q-Rung orthopair fuzzy frank power point aggregation operators with new multi-parametric distance measures

Abstract

The recently proposed q-rung orthopair fuzzy set (q-ROFS) whose main feature is that the qth power of membership degree (MD) and the qth power of non-membership degree (NMD) is equal to or less than 1, is a powerful tool to describe uncertainty. The major contribution of this paper lies to investigate power point average (PPA) aggregation operators with q-rung orthopair fuzzy information based on Frank t-conorm and t-norm. Since the existing power average (PA) operators all rely on the traditional distance measures to measure support degree between the input values, it cannot reflect decision makers’ attitude. In response, this paper introduces firstly a series of distance measures for q-rung orthopair fuzzy numbers (q-ROFNs) based on point operators, from which the corresponding support measures can be obtained. Secondly, based on the proposed point distance measures, new Frank power point average aggregation operators are proposed to aggregate q-rung orthopair fuzzy information. Finally, a novel multiple attribute decision making (MADM) technique is presented based on the proposed Frank power point average aggregation operators. The developed MADM method not only can get more objective information, but also avoid the influence of unduly high or low attribute values on the decision result, providing a new way for decision makers (DMs) under q-rung orthopair fuzzy environment.

Keywords

Multi-attribute decision making frank operational laws

1 Introduction

MADM is the process of ranking alternatives and selecting an optimal one among a set of alternatives assessed on multiple attributes, which is the essence of the decision-making science. MADM has played more and more important role in daily life and has been widely used in economics, management and other fields in recent years [1 –4]. In this process, DMs’ evaluations over alternatives is very important. On the other hand, the second task in this process is to aggregate the described information. Due to the subjective nature of human thinking in real decision-making problems, DMs’ evaluations over alternatives are always imprecise and fuzzy. To appropriately describe the fuzzy information, quite a few tools, such as intuitionistic fuzzy set (IFS) [5], Pythagorean fuzzy set (PFS) [6], complex intuitionistic fuzzy set [7], interval-valued intuitionistic fuzzy sets [8] have been proposed.

Recently, Yager generalized the IFS and PFS and proposed a new concept of q-ROFS whose prominent feature is that the qth power of MD and the qth power of NMD is equal to or less than one [9]. Evidently, q-ROFS relaxes the constraints of both IFS and PFS, which makes it more powerful than IFS and PFS in the aspect of dealing with vagueness. For instance, a decision maker (DM) provides 0.9 and 0.8 as the MD and NMD, respectively. Given 0.9 + 0.8 > 1 and 0.9² + 0.8² > 1, the evaluation attribute value (0.9, 0.8) cannot be expressed by IFS and PFS. In this case, when q = 5, we can get 0.9⁵ + 0.8⁵ < 1 and the evaluation value can be expressed by q-ROFS. Therefore, by adjusting the value of parameter q, q-ROFS allow DMs to independently assign values to MD and NMD. Due to its higher capacity of modelling the fuzziness, quite a few of achievements on q-ROFS have been done, such as distance measures between q-ROFNs [10], integrations of q-rung orthopair fuzzy continuous information [11, 12], continuities, derivatives and differentials of q-ROFS [13], ranking technique [14], knowledge measure [15], combination of q-ROFNs and other fuzzy sets [16, 17], overview on q-ROFS [18], etc. In addition, more scholars have focused on q-rung orthopair fuzzy MADM methods based on aggregation operator theory that can aggregate a collection of individual evaluated values into one. For example, Liu and Wang [19] proposed a family of q-rung orthopair fuzzy weighted averaging and geometric operators. Peng et al. [20] proposed a new exponential operational law on q-ROFNs and then applied it to derive the q-rung orthopair fuzzy weighted exponential aggregation operator. In order to control the uncertainty of valuating data and get intensive information, by associating with classic arithmetic and geometric operators, Xing et al. [21] present a new class of q-rung orthopair fuzzy point weighted aggregation operators. Considering these aggregation operators do not take the interrelationship between q-ROFNs into account, Liu and Liu [22], and Wei et al. [23] successively extended existing operators, such as Bonferroni mean (BM), and Heronian mean to the q-ROFS. Meanwhile, Liu et al. [24] propose a new method based on the q-rung orthopair fuzzy extended BM operator and entropy measure for dealing with heterogeneous relationships among attributes and unknown attribute weight information. Liu et al. [25] put forward a novel MAGDM method based on the q-rung orthopair fuzzy power Maclaurin symmetric mean operator (MSM) to solve the decision-making problems of attribute related and unreasonable evaluating data. However, in real decision-making process, the interrelationships of arguments do not always exist in all the arguments, but in part of the arguments. In order to deal with this situation, Yang and Pang [26] developed several q-rung orthopair fuzzy partitioned BM operators and their weighted forms.

From the above analysis, we can find that researches on MADM methods with q-rung orthopair fuzzy information have achieved great success. However, all these existing methods above are based on Einstein or Algebraic t-norm and t-conorm, which makes them lack of flexibility in the process of aggregation. Currently, Frank t-conorm and t-norm proposed by Frank [27] are the only class of t-conorm and t-norm that satisfy the compatibility. The Frank t-conorm and t-norm have the same advantages as the algebraic, Hamacher, and Einstein t-conorm and t-norm do. On the other hand, they also have an additional parameter that makes aggregation process more flexible than general t-conorm and t-norm. Therefore, it is meaningful to investigate aggregation operators based on Frank t-conorm and t-norm. Besides, different q-rung orthopair fuzzy aggregation operators are used to solve different decision problems. Among them, the PA operator proposed by Yager [28] considers the information about the relationship between the input arguments, and is used to relieve the influences of unreasonable data given by DMs on the decision result. It can aggregate the input data by assigning the weighted vector based on the support degree between the input values. That is to say, the PA operator depends on the support degree of input arguments to assign weights. However, the existing PA operators all depend on the traditional distance measures, such as normalized Hamming distance, Minkowski-type distance, to obtain corresponding support degree between the input values. Considering the DMs’ risk preference, there is a need to define novel PA operators according to different distance measures to obtain more objective information from the original input arguments. Moreover, since Frank t-conorm and t-norm are flexible and powerful in the process of aggregation, it is of great meaning to combine the Frank t-conorm and t-norm and the PA operator for MADM under q-rung orthopair fuzzy environment. Therefore, the motivation of this paper is to propose new q-rung orthopair fuzzy PA aggregation operators based on the Frank t-conorm and t-norm and then apply them to MADM problems. The main contribution and innovation of this paper is the combination of point distance measure and PA operator, which can obtain more objective decision information, and also avoid the influence of too high or too low attribute value on decision result.

Based on the above comprehensive analysis, the goal of this paper are: (1) to introduce new Frank operational laws for q-ROFNs on the basis of Frank t-conorm and t-norm; (2) to define new PPA operator; (3) to develop q-rung orthopair fuzzy Frank power point aggregation operators. (4) to provide a new train of thought for MADM with q-rung orthopair fuzzy information based on the proposed operators.

In order to do this, the rest of the paper is organized as following. Section 2 reviews basic concepts. New Frank operational laws for q-ROFNs are also introduced in this section. Section 3 develops q-rung orthopair fuzzy power point aggregation operators based on Frank operational laws, including q-rung orthopair fuzzy Frank power point averaging (q-ROFFPPA)operator, q-rung orthopair fuzzy Frank power point geometric (q-ROFFPPG) operator. Section 4 introduces a novel method for solving MADM with q-rung orthopair fuzzy information based on the proposed operators. Section 5 provides a numerical example to show the performance of the proposed method.

2 Preliminaries

In this section, we briefly review basic concepts, such as q-ROFS, Frank t-norm and t-conorm, PA operator. Then, we introduce a series of point distance measures for q-ROFNs. Based on it, new PPA operator is also developed in this section.

2.1 q-Rung orthopair fuzzy set

Definition 1 [9]. Let X be an ordinary fixed set, a q-ROFS A defined on X is given by

$A = {〈 x, μ_{A} (x), v_{A} (x) 〉 | x \in X}$ (1) where μ_A ( x ) and v_A ( x ) represent the MD and NMD respectively, satisfying μ_A ( x ) ∈ [0, 1], v_A ( x ) ∈ [0, 1] and 0 ≤ μ_A ( x ) ^q + v_A ( x ) ^q ≤ 1,( q ≥ 1 ). The indeterminacy degree is defined as π_A ( x ) = ( 1 - ( μ_A ( x ) ) ^q - ( v_A ( x ) ) ^q ) ^1/q. For convenience, A = ( μ_A, v_A ) is called a q-ROFN by Liu and Wang [19].

Operational laws for q-ROFNs are also proposed as below.

Definition 2 [19]. Let a = ( μ, v ) , a₁ = ( μ₁, v₁ ), and a₂ = ( μ₂, v₂ ) be three q-ROFNs, and λ be a positive real number, then

(1) $a_{2} \oplus a_{2} = ({(u_{1}^{q} + u_{2}^{q} - u_{1}^{q} u_{2}^{q})}^{1 / q}, v_{1} v_{2})$

(2) $a_{2} \otimes a_{2} = (μ_{1} μ_{2}, {(v_{1}^{q} + v_{2}^{q} - v_{1}^{q} v_{2}^{q})}^{1 / q})$

(3) λa = ( ( 1 - ( 1 - μ^q ) ^λ ) ^1/q, v^λ )

(4) a^λ = ( μ^λ, ( 1 - ( 1 - v^q ) ^λ ) ^1/q ) .

To compare two q-ROFNs, Liu and Wang [19] proposed a comparison method for q-ROFNs.

Definition 3 [19]. Let a = ( μ_a, v_a ) be a q-ROFN, then the score of a is defined asS ( a ) = μ^q - v^q, the accuracy of a is defined asH ( a ) = μ^q + v^q. For any two q-ROFNs, a₁ = ( μ₁, v₁ ) and a₂ = ( μ₂, v₂ ). Then

(1) If S ( a₁ ) > S ( a₂ ), then a₁ > a₂;

(2) If S ( a₁ ) = S ( a₂ ), then If H ( a₁ ) > H ( a₂ ), then a₁ > a₂;

If H ( a₁ ) = H ( a₂ ), then a₁ = a₂

2.2 New Frank operational laws of q-ROFNs

Frank t-norm and t-conorm introduced by Frank [27] are well-known t-conorm and t-norm, which is defined as follows:

$T_{γ} (x, y) = {log}_{θ} (1 + \frac{(θ^{x} - 1) (θ^{y} - 1)}{θ - 1}),$ (2) $\begin{matrix} \forall x, y \in [0, 1], θ \in (1, + \infty) \end{matrix}$

$S_{γ} (x, y) = 1 - {log}_{θ} (1 + \frac{(θ^{1 - x} - 1) (θ^{1 - y} - 1)}{θ - 1}),$ (3) $\begin{matrix} \forall x, y \in [0, 1], θ \in (1, + \infty) \end{matrix}$ where T_γ ( x, y ), S_γ ( x, y ) is a t-conorm and t-norm, respectively.

Based on the above Frank t-norm and t-conorm, we can develop new operational rules on q-ROFNs.

Definition 4. Leta₁ = ( μ₁, v₁ ) and a₂ = ( μ₂, v₂ ) be any two q-ROFNs, then the operational laws based on Frank t-norm and t-conorm are defined as:

$(1) a_{1} \oplus_{F} a_{2} = ({\begin{matrix} (1 - \log_{θ} (1 + \frac{(θ^{1 - μ_{1}^{q}} - 1) (θ^{1 - μ_{2}^{q}} - 1)}{θ - 1})) \\ {(\log_{θ} (1 + \frac{(θ^{v_{1}^{q}} - 1) (θ^{v_{2}^{q}} - 1)}{θ - 1}))}^{\frac{1}{q}} \end{matrix}}^{\frac{1}{q}},)$ $(2) a_{1} \oplus_{F} a_{2} = (\begin{matrix} {(\log_{θ} (1 + \frac{(θ^{μ_{1}^{q}} - 1) (θ^{μ_{2}^{q}} - 1)}{θ - 1}))}^{\frac{1}{q}}, \\ {(1 - \log_{θ} (1 + \frac{(θ^{1 - v_{1}^{q}} - 1) (θ^{1 - v_{2}^{q}} - 1)}{θ - 1}))}^{\frac{1}{q}} \end{matrix})$ $(3) k \cdot_{F} a = (\begin{matrix} {(1 - \log_{θ} (1 + \frac{{(θ^{1 - μ_{1}^{q}} - 1)}^{k}}{{(θ - 1)}^{k - 1}}))}^{\frac{1}{q}}, \\ {(\log_{θ} (1 + \frac{{(θ^{v^{q}} - 1)}^{k}}{{(θ - 1)}^{k - 1}}))}^{\frac{1}{q}} \end{matrix}) k ≻ 0$ $(4) a^{^F^{k}} = (\begin{matrix} {(\log_{θ} (1 + \frac{{(θ^{μ_{1}^{q}} - 1)}^{k}}{{(θ - 1)}^{k - 1}}))}^{\frac{1}{q}}, \\ {(1 - \log_{θ} (1 + \frac{{(θ^{1 - v^{q}} - 1)}^{k}}{{(θ - 1)}^{k - 1}}))}^{\frac{1}{q}} \end{matrix}) k ≻ 0$

Theorem 1. Let a = ( μ, v ), a₁ = ( μ₁, v₁ ), and a₂ = ( μ₂, v₂ ) be any three q-ROFNs, then

(1) a₁ ⊕ _Fa₂ = a₂ ⊕ _Fa₁,

(2) a₁ ⊗ _Fa₂ = a₂ ⊗ _Fa₁,

(3) k ( a₁ ⊕ _Fa₂ ) = ka₂ ⊕ _Fa₁, k ≥ 0,

(4) k₁a ⊕ _Fk₂a = ( k₁ + k₂ ) a, k₁, k₂ ≥ 0,

(5) a^{k
₁} ⊗ _Fa^{k
₂} = ( a ) ^k₁ +k₂, k₁, k₂ ≥ 0,

(6) $a_{1}^{k} \otimes_{F} a_{2}^{k} = {(a_{2} \otimes_{F} a_{1})}^{k}$ , k ≥ 0.

