Linear diophantine fuzzy Aczel-Alsina aggregation operators by using fuzzy-C means clustering algorithm

Abstract

This paper investigates the detailed analysis of linear diophantine fuzzy Aczel-Alsina aggregation operators, enhancing their efficacy and computational efficiency while aggregating fuzzy data by using the fuzzy C-means (FCM) method. The primary goal is to look at the practical uses and theoretical foundations of these operators in the context of fuzzy systems. The aggregation process is optimised using the FCM algorithm, which divides data into clusters iteratively. This reduces computer complexity and enables more dependable aggregation. The mathematical underpinnings of Linear Diophantine Fuzzy Aczel-Alsina aggregation operators are thoroughly examined in this study, along with an explanation of their purpose in handling imprecise and uncertain data. It also investigates the integration of the FCM method, assessing its impact on simplifying the aggregation procedure, reducing algorithmic complexity, and improving the accuracy of aggregating fuzzy data sets. This work illuminates these operators performance and future directions through extensive computational experiments and empirical analysis. It provides an extensive framework that shows the recommended strategy’s effectiveness and use in a variety of real-world scenarios. We obtain our ultimate outcomes through experimental investigation, which we use to inform future work and research. The purpose of the study is to offer academics and practitioners insights on how to improve information fusion techniques and decision-making processes.

Keywords

Linear diophantine fuzzy set Aczel-Alsina operational laws linear diophantine fuzzy Aczel-Alsina aggregation operators fuzzy C-means algorithm

1 Introduction

The methods typically employed in classical mathematics are not always satisfactory for handling real-world problems due to the uncertainties and ambiguity involved. In 1965, Zadeh [33] first proposed the concept of fuzzy sets, while Klir & Yuan [11] presented fuzzy logic in 1996 as an advancement over the traditional crisp set. In the unit closed interval [0, 1], the membership function assigns a real value to every item in the universe and creates a fuzzy set. It is an important mathematical tool for characterising a group of things whose borders are ill-defined. Fuzzy logic, distinguished by a membership function that has a range of 0 to 1, has replaced probability theory as a powerful tool for characterising obscurity, uncertainty, and imprecision in a variety of domains. The ability of fuzzy sets to handle imprecise, uncertain, or subjective information that frequently arises in real-life scenarios leads to a variety of applications across multiple domains. Fuzzy sets are used in control systems and engineering to model imprecise variables. This allows for more adaptable and flexible control mechanisms in processes like industrial automation and temperature regulation. Fuzzy sets help medical professionals make decisions by taking into account the uncertainty that may be present in patient data or symptoms, which helps in medical diagnosis. Fuzzy sets also help adaptive traffic management systems in the transportation and traffic control domains. These systems modify route guidance or signal timing in response to variable conditions, such as traffic density or meteorological forecasts. This has happened since Zadeh’s contribution to fuzzy sets [33]. In order to solve problems in a variety of disciplines, including robotics [44], environments [15], engineering [41], computer science [29], social science [40], and economics [6], researchers have looked into fuzzy sets. You can find studies of a few noteworthy works with fuzzy sets by Anoop, M. B., and Zadeh [32, 36]. Bustince [14] introduced the interval-valued fuzzy set (IVFS) in 2010. Modifications of FS theory [4, 55], application of FS theory in various fields [49] analysis of fuzzy systems [47, 57] are a few appropriate modifications that are listed in the shape. The membership function of this set is defined as denoting the collection of all subsets of [0, 1], for an explanation of the inadequate information. The idea of an intuitionistic fuzzy set (IFS) was created by Atanassov [28] in 1986. The hesitation degree is calculated, and the sum of the membership grade (ψ) and non-membership grade $(V)$ cannot be more than 1, but otherwise, the usage of membership and non-membership grades in an IFS allows the presentation of complex and ambiguous information. PFSs were originally proposed by Yager [45] as a generalization of IFSs and an extension of FSs. Pythagorean fuzzy number (PFN) proposed by Zhang and Xu is much superior to intuitionistic fuzzy number (IFN) [59]. Generalised ortho-pair fuzzy sets were first proposed by Yager [46] and are referred to as q-rung ortho-pair fuzzy sets (q-ROFS). A vital instrument for gathering statistical data is the triangular norm. Within the statistical information system, Menger [30] first examined triangle norms. Additionally, he offered a variety of AOs for the t-norm (TN) and the t-conorm that it associates with (TCN), including algebraic, Einstein, nilpotent, Frank, Archimedean, Hamacher, Dombi, and Lukasiewicz TCN and TN. Numerous academics have examined in-depth the methods used to collect HF data and how they affect decision-making [12, 58]. The 1982 discovery of two novel varieties by Aczel and Alsina (AA) are the Aczel-Alsina t-norm AA-TN and AA-TCN with fundamental characteristics [19]. Walker and Richman looked into the AA-TN and AA-TCN families [8]. Capability for information analysis is one of the attributes shared by AA-TN and AA-TCN. For the purpose of determining which were the best, AA-TN and AA-TCN, Farahbod and Etekeri [9] evaluated the TN and TCN families. Senapati et al. [54] used some basic AA-TN and AA-TCN procedures to find new AOs of IFS AA. Aczel-Alsina operators, which were initially presented by Ye et al. [20], are used by MADM for Z-Numbers. For an overview of several approaches to MADM problems that are covered in the literature, we suggest reading the articles [5, 10, 31, 39, 53, 54]. Schweizer-Sklar [3] introduced the concepts behind the Schweizer-Sklar aggregation tools in 1960. These aggregation techniques are more flexible and enable the provision of trustworthy information for decision-making. For instance, Garg et al. [13] demonstrated several novel aggregation techniques using basic operations under intuitionistic fuzzy information and fundamental Hamacher aggregating tools. As a mathematical foundation for identifying uncertainty, Molodtsov [7] created a unique class of sets that are generally called soft sets. Triangular co-norms, triangle norms, and implications put forth by M.I. Ali and M. Shabir [37] can thus be used to investigate them logically. Riaz and Hashmi [38] recommend using reference factors to further investigate these grades in decision analysis. They introduced the linear Diophantine fuzzy set (LDFS), an innovative fuzzy model that incorporates knowledge about the MSGs and NMSGs of input data through reference or control parameterizations. This feature makes it superior to traditional fuzzy models.

An option where the sum of the membership and non-membership grades is higher than 1 and its total of squares, in some real-world situations, is more than 1 (e.g., 0.8 + 0.9 > 1) may satisfy a decision maker’s (DM) specified attribute. When this happens, both PFS and IFS malfunction. Changes have been made to the restrictions on membership and non-membership grades in order to address these worries. Models of decision-making must include control parameters. The lack of parameterizations in PFS, IFS, and q-ROFS is a disadvantage. LDFS provides the MADM with a multitude of applications for these kinds of real-world problems. This approach allows us to deal with the intuitionistic, Pythagorean, and qrung ortho-pair character of attributes under reference factor influence. For instance, if (0.8 + 0.5 > 1), we can insert reference parameters to make (0.8) (0.3) + (0.5) (0.2) <1, where (0.3, 0.2) are the standards by which membership and non-membership grades are measured, respectively. Due to its similarity to number theory’s well-known linear Diophantine equation ax + by = c, the linear Diophantine fuzzy set (LDFS) is the most appropriate word for the model that has been suggested. To describe how reference parameters work in LDFS since they can’t handle parameterizations in q-ROFSs, PFSs, or IFSs. The suggested model is an improvement over the existing methods, and choosing the grades is entirely up to the decision maker (DM). The physical attributes of the sense of reference are altered by this structure to provide further categorization. The q-linear Diophantine fuzzy set (q-LDFS) is a special combination of the Pythagorean fuzzy set and linear Diophantine fuzzy sets, as well as the q-rung ortho-pair fuzzy set. Almagrabi [1] introduced it and emphasised some of its most important features. Some operators were computed based on PFS, such as Archimedean operators [43], Einstein aggregation operators [60], and Schweizer-Sklar Muirhead mean operators [56] Additionally, in scenarios involving decision-making, for uncertainty to be aggregated effectively, aggregation operators are essential.

FCM clustering [26], also known as fuzzy C-means clustering, is a well-liked algorithm for cluster classification in data analysis and machine learning. Data points can be assigned to numerous clusters with different grades of membership using fuzzy C-means, reflecting the uncertainty or fuzziness in the data, in contrast to traditional crisp clustering methods. Web personalisation technologies that recognise and capitalise on user preferences to provide each user with dynamically tailored content have become increasingly crucial due to the internet’s continuous increase in available information. The logistics development in coastal regions was demonstrated by Xu et al. [34] through the usage of community division. In general, the program clearly encourages the growth of logistics in the regions along the coast. Xu et al. [35] proposed that the restrictions on the Sulfur content of marine fuels under the 1st round of ECA policy have only effectively reduced the SOx concentration in the Bohai Rim and the Yangtze River Delta region, whereas the impact on the Pearl River Delta region isn’t significant.

Using a real-world battery dataset, Zhang et al. [21] suggested the PF-BiGRU-TSAM method. The suggested strategy is superior to certain published techniques, according to experimental data. The idea of power routing was initially presented by Zhang et al. [22]. The dc-dc boost conversion system distributes power to the cells based on their cumulative damage, postponing the failure of cells with more cumulative damage. Conformal transformations, first proposed by J. Zheng and Hu [23], help non-linear kernels find difficult-to-identify minority class regions. These fractional methods proposed by [17, 18, 27, 50]. This helps the classifiers become more adaptive to unbalanced data with complicated distributions. Zheng [24] presented ideas that were better than what the rivals had learned.

Fig. 1

Flowchart of fuzzy-C means algorithm.

The contribution of the method proposed in this article is its practical applicability across several areas. The framework is adaptable and may be used for practical decision-making situations in several domains. The combination of FCM clustering and LDFAA operators allows for the control of ambiguities that are present in multi-attribute decision-making processes.

One special method for managing fuzzy data aggregation is to combine the Fuzzy-C Means clustering algorithm with linear Diophantine fuzzy aggregation operators. This new approach to data aggregation in fuzzy environments is made possible by the merging of ideas from fuzzy clustering techniques with a linear Diophantine fuzzy set. The formulation or modification of a methodology combining fuzzy clustering algorithms and a linear Diophantine fuzzy set is required by the research question. This raises a fresh question about the suitability and efficacy of using such an approach to aggregate fuzzy data. The combination of fuzzy clustering techniques like Fuzzy-C means with linear diophantine fuzzy aggregation operators may not be well covered in the literature as of yet. Therefore, by examining a relatively unexplored area in the realm of fuzzy logic and data analysis, this research issue has the potential to close a gap in the literature. The proposed research aims to shed light on the efficacy, efficiency, and practicality of fuzzy aggregation techniques by examining the Fuzzy-C Means clustering algorithm’s application in the context of linear Diophantine fuzzy aggregation operators. Therefore, the objectives and motivation of this article are as follows: 1.

To invent the Aczel-Alsina operational laws for LDF set based on the Aczel-Alsina t-norm and t-conorm.

To expose the LDFAAWA operator, LDFAAOWA operator, and LDFAAHA operator, and evaluate their fundamental properties and results.

To evaluate the problem of user profiling by fuzzy clustering to enhance the worth of the invented theory.

To show the supremacy and accuracy of the proposed theory with the with the help of a comparative analysis between the proposed techniques and some existing techniques.

The overall scheme of this research paper is organised in this shape: In Section 2, we covered some basic LDFS concepts. In Section 3, it includes several possible Aczel-Alsina operational laws for LDFNs. In Section 4, it includes several possible Aczel-Alsina AOs for LDFNs. In Section 5, we discussed Fuzzy-C means clustering by user profiling. In Section 6, we solved a numerical example by using an algorithm. Some concluding remarks are stated in Section 7.

Fig. 2

Graphical representation of paper scheme.

Table 1

Explanation of symbols

Symbols	Explanation	Symbols	Explanation
$\overset{´}{G}$	Universal set	$\tilde{Y}$	Element of a universal set
$\hat{ϝ}$	Fuzzy set	ψ	Membership function
$V$	Non-membership function	$\ddot{ℑ}$	Intuitionistic fuzzy set
$L$	Linear diophantine fuzzy set	α, β	Parameters of LDFS
$\hat{J}$	Linear diophantine fuzzy number (LDFN)	$\hat{S}$	Score function
$\bar{A}$	Accuracy function	$\ddot{U}$	Quadratic score function
$\ddot{E}$	Expected score function	$\ddot{A}$	Quadratic accuracy function
$T$	Weight vector

2 Basic concepts

In this section, we go over a few fundamental LDFS concepts, functions, and score functions. $\overset{´}{G}$ is the universal set used in this study.

