Abstract
The neutrosophic hesitant fuzzy set is the union of the neutrosophic set and hesitant fuzzy set in which each element has membership, neutrality, and non-membership arrays which verifies infrequency in labeling uncertainty in daily usage. We proposed two novel operators in this analysis that demonstrate evolution, one is the neutrosophic hesitant fuzzy partitioned Maclaurin symmetric mean (NHFPMSM) and the other is the neutrosophic hesitant fuzzy weighted partitioned Maclaurin symmetric mean (NHFWPMSM). These novels aim to draw inspiration from the partitioned Maclaurin symmetric mean concept. The diverse properties and special cases of these operators are demonstrated in this article. We introduce a novel multiple-criteria decision-making method based on the NHFWPMSM operator for choosing the most appropriate substitute from a set of choices ideally. We highlight an organized technique using our approach for selecting the most unique and the best piece of art that focuses on a better composition with appealing colors for innovative art patterns. Moreover, this article shows expert frequency and productiveness through comprehensive contrasting studies and that is how our progressive access excels in existing strategies.
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Introduction
In daily routine, multi-criteria decision-making (MCDM) problems arise across various domains. Decision makers should select the best choice by evaluating diverse options that are based on bounded criteria. Information that is given by decision-makers is usually unclear, incomplete, or clashing due to the difficulty of social themes. The latest research has focused on techniques that tackle these issues in uncertain fields due to elevating the desire for precise decision-making1. In 1965, Zadeh2 suggested the fuzzy set (FS) theory as a beneficial gateway for demonstrating unclear data. However, FS theory has clear bounds in dealing with uncertain information which has immensely spread interest in making it better.
Atanassov3 suggested the non-membership function to broaden FS theory as shown in Fig. 1 and proposed the theory of intuitionistic fuzzy sets (IFS). IFS4 can express membership and non-membership information together. It is an ability that leads to the effective tackling of many applications that are not highlighted by FS. Atanassov and Gargov5 used interval numbers to extend the membership and non-membership functions by introducing the theory of interval-valued intuitionistic fuzzy sets (IVIFS). They put forward this concept to illustrate the evaluation information’s fuzziness nicely. However, in real decision-making, it isn’t always extensive to consider membership and non-membership information. For instance, when a decision maker shows his opinion, he may assign a positive chance of 0.7, a false chance of 0.4, and an uncertain chance of 0.3. At that point, IFS and IVIFS theory can’t handle the situation. So, Smarandache6 defined the neutrosophic set (NS) that can be observed as a generalization of FS and IFS7. It includes three distinct membership functions: truth membership, indeterminacy membership, and falsity membership functions8. However, NS theory first evolved from a philosophical ideology that made its usage in science and engineering quite challenging. Wang9,10 introduced the basics of interval neutrosophic sets (INS) and single-valued neutrosophic sets (SVNS) which are specific cases of NS just to point out this hurdle11,12.
Extension of fuzzy set to neutrosophic set.
A fuzzy set only has a single membership value and this disadvantage makes it challenging for decision-makers to determine the precise value of membership owing to unpredictability. Torra13 extended FS theory into hesitant fuzzy set (HFS) theory in pointing out this challenge14. It enabled decision-makers to give diverse membership values. Furthermore, Chen et al.15 proposed the idea of the interval-valued hesitant fuzzy set (IVHFS) in which the membership values can be expressed using intervals. Zhu et al.16 introduced the non-membership hesitancy function to establish the theory of dual hesitant fuzzy sets (DHFS) after noticing the complex nature of the information provided by decision-makers17. Moreover, an NS causes a chance and opportunity to grip vague information that emerges from the real-world crisis. It is probably quite advisable to deal with unclear and pending knowledge. Demonstrating the degree of authentic membership, indeterminacy, and inexact membership is very beneficial. To tackle this outline, Ye18 was the suggester of the notion of the single-valued neutrosophic hesitant fuzzy set19. Look at for scenario, that the subsequent criteria are being used to evaluate four pieces of artistry: the color palettes \(\Theta _{1}\), emotions \(\Theta _{2}\), tech. skills \(\Theta _{3}\), and ingenuity \(\Theta _{4}\). The imaginative sections can be represented by \(\tilde{P}_{1} = \{\Theta _{1}, \Theta _{4}\}\), and the technical features by \(\tilde{P}_{2} = \{\Theta _{2}, \Theta _{3}\}\). Since both \(\Theta _{1}\) and \(\Theta _{4}\) draw attention to the project’s imaginative elements and are included in \(\tilde{P}_{1}\), we can see that they are connected. In the same way, \(\Theta _{2}\) and \(\Theta _{3}\) are related since they are part of \(\tilde{P}_{2}\) and exhibit the creator’s expertise in the technique. In the entire examination procedure, imagination and technical skills are not necessarily linked, therefore there is no immediate link within \(\tilde{P}_{1}\) and \(\tilde{P}_{2}\). Dutta and Guha20 proposed the partitioned Bonferroni mean (BM) operator, while Liu et al.21 proposed the partitioned Maclaurin symmetric mean (MSM) (PMSM) operator, which relies on IFNs. These two operators assume that the entire array of standards has been broken down into a limited number of kinds and that there is a relationship between standards within just one kind but not between standards from other kinds. After that, Ali22,23 applied this PMSM operator to other typical FSs.
Moreover, under a neutrosophic hesitant fuzzy (NHF) environment, our whole concentration is kept only on the Partitioned Maclaurin Symmetric Mean aggregation operator. The analysis further discovers different neutrosophic hesitant fuzzy aggregation operators and their usage in MCDM24,25,26,27,28,29. While lasting aggregation algorithms apprehend interdependencies over standard, they lack in marking stages where criteria are organized into subdivisions. For example, the study shows the Partitioned Maclaurin Symmetric Mean (PMSM) and Partitioned Bonferroni Mean (PBM) operators, formulated to deal with criteria classified into portions.
Motivation
The Neutrosophic Hesitant Fuzzy Partitioned Maclaurin Symmetric Mean Aggregation Operator (NHFPMSM) integrates the advantages of multiple mathematical models and is particularly useful in circumstances involving a lack of clarity in decision-making.
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Neutrophic FS allows for a great degree of elaboration of questionable information since it encompasses indeterminacy including membership and non-membership degrees. In everyday circumstances wherein information can occasionally not be crisp, this works well. On the other hand, Hesitant fuzzy sets detect hesitancy in skilled computations or decision-making by allowing for an array of probable membership degrees for an aspect. In contrast to typical fuzzy sets, this resemblance can model incompatible, partial, or ambiguous data more well.
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The Maclurin symmetric mean (MSM)30 is a mathematically valid approach to analyzing average information while keeping track of criterion interdependencies. When partitioning is coupled with it, decision-makers (DMs) are able to manage complicated information by dividing it into well-organized subsets while preserving the relationships among the subcriteria. Recognizing the significance of each partition allows DMs to attach different weights to it, making the partitioned form of MSM sufficiently adaptable for simulation.
Thus, the proposed operator allows criteria to have varied degrees of influence over the output, which makes it ideal for uses where criteria are significant and relate to one another.
The proposal points:
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1.
To deal with disadvantages recognized in existing reviews, suggest neutrosophic hesitant fuzzy PMSM (NHFPMSM) and neutrosophic hesitant fuzzy weighted PMSM (NFWPMSM) operators.
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2.
Probe characteristics, theorems, and key examples of the introduced PMSM operators.
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3.
Based on the evaluated NHFWPMSM operator, develop an MCDM technique.
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4.
To show the worth and ideality of the proposed methodology. Make an illustrative example-based study and contrasting procedure.
The following portion mentions the underlying foundations of Neutrosophic hesitant fuzzy sets (NHFS). Also, it formulates the NHFPMSM and NHFWPMSM operators, highlights their key specifications, proposes a decision-making way by such operators, presents a well-explained example-based study, conducts contrasts with existing ones, and yields these brief contexts.
Fundamental concepts
Here, we give some basic ideas analogous to NHFS and PMSM.
Definition 1
13 A HFS \({\mathscr {X}}\) on given \(\Xi\) is described as
where \({\mathfrak {U}}\) is membership function of \(\varepsilon \in \Xi\) in \({\mathscr {X}}\) yield a subset of the unit interval \(\left[ 0,1 \right]\).
A hesitant fuzzy element (HFE) is indicated by \({\mathfrak {U}}= {\mathfrak {U}}\left( \varepsilon \right)\), and a collection of all hesitant fuzzy elements is noted by \({\mathscr {X}}\) for our assistance.
Definition 2
31 The Score function \(\S\) for \({\mathfrak {U}}\) is so-called as:
where \(\#{{\mathfrak {U}}}\) shows the cardinality in \({\mathfrak {U}}\) and \(\gtrdot\) indicate the values belonged to it.
Definition 3
For any two HFEs, Xia and Xu32 propose the following contrast: for any two HFEs, \({{\mathfrak {U}}}_1\) and \({{\mathfrak {U}}}_2\), the following holds:
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If \(\S \left( {{\mathfrak {U}}}_1\right) >\S \left( {{\mathfrak {U}}}_2\right) ,\) then \({{\mathfrak {U}}}_1> {{\mathfrak {U}}}_2\);
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if \(\S \left( {{\mathfrak {U}}}_1\right) =\S \left( {{\mathfrak {U}}}_2\right) ,\) then \({{\mathfrak {U}}}_1= {{\mathfrak {U}}}_2\);
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if \(\S \left( {{\mathfrak {U}}}_1\right) <\S \left( {{\mathfrak {U}}}_2\right) ,\) then \({{\mathfrak {U}}}_1< {{\mathfrak {U}}}_2\).
For any two HFEs \({{\mathfrak {U}}}_1\), and \({{\mathfrak {U}}}_2\), Torra13 and Xia and Xu33 presented the subsequent operations:
Definition 4
Suppose \({{\mathfrak {U}}}_1\), and \({{\mathfrak {U}}}_2\) are any two HFEs and \(\gimel >0\), then
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1.
\({{\mathfrak {U}}}_1\oplus {{\mathfrak {U}}}_2=\bigcup _{r=1,2,\ldots ,\#{{\mathfrak {U}}}_1,s=1,2,\ldots ,\#{{\mathfrak {U}}}_2} \left\{ \gtrdot _r+\ltimes _s-\gtrdot _r\ltimes _s\right\}\);
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2.
