-beneficial criterion, PA will make the current membership degree will be better than the real: μ^A-Pb and the current non-membership degree will be worse than the real: v^A-Pb, from Eq. (9),we obtain: $\begin{matrix} - P {A_{i}}^{k} = \frac{μ - μ^{A - Pc}}{μ^{A - Pc}} \to μ^{A - Pc} = \frac{μ}{1 - P {A_{i}}^{k}} \\ - P {A_{i}}^{k} = \frac{v^{A - Pc} - v}{v^{A - Pc}} \to v^{A - Pc} = \frac{v}{1 + P {A_{i}}^{k}} \end{matrix}$ Likewise, it should be noted that $\frac{μ}{1 - P {A_{i}}^{k}}$ , $\frac{v}{1 - P {A_{i}}^{k}}$ could not be higher than μ_max, v_max, thus, Eq() has also been proved.

CaseC: PC is considered as error factor in the assessment process, for the beneficial criterion, the previous (μ, v) _A-b will be reformulated as: $\begin{matrix} {(μ, v)}_{C - b} = \\ (min (\frac{μ}{1 + {PC}_{j}^{k}}, μ_{\max}), min (\frac{v}{1 - {PC}_{i}^{k}}, v_{\max})) \end{matrix}$

for the non-beneficial criterion, ${(μ_{ij}^{k}, v_{ij}^{k})}_{A - c}$ will be reformulated as: $\begin{matrix} {(μ, v)}_{C - c} = \\ (min (\frac{μ}{1 - {PC}_{j}^{k}}, μ_{\max}), min (\frac{v}{1 + {PC}_{i}^{k}}, v_{\max})) \end{matrix}$

Proof: The proof is similar to above.

Based on the above discussion, if PA and PC are both analyzed, for the beneficial criterion, the preference elimination q-ROFN will be:

$\begin{matrix} {(μ, v)}^{*} = \\ (\begin{matrix} min (\frac{μ}{(1 + {PA}_{i}^{k}) (1 + {PC}_{j}^{k})}, μ_{max}), \\ min (\frac{v}{(1 - {PA}_{i}^{k}) (1 - {PC}_{j}^{k})}, v_{max}) \end{matrix}) \end{matrix}$ (10)

for the non-beneficial criterion, the preference elimination q-ROFN will be:

$\begin{matrix} {(μ, v)}^{*} = \\ (\begin{matrix} min (\frac{μ}{(1 - {PA}_{i}^{k}) (1 - {PC}_{j}^{k})}, μ_{max}), \\ min (\frac{v}{(1 + {PA}_{i}^{k}) (1 + {PC}_{j}^{k})}, v_{max}) \end{matrix}) \end{matrix}$ (11)

Property4.1:Let (μ, v) be a q-ROFN, then the result after preference elimination (μ^*, v^*) is still a q-ROFN:

Proof: $\begin{matrix} {(μ^{*})}^{q} \leq {(μ_{max})}^{q} = μ^{q} + \frac{μ^{q}}{μ^{q} + v^{q}} (1 - μ^{q} - v^{q}) \\ {(v^{*})}^{q} \leq {(v_{max})}^{q} = v^{q} + \frac{v^{q}}{μ^{q} + v^{q}} (1 - μ^{q} - v^{q}) \end{matrix}$ therefore, ${(μ^{*})}^{q} + {(v^{*})}^{q} \leq {(μ_{max})}^{q} + {(v_{max})}^{q} = 1$

5 PROMETHEE Method under q-ROFNs environment

5.1 The dominant function of q-ROFN

At present, the comparison between q-ROFNs is mainly through distance and score function, while there are few studies on the relationship of dominance. Therefore, this section proposes the dominance measure of q-ROFN.

Definition12: Assume α =(μ_α, v_α) and β =(μ_β, v_β) are two q-ROFNs, π_α and π_β are the corresponding hesitation degree. Let S_α and S_β be the score degree of α and β, then the dominance measure of q-ROFN can be defined as follows:

$\begin{matrix} P (α \geq β) = \sqrt{\frac{π_{β}}{π_{α} + π_{β}} \cdot \frac{S_{α}}{4 (S_{α} + S_{β})}} \\ - \sqrt{\frac{π_{α}}{π_{α} + π_{β}} \cdot \frac{S_{β}}{4 (S_{α} + S_{β})}} + \frac{1}{2} \end{matrix}$ (12)

Property5.1: Let α =(μ_α, v_α) , β =(μ_β, v_β) , γ =(μ_γ, v_γ) be three q-ROFNs, then the dominant measure satisfies the following four properties:

(1)0 ≤ P(α ≥ β) ≤ 1

(2) P(α ≥ α) = 0.5

(3) P(α ≥ β) + P(β ≥ α) = 1

$\begin{matrix} (4) P (α \geq β) \geq 0.5, P (β \geq γ) \geq 0.5 \\ \Rightarrow P (α \geq γ) \geq 0.5 \end{matrix}$

Proof: The proofs of (2) and (3) are relatively simple and will not be described in detail here, the following are proofs of (1) and (4):

(1): it can be easily found that $0 \leq \frac{π_{β}}{π_{α} + π_{β}}, \frac{S_{β}}{S_{α} + S_{β}} \leq 1$ therefore, $\begin{matrix} \sqrt{\frac{π_{β}}{π_{α} + π_{β}} \frac{S_{α}}{4 (S_{α} + S_{β})}} \in [0, \frac{1}{2}] \\ \sqrt{\frac{π_{α}}{π_{α} + π_{β}} \frac{S_{β}}{4 (S_{α} + S_{β})}} \in [0, \frac{1}{2}] \end{matrix}$ $\begin{matrix} \sqrt{\frac{π_{β}}{π_{α} + π_{β}} \frac{S_{α}}{4 (S_{α} + S_{β})}} - \sqrt{\frac{π_{α}}{π_{α} + π_{β}} \frac{S_{β}}{4 (S_{α} + S_{β})}} \\ \in [- \frac{1}{2}, \frac{1}{2}] \end{matrix}$ Thus, 0 ≤ P(α ≥ β) ≤ 1 holds. $\begin{matrix} (4) : P (α \geq β) \geq 0.5, P (β \geq γ) \geq 0.5 \Rightarrow \\ \sqrt{\frac{π_{β}}{π_{α} + π_{β}} \frac{S_{α}}{4 (S_{α} + S_{β})}} \geq \sqrt{\frac{π_{α}}{π_{α} + π_{β}} \frac{S_{β}}{4 (S_{α} + S_{β})}}, \\ \sqrt{\frac{π_{γ}}{π_{β} + π_{γ}} \frac{S_{β}}{4 (S_{β} + S_{γ})}} \geq \sqrt{\frac{π_{β}}{π_{β} + π_{γ}} \frac{S_{γ}}{4 (S_{β} + S_{γ})}} \\ \Rightarrow π_{β} S_{α} \geq π_{α} S_{β}, π_{γ} S_{β} \geq π_{β} S_{γ} \\ \Rightarrow π_{β} π_{γ} S_{α} \geq π_{α} π_{γ} S_{β}, π_{α} π_{γ} S_{β} \geq π_{α} π_{β} S_{γ} \\ \Rightarrow π_{β} π_{γ} S_{α} \geq π_{α} π_{β} S_{γ} \\ \Rightarrow π_{γ} S_{α} \geq π_{α} S_{γ} \Rightarrow \\ \sqrt{\frac{π_{γ}}{π_{α} + π_{γ}} \frac{S_{α}}{4 (S_{α} + S_{γ})}} \geq \sqrt{\frac{π_{α}}{π_{α} + π_{γ}} \frac{S_{γ}}{4 (S_{α} + S_{γ})}} \end{matrix}$ Thus, property 5.1 holds.

In decision-making problems, a greater contribution to the distinguishing alternatives should correspond to a greater weight. Therefore, combined with the definition of dominance function, the following weight determination formulas are constructed.

Definition13: The weight vector of criteria for decision maker d_k is denoted as $W^{k} = (w_{1}^{k}, w_{2}^{k}, . . ., w_{m}^{k})$ . Then the criteria weights $w_{j}^{k}$ is defined as follows: $w_{j}^{k} = \frac{\sum_{i < l, i, l = 1}^{n} | P ({α_{ij}}^{k} > {α_{lj}}^{k}) - 0.5 |}{\sum_{j = 1}^{m} \sum_{i < l, i, l = 1}^{n} | P ({α_{ij}}^{k} > {α_{lj}}^{k}) - 0.5 |}$ (13) where α_ij^k and α_lj^k represents the q-ROFN of the alternative x_i and x_l under the criterion c_j.