Definition 1. [33] Let fuzzy set $\hat{ϝ}$ that is subject to the influence of universal set $\overset{´}{G}$ is

\hat{ϝ} = (ψ (\tilde{Y}) | \tilde{Y} \in \overset{´}{G})

(1)

where

ψ (\tilde{Y}) \in [0, 1]

is membership degree (MD) of

\hat{ϝ}

\overset{´}{G}

Definition 2. [28] Let Intuitionistic fuzzy set (IFS) $\ddot{ℑ}$ that is subject to the influence of universal set $\overset{´}{G}$ is

\ddot{ℑ} = (ψ (\tilde{Y}), V (\tilde{Y}) | \tilde{Y} \in \overset{´}{G})

(2)

where

ψ (\tilde{Y}) \in [0, 1]

is membership degree (MD) and

V (\tilde{Y}) \in [0, 1]

\ddot{ℑ}

\overset{´}{G}

with necessary condition

0 \leq ψ (\tilde{Y}) + V (\tilde{Y}) \leq 1

Definition 3. [38] Let LDFS (Linear Diophantine fuzzy set) $L$ in $\overset{´}{G}$ is defined as

L = (\tilde{Y}, 〈 ψ_{L} (\tilde{Y}), V_{L} (\tilde{Y}) 〉, 〈 α_{L} (\tilde{Y}), β_{L} (\tilde{Y}) 〉 : \tilde{Y} \in \overset{´}{G})

(3)

Where

ψ_{L} (\tilde{Y}), V_{L} (\tilde{Y}), α_{L} (\tilde{Y}), β_{L} (\tilde{Y}) \in [0, 1]

are the reference parameters that correspond to the membership grade (MG) and non-membership grade (NMG), respectively, and which meet the following fundamental requirements:

\begin{matrix} 0 \leq α_{L} (\tilde{Y}) + β_{L} (\tilde{Y}) \leq 1 \\ 0 \leq α_{L} (\tilde{Y}) ψ_{L} (\tilde{Y}) + β_{L} (\tilde{Y}) V_{L} (\tilde{Y}) \leq 1 \end{matrix}

(4)

for all

\tilde{Y} \in \overset{´}{G}

. The LDFS

L_{\overset{´}{G}} = {(\tilde{Y}, 〈 1, 0 〉, 〈 1, 0 〉) : \tilde{Y} \in \overset{´}{G}}

(5)

is called the absolute LDFS in

\overset{´}{G}

. The LDFS

L_{\overset{ˇ}{μ}} = {(\tilde{Y}, 〈 0, 1 〉, 〈 0, 1 〉) : \tilde{Y} \in \overset{´}{G}}

(6)

is called the null LDFS in

\overset{´}{G}

. A critical system analysis benefits from reference or control parameters. In addition,

π_{L} (\tilde{Y}) = 1 - (α_{L} (\tilde{Y}) ψ_{L} (\tilde{Y}) + β_{L} (\tilde{Y}) V_{L} (\tilde{Y}))

is known as the indeterminacy degree and the associated reference parameter of

\tilde{Y}

L

Definition 4. [38] A Linear Diophantine fuzzy number is defined in the form

\hat{J} = (〈 ψ_{\hat{J}}, V_{\hat{J}} 〉, 〈 α_{\hat{J}}, β_{\hat{J}} 〉)

satisfying the following conditions:

(i)

$ψ_{\hat{J}}, V_{\hat{J}}, α_{\hat{J}}, β_{\hat{J}} \in [0, 1]$

(ii)

$0 \leq α_{\hat{J}} + β_{\hat{J}} \leq 1$

(iii)

$0 \leq α_{\hat{J}} ψ_{\hat{J}} + β_{\hat{J}} V_{\hat{J}} \leq 1$

Definition 5. Let $\hat{J}$ = $(〈 ψ_{\hat{J}}, V_{\hat{J}} 〉, 〈 α_{\hat{J}}, β_{\hat{J}} 〉)$ be a LDFN, then we indicate the score function (SF) of LDFN as $\hat{S} (\hat{J})$ and we clarify it as

\hat{S} (\hat{J}) = \frac{1}{2} [(ψ_{\hat{J}} - V_{\hat{J}}) + (α_{\hat{J}} - β_{\hat{J}})] .

(7)

Here, LDFN

(\overset{´}{G})

represents a collection of LDFNs on

\overset{´}{G}

. The range of

\hat{S} (\hat{J})

is given as

\hat{S} (\hat{J}) : LDFN (\overset{´}{G}) ⟶ [- 1, 1]

Definition 6. Let $\hat{J}$ = $(〈 ψ_{\hat{J}}, V_{\hat{J}} 〉, 〈 α_{\hat{J}}, β_{\hat{J}} 〉)$ be an LDFN, then we indicate the accuracy function (AF) of LDFN as $\bar{A}$ and we clarify it as

\bar{A} (\hat{J}) = \frac{1}{2} [(\frac{ψ_{\hat{J}} - V_{\hat{J}}}{2}) + (α_{\hat{J}} + β_{\hat{J}})] .

(8)

Here, LDFN

(\overset{´}{G})

represents a collection of LDFNs on

\overset{´}{G}

. The range of

\bar{A} (\hat{J})

is characterized as

\bar{A} (\hat{J}) : LDFN (\overset{´}{G}) ⟶ [0, 1]

Definition 7. Let $\hat{J}$ = $(〈 ψ_{\hat{J}}, V_{\hat{J}} 〉, 〈 α_{\hat{J}}, β_{\hat{J}} 〉)$ be a LDFN, then we indicate the quadratic score function (QSF) of LDFN as $\ddot{U}$ and we clarify it as

\ddot{U} = \frac{1}{2} [({ψ_{\hat{J}}}^{2} - {V_{\hat{J}}}^{2}) + ({α_{\hat{J}}}^{2} - {β_{\hat{J}}}^{2})]

(9)

Here, LDFN

(\overset{´}{G})

represents a collection of LDFNs on

\overset{´}{G}

. The range of

\ddot{U} (\hat{J})

is determined by

\ddot{U} (\hat{J}) : LDFN (\overset{´}{G}) ⟶ [- 1, 1]

Definition 8. Let $\hat{J}$ = $(〈 ψ_{\hat{J}}, V_{\hat{J}} 〉, 〈 α_{\hat{J}}, β_{\hat{J}} 〉)$ be a LDFN then, we indicate the quadratic accuracy function (QAF) of LDFN as $\ddot{A}$ and, we clarify it as

\ddot{A} = \frac{1}{2} [\frac{{ψ_{\hat{J}}}^{2} - {V_{\hat{J}}}^{2}}{2} + ({α_{\hat{J}}}^{2} + {β_{\hat{J}}}^{2})]

(10)

Here, LDFN

(\overset{´}{G})

represents a collection of LDFNs on

\overset{´}{G}

. The range of

\ddot{A} (\hat{J})

is a transformation

\ddot{A} (\hat{J}) : LDFN (\overset{´}{G}) ⟶ [0, 1]

Definition 9. Let $\hat{J}$ = $(〈 ψ_{\hat{J}}, V_{\hat{J}} 〉, 〈 α_{\hat{J}}, β_{\hat{J}} 〉)$ be a LDFN then, we indicate the expected score function (ESF) of LDFN as $\ddot{E}$ and, we clarify it as

\ddot{E} = \frac{1}{2} [\frac{(ψ_{\hat{J}} - V_{\hat{J}} + 1)}{2} + \frac{(α_{\hat{J}} - β_{\hat{J}} + 1)}{2}]

(11)

Here, LDFN

(\overset{´}{G})

represents a collection of LDFNs on

\overset{´}{G}

. The range of

\ddot{E} (\hat{J})

determined by

\ddot{E} (\hat{J}) : LDFN (\overset{´}{G}) ⟶ [0, 1]

Definition 10. Let $\hat{J}$ = $(〈 ψ_{\hat{J}}, V_{\hat{J}} 〉, 〈 α_{\hat{J}}, β_{\hat{J}} 〉)$ be a LDFN and $\ddot{ϑ} > 0$ then, (i)

${\hat{J}}_{1}^{c} = (〈 V_{1}, ψ_{1} 〉, 〈 β_{1}, α_{1} 〉)$

(ii)

$\ddot{ϑ} \hat{J} = (〈 1 - (1 - ψ_{1})^{\ddot{ϑ}}, {V_{1}}^{\ddot{ϑ}} 〉, 〈 1 - (1 - α_{1})^{\ddot{ϑ}}, {β_{1}}^{\ddot{ϑ}} 〉)$

(iii)

${\hat{J}}_{1}^{\ddot{ϑ}} = (〈 {ψ_{1}}^{\ddot{ϑ}}, 1 - (1 - V_{1})^{\ddot{ϑ}} 〉, 〈 α^{\ddot{ϑ}}, 1 - (1 - β_{1})^{\ddot{ϑ}} 〉)$

Definition 11. Let ${\hat{J}}_{1}$ and ${\hat{J}}_{2}$ have two LDFNs. By employing functions, we may associate these two LDFNs in an efficient manner: (i)

If $\hat{S} ({\hat{J}}_{1}) < \hat{S} ({\hat{J}}_{2})$ , then ${\hat{J}}_{1} < {\hat{J}}_{2}$

(ii)

If $\hat{S} ({\hat{J}}_{2}) < \hat{S} ({\hat{J}}_{1})$ , then ${\hat{J}}_{2} < {\hat{J}}_{1}$

(iii)

If $\hat{S} ({\hat{J}}_{2}) = \hat{S} ({\hat{J}}_{1})$ , then ${\hat{J}}_{2} = {\hat{J}}_{1}$ (a)

If $\bar{A} ({\hat{J}}_{1}) < \bar{A} ({\hat{J}}_{2})$ ,then ${\hat{J}}_{1} < {\hat{J}}_{2}$

(b)

If $\bar{A} ({\hat{J}}_{2}) < \bar{A} ({\hat{J}}_{1})$ ,then ${\hat{J}}_{2} < {\hat{J}}_{1}$

(c)

If $\bar{A} ({\hat{J}}_{2}) = \bar{A} ({\hat{J}}_{1})$ ,then ${\hat{J}}_{2} = {\hat{J}}_{1}$

Definition 12. Let $\hat{J}$ = $(〈 ψ_{\hat{J}}, V_{\hat{J}} 〉, 〈 α_{\hat{J}}, β_{\hat{J}} 〉)$ be a group of LDFNs with $TDSFTD i \in \overset{ˇ}{μ}$ , Then, (i)

$\cup_{TDSFTD i \in \overset{ˇ}{μ}} {\hat{J}}_{TDSFTD i} = (〈 \sup_{TDSFTD i \in \overset{ˇ}{μ}} ψ_{TDSFTD i}, \inf_{TDSFTD i \in \overset{ˇ}{μ}} V_{TDSFTD i} 〉, 〈 \sup_{TDSFTD i \in \overset{ˇ}{μ}} α_{TDSFTD i}, \inf_{TDSFTD i \in \overset{ˇ}{μ}} β_{TDSFTD i} 〉)$

(ii)

$\cap_{TDSFTD i \in \overset{ˇ}{μ}} {\hat{J}}_{TDSFTD i} = (〈 \inf_{TDSFTD i \in \overset{ˇ}{μ}} ψ_{TDSFTD i}, \sup_{TDSFTD i \in \overset{ˇ}{μ}} V_{TDSFTD i} 〉, 〈 \inf_{TDSFTD i \in \overset{ˇ}{μ}} α_{TDSFTD i}, \sup_{TDSFTD i \in \overset{ˇ}{μ}} β_{TDSFTD i} 〉)$

Definition 13. Let ${\hat{J}}_{i}$ = $(〈 ψ_{i}, V_{i} 〉, 〈 α_{i}, β_{i} 〉)$ be a LDFNs with TDSFTDi = 1, 2. Then, (i)

${\hat{J}}_{1} \subseteq {\hat{J}}_{2} \Leftrightarrow ψ_{1} \leq ψ_{2}, V_{2} \leq V_{2}, α_{1} \leq α_{2}, β_{2} \leq β_{1}$

(ii)

${\hat{J}}_{1} = {\hat{J}}_{2} \Leftrightarrow ψ_{1} = ψ_{2}, V_{1} = V_{2}, α_{1} = α_{2}, β_{1} = β_{2}$

(iii)

${\hat{J}}_{1} \oplus {\hat{J}}_{2} = (〈 ψ_{1} + ψ_{2} - ψ_{1} ψ_{2}, V_{1} V_{2} 〉, 〈 α_{1} + α_{2} - α_{1} α_{2}, β_{1} β_{2} 〉)$

(iv)

${\hat{J}}_{1} \otimes {\hat{J}}_{2} = (〈 ψ_{1} ψ_{2}, V_{1} + V_{2} - V_{1} V_{2} 〉, 〈 α_{1} α_{2}, β_{1} + β_{2} - β_{1} β_{2} 〉)$

Example 1. Let ${\hat{J}}_{1} = (〈 0.1951, 0.7939 〉, 〈 0.1760, 0.7276 〉)$ and ${\hat{J}}_{2} = (〈 0.1251, 0.8354 〉, 〈 0.0898, 0.8666 〉)$ have two LDFNs. Then, it’s evident that ${\hat{J}}_{1} \subseteq {\hat{J}}_{2}$ . It is possible to confirm that (i)

${\hat{J}}_{1}^{c} = (〈 0.7939, 0.1951 〉, 〈 0.7276, 0.1760 〉)$

(ii)

${\hat{J}}_{1} \cup {\hat{J}}_{2} = (〈 0.1951, 0.7939 〉, 〈 0.1760, 0.7276 〉) = {\hat{J}}_{1}$

(iii)

${\hat{J}}_{1} \cap {\hat{J}}_{2} = (〈 0.1251, 0.8354 〉, 〈 0.0898, 0.8666 〉) = {\hat{J}}_{2}$

(iv)

${\hat{J}}_{1} \oplus {\hat{J}}_{2} = (〈 0.2957, 0.6632 〉, 〈 0.2499, 0.6305 〉)$

(v)

${\hat{J}}_{1} \otimes {\hat{J}}_{2} = (〈 0.0244, 0.9660 〉, 〈 0.0158, 0.9636 〉)$

3 Aczel-Alsina operational laws for LDFNs

Aczel-Alsina Operational laws pertaining to linear Diophantine fuzzy numbers are particular guidelines and precepts that control how aggregation operators behave when dealing with these fuzzy numbers. Diophantine equations are extended to fuzzy numbers with linear Diophantine fuzzy numbers, allowing uncertainty or imprecision in numerical values to be represented. Within this framework, definition and comprehension of t-norms and t-conorms are central to the discussion of Aczel-Alsina operational laws for linear Diophantine fuzzy numbers.