\({{\mathfrak {U}}}_1\otimes {{\mathfrak {U}}}_2=\bigcup _{r=1,2,\ldots ,\#{{\mathfrak {U}}}_1,s=1,2,\ldots ,\#{{\mathfrak {U}}}_2} \left\{ \gtrdot _r \ltimes _s \right\}\);
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3.
\(\gimel {{\mathfrak {U}}}_1=\bigcup _{r=1,2,\ldots ,\#{{\mathfrak {U}}}_1} \left\{ 1-\left( 1-\gtrdot _r\right) ^{\gimel } \right\}\);
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4.
\(\left( {{\mathfrak {U}}}_1\right) ^{\gimel }=\bigcup _{r=1,2,\ldots ,\#{{\mathfrak {U}}}_1} \left\{ {\gtrdot _r}^{\gimel } \right\}\);
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5.
\(\left( {{\mathfrak {U}}}_1\right) ^{\gimel }=\bigcup _{k=1,2,\ldots ,\#{{\mathfrak {U}}}_1} \left\{ 1-\gtrdot _r \right\}\).
Definition 5
34 A NHFS \({\mathscr {Y}}\) on \(\Xi\) is referred as
where \({\mathscr {M}}\left( \varepsilon \right) , {\mathscr {N}}\left( \varepsilon \right) , and {\mathscr {L}}\left( \varepsilon \right)\) shows the membership function, neutral part, and non-membership functions \(\varepsilon \in \Xi\) yield a subset of the unit interval \(\left[ 0,1 \right]\), accordingly, with the constraint:
where \(\gtrdot \left( \varepsilon \right) , \ltimes \left( \varepsilon \right) , and \lessdot \left( \varepsilon \right)\) are the membership value, neutral part, and non-membership value of \(\varepsilon \in \Xi\) in \({\mathscr {Y}}\). A neutrosophic hesitant fuzzy element (NHFE) is indicated by \(\Theta =({\mathscr {M}}\left( \varepsilon \right) ,{\mathscr {N}}\left( \varepsilon \right) ,{\mathscr {L}}\left( \varepsilon \right) )\) for our assistance.
Definition 6
34 The Score function and Accuracy function for NHFE \(\Theta\) are shown as:
where \(\alpha ,\beta , and \gamma\) are number of terms in \(\gtrdot ,\ltimes , and \lessdot\), correspondingly.
Definition 7
34 For any two NHFEs \(\Theta _1\) and \(\Theta _2\), the below mentioned contrast holds:
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i.
\(\S \left( \Theta _1\right) >\S \left( \Theta _2\right)\);
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ii.
\(\S \left( \Theta _1\right) =\S \left( \S _2\right)\) and \(\wp \left( \Theta _1\right) >\wp \left( \Theta _2\right)\).
Definition 8
35 MSM is distinguished as follows:
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i.
If each \(\Psi _k=\Psi \left( k=1,2,\ldots ,n\right) ,\) then \(MSM^{\varrho }\left( \Psi _1,\Psi _2,\ldots ,\Psi _n \right) =\phi\).
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ii.
If \(\Phi _k\le \Psi _k\left( k=1,2,\ldots ,n\right) ,\) then \(MSM^{\varrho }\left( \Phi _1,\Phi _2,\ldots ,\Phi _n \right) \le MSM^{\varrho }\left( \Psi _1,\Psi _2,\ldots ,\Psi _n \right)\).
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iii.
\(\min _{k}\left\{ \Psi _k \right\} \le MSM^{\varrho }\left( \Psi _1,\Psi _2,\ldots ,\Psi _n \right) \le \max _{k}\left\{ \Psi _k \right\} .\)
Proposed NHFPMSM operator and NHFWPMSM operator
Here, we present NHFPMSM and NHFWPMSM operators based on the implementation of the PMSM operator.
NHFPMSM aggregation operator
Definition 9
22 Assume that \(\Theta _1, \Theta _2,\ldots , \Theta _n\) are \('n^{\prime }\) NHFEs. Then, the “Neutrosophic Hesitant Fuzzy Partitioned Maclaurin Symmetric Mean Aggregation (NHFPMSM) Operator is classified as
where \(\alpha\) represent the number of kinds, \(\varrho\) is a parameter which varies from 1 to \(o_\imath\), \(o_\imath\) indicates the number of standards in kinds \(\tilde{p}_\imath\), \(\left( \kappa _1,\kappa _2,\ldots ,\kappa _{\varrho } \right)\) involves all the \(\varrho\)-tuples of \(\left( 1,2,\ldots ,o_{\imath } \right)\), \({\mathscr {C}}^{\varrho }_{o_\imath }\) shows the binomial coefficient, that is \({\mathscr {C}}^{\varrho }_{o_\imath }=\frac{o_\imath !}{\varrho !\left( o_\imath -\varrho \right) !}\).
Theorem 1
Suppose there are \(\Theta _\kappa\) \(\left( 1,2,\ldots ,n\right)\) NHFEs. The aggregated yield is still NHFE which is expressed as below:
Proof
By using the operations of NHFS, we have
Hence, the proof is done. \(\square\)
Theorem 2
(Idempotency). If all NHFEs \(\Theta _{\kappa }(\kappa =1,2,3,\ldots ,n)\) are equal i.e., \(\Theta _{\kappa }\) = \(\Theta (\kappa =1,2,\ldots ,n)\).
Proof
From Eq. (7), we have
\(\square\)
Theorem 3
(Monotonicity) Assume that \(\Theta _{1},\Theta _{2},\ldots ,\Theta _{n}\) are n NHFEs where \(\Theta _{\kappa }=(\gtrdot _{\kappa },\ltimes _{\kappa },\lessdot _{\kappa });\kappa =1,2,\ldots ,n\). Also assume that \(\Theta ^{\prime }_{1},\Theta ^{\prime }_{2},\ldots ,\Theta ^{\prime }_{n}\) where \(\Theta ^{..}_{\kappa }\)=\((\gtrdot ^{\prime }_{\kappa },\ltimes ^{\prime }_{\kappa },\lessdot ^{\prime }_{\kappa }); \kappa =1,2,\ldots ,n\) with the constraint \(\gtrdot _{\kappa }\) \(\ge\) \(\gtrdot ^{\prime }_{\kappa }\), \(\ltimes _{\kappa }\) \(\le\) \(\ltimes ^{\prime }_{\kappa }\) and \(\lessdot _{\kappa }\) \(\le\) \(\lessdot ^{\prime }_{\kappa }\) for all \(\kappa =1,2,\ldots ,n\). Then
Proof
Since \({\varrho }\) \(\ge\) 1 and \({\mathscr {C}}^{\varrho }_{o_\imath }\) \(\ge\) 1. At first, assume that the membership portion \(\gtrdot _{\kappa }\) \(\ge\) \(\gtrdot ^{\prime }_{\kappa }\) for all \(\kappa\), we have \(\prod ^{\varrho }_{j=1} \gtrdot _{\kappa j}\) \(\ge\) \(\prod ^{\varrho }_{j=1} \gtrdot ^{\prime }_{\kappa j}\). Thus
Now assume that the neutral portion \(\ltimes _{\kappa }\) \(\le\) \(\ltimes ^{\prime }_{\kappa }\) for all \(\kappa\) , we have \(\prod ^{\varrho }_{j=1}\left( 1-\ltimes _{\kappa j}\right)\) \(\ge\) \(\prod ^{\varrho }_{j=1}\left( 1-\ltimes ^{\prime }_{\kappa j}\right)\). Thus
Now assume that the non-membership portion \(\lessdot _{\kappa }\) \(\le\) \(\lessdot ^{\prime }_{\kappa }\) for all \(\kappa\) , we have \(\prod ^{\varrho }_{j=1}\left( 1-\lessdot _{\kappa j}\right)\) \(\ge\) \(\prod ^{\varrho }_{j=1}\left( 1-\lessdot ^{\prime }_{\kappa j}\right)\). Thus
Now equate the terms of \(NHFPMSM^{(\varrho )}\left( {\Theta }_1,{\Theta }_2,\ldots ,{\Theta }_n \right)\) with \(NHFPMSM^{(\varrho )}({\Theta }^{\prime }_{1},{\Theta }^{\prime }_{2},\ldots ,{\Theta }_{n})\). Assume that \(\Theta\)=\(\left( \gtrdot _{\Theta },\ltimes _{\Theta },\lessdot _{\Theta }\right)\)=\(NHFPMSM^{(\varrho )}\left( {\Theta }_1,{\Theta }_2,\ldots ,{\Theta }_n \right)\) and suppose that \(\Theta ^{\prime }\)=\((\gtrdot _{\Theta ^{\prime }},\ltimes _{\Theta ^{\prime }},\lessdot _{\Theta ^{\prime }})\)=\(NHFPMSM^{(\varrho )} ({\Theta }^{\prime }_{1},{\Theta }^{\prime }_{2},\ldots ,{\Theta }_{n})\). Thus
clearly, \(NHFPMSM^{(\varrho )}\left( {\Theta }_1,{\Theta }_2,\ldots ,{\Theta }_n \right)\) \(\ge\) \(NHFPMSM^{(\varrho )} \left( {\Theta }^{\prime }_{1},{\Theta }^{\prime }_{2},\ldots ,{\Theta }_{n} \right)\). So, it completes the proof. \(\square\)
Theorem 4
(Boundedness) Consider there is arbitrary NHFE \(\Theta ^{-}=\min _{\kappa }{\Theta _{\kappa }}\) and \(\Theta ^{+}=\max _{\kappa }{\Theta _{\kappa }}\) for \((\kappa =1,2,\ldots ,n)\), then
Proof
As \(\Theta ^{-}\)=\(\min _{\kappa }{\Theta _{\kappa }}\) \(\le\) \({\Theta _{\kappa }}\) from Theorems 2 and 3 ,we can write
Similarly
Thus
\(\square\)
Theorem 5
(Monotonicity via parameter). Let \(\Theta _{1}\),\(\Theta _{2}\),... and \(\Theta _{n}\) be NHFEs, where \(\Theta _{\kappa }\)=\((\gtrdot _{\kappa },\ltimes _{\kappa })\) for \(\kappa\)=1,2,...,n and let \(\varrho = 1,2,\ldots ,\min _{\imath }{{o_\imath }}\). Then, the NHFPMSM operator is monotonically increased when \(\varrho\) is decreased.
