Biparametric Q Rung Orthopair Fuzzy Entropy Measure for Multi Criteria Decision Making Problem

Abstract

In this study we propose a measure of the entropy of the norm (R, S) for q-row orthopair fuzzy sets (qROFS). The proposed entropy measure is validated both theoretically and practically to ensure validity. We also propose a simple methodology for the purpose of solving a multi-criteria decision-analysis problems using the introduced entropy measure. This method takes into account different circumstances of criteria weights, such as unknown weights, as well as other cases when the weights are not fully known. Finally, a demonstration with numerical examples for the proposed entropy has been provided to show how to apply the novel methodologies.

1. Introduction

In different circumstances, the process of settling on a choice is determined by the benefits, which depend on our insight and earlier data. Notwithstanding, because incomplete data are also among the human mistakes, it is very likely to become entangled in the environment’s acquired complexity and lack of framework knowledge. Subsequently, it is truly challenging to find the perfect solution in a given time frame. To handle these vulnerabilities, dubiousness, and inadequacy in the collected data, decision-makers widely use fuzzy sets, as well as their various generalisations, which have been proven to be robust mathematical techniques for modeling inaccuracies in data.

The fuzzy sets theory introduced by Zadeh [1] can be perceived as an attempt to build a corpus of concepts and procedures for dealing with a form of imprecision that develops when the limits of objects are not firmly specified in a systematic way. On account of information addressed as far as fuzzy sets, data conveyed are communicated by a membership function. As a result, the community which has grown around fuzzy set theory have demonstrated a wide range of practical applications [2,3] and extensions [4] of this concept.

The first of the many generalisations of Zadeh’s fuzzy sets is the intuitionistic fuzzy sets concept introduced in 1986 by Atanassov et al. [5]. Since then, intuitionistic fuzzy sets (IFSs) have been used in numerous practical study cases to overcome the uncertainty and indecision inherent in complex decision-making problems. The distinguishing feature of an IFS is that it maps a number from the interval $[0,1]$ to each detail inside the value domain. As a result, we can manipulate a degree of membership, and new degree of non-membership, which is not used in fuzzy sets. IFSs also introduce a degree of uncertainty and the sum of membership values should be equal to one. In the literature, IFSs can be applied to practical applications in the domain of multi-criteria decision-analysis problems, design acknowledgment, dealing with investigation monetary administrations, clinical determination, and so forth [6,7].

However, the IFS has several drawbacks, such as restriction in expressing the domain’s expert- or decision-maker-specific knowledge. If the domain’s expert feels that the membership degree should be 0.7 and the value of non-membership degree is 0.5, then the sum those two values is greater than 1 and therefore breaks the IFS rule. The ability to break this limitation can be useful in certain real-world multi-criteria decision analysis problems. Yager and Abbasov (Yager) [8] correctly pointed out that the sum of the membership degree and the non-membership degree values of a single alternative given by a decision maker might be greater than 1, which means that in some cases the IFS cannot be applied. For functional purposes, which can be seen as following an overall disposition to involve as many data as are accessible, it appears to be convenient, even essential, to likewise consider a hesitation margin.

Yager proposed another generalization of Zadeh’s fuzzy sets, namely Pythagorean fuzzy sets (PFS) [9], which are a straight generalization of IFS. In this extension, the sum of determined membership and non-membership degree values can be less than or equal to 1. This condition is less restrictive than the one which Intuitionistic Fuzzy Sets should meet. That makes Pythagorean fuzzy sets superior to IFSs, because when the sum of squares is less than 1, complex expert knowledge can be presented, for example, when the decision-maker states that the membership should be equal to 0.8 and non-membership function is 0.5 [10].

Recently, Yager et al. introduced the q-rung orthopair fuzzy sets (q-ROFSs) [11,12]. They have been viewed as a proficient instrument to depict ambiguity and uncertainty in multi-criteria decision-analysis problems. This extension also imposes the restriction that the sum of the powers of the value of membership and non-membership degrees should be equal to or less than 1. However, unlike PFS, in q-ROFSs the power should be chosen by an expert or a decision-maker. Therefore, the less-restricted condition for the membership and non-membership functions would be as follows (1):

$${\mu }_M{\left(x_i\right)}^q+{\nu }_M{\left(x_i\right)}^q\le 1,$$

(1)

where q value is larger than, or equal, to 1. For instance, $(0.7,0.1)$ is an intuitionistic membership degree since it satisfies $1\ge 0.7+0.1$. Assuming the non-membership degree value is 0.6, then, at that point, because $1\ge {\left(0.7\right)}^2+{\left(0.6\right)}^2$, (0.7, 0.6) is a membership degree of PFS. Nonetheless, in the event that the level of non-membership is 0.8, this circumstance cannot be depicted by utilizing IFSs or PFSs. Here the q-ROFS comes into play. It is worth noticing that the universe of permitted orthopairs expands as the rung q grows, and we can communicate a more extensive scope of fuzzy data by utilizing q-ROFSs as more orthopairs satisfy the constraint conditions. Therefore, many researchers are working on q-ROFSs because of their capacity to handle uncertainty more effectively in practical decision-making problems [13,14].