T-norms in the context of linear Diophantine fuzzy numbers represent mathematical operations that model the intersection or conjunction (AND) operation between fuzzy numbers. These functions combine two fuzzy numbers and produce an output that represents the intersection of their fuzzy sets. T-norms maintain certain properties, such as commutativity, associativity, and boundary conditions within the context of T-conorms are mathematical operations that model the union or dis-junction (OR) operation between fuzzy numbers in the context of linear Diophantine fuzzy numbers. T-conorms are similar to t-norms in that they take two fuzzy numbers and output the union of their fuzzy sets. Additionally, T-conorms have particular properties and guidelines when it comes to linear Diophantine fuzzy numbers.

Definition 14. We establish the t-norms for Aczel-Alsina. ${\overset{˘}{C}}_{H} (a, b) : [0, 1] \times [0, 1] \to [0, 1]$ and the Aczel-Alsina t-conorms ${\dot{D}}_{H} (a, b) : [0, 1] \times [0, 1] \to [0, 1]$ for all $a, b \in [0, 1]$ and $H \geq 0$ as follows: (i) The t-norms for Aczel-Alsina are described as

{\overset{˘}{C}}_{H} (a, b) = {\begin{matrix} {\overset{˘}{C}}_{dr} (a, b), & if H = 0 \\ \min (a, b), & if H = \infty \\ e^{- ((- ln a)^{H} + (- ln b)^{H})^{\frac{1}{H}}}, & otherwise \end{matrix}

(ii) The t-conorms for Aczel-Alsina are described as

{\dot{D}}_{H} (a, b) = {\begin{matrix} {\dot{D}}_{dr} (a, b), & if H = 0 \\ \max (a, b), & if H = \infty \\ 1 - e^{- ((- ln (1 - a))^{H} + (- ln (1 - b))^{H})^{\frac{1}{H}}}, & otherwise \end{matrix}

Where

{\overset{˘}{C}}_{dr} (a, b)

and

{\dot{D}}_{dr} (a, b)

are the drastic t-norm and the drastic t-conorm, which are represented as

{\overset{˘}{C}}_{dr} (a, b) = {\begin{matrix} a, & if b = 1 \\ b, & if a = 1 \\ 0, & otherwise \end{matrix}

and

{\dot{D}}_{dr} (a, b) = {\begin{matrix} a, & if b = 1 \\ b, & if a = 1 \\ 0, & otherwise \end{matrix}

Definition 15. Let $\hat{J} = (〈 ψ_{TDSFTD i}, V_{TDSFTD i} 〉, 〈 α_{TDSFTD i}, β_{TDSFTD i} 〉)$ be two LDFNs with TDSFTDi = 1, 2 and $\dot{ϒ} > 0$ be any real number; Aczel-Alsina operations for LDFNs is defined as follows:

(i)
${\hat{J}}_{1}^{c} = (〈 V_{1}, ψ_{1} 〉, 〈 β_{1}, α_{1} 〉) .$
(ii)
${\hat{J}}_{1} \cup {\hat{J}}_{2} = (〈 \max {ψ_{1}, ψ_{2}}, \min {V_{1}, V_{2}} 〉, 〈 \max {α_{1}, α_{2}}, \min {β_{1}, β_{2}} 〉) .$
(iii)
${\hat{J}}_{1} \cap {\hat{J}}_{2} = (〈 \min {ψ_{1}, ψ_{2}}, \max {V_{1}, V_{2}} 〉, 〈 \min {α_{1}, α_{2}}, \max {β_{1}, β_{2}} 〉)$
(iv)
${\hat{J}}_{1} \oplus {\hat{J}}_{2} = (〈 1 - e^{- ((- ln (1 - ψ_{1}))^{H} + (- ln (1 - ψ_{2}))^{H})^{\frac{1}{H}}}, e^{- ((- ln V_{1})^{H} + (- ln V_{2})^{H})^{\frac{1}{H}}} 〉), (〈 1 - e^{- ((- ln (1 - α_{1}))^{H} + (- ln (1 - α_{2}))^{H})^{\frac{1}{H}}}, e^{- ((- ln β_{1})^{H} + (- ln β_{2})^{H})^{\frac{1}{H}}} 〉) .$
(v)
${\hat{J}}_{1} \otimes {\hat{J}}_{2} = (〈 e^{- ((- ln ψ_{1})^{H} + (- ln ψ_{2})^{H})^{\frac{1}{H}}}, 1 - e^{- ((- ln (1 - V_{1}))^{H} + (- ln (1 - V_{2}))^{H})^{\frac{1}{H}}} 〉), (〈 e^{- ((- ln α_{1})^{H} + (- ln α_{2})^{H})^{\frac{1}{H}}}, 1 - e^{- ((- ln (1 - β_{1}))^{H} + (- ln (1 - β_{2}))^{H})^{\frac{1}{H}}} 〉) .$
(vi)
$\dot{ϒ} {\hat{J}}_{1} = (〈 1 - e^{- (\dot{ϒ} (- ln (1 - ψ_{1}))^{H})^{\frac{1}{H}}}, e^{- (\dot{ϒ} (- ln V_{1})^{H})^{\frac{1}{H}}} 〉), (〈 1 - e^{- (\dot{ϒ} (- ln (1 - α_{1}))^{H})^{\frac{1}{H}}}, e^{- (\dot{ϒ} (- ln β_{1})^{H})^{\frac{1}{H}}} 〉) .$
(vii)
${({\hat{J}}_{1})}^{\dot{ϒ}} = (〈 e^{- (\dot{ϒ} (- ln ψ_{1})^{H})^{\frac{1}{H}}}, 1 - e^{- (\dot{ϒ} (- ln (1 - V_{1}))^{H})^{\frac{1}{H}}} 〉), (〈 e^{- (\dot{ϒ} (- ln α_{1})^{H})^{\frac{1}{H}}}, 1 - e^{- (\dot{ϒ} (- ln (1 - β_{1}))^{H})^{\frac{1}{H}}} 〉)$ .

Example 2. Let ${\hat{J}}_{1} = (〈 0.8100, 0.3101 〉, 〈 0.7121, 0.1314 〉)$ and ${\hat{J}}_{2} = (〈 0.4211, 0.6121 〉, 〈 0.3241, 0.1242 〉)$ have two LDFNs and $\dot{ϒ} = 0.2$ . Then, it satisfies the Aczel-Alsina operations for LDFNs. (i)
${\hat{J}}_{1}^{c} = (〈 0.3101, 0.8100 〉, 〈 0.1314, 0.7121 〉)$
(ii)
${\hat{J}}_{1} \cup {\hat{J}}_{2} = (〈 0.8100, 0.3101 〉, 〈 0.7121, 0.1314 〉) = {\hat{J}}_{1}$
(iii)
${\hat{J}}_{1} \cap {\hat{J}}_{2} = (〈 0.4211, 0.6121 〉, 〈 0.3241, 0.1242 〉) = {\hat{J}}_{2}$
(iv)
${\hat{J}}_{1} \oplus {\hat{J}}_{2} = (〈 0.8259, 0.2809 〉, 〈 0.7289, 0.0544 〉)$
(v)
${\hat{J}}_{1} \otimes {\hat{J}}_{2} = (〈 0.4105, 0.6383 〉, 〈 0.1960, 0.1758 〉)$
(vi)
$\dot{ϒ} {\hat{J}}_{1} = (〈 0.5241, 0.5923 〉, 〈 0.4269, 0.4034 〉)$
(vii)
$({\hat{J}}_{1})^{\dot{ϒ}} = (〈 0.9100, 0.1529 〉, 〈 0.8591, 0.0610 〉)$

4 Linear diophantine fuzzy Aczel-Alsina aggregation (LDFAAA) operators

New approaches to LDFNs with some notable features, based on Aczel-Alsina operators are presented in this section: LDFAAWA, LDFAAOWA and LDFAAHA.

4.1 LDFAAWA Operator

Definition 16. Let ${\hat{J}}_{TDSFTD i}$ = $(〈 ψ_{TDSFTD i}, V_{TDSFTD i} 〉, 〈 α_{TDSFTD i}, β_{TDSFTD i} 〉)$ be a collection of LDFNs and $T = (T_{1}, T_{2}, \dots, T_{n})^{F}$ be the weight vector (WV) with (TDSFTDi = 1, 2, …, n) such that $T_{TDSFTD i} \in [0, 1)$ and $Y_{TDSFTD i = 1}^{n} T_{TDSFTD i} = 1$ , then the LDFAAWA is a mapping $γ : LDFAAWA ({\overset{´}{G}}^{n}) ⟶ LDFAAWA (\overset{´}{G})$ is referred to as linear diophantine fuzzy Aczel-Alsina weighted average(LDFAAWA) operator and defined as

\begin{matrix} LDFAAWA ({\hat{J}}_{1}, {\hat{J}}_{2}, \dots, {\hat{J}}_{n}) = {oplus}_{TDSFTD i = 1}^{n} T_{TDSFTD i} {\hat{J}}_{TDSFTD i} \\ = T_{1} . {\hat{J}}_{1} \oplus T_{2} . {\hat{J}}_{2} \oplus T_{3} {\hat{J}}_{3} \oplus, . . ., \oplus T_{n} {\hat{J}}_{n} . \end{matrix}

(12)

where,

T

as a WV and

{\hat{J}}_{TDSFTD i}

are the LDFNs.

Theorem 1. Let ${\hat{J}}_{i}$ = $(〈 ψ_{TDSFTD i}, V_{TDSFTD i} 〉, 〈 α_{TDSFTD i}, β_{TDSFTD i} 〉)$ be a collection of LDFNs and $T = (T_{1}, T_{2}, \dots T_{n})^{F}$ be the weight vector (WV) with (TDSFTDi = 1, 2, …, n) such that $T_{TDSFTD i} \in [0, 1)$ and $Y_{TDSFTD i = 1}^{n} T_{TDSFTD i} = 1$ , then the LDFAAWA is a mapping $\bar{Λ} : LDFAAWA ({\overset{´}{G}}^{n}) ⟶ LDFAAWA (\overset{´}{G})$ is referred to as LDFAAWA operator and can be written as

LDFAAWA ({\hat{J}}_{1}, {\hat{J}}_{2}, \dots, {\hat{J}}_{n}) = \oplus_{i = 1}^{n} (T_{i} {\hat{J}}_{i}) = 〈 (1 - e^{- {(\sum_{i = 1}^{n} T_{i} {(- \ln (1 - ψ_{i}))}^{H})}^{\frac{1}{H}}}, e^{- {(\sum_{i = 1}^{n} T_{i} {(- \ln V_{i})}^{H})}^{\frac{1}{H}}}), (1 - e^{- {(\sum_{i = 1}^{n} T_{i} {(- \ln (1 - α_{i}))}^{H})}^{\frac{1}{H}}}, e^{- (\sum_{i = 1}^{n} T_{i} {(- \ln β_{i})}^{H})^{\frac{1}{H}}}) 〉 .

where,

T = (T_{1}, T_{2}, \dots, T_{n})

is the weight vector of

{\hat{J}}_{TDSFTD i} (TDSFTD i = 1, 2, \dots, n)

including

{\hat{J}}_{TDSFTD i} > 0

and

\sum_{TDSFTD i = 0}^{n} T_{TDSFTD i} = 1

Proof. Let ${\hat{J}}_{TDSFTD i}$ = $(〈 ψ_{TDSFTD i}, V_{TDSFTD i} 〉, 〈 α_{TDSFTD i}, β_{TDSFTD i} 〉)$ be a collection of LDFNs and $T = (T_{1}, T_{2}, \dots T_{n})^{F}$ be the weight vector (WV) with (TDSFTDi = 1, 2, …, n) such that $T_{TDSFTD i} \in [0, 1)$ and $Y_{i = 1}^{n} T_{TDSFTD i} = 1$ . Then, LDFAAWA can be proved using the technique of mathematical induction. First, we suppose that the above equation (4.2) is true for n = 2. Therefore,

T_{1} {\hat{J}}_{1} = 〈 (1 - e^{- {(T_{1} {(- \ln (1 - ψ_{1}))}^{H})}^{\frac{1}{H}}}, e^{- {(T_{1} {(- \ln V_{1})}^{H})}^{\frac{1}{H}}}), (1 - e^{- {(T_{1} {(- \ln (1 - α_{1}))}^{H})}^{\frac{1}{H}}}, e^{- {(T_{1} {(- \ln β_{1})}^{H})}^{\frac{1}{H}}}) 〉 .