Proof
By Eq. (7), we have
Suppose that
and
Now we are to show that the function \({\mathfrak {I}}_{\gtrdot }(\varrho )\) is monotonically increased when the parameter \(\varrho\) is decreased. By Maclaurin inequality, we have
Now we can get
Next, we solve by using the contradiction method. Suppose on contrarily that it is monotonically decreased when the \(\varrho\) decreased, \(\mathfrak {I_{\gtrdot }}\) \(\left( \min _{\imath } \left( {o_\imath }\right) \right)\) \(\ge\) ...\(\ge\) \(\mathfrak {I_{\gtrdot }}\) \(\left( 2\right)\) \(\ge\) \(\mathfrak {I_{\gtrdot }}\) \(\left( 1\right)\), thus
Now suppose each standards have identical kinds , especially \(o_\imath\)=\(o(\imath =1,2,\ldots ,\alpha )\) because \(\min _{\imath } (o_\imath )\)=o, we have
since \({\mathfrak {I}}(o_\imath )\) \(\ge\) \({\mathfrak {I}}(1)\). We have
Thus, \((\prod \limits ^{o}_{\kappa _{j}=1}\gtrdot _{\kappa _{j}}) ^\frac{1}{o}\) \(\ge\) \((\frac{\sum \nolimits ^{o}_{\kappa _{j}=1} \gtrdot _{\kappa _{j}}}{{o}})\) which is contradiction against the Theorem 1. Thus, the function \({\mathfrak {I}}(\varrho )\) is monotonically increased when \(\varrho\) decreased. Moreover, we can prove for the function \({\mathfrak {H}}(\varrho )\). So, we have
\(\square\)
Theorem 6
Assume that there are ’n’ NHFEs \(\Theta _{\kappa } (\kappa =1,2,\ldots ,n)\), \(j=1,2,\ldots ,\min _{\imath }{o_{\imath }}\). Then
Now we investigate some unusual cases of the NHFPMSM operator concerning various parameters.
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Case 1:
Assume that \(\alpha _{1}\) is the single kind, the number of sets in \(\alpha _{1} = n\), and the parameter \(\varrho\) are determined by the NHFPMSM operator, we have
$$\begin{aligned} & NHFPMSM^{(\varrho )}\left( {\Theta }_1,{\Theta }_2,\ldots ,{\Theta }_n \right) \nonumber \\ & \quad =\bigcup _{\gtrdot _{\kappa j}\in \Theta _{\kappa j},\ltimes _{\kappa j}\in \Theta _{\kappa j},\lessdot _{\kappa j}\in \Theta _{\kappa j}}\left\{ \begin{array}{c} 1 - \left( \prod \limits ^{1}_{\imath =1}\left( 1 - \left( 1 - \left( \prod \limits _{1<\kappa _{1}< \cdots<o_\imath } \left( 1 - \prod \limits ^{\varrho }_{j=1}\gtrdot _{\kappa j}\right) \right) ^{\frac{1}{{\mathscr {C}}^{\varrho }_{o_\imath }}}\right) ^{\frac{1}{\varrho }}\right) \right) ^{\frac{1}{1}}, \\ \left( \prod \limits ^{1}_{\imath =1}\left( 1 - \left( 1 - \left( \prod \limits _{1<\kappa _{1}< \cdots<o_\imath } \left( 1 - \prod \limits ^{\varrho }_{j=1}\left( 1 - \ltimes _{\kappa j}\right) \right) \right) ^{\frac{1}{{\mathscr {C}}^{\varrho }_{o_\imath }}}\right) ^\frac{1}{\varrho }\right) \right) ^{\frac{1}{1}}, \\ \left( \prod \limits ^{1}_{\imath =1}\left( 1 - \left( 1 - \left( \prod \limits _{1<\kappa _{1}< \cdots<o_\imath } \left( 1 - \prod \limits ^{\varrho }_{j=1}\left( 1 - \lessdot _{\kappa j}\right) \right) \right) ^{\frac{1}{{\mathscr {C}}^{\varrho }_{o_\imath }}}\right) ^\frac{1}{\varrho }\right) \right) ^{\frac{1}{1}} \end{array} \right\} .\nonumber \\ & \quad =\bigcup _{\gtrdot _{\kappa j}\in \Theta _{\kappa j},\ltimes _{\kappa j}\in \Theta _{\kappa j},\lessdot _{\kappa j}\in \Theta _{\kappa j}}\left\{ \begin{array}{c} \bigg (1-\biggl (\prod \limits _{1<\kappa _{1}< \cdots<o_\imath } \biggl (1-\prod \limits ^{\varrho }_{j=1}\gtrdot _{\kappa j}\biggr )\biggr )^{\frac{1}{{\mathscr {C}}^{\varrho }_{o_\imath }}}\bigg )^{\frac{1}{\varrho }}, \\ 1-\bigg (1-\bigg (\prod \limits _{1<\kappa _{1}< \cdots<o_\imath } \bigg (1-\prod \limits ^{\varrho }_{j=1}\bigg (1-\ltimes _{\kappa j}\bigg )\bigg )\bigg )^{\frac{1}{{\mathscr {C}}^{\varrho }_{o_\imath }}} \bigg )^\frac{1}{\varrho },\\ 1-\bigg (1-\bigg (\prod \limits _{1<\kappa _{1}< \cdots <o_\imath } \bigg (1-\prod \limits ^{\varrho }_{j=1}\bigg (1-\lessdot _{\kappa j}\bigg )\bigg )\bigg )^{\frac{1}{{\mathscr {C}}^{\varrho }_{o_\imath }}} \bigg )^\frac{1}{\varrho } \end{array}\right\} \end{aligned}$$(11)This is the called Neutrosophic Hesitant Fuzzy MSM operator.
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Case 2:
Now if \(\alpha =1\) and \(\varrho =1\) then bu using the NHFPMSMS operator , we have
$$\begin{aligned} & NHFPMSM^{(1)}\left( {\Theta }_1,{\Theta }_2,\ldots ,{\Theta }_n \right) \\ & \quad =\bigcup _{\gtrdot _{\kappa j},\ltimes _{\kappa j},\lessdot _{\kappa j}\in \Theta _{\kappa j}}\left\{ \begin{array}{c} 1-\bigg (\prod \limits ^{1}_{\imath =1}\bigg (1-\bigg (1-\biggl (\prod \limits _{1<\kappa _{1}< \cdots<o_\imath } \biggl (1-\prod \limits ^{1}_{j=1}\gtrdot _{\kappa j}\biggr )\biggr )^{\frac{1}{{\mathscr {C}}^{1}_{o_\imath }}}\bigg )^{\frac{1}{1}}\bigg )\bigg )^\frac{1}{1}, \\ \bigg (\prod \limits ^{1}_{\imath =1}\bigg (1-\bigg (1-\bigg (\prod \limits _{1<\kappa _{1}< \cdots<o_\imath } \bigg (1-\prod \limits ^{1}_{j=1}\bigg (1-\ltimes _{\kappa j}\bigg )\bigg )\bigg )^{\frac{1}{{\mathscr {C}}^{1}_{o_\imath }}} \bigg )^\frac{1}{1}\bigg )\bigg )^{\frac{1}{1}},\\ \bigg (\prod \limits ^{1}_{\imath =1}\bigg (1-\bigg (1-\bigg (\prod \limits _{1<\kappa _{1}< \cdots<o_\imath } \bigg (1-\prod \limits ^{1}_{j=1}\bigg (1-\lessdot _{\kappa j}\bigg )\bigg )\bigg )^{\frac{1}{{\mathscr {C}}^{1}_{o_\imath }}} \bigg )^\frac{1}{1}\bigg )\bigg )^{\frac{1}{1}} \end{array}\right\} \\ & \quad =\bigcup _{\gtrdot _{\kappa _{1}},\ltimes _{\kappa _{1}}, \lessdot _{\kappa _{1}}}\left\{ \begin{array}{c} 1-\biggl (\prod \limits _{1<\kappa _{1}<o_\imath } \biggl (1-\gtrdot _{\kappa _{1}}\biggr )\biggr )^{\frac{1}{{\mathscr {C}}^{1}_{o_\imath }}},\\ \bigg (\prod \limits _{1<\kappa _{1}<o_\imath } \bigg (1-\bigg (1-\ltimes _{\kappa _{1}}\bigg )\bigg )\bigg )^{\frac{1}{{\mathscr {C}}^{1}_{o_\imath }}},\\ \bigg (\prod \limits _{1<\kappa _{1}<o_\imath } \bigg (1-\bigg (1-\lessdot _{\kappa _{1}}\bigg )\bigg )\bigg )^{\frac{1}{{\mathscr {C}}^{1}_{o_\imath }}} \end{array}\right\} \end{aligned}$$let \(\left( \kappa _{1}=\kappa \right)\), we have
$$\begin{aligned} & =\bigcup _{\gtrdot _{\kappa _{1}} \in \Theta _{\kappa _{1}},\ltimes _{\kappa _{1}} \in \Theta _{\kappa _{1}}, \lessdot _{\kappa _{1}} \in \Theta _{\kappa _{1}}}\left\{ \begin{array}{c} 1-\biggl (\prod \limits _{\kappa =1}^{o_\imath } \biggl (1-\gtrdot _{\kappa }\biggr )\biggr )^{\frac{1}{o_\imath }},\bigg (\prod \limits _{\kappa =1}^{o_\imath } \ltimes _{\kappa }\bigg )^{\frac{1}{o_\imath }} \bigg (\prod \limits _{\kappa =1}^{o_\imath } \lessdot _{\kappa }\bigg )^{\frac{1}{o_\imath }} \end{array}\right\} \end{aligned}$$which is shortened to Neutrosophic Hesitant Fuzzy Averaging Operator.
