Emergency hospital selection in Peshawar using fractional Diophantine neural network for risky decision-making

Abstract

In emergency medical scenarios, rapid and accurate hospital selection is crucial, especially in densely populated urban areas like Peshawar, Pakistan. This study proposes a novel hybrid decision-making model called the Fractional Diophantine Fuzzy Neural Network (FDFNN), which integrates fractional Diophantine fuzzy information with neural network structures and Hamacher aggregation operators. The key innovation lies in the model’s ability to effectively handle uncertainty, incomplete expert input, and unknown weight vectors—challenges often encountered in real-world multi-criteria decision-making (MCDM) problems. We applied the proposed model to evaluate six hospitals in Peshawar, considering five critical attributes for emergency care. Three Experts information was gathered and used to construct three decision matrices (${\mathcal{E}}_1,{\mathcal{E}}_2$ and ${\mathcal{E}}_3)$, each representing different expert perspectives. The proposed model processed these matrices, aggregating the expert evaluations using the Hamacher operators and accounting for uncertainty in the data. After processing the input data, the model ranked the hospitals and identified the most suitable option for emergency situations based on the selected criteria. Based on expert opinions and the results of the proposed model, Rehman Medical Institute (${\mathbb{Q}}_4$), was identified as the best choice. To validate the effectiveness of the proposed model, we conducted a comparative analysis using established MCDM techniques, including TOPSIS, WASPAS, WS, WP, MOORA, and GRA. The results showed that the proposed model outperformed traditional methods in terms of decision accuracy and reliability. Additionally, the study explored the sensitivity of the model to changes in parameters ($\theta$ and $f$), providing further insights into the robustness of the decision outcomes. Furthermore, we also discuss the advantages and limitations of the proposed method, including its applicability in real-time decision-making scenarios and its potential for large-scale implementation.

Introduction

Peshawar, the capital city of Khyber Pakhtunkhwa province, is one of the largest and most populous urban areas in Pakistan, with a diverse demographic and a growing population of over 2 million residents. The city faces significant challenges in its healthcare infrastructure, particularly in emergency situations. Rapid urbanization, population growth, and increasing road traffic contribute to an urgent need for an efficient and effective emergency response system. In medical emergencies, the speed at which a patient receives treatment is crucial, especially in life-threatening conditions such as heart attacks, strokes, or severe trauma. The choice of hospital can make the difference between life and death; however, selecting the most appropriate hospital is often complicated by various factors, including the availability of specialized medical services, hospital capacity, treatment costs, and real-time traffic conditions. Peshawar's healthcare landscape comprises a mix of public and private hospitals, each with its strengths and weaknesses. Public hospitals, such as LRH and KTH, typically offer a broad range of services but often struggle with overcrowding and resource limitations. Private hospitals like Rehman Medical Institute (RMI) and Northwest General Hospital (NWG) may provide higher-quality care but at a higher cost, which can be a barrier for many patients. In emergencies, where every second counts, it is vital for Emergency Medical Services (EMS) [1] teams to adopt a structured approach to hospital selection. The decision-making process can be made more effective and efficient by using MCGDM [2] approaches, which enable EMS crews to choose suitable hospitals based on the key factors that influence patient outcomes. The MCGDM technique combines individual preference data from multiple decision-makers (DMs) with group preference data to identify the best option among a limited set of alternatives.

Literature review

In the field of modern decision-making, multi-criteria group decision-making (MCGDM) has garnered significant attention in recent years [2, 3, 4–5]. Nevertheless, the rapid advancement of both economic and societal systems has introduced new layers of complexity to collective decision-making processes. A major challenge for researchers lies in effectively capturing the inherent ambiguity, uncertainty, and subjective preferences that characterize diverse perspectives during the evaluation phase. Within MCGDM problems, two elements are particularly critical: the construction of evaluation data and the identification of the optimal alternative. Uncertainty in the representation of evaluation data typically arises in three primary forms: fuzziness, randomness, and incomplete information. To address these issues, scholars have proposed various theoretical models to better express and manage this uncertainty. These include FS [6], IFS [7], PyFS [8], q-ROFSs [9], LDFS [10], q-LDFS [11], N-LDFS [12], among others. While traditional fuzzy models such as FS, IFS, and PyFS have been widely used in decision-making, they face limitations in handling large-scale fractional data and adapting to complex, dynamic environments. The proposed work overcomes these challenges by integrating fractional Diophantine fuzzy sets (FDFS) with neural networks, offering greater flexibility in decision-making and more effectively addressing the uncertainties inherent in emergency hospital selection scenarios. Furthermore, the literature review primarily focuses on two key areas: fuzzy extensions and their applications, and the use of neural networks in addressing decision-making problems.

Fuzzy Extensions and Their Applications

Multi-criteria group decision-making (MCGDM) techniques deals with various complex problems in various fields of sciences and engineering that cannot be resolved using classical methods due to a large number of uncertainties and vagueness present in their data analysis. To handle this, Zadeh [13] presented the idea of fuzzy set theory, where the non-membership degree (NMD) $\gamma$ is calculated by subtracting the membership degree (MD) from 1 and the MD $\mu$ of an element of a set is defined as a characteristic function on the closed interval $[0,1]$. To construct a new framework known as the IFS [14], Atanassove expanded the idea of FS. In this framework, the MD and NMD, or $\mu$ and$\gamma$, are defined separately, but their total is limited to a closed interval $[0,1]$, implying that $(\mu +\gamma )\in [\text{0,1}]$. Furthermore$,1-(\mu +\gamma )$ is the definition of the hesitancy degree. The IFS gained a lot of attention in the past years and was used by many scholars to resolve various real-life problems [15, 16–17]. We are unable to use IFS in some real-world situations where the total of the MD and NMD is greater than one, i.e.$\left((,\mu ,=,0.8,),+,(,\gamma ,=,0.3,),),>,1\right)$. In order to get around these problems, Yager introduced a novel framework known as the Pythagorean fuzzy set (PyFS) [18]. In this framework, the MD and NMD, or $\mu$ and$\gamma$, are defined separately, but their square sum is limited to a closed interval $[0,1]$, implying that $({\mu }^2+{\gamma }^2)\in [\text{0,1}]$. Furthermore, the term $\sqrt[2]{1-\text{sum}(\mu ,\gamma )}$ was referred to as a hesitancy degree. PyFS is used more often in decision-making (DM) scenarios than IFS and FS, but PyFS restricts the decision-makers in many real-world scenarios. To overcome these shortcomings of the PyFS, Yager introduces a novel framework called the q-ROFS [9]. The q-ROFS also has two degrees: MD and NMD, but they also have the requirement that the sum of the $\mathit{qth}$ powers for MD and NMD must always fall inside the close interval [0, 1]. Theoretically and practically, FS, IFS, PyFS, and q-ROFS all have different constraints on the MD and NMD functions. To overcome these constraints on fuzzy sets, Riaz and Hashmi presented a novel concept called LDFS [19] by including the concept of reference parameters (RPs) that IFS, PyFS, and q-ROFS were unable to handle. An innovative extension of the LDFS to the N-LDFS was proposed by Maria et al. [12], along with a description of its advantages. However, due to the lack of analysis about fractional numbers (FNs) in its domain, q-RLDFS also has certain limitations. To address this issue, we extended the q-RLDFS domain to include fractional numbers (FNs) and established the concept of a novel set known as the fractional Diophantine fuzzy set (FDFS), which is a two-degree system referred to as DM, DNM, and RPs subject to the restriction that ${0\le {\left(\mathfrak{P}\right)}^f\mathcal{P}}_{\mathfrak{L}}\left(\mathcal{y}\right)+{\left(\mathfrak{R}\right)}^f{\mathcal{Q}}_{\mathfrak{L}}\left(\mathcal{y}\right)\le 1$ where $f=\frac{p}{q}$, with a conditions $p,q,\in {\mathbb{z}}^{+}$ and$q,\ne 0$. The total of the $\mathit{fth}$ power of the reference parameter corresponding to MG and $\mathit{fth}$ power of the reference parameter corresponding to NMG may be less than or equal to 1. Since it can accommodate a lot of fractional data and gives decision-making experts greater flexibility in the MDs, NMDs, and RPs, the idea of a FDFS is an extension of the current FFS and N-LDFS. The limitations of the traditional fuzzy models are mitigated by FDFS, which increase the decision-making space and facilitate more precise information communication among experts. To describe uncertainty in information science and decision-making models, fractional Diophantine fuzzy information is more practical.

The Application of Neural Networks in Addressing Decision-Making Problems

One important aspect of machine learning (ML) [20] is artificial neural networks, which are used for decision-making. Deep learning (DL) [21] utilizes artificial neural networks (ANNs) to learn from data and perform complex calculations on it. Neural networks consist of layers of connected nodes, with each layer responsible for learning a specific part of the input. For instance, in image recognition, the first layer detects edges, the second layer identifies shapes, and the third layer identifies objects. As the network learns, it adjusts the connections between the nodes to improve its understanding of the data. There are various types of ANNs, such as feed-forward Neural Networks (FFNNs) [22], FBNNs [23], and RNNs [24], which are applied to solve different real-life problems [25, 26]. FFNNs [22] are a type of ANN where the data moves in one direction, from the input layer to the output layer, without cycling back. An FFNN has three main layers: the input layer, hidden layers, and output layer. The input layer takes fuzzy data and sends it to the hidden layers, which process the data using different aggregation methods [27, 28–29] and send the results to the output layer. The fuzzy data interacts with attribute weight vectors, which are determined using methods like entropy [30], TOPSIS [31], and AHP [32]. The output layer generates the final result using activation functions such as Sigmoid [33], ReLU [34], and Softmax [35]. ANNs have been used in various multi-criteria decision-making problems [36, 37–38].

Motivation of the study

Neural networks have developed into an effective tool for solving problems in the real world. It was used by a number of scholars to tackle different MCDM issues. The idea of FFNN was expanded to FFLNN by Saleem et al. [39] and used to choose commercially viable water purification techniques. Shougi et al. [40] designed a continuous linear Diophantine neural network (CLDNN) to overcome obstacles in green supply chain management, using the continuous linear Diophantine fuzzy information. In the recent past, many scholars have worked on the extension of LDFS to overcome these challenges. Numerous studies have been conducted in different sectors, such as in reference [41, 42, 43, 44, 45–46]. Nevertheless, a few studies have been conducted to assess hospital difficulties through the use of MCGDM techniques in the health care industry. Theresa J. Puzhakkara and Shiny Jose [47] use CLDFS for medical diagnosis. Alaa O. Almagrabi et al. [48] use a q-LDFS for emergency decision-making for COVID-19.

Building on the previous research, a variety of tools has been developed to assist in addressing decision-making challenges. However, hybrid structure combining fractional fuzzy sets have not yet been explored or applied to decision-making problems, to the best of our knowledge. To tackle the uncertainties inherent in such problems, this study introduces a novel tool called the fractional Diophantine fuzzy set (FDFS), which merges fractional fuzzy sets with nonlinear Diophantine fuzzy sets. The key objectives of this study are as follows:

To propose a novel decision-making framework that mitigates the drawbacks of traditional fuzzy models, broadens the scope of decision-making, and enhances the accuracy of information sharing among experts, making it more applicable and adaptable to real-world scenarios.

To build a model for managing decision-making in emergencies, particularly in the context of selecting hospitals in critical medical situations, by effectively dealing with the uncertainty and vagueness associated with such decisions.

To design a decision-making framework based on fractional Diophantine fuzzy neural networks (FDNNs), utilizing fractional Diophantine fuzzy data to address complex decision-making tasks.

To investigate essential components such as FDFSs, ranking functions, and distance metrics within the context of FDNNs, and to develop new aggregation operators like FDFHWA, FDFHOWA, and FDFHHWA. These operators will be used to integrate decision data provided by experts in the form of FDFNs, and their beneficial properties will also be explored.

Novelties

This article introduces a novel decision-making model for emergency hospital selection using FDFNNs. In critical medical situations, timely and reliable decisions must be made under uncertainty, where factors such as treatment availability, cost, traffic conditions, and hospital capacity add to the complexity. To address these challenges, the proposed FDFNN model integrates FDFS into a neural network framework, enabling precise evaluation of multiple criteria. A new distance-based weighting mechanism is developed to handle unknown or uncertain expert preferences, while innovative score and accuracy functions improve the ranking of alternatives. The model also incorporates a distance measure to enhance discrimination among options. A real-world case study on hospital selection in Peshawar validates the model’s effectiveness in guiding high-stakes decisions. Furthermore, a sensitivity analysis on the controlling parameters $\theta$ and $f$ confirms the model's robustness and adaptability in dynamic environments. This flexible and powerful approach provides a significant advancement in MCGDM for emergency healthcare services.

Contribution of the study

Emergency decision-making, particularly in the context of selecting suitable hospitals during critical medical situations, requires quick and reliable evaluation under uncertainty. Given the variety of hospitals and the complexity of factors involved—such as treatment availability, cost, traffic conditions, and capacity—there is a pressing need for an advanced decision-making approach that can manage this uncertainty effectively. Multi-criteria group decision-making (MCGDM) techniques are essential for evaluating alternatives in such scenarios. In this study, we propose a novel model called the fractional Diophantine fuzzy neural network (FDFNN), which leverages fractional Diophantine fuzzy information to improve the precision and adaptability of emergency decision-making processes. The main contributions of this work are outlined below:

This study introduces a novel decision-support structure called the Fractional Diophantine Fuzzy Set (FDFS), designed to address the shortcomings of traditional fuzzy models. Unlike existing approaches, FDFS can effectively process large-scale fractional data and offers enhanced flexibility in representing decision membership, non-membership, and reference parameters.

A comprehensive mathematical foundation for FDFS is established through the development of new score and accuracy functions to rank FDFNs. In addition, a distance measure is proposed to quantify the dissimilarity between any two FDFNs, supporting better discrimination among alternatives in decision-making processes.

A novel Fractional Diophantine Fuzzy Neural Network (FDFNN) model is proposed to support complex decision-making tasks. This neural framework not only integrates FDFS-based data but also offers a flexible and powerful structure for solving MCGDM problems, such as emergency hospital selection. A visual indication of the model is provided for better clarity and interpretation.

To address scenarios where expert preferences or criterion weights are uncertain or unavailable, we propose a new distance-based weighting mechanism that objectively estimates these unknowns, ensuring consistency and reliability in the decision-making process.

The effectiveness of the proposed approach is tested through a real-world case study on hospital selection in emergency scenarios in Peshawar. The results demonstrate the model’s capability to guide decision-makers in high-stakes, uncertain environments.

Finally, to verify the stability and robustness of our model, we conduct a sensitivity analysis using various values of the controlling parameters $\theta$ and $f$. This analysis reveals how changes in parameter values influence the final decision, offering insights into the adaptability of the proposed method.

Moreover, Fig. 1 presents the general structure of the article, offering an overview of its primary components and the logical progression of the content. It demonstrates the interconnections between key sections, such as the problem formulation, methodology, results, and conclusions. This diagram serves to clarify the overall framework and aids the reader in comprehending the organization of the article. Tables 1 and 2 provide thorough explanations of the acronyms and symbol used in this article to help readers better comprehend the terms used. These acronyms are essential for simplifying technical language, and the table gives the reader a quick reference for better reading and comprehension.

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Fig. 1

Structure of the Article

Table 1. Acronyms and their descriptions

Acronyms	Descriptions
MCDM	Multi-criteria decision making
MCGDM	Multi-criteria group decision making
FS	Fuzzy set
IFS	Intuitionistic fuzzy set
PyFS	Pythagorean fuzzy set
q-ROFS	q-Rung orthopair fuzzy set
LDFS	Linear Diophantine fuzzy set
N-LDFS	Non-linear Diophantine fuzzy set
FDFS	Fractional Diophantine fuzzy set
FDFNs	Fractional Diophantine fuzzy numbers
DM	Degree of Membership
NMG	Degree of Non-membership
RPs	Reference parameters
DL	Deep learning
AI	Artificial intelligence
ANNS	Artificial neural networks
NNs	Neural networks
FFNNs	Feed-forward neural networks
FBNNs	Feed backward neural networks
RNNs	Recurrent neural networks
FFLNNs	Feed-forward linguistic neural networks
FFDHLNNs	Feed-forward double hierarchy linguistic neural networks
FCNNs	Fuzzy credibility neural networks
DMs	Decision makers
AOs	Aggregation operators
FDFNNs	Fractional Diophantine fuzzy neural networks
FDFHWAO	Fractional Diophantine fuzzy Hamacher weighted aggregation operator
FDFHOWAO	Fractional Diophantine fuzzy Hamacher order weighted aggregation operator
FDFHHWAO	Fractional Diophantine fuzzy Hamacher hybrid weighted aggregation operator

Table 2. Notations

Symbol	Descriptions
$\mathfrak{P}$	Universe of discourse
$\mathfrak{F}$	FS
$\mathcal{K}$	IFS
$\mathbb{K}$	PyFS
$\mathbb{M}$	q-ROFS
$\mathcal{Q}$	LDFS
$\mathbb{N}$	N-LDFS
ℒ	FDFS
$\mathfrak{R}$	FDFNs
${\mathfrak{R}}^1$	Absolute FDFNs
${\mathfrak{R}}^0$	Null FDFNs
${\mu }_{\mathfrak{F}}\left(\mathcal{y}\right)$	Membership degree
${\gamma }_{\mathfrak{F}}\left(\mathcal{y}\right)$	Non-membership degree
$\mathcal{M},\mathcal{N}$	Reference parameters
ScF $\left({\mathfrak{R}}_i\right)$	Score function
ScF $\left({\mathfrak{R}}_i\right)$	Accuracy function
$\lambda >0$	Scalar
$\mathbb{D}\left({\mathfrak{R}}_1,,,{\mathfrak{R}}_2\right)$	Distance between any two FDFNs
$\mathcal{T}\left({\mathcal{P}}_1,,,{\mathcal{P}}_2\right)$	Hamacher t-norm
$\mathcal{S}\left({\mathcal{P}}_1,,,{\mathcal{P}}_2\right)$	Hamacher t-conorm
$\theta$	Hamacher parameter
${\mathfrak{R}}_i$	Collection of fractional Diophantine fuzzy numbers
${\mathcal{E}}_{\mathfrak{s}}$	Decision matrices
${\mathbb{P}}_j$	Attributes
${\mathbb{Q}}_i$	Alternatives
${\mathbb{W}}_j$	Input signal weight vector
${\mathcal{P}}_i$	Hidden layer data of the Fractional Diophantine fuzzy neural network
${\mathcal{Q}}_i$	Output layer data of the Fractional Diophantine fuzzy neural network
$\mathbb{F}\left(\text{ScF},\left({\mathcal{Q}}_i\right)\right)$	Activation function

Preliminaries

We define the basics of FS, IFS, PyFS, q-ROFS, LDFS, and N-LDFS to explore the fundamental ideas and characteristics to acquire the required foundational knowledge before developing a unique concept of a FDFS in this section.