T_{2} {\hat{J}}_{2} = 〈 (1 - e^{- {(T_{2} {(- \ln (1 - ψ_{2}))}^{H})}^{\frac{1}{H}}}, e^{- {(T_{2} {(- \ln V_{2})}^{H})}^{\frac{1}{H}}}), (1 - e^{- {(T_{2} {(- \ln (1 - α_{2}))}^{H})}^{\frac{1}{H}}}, e^{- {(T_{2} {(- \ln β_{2})}^{H})}^{\frac{1}{H}}}) 〉 .

LDFAAWA ({\hat{J}}_{1}, {\hat{J}}_{2}) = T_{1} {\hat{J}}_{1} \oplus T_{2} {\hat{J}}_{2} = 〈 (1 - e^{- {(T_{1} {(- \ln (1 - ψ_{1}))}^{H})}^{\frac{1}{H}}}, e^{- {(T_{1} {(- \ln V_{1})}^{H})}^{\frac{1}{H}}}), (1 - e^{- {(T_{1} {(- \ln (1 - α_{1}))}^{H})}^{\frac{1}{H}}}, e^{- {(T_{1} {(- \ln β_{1})}^{H})}^{\frac{1}{H}}}) 〉 \oplus 〈 (1 - e^{- {(T_{2} {(- \ln (1 - ψ_{2}))}^{H})}^{\frac{1}{H}}}, e^{- {(T_{2} {(- \ln V_{2})}^{H})}^{\frac{1}{H}}}), (1 - e^{- {(T_{2} {(- \ln (1 - α_{2}))}^{H})}^{\frac{1}{H}}}, e^{- {(T_{2} {(- \ln β_{2})}^{H})}^{\frac{1}{H}}}) 〉 . = 〈 (1 - e^{- {(\sum_{i = 1}^{2} T_{i} {(- \ln (1 - ψ_{i}))}^{H})}^{\frac{1}{H}}}, e^{- {(\sum_{i = 1}^{2} T_{i} {(- \ln V_{i})}^{H})}^{\frac{1}{H}}}), (1 - e^{- {(\sum_{i = 1}^{2} T_{i} {(- \ln (1 - α_{i}))}^{H})}^{\frac{1}{H}}}, e^{- {(\sum_{i = 1}^{2} T_{i} {(- \ln β_{i})}^{H})}^{\frac{1}{H}}}) 〉 .

For n = 2, we proved that it is true. Further, we suppose that it holds for n = k.

LDFAAWA ({\hat{J}}_{1}, {\hat{J}}_{2}, \dots, {\hat{J}}_{k}) = \oplus_{i = 1}^{k} (T_{i} {\hat{J}}_{i}) = 〈 (1 - e^{- {(\sum_{i = 1}^{k} T_{i} {(- \ln (1 - ψ_{i}))}^{H})}^{\frac{1}{H}}}, e^{- {(\sum_{i = 1}^{k} T_{i} {(- \ln V_{i})}^{H})}^{\frac{1}{H}}}), (1 - e^{- {(\sum_{i = 1}^{k} T_{i} {(- \ln (1 - α_{i}))}^{H})}^{\frac{1}{H}}}, e^{- {(\sum_{i = 1}^{k} T_{i} {(- \ln β_{i})}^{H})}^{\frac{1}{H}}}) 〉 .

LDFAAWA ({\hat{J}}_{1}, {\hat{J}}_{2}, \dots, {\hat{J}}_{k}, {\hat{J}}_{k + 1}) = \oplus_{i = 1}^{k} (T_{i} {\hat{J}}_{i}) \oplus (T_{k + 1} {\hat{J}}_{k + 1}) = 〈 (1 - e^{- {(\sum_{i = 1}^{k} T_{i} {(- \ln (1 - ψ_{i}))}^{H})}^{\frac{1}{H}}}, e^{- {(\sum_{i = 1}^{k} T_{i} {(- \ln V_{i})}^{H})}^{\frac{1}{H}}}), (1 - e^{- {(\sum_{i = 1}^{k} T_{i} {(- \ln (1 - α_{i}))}^{H})}^{\frac{1}{H}}} e^{- {(\sum_{i = 1}^{k} T_{i} {(- \ln β_{i})}^{H})}^{\frac{1}{H}}}) 〉 \oplus 〈 (1 - e^{- {(T_{k + 1} {(- \ln (1 - ψ_{k + 1}))}^{H})}^{\frac{1}{H}}}, e^{- {(T_{k + 1} {(- \ln V_{k + 1})}^{H})}^{\frac{1}{H}}}), (1 - e^{- {(T_{k + 1} {(- \ln (1 - α_{k + 1}))}^{H})}^{\frac{1}{H}}}, e^{- {(T_{k + 1} {(- \ln β_{k + 1})}^{H})}^{\frac{1}{H}}}) 〉 .

Now, we are to prove that it is also true for n = k + 1, such that:

LDFAAWA ({\hat{J}}_{1}, {\hat{J}}_{2}, \dots, {\hat{J}}_{k + 1}) = 〈 (1 - e^{- {(\sum_{i = 1}^{k + 1} T_{i} {(- \ln (1 - ψ_{i}))}^{H})}^{\frac{1}{H}}}, e^{- {(\sum_{i = 1}^{k + 1} T_{i} {(- \ln V_{i})}^{H})}^{\frac{1}{H}}}), (1 - e^{- {(\sum_{i = 1}^{k + 1} T_{i} {(- \ln (1 - α_{i}))}^{H})}^{\frac{1}{H}}}, e^{- {(\sum_{i = 1}^{k + 1} T_{i} {(- \ln β_{i})}^{H})}^{\frac{1}{H}}}) 〉 .

A few of the advantageous characteristics of an LDFAAWA operator will be covered in the next few paragraphs.

Proposition 1.(Idempotency) If all ${\hat{J}}_{TDSFTD i}$ = $(〈 ψ_{TDSFTD i}, V_{TDSFTD i} 〉, 〈 α_{TDSFTD i}, β_{TDSFTD i} 〉)$ be a collection of LDFNs and $T = (T_{1}, T_{2}, \dots, T_{n})^{F}$ be the weight vector (WV) with (TDSFTDi = 1, 2, …, n) are identical, i.e., ${\hat{J}}_{TDSFTD i} = \hat{J}$ . Then

LDFAAWA ({\hat{J}}_{1}, {\hat{J}}_{2}, \dots, {\hat{J}}_{n}) = \hat{J}

(14)

Proof: We know that

{\hat{J}}_{TDSFTD i} = (〈 ψ_{TDSFTD i}, V_{TDSFTD i} 〉, 〈 α_{TDSFTD i}, β_{TDSFTD i} 〉) = (〈 ψ, V 〉, 〈 α, β 〉)

LDFAAWA ({\hat{J}}_{1}, {\hat{J}}_{2}, ., {\hat{J}}_{n}) = 〈 (1 - e^{- {(\sum_{i = 1}^{n} T_{i} {(- \ln (1 - ψ_{i}))}^{H})}^{\frac{1}{H}}}, e^{- {(\sum_{i = 1}^{n} T_{i} {(- \ln V_{i})}^{H})}^{\frac{1}{H}}}), (1 - e^{- {(\sum_{i = 1}^{n} T_{i} {(- \ln (1 - α_{i}))}^{H})}^{\frac{1}{H}}}, e^{- {(\sum_{i = 1}^{n} T_{i} {(- \ln β_{i})}^{H})}^{\frac{1}{H}}}) 〉 . = 〈 (1 - e^{- {({(- \ln (1 - ψ))}^{H})}^{\frac{1}{H}}}, e^{- {({(- \ln V)}^{H})}^{\frac{1}{H}}}), (1 - e^{- {({(- \ln (1 - α))}^{H})}^{\frac{1}{H}}}, e^{- {({(- \ln β)}^{H})}^{\frac{1}{H}}}) 〉 . = 〈 (1 - e^{\ln (1 - ψ)}, e^{\ln V}), (1 - e^{\ln (1 - α)}, e^{\ln β}) 〉

(〈 ψ, V 〉, 〈 α, β 〉) = \hat{J} .

Therefore, LDFAAWA $({\hat{J}}_{1}, {\hat{J}}_{2}, \dots, {\hat{J}}_{n}) = \hat{J} .$

Proposition 2. (Boundedness) Suppose ${\hat{J}}_{TDSFTD i} = (〈 ψ_{TDSFTD i}, V_{TDSFTD i} 〉, 〈 α_{TDSFTD i}, β_{TDSFTD i} 〉)$ be the set of LDFNs. Suppose ${\hat{J}}^{-} = \min ({\hat{J}}_{1}, {\hat{J}}_{2}, \dots, {\hat{J}}_{n})$ and ${\hat{J}}^{+} = \max ({\hat{J}}_{1}, {\hat{J}}_{2}, \dots, {\hat{J}}_{n}) .$ Then

{\hat{J}}^{-} \leq LDFAAWA ({\hat{J}}_{1}, {\hat{J}}_{2}, \dots, {\hat{J}}_{n}) \leq {\hat{J}}^{+}

(15)

Proof: Suppose ${\hat{J}}_{TDSFTD i} = (〈 ψ_{TDSFTD i}, V_{TDSFTD i} 〉, 〈 α_{TDSFTD i}, β_{TDSFTD i} 〉)$ be the set of LDFNs. Suppose ${\hat{J}}^{-} = \min ({\hat{J}}_{1}, {\hat{J}}_{2}, \dots, {\hat{J}}_{n}) = (〈 ψ^{-}, V^{-} 〉, 〈 α^{-}, β^{-} 〉)$ and ${\hat{J}}^{+} = \max ({\hat{J}}_{1}, {\hat{J}}_{2}, \dots, {\hat{J}}_{n}) = (〈 ψ^{+}, V^{+} 〉, 〈 α^{+}, β^{+} 〉)$ .

Proposition 3. (Monotonicity) Suppose ${\hat{J}}_{TDSFTD i} = (〈 ψ_{TDSFTD i}, V_{TDSFTD i} 〉, 〈 α_{TDSFTD i}, β_{TDSFTD i} 〉)$ be the set of LDFNs. If ${\hat{J}}_{TDSFTD i} \leq {\hat{J}}_{TDSFTD i}$ . Then

LDFAAWA ({\hat{J}}_{1}, {\hat{J}}_{2}, \dots, {\hat{J}}_{n}) \leq LDFAAWA ({\hat{J}}_{1}, {\hat{J}}_{2}, \dots, {\hat{J}}_{n}) .

(16)

Proof: Simple to understand.

We will now address LDFAA ordered weighted averaging (LDFAAOWA) AO using a few fundamental operations that are defined in Definition 15 in order to maintain the arrangement of the options in order of preference throughout the aggregation process.

4.2 LDFAAOWA Operator

Definition 17. Let ${\hat{J}}_{TDSFTD i}$ = $(〈 ψ_{TDSFTD i}, V_{TDSFTD i} 〉, 〈 α_{TDSFTD i}, β_{TDSFTD i} 〉)$ be a collection of LDFNs and $T = (T_{1}, T_{2}, \dots, T_{n})^{F}$ be the weight vector (WV) with $(TDSFTD i = 1, 2, \dots, n) If T_{TDSFTD i} \in [0, 1)$ and $T_{TDSFTD i} = 1$ , then the LDFAAOWA is a mapping $\overset{´}{κ} : LDFAAOWA ({\overset{´}{G}}^{n}) ⟶ LDFAAOWA (\overset{´}{G})$ referred to as linear diophantine fuzzy Aczel-Alsina ordered weighted average operator (LDFAAOWA) is characterised as

\begin{matrix} LDFAAOWA ({\hat{J}}_{1}, {\hat{J}}_{2}, \dots, {\hat{J}}_{n}) = {oplus}_{TDSFTD i = 1}^{n} T_{TDSFTD i} {\hat{J}}_{TDSFTD i} \\ = T_{1} . {\hat{J}}_{1} \oplus T_{2} . {\hat{J}}_{2} \oplus T_{3} {\hat{J}}_{3} \oplus, . . ., \oplus T_{n} {\hat{J}}_{n} . \end{matrix}

(18)

Where

(\tilde{θ} (1), \tilde{θ} (2), \dots, \tilde{θ} (n))

be the permutation of (1, 2, …, n) such that

{\hat{J}}_{\tilde{θ} (TDSFTD i - 1)} \geq {\hat{J}}_{\tilde{θ} (TDSFTD i)} .

