Abstract
A useful tool for expressing fuzziness and uncertainty is the recently developed n,m-rung orthopair fuzzy set (n,m-ROFS). Due to their superior ability to manage uncertain situations compared to theories of q-rung orthopair fuzzy sets, the n,m-rung orthopair fuzzy sets have variety of applications in decision-making in daily life. To deal with ambiguity and unreliability in multi-attribute group decision-making, this study introduces a novel tool called \(k^{n}_{m}\)-rung picture fuzzy set (\(k^{n}_{m}\)-RPFS). The suggested \(k^{n}_{m}\)-RPFS incorporates all of the benefits of n,m-ROFS and represents both the quantitative and qualitative analyses of the decision-makers. The presented model is a fruitful advancement of the q-rung picture fuzzy set (q-RPFS). Furthermore, numerous of its key notions, including as complement, intersection, and union are explained and illustrated with instances. In many decision-making situations, the main benefit of \(k^{n}_{m}\)-rung picture fuzzy sets is the ability to express more uncertainty than q-rung picture fuzzy sets. Then, along with their numerous features, we discover the basic set of operations for the \(k^{n}_{m}\)-rung picture fuzzy sets. Importantly, we present a novel operator, \(k^{n}_{m}\)-rung picture fuzzy weighted power average (\(k^{n}_{m}\)-RPFWPA) over \(k^{n}_{m}\)-rung picture fuzzy sets, and use it to multi-attribute decision-making issues for evaluating alternatives with \(k^{n}_{m}\)-rung picture fuzzy information. Additionally, we use this operator to pinpoint the countries with the highest expat living standards and demonstrate how to choose the best option by comparing aggregate findings and applying score values. Finally, we compare the outcomes of the q-RPFEWA, SFWG, PFDWA, SFDWA, and SFWA operators to those of the \(k^{n}_{m}\)-RPFWPA operator.
Similar content being viewed by others
Avoid common mistakes on your manuscript.
Introduction
Making decisions is a frequent activity in everyday life, with the goal of selecting the best choice from among multiple alternatives. In the initial phases of social evolution, decision-makers typically provided their evaluation results using real numbers. Experts are unable to provide precise real numbers to evaluate the alternatives, because multi-attribute decision-making (MADM) situations are becoming more complex. The vagueness and imprecision of human judgments brought attention to the shortcomings of the crisp set theory. As a result, Zadeh [1] established the theoretical groundwork for the fuzzy set (FS) theory for uncertain knowledge, which enables experts to express their level of satisfaction (membership degree) with respect to a member’s performance inside the unit interval. Many operators in the fuzzy environment have been created to solve decision-making issues. Several operators in a fuzzy environment were examined by Song et al. [2].
Then, the idea of an intuitionistic fuzzy set (IFS), which has a membership degree and a non-membership degree, was first given forth by Atanassov [3]. IFS has drawn a lot of interest, since it first appeared, and many scientists have researched it from both a theoretical and practical standpoint [4,5,6]. Afterward, Yager [7] brought out the idea of a Pythagorean fuzzy set (PYFS). PYFS is more efficient and effective than IFS because of the constraint that the square sum of membership and non-membership degrees must be less than or equal to one. Owing to its features and merits, PYFS has been used in decision-making extensively [8,9,10,11]. Due to its strong capacity for describing the fuzziness and uncertainty of knowledge, multi-attribute group decision-making (MAGDM) has been widely explored and effectively implemented to both management and economics as one of the most significant components of current decision-making theory [12,13,14,15,16,17,18,19]. Decision-makers typically use their intuition and previous knowledge to make decisions in genuine decision-making situations. The requirement is to adequately represent the ambiguous and fuzzy information in the MAGDM process due to the complexity of decision-making issues. PYFSs are unable to handle a number of situations, nevertheless. For some situations where the square sum of membership and non-membership degrees is more than one, PYFSs do not apply. The notion of q-ROFS, whose constraint is the sum of the \(q\textrm{th}\) power of the membership degree and the \(q\textrm{th}\) power of the degree of non-membership is less than or equal to one, was introduced by Yager [20] as a practical solution to these instances. As a result, q-ROFSs loosen the PYFSs constraint and increase the information range. Following this, Liu et al. [21] created a few straightforward weighted averaging operators to combine q-rung orthopair fuzzy numbers (q-ROFNs), and they then used these operators to apply to MAGDM. While, Liu et al. [22] introduced a series of q-rung orthopair fuzzy Bonferroni mean operators.
Al-shami [23] adopted a new approach to cover new cases of vagueness by familiarizing the concept of (2,1)-fuzzy sets. This type of fuzzy sets addresses more situations than IFSs and offers an effective tool for real-life issues that required different ranks of importance. Then, Al-shami and Mhemdi [24] and Ibrahim et al. [25] investigated the idea of (n,m)-rung orthopair fuzzy set (n,m-ROFS), a new fuzzy set extension with the constraint that the sum of the \(n\textrm{th}\) power of the membership degree and the \(m\textrm{th}\) power of the degree of non-membership is less than or equal to one. They also developed the idea of a weighted aggregated operator over (n,m)-ROFS and implemented it to the MADM issues. In this line of research, some valuable contributions have been conducted by several researchers and scholars [26,27,28,29]. Recently, Al-shami et al. [30] have hybridized (n,m)-ROFSs with soft sets to expand the environments of fuzzy and soft sets and handled complicated problems. Hybridizations of fuzzy environments with other uncertainty tools, such as soft sets and neutrosophic sets, have been achieved in [31,32,33,34,35,36,37,38].
We may encounter the following problems, because genuine decision-making situations are too complex. The first problem is that, while q-ROFSs and n,m-ROFSs have been successfully employed in decision-making, there are some instances in which they cannot be applied. For instance, groups of human voters can include those who vote for, abstain from voting, and refuse to participate in a voting. In other sense, when voting, we must cope with more yes, abstain, and no, and refusal responses. Obviously, n,m-ROFSs and q-ROFSs are inapplicable in this situation. Therefore, Cuong [39, 40] advanced the idea of a picture fuzzy set (PFS), which is distinguished by positive, neutral, and negative membership degrees. PFSs have received a lot of academic attention and have been extensively studied since their development [41,42,43,44]. The idea of q-rung picture fuzzy set was created by Li et al. [45]. Garg [46] proposed the idea of picture fuzzy aggregation operators (PFAOs). Wei [47, 48] researched PFAOs as well and offered some opinion on how they can be used in decision-making. The framework of q-rung picture fuzzy (q-RPF) Dombi Hamy mean operators was covered by He et al. [49]. Liu et al. [50] pioneered the specific forms of q-RPF aggregation operators.
In summary, the following are the motivations and goals for this article:
-
1.
To offer the \(k^{n}_{m}\)-RPFS definition and operations on \(k^{n}_{m}\)-RPFS values.
-
2.
Investigation of the development of a innovative decision-making model.
-
3.
The shortcomings and limits of existing operators are overcome by proposed operators, which are more general and perform admirably not only for \(k^{n}_{m}\)-RPFS data but also for q-RPF fuzzy information.
-
4.
In comparison to other modeling techniques, the \(k^{n}_{m}\)-RPFS is more adaptable and effective. This enables individuals to provide more accurate outcomes when faced with decision-making issues.
The rest of this article are organized as follows. “Preliminaries” is devoted to recall the definitions of the types of fuzzy sets that we need through this manuscript. In “\({k}^{n}_{m}\)-Rung picture fuzzy sets”, the definition of \({k}^{n}_{m}\)-RPFS and operational laws for \({k}^{n}_{m}\)-RPFS values are presented. In “\({k}^{n}_{m}\)-Rung picture fuzzy weighted power average”, we discover a new \({k}^{n}_{m}\)-rung picture fuzzy aggregation operator and provide a few of the proposed operator attractive features. In “MADM application to select the best countries for expats”, it is used this operator to suggest an algorithm for a decision-making model and explain one MADM issue, which is deciding which country is best for expats. In “Comparison analysis and discussion”, we go over the comparison analysis of the suggested model with the current operators. In “Conclusions”, it is summarized the findings regarding the proposed theory.
Preliminaries
The basic model that we use in our paper is made of a number of steps, each of which is intriguing in and of itself. We start with the notion of picture fuzzy sets to minimize a longer list of existing models. Let \(\chi \) be a non-empty set and \(\rho (r), \aleph (r), \sigma (r): \chi \rightarrow [0,1]\) represent degree of positive, neutral, and negative memberships of the element \(r\in \chi \), respectively.
Definition 1
[39, 40] A PFS P defined on \(\chi \) is given by \(P = \{\left\langle r, \rho _{P}(r), \aleph _{P}(r), \sigma _{P}(r)\right\rangle : r\in \chi \}\), satisfying \(\rho _{P}(r) + \aleph _{P}(r) + \sigma _{P}(r)\le 1\), \(\forall r\in \chi \). The degree of refusal membership of \(r\in \chi \) to P defined by \(\pi _{P}(r) =1 - (\rho _{P}(r) + \aleph _{P}(r) + \sigma _{P}(r))\).
Definition 2
[51] A spherical fuzzy set (SFS) S defined on \(\chi \) is given by \(S = \{\left\langle r, \rho _{S}(r), \aleph _{S}(r), \sigma _{S}(r)\right\rangle : r\in \chi \}\), satisfying \((\rho _{S}(r))^{2} + (\aleph _{S}(r))^{2} + (\sigma _{S}(r))^{2}\le 1\), \(\forall r\in \chi \). The degree of refusal membership of \(r\in \chi \) to S defined by \(\pi _{S}(r) =\sqrt{1 - ((\rho _{S}(r))^{2} + (\aleph _{S}(r))^{2} + (\sigma _{S}(r))^{2})}\).
Definition 3
[45] A q-RPFS \(\textit{P}\) defined on \(\chi \) is given by
\(\textit{P} = \{\left\langle r, \rho _{\textit{P}}(r), \aleph _{\textit{P}}(r), \sigma _{\textit{P}}(r)\right\rangle : r\in \chi \}\), satisfying \((\rho _{\textit{P}}(r))^{q} + (\aleph _{\textit{P}}(r))^{q} + (\sigma _{\textit{P}}(r))^{q}\le 1\), \(\forall r\in \chi \). The degree of refusal membership of \(r\in \chi \) to \(\textit{P}\) defined by \(\pi _{\textit{P}}(r) =\root q \of {1 - ((\rho _{\textit{P}}(r))^{q} + (\aleph _{\textit{P}}(r))^{q} + (\sigma _{\textit{P}}(r))^{q})}\).
Definition 4
[24, 25] An (m,n)-ROFS \(\textit{P}\) defined on \(\chi \) is given by
\(\textit{P} = \{\left\langle r, \rho _{\textit{P}}(r), \aleph _{\textit{P}}(r)\right\rangle : r\in \chi \}\), satisfying \((\rho _{\textit{P}}(r))^{m} + (\aleph _{\textit{P}}(r))^{n}\le 1\), \(\forall r\in \chi \).
We display the main abbreviations and symbols used through this manuscript in Table 1 and 2, respectively.
\(k^{n}_{m}\)-Rung picture fuzzy sets
This section presents the \(k^{n}_{m}\)-rung picture fuzzy set, the central idea of this study, along with some of its key characteristics. Additionally, we focus on a variety of the aspects of \(k^{n}_{m}\)-rung picture fuzzy sets and propose a number of new operations on them.
Definition 5
A \(k^{n}_{m}\)-rung picture fuzzy set (\(k^{n}_{m}\)-RPFS) \(\varPsi \) defined on \(\chi \) is given by
satisfying
\(\forall r\in \chi \) and \(n,k,m\ge 1\). The degree of refusal membership of \(r\in \chi \) to \(\varPsi \) defined by
To keep things simple, we will mention the symbol \(\varPsi {=} (\rho _{\varPsi },\aleph _{\varPsi },\sigma _{\varPsi })\) for the \(k^{n}_{m}\)-ROFS \(\varPsi {=} \{\langle r,\rho _{\varPsi }(r),\aleph _{\varPsi }(r), \sigma _{\varPsi }(r)\rangle :r\in \chi \}\).
Remark 1
The space of \(k^{n}_{m}\)-rung picture fuzzy grades is larger than the space of the
-
1.
picture fuzzy grades if \(n>1\) or \(k>1\) or \(m>1\).
-
2.
spherical fuzzy grades if \(n,k\ge 2\) and \(m>2\), or \(n,m\ge 2\) and \(k>2\), or \(k,m\ge 2\) and \(n>2\).
-
3.
q-rung picture fuzzy grades if \(n,k\ge q\) and \(m>q\), or \(n,m\ge q\) and \(k>q\), or \(k,m\ge q\) and \(n>q\).
Definition 6
Let \(\varPsi = (\rho _{\varPsi },\aleph _{\varPsi },\sigma _{\varPsi })\), \(\varPsi _1 = (\rho _{\varPsi _1},\aleph _{\varPsi _1},\sigma _{\varPsi _1})\) and \(\varPsi _2 = (\rho _{\varPsi _2},\aleph _{\varPsi _2},\sigma _{\varPsi _2})\) be \(k^{n}_{m}\)-RPFSs. Then
-
1.
\(\varPsi _1\bigwedge \varPsi _2= (\min \{\rho _{\varPsi _1},\rho _{\varPsi _2}\},\min \{\aleph _{\varPsi _1},\aleph _{\varPsi _2}\},\max \{\sigma _{\varPsi _1},\sigma _{\varPsi _2}\})\).
-
2.
\(\varPsi _1\bigvee \varPsi _2= (\max \{\rho _{\varPsi _1},\rho _{\varPsi _2}\},\min \{\aleph _{\varPsi _1},\aleph _{\varPsi _2}\},\min \{\sigma _{\varPsi _1},\sigma _{\varPsi _2}\})\).
-
3.
\(\varPsi ^{c}= ((\sigma _{\varPsi })^\frac{m}{n},\aleph _{\varPsi },(\rho _{\varPsi })^\frac{n}{m})\).
Example 1
Assume that \(\varPsi _1=(0.62,0.85,0.45)\) and \(\varPsi _2=(0.73,0.61,0.69)\) are \(2^{4}_{3}\)-RPFSs for \(\chi = \{r\}\). Then
-
1.
\(\varPsi _1\bigwedge \varPsi _2 = (\min \{\rho _{\varPsi _1},\rho _{\varPsi _2}\},\min \{\aleph _{\varPsi _1},\aleph _{\varPsi _2}\}, \max \{\sigma _{\varPsi _1}, \sigma _{\varPsi _2}\}) = (\min \{0.62,0.73\},\min \{0.85,0.61\}, \max \{0.45, 0.69\}) = (0.62,0.61,0.69)\).
-
2.
\(\varPsi _1\bigvee \varPsi _2 = (\max \{\rho _{\varPsi _1},\rho _{\varPsi _2}\},\min \{\aleph _{\varPsi _1},\aleph _{\varPsi _2}\}, \min \{\sigma _{\varPsi _1}, \sigma _{\varPsi _2}\}) = (\max \{0.62,0.73\},\min \{0.85,0.61\}, \min \{0.45, 0.69\}) = (0.73,0.61,0.45)\).
-
3.
\(\varPsi _{1}^{c}= ((0.45^{3})^\frac{1}{4},0.85,(0.62^{4})^\frac{1}{3}) \approx (0.5494,0.85, 0.5287), \hbox {and} \varPsi _{2}^{c}= ((0.69^{3})^\frac{1}{4},0.61,(0.73^{4})^\frac{1}{3}) \approx -z (0.7571,0.61,0.6573)\).
Remark 2
If \(\varPsi _1 = (\rho _{\varPsi _1},\aleph _{\varPsi _1},\sigma _{\varPsi _1})\) and \(\varPsi _2 = (\rho _{\varPsi _2},\aleph _{\varPsi _2},\sigma _{\varPsi _2})\) are \(k^{n}_{m}\)-RPFSs, then \(\varPsi _1 \bigwedge \varPsi _2\) and \(\varPsi _1 \bigvee \varPsi _2\) are also \(k^{n}_{m}\)-RPFSs.
Theorem 1
If \(\varPsi = (\rho _{\varPsi },\aleph _{\varPsi },\sigma _{\varPsi })\) is a \(k^{n}_{m}\)-RPFS, then \(\varPsi ^{c}\) is also a \(k^{n}_{m}\)-RPFS, and hence \((\varPsi ^{c})^c = \varPsi \).
Proof
Since
then
and hence
Thus, \(\varPsi ^{c}\) is a \(k^{n}_{m}\)-RPFS and it is obvious that
\(\square \)
Theorem 2
Let \(\varPsi _1 = (\rho _{\varPsi _1},\aleph _{\varPsi _1},\sigma _{\varPsi _1})\) and \(\varPsi _2 = (\rho _{\varPsi _2},\aleph _{\varPsi _2},\sigma _{\varPsi _2})\) be \(k^{n}_{m}\)-RPFSs. Then
-
1.
\(\varPsi _1 \bigwedge \varPsi _2 = \varPsi _2 \bigwedge \varPsi _1\).
-
2.
\(\varPsi _1 \bigvee \varPsi _2 = \varPsi _2 \bigvee \varPsi _1\).
Proof
From Definition 6, we have
-
1.
\(\varPsi _1\bigwedge \varPsi _2 = (\min \{\rho _{\varPsi _1},\rho _{\varPsi _2}\},\min \{\aleph _{\varPsi _1},\aleph _{\varPsi _2}\}, \max \{\sigma _{\varPsi _1}, \sigma _{\varPsi _2}\}) = (\min \{\rho _{\varPsi _2},\rho _{\varPsi _1}\},\min \{\aleph _{\varPsi _2},\aleph _{\varPsi _1}\}, \max \{\sigma _{\varPsi _2}, \sigma _{\varPsi _1}\})=\varPsi _2\bigwedge \varPsi _1\).
-
2.
\(\varPsi _1\bigvee \varPsi _2 = (\max \{\rho _{\varPsi _1},\rho _{\varPsi _2}\},\min \{\aleph _{\varPsi _1},\aleph _{\varPsi _2}\}, \min \{\sigma _{\varPsi _1}, \sigma _{\varPsi _2}\}) = (\max \{\rho _{\varPsi _2},\rho _{\varPsi _1}\},\min \{\aleph _{\varPsi _2},\aleph _{\varPsi _1}\}, \min \{\sigma _{\varPsi _2}, \sigma _{\varPsi _1}\})=\varPsi _2\bigvee \varPsi _1\).
\(\square \)
Theorem 3
Let \(\varPsi _1 = (\rho _{\varPsi _1},\aleph _{\varPsi _1},\sigma _{\varPsi _1})\) and \(\varPsi _2 = (\rho _{\varPsi _2},\aleph _{\varPsi _2},\sigma _{\varPsi _2})\) be \(k^{n}_{m}\)-RPFSs. Then
-
1.
for \(\aleph _{\varPsi _1}\ge \aleph _{\varPsi _2}\), we have
-
(a)
\((\varPsi _1\bigwedge \varPsi _2)\bigvee \varPsi _2 = \varPsi _2\).