Definition 2.1:

The FS $\mathfrak{F}$ over non-empty set $\mathfrak{P}$ as defines as follows:$$\mathfrak{F}=\left\{\mathcal{y},,,⟨,{\mu }_{\mathfrak{F}},\left(\mathcal{y}\right),⟩,|,:,\mathcal{y},\in ,\mathfrak{P}\right\}\left(1\right)$$where the transformation ${\mu }_{\mathfrak{F}}:\to [0,1]$ denotes the membership grade (MG) in FS for each $\mathcal{y}\in \mathfrak{P}$.

Definition 2.2:

The IFS $\mathcal{K}$ over non-empty set $\mathfrak{P}$ as defines as follows:$$\mathcal{K}=\left\{\mathcal{y},,,⟨,{\mu }_{\mathcal{K}},\left(\mathcal{y}\right),,,{\gamma }_{\mathcal{K}},\left(\mathcal{y}\right),⟩,:,\mathcal{y},\in ,\mathfrak{P}\right\}\left(2\right)$$where the transformations ${\mu }_{\mathcal{K}}:\to [0,1]$ and ${\gamma }_{\mathcal{K}}:\mathcal{y}:\to [\text{0,1}]$ denoted the MG and NMG for each $\mathcal{y}\in \mathfrak{P}$ respectively, and satisfy the given aspects: $0\le {\mu }_{\mathcal{K}}\left(\mathcal{y}\right)+{\gamma }_{\mathcal{K}}\left(\mathcal{y}\right)\le 1$ and the degree of hesitancy is defined as:$${\mathfrak{K}}_{\mathcal{K}}=\left(1,-,⟨,{\mu }_{\mathcal{K}},\left(\mathcal{y}\right),+,{\gamma }_{\mathcal{K}},\left(\mathcal{y}\right),⟩\right)\left(3\right)$$

We are unable to use IFS in some real-world situations where the total of the MD and NMD is greater than one, i.e. $\left(0.6,+,0.5,>,1\right)$. In order to get around these problems, Yager introduced an innovative concept PyFS.

Definition 2.3:

The Pythagorean fuzzy set (PyFS)$\mathbb{K}$ over non-empty set $\mathfrak{P}$ as defines as follows:$$\mathbb{K}=\left\{\mathcal{y},,,⟨,{\mu }_{\mathbb{K}},\left(\mathcal{y}\right),,,{\gamma }_{\mathbb{K}},\left(\mathcal{y}\right),⟩,:,\mathcal{y},\in ,\mathfrak{P}\right\}\left(4\right)$$where the transformations ${\mu }_{\mathbb{K}}:\mathcal{y}:\to [0,1]$ and ${\gamma }_{\mathbb{K}}:\mathcal{y}:\to [\text{0,1}]$ denoted the MG and NMG for each $\mathcal{y}\in \mathfrak{P}$ respectively, and satisfy the given aspects: $0\le {\left({\mu }_{\mathbb{K}},\left(\mathcal{y}\right)\right)}^2+{\left({\gamma }_{\mathbb{K}},\left(\mathcal{y}\right)\right)}^2\le 1$ and the degree of hesitancy is defined as:$${\mathfrak{K}}_{\mathbb{K}}=\sqrt{\left(1,-,⟨,{\left({\mu }_{\mathbb{K}},\left(\mathcal{y}\right)\right)}^2,+,{\left({\gamma }_{\mathbb{K}},\left(\mathcal{y}\right)\right)}^2,⟩\right)}\left(5\right)$$

We are unable to use PyFS in some real-world situations where the total of the MD and NMD is greater than one, i.e. $\left(0.8,+,0.7,>,1\right)$. In order to get around these problems, Yager introduced an innovative concept known as the q-ROFS.

Definition 2.4:

The q-ROFS $\mathbb{M}$ over non-empty set $\mathfrak{P}$ as defines as follows:$$\mathbb{M}=\left\{\mathcal{y},,,⟨,{\mu }_{\mathbb{M}},\left(\mathcal{y}\right),,,{\gamma }_{\mathbb{M}},\left(\mathcal{y}\right),⟩,:,\mathcal{y},\in ,\mathfrak{P}\right\}\left(6\right)$$where the transformations ${\mu }_{\mathbb{M}}:\mathcal{y}:\to [\text{0,1}]$ and ${\gamma }_{\mathbb{M}}:\mathcal{y}:\to [\text{0,1}]$ denoted the MG and NMG for each $\mathcal{y}\in \mathfrak{P}$ respectively, and satisfy the given aspects: $0\le {\left({\mu }_{\mathbb{M}},\left(\mathcal{y}\right)\right)}^q+{\left({\gamma }_{\mathbb{M}},\left(\mathcal{y}\right)\right)}^q\le 1$ and the degree of hesitancy is defined as:$${\mathfrak{K}}_{\mathbb{M}}=\sqrt[q]{\left(1,-,⟨,{\left({\mu }_{\mathbb{M}},\left(\mathcal{y}\right)\right)}^q,+,{\left({\gamma }_{\mathbb{M}},\left(\mathcal{y}\right)\right)}^q,⟩\right)}\left(7\right)$$q-ROFS is a more generalized form as compared to other fuzzy extensions that increase the q-rung spaces and give greater flexibility to decision-makers in uncertain situations to make a decision, graphically shown in Fig. 2. We are unable to use q-ROFS in some real-world situations where the total of the MD and NMD is greater than one, i.e. $\left(1,+,1,>,1\right)$. In order to get around these problems, Riaz and Hashmi introduced an innovative concept called the LDFS.

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Fig. 2

Comparisons of spaces of the IFS, PyFS, FFS, quadratic PyFS, and q-ROFS

Definition 2.5:

Let $\mathfrak{P}$ be a non-empty fixed set then the LDFS $\mathcal{Q}$ over non-empty set $\mathfrak{P}$ as defines as follows:$$\mathcal{Q}=\left\{\mathcal{y},,,\left(⟨{\mu }_{\mathcal{Q}}\left(\mathcal{y}\right),{\gamma }_{\mathcal{Q}}\left(\mathcal{y}\right)⟩,,,⟨\mathcal{M},\mathcal{N}⟩\right),:,\mathcal{y},\in ,\mathfrak{P}\right\}\left(8\right)$$where $⟨{\mu }_{\mathcal{Q}}\left(\mathcal{y}\right),{\gamma }_{\mathcal{Q}}\left(\mathcal{y}\right)⟩,⟨\mathcal{M},\mathcal{N}⟩\in [0,1]$ represents the MD, NMD and the RPs respectively, and satisfy the given aspects:$0\le \left(\mathcal{M}\right){\mu }_{\mathcal{Q}}\left(\mathcal{y}\right)+\left(\mathcal{N}\right){\gamma }_{\mathcal{Q}}\left(\mathcal{y}\right)\le 1\forall \mathcal{y}\in \mathfrak{P}$ with $0\le \left(\mathcal{M}\right)+\left(\mathcal{N}\right)\le 1$. The degree of hesitancy is defined as:$${\pi }_{{\mathfrak{K}}_{\mathcal{Q}}}=\left(1,-,\left(\mathcal{M}\right),{\mu }_{\mathcal{Q}},\left(\mathcal{y}\right),-,\left(\mathcal{N}\right),{\gamma }_{\mathcal{Q}},\left(\mathcal{y}\right)\right)\left(9\right)$$where the indeterminacy degree's reference parameter is denoted by $\pi .$

Definition 2.6:

Let $\mathfrak{P}$ be a non-empty fixed set then the N-LDFS $\mathbb{N}$ over non-empty set $\mathfrak{P}$ as defines as follows:$$\mathbb{N}=\left\{\mathcal{y},,,\left(⟨{\mu }_{\mathbb{N}}\left(\mathcal{y}\right),{\gamma }_{\mathbb{N}}\left(\mathcal{y}\right)⟩,,,⟨\mathcal{M},\mathcal{N}⟩\right),:,\mathcal{y},\in ,\mathfrak{P}\right\}\left(10\right)$$where $⟨{\mu }_{\mathbb{N}}\left(\mathcal{y}\right),{\gamma }_{\mathbb{N}}\left(\mathcal{y}\right)⟩,⟨\mathcal{M},\mathcal{N}⟩\in [0,1]$ represents the MD, the NMD and the RPs respectively, and satisfy the given aspects:$0\le {\left(\mathcal{M}\right)}^q{\mu }_{\mathbb{N}}\left(\mathcal{y}\right)+{\left(\mathcal{N}\right)}^q{\gamma }_{\mathbb{N}}\left(\mathcal{y}\right)\le 1\forall \mathcal{y}\in \mathfrak{P}$ with $0\le {\left(\mathcal{M}\right)}^q+{\left(\mathcal{N}\right)}^q\le 1$. The degree of hesitancy is defined as:$${\dot{\pi }}_{{\mathfrak{K}}_{\mathbb{Q}}}=\sqrt[q]{1-\left({\left(\mathcal{M}\right)}^q,{\mu }_{\mathbb{N}},\left(\mathcal{y}\right),+,{\left(\mathcal{N}\right)}^q,{\gamma }_{\mathbb{N}},\left(\mathcal{y}\right)\right)}\left(11\right)$$where the indeterminacy degree's reference parameter is denoted by $\dot{\pi }$.

Fractional diophantine fuzzy set

The notion of a fractional Diophantine fuzzy set (FDFS) is presented in this section. The suggested model is comparable to the widely recognized linear Diophantine equation $\text{ax}+\text{by}=\text{c}$ in number theory; however, by including the $\mathit{fth}$ power with the reference parameters, we were able to generalize this idea. Many scholars use N-LDFS to address MCDM difficulties and offer greater flexibility in solving these problems. However, N-LDFS has certain limitations due to the lack of extensive analysis of fractional numbers (FNs) in their field. To address this issue, we established the idea of a novel set called the fractional Diophantine fuzzy set (FDFS) and extended the N-LDFS domain to encompass fractional numbers (FNs). Now we define a FDFS as follows:

Definition 3.1:

Let ℬ be a non-empty fixed set, then a fractional Diophantine fuzzy set ℒ on ℬ as defined as follows:

ℒ $=\left\{\mathcal{y},,,\left(⟨{\mu }_{\mathfrak{L}}\left(\mathcal{y}\right),{\gamma }_{\mathfrak{L}}\left(\mathcal{y}\right)⟩,,,⟨\mathcal{M},\mathcal{N}⟩\right),:,\mathcal{y},\in ,\mathfrak{P}\right\}$, $\left(12\right)$

where$⟨{\mu }_{\mathfrak{L}}\left(\mathcal{y}\right),{\gamma }_{\mathfrak{L}}\left(\mathcal{y}\right)⟩$, $⟨\mathcal{M},\mathcal{N}⟩\in [0,1]$ represents the MD, the NMD and the RPs respectively, and satisfy the given aspects: $0\le {\left(\mathcal{M}\right)}^f{\mu }_{\mathfrak{L}}\left(\mathcal{y}\right)+{\left(\mathcal{N}\right)}^f{\gamma }_{\mathfrak{L}}\left(\mathcal{y}\right)\le 1\forall \mathcal{y}\in \mathfrak{P},f=\frac{p}{q},\left(p,,,q,,,\in ,{\mathbb{z}}^{+}\right)$ and $q\ne 0$ with $0\le {\left(\mathcal{M}\right)}^f+{\left(\mathcal{N}\right)}^f\le 1,f=\frac{p}{q},\left(p,,,q,,,\in ,{\mathbb{z}}^{+}\right)$ and$q\ne 0$. The total of the $\mathit{fth}$ power of the reference parameter corresponding to MG and $\mathit{fth}$ power of the reference parameter corresponding to NMG may be less than or equal to 1. Since it can accommodate a lot of fractional data and gives decision-making experts greater flexibility in the DM, DNM, and RPs, the idea of a FDFS is an extension of the current FFS and N-LDFS. The limitations of the traditional fuzzy models are mitigated by FDFS, which increase the decision-making space and facilitate more precise information communication among experts. To describe uncertainty in information science and decision-making models, fractional Diophantine fuzzy information is more practical, as shown in Fig. 3.

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Fig. 3

Figures (a), (b), and (c) show comparison of spaces among LDFS, quadratic n-LDFS, cubic n-LDFS, bi-quadratic n-LDFS, and FDFS

Definition 3.2:

The fractional Diophantine fuzzy numbers (FDFNs) $\mathfrak{R}$ over non-empty set $\mathfrak{P}$ as defines as follows:$$\mathfrak{R}=\left(⟨{\mu }_{\mathfrak{R}},{\gamma }_{\mathfrak{R}}⟩,,,⟨\mathcal{M},\mathcal{N}⟩\right)\left(13\right)$$

The collection of FDFNs is denoted by $\mathfrak{R}$, having the following limitations:

• $0\le {\left(\mathcal{M}\right)}^f{\mu }_{\mathfrak{R}}\left(\mathcal{y}\right)+{\left(\mathcal{N}\right)}^f{\gamma }_{\mathfrak{R}}\left(\mathcal{y}\right)\le 1\forall \mathcal{y}\in \mathfrak{P},f=\frac{p}{q},\left(p,,,q,\in ,{\mathbb{z}}^{+}\right)$ and$q\ne 0$.

• $0\le {\left(\mathcal{M}\right)}^f+{\left(\mathcal{N}\right)}^f\le 1,f=\frac{p}{q},\left(p,,,q,,,\in ,{\mathbb{z}}^{+}\right)$ and$q\ne 0$.

• $0\le \left(\mathcal{M}\right),{\mu }_{\mathfrak{R}}\left(\mathcal{y}\right),\left(\mathcal{N}\right),{\gamma }_{\mathfrak{R}}\left(\mathcal{y}\right)\le 1.$

Definition 3.3:

The absolute FDFS ${\mathfrak{R}}^1$ over non-empty set $\mathfrak{P}$ as defines as follows:$${\mathfrak{R}}^1=\left\{\left(\mathcal{y},,,⟨,\text{1,0},⟩,,,⟨,\text{1,0},⟩\right),:,\mathcal{y},\in ,\mathfrak{P}\right\}\left(14\right)$$

Definition 3.4:

The null FDFS ${\mathfrak{R}}^0$ over non-empty set $\mathfrak{P}$ as defines as follows:$${\mathfrak{R}}^0=\left\{\left(\mathcal{y},,,⟨,\text{0,1},⟩,,,⟨,\text{0,1},⟩\right),:,\mathcal{y},\in ,\mathfrak{P}\right\}\left(15\right)$$

We have the following scenarios when $p,q$ have different values:

Case 1: Where $f=\frac{p}{q}\ge 1$ and $p=q$, then, the FDFS is reduced to LDFS.

Case 2: Where $f=\frac{p}{q}\ge 1$ and $p=2q$, then, the FDFS is reduced to quadratic LDFS.

Case 3: Where $f=\frac{p}{q}\ge 1$ and $p=3q$, then, the FDFS is reduced to cubic LDFS.

Case 4: Where $f=\frac{p}{q}\ge 1$ and $p=4q$, then, the FDFS is reduced to bi-quadratic LDFS.

Case 5: Where $f=\frac{p}{q}\ge 1$ and $p=nq$, then, the FDFS is reduced to N-LDFS and so on as shown in Fig. 4. That FDFS offers advantages over $f$ value fluctuations. Observably, the Diophantine region grows with increasing fractional $f$, allowing the border limitations to transmit a greater variety of fractional fuzzy data.

[See PDF for image]

Fig. 4

Flow chart of FDFS concept

Buying of a mobile phone

When selecting a mobile phone, a consumer faces a variety of choices, including popular brands such as Oppo, Tecno, Infinix, Vivo, and so on. The individual has three options to choose the best mobile phone for herself from the list. To make an informed decision, the individual considers factors such as budget and personal satisfaction. Let ${\text{X}}_1=$ Oppo, ${\text{X}}_2=$ Tecno, and ${\text{X}}_3=$ Infinix, which are all prominent mobile phone brands among the options available for selection. To generate the source data effectively, we can utilize FDFS, where the reference parameters $\mathcal{M}$ and $\mathcal{N}$ correspond to "top-notch technology" and "mediocre technology," respectively. The membership and non-membership grades $\left({\mu }_{\mathfrak{R}},,,{\gamma }_{\mathfrak{R}}\right)$ indicate levels of corporate satisfaction and dissatisfaction. These parameters typically incorporate relationships between linguistic phrases and various traits or attributes. Depending on specific requirements or situational demands, the characteristics of these reference parameters can be modified. For instance, we might set $\mathcal{M}=$ to affordable and $\mathcal{N}=$ to premium-priced, or define $\mathcal{M}=$ superior camera capabilities versus $\mathcal{N}=$ basic camera features. This flexibility in the $\mathit{fth}$ powers of the reference parameters enhances both the membership grade and non-membership grade areas, broadening the range of possible solutions. Accordingly, Tables 3 and 4 illustrate the application of FDFS in selecting the most suitable mobile phone option from a variety of choices.