This highly permissive condition on membership and non-membership values in particular allows the decision-maker to propose these values almost independently, with no drastic restrictions on them, with the sole limitation being that if one is unity, the other should be zero. The fundamental properties and operations on q-ROFS were proposed by Yager et al. [12] and were used in knowledge portrayal. Ali [15] proposed one more perspective on q-ROFS by utilizing orbits. Different aggregation operators, such as the q-ROFWG and q-ROFWA operators (Liu and Wang) [16], q-ROFPMSM and q-ROFPA operators (Liu et al.) [17], q-ROFWGBM and q-ROFGBM operators [18], IFMM operator (Liu and Li) [19], q-ROFWPHM operator (Liu et al.) [20], and q-ROFIDHM operator (Xing et al.) [21], have been introduced and used in q-ROFS.

Hussain et al. [22] merged q-ROFS and soft sets, defining their aggregation operators in the process. Banerjee et al. [23] proposed the qROFS SMAA-QUALIFLEX technique. An algorithm for emergency decision-making was proposed by Peng et al. [24], using q-ROFSs. Zhong et al. [25] defined the q-ROFSs Dombi power partitioned Heronian mean operators.

With their practical applications and analytical thinking, Bajaj [26] developed a novel entropy function under an R-norm IFS environment as well as a weighted R-norm IFS divergence measure. Gandotra et al. [27] used parametric entropy measure under the $\alpha$-cut and $(\alpha ,\beta )$-cut-based distance metric for different parameters’ values to investigate multi-criteria decision-making problems and introduced a ranking technique for the available options. Guleria and Bajaj [28] introduced a parametric (R, S)-norm information measure under the Pythagorean fuzzy set environment, and provided a confirmation of its credibility.

Garg created another PDM for interval-valued q-ROFS with amazing properties [29]. Some new weighted averaging aggregation operators for q-ROFSs and, henceforth, a multi-attribute group decision making technique to solve practical decision-making problems were given by Garg and Chen [30].

The similarity measure is an essential research direction in fuzzy set theory. It is employed in a variety of applications, including medical diagnosis and general pattern recognition. Peng et al. [31] developed the distance, similarity, entropy, and inclusion measures for q-ROFSs.

Liu et al. [32] proposed a cosine comparability measure and modification of Euclidean distance measure of q-ROFSs, and researched their properties.

When we have accurate information, the knowledge associated with q-rung orthopair fuzzy values (q-ROFVs) increases. When the ambiguity and uncertainty factor increase, the knowledge associated with q-ROFVs falls. A method to evaluate the knowledge associated with q-ROFS was suggested by Khan et al. [33], who also proposed a q-ROFS ranking system [34].

The main contribution of this paper is the proposition of a novel (R, S)-norm q-rung orthopair fuzzy set information measure. We prove that the submitted information measure is a true entropy function and is usable in a practical MCDM problem. We demonstrate it using a numerical example of a study case of choosing the best vehicle to purchase based on such objectives as solace, mileage, security, and design.

The other part of the paper is structured as follows: the essential definitions for AIFS, PYFS, and q-ROFS are found in Section 2, an aphoristic meaning of information measure, and its connected properties are examined with graphical clarifications in Section 3. Section 4 provides a numerical example that illustrates how the presented algorithm works, and in the concluding Section 5, we summarize our work and propose some future research directions.

2. Preliminaries

In the following section, we present the review of the concepts in the literature related to intuitionistic fuzzy sets and Pythagorean fuzzy sets.

Definition 1

(Atanassov [5]). Let us take Y as a universal set for further calculations. An intuitionistic fuzzy set, M, on Y has the form:

$$\mathrm{M}=\left\{〈\mathrm{y},{\mu }_{\mathrm{M}}\left(\mathrm{y}\right),{\nu }_{\mathrm{M}}\left(\mathrm{y}\right)〉\mid \mathrm{y}\in \mathrm{Y}\right\},$$

(2)

where the degree of membership is ${\mu }_M\left(y\right):Y\to [0,1]$, and the function ${\nu }_M\left(y\right):Y\to [0,1]$ is the degree of non-membership of the element $y\in Y$ to M, which satisfies (3).

$${\mu }_M\left(y\right)+{\nu }_M\left(y\right)\le 1,\forall y\in Y$$

(3)

The degree of Y’s hesitation to approach M is defined as ${\pi }_M\left(y\right)=1-{\mu }_M\left(y\right)-{\nu }_M\left(y\right)$ for all $y\in Y$. The collection of intuitionistic fuzzy sets on universal sets Y is referred to as IFS (Y).

