Abstract
Decision-making in uncertain and imprecise environments often requires robust mathematical frameworks capable of effectively representing and aggregating conflicting information. Circular Intuitionistic Fuzzy Sets (C-IFSs) are powerful tools for modeling such complexities by combining intuitionistic and cubic fuzzy set properties. In this study, we extend the applicability of C-IFSs by integrating them with the Hamacher operational framework, offering a more flexible and adaptive approach to aggregation. To this end, we propose six novel aggregation operators: Circular Intuitionistic Fuzzy Hamacher Weighted Average (CIFHWA), Circular Intuitionistic Fuzzy Hamacher Ordered Weighted Average (CIFHOWA), Circular Intuitionistic Fuzzy Hamacher Hybrid Weighted Average (CIFHHWA), Circular Intuitionistic Fuzzy Hamacher Weighted Geometric (CIFHWG), Circular Intuitionistic Fuzzy Hamacher Ordered Weighted Geometric (CIFHOWG), and Circular Intuitionistic Fuzzy Hamacher Hybrid Weighted Geometric (CIFHHWG). These operators are designed to address multi-criteria decision-making challenges with improved precision. We further develop score and accuracy functions to rank C-IFSs and propose a neural-based scheme employing cubic correlation coefficients to enhance computational efficiency. A detailed numerical example validates the framework’s effectiveness, illustrating its practical utility. Additionally, comparative analyses with existing techniques and sensitivity and robustness examinations demonstrate its superiority in handling complex decision-making scenarios. The findings underscore the advantages of using Hamacher-based operations with C-IFSs, offering a novel contribution to fuzzy decision-making and opening avenues for future research in uncertain data analysis.
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Introduction
Artificial Neural Networks have been a significant emphasis in artificial intellect since the 1980s. They classify the humanoid brain’s neural net from a data-handling viewpoint, founding a basic model and building various systems based on dissimilar joining designs1. Neural schemes are also recognized as manufactured neural schemes and are a subsection of mechanism learning and are at the emotion of deep learning controls. Their designation and construction are pushed by the humanoid intelligence, imitating the means that organic neurons flag to one additional. Fake neural schemes include hub sheets containing an input layer of one or more enclosed up coatings, and a harvest layer. Each neuron interfaces with another and has a related weight and edge. Neural schemes depend on making data to memorize and brand their precision over the period. Once these knowledge controls are fine-tuned for precision, authorize us to categorize and bunch data in a tall haste. Projects in discourse greeting can take notes versus times when likened to the physical familiar proof by humanoid specialists in deep neural networking Fig. 1.
Structure of deep neural networks.
An output, known as the start function, is represented by each node in the network. The artificial neural network’s (ANN) memory is the connections between nodes, which transmit signals and are each given a weight that affects the network’s performance2. Artificial neural networks (ANNs) were created to replicate how the human brain works to do tasks where traditional approaches were unsuccessful. This field has advanced from eschewing earlier methods to more properly representing biological processes. Artificial Neural Networks (ANNs) can learn and model complex relationships. Typically organized with neurons connected in various patterns, enabling some neurons to contribute to the output of others. The functioning of the neural network involves interconnected neurons, with each neuron linked to others through connections that resemble the biological axon-synapse-dendrite structure. These connected neurons collaborate to process and analyze data. Each connection has a weight, determining the impact one neuron has on another, allowing the weights to govern the interactions between neurons and adjust the network’s performance.
Figure 2, Neuron and myelinated axon with the flag is given as.
Neuron and myelinated axon, with flag stream from inputs at dendrites to yields at axon terminals.
The neurons are routinely arranged into numerous levels, primarily in deep intelligence. Neurons of one veneer boundary as it were to neurons of the punctually successful and promptly captivating after coatings. The sheet that gets outdoor data is the contribution layer. The sheet that crops the dangerous consequence is the harvest coating. Single sheets and unaltered schemes are too used. Two layers of frequent association projects are imaginable. The container is ‘fully connected’, with respective neurons in one deposit interfacing to each neuron inside the sheet. They can be assembled, where a pucker of neurons in one sheet interfaces to a sole neuron within the sheet, in this method reducing the number of neurons in that pane. Abiodun et al.3 proposed the neuronal plans. Wu et al.4 proposed an explicit search of its use in data, prescription, economy, control, transportation, and brain power research. Islam et al.5 proposed the real of manufactured neuronal organisms. Assembled neuronic assemblies can be connected to an arranged number of real-world assets of impressive density.
Figure 3 of neural networks is given aa.
Neural networks.
Literature review
Cakr and Tas6 developed an original C-IFS MCDM technique, assimilating criteria weighting with alternate position procedures for improved decision-making. In our veritable lifetime glitches, we constantly express diverse sorts of official glitches, where our simple and unique accentuation is to ponder precisely how to sort a more beneficial choice. So, attribute decision-making creates a noteworthy preparation within the choice disciplines. Even though there is ceaseless unclearness and questions in our life expectancy subjects we cannot make choices in solitary by implying fresh information. Zadeh7 built up unique cognitive fuzzy sets where FSs consume a certainty score whose variety could be a temporary component and had remained common in various zones for treatment questions emerging from imprecision and inadequate belongingness8,9. Liu10 introduced two approaches for multicriteria group decision-making founded on these operators. Hussian et al.11 proposed the framework for multi-criteria choice-making and an organized shrewd handle for choice-making setup by utilizing the proposed approach. Ahmad et al.12 displayed the Hamacher accumulation administrators. Ahmad et al.13 characterized the Hamacher t-norm and t-conorm operational laws. Thilagavathy et al.14 realistic the choice creation method named WASPAS. Ali et al.15 anticipated the intuitionistic fuzzy weighted Archimedean Heronian accumulation administrator. Petal et al.16 anticipated the connection between some plan organization issues to demonstrate its viability and a novel confront appreciation procedure. Bharati et al.17 arranged measured triangular intuitionistic fuzzy numbers. Jebadass and Balasubramaniam18 arranged the strategies checking histogram equalization, distinction deficient versatile histogram equalization, histogram necessity, degrading, and fragmentary arrange doping advancement. Alolaiyan et al.19 arranged efficiently several irreplaceable belongings of the novel likeness sum and approved its proficiency through arithmetical craftsmanship. Deveci et al.20 proposed the insufficiencies of the traditionalist method set up if a scientific safe and another position strategy is based on the overexcited volume metric. Wang et al.21 projected the notion of a fuzzy rough set perfect in the IFFD. Alkan and Kahraman22 predict the non-linear nonstop intuitionistic fuzzy sets and practice in multi-criteria decision creation copies. Wu et al.23 proposed objectivity amount gratified the clear senses of severe intuitionistic fuzzy comparison. Gohain et al.24 deliberated the project appreciation and gathering glitches of the structure physical group. Rahman et al.25 projected the describe the energetic basics, null set, and supplement set. Yu et al.26 deliberate the belvedere of statistics dispersal trials, study theme has gradually interpreted from theatrical structure to practical request, and many subjects such as self-citation, unintended quote, ranking, and locations that may touch the importance of the chief track. Pattanayak et al.27 recognized the probabilistic intuitionistic fuzzy set by sighting the ratio tendency change of crisp statement sideways with the horrid of mutual membership standards.
