A novel decision-making approach for the selection of best deep learning techniques under logarithmic fractional fuzzy set information

In fields like data mining, pattern recognition, computer vision, and other Multi-Criteria Decision-Making (MCDM) areas, many real-world problems are characterized by their intricate and uncertain nature. Over time, different theories have been developed to address these difficulties more effectively. Among these, Zadeh¹ proposed the concept of fuzzy set (FS) theory, where each element of a universal set is assigned a membership degree (MD) ranging from [0, 1]. Nevertheless, FS theory had certain constraints, as it focused solely on the Membership Degree (MD) without incorporating a Non-Membership Degree (NMD). To address this limitation, Atanassov² proposed Intuitionistic Fuzzy Sets (IFS), which provide a framework where both (MD) and (NMD) are considered, with the condition that their sum does not exceed. While (IFS) marked significant progress, it encountered limitations in solving more complex problems. To overcome these problems, Abdullah³ proposed Fractional Fuzzy Sets (FFS), a more flexible approach to persistent problems. (FFS) introduced (MD) and (NMD), with their combined fractional power constrained within. Building on these advancements, Li et al.⁴ explored logarithmic operational laws (Log-OLs) for (IFS) and Intuitionistic Fuzzy Numbers (IFNs), offering valuable theoretical contributions. Additionally, Garge et al.⁵ developed new Log-OLs and aggregation operators (AoPs) tailored for Pythagorean Fuzzy Sets (PhyFS), demonstrating their practical applications in addressing real-world problems.

Some work has been done in the area of FS, IFS, and FFS. However, in this article, we developed a number of AoPs based on the operational laws (OLs) of logarithmic fractional fuzzy numbers (Log-FFNs). AoPs can be used to collect expert information related to MCGDM problems. For best results, there are a number of score functions we can use to rank the MCGDM problems. This section presents the proposed aggregation operators (AoPs), including the logarithmic fractional fuzzy weighted averaging (Log-FFWA) operator, the logarithmic fractional fuzzy ordered weighted averaging (Log-FFOWA) operator, and the logarithmic fractional fuzzy hybrid weighted averaging (Log-FFHWA) operator, all of which are linked to fuzzy sets. In addition, it provides generalization data. We will now discuss the main idea of our work.

1. (1)

   The goal of our proposed work is to address existing fuzzy set models, such as FS, IFS, PythFS, FFS, and q-ROFS, which fall short in real-world decision-making problems due to their sole dependence on MD and NMD.

2. (2)

   The second goal of our proposed model is to present a complete set of Aggregation Operators (AoPs) grounded in the operational principles of Log-FFS. In particular, we define the Logarithmic Fractional Fuzzy Weighted Averaging (Log-FFWA), Logarithmic Fractional Fuzzy Ordered Weighted Averaging (Log-FFOWA), and Logarithmic Fractional Fuzzy Hybrid Weighted Averaging (Log-FFHWA) operators, which are tailored to aggregate expert opinions, thereby enhancing the decision-making (DM) process effectively.

3. (3)

   Our third goal is to determine the most effective deep learning techniques by employing advanced decision-making methods, specifically, using extended TOPSIS and COPRAS methods.

A newly developed Intuitionistic Fuzzy Set (IFS) approach was proposed by Garg et al.⁶, which was subsequently applied to real-world problems. Thao et al.⁷ explored various distance and similarity measures within the IFS framework. Rani et al.⁸ proposed new distance measures for IFS, leveraging the properties of triangle centers in isosceles triangular fuzzy numbers, and examined their practical applications. Gohain et al.⁹ focused on distance measures for IFS and their real-world applications. In contrast, Kahraman et al.¹⁰ extended the information axiom to select logarithmic distance measures, moving from the conventional method to IFS. Arora et al.¹¹ evaluated the efficiency of logarithmic entropy measures for PythFS, focusing on diseases arising from Post-COVID complications through the application of the TOPSIS method. Garg et al.⁵ proposed a novel set of Log-OLs and Aggregation Operators (AoPs) for PythFS and demonstrated their real-world applications. Zeng et al.¹² developed Q-rung ortho-pair fuzzy weighted induced logarithmic distance measures and examined their practical uses. Abdullah et al.¹³ developed spherical fuzzy logarithmic aggregation operators grounded in entropy measures, emphasizing their and utility in real-life problems. Alfaro-García et al.¹⁴ suggested new logarithmic aggregation operators along with corresponding distance measures. Abdullah et al.¹⁵ developed a decision support system for choosing energy sources based on credibility fuzzy data. Rahman et al.¹⁶ proposed logarithmic intuitionistic fuzzy Einstein aggregation operators, which take into consideration varying levels of confidence. Qiyas et al.¹⁷ explored the use of logarithmic cubic aggregation operators to evaluate the effects of online learning during the COVID-19 pandemic. Pérrez-Romero et al.¹⁸ developed covariance logarithmic aggregation operators for application in decision-making tasks. Rahman et al.¹⁹ introduced innovative logarithmic aggregation operators and applied them to group decision-making utilizing t-norms and t-conorms. Haque et al.²⁰ proposed logarithmic operational laws and aggregation operators for trapezoidal neutrosophic numbers, using them in Multi-Criteria Group Decision-Making (MCGDM) to identify the most harmful viruses. Yahya et al.²¹ examined Fank aggregation operators and their role in probabilistic hesitant fuzzy multiple attribute decision-making.

In Multi-Criteria Group Decision-Making (MCGDM) problems, expert opinions can be integrated using a variety of algorithms. However, conventional Aggregation Operators (AoPs) have not fully addressed some of the challenges inherent in Multi-Criteria Decision-Making (MCDM). Various methods can be used to determine the best solutions to MCDM problems. In this context, the extended TOPSIS and COPRAS methods are employed to derive the optimal solution. These MCDM techniques, particularly the extended TOPSIS and COPRAS methods, are adept at handling both quantitative and qualitative data. Zeng et al.²² explored a decision-making approach grounded on interval-valued intuitionistic fuzzy sets, integrating nonlinear programming with the TOPSIS framework. Alkan et al.²³ applied the q-rung ortho-pair fuzzy TOPSIS technique to evaluate government responses during the COVID-19 pandemic. Yahya et al.²⁴ introduced an innovative multi-criteria group decision-making framework by combining the intuitionistic fuzzy rough Frank aggregation operator with the EDAS method.