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Case 3:
Now if \(\alpha =1\) and \(\varrho =2\) then by using NHFPMSMS operator, we have
$$\begin{aligned} & NHFPMSM^{(2)}\left( {\Theta }_1,{\Theta }_2,\ldots ,{\Theta }_n \right) \nonumber \\ & \quad =\bigcup _{\gtrdot _{\kappa j}\in \Theta _{\kappa j},\ltimes _{\kappa j}\in \Theta _{\kappa j},\lessdot _{\kappa j}\in \Theta _{\kappa j}}\left\{ \begin{array}{c} 1-\left( \prod \limits ^{1}_{\imath =1}\left( 1-\left( 1-\left( \prod \limits _{1<\kappa _{1}< \cdots<o_\imath } \left( 1-\prod \limits ^{2}_{j=1}\gtrdot _{\kappa j}\right) \right) ^{\frac{1}{{\mathscr {C}}^{1}_{o_\imath }}}\right) ^{\frac{1}{2}}\right) \right) ^\frac{1}{1}, \\ \left( \prod \limits ^{1}_{\imath =1}\left( 1-\left( 1-\left( \prod \limits _{1<\kappa _{1}< \cdots<o_\imath } \left( 1-\prod \limits ^{2}_{j=1}\left( 1-\ltimes _{\kappa j}\right) \right) \right) ^{\frac{1}{{\mathscr {C}}^{1}_{o_\imath }}} \right) ^\frac{1}{2}\right) \right) ^{\frac{1}{1}},\\ \left( \prod \limits ^{1}_{\imath =1}\left( 1-\left( 1-\left( \prod \limits _{1<\kappa _{1}< \cdots<o_\imath } \left( 1-\prod \limits ^{2}_{j=1}\left( 1-\lessdot _{\kappa j}\right) \right) \right) ^{\frac{1}{{\mathscr {C}}^{1}_{o_\imath }}} \right) ^\frac{1}{2}\right) \right) ^{\frac{1}{1}} \end{array}\right\} \nonumber \\ & \quad =\bigcup _{\gtrdot _{\kappa j}\in \Theta _{\kappa j},\ltimes _{\kappa j}\in \Theta _{\kappa j},\lessdot _{\kappa j}\in \Theta _{\kappa j}}\left\{ \begin{array}{c} \left( 1-\left( \prod \limits _{1<\kappa _{1}<i_{2}<o_\imath } \left( 1-\gtrdot _{\kappa _{1}}\gtrdot _{\kappa _{2}}\right) \right) ^{\frac{1}{{\mathscr {C}}^{2}_{o_\imath }}}\right) ^{\frac{1}{2}}, \\ 1-\left( 1-\left( \prod \limits _{1<\kappa _{1}<\kappa _{2}<o_\imath } \left( 1-\left( 1-\ltimes _{\kappa _{1}}\ltimes _{\kappa _{2}}\right) \right) \right) ^{\frac{1}{{\mathscr {C}}^{2}_{o_\imath }}} \right) ^\frac{1}{2},\\ 1-\left( 1-\left( \prod \limits _{1<\kappa _{1}<\kappa _{2}<o_\imath } \left( 1-\left( 1-\lessdot _{\kappa _{1}}\lessdot _{\kappa _{2}}\right) \right) \right) ^{\frac{1}{{\mathscr {C}}^{2}_{o_\imath }}} \right) ^\frac{1}{2} \end{array}\right\} \nonumber \\ & \quad =\bigcup _{\gtrdot _{\kappa j}\in \Theta _{\kappa j},\ltimes _{\kappa j}\in \Theta _{\kappa j},\lessdot _{\kappa j}\in \Theta _{\kappa j}}\left\{ \begin{array}{c} \left( 1-\left( \prod \limits _{1<\kappa _{1}<\kappa _{2}<o_\imath } \left( 1-\gtrdot _{\kappa _{1}}\gtrdot _{\kappa _{2}}\right) \right) ^{\frac{2}{o_{\imath }(o_{\imath }-1)}}\right) ^{\frac{1}{2}},\\ 1-\left( 1-\left( \prod \limits _{1<\kappa _{1}<\kappa _{2}<o_\imath } \left( \ltimes _{\kappa _{1}}\ltimes _{\kappa _{2}}\right) \right) ^{\frac{2}{o_{\imath }(o_{\imath }-1)}} \right) ^\frac{1}{2},\\ 1-\left( 1-\left( \prod \limits _{1<\kappa _{1}<\kappa _{2}<o_\imath } \left( \lessdot _{\kappa _{1}}\lessdot _{\kappa _{2}}\right) \right) ^{\frac{2}{o_{\imath }(o_{\imath }-1)}} \right) ^\frac{1}{2} \end{array}\right\} \nonumber \\ & \quad =\bigcup _{\gtrdot _{\kappa j}\in \Theta _{\kappa j},\ltimes _{\kappa j}\in \Theta _{\kappa j},\lessdot _{\kappa j}\in \Theta _{\kappa j}}\left\{ \begin{array}{c} \left( 1-\left( \prod \limits _{\kappa _{1},\kappa _{2}=1;\kappa _{1}=\kappa _{2}}^{o_\imath } \left( 1-\gtrdot _{\kappa _{1}}\gtrdot _{\kappa _{2}}\right) \right) ^{\frac{1}{2}.\frac{2}{o_{\imath }(o_{\imath }-1)}}\right) ^{\frac{1}{2}},\\ 1-\left( 1-\left( \prod \limits _{\kappa _{1},\kappa _{2}=1;\kappa _{1}=\kappa _{2}}^{o_\imath } \left( \ltimes _{\kappa _{1}}\ltimes _{\kappa _{2}}\right) \right) ^{\frac{1}{2}.\frac{2}{o_{\imath }(o_{\imath }-1)}} \right) ^\frac{1}{2},\\ 1-\left( 1-\left( \prod \limits _{\kappa _{1},\kappa _{2}=1;\kappa _{1}=\kappa _{2}}^{o_\imath } \left( \lessdot _{\kappa _{1}}\lessdot _{\kappa _{2}}\right) \right) ^{\frac{1}{2}.\frac{2}{o_{\imath }(o_{\imath }-1)}} \right) ^\frac{1}{2} \end{array}\right\} \nonumber \\ & \quad =\bigcup _{\gtrdot _{\kappa j}\in \Theta _{\kappa j},\ltimes _{\kappa j}\in \Theta _{\kappa j},\lessdot _{\kappa j}\in \Theta _{\kappa j}}\left\{ \begin{array}{c} \left( 1-\left( \prod \limits _{\kappa _{1},\kappa _{2}=1;\kappa _{1}=\kappa _{2}}^{o_\imath } \left( 1-\gtrdot _{\kappa _{1}}\gtrdot _{\kappa _{2}}\right) \right) ^{\frac{1}{o_{\imath }(o_{\imath }-1)}}\right) ^{\frac{1}{2}}, \\ 1-\left( 1-\left( \prod \limits _{\kappa _{1},\kappa _{2}=1;\kappa _{1}=\kappa _{2}}^{o_\imath } \left( \ltimes _{\kappa _{1}}\ltimes _{\kappa _{2}}\right) \right) ^{\frac{1}{o_{\imath }(o_{\imath }-1)}} \right) ^\frac{1}{2},\\ 1-\left( 1-\left( \prod \limits _{\kappa _{1},\kappa _{2}=1;\kappa _{1}=\kappa _{2}}^{o_\imath } \left( \lessdot _{\kappa _{1}}\lessdot _{\kappa _{2}}\right) \right) ^{\frac{1}{o_{\imath }(o_{\imath }-1)}} \right) ^\frac{1}{2} \end{array}\right\} , \end{aligned}$$(12)which is shorten to Neutrosophic Hesitant Fuzzy Bonferronin Mean Operator \(DHFBM^{(1,1)}\left( {\Theta }_1,{\Theta }_2,\ldots ,{\Theta }_n \right)\)36.
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Case 4:
Now if \(\alpha =1\) and \(\varrho =o_{\imath }\) then by using the NHFPMSMS operator, we have
$$\begin{aligned} & NHFPMSM^{(n)}\left( {\Theta }_1,{\Theta }_2,\ldots ,{\Theta }_n \right) \nonumber \\ & \quad =\bigcup _{\gtrdot _{\kappa j}\in \Theta _{\kappa j},\ltimes _{\kappa j}\in \Theta _{\kappa j},\lessdot _{\kappa j}\in \Theta _{\kappa j}}\left\{ \begin{array}{c} 1-\left( \prod \limits ^{1}_{\imath =1}\left( 1-\left( 1-\left( \prod \limits _{1<\kappa _{1}< \cdots<o_\imath } \left( 1-\prod \limits ^{o_\imath }_{j=1}\gtrdot _{\kappa j}\right) \right) ^{\frac{1}{{\mathscr {C}}^{o_\imath }_{o_\imath }}}\right) ^{\frac{1}{o_\imath }}\right) \right) ^\frac{1}{1}, \\ \left( \prod \limits ^{1}_{\imath =1}\left( 1-\left( 1-\left( \prod \limits _{1<\kappa _{1}< \cdots<o_\imath } \left( 1-\prod \limits ^{o_\imath }_{j=1}\left( 1-\ltimes _{\kappa j}\right) \right) \right) ^{\frac{1}{{\mathscr {C}}^{o_\imath }_{o_\imath }}} \right) ^\frac{1}{o_\imath }\right) \right) ^{\frac{1}{1}},\\ \left( \prod \limits ^{1}_{\imath =1}\left( 1-\left( 1-\left( \prod \limits _{1<\kappa _{1}< \cdots <o_\imath } \left( 1-\prod \limits ^{o_\imath }_{j=1}\left( 1-\lessdot _{\kappa j}\right) \right) \right) ^{\frac{1}{{\mathscr {C}}^{o_\imath }_{o_\imath }}} \right) ^\frac{1}{o_\imath }\right) \right) ^{\frac{1}{1}} \end{array}\right\} \nonumber \\ & \quad =\bigcup _{\gtrdot _{\kappa j}\in \Theta _{\kappa j},\ltimes _{\kappa j}\in \Theta _{\kappa j},\lessdot _{\kappa j}\in \Theta _{\kappa j}}\left\{ \begin{array}{c} \left( 1- \left( 1-\prod \limits ^{o_\imath }_{j=1}\gtrdot _{\kappa j}\right) \right) ^{\frac{1}{o_\imath }},1-\left( 1- \left( 1-\prod \limits ^{\varrho }_{j=1}\left( 1-\ltimes _{\kappa j}\right) \right) \right) ^\frac{1}{\varrho },\\ 1-\left( 1- \left( 1-\prod \limits ^{\varrho }_{j=1}\left( 1-\lessdot _{\kappa j}\right) \right) \right) ^\frac{1}{\varrho } \end{array}\right\} \nonumber \\ & \quad =\bigcup _{\gtrdot _{\kappa j}\in \Theta _{\kappa j},\ltimes _{\kappa j}\in \Theta _{\kappa j},\lessdot _{\kappa j}\in \Theta _{\kappa j}}\left\{ \begin{array}{c} \left( 1- \left( 1-\prod \limits ^{o_\imath }_{j=1}\gtrdot _{\kappa j}\right) \right) ^{\frac{1}{o_\imath }},1-\left( \prod \limits ^{\varrho }_{j=1}\left( 1-\ltimes _{\kappa j}\right) \right) ^\frac{1}{o_\imath },\\ 1-\left( \prod \limits ^{\varrho }_{j=1}\left( 1-\lessdot _{\kappa j}\right) \right) ^\frac{1}{o_\imath } \end{array}\right\} . \end{aligned}$$(13)This is a Neutrosophic Hesitant Fuzzy Geometric Mean Operator.