Figure 4 history is given as.
History.
Khan et al.28 proposed the Cq-ROF Aczel-Alsina power-weighted averaging (Cq-ROFAAPWA) and aggregation Cq-ROF Aczel-Alsina power-weighted geometric (Cq-ROFAAPWG) operators. Khan et al.29 proposed the newly proposed approach T-spherical fuzzy (T-SF) confidence level weighted averaging T-SFWAc and T-SF confidence level weighted geometric T-SFWG. Zhang et al.30 devoted to introducing the interval-valued q-ROF Aczel- Alsina power-weighted averaging (IV-q-ROFAAPWA) and interval-valued q-ROF Aczel-Alsina power-weighted geometric (IV-q-ROFAAPWG) operators. Khan et al.31 proposed the q-Rung orthopair fuzzy (q-ROF) AA power-weighted averaging (q-ROFAAPWA) and q-ROF AA power-weighted geometric (q-ROFAAPWG) operators. Khan et al.32 proposed the based on the proposed AOs and utilized in medical diagnosis (MD) problems. Khan et al.33 proposed the complex T-spherical fuzzy (TSF) power-weighted averaging and complex TSF power-weighted geometric (CTSFPWG) operators.
Boltürk et al.34 introduced the intuitionistic fuzzy sets explaining the ambiguity of membership and non-membership levels done a circle whose length. Fahmi et al.35 exhibited the resolution to complex real-life problems, a system for the MADM problem. Fahmi et al.36 proposed a mathematical example to explain the proposed performance phase by stage. Fahmi et al.37 demonstrated the aggregation and geometric operators. Fahmi et al.38 attained the connection between the proposed operators. Fahmi et al.39 extended the typical VIKOR technique to solve the MCDM technique created on triangular cubic fuzzy numbers. These operators use different rankings40,41,42.
Gap
In this paper, we pressure these first industrialized Circular intuitionistic fuzzy sets and create their Hamacher AOs with properties operated in.
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1.
To normalize the Hamacher operational laws for vaguely binary program Circular intuitionistic fuzzy sets.
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2.
To achieve the Circular intuitionistic fuzzy sets Hamacher average, Circular intuitionistic fuzzy sets Hamacher geometric operators.
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3.
To define the proposed method based on Circular intuitionistic fuzzy sets.
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4.
To produce some illustrations for comparing discrete operators with existing operators.
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5.
The complexity of the socioeconomic location inspires us to make decisions with faith in real data. Indecision often rises in real ecosphere glitches unresolved to many multifaceted fetters, insufficiency of data, and nonappearance of retro and variability. Given the assessments of the aforesaid work, no detectives have yet obtained the impression of a circular intuitionistic fuzzy variable neural scheme formation method for it. In teaching to do so, we’ll label the circular intuitionistic fuzzy mutable and related procedures.
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6.
The estimate of the premium supernumerary in a circular intuitionistic fuzzy location is a very difficult neural scheme and has numerous vague factors. In the current neural scheme methods, estimate data is just portrayed by circular intuitionistic fuzzy variables, which May rapidly data scar. Thus, we vital an extra overall model to complicate the likelihood of substitutes.
Aim of the study
The prime aim of this training is to improve a progressive decision-making outline that influences Circular intuitionistic fuzzy sets and Hamacher operators to progress the accuracy and heftiness of multi-criteria decision-making under vagueness. Exactly, the study speeches the following research areas:
To propose original aggregation operators, with CIFHWA, CIFHOWA, CIFHHWA, CIFHWG, CIFHOWG, and CIFHHWG, that integrate Hamacher operators with C-IFSs.
To progress a score and accuracy function for position decision options based on C-IFSs, cultivating the decision process in indeterminate and vague settings.
To implement a neural scheme created on circular correlation coefficients (CCF), which enhances computational competence while conserving the accuracy of the decision-making procedure.
Novelty of the approach
The novelty of our method denigrations in the mixture of three key mechanisms:
Circular Intuitionistic Fuzzy Sets (C-IFSs) We spread traditional IFSs by presenting circularity in membership and non-membership functions if a more complete perfect for doubt. This novel extension permits more compound and realistic symbols of decision-making difficulties, where criteria affect each other in recurring patterns.
Hamacher Aggregation Operators By joining Hamacher operators obsessed with the decision-making procedure, we propose a more overall and supple aggregation device. This invention allows the model to handle varying degrees of effect among decision criteria, which is vital in real-world MCDM states.
Neural Scheme with Circular Correlation Coefficients (CCF) The incorporation of a neural method with cubic correlation coefficients speeches the computational contests regularly faced in high-dimensional decision-making problems. This novel neural scheme enhances the overall efficiency of the managerial context, qualifying it to measure more excellently with larger datasets.
Contribution
This education contributes to the existing works in some important conducts:
Advancement in Fuzzy Decision-Making The overview of C-IFSs combined with Hamacher aggregation operator’s suggestions an additional vigorous and flexible method to fuzzy decision-making, refining the limitations of traditional fuzzy sets and their aggregation approaches.
Enhanced Decision-Making Framework By mixing C-IFSs with neural networks and suggesting novel aggregation operators, this paper presents a progressive decision-making outline that outdoes previous methods, chiefly in standings of accuracy, suppleness, and computational competence.
New Tools for Ranking Alternatives The planned score and accuracy functions for C-IFSs deliver a novel apparatus for position alternatives in decision-making problems, educating the status exactness in setups with undefined or inconsistent criteria.
Practical Applications The procedure has extensive everyday applications, mostly in areas demanding multi-criteria decision-making under vagueness, such as production design, investment, healthcare, and reproduction cleverness.
The time out of this paper is ordered as follows. Section “Background” concisely examines some theory data of intuitionistic fuzzy sets. In Section “CIFNs aggregation operator based on Hamacher”, the CIF-Hamacher aggregation operators are obtainable with their details. In Section “Neural scheme based on CCF”, the CIF-Hamacher technique is practical to the neural scheme. In Section “Case history”, we define the case study, numerical analysis, and comparison analysis. In Section “Discussion”, the results and discussions are presented. In Section “Conclusion”, we define the conclusion in our instructions.
Background
Definition 2.1
7 Let us consider that \(\Phi \ne X\) and by a fuzzy set \(\gamma =\left\{\begin{array}{c}\langle x,{\mu }_{\gamma \left(x\right)}\rangle \\ : x\in X\end{array}\right\},\) \({\mu }_{\gamma \left(x\right)}\) is a mapping from \(X\) to \([\text{0,1}]\) present membership task of a component \(x\) in \(X\).