Definition14: The weight vector of decision makers is denoted as $\tilde{W} = ({\tilde{w}}^{1}, {\tilde{w}}^{2}, . . ., {\tilde{w}}^{n})$ . Then the weight ${\tilde{w}}^{k}$ is defined as follows: ${\tilde{w}}^{k} = \frac{\sum_{j = 1}^{m} \sum_{i < l, i, l = 1}^{n} | P ({α_{ij}}^{k} > {α_{lj}}^{k}) - 0.5 |}{\sum_{k = 1}^{p} \sum_{j = 1}^{m} \sum_{i < l, i, l = 1}^{n} | P ({α_{ij}}^{k} > {α_{lj}}^{k}) - 0.5 |}$ (14)

5.2 The decision making procedure

Summarizing the results above, we come up with a novel method that can be applied to MCGDM under q-ROFNs environment, where the information about the weights is unknown completely. The decision making procedure can be established as follows

Step1: Obtain the evaluation information of the DMs and transform to q-ROFN.

Step2: Determine the experts’ hidden inherent preference according to definition 8-10.

Step3: Preference elimination for initial evaluation data by Eq. (10) and Eq. (11), the processed decision matrix is recorded as: ${(A^{k})}^{*} = [\begin{matrix} {(μ_{11}^{k}, v_{11}^{k})}^{*} & {(μ_{12}^{k}, v_{12}^{k})}^{*} & \dots & {(μ_{1 m}^{k}, v_{1 m}^{k})}^{*} \\ {(μ_{21}^{k}, v_{21}^{k})}^{*} & {(μ_{22}^{k}, v_{22}^{k})}^{*} & \dots & {(μ_{2 m}^{k}, v_{2 m}^{k})}^{*} \\ \cdot & \cdot & \cdot & \cdot \\ \cdot & \cdot & \cdot & \cdot \\ \cdot & \cdot & \cdot & \cdot \\ {(μ_{n 1}^{k}, v_{n 1}^{k})}^{*} & {(μ_{n 2}^{k}, v_{n 2}^{k})}^{*} & \dots & {(μ_{nm}^{k}, v_{nm}^{k})}^{*} \end{matrix}]$ where (A^k) ^* represents the decision matrix of d_k after preference elimination

Fig.3

The procedure of proposed method.

Step4: To ensure the consistence of the type of each criteria, the decision matrix is further transformed to $R^{k} = {[r_{ij}^{k}]}_{n \times m}$ by the following method: $r_{ij}^{k} = {\begin{matrix} {(μ_{ij}^{k}, v_{ij}^{k})}^{*}, & for benefit attribute of c_{j} \\ {(v_{ij}^{k}, μ_{ij}^{k})}^{*}, & for cost attribute of c_{j} \end{matrix}$

Step5: Determine the expert weights and the criteria weights by Eq.(13) and Eq. (14).

Step6: Calculate the inflow, outflow and the net flow of alternative x_i, i = 1, 2, . . . , n. $φ^{+} (x_{i}) = \frac{\underset{l = 1}{\sum^{p}} {\bar{w}}_{l} Π (x_{i}, x_{k})}{\sum_{i = 1}^{n} \underset{l = 1}{\sum^{p}} {\bar{w}}_{l} Π (x_{i}, x_{k})}$ (15) $φ^{(} x_{i}) = \frac{\underset{l = 1}{\sum^{p}} {\bar{w}}_{l} Π (x_{k}, x_{i})}{\sum_{i = 1}^{n} \underset{l = 1}{\sum^{p}} {\bar{w}}_{l} Π (x_{k}, x_{i})}$ (16) $φ (x_{i}) = φ^{+} (x_{i}) - φ^{(} x_{i})$ (17) where $\begin{matrix} Π (x_{i}, x_{k}) = \end{matrix}$ $\begin{matrix} \sum_{j} w_{j}^{l} \underset{k = 1, k \neq i}{\sum^{n}} max {(P (r_{ij}^{l} \geq r_{kj}^{l}) - 0.5), 0} \\ Π (x_{k}, x_{i}) = \\ \sum_{j} w_{j}^{l} \underset{k = 1, k \neq i}{\sum^{n}} max {(P (r_{kj}^{l} \geq r_{ij}^{l}) - 0.5), 0} \end{matrix}$

Step7: Determine the final partial order relation of the alternatives.

(1) x_i ≻ x_j if φ(x_i) > φ(x_j)

(2) x_i = x_j if φ(x_i) = φ(x_j)

In the proposed method, Step1-3 are equivalent to pre-processing the data in advance, including the determination and elimination of hidden inherent preference, Step4-7 is the subsequent decision-making process, which calculates the total dominance degree of each alternative and determine the final partial order relation. Particularly, if the decision-making process is started directly from Step 4, the ranking result without eliminating preferences will be obtained. The procedure of proposed method are shown in Figure 3 fig:decision-making-procdure.