Table 3. FDFS 1st Company

$\mathfrak{R}$	$\left(⟨{\mu }_{\mathfrak{R}},{\gamma }_{\mathfrak{R}}⟩,,,⟨\mathcal{M},\mathcal{N}⟩\right)$
${\text{X}}_1$	$\left(⟨,\text{0.6,0.7},⟩,,,⟨,0.3,,,0.6,⟩\right)$
${\text{X}}_2$	$\left(⟨,\text{0.8,0.3},⟩,,,⟨,0.4,,,0.5,⟩\right)$
${\text{X}}_3$	$\left(⟨,\text{0.5,1},⟩,,,⟨,0.4,,,0.3,⟩\right)$

Table 4. FDFS 2nd Company

$\mathfrak{R}$	$\left(⟨{\mu }_{\mathfrak{R}},{\gamma }_{\mathfrak{R}}⟩,,,⟨\mathcal{M},\mathcal{N}⟩\right)$
${\text{X}}_1$	$\left(⟨,\text{1,0.9},⟩,,,⟨,0.2,,,0.7,⟩\right)$
${\text{X}}_2$	$\left(⟨,\text{1,1},⟩,,,⟨,0.3,,,0.4,⟩\right)$
${\text{X}}_3$	$\left(⟨,\text{0.2,0.3},⟩,,,⟨,0.8,,,0.6,⟩\right)$

Buying of a laptop

When choosing a laptop, consumers have many options from well-known brands like HP, Dell, Acer, Apple, Lenovo, and so on. Suppose someone has four choices: Let ${\text{A}}_1=$ HP, ${\text{A}}_2=$ Dell,${\text{A}}_3=$ Acer, and ${\text{A}}_4=$ Lenovo. To make a smart decision, the person considers important factors such as their budget and personal preferences. To generate the source data effectively, we can utilize FDFS, where the reference parameters $\mathcal{M}$ and $\mathcal{N}$ correspond to highly recommended technology and less recommended technology, respectively. The membership and non-membership grades $\left({\mu }_{\mathfrak{R}},,,{\gamma }_{\mathfrak{R}}\right)$ indicate levels of corporate satisfaction and dissatisfaction. These parameters typically incorporate relationships between linguistic phrases and various traits or attributes. Depending on what the person wants, they can adjust the meanings of $\mathcal{M}$ and $\mathcal{N}$. For example, $\mathcal{M}$ could be set to represent affordable while $\mathcal{N}$ represents expensive, or $\mathcal{M}$ might mean excellent battery life compared to $\mathcal{N}$, which could mean average battery life. The beauty of these $\mathit{fth}$ powers of the reference parameter is that they increase the MG and NMG areas while also expanding the parameterization range of the issues. According to these reference parameters, Tables 5 and 6 show the FDFS to purchase the best Laptop for herself among various options.

Table 5. FDFS 1st Company

$\mathfrak{R}$	$\left(⟨{\mu }_{\mathfrak{R}},{\gamma }_{\mathfrak{R}}⟩,,,⟨\mathcal{M},\mathcal{N}⟩\right)$
${\text{A}}_1$	$\left(⟨,\text{1,0.6},⟩,,,⟨,0.4,,,0.5,⟩\right)$
${\text{A}}_2$	$\left(⟨,\text{0.8,1},⟩,,,⟨,0.3,,,0.7,⟩\right)$
${\text{A}}_3$	$\left(⟨,\text{0.7,0.5},⟩,,,⟨,0.6,,,0.4,⟩\right)$
${\text{A}}_4$	$\left(⟨,\text{1,1},⟩,,,⟨,0.5,,,0.5,⟩\right)$

Table 6. FDFS 2nd Company

$\mathfrak{R}$	$\left(⟨{\mu }_{\mathfrak{R}},{\gamma }_{\mathfrak{R}}⟩,,,⟨\mathcal{M},\mathcal{N}⟩\right)$
${\text{A}}_1$	$\left(⟨,\text{1,1},⟩,,,⟨,0.2,,,0.7,⟩\right)$
${\text{A}}_2$	$\left(⟨,\text{0.5,0},,,6,⟩,,,⟨,0.2,,,0.6,⟩\right)$
${\text{A}}_3$	$\left(⟨,\text{1,0.8},⟩,,,⟨,0.3,,,0.4,⟩\right)$
${\text{A}}_4$	$\left(⟨,\text{0.8,1},⟩,,,⟨,0.5,,,0.3,⟩\right)$

Corollary 1:

For $p=nq$, any IFS are an N-LDFS, and any N-LDFS is an FDFS. But the converse is not true, as shown in Fig. 5.

[See PDF for image]

Fig. 5

Fuzzy extensions from FS to FDFS

Corollary 2:

For $p=q$, any IFS are a LDFS, and any LDFS is an FDFS. But the converse is not true, as shown in Fig. 5.

Definition 3.2.1:

The fractional Diophantine fuzzy numbers (FDFNs) ${\mathfrak{R}}_i=\left(⟨{\mu }_{{\mathfrak{R}}_i},{\gamma }_{{\mathfrak{R}}_i}⟩,,,⟨{\mathcal{M}}_i,{\mathcal{N}}_i⟩\right)$ is a collection of FDFNs over non-empty set $\mathfrak{P}$, then the score function (ScF) for any fractional Diophantine fuzzy number is a mapping such as: ScF $\left({\mathfrak{R}}_i\right)=FDFNs\left({\mathfrak{R}}_i\right)\to [\text{0,1}]$ and define as;$$\left({\mathfrak{R}}_i\right)=\left\{\frac{\left({\mu }_{{\mathfrak{R}}_i}{\gamma }_{{\mathfrak{R}}_i}+1\right)+{\left({\mathcal{M}}_i\right)}^f-{\left({\mathcal{N}}_i\right)}^f+1}{4}\right\};f=\frac{p}{q}\ge 1\left(16\right)$$

Definition 3.2.2:

The fractional Diophantine fuzzy numbers (FDFNs) ${\mathfrak{R}}_i=\left(⟨{\mu }_{{\mathfrak{R}}_i},{\gamma }_{{\mathfrak{R}}_i}⟩,,,⟨{\mathcal{M}}_i,{\mathcal{N}}_i⟩\right)$ is a collection of FDFNs over non-empty set $\mathfrak{P}$, then the accuracy function (AcF) for any fractional Diophantine fuzzy number is a mapping such as: AcF $\left({\mathfrak{R}}_i\right)=FDFNs\left({\mathfrak{R}}_i\right)\to [\text{0,1}]$ and define as;$$\left({\mathfrak{R}}_i\right)=\left\{\frac{\left({\mu }_{{\mathfrak{R}}_i},-,{\gamma }_{{\mathfrak{R}}_i}\right)}{4},+,\frac{\left({\left({\mathcal{M}}_i\right)}^f,-,{\left({\mathcal{N}}_i\right)}^f\right)}{2}\right\};f=\frac{p}{q}\ge 1\left(17\right)$$

Definition 3.2.3:

We can easily compare two FDFNs by using the score and accuracy functions as follows:

1. If ScF $\left({\mathfrak{R}}_1\right)>$ ScF $\left({\mathfrak{R}}_2\right)$, then ${\mathfrak{R}}_1>{\mathfrak{R}}_2$;

2. If ScF $\left({\mathfrak{R}}_1\right)<$ ScF $\left({\mathfrak{R}}_2\right)$, then ${\mathfrak{R}}_1<{\mathfrak{R}}_2$;

3. If ScF $\left({\mathfrak{R}}_1\right)=$ ScF $\left({\mathfrak{R}}_2\right)$, then:

1. If AcF $\left({\mathfrak{R}}_1\right)>$ AcF $\left({\mathfrak{R}}_2\right)$, then ${\mathfrak{R}}_1>{\mathfrak{R}}_2$;

2. If AcF $\left({\mathfrak{R}}_1\right)<$ AcF $\left({\mathfrak{R}}_2\right)$, then ${\mathfrak{R}}_1<{\mathfrak{R}}_2$;

3. If AcF $\left({\mathfrak{R}}_1\right)=$ AcF $\left({\mathfrak{R}}_2\right)$, then ${\mathfrak{R}}_1\sim {\mathfrak{R}}_2$.

Definition 3.2.4:

Suppose $\mathfrak{P}$ is a fixed, non-empty set and ${\mathfrak{R}}_i=\left\{\left(⟨{\mu }_{{\mathfrak{R}}_i},{\gamma }_{{\mathfrak{R}}_i}⟩,,,⟨{\mathcal{M}}_i,{\mathcal{N}}_i⟩\right),:,i,=,\text{1,2}\right\}$, be any two FDFNs over $\mathfrak{P}$, then the distance between any two FDFNs define as follows;$$\mathbb{D}\left({\mathfrak{R}}_1,,,{\mathfrak{R}}_2\right)=\left\{\frac{\left|{\mu }_{{\mathfrak{R}}_1},-,{\mu }_{{\mathfrak{R}}_2}\right|+\left|{\gamma }_{{\mathfrak{R}}_1},-,{\gamma }_{{\mathfrak{R}}_2}\right|+\left|{\left({\mathcal{M}}_1\right)}^f,-,{\left({\mathcal{M}}_2\right)}^f\right|+\left|{\left({\mathcal{N}}_1\right)}^f,-,{\left({\mathcal{N}}_2\right)}^f\right|}{4}\right\};\left(18\right)$$

Hamacher operational laws for fractional diophantine fuzzy numbers

This portion discusses the derisible characteristics of fractional Diophantine fuzzy numbers and develops new Hamacher operational rules for them.

Definition 4.1:

Let ${\mathcal{P}}_1$ and ${\mathcal{P}}_2$ are any two real numbers, then the Hamacher t-norm and t-conorm [49] for any two real numbers define as follows;$$\mathcal{T}\left({\mathcal{P}}_1,,,{\mathcal{P}}_2\right)=\frac{{\mathcal{P}}_1{\mathcal{P}}_2}{\mathrm{\theta }+\left(1,-,\mathrm{\theta }\right).\left({\mathcal{P}}_1,+,{\mathcal{P}}_2,-,{\mathcal{P}}_1,{\mathcal{P}}_2\right)},\mathrm{\theta }\ge 1\left(19\right)$$$$\mathcal{S}\left({\mathcal{P}}_1,,,{\mathcal{P}}_2\right)=\frac{{\mathcal{P}}_1+{\mathcal{P}}_2-{\mathcal{P}}_1{\mathcal{P}}_2-\left(1,-,\mathrm{\theta }\right){\mathcal{P}}_1{\mathcal{P}}_2}{1-\left(1,-,\mathrm{\theta }\right){\mathcal{P}}_1{\mathcal{P}}_2},\mathrm{\theta }\ge 1\left(20\right)$$

The algebraic and Einstein t-norms and t-conorms are generalized by the Hamacher t-norm and t-conorm. By putting $\mathrm{\theta }=1$, we get algebraic t-norm and t-conorm [50], which is defined as follows:$$\mathcal{T}\left({\mathcal{P}}_1,,,{\mathcal{P}}_2\right)={\mathcal{P}}_1{\mathcal{P}}_2\left(21\right)$$$$\mathcal{S}\left({\mathcal{P}}_1,,,{\mathcal{P}}_2\right)={\mathcal{P}}_1+{\mathcal{P}}_2-{\mathcal{P}}_1{\mathcal{P}}_2\left(22\right)$$

By putting $\mathrm{\theta }=2$, we get Einstein t-norm and t-conorm [51], which is defined as follows:$$\mathcal{T}\left({\mathcal{P}}_1,,,{\mathcal{P}}_2\right)=\frac{{\mathcal{P}}_1{\mathcal{P}}_2}{1+\left(1,-,{\mathcal{P}}_1\right).\left(1,-,{\mathcal{P}}_2\right)}\left(23\right)$$$$\mathcal{S}\left({\mathcal{P}}_1,,,{\mathcal{P}}_2\right)=\frac{{\mathcal{P}}_1+{\mathcal{P}}_2}{1+{\mathcal{P}}_1{\mathcal{P}}_2}\left(24\right)$$

Using the concepts of the Hamacher t-norm and t-conorm, we define the following Hamacher operational laws for fractional Diophantine fuzzy numbers.

Definition 4.2:

Suppose $\mathfrak{P}$ is a fixed, non-empty set and ${\mathfrak{R}}_i=\left\{\left(⟨{\mu }_{{\mathfrak{R}}_i},{\gamma }_{{\mathfrak{R}}_i}⟩,,,⟨{\mathcal{M}}_i,{\mathcal{N}}_i⟩\right),:,i,=,\text{1,2}\right\}$, be any two FDFNs over $\mathfrak{P}$ and the scalar $\lambda >0$, then the Hamacher operational laws for fractional Diophantine fuzzy numbers is define as follows:

• ${{\mathfrak{R}}_1}^c=\left(⟨{\gamma }_{{\mathfrak{R}}_1},{\mu }_{{\mathfrak{R}}_1}⟩,,,⟨{{\mathcal{N}}_1}^f,{{\mathcal{M}}_1}^f⟩\right)$

• ${\mathfrak{R}}_1\otimes {\mathfrak{R}}_2=\left\{\begin{array}{c}\left(\frac{{\mu }_{{\mathfrak{R}}_1}{\mu }_{{\mathfrak{R}}_2}}{\mathrm{\theta }+\left(1,-,\mathrm{\theta }\right).\left({\mu }_{{\mathfrak{R}}_1},+,{\mu }_{{\mathfrak{R}}_2},-,{\mu }_{{\mathfrak{R}}_1},{\mu }_{{\mathfrak{R}}_2}\right)},,,\frac{{\gamma }_{{\mathfrak{R}}_1}+{\gamma }_{{\mathfrak{R}}_2}-{\gamma }_{{\mathfrak{R}}_1}{\gamma }_{{\mathfrak{R}}_2}-\left(1,-,\mathrm{\theta }\right){\gamma }_{{\mathfrak{R}}_1}{\gamma }_{{\mathfrak{R}}_2}}{1-\left(1,-,\mathrm{\theta }\right){\gamma }_{{\mathfrak{R}}_1}{\gamma }_{{\mathfrak{R}}_2}}\right),\\ \left(\sqrt[f]{\frac{{{\mathcal{M}}_1}^f{{\mathcal{M}}_2}^f}{\mathrm{\theta }+\left(1,-,\mathrm{\theta }\right).\left({{\mathcal{M}}_1}^f,+,{{\mathcal{M}}_2}^f,-,{{\mathcal{M}}_1}^f,{{\mathcal{M}}_2}^f\right)}},,,\sqrt[f]{\frac{{{\mathcal{N}}_1}^f+{{\mathcal{N}}_2}^f-{{\mathcal{N}}_1}^f{{\mathcal{N}}_2}^f-\left(1,-,\mathrm{\theta }\right){{\mathcal{N}}_1}^f{{\mathcal{N}}_2}^f}{1-\left(1,-,\mathrm{\theta }\right){{\mathcal{N}}_1}^f{{\mathcal{N}}_2}^f}}\right)\end{array}\right\}$

• ${\mathfrak{R}}_1\oplus {\mathfrak{R}}_2=\left\{\begin{array}{c}\left(\frac{{\mu }_{{\mathfrak{R}}_1}+{\mu }_{{\mathfrak{R}}_2}-{\mu }_{{\mathfrak{R}}_1}{\mu }_{{\mathfrak{R}}_2}-\left(1,-,\mathrm{\theta }\right){\mu }_{{\mathfrak{R}}_1}{\mu }_{{\mathfrak{R}}_2}}{1-\left(1,-,\mathrm{\theta }\right){\mu }_{{\mathfrak{R}}_1}{\mu }_{{\mathfrak{R}}_2}},,,\frac{{\gamma }_{{\mathfrak{R}}_1}{\gamma }_{{\mathfrak{R}}_2}}{\mathrm{\theta }+\left(1,-,\mathrm{\theta }\right).\left({\gamma }_{{\mathfrak{R}}_1},+,{\gamma }_{{\mathfrak{R}}_2},-,{\gamma }_{{\mathfrak{R}}_1},{\gamma }_{{\mathfrak{R}}_2}\right)}\right),\\ \left(\sqrt[f]{\frac{{{\mathcal{M}}_1}^f+{{\mathcal{M}}_2}^f-{{\mathcal{M}}_1}^f{{\mathcal{M}}_2}^f-\left(1,-,\mathrm{\theta }\right){{\mathcal{M}}_1}^f{{\mathcal{M}}_2}^f}{1-\left(1,-,\mathrm{\theta }\right){{\mathcal{M}}_1}^f{{\mathcal{M}}_2}^f}},,,\sqrt[f]{\frac{{{\mathcal{N}}_1}^f{{\mathcal{N}}_2}^f}{\mathrm{\theta }+\left(1,-,\mathrm{\theta }\right).\left({{\mathcal{N}}_1}^f,+,{{\mathcal{N}}_2}^f,-,{{\mathcal{N}}_1}^f,{{\mathcal{N}}_2}^f\right)}},,\right)\end{array}\right\}$