Definition 2

(Yager [9]). Let Y be a nonempty finite set. The mathematical equation that follows can be used to represent a Pythagorean fuzzy set (PFS) M on Y

$$M=\left\{\left(y,{\mu }_M\left(y\right),{\nu }_M\left(y\right)\right):y\in Y\right\},$$

(4)

where ${\mu }_M\left(y\right):Y\to [0,1]$ is the degree of membership, and the function ${\nu }_M\left(y\right):Y\to [0,1]$ is the degree of non-membership of the element $y\in Y$ to M, which satisfies (5).

$${\mu }_M^2\left(y\right)+{\nu }_M^2\left(y\right)\le 1,\forall y\in Y$$

(5)

Pythagorean fuzzy set M’s hesitancy for all $y\in Y$ is defined as (6).

$${\pi }_M\left(y\right)=\sqrt{1-{\mu }_M^2\left(y\right)-{\nu }_M^2\left(y\right)}$$

(6)

M’s Pythagorean fuzzy complement set is given by $M^C=\left\{\left(y,{\nu }_M\left(y\right),{\mu }_M\left(y\right)\right)\vee y\in Y\right\}$.

Definition 3

(Yager [11]). A q-ROFS on a universal set Y is defined as follows (7):

$$M=\left\{\left(y,{\mu }_M\left(y\right),{\nu }_M\left(y\right)\right):y\in Y\right\},$$

(7)

where ${\mu }_M$ is membership function and ${\nu }_M$ non-membership function from the universal set Y to the closed interval $[0,1]$. The power qth of the aforementioned functions when added give a sum less than or equal to 1 in q-ROFS. That is (8),

$${\mu }_M{\left(x_i\right)}^q+{\nu }_M{\left(x_i\right)}^q\le 1$$

(8)

The hesitancy degree will be given by ${\pi }_M{\left(x_i\right)}^q={\left(1-\left({\mu }_M{\left(x_i\right)}^q+{\nu }_M{\left(x_i\right)}^q\right)\right)}^{\frac{1}{q}}$.

Definition 4.

Let M and N two q-ROFNs. The $H\left(M\right)$ is called the entropy of q-ROFN if and only if the axioms provided below are true:

• (q-ROFS1) Sharpness property: $H\left(M\right)=0$ iff M is a crisp set i.e., ${\mu }_M\left(x_i\right)=0,$ ${\nu }_M\left(x_i\right)=1$; or ${\mu }_M\left(x_i\right)=1, {\nu }_M\left(x_i\right)=0$.
• (q-ROFS2) Maximality property: $H\left(M\right)$ is maximum iff ${\mu }_{\mathrm{M}}\left({\mathrm{x}}_i\right)={\nu }_{\mathrm{M}}\left({\mathrm{x}}_i\right)={\pi }_{\mathrm{M}}\left({\mathrm{x}}_i\right)=\frac{1}{\sqrt{3}}\forall x_{\mathrm{i}}\in X$.
• (q-ROFS3) Symmetry property: $H\left(M\right)=H\left(M^C\right)$.
• (q-ROFS4) Resolution property: $H\left(M\right)\le H\left(N\right)$ iff $M\subseteq N$, i.e., ${\mu }_M\left(x_i\right)\le {\nu }_N\left(x_i\right)$ and ${\mu }_M\left(x_i\right)\ge {\nu }_N\left(x_i\right)$ for ${\mu }_N\left(x_i\right)\le {\nu }_N\left(x_i\right)$ or if ${\mu }_M\left(x_i\right)\ge {\nu }_N\left(x_i\right)$ and ${\nu }_M\left(x_i\right)\le {\nu }_N\left(x_i\right)$ for ${\mu }_N\left(x_i\right)\ge {\nu }_N\left(x_i\right)$.

Definition 5

(Yager [11]). If $Q_1={({\mu }_{Q_1},{\nu }_{Q_1})}^q$ and $Q_2={({\mu }_{Q_2},{\nu }_{Q_2})}^q$ are any two q-ROFNS, then the operation rules between them are as follows:

• $Q_1\bigcup Q_2={\left(max\{{\mu }_{Q_1},{\mu }_{Q_2}\},min\{{\mu }_{Q_1},{\mu }_{Q_2}\}\right)}^q$,
• $Q_1\bigcap Q_2={\left(min\{{\mu }_{Q_1},{\mu }_{Q_2}\},max\{{\mu }_{Q_1},{\mu }_{Q_2}\}\right)}^q$,
• $Q^C={({\nu }_{Q_1},{\mu }_{Q_1})}^q$, where $Q_1^C$ is the inverse of $Q_1$,
• $Q_1\preccurlyeq Q_2$ if and only if ${\mu }_{Q_1}\le {\mu }_{Q_2}$, ${\nu }_{Q_1}\ge {\nu }_{Q_2}$.

Definition 6.