Theorem 2.Let ${\hat{J}}_{TDSFTD i}$ = $(〈 ψ_{TDSFTD i}, V_{TDSFTD i} 〉, 〈 α_{TDSFTD i}, β_{TDSFTD i} 〉)$ be a collection of LDFNs and $T = (T_{1}, T_{2}, \dots T_{n})^{F}$ be the weight vector (WV) with (TDSFTDi = 1, 2, …, n) such that $T_{TDSFTD i} \in [0, 1)$ and $Y_{TDSFTD i = 1}^{n} T_{TDSFTD i} = 1$ , then the LDFAAOWA is a mapping $\overset{´}{κ} : LDFAAOWA ({\overset{´}{G}}^{n}) ⟶ LDFAAOWA (\overset{´}{G})$ referred to as LDFAAOWA operator and can be written as

LDFAAOWA ({\hat{J}}_{1}, {\hat{J}}_{2}, \dots, {\hat{J}}_{n}) = \oplus_{i = 1}^{n} (T_{i} {\hat{J}}_{i}) = 〈 (1 - e^{- {(\sum_{i = 1}^{n} T_{i} {(- \ln (1 - ψ_{i}))}^{H})}^{\frac{1}{H}}}, e^{- {(\sum_{i = 1}^{n} T_{i} {(- \ln V_{i})}^{H})}^{\frac{1}{H}}}), (1 - e^{- {(\sum_{i = 1}^{n} T_{i} {(- \ln (1 - α_{i}))}^{H})}^{\frac{1}{H}}}, e^{- {(\sum_{i = 1}^{n} T_{i} {(- \ln β_{i})}^{H})}^{\frac{1}{H}}}) 〉 .

The properties mentioned in Proposition 1, 2, and 3 are analogously satisfied by the LDFAAOWA operator.

Proposition 4.(Idempotency) If all ${\hat{J}}_{TDSFTD i}$ = $(〈 ψ_{TDSFTD i}, V_{TDSFTD i} 〉, 〈 α_{TDSFTD i}, β_{TDSFTD i} 〉)$ be a collection of LDFNs and $T = (T_{1}, T_{2}, \dots, T_{n})^{F}$ be the weight vector (WV) with (TDSFTDi = 1, 2, …, n) are identical, i.e, ${\hat{J}}_{TDSFTD i} = \hat{J}$ . Then

LDFAAOWA ({\hat{J}}_{1}, {\hat{J}}_{2}, \dots, {\hat{J}}_{n}) = \hat{J} .

(19)

Proposition 5.(Boundedness) Suppose ${\hat{J}}_{TDSFTD i}$ = $(〈 ψ_{TDSFTD i}, V_{TDSFTD i} 〉, 〈 α_{TDSFTD i}, β_{TDSFTD i} 〉)$ be the set of LDFNs. Suppose ${\hat{J}}^{-} = \min ({\hat{J}}_{1}, {\hat{J}}_{2}, \dots, {\hat{J}}_{n})$ and ${\hat{J}}^{+} = \max ({\hat{J}}_{1}, {\hat{J}}_{2}, \dots, {\hat{J}}_{n}) .$ Then

{\hat{J}}^{-} \leq LDFAAOWA ({\hat{J}}_{1}, {\hat{J}}_{2}, \dots, {\hat{J}}_{n}) \leq {\hat{J}}^{+}

(20)

Proposition 6.(Monotonicity) Suppose ${\hat{J}}_{TDSFTD i} = (〈 ψ_{TDSFTD i}, V_{TDSFTD i} 〉, 〈 α_{TDSFTD i}, β_{TDSFTD i} 〉)$ be the set of LDFNs. If ${\hat{J}}_{TDSFTD i} \leq {\hat{J}}_{TDSFTD i}$ . Then

LDFAAOWA ({\hat{J}}_{1}, {\hat{J}}_{2}, \dots, {\hat{J}}_{n}) \leq LDFAAOWA ({\hat{J}}_{1}, {\hat{J}}_{2}, \dots, {\hat{J}}_{n}) .

(21)

It is evident from Definitions 16 and 17 that the LDFAAWA operator and the LDFAAOWA operator, respectively, give weight to the LDFNs and their ordered placements. The LDF AA hybrid averaging (LDFAAHA) operator is obtained by combining these two operators.

4.3 LDFAAHA Operator

Definition 18. Let ${\hat{J}}_{TDSFTD i} = (〈 ψ_{TDSFTD i}, V_{TDSFTD i} 〉, 〈 α_{TDSFTD i}, β_{TDSFTD i} 〉)$ be a collection of LDFNs, then the LDFAAHA is a mapping $\nabla : LDFAAHA ({\overset{´}{G}}^{n}) ⟶ LDFAAHA (\overset{´}{G})$ referred to as linear diophantine fuzzy Aczel-Alsina hybrid average (LDFAAHA) operator and defined as

\begin{matrix} LDFAAHA ({\hat{J}}_{1}, {\hat{J}}_{2}, \dots, {\hat{J}}_{n}) = {oplus}_{TDSFTD i = 1}^{n} \dot{Ξ} {\hat{J}}_{TDSFTD i} \\ = {\dot{Ξ}}_{1} . {\hat{J}}_{1} \oplus {\dot{Ξ}}_{2} . {\hat{J}}_{2} \oplus {\hat{Ξ}}_{3} {\hat{J}}_{3} \oplus, . . ., \oplus {\dot{Ξ}}_{n} {\hat{J}}_{n} . \end{matrix}

(22)

where,

\dot{Ξ}

is a WV and

\tilde{\hat{J}} = n T_{i} {\hat{J}}_{TDSFTD i}

with n as a balancing coefficient.

The theorem for the LDFAAHA operator can now be obtained in the manner shown below.

Theorem 3. Let ${\hat{J}}_{TDSFTD i}$ = $(〈 ψ_{TDSFTD i}, V_{TDSFTD i} 〉, 〈 α_{TDSFTD i}, β_{TDSFTD i} 〉)$ be a collection of LDFNs, then the LDFAAHA is a mapping $\nabla : LDFAAHA ({\overset{´}{G}}^{n}) ⟶ LDFAAHA (\overset{´}{G})$ is called LDFAAHA operator and can be written as

LDFAAHA ({\hat{J}}_{1}, {\hat{J}}_{2}, ., {\hat{J}}_{n}) = \oplus_{i = 1}^{n} ({\dot{Ξ}}_{i} {\hat{J}}_{i}) = 〈 (1 - e^{- {(\sum_{i = 1}^{n} {\dot{Ξ}}_{i} {(- \ln (1 - ψ_{i}))}^{H})}^{\frac{1}{H}}}, e^{- {(\sum_{i = 1}^{n} {\dot{Ξ}}_{i} {(- \ln V_{i})}^{H})}^{\frac{1}{H}}}), (1 - e^{- {(\sum_{i = 1}^{n} {\dot{Ξ}}_{i} {(- \ln (1 - α_{i}))}^{H})}^{\frac{1}{H}}}, e^{- {(\sum_{i = 1}^{n} {\dot{Ξ}}_{i} {(- \ln β_{i})}^{H})}^{\frac{1}{H}}}) 〉 .

Proof This proof is similar to Theorem 1.

Proposition 7. (Idempotency) If all ${\hat{J}}_{TDSFTD i}$ = $(〈 ψ_{TDSFTD i}, V_{TDSFTD i} 〉, 〈 α_{TDSFTD i}, β_{TDSFTD i} 〉)$ be a collection of LDFNs and $T = (T_{1}, T_{2}, \dots, T_{n})^{F}$ be the weight vector (WV) with (TDSFTDi = 1, 2, …, n) are identical, i.e, ${\hat{J}}_{TDSFTD i} = \hat{J}$ . Then

LDFAAHA ({\hat{J}}_{1}, {\hat{J}}_{2}, \dots, {\hat{J}}_{n}) = \hat{J} .

(23)

Proposition 8. (Boundedness) Suppose

{\hat{J}}_{TDSFTD i}

(〈 ψ_{TDSFTD i}, V_{TDSFTD i} 〉, 〈 α_{TDSFTD i}, β_{TDSFTD i} 〉)

be the set of LDFNs. Suppose

{\hat{J}}^{-} = \min ({\hat{J}}_{1}, {\hat{J}}_{2}, \dots, {\hat{J}}_{n})

and

{\hat{J}}^{+} = \max ({\hat{J}}_{1}, {\hat{J}}_{2}, \dots, {\hat{J}}_{n}) .

Then

{\hat{J}}^{-} \leq LDFAAHA ({\hat{J}}_{1}, {\hat{J}}_{2}, \dots, {\hat{J}}_{n}) \leq {\hat{J}}^{+}

(24)

Proposition 9(Monotonicity) Suppose ${\hat{J}}_{TDSFTD i} = (〈 ψ_{TDSFTD i}, V_{TDSFTD i} 〉, 〈 α_{TDSFTD i}, β_{TDSFTD i} 〉)$ be the set of LDFNs. If ${\hat{J}}_{TDSFTD i} \leq \hat{J_{TDSFTD i}}$ . Then

LDFAAHA ({\hat{J}}_{1}, {\hat{J}}_{2}, \dots, {\hat{J}}_{n}) \leq LDFAAHA ({\hat{J}}_{1}, {\hat{J}}_{2}, \dots, {\hat{J}}_{n}) .

(25)

Theorem 4.Special cases of the LDFAAHA operators are the LDFAAWA and LDFAAOWA operators.

Proof. Suppose $\dot{Ξ} = (1 / ð, 1 / ð, \dots, 1 / ð)^{F}$ , Then

\begin{matrix} LDFAAHA ({\hat{J}}_{1}, {\hat{J}}_{2}, \dots, {\hat{J}}_{n}) = {\dot{Ξ}}_{1} . {\hat{J}}_{1} \oplus {\dot{Ξ}}_{2} . {\hat{J}}_{2} \oplus, \dots, \\ \oplus {\dot{Ξ}}_{n} {\hat{J}}_{n} = \frac{1}{ð} ({\hat{J}}_{1} \oplus {\hat{J}}_{2} \oplus, \dots, {\hat{J}}_{n}) \\ = T {\hat{J}}_{1} \oplus T {\hat{J}}_{2}, \dots, T {\hat{J}}_{n} = LDFAAWA ({\hat{J}}_{1}, {\hat{J}}_{2}, \dots, {\hat{J}}_{n}) . \end{matrix}

(26)

Suppose

T = (1 / ð . 1 / ð, \dots, 1 / ð)^{F}

and

{\hat{J}}_{TDSFTD i} = {\hat{J}}_{TDSFTD i} (TDSFTD i = 1, 2, \dots, n)

. Then

\begin{matrix} LDFAAHA ({\hat{J}}_{1}, {\hat{J}}_{2}, \dots, {\hat{J}}_{n}) = {\dot{Ξ}}_{1} . {\hat{J}}_{1} \oplus {\dot{Ξ}}_{2} . {\hat{J}}_{2} \oplus, \dots, \\ \oplus {\dot{Ξ}}_{n} {\hat{J}}_{n} = {\dot{Ξ}}_{1} . {\hat{J}}_{1} \oplus {\dot{Ξ}}_{2} . {\hat{J}}_{2} \oplus, \dots, \oplus {\dot{Ξ}}_{n} {\hat{J}}_{n} \\ = LDFAAOWA ({\hat{J}}_{1}, {\hat{J}}_{2}, \dots, {\hat{J}}_{n}) . \end{matrix}

(27)

This completes the proof.

5 User profiling by fuzzy clustering

In this paper, Fuzzy clustering algorithms are used in user profiling to group users according to their traits, behaviors, or preferences in a way that permits them to have overlapping memberships in several user groups. Conventional clustering methods allocate each data point (user) to exactly one cluster, but fuzzy clustering, such as fuzzy C-means, allows for partial memberships. This allows users to be part of multiple profiles at once in user profiling, capturing the subtleties and ambiguities in their choices or actions. Fuzzy clustering,

for example, could be used in an e-commerce environment to classify users according to their buying patterns. Instead of strictly fitting into one buyer personal, users may display traits from several buyer personal (such as "luxury seeker," "frequent shopper," or "budget-conscious") to varying degrees. This method recognizes the complexity of user preferences, which allows for tailored recommendations, focused marketing campaigns, or unique user experiences. Businesses can more effectively customize their services and user interactions by using fuzzy clustering for user profiling, taking into account the varied and dynamic nature of consumer behavior. In linear Diophantine equations, the integer variables have linear relationships with one another. Using linear diophantine equations to create mathematical models that represent user characteristics or behaviors could be one way to apply these equations to user profiling. In data-driven user profiling scenarios, however, the traditional method may not be to directly use Linear Diophantine equations for user profiling.

Linear Diophantine equations may not always be used directly in user profiling to cluster or profile users. Alternatively, methods for aggregating or summarizing user preferences or behavior patterns, such as Linear Diophantine Fuzzy Sets or Linear Diophantine Aczel-Alsina aggregation operators, might be taken into consideration. User profiles are located through a clustering procedure following the derivation of the matrix of interest degrees. Every user profile will feature users with comparable interests and browsing patterns. This work used the well recognised Fuzzy C-Means (FCM) clustering approach to arrange vectors b_TDSFTDi into overlapping clusters that correspond to user profiles [16].