2.3 New point distance measures for q-ROFNs

Distance measures, as an effective tool to compare fuzzy information, have been widely used in MADM problems. Recently, Minkowski-type distance for q-ROFNs is proposed by Du [10].

Definition 5 [10]. Let a₁ = (μ₁, v₁) and a₂ = (μ₂, v₂) be any two q-ROFNs (q > 1), then the Minkowski-type distance between a₁and a₂ is given by

$d (a_{1}, a_{2}) = {(\frac{1}{2} {| μ_{1} - μ_{2} |}^{q} + \frac{1}{2} {| v_{1} - v_{2} |}^{q})}^{1 / - q}$ (4)

Example 1. Assume that a₁ = ( 0.8, 0.3 ) a₂ = ( 0.6, 0.4 ), a₃ = ( 0.9, 0.4 ) and a₄ = ( 0.7, 0.5 ) be four q-ROFNs and the parameter q is equal to three. Based on the Definition 5, we can obtain the Minkowski-type distance d ( a₁, a₂ ) = 0.1651 and d ( a₃, a₄ ) = 0.1651.

Due to omitting indeterminacy degrees, the calculation result of the distance measure obtained by Du [19] is inconsistent with commonsense. Although the values of the indeterminacy are well known as the consequence of the MD and NMD, these values should not be discarded from the formulas. Szmidt and Kacprzyk [29] give an illustration from the point of both analytical and geometrical about why the three parameters should be considered. In order to overcome the drawback, we propose a modified distance measure taking into account the MD, NMD, and indeterminacy degrees.

Definition 6: Let a₁ = (μ₁, v₁) and a₂ = (μ₂, v₂) be any two q-ROFNs, then the distance between a₁ and a₂ is defined as follows:

$d (a_{1}, a_{2}) = \frac{1}{2} (| μ_{1}^{q} - μ_{2}^{q} | + | ν_{1}^{q} - ν_{2}^{q} | + | π_{1}^{q} - π_{2}^{q} |)$ (5)

Considering the DMs’ risk preference, there is a need to define new distance measures. As point operations can redistribute the MD, NMD in q-ROFNs according to different principle, we introduce the q-rung orthopair fuzzy point operations defined by Xing et al. [21] to reflect DMs’ attitude, and then combine the point operators with distance measures to develop new point distance measures for q-ROFNs.

Definition 7. [21] For a q-ROFN a = ( μ, v ), q ≥ 1, let ξ, ς ∈ [0, 1] , then q-rung orthopair fuzzy point operators is defined as follows:

$D_{ξ} (a) = {{(μ^{q} + ξ π^{q})}^{1 / q}, {(v^{q} + (1 - ξ) π^{q})}^{1 / q}}$

$F_{ξ, ς} (a) = {{(μ^{q} + ξ π^{q})}^{1 / q}, {(v^{q} + ς π^{q})}^{1 / q}},$

where ξ + ς ≤ 1

$G_{ξ, ς} (a) = {{(ξ μ^{q})}^{1 / q}, {(ς v^{q})}^{1 / q}}$

$H_{ξ, ς} (a) = {{(ξ μ^{q})}^{1 / q}, {(v^{q} + ς π^{q})}^{1 / q}}$

$J_{ξ, ς} (a) = {{(μ^{q} + ξ π^{q})}^{1 / q}, {(ς v^{q})}^{1 / q}}$

As Xing et al. [21] point that D_ξ ( a ) and F_ξ,ς ( a ) can reduce the uncertainty of q-ROFS, G_ξ,ς ( a ) can increase the uncertainty of q-ROFS. Meanwhile, H_ξ,ς ( a ) reduce the MD and increase the NMD, which can denote a pessimistic attitude. Similarly, J_ξ,ς ( a ) can denote an optimistic attitude.

Based on the above point operators, we propose new point distance measures for q-ROFNs in the following.

Definition 8: Let a₁ = (μ₁, v₁) and a₂ = (μ₂, v₂) be any two q-ROFNs, $t_{μ} = μ_{1}^{q} - μ_{2}^{q},$ $t_{v} = ν_{1}^{q} - ν_{2}^{q},$ $t_{π} = π_{1}^{q} -$ $π_{2}^{q} = - t_{μ} - t_{ν},$ q ≥ 1, ξ, ς ∈ [0, 1], then the point distance measures between a₁ and a₂ based on D_ξ ( a ) , F_ξ,ς ( a ), G_ξ,ς ( a ) , H_ξ,ς ( a ) , J_ξ,ς ( a ) operators, respectively, are defined as follows:

d_{D
_ξ} ( a₁, a₂ ) = | ( 1 - ξ ) t_μ - ξt_v|

$d_{F_{ξ, ς}} (a_{1}, a_{2}) = \frac{1}{2} (\begin{matrix} | (1 - ξ) t_{μ} - ξ t_{v} | + \\ | (1 - ς) t_{v} - ς t_{μ} | + | (1 - ξ - ς) (t_{μ} + t_{v}) | \end{matrix}),$

where ξ + ς ≤ 1

$d_{G_{ξ, ς}} (a_{1}, a_{2}) = \frac{1}{2} (| ξ t_{μ} | + | ς t_{v} | + | ξ t_{μ} + ς t_{v} |)$

$d_{H_{ξ, ς}} (a_{1}, a_{2}) = \frac{1}{2} (\begin{matrix} | ξ t_{μ} | + | ς t_{μ} - (1 - ς) t_{v} | + \\ | (ξ - ς) t_{μ} + (1 - ς) t_{v} | \end{matrix})$

$d_{J_{ξ, ς}} (a_{1}, a_{2}) = \frac{1}{2} (\begin{matrix} | (1 - ξ) t_{μ} - ξ t_{v} | + | ς t_{v} | + \\ | (1 - ξ) t_{μ} + (ς - ξ) t_{v} | \end{matrix})$

Remark 2: By giving different values of the parameters ξ and ς, we get the following special cases. $\begin{matrix} \begin{matrix} d_{F_{0, 0}} (a_{1}, a_{2}) = d_{G_{1, 1}} (a_{1}, a_{2}) = d_{H_{1, 0}} (a_{1}, a_{2}) \\ = d_{J_{0, 1}} (a_{1}, a_{2}) = \frac{1}{2} (| t_{μ} | + | t_{v} | + | t_{μ} + t_{v} |) \end{matrix} \end{matrix}$ $\begin{matrix} = \frac{1}{2} (| μ_{1}^{q} - μ_{2}^{q} | + | ν_{1}^{q} - ν_{2}^{q} | + | π_{1}^{q} - π_{2}^{q} |) \end{matrix}$

Which is the distance measure defined in Definition 6.

It is easy to verify that the above point distance measures satisfy the following properties.

Theorem 2: Let a₁ = (μ₁, v₂) , a₂ = (μ₂, ν₂) , and a₃ = (μ₃, v₃) be any three q-ROFNs, then the point distance measures for q-ROFN s satisfies following properties:

(1)0 ≤ d_Δ ( a₁, a₂ ) ≤ 1;

(2)d_Δ ( a₁, a₂ ) = d_Δ ( a₂, a₁ );

(3) If a₁ = a₂, then d_Δ ( a₁, a₂ ) = 0;

(4) If a₁ ≤ a₂ ≤ a₃, then $\begin{matrix} d_{Δ} (a_{1}, a_{2}) \leq d_{Δ} (a_{1}, a_{3}) and d_{Δ} (a_{2}, a_{3}) \leq d_{Δ} (a_{1}, a_{3}) . \end{matrix}$

Where Δ denotes D_ξ ( a ) , F_ξ,ς ( a ) , G_ξ,ς ( a ), H_ξ,ς ( a ), J_ξ,ς ( a )

Example 2. Assume that a₁ = (0.8,0.3) and a₂ = (0.6,0.4) be two q-ROFNs. Then the point distance between a₁ and a₂ can be calculated according to Definition 8. Firstly, we can obtain $t_{μ} = μ_{1}^{q} - μ_{2}^{q} = 0.296,$ $t_{v} = ν_{1}^{q} - ν_{2}^{q} = - 0.037,$ and $t_{π} = π_{1}^{q} - π_{2}^{q} = - t_{μ} - t_{ν} = - 0.259$ , then different point distances between a₁ and a₂ are as follows: (Suppose q = 3, ξ = 0.4, ς = 0.1): $\begin{matrix} d_{D_{0.4}} (a_{1}, a_{2}) & = | 0.6 \times 0.296 - 0.4 \times (- 0.037) | \\ = 0.1924 \end{matrix}$

$\begin{matrix} d_{F_{0.4, 0.1}} (a_{1}, a_{2}) \\ = \frac{1}{2} (\begin{matrix} | 0.6 \times 0.296 - 0.4 \times (- 0.037) | + \\ | - 0.9 \times 0.037 - 0.1 \times 0.296 | + | - 0.5 \times 0.259 | \end{matrix}) \\ = 0.1924 \end{matrix}$

$\begin{matrix} d_{G_{0.4, 0.1}} (a_{1}, a_{2}) \\ = \frac{1}{2} (\begin{matrix} | 0.4 \times 0.296 | + | - 0.1 \times 0.037 | + \\ | 0.4 \times 0.296 - 0.1 \times 0.037 | \end{matrix}) \\ = 0.1184 \end{matrix}$ $\begin{matrix} d_{H_{0.4, 0.1}} (a_{1}, a_{2}) \\ = \frac{1}{2} (\begin{matrix} | 0.4 \times 0.296 | + | \begin{matrix} 0.1 \times 0.296 + \\ 0.9 \times 0.037 \end{matrix} | \\ + | 0.3 \times 0.296 + 0.9 \times 0.037 | \end{matrix}) = 0.1184 \end{matrix}$ $\begin{matrix} d_{J_{0.4, 0.1}} (a_{1}, a_{2}) \\ = \frac{1}{2} (\begin{matrix} | 0.6 \times 0.296 + 0.4 \times 0.037 | + \\ | 0.1 \times 0.037 | + | \begin{matrix} 0.6 \times 0.296 + \\ 0.3 \times 0.037 \end{matrix} | \end{matrix}) = 0.1924 \end{matrix}$

From the above analysis, we can see that the point distance measures defined are not only formally correct, but also since they use all the available information, they give in effect results that are correct and consistent with the essence of q-ROFS. On the other hand, the point distance measures can redistribute the MD and NMD of q-ROFS according to different principle, and thus can obtain more objective information from the original one.

By combining with point operators, the new point distances d_{D
_ξ} and d_{F
_ξ,ς} can reduce the uncertainty of q-ROFNs. Similarly, the point distances d_{G
_ξ,ς} increases the uncertainty of q-ROFNs. Meanwhile, the point distances d_{H
_ξ,ς} represents a pessimistic distance measure by increasing the membership information. Similarly, d_{J
_ξ,ς} represents an optimistic distance measure by reducing the membership information. Thus, from the point view of actual needs, DMs can appropriately select different point distance measures proposed in this paper to express their pessimistic or optimistic attitude.

2.4 Power point average operator

The PA operator was firstly proposed by Yager [28]. The advantage of PA operator is that it can reduce the effect of unduly high and low arguments on the final results.

Definition 9 [28]. Let a_i ( i = 1, 2, . . . , n ) be a collection of crisp numbers. The PA operator is defined as:

$\begin{matrix} PA (a_{1}, a_{2}, . . ., a_{n}) \\ = \sum_{i = 1}^{n} ((1 + T (a_{i})) a_{i} / \sum_{j = 1}^{n} (1 + T (a_{j}))) \end{matrix}$ (6) where $T (a_{i}) = \sum_{j = 1, j \neq i}^{n} Sup (a_{i}, a_{j})$ , Sup ( a_i, a_j ) is the support for a_i and a_j. The support satisfies the following properties.

(1) Sup ( a_i, a_j ) ∈ [0, 1]

(2) Sup ( a_i, a_j ) = Sup ( a_j, a_i )

(3) If d ( a_i, a_j ) < d ( a_l, a_k ), then Sup ( a_i, a_j ) > Sup ( a_l, a_k ), where d ( a_i, a_j ) is the distance betweena_i and a_j.

Obviously, the distance measure plays an important role in the PA operator. Before using the PA operator, the key step is to determine the support Sup (a_i, a_j), which depends on the distance of a_i from other values. The smaller the distance, the closer the two values and the more they support each other. Considering that the point distance measures can redistribute the MD and NMD of q-ROFS according to different principle and reflect the DMs’ risk preference, in the following, we define a kind of new PPA operator.

Definition 10. Let a_i ( i = 1, 2, . . . , n ) be a collection of crisp numbers. The PPA operator is defined as:

$\begin{matrix} PPA (a_{1}, a_{2}, . . ., a_{n}) \\ = \sum_{i = 1}^{n} ((1 + T_{Δ} (a_{i})) a_{i} / \sum_{j = 1}^{n} (1 + T_{Δ} (a_{j}))) \end{matrix}$ (7) where $T_{Δ} (a_{i}) = \sum_{j = 1, j \neq i}^{n} S {up}_{Δ} (a_{i}, a_{j})$ , Sup_Δ ( a_i, a_j ) is the point support for a_i and a_j. The point support satisfies the following properties.

(1) Sup_Δ ( a_i, a_j ) ∈ [0, 1]

(2) Sup_Δ ( a_i, a_j ) = Sup_Δ ( a_j, a_i )

(3) If d_Δ ( a_i, a_j ) < d_Δ ( a_l, a_k ), then Sup_Δ ( a_i, a_j ) > Sup_Δ ( a_l, a_k ), where d_Δ ( a_i, a_j ) is the point distance between a_i and a_j defined in Definition 8.

3 q-Rung Orthopair Fuzzy Frank Power Point Aggregation Operators

Based on the Frank operational laws of q-ROFNs and the PPA operator in Sect. 2, this section introduces q-rung orthopair fuzzy Frank power point aggregation operators based on the Frank t-conorm and t-norm.