Take $M,N\in$ q-ROPFS on Y. The distance measure, d, is defined as a mapping $d:$ q-ROPFS × q-ROPFS $\to [0,1]$ that meets the following criteria:

• $0\le d(M,N)\le 1$,
• $d(M,N)=d(N,M)$,
• $d(M,N)=0$ iff $M=N$,
• if $M\le N\le R$ then $d(M,N)\le d(M,R)$ and $d(N,R)\le d(P,R)$.

Various standard distance measures are presented as follows:

• Hamming Distance: $d^H$ q-ROFS(M, N) =

  $$\frac{1}{2\left|x\right|}\sum _{i=1}^n\left(\mid {\mu }_M{\left(x_i\right)}^{\mathrm{q}}-{\mu }_N{\left(x_i\right)}^{\mathrm{q}}|+|{\nu }_M{\left(x_i\right)}^{\mathrm{q}}-{\nu }_N{\left(x_i\right)}^{\mathrm{q}}|+|{\pi }_M{\left(x_i\right)}^{\mathrm{q}}-{\pi }_N{\left(x_i\right)}^{\mathrm{q}}\mid \right)$$

  (9)
• Normalized Hamming Distance: $d^{NH}$ q-ROFS(M, N) =

  $$\frac{1}{2n\left|x\right|}\sum _{i=1}^n\left(\mid {\mu }_M{\left(x_i\right)}^{\mathrm{q}}-{\mu }_N{\left(x_i\right)}^{\mathrm{q}}|+|{\nu }_M{\left(x_i\right)}^{\mathrm{q}}-{\nu }_N{\left(x_i\right)}^{\mathrm{q}}|+|{\pi }_M{\left(x_i\right)}^{\mathrm{q}}-{\pi }_N{\left(x_i\right)}^{\mathrm{q}}\mid \right)$$

  (10)

3. Entropy Measure of q-ROFS

In the following section we describe the proposition of a new type of bi-parametric entropy measure for q-rung Orthopair Fuzzy sets. From the definition, the function is considered an entropy function if it is real-valued and satisfies the axioms presented in Definition 4. In this Section we also prove that our proposed measure fulfils the definition of the entropy function.

We propose the q-rung orthopair fuzzy entropy analogous to measure (11) in the q-rung orthopair fuzzy information environment:

$$H_S^R\left(M\right)=\{\begin{array}{l}\frac{(\mathrm{R}\times S)}{(R-S)}\sum _{i=1}^n\frac{1}{n}\left[{\left({\mu }_M{\left(x_i\right)}^{qS}+{\nu }_M{\left(x_i\right)}^{qS}+{\pi }_M{\left(x_i\right)}^{qS}\right)}^{\frac{1}{S}}-{\left({\mu }_M{\left(x_i\right)}^{qR}+{\nu }_M{\left(x_i\right)}^{qR}+{\pi }_M{\left(x_i\right)}^{qR}\right)}^{\frac{1}{R}}\right],\hfill \\ \mathrm{where}R,S>0;\mathrm{either}0<S<1\mathrm{and}1<R<\infty \mathrm{or}0<R<1\mathrm{and}1<S<\infty \\ \frac{\mathrm{R}}{(\mathrm{R}-1)}\sum _{i=1}^{\mathrm{n}}\frac{1}{\mathrm{n}}\left[1-{\left({\mu }_M{\left({\mathrm{x}}_{\mathrm{i}}\right)}^{\mathrm{qR}}+{\nu }_{\mathrm{M}}{\left({\mathrm{x}}_{\mathrm{i}}\right)}^{\mathrm{qR}}+{\pi }_M{\left({\mathrm{x}}_{\mathrm{i}}\right)}^{\mathrm{qR}}\right)}^{\frac{1}{\mathrm{R}}}\right],\\ \mathrm{where}S=1,R>0,R\ne 1\\ -\frac{1}{n}\sum _{i=1}^n\left[\left({{\mu }_M\left(x_i\right)}^q\log \left({{\mu }_M\left(x_i\right)}^q\right)\right)+\left({{\nu }_M\left(x_i\right)}^q\log \left({{\nu }_M\left(x_i\right)}^q\right)\right)+\left({{\pi }_M\left(x_i\right)}^q\log \left({{\pi }_M\left(x_i\right)}^q\right)\right)\right],\\ \mathrm{where}R=1,S\to 1\mathrm{or\; where} S=1,R\to 1\end{array}$$

(11)

Theorem 1.

The proposed entropy measure is a valid q-ROFS information measure which is proven below.

Proof. 