• $\lambda {\mathfrak{R}}_1=\left\{\begin{array}{c}\left(\frac{{\left(1,+,\left(\mathrm{\theta },-,1\right),{\mu }_{{\mathfrak{R}}_1}\right)}^{\lambda }-{\left(1,-,{\mu }_{{\mathfrak{R}}_1}\right)}^{\lambda }}{{\left(1,+,\left(\mathrm{\theta },-,1\right),{\mu }_{{\mathfrak{R}}_1}\right)}^{\lambda }+\left(\mathrm{\theta },-,1\right){\left(1,-,{\mu }_{{\mathfrak{R}}_1}\right)}^{\lambda }},,,\frac{\mathrm{\theta }{\left({\gamma }_{{\mathfrak{R}}_1}\right)}^{\lambda }}{{\left(1,+,\left(\mathrm{\theta },-,1\right),\left(1,-,{\gamma }_{{\mathfrak{R}}_1}\right)\right)}^{\lambda }+\left(\mathrm{\theta },-,1\right){\left({\gamma }_{{\mathfrak{R}}_1}\right)}^{\lambda }}\right),\\ \left(\sqrt[f]{\frac{{\left(1,+,\left(\mathrm{\theta },-,1\right),{{\mathcal{M}}_1}^f\right)}^{\lambda }-{\left(1,-,{{\mathcal{M}}_1}^f\right)}^{\lambda }}{{\left(1,+,\left(\mathrm{\theta },-,1\right),{{\mathcal{M}}_1}^f\right)}^{\lambda }+\left(\mathrm{\theta },-,1\right){\left(1,-,{{\mathcal{M}}_1}^f\right)}^{\lambda }}},,,\sqrt[f]{\frac{\mathrm{\theta }{\left({{\mathcal{N}}_1}^f\right)}^{\lambda }}{{\left(1,+,\left(\mathrm{\theta },-,1\right),\left(1,-,{{\mathcal{N}}_1}^f\right)\right)}^{\lambda }+\left(\mathrm{\theta },-,1\right){\left({{\mathcal{N}}_1}^f\right)}^{\lambda }}}\right)\end{array}\right\}$

• ${{\mathfrak{R}}_1}^{\lambda }=\left\{\begin{array}{c}\left(\frac{\mathrm{\theta }{\left({\mu }_{{\mathfrak{R}}_1}\right)}^{\lambda }}{{\left(1,+,\left(\mathrm{\theta },-,1\right),\left(1,-,{\mu }_{{\mathfrak{R}}_1}\right)\right)}^{\lambda }+\left(\mathrm{\theta },-,1\right){\left({\mu }_{{\mathfrak{R}}_1}\right)}^{\lambda }},,,\frac{{\left(1,+,\left(\mathrm{\theta },-,1\right),{\gamma }_{{\mathfrak{R}}_1}\right)}^{\lambda }-{\left(1,-,{\gamma }_{{\mathfrak{R}}_1}\right)}^{\lambda }}{{\left(1,+,\left(\mathrm{\theta },-,1\right),{\gamma }_{{\mathfrak{R}}_1}\right)}^{\lambda }+\left(\mathrm{\theta },-,1\right){\left(1,-,{\gamma }_{{\mathfrak{R}}_1}\right)}^{\lambda }}\right),\\ \left(\sqrt[f]{\frac{\mathrm{\theta }{\left({{\mathcal{M}}_1}^f\right)}^{\lambda }}{{\left(1,+,\left(\mathrm{\theta },-,1\right),\left(1,-,{{\mathcal{M}}_1}^f\right)\right)}^{\lambda }+\left(\mathrm{\theta },-,1\right){\left({{\mathcal{M}}_1}^f\right)}^{\lambda }}},,,\sqrt[f]{\frac{{\left(1,+,\left(\mathrm{\theta },-,1\right),{{\mathcal{N}}_1}^f\right)}^{\lambda }-{\left(1,-,{{\mathcal{N}}_1}^f\right)}^{\lambda }}{{\left(1,+,\left(\mathrm{\theta },-,1\right),{{\mathcal{N}}_1}^f\right)}^{\lambda }+\left(\mathrm{\theta },-,1\right){\left(1,-,{{\mathcal{N}}_1}^f\right)}^{\lambda }}}\right)\end{array}\right\}$

Using the concept of the Hamacher operational laws for fractional Diophantine fuzzy numbers, we define a series of fractional Diophantine fuzzy Hamacher weighted AOs in the next section.

Fractional diophantine fuzzy hamacher weighted aggregation (FDFHWA) operators

In this section, we define FDFHWA, FDFHOWA and FDFHHWA aggregation operators for fractional Diophantine fuzzy numbers.

Definition 5.1:

Suppose $\mathfrak{P}$ is a fixed, non-empty set and ${\mathfrak{R}}_i=\left\{\left(⟨{\mu }_{{\mathfrak{R}}_i},{\gamma }_{{\mathfrak{R}}_i}⟩,,,⟨{\mathcal{M}}_i,{\mathcal{N}}_i⟩\right),:,i,=,\text{1,2},,,3,,,\cdots ,,,n\right\}$, be a group of FDFNs over $\mathfrak{P}$, then the FDFHWA operator is a transformation from ${\mathrm{\Psi }}^n\to \mathrm{\Psi }$ and define as follows;

FDFHWA $\left({\mathfrak{R}}_1,,,{\mathfrak{R}}_2,,,{\mathfrak{R}}_3,,,\cdots ,,,{\mathfrak{R}}_n\right)={\oplus }_{i=1}^n\left({\mathbb{W}}_i,{\mathfrak{R}}_i\right)$$$=\left\{\begin{array}{c}\left(\begin{array}{c}\frac{{\prod }_{i=1}^n{\left(1,+,\left(\mathrm{\theta },-,1\right),{\mu }_{{\mathfrak{R}}_i}\right)}^{{\mathbb{W}}_i}-{\prod }_{i=1}^n{\left(1,-,{\mu }_{{\mathfrak{R}}_i}\right)}^{{\mathbb{W}}_i}}{{\prod }_{i=1}^n{\left(1,+,\left(\mathrm{\theta },-,1\right),{\mu }_{{\mathfrak{R}}_i}\right)}^{{\mathbb{W}}_i}+\left(\mathrm{\theta },-,1\right){\prod }_{i=1}^n{\left(1,-,{\mu }_{{\mathfrak{R}}_i}\right)}^{{\mathbb{W}}_i}},\\ \frac{\mathrm{\theta }{\prod }_{i=1}^n{\left({\gamma }_{{\mathfrak{R}}_i}\right)}^{{\mathbb{W}}_i}}{{\prod }_{i=1}^n{\left(1,+,\left(\mathrm{\theta },-,1\right),\left(1,-,{\gamma }_{{\mathfrak{R}}_i}\right)\right)}^{{\mathbb{W}}_i}+\left(\mathrm{\theta },-,1\right){\prod }_{i=1}^n{\left({\gamma }_{{\mathfrak{R}}_i}\right)}^{{\mathbb{W}}_i}}\end{array}\right),\\ \left(\begin{array}{c}\sqrt[f]{\frac{{\prod }_{i=1}^n{\left(1,+,\left(\mathrm{\theta },-,1\right),{{\mathcal{M}}_i}^f\right)}^{{\mathbb{W}}_i}-{\prod }_{i=1}^n{\left(1,-,{{\mathcal{M}}_i}^f\right)}^{{\mathbb{W}}_i}}{{\prod }_{i=1}^n{\left(1,+,\left(\mathrm{\theta },-,1\right),{{\mathcal{M}}_i}^f\right)}^{{\mathbb{W}}_i}+\left(\mathrm{\theta },-,1\right){\prod }_{i=1}^n{\left(1,-,{{\mathcal{M}}_i}^f\right)}^{{\mathbb{W}}_i}}},\\ \sqrt[f]{\frac{\mathrm{\theta }{\prod }_{i=1}^n{\left({{\mathcal{N}}_i}^f\right)}^{{\mathbb{W}}_i}}{{\prod }_{i=1}^n{\left(1,+,\left(\mathrm{\theta },-,1\right),\left(1,-,{{\mathcal{N}}_i}^f\right)\right)}^{{\mathbb{W}}_i}+\left(\mathrm{\theta },-,1\right){\prod }_{i=1}^n{\left({{\mathcal{N}}_i}^f\right)}^{{\mathbb{W}}_i}}}\end{array}\right)\end{array}\right\}\left(25\right)$$where $\mathrm{\theta }\ge 1$ and $\mathbb{W}=\left({\mathbb{w}}_1,,,{\mathbb{w}}_2,,,{\mathbb{w}}_3,,,\cdots ,{\mathbb{w}}_n\right)$ is the weight vector corresponding to the collection of FDFNs over $\mathfrak{P}$, and satisfy the given aspects ${\mathbb{W}}_i>0$ and ${\sum }_{i=1}^n{\mathbb{W}}_i=1$. Using the mathematical induction and FDFS operations we can easily proof this operator. In FDFHWA operator, ${\mu }_{{\mathfrak{R}}_i}$ denoted the MD and ${\gamma }_{{\mathfrak{R}}_i}$ denoted the NMD functions, ${\mathcal{M}}_i,{\mathcal{N}}_i$ denoted the RPs for both ${\mu }_{{\mathfrak{R}}_i}$ and ${\gamma }_{{\mathfrak{R}}_i}$ respectively.

Theorem 1

Suppose $\mathfrak{P}$ is a fixed, non-empty set and ${\mathfrak{R}}_i=\left\{\left(⟨{\mu }_{{\mathfrak{R}}_i},{\gamma }_{{\mathfrak{R}}_i}⟩,,,⟨{\mathcal{M}}_i,{\mathcal{N}}_i⟩\right),:,i,=,\text{1,2},,,3,,,\cdots ,,,n\right\}$ be a collection of FDFNs over $\mathfrak{P}$; then the number obtained by using the FDFHWA operator is again a FDFN.

FDFHWA $\left({\mathfrak{R}}_1,,,{\mathfrak{R}}_2,,,{\mathfrak{R}}_3,,,\cdots ,,,{\mathfrak{R}}_n\right)={\oplus }_{i=1}^n\left({\mathbb{W}}_i,{\mathfrak{R}}_i\right)$$$=\left\{\begin{array}{c}\left(\begin{array}{c}\frac{{\prod }_{i=1}^n{\left(1,+,\left(\mathrm{\theta },-,1\right),{\mu }_{{\mathfrak{R}}_i}\right)}^{{\mathbb{W}}_i}-{\prod }_{i=1}^n{\left(1,-,{\mu }_{{\mathfrak{R}}_i}\right)}^{{\mathbb{W}}_i}}{{\prod }_{i=1}^n{\left(1,+,\left(\mathrm{\theta },-,1\right),{\mu }_{{\mathfrak{R}}_i}\right)}^{{\mathbb{W}}_i}+\left(\mathrm{\theta },-,1\right){\prod }_{i=1}^n{\left(1,-,{\mu }_{{\mathfrak{R}}_i}\right)}^{{\mathbb{W}}_i}},\\ \frac{\mathrm{\theta }{\prod }_{i=1}^n{\left({\gamma }_{{\mathfrak{R}}_i}\right)}^{{\mathbb{W}}_i}}{{\prod }_{i=1}^n{\left(1,+,\left(\mathrm{\theta },-,1\right),\left(1,-,{\gamma }_{{\mathfrak{R}}_i}\right)\right)}^{{\mathbb{W}}_i}+\left(\mathrm{\theta },-,1\right){\prod }_{i=1}^n{\left({\gamma }_{{\mathfrak{R}}_i}\right)}^{{\mathbb{W}}_i}}\end{array}\right),\\ \left(\begin{array}{c}\sqrt[f]{\frac{{\prod }_{i=1}^n{\left(1,+,\left(\mathrm{\theta },-,1\right),{{\mathcal{M}}_i}^f\right)}^{{\mathbb{W}}_i}-{\prod }_{i=1}^n{\left(1,-,{{\mathcal{M}}_i}^f\right)}^{{\mathbb{W}}_i}}{{\prod }_{i=1}^n{\left(1,+,\left(\mathrm{\theta },-,1\right),{{\mathcal{M}}_i}^f\right)}^{{\mathbb{W}}_i}+\left(\mathrm{\theta },-,1\right){\prod }_{i=1}^n{\left(1,-,{{\mathcal{M}}_i}^f\right)}^{{\mathbb{W}}_i}}},\\ \sqrt[f]{\frac{\mathrm{\theta }{\prod }_{i=1}^n{\left({{\mathcal{N}}_i}^f\right)}^{{\mathbb{W}}_i}}{{\prod }_{i=1}^n{\left(1,+,\left(\mathrm{\theta },-,1\right),\left(1,-,{{\mathcal{N}}_i}^f\right)\right)}^{{\mathbb{W}}_i}+\left(\mathrm{\theta },-,1\right){\prod }_{i=1}^n{\left({{\mathcal{N}}_i}^f\right)}^{{\mathbb{W}}_i}}}\end{array}\right)\end{array}\right\}\left(26\right)$$where $\mathrm{\theta }\ge 1$ and $\mathbb{W}=\left({\mathbb{w}}_1,,,{\mathbb{w}}_2,,,{\mathbb{w}}_3,,,\cdots ,{\mathbb{w}}_n\right)$ is the weight vector corresponding to the collection of FDFNs over $\mathfrak{P}$, and satisfy the given aspects ${\mathbb{W}}_i>0$ and ${\sum }_{i=1}^n{\mathbb{W}}_i=1$. In FDFHWA operator, ${\mu }_{{\mathfrak{R}}_i}$ denoted the MD and ${\gamma }_{{\mathfrak{R}}_i}$ denoted the DNM functions, ${\mathcal{M}}_i,{\mathcal{N}}_i$ denoted the RPs for both ${\mu }_{{\mathfrak{R}}_i}$ and ${\gamma }_{{\mathfrak{R}}_i}$ respectively.

Proof 1

Using the mathematical induction and FDFS operations we can easily proof this operator.

1. Firstly, we proof this result for $i=2$ then;

   FDFHWA $\left({\mathfrak{R}}_1,,,{\mathfrak{R}}_2\right)={\oplus }_{i=1}^2\left({\mathbb{W}}_i,{\mathfrak{R}}_i\right)$$$=\left[\left\{\begin{array}{c}\left(\begin{array}{c}\frac{{\left(1,+,\left(\mathrm{\theta },-,1\right),{\mu }_{{\mathfrak{R}}_1}\right)}^{{\mathbb{W}}_1}-{\left(1,-,{\mu }_{{\mathfrak{R}}_1}\right)}^{{\mathbb{W}}_1}}{{\left(1,+,\left(\mathrm{\theta },-,1\right),{\mu }_{{\mathfrak{R}}_1}\right)}^{{\mathbb{W}}_1}+\left(\mathrm{\theta },-,1\right){\left(1,-,{\mu }_{{\mathfrak{R}}_1}\right)}^{{\mathbb{W}}_1}},\\ \frac{\mathrm{\theta }{\left({\gamma }_{{\mathfrak{R}}_1}\right)}^{{\mathbb{W}}_1}}{{\left(1,+,\left(\mathrm{\theta },-,1\right),\left(1,-,{\gamma }_{{\mathfrak{R}}_1}\right)\right)}^{{\mathbb{W}}_1}+\left(\mathrm{\theta },-,1\right){\left({\gamma }_{{\mathfrak{R}}_1}\right)}^{{\mathbb{W}}_1}}\end{array}\right),\\ \left(\begin{array}{c}\sqrt[f]{\frac{{\left(1,+,\left(\mathrm{\theta },-,1\right),{{\mathcal{M}}_1}^f\right)}^{{\mathbb{W}}_1}-{\left(1,-,{{\mathcal{M}}_1}^f\right)}^{{\mathbb{W}}_1}}{{\left(1,+,\left(\mathrm{\theta },-,1\right),{{\mathcal{M}}_1}^f\right)}^{{\mathbb{W}}_1}+\left(\mathrm{\theta },-,1\right){\left(1,-,{{\mathcal{M}}_1}^f\right)}^{{\mathbb{W}}_1}}},\\ \sqrt[f]{\frac{\mathrm{\theta }{\left({{\mathcal{N}}_1}^f\right)}^{{\mathbb{W}}_1}}{{\left(1,+,\left(\mathrm{\theta },-,1\right),\left(1,-,{{\mathcal{N}}_1}^f\right)\right)}^{{\mathbb{W}}_1}+\left(\mathrm{\theta },-,1\right){\left({{\mathcal{N}}_1}^f\right)}^{{\mathbb{W}}_1}}}\end{array}\right)\end{array}\right\},\oplus ,\left\{\begin{array}{c}\left(\begin{array}{c}\frac{{\left(1,+,\left(\mathrm{\theta },-,1\right),{\mu }_{{\mathfrak{R}}_2}\right)}^{{\mathbb{W}}_2}-{\left(1,-,{\mu }_{{\mathfrak{R}}_2}\right)}^{{\mathbb{W}}_2}}{{\left(1,+,\left(\mathrm{\theta },-,1\right),{\mu }_{{\mathfrak{R}}_2}\right)}^{{\mathbb{W}}_2}+\left(\mathrm{\theta },-,1\right){\left(1,-,{\mu }_{{\mathfrak{R}}_2}\right)}^{{\mathbb{W}}_2}},\\ \frac{\mathrm{\theta }{\left({\gamma }_{{\mathfrak{R}}_2}\right)}^{{\mathbb{W}}_2}}{{\left(1,+,\left(\mathrm{\theta },-,1\right),\left(1,-,{\gamma }_{{\mathfrak{R}}_2}\right)\right)}^{{\mathbb{W}}_2}+\left(\mathrm{\theta },-,1\right){\left({\gamma }_{{\mathfrak{R}}_2}\right)}^{{\mathbb{W}}_2}}\end{array}\right),\\ \left(\begin{array}{c}\sqrt[f]{\frac{{\left(1,+,\left(\mathrm{\theta },-,1\right),{{\mathcal{M}}_2}^f\right)}^{{\mathbb{W}}_2}-{\left(1,-,{{\mathcal{M}}_2}^f\right)}^{{\mathbb{W}}_2}}{{\left(1,+,\left(\mathrm{\theta },-,1\right),{{\mathcal{M}}_2}^f\right)}^{{\mathbb{W}}_2}+\left(\mathrm{\theta },-,1\right){\left(1,-,{{\mathcal{M}}_2}^f\right)}^{{\mathbb{W}}_2}}},\\ \sqrt[f]{\frac{\mathrm{\theta }{\left({{\mathcal{N}}_2}^f\right)}^{{\mathbb{W}}_2}}{{\left(1,+,\left(\mathrm{\theta },-,1\right),\left(1,-,{{\mathcal{N}}_2}^f\right)\right)}^{{\mathbb{W}}_2}+\left(\mathrm{\theta },-,1\right){\left({{\mathcal{N}}_2}^f\right)}^{{\mathbb{W}}_2}}}\end{array}\right)\end{array}\right\}\right]$$$$=\left\{\begin{array}{c}\left(\begin{array}{c}\frac{{\prod }_{i=1}^2{\left(1,+,\left(\mathrm{\theta },-,1\right),{\mu }_{{\mathfrak{R}}_i}\right)}^{{\mathbb{W}}_i}-{\prod }_{i=1}^2{\left(1,-,{\mu }_{{\mathfrak{R}}_i}\right)}^{{\mathbb{W}}_i}}{{\prod }_{i=1}^2{\left(1,+,\left(\mathrm{\theta },-,1\right),{\mu }_{{\mathfrak{R}}_i}\right)}^{{\mathbb{W}}_i}+\left(\mathrm{\theta },-,1\right){\prod }_{i=1}^2{\left(1,-,{\mu }_{{\mathfrak{R}}_i}\right)}^{{\mathbb{W}}_i}},\\ \frac{\mathrm{\theta }{\prod }_{i=1}^2{\left({\gamma }_{{\mathfrak{R}}_i}\right)}^{{\mathbb{W}}_i}}{{\prod }_{i=1}^2{\left(1,+,\left(\mathrm{\theta },-,1\right),\left(1,-,{\gamma }_{{\mathfrak{R}}_i}\right)\right)}^{{\mathbb{W}}_i}+\left(\mathrm{\theta },-,1\right){\prod }_{i=1}^2{\left({\gamma }_{{\mathfrak{R}}_i}\right)}^{{\mathbb{W}}_i}}\end{array}\right),\\ \left(\begin{array}{c}\sqrt[f]{\frac{{\prod }_{i=1}^2{\left(1,+,\left(\mathrm{\theta },-,1\right),{{\mathcal{M}}_i}^f\right)}^{{\mathbb{W}}_i}-{\prod }_{i=1}^2{\left(1,-,{{\mathcal{M}}_i}^f\right)}^{{\mathbb{W}}_i}}{{\prod }_{i=1}^2{\left(1,+,\left(\mathrm{\theta },-,1\right),{{\mathcal{M}}_i}^f\right)}^{{\mathbb{W}}_i}+\left(\mathrm{\theta },-,1\right){\prod }_{i=1}^2{\left(1,-,{{\mathcal{M}}_i}^f\right)}^{{\mathbb{W}}_i}}},\\ \sqrt[f]{\frac{\mathrm{\theta }{\prod }_{i=1}^2{\left({{\mathcal{N}}_i}^f\right)}^{{\mathbb{W}}_i}}{{\prod }_{i=1}^2{\left(1,+,\left(\mathrm{\theta },-,1\right),\left(1,-,{{\mathcal{N}}_i}^f\right)\right)}^{{\mathbb{W}}_i}+\left(\mathrm{\theta },-,1\right){\prod }_{i=1}^2{\left({{\mathcal{N}}_i}^f\right)}^{{\mathbb{W}}_i}}}\end{array}\right)\end{array}\right\}$$