NHFWPMSM aggregation operators
Here, we initiated the NHFWPMSM operator for NHFEs.
Definition 10
22 Assume that \({\Theta }_1,{\Theta }_2,\ldots ,{\Theta }_n\) are set of n NHFEs, then NHFWPMSM operator is described as
where \(\alpha\) represent the number of kinds, \(\varrho\) is a parameter which varies from 1 to \(o_\imath\), \(o_\imath\) indicates the number of standards in kinds \(\tilde{p}_\imath\), \(\left( \kappa _1,\kappa _2,\ldots ,\kappa _{\varrho } \right)\) involves all the \(\varrho\)-tuples of \(\left( 1,2,\ldots ,o_{\imath } \right)\), \({\mathscr {C}}^{\varrho }_{o_\imath }\) shows the binomial coefficient, that is \({\mathscr {C}}^{\varrho }_{o_\imath }=\frac{o_\imath !}{\varrho !\left( o_\imath -\varrho \right) !}\) and \(\varpi _{j}\) \(\ge\) 0 indicate the weights analogous to each attribute, fulfills \(\sum \nolimits ^{n}_{j=1}\varpi _{j}\)=1.
Theorem 7
Assume that there are \({\Theta }_i \left( 1,2,\ldots ,n\right)\) NHFEs. The aggregated yield is still NHFEs which is expressed as below:
Proof
It can be proven on the same lines as Theorem 1. \(\square\)
Example 1
Suppose that \(\Theta _{1}\) \(=\) \(\left\{ \left\{ 0.8,0.9\right\} ,\left\{ 0.4,0.8\right\} ,\left\{ 0.5\right\} \right\}\), \(\Theta _{2}\) \(=\) \(\left\{ \left\{ 0.7,0.8\right\} ,\left\{ 0.4\right\} ,\left\{ 0.3,0.4\right\} \right\}\) , \(\Theta _{3}\) \(=\) \(\left\{ \left\{ 0.3,0.6\right\} ,\left\{ 0.5,0.6\right\} ,\left\{ 0.4,0.5\right\} \right\}\) and \(\Theta _{4}\) \(=\) \(\left\{ \left\{ 0.9\right\} ,\left\{ 0.2,0.4\right\} ,\left\{ 0.7,0.8\right\} \right\}\) are four NHFEs types for two kinds \(\tilde{P}_{1}\) and \(\tilde{P}_{2}\) with \(\tilde{P}_{1}\)= \(\left\{ \Theta _{1},\Theta _{3}\right\}\) and \(\tilde{P}_{2}\) = \(\left\{ \Theta _{2},\Theta _{4}\right\}\) . Using the NHFPMSM operator to aggregate this NHFEs. let \(\varrho\) = 2 , then
Theorem 8
(Idempotency) If all NHFEs \(\Theta _{\kappa }(\kappa =1,2,3,\ldots ,n)\) are equal i.e., \(\Theta _{\kappa } = \Theta (\kappa =1,2,\ldots ,n)\). Then
Theorem 9
(Monotonicity) Assume that \(\Theta _{\kappa }\)=\((\gtrdot _{\kappa },\ltimes _{\kappa },\lessdot _{\kappa })\) and \(\Theta ^{\prime }_{\kappa }\)=\((\gtrdot ^{\prime }_{\kappa },\ltimes ^{\prime }_{\kappa },\lessdot ^{\prime }_{\kappa })\) are two element of NHFEs , for (\(\kappa =1,2,\ldots ,n\)) such that \(\gtrdot _{\kappa } \ge \gtrdot ^{\prime }_{\kappa }, \ltimes _{\kappa } \le \ltimes ^{\prime }_{\kappa }\) and \(\lessdot _{\kappa } \, \le \lessdot ^{\prime }_{\kappa }\) all \((\gtrdot _{\kappa },\ltimes _{\kappa },\lessdot _{\kappa })\) \(\in\) \(\Theta _{\kappa }\) and \((\gtrdot ^{\prime }_{\kappa },\ltimes ^{\prime }_{\kappa },\lessdot ^{\prime }_{\kappa })\) \(\in\) \(\Theta ^{\prime }_{\kappa }\) , then
Theorem 10
(Boundedness) Consider there is arbitrary NHFE \(\Theta ^{-}=\min _{\kappa }{\Theta _{\kappa }}\) and \(\Theta ^{+}=\max _{\kappa }{\Theta _{\kappa }}\) for \((\kappa =1,2,\ldots ,n)\), then
Theorem 11
(Monotonicity via parameter) Let \(\Theta _{1}\),\(\Theta _{2}\),...and \(\Theta _{n}\) be NHFEs, where \(\Theta _{\kappa }=(\gtrdot _{\kappa },\ltimes _{\kappa })\) for \(\kappa =1,2,\ldots ,n\) and let \(\varrho = 1,2,\ldots\), \(\min _{\imath }{{o_\imath }}\). Then, the NHFWPMSM operator is monotonically increased when \(\varrho\) is decreased.
Theorem 12
Assume that there are ’n’ NHFEs \(\Theta _{\kappa } (\kappa =1,2,\ldots ,n), j=1,2,\ldots ,\min _{\imath }{o_{\imath }}\). Then
Now we investigate some unusual cases of the NHFWPMSM operator concerning various parameters.
-
Case 1:
Assume that \(\alpha _{1}\) is the single kind, the number of sets in \(\alpha _{1} = n\), and the parameter \(\varrho\) are determined by the NHFWPMSM operator, we have
$$\begin{aligned} & NHFWPMSM^{(\varrho )}\left( {\Theta }_1,{\Theta }_2,\ldots ,{\Theta }_n \right) \nonumber \\ & \quad =\bigcup _{\gtrdot _{\kappa j}\in \Theta _{\kappa j},\ltimes _{\kappa j}\in \Theta _{\kappa j},\lessdot _{\kappa j}\in \Theta _{\kappa j}} \left\{ \begin{array}{c} 1-\left( \prod \limits ^{1}_{\imath =1}\left( 1-\left( 1-\left( \prod \limits _{1<\kappa _{1}< \cdots<o_\imath } \left( 1-\prod \limits ^{\varrho }_{j=1}\left( 1-(1-\gtrdot _{\kappa j})^\varpi _{\kappa _{j}}\right) \right) \right) ^{\frac{1}{{\mathscr {C}}^{\varrho }_{o_\imath }}} \right) ^{\frac{1}{\varrho }}\right) \right) ^\frac{1}{1}, \\ \left( \prod \limits ^{1}_{\imath =1}\left( 1-\left( 1-\left( \prod \limits _{1<\kappa _{1}< \cdots<o_\imath } \left( 1-\prod \limits ^{\varrho }_{j=1}\left( 1-\ltimes ^{\varpi _{\kappa j}}_{ij}\right) \right) \right) ^{\frac{1}{{\mathscr {C}}^{\varrho }_{o_\imath }}}\right) ^\frac{1}{\varrho }\right) \right) ^{\frac{1}{1}},\\ \left( \prod \limits ^{1}_{\imath =1}\left( 1-\left( 1-\left( \prod \limits _{1<\kappa _{1}< \cdots<o_\imath } \left( 1-\prod \limits ^{\varrho }_{j=1}\left( 1-\lessdot ^{\varpi _{\kappa j}}_{ij}\right) \right) \right) ^{\frac{1}{{\mathscr {C}}^{\varrho }_{o_\imath }}}\right) ^\frac{1}{\varrho }\right) \right) ^{\frac{1}{1}} \end{array}\right\} .\nonumber \\ & \quad =\bigcup _{\gtrdot _{\kappa j}\in \Theta _{\kappa j},\ltimes _{\kappa j}\in \Theta _{\kappa j},\lessdot _{\kappa j}\in \Theta _{\kappa j}} \left\{ \begin{array}{c} \left( 1-\left( \prod \limits _{1<\kappa _{1}< \cdots<o_\imath } \left( 1-\prod \limits ^{\varrho }_{j=1}\left( 1-(1-\gtrdot _{\kappa j})^\varpi _{\kappa _{j}}\right) \right) \right) ^{\frac{1}{{\mathscr {C}}^{\varrho }_{o_\imath }}} \right) ^{\frac{1}{\varrho }}, \\ 1-\left( 1-\left( \prod \limits _{1<\kappa _{1}< \cdots<o_\imath } \left( 1-\prod \limits ^{\varrho }_{j=1}\left( 1-\ltimes ^{\varpi _{\kappa j}}_{ij}\right) \right) \right) ^{\frac{1}{{\mathscr {C}}^{\varrho }_{o_\imath }}}\right) ^\frac{1}{\varrho },\\ 1-\left( 1-\left( \prod \limits _{1<\kappa _{1}< \cdots <o_\imath } \left( 1-\prod \limits ^{\varrho }_{j=1}\left( 1-{\updownarrow }^{\varpi _{\kappa j}}_{ij}\right) \right) \right) ^{\frac{1}{{\mathscr {C}}^{\varrho }_{o_\imath }}}\right) ^\frac{1}{\varrho } \end{array}\right\} , \end{aligned}$$(20)which is shortened to Neutrosophic Hesitant Fuzzy Weighted MSM Operator.