6 An illustrative example

With the rapid progress of science and technology, problems such as energy shortage and environmental pollution are gradually intensifying. In order to achieve carbon reduction goals, the new energy vehicle market has developed rapidly. Nevertheless, as an important component of new energy vehicles, the development speed of electric vehicles (EV)is limited by charging facilities and other supporting facilities. Nowadays, the problems of electric vehicles charging station (EVCS) are mainly reflected in poor efficiency, inconvenient settlement, poor compatibility and serious impact by bad weather. Therefore, the design of charging station is particularly important.

To determine an optimal EVCS, there are five alternatives denoted as X ={ x₁, x₂, x₃, x₄, x₅ }, four criteria denoted as C ={ c₁, c₂, c₃, c₄ }, where c₁ refers to total compatibility volume of available motorcycle type, c₂ refers to charging efficiency, c₃ refers to the transaction settlement convenience and c₄ refers to the security. Through a questionnaire, we collect three experts D ={ d₁, d₂, d₃ } to give their opinions about the five alternatives in terms of these four criteria by q-ROFN. What’s more, the weight vector are completely unknown.

Step1: Construct the decision matrices, which are shown in Table 2-4.

Step4: Determine the criteria weight vector W^l =(w₁^l, w₂^l, w₃^l, w₄^l) , l = 1, 2, 3 and the weight vector of decision makers $\bar{W}$ by Eq (13) and (14): $\begin{matrix} W^{1} = (0.12, 0.44, 0.14, 0.30) \\ W^{2} = (0.35, 0.16, 0.30, 0.19) \\ W^{3} = (0.23, 0.05, 0.54, 0.18) \\ \bar{W} = (0.45, 0.21, 0.34) \end{matrix}$ Step5: The inflow, outflow and the net flow are calculated by Eq (17) and the result are shown in Table 9.

Table 9
The inflow, outfolw and net flow of five altermatives

x ₁ x ₂ x ₃ x ₄ x ₅

inflow 0.1210 0.4236 0.0535 0.1020 0.2919

outfolw 0.1998 0.0712 0.3254 0.2735 0.1302

net flow -0.0788 0.3524 -0.2719 -0.1635 0.1617

	x ₁	x ₂	x ₃	x ₄	x ₅
inflow	0.1210	0.4236	0.0535	0.1020	0.2919
outfolw	0.1998	0.0712	0.3254	0.2735	0.1302
net flow	-0.0788	0.3524	-0.2719	-0.1635	0.1617

Step6: Rank the five alternatives: $x_{2} ≻ x_{5} ≻ x_{1} ≻ x_{4} ≻ x_{3}$

7 Analysis and discussion

To demonstrate the feasibility of the proposed method, below we perform some comparative analyses with the some existing methods and the proposed method. In order to reduce the uncertain influence of variables on the results, the criteria weight and expert weight is still the weight obtained in this paper.

7.1 Compared with the q-ROFWA aggregation operator-based method

Step 1: Normalized the decision matrices of every DM $R^{k} = {[r_{ij}^{k}]}_{n \times m}, k = 1, 2, 3$ , which are shown in Tables 6, 7 and 8, respectively.

Step 2: Determine the criteria weight vector W^k =(w₁^k, w₂^k, w₃^k, w₄^k) , l = 1, 2, 3 and the weight vector of decision makers $\bar{W} = {w_{1}, w_{2}, w_{3}}$ by Eq (13) and Eq (14). $\begin{matrix} W^{1} = (0.12, 0.44, 0.14, 0.30) \\ W^{2} = (0.35, 0.16, 0.30, 0.19) \\ W^{3} = (0.23, 0.05, 0.54, 0.18) \\ \bar{W} = (0.45, 0.21, 0.34) \end{matrix}$

Step 3: The comprehensive values r_i, i = 1, 2, 3, 4, 5 of each alternative is calculated by the q-ROFWA operator, taht is $r_{i} = \oplus_{k = 1}^{3} w_{k} (\oplus_{j = 1}^{4} w_{j}^{k} r_{ij}^{k})$ $\begin{matrix} r_{1} = (0.5432, 0.3094), r_{2} = (0.6839, 0.2559) \\ r_{3} = (0.4685, 0.3545), r_{4} = (0.5196, 0.3025) \\ r_{5} = (0.6437, 0.2804) \end{matrix}$

Step 4: The score of each alternative can be obtained by Eq (5). $\begin{matrix} E (r_{1}) = 0.5965; E (r_{2}) = 0.6942; E (r_{3}) = 0.5544 \\ E (r_{4}) = 0.5846; E (r_{5}) = 0.6626 \end{matrix}$

Step 5: Get the ranking of the alternatives

The ranking results by q-ROFWA operator is x₂ ≻ x₅ ≻ x₁ ≻ x₄ ≻ x₃

7.2 Compared with the fuzzy MABAC method

The MABAC model is an useful method, which uses a border approximation area (BAA) and then calculate the distance between each alternative and the BAA to get each alternative’s sum value. Below we use q-rung orthopair fuzzy MABAC model [28] to solve the above case.