To confirm that the proposed entropy function is a valid information measure it is essential to show that it satisfies all the axioms of entropy measure presented above. □

• (q-ROFS1) Sharpness property: If $H_S^R\left(M\right)=0$, then

  $${\left({\mu }_M{\left({\mathrm{x}}_{\mathrm{i}}\right)}^{\mathrm{qS}}+{\nu }_{\mathrm{M}}{\left({\mathrm{x}}_{\mathrm{i}}\right)}^{\mathrm{qS}}+{\pi }_{\mathrm{M}}{\left({\mathrm{x}}_{\mathrm{i}}\right)}^{\mathrm{qS}}\right)}^{\frac{1}{\mathrm{S}}}-{\left({\mu }_{\mathrm{M}}{\left({\mathrm{x}}_{\mathrm{i}}\right)}^{\mathrm{qR}}+{\nu }_{\mathrm{M}}{\left({\mathrm{x}}_{\mathrm{i}}\right)}^{\mathrm{qR}}+{\pi }_{\mathrm{M}}{\left({\mathrm{x}}_{\mathrm{i}}\right)}^{\mathrm{qR}}\right)}^{\frac{1}{\mathrm{R}}}=0$$

  (12)
  Since $R,S>1$ and $R,S\ne 1$ then the previously mentioned condition is true in the following cases:

  (a)

  Either ${\mu }_M\left(x_i\right)=1,$ i.e., ${\nu }_M\left(x_i\right)={\pi }_M\left(x_i\right)=0$, or

  (b)

  ${\nu }_M\left(x_i\right)=1,$ i.e., ${\mu }_M\left(x_i\right)={\pi }_M\left(x_i\right)=0$, or

  (c)

  ${\pi }_M\left(x_i\right)=1,$ i.e., ${\mu }_M\left(x_i\right)={\nu }_M\left(x_i\right)=0$.

  These three examples indicate that M is a crisp set, and therefore $H_R^S\left(M\right)=0$ is returned.
• (q-ROFS2) Maximality property: Mathematically, we confirm the concavity of the $H_S^R\left(M\right)$ by calculating its Hessian at the critical point, i.e., ${\left(\frac{1}{3}\right)}^{\frac{1}{q}}$ with particular values of R and S. The Hessian of $H_S^R\left(M\right)$ is as $\left[R>1(=3),S<1(=0.3)\mathrm{and}q=2\right]$:

  $$H_S^R\left(M\right)=\left[\begin{array}{ccc}-10.4589\hfill & 2.232816\hfill & 2.232816\hfill \\ 2.232816\hfill & -10.4589\hfill & 2.232816\hfill \\ 2.232816\hfill & 2.232816\hfill & -10.4589\hfill \end{array}\right]$$

  (13)
  It should be observed that $H_S^R\left(M\right)$ is a negative semi-definite matrix for all conceivable combinations of R and S, implying that the function is concave. As a result, the maximality property is determined by the function’s concavity.
• (q-ROFS3) Symmetry property: The definition makes it clear that

  $$H_S^R\left(M\right)=H_S^R\left(M^C\right)$$

  (14)
• (q-ROFS4) Resolution property: We have

  $$\begin{array}{c}\mid \left({\mu }_{\mathrm{M}}\left({\mathrm{x}}_{\mathrm{i}}\right)-1/\sqrt[\mathrm{q}]{3}\right)|+|\left({\nu }_{\mathrm{M}}\left({\mathrm{x}}_{\mathrm{i}}\right)-1/\sqrt[\mathrm{q}]{3}\right)|+|\left({\pi }_{\mathrm{M}}\left({\mathrm{x}}_{\mathrm{i}}\right)-1/\sqrt[\mathrm{q}]{3}\right)|\ge \\ \ge |\left({\mu }_{\mathrm{N}}\left({\mathrm{x}}_{\mathrm{i}}\right)-1/\sqrt[\mathrm{q}]{3}\right)|+|\left({\nu }_{\mathrm{N}}\left({\mathrm{x}}_{\mathrm{i}}\right)-1/\sqrt[q]{3}\right)|+|\left({\pi }_{\mathrm{N}}\left({\mathrm{x}}_{\mathrm{i}}\right)-1/\sqrt[\mathrm{q}]{3}\right)\mid \end{array}$$

  (15)

  and

  $$\begin{array}{c}{\left({\mu }_{\mathrm{M}}\left({\mathrm{x}}_{\mathrm{i}}\right)-1/\sqrt[q]{3}\right)}^q+{\left({\nu }_{\mathrm{M}}\left({\mathrm{x}}_{\mathrm{i}}\right)-1/\sqrt[q]{3}\right)}^q+{\left({\pi }_{\mathrm{M}}\left({\mathrm{x}}_{\mathrm{i}}\right)-1/\sqrt[q]{3}\right)}^q\ge \\ \ge {\left({\mu }_{\mathrm{N}}\left({\mathrm{x}}_{\mathrm{i}}\right)-1/\sqrt[\mathrm{q}]{3}\right)}^q+{\left({\nu }_{\mathrm{N}}\left({\mathrm{x}}_{\mathrm{i}}\right)-1/\sqrt[q]{3}\right)}^q+{\left({\pi }_N\left(x_i\right)-1/\sqrt[q]{3}\right)}^q\end{array}$$

  (16)