-
Case 2:
Now if \(\alpha =1\) and \(\varrho =1\) then bu using the NHFWPMSMS operator , we have
$$\begin{aligned} & NHFWPMSM^{(\varrho )}\left( {\Theta }_1,{\Theta }_2,\ldots ,{\Theta }_n \right) \nonumber \\ & \quad =\bigcup _{\gtrdot _{\kappa j}\in \Theta _{\kappa j},\ltimes _{\kappa j}\in \Theta _{\kappa j},\lessdot _{\kappa j}\in \Theta _{\kappa j}} \left\{ \begin{array}{c} 1-\left( \prod \limits ^{1}_{\imath =1}\left( 1-\left( 1-\left( \prod \limits _{1<\kappa _{1}< \cdots<o_\imath } \left( 1-\prod \limits ^{1}_{j=1}\left( 1-(1-\gtrdot _{\kappa j})^\varpi _{\kappa _{j}}\right) \right) \right) ^{\frac{1}{{\mathscr {C}}^{1}_{o_\imath }}} \right) ^{\frac{1}{1}}\right) \right) ^\frac{1}{1}, \\ \left( \prod \limits ^{1}_{\imath =1}\left( 1-\left( 1-\left( \prod \limits _{1<\kappa _{1}< \cdots<o_\imath } \left( 1-\prod \limits ^{1}_{j=1}\left( 1-\ltimes ^{\varpi _{\kappa j}}_{ij}\right) \right) \right) ^{\frac{1}{{\mathscr {C}}^{1}_{o_\imath }}}\right) ^\frac{1}{1}\right) \right) ^{\frac{1}{1}},\\ \left( \prod \limits ^{1}_{\imath =1}\left( 1-\left( 1-\left( \prod \limits _{1<\kappa _{1}< \cdots<o_\imath } \left( 1-\prod \limits ^{1}_{j=1}\left( 1-\lessdot ^{\varpi _{\kappa j}}_{ij}\right) \right) \right) ^{\frac{1}{{\mathscr {C}}^{1}_{o_\imath }}}\right) ^\frac{1}{1}\right) \right) ^{\frac{1}{1}} \end{array}\right\} .\nonumber \\ & \quad =\bigcup _{\gtrdot _{\kappa j}\in \Theta _{\kappa j},\ltimes _{\kappa j}\in \Theta _{\kappa j},\lessdot _{\kappa j}\in \Theta _{\kappa j}} \left\{ \begin{array}{c} 1-\left( \prod \limits _{1<\kappa _{1}< \cdots<o_\imath } \left( 1-\gtrdot _{\kappa j}\right) ^\varpi _{\kappa _{j}} \right) ^{\frac{1}{o_\imath }}, \left( \prod \limits _{1<\kappa _{1}< \cdots<o_\imath } \left( \ltimes ^{\varpi _{\kappa j}}_{ij}\right) \right) ^{\frac{1}{o_\imath }},\\ \left( \prod \limits _{1<\kappa _{1}< \cdots <o_\imath } \left( \lessdot ^{\varpi _{\kappa j}}_{ij}\right) \right) ^{\frac{1}{o_\imath }} \end{array}\right\} , \end{aligned}$$(21)which is shortened to Neutrosophic Hesitant Weighted Averaging Operator.
-
Case 3:
Now if \(\alpha =1\) and \(\varrho =2\) then by using NHFWPMSMS operator, we have
$$\begin{aligned} & NHFWPMSM^{(\varrho )}\left( {\Theta }_1,{\Theta }_2,\ldots ,{\Theta }_n \right) \nonumber \\ & \quad =\bigcup _{\gtrdot _{\kappa j}\in \Theta _{\kappa j},\ltimes _{\kappa j}\in \Theta _{\kappa j},\lessdot _{\kappa j}\in \Theta _{\kappa j}} \left\{ \begin{array}{c} 1-\left( \prod \limits ^{1}_{\imath =1}\left( 1-\left( 1-\left( \prod \limits _{1<\kappa _{1}< \cdots<o_\imath } \left( 1-\prod \limits ^{2}_{j=1}\left( 1-(1-\gtrdot _{\kappa j})^\varpi _{\kappa _{j}}\right) \right) \right) ^{\frac{1}{{\mathscr {C}}^{2}_{o_\imath }}} \right) ^{\frac{1}{2}}\right) \right) ^\frac{1}{1}, \\ \left( \prod \limits ^{1}_{\imath =1}\left( 1-\left( 1-\left( \prod \limits _{1<\kappa _{1}< \cdots<o_\imath } \left( 1-\prod \limits ^{2}_{j=1}\left( 1-\ltimes ^{\varpi _{\kappa j}}_{ij}\right) \right) \right) ^{\frac{1}{{\mathscr {C}}^{2}_{o_\imath }}}\right) ^\frac{1}{2}\right) \right) ^{\frac{1}{1}},\\ \left( \prod \limits ^{1}_{\imath =1}\left( 1-\left( 1-\left( \prod \limits _{1<\kappa _{1}< \cdots<o_\imath } \left( 1-\prod \limits ^{2}_{j=1}\left( 1-\lessdot ^{\varpi _{\kappa j}}_{ij}\right) \right) \right) ^{\frac{1}{{\mathscr {C}}^{2}_{o_\imath }}}\right) ^\frac{1}{2}\right) \right) ^{\frac{1}{1}} \end{array}\right\} .\nonumber \\ & \quad =\bigcup _{\gtrdot _{\kappa j}\in \Theta _{\kappa j},\ltimes _{\kappa j}\in \Theta _{\kappa j},\lessdot _{\kappa j}\in \Theta _{\kappa j}}\{\begin{array}{c} \left( 1-\left( \prod \limits _{1<\kappa _{1}<\kappa _{2}<o_\imath } \left( (1-\gtrdot _{\kappa _{1}})^\varpi _{\kappa _{1}}(1-\gtrdot _{\kappa _{2}})^\varpi _{\kappa _{2}}\right) \right) ^{\frac{2}{n(n-1)}} \right) ^{\frac{1}{2}}, \\ 1-\left( 1-\left( \prod \limits _{1<\kappa _{1}<\kappa _{2}<o_\imath } \left( \ltimes ^{\varpi _{\kappa _{1}}}_{\kappa _{1}}\ltimes ^{\varpi _{\kappa _{2}}}_{\kappa _{2}}\right) \right) ^{\frac{2}{n(n-1)}}\right) ^\frac{1}{2},\\ 1-\left( 1-\left( \prod \limits _{1<\kappa _{1}<\kappa _{2}<o_\imath } \left( \lessdot ^{\varpi _{\kappa _{1}}}_{\kappa _{1}}\lessdot ^{\varpi _{\kappa _{2}}}_{\kappa _{2}}\right) \right) ^{\frac{2}{n(n-1)}}\right) ^\frac{1}{2} \end{array} \}\nonumber \\ & \quad =\bigcup _{\gtrdot _{\kappa j}\in \Theta _{\kappa j},\ltimes _{\kappa j}\in \Theta _{\kappa j},\lessdot _{\kappa j}\in \Theta _{\kappa j}} \{\begin{array}{c} \left( 1-\left( \prod \limits _{\kappa _{1},\kappa _{2}=1;\kappa _{1}=\kappa _{2}}^{o_{\imath }} \left( (1-\gtrdot _{\kappa _{1}})^\varpi _{\kappa _{1}}(1-\gtrdot _{\kappa _{2}})^\varpi _{\kappa _{2}}\right) \right) ^{\frac{1}{2}\frac{2}{o_{\imath }(o_{\imath }-1)}} \right) ^{\frac{1}{2}}, \\ 1-\left( 1-\left( \prod \limits _{\kappa _{1},\kappa _{2}=1;\kappa _{1}=\kappa _{2}}^{o_{\imath }} \left( \ltimes ^{\varpi _{\kappa _{1}}}_{\kappa _{1}}\ltimes ^{\varpi _{\kappa _{2}}}_{\kappa _{2}}\right) \right) ^{\frac{1}{2}.\frac{2}{o_{\imath }(o_{\imath }-1)}}\right) ^\frac{1}{2},\\ 1-\left( 1-\left( \prod \limits _{\kappa _{1},\kappa _{2}=1;\kappa _{1}=\kappa _{2}}^{o_{\imath }} \left( \lessdot ^{\varpi _{\kappa _{1}}}_{\kappa _{1}}\lessdot ^{\varpi _{\kappa _{2}}}_{\kappa _{2}}\right) \right) ^{\frac{1}{2}.\frac{2}{o_{\imath }(o_{\imath }-1)}}\right) ^\frac{1}{2} \end{array}\}\nonumber \\ & \quad =\bigcup _{\gtrdot _{\kappa j}\in \Theta _{\kappa j},\ltimes _{\kappa j}\in \Theta _{\kappa j},\lessdot _{\kappa j}\in \Theta _{\kappa j}} \{\begin{array}{c} \left( 1-\left( \prod \limits _{\kappa _{1},\kappa _{2}=1;\kappa _{1}=\kappa _{2}}^{o_{\imath }} \left( (1-\gtrdot _{\kappa _{1}})^\varpi _{\kappa _{1}}(1-\gtrdot _{\kappa _{2}})^\varpi _{\kappa _{2}}\right) \right) ^{\frac{1}{o_\imath (o_{\imath }-1)}} \right) ^{\frac{1}{2}}, \\ 1-\left( 1-\left( \prod \limits _{\kappa _{1},\kappa _{2}=1;\kappa _{1}=\kappa _{2}}^{o_{\imath }} \left( \ltimes ^{\varpi _{\kappa _{1}}}_{\kappa _{1}}\ltimes ^{\varpi _{\kappa _{2}}}_{\kappa _{2}}\right) \right) ^{\frac{1}{o_{\imath }(o_{\imath }-1)}}\right) ^\frac{1}{2},\\ 1-\left( 1-\left( \prod \limits _{\kappa _{1},\kappa _{2}=1;\kappa _{1}=\kappa _{2}}^{o_{\imath }} \left( \lessdot ^{\varpi _{\kappa _{1}}}_{\kappa _{1}}\lessdot ^{\varpi _{\kappa _{2}}}_{\kappa _{2}}\right) \right) ^{\frac{1}{o_{\imath }(o_{\imath }-1)}}\right) ^\frac{1}{2} \end{array}\}, \end{aligned}$$(22)which is shortened to Neutrosophic Hesitant Fuzzy Weighted Bonferroni Mean Operator
\(DHFWBM^{(1,1)}\left( \Theta _{1},\Theta _{2},\ldots ,\Theta _{n}\right)\).