   Hence the result true for $i=2$.

2. Now we assume that the result true for $i=k$, then;

   FDFHWA $\left({\mathfrak{R}}_1,,,{\mathfrak{R}}_2,,,{\mathfrak{R}}_3,,,{\mathfrak{R}}_4,\cdots ,,,{\mathfrak{R}}_k\right)={\oplus }_{i=1}^k\left({\mathbb{W}}_i,{\mathfrak{R}}_i\right)$$$=\left\{\begin{array}{c}\left(\begin{array}{c}\frac{{\prod }_{i=1}^k{\left(1,+,\left(\mathrm{\theta },-,1\right),{\mu }_{{\mathfrak{R}}_i}\right)}^{{\mathbb{W}}_i}-{\prod }_{i=1}^k{\left(1,-,{\mu }_{{\mathfrak{R}}_i}\right)}^{{\mathbb{W}}_i}}{{\prod }_{i=1}^k{\left(1,+,\left(\mathrm{\theta },-,1\right),{\mu }_{{\mathfrak{R}}_i}\right)}^{{\mathbb{W}}_i}+\left(\mathrm{\theta },-,1\right){\prod }_{i=1}^k{\left(1,-,{\mu }_{{\mathfrak{R}}_i}\right)}^{{\mathbb{W}}_i}},\\ \frac{\mathrm{\theta }{\prod }_{i=1}^k{\left({\gamma }_{{\mathfrak{R}}_i}\right)}^{{\mathbb{W}}_i}}{{\prod }_{i=1}^k{\left(1,+,\left(\mathrm{\theta },-,1\right),\left(1,-,{\gamma }_{{\mathfrak{R}}_i}\right)\right)}^{{\mathbb{W}}_i}+\left(\mathrm{\theta },-,1\right){\prod }_{i=1}^k{\left({\gamma }_{{\mathfrak{R}}_i}\right)}^{{\mathbb{W}}_i}}\end{array}\right),\\ \left(\begin{array}{c}\sqrt[f]{\frac{{\prod }_{i=1}^k{\left(1,+,\left(\mathrm{\theta },-,1\right),{{\mathcal{M}}_i}^f\right)}^{{\mathbb{W}}_i}-{\prod }_{i=1}^k{\left(1,-,{{\mathcal{M}}_i}^f\right)}^{{\mathbb{W}}_i}}{{\prod }_{i=1}^k{\left(1,+,\left(\mathrm{\theta },-,1\right),{{\mathcal{M}}_i}^f\right)}^{{\mathbb{W}}_i}+\left(\mathrm{\theta },-,1\right){\prod }_{i=1}^k{\left(1,-,{{\mathcal{M}}_i}^f\right)}^{{\mathbb{W}}_i}}},\\ \sqrt[f]{\frac{\mathrm{\theta }{\prod }_{i=1}^k{\left({{\mathcal{N}}_i}^f\right)}^{{\mathbb{W}}_i}}{{\prod }_{i=1}^k{\left(1,+,\left(\mathrm{\theta },-,1\right),\left(1,-,{{\mathcal{N}}_i}^f\right)\right)}^{{\mathbb{W}}_i}+\left(\mathrm{\theta },-,1\right){\prod }_{i=1}^k{\left({{\mathcal{N}}_i}^f\right)}^{{\mathbb{W}}_i}}}\end{array}\right)\end{array}\right\}$$

3. Now we show that the result true for $i=k+1$, then;

   FDFHWA $\left({\mathfrak{R}}_1,,,{\mathfrak{R}}_2,,,{\mathfrak{R}}_3,,,{\mathfrak{R}}_4,\cdots ,,,{\mathfrak{R}}_{k+1}\right)={\oplus }_{i=1}^{k+1}\left({\mathbb{W}}_i,{\mathfrak{R}}_i\right)$$$=\left\{\begin{array}{c}\left(\begin{array}{c}\frac{{\prod }_{i=1}^k{\left(1,+,\left(\mathrm{\theta },-,1\right),{\mu }_{{\mathfrak{R}}_i}\right)}^{{\mathbb{W}}_i}-{\prod }_{i=1}^k{\left(1,-,{\mu }_{{\mathfrak{R}}_i}\right)}^{{\mathbb{W}}_i}}{{\prod }_{i=1}^k{\left(1,+,\left(\mathrm{\theta },-,1\right),{\mu }_{{\mathfrak{R}}_i}\right)}^{{\mathbb{W}}_i}+\left(\mathrm{\theta },-,1\right){\prod }_{i=1}^k{\left(1,-,{\mu }_{{\mathfrak{R}}_i}\right)}^{{\mathbb{W}}_i}},\\ \frac{\mathrm{\theta }{\prod }_{i=1}^k{\left({\gamma }_{{\mathfrak{R}}_i}\right)}^{{\mathbb{W}}_i}}{{\prod }_{i=1}^k{\left(1,+,\left(\mathrm{\theta },-,1\right),\left(1,-,{\gamma }_{{\mathfrak{R}}_i}\right)\right)}^{{\mathbb{W}}_i}+\left(\mathrm{\theta },-,1\right){\prod }_{i=1}^k{\left({\gamma }_{{\mathfrak{R}}_i}\right)}^{{\mathbb{W}}_i}}\end{array}\right),\\ \left(\begin{array}{c}\sqrt[f]{\frac{{\prod }_{i=1}^k{\left(1,+,\left(\mathrm{\theta },-,1\right),{{\mathcal{M}}_i}^f\right)}^{{\mathbb{W}}_i}-{\prod }_{i=1}^k{\left(1,-,{{\mathcal{M}}_i}^f\right)}^{{\mathbb{W}}_i}}{{\prod }_{i=1}^k{\left(1,+,\left(\mathrm{\theta },-,1\right),{{\mathcal{M}}_i}^f\right)}^{{\mathbb{W}}_i}+\left(\mathrm{\theta },-,1\right){\prod }_{i=1}^k{\left(1,-,{{\mathcal{M}}_i}^f\right)}^{{\mathbb{W}}_i}}},\\ \sqrt[f]{\frac{\mathrm{\theta }{\prod }_{i=1}^k{\left({{\mathcal{N}}_i}^f\right)}^{{\mathbb{W}}_i}}{{\prod }_{i=1}^k{\left(1,+,\left(\mathrm{\theta },-,1\right),\left(1,-,{{\mathcal{N}}_i}^f\right)\right)}^{{\mathbb{W}}_i}+\left(\mathrm{\theta },-,1\right){\prod }_{i=1}^k{\left({{\mathcal{N}}_i}^f\right)}^{{\mathbb{W}}_i}}}\end{array}\right)\end{array}\right\}$$$$=\left[\left\{\begin{array}{c}\left(\begin{array}{c}\frac{{\prod }_{i=1}^k{\left(1,+,\left(\mathrm{\theta },-,1\right),{\mu }_{{\mathfrak{R}}_i}\right)}^{{\mathbb{W}}_i}-{\prod }_{i=1}^k{\left(1,-,{\mu }_{{\mathfrak{R}}_i}\right)}^{{\mathbb{W}}_i}}{{\prod }_{i=1}^k{\left(1,+,\left(\mathrm{\theta },-,1\right),{\mu }_{{\mathfrak{R}}_i}\right)}^{{\mathbb{W}}_i}+\left(\mathrm{\theta },-,1\right){\prod }_{i=1}^k{\left(1,-,{\mu }_{{\mathfrak{R}}_i}\right)}^{{\mathbb{W}}_i}},\\ \frac{\mathrm{\theta }{\prod }_{i=1}^k{\left({\gamma }_{{\mathfrak{R}}_i}\right)}^{{\mathbb{W}}_i}}{{\prod }_{i=1}^k{\left(1,+,\left(\mathrm{\theta },-,1\right),\left(1,-,{\gamma }_{{\mathfrak{R}}_i}\right)\right)}^{{\mathbb{W}}_i}+\left(\mathrm{\theta },-,1\right){\prod }_{i=1}^k{\left({\gamma }_{{\mathfrak{R}}_i}\right)}^{{\mathbb{W}}_i}}\end{array}\right),\\ \left(\begin{array}{c}\sqrt[f]{\frac{{\prod }_{i=1}^k{\left(1,+,\left(\mathrm{\theta },-,1\right),{{\mathcal{M}}_i}^f\right)}^{{\mathbb{W}}_i}-{\prod }_{i=1}^k{\left(1,-,{{\mathcal{M}}_i}^f\right)}^{{\mathbb{W}}_i}}{{\prod }_{i=1}^k{\left(1,+,\left(\mathrm{\theta },-,1\right),{{\mathcal{M}}_i}^f\right)}^{{\mathbb{W}}_i}+\left(\mathrm{\theta },-,1\right){\prod }_{i=1}^k{\left(1,-,{{\mathcal{M}}_i}^f\right)}^{{\mathbb{W}}_i}}},\\ \sqrt[f]{\frac{\mathrm{\theta }{\prod }_{i=1}^k{\left({{\mathcal{N}}_i}^f\right)}^{{\mathbb{W}}_i}}{{\prod }_{i=1}^k{\left(1,+,\left(\mathrm{\theta },-,1\right),\left(1,-,{{\mathcal{N}}_i}^f\right)\right)}^{{\mathbb{W}}_i}+\left(\mathrm{\theta },-,1\right){\prod }_{i=1}^k{\left({{\mathcal{N}}_i}^f\right)}^{{\mathbb{W}}_i}}}\end{array}\right)\end{array}\right\},\oplus ,\left\{\begin{array}{c}\left(\begin{array}{c}\frac{{\left(1,+,\left(\mathrm{\theta },-,1\right),{\mu }_{{\mathfrak{R}}_{k+1}}\right)}^{{\mathbb{W}}_{k+1}}-{\left(1,-,{\mu }_{{\mathfrak{R}}_{k+1}}\right)}^{{\mathbb{W}}_{k+1}}}{{\left(1,+,\left(\mathrm{\theta },-,1\right),{\mu }_{{\mathfrak{R}}_{k+1}}\right)}^{{\mathbb{W}}_{k+1}}+\left(\mathrm{\theta },-,1\right){\left(1,-,{\mu }_{{\mathfrak{R}}_{k+1}}\right)}^{{\mathbb{W}}_{k+1}}},\\ \frac{\mathrm{\theta }{\left({\gamma }_{{\mathfrak{R}}_{k+1}}\right)}^{{\mathbb{W}}_{k+1}}}{{\left(1,+,\left(\mathrm{\theta },-,1\right),\left(1,-,{\gamma }_{{\mathfrak{R}}_{k+1}}\right)\right)}^{{\mathbb{W}}_{k+1}}+\left(\mathrm{\theta },-,1\right){\left({\gamma }_{{\mathfrak{R}}_{k+1}}\right)}^{{\mathbb{W}}_{k+1}}}\end{array}\right),\\ \left(\begin{array}{c}\sqrt[f]{\frac{{\left(1,+,\left(\mathrm{\theta },-,1\right),{{\mathcal{M}}_{k+1}}^f\right)}^{{\mathbb{W}}_{k+1}}-{\left(1,-,{{\mathcal{M}}_{k+1}}^f\right)}^{{\mathbb{W}}_{k+1}}}{{\left(1,+,\left(\mathrm{\theta },-,1\right),{{\mathcal{M}}_{k+1}}^f\right)}^{{\mathbb{W}}_{k+1}}+\left(\mathrm{\theta },-,1\right){\left(1,-,{{\mathcal{M}}_{k+1}}^f\right)}^{{\mathbb{W}}_{k+1}}}},\\ \sqrt[f]{\frac{\mathrm{\theta }{\left({{\mathcal{N}}_{k+1}}^f\right)}^{{\mathbb{W}}_{k+1}}}{{\left(1,+,\left(\mathrm{\theta },-,1\right),\left(1,-,{{\mathcal{N}}_{k+1}}^f\right)\right)}^{{\mathbb{W}}_{k+1}}+\left(\mathrm{\theta },-,1\right){\left({{\mathcal{N}}_{k+1}}^f\right)}^{{\mathbb{W}}_{k+1}}}}\end{array}\right)\end{array}\right\}\right]$$$$=\left\{\begin{array}{c}\left(\begin{array}{c}\frac{{\prod }_{i=1}^{k+1}{\left(1,+,\left(\mathrm{\theta },-,1\right),{\mu }_{{\mathfrak{R}}_i}\right)}^{{\mathbb{W}}_i}-{\prod }_{i=1}^{k+1}{\left(1,-,{\mu }_{{\mathfrak{R}}_i}\right)}^{{\mathbb{W}}_i}}{{\prod }_{i=1}^{k+1}{\left(1,+,\left(\mathrm{\theta },-,1\right),{\mu }_{{\mathfrak{R}}_i}\right)}^{{\mathbb{W}}_i}+\left(\mathrm{\theta },-,1\right){\prod }_{i=1}^{k+1}{\left(1,-,{\mu }_{{\mathfrak{R}}_i}\right)}^{{\mathbb{W}}_i}},\\ \frac{\mathrm{\theta }{\prod }_{i=1}^{k+1}{\left({\gamma }_{{\mathfrak{R}}_i}\right)}^{{\mathbb{W}}_i}}{{\prod }_{i=1}^{k+1}{\left(1,+,\left(\mathrm{\theta },-,1\right),\left(1,-,{\gamma }_{{\mathfrak{R}}_i}\right)\right)}^{{\mathbb{W}}_i}+\left(\mathrm{\theta },-,1\right){\prod }_{i=1}^{k+1}{\left({\gamma }_{{\mathfrak{R}}_i}\right)}^{{\mathbb{W}}_i}}\end{array}\right),\\ \left(\begin{array}{c}\sqrt[f]{\frac{{\prod }_{i=1}^{k+1}{\left(1,+,\left(\mathrm{\theta },-,1\right),{{\mathcal{M}}_i}^f\right)}^{{\mathbb{W}}_i}-{\prod }_{i=1}^{k+1}{\left(1,-,{{\mathcal{M}}_i}^f\right)}^{{\mathbb{W}}_i}}{{\prod }_{i=1}^{k+1}{\left(1,+,\left(\mathrm{\theta },-,1\right),{{\mathcal{M}}_i}^f\right)}^{{\mathbb{W}}_i}+\left(\mathrm{\theta },-,1\right){\prod }_{i=1}^{k+1}{\left(1,-,{{\mathcal{M}}_i}^f\right)}^{{\mathbb{W}}_i}}},\\ \sqrt[f]{\frac{\mathrm{\theta }{\prod }_{i=1}^{k+1}{\left({{\mathcal{N}}_i}^f\right)}^{{\mathbb{W}}_i}}{{\prod }_{i=1}^{k+1}{\left(1,+,\left(\mathrm{\theta },-,1\right),\left(1,-,{{\mathcal{N}}_i}^f\right)\right)}^{{\mathbb{W}}_i}+\left(\mathrm{\theta },-,1\right){\prod }_{i=1}^{k+1}{\left({{\mathcal{N}}_i}^f\right)}^{{\mathbb{W}}_i}}}\end{array}\right)\end{array}\right\}$$

Thus, the outcome is valid for $i=k+1$, this completes the proof.