-
(b)
\((\varPsi _1\bigvee \varPsi _2)\bigwedge \varPsi _2 = \varPsi _2\).
-
(a)
-
2.
for \(\aleph _{\varPsi _1}<\aleph _{\varPsi _2}\), we have
-
(a)
\((\varPsi _1\bigwedge \varPsi _2)\bigvee \varPsi _2 \ne \varPsi _2\).
-
(b)
\((\varPsi _1\bigvee \varPsi _2)\bigwedge \varPsi _2 \ne \varPsi _2\).
-
(a)
Proof
-
1.
Let \(\aleph _{\varPsi _1}\ge \aleph _{\varPsi _2}\), then
-
(a)
\((\varPsi _1\bigwedge \varPsi _2)\bigvee \varPsi _2 =(\min \{\rho _{\varPsi _1},\rho _{\varPsi _2}\},\min \{\aleph _{\varPsi _1},\aleph _{\varPsi _2}\}\), \(\max \{\sigma _{\varPsi _1},\sigma _{\varPsi _2}\})\bigvee (\rho _{\varPsi _2},\aleph _{\varPsi _2},\sigma _{\varPsi _2}) =(\max \{\min \{\rho _{\varPsi _1},\rho _{\varPsi _2}\}, \rho _{\varPsi _2}\},\min \{\min \{\aleph _{\varPsi _1},\aleph _{\varPsi _2}\}, \aleph _{\varPsi _2}\},\) \(\min \{\max \{\sigma _{\varPsi _1},\sigma _{\varPsi _2}\},\sigma _{\varPsi _2}\}) = (\rho _{\varPsi _2},\aleph _{\varPsi _2},\sigma _{\varPsi _2})=\varPsi _{2}\).
-
(b)
\((\varPsi _1\bigvee \varPsi _2)\bigwedge \varPsi _2 =(\max \{\rho _{\varPsi _1},\rho _{\varPsi _2}\},\min \{\aleph _{\varPsi _1},\aleph _{\varPsi _2}\},\) \(\min \{\sigma _{\varPsi _1},\sigma _{\varPsi _2}\})\bigwedge (\rho _{\varPsi _2},\aleph _{\varPsi _2},\sigma _{\varPsi _2}) =(\min \{\max \{\rho _{\varPsi _1},\rho _{\varPsi _2}\}, \rho _{\varPsi _2}\},\min \{\min \{\aleph _{\varPsi _1},\aleph _{\varPsi _2}\}, \aleph _{\varPsi _2}\},\) \(\max \{\min \{\sigma _{\varPsi _1},\sigma _{\varPsi _2}\},\sigma _{\varPsi _2}\}) = (\rho _{\varPsi _2},\aleph _{\varPsi _2},\sigma _{\varPsi _2})=\varPsi _{2}\).
-
(a)
-
2.
Let \(\aleph _{\varPsi _1}<\aleph _{\varPsi _2}\), then
-
(a)
\((\varPsi _1\bigwedge \varPsi _2)\bigvee \varPsi _2 =(\min \{\rho _{\varPsi _1},\rho _{\varPsi _2}\},\min \{\aleph _{\varPsi _1},\aleph _{\varPsi _2}\}, \) \(\max \{\sigma _{\varPsi _1},\sigma _{\varPsi _2}\})\bigvee (\rho _{\varPsi _2},\aleph _{\varPsi _2},\sigma _{\varPsi _2}) =(\max \{\min \{\rho _{\varPsi _1},\rho _{\varPsi _2}\}, \rho _{\varPsi _2}\},\min \{\min \{\aleph _{\varPsi _1},\aleph _{\varPsi _2}\}, \aleph _{\varPsi _2}\},\min \{\max \{\sigma _{\varPsi _1},\sigma _{\varPsi _2}\},\sigma _{\varPsi _2}\}) = (\rho _{\varPsi _2},\aleph _{\varPsi _1},\sigma _{\varPsi _2})\ne \varPsi _{2}\).
-
(b)
\((\varPsi _1\bigvee \varPsi _2)\bigwedge \varPsi _2 =(\max \{\rho _{\varPsi _1},\rho _{\varPsi _2}\},\min \{\aleph _{\varPsi _1},\aleph _{\varPsi _2}\},\) \( \min \{\sigma _{\varPsi _1},\sigma _{\varPsi _2}\})\bigwedge (\rho _{\varPsi _2},\aleph _{\varPsi _2},\sigma _{\varPsi _2}) =(\min \{\max \{\rho _{\varPsi _1},\rho _{\varPsi _2}\}, \rho _{\varPsi _2}\},\min \{\min \{\aleph _{\varPsi _1},\aleph _{\varPsi _2}\}, \aleph _{\varPsi _2}\},\) \(\max \{\min \{\sigma _{\varPsi _1},\sigma _{\varPsi _2}\},\sigma _{\varPsi _2}\}) = (\rho _{\varPsi _2},\aleph _{\varPsi _1},\sigma _{\varPsi _2})\ne \varPsi _{2}\). \(\square \)
-
(a)
Theorem 4
Let \(\varPsi _1 = (\rho _{\varPsi _1},\aleph _{\varPsi _1},\sigma _{\varPsi _1})\), \(\varPsi _2 = (\rho _{\varPsi _2},\aleph _{\varPsi _2},\sigma _{\varPsi _2})\) and \(\varPsi _3 = (\rho _{\varPsi _3},\aleph _{\varPsi _3},\sigma _{\varPsi _3})\) be \(k^{n}_{m}\)-RPFSs. Then
-
1.
\(\varPsi _1 \bigwedge (\varPsi _2\bigwedge \varPsi _3) = (\varPsi _1\bigwedge \varPsi _2) \bigwedge \varPsi _3\).
-
2.
\(\varPsi _1 \bigvee (\varPsi _2\bigvee \varPsi _3) = (\varPsi _1\bigvee \varPsi _2) \bigvee \varPsi _3\).
Proof
For the three \(k^{n}_{m}\)-RPFSs \(\varPsi _1, \varPsi _2\), and \(\varPsi _3\), based on Definition 6, we have
-
1.
\(\varPsi _1\bigwedge (\varPsi _2\bigwedge \varPsi _3) = (\rho _{\varPsi _1},\aleph _{\varPsi _1},\sigma _{\varPsi _1})\bigwedge \) \( (\min \{\rho _{\varPsi _2},\rho _{\varPsi _3}\},\min \{\aleph _{\varPsi _2},\aleph _{\varPsi _3}\},\max \{\sigma _{\varPsi _2},\sigma _{\varPsi _3}\}) = (\min \{\rho _{\varPsi _1},\min \{\rho _{\varPsi _2},\rho _{\varPsi _3}\}\},\min \{\aleph _{\varPsi _1},\min \{\aleph _{\varPsi _2}, \aleph _{\varPsi _3}\}\},\) \(\max \{\sigma _{\varPsi _1},\max \{\sigma _{\varPsi _2},\sigma _{\varPsi _3}\}\}) = (\min \{\min \{\rho _{\varPsi _1},\rho _{\varPsi _2}\},\rho _{\varPsi _3}\}, \min \{\min \{\aleph _{\varPsi _1},\)\( \aleph _{\varPsi _2}\}, \aleph _{\varPsi _3}\},\) \( \max \{\max \{\sigma _{\varPsi _1},\sigma _{\varPsi _2}\},\sigma _{\varPsi _3}\}) = (\min \{\rho _{\varPsi _1},\rho _{\varPsi _2}\}, \min \{\aleph _{\varPsi _1},\aleph _{\varPsi _2}\},\max \{\sigma _{\varPsi _1},\sigma _{\varPsi _2}\})\bigwedge (\rho _{\varPsi _3},\aleph _{\varPsi _3},\sigma _{\varPsi _3}) =(\varPsi _1\bigwedge \varPsi _2) \bigwedge \varPsi _3\).
-
2.
\(\varPsi _1\bigvee (\varPsi _2\bigvee \varPsi _3) = (\rho _{\varPsi _1},\aleph _{\varPsi _1},\sigma _{\varPsi _1})\bigvee (\max \{\rho _{\varPsi _2},\rho _{\varPsi _3}\}, \min \{\aleph _{\varPsi _2},\aleph _{\varPsi _3}\},\min \{\sigma _{\varPsi _2},\sigma _{\varPsi _3}\}) =(\max \{\rho _{\varPsi _1},\max \{\rho _{\varPsi _2},\rho _{\varPsi _3}\}\}, \min \{\aleph _{\varPsi _1},\min \{\aleph _{\varPsi _2},\aleph _{\varPsi _3}\}\},\) \(\min \{\sigma _{\varPsi _1},\min \{\sigma _{\varPsi _2},\sigma _{\varPsi _3}\}\}) =(\max \{\max \{\rho _{\varPsi _1},\rho _{\varPsi _2}\},\rho _{\varPsi _3}\}, \min \{\min \{\aleph _{\varPsi _1}, \aleph _{\varPsi _2}\}, \aleph _{\varPsi _3}\},\) \(\min \{\min \{\sigma _{\varPsi _1},\sigma _{\varPsi _2}\},\sigma _{\varPsi _3}\}){=}(\max \{\rho _{\varPsi _1},\rho _{\varPsi _2}\}, \min \{\aleph _{\varPsi _1},\aleph _{\varPsi _2}\},\min \{\sigma _{\varPsi _1},\sigma _{\varPsi _2}\})\bigvee (\rho _{\varPsi _3},\aleph _{\varPsi _3},\sigma _{\varPsi _3}) = (\varPsi _1\bigvee \varPsi _2) \bigvee \varPsi _3\).\(\square \)
Theorem 5
Let \(\varPsi _1 = (\rho _{\varPsi _1},\aleph _{\varPsi _1},\sigma _{\varPsi _1})\), \(\varPsi _2 = (\rho _{\varPsi _2},\aleph _{\varPsi _2},\sigma _{\varPsi _2})\) and \(\varPsi _3 = (\rho _{\varPsi _3},\aleph _{\varPsi _3},\sigma _{\varPsi _3})\) be \(k^{n}_{m}\)-RPFS. Then
-
1.
\((\varPsi _1\bigvee \varPsi _2)\bigwedge \varPsi _3=(\varPsi _1\bigwedge \varPsi _3)\bigvee (\varPsi _2\bigwedge \varPsi _3)\).
-
2.
\((\varPsi _1\bigwedge \varPsi _2)\bigvee \varPsi _3=(\varPsi _1\bigvee \varPsi _3)\bigwedge (\varPsi _2\bigvee \varPsi _3)\).
Proof
For the \(k^{n}_{m}\)-RPFSs \(\varPsi _1, \varPsi _2\) and \(\varPsi _3\), based on Definition 6, we have
-
1.
\((\varPsi _1\bigvee \varPsi _2)\bigwedge \varPsi _3=(\max \{\rho _{\varPsi _1},\rho _{\varPsi _2}\},\min \{\aleph _{\varPsi _1},\aleph _{\varPsi _2}\},\min \{\sigma _{\varPsi _1},\sigma _{\varPsi _2}\})\bigwedge (\rho _{\varPsi _3},\aleph _{\varPsi _3},\sigma _{\varPsi _3}) =(\min \{\max \{\rho _{\varPsi _1},\rho _{\varPsi _2}\}, \rho _{\varPsi _3}\},\min \{\min \{\aleph _{\varPsi _1},\aleph _{\varPsi _2}\}, \aleph _{\varPsi _3}\}, \max \{\min \{\sigma _{\varPsi _1},\sigma _{\varPsi _2}\},\sigma _{\varPsi _3}\})\), and \((\varPsi _1\bigwedge \varPsi _3)\bigvee (\varPsi _2\bigwedge \varPsi _3)=(\min \{\rho _{\varPsi _1},\rho _{\varPsi _3}\},\min \{\aleph _{\varPsi _1}, \aleph _{\varPsi _3}\},\max \{\sigma _{\varPsi _1},\sigma _{\varPsi _3}\}) \bigvee (\min \{\rho _{\varPsi _2},\rho _{\varPsi _3}\},\min \{\aleph _{\varPsi _2},\aleph _{\varPsi _3}\},\max \{\sigma _{\varPsi _2},\sigma _{\varPsi _3}\}) =(\max \{\min \{\rho _{\varPsi _1},\rho _{\varPsi _3}\}, \min \{\rho _{\varPsi _2},\rho _{\varPsi _3}\}\}\), \(\min \{\min \{\aleph _{\varPsi _1},\aleph _{\varPsi _3}\}, \min \{\aleph _{\varPsi _2},\aleph _{\varPsi _3}\}\}, \min \{\max \{\sigma _{\varPsi _1},\sigma _{\varPsi _3}\},\max \{\sigma _{\varPsi _2},\sigma _{\varPsi _3}\}\})\). Then
$$\begin{aligned}{} & {} {\min }\{{\max }\{\rho _{\varPsi _1},\rho _{\varPsi _2}\}, \rho _{\varPsi _3}\}\\{} & {} \quad =\left\{ \begin{array}{lll} \rho _{\varPsi _2} &{} \text{ if }\, \rho _{\varPsi _1} \le \rho _{\varPsi _2}\le \rho _{\varPsi _3},\\ \rho _{\varPsi _1} &{} \text{ if }\, \rho _{\varPsi _2} \le \rho _{\varPsi _1}\le \rho _{\varPsi _3},\\ \rho _{\varPsi _3} &{} \text{ if }\, \rho _{\varPsi _1} \le \rho _{\varPsi _3}\le \rho _{\varPsi _2},\\ \rho _{\varPsi _3} &{} \text{ if }\, \rho _{\varPsi _3} \le \rho _{\varPsi _1}\le \rho _{\varPsi _2},\\ \rho _{\varPsi _3} &{} \text{ if }\, \rho _{\varPsi _2} \le \rho _{\varPsi _3}\le \rho _{\varPsi _1},\\ \rho _{\varPsi _3} &{} \text{ if }\, \rho _{\varPsi _3} \le \rho _{\varPsi _2}\le \rho _{\varPsi _1}, \end{array}\right. \\{} & {} {\min }\{{\min }\{\aleph _{\varPsi _1},\aleph _{\varPsi _2}\}, \aleph _{\varPsi _3}\}\\{} & {} \quad =\left\{ \begin{array}{lll} \aleph _{\varPsi _1} &{} \text{ if }\, \aleph _{\varPsi _1} \le \aleph _{\varPsi _2}\le \aleph _{\varPsi _3},\\ \aleph _{\varPsi _2} &{} \text{ if }\,\aleph _{\varPsi _2} \le \aleph _{\varPsi _1}\le \aleph _{\varPsi _3},\\ \aleph _{\varPsi _1} &{} \text{ if }\,\aleph _{\varPsi _1} \le \aleph _{\varPsi _3}\le \aleph _{\varPsi _2},\\ \aleph _{\varPsi _3} &{} \text{ if }\,\aleph _{\varPsi _3} \le \aleph _{\varPsi _1}\le \aleph _{\varPsi _2},\\ \aleph _{\varPsi _2} &{} \text{ if }\,\aleph _{\varPsi _2} \le \aleph _{\varPsi _3}\le \aleph _{\varPsi _1},\\ \aleph _{\varPsi _3} &{} \text{ if }\,\aleph _{\varPsi _3} \le \aleph _{\varPsi _2}\le \aleph _{\varPsi _1}, \end{array}\right. \\{} & {} {\max }\{{\min }\{\sigma _{\varPsi _1},\sigma _{\varPsi _2}\},\sigma _{\varPsi _3}\}\\{} & {} \quad = \left\{ \begin{array}{lll} \sigma _{\varPsi _3} &{} \text{ if }\, \sigma _{\varPsi _1} \le \sigma _{\varPsi _2}\le \sigma _{\varPsi _3},\\ \sigma _{\varPsi _3} &{} \text{ if }\,\sigma _{\varPsi _2} \le \sigma _{\varPsi _1}\le \sigma _{\varPsi _3},\\ \sigma _{\varPsi _3} &{} \text{ if }\,\sigma _{\varPsi _1} \le \sigma _{\varPsi _3}\le \sigma _{\varPsi _2},\\ \sigma _{\varPsi _1} &{} \text{ if }\,\sigma _{\varPsi _3} \le \sigma _{\varPsi _1}\le \sigma _{\varPsi _2},\\ \sigma _{\varPsi _3} &{} \text{ if }\,\sigma _{\varPsi _2} \le \sigma _{\varPsi _3}\le \sigma _{\varPsi _1},\\ \sigma _{\varPsi _2} &{} \text{ if }\,\sigma _{\varPsi _3} \le \sigma _{\varPsi _2}\le \sigma _{\varPsi _1}, \end{array}\right. \\{} & {} {\max }\{{\min }\{\rho _{\varPsi _1},\rho _{\varPsi _3}\}, {\min }\{\rho _{\varPsi _2},\rho _{\varPsi _3}\}\}\\{} & {} \quad = \left\{ \begin{array}{lll} \rho _{\varPsi _2} &{} \text{ if }\, \rho _{\varPsi _1} \le \rho _{\varPsi _2}\le \rho _{\varPsi _3},\\ \rho _{\varPsi _1} &{} \text{ if }\,\rho _{\varPsi _2} \le \rho _{\varPsi _1}\le \rho _{\varPsi _3},\\ \rho _{\varPsi _3} &{} \text{ if }\,\rho _{\varPsi _1} \le \rho _{\varPsi _3}\le \rho _{\varPsi _2},\\ \rho _{\varPsi _3} &{} \text{ if }\,\rho _{\varPsi _3} \le \rho _{\varPsi _1}\le \rho _{\varPsi _2},\\ \rho _{\varPsi _3} &{} \text{ if }\,\rho _{\varPsi _2} \le \rho _{\varPsi _3}\le \rho _{\varPsi _1},\\ \rho _{\varPsi _3} &{} \text{ if }\,\rho _{\varPsi _3} \le \rho _{\varPsi _2}\le \rho _{\varPsi _1}, \end{array}\right. \\{} & {} {\min }\{{\min }\{\aleph _{\varPsi _1},\aleph _{\varPsi _3}\}, {\min }\{\aleph _{\varPsi _2},\aleph _{\varPsi _3}\}\}\\{} & {} \quad = \left\{ \begin{array}{lll} \aleph _{\varPsi _1} &{} \text{ if }\, \aleph _{\varPsi _1} \le \aleph _{\varPsi _2}\le \aleph _{\varPsi _3},\\ \aleph _{\varPsi _2} &{} \text{ if }\,\aleph _{\varPsi _2} \le \aleph _{\varPsi _1}\le \aleph _{\varPsi _3},\\ \aleph _{\varPsi _1} &{} \text{ if }\,\aleph _{\varPsi _1} \le \aleph _{\varPsi _3}\le \aleph _{\varPsi _2},\\ \aleph _{\varPsi _3} &{} \text{ if }\,\aleph _{\varPsi _3} \le \aleph _{\varPsi _1}\le \aleph _{\varPsi _2},\\ \aleph _{\varPsi _2} &{} \text{ if }\,\aleph _{\varPsi _2} \le \aleph _{\varPsi _3}\le \aleph _{\varPsi _1},\\ \aleph _{\varPsi _3} &{} \text{ if }\,\aleph _{\varPsi _3} \le \aleph _{\varPsi _2}\le \aleph _{\varPsi _1}, \end{array}\right. \end{aligned}$$and
$$\begin{aligned}{} & {} {\min }\{{\max }\{\sigma _{\varPsi _1},\sigma _{\varPsi _3}\},{\max }\{\sigma _{\varPsi _2},\sigma _{\varPsi _3}\}\}\\{} & {} \quad = \left\{ \begin{array}{lll} \sigma _{\varPsi _3} &{} \text{ if }\, \sigma _{\varPsi _1} \le \sigma _{\varPsi _2}\le \sigma _{\varPsi _3},\\ \sigma _{\varPsi _3} &{} \text{ if }\,\sigma _{\varPsi _2} \le \sigma _{\varPsi _1}\le \sigma _{\varPsi _3},\\ \sigma _{\varPsi _3} &{} \text{ if }\,\sigma _{\varPsi _1} \le \sigma _{\varPsi _3}\le \sigma _{\varPsi _2},\\ \sigma _{\varPsi _1} &{} \text{ if }\,\sigma _{\varPsi _3} \le \sigma _{\varPsi _1}\le \sigma _{\varPsi _2},\\ \sigma _{\varPsi _3} &{} \text{ if }\,\sigma _{\varPsi _2} \le \sigma _{\varPsi _3}\le \sigma _{\varPsi _1},\\ \sigma _{\varPsi _2} &{} \text{ if }\,\sigma _{\varPsi _3} \le \sigma _{\varPsi _2}\le \sigma _{\varPsi _1}. \end{array}\right. \end{aligned}$$Thus, \(\min \{\max \{\rho _{\varPsi _1},\rho _{\varPsi _2}\}, \rho _{\varPsi _3}\}=\max \{\min \{\rho _{\varPsi _1},\rho _{\varPsi _3}\}, \min \{\rho _{\varPsi _2},\rho _{\varPsi _3}\}\}\), \(\min \{\min \{\aleph _{\varPsi _1},\aleph _{\varPsi _2}\}, \aleph _{\varPsi _3}\}=\min \{\min \{\aleph _{\varPsi _1},\aleph _{\varPsi _3}\}, \min \{\aleph _{\varPsi _2},\aleph _{\varPsi _3}\}\}\), and \(\max \{\min \{\sigma _{\varPsi _1},\sigma _{\varPsi _2}\},\sigma _{\varPsi _3}\}= \min \{\max \{\sigma _{\varPsi _1},\sigma _{\varPsi _3}\},\max \{\sigma _{\varPsi _2},\sigma _{\varPsi _3}\}\}\). Therefore, \((\varPsi _1\bigvee \varPsi _2)\bigwedge \varPsi _3=(\varPsi _1\bigwedge \varPsi _3)\bigvee (\varPsi _2\bigwedge \varPsi _3)\).