Additionally, Wang et al.²⁵ developed a fuzzy TOPSIS model that combines both subjective and objective weighting mechanisms to enhance decision accuracy. Nwe et al.²⁶ proposed a resampling technique to handle imbalanced datasets with skewed distributions. Solangi et al.²⁷ applied the fuzzy TOPSIS method to evaluate and address challenges in the development of renewable energy in Pakistan. Chodha et al.²⁸ used TOPSIS combined with Entropy-based MCDM techniques for the selection of industrial arc welding robots. Zhang et al.²⁹ focused on the COPRAS method in multi-attribute group decision-making within a picture-fuzzy context and explored its practical applications. Mishra et al.³⁰ advanced the COPRAS technique by incorporating interval-valued hesitant Fermatean fuzzy sets and explored its practical applications. Akram et al.³¹ further refined the COPRAS method by integrating linguistic Fermatean fuzzy sets with Hamy mean operators. Garg et al.³² proposed a novel approach by combining complex intuitionistic fuzzy soft sets with the SWARA-COPRAS framework. Bhuvanesh et al.³³ introduced a decision-making model for lean tool selection, leveraging fuzzy TOPSIS and COPS methodologies. Xiang et al.³⁴ utilized the fuzzy SWARA-COPRAS method to evaluate and select coal transportation companies. Krishankumar et al.³⁵ designed an integrated COPRAS-based decision-making model using probabilistic hesitant fuzzy set data. Wu et al.³⁶ presented a hybrid approach combining interval type-2 fuzzy best-worst and extended VIKOR methods. Hu et al.³⁷ developed a unique doctor ranking system based on the VIKOR technique. Gou et al.³⁸ enhanced the VIKOR method by incorporating probabilistic double hierarchy linguistic term sets. Khan et al.³⁹ proposed an improved VIKOR method using generalized dissimilarity measures for Pythagorean fuzzy sets. Riaz et al.⁴⁰ introduced the cubic bipolar fuzzy-VIKOR method, integrating innovative distance and entropy measures with Einstein averaging aggregation operators. Cheng et al.⁴¹ proposed an enhanced VIKOR method utilizing q-rung ortho-pair fuzzy sets to address sustainable enterprise risks in small and medium-sized manufacturing firms. Akram et al.⁴² focused on group decision-making by applying the complex spherical fuzzy VIKOR method.

Motivation of the study

Many real-world decision-making challenges cannot be addressed effectively using the FS, IFS, PythFS, q-ROFS, and FFS frameworks due to their inherent limitations. However, the rationale for our approach is explored in more detail here since the FS and its generalization explained the MD. In this paper, we defined Log-FFS, and as well as under the Log-FFS information, the aggregation operators are developed. Also, to define the Logarithm operational laws under Log-FFS information, we used the basic operational laws of Log-FFWA, Log-FFOWA, and Log-FFHWA aggregation operators.

Contribution of the proposed work

This research introduces a novel approach, as no previous work has focused on fractional fuzzy sets and logarithmic fuzzy sets. In this paper, we define a new set called Log-FFS and explore its fundamental properties. The key objectives of this work are outlined below:

1. (a)

   We propose a novel set referred to as Log-FFS.

2. (b)

   We formulate new logarithmic operational laws to process Log-FFS data effectively.

3. (c)

   Based on these logarithmic operational laws, we introduce a variety of averaging Aggregation Operators (AoPs) grounded in Log-FFS data.

4. (d)

   The approaches formulated in this study are well-suited for addressing practical decision-making problems, particularly in the realm of Deep Learning Techniques, to identify the most effective solutions.

Outlines of the article

The structure of the article is as follows: Abbreviation section as the initial section. “Introduction” section introduces the introduction, while “Basic definitions” section presents basic definitions. “Logarithmic fractional fuzzy set” section discusses the Logarithmic fractional fuzzy set, and “Developed operational laws for logarithmic fractional fuzzy set” section introduces the Operational laws for Logarithmic fractional fuzzy set. “Developed AoPs under logarithmic fractional fuzzy set information” section focuses on the development of AoPs under Logarithmic fractional fuzzy set information, followed by “Logarithmic fractional fuzzy weighted average aggregation operator” section, which covers the Logarithmic fractional fuzzy weighted average aggregation operator. In “Logarithmic fractional fuzzy ordered weighted average aggregation operator” section, the Logarithmic fractional fuzzy ordered weighted average aggregation operator is explored. “Logarithmic fractional fuzzy hybrid weighted average aggregation operator” section details on Logarithmic fractional fuzzy hybrid weighted average aggregation operator, and “Algorithm of the extended TOPSIS approach” and “Algorithm of the extended COPRAS approach” sections give the algorithm Of The extended TOPSIS and COPRAS approaches, respectively. “A real life decision making problem” section delves into real-world decision-making applications. Computational results for the proposed TOPSIS and COPRAS approaches are presented in “Computational results for the extended TOPSIS approach” and “Computational results for the extended COPRAS approach” sections respectively, while “Verification by VIKOR approach under Log-FFS information” section covers the verification by VIKOR approach under Log-FFS information. “Computational results of extended VIKOR approach” section validates the proposed methods through comparison with the VIKOR approach. Comparison and discussion are listed in “Comparison and discussion” section. Finally, “Concluding remarks and future directions” section concludes the paper and discusses future directions.

In this section, we present a series of definitions and their corresponding properties. These encompass essential concepts like FS, IFS, PythFS, FeFS, Q-ROFS, and FFS. A clear understanding of these definitions will lay the groundwork for the following analysis.

Consider a \(A \ne \emptyset\). The set \(L\) can be described as \(L = \{ (a, \Omega _L(a)) \mid a \in A \}\), which is termed a fuzzy set (FS), where for each element \(a \in A\), the function \(\Omega _L(a) \in [0, 1]\) represents a PMD. Consider the case where \(A\) is a non-empty set. In this problem, the set \(L = \{ (a, \Omega _L(a), \lambda _L(a)) \mid a \in A \}\) is identified as an intuitionistic fuzzy set (IFS), where \(\Omega _L(a)\) and \(\lambda _L(a) \in [0, 1]\) correspond to PMD and NMD for each element \(a \in A\) in \(L\). Additionally, these values satisfy the condition that for all \(a \in A\), the sum of the positive and negative membership degrees must be bounded, i.e.,

Let \(A\) be a non-empty set, and define the set \(L = \{ (a, \Omega _L(a), \lambda _L(a)) \mid a \in A \}\), which is classified as a Pythagorean fuzzy set (PythFS). In this set, \(\Omega _L(a)\) and \(\lambda _L(a)\) represent PMD and NMD for each element \(a \in A\), respectively. These degrees of membership follow the inequality

for every element \(a \in A\).