Step 1: Normalized the decision matrices of every DM $R^{k} = {[r_{ij}^{k}]}_{n \times m}, k = 1, 2, 3$ , which are shown in Tables 6, 7 and 8, respectively.

Step 2: Integrate evaluation information from all DMs according to DM’s weights (Suppose that the weight vector of the criteria is also (0.45, 0.21, 0.34)), $r_{ij} = \oplus_{k = 1}^{3} w_{k} r_{ij}^{k}$ represents the aggregated evaluation and the results are listed in Table 10.

Table 10
The comprehensive evaluation information of DMs

c ₁ c ₂ c ₃ c ₄

x ₁ (0.42,0.40) (0.47,0.30) (0.53,0.31) (0.58,0.31)

x ₂ (0.52,0.29) (0.53,0.33) (0.71,0.23) (0.69,0.29)

x ₃ (0.43,0.47) (0.35,0.38) (0.54,0.25) (0.52,0.34)

x ₄ (0.53,0.29) (0.58,0.28) (0.40,0.28) (0.43,0.35)

x ₅ (0.52,0.27) (0.73,0.27) (0.45,0.36) (0.54,0.35)

	c ₁	c ₂	c ₃	c ₄
x ₁	(0.42,0.40)	(0.47,0.30)	(0.53,0.31)	(0.58,0.31)
x ₂	(0.52,0.29)	(0.53,0.33)	(0.71,0.23)	(0.69,0.29)
x ₃	(0.43,0.47)	(0.35,0.38)	(0.54,0.25)	(0.52,0.34)
x ₄	(0.53,0.29)	(0.58,0.28)	(0.40,0.28)	(0.43,0.35)
x ₅	(0.52,0.27)	(0.73,0.27)	(0.45,0.36)	(0.54,0.35)

Step 3: Since MABAC method requires the weight of criteria to be consistent, $w_{j} = \sum_{k = 1}^{3} w_{k} w_{j}^{k}$ is assumed to be true. According to the weight of criteria, the weighted comprehensive matrix WR_ij = w_j · r_ij =(μ_ij, v_ij) , i = 1, 2, . . , 5 ; j = 1, 2, . . . , 4. The results are listed in Table 11.

Table 11

The weighted comprehensive matrix

	c ₁	c ₂	c ₃	c ₄
x ₁	(0.25,0.83)	(0.30,0.74)	(0.37,0.69)	(0.36,0.76)
x ₂	(0.32,0.77)	(0.34,0.76)	(0.50,0.63)	(0.44,0.75)
x ₃	(0.26,0.86)	(0.22,0.79)	(0.38,0.65)	(0.32,0.78)
x ₄	(0.32,0.78)	(0.38,0.73)	(0.27,0.67)	(0.27,0.78)
x ₅	(0.31,0.76)	(0.48,0.72)	(0.31,0.72)	(0.34,0.78)

Step 4: Compute the values of BAA matrix G = [g_j] _1×n.where $\begin{matrix} g_{j} = {(\underset{i = 1}{\prod^{5}} W R_{ij})}^{1 / 5} \\ = {{(\underset{i = 1}{\prod^{5}} μ_{ij})}^{1 / 5}, \sqrt[q]{1 - \underset{i = 1}{\prod^{5}} {(1 - v_{ij}^{q})}^{1 / 5}}} \end{matrix}$ The results are listed as follows: $\begin{matrix} g_{1} = (0.29, 0.80), g_{2} = (0.33, 0.75) \\ g_{3} = (0.36, 0.68), g_{4} = (0.34, 0.77) \end{matrix}$

Step 5: Compute the distance d_ij, i = 1, . . , 5 ; j = 1, . . , 4 between each alternatives and the BAA, the distance is defined as: $d_{ij} = {\begin{matrix} d (W R_{ij}, g_{j}), & if W R_{ij} > g_{j} \\ 0, & if W R_{ij} = g_{j} \\ - d (W R_{ij}, g_{j}) & if W R_{ij} < g_{j} \end{matrix}$ where $\begin{matrix} d (W R_{ij}, g_{i}) = \frac{1}{2} | {(μ_{W R_{ij}})}^{q} - {(μ_{g_{i}})}^{q} | \\ + | {(v_{W R_{ij}})}^{q} - {(v_{g_{i}})}^{q} | + | {(π_{W R_{ij}})}^{q} - {(π_{g_{i}})}^{q} | \end{matrix}$ The results are set out in Table 12.