5.1 Proposed method based on LDFAA operators Fuzzy C-Means (FCM) clustering Algorithm

The FCM clustering based on LDFAA algorithm works as follows:

Step 1.
Input the LDF data in the form of LDFNs Matrix. Take two matrices of LDFNs in the form of $D = (\dot{Z})_{m \times n}$
$[\begin{matrix} {\dot{x}}_{11} & {\dot{x}}_{12} & {\dot{x}}_{13} & \dots & {\dot{x}}_{1 n} \\ {\dot{x}}_{21} & {\dot{x}}_{22} & {\dot{x}}_{23} & \dots & {\dot{x}}_{2 n} \\ ⋮ & ⋮ & ⋮ & ⋱ & ⋮ \\ {\dot{x}}_{m 1} & {\dot{x}}_{m 2} & {\dot{x}}_{m 3} & \dots & {\dot{x}}_{mn} \end{matrix}]$

Step 2.
Input the weight vector of the given LDFNs matrix, where, $T = (T_{1}, T_{2}, \dots, T_{n})^{F}$ be the weight vector (WV) with (TDSFTDi = 1, 2, …, n) such that $T_{TDSFTD i} \in [0, 1)$ and sum of all the weight vectors must be equal to 1,
Step 3.
Calculate aggregated values by using LDFAAWA operator.

$LDFAAWA ({\hat{J}}_{1}, {\hat{J}}_{2}, \dots, {\hat{J}}_{n}) = \oplus_{i = 1}^{n} (T_{i} {\hat{J}}_{i}) = 〈 (1 - e^{- {(\sum_{i = 1}^{n} T_{i} {(- \ln (1 - ψ_{i}))}^{H})}^{\frac{1}{H}}}, e^{- {(\sum_{i = 1}^{n} T_{i} {(- \ln V_{i})}^{H})}^{\frac{1}{H}}}), (1 - e^{- {(\sum_{i = 1}^{n} T_{i} {(- \ln (1 - α_{i}))}^{H})}^{\frac{1}{H}}}, e^{- {(\sum_{i = 1}^{n} T_{i} {(- \ln β_{i})}^{H})}^{\frac{1}{H}}}) 〉 .$
Step 4.
The FCM Algorithms works on LDFAAWA $U = [U_{ic}]_{TDSFTD i = 1 \dots n}^{c = 1 \dots c}$ matrix.
Step 5.
When using fuzzy K-means clustering, a sample’s membership and non-membership can be graded or fuzzy, depending on the cluster’s parameters α, β. .
Membership and non-membership of LDFAA sample $x_{j}$ and $y_{j}$ in clusters is defined by ψ_TDSFTDiα_TDSFTDi and $V_{TDSFTD i} β_{TDSFTD i}$ .
.
$ψ_{TDSFTD i j} α_{TDSFTD i j} \in [0, 1]$ and $ψ_{TDSFTD i j} β_{TDSFTD i j} \in [0, 1], 0 \leq α + β \leq 1$

Step 6.
Calculate the cluster centers to find the centroid
$C_{j} = \frac{〈 Σ_{TDSFTD i = 1}^{n} ψ_{TDSFTD i j}^{b} x_{TDSFTD i}, Σ_{TDSFTD i = 1}^{n} V_{TDSFTD i j}^{b} y_{j} 〉, 〈 Σ_{TDSFTD i = 1}^{n} α_{TDSFTD i j}^{b} x_{TDSFTD i}, Σ_{i = 1}^{b} β_{TDSFTD i j}^{b} y_{TDSFTD i} 〉}{〈 Σ_{TDSFTD i = 1}^{n} ψ_{TDSFTD i j}^{b}, Σ_{TDSFTD i = 1}^{n} V_{TDSFTD i j}^{b} 〉, 〈 Σ_{TDSFTD i = 1}^{n} α_{TDSFTD i j}^{b}, Σ_{TDSFTD i = 1}^{b} β_{TDSFTD i j}^{b} 〉}$
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Step 7.
Find out the distance of each point from the centroid.
$\begin{matrix} {\dot{D}}_{TDSFTD i c} = (〈 ∥ x_{TDSFTD i c} - c_{TDSFTD i c} ∥, ∥ y_{TDSFTD i c} - c_{TDSFTD i c} ∥ 〉, 〈 ∥ x_{TDSFTD i c} - P c_{TDSFTD i c} ∥, \\ ∥ y_{TDSFTD i c} - P c_{TDSFTD i c} ∥ 〉) \end{matrix}$
(27)
where $P$ stands for the non-membership and membership parameters in the case where b ≥ 1.
Step 8.
Updating the membership values and non-membership values and also their parameters.
$\begin{matrix} {\tilde{ϖ}}_{TDSFTD i} = (〈 \frac{1}{Σ_{TDSFTD i = 1}^{n} (\frac{∥ x_{TDSFTD i} - c_{TDSFTD i} ∥}{∥ x_{TDSFTD i} - c_{k} ∥})^{\frac{2}{b - 1}}}, \frac{1}{Σ_{TDSFTD i = 1}^{n} (\frac{∥ y_{TDSFTD i} - c_{TDSFTD i} ∥}{∥ y_{TDSFTD i} - c_{k} ∥})^{\frac{2}{b - 1}}} 〉, \\ 〈 \frac{1}{Σ_{TDSFTD i = 1}^{n} (\frac{∥ x_{TDSFTD i} - P c_{TDSFTD i} ∥}{∥ x_{TDSFTD i} - P c_{k} ∥})^{\frac{2}{b - 1}}}, \frac{1}{Σ_{TDSFTD i = 1}^{n} (\frac{∥ y_{TDSFTD i} - P c_{TDSFTD i} ∥}{∥ y_{i} - P c_{k} ∥})^{\frac{2}{b - 1}}} 〉) . \end{matrix}$
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Step 9.
Utilising the anticipated score function (ESF), quadratic score function (QSF), and score function (SF), calculate the score for each characteristic after identifying the resultant set we have $\hat{S} (\hat{J}) = \frac{1}{2} [(ψ_{\hat{J}} - V_{\hat{J}}) + (α_{\hat{J}} - β_{\hat{J}})],$ $\ddot{U} = \frac{1}{2} [({ψ_{\hat{J}}}^{2} - {V_{\hat{J}}}^{2}) + ({α_{\hat{J}}}^{2} - {β_{\hat{J}}}^{2})],$ $\ddot{E} = \frac{1}{2} [\frac{(ψ_{\hat{J}} - V_{\hat{J}} + 1)}{2} + \frac{(α_{\hat{J}} - β_{\hat{J}} + 1)}{2}] .$
Step 10.
Find ranking by using score function, quadratic score function and expected score function.
Step 11.
Select the reference set attribute that has the highest score.

Fig. 3
Flowchart for the Proposed Algorithm.

Table 2
LDF-decision matrix

$L_{1}$ $L_{2}$ $L_{3}$ $L_{4}$

${\hat{J}}_{1}$ (〈0.81, 0.31〉, 〈0.81, 0.13〉) (〈0.16, 0.79〉, 〈0.14, 0.76〉) (〈0.56, 0.27〉, 〈0.37, 0.41〉) (〈0.81, 0.47〉, 〈0.52, 0.39〉)

${\hat{J}}_{2}$ (〈0.71, 0.22〉, 〈0.67, 0.29〉) (〈0.45, 0.31〉, 〈0.11, 0.37〉) (〈0.56, 0.31〉, 〈0.25, 0.61〉) (〈0.47, 0.81〉, 〈0.32, 0.51〉)

${\hat{J}}_{3}$ (〈0.11, 0.32〉, 〈0.56, 0.33〉) (〈0.71, 0.52〉, 〈0.56, 0.21〉) (〈0.81, 0.16〉, 〈0.31, 0.71〉) (〈0.71, 0.56〉, 〈0.15.0.11〉)

${\hat{J}}_{4}$ (〈0.29, 0.23〉, 〈0.67, 0.13〉) (〈0.27, 0.12〉, 〈0.45, 0.35〉) (〈0.27, 0.67〉, 〈0.11, 0.32〉) (〈0.36, 0.73〉, 〈0.31, 0.63〉)

Table 3
LDF-decision matrix

$L_{1}$ $L_{2}$ $L_{3}$ $L_{4}$

${\hat{J}}_{1}$ (〈0.79, 0.36〉, 〈0.16, 0.56〉) (〈0.83, 0.21〉, 〈0.76, 0.13〉) (〈0.36, 0.43〉, 〈0.57, 0.31〉) (〈0.11, 0.32〉, 〈0.27, 0.67〉)

${\hat{J}}_{2}$ (〈0.93, 0.63〉, 〈0.53, 0.21〉) (〈0.71, 0.53〉, 〈0.37, 0.21〉) (〈0.11, 0.22〉, 〈0.67, 0.29〉) (〈0.33, 0.56〉, 〈0.11, 0.32〉)

${\hat{J}}_{3}$ (〈0.56, 0.79〉, 〈0.23, 0.51〉) (〈0.56, 0.63〉, 〈0.27, 0.32〉) (〈0.14, 0.76〉, 〈0.16, 0.76〉) (〈0.21, 0.56〉, 〈0.52, 0.71〉)

${\hat{J}}_{4}$ (〈0.71, 0.81〉, 〈0.26, 0.76〉) (〈0.37, 0.71〉, 〈0.37, 0.21〉) (〈0.45, 0.35〉, 〈0.27, 0.12〉) (〈0.41, 0.37〉, 〈0.27, 0.56〉)

Table 4
LDFAAWA-decision matrix

$L_{1}$ $L_{2}$ $L_{3}$ $L_{4}$

${\hat{J}}_{1}$ (〈0.3239, 0.6531〉, 〈0.0549, 0.6011〉) (〈0.3147, 0.7054〉, 〈0.2639, 0.6379〉) (〈0.2181, 0.4182〉, 〈0.2240, 0.6698〉) (〈0.2919, 0.6921〉, 〈0.1842, 0.7705〉)

${\hat{J}}_{2}$ (〈0.6571, 0.5811〉, 〈0.4006, 0.5538〉) (〈0.3960, 0.6089〉, 〈0.1469, 0.4957〉) (〈0.2270, 0.4782〉, 〈0.3185, 0.6214〉) (〈0.2475, 0.8048〉, 〈0.1288, 0.6077〉)

${\hat{J}}_{3}$ (〈0.1998, 0.7210〉, 〈0.2269, 0.6544〉) (〈0.3872, 0.7668〉, 〈0.2367, 0.5260〉) (〈0.3500, 0.6058〉, 〈0.1216, 0.8635〉) (〈0.2956, 0.7589〉, 〈0.1920, 0.5452〉)

${\hat{J}}_{4}$ (〈0.1951, 0.7939〉, 〈0.1760, 0.7276〉) (〈0.1011, 0.7130〉, 〈0.1354, 0.6987〉) (〈0.1177, 0.8194〉, 〈0.0575, 0.6391〉) (〈0.1251, 0.8354〉, 〈0.0898, 0.8666〉)

Table 5
Membership clusters of visitor ( $x_{j}$ )

Matches Homepage Entertainment Picture

1 2 3 2

2 1 2 3

3 2 1 4

4 2 4 1

Table 6
Non-membership clusters of visitor ( $y_{j}$ )

Matches Homepage Entertainment Picture

4 1 2 4

1 2 1 1

3 2 3 3

2 2 3 4

Table 7
Centroid of the FCM of LDFAAWA

$L_{1}$ $L_{2}$ $L_{3}$ $L_{4}$

${\hat{J}}_{1}$ (〈2.0182, 2.5408〉, 〈2.4493, 3.6377〉) (〈1.2975, 2.0522〉, 〈1.3477, 2.8666〉) (〈2.2297, 2.4244〉, 〈2.0549, 3.8708〉) (〈2.7707, 3.1992〉, 〈2.9455, 5.1128〉)

${\hat{J}}_{2}$ (〈2.1719, 2.6172〉, 〈2.1401, 2.5960〉) (〈1.6232, 1.7468〉, 〈1.8700, 1.6693〉) (〈1.9271, 2.3693〉, 〈2.3045, 2.3008〉) (〈3.0738, 3.1347〉, 〈2.6640, 3.2194〉)

${\hat{J}}_{3}$ (〈2.4362, 2.3385〉, 〈1.8302, 2.6319〉) (〈1.5165, 1.8787〉, 〈1.4018, 1.7744〉) (〈1.7994, 2.5616〉, 〈2.2471, 2.3861〉) (〈3.2009, 3.2698〉, 〈2.7525, 3.0426〉)

${\hat{J}}_{4}$ (〈2.1347, 2.3694〉, 〈2.2002, 2.5545〉) (〈1.7550, 1.8004〉, 〈1.8270, 1.7048〉) (〈2.1174, 2.2608〉, 〈2.1383, 2.9279〉) (〈2.4493, 2.9550〉, 〈2.8626, 3.3013〉)