Next, for a set \(A\), where \(A \ne \emptyset\), consider the set \(L = \{ (a, \Omega _L(a), \lambda _L(a)) \mid a \in A \}\), which is categorized as a generalized fuzzy set (FeFS). For each \(a \in A\), the values \(\Omega _L(a)\) and \(\lambda _L(a)\) are the PMD and NMD, respectively, and they satisfy the condition that

Consider a non-empty set \(A\), and define the set \(L = \{ (a, \Omega _L(a), \lambda _L(a)) \mid a \in A \}\), which can also be classified as a \(q\)-Rough fuzzy set (q-ROFS). Here, \(\Omega _L(a)\) and \(\lambda _L(a)\) represent the PMD and NMD for each element \(a \in A\), and they satisfy the following condition:

for all \(a \in A\), where \(q\) is a positive real number.

Finally, define a fuzzy set \(F\) on \(A\) using two mappings, \(\Omega _L: A \rightarrow [0, 1]\) and \(\lambda _L: A \rightarrow [0, 1]\), where \(\Omega _L(a)\) represents the PMD and \(\lambda _L(a)\) represents NMD. These functions must satisfy the condition that

for each \(a \in A\), where \(f = \frac{p}{q}\). This defines a fuzzy set in the context of rough set theory, where the condition is expressed as

with the restriction that

for all \(a \in A\).

Note that the real number system has no meaning for \(Log_{\daleth }0\) and \(Log_{\daleth }1\) is not well defined. Therefore, we assume that the Fractional Fuzzy Number (FFNs) L is greater than zero. Also, the base of \(Log_{\daleth }\ne 1\) is always positive and in the open interval (0, 1). Consider a non-empty set \(A\). Let the set

be defined with the condition \(0 \le \Omega _L^f(a) + \lambda _L^f(a) \le 1\), which classifies it as a fuzzy set (FS). The Log-FFNs (Logarithmic Fuzzy Fuzzy Networks) are then defined as:

Consider a non-empty set \(A\). Let the set \(L = \left\{ (a, \Omega _L(a), \lambda _L(a)) \mid a \in L \right\}\) be defined with the condition \(0 \le \Omega _L^f(a) + \lambda _L^f(a) \le 1\), where \(\Omega _L(a)\) and \(\lambda _L(a)\) represent the PMD and NMD of element \(a \in A\), respectively. The PMD is represented as \(\left( 1 - \log _{\daleth } \Omega _L(a) \right) : A \longrightarrow [0,1]\) such that \(\left( 1 - \log _{\daleth } \Omega _L(a) \right) \in [0,1]\) for all \(a \in A\). The NMD is represented as \(\log _{\daleth } \left( 1 - \lambda _L(a) \right) : A \longrightarrow [0,1]\) such that \(\log _{\daleth } \left( 1 - \lambda _L(a) \right) \in [0,1]\) for all \(a \in A\).

Then, a fractional fuzzy set (FFS) \(F\) on \(A\) is defined by the pair of mappings \(\Omega _{L}:A\longrightarrow \left[ 0,1\right]\) and \(\lambda _{L}:A\longrightarrow \left[ 0,1\right] .\) Where the pair of mapping \(log_{\daleth }\Omega _{L}:A\longrightarrow \left[ 0,1\right]\) and \(log_{\daleth }\lambda _{L}:A\longrightarrow \left[ 0,1\right]\) are the PMD and NMD respectively such that \(0\le \log _{\daleth }\Omega _{L}^{f}\left( a\right) +\lambda _{\daleth L}^{f}\left( a\right) \le 1,\) for each \(a\in A,\) where \(\left( f= \frac{p}{q}\text { and }\daleth \ne 1\right) .\) \(\log _{\daleth }L\) \(=\left\{ \begin{array}{c} (a,\log \Omega _{L}\left( a\right) , \\ \log \lambda _{L}\left( a\right) )|a\in L \end{array} \right\}\) such that \(0\le \log _{\daleth }\Omega _{L}^{f}+\log . _{\daleth }\lambda _{L}^{f}\le 1\).

In this section, we developed some novel Log-OLs for Log-FFNs and discussed their properties. Consider \(L_{1}\left( a\right)\) and \(L_{2}\left( a\right)\) are the two Log-FFNs. Then \(L_{1}\left( a\right)\) and \(L_{2}\left( a\right)\) are defined as, \(L_{1}(a)=\left\{ \Omega _{L_{1}}\left( a\right) ,\lambda _{L_{1}}\left( a\right) \right\}\) and \(L_{2}(a)=\left\{ \Omega _{L_{2}}\left( a\right) ,\lambda _{L_{2}}\left( a\right) \right\} .\) Now we have to define the logarithmic operations thats are,

1. (1)

   \(log_{\daleth }L_{1}\oplus log_{\daleth }L_{2}=\left\{ \left( 1-\log _{\daleth }\Omega _{L_{1}}\right) .\left( \log _{\daleth }\Omega _{L_{2}}\right) ,\log _{\daleth }(1-\lambda _{L_{1}}).\log _{\daleth }(1-\lambda _{L_{2}})\right\} .\)

2. (2)

   \(log_{\daleth }L_{1}\otimes log_{\daleth }L_{2}=\left\{ \begin{array}{c} (\left( 1-\log _{\daleth }\Omega _{L_{1}}\right) .\left( 1-\log _{\daleth }\Omega _{L_{2}}\right) , \\ 1-(1-\log _{\daleth }(1-\lambda _{L_{1}})).\left( 1-\log _{\daleth }(1-\lambda _{L_{2}})\right) . \end{array} \right\}\)

3. (3)

   \(\gamma .log_{\daleth }L=\left\{ 1-(log_{\daleth }\Omega _{L})^{\gamma },log_{\daleth }(1-\lambda _{L})^{\gamma }\right\}\)

4. (4)

   \(log_{\daleth }L^{\gamma }=\left\{ (1-\log _{\daleth }\Omega _{L})^{\gamma },1-(1-log_{\daleth }(1-\lambda _{L}))^{\gamma }\right\}\).