If ${\mu }_M\left(x_i\right)\le {\nu }_N\left(x_i\right)$ and ${\nu }_M\left(x_i\right)\le {\nu }_N\left(x_i\right)$ with $max\{{\nu }_N\left(x_i\right),{\nu }_N\left(x_i\right)\}\le 1/\sqrt[q]{3}$, then ${\mu }_M\left(x_{\mathrm{i}}\right)\le {\mu }_N\left(x_i\right)\le 1/\sqrt[q]{3}$, ${\nu }_{\mathrm{M}}\left({\mathrm{x}}_{\mathrm{i}}\right)\le {\nu }_{\mathrm{N}}\left({\mathrm{x}}_{\mathrm{i}}\right)\le 1/\sqrt[q]{3}$, $\pi \left({\mathrm{x}}_{\mathrm{i}}\right)>{\pi }_N\left({\mathrm{x}}_{\mathrm{i}}\right)\ge 1/\sqrt[q]{3}$.

This implies that the above result holds.

Similarly if ${\mu }_M\left(x_i\right)\ge {\mu }_N\left(x_i\right)$ and ${\nu }_M\left(x_i\right)\ge {\nu }_N\left(x_i\right)$, which implies that $max\{{\mu }_N\left(x_i\right),{\nu }_N\left(x_i\right)\}\ge 1/\sqrt[q]{3}$ then the result also holds.

Therefore, by using the previously obtained result we show that $H_S^R\left(M\right)$ fulfils the resolution condition. Hence $H_S^R\left(M\right)$ satisfies all the axioms of q-ROFS and it can be used as a valid measure of q-ROFS information.

An algorithm has been given to address the multi-criteria decision-making problem as follows: the primary goal of the multi-criteria decision-making problem is to choose the best or optimal alternative from the m feasible alternatives that are accessible, that is, $Z=\{z_1,z_2,\dots ,z_m\}$ based on certain laid-down criteria n and $O=\{o_1,o_2,\dots ,o_m\}$. For this, first we take the appraisal values of an alternative $z_i(i=1,2,3,\dots ,m)$ with respect to the criteria $o_j(j=1,2,3,\dots ,n)$, given by $z_{ij}=(p_{ij},q_{ij})$, satisfying $0\le p_{ij}\le 1$, $0\le q_{ij}\le 1$ and $0\le p_{ij}+q_{ij}\le 1$ with $i=1,2,3,\dots ,m$ and $j=1,2,3,\dots ,n$. Hence, the above problem can be represented by modeling it through the following Pythagorean fuzzy decision matrix (17):

$$R={\left(p_{ij},q_{ij}\right)}_{m\times n}=\left(z_{ij}\right)=\begin{array}{ccccc}& o_1& o_2& \dots & o_n\\ z_1& \left(p_{11},q_{11}\right)& \left(p_{12},q_{12}\right)& \dots & \left(p_{1n},q_{1n}\right)\\ z_2& \left(p_{21},q_{21}\right)& \left(p_{22},q_{22}\right)& \dots & \left(p_{2n},q_{2n}\right)\\ ⋮& ⋮& ⋮& \ddots & ⋮\\ z_m& \left(p_{m1},q_{m2}\right)& \left(p_{m2},q_{m2}\right)& \dots & \left(p_{mn},q_{mn}\right)\end{array}$$

(17)

Let $w={(w_1,w_2,\dots ,w_n)}^T$ be the vector of criteria weights where $0\le w_j\le 1$ and ${\sum }_{j=1}^n{w}_j=1$ is the degree of importance of the $j$th criteria. However, in some decision-making problems the criteria weights can be completely unknown or partially known. This problem usually appears due to a lack of knowledge, time, or data, or because of limited experience in the problem domain.

In this section, we present and then discuss the method to determine the criteria weights using the proposed q-ROFS entropy measure. In cases when weights of the criteria are fully unknown, then it is possible to calculate the weights using the proposed PFS entropy as shows Formula (18):

$$w_j=\frac{1-e_j}{n-{\sum }_{j=1}^n{e}_j}j=1,2,\cdots ,n,$$

(18)

where $e_j=\frac{1}{m}{\sum }_{i=1}^m{H}_S^R\left(z_{ij}\right)$ and $H_S^R$ is the proposed Pythagorean fuzzy entropy for $z_{ij}=(p_{ij},q_{ij})$ calculated using (19).

$$H_S^R\left(M\right)=\frac{(\mathrm{R}\times \mathrm{S})}{(\mathrm{R}-\mathrm{S})}\sum _{\mathrm{i}=1}^{\mathrm{n}}\frac{1}{\mathrm{n}}\left[{\left({\mu }_{\mathrm{M}}{\left(z_{ij}\right)}^{\mathrm{qS}}+{\nu }_{\mathrm{M}}{\left(z_{ij}\right)}^{\mathrm{qS}}+{\pi }_{\mathrm{M}}{\left(z_{ij}\right)}^{\mathrm{qS}}\right)}^{\frac{1}{\mathrm{S}}}-{\left({\mu }_{\mathrm{M}}{\left(z_{ij}\right)}^{\mathrm{qR}}+{\nu }_{\mathrm{M}}{\left(z_{ij}\right)}^{\mathrm{qR}}+{\pi }_{\mathrm{M}}{\left(z_{ij}\right)}^{\mathrm{qR}}\right)}^{\frac{1}{\mathrm{R}}}\right]$$

(19)

The procedure of the proposed algorithm for decision-making under q-ROFS environment using the proposed entropy measure is shown in Figure 1.