-
Case 4:
Now if \(\alpha =1\) and \(\varrho =o_{\imath }\) then by using the NHFWPMSMS operator , we have
$$\begin{aligned} & NHFWPMSM^{(\varrho )}\left( {\Theta }_1,{\Theta }_2,\ldots ,{\Theta }_n \right) \nonumber \\ & \quad =\bigcup _{\gtrdot _{\kappa j}\in \Theta _{\kappa j},\ltimes _{\kappa j}\in \Theta _{\kappa j},\lessdot _{\kappa j}\in \Theta _{\kappa j}} \left\{ \begin{array}{c} 1-\left( \prod \limits ^{1}_{\imath =1}\left( 1-\left( 1-\left( \prod \limits _{1<\kappa _{1}< \cdots<o_\imath } \left( 1-\prod \limits ^{o_{\imath }}_{j=1}\left( 1-(1-\gtrdot _{\kappa j})^\varpi _{\kappa _{j}}\right) \right) \right) ^{\frac{1}{{\mathscr {C}}^{o_{\imath }}_{o_\imath }}} \right) ^{\frac{1}{o_{\imath }}}\right) \right) ^\frac{1}{1}, \\ \left( \prod \limits ^{1}_{\imath =1}\left( 1-\left( 1-\left( \prod \limits _{1<\kappa _{1}< \cdots<o_\imath } \left( 1-\prod \limits ^{o_{\imath }}_{j=1}\left( 1-\ltimes ^{\varpi _{\kappa j}}_{ij}\right) \right) \right) ^{\frac{1}{{\mathscr {C}}^{o_{\imath }}_{o_\imath }}}\right) ^\frac{1}{o_{\imath }}\right) \right) ^{\frac{1}{1}},\\ \left( \prod \limits ^{1}_{\imath =1}\left( 1-\left( 1-\left( \prod \limits _{1<\kappa _{1}< \cdots <o_\imath } \left( 1-\prod \limits ^{o_{\imath }}_{j=1}\left( 1-\lessdot ^{\varpi _{\kappa j}}_{ij}\right) \right) \right) ^{\frac{1}{{\mathscr {C}}^{o_{\imath }}_{o_\imath }}}\right) ^\frac{1}{o_{\imath }}\right) \right) ^{\frac{1}{1}} \end{array}\right\} .\nonumber \\ & \quad =\bigcup _{\gtrdot _{\kappa j}\in \Theta _{\kappa j},\ltimes _{\kappa j}\in \Theta _{\kappa j},\lessdot _{\kappa j}\in \Theta _{\kappa j}} \left\{ \begin{array}{c} \left( \prod \limits ^{o_{\imath }}_{j=1}\left( 1-(1-\gtrdot _{\kappa j})^\varpi _{\kappa _{j}}\right) \right) ^{\frac{1}{o_{\imath }}},1-\left( \prod \limits ^{o_{\imath }}_{j=1}\left( 1-\ltimes ^{\varpi _{\kappa j}}_{ij}\right) \right) ^\frac{1}{o_{\imath }}, 1-\left( \prod \limits ^{o_{\imath }}_{j=1}\left( 1-\lessdot ^{\varpi _{\kappa j}}_{ij}\right) \right) ^\frac{1}{o_{\imath }} \end{array}\right\} \nonumber \\ & \quad =\bigcup _{\gtrdot _{\kappa j}\in \Theta _{\kappa j},\ltimes _{\kappa j}\in \Theta _{\kappa j},\lessdot _{\kappa j}\in \Theta _{\kappa j}} \left\{ \begin{array}{c} \left( \prod \limits ^{o_{\imath }}_{j=1}\left( 1-(1-\gtrdot _{\kappa j})^\varpi _{\kappa _{j}}\right) \right) ^{\frac{1}{o_{\imath }}},1-\left( \prod \limits ^{o_{\imath }}_{j=1}\left( 1-\ltimes ^{\varpi _{\kappa j}}_{ij}\right) \right) ^\frac{1}{o_{\imath }},\\ 1-\left( \prod \limits ^{o_{\imath }}_{j=1}\left( 1-\lessdot ^{\varpi _{\kappa j}}_{ij}\right) \right) ^\frac{1}{o_{\imath }} \end{array}\right\} , \end{aligned}$$(23)which is reduced to a Neutrosophic Hesitant Fuzzy Weighted Geometric Mean Operator.
A mechanism for solving MCDM problems entrenched by NHFWPMSM operator
This analysis is designed to order the MCDM mechanism over the freshly approaching NHF operators. Assume that \(\Delta = \left\{ \Delta _{1},\Delta _{2},\ldots ,\Delta _{m}\right\}\) be the alternatives set for each attribute, \(\digamma =\left\{ \digamma _{1},\digamma _{2},\ldots ,\digamma _{n}\right\}\) be the set of attributes, and \(\varpi\)=\(\left\{ \varpi _{1},\varpi _{2},\ldots ,\varpi _{n}\right\}\) are weight-vectors of the standards set M. For \(\sum \nolimits _{k=1}^{n}\varpi _{k}\)=1 and \(\varpi _{k}\) \(\in [0,1],\) the analogous weights of various standards in the decision-making method are presented by the weight vector \(\varpi\). Suppose that \(\Delta =\left\{ \Delta _{1},\Delta _{2},\ldots ,\Delta _{m}\right\}\) is divided into \(\alpha\) specific kinds \(\left\{ \tilde{P}_{1},\tilde{P}_{2},...\tilde{P}_{\alpha }\right\}\). There exists a relation between standards of the same kind but nothing between standards inside others. The essential aspects of the suggested mechanism are described below in accordance with Fig. 2.
Flowchart mechanism for addressing MCDM problems entrenched by NHFWPMSM operators.
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Step 1. Forming of the NHF decision matrix: Considering the prior scenario, the MCDM issue is formed by the subsequent decision matrix:
$$\begin{aligned} {\textbf{Z}}_{m\times n} = \begin{pmatrix} \Theta _{11} & \dots & \Theta _{1j} & \dots & \Theta _{1n} \\ \vdots & \ddots & \ \vdots & \ddots & \vdots \\ \Theta _{s1} & \dots & \Theta _{sj} & \dots & \Theta _{sn} \\ \vdots & \ddots & \ \vdots & \ddots & \vdots \\ \Theta _{m1} & \dots & \Theta _{mj} & \dots & \Theta _{mn} \end{pmatrix} \end{aligned}$$(24) -
Step 2. Normality: At that phase, some benefit and expense standards are used for modifying the NHF decision-matrix \({\textbf{Z}}_{m \times n}\). These two standards have opposed responses to alteration in values i.e., the benefit standard works superiorly while the cost standard works poorly. So, we adopt the normality approach to ensure satisfactory needs and adapt the cost standard to the benefit standard.
$$\begin{aligned} \Theta ^{\prime }_{sj} = {\left\{ \begin{array}{ll} \Theta _{sj}, & \digamma _{j} \text { is a befit standard,}\\ \big (\Theta _{sj}\big )^{c}, & \digamma _{j} \text { is cost standard.} \end{array}\right. } \end{aligned}$$(25) -
Step 3. Aggregation: The suggested NHFWPMSM operator is applied to get the score of every alternative \(\Delta _{j},\) as shown below:
$$\begin{aligned} & \Theta _{\kappa }=NHFWPMSM^{(\varrho )}\left( {\Theta }_1,{\Theta }_2,\ldots ,{\Theta }_n \right) \nonumber \\ & \quad =\bigcup _{\gtrdot _{\kappa j}\in \Theta _{\kappa j},\ltimes _{\kappa j}\in \Theta _{\kappa j},\lessdot _{\kappa j}\in \Theta _{\kappa j}} \left\{ \begin{array}{c} 1-\left( \prod \limits ^{\alpha }_{\imath =1}\left( 1-\left( 1-\left( \prod \limits _{1<\kappa _{1}< \cdots<o_\imath } \left( 1-\prod \limits ^{\varrho }_{j=1}\left( 1-(1-\gtrdot _{\kappa j})^\varpi _{\kappa _j}\right) \right) \right) ^{\frac{1}{{\mathscr {C}}^{\varrho }_{o_\imath }}} \right) ^{\frac{1}{\varrho }}\right) \right) ^\frac{1}{\alpha }, \\ \left( \prod \limits ^{\alpha }_{\imath =1}\left( 1-\left( 1-\left( \prod \limits _{1<\kappa _{1}< \cdots<o_\imath } \left( 1-\prod \limits ^{\varrho }_{j=1}\left( 1-\ltimes ^{\varpi _{\kappa j}}_{\kappa j}\right) \right) \right) ^{\frac{1}{{\mathscr {C}}^{\varrho }_{o_\imath }}}\right) ^\frac{1}{\varrho }\right) \right) ^{\frac{1}{\alpha }},\\ \left( \prod \limits ^{\alpha }_{\imath =1}\left( 1-\left( 1-\left( \prod \limits _{1<\kappa _{1}< \cdots <o_\imath } \left( 1-\prod \limits ^{\varrho }_{j=1}\left( 1-\lessdot ^{\varpi _{\kappa j}}_{\kappa j}\right) \right) \right) ^{\frac{1}{{\mathscr {C}}^{\varrho }_{o_\imath }}}\right) ^\frac{1}{\varrho }\right) \right) ^{\frac{1}{\alpha }} \end{array}\right\} . \end{aligned}$$(26)where \(\alpha\) represent the number of kinds, \(\varrho\) is a parameter which varies from 1 to \(o_\imath\), \(o_\imath\) indicates the number of standards in kinds \(t_\imath\), \(\left( \kappa _1,\kappa _2,\ldots ,\kappa _{\varrho } \right)\) involves all the \(\varrho\)-tuples of \(\left( 1,2,\ldots ,o_{\imath } \right)\), \({\mathscr {C}}^{\varrho }_{o_\imath }\) shows the binomial coefficient, that is \({\mathscr {C}}^{\varrho }_{o_\imath }=\frac{o_\imath !}{\varrho !\left( o_\imath -\varrho \right) !}\) and \(\varpi _{j}\) \(\ge\) 0 indicate the weights analogous to each attribute, fulfills \(\sum \nolimits ^{n}_{j=1}\varpi _{j}\)=1.