Score and accuracy functions for logarithmic fractional fuzzy set

Let \(\text {Log}_{\daleth } L = \left\{ 1 - \log _{\daleth } \Omega _{L_1}, \log _{\daleth } (1 - \lambda _{L_1}) \right\}\) be the Log-FFNs. Then, the score function and accuracy function are defined, respectively, as

and

This section aims to introduce novel logarithmic aggregation operators formulated using fractional fuzzy information. We examine different aggregation operators (AoPs), including Log-FFWA, Log-FFOWA, and Log-FFHWA, which are developed based on the operational laws defined for Log-FFS data. Additionally, we analyze the key properties of these proposed aggregation operators (AoPs).

Let \(L_{k}=(\Omega _{Lk},\lambda _{Lk}),(k=1,2,3,...n)\) be the family of Log-FFNs with \(0<\daleth _{k}\le \Omega _{Lk}\le 1\) and \(\daleth _{k}\ne 1.\) Then, the mapping \(Log-FFWA:\Phi ^{n}\longrightarrow \Phi\) is defined as,

The Logarithmic Fractional Fuzzy Weighted Average \(\text {Log-FFWA}\)) operator is defined as a function that operates on a set of logarithmic fractional fuzzy numbers (Log-FFNs). These Log-FFNs are associated with a weight vector \(\eta = (\eta _1, \eta _2, \dots , \eta _n)^{T}\), where each individual weight \(\eta _k\) lies within the interval [0, 1]. Additionally, the weights are constrained such that their sum equals unity, i.e., \(\sum \limits _{k=1}^{n} \eta _k = 1\).

Theorem 1

Let \(L_{k}=(\Omega _{Lk},\lambda _{Lk}),(k=1,2,3,...n)\) be the family of Log-FFNs with \(0<\daleth _{k}\le \Omega _{Lk}\le 1\) and \(\daleth _{k}\ne 1.\) Then the the aggregated values Log-FFNs are also Log-FFNs.

where \(\eta = (\eta _1, \eta _2, \dots , \eta _n)^{T}\) represents the weighted vectors as the Log-FFNs, such that \(\eta _k \in [0,1]\) and \(\sum \limits _{k=1}^{n} \eta _k = 1.\)

Proof

It is obvious. \(\square\)

The fundamental characteristics of the proposed AoPs are define as follows:

1. (1)

   Idempotent property Consider the set of Log-FFNs \(L_k = (\Omega _{Lk}, \lambda _{Lk})\) for \(k = 1, 2, 3, \dots , n\), and let \(\eta = (\eta _1, \eta _2, \dots , \eta _n)^T\) represent the weighted vector of Log-FFNs, the following relation holds: \(\text {Log-FFWA}\left( (L_1, \dots , L_n) \right) = \text {Log}_{\daleth } L, \quad \text {where} \quad \daleth _1 = \daleth _2 = \dots = \daleth _n = \daleth\).

2. (2)

   Boundedness property For the collection of Log-FFNs \(L_{k}=(\Omega _{Lk},\lambda _{Lk})\), for \((k=1,2,3,...n)\), and the weighted vector \(\eta =(\eta _{1},\eta _{2},...,\eta _{n})^{T}\) representing the Log-FFNs, we define \(L^{-}=\min (L_{1},...,L_{n})\) and \(L^{+}=\min (L_{1},...,L_{n})\) then,

   $$\begin{aligned} L^{-}\le Log-FFWA(L_{1},L_{2}, {\ldots },L_{n})\le L^{+}, \end{aligned}$$

   (6.3)

   where \(\daleth _{1}=\daleth _{2}={\ldots } =\daleth _{n}=\daleth\).

3. (3)

   Monotonicity Property For the collection of Log-FFNs \(L_{k},L_{k}^{\divideontimes }\) for \((k=1,2,...n)\), and let \(\eta =(\eta _{1},\eta _{2},...,\eta _{n})^{T}\) represent the weighted vector of Log-FFNs. Then, the following inequality holds:

   $$\begin{aligned} & Log-FFWA\left( (L_{1},...,L_{n})\right) \\\le & Log-FFWA\left( (L_{1}^{\divideontimes },...,L_{n}^{\divideontimes })\right) . \nonumber \end{aligned}$$

   (6.4)

Let \(L_{k}=(\Omega _{\left( Lk\right) _{\varkappa }},\lambda _{\left( Lk\right) _{\varkappa }}),(k=1,2,3,...n)\) be the family of Log-FFNs with \(0<\daleth _{k}\le \Omega _{Lk}\le 1\) and \(\daleth _{k}\ne 1.\) Then the mapping \(Log-FFOWA:\Phi ^{n}\longrightarrow \Phi\) is defined as;

The Logarithmic Fractional Fuzzy Ordered Weighted Averaging \(\text {Log-FFOWA}\) operator is a function designed to process a collection of logarithmic fractional fuzzy numbers (Log-FFNs). It utilizes a weight vector \(\eta = (\eta _1, \eta _2, \dots , \eta _n)^T\), where each weight \(\eta _k\) is constrained to the interval [0, 1]. Furthermore, the weights are normalized such that their sum, i.e., \(\sum _{k=1}^{n} \eta _k = 1\).