The detailed description of the introduced methodology are enumerated step by step below:

Step 1: Use the domain’ expert knowledge to define the decision matrix with m alternatives and n criteria $R={(p_{ij},q_{ij})}_{m\times n}=o_j\left(z_i\right)$, where the elements $o_j\left(z_i\right),(i=1,2,\dots ,m;$$j=1,2,\dots ,n)$ are the performances of the alternative $z_i\in Z$ with regard to the criteria $o_j\in O$.

Step 2: Compute the normalized decision matrix using matrix from the step 1.

Step 3: Use proposed entropy measure (20) to calculate the criteria weight vector

$$H_S^R\left(M\right)=\frac{(\mathrm{R}\times \mathrm{S})}{(\mathrm{R}-\mathrm{S})}\sum _{\mathrm{i}=1}^{\mathrm{n}}\frac{1}{\mathrm{n}}\left[{\left({\mu }_{\mathrm{M}}{\left(x_i\right)}^{\mathrm{qS}}+{\nu }_{\mathrm{M}}{\left(x_i\right)}^{\mathrm{qS}}+{\pi }_{\mathrm{M}}{\left(x_i\right)}^{\mathrm{qS}}\right)}^{\frac{1}{\mathrm{S}}}-{\left({\mu }_{\mathrm{M}}{\left(x_i\right)}^{\mathrm{qR}}+{\nu }_{\mathrm{M}}{\left(x_i\right)}^{\mathrm{qR}}+{\pi }_{\mathrm{M}}{\left(x_i\right)}^{\mathrm{qR}}\right)}^{\frac{1}{\mathrm{R}}}\right]$$

(20)

Step 4: Determine the best solution $z^{+}$ as well as the least-preferred solution $z^{-}$ using Equations (22) and (21) accordingly:

$$z^{+}=\{({\alpha }_1^{+},{\beta }_1^{+}),({\alpha }_2^{+},{\beta }_2^{+}),\dots ,({\alpha }_n^{+},{\beta }_n^{+})\},$$

(21)

where $({\alpha }_j^{+},{\beta }_j^{+})=\left(sup({\mu }_M\left(z_i\right),inf{\nu }_M\left(z_i\right))\right))$, $z_j\in Z$, $j=1,2,\dots ,n$.

$$z^{-}=\{({\alpha }_1^{-},{\beta }_1^{-}),({\alpha }_2^{-},{\beta }_2^{-}),\dots ,({\alpha }_n^{-},{\beta }_n^{-})\},$$

(22)

where $({\alpha }_j^{-},{\beta }_j^{-})=\left(inf({\mu }_M\left(z_i\right),sup\left({\nu }_M\left(z_i\right)\right))\right)$, $z_j\in Z$, $j=1,2,\dots ,n$.

Step 5: Evaluate the distance between the alternative $z_i$ and the most preferred $z^{+}$ and least preferred $z^{-}$ solutions using Definition 6 or Equations (23) and (24) below:

$$l\left(z_i,z^{+}\right)=\frac{1}{2}\left[\sum _{j=1}^n{w}_j\left|{\left({\alpha }_{ij}\right)}^q-{\left({\alpha }_j^{+}\right)}^q\right|+\left|{\left({\beta }_{ij}\right)}^q-{\left({\beta }_j^{+}\right)}^q\right|+\left|{\left({\pi }_{ij}\right)}^q-{\left({\pi }_j^{+}\right)}^q\right|\right]$$

(23)

$$l\left(z_i,z^{-}\right)=\frac{1}{2}\left[\sum _{j=1}^n{w}_j\left|{\left({\alpha }_{ij}\right)}^q-{\left({\alpha }_j^{-}\right)}^q\right|+\left|{\left({\beta }_{ij}\right)}^q-{\left({\beta }_j^{-}\right)}^q\right|+\left|{\left({\pi }_{ij}\right)}^q-{\left({\pi }_j^{-}\right)}^q\right|\right]$$

(24)

Step 6: Determine the degree of relative closeness $l_i$’s (25).

$$l_i=\frac{l\left(z_i,z^{-}\right)}{l\left(z_i,z^{+}\right)+l\left(z_i,z^{-}\right)}$$

(25)

Step 7: Determine the final ranking based on the degree of relative of closeness obtained in step 6. A better alternative has a larger value of relative closeness.