Theorem 2 (Idempotency).

Let $\mathfrak{P}$ be a fixed set that is not empty and ${\mathfrak{R}}_i=\left\{\left(⟨{\mu }_{{\mathfrak{R}}_i},{\gamma }_{{\mathfrak{R}}_i}⟩,,,⟨{\mathcal{M}}_i,{\mathcal{N}}_i⟩\right),:,i,=,\text{1,2},,,3,,,\cdots ,,,n\right\}$ is a set of FDFNs over $\mathfrak{P}$ which all are same i.e. ${\mathfrak{R}}_i=\mathfrak{R}$ then;$$\left({\mathfrak{R}}_1,,,{\mathfrak{R}}_2,,,{\mathfrak{R}}_3,,,\cdots ,,,{\mathfrak{R}}_n\right)=\mathfrak{R}\left(27\right)$$

Theorem 3 (boundedness).

Let $\mathfrak{P}$ be a fixed set that is not empty and ${\mathfrak{R}}_i=\left\{\left(⟨{\mu }_{{\mathfrak{R}}_i},{\gamma }_{{\mathfrak{R}}_i}⟩,,,⟨{\mathcal{M}}_i,{\mathcal{N}}_i⟩\right),:,i,=,\text{1,2},,,3,,,\cdots ,,,n\right\}$ is a set of FDFNs over $\mathfrak{P}$ and ${{\mathfrak{R}}^{-}}_i=min\left(⟨{\mu }_{{\mathfrak{R}}_i},{\gamma }_{{\mathfrak{R}}_i}⟩,,,⟨{\mathcal{M}}_i,{\mathcal{N}}_i⟩\right)$, also ${{\mathfrak{R}}^{+}}_i=max\left(⟨{\mu }_{{\mathfrak{R}}_i},{\gamma }_{{\mathfrak{R}}_i}⟩,,,⟨{\mathcal{M}}_i,{\mathcal{N}}_i⟩\right)$ then;$${{\mathfrak{R}}^{-}}_i\le \text{FDFHWA}\left({\mathfrak{R}}_1,,,{\mathfrak{R}}_2,,,{\mathfrak{R}}_3,,,\cdots ,,,{\mathfrak{R}}_n\right)\le {{\mathfrak{R}}^{+}}_i\left(28\right)$$

Theorem 3 (Monotonicity).

Let $\mathfrak{P}$ be a fixed set that is not empty and ${\mathfrak{R}}_i=\left\{\left(⟨{\mu }_{{\mathfrak{R}}_i},{\gamma }_{{\mathfrak{R}}_i}⟩,,,⟨{\mathcal{M}}_i,{\mathcal{N}}_i⟩\right),:,i,=,\text{1,2},,,3,,,\cdots ,,,n\right\}$ is a set of FDFNs over $\mathfrak{P}$ and $\mathfrak{R}_{}^{\ast }_i^{}\ge {\mathfrak{R}}_i$ then;$$\text{FDFHWA}\left(\mathfrak{R}_{}^{\ast }_1^{},,,\mathfrak{R}_{}^{\ast }_2^{},,,\mathfrak{R}_{}^{\ast }_3^{},,,\cdots ,,,\mathfrak{R}_{}^{\ast }_n^{}\right)\ge \text{FDFHWA}\left({\mathfrak{R}}_1,,,{\mathfrak{R}}_2,,,{\mathfrak{R}}_3,,,\cdots ,,,{\mathfrak{R}}_n\right)\left(29\right)$$

Definition 5.2:

Suppose $\mathfrak{P}$ is a fixed, non-empty set and ${\mathfrak{R}}_i=\left\{\left(⟨{\mu }_{{\mathfrak{R}}_i},{\gamma }_{{\mathfrak{R}}_i}⟩,,,⟨{\mathcal{M}}_i,{\mathcal{N}}_i⟩\right),:,i,=,\text{1,2},,,3,,,\cdots ,,,n\right\}$, is a set of FDFNs over $\mathfrak{P}$, then the FDFHOWA operator is a transformation from ${\mathrm{\Psi }}^n\to \mathrm{\Psi }$ and define as follows;

FDFHOWA $\left({\mathfrak{R}}_1,,,{\mathfrak{R}}_2,,,{\mathfrak{R}}_3,,,\cdots ,,,{\mathfrak{R}}_n\right)={\oplus }_{i=1}^n\left({\mathbb{W}}_i,{\mathfrak{R}}_{\phi (i)}\right)$$$=\left\{\begin{array}{c}\left(\begin{array}{c}\frac{{\prod }_{i=1}^n{\left(1,+,\left(\mathrm{\theta },-,1\right),{\mu }_{{\mathfrak{R}}_{\phi (i)}}\right)}^{{\mathbb{W}}_i}-{\prod }_{i=1}^n{\left(1,-,{\mu }_{{\mathfrak{R}}_{\phi (i)}}\right)}^{{\mathbb{W}}_i}}{{\prod }_{i=1}^n{\left(1,+,\left(\mathrm{\theta },-,1\right),{\mu }_{{\mathfrak{R}}_{\phi (i)}}\right)}^{{\mathbb{W}}_i}+\left(\mathrm{\theta },-,1\right){\prod }_{i=1}^n{\left(1,-,{\mu }_{{\mathfrak{R}}_{\phi (i)}}\right)}^{{\mathbb{W}}_i}},\\ \frac{\mathrm{\theta }{\prod }_{i=1}^n{\left({\gamma }_{{\mathfrak{R}}_{\phi (i)}}\right)}^{{\mathbb{W}}_i}}{{\prod }_{i=1}^n{\left(1,+,\left(\mathrm{\theta },-,1\right),\left(1,-,{\gamma }_{{\mathfrak{R}}_{\phi (i)}}\right)\right)}^{{\mathbb{W}}_i}+\left(\mathrm{\theta },-,1\right){\prod }_{i=1}^n{\left({\gamma }_{{\mathfrak{R}}_{\phi (i)}}\right)}^{{\mathbb{W}}_i}}\end{array}\right),\\ \left(\begin{array}{c}\sqrt[f]{\frac{{\prod }_{i=1}^n{\left(1,+,\left(\mathrm{\theta },-,1\right),{{\mathcal{M}}_{\phi (i)}}^f\right)}^{{\mathbb{W}}_i}-{\prod }_{i=1}^n{\left(1,-,{{\mathcal{M}}_{\phi (i)}}^f\right)}^{{\mathbb{W}}_i}}{{\prod }_{i=1}^n{\left(1,+,\left(\mathrm{\theta },-,1\right),{{\mathcal{M}}_{\phi (i)}}^f\right)}^{{\mathbb{W}}_i}+\left(\mathrm{\theta },-,1\right){\prod }_{i=1}^n{\left(1,-,{{\mathcal{M}}_{\phi (i)}}^f\right)}^{{\mathbb{W}}_i}}},\\ \sqrt[f]{\frac{\mathrm{\theta }{\prod }_{i=1}^n{\left({{\mathcal{N}}_{\phi (i)}}^f\right)}^{{\mathbb{W}}_i}}{{\prod }_{i=1}^n{\left(1,+,\left(\mathrm{\theta },-,1\right),\left(1,-,{{\mathcal{N}}_{\phi (i)}}^f\right)\right)}^{{\mathbb{W}}_i}+\left(\mathrm{\theta },-,1\right){\prod }_{i=1}^n{\left({{\mathcal{N}}_{\phi (i)}}^f\right)}^{{\mathbb{W}}_i}}}\end{array}\right)\end{array}\right\}\left(30\right)$$where $\mathrm{\theta }\ge 1$ and $\mathbb{W}=\left({\mathbb{w}}_1,,,{\mathbb{w}}_2,,,{\mathbb{w}}_3,,,\cdots ,{\mathbb{w}}_n\right)$ is the weight vector corresponding to the collection of FDFNs over $\mathfrak{P}$, and satisfy the given aspects ${\mathbb{W}}_i>0$ and ${\sum }_{i=1}^n{\mathbb{W}}_i=1$. $\phi (1),\phi \left(2\right),\phi \left(3\right),\cdots ,\phi \left(n\right)$, is the permutation corresponding to the collection of FDFNs over ℬ, and satisfy ${\mathfrak{D}}_{\delta \left(1,-,i\right)}\ge {\mathfrak{D}}_{\delta \left(i\right)}\forall i=\text{1,2},3\cdots n$. Using the mathematical induction and FDFS operations we can easily proof this operator. We can easily proofs all the properties given in Eqs. 27, 28, and 29 for the FDFHOWA operators.

Definition 5.3:

Suppose $\mathfrak{P}$ is a fixed, non-empty set and ${\mathfrak{R}}_i=\left\{\left(⟨{\mu }_{{\mathfrak{R}}_i},{\gamma }_{{\mathfrak{R}}_i}⟩,,,⟨{\mathcal{M}}_i,{\mathcal{N}}_i⟩\right),:,i,=,\text{1,2},,,3,,,\cdots ,,,n\right\}$, is a set of FDFNs over $\mathfrak{P}$, then the FDFHOWA operator is a transformation from ${\mathrm{\Psi }}^n\to \mathrm{\Psi }$ and define as follows;

FDFHOWA $\left({\mathfrak{R}}_1,,,{\mathfrak{R}}_2,,,{\mathfrak{R}}_3,,,\cdots ,,,{\mathfrak{R}}_n\right)={\oplus }_{i=1}^n\left({\mathbb{W}}_i,{\widehat{\mathfrak{R}}}_{\phi (i)}\right)$$$=\left\{\begin{array}{c}\left(\begin{array}{c}\frac{{\prod }_{i=1}^n{\left(1,+,\left(\mathrm{\theta },-,1\right),{\mu }_{{\widehat{\mathfrak{R}}}_{\phi (i)}}\right)}^{{\mathbb{W}}_i}-{\prod }_{i=1}^n{\left(1,-,{\mu }_{{\widehat{\mathfrak{R}}}_{\phi (i)}}\right)}^{{\mathbb{W}}_i}}{{\prod }_{i=1}^n{\left(1,+,\left(\mathrm{\theta },-,1\right),{\mu }_{{\widehat{\mathfrak{R}}}_{\phi (i)}}\right)}^{{\mathbb{W}}_i}+\left(\mathrm{\theta },-,1\right){\prod }_{i=1}^n{\left(1,-,{\mu }_{{\widehat{\mathfrak{R}}}_{\phi (i)}}\right)}^{{\mathbb{W}}_i}},\\ \frac{\mathrm{\theta }{\prod }_{i=1}^n{\left({\gamma }_{{\widehat{\mathfrak{R}}}_{\phi (i)}}\right)}^{{\mathbb{W}}_i}}{{\prod }_{i=1}^n{\left(1,+,\left(\mathrm{\theta },-,1\right),\left(1,-,{\gamma }_{{\widehat{\mathfrak{R}}}_{\phi (i)}}\right)\right)}^{{\mathbb{W}}_i}+\left(\mathrm{\theta },-,1\right){\prod }_{i=1}^n{\left({\gamma }_{{\widehat{\mathfrak{R}}}_{\phi (i)}}\right)}^{{\mathbb{W}}_i}}\end{array}\right),\\ \left(\begin{array}{c}\sqrt[f]{\frac{{\prod }_{i=1}^n{\left(1,+,\left(\mathrm{\theta },-,1\right),{{\widehat{\mathcal{M}}}_{\phi (i)}}^f\right)}^{{\mathbb{W}}_i}-{\prod }_{i=1}^n{\left(1,-,{{\widehat{\mathcal{M}}}_{\phi (i)}}^f\right)}^{{\mathbb{W}}_i}}{{\prod }_{i=1}^n{\left(1,+,\left(\mathrm{\theta },-,1\right),{{\widehat{\mathcal{M}}}_{\phi (i)}}^f\right)}^{{\mathbb{W}}_i}+\left(\mathrm{\theta },-,1\right){\prod }_{i=1}^n{\left(1,-,{{\widehat{\mathcal{M}}}_{\phi (i)}}^f\right)}^{{\mathbb{W}}_i}}},\\ \sqrt[f]{\frac{\mathrm{\theta }{\prod }_{i=1}^n{\left({{\widehat{\mathcal{N}}}_{\phi (i)}}^f\right)}^{{\mathbb{W}}_i}}{{\prod }_{i=1}^n{\left(1,+,\left(\mathrm{\theta },-,1\right),\left(1,-,{{\widehat{\mathcal{N}}}_{\phi (i)}}^f\right)\right)}^{{\mathbb{W}}_i}+\left(\mathrm{\theta },-,1\right){\prod }_{i=1}^n{\left({{\widehat{\mathcal{N}}}_{\phi (i)}}^f\right)}^{{\mathbb{W}}_i}}}\end{array}\right)\end{array}\right\}\left(31\right)$$where $\mathrm{\theta }\ge 1$ and $\mathbb{W}=\left({\mathbb{w}}_1,,,{\mathbb{w}}_2,,,{\mathbb{w}}_3,,,\cdots ,{\mathbb{w}}_n\right)$ is the weight vector corresponding to the collection of FDFNs over $\mathfrak{P}$, and satisfy the given aspects ${\mathbb{W}}_i>0$ and ${\sum }_{i=1}^n{\mathbb{W}}_i=1$.${\widehat{\mathfrak{R}}}_{\phi (i)}$, denoted the $\mathit{ith}$ biggest weighted fractional Diophantine fuzzy values. We can easily proofs all the properties given in equations $27,28$, and $29$ for the FDFHOWA operators.

Algorithm of the proposed methodology

In this section of this article, we proposed an algorithm for our proposed decision-making framework. In order to determine the outcome of the expert information by using our proposed decision-making model, we assume that ${\mathbb{P}}_j=\left({\mathbb{P}}_1,,,{\mathbb{P}}_2,,,{\mathbb{P}}_3,,,\cdots ,,,{\mathbb{P}}_m\right)$ denoted the collection of $m$ numbers of attributes and ${\mathbb{Q}}_i=\left({\mathbb{Q}}_1,,,{\mathbb{Q}}_2,,,{\mathbb{Q}}_3,,,\cdots ,,,{\mathbb{Q}}_n\right)$ denoted the collection of $n$ numbers of alternatives chosen by the decision-maker.$\mathbb{W}=\left({\mathbb{w}}_1,,,{\mathbb{w}}_2,,,{\mathbb{w}}_3,,,\cdots ,{\mathbb{w}}_m\right)$, is the weight vector corresponding to the set of $m$ numbers of attributes, and satisfy the given aspects ${\mathbb{W}}_j>0$ and ${\sum }_{j=1}^m{\mathbb{W}}_j=1$. We present an innovative decision-making framework called the fractional Diophantine fuzzy neural network based on the fractional Diophantine fuzzy Hamacher aggregation operator to resolve MCGDM problems in an enhanced way. Our proposed decision-making model consists of the following layers:

Input layer: In the input layer, we perform the following actions:

Step 1: In this step, we collect expert data from the experts who want to share their information in the context of FDFNs about the collection of $n$ numbers of alternatives chosen by the decision-maker on the basis of $m$ numbers of attributes and construct common decision matrices as follows:$${\mathcal{E}}_{\mathfrak{s}}={\left[{\mathfrak{R}}_{\mathit{ij}}\right]}_{n\times m}\left[\begin{array}{cccc}{\mathfrak{R}}_{11}& {\mathfrak{R}}_{12}& \cdots & {\mathfrak{R}}_{1m}\\ {\mathfrak{R}}_{21}& {\mathfrak{R}}_{22}& \cdots & {\mathfrak{R}}_{2m}\\ {\mathfrak{R}}_{31}& {\mathfrak{R}}_{32}& \cdots & {\mathfrak{R}}_{3m}\\ ⋮& ⋮& \ddots & ⋮\\ {\mathfrak{R}}_{n1}& {\mathfrak{R}}_{n2}& \cdots & {\mathfrak{R}}_{\mathit{nm}}\end{array}\right]\left(32\right)$$where ${\mathcal{E}}_{\mathfrak{s}}$ denote the numbers of expert who want to share their information and${\mathfrak{R}}_{\mathit{ij}}=\left\{\left(⟨{\mu }_{{\mathfrak{R}}_{\mathit{ij}}},{\gamma }_{{\mathfrak{R}}_{\mathit{ij}}}⟩,,,⟨{\mathcal{M}}_{\mathit{ij}},{\mathcal{N}}_{\mathit{ij}}⟩\right),:,i,=,\text{1,2},,,3,,,\cdots ,n,,,j,=,\text{1,2},,,3,,,\cdots ,m\right\}$, represents the collection of FDFNs.