Definition 2.2
8 A IFS \(A\) in \(E\) is a thing consuming the form of a universe \(E :\) \(A=\left\{\langle {\varsigma }_{A}(x),{\chi }_{A}(x)\rangle : x\in E\right\},\) where \(0\le {\varsigma }_{A}(x)+{\chi }_{A}(x)\le 1,\) \({\varsigma }_{A}(x): E\to [\text{0,1}],{\varsigma }_{A}(x): E\to [\text{0,1}].\) The degree of indeterminacy is defined as \({\pi }_{A}(x)=1-{\varsigma }_{A}(x)-{\chi }_{A}(x).\)
Definition. 2.2 8. Let \({a}_{1}=\langle {\varsigma }_{1}(x),{\chi }_{1}(x)\rangle\) and \({a}_{2}=\langle {\varsigma }_{2}(x),{\chi }_{2}(x)\rangle\) be two IFNs, \(\lambda >0\), then
CIFN and operational laws
Definition 2.1.1
9 A CIFS \(A\) in \(E\) is an object for a universe \(E :\) \(A=\left\{\begin{array}{c}\langle {\varsigma }_{A}(x),\\ {\chi }_{A}(x);r\rangle :\\ x\in E\end{array}\right\},\) where \(0\le {\varsigma }_{A}(x)+{\chi }_{A}(x)\le 1,r\in [\text{0,1}]\) , \({\varsigma }_{A}(x): E\to [\text{0,1}],{\varsigma }_{A}(x): E\to [\text{0,1}],r\) define the radius of the circle around each component \(x,x\in E\) to the set \(A\subseteq E.\) The degree of indeterminacy is defined as \({\pi }_{A}(x)=1-{\varsigma }_{A}(x)-{\chi }_{A}(x).\)
Definition 2.1.2
9 Let \({G}_{1}=\langle {\xi }_{1}(h),{\chi }_{1}(h);{r}_{1}\rangle\) and \({G}_{2}=\langle {\xi }_{2}(h),{\chi }_{2}(h);{r}_{2}\rangle\) be two CIFNs based on Hamacher, \(\lambda >0\), then
C-IFSs based on Hamacher
Definition 2.2.1
Let \({a}_{1}=\left\{{\varsigma }_{1},{\chi }_{1};{r}_{1}\right\}\) and \({a}_{2}=\left\{{\varsigma }_{2},{\chi }_{2},{r}_{2}\right\}\) be two C-IFSs based on Hamacher, \(\lambda >0\), then.
Definition 2.2.2
Let \({a}_{j}=\left\{{\varsigma }_{j},{\chi }_{j};{r}_{j}\right\}\) be the C-IFSs, then the score function \(S\) is given below.
CIFNs aggregation operator based on Hamacher
This section proposes CIFHWA, CIFHOWA, CIFHHWA CIFHWG, CIFHOWG and CIFHHWG operators.
CIFHWA operator
Definition 3.1.1
The collection of CIFNs are \({d}_{j}=\left\{{\varsigma }_{j},{\chi }_{j};{r}_{j}\right\}\) and the weight vector is \(G=({G}_{1},{G}_{2},\ldots,{G}_{n}{)}^{T}\) with \({G}_{j}\in [\text{0,1}]\) and \(\sum_{j=1}^{n}{G}_{j}=1\) . Then CIFHWA \(\left({d}_{1},{d}_{2},\ldots,{d}_{n}\right)={\oplus }_{j=1}^{n}{G}_{j}{d}_{j}\) is said CIFHWA operator.
Theorem 3.1.2
The collection of CIFNs are \({a}_{j}=\left\{{\varsigma }_{1},{\chi }_{1};{r}_{1}\right\}\) and the weight vector is \(\Gamma =({\Gamma }_{1},{\Gamma }_{2},\ldots,{\Gamma }_{n}{)}^{T}\) with \({\Gamma }_{j}\in [\text{0,1}]\) and \(\sum_{j=1}^{n}{\Gamma }_{j}=1\). Then it is said that the CIFHWA operator and CIFHWA \(\left({a}_{1},{a}_{2},\ldots,{a}_{n}\right)=\left[\begin{array}{c}\frac{\prod_{J=1}^{n}(1+(\Gamma -1)({\varsigma }_{j}){)}^{\Gamma }-\prod_{J=1}^{n}(1-{\varsigma }_{j}){)}^{\Gamma }}{\prod_{J=1}^{n}(1+(\Gamma -1)(1-{\varsigma }_{j}{)}^{\Gamma }+(\Gamma -1)\prod_{J=1}^{n}(({\varsigma }_{j}){)}^{\Gamma }},\\ \frac{\Gamma \prod_{J=1}^{n}({\chi }_{j}{)}^{\Gamma }}{(1+(\Gamma -1)\prod_{J=1}^{n}({\chi }_{j}){)}^{\Gamma }+(\Gamma -1)\prod_{J=1}^{n}(1-({\chi }_{j}){)}^{\Gamma }};\\ \frac{\prod_{J=1}^{n}(1+(\Gamma -1)({r}_{j}){)}^{\Gamma }-\prod_{J=1}^{n}(1-{r}_{j}){)}^{\Gamma }}{\prod_{J=1}^{n}(1+(\Gamma -1)(1-{r}_{j}{)}^{\Gamma }+(\Gamma -1)\prod_{J=1}^{n}(({r}_{j}){)}^{\Gamma }}\end{array}\right]\)
Proof Since \(n\) is true and \(n=1\)
Since \(n=k\)
CIFHWA \(\left({a}_{1},{a}_{2},\ldots,{a}_{n}\right)=\)
Since \(n=k+1\)
CIFHWA \(\left({a}_{1},{a}_{2},\ldots,{a}_{k+1}\right)=\)
CIFHWA \(\left({a}_{1},{a}_{2},\ldots,{a}_{k+1}\right)+CIFHWA\left({a}_{1},{a}_{2},\ldots,{a}_{k}\right)=\)
Theorem 3.1.3
(Idempotency) If \({d}_{k}=\left\{{\varsigma }_{k},{\chi }_{k};{r}_{k}\right\}\) for all \(k=\text{1,2},3,\ldots,n,\) then CIFHWA \((Y,Y,Y,\ldots,Y)=Y.\)
Theorem 3.1.4
(Commutativity) If \(({G}_{1}{\prime},{G}_{2}{\prime},\ldots,{G}_{n}{\prime})\) is any permutation of \(({G}_{1},{G}_{2},\ldots,{G}_{n})\) , then CIFHWA \(({G}_{1}{\prime},{G}_{2}{\prime},\ldots,{G}_{n}{\prime})=\) CIFHWA \(({G}_{1},{G}_{2},\ldots,{G}_{n})\).