3.1 q-ROFFPPA aggregation operator

Definition 11. Let a_i = ( μ_i, v_i )(i = 1, 2, …, n) be a collection of q-ROFNs, taking ξ,ς ∈ [0, 1], then a q-ROFFPPA operator is defined as:

$\begin{matrix} q - ROFFPPA (a_{1}, a_{2} \dots a_{n}) \\ = \sum_{i = 1}^{n} \oplus^{F} \frac{(1 + T_{Δ} (a_{i}))}{\sum_{j = 1}^{n} (1 + T_{Δ} (a_{j}))} a_{i} \end{matrix}$ (8) where $T_{Δ} (a_{i}) = \sum_{j = 1, j \neq i}^{n} S {up}_{Δ} (a_{i}, a_{j})$ denotes the point support of the ith q-ROFN by all other q-ROFNs, Sup_Δ ( a_i, a_j ) is the point support for a_i and a_j. The point support satisfies the following properties. (a) Sup_Δ ( a_i, a_j ) ∈ [0, 1] , (b) Sup_Δ ( a_i, a_j ) = Sup_Δ ( a_j, a_i ), (c) If d_Δ ( a_i, a_j ) < d_Δ ( a_l, a_k ), then Sup_Δ ( a_i, a_j ) > Sup_Δ ( a_l, a_k ), where d_Δ ( a_i, a_j ) is the point distance between a_i and a_j defined in Definition 8.

According to Frank operational laws defined in Definition 4, we can obtain the following Theorem.

Theorem 3. Let a_i = ( μ_i, v_i )(i = 1, 2, …, n) be a collection of q-ROFNs, taking ξ, ς ∈ [0, 1], then the aggregated value by using the q-ROFFPPA operator is still a q-ROFN and $q - R O F F P P A (a_{1}, a_{2}, \dots, a_{n}) = (\begin{matrix} {(1 - \log_{θ} (1 + Π_{i = 1}^{n} {(θ^{1 - μ_{i}^{q}} - 1)}^{\frac{(1 + T_{△} (a_{i}))}{\sum_{j = 1}^{n} (1 + T_{△} (a_{j}))}}))}^{\frac{1}{q}}, \\ {(\log_{θ} (1 + Π_{i = 1}^{n} {(θ^{v_{i}^{q}} - 1)}^{^{\frac{(1 + T_{△} (a_{i}))}{\sum_{j = 1}^{n} (1 + T_{△} (a_{j}))}}}))}^{\frac{1}{q}} \end{matrix})$ (9)

Proof. It is easily to prove q-ROFFPPA operator is a q-ROFN, so we omit here. Then, we prove Equation (9) by using mathematical induction on n.

When n = 1, according to Theorem 1, we get $\begin{matrix} q - ROFFPPA (a_{1}) & = \frac{(1 + T_{Δ} (a_{1}))}{\sum_{j = 1}^{1} (1 + T_{Δ} (a_{j}))} \cdot_{F} a_{1} \\ = a_{1} = (μ_{1}, v_{1}) \end{matrix}$

Since $\frac{(1 + T_{Δ} (a_{1}))}{\sum_{j = 1}^{1} (1 + T_{Δ} (a_{j}))} = 1,$ we have

$q - R O F F P P A (a_{1}) = (\begin{matrix} {(1 - \log_{θ} (1 + Π_{i = 1}^{1} {(θ^{1 - μ_{i}^{q}} - 1)}^{\frac{(1 + T_{△} (a_{i}))}{\sum_{j = 1}^{1} (1 + T_{△} (a_{j}))}}))}^{\frac{1}{q}}, \\ {(\log_{θ} (1 + Π_{i = 1}^{1} {(θ^{v_{i}^{q}} - 1)}^{^{\frac{(1 + T_{△} (a_{i}))}{\sum_{j = 1}^{1} (1 + T_{△} (a_{j}))}}}))}^{\frac{1}{q}} \end{matrix})$ which means that Equation (9) holds for n = 1.

If Equation (9) holds for n = k, that is

$q - R O F F P P A (a_{1}, a_{2}, \dots, a_{k}) = (\begin{matrix} {(1 - \log_{θ} (1 + Π_{i = 1}^{k} {(θ^{1 - μ_{i}^{q}} - 1)}^{\frac{(1 + T_{△} (a_{i}))}{\sum_{j = 1}^{k} (1 + T_{△} (a_{j}))}}))}^{\frac{1}{q}}, \\ {(\log_{θ} (1 + Π_{i = 1}^{k} {(θ^{v_{i}^{q}} - 1)}^{^{\frac{(1 + T_{△} (a_{i}))}{\sum_{j = 1}^{k} (1 + T_{△} (a_{j}))}}}))}^{\frac{1}{q}} \end{matrix})$

When n = k + 1, we have

$\begin{matrix} \begin{matrix} q - ROFFPPA (a_{1}, a_{2}, . . ., a_{k}, a_{k + 1}) = \\ q - ROFFPPA (a_{1}, a_{2}, . . ., a_{k}) \oplus_{F} \frac{(1 + T (a_{k + 1}))}{\sum_{j = 1}^{k + 1} (1 + T (a_{j}))} a_{k + 1} \end{matrix} \end{matrix}$ $= (\begin{matrix} {(1 - \log_{θ} (1 + Π_{i = 1}^{k} {(θ^{1 - μ_{i}^{q}} - 1)}^{\frac{(1 + T △ (a_{i}))}{\sum_{j = 1}^{k} (1 + T △ (a_{j}))}}))}^{\frac{1}{q}}, \\ {(\log_{θ} (1 + Π_{i = 1}^{k} {(θ^{v_{i}^{q}} - 1)}^{\frac{(1 + T △ (a_{i}))}{\sum_{j = 1}^{k} (1 + T △ (a_{j}))}}))}^{\frac{1}{q}} \end{matrix}) \oplus F$ $(\begin{matrix} {(1 - \log_{θ} (1 + \frac{{(θ^{1 - μ_{k + 1}^{q}} - 1)}^{\frac{(1 + T △ (a_{k + 1}))}{\sum_{j = 1}^{k + 1} (1 + T (a_{j}))}}}{{(θ - 1)}^{^{\frac{(1 + T △ (a_{k + 1}))}{\sum_{j = 1}^{k + 1} (1 + T △ (a_{j}))} - 1}}}))}^{\frac{1}{q}}, \\ {(\log_{θ} (1 + \frac{{(θ^{v_{k + 1}^{q}} - 1)}^{\frac{(1 + T △ (a_{k + 1}))}{\sum_{j = 1}^{k + 1} (1 + T (a_{j}))}}}{{(θ - 1)}^{^{\frac{(1 + T △ (a_{k + 1}))}{\sum_{j = 1}^{k + 1} (1 + T △ (a_{j}))} - 1}}}))}^{\frac{1}{q}} \end{matrix})$ $= (\begin{matrix} {(1 - \log_{θ} (1 + \frac{\prod_{i = 1}^{k + 1} {(θ^{1 - μ_{i}^{q}} - 1)}^{\frac{(1 + T △ (a_{k + 1}))}{\sum_{j = 1}^{k + 1} (1 + T (a_{j}))}}}{{(θ - 1)}^{\sum_{i = 1}^{k + 1} \frac{(1 + T △ (a_{k + 1}))}{\sum_{j = 1}^{k + 1} (1 + T △ (a_{j}))} - 1}}))}^{\frac{1}{q}}, \\ {(\log_{θ} (1 + \frac{\prod_{i = 1}^{k + 1} {(θ^{v_{i}^{q}} - 1)}^{\frac{(1 + T △ (a_{k + 1}))}{\sum_{j = 1}^{k + 1} (1 + T (a_{j}))}}}{{(θ - 1)}^{\sum_{i = 1}^{k + 1} \frac{(1 + T △ (a_{k + 1}))}{\sum_{j = 1}^{k + 1} (1 + T △ (a_{j}))} - 1}}))}^{\frac{1}{q}} \end{matrix})$

Since $\sum_{i = 1}^{k + 1} \frac{(1 + T_{Δ} (a_{k + 1}))}{\sum_{j = 1}^{k + 1} (1 + T_{Δ} (a_{j}))} = 1,$ we get

$\begin{array}{l} q - R O F F P P A (a_{1}, a_{2}, \dots, a_{k + 1}) \\ = (\begin{matrix} {(1 - \log_{θ} (1 + \prod_{i = 1}^{k + 1} (θ^{1 - μ_{i}^{q}} - 1) \frac{n (1 + T_{Δ} (a_{i}))}{\sum_{j = 1}^{n} (1 + T_{Δ} (a_{j}))}))}^{1 / q}, \\ {(\log_{θ} (1 + \prod_{i = 1}^{k + 1} (θ^{v_{i}^{q}} - 1) \frac{n (1 + T_{Δ} (a_{i}))}{\sum_{j = 1}^{n} (1 + T_{Δ} (a_{j}))}))}^{1 / q} \end{matrix}) \end{array}$

which means that Equation (9) holds for n = k + 1.

Therefore, Equation (9) holds for all n. Thus, the proof of Theorem 4 is completed.□

Theorem 4. Let a_i = ( μ_i, v_i )(i = 1, 2, …, n) be a collection of q-ROFNs, then

$\begin{matrix} q - ROFFPPA (λ \cdot_{F} a_{1}, λ \cdot_{F} a_{2}, . . ., λ \cdot_{F} a_{n}) \\ = λ \cdot_{F} q - ROFFPPA (a_{1}, a_{2}, . . ., a_{n}) \end{matrix}$ (10)

Proof. By the operational laws defined in Definition 4, we get

$\begin{matrix} q - ROFFPPA (λ \cdot_{F} a_{1}, λ \cdot_{F} a_{2}, . . ., λ \cdot_{F} a_{n}) \end{matrix}$ $= {\begin{cases} {(1 - \log_{θ} (1 + \frac{{\prod_{i = 1}^{n} (θ^{1 - μ_{i}^{q}} - 1)}^{\frac{λ n (1 + T △ (a_{i}))}{\sum_{j = 1}^{n} (1 + T △ (a_{j}))}}}{{(θ - 1)}^{(λ - 1)}}))}^{\frac{1}{q}}, {(\log_{θ} (1 + \frac{{\prod_{i = 1}^{n} (θ^{v_{i}^{q}} - 1)}^{\frac{λ n (1 + T △ (a_{i}))}{\sum_{j = 1}^{n} (1 + T △ (a_{j}))}}}{{(θ - 1)}^{(λ - 1)}}))}^{\frac{1}{q}} \end{cases}}$

And λ · _Fq - ROFFPPA ( a₁, a₂, . . . , a_n )

$= λ \cdot_{F} (\begin{matrix} {(1 - \log_{θ} (1 + {\prod_{i = 1}^{n} (θ^{1 - μ_{i}^{q}} - 1)}^{\frac{n (1 + T △ (a_{i}))}{\sum_{j = 1}^{n} (1 + T △ (a_{j}))}}))}^{\frac{1}{q}}, \\ {(\log_{θ} (1 + {\prod_{i = 1}^{n} (θ^{v_{i}^{q}} - 1)}^{\frac{n (1 + T △ (a_{i}))}{\sum_{j = 1}^{n} (1 + T △ (a_{j}))}}))}^{\frac{1}{q}} \end{matrix})$

$= {\begin{cases} {(1 - \log_{θ} (1 + \frac{{\prod_{i = 1}^{n} (θ^{1 - μ_{i}^{q}} - 1)}^{\frac{λ n (1 + T △ (a_{i}))}{\sum_{j = 1}^{n} (1 + T △ (a_{j}))}}}{{(θ - 1)}^{(λ - 1)}}))}^{\frac{1}{q}}, {(\log_{θ} (1 + \frac{{\prod_{i = 1}^{n} (θ^{v_{i}^{q}} - 1)}^{\frac{λ n (1 + T △ (a_{i}))}{\sum_{j = 1}^{n} (1 + T △ (a_{j}))}}}{{(θ - 1)}^{(λ - 1)}}))}^{\frac{1}{q}} \end{cases}}$ $\begin{matrix} = q - ROFFPPA (λ \cdot_{F} a_{1}, λ \cdot_{F} a_{2}, . . ., λ \cdot_{F} a_{n}) \end{matrix}$

Theorem 5. Let a_i = ( μ_i, v_i )(i = 1, 2, …, n) be a collection of q-ROFNs, then $\begin{matrix} q - ROFFPPA (a_{1} \oplus_{F} a, . . ., a_{n} \oplus_{F} a) \end{matrix}$

$= q - ROFFPPA (a_{1}, . . ., a_{n}) \oplus_{F} a$ (11)

Proof. By the operational law defined in Theme 4, we have

$a_{i} \oplus _{F} a = (\begin{matrix} {(1 - \log_{θ} (1 + \frac{(θ^{1 - μ_{i}^{q}} - 1) (θ^{1 - μ_{i}^{q}} - 1)}{θ - 1}))}^{\frac{1}{q}}, \\ {(\log_{θ} (1 + \frac{(θ^{v_{i}^{q}} - 1) (θ^{v^{q}} - 1)}{θ - 1}))}^{\frac{1}{q}} \end{matrix})$

Thus q - ROFFPPA ( a₁ ⊕ _Fa, . . . , a_n ⊕ _Fa ) = $= (\begin{matrix} {(1 - \log_{θ} (1 + \frac{(θ^{1 - μ^{q}} - 1) \prod_{i = 1}^{n} {(θ^{1 - μ_{i}^{q}} - 1)}^{\frac{n (1 + T △ (a_{i}))}{\sum_{j = 1}^{n} (1 + T △ (a_{j}))}}}{θ - 1}))}^{\frac{1}{q}} \\ {(\log_{θ} (1 + \frac{(θ^{v^{q}} - 1) \prod_{i = 1}^{n} {(θ^{v_{i}^{q}} - 1)}^{\frac{n (1 + T △ (a_{i}))}{\sum_{j = 1}^{n} (1 + T △ (a_{j}))}}}{θ - 1}))}^{\frac{1}{q}} \end{matrix})$ and q - ROFFPPA ( a₁, a₂, . . . , a_n ) ⊕ _Fa = $= (\begin{matrix} {(1 - \log_{θ} (1 + \frac{(θ^{1 - μ^{q} - 1}) \prod_{i = 1}^{n} (θ^{1 - μ^{q} - 1}) \frac{n (1 + T_{Δ} (a_{i}))}{\sum_{j = 1}^{n} (1 + T_{Δ} (a_{j}))}}{θ - 1}))}^{1 / q}, \\ {(\log_{θ} (1 + \frac{(θ^{v^{q}} - 1) \prod_{i = 1}^{n} (θ^{v_{i}^{q}} - 1) \frac{(n (1 + T_{Δ} (a_{i})))}{\sum_{j = 1}^{n} (1 + T_{Δ} (a_{j}))}}{θ - 1}))}^{1 / q} \end{matrix})$ $= (\begin{matrix} {(1 - \log_{θ} (1 + \frac{(θ^{1 - μ^{q}} - 1) \prod_{i = 1}^{n} (θ^{1 - μ_{i}^{q}} - 1) \frac{(n (1 + T_{Δ} (a_{i})))}{\sum_{j = 1}^{n} (1 + T_{Δ} (a_{j}))}}{θ - 1}))}^{1 / q}, \\ {(\log_{θ} (1 + \frac{θ^{v^{q}} - 1 \prod_{i = 1}^{n} (θ^{v_{i}^{q}} - 1) \frac{n (1 + T_{Δ} (a_{i}))}{\sum_{j = 1}^{n} (1 + T_{Δ} (a_{j}))}}{θ - 1}))}^{1 / q} \end{matrix})$ $\begin{matrix} = q - ROFFPPA (a_{1} \oplus_{F} a, . . ., a_{n} \oplus_{F} a) \end{matrix}$

Based on the Theorems 4 and 5, it is easy to obtain the following Theorem 6.