Step 2: We normalized expert information in the environment of fractional Diophantine fuzzy numbers by using the following equation in this Phase.$${\widehat{\mathcal{E}}}_{\mathfrak{s}}=\left\{\begin{array}{c}{\mathfrak{R}}_{\mathit{ij}},forbenefitcriteria\\ {\left({\mathfrak{R}}_{\mathit{ij}}\right)}^c,forcostcriteria\end{array}\right)\left(33\right)$$

Step 3: We determine the input signal weight vector corresponding to each attribute by using the following equations $\left(21\right)$ and $\left(22\right)$.$${\mathfrak{S}}_j=\frac{e^{-\mathbb{D}\left({\mathfrak{R}}_{\mathit{ij}},,,{\left({\mathfrak{R}}_{\mathit{ij}}\right)}^c\right)}}{\frac{1}{m}{\sum }_{j=1}^m{e}^{-\mathfrak{Q}\left({\mathfrak{R}}_{\mathit{ij}},,,{\left({\mathfrak{R}}_{\mathit{ij}}\right)}^c\right)}}\left(34\right)$$where $\mathbb{D}\left({\mathfrak{R}}_{\mathit{ij}},,,{\left({\mathfrak{R}}_{\mathit{ij}}\right)}^c\right)$ represents the fractional Diophantine fuzzy distance between the fractional Diophantine fuzzy number ${\mathfrak{R}}_{\mathit{ij}}$ and its complement ${\left({\mathfrak{R}}_{\mathit{ij}}\right)}^c$.$$\mathbb{W}=\frac{{\mathfrak{S}}_j}{{\sum }_{j=1}^m{\mathfrak{S}}_j}\left(35\right)$$

Hidden layer: We perform the following operations in the hidden layer:

Step 1: Using the FDFHWA operator and the input signal weight vectors, we determine the hidden layer data of the proposed decision-making model in this step as follows:$${\mathcal{P}}_i=\left\{\begin{array}{c}\left(\begin{array}{c}\frac{{\prod }_{i=1}^n{\left(1,+,\left(\mathrm{\theta },-,1\right),{\mu }_{{\mathfrak{R}}_{\mathit{ij}}}\right)}^{{\mathbb{W}}_j}-{\prod }_{i=1}^n{\left(1,-,{\mu }_{{\mathfrak{R}}_{\mathit{ij}}}\right)}^{{\mathbb{W}}_j}}{{\prod }_{i=1}^n{\left(1,+,\left(\mathrm{\theta },-,1\right),{\mu }_{{\mathfrak{R}}_{\mathit{ij}}}\right)}^{{\mathbb{W}}_j}+\left(\mathrm{\theta },-,1\right){\prod }_{i=1}^n{\left(1,-,{\mu }_{{\mathfrak{R}}_{\mathit{ij}}}\right)}^{{\mathbb{W}}_j}},\\ \frac{\mathrm{\theta }{\prod }_{i=1}^n{\left({\gamma }_{{\mathfrak{R}}_{\mathit{ij}}}\right)}^{{\mathbb{W}}_j}}{{\prod }_{i=1}^n{\left(1,+,\left(\mathrm{\theta },-,1\right),\left(1,-,{\gamma }_{{\mathfrak{R}}_{\mathit{ij}}}\right)\right)}^{{\mathbb{W}}_j}+\left(\mathrm{\theta },-,1\right){\prod }_{i=1}^n{\left({\gamma }_{{\mathfrak{R}}_{\mathit{ij}}}\right)}^{{\mathbb{W}}_j}}\end{array}\right),\\ \left(\begin{array}{c}\sqrt[f]{\frac{{\prod }_{i=1}^n{\left(1,+,\left(\mathrm{\theta },-,1\right),{{\mathcal{M}}_{\mathit{ij}}}^f\right)}^{{\mathbb{W}}_j}-{\prod }_{i=1}^n{\left(1,-,{{\mathcal{M}}_{\mathit{ij}}}^f\right)}^{{\mathbb{W}}_j}}{{\prod }_{i=1}^n{\left(1,+,\left(\mathrm{\theta },-,1\right),{{\mathcal{M}}_{\mathit{ij}}}^f\right)}^{{\mathbb{W}}_j}+\left(\mathrm{\theta },-,1\right){\prod }_{i=1}^n{\left(1,-,{{\mathcal{M}}_{\mathit{ij}}}^f\right)}^{{\mathbb{W}}_j}}},\\ \sqrt[f]{\frac{\mathrm{\theta }{\prod }_{i=1}^n{\left({{\mathcal{N}}_{\mathit{ij}}}^f\right)}^{{\mathbb{W}}_j}}{{\prod }_{i=1}^n{\left(1,+,\left(\mathrm{\theta },-,1\right),\left(1,-,{{\mathcal{N}}_{\mathit{ij}}}^f\right)\right)}^{{\mathbb{W}}_j}+\left(\mathrm{\theta },-,1\right){\prod }_{i=1}^n{\left({{\mathcal{N}}_{\mathit{ij}}}^f\right)}^{{\mathbb{W}}_j}}}\end{array}\right)\end{array}\right\}\left(36\right)$$where $\mathrm{\theta }\ge 1$ and $\mathbb{W}=\left({\mathbb{w}}_1,,,{\mathbb{w}}_2,,,{\mathbb{w}}_3,,,\cdots ,{\mathbb{w}}_n\right)$ is the input signal weight vector corresponding to the collection $m$ numbers of attributes, and satisfy the given aspects ${\mathbb{W}}_j>0$ and ${\sum }_{j=1}^m{\mathbb{W}}_j=1$. In the above equitation of the fractional Diophantine neural network, ${\mu }_{{\mathfrak{R}}_{\mathit{ij}}}$ denoted the MD and ${\gamma }_{{\mathfrak{R}}_{\mathit{ij}}}$ denoted the DNM functions, ${\mathcal{M}}_{\mathit{ij}},{\mathcal{N}}_{\mathit{ij}}$ denoted the RPs for both ${\mu }_{{\mathfrak{R}}_{\mathit{ij}}}$ and ${\gamma }_{{\mathfrak{R}}_{\mathit{ij}}}$ respectively.

Step 2: In this phase, we use the fractional Diophantine fuzzy distance measure provided in equations $\left(30\right)$ and $\left(31\right)$ to determine the hidden layer signal weight vector.

Output layer: We perform the following operations in the output layer:

Step 1: Using the FDFHWA operator and the hidden layer signal weight vectors, we determine the output layer data of the proposed decision-making model in this phase as follows:$${\mathcal{Q}}_i=\left\{\begin{array}{c}\left(\begin{array}{c}\frac{{\prod }_{i=1}^n{\left(1,+,\left(\mathrm{\theta },-,1\right),{\mu }_{{\mathfrak{R}}_{\mathit{ij}}}\right)}^{{\dot{\mathbb{w}}}_j}-{\prod }_{i=1}^n{\left(1,-,{\mu }_{{\mathfrak{R}}_{\mathit{ij}}}\right)}^{{\dot{\mathbb{w}}}_j}}{{\prod }_{i=1}^n{\left(1,+,\left(\mathrm{\theta },-,1\right),{\mu }_{{\mathfrak{R}}_{\mathit{ij}}}\right)}^{{\dot{\mathbb{w}}}_j}+\left(\mathrm{\theta },-,1\right){\prod }_{i=1}^n{\left(1,-,{\mu }_{{\mathfrak{R}}_{\mathit{ij}}}\right)}^{{\dot{\mathbb{w}}}_j}},\\ \frac{\mathrm{\theta }{\prod }_{i=1}^n{\left({\gamma }_{{\mathfrak{R}}_{\mathit{ij}}}\right)}^{{\dot{\mathbb{w}}}_j}}{{\prod }_{i=1}^n{\left(1,+,\left(\mathrm{\theta },-,1\right),\left(1,-,{\gamma }_{{\mathfrak{R}}_{\mathit{ij}}}\right)\right)}^{{\dot{\mathbb{w}}}_j}+\left(\mathrm{\theta },-,1\right){\prod }_{i=1}^n{\left({\gamma }_{{\mathfrak{R}}_{\mathit{ij}}}\right)}^{{\dot{\mathbb{w}}}_j}}\end{array}\right),\\ \left(\begin{array}{c}\sqrt[f]{\frac{{\prod }_{i=1}^n{\left(1,+,\left(\mathrm{\theta },-,1\right),{{\mathcal{M}}_{\mathit{ij}}}^f\right)}^{{\dot{\mathbb{w}}}_j}-{\prod }_{i=1}^n{\left(1,-,{{\mathcal{M}}_{\mathit{ij}}}^f\right)}^{{\dot{\mathbb{w}}}_j}}{{\prod }_{i=1}^n{\left(1,+,\left(\mathrm{\theta },-,1\right),{{\mathcal{M}}_{\mathit{ij}}}^f\right)}^{{\dot{\mathbb{w}}}_j}+\left(\mathrm{\theta },-,1\right){\prod }_{i=1}^n{\left(1,-,{{\mathcal{M}}_{\mathit{ij}}}^f\right)}^{{\dot{\mathbb{w}}}_j}}},\\ \sqrt[f]{\frac{\mathrm{\theta }{\prod }_{i=1}^n{\left({{\mathcal{N}}_{\mathit{ij}}}^f\right)}^{{\dot{\mathbb{w}}}_j}}{{\prod }_{i=1}^n{\left(1,+,\left(\mathrm{\theta },-,1\right),\left(1,-,{{\mathcal{N}}_{\mathit{ij}}}^f\right)\right)}^{{\dot{\mathbb{w}}}_j}+\left(\mathrm{\theta },-,1\right){\prod }_{i=1}^n{\left({{\mathcal{N}}_{\mathit{ij}}}^f\right)}^{{\dot{\mathbb{w}}}_j}}}\end{array}\right)\end{array}\right\}\left(37\right)$$where $\mathrm{\theta }\ge 1$ and $\dot{\mathbb{W}}=\left({\dot{\mathbb{w}}}_1,,,{\dot{\mathbb{w}}}_2,,,{\dot{\mathbb{w}}}_3,,,\cdots ,{\dot{\mathbb{w}}}_n\right)$ is the hidden layer signal weight vector corresponding to the collection $m$ numbers of attributes, and satisfy the given aspects ${\dot{\mathbb{W}}}_i>0$ and ${\sum }_{i=1}^n{\dot{\mathbb{W}}}_i=1$. ${\mathcal{Q}}_i,$ denoted the output layer data of the fractional Diophantine fuzzy neural networks.

Step 2: In this phase, we use the score function represented by ScF $\left({\mathcal{Q}}_i\right)$ and provided in equation $\left(16\right)$ to determine the score values of the output layer data of the proposed decision-making model.

Step 3: In this step, we find the final outcome of the proposed model by applying the following soft plus activation function on the score values of the output layer data.$$\mathbb{F}\left(\text{ScF},\left({\mathcal{Q}}_i\right)\right)={\text{log}}_e\left(1,+,e^{\left(\text{ScF},\left({\mathcal{Q}}_i\right)\right)}\right),\left(38\right)$$where $\text{ScF}\left({\mathcal{Q}}_i\right)$ denotes the score values of the output layer data of the proposed model and the soft plus activation function are represented graphically in Fig. 6. The continuous and differentiable nature of the Soft plus function ensures that the output values are non-negative and smoothly varying. When mapped to a preference matrix, this guarantees that relative magnitudes and rankings between alternatives are preserved without any significant loss of information. The transformation from continuous outputs to ranked preferences retains the ordinal relationships between alternatives, ensuring that subtle differences in decision values are reflected accurately. This characteristic is crucial in emergency decision-making, where small distinctions can be critical for the final decision-making process.

[See PDF for image]

Fig. 6

Soft plus Activation Function

Graphical indication of the proposed methodology

In this study, we propose a novel decision-making framework based on the Fractional Diphontine Neural Network (FDNN), which extends the traditional fuzzy neural network architecture to effectively address complex multi-criteria decision-making (MCDM) problems. The FDNN leverages Fractional Diphontine Fuzzy Numbers (FDFNs) to model the inherent uncertainty and imprecision in expert evaluations and employs Hamacher aggregation operators to integrate information throughout the network. The model consists of three main layers: input, hidden, and output. Initially, expert data is collected and represented in the form of decision matrices using FDFNs. These matrices are then normalized using a simple normalization technique described in Eq. (33) to ensure the comparability of different criteria. The normalized data is assigned signal weight vectors in the input layer, which reflect the relative importance of the criteria. These weighted inputs are processed through the Hamacher aggregation operators and passed to the hidden layer, where further aggregation takes place using the hidden layer's signal weights. This process captures non-linear dependencies and interactions among the input criteria. The resulting outputs from the hidden layer are then transmitted to the output layer, where they are again aggregated using Hamacher operators and converted into fuzzy outputs in the form of FDFNs. Since these fuzzy outputs cannot be directly used for decision-making, a score function is applied to defuzzify them into crisp numerical values. Finally, to enhance decision interpretability and maintain continuity, the Softplus activation function is applied to the score values in the output layer.

This smooth, differentiable function ensures stable gradient behavior and produces non-negative decision outputs suitable for expert-driven applications. Overall, the FDNN provides a robust and structured approach to synthesizing uncertain expert knowledge and generating reliable decision outcomes. A visual summary of this process is provided in Fig. 7.

[See PDF for image]

Fig. 7

Flow chart of the proposed decision making model

Application of the proposed decision-making model for hospital selection in emergency cases in peshawar

Peshawar, the capital city of Khyber Pakhtunkhwa province, is one of the largest and most populous urban areas in Pakistan, with a diverse demographic and an increasing population of over 2 million residents. The city faces significant challenges in its healthcare infrastructure, particularly in emergencies. Rapid urbanization, population growth, and increasing road traffic contribute to an urgent need for an efficient and effective emergency response system. In emergency medical situations, the speed at which a patient receives treatment is critical, especially for life-threatening conditions like heart attacks, strokes, or severe trauma. The choice of hospital can make the difference between life and death; however, selecting the most appropriate facility is often complicated by various factors, including the availability of specialized medical services, hospital capacity, treatment costs, and real-time traffic conditions. Peshawar’s healthcare landscape comprises a mix of public and private hospitals, each with its strengths and weaknesses. Public hospitals, such as Lady Reading Hospital (LRH) and Khyber Teaching Hospital (KTH), typically offer a broad range of services but often struggle with overcrowding and resource limitations. Private hospitals like Rehman Medical Institute (RMI) and Northwest General Hospital (NWG) may provide higher-quality care but at a higher cost, which can be a barrier for many patients. In emergencies, where every second counts, it is vital for Emergency Medical Services (EMS) teams to adopt a structured approach to hospital selection. Utilizing Multi-Criteria Group Decision-Making (MCGDM) techniques can enhance the efficiency and effectiveness of the decision-making process, allowing EMS teams to select appropriate hospitals based on critical attributes that influence patient outcomes.

The challenge of selecting the right hospital in an emergency context is particularly acute in urban areas like Peshawar, where multiple factors can significantly affect the outcome of medical interventions. Quick access to appropriate medical care is essential, and EMS teams must navigate a complex environment of varying hospital capabilities and traffic conditions. This case study explores the process of hospital selection for emergency cases in Peshawar using an MCGDM approach, specifically the fractional Diophantine fuzzy neural network. For this case study, we analyze six hospitals in Peshawar, each with varying levels of service and capacity:

• ${\mathbb{Q}}_1=$ Lady Reading Hospital (LRH),

• ${\mathbb{Q}}_2=$ Khyber Teaching Hospital (KTH),

• ${\mathbb{Q}}_3=$ Northwest General Hospital (NWG),

• ${\mathbb{Q}}_4=$ Rehman Medical Institute (RMI),

• ${\mathbb{Q}}_5=$ Prime Teaching Hospital (PTH)

• ${\mathbb{Q}}_6=$ Hayatabad Medical Complex (HMC).

To select the most appropriate hospital in case of emergency, experts evaluate each hospital option based on five attributes: ${\mathbb{P}}_1=$ Travel Time,${\mathbb{P}}_2=$ Quality of Emergency Care,${\mathbb{P}}_3=$ Hospital Capacity,${\mathbb{P}}_4=$ Cost of Treatment and ${\mathbb{P}}_5=$ Traffic Conditions. ${\mathbb{P}}_4$ is a cost type attribute, while other attributes are benefit type. Rather than generating synthetic emergency scenarios, the evaluation is based on real-world conditions specific to Peshawar. Expert input incorporated practical considerations such as typical ambulance response times, hospital occupancy levels, and traffic congestion patterns during emergencies.

Results and discussion

The numerical results obtained by applying our proposed model for selecting the most suitable hospital for emergency situations in Peshawar are discussed in this section of the article. Emergency hospitals play a crucial role in providing timely and effective medical care during critical health events. With the increasing demand for quality healthcare and the need for quick medical intervention, choosing the best hospital becomes a challenging decision-making task for experts and policymakers. Therefore, we proposed an innovative model for hospital selection using fractional Diophantine fuzzy numbers (FDFNs). The FDFN approach extends traditional fuzzy models by allowing more flexible and accurate handling of uncertain and imprecise expert information. This model overcomes the limitations of classical fuzzy sets by expanding the decision-making space and enabling experts to express their opinions more precisely. The use of FDFNs provides a more practical and effective way to represent uncertainty subsection, we in healthcare decision-making, leading to better and more reliable hospital selection outcomes.

7.1 Results: In this discuss the results obtained using our proposed decision model. We introduce a new approach called the fractional Diophantine fuzzy neural network, which is based on the fractional Diophantine Hamacher Aggregation Operator. This model is designed to better address complex decision-making problems. It enhances traditional feed-forward neural networks by using three layers: input, hidden, and output. Each layer processes fractional Diophantine fuzzy data to effectively handle uncertainty and unclear information from experts. Furthermore, the layers of the proposed model are discussed below.

Input layer: We perform the following operations in the input layer.

Step 1: In this step, we gather expert information and construct three decision matrices (${\mathcal{E}}_1,{\mathcal{E}}_2$ and ${\mathcal{E}}_3)$ in the context of FDFNs. These matrices are based on the data provided by experts regarding the selection of hospitals for emergency situations in Peshawar, involving six alternatives and five attributes.

Step 2: In this step, we normalized expert information because we consider ${\mathbb{P}}_4$ is a non-beneficial criterion, while the other criterions are beneficial one.

Step 3: We assume every attribute to be input signals for the fractional Diophantine fuzzy neural network. Furthermore, we determine the input signal weight vector corresponding to each attribute with the help of the fractional Diophantine fuzzy distance measure given in Eqs. $\left(34\right)$ and $\left(35\right)$. The final results are given in Table 7.

Table 7. Input signal weight vector of the expert information

	${\mathbb{W}}_1$	${\mathbb{W}}_2$	${\mathbb{W}}_3$
${\mathbb{P}}_1$	0.2118	0.1947	0.2005
${\mathbb{P}}_2$	0.2123	0.196	0.2023
${\mathbb{P}}_3$	0.2063	0.2166	0.2154
${\mathbb{P}}_4$	0.189	0.1811	0.1954
${\mathbb{P}}_5$	0.1806	0.2116	0.1864

Hidden layer: In the hidden layer, the FDFEWA operator, along with a weight vector derived from the input signals, performs sophisticated calculations on the expert data to enhance its characteristics. We perform the following operations in the hidden layer:

Step 1: The FDFHWA operator and the input signal weight vectors are used to locate the hidden layer data of the proposed decision-making model. Table 8 displays the final conclusions.