-
2.
The evidence resembles (1). \(\square \)
Theorem 6
Let \(\varPsi _1 = (\rho _{\varPsi _1},\aleph _{\varPsi _1},\sigma _{\varPsi _1})\) and \(\varPsi _2 = (\rho _{\varPsi _2},\aleph _{\varPsi _2},\sigma _{\varPsi _2})\) be \(k^{n}_{m}\)-RPFSs. Then
-
1.
\((\varPsi _1 \bigwedge \varPsi _2)^{c} = \varPsi _1^{c} \bigvee \varPsi _2^{c}\).
-
2.
\((\varPsi _1 \bigvee \varPsi _2)^{c} = \varPsi _1^{c} \bigwedge \varPsi _2^{c}\).
Proof
For the \(k^{n}_{m}\)-RPFSs \(\varPsi _1\) and \(\varPsi _2\), based on Definition 6, we have
-
1.
\((\varPsi _1 \bigwedge \varPsi _2)^{c}{=}(\min \{\rho _{\varPsi _1},\rho _{\varPsi _2}\},\min \{\aleph _{\varPsi _1},\aleph _{\varPsi _2}\},\max \{\sigma _{\varPsi _1},\sigma _{\varPsi _2}\})^{c}{=}(\max \{(\sigma _{\varPsi _1})^\frac{m}{n},(\sigma _{\varPsi _2})^\frac{m}{n}\}, \min \{\aleph _{\varPsi _1},\aleph _{\varPsi _2}\},\min \{(\rho _{\varPsi _1})^\frac{n}{m},(\rho _{\varPsi _2})^\frac{n}{m}\}){=}((\sigma _{\varPsi _1})^\frac{m}{n},\aleph _{\varPsi _1},(\rho _{\varPsi _1})^\frac{n}{m})\bigvee ((\sigma _{\varPsi _2})^\frac{m}{n},\aleph _{\varPsi _2},(\rho _{\varPsi _2})^\frac{n}{m}){=}\varPsi _1^{c}{\bigvee }\varPsi _2^{c}\).
-
2.
\((\varPsi _1 \bigvee \varPsi _2)^{c} = (\max \{\rho _{\varPsi _1},\rho _{\varPsi _2}\},\min \{\aleph _{\varPsi _1},\aleph _{\varPsi _2}\},\min \{\sigma _{\varPsi _1},\sigma _{\varPsi _2}\})^{c} = (\min \{(\sigma _{\varPsi _1})^\frac{m}{n},(\sigma _{\varPsi _2})^\frac{m}{n}\}, \min \{\aleph _{\varPsi _1},\aleph _{\varPsi _2}\},\max \{(\rho _{\varPsi _1})^\frac{n}{m},(\rho _{\varPsi _2})^\frac{n}{m}\}) = ((\sigma _{\varPsi _1})^\frac{m}{n},\aleph _{\varPsi _1},(\rho _{\varPsi _1})^\frac{n}{m})\bigwedge ((\sigma _{\varPsi _2})^\frac{m}{n},\aleph _{\varPsi _2},(\rho _{\varPsi _2})^\frac{n}{m}) = \varPsi _1^{c} \bigwedge \varPsi _2^{c}\). \(\square \)
Definition 7
Let \(\varPsi = (\rho _{\varPsi },\aleph _{\varPsi },\sigma _{\varPsi })\), \(\varPsi _1 = (\rho _{\varPsi _1},\aleph _{\varPsi _1},\sigma _{\varPsi _1})\) and \(\varPsi _2 = (\rho _{\varPsi _2},\aleph _{\varPsi _2},\sigma _{\varPsi _2})\) be \(k^{n}_{m}\)-RPFSs. Then
-
1.
\(\varPsi _1 \oplus \varPsi _2 = \left( \frac{\rho _{\varPsi _1}\rho _{\varPsi _2}}{(1+ \rho _{\varPsi _1}^{n}\rho _{\varPsi _2}^{n})^\frac{1}{n}}, \frac{\aleph _{\varPsi _1}\aleph _{\varPsi _2}}{(1+(1- \aleph _{\varPsi _1}^{k})(1- \aleph _{\varPsi _2}^{k}))^\frac{1}{k}},\right. \left. \qquad \qquad \qquad \quad \frac{\sigma _{\varPsi _1}\sigma _{\varPsi _2}}{(1+(1- \sigma _{\varPsi _1}^{m})(1- \sigma _{\varPsi _2}^{m}))^\frac{1}{m}}\right) \),
-
2.
\(\varPsi _1 \otimes \varPsi _2 = \left( \frac{\rho _{\varPsi _1}\rho _{\varPsi _2}}{(1+(1- \rho _{\varPsi _1}^{n})(1- \rho _{\varPsi _2}^{n}))^\frac{1}{n}}, \frac{\aleph _{\varPsi _1}\aleph _{\varPsi _2}}{(1+(1- \aleph _{\varPsi _1}^{k})(1- \aleph _{\varPsi _2}^{k}))^\frac{1}{k}},\right. \left. \qquad \quad \qquad \qquad \frac{\sigma _{\varPsi _1}\sigma _{\varPsi _2}}{(1+ \sigma _{\varPsi _1}^{m}\sigma _{\varPsi _2}^{m})^\frac{1}{m}}\right) \),
-
3.
\(\ell \varPsi = \left( (\frac{(1+\rho _{\varPsi }^{n})^{\ell }- (1-\rho _{\varPsi }^{n})^{\ell }}{(1+\rho _{\varPsi }^{n})^{\ell }+ (1-\rho _{\varPsi }^{n})^{\ell }})^\frac{1}{n}, \frac{\aleph _{\varPsi }^{\ell }}{((2-\aleph _{\varPsi }^{k})^{\ell }+ (1-\aleph _{\varPsi }^{k})^{\ell })^\frac{1}{k}},\right. \left. \qquad \quad \qquad \qquad \frac{\sigma _{\varPsi }^{\ell }}{((2-\sigma _{\varPsi }^{m})^{\ell }+ (\sigma _{\varPsi }^{m})^{\ell })^\frac{1}{m}}\right) \), and
-
4.
\( \varPsi ^{\ell } = \left( \frac{\rho _{\varPsi }^{\ell }}{((2-\rho _{\varPsi }^{n})^{\ell }+ (\rho _{\varPsi }^{n})^{\ell })^\frac{1}{n}}, \frac{\aleph _{\varPsi }^{\ell }}{((2-\aleph _{\varPsi }^{k})^{\ell }+ (1-\aleph _{\varPsi }^{k})^{\ell })^\frac{1}{k}},\right. \left. \qquad \qquad \qquad \quad (\frac{(1+\sigma _{\varPsi }^{m})^{\ell }- (1-\sigma _{\varPsi }^{m})^{\ell }}{(1+\sigma _{\varPsi }^{m})^{\ell }+ (1-\sigma _{\varPsi }^{m})^{\ell }})^\frac{1}{m}\right) \),
where \(\ell \) is a positive real number (\(\ell > 0\)).
Example 2
Assume that \(\varPsi _1=(0.71, 0.94, 0.57)\) and \(\varPsi _2=( 0.82, , 0.70, 0.87)\) are \(4^{5}_{7}\)-RPFSs for \(\chi = \{r\}\). Then
-
1.
\(\varPsi _1 \oplus \varPsi _2 = \left( \frac{(0.71)(0.82)}{(1+ (0.71^{5})(0.82^{5}))^\frac{1}{5}}, \frac{(0.94)(0.70)}{(1+(1- 0.94^{4})(1- 0.70^{4}))^\frac{1}{4}},\right. \left. \qquad \quad \frac{(0.57)(0.87)}{(1+(1- 0.57^{7})(1- 0.87^{7}))^\frac{1}{7}}\right) \qquad \quad \approx (0.5747, 0.6331, 0.4633)\).
-
2.
\(\varPsi _1 \otimes \varPsi _2 = \left( \frac{(0.71)(0.82)}{(1+(1- 0.71^{5})(1- 0.82^{5}))^\frac{1}{5}}, \right. \left. \qquad \quad \frac{(0.94)(0.70)}{(1+(1- 0.94^{4})(1- 0.70^{4}))^\frac{1}{4}}, \frac{(0.57)(0.87)}{(1+ (0.57^{7})(0.87^{7}))^\frac{1}{7}}\right) \qquad \quad \approx (0.5357, 0.6331, 0.4954)\).
-
3.
\(\ell \varPsi _1 = \left( (\frac{(1+0.71^{5})^{2}- (1-0.71^{5})^{2}}{(1+0.71^{5})^{2}+ (1-0.71^{5})^{2}})^\frac{1}{5},\right. \left. \qquad \quad \frac{0.94^{2}}{((2-0.94^{4})^{2}+ (1-0.94^{4})^{2})^\frac{1}{4}}, \frac{0.57^{2}}{((2-0.57^{7})^{2}+ (0.57^{7})^{2})^\frac{1}{7}}\right) \approx (0.8104, 0.7939, 0.2673)\).
-
4.
\(\varPsi ^{\ell }_1 = \left( \frac{0.71^{2}}{((2-0.71^{5})^{2}+ (0.71^{5})^{2})^\frac{1}{5}}, \frac{0.94^{2}}{((2-0.94^{4})^{2}+ (1-0.94^{4})^{2})^\frac{1}{4}}, \right. \left. \qquad \quad (\frac{(1+0.57^{7})^{2}- (1-0.57^{7})^{2}}{(1+0.57^{7})^{2}+ (1-0.57^{7})^{2}})^\frac{1}{7}\right) \approx (0.3960, 0.7939, 0.6293)\), for \(\ell = 2\).
Theorem 7
If \(\varPsi _1 = (\rho _{\varPsi _1},\aleph _{\varPsi _1},\sigma _{\varPsi _1})\) and \(\varPsi _2 = (\rho _{\varPsi _2},\aleph _{\varPsi _2}, \sigma _{\varPsi _2})\) are \(k^{n}_{m}\)-RPFSs, then \(\varPsi _1 \oplus \varPsi _2\) and \(\varPsi _1 \otimes \varPsi _2\) are also \(k^{n}_{m}\)-RPFSs.
Proof
For \(k^{n}_{m}\)-RPFSs \(\varPsi _1 = (\rho _{\varPsi _1},\aleph _{\varPsi _1},\sigma _{\varPsi _1})\) and \(\varPsi _2 = (\rho _{\varPsi _2},\aleph _{\varPsi _2},\sigma _{\varPsi _2})\), the following relations are evident:
which indicates that
implies
and hence
Therefore
Similarly
and
Since
then
and hence
Similarly
and
And also we have
which implies that
Similarly
Then, we have
-
1.
\(\left( \dfrac{\rho _{\varPsi _1}\rho _{\varPsi _2}}{(1+ \rho _{\varPsi _1}^{n}\rho _{\varPsi _2}^{n})^{\frac{1}{n}}}\right) ^{n} + \left( \dfrac{\aleph _{\varPsi _1}\aleph _{\varPsi _2}}{(1+(1- \aleph _{\varPsi _1}^{k})(1- \aleph _{\varPsi _2}^{k}))^\frac{1}{k}}\right) ^{k}\)
$$\begin{aligned}{} & {} \qquad + \left( \frac{\sigma _{\varPsi _1}\sigma _{\varPsi _2}}{(1+(1- \sigma _{\varPsi _1}^{m})(1- \sigma _{\varPsi _2}^{m}))^\frac{1}{m}}\right) ^{m}\\{} & {} \quad = \frac{\rho _{\varPsi _1}^{n}\rho _{\varPsi _2}^{n}}{1+ \rho _{\varPsi _1}^{n}\rho _{\varPsi _2}^{n}} + \frac{\aleph _{\varPsi _1}^{k}\aleph _{\varPsi _2}^{k}}{1+(1- \aleph _{\varPsi _1}^{k})(1- \aleph _{\varPsi _2}^{k})} \\{} & {} \qquad + \frac{\sigma _{\varPsi _1}^{m}\sigma _{\varPsi _2}^{m}}{1+(1- \sigma _{\varPsi _1}^{m})(1- \sigma _{\varPsi _2}^{m})}\\{} & {} \quad \le \frac{\rho _{\varPsi _1}^{n}\rho _{\varPsi _2}^{n}+\rho _{\varPsi _1}^{n}}{1+ \rho _{\varPsi _1}^{n}\rho _{\varPsi _2}^{n}} + \frac{\aleph _{\varPsi _1}^{k}\aleph _{\varPsi _2}^{k}}{1+ \rho _{\varPsi _1}^{n}\rho _{\varPsi _2}^{n}} + \frac{\sigma _{\varPsi _1}^{m}\sigma _{\varPsi _2}^{m}}{1+ \rho _{\varPsi _1}^{n}\rho _{\varPsi _2}^{n}}\\{} & {} \quad = \frac{\rho _{\varPsi _1}^{n}\rho _{\varPsi _2}^{n}+\rho _{\varPsi _1}^{n} + \aleph _{\varPsi _1}^{k}\aleph _{\varPsi _2}^{k}+ \sigma _{\varPsi _1}^{m}\sigma _{\varPsi _2}^{m}}{1+ \rho _{\varPsi _1}^{n}\rho _{\varPsi _2}^{n}}\\{} & {} \quad \le \frac{\rho _{\varPsi _1}^{n}\rho _{\varPsi _2}^{n}+\rho _{\varPsi _1}^{n} + \aleph _{\varPsi _1}^{k}+ \sigma _{\varPsi _1}^{m}}{1+ \rho _{\varPsi _1}^{n}\rho _{\varPsi _2}^{n}}\\{} & {} \quad \le \frac{\rho _{\varPsi _1}^{n}\rho _{\varPsi _2}^{n}+1}{1+ \rho _{\varPsi _1}^{n}\rho _{\varPsi _2}^{n}} = 1. \end{aligned}$$ -
2.