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Step 4. Scoring: From Eq. (4) , the Score value of the resulting NHFEs \(\Theta _{\kappa } (\kappa =1,2,\ldots ,m)\) are evaluated (When score values are equal, then we use the Accuracy function instead of Score function as in the Eq. 5).
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Step 5. Ordering: In the last phase, the most appropriate choice is picked after the ordering of the score values (or the values obtain from Accuracy function).
Illustrative case study
The applied case study of electing the most unique and the best piece of art prioritizing a better composition with an appealing color range for innovative artwork patterns is supposed here. The process of choosing the best artwork for a room requires a careful analysis of many factors to create a setting that is supportive of child growth and education. This idea supplies an organized framework that can be adjusted for various learning or growth settings. Inside the field of art, we analyze four distinct alternatives: Large-scale painting \(\Delta _{1}\), Sculpture \(\Delta _{2}\), Photography \(\Delta _{3}\), and Mixed media artwork \(\Delta _{4}\). We then recognize four attributes: composition \(\digamma _{1}\), color palette \(\digamma _{2}\), subject matter \(\digamma _{3}\), and emotional impact \(\digamma _{4}\). Allocating the weight vector \(\varpi =(0.3, 0.2, 0.1, 0.4)\) to show the preference of each standard, the decision-makers utilize these advantages.To expedite the evaluation, standards \(\digamma _{i}\) are classified into two distinct partitions \(\tilde{P}_{1}=\{\digamma _{1},\digamma _{3}\}\) and \(\tilde{P}_{2}=\{\digamma _{2},\digamma _{4}\}\). The trained experts apply these standards with their associated weights to determine the four alternatives concerning the identified standards. Table 1 depicts the neutrosophic hesitant fuzzy assigning vector.
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Step 1 : The development of NHF decision-matrix
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Step 2 : There is no need for normalization. As all the considered standards are of the same type (benefit).
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Step 3 : By using Equation (26), when we take \(\varrho =2\) then the aggregated values of each alternative \(\Delta _{\kappa }(\kappa =1,2,\ldots ,m)\) can be evaluated as:
$$\begin{aligned} & \Delta _{1} = \left\{ \begin{array}{c}\left\{ \begin{array}{c} 0.2471,0.2759,0.2763,0.3040,\\ 0.2541, 0.2826, 0.2879, 0.3151 \end{array}\right\} ,\left\{ \begin{array}{c} 0.7918,0.8217,0.7997,0.8299,\\ 0.8191, 0.8500, 0.8230, 0.8540 \end{array}\right\} ,\\ \left\{ \begin{array}{c} 0.8513,0.8683,0.8612,0.8761,\\ 0.8591, 0.8763, 0.8692, 0.8841 \end{array}\right\} \end{array}\right\} \\ & \Delta _{2} = \left\{ \begin{array}{c}\left\{ \begin{array}{c} 0.1428,0.2016,0.1650,0.2222,\\ 0.1555,0.2134,0.1879,0.2435 \end{array}\right\} ,\left\{ \begin{array}{c} 0.7000,0.7733,0.7202,0.7956,\\ 0.7150,0.7898,0.7326,0.8092 \end{array}\right\} ,\\ \left\{ \begin{array}{c} 0.6725,0.7053,0.7406,0.7634,\\ 0.6920, 0.7257, 0.7620, 0.7855 \end{array}\right\} \end{array}\right\} \\ & \Delta _{3} = \left\{ \begin{array}{c}\left\{ \begin{array}{c} 0.2242,0.3356,0.2512,0.3588,\\ 0.2337,0.3437,0.2645,0.3702 \end{array}\right\} ,\left\{ \begin{array}{c} 0.8221, 0.8509, 0.8445, 0.8741,\\ 0.8322,0.8614,0.8519,0.8817 \end{array}\right\} ,\\ \left\{ \begin{array}{c} 0.7716, 0.8180, 0.8053, 0.8380,\\ 0.8051,0.8536,0.8403,0.8745 \end{array}\right\} \end{array}\right\} \\ & \Delta _{4} = \left\{ \begin{array}{c}\left\{ \begin{array}{c} 0.1729,0.2133,0.2010,0.2401,\\ 0.1906,0.2302,0.2329,0.2704 \end{array}\right\} ,\left\{ \begin{array}{c} 0.7540,0.8077,0.7659,0.8205,\\ 0.7832, 0.8390, 0.7912, 0.8476 \end{array}\right\} , \\ \left\{ \begin{array}{c} 0.7731,0.8077,0.8122,0.8381,\\ 0.7799,0.8148,0.8194,0.8455 \end{array}\right\} \end{array}\right\} \end{aligned}$$ -
Step 4: By using Eq.(4), the Score value of each alternatives \(\Delta _{\kappa }(\kappa =1,2,...4)\) are evaluated as: \(\S (\Delta _{1})=0.1962\), \(\S (\Delta _{2})=0.2354\) ,\(\S (\Delta _{3})=0.2065\), \(\S (\Delta _{4})=0.2022\)
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Step 5 : From above, the final order can be expressed as \(\Delta _{2}>\Delta _{3}>\Delta _{4}>\Delta _{1}\). Thus, the best choice is \(\Delta _{2}\).
Here, we have an interesting graph of heatmap in which we can clearly express our best choice which is \(\Delta _{2}\) via Fig. 3.
Desired ordering and optimal choice by using the suggested aggregations.
Assessing model weight stability
Analyze sensitivity is a type of economic framework that is employed to see how changes in input components affect the results. The sequence obtained from the NHFWPMSM operator has somehow different choices, but \(\Delta _{2}\) is still the optimal choice.
Model weight sensitivity is a swift approach used in MCDM problems. Most of the time, attributes are assigned varying weights based on preferences, proficiency, or other domains. The examination entails changing the weights and analyzing the impact on the final decision-making step. The Table 2 and Fig. 4 interpret that the alternate weight assignment contributes to no change in the sequence of alternatives, with \(\Delta _{2}\) being the optimal.
Graphical assessment of weight sensitivity using NHFWPMSM operator.
Comparative research
Contrasting examination with many other lasting aggregation operators comprising of the single-valued neutrosophic hesitant fuzzy weighted average operator (SVNHFWA), single-valued neutrosophic hesitant fuzzy weighted geometric operator (SVNHFWG)18, single-valued neutrosophic hesitant fuzzy dombi weighted average Operator (SVNHFDWA), single-valued neutrosophic hesitant fuzzy dombi weighted geometric Operator (SVNHFDWG)37 are subsequently took forward to demonstrate the advantages of the described methodology. Ultimate results achieved by engaging all of the narrated aggregation operators to the foregoing instance are displayed in Table 3 and Fig. 5.
Graphical comparison under NHFWPMSM operator.
By the outputs illustrated in Table 3, identical classification of the four choices is immensely influenced through the access provided in the current article and the analysis based on SVNHFWA and SVNHFWG operators18, SVNHFDWA and SVNHFDWG operators37. The authenticity of the operator mapped out in the latest study is demonstrated by this. Selections attained from these operators is \(\Delta _{2}>\Delta _{3}>\Delta _{4} >\Delta _{1}\), where \(\Delta _{2}\) is still the most optimal choice.
Validation
The Spearman’s rank correlation coefficient \({\overline{\rho }}\) (see in Table 4), which is 0.8 across each contrast, indicates a highly +ve correlation among the ranks of prior operators and the suggested NHFWPMSM operator. It can be evident from this overview that the rankings derived with the help of the various aggregate operators and the NHFWPMSM operator are very comparable.
Discussion
When we compared our proposed operator (NHFWPMSM) to the existing ones, we got an identical optimal choice. However, our operator is more appropriate than existing ones based on the NHFS framework because of the adaptability in which the DMs can choose to assess each partition differently by how valuable it is.
The high computational cost of the suggested NHFWPMSM operator is a drawback that decreases its effectiveness when used with massive data sets. Specific neutrosophic and hesitant fuzzy values are necessary for it to work, but they can be hard to come by and vulnerable to variability. Furthermore, in dynamic DM situations, the partitioning strategy could miss more subtle links, and the operator may oversimplify intricate links across standards.
Conclusions
We had an impactful analysis of an MCDM technique within the NHF zone in the current proposal (Beginning of an effective NHFWPMSM aggregate operator) as NHFS is an exemplary tool for dealing with complex and unsure content. Many properties of the framed operators are examined in detail and shown that some present operators (consider their partitioned relation along with the link between criteria) are key examples of the provided operators. Also, we came MCDM strategy depending on the proposed NHF operators. In addition, a numerical depiction is given for selecting the most unique and the best piece of art preferring a composition with an attractive color extent for productive artwork patterns to show the usage of the suggested method. The devised way was compared to existing methods later saying the proposed method is more accurate for tackling uncertain data and capable of applying different many partitions among criteria. Summing up, an extension of these operators to efficient areas like cubic neutrosophic hesitant (CNH), q-rung picture hesitant, p,q-cubic quasi-rung fuzzy, and many others is possible. On the other hand, the utilization of existing aggregation operators can significantly improve the effectiveness of certain MCDM strategies, such as AHP, TODIM, PROMETHEE, and ELECTRE38,39,40. We desire that our presented analysis with create up-to-date techniques for highlighting decision-making challenges across different fuzzy scenarios.
Data availibility
The datasets used and analyzed during the current study are available from the corresponding author on reasonable request.
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Jawad Ali, Usman Khalid, Muhammad Ahsan Binyamin, and Muhammed Ibrahem Syam contributed equally to this work. Jawad Ali was responsible for the conceptualization and design of the study. Usman Khalid carried out the data collection and analysis. Muhammad Ahsan Binyamin contributed to the interpretation of data and drafting of the manuscript. Muhammed Ibrahem Syam supervised the study and reviewed the manuscript. All authors read and approved the final manuscript.
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Ali, J., Khalid, U., Binyamin, M.A. et al. Innovative art selection through neutrosophic hesitant fuzzy partitioned Maclaurin symmetric mean aggregation operator. Sci Rep 15, 8474 (2025). https://doi.org/10.1038/s41598-025-92432-8
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DOI: https://doi.org/10.1038/s41598-025-92432-8