Theorem 3.1.5
(Boundedness) If \({Y}^{-}=min({b}_{1},{b}_{2},\ldots,{b}_{n}),\) \({Y}^{+}\) \(=\) \(\mathit{max}({b}_{1},{b}_{2},\ldots,{b}_{n}),\) then \({Y}^{-}\le\) CIFHWA \(({b}_{1},{b}_{2},\ldots,{b}_{n})\le\) \({Y}^{+}.\)
CIFHOWA operator
Definition 3.2.1
The gathering of CIFNs are \({{\varvec{e}}}_{{\varvec{j}}}=\left\{{\boldsymbol{\varsigma }}_{{\varvec{j}}},{{\varvec{\chi}}}_{{\varvec{j}}};{{\varvec{r}}}_{{\varvec{j}}}\right\}\) and the weight vector is \({\varvec{L}}=({{\varvec{L}}}_{1},{{\varvec{L}}}_{2},\ldots,{{\varvec{L}}}_{{\varvec{n}}}{)}^{{\varvec{T}}}\) with \({{\varvec{L}}}_{{\varvec{j}}}\in [\text{0,1}]\) and \({\sum }_{{\varvec{j}}=1}^{{\varvec{n}}}{\varvec{L}}=1\). Then CIFHOWA \(\left({{\varvec{e}}}_{1},{{\varvec{e}}}_{2},\ldots,{{\varvec{e}}}_{{\varvec{n}}}\right)={\oplus }_{{\varvec{j}}=1}^{{\varvec{n}}}{{\varvec{L}}}_{{\varvec{j}}}{{\varvec{e}}}_{{\varvec{j}}}\) is said the CIFHOWA operator.
Theorem 3.2.2
The collection of CIFNs are \({G}_{j}=\left\{{\varsigma }_{j},{\chi }_{j};{r}_{j}\right\}\) and the weight vector is \(\Gamma =({\Gamma }_{1},{\Gamma }_{2},\ldots,{\Gamma }_{n}{)}^{T}\) with \({\Gamma }_{j}\in [\text{0,1}]\) and \({\sum }_{j=1}^{n}{\Gamma }_{j}=1\). Then it is said CIFHOWA operator and CIFHOWA \(\left({G}_{1},{G}_{2},\ldots,{G}_{n}\right)=\) \(\left[\begin{array}{c}\frac{\prod_{J=1}^{n}(1+(\Gamma -1)({\varsigma }_{j}){)}^{\Gamma }-\prod_{J=1}^{n}(1-{\varsigma }_{j}){)}^{\Gamma }}{\prod_{J=1}^{n}(1+(\Gamma -1)(1-{\varsigma }_{j}{)}^{\Gamma }+(\Gamma -1)\prod_{J=1}^{n}(({\varsigma }_{j}){)}^{\Gamma }},\\ \frac{\Gamma \prod_{J=1}^{n}({\chi }_{j}{)}^{\Gamma }}{(1+(\Gamma -1)\prod_{J=1}^{n}({\chi }_{j}){)}^{\Gamma }+(\Gamma -1)\prod_{J=1}^{n}(1-({\chi }_{j}){)}^{\Gamma }};\\ \frac{\prod_{J=1}^{n}(1+(\Gamma -1)({r}_{j}){)}^{\Gamma }-\prod_{J=1}^{n}(1-{r}_{j}){)}^{\Gamma }}{\prod_{J=1}^{n}(1+(\Gamma -1)(1-{r}_{j}{)}^{\Gamma }+(\Gamma -1)\prod_{J=1}^{n}(({r}_{j}){)}^{\Gamma }}\end{array}\right]\)
Theorem 3.2.3
(Idempotency) If \({Y}_{j}=\left\{{\varsigma }_{j},{\chi }_{j};{r}_{j}\right\}\) for all \(j=\text{1,2},3,\ldots,n,\) then CIFHOWA \((Y,Y,Y,\ldots,Y)=Y.\)
Theorem 3.2.4
(Commutativity) If \(({G}_{1}{\prime},{G}_{2}{\prime},\ldots,{G}_{n}{\prime})\) is any permutation of \(({G}_{1},{G}_{2},\ldots,{G}_{n})\) , then CIFHOWA \(({G}_{1}{\prime},{G}_{2}{\prime},\ldots,{G}_{n}{\prime})=\) CIFHOWA \(({G}_{1},{G}_{2},\ldots,{G}_{n}).\)
Theorem 3.2.5
(Boundedness) If \({Y}^{-}=min({R}_{1},{R}_{2},\ldots,{R}_{n}),\) \({Y}^{+}\) \(=\) \(\mathit{max}({R}_{1},{R}_{2},\ldots,{R}_{n}),\) then \({Y}^{-}\le\) CIFHOWA \(({R}_{1},{R}_{2},\ldots,{R}_{n})\le\) \({Y}^{+}.\)
CIFHHWA operator
Definition 3.3.1
The gathering of CIFNs are \({h}_{j}=\left\{{\varsigma }_{j},{\chi }_{j};{r}_{j}\right\}\) and the weight vector is \(U=({U}_{1},{U}_{2},\ldots,{U}_{n}{)}^{T}\) with \({U}_{j}\in [\text{0,1}]\) and \({\sum }_{j=1}^{n}{\text{U}}_{j}=1\). Then CIFHHWA \(\left({h}_{1},{h}_{2},\ldots,{h}_{n}\right)={\oplus }_{j=1}^{n}{U}_{j}{h}_{j}\) is said the CIFHHWA operator.