\(\left( \dfrac{\rho _{\varPsi _1}\rho _{\varPsi _2}}{(1+(1- \rho _{\varPsi _1}^{n})(1- \rho _{\varPsi _2}^{n}))^\frac{1}{n}}\right) ^{n}\)
$$\begin{aligned}{} & {} \qquad + \left( \frac{\aleph _{\varPsi _1}\aleph _{\varPsi _2}}{(1+(1- \aleph _{\varPsi _1}^{k})(1- \aleph _{\varPsi _2}^{k}))^\frac{1}{k}}\right) ^{k} \\{} & {} \qquad +\left( \frac{\sigma _{\varPsi _1}\sigma _{\varPsi _2}}{(1+ \sigma _{\varPsi _1}^{m}\sigma _{\varPsi _2}^{m})^\frac{1}{m}}\right) ^{m}\\{} & {} \quad =\frac{\rho _{\varPsi _1}^{n}\rho _{\varPsi _2}^{n}}{1+(1- \rho _{\varPsi _1}^{n})(1- \rho _{\varPsi _2}^{n})} \\{} & {} \qquad + \frac{\aleph _{\varPsi _1}^{k}\aleph _{\varPsi _2}^{k}}{1+(1- \aleph _{\varPsi _1}^{k})(1- \aleph _{\varPsi _2}^{k})}+ \frac{\sigma _{\varPsi _1}^{m}\sigma _{\varPsi _2}^{m}}{1+ \sigma _{\varPsi _1}^{m}\sigma _{\varPsi _2}^{m}}\\{} & {} \quad \le \frac{\rho _{\varPsi _1}^{n}\rho _{\varPsi _2}^{n}}{1+ \sigma _{\varPsi _1}^{m}\sigma _{\varPsi _2}^{m}} + \frac{\aleph _{\varPsi _1}^{k}\aleph _{\varPsi _2}^{k}}{1+ \sigma _{\varPsi _1}^{m}\sigma _{\varPsi _2}^{m}}+ \frac{\sigma _{\varPsi _1}^{m}\sigma _{\varPsi _2}^{m}+\sigma _{\varPsi _1}^{m}}{1+ \sigma _{\varPsi _1}^{m}\sigma _{\varPsi _2}^{m}}\\{} & {} \quad =\frac{\rho _{\varPsi _1}^{n}\rho _{\varPsi _2}^{n}+ \aleph _{\varPsi _1}^{k}\aleph _{\varPsi _2}^{k}+ \sigma _{\varPsi _1}^{m}\sigma _{\varPsi _2}^{m}+\sigma _{\varPsi _1}^{m}}{1+ \sigma _{\varPsi _1}^{m}\sigma _{\varPsi _2}^{m}}\\{} & {} \quad \le \frac{\rho _{\varPsi _1}^{n}+ \aleph _{\varPsi _1}^{k}+ \sigma _{\varPsi _1}^{m}\sigma _{\varPsi _2}^{m}+\sigma _{\varPsi _1}^{m}}{1+ \sigma _{\varPsi _1}^{m}\sigma _{\varPsi _2}^{m}}\\{} & {} \quad =\frac{1+ \sigma _{\varPsi _1}^{m}\sigma _{\varPsi _2}^{m}}{1+ \sigma _{\varPsi _1}^{m}\sigma _{\varPsi _2}^{m}} = 1. \end{aligned}$$
These indicate that both of \(\varPsi _1 \oplus \varPsi _2\) and \(\varPsi _1 \otimes \varPsi _2\) are \(k^{n}_{m}\)-RPFSs. \(\square \)
Theorem 8
Let \(\varPsi = (\rho _{\varPsi },\aleph _{\varPsi },\sigma _{\varPsi })\) be a \(k^{n}_{m}\)-RPFS and \(\ell > 0\). Then, \(\ell \varPsi \) and \(\varPsi ^{\ell }\) are also \(k^{n}_{m}\)-RPFSs.
Proof
Since
then
which implies that
Again since
then
thus
Similarly
Therefore
-
1.
\( \left( \left( \dfrac{(1+\rho _{\varPsi }^{n})^{\ell }- (1-\rho _{\varPsi }^{n})^{\ell }}{(1+\rho _{\varPsi }^{n})^{\ell }+ (1-\rho _{\varPsi }^{n})^{\ell }}\right) ^\frac{1}{n}\right) ^{n}\)
$$\begin{aligned}{} & {} \qquad + \left( \frac{\aleph _{\varPsi }^{\ell }}{((2-\aleph _{\varPsi }^{k})^{\ell }+ (1-\aleph _{\varPsi }^{k})^{\ell })^\frac{1}{k}}\right) ^{k} \\{} & {} \qquad +\left( \frac{\sigma _{\varPsi }^{\ell }}{((2-\sigma _{\varPsi }^{m})^{\ell }+ (\sigma _{\varPsi }^{m})^{\ell })^\frac{1}{m}}\right) ^{m}\\{} & {} \quad = \frac{(1+\rho _{\varPsi }^{n})^{\ell }- (1-\rho _{\varPsi }^{n})^{\ell }}{(1+\rho _{\varPsi }^{n})^{\ell }+ (1-\rho _{\varPsi }^{n})^{\ell }}\\{} & {} \qquad +\frac{(\aleph _{\varPsi }^{\ell })^{k}}{(2-\aleph _{\varPsi }^{k})^{\ell }+ (1-\aleph _{\varPsi }^{k})^{\ell }} + \frac{(\sigma _{\varPsi }^{\ell })^{m}}{(2-\sigma _{\varPsi }^{m})^{\ell }+ (\sigma _{\varPsi }^{m})^{\ell }}\\{} & {} \quad \le \frac{(1+\rho _{\varPsi }^{n})^{\ell }- (\aleph _{\varPsi }^{k})^{\ell }}{(1+\rho _{\varPsi }^{n})^{\ell }+ (\sigma _{\varPsi }^{m})^{\ell }}+ \frac{(\aleph _{\varPsi }^{\ell })^{k}}{(1+\rho _{\varPsi }^{n})^{\ell }+ (\sigma _{\varPsi }^{m})^{\ell }} \\{} & {} \qquad + \frac{(\sigma _{\varPsi }^{\ell })^{m}}{(1+\rho _{\varPsi }^{n})^{\ell }+ (\sigma _{\varPsi }^{m})^{\ell }} = 1. \end{aligned}$$ -
2.
\(\left( \dfrac{\rho _{\varPsi }^{\ell }}{\left( (2-\rho _{\varPsi }^{n})^{\ell }+ (\rho _{\varPsi }^{n})^{\ell }\right) ^\frac{1}{n}}\right) ^{n}\)
$$\begin{aligned}{} & {} \qquad + \left( \frac{\aleph _{\varPsi }^{\ell }}{\left( (2-\aleph _{\varPsi }^{k})^{\ell }+ (1-\aleph _{\varPsi }^{k})^{\ell }\right) ^\frac{1}{k}}\right) ^{k} \\{} & {} \qquad +\left( \left( \frac{(1+\sigma _{\varPsi }^{m})^{\ell }- (1-\sigma _{\varPsi }^{m})^{\ell }}{(1+\sigma _{\varPsi }^{m})^{\ell }+ (1-\sigma _{\varPsi }^{m})^{\ell }}\right) ^\frac{1}{m}\right) ^{m}\\{} & {} \quad = \frac{(\rho _{\varPsi }^{\ell })^{n}}{(2-\rho _{\varPsi }^{n})^{\ell }+ (\rho _{\varPsi }^{n})^{\ell }} +\frac{(\aleph _{\varPsi }^{\ell })^{k}}{(2-\aleph _{\varPsi }^{k})^{\ell }+ (1-\aleph _{\varPsi }^{k})^{\ell }} \\{} & {} \qquad +\frac{(1+\sigma _{\varPsi }^{m})^{\ell }- (1-\sigma _{\varPsi }^{m})^{\ell }}{(1+\sigma _{\varPsi }^{m})^{\ell }+ (1-\sigma _{\varPsi }^{m})^{\ell }}\\{} & {} \quad \le \frac{(\rho _{\varPsi }^{\ell })^{n}}{(1+\sigma _{\varPsi }^{m})^{\ell }+ (\rho _{\varPsi }^{n})^{\ell }} +\frac{(\aleph _{\varPsi }^{\ell })^{k}}{(1+\sigma _{\varPsi }^{m})^{\ell }+ (\rho _{\varPsi }^{n})^{\ell }} \\{} & {} \qquad + \frac{(1+\sigma _{\varPsi }^{m})^{\ell }- (\aleph _{\varPsi }^{k})^{\ell }}{(1+\sigma _{\varPsi }^{m})^{\ell }+ (\rho _{\varPsi }^{n})^{\ell }} = 1. \end{aligned}$$
It is obvious that
Hence, \(\ell \varPsi \) and \(\varPsi ^{\ell }\) are \(k^{n}_{m}\)-RPFSs. \(\square \)
Theorem 9
Let \(\varPsi = (\rho _{\varPsi _1},\aleph _{\varPsi _1},\sigma _{\varPsi _1})\) and \(\varPsi _2 = (\rho _{\varPsi _2},\aleph _{\varPsi _2},\sigma _{\varPsi _2})\) be \(k^{n}_{m}\)-RPFSs. Then
-
1.
\(\varPsi _1 \oplus \varPsi _2 = \varPsi _2 \oplus \varPsi _1\).
-
2.
\(\varPsi _1 \otimes \varPsi _2 = \varPsi _2 \otimes \varPsi _1\).
Proof
From Definition 7, we can obtain
-
1.
\(\varPsi _1 \oplus \varPsi _2 = \left( \frac{\rho _{\varPsi _1}\rho _{\varPsi _2}}{(1+ \rho _{\varPsi _1}^{n}\rho _{\varPsi _2}^{n})^\frac{1}{n}}, \frac{\aleph _{\varPsi _1}\aleph _{\varPsi _2}}{(1+(1- \aleph _{\varPsi _1}^{k})(1- \aleph _{\varPsi _2}^{k}))^\frac{1}{k}},\right. \left. \qquad \frac{\sigma _{\varPsi _1}\sigma _{\varPsi _2}}{(1+(1- \sigma _{\varPsi _1}^{m})(1- \sigma _{\varPsi _2}^{m}))^\frac{1}{m}}\right) = \left( \frac{\rho _{\varPsi _2}\rho _{\varPsi _1}}{(1+ \rho _{\varPsi _2}^{n}\rho _{\varPsi _1}^{n})^\frac{1}{n}},\right. \left. \qquad \frac{\aleph _{\varPsi _2}\aleph _{\varPsi _1}}{(1+(1- \aleph _{\varPsi _2}^{k})(1- \aleph _{\varPsi _1}^{k}))^\frac{1}{k}},\frac{\sigma _{\varPsi _2}\sigma _{\varPsi _1}}{(1+(1- \sigma _{\varPsi _2}^{m})(1- \sigma _{\varPsi _1}^{m}))^\frac{1}{m}}\right) = \varPsi _2 \oplus \varPsi _1\).
-
2.
\(\varPsi _1 \otimes \varPsi _2 = \left( \frac{\rho _{\varPsi _1}\rho _{\varPsi _2}}{(1+(1- \rho _{\varPsi _1}^{n})(1- \rho _{\varPsi _2}^{n}))^\frac{1}{n}}, \frac{\aleph _{\varPsi _1}\aleph _{\varPsi _2}}{(1+(1- \aleph _{\varPsi _1}^{k})(1- \aleph _{\varPsi _2}^{k}))^\frac{1}{k}},\right. \left. \qquad \frac{\sigma _{\varPsi _1}\sigma _{\varPsi _2}}{(1+ \sigma _{\varPsi _1}^{m}\sigma _{\varPsi _2}^{m})^\frac{1}{m}}\right) = \left( \frac{\rho _{\varPsi _2}\rho _{\varPsi _1}}{(1+(1- \rho _{\varPsi _2}^{n})(1- \rho _{\varPsi _1}^{n}))^\frac{1}{n}},\right. \left. \qquad \frac{\aleph _{\varPsi _2}\aleph _{\varPsi _1}}{(1+(1- \aleph _{\varPsi _2}^{k})(1- \aleph _{\varPsi _1}^{k}))^\frac{1}{k}}, \frac{\sigma _{\varPsi _2}\sigma _{\varPsi _1}}{(1+ \sigma _{\varPsi _2}^{m}\sigma _{\varPsi _1}^{m})^\frac{1}{m}}\right) = \varPsi _2 \otimes \varPsi _1\).
\(\square \)
Theorem 10
Let \(\varPsi _1 = (\rho _{\varPsi _1},\aleph _{\varPsi _1},\sigma _{\varPsi _1})\), \(\varPsi _2 = (\rho _{\varPsi _2},\aleph _{\varPsi _2}, \sigma _{\varPsi _2})\) and \(\varPsi _3 = (\rho _{\varPsi _3},\aleph _{\varPsi _3},\sigma _{\varPsi _3})\) be \(k^{n}_{m}\)-RPFSs. Then
-
1.
\(\varPsi _1\oplus \varPsi _2\oplus \varPsi _3= \varPsi _2\oplus \varPsi _1\oplus \varPsi _3\).
-
2.
\(\varPsi _1\otimes \varPsi _2\otimes \varPsi _3= \varPsi _2\otimes \varPsi _1\otimes \varPsi _3\).
Proof
(1) \(\varPsi _1\oplus \varPsi _2\oplus \varPsi _3 = (\rho _{\varPsi _1},\aleph _{\varPsi _1},\sigma _{\varPsi _1})\oplus (\rho _{\varPsi _2},\aleph _{\varPsi _2},\sigma _{\varPsi _2})\oplus \)
where
and
(2) We can perform proof similarly to (1). \(\square \)
Theorem 11
Let \(\varPsi = (\rho _{\varPsi },\aleph _{\varPsi },\sigma _{\varPsi })\), \(\varPsi _1 = (\rho _{\varPsi _1},\aleph _{\varPsi _1},\sigma _{\varPsi _1})\) and \(\varPsi _2 = (\rho _{\varPsi _2},\aleph _{\varPsi _2},\sigma _{\varPsi _2})\) be \(k^{n}_{m}\)-RPFSs, and \(\ell > 0\). Then
-
1.
\((\varPsi _1 \oplus \varPsi _2)^{c} = \varPsi _1^{c} \otimes \varPsi _2^{c}\).
-
2.
\((\varPsi _1 \otimes \varPsi _2)^{c} = \varPsi _1^{c} \oplus \varPsi _2^{c}\).
-
3.
\((\varPsi ^{c})^{\ell } = (\ell \varPsi )^{c}\).
-
4.
\(\ell (\varPsi )^{c} = (\varPsi ^{\ell })^{c}\).
Proof
For the \(k^{n}_{m}\)-RPFSs \(\varPsi , \varPsi _1, \varPsi _2\), and \(\ell > 0\), based on Definitions 6 and 7, we have
-
1.
\((\varPsi _1 \oplus \varPsi _2)^{c}\)
$$\begin{aligned}{} & {} =\left( \frac{\rho _{\varPsi _1}\rho _{\varPsi _2}}{(1+ \rho _{\varPsi _1}^{n}\rho _{\varPsi _2}^{n})^\frac{1}{n}},\frac{\aleph _{\varPsi _1}\aleph _{\varPsi _2}}{(1+(1- \aleph _{\varPsi _1}^{k})(1- \aleph _{\varPsi _2}^{k}))^\frac{1}{k}},\right. \\{} & {} \left. \quad \frac{\sigma _{\varPsi _1}\sigma _{\varPsi _2}}{(1+(1- \sigma _{\varPsi _1}^{m})(1-\sigma _{\varPsi _2}^{m}))^\frac{1}{m}}\right) ^{c}\\{} & {} \quad = \left( \left( \frac{\sigma _{\varPsi _1}\sigma _{\varPsi _2}}{(1+(1- \sigma _{\varPsi _1}^{m})(1- \sigma _{\varPsi _2}^{m}))^\frac{1}{m}}\right) ^\frac{m}{n},\right. \\{} & {} \left. \quad \frac{\aleph _{\varPsi _1}\aleph _{\varPsi _2}}{(1+(1- \aleph _{\varPsi _1}^{k})(1- \aleph _{\varPsi _2}^{k}))^\frac{1}{k}}, \left( \frac{\rho _{\varPsi _1}\rho _{\varPsi _2}}{(1+ \rho _{\varPsi _1}^{n}\rho _{\varPsi _2}^{n})^\frac{1}{n}}\right) ^\frac{n}{m}\right) \\{} & {} \quad = \left( \frac{\sigma _{\varPsi _1}^\frac{m}{n}\sigma _{\varPsi _2}^\frac{m}{n}}{(1+(1- \sigma _{\varPsi _1}^{m})(1- \sigma _{\varPsi _2}^{m}))^\frac{1}{n}},\right. \\{} & {} \left. \quad \frac{\aleph _{\varPsi _1}\aleph _{\varPsi _2}}{(1+(1- \aleph _{\varPsi _1}^{k})(1- \aleph _{\varPsi _2}^{k}))^\frac{1}{k}}, \frac{\rho _{\varPsi _1}^\frac{n}{m}\rho _{\varPsi _2}^\frac{n}{m}}{(1+ \rho _{\varPsi _1}^{n}\rho _{\varPsi _2}^{n})^\frac{1}{m}}\right) \\{} & {} \quad = \left( (\sigma _{\varPsi _1})^\frac{m}{n},\aleph _{\varPsi _1},(\rho _{\varPsi _1})^\frac{n}{m}\right) \otimes \left( (\sigma _{\varPsi _2})^\frac{m}{n},\aleph _{\varPsi _2},(\rho _{\varPsi _2})^\frac{n}{m}\right) \\{} & {} \quad = \varPsi _1^{c} \otimes \varPsi _2^{c}. \end{aligned}$$ -
2.
\((\varPsi _1 \otimes \varPsi _2)^{c} = \left( \dfrac{\rho _{\varPsi _1}\rho _{\varPsi _2}}{(1+(1- \rho _{\varPsi _1}^{n})(1- \rho _{\varPsi _2}^{n}))^\frac{1}{n}},\right. \)
$$\begin{aligned}{} & {} \left. \quad \frac{\aleph _{\varPsi _1}\aleph _{\varPsi _2}}{(1+(1- \aleph _{\varPsi _1}^{k})(1- \aleph _{\varPsi _2}^{k}))^\frac{1}{k}},\frac{\sigma _{\varPsi _1}\sigma _{\varPsi _2}}{(1+ \sigma _{\varPsi _1}^{m}\sigma _{\varPsi _2}^{m})^\frac{1}{m}}\right) ^{c}\\{} & {} \quad =\left( \left( \frac{\sigma _{\varPsi _1}\sigma _{\varPsi _2}}{(1+ \sigma _{\varPsi _1}^{m}\sigma _{\varPsi _2}^{m})^\frac{1}{m}}\right) ^\frac{m}{n}, \frac{\aleph _{\varPsi _1}\aleph _{\varPsi _2}}{(1+(1- \aleph _{\varPsi _1}^{k})(1- \aleph _{\varPsi _2}^{k}))^\frac{1}{k}},\right. \\{} & {} \qquad \left. \left( \frac{\rho _{\varPsi _1}\rho _{\varPsi _2}}{(1+(1- \rho _{\varPsi _1}^{n})(1- \rho _{\varPsi _2}^{n}))^\frac{1}{n}}\right) ^\frac{n}{m}\right) \\{} & {} \quad =\left( \frac{\sigma _{\varPsi _1}^\frac{m}{n}\sigma _{\varPsi _2}^\frac{m}{n}}{(1+ \sigma _{\varPsi _1}^{m}\sigma _{\varPsi _1}^{m})^\frac{1}{n}}, \frac{\aleph _{\varPsi _1}\aleph _{\varPsi _2}}{(1+(1- \aleph _{\varPsi _1}^{k})(1- \aleph _{\varPsi _2}^{k}))^\frac{1}{k}},\right. \\{} & {} \qquad \left. \frac{\rho _{\varPsi _1}^\frac{n}{m}\rho _{\varPsi _2}^\frac{n}{m}}{(1+(1- \rho _{\varPsi _1}^{n})(1- \rho _{\varPsi _2}^{n}))^\frac{1}{m}}\right) \\{} & {} \quad = \left( (\sigma _{\varPsi _1})^\frac{m}{n},\aleph _{\varPsi _1},(\rho _{\varPsi _1})^\frac{n}{m}\right) \oplus \left( (\sigma _{\varPsi _2})^\frac{m}{n},\aleph _{\varPsi _2},(\rho _{\varPsi _2})^\frac{n}{m}\right) \\{} & {} \quad = \varPsi _1^{c} \oplus \varPsi _2^{c}. \end{aligned}$$ -
3.