Table 8
Distance of each point from the centroid

$L_{1}$ $L_{2}$ $L_{3}$ $L_{4}$

${\hat{J}}_{1}$ (〈1.6485, 2.0143〉, 〈2.0768, 2.8902〉) (〈1.9543, 1.8316〉, 〈1.4971, 1.7660〉) (〈2.2759, 2.0932〉, 〈1.4763, 1.8807〉) (〈1.5259, 2.1139〉, 〈1.7195, 1.9837〉)

${\hat{J}}_{2}$ (〈0.4406, 3.0401〉, 〈0.5733, 5.7327〉) (〈0.6547, 3.0185〉, 〈0.9910, 3.0453〉) (〈0.7332, 3.0654〉, 〈0.5592, 2.9677〉) (〈0.9514, 2.7036〉, 〈0.8729, 3.3935〉)

${\hat{J}}_{3}$ (〈2, 1167, 0.7645〉, 〈1.7185, 2.5258〉) (〈1.5953, 0.6528〉, 〈2.0598, 0.8997〉) (〈1.3524, 0.8469〉, 〈2.1996, 0.7517〉) (〈2.1123, 0.9931〉, 〈1.8052, 0.6177〉)

${\hat{J}}_{4}$ (〈3.2696, 1.1259〉, 〈3.2246, 2.3301〉) (〈3.4758, 1.2615〉, 〈3.0198, 1.2501〉) (〈3.5165, 0.9244〉, 〈3.3479, 1.3204〉) (〈3.0305, 1.3505〉, 〈3.1959, 0.9423〉)

6 Numerical example

	$L_{1}$	$L_{2}$	$L_{3}$	$L_{4}$
${\hat{J}}_{1}$	(〈0.81, 0.31〉, 〈0.81, 0.13〉)	(〈0.16, 0.79〉, 〈0.14, 0.76〉)	(〈0.56, 0.27〉, 〈0.37, 0.41〉)	(〈0.81, 0.47〉, 〈0.52, 0.39〉)
${\hat{J}}_{2}$	(〈0.71, 0.22〉, 〈0.67, 0.29〉)	(〈0.45, 0.31〉, 〈0.11, 0.37〉)	(〈0.56, 0.31〉, 〈0.25, 0.61〉)	(〈0.47, 0.81〉, 〈0.32, 0.51〉)
${\hat{J}}_{3}$	(〈0.11, 0.32〉, 〈0.56, 0.33〉)	(〈0.71, 0.52〉, 〈0.56, 0.21〉)	(〈0.81, 0.16〉, 〈0.31, 0.71〉)	(〈0.71, 0.56〉, 〈0.15.0.11〉)
${\hat{J}}_{4}$	(〈0.29, 0.23〉, 〈0.67, 0.13〉)	(〈0.27, 0.12〉, 〈0.45, 0.35〉)	(〈0.27, 0.67〉, 〈0.11, 0.32〉)	(〈0.36, 0.73〉, 〈0.31, 0.63〉)

	$L_{1}$	$L_{2}$	$L_{3}$	$L_{4}$
${\hat{J}}_{1}$	(〈0.79, 0.36〉, 〈0.16, 0.56〉)	(〈0.83, 0.21〉, 〈0.76, 0.13〉)	(〈0.36, 0.43〉, 〈0.57, 0.31〉)	(〈0.11, 0.32〉, 〈0.27, 0.67〉)
${\hat{J}}_{2}$	(〈0.93, 0.63〉, 〈0.53, 0.21〉)	(〈0.71, 0.53〉, 〈0.37, 0.21〉)	(〈0.11, 0.22〉, 〈0.67, 0.29〉)	(〈0.33, 0.56〉, 〈0.11, 0.32〉)
${\hat{J}}_{3}$	(〈0.56, 0.79〉, 〈0.23, 0.51〉)	(〈0.56, 0.63〉, 〈0.27, 0.32〉)	(〈0.14, 0.76〉, 〈0.16, 0.76〉)	(〈0.21, 0.56〉, 〈0.52, 0.71〉)
${\hat{J}}_{4}$	(〈0.71, 0.81〉, 〈0.26, 0.76〉)	(〈0.37, 0.71〉, 〈0.37, 0.21〉)	(〈0.45, 0.35〉, 〈0.27, 0.12〉)	(〈0.41, 0.37〉, 〈0.27, 0.56〉)

	$L_{1}$	$L_{2}$	$L_{3}$	$L_{4}$
${\hat{J}}_{1}$	(〈0.3239, 0.6531〉, 〈0.0549, 0.6011〉)	(〈0.3147, 0.7054〉, 〈0.2639, 0.6379〉)	(〈0.2181, 0.4182〉, 〈0.2240, 0.6698〉)	(〈0.2919, 0.6921〉, 〈0.1842, 0.7705〉)
${\hat{J}}_{2}$	(〈0.6571, 0.5811〉, 〈0.4006, 0.5538〉)	(〈0.3960, 0.6089〉, 〈0.1469, 0.4957〉)	(〈0.2270, 0.4782〉, 〈0.3185, 0.6214〉)	(〈0.2475, 0.8048〉, 〈0.1288, 0.6077〉)
${\hat{J}}_{3}$	(〈0.1998, 0.7210〉, 〈0.2269, 0.6544〉)	(〈0.3872, 0.7668〉, 〈0.2367, 0.5260〉)	(〈0.3500, 0.6058〉, 〈0.1216, 0.8635〉)	(〈0.2956, 0.7589〉, 〈0.1920, 0.5452〉)
${\hat{J}}_{4}$	(〈0.1951, 0.7939〉, 〈0.1760, 0.7276〉)	(〈0.1011, 0.7130〉, 〈0.1354, 0.6987〉)	(〈0.1177, 0.8194〉, 〈0.0575, 0.6391〉)	(〈0.1251, 0.8354〉, 〈0.0898, 0.8666〉)

	$L_{1}$	$L_{2}$	$L_{3}$	$L_{4}$
${\hat{J}}_{1}$	(〈2.0182, 2.5408〉, 〈2.4493, 3.6377〉)	(〈1.2975, 2.0522〉, 〈1.3477, 2.8666〉)	(〈2.2297, 2.4244〉, 〈2.0549, 3.8708〉)	(〈2.7707, 3.1992〉, 〈2.9455, 5.1128〉)
${\hat{J}}_{2}$	(〈2.1719, 2.6172〉, 〈2.1401, 2.5960〉)	(〈1.6232, 1.7468〉, 〈1.8700, 1.6693〉)	(〈1.9271, 2.3693〉, 〈2.3045, 2.3008〉)	(〈3.0738, 3.1347〉, 〈2.6640, 3.2194〉)
${\hat{J}}_{3}$	(〈2.4362, 2.3385〉, 〈1.8302, 2.6319〉)	(〈1.5165, 1.8787〉, 〈1.4018, 1.7744〉)	(〈1.7994, 2.5616〉, 〈2.2471, 2.3861〉)	(〈3.2009, 3.2698〉, 〈2.7525, 3.0426〉)
${\hat{J}}_{4}$	(〈2.1347, 2.3694〉, 〈2.2002, 2.5545〉)	(〈1.7550, 1.8004〉, 〈1.8270, 1.7048〉)	(〈2.1174, 2.2608〉, 〈2.1383, 2.9279〉)	(〈2.4493, 2.9550〉, 〈2.8626, 3.3013〉)

	$L_{1}$	$L_{2}$	$L_{3}$	$L_{4}$
${\hat{J}}_{1}$	(〈1.6485, 2.0143〉, 〈2.0768, 2.8902〉)	(〈1.9543, 1.8316〉, 〈1.4971, 1.7660〉)	(〈2.2759, 2.0932〉, 〈1.4763, 1.8807〉)	(〈1.5259, 2.1139〉, 〈1.7195, 1.9837〉)
${\hat{J}}_{2}$	(〈0.4406, 3.0401〉, 〈0.5733, 5.7327〉)	(〈0.6547, 3.0185〉, 〈0.9910, 3.0453〉)	(〈0.7332, 3.0654〉, 〈0.5592, 2.9677〉)	(〈0.9514, 2.7036〉, 〈0.8729, 3.3935〉)
${\hat{J}}_{3}$	(〈2, 1167, 0.7645〉, 〈1.7185, 2.5258〉)	(〈1.5953, 0.6528〉, 〈2.0598, 0.8997〉)	(〈1.3524, 0.8469〉, 〈2.1996, 0.7517〉)	(〈2.1123, 0.9931〉, 〈1.8052, 0.6177〉)
${\hat{J}}_{4}$	(〈3.2696, 1.1259〉, 〈3.2246, 2.3301〉)	(〈3.4758, 1.2615〉, 〈3.0198, 1.2501〉)	(〈3.5165, 0.9244〉, 〈3.3479, 1.3204〉)	(〈3.0305, 1.3505〉, 〈3.1959, 0.9423〉)

To show that FCM is useful for grouping Web users with similar interests, we ran some test simulations. We employed access logs from a website with a 15-years old average that was intended for younger users. On several websites, we employ FCM to group the data. In this example, we demonstrate how frequently visitors visit pages such as the matches page, the homepage, the entertainment page, and the picture page. This data aids in categorizing the data in different ways, such as determining which visitor is more interested in a given field. These days, a lot of people visit websites to gather data, so through this system, we are able to determine the number of people who are interested in these fields. In particular, we gather data on children under the age of fifteen. In FCM, we only use membership values; however, in this case, we also use non-membership values based on the LDFNS data Table 4. As a result, we are able to accurately gather the results of our work, assist in organizing the data according to their respective fields, and determine the true interest of visitors as well as those who are not interested and who visit these sites frequently. Visitors show interest in pages that employ the most cluster vectors to group their interests.

Table 9
Update membership and non-membership clusters of visitor behaviour

Visited pages	Interest degrees
Matches	(〈0.05685, 0.06170〉, 〈0.05799, 0.05397〉)
Homepage	(〈0.04633, 0.06184〉, 〈0.04915, 0.04727〉)
Entertainment	(〈0.05414, 0.05408〉, 〈0.06010, 0.02125〉)
Picture	(〈0.06163, 0.06276〉, 〈0.06216, 0.03491〉)

Step 1.

Obtain matrix of LDFNS $D = (\dot{Z})_{m \times n}$ in Table 2 and Table 3. In this stage, we deal with two matrices, apply the operators, and then combine them into a single form.

Step 2.

Assume that the following values make up the weight vector: $T_{1} = 0.2$ , $T_{2} = 0.4$ , $T_{3} = 0.3$ and ${\tilde{§}}_{4} = 0.1$ the sum of all these weight vectors must be equals to 1.

Step 3.

Aggregate the LDF values using LDFAAWA operator given in Table 4.

Step 4.

Using the aggregate matrix of LDFAAWA, Table 4 illustrates how the FCM algorithms operate. The values in Table 4 correspond to the values of website visitors.

Step 5.

We maintained the interest of visitors in the matches, home page, entertainment, and picture in this step. We thus take membership clusters and non-membership clusters, which correspond to the website pages’ descriptions. Because of these clusters, we are able to classify the data in a way that makes sense and shows us which visitors are most interested in each of these different web-page groups. A parameter that in some manner, balances the data and aids in data classification.

Step 6.

Centroid of the clusters vectors are given in Table 7. By using formula of the centroid $C_{i j}$ is given in Eq (5.1). We were able to obtain the data’s center values and arrange them into groups as a result of this step.

Step 7.

We can determine the distance between each cluster point and our internet web-page by utilizing the distance formula Eq (5.2). This step leads us to the profile that has received the most views. Table 8 presents the outcome of utilizing the distance formula for the given data. In this step, we go through user profiling and the child’s interest through the data center $\dot{D}$ distance develops through the websites where the data is grouped and contributed in various ways.

Step 8.

The profile in Table 9 has been updated with the membership and non-membership cluster values.

Step 9.

This step involves computing the score function (SF), quadratic score function, and expected score function. To find these score values, we first developed the proposed method to find the score values of several operators. Next, we proceed to the updated cluster values listed in Table 13. Table 10 provides predicted score values; Table 11 provides operators for quadratic score values; and Table 12 provides expected score values. These score values are highly helpful in determining the final result of any proposed method. In order to determine which profile is most interested, we also use these three scoring functions in our FCM algorithm. This aids in our determination of the most viewed profile.