Theorem 3.3.2
The collection of CIFNs are \({f}_{j}=\left\{{\varsigma }_{1},{\chi }_{1};{r}_{1}\right\}\) and the weight vector is \(\Gamma =({\Gamma }_{1},{\Gamma }_{2},\ldots,{\Gamma }_{n}{)}^{T}\) with \({\Gamma }_{j}\in [\text{0,1}]\) and \({\sum }_{j=1}^{n}{\Gamma }_{j}=1\). Then it is said CIFHHWA operator and CIFHHWA \(\left({f}_{1},{f}_{2},\ldots,{f}_{n}\right)\) \(=\) \(\left[\begin{array}{c}\frac{\prod_{J=1}^{n}(1+(\Gamma -1)({\varsigma }_{j}){)}^{\Gamma }-\prod_{J=1}^{n}(1-{\varsigma }_{j}){)}^{\Gamma }}{\prod_{J=1}^{n}(1+(\Gamma -1)(1-{\varsigma }_{j}{)}^{\Gamma }+(\Gamma -1)\prod_{J=1}^{n}(({\varsigma }_{j}){)}^{\Gamma }},\\ \frac{\Gamma \prod_{J=1}^{n}({\chi }_{j}{)}^{\Gamma }}{(1+(\Gamma -1)\prod_{J=1}^{n}({\chi }_{j}){)}^{\Gamma }+(\Gamma -1)\prod_{J=1}^{n}(1-({\chi }_{j}){)}^{\Gamma }};\\ \frac{\prod_{J=1}^{n}(1+(\Gamma -1)({r}_{j}){)}^{\Gamma }-\prod_{J=1}^{n}(1-{r}_{j}){)}^{\Gamma }}{\prod_{J=1}^{n}(1+(\Gamma -1)(1-{r}_{j}{)}^{\Gamma }+(\Gamma -1)\prod_{J=1}^{n}(({r}_{j}){)}^{\Gamma }}\end{array}\right]\)
Theorem 3.3.3
(Idempotency) If \({f}_{j}=\left\{{\varsigma }_{j},{\chi }_{j};{r}_{j}\right\}\) for all \(j=\text{1,2},3,\ldots,n,\) then CIFHHWA \((f,f,f,\ldots,f)=f.\)
Theorem 3.3.4
(Commutativity) If \(({D}_{1}{\prime},{D}_{2}{\prime},\ldots,{D}_{n}{\prime})\) is any permutation of \(({D}_{1},{D}_{2},\ldots,{D}_{n})\) , then CIFHHWA \(({D}_{1}{\prime},{D}_{2}{\prime},\ldots,{D}_{n}{\prime})=\) CIFHHWA \(({D}_{1},{D}_{2},\ldots,{D}_{n}).\)
Theorem 3.3.5
(Boundedness) If \({Y}^{-}=min({b}_{1},{b}_{2},\ldots,{b}_{n}),\) \({Y}^{+}\) \(=\) \(\mathit{max}({b}_{1},{b}_{2},\ldots,{b}_{n}),\) then \({Y}^{-}\le\) CIFHHWA \(({b}_{1},{b}_{2},\ldots,{b}_{n})\le {Y}^{+}\)
CIFHWG operator
Definition 3.4.1
The gathering of CIFNs are \({f}_{j}=\left\{{\varsigma }_{j},{\chi }_{j};{r}_{j}\right\}\) and the weight vector is \(G=({G}_{1},{G}_{2},\ldots,{G}_{n}{)}^{T}\) with \({G}_{j}\in [\text{0,1}]\) and \({\sum }_{j=1}^{n}{\text{G}}_{j}=1\). Then CIFHWG \(\left({f}_{1},{f}_{2},\ldots,{f}_{n}\right)={\otimes }_{j=1}^{n}{f}_{j}^{{G}_{j}}\) is said the CIFHWG operator.
Theorem 3.4.2
The collection of CIFNs are \({a}_{j}=\left\{{\varsigma }_{j},{\chi }_{j};{r}_{j}\right\}\) and the weight vector is \(\eta =({L}_{1},{L}_{2},\ldots,{L}_{n}{)}^{T}\) with \({L}_{j}\in [\text{0,1}]\) and \({\sum }_{j=1}^{n}{\text{L}}_{j}=1\). Then it is said that the CIFHWG operator and CIFHWG \(\left({a}_{1},{a}_{2},\ldots,{a}_{n}\right)=\)
Proof Since \(n=1\)
Since \(n=k\)
Since \(n = k + 1\)
Theorem 3.4.3
(Idempotency) If \({f}_{k}=\left\{{\varsigma }_{k},{\chi }_{k};{r}_{k}\right\}\) for all \(k=\text{1,2},3,\ldots,n,\) then CIFHWG \((Y,Y,Y,\ldots,Y)=Y\).
Theorem 3.4.4
(Commutativity) If \(({G}_{1}{\prime},{G}_{2}{\prime},\ldots,{G}_{n}{\prime})\) is any permutation of \(({G}_{1},{G}_{2},\ldots,{G}_{n})\), then CIFHWG \(({G}_{1}{\prime},{G}_{2}{\prime},\ldots,{G}_{n}{\prime})=\) CIFHWG \(({G}_{1},{G}_{2},\ldots,{G}_{n})\).
Theorem 3.4.5
(Boundedness) If \({Y}^{-}=min({b}_{1},{b}_{2},\ldots,{b}_{n}),{Y}^{+}\) \(=\) \(\mathit{max}({b}_{1},{b}_{2},\ldots,{b}_{n}),\) then \({Y}^{-}\le\) CIFHWG \(({b}_{1},{b}_{2},\ldots,{b}_{n})\le\) \({Y}^{+}\).
CIFHOWG operator
Definition 3.5.1
The gathering of CIFNs are \({f}_{j}=\left\{{\varsigma }_{j},{\chi }_{j};{r}_{j}\right\}\) and the weight vector is \(H=({H}_{1},{H}_{2},\ldots,{H}_{n}{)}^{T}\) with \({H}_{j}\in [\text{0,1}]\) and \(\sum_{j=1}^{n}{H}_{j}=1.\) Then CIFHOWG \(\left({f}_{1},{f}_{2},\ldots,{f}_{n}\right)={\otimes }_{j=1}^{n}{f}_{j}^{{H}_{j}}\) is said the CIFHOWG operator.
Theorem 3.5.2
The collection of CFFNs are \({a}_{j}=\left\{{\varsigma }_{j},{\chi }_{j};{r}_{j}\right\}\) and the weight vector is \(L=({L}_{1},{L}_{2},\ldots,{L}_{n}{)}^{T}\) with \({L}_{j}\in [\text{0,1}]\) and \(\sum_{j=1}^{n}{L}_{j}=1.\) Then it is said CIFHOWG operator and CIFHOWG \(\left({a}_{1},{a}_{2},\ldots,{a}_{n}\right)=\)
Theorem 3.5.3
(Idempotency) If \({Y}_{j}=\left\{{\varsigma }_{j},{\chi }_{j};{r}_{j}\right\}\) for all \(j=\text{1,2},3,\ldots,n,\) then CCHFOWG \((Y,Y,Y,\ldots,Y)=Y.\)
Theorem 3.5.4
(Commutativity) If \(({Q}_{1}{\prime},{Q}_{2}{\prime},\ldots,{Q}_{n}{\prime})\) is any permutation of \(({Q}_{1},{Q}_{2},\ldots,{Q}_{n})\) , then CFHFOWG \(({Q}_{1}{\prime},{Q}_{2}{\prime},\ldots,{Q}_{n}{\prime})=\) CCHFOWG \(({Q}_{1},{Q}_{2},\ldots,{Q}_{n})\)
Theorem 3.5.5
(Boundedness) If \({Y}^{-}=min({b}_{1},{b}_{2},\ldots,{b}_{n}),\) \({Y}^{+}\) \(=\) \(\mathit{max}({b}_{1},{b}_{2},\ldots,{b}_{n}),\) then \({Y}^{-}\le\) CCHFOWG \(({b}_{1},{b}_{2},\ldots,{b}_{n})\le\) \({Y}^{+}.\)
CIFHHWG operator
Definition 3.6.1
The gathering of CCFNs are \({h}_{j}=\left\{{\varsigma }_{j},{\chi }_{j};{r}_{j}\right\}\) and the weight vector is \(G=({G}_{1},{G}_{2},\ldots,{G}_{n}{)}^{T}\) with \({G}_{j}\in [\text{0,1}]\) and.