Table 12

The distance between WCM and the BAA

x ₁	-0.0476	0.0258	-0.0301	0.0125
x ₂	0.0565	-0.0218	0.0829	0.0462
x ₃	-0.1904	-0.0684	0.0364	-0.0082
x ₄	0.0520	0.0327	-0.0374	-0.0210
x ₅	0.0772	0.0762	-0.0719	-0.0219

Step 6: Sum the values of each alternative $S_{i} = \sum_{j = 1}^{n} d_{ij}$ and get the ranking results. $\begin{matrix} S_{1} = - 0.0393, S_{2} = 0.1638, S_{3} = - 0.1496 \\ S_{4} = 0.0264, S_{5} = 0.0595 \end{matrix}$ Thus, the ranking results by fuzzy MABAC method is x₂ ≻ x₅ ≻ x₄ ≻ x₁ ≻ x₃

According to the above comparison, it can be easily find that the three methods have the same optimal ranking results, which shows the feasibility of the proposed method. In addition, compared with the proposed method, it can be seen that the ranking results of aggregation operator-based method is less discriminative. Although the fuzzy MABAC method overcomes the problem of low discrimination, its calculation is more complicated and it requires the weight of the criterion to be consistent, that is the criteria weight is the same for each expert by default, which does not necessarily correspond to reality. However, numerous studies have shown that the operation of the PROMETHEE method is easy and convenient which just requires the criterion value of each project and weight vector of criteria. Different DMs are considered to have different criteria weights in this paper. therefore, PROMETHEE method is a better choice to get the ranking results.

7.3 Sensitivity analysis

In order to further explore the impact of the value of q on the decision results, according to our proposed method, we can obtain the ranking relation of the five alternatives when q get other values. The results are presented in Figure 4.

Fig. 4

Ranking results of alternatives when q ∈ [1, 10].

It can be easily find that with the change of q values, the ranking of the alternatives has also changed slightly, but x₃ and x₄ are always ranked behind, what’s more, it is noted that the best alternative is still x₂ when q ≥ 3, which shows the robustness of the proposed method.

7.4 The necessity of preference elimination

To manifest the necessity of preference elimination, for different q values, the ranking results obtained by the complete decision procedure are compared with those obtained by deleting Step 2 and Step 3, the results are shown in the following table.

Fig. 5

Ranking comparison of before and after perferenceelimination.

As can be seen from Table 13, there is a big difference in the ranking results before and after eliminating preferences. Analyze from the numerical data

Table 13

Ranking results before and after preference elimination

q	Without preference elimination	Preference elimination
1	x₂ ≻ x₅ ≻ x₁ ≻ x₃ ≻ x₄	x₂ ≻ x₅ ≻ x₄ ≻ x₃ ≻ x₁
2	x₅ ≻ x₂ ≻ x₁ ≻ x₄ ≻ x₃	x₂ ≻ x₅ ≻ x₄ ≻ x₃ ≻ x₁
3	x₂ ≻ x₅ ≻ x₁ ≻ x₄ ≻ x₃	x₂ ≻ x₄ ≻ x₅ ≻ x₃ ≻ x₁
4	x₂ ≻ x₅ ≻ x₁ ≻ x₄ ≻ x₃	x₂ ≻ x₄ ≻ x₅ ≻ x₃ ≻ x₁
5	x₂ ≻ x₅ ≻ x₁ ≻ x₄ ≻ x₃	x₂ ≻ x₄ ≻ x₅ ≻ x₃ ≻ x₁
6	x₂ ≻ x₁ ≻ x₅ ≻ x₄ ≻ x₃	x₂ ≻ x₄ ≻ x₅ ≻ x₃ ≻ x₁
7	x₂ ≻ x₁ ≻ x₅ ≻ x₃ ≻ x₄	x₄ ≻ x₅ ≻ x₂ ≻ x₃ ≻ x₁
8	x₂ ≻ x₁ ≻ x₅ ≻ x₃ ≻ x₄	x₄ ≻ x₅ ≻ x₂ ≻ x₃ ≻ x₁
9	x₂ ≻ x₁ ≻ x₅ ≻ x₃ ≻ x₄	x₄ ≻ x₅ ≻ x₂ ≻ x₃ ≻ x₁
10	x₂ ≻ x₁ ≻ x₅ ≻ x₄ ≻ x₃	x₄ ≻ x₅ ≻ x₂ ≻ x₃ ≻ x₁