Table 10

The score detail under LDFAAWA operator

	Score values
Score function	${\hat{S}}_{N_{1}}$	${\hat{S}}_{N_{2}}$	${\hat{S}}_{N_{3}}$	${\hat{S}}_{N_{4}}$
	-0.0448	-0.2744	-0.4422	-0.7751
Ranking sequence	$N_{1}$ > $N_{2}$ > $N_{3}$ > $N_{4}$
Quadratic function	${\ddot{U}}_{N_{1}}$	${\ddot{U}}_{N_{2}}$	${\ddot{U}}_{N_{3}}$	${\ddot{U}}_{N_{4}}$
	-0.4295	-0.2310	-0.4087	-0.0484
Ramking sequence	$N_{4}$ > $N_{2}$ > $N_{3}$ > $N_{1}$
Expected score function	${\ddot{E}}_{N_{1}}$	${\ddot{E}}_{N_{2}}$	${\ddot{E}}_{N_{3}}$	${\ddot{E}}_{N_{4}}$
	0.2759	0.3627	0.2788	0.1124
Ranking sequence	$N_{2}$ > $N_{3}$ > $N_{1}$ > $N_{4}$

Table 11

The score detail under LDFAAOWA operator

	Score values
Score function	${\hat{S}}_{N_{1}}$	${\hat{S}}_{N_{2}}$	${\hat{S}}_{N_{3}}$	${\hat{S}}_{N_{4}}$
	-0.0400	-0.07661	-0.0182	-0.2311
Ranking sequence		$N_{1}$ > $N_{2}$ > $N_{3}$ > $N_{4}$
Quadratic function	${\ddot{U}}_{N_{1}}$	${\ddot{U}}_{N_{2}}$	${\ddot{U}}_{N_{3}}$	${\ddot{U}}_{N_{4}}$
	-0.2502	-0.3121	-0.4011	-0.0212
Ranking sequence		$N_{4}$ > $N_{1}$ > $N_{2}$ > $N_{3}$
Expected score function	${\ddot{E}}_{N_{1}}$	${\ddot{E}}_{N_{2}}$	${\ddot{E}}_{N_{3}}$	${\ddot{E}}_{N_{4}}$
	0.3134	0.0567	0.1212	0.1101
Ranking sequence		$N_{1}$ > $N_{3}$ > $N_{4}$ > $N_{2}$

Table 12

The score detail under LDFAAHA operator

	Score values
Score function	${\hat{S}}_{N_{1}}$	${\hat{S}}_{N_{2}}$	${\hat{S}}_{N_{3}}$	${\hat{S}}_{N_{4}}$
	-0.0221	-0.1123	-0.1233	-0.2351
Ranking sequence		$N_{1}$ > $N_{2}$ > $N_{3}$ > $N_{4}$
Quadratic function	${\ddot{U}}_{N_{1}}$	${\ddot{U}}_{N_{2}}$	${\ddot{U}}_{N_{3}}$	${\ddot{U}}_{N_{4}}$
	-0.2501	-0.2786	-0.3341	-0.03211
Ranking sequence		$N_{4}$ > $N_{1}$ > $N_{2}$ > $N_{3}$
Expected score function	${\ddot{E}}_{N_{1}}$	${\ddot{E}}_{N_{2}}$	${\ddot{E}}_{N_{3}}$	${\ddot{E}}_{N_{4}}$
	0.4332	0.0112	0.1212	0.1161
Ranking sequence		$N_{1}$ > $N_{3}$ > $N_{4}$ > $N_{2}$

Table 13

Page has the highest level of interest across all four clusters

	Interest degree
Score function	Matches	Homepage	Entertainment	Picture
	-0.00038	-0.00682	0.02609	0.01306
Ranking sequence		$N_{3}$ > $N_{4}$ > $N_{1}$ > $N_{2}$
Quadratic function	Matches	Homepage	Entertainment	Picture
	-0.00006	0.00148	0.00158	0.00125
Ranking sequence		$N_{3}$ > $N_{2}$ > $N_{4}$ > $N_{1}$
Expected score function	Matches	Homepage	Entertainment	Picture
	0.49979	0.49659	0.50972	0.50653
Ranking sequence		$N_{3}$ > $N_{4}$ > $N_{1}$ > $N_{2}$

Table 13. illustrates how we calculated the four clusters with the highest level of interest, taking into account non-membership values as well. By using the score values implemented in FCM.

Step 10.

Determine which score function (SF), quadratic score function (QSF), and expected score function are ranked highest in Table 14. We used our created operators to rank the values. However, Table 15 shows the clusters where visitors view the most profiles after computing the ranking of the interest degree using score values in Table 15.

Table 14

Ranking

Developed operators	Score	Quadratic score	Expected score
LDFAAWA	$N_{1}$ > $N_{2}$ > $N_{3}$ > $N_{4}$	$N_{4}$ > $N_{2}$ > $N_{3}$ > $N_{1}$	$N_{2}$ > $N_{3}$ > $N_{1}$ > $N_{4}$
LDFAAOWA	$N_{1}$ > $N_{2}$ > $N_{3}$ > $N_{4}$	$N_{4}$ > $N_{1}$ > $N_{2}$ > $N_{3}$	$N_{1}$ > $N_{3}$ > $N_{4}$ > $N_{2}$
LDFAAHA	$N_{1}$ > $N_{2}$ > $N_{3}$ > $N_{4}$	$N_{4}$ > $N_{1}$ > $N_{2}$ > $N_{3}$	$N_{1}$ > $N_{3}$ > $N_{4}$ > $N_{2}$

Table 15

Ranking

Clusters	Score	Quadratic score	Expected score
Interest degree	$N_{3}$ > $N_{4}$ > $N_{1}$ > $N_{2}$	$N_{3}$ > $N_{2}$ > $N_{4}$ > $N_{1}$	$N_{3}$ > $N_{4}$ > $N_{1}$ > $N_{2}$

N_{1}

denotes the matches,

N_{2}

denotes the homepage,

N_{3}

denotes the entertainment and

N_{4}

denotes the picture in interest degree data.

Step 11.

The Table 14 above that the $N_{1}$ and $N_{4}$ is the optimal alternative attribute in this step. Furthermore, Table 15 shows that $N_{3}$ , the Entertainment profile, is the most viewed and interested profile in the fuzzy profiling cluster. This indicates that children under the age of fifteen view the $N_{3}$ profile the most.

Fig. 4

The visual depiction of Table 10.

Fig. 5

The visual depiction of Table 11.

Fig. 6

The visual depiction of Table 12.

Fig. 7

The visual depiction of Table 13.

Fig. 8

The visual depiction of Table 14.

Fig. 9

The visual depiction of Table 15.

6.1 Graphical representation

6.2 Comparison analysis

By comparing the performances of the Linear Diophantine Aczel-Alsina aggregation operators with alternative aggregation approaches, comprehensive insights into their individual performances have been achieved. Comparing the outcomes with the particular aggregation operators, we find that there are some significant differences in how we handle vague and unknown data sets. The effusiveness of the Linear Diophantine Aczel-Alsina aggregation operators was demonstrated in certain data types or scenarios, exhibiting specific special features or strengths such as computational efficiency, accuracy, adaptability, etc. On the other hand, other operators demonstrated this in specific contexts. The lovely thing about this structure is that it establishes criteria for categorization based on characterizations and fosters independence between grades that are membership and non-membership. Table 16 compares the aggregation operators that are offered with a few operators that are already in place.

Table 16
A comparative analysis between the suggested operators and a few legacy operators

Authors AOs Ranking The ideal substitutes

Wang and Liu [31] IFEWA $N_{1}$ > $N_{3}$ > $N_{2}$ > $N_{4}$ $N_{1}$

IFEOWA $N_{1}$ > $N_{2}$ > $N_{4}$ > $N_{3}$ $N_{1}$

Xu [5] IFWA $N_{1}$ > $N_{4}$ > $N_{3}$ > $N_{2}$ $N_{1}$

IFOWA $N_{1}$ > $N_{2}$ > $N_{3}$ > $N_{4}$ $N_{1}$

IFHA $N_{2}$ > $N_{3}$ > $N_{1}$ > $N_{4}$ $N_{2}$

Riaz and Hashmi [38] LDFWA $N_{2}$ > $N_{4}$ > $N_{3}$ > $N_{1}$ $N_{2}$

LDFOWA $N_{2}$ > $N_{3}$ > $N_{1}$ > $N_{4}$ $N_{2}$

LDFHA $N_{2}$ > $N_{1}$ > $N_{4}$ > $N_{3}$ $N_{2}$

Proposed LDFAAWA $N_{1}$ > $N_{2}$ > $N_{3}$ > $N_{4}$ $N_{1}$

LDFAAOWA $N_{1}$ > $N_{2}$ > $N_{3}$ > $N_{4}$ $N_{1}$

LDFAAHA $N_{1}$ > $N_{2}$ > $N_{3}$ > $N_{4}$ $N_{1}$

Fig. 10

The visual depiction of Table 16.

6.3 Experimental study

This is an overview of the experimental settings in which online users are clustered according to their interests using access logs and Fuzzy C-Means (FCM). Access logs from a 15-year-old website intended for younger users. The attributes of each user’s interactions with various website pages (matches, homepages, entertainment, and picture pages) are represented as data points. Choose between random initialization and smart seeding strategies for cluster centroids’ initialization. Dividing the dataset into groups for testing and training in order to assess it. Using the dataset to run the Fuzzy C-Means method, which updates the cluster centroids iteratively and determines membership values, non-membership, and parameters. Adding non-membership values to the FCM process in order to represent user apathy or decreased affiliation with particular clusters, as found in the LDFNS data table. Regulating elements, including pre-processing stages, dataset selection, and algorithm parameters, to ensure consistency. Examining the FCM-obtained clusters for analysis, analysing trends, and determining user groups with specific interests. Using tables and visualisations to present the outcomes of the experiment. FCM may have played a larger part in the creation of more resilient and flexible clustering models by expanding its function in machine learning algorithms, particularly in clustering tasks.

7 Conclusion

Conclusively, this research has explored the domain of Linear Diophantine Fuzzy sets in relation to Aczel-Alsina aggregation operators, utilising the novel integration with the Fuzzy C-means algorithm. With thorough analysis and empirical experimentation, this work has demonstrated the feasibility and effectiveness of this combination in managing ambiguous and imprecise data in a range of fuzzy systems. 1.

We invented the Aczel-Alsina operational laws for the LDF set based on the Aczel-Alsina t-norm and t-conorm.

We exposed the LDFAAWA operator, LDFAAO-WA operator and LDFAAHA operator and evaluated their fundamental properties and results.

After the evaluation of these operators, we evaluate the problem of user profiling by fuzzy clustering to enhance the worth of the invention. theory.

We used a comparative study between the suggested procedures and a few current techniques to demonstrate the superiority and accuracy of the suggested theory.

Adding the Fuzzy C-means algorithm to these aggregating operators improves their accuracy and computational efficiency, offering a practical solution with potential applications in a variety of industries. In the near future, we will employ artificial intelligence in the context of suggested techniques through the use of robots and automation, virtual assistants and chat bots, decision-making and data analysis, individualised learning and training, and neural networks. In order to increase the value of the developed theory, we also intend to alter the suggested methods for a few extensions, such as Generalised linear diophantine fuzzy Aczel-Alsina aggregation operators and Linear diophantine fuzzy Aczel-Alsina Geometric aggregation operators, to improve the worth of the invented theory. On the other hand, to handle ambiguity in the decision-making technique, the described approach can be extended to various kinds of imprecise and ambiguous contexts in future studies. The approach is also applicable to a broad range of research domains, including financial analytics [2], environmental monitoring and sustainability [48], advanced supply chain modelling methods [25] and Internet of Things services [42].

Conflict of interest

About the publishing of the study article, there are no conflicts of interest, according to the authors.

Acknowledgments

Aziz Khan and Thabet Abdeljawad would like to thank Prince Sultan University for paying the APC and support through TAS research lab.

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Authors	AOs	Ranking	The ideal substitutes
Wang and Liu [31]	IFEWA	$N_{1}$ > $N_{3}$ > $N_{2}$ > $N_{4}$	$N_{1}$
	IFEOWA	$N_{1}$ > $N_{2}$ > $N_{4}$ > $N_{3}$	$N_{1}$
Xu [5]	IFWA	$N_{1}$ > $N_{4}$ > $N_{3}$ > $N_{2}$	$N_{1}$
	IFOWA	$N_{1}$ > $N_{2}$ > $N_{3}$ > $N_{4}$	$N_{1}$
	IFHA	$N_{2}$ > $N_{3}$ > $N_{1}$ > $N_{4}$	$N_{2}$
Riaz and Hashmi [38]	LDFWA	$N_{2}$ > $N_{4}$ > $N_{3}$ > $N_{1}$	$N_{2}$
	LDFOWA	$N_{2}$ > $N_{3}$ > $N_{1}$ > $N_{4}$	$N_{2}$
	LDFHA	$N_{2}$ > $N_{1}$ > $N_{4}$ > $N_{3}$	$N_{2}$
Proposed	LDFAAWA	$N_{1}$ > $N_{2}$ > $N_{3}$ > $N_{4}$	$N_{1}$
	LDFAAOWA	$N_{1}$ > $N_{2}$ > $N_{3}$ > $N_{4}$	$N_{1}$
	LDFAAHA	$N_{1}$ > $N_{2}$ > $N_{3}$ > $N_{4}$	$N_{1}$

Linear diophantine fuzzy Aczel-Alsina aggregation operators by using fuzzy-C means clustering algorithm

Abstract

Keywords

1 Introduction

4.1 LDFAAWA Operator

5.1 Proposed method based on LDFAA operators Fuzzy C-Means (FCM) clustering Algorithm

Table 9 Update membership and non-membership clusters of visitor behaviour

6.2 Comparison analysis

7 Conclusion

Conflict of interest

Acknowledgments

References

Table 9
Update membership and non-membership clusters of visitor behaviour