\((\varPsi ^{c})^{\ell } =((\sigma _{\varPsi })^\frac{m}{n},\aleph _{\varPsi },(\rho _{\varPsi })^\frac{n}{m})^{\ell }\)
$$\begin{aligned}{} & {} \quad =\left( \frac{(\sigma _{\varPsi }^\frac{m}{n})^{\ell }}{((2-\sigma _{\varPsi }^{m})^{\ell }+ (\sigma _{\varPsi }^{m})^{\ell })^\frac{1}{n}},\frac{\aleph _{\varPsi }^{\ell }}{((2-\aleph _{\varPsi }^{k})^{\ell }+ (1-\aleph _{\varPsi }^{k})^{\ell })^\frac{1}{k}},\right. \\{} & {} \qquad \left. \left( \frac{(1+\rho _{\varPsi }^{n})^{\ell }- (1-\rho _{\varPsi }^{n})^{\ell }}{(1+\rho _{\varPsi }^{n})^{\ell }+ (1-\rho _{\varPsi }^{n})^{\ell }}\right) ^\frac{1}{m}\right) \\{} & {} \quad =\left( \left( \frac{\sigma _{\varPsi }^{\ell }}{((2-\sigma _{\varPsi }^{m})^{\ell }+ (\sigma _{\varPsi }^{m})^{\ell })^\frac{1}{m}}\right) ^\frac{m}{n},\right. \\{} & {} \left. \quad \frac{\aleph _{\varPsi }^{\ell }}{((2-\aleph _{\varPsi }^{k})^{\ell }+ (1-\aleph _{\varPsi }^{k})^{\ell })^\frac{1}{k}},\right. \\{} & {} \qquad \left. \left( \left( \frac{(1+\rho _{\varPsi }^{n})^{\ell }- (1-\rho _{\varPsi }^{n})^{\ell }}{(1+\rho _{\varPsi }^{n})^{\ell }+ (1-\rho _{\varPsi }^{n})^{\ell }}\right) ^\frac{1}{n}\right) ^\frac{n}{m}\right) \\{} & {} \quad =\left( \left( \frac{(1+\rho _{\varPsi }^{n})^{\ell }- (1-\rho _{\varPsi }^{n})^{\ell }}{(1+\rho _{\varPsi }^{n})^{\ell }+ (1-\rho _{\varPsi }^{n})^{\ell }}\right) ^\frac{1}{n},\right. \\{} & {} \qquad \left. \frac{\aleph _{\varPsi }^{\ell }}{((2-\aleph _{\varPsi }^{k})^{\ell }+ (1-\aleph _{\varPsi }^{k})^{\ell })^\frac{1}{k}}, \frac{\sigma _{\varPsi }^{\ell }}{((2-\sigma _{\varPsi }^{m})^{\ell }+ (\sigma _{\varPsi }^{m})^{\ell })^\frac{1}{m}}\right) ^{c}\\{} & {} \quad =(\ell \varPsi )^{c}. \end{aligned}$$ -
4.
\(\ell (\varPsi )^{c}= \ell \left( (\sigma _{\varPsi })^\frac{m}{n},\aleph _{\varPsi },(\rho _{\varPsi })^\frac{n}{m}\right) \)
$$\begin{aligned}{} & {} \quad =\left( \left( \frac{(1+\sigma _{\varPsi }^{m})^{\ell }- (1-\sigma _{\varPsi }^{m})^{\ell }}{(1+\sigma _{\varPsi }^{m})^{\ell }+ (1-\sigma _{\varPsi }^{m})^{\ell }}\right) ^\frac{1}{n},\right. \\{} & {} \qquad \left. \frac{\aleph _{\varPsi }^{\ell }}{((2-\aleph _{\varPsi }^{k})^{\ell }+ (1-\aleph _{\varPsi }^{k})^{\ell })^\frac{1}{k}},\right. \\{} & {} \left. \quad \frac{(\rho _{\varPsi }^\frac{n}{m})^{\ell }}{((2-\rho _{\varPsi }^{n})^{\ell }+ (\rho _{\varPsi }^{n})^{\ell })^\frac{1}{m}}\right) \\{} & {} \quad =\left( \left( \frac{(1+\sigma _{\varPsi }^{m})^{\ell }- (1-\sigma _{\varPsi }^{m})^{\ell }}{(1+\sigma _{\varPsi }^{m})^{\ell }+ (1-\sigma _{\varPsi }^{m})^{\ell }}\right) ^\frac{1}{n},\right. \\{} & {} \qquad \left. \frac{\aleph _{\varPsi }^{\ell }}{\left( (2-\aleph _{\varPsi }^{k})^{\ell }+ (1-\aleph _{\varPsi }^{k})^{\ell }\right) ^\frac{1}{k}},\right. \\{} & {} \left. \quad \frac{(\rho _{\varPsi }^{\ell })^\frac{n}{m}}{((2-\rho _{\varPsi }^{n})^{\ell }+ (\rho _{\varPsi }^{n})^{\ell })^\frac{1}{m}}\right) \\{} & {} \quad =\left( \frac{\rho _{\varPsi }^{\ell }}{((2-\rho _{\varPsi }^{n})^{\ell }+ (\rho _{\varPsi }^{n})^{\ell })^\frac{1}{n}},\right. \\{} & {} \left. \quad \frac{\aleph _{\varPsi }^{\ell }}{((2-\aleph _{\varPsi }^{k})^{\ell }+ (1-\aleph _{\varPsi }^{k})^{\ell })^\frac{1}{k}},\right. \\{} & {} \qquad \left. \left( \frac{(1+\sigma _{\varPsi }^{m})^{\ell }- (1-\sigma _{\varPsi }^{m})^{\ell }}{(1+\sigma _{\varPsi }^{m})^{\ell }+ (1-\sigma _{\varPsi }^{m})^{\ell }}\right) ^\frac{1}{m}\right) ^{c}\\{} & {} \quad = (\varPsi ^{\ell })^{c}. \end{aligned}$$
\(\square \)
Theorem 12
Let \(\varPsi _1 = (\rho _{\varPsi _1},\aleph _{\varPsi _1},\sigma _{\varPsi _1})\), \(\varPsi _2 = (\rho _{\varPsi _2},\aleph _{\varPsi _2}, \sigma _{\varPsi _2})\) and \(\varPsi _3 = (\rho _{\varPsi _3},\aleph _{\varPsi _3},\sigma _{\varPsi _3})\) be \(k^{n}_{m}\)-RPFSs. Then
-
1.
\((\varPsi _1\bigwedge \varPsi _2)\oplus \varPsi _3=(\varPsi _1\oplus \varPsi _3)\bigwedge (\varPsi _2\oplus \varPsi _3)\).
-
2.
\((\varPsi _1\bigvee \varPsi _2)\oplus \varPsi _3=(\varPsi _1\oplus \varPsi _3)\bigvee (\varPsi _2\oplus \varPsi _3)\).
-
3.
\((\varPsi _1\bigwedge \varPsi _2)\otimes \varPsi _3=(\varPsi _1\otimes \varPsi _3)\bigwedge (\varPsi _2\otimes \varPsi _3)\).
-
4.
\((\varPsi _1\bigvee \varPsi _2)\otimes \varPsi _3=(\varPsi _1\otimes \varPsi _3)\bigvee (\varPsi _2\otimes \varPsi _3)\).
Proof
We will provide proof to back up our claim (1). The other claims are verified similarly. From Definitions 6 and 7, we have
where
Now
If \(\rho _{\varPsi _1}\le \rho _{\varPsi _2}\), then \(\rho _{\varPsi _1}^{n}\le \rho _{\varPsi _2}^{n}\), and hence
Similarly, if \(\rho _{\varPsi _2}\le \rho _{\varPsi _1}\), then
If \(\aleph _{\varPsi _1}\le \aleph _{\varPsi _2}\), then \(\aleph _{\varPsi _1}^{k}\le \aleph _{\varPsi _2}^{k}\) and \(\aleph _{\varPsi _2}^{k} -\aleph _{\varPsi _1}^{k}\ge 0\). Since \(\aleph _{\varPsi _3}^{k}-1\le 1\), then
Similarly
-
1.
if \(\aleph _{\varPsi _2}\le \aleph _{\varPsi _1}\), then \( \frac{\aleph _{\varPsi _2}\aleph _{\varPsi _3}}{(1 + (1-\aleph _{\varPsi _2}^{k})(1-\aleph _{\varPsi _3}^{k}))^\frac{1}{k}} \le \frac{\aleph _{\varPsi _1}\aleph _{\varPsi _3}}{(1 + (1 -\aleph _{\varPsi _1}^{k})(1-\aleph _{\varPsi _3}^{k}))^\frac{1}{k}}\),
-
2.
if \(\sigma _{\varPsi _1}\le \sigma _{\varPsi _2}\), then \(\frac{\sigma _{\varPsi _1}\sigma _{\varPsi _3}}{(1 + (1 -\sigma _{\varPsi _1}^{m})(1-\sigma _{\varPsi _3}^{m}))^\frac{1}{m}}\le \frac{\sigma _{\varPsi _2}\sigma _{\varPsi _3}}{(1 + (1-\sigma _{\varPsi _2}^{m})(1-\sigma _{\varPsi _3}^{m}))^\frac{1}{m}}\),
-
3.
if \(\sigma _{\varPsi _2}\le \sigma _{\varPsi _1}\), then \( \frac{\sigma _{\varPsi _2}\sigma _{\varPsi _3}}{(1 + (1-\sigma _{\varPsi _2}^{m})(1-\sigma _{\varPsi _3}^{m}))^\frac{1}{m}} \le \frac{\sigma _{\varPsi _1}\sigma _{\varPsi _3}}{(1 + (1 -\sigma _{\varPsi _1}^{m})(1-\sigma _{\varPsi _3}^{m}))^\frac{1}{m}}\).
Therefore
Thus, \((\varPsi _1\bigwedge \varPsi _2)\oplus \varPsi _3=(\varPsi _1\oplus \varPsi _3)\bigwedge (\varPsi _2\oplus \varPsi _3)\). \(\square \)
Theorem 13
Let \(\varPsi _1 = (\rho _{\varPsi _1},\aleph _{\varPsi _1},\sigma _{\varPsi _1})\) and \(\varPsi _2 = (\rho _{\varPsi _2},\aleph _{\varPsi _2},\sigma _{\varPsi _2})\) be \(k^{n}_{m}\)-RPFSs, and \(\ell > 0\). Then
-
1.
\(\ell (\varPsi _1\bigvee \varPsi _2) = \ell \varPsi _1\bigvee \ell \varPsi _2\).
-
2.
\((\varPsi _1\bigvee \varPsi _2)^{\ell } = \varPsi _1^{\ell }\bigvee \varPsi _2^{\ell }\).
Proof
For the \(k^{n}_{m}\)-RPFSs \(\varPsi _1, \varPsi _2\), and \(\ell > 0\), based on Definitions 6 and 7, we have
-
1.
\(\ell (\varPsi _1\bigvee \varPsi _2)= \ell (\max \{\rho _{\varPsi _1},\rho _{\varPsi _2}\},\min \{\aleph _{\varPsi _1},\aleph _{\varPsi _2}\}\),
$$\begin{aligned}{} & {} \min \{\sigma _{\varPsi _1},\sigma _{\varPsi _2}\})\\{} & {} \quad = \left( \left( \frac{(1+\max \{\rho _{\varPsi _1},\rho _{\varPsi _2}\}^{n})^{\ell }- (1-\max \{\rho _{\varPsi _1},\rho _{\varPsi _2}\}^{n})^{\ell }}{(1+\max \{\rho _{\varPsi _1},\rho _{\varPsi _2}\}^{n})^{\ell }+ (1-\max \{\rho _{\varPsi _1},\rho _{\varPsi _2}\}^{n})^{\ell }}\right) ^\frac{1}{n}, \right. \\{} & {} \quad \left. \frac{\min \{\aleph _{\varPsi _1},\aleph _{\varPsi _2}\}^{\ell }}{((2-\min \{\aleph _{\varPsi _1},\aleph _{\varPsi _2}\}^{k})^{\ell }+ (1-\min \{\aleph _{\varPsi _1},\aleph _{\varPsi _2}\}^{k})^{\ell })^\frac{1}{k}},\right. \\{} & {} \qquad \left. \frac{\min \{\sigma _{\varPsi _1},\sigma _{\varPsi _2}\}^{\ell }}{((2-\min \{\sigma _{\varPsi _1},\sigma _{\varPsi _2}\}^{m})^{\ell }+ (\min \{\sigma _{\varPsi _1},\sigma _{\varPsi _2}\}^{m})^{\ell })^\frac{1}{m}}\right) , \end{aligned}$$where
$$\begin{aligned}{} & {} \left( \frac{(1+\max \{\rho _{\varPsi _1},\rho _{\varPsi _2}\}^{n})^{\ell }- (1-\max \{\rho _{\varPsi _1},\rho _{\varPsi _2}\}^{n})^{\ell }}{(1+\max \{\rho _{\varPsi _1},\rho _{\varPsi _2}\}^{n})^{\ell }+ (1-\max \{\rho _{\varPsi _1},\rho _{\varPsi _2}\}^{n})^{\ell }}\right) ^\frac{1}{n}\\{} & {} \quad =\left\{ \begin{array}{lll} \frac{(1+\rho _{\varPsi _2}^{n})^{\ell }- (1-\rho _{\varPsi _2}^{n})^{\ell }}{(1+\rho _{\varPsi _2}^{n})^{\ell }+ (1+\rho _{\varPsi _2}^{n})^{\ell }})^\frac{1}{n} &{} \text{ for }\, \rho _{\varPsi _1} \le \rho _{\varPsi _2},\\ \frac{(1+\rho _{\varPsi _1}^{n})^{\ell }- (1-\rho _{\varPsi _1}^{n})^{\ell }}{(1+\rho _{\varPsi _1}^{n})^{\ell }+ (1+\rho _{\varPsi _1}^{n})^{\ell }})^\frac{1}{n} &{} \text{ for }\, \rho _{\varPsi _2} \le \rho _{\varPsi _1}, \end{array}\right. \\{} & {} \quad \frac{\min \{\aleph _{\varPsi _1},\aleph _{\varPsi _2}\}^{\ell }}{((2-\min \{\aleph _{\varPsi _1},\aleph _{\varPsi _2}\}^{k})^{\ell }+ (1-\min \{\aleph _{\varPsi _1},\aleph _{\varPsi _2}\}^{k})^{\ell })^\frac{1}{k}}\\{} & {} \quad =\left\{ \begin{array}{lll} \frac{\aleph _{\varPsi _1}^{\ell }}{((2-\aleph _{\varPsi _1}^{k})^{\ell }+ (1-\aleph _{\varPsi _1}^{k})^{\ell })^\frac{1}{k}} &{} \text{ for } \aleph _{\varPsi _1} \le \aleph _{\varPsi _2},\\ \frac{\aleph _{\varPsi _2}^{\ell }}{((2-\aleph _{\varPsi _2}^{k})^{\ell }+ (1-\aleph _{\varPsi _2}^{k})^{\ell })^\frac{1}{k}} &{} \text{ for }\, \aleph _{\varPsi _2} \le \aleph _{\varPsi _1}, \end{array}\right. \\{} & {} \quad \frac{\min \{\sigma _{\varPsi _1},\sigma _{\varPsi _2}\}^{\ell }}{((2-\min \{\sigma _{\varPsi _1},\sigma _{\varPsi _2}\}^{m})^{\ell }+ (\min \{\sigma _{\varPsi _1},\sigma _{\varPsi _2}\}^{m})^{\ell })^\frac{1}{m}} \\{} & {} \quad =\left\{ \begin{array}{lll} \frac{\sigma _{\varPsi _1}^{\ell }}{((2-\sigma _{\varPsi _1}^{m})^{\ell }+ (\sigma _{\varPsi _1}^{m})^{\ell })^\frac{1}{m}} &{} \text{ for }\, \sigma _{\varPsi _1} \le \sigma _{\varPsi _2},\\ \frac{\sigma _{\varPsi _2}^{\ell }}{((2-\sigma _{\varPsi _2}^{m})^{\ell }+ (\sigma _{\varPsi _2}^{m})^{\ell })^\frac{1}{m}} &{} \text{ for }\, \sigma _{\varPsi _2} \le \sigma _{\varPsi _1}. \end{array}\right. \end{aligned}$$Now
$$\begin{aligned} \ell \varPsi _1\bigvee \ell \varPsi _2= & {} \left( \left( \frac{(1+\rho _{\varPsi _1}^{n})^{\ell }- (1-\rho _{\varPsi _1}^{n})^{\ell }}{(1+\rho _{\varPsi _1}^{n})^{\ell }+ (1-\rho _{\varPsi _1}^{n})^{\ell }}\right) ^\frac{1}{n},\right. \\{} & {} \left. \frac{\aleph _{\varPsi _1}^{\ell }}{((2-\aleph _{\varPsi _1}^{k})^{\ell }+ (1-\aleph _{\varPsi _1}^{k})^{\ell })^\frac{1}{k}},\right. \\{} & {} \left. \frac{\sigma _{\varPsi _1}^{\ell }}{((2-\sigma _{\varPsi _1}^{m})^{\ell }+ (\sigma _{\varPsi _1}^{m})^{\ell })^\frac{1}{m}}\right) \\{} & {} \bigvee \left( \left( \frac{(1+\rho _{\varPsi _2}^{n})^{\ell }- (1-\rho _{\varPsi _2}^{n})^{\ell }}{(1+\rho _{\varPsi _2}^{n})^{\ell }+ (1-\rho _{\varPsi _2}^{n})^{\ell }}\right) ^\frac{1}{n},\right. \\{} & {} \left. \frac{\aleph _{\varPsi _2}^{\ell }}{((2-\aleph _{\varPsi _2}^{k})^{\ell }+ (1-\aleph _{\varPsi _2}^{k})^{\ell })^\frac{1}{k}},\right. \\{} & {} \left. \frac{\sigma _{\varPsi _2}^{\ell }}{((2-\sigma _{\varPsi _2}^{m})^{\ell }+ (\sigma _{\varPsi _2}^{m})^{\ell })^\frac{1}{m}}\right) \\{} & {} =\left( \max \left\{ \left( \frac{(1+\rho _{\varPsi _1}^{n})^{\ell }- (1-\rho _{\varPsi _1}^{n})^{\ell }}{(1+\rho _{\varPsi _1}^{n})^{\ell }+ (1-\rho _{\varPsi _1}^{n})^{\ell }}\right) ^\frac{1}{n},\right. \right. \\{} & {} \left. \left. \left( \frac{(1+\rho _{\varPsi _2}^{n})^{\ell }- (1-\rho _{\varPsi _2}^{n})^{\ell }}{(1+\rho _{\varPsi _2}^{n})^{\ell }+ (1-\rho _{\varPsi _2}^{n})^{\ell }}\right) ^\frac{1}{n}\right\} ,\right. \\{} & {} \left. \min \left\{ \frac{\aleph _{\varPsi _1}^{\ell }}{((2-\aleph _{\varPsi _1}^{k})^{\ell }+ (1-\aleph _{\varPsi _1}^{k})^{\ell })^\frac{1}{k}},\right. \right. \\{} & {} \left. \left. \frac{\aleph _{\varPsi _2}^{\ell }}{((2-\aleph _{\varPsi _2}^{k})^{\ell }+ (1-\aleph _{\varPsi _2}^{k})^{\ell })^\frac{1}{k}}\right\} , \right. \\{} & {} \left. \min \left\{ \frac{\sigma _{\varPsi _1}^{\ell }}{((2-\sigma _{\varPsi _1}^{m})^{\ell }+ (\sigma _{\varPsi _1}^{m})^{\ell })^\frac{1}{m}}, \right. \right. \\{} & {} \left. \left. \frac{\sigma _{\varPsi _2}^{\ell }}{((2-\sigma _{\varPsi _2}^{m})^{\ell }+ (\sigma _{\varPsi _2}^{m})^{\ell })^\frac{1}{m}}\right\} \right) . \end{aligned}$$If \(\rho _{\varPsi _1}\le \rho _{\varPsi _2}\), then \(2\rho _{\varPsi _1}^{n} \le 2\rho _{\varPsi _2}^{n}\), and hence
$$\begin{aligned}{} & {} (1+\rho _{\varPsi _1}^{n}) (1-\rho _{\varPsi _2}^{n})\le (1+\rho _{\varPsi _2}^{n}) (1-\rho _{\varPsi _1}^{n})\\{} & {} \quad \Rightarrow (1+\rho _{\varPsi _1}^{n})^{\ell } (1-\rho _{\varPsi _2}^{n})^{\ell }\le (1+\rho _{\varPsi _2}^{n})^{\ell } (1-\rho _{\varPsi _1}^{n})^{\ell }\\{} & {} \quad \Rightarrow ad\le b c\\{} & {} \quad \Rightarrow 2ad\le 2b c\\{} & {} \quad \Rightarrow (a-c)(b + d)\le (b - d) (a+c)\\{} & {} \quad \Rightarrow ((1+\rho _{\varPsi _1}^{n})^{\ell }- (1-\rho _{\varPsi _1}^{n})^{\ell })((1+\rho _{\varPsi _2}^{n})^{\ell } + (1-\rho _{\varPsi _2}^{n})^{\ell })\\{} & {} \quad \le ((1+\rho _{\varPsi _2}^{n})^{\ell } - (1-\rho _{\varPsi _2}^{n})^{\ell }) ((1+\rho _{\varPsi _1}^{n})^{\ell } + (1-\rho _{\varPsi _1}^{n})^{\ell })\\{} & {} \quad \Rightarrow \left( \frac{(1+\rho _{\varPsi _1}^{n})^{\ell }- (1-\rho _{\varPsi _1}^{n})^{\ell }}{(1+\rho _{\varPsi _1}^{n})^{\ell } + (1-\rho _{\varPsi _1}^{n})^{\ell }}\right) ^\frac{1}{n} \\{} & {} \qquad \le \left( \frac{(1+\rho _{\varPsi _2}^{n})^{\ell } -(1-\rho _{\varPsi _2}^{n})^{\ell }}{(1+\rho _{\varPsi _2}^{n})^{\ell } + (1-\rho _{\varPsi _2}^{n})^{\ell }}\right) ^\frac{1}{n} . \end{aligned}$$Similarly, if \(\rho _{\varPsi _2}\le \rho _{\varPsi _1}\) then \( (\frac{(1+\rho _{\varPsi _2}^{n})^{\ell } - (1-\rho _{\varPsi _2}^{n})^{\ell }}{(1+\rho _{\varPsi _2}^{n})^{\ell } + (1-\rho _{\varPsi _2}^{n})^{\ell }})^\frac{1}{n} \le (\frac{(1+\rho _{\varPsi _1}^{n})^{\ell }- (1-\rho _{\varPsi _1}^{n})^{\ell }}{(1+\rho _{\varPsi _1}^{n})^{\ell } + (1-\rho _{\varPsi _1}^{n})^{\ell }})^\frac{1}{n}\), where, \(a= (1+\rho _{\varPsi _1}^{n})^{\ell }\), \(d=(1-\rho _{\varPsi _2}^{n})^{\ell }\), \(b=(1+\rho _{\varPsi _2}^{n})^{\ell }\) and \(c=(1-\rho _{\varPsi _1}^{n})^{\ell }\).