\(\sum_{j=1}^{n}{G}_{j}=1.\left({r}_{1},{r}_{2},\ldots,{r}_{n}\right)={\otimes }_{j=1}^{n}{h}_{j}^{{G}_{j}}\) Then CIFHHWG is said the CIFHHWG operator.
Theorem 3.6.2
The collection of CCFNs are \({a}_{j}=\left\{{\varsigma }_{j},{\chi }_{j};{r}_{j}\right\}\) and the weight vector is \(L=({L}_{1},{L}_{2},\ldots,{L}_{n}{)}^{T}\) with \({L}_{j}\in [\text{0,1}]\) and \(\sum_{j=1}^{n}{L}_{j}=1\) . Then it is said CIFHHWG operator and CIFHHWG \(\left({a}_{1},{a}_{2},\ldots,{a}_{n}\right)=\)
Theorem 3.6.3
(Idempotency) If \({Y}_{j}=\left\{{\varsigma }_{j},{\chi }_{j};{r}_{j}\right\}\) for all \(j=\text{1,2},3,\ldots,n,\) then CCHFHWG \((Y,Y,Y,\ldots,Y)=Y.\)
Theorem 3.6.4
(Commutativity) If \(({H}_{1}{\prime},{H}_{2}{\prime},\ldots,{H}_{n}{\prime})\) is any permutation of \(({H}_{1},{H}_{2},\ldots,{H}_{n})\) , then CCHFHWG \(({H}_{1}{\prime},{H}_{2}{\prime},\ldots,{H}_{n}{\prime})=\) CCHFHWG \(({H}_{1},{H}_{2},\ldots,{H}_{n})\)
Theorem 3.6.5
(Boundedness) If \({Y}^{-}=min({b}_{1},{b}_{2},\ldots,{b}_{n}),\) \({Y}^{+}\) \(=\) \(\mathit{max}({b}_{1},{b}_{2},\ldots,{b}_{n}),\) then \({Y}^{-}\le\) CCHFHWG \(({b}_{1},{b}_{2},\ldots,{b}_{n})\le\)\(Y^{ + } .\)
Neural scheme based on CCF
In this section, we define two cases, case one proposes the CIFHWA, and case two proposes the CIFHWG operators.
Case 1
Step 1: Describe the CIF decision matrix.
Step 2: Describe the CIFHWA operator and \(\Gamma =\left({\Gamma }_{1},{\Gamma }_{2},\ldots,{\Gamma }_{n}\right).\)
we define the Cubic Intuitionistic Fuzzy Hamacher Weighted Averaging (CIFHWA) Operator and introduce the parameters \(\Gamma =\left({\Gamma }_{1},{\Gamma }_{2},\ldots,{\Gamma }_{n}\right).\), which influences the aggregation process. The operator combines decision criteria for alternatives in a multi-criteria decision-making process using circular intuitionistic fuzzy sets.
Step 3: Describe the CIFHWA operator and \(\Gamma =\left({\Gamma }_{1},{\Gamma }_{2},\ldots,{\Gamma }_{n}\right).\)
Step 4: This regularization term can be written as the sum of squares of the network weights
where, F: Total objective function, \(\beta\): Weight for the error term, \(\alpha\): Regularization coefficient, t: Target output vector, a: Actual output vector, x: Network weights, \({r}_{i}\): Additional penalty term.
Step 5: Compute new estimates for the regularization parameters \(\alpha =\frac{\zeta }{2E},\beta =\frac{N-\zeta }{2E}\)
Where, \(\alpha ,\beta\) Updated regularization parameters, \(\zeta\): Effective number of parameters, E: Error term from the output error, G: Sum of squared network weights, N: Total number of samples or parameters.
Step 6: Calculate the effective number of parameters \(\zeta ={\sum }_{i=1}^{n}\frac{\beta {\zeta }_{i}}{\beta {\zeta }_{i}+2\alpha }\)
Where: \(\beta ,\alpha\) and \(\zeta\) are constants or coefficients related to the model, n is the total number of parameters considered.
Step 7: Calculate the performance analysis
where: \(a,p,q\) are performance-related variables or coefficients.
Step 8: Describe the CIFHWA operator again and \(\Gamma =\left({\Gamma }_{1},{\Gamma }_{2},\ldots,{\Gamma }_{n}\right).\)
Step 9: Find the score function.
Step 10: Find the ranking.
Case 2
Step 1: Describe the CIF decision matrix.
Step 2: Describe the CIFHWG operator and \(\lambda =\left({\lambda }_{1},{\lambda }_{2},\ldots,{\lambda }_{n}\right).\)
Step 3: Describe the CIFHWG operator and \(\lambda =\left({\lambda }_{1},{\lambda }_{2},\ldots,{\lambda }_{n}\right).\)
Step 4 This regularization term can be written as the sum of squares of the network weights
where, F: Total objective function, \(\beta\): Weight for the error term, \(\alpha\): Regularization coefficient, t: Target output vector, a: Actual output vector, x: Network weights, \({r}_{i}\): Additional penalty term.
Step 5: Compute new estimates for the regularization parameters \(\alpha =\frac{\zeta }{2E},\beta =\frac{N-\zeta }{2E}\)
Where, \(\alpha ,\beta\) Updated regularization parameters, \(\zeta\): Effective number of parameters, E: Error term from the output error, G: Sum of squared network weights, N: Total number of samples or parameters.
Step 6: Calculate the effective number of parameters \(\zeta ={\sum }_{i=1}^{n}\frac{\beta {\zeta }_{i}}{\beta {\zeta }_{i}+2\alpha }\)
Where: \(\beta ,\alpha\) and \(\zeta\) are constants or coefficients related to the model, n is the total number of parameters considered.
Step 7: Calculate the performance analysis
where: \(a,p,q\) are performance-related variables or coefficients.
Step 8: Describe the CIFHWG operator again and \(\lambda =\left({\lambda }_{1},{\lambda }_{2},\ldots,{\lambda }_{n}\right).\)
Step 9: Find the score function.
Step 10: Find the ranking.