Theorem 2

Let \(L_{k}=(\Omega _{\left( Lk\right) _{\varkappa }},\lambda _{\left( Lk\right) _{\varkappa }}),(k=1,2,3,...n)\) be the family of Log-FFNs with \(0<\daleth _{k}\le \Omega _{\left( Lk\right) _{\varkappa }}\le 1\) and \(\daleth _{k}\ne 1.\)Then, the aggregated values of Log-FFNs are also Log-FFNs.

where \(\eta =(\eta _{1},\eta _{2},...,\eta _{n})^{T}\) represent the weighted vectors are the Log-FFNs, such that \(\eta _{k}\in \left[ 0,1\right]\) and \(\sum \limits _{k=1}^{n}\eta _{k}=1.\)

Proof

It is obvious. \(\square\)

The fundamental characteristics of the proposed AoPs are described as follows:

1. (1)

   Idempotent Property For the collection of Log-FFNs \(L_{k}=(\Omega _{\left( Lk\right) _{\varkappa }},\lambda _{\left( Lk\right) _{\varkappa }}),(k=1,2,3,...n)\) and \(\eta =(\eta _{1},\eta _{2},...,\eta _{n})^{T}\) represent the weighted vector of Log-FFNs.Then,

   $$\begin{aligned} Log-FFOWA(L_{1_{\varkappa }},...,L_{n_{\varkappa }})=Log_{\daleth }L_{\varkappa }, \end{aligned}$$

   (7.3)

   where \(\daleth _{1}=\daleth _{2}={\cdots }=\daleth _{n}=\daleth\).

2. (2)

   Boundedness Property For the collection of Log-FFNs \(L_{k}=(\Omega _{\left( Lk\right) _{\varkappa }},\lambda _{\left( Lk\right) _{\varkappa }}),(k=1,2,3,...n)\) and \(\eta =(\eta _{1},\eta _{2},...,\eta _{n})^{T}\) represent the weighted vector of Log-FFNs . Then \(L^{-}=\min (L_{1},...,L_{n})\) and \(L^{+}=\min (L_{1},...,L_{n})\) then,

   $$\begin{aligned} L^{-}\le Log-FFOWA(L_{1},L_{2},{\cdots },L_{n})\le L^{+}, \end{aligned}$$

   (7.4)

   where \(\daleth _{1}=\daleth _{2}={\cdots }=\daleth _{n}=\daleth\).

3. (3)

   Monotonicity Property For the collection of Log-FFNs \(L_{k},L_{k}^{\divideontimes }=(k=1,2,3,...n)\) and \(\eta =(\eta _{1},\eta _{2},...,\eta _{n})^{T}\) represent the weighted vector of Log-FFNs. Then,

   $$\begin{aligned} & Log-FFOWA\left( (L_{1},...,L_{n})\right) \\\le & Log-FFOWA\left( (L_{1}^{\divideontimes },...,L_{n}^{\divideontimes })\right) . \nonumber \end{aligned}$$

   (7.5)

Let \(L_{k}=(\Omega _{\left( Lk\right) _{\varkappa }}^{\partial },\lambda _{\left( Lk\right) _{\varkappa }}^{\partial }),(k=1,2,3,...n)\) be the family of Log-FFNs with \(0<\daleth _{k}\le \Omega _{Lk}\le 1\) and \(\daleth _{k}\ne 1.\)Then, the mapping \(Log-FFHWA:\Phi ^{n}\longrightarrow \Phi\) is defined as,

The Logarithmic Fractional Fuzzy Hybrid Weighted Averaging \(Log-FFHWA\) operator is a function designed to aggregate logarithmic fractional fuzzy numbers (Log-FFNs). It employs a weight vector \(\eta = (\eta _1, \eta _2, \ldots , \eta _n)^T\), where each weight \(\eta _k\) is defined within the interval [0, 1]. Additionally, the weights are normalized to ensure their sum equals one, i.e., \(\sum _{k=1}^{n} \eta _k = 1\).

Theorem 3

Let \(L_{k}=(\Omega _{\left( Lk\right) _{\varkappa }}^{\partial },\lambda _{\left( Lk\right) _{\varkappa }}^{\partial }),(k=1,2,3,...n)\) be the family of Log-FFNs with \(0<\daleth _{k}\le \Omega _{Lk}\le 1\) and \(\daleth _{k}\ne 1.\) Then, the aggregated values of Log-FFNs are also Log-FFNs.

where \(\eta =(\eta _{1},\eta _{2},...,\eta _{n})^{T}\) represent the weighted vectors are the Log-FFNs, such that \(\eta _{k}\in \left[ 0,1\right]\) and \(\sum \limits _{k=1}^{n}\eta _{k}=1.\)

Proof

It is obvious. \(\square\)

The proposed AoPs are defined by the following basic properties.

1. (1)

   Idempotent Property Let \(L_{k}=(\Omega _{\left( Lk\right) _{\varkappa }}^{\partial },\lambda _{\left( Lk\right) _{\varkappa }}^{\partial })\), for \((k=1,2,3,...n)\) represent the collection of Log-FFNs, and let \(\eta =(\eta _{1},\eta _{2},...,\eta _{n})^{T}\) denote the weighted vector of Log-FFNs.Then,

   $$\begin{aligned} Log-FFHWA(L_{1_{\varkappa }}^{\partial },...,L_{n_{\varkappa }}^{\partial })=Log_{\daleth }L_{\varkappa }^{\partial }, \end{aligned}$$

   (8.3)

   where \(\daleth _{1}=\daleth _{2}={\cdots }=\daleth _{n}=\daleth\).

2. (2)

   Boundedness Property For the collection of Log-FFNs \(L_{k}=(\Omega _{\left( Lk\right) _{\varkappa }}^{\partial },\lambda _{\left( Lk\right) _{\varkappa }}^{\partial })\), where \((k=1,2,3,...n)\) and \(\eta =(\eta _{1},\eta _{2},...,\eta _{n})^{T}\) represent the weighted vector of Log-FFNs, Then \(L^{-}=\min (L_{1},...,L_{n})\) and \(L^{+}=\min (L_{1},...,L_{n})\) as follows:

   $$\begin{aligned} L^{-}\le Log-FFHWA(L_{1},L_{2},{\cdots },L_{n})\le L^{+}, \end{aligned}$$

   (8.4a)

   where \(\daleth _{1}=\daleth _{2}={\cdots }=\daleth _{n}=\daleth\).

3. (3)

   Monotonicity Property Consider the collection of Log-FFNs \(L_{k},L_{k}^{\divideontimes }\) for \((k=1,2,3,...n)\), and let \(\eta =(\eta _{1},\eta _{2},...,\eta _{n})^{T}\) represent the weighted vector of Log-FFNs. Then,

   $$\begin{aligned} & Log-FFHWA\left( (L_{1},...,L_{n})\right) \\\le & Log-FFHWA\left( (L_{1}^{\divideontimes },...,L_{n}^{\divideontimes })\right) . \nonumber \end{aligned}$$

   (8.5)

The following discussion presents the steps of the proposed extended TOPSIS method.