If \(\aleph _{\varPsi _1}\le \aleph _{\varPsi _2}\), then \(\aleph _{\varPsi _1}^{k}\le \aleph _{\varPsi _2}^{k}\) and hence \(-\aleph _{\varPsi _1}^{k}\ge -\aleph _{\varPsi _2}^{k}\) implies \((2-\aleph _{\varPsi _1}^{k})^{\ell }\ge (2-\aleph _{\varPsi _2}^{k})^{\ell }\) and \((1-\aleph _{\varPsi _1}^{k})^{\ell }\ge (1-\aleph _{\varPsi _2}^{k})^{\ell }\)
$$\begin{aligned}{} & {} \Rightarrow (2-\aleph _{\varPsi _1}^{k})^{\ell } +(1-\aleph _{\varPsi _1}^{k})^{\ell }\ge (2-\aleph _{\varPsi _2}^{k})^{\ell } + (1-\aleph _{\varPsi _2}^{k})^{\ell }\\{} & {} \quad \Rightarrow \frac{\aleph _{\varPsi _2}^{\ell }}{((2-\aleph _{\varPsi _2}^{k})^{\ell } + (1-\aleph _{\varPsi _2}^{k})^{\ell })^\frac{1}{k}}\\{} & {} \quad \ge \frac{\aleph _{\varPsi _1}^{\ell }}{((2-\aleph _{\varPsi _2}^{k})^{\ell } + (1-\aleph _{\varPsi _2}^{k})^{\ell })^\frac{1}{k}}\\{} & {} \quad \ge \frac{\aleph _{\varPsi _1}^{\ell }}{((2-\aleph _{\varPsi _1}^{k})^{\ell } +(1-\aleph _{\varPsi _1}^{k})^{\ell })^\frac{1}{k}}\\{} & {} \quad \Rightarrow \frac{\aleph _{\varPsi _1}^{\ell }}{((2-\aleph _{\varPsi _1}^{k})^{\ell }+ (1-\aleph _{\varPsi _1}^{k})^{\ell })^\frac{1}{k}}\\{} & {} \quad \le \frac{\aleph _{\varPsi _2}^{\ell }}{((2-\aleph _{\varPsi _2}^{k})^{\ell }+ (1-\aleph _{\varPsi _2}^{k})^{\ell })^\frac{1}{k}}. \end{aligned}$$Similarly, if \(\aleph _{\varPsi _2}\le \aleph _{\varPsi _1}\), then \( \frac{\aleph _{\varPsi _1}^{\ell }}{((2-\aleph _{\varPsi _1}^{k})^{\ell }+ (1-\aleph _{\varPsi _1}^{k})^{\ell })^\frac{1}{k}}\ge \frac{\aleph _{\varPsi _2}^{\ell }}{((2-\aleph _{\varPsi _2}^{k})^{\ell }+ (1-\aleph _{\varPsi _2}^{k})^{\ell })^\frac{1}{k}}\).
If \(\sigma _{\varPsi _1}\le \sigma _{\varPsi _2}\), then \(\sigma _{\varPsi _1}^{m}\le \sigma _{\varPsi _2}^{m}\), and hence
$$\begin{aligned}{} & {} (2\sigma _{\varPsi _1}^{m}-\sigma _{\varPsi _1}^{m}\sigma _{\varPsi _2}^{m})^{\ell }\le (2\sigma _{\varPsi _2}^{m}-\sigma _{\varPsi _1}^{m}\sigma _{\varPsi _2}^{m})^{\ell }\\{} & {} \quad \Rightarrow (\sigma _{\varPsi _1}^{m})^{\ell }(2-\sigma _{\varPsi _2}^{m})^{\ell }\le (\sigma _{\varPsi _2}^{m})^{\ell }(2-\sigma _{\varPsi _1}^{m})^{\ell }\\{} & {} \quad \Rightarrow (\sigma _{\varPsi _1}^{m})^{\ell }(2-\sigma _{\varPsi _2}^{m})^{\ell }+ (\sigma _{\varPsi _1}^{m}\sigma _{\varPsi _2}^{m})^{\ell }\\{} & {} \quad \le (\sigma _{\varPsi _2}^{m})^{\ell }(2-\sigma _{\varPsi _1}^{m})^{\ell } + (\sigma _{\varPsi _1}^{m}\sigma _{\varPsi _2}^{m})^{\ell }\\{} & {} \quad \Rightarrow (\sigma _{\varPsi _1}^{\ell })^{m}((2-\sigma _{\varPsi _2}^{m})^{\ell }+ (\sigma _{\varPsi _2}^{m})^{\ell })\\{} & {} \quad \le (\sigma _{\varPsi _2}^{\ell })^{m}((2-\sigma _{\varPsi _1}^{m})^{\ell }+ (\sigma _{\varPsi _1}^{m})^{\ell })\\{} & {} \quad \Rightarrow \frac{\sigma _{\varPsi _1}^{\ell }}{((2-\sigma _{\varPsi _1}^{m})^{\ell }+ (\sigma _{\varPsi _1}^{m})^{\ell })^\frac{1}{m}}\\{} & {} \quad \le \frac{\sigma _{\varPsi _2}^{\ell }}{((2-\sigma _{\varPsi _2}^{m})^{\ell }+ (\sigma _{\varPsi _2}^{m})^{\ell })^\frac{1}{m}}. \end{aligned}$$Similarly, if \(\sigma _{\varPsi _2}\le \sigma _{\varPsi _1}\), then \(\frac{\sigma _{\varPsi _1}^{\ell }}{((2-\sigma _{\varPsi _1}^{m})^{\ell }+ (\sigma _{\varPsi _1}^{m})^{\ell })^\frac{1}{m}}\ge \frac{\sigma _{\varPsi _2}^{\ell }}{((2-\sigma _{\varPsi _2}^{m})^{\ell }+ (\sigma _{\varPsi _2}^{m})^{\ell })^\frac{1}{m}}\). Therefore, \(\ell (\varPsi _1\bigvee \varPsi _2) = \ell \varPsi _1\bigvee \ell \varPsi _2\).
-
2.
We can perform proof similarly to (1).\(\square \)
To rank \(k^{n}_{m}\)-RPFSs, we offer the \(k^{n}_{m}\)-RPFS scoring function and accuracy function:
Definition 8
Let \(\varPsi = (\rho _{\varPsi },\aleph _{\varPsi },\sigma _{\varPsi })\) be a \(k^{n}_{m}\)-RPFS. Then
-
1.
a score function of \(\varPsi \) is represented by the following:
$$\begin{aligned} S(\varPsi ) = \rho _{\varPsi }^{n}-\aleph _{\varPsi }^{k}-\sigma _{\varPsi }^{m}. \end{aligned}$$ -
2.
an accuracy function of \(\varPsi \) is represented by the following:
$$\begin{aligned} A(\varPsi ) = \rho _{\varPsi }^{n}+\aleph _{\varPsi }^{k}+\sigma _{\varPsi }^{m}. \end{aligned}$$
Example 3
Consider the \(k^{n}_{m}\)-RPFS \(\varPsi = (0.63, 0.75, 0.41)\), then
and
Theorem 14
Let \(\varPsi = (\rho _{\varPsi },\aleph _{\varPsi },\sigma _{\varPsi })\) be any \(k^{n}_{m}\)-RPFS, then \(S(\varPsi )\in [-1, 1]\).
Proof
Since for any \(k^{n}_{m}\)-RPFS \(\varPsi \), we have \(\rho _{\varPsi }^{n}+\aleph _{\varPsi }^{k}+\sigma _{\varPsi }^{m}\le 1\). Hence, \(\rho _{\varPsi }^{n}-\aleph _{\varPsi }^{k}-\sigma _{\varPsi }^{m}\le \rho _{\varPsi }^{n}\le 1\) and \(\rho _{\varPsi }^{n}-\aleph _{\varPsi }^{k}-\sigma _{\varPsi }^{m}\ge -\rho _{\varPsi }^{n}-\aleph _{\varPsi }^{k}-\sigma _{\varPsi }^{m}\ge -1\) . Thus, \(-1\le \rho _{\varPsi }^{n}-\aleph _{\varPsi }^{k}-\sigma _{\varPsi }^{m}\le 1\), namely \(S(\varPsi )\in [-1, 1]\). In particular, if \(\varPsi = (0, 0, 1)\) or (0, 1, 0), then \(S(\varPsi ) = -1\) and if \(\varPsi = (1, 0, 0)\), then \(S(\varPsi ) = 1\). \(\square \)
Remark 3
For any \(k^{n}_{m}\)-RPFS \(\varPsi = (\rho _{\varPsi },\aleph _{\varPsi },\sigma _{\varPsi })\), we have \(A(\varPsi )\in [0, 1]\).
Below is the law that compares \(k^{n}_{m}\)-RPFSs:
Remark 4
Consider two \(k^{n}_{m}\)-RPFSs \(\varPsi _{1} = (\rho _{\varPsi _{1}},\aleph _{\varPsi _{1}},\sigma _{\varPsi _{1}})\) and \(\varPsi _{2} = (\rho _{\varPsi _{2}},\aleph _{\varPsi _{2}},\sigma _{\varPsi _{2}})\), then the comparison method is intended as
-
1.
if \(S(\varPsi _{1})< S(\varPsi _{2})\), then \(\varPsi _{1} < \varPsi _{2}\),
-
2.
if \(S(\varPsi _{1})> S(\varPsi _{2})\), then \(\varPsi _{1} > \varPsi _{2}\),
-
3.
if \(S(\varPsi _{1}) = S(\varPsi _{2})\), then
-
(a)
if \(A(\varPsi _{1})< A(\varPsi _{2})\), then \(\varPsi _{1} < \varPsi _{2}\),
-
(b)
if \(A(\varPsi _{1})> A(\varPsi _{2})\), then \(\varPsi _{1} > \varPsi _{2}\),
-
(c)
if \(A(\varPsi _{1}) = A(\varPsi _{2})\), then \(\varPsi _{1}\approx \varPsi _{2}\).
-
(a)
\(k^{n}_{m}\)-Rung picture fuzzy weighted power average
In this section, we present the \(k^{n}_{m}\)-rung picture fuzzy weighted power average operator and highlight its key characteristics. We specifically verify this operator’s boundedness, monotonicity, and idempotency features.
Definition 9
Let \(\varPsi _{i} = (\rho _{\varPsi _{i}},\aleph _{\varPsi _{i}},\sigma _{\varPsi _{i}})\) \((i = 1, 2, ..., v)\) be a value of \(k^{n}_{m}\)-RPFSs and \(\theta = (\theta _1, \theta _2, ..., \theta _v)^{T}\) be weight vector of \(\varPsi _{i}\) with \(\theta _i > 0\) and \(\sum _{i=1}^{v}\theta _i = 1\). Then, a \(k^{n}_{m}\)-rung picture fuzzy weighted power average (\(k^{n}_{m}\)-RPFWPA) operator is a function \(k^{n}_{m}\)-RPFWPA\(: \varPsi ^{v}\rightarrow \varPsi \), where \(k^{n}_{m}\)-RPFWPA\((\varPsi _1, \varPsi _2, ..., \varPsi _v)=((\sum _{i=1}^{v}\theta _i\rho _{\varPsi _{i}}^{n})^{\frac{1}{n}},\) \( (\sum _{i=1}^{v}\theta _i\aleph _{\varPsi _{i}}^{k})^{\frac{1}{k}}, (\sum _{i=1}^{v}\theta _i\sigma _{\varPsi _{i}}^{m})^{\frac{1}{m}})\).
Example 4
Consider \(\varPsi _1 = (0.2, 0.3, 0.4), \varPsi _2 = (0.5, 0.7,\) \( 0.6), \varPsi _3 = (0.8, 0.1, 0.3)\) and \(\varPsi _4 = (0.1, 0.2, 0.5)\) are four \(k^{n}_{m}\)-RPFSs, and \(\theta = (0.3, 0.1, 0.4, 0.2)^{T}\) be a weight vector of \(\varPsi _{i}\) (i= 1, 2, 3, 4). Then, \(k^{n}_{m}\)-RPFWPA\((\varPsi _1, \varPsi _2, \varPsi _3, \varPsi _4)= ((0.3\times 0.2^{n} + 0.1 \times 0.5^{n} + 0.4 \times 0.8^{n} + 0.2 \times 0.1^{n})^{\frac{1}{n}}, (0.3\times 0.3^{k} + 0.1 \times 0.7^{k} + 0.4 \times 0.1^{k} + 0.2 \times 0.2^{k})^{\frac{1}{k}}, (0.3\times 0.4^{m} + 0.1 \times 0.6^{m} + 0.4 \times 0.3^{m} + 0.2 \times 0.5^{m})^{\frac{1}{m}}) \)
Theorem 15
Let \(\varPsi _{i} = (\rho _{\varPsi _{i}},\aleph _{\varPsi _{i}},\sigma _{\varPsi _{i}})\) \((i = 1, 2, ..., v)\) be a value of \(k^{n}_{m}\)-RPFSs and \(\theta = (\theta _1, \theta _2, ..., \theta _v)^{T}\) be weight vector of \(\varPsi _{i}\) with \(\theta _i > 0\) and \(\sum _{i=1}^{v}\theta _i = 1\). Then, \(k^{n}_{m}\)-RPFWPA\((\varPsi _1, \varPsi _2, ..., \varPsi _v)\) is a \(k^{n}_{m}\)-RPFS.
Proof
For any \(k^{n}_{m}\)-RPFS \(\varPsi _i = (\rho _{\varPsi _{i}},\aleph _{\varPsi _{i}},\sigma _{\varPsi _{i}})\), we have
and
Then, we get
and so
implies that
Therefore
It is obvious that
and
This indicates that \(k^{n}_{m}\)-RPFWPA\((\varPsi _1, \varPsi _2, ..., \varPsi _v)\) is a \(k^{n}_{m}\)-RPFS. \(\square \)
Theorem 16
Let \(\varPsi _{i}= (\rho _{\varPsi _{i}},\aleph _{\varPsi _{i}},\sigma _{\varPsi _{i}})\) and \(L_{i} = (\rho _{L_{i}},\aleph _{L_{i}}, \sigma _{L_{i}}) (i = 1, 2, ..., v)\) be two values of \(k^{n}_{m}\)-RPFSs, and \(\theta = (\theta _1, \theta _2, ..., \theta _v)^{T}\) be a weight vector of \(\varPsi _{i}\) with \(\sum _{i=1}^{v}\theta _i = 1\).