Theorem 6. Let a_i = ( μ_i, v_i )(i = 1, 2, …, n) be a collection of q-ROFNs, then

$\begin{matrix} q - ROFFPPA (λ \cdot_{F} a_{1} \oplus_{F} a, . . ., λ \cdot_{F} a_{n} \oplus_{F} a) \\ = λ \cdot_{F} q - ROFFPPA (a_{1}, . . ., a_{n}) \oplus_{F} a \end{matrix}$ (12)

Theorem 7. Let and a_{η
_i} = ( μ_{η
_i}, v_{η
_i} ) be two collections of q-ROFNs, then

$\begin{matrix} q - ROFFPPA (a_{γ_{1}} \oplus_{F} a_{η_{1}}, . . ., a_{γ_{n}} \oplus_{F} a_{η_{n}}) \\ = \begin{matrix} q - ROFFPPA (a_{γ_{1}}, . . ., a_{γ_{n}}) \\ \oplus_{F} q - ROFFPPA (a_{η_{1}}, . . ., a_{η_{n}}) \end{matrix} \end{matrix}$ (13)

It can be easily proved that the q-ROFFPPA operator has the following properties.

Theorem 8. (Monotonicity) Let and a_{η
_i} = ( μ_{η
_i}, v_{η
_i} ) ( i = 1, 2, . . . , n ) be two collections of q-ROFNs, if μ_{γ_i} ≤ μ_{η
_i} and v_{γ_i} ≥ v_{η
_i} holds for all i, then

$\begin{matrix} q - ROFFPPA (a_{γ_{1}}, . . ., a_{γ_{n}}) \leq q \\ - ROFFPPA (a_{η_{1}}, . . ., a_{η_{n}}) \end{matrix}$ (14)

Theorem 9. (Boundedness) Let a_i = ( μ_i, v_i ) (i = 1, 2 . . . .n) be a collection of q-ROFNs, ifa⁺ = ( max ( μ_i ) , min ( v_i ) ) and a^- = ( min ( μ_i ) , max ( v_i ) ), then

$a^{-} \leq q - ROFFPPA (a_{1}, a_{2}, . . ., a_{n}) \leq a^{+}$ (15)

Theorem 10. (Idempotency). If a_i = ( μ_i, v_i )(i = 1, 2 . . . .n) are equal, i.e.a_i = a = ( μ, v ) ( i = 1, 2, . . . , n ), then

$q - ROFFPPA (a_{1}, a_{2}, . . ., a_{n}) = a$ (16)

According to different values of the parameter θ, the following special cases can be obtained.

Theorem 11. Let a_i = ( μ_i, v_i ) ( i = 1, 2, . . . , n ) be a collection of q-ROFNs, then

(1) If θ → 1, then the q-ROFFPPA operator is reduced to a q-rung orthopair fuzzy power point operator (q-ROFPPA) based on the Algebraic t-conorm and t-norm, which is defined as: $\begin{array}{l} \lim_{θ \to 1} q - R O F F P P A \\ = (\begin{matrix} {(1 - \prod_{i = 1}^{n} {(1 - μ_{i}^{q})}^{(1 + T_{Δ} (a_{i})) / \sum_{j = 1}^{n} (1 + T_{Δ} (a_{j}))})}^{1 / q}, \\ \prod_{i = 1}^{n} {(v_{i})}^{(1 + T_{Δ} (a_{i})) / \sum_{j = 1}^{n} (1 + T_{Δ} (a_{j}))} \end{matrix}) . \end{array}$

(2) If θ→ + ∞, then the q-ROFFPPA operator is reduced to the traditional arithmetic power point average operator, which is defined as: $\begin{array}{l} \lim_{θ \to + \infty} q - R O F F P P A \\ = (\begin{matrix} {(1 - \sum_{i = 1}^{n} \frac{(1 + T_{Δ} (a_{i}))}{\sum_{j = 1}^{n} (1 + T_{Δ} (a_{j}))} μ_{i}^{q})}^{1 / q}, \\ {(\sum_{i = 1}^{n} \frac{(1 + T_{Δ} (a_{i}))}{\sum_{j = 1}^{n} (1 + T_{Δ} (a_{j}))} v_{i}^{q})}^{1 / q} \end{matrix}) . \end{array}$

Proof. (1) As $\begin{array}{l} \lim_{θ \to 1} q - R O F F P P A = \\ \lim_{θ \to 1} (\begin{matrix} {(1 - \log_{θ} (1 + \prod_{i = 1}^{n} (θ^{1 - μ_{i}^{q}} - 1) \frac{n (1 + T_{Δ} (a_{i}))}{\sum_{j = 1}^{n} (1 + T_{Δ} (a_{j}))}))}^{1 / q}, \\ {(\log_{θ} (1 + \prod_{i = 1}^{n} (θ^{v_{i}^{q}} - 1) \frac{n (1 + T_{Δ} (a_{i}))}{\sum_{j = 1}^{n} (1 + T_{Δ} (a_{j}))}))}^{1 / q} \end{matrix}) \end{array}$

We first prove

that

$\begin{array}{l} \lim_{θ \to 1} {(\log_{θ} (1 + \prod_{i = 1}^{n} {(θ^{v_{i}^{q}} - 1)}^{^{(1 + T_{Δ} (a_{i}))} / \sum_{j = 1}^{n} (1 + T_{Δ} (a_{j}))}))}^{1 / q} \\ = \prod_{i = 1}^{n} {(v_{i})}^{^{(1 + T_{Δ} (a_{i}))} / \sum_{j = 1}^{n} (1 + T_{Δ} (a_{j}))} \end{array}$

Since θ → 1, we get ${(θ^{ν_{i}^{q}} - 1)}^{(1 + T_{Δ} (a_{i})) / - \sum_{j = 1}^{n} (1 + T_{Δ} (a_{j}))} \to 0$ . Thus, based on logarithmic transform and the law of equivalent infinitesimal replacement ln( 1 + x ) ∼ x ( x ≻ 0 ) , we get

$\begin{array}{l} {(\log_{θ} (1 + \prod_{i = 1}^{n} {(θ^{v_{i}^{q} - 1 - 1})}^{(1 + T_{Δ} (a_{i})) / \sum_{j = 1}^{n} (1 + T_{Δ} (a_{j}))}))}^{1 / q} \\ = {(\frac{I n (1 + \prod_{i = 1}^{n} {(θ^{v_{i}^{q} - 1 - 1})}^{(1 + T_{Δ} (a_{i})) / \sum_{j = 1}^{n} (1 + T_{Δ} (a_{j}))})}{I n θ})}^{1 / q} \end{array}$ $\sim {(\frac{\prod_{i = 1}^{n} {(θ^{v_{i}^{q}} - 1 - 1)}^{(1 + T_{Δ} (a_{i})) / \sum_{j = 1}^{n} (1 + T_{Δ} (a_{j}))}}{I n θ})}^{1 / q}$

By Taylor expansion, we have $θ^{ν_{i}^{q}} - 1 = ν_{i}^{q} ln θ + \frac{ν_{i}^{q}}{2} {(ln θ)}^{2} + \dots = ν_{i}^{q} ln θ + O (ln θ)$

Thus, we get

$\begin{array}{l} \lim_{θ \to 1} {(\log_{θ} (1 + {\prod_{i = 1}^{n} (θ^{v_{i}^{q}} - 1)}^{(1 + T_{Δ} (a_{i})) / \sum_{j = 1}^{n} (1 + T_{Δ} (a_{j}))}))}^{1 / q} = \\ \lim_{θ \to 1} {(\frac{I n (1 + {\prod_{i = 1}^{n} (θ^{v_{i}^{q}} - 1)}^{(1 + T_{Δ} (a_{i})) / \sum_{j = 1}^{n} (1 + T_{Δ} (a_{j}))})}{I n θ})}^{1 / q} \end{array}$

$= \lim_{θ \to 1} {(\frac{{\prod_{i = 1}^{n} (θ^{v_{i - 1}^{q}})}^{(1 + T_{Δ} (a_{i})) / \sum_{j = 1}^{n} (1 + T_{Δ} (a_{j}))}}{I n θ})}^{1 / q} =$ $\lim_{θ \to 1} {(\frac{\prod_{i = 1}^{n} {(θ^{v_{i}^{q}} I n θ)}^{(1 + T_{Δ} (a_{i})) / \sum_{j = 1}^{n} (1 + T △ (a_{j}))}}{I n θ})}^{1 / q}$ $\begin{matrix} = \prod_{i = 1}^{n} {(ν_{i}^{q})}^{(1 + T_{Δ} (a_{i})) / - \sum_{j = 1}^{n} (1 + T_{Δ} (a_{j}))} \end{matrix}$

Based on the above, we also have

$\begin{array}{l} \begin{matrix} \lim \\ θ \to 1 \end{matrix} (1 - \log_{θ} {(\begin{matrix} 1 + \\ {\prod_{i = 1}^{n} (θ^{1 - μ_{i - 1}^{q}})}^{1 + T_{Δ} (a_{i})) / \sum_{j = 1}^{n} (1 + T_{Δ} (a_{j}))} \end{matrix})}^{1 / q}) \\ = {(1 - \prod_{i = 1}^{n} {(1 - μ_{i}^{q})}^{(1 + T △ (a_{j})) / \sum_{j = 1}^{n} (1 + T_{Δ} (a_{j}))})}^{1 / q} \end{array}$

Thus,

$\begin{array}{l} \lim_{θ \to + \infty} q - R O F F P P A \\ = (\begin{matrix} {(1 - \prod_{i = 1}^{n} {(1 - μ_{i}^{q})}^{(1 + T_{Δ} (a_{i})) / \sum_{j = 1}^{n} (1 + T_{Δ} (a_{j}))})}^{1 / q}, \\ \prod_{i = 1}^{n} {(v_{i})}^{(1 + T_{Δ} (a_{i})) / \sum_{j = 1}^{n} (1 + T_{Δ} (a_{j}))} \end{matrix}) . \end{array}$

Firstly, we prove that

$\begin{array}{l} \lim_{θ \to \infty} {(\log_{θ} (1 + \prod_{i = 1}^{n} {(θ^{v_{i}^{q}} - 1)}^{(1 + T_{Δ} (a_{i})) / \sum_{j = 1}^{n} (1 + T_{Δ} (a_{j}))}))}^{1 / q} . \\ = {(\sum_{i = 1}^{n} \frac{n (1 + T_{Δ} (a_{i}))}{\sum_{j = 1}^{n} (1 + T_{Δ} (a_{j}))} v^{_{i}^{q}})}^{1 / q} \end{array}$

As ${log}_{θ} (1 + \prod_{i = 1}^{n} {(θ^{v_{i}^{q}} - 1)}^{(1 + T_{Δ} (a_{i})) / - \sum_{j = 1}^{n} (1 + T_{Δ} (a_{j}))})$ is continuous, we get

$\begin{array}{l} \lim_{θ \to + \infty} {(\log_{θ} (1 + \prod_{i = 1}^{n} {(θ^{v_{i}^{q}} - 1)}^{(1 + T_{Δ} (a_{i})) / \sum_{j = 1}^{n} (1 + T_{Δ} (a_{j}))}))}^{\frac{1}{q}} = \\ {(\lim_{θ \to + \infty} \log_{θ} (1 + \prod_{i = 1}^{n} {(θ^{v_{i}^{q}} - 1)}^{(1 + T_{Δ} (a_{i})) / \sum_{j = 1}^{n} (1 + T_{Δ} (a_{j}))}))}^{\frac{1}{q}} \end{array}$