1. (1)

   Representation of Expert Information

   \(\hbox {Step}^{*}\)-1 The expert’s information a set of alternatives and criteria.

   $$\begin{aligned} D=\left[ \eta _{k}G\left( \varphi _{k}\right) \right] _{m\times n}. \end{aligned}$$
2. (2)

   Aggregation of Expert Information

   \(\hbox {Step}^{*}\)-2 Represent the aggregated data of expert information using the proposed AoPs like Log-FFWA, Log-FFOWA and Log-FFHWA.

   $$\begin{aligned} & Log-FFWA(L_{1},...,L_{n}) \\ & \quad =\left\{ \begin{array}{c} 1-\prod \limits _{k=1}^{n}\left( Log_{\daleth _{k}}\Omega _{Lk}\right) ^{\eta _{k}}, \\ \prod \limits _{k=1}^{n}\left( 1-Log_{\daleth _{k}}\lambda _{Lk}\right) ^{\eta _{k}} \end{array} \right\} ,\\ & Log-FFOWA(L_{1_{\varkappa }},...,L_{n_{\varkappa }}) \\ & \quad = \left\{ \begin{array}{c} 1-\prod \limits _{k=1}^{n}\left( Log_{\daleth _{k}}\Omega \left( _{Lk}\right) _{\varkappa }\right) ^{\eta _{k}}, \\ \prod \limits _{k=1}^{n}\left( 1-Log_{\daleth _{k}}\left( \lambda \left( _{Lk}\right) _{\varkappa }\right) \right) ^{\eta _{k}} \end{array} \right\} ,\\ & Log-FFHWA(L_{1_{\varkappa }}^{\partial },...,L_{n_{\varkappa }}^{\partial }) \\ & \quad = \left\{ \begin{array}{c} 1-\prod \limits _{k=1}^{n}\left( Log_{\daleth _{k}}\Omega ^{\partial }\left( _{Lk}\right) _{\varkappa }\right) ^{\eta _{k}}, \\ \prod \limits _{k=1}^{n}\left( 1-Log_{\daleth _{k}}\left( \lambda ^{\partial }\left( _{Lk}\right) _{\varkappa }\right) \right) ^{\eta _{k}}. \end{array} \right\} \end{aligned}$$
3. (3)

   To Calculate the Positive Ideal Solution (PIS) and Negative Ideal Solution (NIS)

   \(\hbox {Step}^{*}\)-3 Compute PIS and NIS, respectively.

   $$\begin{aligned} PIS(\Omega _{kj}^{+})= & \max _{j}\left( \Omega _{k},\lambda _{k}\right) , \end{aligned}$$

   (9.1)
   $$\begin{aligned} NIS(\Omega _{kj}^{-})= & \min _{j}\left( \Omega _{k},\lambda _{k}\right) . \end{aligned}$$

   (9.2)
4. (4)

   To Calculate the Distance measure with PIS and NIS

   \(\hbox {Step}^{*}\)-4 Compute PIS and NIS, respectively.

5. (5)

   To Calculate the Weighted Values

   \(\hbox {Step}^{*}\)-5 Compute the weighted values using the following formulas.

   $$\begin{aligned} \beta _{kj}^{+}= & \sum _{k=1}^{n}(\Omega _{kj}^{+})\times \eta _{k}, \end{aligned}$$

   (9.3)
   $$\begin{aligned} \beta _{kj}^{-}= & \sum _{k=1}^{n-}(\lambda _{kj}^{-})\times \eta _{k}. \end{aligned}$$

   (9.4)
6. (6)

   To Calculate the Closeness Indices

   \(\hbox {Step}^{*}\)-6 Compute the closeness indices.

   $$\begin{aligned} \alpha _{k}=\frac{\beta _{kj}^{+}}{\beta _{kj}^{+}+\beta _{kj}^{-}}. \end{aligned}$$

   (9.5)

The steps of the proposed extended TOPSIS method are outlined as follows:

1. (1)

   Representation of Expert Information

   \(\hbox {Step}^{*}\)-1 Same as \(\hbox {Step}^{*}\)-1 of extended TOPSIS approach.

2. (2)

   Aggregation of Expert information

   \(\hbox {step}^{*}\)-2 The same as \(\hbox {step}^{*}\)-2 in the extended TOPSIS approach.

3. (3)

   Calculation of Benefit and Cost types criteria

   \(\hbox {Step}^{*}\)-3 The set of cost-type criteria is denoted as \(R_{\text {cost}} = \{1, 2, 3, \dots , l\}\), whereas the set of benefit-type criteria is represented as \(R_{\text {benefit}} = \{1+l, 2+l, 3+l, \dots , n\}\), respectively.

   $$\begin{aligned} \Omega _{k}= & \bigoplus \limits _{k=1}^{n}\eta _{k}\nu _{j}, \\ \lambda _{k}= & \bigoplus \limits _{k=1}^{n}\eta _{k}\nu _{j}. \nonumber \end{aligned}$$

   (10.1)
4. (4)

   Calculation of degree of Relative weights (\(r_{k}\)) \(\hbox {step}^{*}\)-4

   Calculate the degree of relative weights.

   $$\begin{aligned} r_{k}= & S(\Omega _{k})+\frac{\min _{k}S(\lambda _{k})\sum \limits _{j=1}^{p}S(\lambda _{k})}{S(\lambda _{k})\sum \limits _{j=1}^{p}\frac{\min _{k}S(\lambda _{k})}{S(\lambda _{k})}}, \end{aligned}$$

   (10.2)
   $$\begin{aligned} r_{k}= & S(\Omega _{k})+\frac{\sum \limits _{j=1}^{p}S(\lambda _{k})}{S(\lambda _{k})\sum \limits _{j=1}^{p}\frac{1}{S(\lambda _{k})}}. \end{aligned}$$

   (10.3)
5. (5)

   Calculation of Priority Order (\(B^{*}\))

   \(\hbox {Step}^{*}\)-5 Calculate the priority order.

   $$\begin{aligned} B^{*}=\max _{k}(r_{k}). \end{aligned}$$

   (10.4)
6. (6)

   Determination of utility degree (\(Ck_{k}\))

   \(\hbox {Step}^{*}\)-6 Calculate the utility degrees.

   $$\begin{aligned} Ck_{k}=\frac{r_{k}}{r_{\max }}\times 100\%. \end{aligned}$$

   (10.5)
7. (7)

   Ranking of alternatives

   \(\hbox {Step}^{*}\)-7 Rank all the alternatives based on their utility degrees.