TP E_{i} = φ (x_{i}) - φ ({x_{i}}^{*})

is denoted as the total preference effect on x_i. where φ(x_i^*) and φ(x_i) represent the total net flow before and after elimination respectively. Then TPE_i > 0 represents that the positive effects of preference outweigh the negative effects while TPE_i < 0 represents that the negative effects of preference outweigh the positive effects. Assume q = 3, based on Table 2-4 and Table 6-8, TPE

Table 2

The q-ROFN decision matrix provided by d₁

	c ₁	c ₂	c ₃	c ₄
x ₁	(0.5,0.4)	(0.5,0.3)	(0.5,0.3)	(0.5,0.4)
x ₂	(0.6,0.2)	(0.6,0.3)	(0.6,0.3)	(0.6,0.3)
x ₃	(0.5,0.4)	(0.3,0.5)	(0.6,0.2)	(0.4,0.4)
x ₄	(0.6,0.2)	(0.6,0.2)	(0.4,0.2)	(0.3,0.6)
x ₅	(0.4,0.3)	(0.7,0.2)	(0.4,0.5)	(0.4,0.5)

Table 3

The q-ROFN decision matrix provided by d₂

	c ₁	c ₂	c ₃	c ₄
x ₁	(0.4,0.2)	(0.6,0.2)	(0.4,0.4)	(0.5,0.3)
x ₂	(0.5,0.3)	(0.6,0.2)	(0.6,0.2)	(0.5,0.4)
x ₃	(0.4,0.4)	(0.6,0.2)	(0.5,0.3)	(0.7,0.2)
x ₄	(0.5,0.4)	(0.7,0.2)	(0.5,0.2)	(0.7,0.2)
x ₅	(0.6,0.3)	(0.7,0.2)	(0.4,0.2)	(0.4,0.2)

Table 4

The q-ROFN decision matrix provided by d₃

	c ₁	c ₂	c ₃	c ₄
x ₁	(0.4,0.5)	(0.5,0.2)	(0.5,0.3)	(0.5,0.2)
x ₂	(0.5,0.4)	(0.5,0.3)	(0.6,0.2)	(0.7,0.2)
x ₃	(0.4,0.5)	(0.5,0.2)	(0.4,0.3)	(0.5,0.3)
x ₄	(0.5,0.3)	(0.5,0.3)	(0.3,0.5)	(0.5,0.2)
x ₅	(0.6,0.2)	(0.6,0.3)	(0.4,0.4)	(0.6,0.3)

Step2: Calculate the hidden inherent preference of DMs according to definition 8-10:

Table 5

the hidden preference vector of each DM

	PA	PC
d ₁	(0.01,0.06,-0.06,-0.08,-0.12)	(0.09,-0.06,0.05,-0.28)
d ₂	(0.01,-0.06,0.19,0.23,0.06)	(-0.01,0.24,0.03,0.17)
d ₃	(0.08,-0.03,0.11,0.11,0.06)	(-0.01,0.37,-0.27,0.24)

Step3:Preference elimination for initial information according to Eq (10) and (11):

Table 6

Preference elimination for d₁

	c ₁	c ₂	c ₃	c ₄
x ₁	(0.46,0.44)	(0.53,0.28)	(0.47,0.32)	(0.69,0.32)
x ₂	(0.52,0.23)	(0.60,0.30)	(0.54,0.34)	(0.78,0.25)
x ₃	(0.49,0.41)	(0.34,0.44)	(0.61,0.20)	(0.59,0.29)
x ₄	(0.60,0.20)	(0.70,0.17)	(0.41,0.20)	(0.45,0.43)
x ₅	(0.42,0.29)	(0.85,0.17)	(0.43,0.47)	(0.63,0.35)

Table 7

Preference elimination for d₂

	c ₁	c ₂	c ₃	c ₄
x ₁	(0.40,0.20)	(0.48,0.27)	(0.39,0.41)	(0.42,0.37)
x ₂	(0.54,0.28)	(0.51,0.25)	(0.62,0.19)	(0.45,0.46)
x ₃	(0.34,0.49)	(0.41,0.33)	(0.41,0.38)	(0.50,0.28)
x ₄	(0.41,0.51)	(0.46,0.28)	(0.40,0.27)	(0.49,0.28)
x ₅	(0.57,0.31)	(0.53,0.28)	(0.37,0.22)	(0.32,0.26)