-
1.
(Idempotency) If \(\varPsi _{i} = \varPsi = (\rho _{\varPsi },\aleph _{\varPsi },\sigma _{\varPsi })\) \(\forall i\), then \(k^{n}_{m}\)-RPFWPA\((\varPsi _1, \varPsi _2, ..., \varPsi _v)= \varPsi \).
-
2.
(Boundedness) Suppose that \(\rho _{\varPsi }^{-}= \min _{1\le i\le v}\{ \rho _{\varPsi _{i}}\}\), \(\rho _{\varPsi }^{+}= \max _{1\le i\le v}\{ \rho _{\varPsi _{i}}\}\), \(\aleph _{\varPsi }^{-}= \min _{1\le i\le v}\{ \aleph _{\varPsi _{i}}\}\), \(\sigma _{\varPsi }^{-}= \min _{1\le i\le v}\{ \sigma _{\varPsi _{i}}\}\), \(\sigma _{\varPsi }^{+}= \max _{1\le i\le v}\{ \sigma _{\varPsi _{i}}\}\) and \(\aleph _{\varPsi }= (\sum _{i=1}^{v}\theta _i\aleph _{\varPsi _{i}}^{k})^\frac{1}{k}\). Then
$$\begin{aligned}{} & {} (\rho _{\varPsi }^{-}, \aleph _{\varPsi }, \sigma _{\varPsi }^{+})\le k^{n}_{m}-\textrm{RPFWPA}(\varPsi _1, \varPsi _2, ..., \varPsi _v) \\{} & {} \le (\rho _{\varPsi }^{+}, \aleph _{\varPsi }, \sigma _{\varPsi }^{-}). \end{aligned}$$ -
2.
(Monotonicity) If \(S(\varPsi _{i})\le S(L_{i})\) \(\forall i\), then
$$\begin{aligned}{} & {} k^{n}_{m}-\textrm{RPFWPA}(\varPsi _1, \varPsi _2, ..., \varPsi _v)\\{} & {} \quad \le \,k^{n}_{m}-\textrm{RPFWPA}(L_1,L_2, ..., L_v). \end{aligned}$$
Proof
-
1.
Since \(\varPsi _{i} = \varPsi = (\rho _{\varPsi }, \aleph _{\varPsi },\sigma _{\varPsi })\forall i\), then \(k^{n}_{m}\)-RPFWPA \((\varPsi _1, \varPsi _2, ..., \varPsi _v)=\Big (\Big (\sum _{i=1}^{v}\theta _i\rho _{\varPsi _{i}}^{n}\Big )^{\frac{1}{n}}, \Big (\!\sum _{i=1}^{v}\theta _i\aleph _{\varPsi _{i}}^{k}\!\Big )^{\frac{1}{k}}, \Big (\!\sum _{i=1}^{v}\theta _i\sigma _{\varPsi _{i}}^{m}\!\Big )^{\frac{1}{m}}\!\Big ){=}\Big (\!\Big (\!\sum _{i=1}^{v}\theta _i\rho _{\varPsi _{}}^{n}\!\Big )^{\frac{1}{n}}, \Big (\sum _{i=1}^{v}\theta _i\aleph _{\varPsi _{}}^{k}\Big )^{\frac{1}{k}}, \Big (\sum _{i=1}^{v}\theta _i\sigma _{\varPsi _{}}^{m})^{\frac{1}{m}}\Big )= \Big (\rho _{\varPsi _{}}\Big (\sum _{i=1}^{v}\theta _i\Big )^{\frac{1}{n}}, \aleph _{\varPsi _{}}\Big (\sum _{i=1}^{v}\theta _i\Big )^{\frac{1}{k}}, \sigma _{\varPsi _{}}\Big (\sum _{i=1}^{v}\theta _i\Big )^{\frac{1}{m}}\Big )= (\rho _{\varPsi }, \aleph _{\varPsi },\sigma _{\varPsi })= \varPsi \).
-
2.
For any \(\varPsi _{i} = (\rho _{\varPsi _{i}},\aleph _{\varPsi _{i}},\sigma _{\varPsi _{i}})\) \((i = 1, 2, ..., v)\), we can get \(\rho _{\varPsi }^{-}\le \rho _{\varPsi _{i}} \le \rho _{\varPsi }^{+}\) and \(\sigma _{\varPsi }^{-}\le \sigma _{\varPsi _{i}} \le \sigma _{\varPsi }^{+}\). Then, the inequalities for membership value are
$$\begin{aligned} {\rho _{\varPsi }^{-}}= & {} \left( \sum _{i=1}^{v}\theta _i{\rho _{\varPsi }^{-}}^{n}\right) ^{\frac{1}{n}}\le \left( \sum _{i=1}^{v}\theta _i\rho _{\varPsi _{i}}^{n}\right) ^{\frac{1}{n}}\\\le & {} \left( \sum _{i=1}^{v}\theta _i{\rho _{\varPsi }^{+}}^{n}\right) ^{\frac{1}{n}} = \rho _{\varPsi }^{+}. \end{aligned}$$Similarly, for non-membership value
$$\begin{aligned} {\sigma _{\varPsi }^{-}}= & {} \left( \sum _{i=1}^{v}\theta _i{\sigma _{\varPsi }^{-}}^{m}\right) ^{\frac{1}{m}}\le \left( \sum _{i=1}^{v}\theta _i\sigma _{\varPsi _{i}}^{m}\right) ^{\frac{1}{m}}\\\le & {} \left( \sum _{i=1}^{v}\theta _i{\sigma _{\varPsi }^{+}}^{m}\right) ^{\frac{1}{m}}= \sigma _{\varPsi }^{+}. \end{aligned}$$Now
$$\begin{aligned}{} & {} S(k^{n}_{m}-\textrm{RPFWPA}(\varPsi _1, \varPsi _2, ..., \varPsi _v))\\{} & {} \quad = \sum _{i=1}^{v}\theta _i\rho _{\varPsi _{i}}^{n}- \sum _{i=1}^{v}\theta _i\aleph _{\varPsi _{i}}^{k}- \sum _{i=1}^{v}\theta _i\sigma _{\varPsi _{i}}^{m}\le \sum _{i=1}^{v}\theta _i{\rho _{\varPsi }^{+}}^{n} \\{} & {} \quad - \sum _{i=1}^{v}\theta _i\aleph _{\varPsi _{i}}^{k} - \sum _{i=1}^{v}\theta _i{\sigma _{\varPsi }^{-}}^{m} ={\rho _{\varPsi }^{+}}^{n} - \aleph _{\varPsi }^{k} - {\sigma _{\varPsi }^{-}}^{m}\\{} & {} \quad =S(\rho _{\varPsi }^{+}, \aleph _{\varPsi }, \sigma _{\varPsi }^{-}), \end{aligned}$$and
$$\begin{aligned}{} & {} S(k^{n}_{m}-\textrm{RPFWPA}(\varPsi _1, \varPsi _2, ..., \varPsi _v))\\{} & {} \quad = \sum _{i=1}^{v}\theta _i\rho _{\varPsi _{i}}^{n}- \sum _{i=1}^{v}\theta _i\aleph _{\varPsi _{i}}^{k}- \sum _{i=1}^{v}\theta _i\sigma _{\varPsi _{i}}^{m}\\{} & {} \quad \ge \sum _{i=1}^{v}\theta _i{\rho _{\varPsi }^{-}}^{n} - \sum _{i=1}^{v}\theta _i\aleph _{\varPsi _{i}}^{k} - \sum _{i=1}^{v}\theta _i{\sigma _{\varPsi }^{+}}^{m}\\{} & {} \quad ={\rho _{\varPsi }^{-}}^{n} - \aleph _{\varPsi }^{k} - {\sigma _{\varPsi }^{+}}^{m} =S(\rho _{\varPsi }^{-}, \aleph _{\varPsi }, \sigma _{\varPsi }^{+}). \end{aligned}$$Therefore
$$\begin{aligned} (\rho _{\varPsi }^{-}, \aleph _{\varPsi }, \sigma _{\varPsi }^{+})\le & {} k^{n}_{m}-\textrm{RPFWPA}(\varPsi _1, \varPsi _2, ..., \varPsi _v) \\\le & {} (\rho _{\varPsi }^{+}, \aleph _{\varPsi }, \sigma _{\varPsi }^{-}). \end{aligned}$$ -
3.
Since for all i we have \(S(\varPsi _{i})\le S(L_{i})\), then
$$\begin{aligned}{} & {} \rho _{\varPsi _{1}}^{n}- \aleph _{\varPsi _{1}}^{k}-\sigma _{\varPsi _{1}}^{m}\le \rho _{L_{1}}^{n}- \aleph _{L_{1}}^{k}-\sigma _{L_{1}}^{m}\\{} & {} \rho _{\varPsi _{2}}^{n}- \aleph _{\varPsi _{2}}^{k}-\sigma _{\varPsi _{2}}^{m}\le \rho _{L_{2}}^{n}- \aleph _{L_{2}}^{k}-\sigma _{L_{2}}^{m}\\{} & {} \quad .\\{} & {} \quad .\\{} & {} \quad .\\{} & {} \quad \rho _{\varPsi _{v}}^{n}- \aleph _{\varPsi _{v}}^{k}-\sigma _{\varPsi _{v}}^{m}\le \rho _{L_{v}}^{n}- \aleph _{L_{v}}^{k}-\sigma _{L_{v}}^{m}, \end{aligned}$$and so
$$\begin{aligned}{} & {} S(k^{n}_{m}-\textrm{RPFWPA}(\varPsi _1, \varPsi _2, ..., \varPsi _v))\\{} & {} \quad = \sum _{i=1}^{v}\theta _i\rho _{\varPsi _{i}}^{n}- \sum _{i=1}^{v}\theta _i\aleph _{\varPsi _{i}}^{k}- \sum _{i=1}^{v}\theta _i\sigma _{\varPsi _{i}}^{m}\\{} & {} \quad \le \sum _{i=1}^{v}\theta _i\rho _{L_{i}}^{n}- \sum _{i=1}^{v}\theta _i\aleph _{L_{i}}^{k}- \sum _{i=1}^{v}\theta _i\sigma _{L_{i}}^{m} \\{} & {} \quad =S(k^{n}_{m}-\textrm{RPFWPA}(L_1, L_2, ..., L_v)). \end{aligned}$$Thus, \(k^{n}_{m}\)-RPFWPA\((\varPsi _1, \varPsi _2, ..., \varPsi _v) \le \) \(k^{n}_{m}\)-RPFWPA \((L_1, L_2, ..., L_v)\).
\(\square \)
Theorem 17
Let \(\varPsi _{i} = (\rho _{\varPsi _{i}},\aleph _{\varPsi _{i}},\sigma _{\varPsi _{i}})\) \((i = 1, 2, ..., v)\) be a value of \(k^{n}_{m}\)-RPFSs, \(\varPsi = (\rho _{\varPsi },\aleph _{\varPsi },\sigma _{\varPsi })\) be \(k^{n}_{m}\)-RPFS and \(\theta = (\theta _1, \theta _2, ..., \theta _v)^{T}\) be a weight vector of \(\varPsi _{i}\) with \(\sum _{i=1}^{v}\theta _i = 1\). Then
-
1.
\(A(k^{n}_{m}\)-RPFWPA\((\varPsi _1 \oplus \varPsi , \varPsi _2 \oplus \varPsi , ..., \varPsi _v \oplus \varPsi )) \le A(k^{n}_{m}\)-RPFWPA\((\varPsi _1, \varPsi _2, ..., \varPsi _v))\).
-
2.
\(A(k^{n}_{m}\)-RPFWPA\((\varPsi _1 \otimes \varPsi , \varPsi _2 \otimes \varPsi , ..., \varPsi _v \otimes \varPsi )) \le A(k^{n}_{m}\)-RPFWPA\((\varPsi _1, \varPsi _2, ..., \varPsi _v))\).
Proof
-
1.
Since \(\rho _{\varPsi }^{n}\le 1+ \rho _{\varPsi _1}^{n}\rho _{\varPsi }^{n}\), then \(\frac{\rho _{\varPsi }^{n}}{1+ \rho _{\varPsi _1}^{n}\rho _{\varPsi }^{n}}\le 1\), and hence
$$\begin{aligned} \theta _1\left( \frac{\rho _{\varPsi _1}^{n}\rho _{\varPsi }^{n}}{1+ \rho _{\varPsi _1}^{n}\rho _{\varPsi }^{n}}\right) \le \theta _1\rho _{\varPsi _1}^{n}. \end{aligned}$$Therefore, we have
$$\begin{aligned}{} & {} \theta _1\left( \frac{\rho _{\varPsi _2}^{n}\rho _{\varPsi }^{n}}{1+ \rho _{\varPsi _2}^{n}\rho _{\varPsi }^{n}}\right) \le \theta _1\rho _{\varPsi _2}^{n}\\{} & {} \quad .\\{} & {} \quad .\\{} & {} \quad .\\{} & {} \quad \theta _v\left( \frac{\rho _{\varPsi _v}^{n}\rho _{\varPsi }^{n}}{1+ \rho _{\varPsi _v}^{n}\rho _{\varPsi }^{n}}\right) \le \theta _v\rho _{\varPsi _v}^{n}. \end{aligned}$$Thus
Table 3 \(k^{n}_{m}\)-RPF values Table 4 Aggregated \(k^{n}_{m}\)-RPF information matrix $$\begin{aligned} \sum _{i=1}^{v}\theta _i \left( \frac{\rho _{\varPsi _i}^{n}\rho _{\varPsi }^{n}}{1+ \rho _{\varPsi _i}^{n}\rho _{\varPsi }^{n}}\right) \le \sum _{i=1}^{v}\theta _i\rho _{\varPsi _{i}}^{n}. \end{aligned}$$Similarly, we have
$$\begin{aligned} \sum _{i=1}^{v}\theta _i \left( \frac{\aleph _{\varPsi _i}^{k}\aleph _{\varPsi }^{k}}{1+(1- \aleph _{\varPsi _i}^{k})(1- \aleph _{\varPsi }^{k})}\right) \le \sum _{i=1}^{v}\theta _i\aleph _{\varPsi _{i}}^{k}, \end{aligned}$$and
$$\begin{aligned} \sum _{i=1}^{v}\theta _i \left( \frac{\sigma _{\varPsi _i}^{m}\sigma _{\varPsi }^{m}}{1+(1- \sigma _{\varPsi _i}^{m})(1- \sigma _{\varPsi }^{m})}\right) \le \sum _{i=1}^{v}\theta _i\sigma _{\varPsi _{i}}^{m}. \end{aligned}$$Now, we have \(k^{n}_{m}\)-RPFWPA\((\varPsi _1 \oplus \varPsi , \varPsi _2 \oplus \varPsi , ..., \varPsi _v \oplus \varPsi ) =((\sum _{i=1}^{v}\theta _i (\frac{\rho _{\varPsi _i}^{n}\rho _{\varPsi }^{n}}{1+ \rho _{\varPsi _i}^{n}\rho _{\varPsi }^{n}}))^\frac{1}{n}, (\sum _{i=1}^{v}\theta _i (\frac{\aleph _{\varPsi _i}^{k}\aleph _{\varPsi }^{k}}{1+(1- \aleph _{\varPsi _i}^{k})(1- \aleph _{\varPsi }^{k})}))^\frac{1}{k}, (\sum _{i=1}^{v}\theta _i (\frac{\sigma _{\varPsi _i}^{m}\sigma _{\varPsi }^{m}}{1+(1- \sigma _{\varPsi _i}^{m})(1- \sigma _{\varPsi }^{m})}))^\frac{1}{m})\), then \(A(k^{n}_{m}\)-RPFWPA\((\varPsi _1 \oplus \varPsi , \varPsi _2 \oplus \varPsi , ..., \varPsi _v \oplus \varPsi )) =\sum _{i=1}^{v}\theta _i (\frac{\rho _{\varPsi _i}^{n}\rho _{\varPsi }^{n}}{1+ \rho _{\varPsi _i}^{n}\rho _{\varPsi }^{n}})+ \sum _{i=1}^{v}\theta _i (\frac{\aleph _{\varPsi _i}^{k}\aleph _{\varPsi }^{k}}{1+(1- \aleph _{\varPsi _i}^{k})(1- \aleph _{\varPsi }^{k})}) + \sum _{i=1}^{v}\theta _i \left( \!\frac{\sigma _{\varPsi _i}^{m}\sigma _{\varPsi }^{m}}{1+(1- \sigma _{\varPsi _i}^{m})(1- \sigma _{\varPsi }^{m})}\!\right) {\le }\sum _{i=1}^{v}\theta _i\rho _{\varPsi _{i}}^{n}{+}\sum _{i=1}^{v}\theta _i \aleph _{\varPsi _{i}}^{k} + \sum _{i=1}^{v}\theta _i\sigma _{\varPsi _{i}}^{m} = A(k^{n}_{m}\)-RPFWPA\((\varPsi _1, \varPsi _2, ..., \varPsi _v))\).
-
2.
We can proof in a similar fashion to (1).
\(\square \)
The proof of the following result is similar to the proof of Theorem 17, so we omit it.
Theorem 18
Let \(\varPsi _{i} = (\rho _{\varPsi _{i}},\aleph _{\varPsi _{i}},\sigma _{\varPsi _{i}})\) and \(L_{i} = (\rho _{L_{i}},\sigma _{L_{i}})\) \((i = 1, 2, ..., v)\) be two values of \(k^{n}_{m}\)-RPFSs, and \(\theta = (\theta _1, \theta _2, ..., \theta _v)^{T}\) be a weight vector of them with \(\sum _{i=1}^{v}\theta _i = 1\). Then
-
1.
\(A(k^{n}_{m}\)-RPFWPA\((\varPsi _1 \oplus L_1, \varPsi _2 \oplus L_2, ..., \varPsi _v \oplus L_v)) \le A(k^{n}_{m}\)-RPFWPA\((\varPsi _1, \varPsi _2, ..., \varPsi _v))\).
-
2.
\(A(k^{n}_{m}\)-RPFWPA\((\varPsi _1 \oplus L_1, \varPsi _2 \oplus L_2, ..., \varPsi _v \oplus L_v)) \le A(k^{n}_{m}\)-RPFWPA\((L_1, L_2, ..., L_v))\).
-
3.
\(A(k^{n}_{m}\)-RPFWPA\((\varPsi _1 \otimes L_1, \varPsi _2 \otimes L_2, ..., \varPsi _v \otimes L_v)) \le A(k^{n}_{m}\)-RPFWPA\((\varPsi _1, \varPsi _2, ..., \varPsi _v))\).
-
4.
\(A(k^{n}_{m}\)-RPFWPA\((\varPsi _1 \otimes L_1, \varPsi _2 \otimes L_2, ..., \varPsi _v \otimes L_v)) \le A(k^{n}_{m}\)-RPFWPA\((L_1, L_2, ..., L_v))\).