The proposed approaches have been applied to a practical case, aiming to determine the most suitable Deep Learning (DL) techniques using Log-FFS data. In addition, specific properties serve as the key criteria for evaluating these DP techniques. Various DP techniques, treated as alternatives, are also examined in detail. A comprehensive analysis of the real-world example is provided below:

1. (1)

   \(\textit{Multi-Layer Perceptron (MLP)}\) A Multilayer Perceptron (MLP) is a type of artificial neural network (ANN) that typically employs a feedforward architecture. It consists of multiple layers of perceptrons, including an input layer, one or more hidden layers, and an output layer. The arrangement, configuration, and structure of these layers determines the performance and functionality of an MLP.

2. (2)

   \(\textit{Convolutional Neural Network (CNN or ConvNet)}\) A Convolutional Neural Network (CNN) is a specialized type of artificial neural network specifically designed to process visual data. CNNs are highly effective at detecting two-dimensional (2D) patterns, making them widely applicable in areas such as image recognition, medical image analysis, image segmentation, natural language processing, and numerous other domains.

3. (3)

   \(\textit{Recurrent Neural Network (RNN)}\) A Recurrent Neural Network (RNN) is a prominent class of neural networks tailored for sequential or time-series data. RNNs operate by incorporating the output from previous time steps as input for the current step. This unique feature is enabled by cyclic connections between nodes, allowing RNNs to maintain and utilize information over extended periods.

4. (4)

   \(\textit{Generative Adversarial Network (GAN)}\) A GAN is a machine learning framework used for generative tasks, utilizing deep learning techniques such as convolutional neural networks. The functioning of a GAN includes multiple steps, which highlight its importance as a prominent alternative in the field.

5. (5)

   \(\textit{Deep Belief Network (DBN)}\) A Deep Belief Network (DBN) is a neural network architecture that combines principles of generative graphical models. This study, proposes the fifth alternative approach for consideration.

Several criteria are thoroughly analyzed, forming the foundation for choosing the most suitable alternative using the proposed method. These criteria are denoted as \(h_k = \{h_1, h_2, h_3, h_4, h_5\}\), which include data dependence, hardware dependence, feature engineering, model training and testing, and execution time.

1. (1)

   \(\textit{Data Dependence}\) This criterion is regarded as the first type, where we discuss the data in detail, including aspects such as data volume, data collection methods, and data types. In deep learning processes (and their various forms), the effectiveness of algorithms often relies on large datasets to solve different types of problems.

2. (2)

   \(\textit{Hardware Dependence}\) The hardware-dependent criteria is considered the second type in this paper. In deep learning processes, extensive computations demand more precise hardware, typically GPUs rather than CPUs. GPUs are equipped with layers that allow data to be divided and processed for subsequent computations.

3. (3)

   \(\textit{Feature Engineering}\) Feature engineering is regarded as the third type of criterion, and within this criterion, we discuss aspects such as the characteristics, properties, and attributes of any given data.

4. (4)

   \(\textit{Model Training and Testing}\) Model training and testing are regarded as the fourth criterion. This criterion addresses the processes of training and testing a dataset, which may involve numerous parameters, often requiring significant time for evaluation. In deep learning, the testing phase is usually time-intensive due to the complexity of the data, whereas in machine learning, both training and testing typically take less time.

5. (5)

   \(\textit{Execution Time}\) Execution time is considered the fifth criterion, and in this criterion, we discuss the time required for execution in both deep learning and machine learning. In machine learning, the execution time is relatively short, while in deep learning, the process typically requires a significantly longer time to execute any execute any data.

The computational results of the extended TOPSIS method are presented in this section as follows:

\(\hbox {Step}^{*}\) -1:

The expert data is provided in Tables 1, 2, and 3.

\(\hbox {Step}^{*}\)-2:

The results of expert weight vectors and criteria weight vectors are as follows:

$$\begin{aligned} \eta= & \left\{ 0.25,0.35,0.4\right\} \\ \eta= & \left\{ 0.15,0.25,0.2,0.3,0.1\right\} . \end{aligned}$$

\(\hbox {Step}^{*}\)-3:

Tables 4 and 5 shows the aggregated expert information, respectively.

\(\hbox {Step}^{*}\)-4:

Tables 5 and 6 display the computed results for the Positive Ideal Solution (PIS) and Negative Ideal Solution (NIS), respectively.

\(\hbox {Step}^{*}\)-5:

Tables 7 and 8 show the results of the distance measures from each aggregated value, computed relative to the PIS and NIS, respectively.

\(\hbox {Ste}^{*}\)-6:

The result of weighted values are as follows.

$$\begin{aligned} \beta _{kj}^{+}= & .0203,0.0188,0.0373,0.0251,0.0256 \\ \beta _{kj}^{-}= & .0302,0.0232,0.0436,0.0403,0.0294. \end{aligned}$$

\(\hbox {Step}^{*}\)-7:

Table 9 presents the results of the closeness indices, as shown below.

\(\hbox {Step}^{*}\)-1:

Identical to \(\hbox {step}^{*}\)-1 of extended TOPSIS approach.

\(\hbox {Step}^{*}\)-2:

Identical to \(\hbox {step}^{*}\)-2 of extended TOPSIS approach.

\(\hbox {Step}^{*}\)p-3:

Table 10 presents the result of \(r_{k}\).

\(\hbox {Step}^{*}\)-4:

Table 11 presents the result of \(B^{*}\).

5:

Table 12 displays the results for \(Ck_{k}.\)

To assess the accuracy and validity of our proposed algorithms, we compare them with the VIKOR method under Log-FFNs in this section. The comparison steps are presented below:

\(\hbox {Step}^{*}\)-1:

: Identical to Algorithm-I.

\(\hbox {Step}^{*}\)-2:

: Identical to Algorithm-I.

\(\hbox {Step}^{*}\)-3:

: Identical to Algorithm-I.