According to the L’Hospital’s rule, we get

$\begin{array}{l} (\begin{matrix} \lim \\ θ \to + \infty \end{matrix} \log_{θ} (1 + \prod_{i = 1}^{n} {{(θ_{i}^{v^{q}} - 1)}^{(1 + T_{Δ} (a_{i})) / \sum_{j = 1}^{n} (1 = T_{Δ} (a_{j}))}))}^{1 / q} \\ = {(\begin{matrix} \lim \\ θ \to + \infty \end{matrix} \frac{I n (1 + {\prod_{i = 1}^{n} (θ_{i}^{v^{q}} - 1)}^{(1 + T_{Δ} (a_{i})) / \sum_{j = 1}^{n} (1 = T_{Δ} (a_{j}))})}{I n θ})}^{1 / q} \end{array}$ $\begin{array}{l} = {(\begin{matrix} \lim \\ θ \to + \infty \end{matrix} \frac{\sum_{i = 1}^{n} \frac{n (1 + T_{Δ} (a_{i}))}{\sum_{j = 1}^{n} (1 + T_{Δ} (a_{j}))} \frac{v_{i}^{q}}{1 - 1 / θ_{i}^{v^{q}}}}{(1 + 1 / \prod_{i = 1}^{n} {(θ_{i}^{v^{q}} - 1)}^{(1 + T_{Δ} (a_{i})) / \sum_{j = 1}^{n} (1 + T_{Δ} (a_{j}))})})}^{\frac{1}{q}} \\ = {(\sum_{i = 1}^{n} \frac{n (1 + T △ (a_{i}))}{\sum_{j = 1}^{n} (1 + T △ (a_{j}))} v_{i}^{q})}^{\frac{1}{q}} \end{array}$

Similarity, we get $\begin{array}{l} \frac{\lim}{θ \to + \infty} {(1 - \log_{θ} (1 + \prod_{1 = 1}^{n} {(θ^{1 - μ_{i}^{q}} - 1)}^{(1 + T △ (a_{i})) / \sum_{j = 1}^{n} (1 + T △_(a_{j}))}))}^{\frac{1}{q}} \\ = (1 - \sum_{i = 1}^{n} \frac{n (1 + T_{Δ} (a_{i}))}{\sum_{j = 1}^{n} (1 + T_{Δ} (a_{j}))} μ_{i}^{q}) 1 / q \end{array}$

Thus $\begin{matrix} \begin{matrix} \lim_{θ \to + \infty} q - R O F F P P A \\ = (\begin{matrix} {(1 - \sum_{i = 1}^{n} \frac{n (1 + T_{Δ} (a_{i}))}{\sum_{j = 1}^{n} (1 + T_{Δ} (a))} μ_{i}^{q})}^{\frac{1}{q}}, \\ {(\sum_{i = 1}^{n} \frac{n (1 + T_{Δ} (a_{i}))}{\sum_{j = 1}^{n} (1 + T_{Δ} (a_{j}))} v_{i}^{q})}^{\frac{1}{q}} \end{matrix}) . \end{matrix} \end{matrix}$

Theorem 12. For arguments a_i = ( μ_i, v_i ) ( i = 1, 2, . . . , n ) and θ ≻ 1,the q-ROFFPPA operator is monotonically decreasing with respect to θ.

Proof. Let $f (θ) = {log}_{θ} (1 + \prod_{i = 1}^{n} {(θ^{1 - μ_{i}^{q}} - 1)}^{(1 + T_{Δ} (a_{i})) / - \sum_{j = 1}^{n} (1 + T_{Δ} (a_{j}))})$ and $g (θ) = {log}_{θ} (1 + \prod_{i = 1}^{n} {(θ^{ν_{i}^{q}} - 1)}^{(1 + T_{Δ} (a_{i})) / - \sum_{j = 1}^{n} (1 + T_{Δ} (a_{j}))}),$ we have $q - R O F F P P A = = ({(1 - f (θ))}^{1 / q}, {(g (θ))}^{1 / q})$

Thus, the score value of q-ROFFPPA operator isS_{q-ROFFPPA(θ)} = 1 - f ( θ ) - g ( θ ). Then, we just need to prove thatS_{q-ROFFPPA(θ)}is monotonically decreasing with respect to θ. Taking the derivative of S_{q-ROFFPPA(θ)} with respect to θ, we obtain $\begin{matrix} \frac{d}{d θ} (S_{q - ROFFPPA (θ)}) = \frac{d}{d θ} ((1 - f (θ)) - (g (θ))) \end{matrix}$ $\begin{matrix} = - (f (θ) \frac{df (θ)}{d θ} + g (θ) \frac{dg (θ)}{d θ}) \end{matrix}$

Taking the derivative off ( θ ) with respect to θ, we obtain

$\begin{matrix} \frac{df (θ)}{d θ} = \frac{d}{d θ} ({log}_{θ} (1 + \prod_{i = 1}^{n} {(θ^{1 - μ_{i}^{q}} - 1)}^{(1 + T_{Δ} (a_{i})) / - \sum_{j = 1}^{n} (1 + T_{Δ} (a_{j}))})) = \end{matrix}$ $\begin{matrix} \frac{\prod_{i = 1}^{n} {(θ^{1 - μ_{i}^{q}} - 1)}^{(1 + T_{Δ} (a_{i})) / - \sum_{j = 1}^{n} (1 + T_{Δ} (a_{j}))} \sum_{i = 1}^{n} \frac{n (1 + T_{Δ} (a_{i}))}{\sum_{j = 1}^{n} (1 + T_{Δ} (a_{j}))} \frac{(1 - μ_{i}^{q}) θ^{- μ_{i}^{q}}}{θ^{μ_{i}^{2}} - 1}}{(1 + \prod_{i = 1}^{n} {(θ^{1 - μ_{i}^{q}} - 1)}^{(1 + T_{Δ} (a_{i})) / - \sum_{j = 1}^{n} (1 + T_{Δ} (a_{j}))}) ln θ} \end{matrix}$

Since $θ ≻ 1, 0 \leq μ_{i}^{q}, \frac{n (1 + T_{Δ} (a_{i}))}{\sum_{j = 1}^{n} (1 + T_{Δ} (a_{j}))} \leq 1$ , we get $\begin{matrix} \frac{df (θ)}{d θ} ≻ 0 . \end{matrix}$

Similarly, we get $\frac{dg (θ)}{d θ} ≻ 0$

Sinceθ ≻ 1, we have f ( θ ) ≥ 0, g ( θ ) ≥ 0 . Thus, $\frac{d}{d θ} (S_{q - ROFFPPA (θ)}) = - (f (θ) \frac{df (θ)}{d θ} + g (θ) \frac{dg (θ)}{d θ}) \leq 0$

Furthermore, S_{q-ROFFPPA(θ)} is monotonically decreasing with respect to θ,which denotes q-ROFFPPA operator is monotonically decreasing with respect to θ.

Remark 3. Obviously, q-ROFFPPA operator has monotonic property when we take θ ≻ 1, meaning that DMs can choose flexibly parameter θ based on their attitude. In practical decision situations, if the DM’s attitude is optimistic, then we let θ → 1, if the DM’s attitude is pessimistic, then we let θ→ + ∞.

3.2 q-ROFFPPG aggregation operator

Definition 12. Let a_i = ( μ_i, v_i ) ( i = 1, 2, . . . , n ) be a collection of q-ROFNs, then a q-ROFFPPG operator is defined as:

$\begin{array}{l} q - R O F F P P G (a_{1}, a_{2} \dots a_{n}) \\ = \prod_{i = 1}^{n} {^{\otimes_{F}}}_{(a_{i})} \frac{(1 + T_{Δ} (a_{i}))}{\sum_{j = 1}^{n} (1 + T_{Δ} (a_{j}))} \end{array}$ (17) where $T (a_{i}) = \sum_{j = 1, j \neq i}^{n} S {up}_{Δ} (a_{i}, a_{j})$ denotes the support of the ith q-ROFN by all other q-ROFNs, Sup_Δ ( a_i, a_j ) is the support fora_iand a_j . The support satisfies the properties defined in Definition 10.

The proof of the following theorem is similar to that of Theorem 4, we do not duplicate it here.

Theorem 14. Let a_i = ( μ_i, v_i ) ( i = 1, 2, . . . , n ) be a collection of q-ROFNs, then the aggregated value by using the q-ROFFPPG is still a q-ROFN and

$\begin{matrix} q - ROFFPPG (a_{1}, a_{2} \dots a_{n}) \end{matrix}$

$= (\begin{matrix} {(\log_{θ} (1 + \prod_{i = 1}^{n} (θ^{μ_{i}^{q}} - 1) \frac{(1 + T_{Δ} (a_{i}))}{\sum_{j = 1}^{n} (1 + T_{Δ} (a_{j}))}))}^{1 / q}, \\ (1 - \log_{θ} (1 + \prod_{i = 1}^{n} (θ^{1 - v_{i}^{q}} - 1) \frac{(1 + T_{Δ} (a_{i}))}{\sum_{j = 1}^{n} {(1 + T_{Δ} (a_{j}))}^{1 / q}})) \end{matrix})$ (18)

Theorem 15. Let anda_{η
_i} = ( μ_{η
_i}, v_{η
_i} ) ( i = 1, 2, . . . , n ) be two collections of q-ROFNs, then $\begin{matrix} q - ROFFPPG (a_{γ_{1}} \otimes_{F} a_{η_{1}}, . . ., a_{γ_{n}} \otimes_{F} a_{η_{n}}) = \end{matrix}$

$\begin{matrix} q - ROFFPPG (a_{γ_{1}}, . . ., a_{γ_{n}}) \\ \otimes_{F} q - ROFFPPG (a_{η_{1}}, . . ., a_{η_{n}}) \end{matrix}$ (19)

Theorem 16. Let a_i = ( μ_i, v_i ) be a collection of q-ROFNs, then

$\begin{matrix} q - ROFFPPG (a_{1}^{\land_{F} λ}, . . ., a_{n}^{\land_{F} λ}) \\ = {(q - ROFFPPG (a_{1},, . . ., a_{n}))}^{\land_{F} λ} \end{matrix}$ (20)

Based on the Theorems 15 and 16, it is easy to obtain the following Theorem 17.

Theorem 17. Let a_i = ( μ_i, v_i ) ( i = 1, 2, . . . , n ) be a collection of q-ROFNs, then $\begin{matrix} q - ROFFPPG (a_{1}^{\land_{F} λ} \otimes_{F} a_{2}^{\land_{F} λ}, . . ., a_{n}^{\land_{F} λ} \otimes_{F} a) \end{matrix}$

$= q - ROFFPPG (a_{1}^{\land_{F} λ}, a_{2}^{\land_{F} λ}, . . ., a_{n}^{\land_{F} λ}) \otimes_{F} a$ (21)

The q-ROFFPPG operator has properties similar to the q-ROFFPPA operator, such as idempotency, monotonicity and boundedness under some conditions.

According to different values of the parameter θ, we can get the following special cases.

Theorem 18. Let a_i = ( μ_i, v_i ) ( i = 1, 2, . . . , n ) be a collection of q-ROFNs, then

(1) If θ → 1, then the q-ROFFPP G operator is reduced to a q-rung orthopair fuzzy power point geometric operator (q-ROFFPPG) based on the Algebraic t-conorm and t-norm, which is defined as: $\lim_{θ \to 1} q - R O F F P P G = (\begin{matrix} \prod_{i = 1}^{n} {(μ_{i})}^{(1 + T_{Δ} (a_{i})) / \sum_{j = 1}^{n} (1 + T_{Δ} (a_{j}))}, \\ {(1 - \prod_{i = 1}^{n} {(1 - v_{i}^{q})}^{(1 + T_{Δ} (a_{i})) / \sum_{j = 1}^{n} (1 + T_{Δ} (a_{j}))})}^{1 / q} \end{matrix}) .$

(2) If θ→ + ∞, then the q-ROFFPPG operator is reduced to the traditional arithmetic power point average operator, which is defined as: $\begin{array}{l} \lim_{θ \to + \infty} q - R O F F P P G = (\begin{matrix} {(\sum_{i = 1}^{n} \frac{(1 + T_{Δ} (a_{i}))}{\sum_{j = 1}^{n} (1 + T_{Δ} (a_{j}))} μ_{i}^{q})}^{1 / q}, \\ {(1 - \sum_{i = 1}^{n} \frac{(1 + T_{Δ} (a_{i}))}{\sum_{j = 1}^{n} (1 + T_{Δ} (a_{j)}))} v_{i}^{q})}^{1 / q} \end{matrix}) . \end{array}$

Theorem 19. For given arguments a_i = ( μ_i, v_i ) ( i = 1, 2, . . . , n ) and θ ≻ 1,the q-ROFFPPG operator is monotonically increasing with respect to θ.

Theorem 20. Let a_i = ( μ_i, v_i ) (i = 1, 2, …, n) be a collection of q-ROFNs, q ≥ 1, then the correlations of the proposed q-ROFFPPA and q-ROFFPPG operators is as below:

q - ROFFPPG ( a₁, a₂, . . . , a_n ) ^c $= q - ROFFPPA (a_{1}^{c}, a_{2}^{c}, . . ., a_{n}^{c})$

q - ROFFPPA ( a₁, a₂, . . . , a_n ) ^c $= q - ROFFPPG (a_{1}^{c}, a_{2}^{c}, . . ., a_{n}^{c})$

q - ROFFPPA ( a₁, a₂, . . . , a_n ) ≥q - ROFFPPG ( a₁, a₂, . . . , a_n ) , θ ≻ 1

Proof. (1) and (2) are obvious. We only prove (3) holds.