MADM application to select the best countries for expats
In this section, we use the \(k^{n}_{m}\)-RPFWPA operator to determine the best country for expats between several countries. The data in this approach are provided by the decision-makers in the form of \(k^{n}_{m}\)-rung picture fuzzy sets.
Let \(C = \{C_1, C_2, ..., C_u\}\) be a set that represents the available options. The weights of the v attributes \(K_1, K_2, ..., K_v\), explaining the key elements affecting the MADM problem, are represented by the weight vector \(\theta =\{\theta _1, \theta _2, ..., \theta _v\}\), where \(\theta _i > 0\) and \(\sum _{i=1}^{v}\theta _i = 1\). Suppose that \(\tilde{M}= (\rho _{ji},\aleph _{ji},\sigma _{ji})_{u\times v}\) is the \(k^{n}_{m}\)-RPF decision matrix, where \(\rho _{ji}\) indicates the positive membership degree to which the option \(C_j\) fulfills the attribute \(K_i\) specified by the decision maker, \(\aleph _{ji}\) indicates the neutral membership degree that the option \(C_j\) does not fulfill the attribute \(K_i\), and \(\sigma _{ji}\) indicates the negative membership degree that the option \(C_j\) does not fulfill the attribute \(K_i\) specified by the decision maker, where \((\rho _{ji})^{n} + (\aleph _{ji})^{k} + (\sigma _{ji})^{m} \le 1\). In light of this, for solving an MADM problem, we give the following Algorithm 1.
Steps to solve MADM problem by \(k^{n}_{m}\)-RPFWPA operator.
Application example
People are becoming more and more captivated by the thought of uprooting and relocating overseas as the world becomes more connected and simple to navigate. Since the beginning of civilization, people have aspired to advance, discover the world, and settle in new places. Thousands of people desire to immigrate to a new country each year, demonstrating that this attitude is still present today.
The selection of an appropriate country is an important step to move to a new country. Suppose Austria, Sweden, The Netherlands, Finland, and Norway are a possible countries for selection, and \(K = \{K_1, K_2, K_3, K_4, K_5, K_6, K_7, K_8, \) \( K_9, K_{10}, K_{11}, K_{12}, K_{13}\}\) is a set of 13 keys for the selection of countries, where
\(K_1: \) Unemployment rate,
\(K_2: \) Average income per capita,
\(K_3: \) Level of crime,
\(K_4: \) Amount of space,
\(K_5: \) Decent working hours,
\(K_6: \) Overall labor rights,
\(K_7: \) City center apartment cost,
\(K_8: \) Rural apartment cost,
\(K_9: \) Air quality,
\(K_{10}: \) Drinking water quality and accessibility,
\(K_{11}: \) Garbage disposal,
\(K_{12}: \) Quality of parks, and
\(K_{13}: \) Gender equality.
Table 3 shows how we built the \(k^{n}_{m}\)-rung picture fuzzy decision matrices. Now, we applying the \(k^{n}_{m}\)-RPFWPA operator with weight vectors \(\theta = (0.11, 0.05, 0.08, 0.08, 0.05, 0.04, 0.05, 0.09, 0.08, 0.09, 0.08, \) \( 0.09, 0.11)^{T}\), and putting \(n=m= 1, 2, k= 2, 3\), and \(n=m= 3, k= 4\) as follows in Table 4. Finally, Table 5 shows the score value for each option, and Table 6 shows their ranking.
We have used various values of n, k, and m to rank the options to explain the impact of the parameters n, k and m on MADM outcomes. In Table 6, the results of the ranking of the options based on the \(k^{n}_{m}\)-RPFWPA operator are shown. The possibilities were ranked as \(C_4> C_2> C_5> C_1 > C_3\); here, Finland is the best choice and Sweden is the best second choice. Thus, the overall best rank is Finland.
Comparison analysis and discussion
In this section, we contrast the suggested operator with other existing operators to show the benefits of the proposed operator. In this case, we use data from a particular application in accordance with the operators by which we must compare our suggested model to ensure the precision and effectiveness of our generated operator. To compare the \(k^{n}_{m}\)-RPFWPA operator with the 3-RPFEWA [52], SFWG and SFWA [53], SFDWA [54], and PFDWA [55] operators. For the sake of brevity, we have used the techniques described here on the application offered in [52]. Table 7 demonstrates that for various values of n, k, and m, the proposed \(k^{n}_{m}\)-RPFWPA operator and the the various existing operators yield the same optimal decision object, and the ultimate order is \(J_2> J_3> J_1 > J_4\). It is evident that \(J_2\) is the perfect choice for business’s location. This suggests that our approach is reliable and usable in decision-making issues. The \(k^{n}_{m}\)-RPFWPA covers more MADM problems utilizing various values of the parameters n, k, and m that are appropriate for MADM problems, since it is the more flexible, accomplished, and broad model. For this reason, we advocate the theory of \(k^{n}_{m}\)-RPFWPA, which has a larger variety of applications than the current operators and is therefore preferable to them.
By the above-mentioned practical example, the proposed operators overcome shortcomings and limits of existing operators, which means that they are more general and perform admirably not only for \(k^{n}_{m}\)-RPFS data but also for q-RPF fuzzy information. That is, in comparison to other modeling techniques, the \(k^{n}_{m}\)-RPFS is more adaptable and effective which enables individuals to provide more accurate outcomes when faced with decision-making issues.
Conclusions
As it is well known that all the existing extensions of PFSs are particular cases of q-RPFSs. However, there exist some situations and cases requiring evaluations with different importance (whence calibration) for the degrees of positive membership, neutral membership, and negative membership which cannot be met by the forgoing extensions of PFSs. To achieve this target, we are writing this manuscript.
First, we have suggested a novel generalized q-RPFS called \(k^{n}_{m}\)-rung picture fuzzy set and contrasted it to other types of picture fuzzy sets. Several operators on \(k^{n}_{m}\)-rung picture fuzzy sets have been presented and their features have been demonstrated. Then, the application of mathematical operations known as "aggregate operators" has been brought them to the status of vital tools for fusing several inputs into a single, practical output. As a result, we have successfully designed a novel aggregation operator for \(k^{n}_{m}\)-RPFS. We have created the \(k^{n}_{m}\)-RPFWPA operator and investigated its many features such as idempotency, boundedness, and monotonicity. Finally, we used the given operator to develop a novel technique for solving MAGDM problems. To demonstrate the validity of the proposed method, we utilized it to identify the best country for expats. In addition, we did a comparison analysis to demonstrate the efficacy and superiority of the suggested method. The proposed technique can be used to solve various real-world decision-making problems due to the strong ability of \(k^{n}_{m}\)-RPFSs to describe fuzziness and represent decision-makers’ evaluations over options, as well as the power of the \(k^{n}_{m}\)-RPFWPA operator.
Our future road encourages further studies to investigate more aggregation operators for integrating \(k^{n}_{m}\)-rung picture data in light of the benefits of \(k^{n}_{m}\)-RPFSs. Additionally, we should look at more MAGDM algorithms that use \(k^{n}_{m}\)-rung picture information. Also, \(k^{n}_{m}\)-rung picture fuzzy sets can be expanded to \(k^{n}_{m}\)-rung picture fuzzy soft set, and numerous applications can be introduced on it. Since the solutions to many complicated issues, especially with the progression of technology and communication, require resorting to more than one analytical tool, we shall integrate the current type of picture fuzzy sets with some rough sets models [56, 57] and abstract structures [58,59,60] to produce new frameworks of uncertainty.
References
Zadeh LA (1965) Fuzzy sets. Inf Control 8:338–353
Song Q, Kandel, Schneider AM (2003) Parameterized fuzzy operators in fuzzy decision-making. Int J Intell Syst 18:971–987
Atanassov KT (1986) Intuitionistic fuzzy sets. Fuzzy Sets Syst 20:87–96
Liu PD (2017) Multiple attribute decision-making methods based on normal intuitionistic fuzzy interaction aggregation operators. Symmetry 9:261
Liu PD, Mahmood T, Khan Q (2017) Multi-attribute decision-making based on prioritized aggregation operator under hesitant intuitionistic fuzzy linguistic environment. Symmetry 9:270
Wang SW, Liu J (2017) Extension of the TODIM method to intuitionistic linguistic multiple attribute decision making. Symmetry 9:95
Yager RR (2013) Pythagorean membership grades in multicriteria decision making. In: Technical report MII-3301, Machine Intelligence Institute, Iona College, New Rochelle
Wei GW, Lu M, Tang XY, Wei Y (2018) Pythagorean hesitant fuzzy Hamacher aggregation operators and their application to multiple attribute decision making. Int J Intell Syst 33:1197–1233
Liang D, Xu ZS (2017) The new extension of TOPSIS method for multiple criteria decision making with hesitant Pythagorean fuzzy sets. Appl Soft Comput 60:167–179
Zhang RT, Wang J, Zhu XM, Xia MM, Yu M (2017) Some generalized Pythagorean fuzzy Bonferroni mean aggregation operators with their application to multi-attribute group decision-making. Complexity 2017:5937376
Ren PJ, Xu ZS, Gou XJ (2016) Pythagorean fuzzy TODIM approach to multi-criteria decision making. Appl Soft Comput 42:246–259
Liu PD, Teng F (2018) Some Muirhead mean operators for probabilistic linguistic term sets and their applications to multiple attribute decision-making. Appl Soft Comput 68:396–431
Liang X, Jiang YP, Liu PD (2018) Stochastic multiple-criteria decision making with 2-tuple aspirations: a method based on disappointment stochastic dominance. Int Trans Oper Res 25:913–940
Li Y, Liu PD (2015) Some Heronian mean operators with 2-tuple linguistic information and their application to multiple attribute group decision making. Technol Econ Dev Econ 21:797–814
Garg H (2018) Some methods for strategic decision-making problems with immediate probabilities in Pythagorean fuzzy environment. Int J Intell Syst 33:687–712
Mahmood T, Liu PD, Ye J, Khan Q (2018) Several hybrid aggregation operators for triangular intuitionistic fuzzy set and their application in multi-criteria decision making. Granul Comput 3:153–168
Kumar PS (2020) Algorithms for solving the optimization problems using fuzzy and intuitionistic fuzzy set. Int J Syst Assur Eng Manag 11:189–222
Kumar PS (2019) Intuitionistic fuzzy solid assignment problems: a software-based approach. Int J Syst Assur Eng Manag 10:661–675
Kumar PS (2016) A simple method for solving type-2 and type-4 fuzzy transportation problems. Int J Fuzzy Logic Intell Syst 16(4):225–237
Yager RR (2017) Generalized orthopair fuzzy sets. IEEE Trans Fuzzy Syst 25:1222–1230
Liu PD, Wang P (2018) Some q-rung orthopair fuzzy aggregation operators and their applications to multiple-attribute decision making. Int J Intell Syst 33:259–280
Liu PD, Liu JL (2018) Some q-rung orthopair fuzzy Bonferroni mean operators and their application to multi-attribute group decision making. Int J Intell Syst 33:315–347
Al-shami TM (2023) (2,1)-Fuzzy sets: properties, weighted aggregated operators and their applications to multi-criteria decision-making methods. Complex Intell Syst 9:1687–1705
Al-shami TM, Mhemdi A (2023) Generalized frame for orthopair fuzzy sets: \((m,n)\)-fuzzy sets and their applications to multi-criteria decision-making methods. Information (Switzerland) 14(1):56
Ibrahim HZ, Alshammari I (2022) n, m-Rung orthopair fuzzy sets with applications to multicriteria decision making. IEEE Access 10:99562–99572
Al-shami TM, Ibrahim HZ, Azzam AA, EL-Maghrabi AI, (2022) SR-fuzzy sets and their applications to weighted aggregated operators in decision-making. J Funct Spaces 2022:3653225
Ibrahim HZ, Al-shami TM, Elbarbary OG (2021) (3, 2)-Fuzzy sets and their applications to topology and optimal choices. Comput Intell Neurosci 2021:1272266
Ibrahim HZ, Al-shami TM, Mhemdi A (2023) Applications of \(n{\rm th}\) power root fuzzy sets in multicriteria decision making. J Math 2023:1487724
Al-shami TM, Ibrahim HZ, Mhemdi A, Abu-Gdairi R (2022) \(n{\rm th}\) power root fuzzy sets and its topology. Int J Fuzzy Logic Intell Syst 22(4):350–365
Al-shami TM, Alcantud JCR, Mhemdi A (2023) New generalization of fuzzy soft sets: \((a, b)\)-fuzzy soft sets. AIMS Math 8(2):2995–3025
Jafar MN, Saeed M (2022) Matrix theory for neutrosophic hypersoft set and applications in multiattributive multicriteria decision-making problems. J Math 2022:6666408
Jafar MN, Saeed M, Saqlain, Yang M-S (2021) Trigonometric similarity measures for neutrosophic hypersoft sets with application to renewable energy source selection. IEEE Access 9:129178–129187. https://doi.org/10.1109/ACCESS.2021.3112721
Jafar MN, Khan MR, Sultan H, Ahmed N (2020) Interval valued fuzzy soft sets and algorithm of IVFSS applied to the risk analysis of prostate cancer. Int J Comput Appl 177(38):18–26
Saleh S, Abu-Gdairi Radwan, Al-shami TM, Mohammed S, Abdo S (2022) On categorical property of fuzzy soft topological spaces. Appl Math Inf Sci 16(4):635–641
Saleh S, Al-shami TM, Mhemdi A (2023) On some new types of fuzzy soft compact spaces. J Math 2023:5065592
Malik N, Shabir M, Al-shami TM, Gul R, Mhemdi A (2023) Medical decision-making techniques based on bipolar soft information. AIMS Math 8(8):18185–18205
Malik N, Shabir M, Al-shami TM, Gul R, Arar M, Hosny M (2023) Rough bipolar fuzzy ideals in semigroups. Complex Intell Syst. https://doi.org/10.1007/s40747-023-01132-1
Gul R, Shabir M, Al-shami TM, Hosny M (2023) A Comprehensive study on \((\alpha , \beta )\)-multi-granulation bipolar fuzzy rough sets under bipolar fuzzy preference relation. AIMS Math 8(11):25888–25921
Cuong BC (2013) Picture fuzzy sets first results. Part 1, Seminar neuro-fuzzy systems with applications, Preprint 03/2013, Institute of Mathematics, Hanoi
Cuong BC (2013) Picture fuzzy sets first results. Part 2, Seminar neuro-fuzzy systems with applications, Preprint 04/2013, Institute of Mathematics, Hanoi
Wei GW (2017) Picture uncertain linguistic Bonferroni mean operators and their application to multiple attribute decision making. Kybernetes 46:1777–1800
Wei GW (2016) Picture fuzzy cross-entropy for multiple attribute decision making problems. J Bus Econ Manag 17:491–502
Li DX, Dong H, Jin X (2017) Model for evaluating the enterprise marketing capability with picture fuzzy information. J Intell Fuzzy Syst 33:3255–3263
Thong PH (2016) Picture fuzzy clustering: a new computational intelligence method. Soft Comput 20:3549–3562
Li L, Zhang R, Wang J, Shang X, Bai K (2018) A novel approach to multi-attribute group decision making with q-rung picture linguistic information. Symmetry 10(5):172
Garg H (2017) Some picture fuzzy aggregation operators and their applications to multi-criteria decision making. Arab J Sci Eng 42(12):5275–5290
Wei GW (2017) Picture fuzzy aggregation operators and their application to multiple attribute decision making. J Intell Fuzzy Syst 33:713–724
Wei GW (2018) Picture fuzzy Hamacher aggregation operators and their application to multiple attribute decision making. Fund Inf 157:271–320
He J, Wang X, Zhang R, Li L (2019) Some q-rung picture fuzzy Dombi Hamy mean operators with their application to project assessment. Mathematics 7(5):468
Liu P, Shahzadi G, Akram M (2020) Specific types of q-rung picture fuzzy Yager aggregation operators for decision making. Int J Comput Intell Syst 13(1):1072–1091
Gundogdu FK, Kahraman C (2019) Spherical fuzzy sets and spherical fuzzy TOPSIS method. J Intell Fuzzy Syst 36(1):337–352
Akram M, Shahzadi G, Alcantud JCR (2022) Multi-attribute decision-making with q-rung picture fuzzy information. Granul Comput 7:197–215
Ashraf S, Abdullah S (2019) Spherical aggregation operators and their application in multi-attribute group decision making. Int J Intell Syst 34(3):493–523
Ashraf S, Abdullah S, Mahmood T (2020) Spherical fuzzy Dombi aggregation operators and their application in group decision making problems. J Ambient Intell Humaniz Comput 11:2731–2749
Jana C, Senapati T, Pal M, Yager RR (2019) Picture fuzzy Dombi aggregation operators: application to MADM process. Appl Soft Comput 74:99–109
Al-shami TM, Mhemdi A (2023) Approximation spaces inspired by subset rough neighborhoods with applications. Demonstr Math 56(1):20220223. https://doi.org/10.1515/dema-2022-0223
Mustafa H, Al-shami TM, Wassef R (2023) Rough set paradigms via containment neighborhoods and ideals. Filomat 37(14):4683–4702
Gulzar M, Alghazzawi D, Mateen MH, Kausar N (2020) A certain class of t-intuitionistic fuzzy subgroups. IEEE Access 8:163260–163268. https://doi.org/10.1109/ACCESS.2020.3020366
Gulzar M, Mateen MH, Alghazzawi D, Kausar N (2020) A novel applications of complex intuitionistic fuzzy sets in group theory. IEEE Access 8:196075–196085. https://doi.org/10.1109/ACCESS.2020.3034626
Ameen ZA, Al-shami TM, Azzam AA, Mhemdi A (2022) A novel fuzzy structure: infra-fuzzy topological spaces. J Funct Spaces 2022:9778069
Funding
The fourth author extends her appreciation to the Deanship of Scientific Research at King Khalid University for funding this work through a research groups program under Grant no. RGP2/310/44.
Author information
Authors and Affiliations
Corresponding author
Ethics declarations
Conflict of interest
The authors declare no conflict of interest.
Human participants
This article does not contain any studies with human participants performed by any of the authors.
Informed consent
The authors have read and agreed to submit this version of the manuscript.
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.
About this article
Cite this article
Ibrahim, H.Z., Al-shami, T.M., Arar, M. et al. \(k^{n}_{m}\)-Rung picture fuzzy information in a modern approach to multi-attribute group decision-making. Complex Intell. Syst. 10, 2605–2625 (2024). https://doi.org/10.1007/s40747-023-01277-z
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s40747-023-01277-z
Keywords
- \(k^{n}_{m}\)-Rung picture fuzzy set
- \(k^{n}_{m}\)-Rung picture fuzzy weighted power average operator
- Multi-attribute decision-making problem