\(\hbox {Step}^{*}\)-4:

Compute the distance measures between the values in the aggregated matrix and compare them with Positive Ideal Solution (PIS) and Negative Ideal Solution (NIS), respectively.

$$\begin{aligned} d\left( (\eta _{k}^{+}G\left( \varphi _{k}\right) ,\eta _{k}^{-}G\left( \varphi _{k}\right) \right) =\frac{1}{n}\sum \limits _{i=1}^{n}\left( |Log_{\daleth _{k}}^{+}\lambda _{Lk}-Log_{\daleth _{k}}^{-}\Omega _{Lk}|+|Log_{\daleth _{k}}^{+}\lambda _{Lk}-Log_{\daleth _{k}}^{-}\Omega _{Lk}|\right) . \end{aligned}$$

(14.1)

\(\hbox {Step}^{*}\)-5:

Evaluate the group utility values \(A_{i}\), individual regret values \(B_{i}\), and compromise values \(D_{i}\).

$$\begin{aligned} A_{i}=\sum \limits _{i=1}^{n}w_{j}\left( \frac{d(Log_{\daleth _{k}}^{+}\lambda _{Lk}-Log_{\daleth _{k}}^{-}\Omega _{Lk})}{d(Log_{\daleth _{k}}^{+}\lambda _{Lk}-Log_{\daleth _{k}}^{-}\Omega _{Lk})}\right) \end{aligned}$$

(14.2)

.

where \(A_{i}^{-}=\max _{i}A_{i},\) \(A_{i}^{*}=\min _{i}A_{i},\) \(B_{i}^{-}=\max _{i}B_{i},B_{i}^{*}=\min _{i}B_{i}.\)

\(\hbox {Step}^{*}\)-6:

This step involves ranking all the alternatives based on their compromise values \(D_{i}\)

In this section, we present the computational results of the developed Extended VIKOR methodology under Log-FFS data.

\(\hbox {Step}^{*}\)-1:

Identical to Methodology-I.

\(\hbox {Step}^{*}\)-2:

Identical to Methodology-I.

\(\hbox {Step}^{*}\)-3:

Identical to Methodology-II.

\(\hbox {Step}^{*}\)-4:

Identical to Methodology-II. The results for \(A_{i}\), \(B_{i}\), and \(D_{i}\) are presented in Table 13, respectively.

Comparison and discussion with existing AoPs

This section presents an assessment of the performance of our proposed method, comparing it with other existing approaches. In addition, to validate the accuracy of the proposed method’s accuracy, we compare our results with those obtained from well-established techniques.

Comparison with FFRWA

We compared our proposed Log-FFWA aggregation operator with the existing FFRWA aggregation operator³, and both methods produced identical results, as demonstrated in Table 15.

Result of comparison

In the comparison section, we have seen that the best optimal value in the ranking is the same in our developed aggregated values, TOPSIS and COPRAS approaches, and in the verified developed VIKOR method given in Table 14. This means that the ranking of our developed work is correct and the same.

In this study, we introduced a novel decision-making framework based on Logarithmic Fractional Fuzzy Sets (Log-FFS), incorporating three newly developed aggregation operators: Log-FFWA, Log-FFOWA, and Log-FFHWA. These operators were integrated into a multi-criteria decision-making (MCDM) environment to evaluate and rank deep learning techniques under uncertainty and vagueness.

The effectiveness of the proposed Log-FFS-based approach was validated through comparative analyses using three established MCDM methods: VIKOR, TOPSIS, and COPRAS. The ranking results were consistent across all three methods, with alternative \(h_5\) consistently emerging as the top choice. This consistency confirms the robustness, stability, and reliability of the proposed framework in handling imprecise, conflicting, and uncertain decision data.

This research contributes to the field by extending fractional fuzzy models into the logarithmic ___domain, thereby enhancing their computational flexibility and decision precision. While the current application focused on the evaluation of deep learning models, the proposed methodology demonstrates strong potential for broader applications, including health care decision-making, sustainable energy planning, and financial risk assessment all of which often involve multiple conflicting criteria and uncertain data.

Drawbacks of the proposed method

However, several drawbacks of the proposed method warrant consideration:

1. (1)

   Computational complexity The proposed method requires significant computational resources, especially when dealing with large-scale problems with numerous alternatives and criteria. As the number of decision-making alternatives and criteria increases, the time complexity and computational demand grow, potentially making the method less scalable.

2. (2)

   Parameter tuning and expert dependence The framework relies on expert-defined membership functions and careful parameter tuning. This introduces a level of subjectivity, as expert judgments can vary and may lead to inconsistencies in the decision-making process, particularly when applied across different domains.

3. (3)

   Interpretation of Fuzzy Outputs The aggregated fuzzy outputs generated by the Log-FFS method may be difficult to interpret, particularly for practitioners who are not familiar with fuzzy logic. This can make it challenging to apply the framework in practical settings, especially when decisions must be communicated to non-expert stake holders.

4. (4)

   Sensitivity to data quality Like many fuzzy logic-based approaches, the Log-FFS framework is sensitive to the quality of the input data. Inaccurate or incomplete data could lead to unreliable decision outcomes, highlighting the importance of high-quality input for effective decision-making.

While these challenges are manageable, they underscore the importance of continued research to simplify the model’s implementation, improve computational efficiency, and enhance user accessibility. Addressing these drawbacks will make the framework more practical and applicable to a wider range of real-world problems.

Future work

Future research will compare the proposed Logarithmic Fractional Fuzzy Set (Log-FFS) approach with other Multi-Criteria Decision-Making (MCDM) methods (e.g., AHP, ELECTRE, PROMETHEE) to assess its performance and robustness. The study will also explore Log-FFS in real-world applications with high uncertainty and complex decision structures, such as health care (treatment planning), renewable energy (source prioritization), financial risk assessment, and industrial processes (supplier selection, quality control).

Further investigations will develop aggregation operators (e.g., Dombi, Einstein, Hamacher norms) within Log-FFS and integrate it with advanced MCDM techniques (CoCoSo, CODAS, COPRAS) to enhance flexibility. The framework’s versatility also extends to S-box encryption, neural networks, machine learning, robot selection, and cluster analysis, demonstrating its broad applicability in uncertainty-driven decision-making.