Case history
In this section, the proposed cohesive model is practical to dealer assessment and instruction distribution with admiration to ecological standards in a tissue tabloid industrial firm which is named XYZ in this paper. XYZ crops dissimilar kinds of tissue tabloid counting facemask tissues, weekly towels, bench bibs, etc. The commercial facilities have more than 160 people counting managers and practical workers. The manufacture size of XYZ is about 5000 loads apiece annum, and the entire remaining auction income of the business was around $31 million in 2016. XYZ has marketplaces in more than 30 republics counting 62 clientele 27 retailers and 35 sovereign suppliers). It has been an expert in numerous values like ISO 9001:2015, ISO 10002:2014, and ISO 10668. Because of the likely result of the paper crops on the setting, XYZ has begun a strategy meanwhile 2015 to advance its procedures with deference to ecological features and get an ISO 14001 guarantee. These business procurements are rare and substantial yearly, and the weight of the uncooked solid is roughly identical to the manufactured bulk.
As a fragment of its conservational plan, XYZ streams the desired raw solid of a day from around traders bestowing to conservation norms. In the earlier 10 years, the corporation has obtained the red factual from more than 14 dealers, but approximately of them have not encountered the obligatory excellence of XYZ. Therefore, the business obvious to boundary the number of dealers and select merchants which have advanced taxes of excellence, inferior taxes of tardiness, and devotion to ecological values. For this aim, the panel of management of the business shaped a collection of four specialists (Dm1 to Dm4) from dissimilar sections (manufacture, investigation and growth, purchasing, advertising, and humanoid reserve organization).
Numerical example
In this subsection, the ecological assessment standards and their meanings. Standards meaning the proposed cohesive model is practical to this instance as follows:
\(LHR :\) Ecological contamination.
This standard is connected to the projected equal production of airborne contaminants, injurious materials, solid wildernesses, and production of air impurities waste marine, which are statements by a provider in its making development.
\(KHI :\) Reserve feeding.
This criterion is correlated to the appraised equal of rare factual eating, liveliness feeding, and river eating throughout the progression of construction.
\(FSD :\) Biological revolution.
This criterion is interrelated to the advance of progressions and harvests that can help to maintainable growth using the profitable request of information to spread straight or unintended environmental developments.
\(ISB :\) Organic organization.
This criterion is connected to the preparation of capital for emerging, physical construction, and applying rules for ecological defense. ISO 14000 and ISO 14001 are the greatest extensive secondhand values in an ecological organization method. We define the two methods; the first proposed method is defined in the CIFHWA operator, and the second proposed method is defined in the CIFHWG operator.
Case 1
Step 1: Calculate the CIF decision matrix in Tables 1 and 2.
Step 2 Provide the CIFHWA operator in Table 3 and \(\xi =\left(\text{0.26,0.21,0.25,0.28}\right).\)
Step 3 Provide the CIFHWA operator in Table 4 and \(\xi =\left(\text{0.24,0.21,0.25,0.31}\right).\)
Step 4 This regularization term can be written as the sum of squares of the network weights
Step 5: Compute new estimates for the regularization parameters and Table 5.
Step 6: Calculate the effective number of parameters
Step 7: Calculate the performance analysis Tables 6, 7 and 8.
Step 8: Provide the again CIFHWA operator in Table 9 and \({s}_{1}=0.2034,{s}_{2}=0.2734,{s}_{3}=0.1867,{s}_{4}=0.6534.\)
Step 9: Find the score function
Step 10: The ranking is \(Z{C}_{4}>Z{C}_{3}>Z{C}_{2}>Z{C}_{1}\) and best is \(Z{C}_{4}\) .
Figure 5 proposed method is given as.
Case 1 of score function of different values.
Case 2
Step 1: Calculate the decision-making Tables 1 and 2
Step 2: Provide the CIFHWG operator in Table 10 and \(\xi =\left(\text{0.26,0.21,0.25,0.28}\right).\)
Step 3 Provide the CIFHWG operator in Table 11 and \(\xi =\left(\text{0.26,0.21,0.25,0.28}\right).\)
Step 4: This regularization term can be written as the sum of squares of the network weights
Step 5: Compute new estimates for the regularization parameters and Table 12.
Step 6: Calculate the effective number of parameters
Step 7: Calculate the performance analysis in Tables 13, 14, and 15.
Step 8: Provide the CIFHWG operator again and in Table 16, the weight vector is \({s}_{1}=0.2198,{s}_{2}=0.2706,{s}_{3}=0.1093,{s}_{4}=0.8765\)
Step 9: Find the score function
Step 10: The ranking is \(Z{C}_{4}>Z{C}_{3}>Z{C}_{2}>Z{C}_{1}\) and best is \(Z{C}_{4}\) .
Figure 6 provides the CIFHWG operator.
Score function of the CIFHWG operator.
Comparative examinations
In this design, several sets of criteria weights are characterized by the number of criteria. In other words, in case we have m criteria, we should define m sets of criteria weights for the affectability examination. In each of the sets, one measure has the most significance (weight), another has the slightest significance, and the other criteria have importance or weights between the foremost and slightest imperative criteria. Using this design, we can look at varying the weights of criteria effectively and simply. The weight of each measure in each of the sets is defined in Tables 17 and 18.
Figure 7 is given as below.
Is a different comparison analysis.
Table 18 Comparison matrix is given in below.
Figure 8 of comparison analysis.
Is a different theory.
Interpretation of results
The experimental results show that the proposed outline meaningfully recovers the decision-making procedure likened to existing systems. The six aggregation operators industrialized in this education suggestion improved suppleness and correctness compared to normal aggregation methods, such as the Weighted Average (WA) and Ordered Weighted Average (OWA). The neural scheme founded on CIF improves computational competence, enabling the perfect to measure efficiently in real-time decision-making situations.
The experiment consequences are assumed below in Table 19.
Figure 9 of Interpretation of Results is given as.
Of different Interpretation of Results.
Limitations of the proposed approach
In this subsection, we define the limitations of the proposed method in Table 20.
Preceding methods, such as individuals using standard intuitionistic fuzzy sets or humble weighted averages, frequently fail to completely imprison the multifaceted relations between criteria or to knob indecision successfully. By incorporating Hamacher operators, which deliver a universal method of aggregation, our process agrees for the better showing of interdependencies between decision criteria, attracting the decision-making course.
Discussion
In this section, we define the discussion and results, Theoretical implications, Theoretical implications and Practical Implications.
Discussion and results
The proposed method starts with major the situation and choosing the fitting specialists, a concern consigned to the executives. This stage is serious, as decision-makers can have variable viewpoints and skills on the theme. The procedure executive allows weights to the specialists’ assessments founded on their contribution or might opt to extravagance all specialists, similarly, supporting with the expectations of the proposed model.
The results of this study prove that the proposed decision-making outline, which mixes circular intuitionistic fuzzy sets (C-IFSs) with Hamacher operators and neural schemes, proposes an important progression over existing approaches in the works. The improved aggregation operators CIFHWA, CIFHOWA, CIFHHWA, CIFHWG, CIFHOWG, and CIFHHWG demonstrate greater presentation in taking the shades of indecision and fuzziness, which are often incompetently addressed by outmoded fuzzy aggregation approaches. When compared with preceding educations, our results indicate that the proposed outline meaningfully recovers the suppleness and correctness of decision-making in multi-criteria decision-making difficulties.