Table 8

Preference elimination for d₃

	c ₁	c ₂	c ₃	c ₄
x ₁	(0.38,0.54)	(0.34,0.34)	(0.63,0.26)	(0.37,0.29)
x ₂	(0.52,0.38)	(0.38,0.46)	(0.84,0.15)	(0.58,0.26)
x ₃	(0.37,0.55)	(0.33,0.35)	(0.49,0.27)	(0.36,0.44)
x ₄	(0.46,0.33)	(0.33,0.53)	(0.37,0.44)	(0.36,0.30)
x ₅	(0.57,0.21)	(0.41,0.48)	(0.51,0.34)	(0.45,0.42)

_i, i = 1, 2, . . , 5 are obtained $\begin{matrix} TP E_{1} = - 0.0685, TP E_{2} = 0.1133, TP E_{3} = 0.1614 \\ TP E_{4} = 0.1769, TP E_{5} = - 0.1565 \end{matrix}$ Therefore, if the ranking results obtained by our method are restored, without eliminating preferences, {x₁, x₅} should be ranked lower than the current ranking while {x₂, x₃, x₄} should be ranked higher than the current ranking, which indicates that the influence of preference cannot be ignored. They are visually displayed in Figure 5.

8 Conclusion

With the development of technology and the increase of social demands, more and more individuals can participate in the decision-making process, which puts forward higher requirements for the accuracy of evaluation information. However, due to different experience and psychological cognition of decision-makers, the HIP of evaluation information is obviously different. Therefore, to improve the reasonability of decision-making, under the environment of q-ROFNs, this paper first introduces a rigorous score function to be prepared for the subsequent determination of HIP. Then based on the rigorous score function, some progressive concepts are proposed, including AVS, PSA, NSA, which make sure that the determination and quantification of HIP is reasonable. Furthermore, a risk model is established to eliminate the HIP by analyzing the possible impact. Finally, we combined the PROMETHEE method with q-ROFNs and applied the new PROMETHEE method to a practical decision-making problem to indicate the validity of the proposed method.

In decision problems, information can be influenced by potential risk factors and leads to deviation, to describe associated risk with the fuzzy data, some concepts like R-numbers[30] and R-Sets[31] are proposed. However, these models give risk values subjectively, which leads to a great deal of subjectivity. Compared with these existing methods, in this paper, all the risks that could lead to deviations are aggregated into HIP and HIP is determinated from the initial evaluation, which ensures the objectivity of the extraction. More deeply, this is a preprocessing process on the initial data, which leads to a more reasonable information. What’s more, the decision results we obtained are based on the dominant relation between q-ROFNs, which will reduce more uncertainties compared with the traditional aggregation operators. Hence, the method proposed in this paper has a great applicability.

The main contributions of the study can be summarized as three points.

Some progressive concepts are proposed based on a novel rigorous score function to determine the HIP.

To improve the reasonability and validity of decision results, HIP is determined and eliminated by a risk model.

We introduce the dominant relation between q-ROFNs to provide a novel PROMETHEE method with q-ROFSs.

Footnotes

Acknowledgment

The authors are highly thankful to any anonymous referee for their careful reading and insightful comments, their constructive opinions and suggestions will generate an improved version of this article. The work is supported by the National Nature Science Foundation of China (No.72171002, No.61806001) and the Project of Graduate Academic Innovation of Anhui University.

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The determination and elimination of hidden inherent preference using q-ROFNs multicriteria group decision making problem

Abstract

Keywords

1 Introduction

2 Preliminaries

4.1 Problem description

5.1 The dominant function of q-ROFN

Table 9 The inflow, outfolw and net flow of five altermatives x 1 x 2 x 3 x 4 x 5 inflow 0.1210 0.4236 0.0535 0.1020 0.2919 outfolw 0.1998 0.0712 0.3254 0.2735 0.1302 net flow -0.0788 0.3524 -0.2719 -0.1635 0.1617

7.1 Compared with the q-ROFWA aggregation operator-based method

7.2 Compared with the fuzzy MABAC method

Footnotes

Acknowledgment

References

Table 9
The inflow, outfolw and net flow of five altermatives

x ₁ x ₂ x ₃ x ₄ x ₅

inflow 0.1210 0.4236 0.0535 0.1020 0.2919

outfolw 0.1998 0.0712 0.3254 0.2735 0.1302

net flow -0.0788 0.3524 -0.2719 -0.1635 0.1617