4. Numerical Example

A numerical example has been considered in order to depict the illustration of the proposed algorithm.

Assume a car organization produces four distinct vehicles $z_1,z_2,z_3$, and $z_4$. Assume a client needs to purchase a vehicle in light of the four given measures:

• $o_1$—solace,
• $o_2$—mileage,
• $o_3$—security,
• $o_4$—interior design.

We expect the evaluation upside of the other options, as for every rule given by the master, they are addressed by PFS as shown in

Table 1. Decision matrix built by an expert.

	${\mathit{o}}_1$	${\mathit{o}}_2$	${\mathit{o}}_3$	${\mathit{o}}_4$
$z_1$	(0.9, 0.3)	(0.7, 0.6)	(0.5, 0.8)	(0.6, 0.3)
$z_2$	(0.4, 0.7)	(0.9, 0.2)	(0.8, 0.1)	(0.5, 0.3)
$z_3$	(0.8, 0.4)	(0.7, 0.5)	(0.6, 0.2)	(0.7, 0.4)
$z_4$	(0.7, 0.2)	(0.8, 0.2)	(0.8, 0.4)	(0.6, 0.6)

.

Next, we use the previously described procedure to calculate alternatives’ ranking.

Step 1: Determine the decision matrix from the collected data. The decision matrix is shown in Table 1. Value $z_{ij}=(p_{ij},q_{ij})$ determines the appraisal of the alternative $z_i$. The values of membership $p_{ij}$ and non-membership $q_{ij}$ satisfy $0\le p_{ij}\le 1$, $0\le q_{ij}\le 1$ and $0\le p_{ij}+q_{ij}\le 1$.

Step 2: Since the informational data are homogenous, in this case we do not need to normalize the decision matrix. The normalized decision matrix is the same as the decision matrix shown in Table 1 and will be used in further calculation in the same form.

Step 3: The criteria weights vector is calculated using Equation (20). The resulting weights are shown in Equation (26). According to the calculations, criterion $o_2$ is the most important one, and criteria $o_3$ and $o_4$ are least important.

$$w={(w_1,w_2,w_3,w_4)}^T={(0.249418,0.307786,0.232969,0.209827)}^T$$

(26)

Step 4: The most preferred ($z^{+}$) and the least preferred ($z^{-}$) solutions are based on the values in the decision matrix and defined as (27):

$$\begin{array}{c}{z}^{-}=\{(0.4,0.7),(0.7,0.6),(0.5,0.8),(0.6,0.6)\}{z}^{+}=\{(0.9,0.3),(0.9,0.2),(0.8,0.1),(0.7,0.4)\}\end{array}$$

(27)

Step 5: The distances between each of $z_i$ from $z^{+}$ and $z^{-}$ are calculated using Formulas (23) and (24) and are presented in

Table 2. The distances between each of $z_i$ from $z^{+}$ and $z^{-}$.

${\mathit{z}}_{\mathit{i}}$	${\mathit{z}}^{+}$	${\mathit{z}}^{-}$
$z_1$	0.19701	0.083599
$z_2$	0.07313	0.049739
$z_3$	0.020283	0.071691
$z_4$	0.15843	0.037391

.

Step 6: The values of relative degree of closeness $l_i$ are shown in

Table 3. The values of relative degree of closeness.

	${\mathit{l}}_1$	${\mathit{l}}_2$	${\mathit{l}}_3$	${\mathit{l}}_4$
$l_i$	−0.24357	−4.02108	−0.42271	−0.55273

. Those values are calculated using Formula (25). It is apparent that $l_2$ has the smallest value, which means it performs poorly, and $l_1$ has the largest value which means that alternative $z_1$ is the closest one to the ideal solution.

Step 7: The alternatives are ranked in order of relative closeness from the biggest to the smallest value: $z_1>z_3>z_4>z_2$ and $z_1$ is the best option.

To calculate this ranking in the Equation (11) we used values of $R=3$ and $S=0.3$ which are arbitrary chosen. The ranking’s consistency technique for other values of parameters R and S should be investigated and examined by conducting a simulation study with various parameter values depending on the demand. However, in this paper we concentrated mainly on the theoretical proposition of the new approach. Therefore doing such simulations is out of the scope of this study.

5. Conclusions

An $(R,S)$ norm entropy measure has been successfully proposed under the concept of q-ROFS. The validity of the proposed entropy measure has also been proved theoretically. The proposed entropy has been used to solve a multi-criteria decision-making problem with the help of a numerical example that illustrates the application of the proposed methodologies to the problem of choosing the best vehicle based on an expert’s evaluation. The findings indicate that this approach may provide a practical and adaptable technique to handle with complete customized decision-making applications information about the criteria used by experts. This proposition also introduces several future research directions, such as a comparison with other information measure methods for q-rung orthopair fuzzy sets on practical examples or a simulation that shows how the different values R and S affect the final ranking in the proposed methodology.