In order to prove $\begin{matrix} q - ROFFPPA (a_{1}, a_{2}, . . ., a_{n}) \\ \geq q - ROFFPPG (a_{1}, a_{2}, . . ., a_{n}) \end{matrix}$ we should prove that $\begin{matrix} {log}_{θ} (1 + \prod_{i = 1}^{n} {(θ^{μ_{i}^{q}} - 1)}^{\frac{n (1 + T_{Δ} (a_{i}))}{\sum_{j = 1}^{n} (1 + T_{Δ} (a_{j}))}}) \\ \leq 1 - {log}_{θ} (1 + \prod_{i = 1}^{n} {(θ^{1 - μ_{i}^{q}} - 1)}^{\frac{n (1 + T_{Δ} (a_{i}))}{\sum_{j = 1}^{n} (1 + T_{Δ} (a_{j}))}}) \\ 1 - {log}_{θ} (1 + \prod_{i = 1}^{n} {(θ^{1 - ν_{i}^{q}} - 1)}^{\frac{n (1 + T (a_{i}))}{\sum_{j = 1}^{n} (1 + T (a_{j}))}}) \\ \geq {log}_{θ} (1 + \prod_{i = 1}^{n} {(θ^{ν_{i}^{q}} - 1)}^{\frac{n (1 + T (a_{i}))}{\sum_{j = 1}^{n} (1 + T (a_{j}))}}) \end{matrix}$

As $θ ≻ 1 \Rightarrow θ^{1 - μ_{i}^{q}} ≻ 1,$ so we get $\begin{matrix} \prod_{i = 1}^{n} {(θ^{1 - μ_{i}^{q}} - 1)}^{\frac{n (1 + T_{Δ} (a_{i}))}{\sum_{j = 1}^{n} (1 + T_{Δ} (a_{j}))}} \\ \leq \sum_{i = 1}^{n} \frac{n (1 + T_{Δ} (a_{i}))}{\sum_{j = 1}^{n} (1 + T_{Δ} (a_{j}))} (θ^{1 - μ_{i}^{q}} - 1) \end{matrix}$

Then $\begin{matrix} 1 - {log}_{θ} (1 + \prod_{i = 1}^{n} {(θ^{1 - μ_{i}^{q}} - 1)}^{\frac{n (1 + T_{Δ} (a_{i}))}{\sum_{j = 1}^{n} (1 + T_{Δ} (a_{j}))}}) - {log}_{θ} (1 + \prod_{i = 1}^{n} {(θ^{μ_{i}^{q}} - 1)}^{\frac{n (1 + T_{Δ} (a_{i}))}{\sum_{j = 1}^{n} (1 + T_{Δ} (a_{j}))}}) \end{matrix}$ $\begin{matrix} = {log}_{θ} \frac{θ}{(1 + \prod_{i = 1}^{n} {(θ^{1 - μ_{i}^{q}} - 1)}^{\frac{n (1 + T_{Δ} (a_{i}))}{\sum_{j = 1}^{n} (1 + T_{Δ} (a_{j}))}}) (1 + \prod_{i = 1}^{n} {(θ^{μ_{i}^{q}} - 1)}^{\frac{n (1 + T_{Δ} (a_{i}))}{\sum_{j = 1}^{n} (1 + T_{Δ} (a_{j}))}})} \end{matrix}$ $\begin{matrix} \geq {log}_{θ} \frac{θ}{(1 + \sum_{i = 1}^{n} \frac{n (1 + T_{Δ} (a_{i}))}{\sum_{j = 1}^{n} (1 + T_{Δ} (a_{j}))} (θ^{1 - μ_{i}^{q}} - 1)) (1 + \sum_{i = 1}^{n} \frac{n (1 + T_{Δ} (a_{i}))}{\sum_{j = 1}^{n} (1 + T_{Δ} (a_{j}))} (θ^{μ_{i}^{q}} - 1))} \end{matrix}$ $\begin{matrix} = {log}_{θ} \frac{θ}{(\sum_{i = 1}^{n} \frac{n (1 + T_{Δ} (a_{i}))}{\sum_{j = 1}^{n} (1 + T_{Δ} (a_{j}))} θ^{1 - μ_{i}^{q}}) (\sum_{i = 1}^{n} \frac{n (1 + T_{Δ} (a_{i}))}{\sum_{j = 1}^{n} (1 + T_{Δ} (a_{j}))} θ^{μ_{i}^{q}})} \geq 0 \end{matrix}$

Similarly, we can get $\begin{matrix} \begin{matrix} 1 - {log}_{θ} (1 + \prod_{i = 1}^{n} {(θ^{1 - ν_{i}^{q}} - 1)}^{\frac{n (1 + T (a_{i}))}{\sum_{j = 1}^{n} (1 + T (a_{j}))}}) \geq \\ {log}_{θ} (1 + \prod_{i = 1}^{n} {(θ^{ν_{i}^{q}} - 1)}^{\frac{n (1 + T (a_{i}))}{\sum_{j = 1}^{n} (1 + T (a_{j}))}}) \end{matrix} \end{matrix}$

Thus, when θ ≻ 1, we have $\begin{matrix} q - ROFFPPG (a_{1}, a_{2}, . . ., a_{n}) \end{matrix}$ $\begin{matrix} \leq q - ROFFPPA (a_{1}, a_{2}, . . ., a_{n}) \end{matrix}$

4 A Novel Method to MADM with q-Rung Orthopair Fuzzy Information

In the present section, we introduce a novel approach to MADM problems under q-rung orthopair fuzzy environment. A MADM problem with q-rung orthopair fuzzy information can be described as: let X ={ x₁, x₂, . . . , x_m } be a set of alternatives, G ={ G₁, G₂, . . . , G_s } be a set of attributes. For attribute G_j ( j = 1, 2, . . . , s ) of alternative x_i ( i = 1, 2, . . . , m ), DMs are required to use a q-ROFN to express their preference information, which can be denoted as a_ij = ( μ_ij, v_ij ) . Therefore, a q-rung orthopair fuzzy decision matrix can be obtained a = ( a_ij ) _m×s. In the following, with the aid of the above- proposed q-rung orthopair fuzzy aggregation operators, a novel approach to solve this problem is introduced. For clarity, the scheme of the application of q-ROFSs to MCDM with the aid of the proposed aggregation operators is shown in Fig. 1.

Fig. 1

Flow chart of MADM with q-Rung orthopair fuzzy based on q-ROFFPPA and q-ROFFPPG operator.

For Fig. 1, the detailed decision making process is designed as follows:

Step 1. Generally, the decision matrix should be normalized by

$a_{ij} = {\begin{matrix} {(μ_{ij}, v_{ij})}_{q} G_{j} \in I_{1} \\ {(v_{ij}, μ_{ij})}_{q} G_{j} \in I_{2} \end{matrix}$ (22) whereI₁ and I₂ represent benefit attributes and cost attributes, respectively.

Step 2: Calculate the point distance d_Δ ( a_ik, a_il ) between a_ik and a_il according to Definition 8, and the point supports among attributeG_j (j = 1, 2, . . . , n) of alternativeX_i

$\sup_{Δ} (a_{ik}, a_{il}) = 1 - d_{Δ} (a_{ik}, a_{il})$ (23) where Δ denotes D_ξ ( a_ij ) , F_ξ,ς ( a_ij ) , G_ξ,ς ( a_ij ) , H_ξ,ς ( a_ij ) , J_ξ,ς ( a_ij ) , i = 1, 2, . . . , m ; k, l = 1, 2, . . . , n.

Step 3: Calculate T_Δ ( a_ij ) of alternative X_i with respect to attributeG_j (j = 1, 2, . . . , n)

$T △ (a i j) = \sum_{\underset{l \neq k}{l = 1}}^{n} S u p △ (a i k, a i l)$ (24) where i = 1, 2, . . . , m ; k, l = 1, 2, . . . , n and j = 1, 2, . . . . , m.

Step 4: Calculate the weight vector w_ij of the associated q-ROFN a_ij according to PPA operator,

$w_{ij} = \frac{(1 + T_{Δ} (a_{ij}))}{\sum_{k = 1}^{n} (1 + T_{Δ} (a_{ik}))}$ (25)

Step 5. For alternative x_i ( i = 1, 2, . . . , m ), utilize the q-ROFFPPA operator, $\begin{array}{l} q - R O F F P P A (a_{i 1}, a_{i 2} \dots a_{i n}) \\ = \sum_{i = 1}^{n}^{\otimes_{F}} \frac{(1 = T_{Δ} (a_{i}))}{\sum_{j = 1}^{n} (1 + T_{Δ} (a_{j}))} a_{i} \end{array}$ or the q-ROFFPPG operator $\begin{array}{l} q - R O F F P P G (a_{1}, a_{2} \dots a_{n}) \\ = \prod_{i = 1}^{n} \otimes_{F (a_{i})} \frac{(1 + T_{Δ} (a_{i}))}{\sum_{j = 1}^{n} (1 + T_{Δ} (a_{j}))} \end{array}$ to aggregate all the attributes values. Therefore, we can get overall valuesx_i ( i = 1, 2, . . . , m ) of alternatives.

Step 6. Rank x_i ( i = 1, 2, . . . , m ) based on the score function S ( a_i ) or accuracy function H ( a_i ) by Definition 3.

Step 7. Rank alternatives x_i ( i = 1, 2, . . . , m ) according to the rank of the corresponding overall values.

5 Numerical example

Let us consider a real-life MADM problem to show the performance of the proposed approach as well as comparative analysis.

Example 3. At present, in China, large hospitals are overcrowded, and the burden of disease is getting heavier. As the part of China’s 13th Five Year Plan (2016–2020) for deepening medical and health system reform, the hierarchical medical treatment system aims to provide an effective response to the challenges of insufficient and unbalanced medical resources in China. However, the hierarchical medical treatment system in China has not been effectively carried out and remains to be further perfected. In order to push forward this system, analyzing the implementation effects of hierarchical diagnosis and treatment of the new rural cooperative medical system (NRCMS) is a key action. Therefore, in the present case analysis, we focus on evaluating the effectiveness of hierarchical diagnosis and treatment of the NRCMS to assist the hierarchical medical system in China. The effectiveness assessment of hierarchical diagnosis and treatment of the NRCMS can be viewed as a MADM problem. The specific statement about the MADM problem is described as follows:

To explore the basic situation of the execution of the hierarchical diagnosis and treatment policy, four primary health care institutions in the hierarchical medical treatment system, denoted by x_i (i = 1, 2, 3, 4), need to be evaluated after the hierarchical diagnosis and treatment’s operation. The four primary health care institutions are evaluated from the following four assessment indicators (attributes): G₁: distribution of outpatient visit; G₂: the number of two-way referrals and the referrals diseases; G₃: satisfaction degree of patients; G₄: average fee per patient. Suppose that the DM gives the rating values for the four primary health care institutions with respect to attributes G _j by the q-ROFNs, and the decision matrix is shown in Table 1.

Table 1
The q-rung orthopair fuzzy decision matrix

G₁ G₂ G₃ G₄

x₁ (0.5,0.4) (0.5,0.3) (0.2,0.6) (0.5,0.4)

x₂ (0.6,0.2) (0.8,0.1) (0.6,0.2) (0.5,0.3)

x₃ (0.5,0.4) (0.2,0.6) (0.6,0.2) (0.4,0.4)

x₄ (0.4,0.3) (0.7,0.2) (0.4,0.5) (0.4,0.5)

	G₁	G₂	G₃	G₄
x₁	(0.5,0.4)	(0.5,0.3)	(0.2,0.6)	(0.5,0.4)
x₂	(0.6,0.2)	(0.8,0.1)	(0.6,0.2)	(0.5,0.3)
x₃	(0.5,0.4)	(0.2,0.6)	(0.6,0.2)	(0.4,0.4)
x₄	(0.4,0.3)	(0.7,0.2)	(0.4,0.5)	(0.4,0.5)

5.1 Application of the proposed MADM method

Step 1. As all the attributes are benefit attribute, the decision matrix does not need to be normalized.

Step 2. Calculate the point distance d_Δ ( a_ik, a_il ) betweena_ik and a_il, based on the Definition 6.

When i = 1, then $\begin{matrix} d_{Δ} (a_{11}, a_{12}) & = 0.0148, d_{Δ} (a_{11}, a_{13}) \\ = 0.1310, d_{Δ} (a_{11}, a_{14}) = 0 \end{matrix}$ $\begin{matrix} d_{Δ} (p_{12}, p_{13}) & = 0.1458, d_{Δ} (p_{12}, p_{14}) \\ = 0.0148, d_{Δ} (p_{13}, p_{14}) = 0.1310 \end{matrix}$

When i = 2, then $\begin{matrix} d_{Δ} (a_{11}, a_{12}) & = 0.1804, d_{Δ} (a_{11}, a_{13}) \\ = 0, d_{Δ} (a_{11}, a_{14}) = 0.0622 \end{matrix}$ $\begin{matrix} d_{Δ} (p_{12}, p_{13}) & = 0.1804, d_{Δ} (p_{12}, p_{14}) \\ = 0.2426, d_{Δ} (p_{13}, p_{14}) = 0.0622 \end{matrix}$

When i = 3, then $\begin{matrix} d_{Δ} (a_{11}, a_{12}) & = 0.1310, d_{Δ} (a_{11}, a_{13}) \\ = 0.0770, d_{Δ} (a_{11}, a_{14}) = 0.0366 \end{matrix}$ $\begin{matrix} d_{Δ} (p_{12}, p_{13}) & = 0.2080, d_{Δ} (p_{12}, p_{14}) \\ = 0.0944, d_{Δ} (p_{13}, p_{14}) = 0.1136 \end{matrix}$

When i = 4, then $\begin{matrix} d_{Δ} (a_{11}, a_{12}) & = 0.1750, d_{Δ} (a_{11}, a_{13}) \\ = 0.0392, d_{Δ} (a_{11}, a_{14}) = 0.0.392 \end{matrix}$ $\begin{matrix} d_{Δ} (p_{12}, p_{13}) & = 0.2142, d_{Δ} (p_{12}, p_{14}) \\ = 0.2142, d_{Δ} (p_{13}, p_{14}) = 0 \end{matrix}$

Step 2. Calculate the point support Sup (a_ij, a_il) (i = 1, 2, 3, 4 ; j, l = 1, 2, . . . , 4) of alternatives with respect to attribute X_i (i = 1, 2, 3, 4) between other attributes based on Equation (23) (for simplicity, the Sup_Δ (a_ij, a_il) is denoted by $S_{jl}^{i}$ ) and obtain $\begin{matrix} \begin{matrix} S_{12}^{1} = S_{21}^{1} = 0.9852, S_{13}^{1} = S_{31}^{1} = 0.8690, \\ S_{14}^{1} = S_{41}^{1} = 1, S_{23}^{1} = S_{32}^{1} = 0.8542, \\ S_{24}^{1} = S_{42}^{1} = 0.9852, S_{34}^{1} = S_{43}^{1} = 0.8690, \\ S_{12}^{2} = S_{21}^{2} = 0.8196, S_{13}^{2} = S_{31}^{2} = 1, \\ S_{14}^{2} = S_{41}^{2} = 0.9378, S_{23}^{2} = S_{32}^{2} = 0.8196, \\ S_{24}^{2} = S_{42}^{2} = 0.7574, S_{34}^{2} = S_{43}^{2} = 0.9378 \\ S_{12}^{3} = S_{21}^{3} = 0.8690, S_{13}^{3} = S_{31}^{3} = 0.9230, \\ S_{14}^{3} = S_{41}^{3} = 0.9634, S_{23}^{3} = S_{32}^{3} = 0.7920, \\ S_{24}^{3} = S_{42}^{3} = 0.9056, S_{34}^{3} = S_{43}^{3} = 0.8864 \\ S_{12}^{4} = S_{21}^{4} = 0.8250, S_{13}^{4} = S_{31}^{4} = 0.9608, \\ S_{14}^{4} = S_{41}^{4} = 0.9608, S_{23}^{4} = S_{32}^{4} = 0.7858, \\ S_{24}^{4} = S_{42}^{4} = 0.7858, S_{34}^{4} = S_{43}^{4} = 1 \end{matrix} \end{matrix}$ .