Table 21 results and discussion is given below.
In addition, the overview of neural schemes, exactly the use of a circular intuitionistic fuzzy set, enhances the computational competence of the decision-making process. This speech is one of the major limitations in existing approaches, where computational difficulty can often limit their applicability in real-time decision-making schemes. Our outline offers a talented answer for practical requests by balancing computational viability with the essential for precise decision-making.
In standings of theoretic contributions, this study bonds the gap between customary fuzzy set philosophy and contemporary computational intellect techniques, contributing a novel mixture of aggregation operators and neural network-based optimizations. The aptitude to vigorous substitutes additional precisely and compliantly, while management uncertainty and fuzziness, marks an important contribution to the arena of fuzzy decision-making and multi-criteria examination. While our outline demonstrates strong compensations, coming research might focus on spreading this practice to active decision-making settings and mixing it with real-time information watercourses. Moreover, the possibility of crossbreeding this approach with additional mechanism knowledge techniques, such as profound learning, could additionally improve its competence in large-scale, high-dimensional decision-making difficulties. This discussion piece now obviously compares the proposed technique to existing studies, highlighting the meaning of the results and the real contribution of the object to the field. It also proposes directions for upcoming research to advance the practice.
Theoretical implications
In this study, CIFN procedures based on group decision-making are smeared to explain a Hamacher tricky in dealer assortment. The CIFN technique is consumed to regulate the weights of the criteria, and the CIF Hamacher is exhausted to choose the maximum appropriate round dealer. CIFN permits the symbol of price varieties instead of exact statistics, taking the imprecision in specialists’ sentences. This types the proposed method more truthful and appropriate for physicians who goal to choice round suppliers.
To authenticate the attitude, the results are associated with other MCDM approaches. The weights resulting from the CIF Hamacher are used with CIF Hamacher approaches to recalculate the positions of possible dealers. The rankings completed by all four approaches are reliable. Moreover, a one-at-a-time sensitivity examination is done to assess how vagaries in criteria weights shake the consequences. The findings demonstrate that the proposed method is additionally bendable and actual in reproducing decision-makers’ favorites when assessing round dealers in CIFNS settings.
Managerial implications
From the managerial viewpoint, a proposed joined classic containing financial, social, and round standards will allow bosses to comprehend the idea of sustainability and, assist them choose round dealers, which in go donate to refining suppliers’ sustainability presentation Yu et al.26. According to the findings, while evaluating circular suppliers, decision-makers should give priority to the economic factor first, with the social and circular factors coming in second and third, respectively. Stated differently, economic factors are considered the most crucial in the process of evaluating suppliers Pattanayak et al.27. An additional main finding is that communal criteria raise the consciousness of business directors concerning work security, privileges of staff, worker exercise and growth programs, and see-through data distribution. Therefore, the directors would emphasize communal criteria since they directly affect the long-term sustainability presentation of dealers as well as investors’ prospects Pattanayak et al.27. The case company’s emphasis on the circular dimension, which encompasses eco-design, the creation of eco-friendly products, adherence to environmental regulations, product disclosures, and the use of cleaner production technologies, is one of the main conclusions. Managers should give recycling, reusing, and remanufacturing materials and resources top priority to optimize the circular benefits. These initiatives not only lessen pollution and waste but also preserve resources, encourage sustainability, and offer competitive advantages in the social, economic, and environmental spheres. Furthermore, the evaluation criteria showed that quality was the most important factor, but product declarations and environmental requirements were ranked lowest, suggesting that they had little influence on the choice made in the end. Although some alternatives may perform better on social and environmental criteria, this research highlights the importance of economic considerations when choosing circular suppliers.
Practical implications
From a practical view, the primary neutral of this education is to regulate the assessment standards for CIFN, founded on an appraisal of the pertinent works and specialists’ assessments. Another impartial is to suggest a united group decision-making model in a CIFN setting that will assist physicians to choose suitable round dealers. Added criteria can be auxiliary or uninvolved from the perfect, contingent on the supplies of physicians. Thus, easily implement the assessment criteria vessel for practical requests. The result of this education is authorized discussions with physicians. Specialists from the business and dealers avowed that the fallouts were precise and applied. Dealers are exactly renowned that the proposed technique suggestions appreciated perceptions into identifying their faintness and attract their sustainability act. These findings support preceding trainings by Yu et al. 26 and Pattanayak et al. 27, which highlight the standing of executives raising long-term associations with providers to support capabilities and discourse weaknesses effectually.
Conclusion
This study introduces a robust decision-making framework based on Circular Intuitionistic Fuzzy Sets (C-IFSs) within the Hamacher operational framework. By developing six novel aggregation operators Cubic Intuitionistic Fuzzy Hamacher Weighted Average (CIFHWA), Ordered Weighted Average (CIFHOWA), Hybrid Weighted Average (CIFHHWA), Weighted Geometric (CIFHWG), Ordered Weighted Geometric (CIFHOWG), and Hybrid Weighted Geometric (CIFHHWG) we provide flexible tools for handling multi-criteria decision-making (MCDM) problems. The proposed score and accuracy functions further enhance the ranking and evaluation of alternatives in fuzzy environments. Additionally, the neural scheme based on Circular correlation coefficients (CCF) integrates computational intelligence into the decision-making process, enabling adaptive and efficient solutions. Through a detailed numerical example, the practicality and effectiveness of the proposed methodology are demonstrated. Comparative analyses with existing techniques and sensitivity examinations validate the robustness, reliability, and superior performance of the model. The findings underline the significant advantages of combining C-IFSs with Hamacher operations and neural-based approaches for addressing uncertainty and imprecision in real-world decision-making scenarios. This study opens new avenues for research and applications in fuzzy systems, artificial intelligence, and decision-making under uncertainty.
In the future, we will concentrate on analyzing Hamacher operators for circular q-rung orthopair fuzzy data, taking delays into account. Furthermore, we will strive to implement these operators in28,29,30,31,32,33 and practical applications such as pattern recognition, machine learning, data mining, and statistical analysis to enhance their effectiveness and applicability.
Data availability
Data will be made available on request to corresponding author.
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Acknowledgements
Aziz Khan and Thabet Abdeljawad would like to thank Prince Sultan University for paying the APC and the support through TAS research lab.
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Fahmi, A., Khan, A., Maqbool, Z. et al. Circular intuitionistic fuzzy Hamacher aggregation operators for multi-attribute decision-making. Sci Rep 15, 5618 (2025). https://doi.org/10.1038/s41598-025-88845-0
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DOI: https://doi.org/10.1038/s41598-025-88845-0
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