Step 3. According on Equation (24), calculate the T_Δ (a_ij) (i = 1, 2, 3, 4 ; j = 1, 2, . . . , 5) of alternatives with respect to attribute X_i (i = 1, 2, 3, 4), and obtain $\begin{matrix} T_{Δ} (a_{11}) = 2.8542, T_{Δ} (a_{12}) = 2.8246, \end{matrix}$ $\begin{matrix} T_{Δ} (a_{13}) = 2.7232, T_{Δ} (a_{14}) = 2.8542 \end{matrix}$ $\begin{matrix} T_{Δ} (a_{21}) = 2.7574, T_{Δ} (a_{22}) = 3.3966, \end{matrix}$ $\begin{matrix} T_{Δ} (a_{23}) = 2.7574, T_{Δ} (a_{24}) = 2.6330 \end{matrix}$ $\begin{matrix} T_{Δ} (a_{31}) = 2.7554, T_{Δ} (a_{32}) = 2.5666, \end{matrix}$ $\begin{matrix} T_{Δ} (a_{33}) = 2.6784, T_{Δ} (a_{34}) = 2.7554 \end{matrix}$ $\begin{matrix} T_{Δ} (a_{41}) = 2.7466, T_{Δ} (a_{42}) = 2.3966, \end{matrix}$ $\begin{matrix} T_{Δ} (a_{43}) = 2.7858, T_{Δ} (a_{44}) = 2.7466 . \end{matrix}$

Step 4. Calculate the power weighting w_i (i = 1, 2, 3, 4) of alternatives with respect to attributeX_i (i = 1, 2, 3, 4) based on Equation (25) and obtain $\begin{matrix} w_{11} = 0.2536, w_{12} = 0.2509, \\ w_{13} = 0.2419, w_{14} = 0.2536 \end{matrix}$ $\begin{matrix} w_{21} = 0.2615, w_{22} = 0.2273, \\ w_{23} = 0.2615, w_{24} = 0.2497 \end{matrix}$ $\begin{matrix} w_{31} = 0.2562, w_{32} = 0.2386, \\ w_{33} = 0.2490, w_{34} = 0.2562 \end{matrix}$ $\begin{matrix} w_{41} = 0.2573, w_{42} = 0.2245, \\ w_{43} = 0.2610, w_{44} = 0.2573 \end{matrix}$

Step 5. Aggregate the attributes values for alternative x_i ( i = 1, 2, 3, 4 ) by using the q-ROFFPPA (θ = 2) operator. Therefore, we can get a set of overall values. $\begin{matrix} a_{1} = (0.4605, 0.9677), a_{2} = (0.6486, 0.9963), \\ a_{3} = (0.4740, 0.9695), a_{4} = (0.5106, 0.9745) \end{matrix}$

Step 6. Calculate scores of a_i ( i = 1, 2, 3, 4 ) according to the Definition 3, we can get $\begin{matrix} s (a_{1}) = - 0.8087 s (a_{2}) = - 0.7159 \\ s (a_{3}) = - 0.8047 s (a_{4}) = - 0.7922 . \end{matrix}$

Therefore, the rank of the overall values is a₂ ≻ a₄ ≻ a₃ ≻ a₁.

Step 7. Rank alternative x_i ( i = 1, 2, 3, 4 ) according to the rank of a_i ( i = 1, 2, 3, 4 ). We can get x₂ ≻ x₄ ≻ x₃ ≻ x₁.

Thus, according to the calculation results, the second primary health care institution is the best in implementing the hierarchical diagnosis and treatment policy. Meanwhile, the first primary health care institution is not so well. Thus, it is necessary for the first primary health care institution to strengthen the publicity of hierarchical diagnosis and treatment policies and improve the residents’ knowledge of the hierarchical diagnosis and treatment policies.

5.2 Influence of parameter θ, q on the final result

To reflect the influence of parameters q and θ on the ranking results, we utilize different parameters q to solve the MADM problem in Example 3 by the proposed q-ROFFPPA and q-ROFFPPG operators, respectively. Results are shown as Fig. 2 and Table 2.

Fig. 2

Score values of the alternatives whenq ∈ ( 2, 10 ) based on the q-ROFFPPA operator.

Table 2

Ranking results by assigning different values to parameter θ in q-ROFFPPA operator

Parameter θ	Score values of overall values	Ranking Results
θ = a₁ → 1	s ( a₁ ) = 0.0977 s ( a₂ ) = 0.2765 s ( a₃ ) = 0.1073 s ( a₄ ) = 0.1354	x₂ ≻ x₄ ≻ x₃ ≻ x₁
θ = a₂ = 2	s ( a₁ ) = -0.8087 s ( a₂ ) = -0.7159 s ( a₃ ) = -0.8047 s ( a₄ ) = -0.7922	x₂ ≻ x₄ ≻ x₃ ≻ x₁
θ = a₃ = 3	s ( a₁ ) = -0.8093 s ( a₂ ) = -0.7178 s ( a₃ ) = -0.8056 s ( a₄ ) = -0.7936	x₂ ≻ x₄ ≻ x₃ ≻ x₁
θ = a₃ = 5	s ( a₁ ) = -0.8099 s ( a₂ ) = -0.7197 s ( a₃ ) = -0.8066 s ( a₄ ) = -0.7950	x₂ ≻ x₄ ≻ x₃ ≻ x₁
θ = a₄ = 10	s ( a₁ ) = -0.8105 s ( a₂ ) = -0.7219 s ( a₃ ) = -0.8075 s ( a₄ ) = -0.7965	x₂ ≻ x₄ ≻ x₃ ≻ x₁
θ = a₄ = 30	s ( a₁ ) = -0.8112 s ( a₂ ) = -0.7244 s ( a₃ ) = -0.8085 s ( a₄ ) = -0.7981	x₂ ≻ x₄ ≻ x₃ ≻ x₁
θ = a₅ = 50	s ( a₁ ) = -0.8113 s ( a₂ ) = -0.7252 s ( a₃ ) = -0.8088 s ( a₄ ) = -0.7985	x₂ ≻ x₄ ≻ x₃ ≻ x₁

Figure 2 shows the scores values of four primary health care institutions obtained by the q-ROFFPPA operator. From Fig. 2, we can see that the scores of the overall values are different when assigning different parameters q to the q-ROFFPPA operator. However, the ranking results are alwaysx₂ ≻ x₄ ≻ x₃ ≻ x₁. Furthermore, x₂ is always the best primary health care institution. In addition, the score values of the overall assessments by utilizing the q-ROFFPPA operator become smaller as q increases on [2, 10]. Therefore, the parameter q can be viewed as DMs’ attitude. The more optimistic the DMs are, the smaller values assigned to q and the more pessimistic the DMs are, the greater values assigned to q. Moreover, as the values of q become greater and greater, the score values are very close to a fixed value, no matter the value of q.

In the following, we investigate the influence of parameter θ on score values and ranking results obtained by the q-ROFFPPA operator. Here, we take q = 3. Details can be found in Table 2.

From Table 2, we find that with different parametersθin the q-ROFFPPA operator, the scores vary. However, no matter what the parameterθ, the ranking result is alwaysx₂ ≻ x₄ ≻ x₃ ≻ x₁, and thus the best primary health care institution is alwaysx₂. By further analyzing Table 2, we can easily find that the score values of alternatives obtained by q-ROFFPPA operator become smaller asθincreases. Therefore, DMs can appropriately select the value of θaccording to their preference and actual needs.

5.3 Comparison analysis

In this section, some comparisons are conducted to illustrate the effectiveness and advantages of the q-ROFFPPA and q-ROFFPPA operators. Considering the q-ROFFPPA and q-ROFFPPG operators that combine Frank operational rules with PPA operator under q-rung orthopair fuzzy environment, we select the following existing approaches as reference approaches to solve the above Example 3.

(1) Comparing with the methods proposed by Xu [30], Ma and Xu [31], and Liu and Wang [19]

In this subsection, we utilize Xu’s method [30], based on intuitionistic fuzzy weighted averaging (IFWA) operator, Ma and Xu’s method [31] based on the Pythagorean fuzzy weighted averaging (PFWA) operator, and Liu and Wang’s method [19] based on the q-rung orthopair fuzzy weighted averaging (q-ROFWA) operator to solve the above example. Results are shown in Table 3.

Table 3
Ranking results based on different methods for Example 3

Approaches Scores values of a_i (i = 1, 2, 3, 4) Ranking results

Xu’s method [30] based on IFWA operator s ( a₁ ) = 0.025,s ( a₂ ) = 0.4582, s ( a₃ ) = 0.0711, s ( a₄ ) = 0.1455 x₂ ≻ x₄ ≻ x₃ ≻ x₁

Ma and Xu’s method [31] based on PFWA operator s ( a₁ ) = 0.0121,s ( a₂ ) = 0.1780, s ( a₃ ) = 0.0254, s ( a₄ ) = 0.0518 x₂ ≻ x₄ ≻ x₃ ≻ x₁

Liu and Wang’s method [19] based on q-ROFWA operator s ( a₁ ) = 0.0011,s ( a₂ ) = 0.0230, s ( a₃ ) = 0.0013, s ( a₄ ) = -0.0029 x₂ ≻ x₄ ≻ x₃ ≻ x₁

Our approach based on q- q-ROFFPPA operator (q = 3) s ( a₁ ) = -0.8087 s ( a₂ ) = -0.7159 s ( a₃ ) = -0.8047 s ( a₄ ) = -0.7922 x₂ ≻ x₄ ≻ x₃ ≻ x₁

Approaches	Scores values of a_i (i = 1, 2, 3, 4)	Ranking results
Xu’s method [30] based on IFWA operator	s ( a₁ ) = 0.025,s ( a₂ ) = 0.4582, s ( a₃ ) = 0.0711, s ( a₄ ) = 0.1455	x₂ ≻ x₄ ≻ x₃ ≻ x₁
Ma and Xu’s method [31] based on PFWA operator	s ( a₁ ) = 0.0121,s ( a₂ ) = 0.1780, s ( a₃ ) = 0.0254, s ( a₄ ) = 0.0518	x₂ ≻ x₄ ≻ x₃ ≻ x₁
Liu and Wang’s method [19] based on q-ROFWA operator	s ( a₁ ) = 0.0011,s ( a₂ ) = 0.0230, s ( a₃ ) = 0.0013, s ( a₄ ) = -0.0029	x₂ ≻ x₄ ≻ x₃ ≻ x₁
Our approach based on q- q-ROFFPPA operator (q = 3)	s ( a₁ ) = -0.8087 s ( a₂ ) = -0.7159 s ( a₃ ) = -0.8047 s ( a₄ ) = -0.7922	x₂ ≻ x₄ ≻ x₃ ≻ x₁

The ranking result obtained by our method based on the q-ROFFPPA operator is the same as the ones obtained by Xu’s method [30], Ma and Xu’s method [31], and Liu and Wang’s method [19], i.e. x₂ ≻ x₄ ≻ x₃ ≻ x₁, which verifies the validity of the new method.

However, since the ranking results obtained by these methods are the same, this result cannot reflect the advantages of our method very well. To further illustrate the main advantages of our method, we do a further analysis by adjusting some of the data in the above example.

Example 4. In the process of decision making, DMs may provide some unreasonable evaluation values owning to the difference of background and knowledge. In order to model above situation, we change the attribute values a₂₁ and a₂₄ in Table 4 from (0.6, 0.2) to a too small value (0.01, 0.03) and from (0.5, 0.3) to (0.04, 0.05), respectively, then we solve the MADM problem by using Xu’s method, Ma and Xu’s method, Liu and Wang’s method, and the proposed approach. The score values and ranking results provided by above approaches are shown in Table 4.

Table 4

Ranking results based on different methods for Example 4

Approaches	Score values ofa_i (i = 1, 2, 3, 4)	Ranking results
Xu’s method [30] based on IFWA operator	s ( a₁ ) = 0.0257,s ( a₂ ) = 0.4009, s ( a₃ ) = 0.0711, s ( a₄ ) = 0.1455	x₂ ≻ x₄ ≻ x₃ ≻ x₁
Ma and Xu’s method [31] based on PFWA operator	s ( a₁ ) = 0.0121,s ( a₂ ) = 0.0945, s ( a₃ ) = 0.0254, s ( a₄ ) = 0.0518	x₂ ≻ x₄ ≻ x₃ ≻ x₁
Liu and Wang’s method [19] based on q-ROFWA operator	s ( a₁ ) = 0.0011,s ( a₂ ) = 0.0097, s ( a₃ ) = 0.0013, s ( a₄ ) = -0.0029	x₂ ≻ x₄ ≻ x₃ ≻ x₁
Our approach based on q- q-ROFFPPA operator (q = 3)	s ( a₁ ) = -0.8087 s ( a₂ ) = -0.8058 s ( a₃ ) = -0.8047 s ( a₄ ) = -0.7922	x₄ ≻ x₃ ≻ x₂ ≻ x₁

In Example 4, we adjust the dataa₂₁ and a₂₄ in Table 4 from (0.6, 0.2) to a too small value (0.01, 0.03) and from (0.5, 0.3) to (0.04, 0.05), respectively. From Table 4, we can find that the optimal alternative obtained by our proposed approach is changed from x₂ into x₄, while the ranking results obtained by other three methods are unchanged. For the above results, we can provide an explanation as follows. When we changed the attribute vales a₂₁ and a₂₄ to a too small value (0.01, 0.03) and (0.04, 0.05), because the proposed approach in this paper uses the power point weighted vector by measuring the point distance of the arguments, it could avoid the influence of those unduly high or low attribute values on the decision result. However, the methods based on IFWA, PFWA, q-ROFWA operators aggregate attribute values by using simple weighted averaging operators and do not consider the special situation. Thus, in this example, although the score values of the second primary health care institution A₂ obtained by methods based on IFWA, PFWA, and q-ROFWA operators are becoming smaller, the ranking result obtained by these methods is unchanged whereas the best alternative identified by the proposed method become x₂ from x₄.

Thus, the proposed approach is more reasonable in solving practical decision-making problems where the decision data of some alternatives is unduly high or unduly low.

(2) Comparing with Xu’s method [32], Wei and Lu’s method [33] and Zhang et al.’s method [34]

In this subsection, we utilize Xu’s method [32] based on intuitionistic fuzzy power weighted averaging (IFPWA) operator, Wei and Lu’s method [33] based on the Pythagorean fuzzy power weighted average (PFPWA) operator, Zhang et al.’s method [34] based on intuitionistic fuzzy frank power weighted averaging (IFFPWA) operator to solve the above example. Results are shown in Table 5.

Table 5

Ranking results based on different methods for Example 3

Approaches	Score values ofa_i (i = 1, 2, 3, 4)	Ranking results
Xu’s method [32] based on IFPWA operator	s ( a₁ ) = 0.0322,s ( a₂ ) = 0.4530, s ( a₃ ) = 0.0804, s ( a₄ ) = 0.1250	x₂ ≻ x₄ ≻ x₃ ≻ x₁
Wei and Lu’s method [33] based on PFPWA operator	s ( a₁ ) = 0.0137,s ( a₂ ) = 0.1732, s ( a₃ ) = 0.0273, s ( a₄ ) = 0.0448	x₂ ≻ x₄ ≻ x₃ ≻ x₁
Zhang et al.’s method [34] based on IFFPWA operator	s ( a₁ ) = -0.1294,s ( a₂ ) = -0.1582, s ( a₃ ) = -0.1462, s ( a₄ ) = -0.1256,	x₂ ≻ x₄ ≻ x₃ ≻ x₁
Our approach based on q- q-ROFFPPA operator (q = 3)	s ( a₁ ) = -0.8087 s ( a₂ ) = -0.7159 s ( a₃ ) = -0.8047 s ( a₄ ) = -0.7922	x₂ ≻ x₄ ≻ x₃ ≻ x₁

From Table 5, we can find that the ranking result obtained by our method based on q-ROFFPPA operator, Xu’s method [32] based on IFPWA operator, Wei and Lu’s method [33] based on PFPWA operator and Zhang et al.’s method [34] based on IFFPWA operator produce is the same, i.e. x₂ ≻ x₄ ≻ x₃ ≻ x₁. The reason is that they all adopt power weighted vector by measuring the distance of the arguments. The difference is our proposed method is based on the point distance, and thus can be viewed as DM’s pessimistic or optimistic attitude. That’s why the score values of alternatives obtained by methods based on q-ROFFPPA operator are smaller than the Xu’s method [32], Wei and Lu’s method [33] and Zhang et al.’s method [34].

However, Xu’s method, Wei and Lu’s method, Zhang et al.’s method also have some limitations. That is, they are based on the IFNs and PFNs, and thus cannot deal with the situation where the sum or square sum of the MD and the NMD is bigger than 1, whereas the proposed method can.

To further explain the advantages of the proposed approach in modeling fuzzy information comparing with Xu’s method [32], Wei and Lu’s method [33], Zhang et al.’s method [34], we give another real-life example.

Example 5. To improve the quality of medical care of clinical departments in a specialist hospital, four clinical departments of the hospital denoted by x_i (i = 1, 2, 3, 4), need to be objectively evaluated. The four clinical departments are evaluated from four main aspects: G₁: hospital infection rate; G₂: length of hospital stay; G₃: time and cost consumption index; G₄: adverse events rate of hospitalized patient; Suppose that a DM gives the rating values by using q-ROFNs, and the decision matrix is shown in Table 6.

Table 6

The decision matrix of Example 5

	G₁	G₂	G₃	G₄
x₁	(0.9,0.3)	(0.8,0.1)	(0.6,0.6)	(0.6,0.3)
x₂	(0.7,0.4)	(0.8,0.7)	(0.8,0.2)	(0.4,0.3)
x₃	(0.8,0.4)	(0.7,0.5)	(0.4,0.2)	(0.6,0.4)
x₄	(0.8,0.2)	(0.8,0.2)	(0.9,0.7)	(0.7,0.4)

Then, we apply the proposed MADM approach and Xu’s method [32], Wei and Lu’s method [33], Zhang et al.’s method [34] to address the MADM problem, respectively. Details can be found in Table 7.

Table 7

Ranking results based on different methods for Example 5

Approaches	Score values ofa_i (i = 1, 2, 3, 4)	Ranking results
Xu’s method [32] based on IFPWA operator	Cannot be calculated	No
Wei and Lu’s method [33] based on PFPWA operator	Cannot be calculated	No
Zhang et al.’s method [34] based on IFFPWA operator	Cannot be calculated	No
Our approach based on q-ROFFPPA operator (q = 3)	s ( a₁ ) = -0.4795 s ( a₂ ) = -0.4952 s ( a₃ ) = -0.6372 s ( a₄ ) = -0.3376	x₄ ≻ x₁ ≻ x₂ ≻ x₃

From Table 7, we know the elements a₂₂ and a₄₃ are (0.8, 0.7), (0.9, 0.7), respectively. Given 0.8² + 0.7² > 1and 0.9² + 0.7² > 1, the evaluation attribute values (0.8, 0.7) and (0.9, 0.7) cannot be expressed by IFNs and PFNs. Thus, as shown in Table 7, Xu’s method [32], Wei and Lu’s method [33], Zhang et al.’s method [34] cannot solve the above problem as the MD and NMD do not satisfy the constraint condition of IFNs and PFNs. However, the proposed approach based on the q-ROFFPPA operator can still work as (0.8, 0.7) and (0.9, 0.7) can be represented by q-ROFNs by adjusting the value of q. According to the calculation results above, the quality of medical service in each department can be accurately reflected. The top one department is x₄ with the scores of -0.3376, and the quality of medical service is better than that of other departments. The bottom one department is x₃, and the quality of medical services is poor. Thus, for the third department, in the premise of guaranteeing quality, further attention should be paid to reduce the cost and length of hospital stay, strengthen the management of hospital infection and improve the emergency response system of adverse events.

Our approach provides technical support for hospital departments to make scientific and effective improvement measures and departmental delicacy management. Therefore, the applicable range of our approach is wider than the methods based on the IFPW, PFWA and IFFPWA operators.

(3) Summary about the proposed operators’ superiorities

To further illustrate the advantages of the proposed aggregation operators, we present the characteristics of proposed operators and operators in Refs. [19 , 30–36]. Results can be found in Table 8.

Table 8

Characteristics of different aggregation operators

Aggregation operators	Monotonically with respect to the parameter	Whether allows the input arguments support each other	Whether can reallocate the degree of hesitance	Whether there is a parameter to reflect preferences	Whether model uncertainty more powerful
IFWA [30]	No	No	No	No	No
PFWA [31]	No	No	No	No	No
q-ROFWA [19]	No	No	No	No	Yes
IFPWA [32]	No	Yes	No	No	No
PFPWA [33]	No	Yes	No	No	No
IFFPWA[34]	Yes	Yes	No	Yes	No
ST-q-ROFWA [35]	Yes	Yes	No	Yes	Yes
q-ROFWNA [36]	No	Yes	Yes	Yes	Yes
q-ROFFPPA	Yes	Yes	Yes	Yes	Yes
q-ROFFPPG	Yes	Yes	Yes	Yes	Yes

From Table 8, we can find our operators have the following advantages compared with the existing operators introduced in Refs. [19 , 30–34]:

From the point view of operational laws, the IFWA, PFWA, q-ROFWA, IFPWA, PFPWA and q-ROFGPWA operators in Refs. [19 , 30–34] use traditional operational rules and thus there is no a parameter to reflect DMs’ attitude, whereas the q-ROFFPPA and q-ROFFPPG operators proposed in this paper adopt the Frank operational laws. Thus, the advantage of the proposed method is that it is monotonic with respect to the parameter θ, which means that DMs have the flexibility to choose values of parameter θ based on his/her attitude. In practical decision situations, if the DM’s attitude is optimistic, then we let θ → 1, if the DM’s attitude is pessimistic, then we let θ→ + ∞.

From the point view of aggregation operators, IFWA, PFWA, q-ROFWA, ST-q-ROFWA, q-ROFWNA operators do not adopt the power weighted vector and thus cannot relieve the influence of the unreasonable data, whereas IFPWA and PFPWA operators [32, 33] and the proposed operators in this paper consider this situation. They allow the values being aggregated to support and reinforce each other based on the support degree between the input arguments, thereby making the opinion more reasonable. However, IFPWA and PFPWA operators [32, 33] use traditional distance to measure the support degrees, whereas the q-ROFFPPA and q-ROFFPPG operators proposed in this paper adopt the point distance measures which can reallocate the MD and NMD of q-ROFNs according to different principle. Thus, it can get more objective result from the original one and give in effect results that are consistent with the essence of q-ROFS. Moreover, our approach based on q-ROFFPPA and q-ROFFPPG operators adopt the Frank operational laws, which is the same as the IFFPWA operator [34]. However, IFFPWA operator is a special case of our proposed q-ROFFPPA (ξ, ς = 0, q=1) operator.

From the point view of information expression, the scope of application of our approach is very wide. Although the IFFPWA operator [34] cannot only reduce the influence of the unduly high or low arguments but also adopt the Frank operational laws, it can only solve MADM problems expressed by IFNs. Moreover, compared with the IFFPWA operator [34], the proposed q-ROFFPPA and q-ROFFPPG operators are more functional as they can reallocate the degree of hesitance by the defined point distance. Thus, DMs can appropriately select different point distance measures proposed in this paper to express their pessimistic or optimistic attitude. Additionally, DMs can appropriately select the values of θ, q in the q-ROFFPPA and q-ROFFPPG operators according to actual needs.

In summary, because the proposed method can combine the Frank operational laws with the PPA operator, it provides a flexible and general tool to deal with MADM problems. Based on the comparisons and analysis above, our method is more functional and reasonable.

6 Conclusions

In this paper, we proposed q-rung orthopair fuzzy Frank power point aggregation operators by combining Frank operational laws and PPA operators. Then, based on q-rung orthopair fuzzy Frank power point aggregation operators, we developed a framework to the MADM problem under q-rung orthopair fuzzy environment. The strengths of the proposed method were also discussed via comparative analysis.

The main academic contributions of this paper are four aspects. First, Frank operational rules of q-ROFNs are provided. The operational rules have an additional parameter that makes aggregation operators monotonically decreasing with respect to the parameter, which makes aggregation process more flexible. Second, a series of distance measures based on point operators for q-ROFS are developed. Based on it, new PPA operators that can control the degree of hesitance by the defined point distance are proposed. Third, q-rung orthopair fuzzy aggregation operators are proposed, including q-ROFFPPA and q-ROFFPPG operators. By using the power point weighted vector, the proposed operators not only can get more objective information, but also avoid the influence of unduly high or low attribute values on the decision result. Fourth, a novel approach to MADM with q-rung orthopair fuzzy information is proposed. The proposed method has a strong practicability and dependability and can be further applied to various practical decision-making problems, such as healthcare management, supplier selection, emergency decision making, and so on.

For future study, it is worth integrating the PPA operator with other operational laws, such as Sine trigonometric operational laws [37], etc. In addition, further research can extend the proposed operators to other fuzzy sets, such as interval-valued intuitionistic fuzzy set [38], interval-valued Pythagorean fuzzy set [39] and linguistic interval-valued Pythagorean fuzzy set [40], and so on.

Footnotes

Acknowledgments

This work was supported by Fundamental Research Funds for the Central Universities.

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q-Rung orthopair fuzzy frank power point aggregation operators with new multi-parametric distance measures

Abstract

Keywords

1 Introduction

2 Preliminaries

2.1 q-Rung orthopair fuzzy set

3.1 q-ROFFPPA aggregation operator

Table 1 The q-rung orthopair fuzzy decision matrix G1 G2 G3 G4 x1 (0.5,0.4) (0.5,0.3) (0.2,0.6) (0.5,0.4) x2 (0.6,0.2) (0.8,0.1) (0.6,0.2) (0.5,0.3) x3 (0.5,0.4) (0.2,0.6) (0.6,0.2) (0.4,0.4) x4 (0.4,0.3) (0.7,0.2) (0.4,0.5) (0.4,0.5)

5.2 Influence of parameter θ, q on the final result

Footnotes

Acknowledgments

References

Table 1
The q-rung orthopair fuzzy decision matrix

G₁ G₂ G₃ G₄

x₁ (0.5,0.4) (0.5,0.3) (0.2,0.6) (0.5,0.4)

x₂ (0.6,0.2) (0.8,0.1) (0.6,0.2) (0.5,0.3)

x₃ (0.5,0.4) (0.2,0.6) (0.6,0.2) (0.4,0.4)

x₄ (0.4,0.3) (0.7,0.2) (0.4,0.5) (0.4,0.5)