Table 8. Hidden layer data of the fractional Diophantine fuzzy neural network

${\mathcal{P}}_i$	${\mathbb{P}}_1$	${\mathbb{P}}_2$	${\mathbb{P}}_3$
${\mathbb{Q}}_1$	$⟨0.372,0.887⟩,⟨0.510,0.708⟩$	$⟨0.360,0.802⟩,⟨0.616,0.665⟩$	$⟨0.333,0.628⟩,⟨0.480,0.621⟩$
${\mathbb{Q}}_2$	$⟨0.360,0.789⟩,⟨0.573,0.709⟩$	$⟨0.326,0.742⟩,⟨0.481,0.642⟩$	$⟨0.5,0.724⟩,⟨0.387,0.641⟩$
${\mathbb{Q}}_3$	$⟨0.408,0.806⟩,⟨0.723,0.552⟩$	$⟨0.362,0.658⟩,⟨0.665,0.527⟩$	$⟨0.326,0.861⟩,⟨0.630,0.494⟩$
${\mathbb{Q}}_4$	$⟨0.428,0.789⟩,⟨0.707,0.622⟩$	$⟨0.374,0.752⟩,⟨0.694,0.528⟩$	$⟨0.337,0.776⟩,⟨0.649,0.598⟩$
${\mathbb{Q}}_5$	$⟨0.5,0.771⟩,⟨0.626,0.636⟩$	$⟨0.381,0.864⟩,⟨0.667,0.790⟩$	$⟨0.390,0.817⟩,⟨0.678,0.695⟩$
${\mathbb{Q}}_6$	$⟨0.5,0.887⟩,⟨0.715,0.654⟩$	$⟨0.359,0.869⟩,⟨0.540,0.777⟩$	$⟨0.380,0.818⟩,⟨0.589,0.786⟩$

${\mathcal{P}}_i$	${\mathbb{P}}_4$	${\mathbb{P}}_5$
${\mathbb{Q}}_1$	$⟨0.5,0.776⟩,⟨0.635,0.771⟩$	$⟨0.5,0.698⟩,⟨0.716,0.715⟩$
${\mathbb{Q}}_2$	$⟨0.217,0.757⟩,⟨0.576,0.801⟩$	$⟨0.5,0.699⟩,⟨0.535,0.742⟩$
${\mathbb{Q}}_3$	$⟨0.5,0.747⟩,⟨0.560,0.792⟩$	$⟨0.372,0.710⟩,⟨0.288,0.864⟩$
${\mathbb{Q}}_4$	$⟨0.244,0.844⟩,⟨0.638,0.796⟩$	$⟨0.248,0.716⟩,⟨0.709,0.660⟩$
${\mathbb{Q}}_5$	$⟨0.395,0.897⟩,⟨0.733,0.666⟩$	$⟨0.375,0.866⟩,⟨0.508,0.870⟩$
${\mathbb{Q}}_6$	$⟨0.342,0.758⟩,⟨0.616,0.808⟩$	$⟨0.338,0.674⟩,⟨0.751,0.624⟩$

Step 2: In this phase, we use the fractional Diophantine fuzzy distance measure provided in Eqs. $\left(30\right)$ and $\left(31\right)$ to determine the hidden layer signal weight vector. The hidden layer signal weight vector denoted $\dot{\mathbb{W}}$ by and given as: $\dot{\mathbb{W}}=\left(0.2028,,,0.201,,,0.2025,,,0.1922,,,0.2015\right)$.

Output layer: The output layer is responsible for generating the final output of the proposed decision-making model. It receives processed data from the hidden layers and applies an activation function, such as the soft plus activation function, to the score values in order to generate the final results based on expert input. We perform the following operations in the output layer:

Step 1: The FDFHWA operator and the input signal weight vectors are used to locate the output layer data of the proposed decision-making model. Table 9 displays the final conclusions.

Table 9. Output layer data of the fractional Diophantine fuzzy neural network

	${\mathbb{Q}}_1$	${\mathbb{Q}}_2$	${\mathbb{Q}}_3$
${\mathcal{Q}}_i$	$⟨0.369,0.760⟩,⟨0.673,0.695⟩$	$⟨0.356,0.742⟩,⟨0.589,0.706⟩$	$⟨0.361,0.758⟩,⟨0.684,0.637⟩$
	${\mathbb{Q}}_4$	${\mathbb{Q}}_5$	${\mathbb{Q}}_6$
${\mathcal{Q}}_i$	$⟨0.330,0.775⟩,⟨0.730,0.636⟩$	$⟨0.367,0.843⟩,⟨0.709,0.732⟩$	$⟨0.804,0.817⟩,⟨0.713,0.728⟩$

Step 2: Using the score function provided in Eq. $(16)$, we determine the score values of the output layer data of the proposed decision-making model in this phase. Table 10 shows the final results of the proposed model.

Table 10. Final output of the fractional Diophantine fuzzy neural network

	${\mathbb{Q}}_1$	${\mathbb{Q}}_2$	${\mathbb{Q}}_3$	${\mathbb{Q}}_4$	${\mathbb{Q}}_5$	${\mathbb{Q}}_6$
$\text{ScF}\left({\mathcal{Q}}_i\right)$	0.3967	0.3794	0.4113	0.4131	0.3736	0.3832
$\mathbb{F}\left(\text{ScF},\left({\mathcal{Q}}_i\right)\right)$	0.9106	0.9003	0.9194	0.9205	0.8969	0.9026

Step 3: In this step, we find the final outcome of the proposed decision-making model by applying the following soft plus activation function on the score values of the output layer data. The final results are given in Table 10.

According to our proposed decision-making model, Rehman Medical Institute (RMI) was the top choice, offering high-quality emergency care and minimal wait times, despite its cost. This case highlights the importance of balancing proximity with care quality and capacity. The selection process is given in Fig. 8.

[See PDF for image]

Fig. 8

Fractional Diophantine fuzzy neural network for the hospital selection in emergency cases in Peshawar

Comparison and discussion

To validate the applicability and effectiveness of the proposed decision-making model, we conduct a comparative analysis against existing MCGDM techniques and machine learning models, including Multi-Layer Perceptrons (MLPs) and Random Forest, in this section. For this, we use the same expert data collected for selecting the best hospital for emergency situations in Peshawar. We first construct three decision matrices based on FDFNs, which include six alternatives chosen by decision-makers based on five attributes. Since expert weight plays an important role in combining information from multiple sources, we use the fractional Diophantine fuzzy distance measure method to determine the expert weight vector: $\mathbb{W}=\left(0.3380,,,0.3329,,,0.3291\right).$ Next, we aggregate the expert data from multiple sources using the FDFHWA operator along with the expert weight vector for each matrix. The final aggregated expert information is presented in Table 11.

Table 11. Aggregated expert data in the context of FDFNs

${\mathcal{P}}_i$	${\mathbb{P}}_1$	${\mathbb{P}}_2$	${\mathbb{P}}_3$
${\mathbb{Q}}_1$	$⟨0.473,0.817⟩,⟨0.558,0.471⟩$	$⟨0.466,0.678⟩,⟨0.667,0.429⟩$	$⟨0.437,0.420⟩,⟨0.522,0.403⟩$
${\mathbb{Q}}_2$	$⟨0.468,0.639⟩,⟨0.625,0.476⟩$	$⟨0.437,0.572⟩,⟨0.525,0.393⟩$	$⟨0.520,0.558⟩,⟨0.422,0.425⟩$
${\mathbb{Q}}_3$	$⟨0.501,0.659⟩,⟨0.774,0.300⟩$	$⟨0.464,0.425⟩,⟨0.719,0.280⟩$	$⟨0.420,0.778⟩,⟨0.678,0.272⟩$
${\mathbb{Q}}_4$	$⟨0.513,0.643⟩,⟨0.760,0.374⟩$	$⟨0.477,0.568⟩,⟨0.746,0.279⟩$	$⟨0.441,0.640⟩,⟨0.700,0.374⟩$
${\mathbb{Q}}_5$	$⟨0.526,0.603⟩,⟨0.680,0.376⟩$	$⟨0.485,0.774⟩,⟨0.720,0.581⟩$	$⟨0.482,0.705⟩,⟨0.726,0.486⟩$
${\mathbb{Q}}_6$	$⟨0.536,0.806⟩,⟨0.768,0.406⟩$	$⟨0.464,0.786⟩,⟨0.588,0.569⟩$	$⟨0.479,0.707⟩,⟨0.636,0.611⟩$

${\mathcal{P}}_i$	${\mathbb{P}}_4$	${\mathbb{P}}_5$
${\mathbb{Q}}_1$	$⟨0.536,0.596⟩,⟨0.700,0.580⟩$	$⟨0.532,0.465⟩,⟨0.766,0.462⟩$
${\mathbb{Q}}_2$	$⟨0.332,0.558⟩,⟨0.637,0.610⟩$	$⟨0.536,0.471⟩,⟨0.587,0.512⟩$
${\mathbb{Q}}_3$	$⟨0.532,0.538⟩,⟨0.617,0.599⟩$	$⟨0.488,0.503⟩,⟨0.317,0.715⟩$
${\mathbb{Q}}_4$	$⟨0.371,0.700⟩,⟨0.700,0.600⟩$	$⟨0.384,0.525⟩,⟨0.762,0.403⟩$
${\mathbb{Q}}_5$	$⟨0.502,0.811⟩,⟨0.793,0.403⟩$	$⟨0.480,0.756⟩,⟨0.556,0.723⟩$
${\mathbb{Q}}_6$	$⟨0.470,0.571⟩,⟨0.679,0.604⟩$	$⟨0.459,0.431⟩,⟨0.801,0.366⟩$

After that, we calculate the attribute weight vector of the combined expert data using the same aggregation operator to ensure a balanced evaluation. The resulting attribute weight vector is: ${\mathbb{W}}_j=(0.1953,0.1994,0.2019,0.2009,0.2025)$. Finally, we determine the optimal hospital for emergency care by applying various MCGDM techniques such as TOPSIS, WASPAS, WS, WP, MOORA, and GRA, as well as machine learning models like MLPs and Random Forest, to the aggregated expert data and its attribute weight vector. The results are summarized in Table 12, and the final rankings are shown in Table 13. The findings of the proposed model are highly consistent with those obtained from other methods.

Table 12. Outcome of the expert data by using MCGDM techniques

	${\mathbb{Q}}_1$	${\mathbb{Q}}_2$	${\mathbb{Q}}_3$	${\mathbb{Q}}_4$	${\mathbb{Q}}_5$	${\mathbb{Q}}_6$
TOPSIS technique	0.5806	0.5696	0.5786	0.6525	0.3895	0.5253
WASPAS technique	0.4968	0.4815	0.4940	0.4995	0.4679	0.4855
GRA technique	0.3475	0.4304	0.4214	0.6105	0.4194	0.4747
MLP technique	0.4611	0.4840	0.4991	0.5054	0.4903	0.4461
Random Forest method	0.4978	0.4817	0.4947	0.5000	0.4686	0.4867
VIKOOR technique	0.5918	0.5807	0.4947	0.7012	0.6686	0.4837
WS technique	0.4978	0.4817	0.4947	0.5000	0.4686	0.4867
MOORA technique	0.3465	0.3224	0.3432	0.4348	0.3100	0.3304
WP technique	0.4953	0.4811	0.4929	0.4987	0.4668	0.4837

Table 13. Ranking of the alternatives using MCDM techniques and our proposed technique

	Ranking of the alternatives	Best one
Proposed technique	${\mathbb{Q}}_4>{\mathbb{Q}}_3>{\mathbb{Q}}_1>{\mathbb{Q}}_6>{\mathbb{Q}}_2>{\mathbb{Q}}_5$	${\mathbb{Q}}_4$
TOPSIS technique [52]	${\mathbb{Q}}_4>{\mathbb{Q}}_1>{\mathbb{Q}}_3>{\mathbb{Q}}_6>{\mathbb{Q}}_2>{\mathbb{Q}}_5$	${\mathbb{Q}}_4$
WASPAS technique [53]	${\mathbb{Q}}_4>{\mathbb{Q}}_1>{\mathbb{Q}}_3>{\mathbb{Q}}_2>{\mathbb{Q}}_6>{\mathbb{Q}}_5$	${\mathbb{Q}}_4$
GRA technique [54]	${\mathbb{Q}}_4>{\mathbb{Q}}_6>{\mathbb{Q}}_2>{\mathbb{Q}}_3>{\mathbb{Q}}_5>{\mathbb{Q}}_1$	${\mathbb{Q}}_4$
MLP technique [55]	${\mathbb{Q}}_4>{\mathbb{Q}}_3>{\mathbb{Q}}_5>{\mathbb{Q}}_2>{\mathbb{Q}}_1>{\mathbb{Q}}_6$	${\mathbb{Q}}_4$
Random Forest technique [56]	${\mathbb{Q}}_4>{\mathbb{Q}}_1>{\mathbb{Q}}_3>{\mathbb{Q}}_6>{\mathbb{Q}}_2>{\mathbb{Q}}_5$	${\mathbb{Q}}_4$
VIKOOR technique [57]	${\mathbb{Q}}_4>{\mathbb{Q}}_5>{\mathbb{Q}}_1>{\mathbb{Q}}_2>{\mathbb{Q}}_3>{\mathbb{Q}}_6$	${\mathbb{Q}}_4$
WS technique [58]	${\mathbb{Q}}_4>{\mathbb{Q}}_1>{\mathbb{Q}}_3>{\mathbb{Q}}_6>{\mathbb{Q}}_2>{\mathbb{Q}}_5$	${\mathbb{Q}}_4$
MOORA technique [59]	${\mathbb{Q}}_4>{\mathbb{Q}}_1>{\mathbb{Q}}_3>{\mathbb{Q}}_6>{\mathbb{Q}}_2>{\mathbb{Q}}_5$	${\mathbb{Q}}_4$
WP technique [60]	${\mathbb{Q}}_4>{\mathbb{Q}}_1>{\mathbb{Q}}_3>{\mathbb{Q}}_6>{\mathbb{Q}}_2>{\mathbb{Q}}_5$	${\mathbb{Q}}_4$

The proposed model demonstrates greater stability and reliability compared to other MCGDM and machine learning approaches. Despite differences in the evaluation methods, the ranking of hospital alternatives remains largely consistent across techniques. This consistency indicates that the proposed decision-making model is well-suited for practical emergency hospital selection problems, providing dependable and trustworthy results. Based on the results of the proposed model and comparative techniques, Rehman Medical Institute (RMI) is identified as the best hospital for emergency situations in Peshawar. RMI offers high-quality emergency care with minimal waiting times, which is crucial for urgent cases such as heart attacks, strokes, or severe trauma. Although the cost at RMI is higher compared to public hospitals, its superior medical services and faster emergency response make it the preferred choice. These findings are illustrated in Fig. 9 and Table 13.

[See PDF for image]

Fig. 9

Comparison among our proposed decision-making model and MCGDM techniques

The proposed model also shows a high level of agreement in ranking alternatives when compared to other methods. While the computational effort may vary across approaches, the results confirm that this model is both practical and reliable for making informed decisions in emergency hospital selection.

Sensitive analysis

In this section, we examine the influence of the Hamacher parameter $\mathrm{\theta }$ and parameter $f$ on the decision results obtained by using the proposed decision-making model. To extract expert information, we employ the proposed decision-making model, which is based on the fractional Diophantine fuzzy Hamacher aggregation operators and fractional Diophantine fuzzy numbers, both of which are directly influenced by changes in the values of $\mathrm{\theta }$ and $f$. Consequently, variations in these parameters significantly impact the decision outcomes produced by our model.

When varying the Hamacher parameter $\mathrm{\theta }$ within the interval $[1,39]$, the ranking of alternatives remains stable, with ${\mathbb{Q}}_4$ consistently identified as the best choice for decision-makers, although slight fluctuations occur in the numerical values of the alternatives. However, as $\mathrm{\theta }$ increases to the interval $[40,60]$, the value of ${\mathbb{Q}}_4$ decreases while ${\mathbb{Q}}_3$ increases, causing a reversal in ranking where ${\mathbb{Q}}_3$ surpasses ${\mathbb{Q}}_4$. For values of $\mathrm{\theta }$ in the interval $[61,100]$, small changes in numerical values are observed, but the ranking returns to its original order, with ${\mathbb{Q}}_4$ remaining the top alternative, as shown in Fig. 10.

[See PDF for image]

Fig. 10

Influence of the Hamacher parameter $\mathit{\theta }$ on the decision results

Similarly, when varying parameter $f$ within $[1,100]$, three distinct behaviors emerge. In the interval $[1,10.5]$, minor changes occur in alternative values without affecting the ranking, with ${\mathbb{Q}}_4$ maintaining its position as the top choice. Between $(10.5,30.1]$, the value of ${\mathbb{Q}}_4$ declines while ${\mathbb{Q}}_3$ rises, resulting in ${\mathbb{Q}}_3$ overtaking ${\mathbb{Q}}_4$. Finally, in the interval $(\text{30.1,100}]$, the value of ${\mathbb{Q}}_2$ increases significantly and stabilizes, leading to a change in the ranking where ${\mathbb{Q}}_2$ becomes the best alternative, as illustrated in Fig. 11.

[See PDF for image]

Fig. 11

Influence of the Hamacher parameter $\mathit{f}$ on the decision results

When both parameters $\theta$ and $f$ are varied simultaneously within the interval [1, 9], slight changes occur in the numerical values of alternatives, but the ranking remains unchanged, with ${\mathbb{Q}}_4$ still favored by decision experts, as presented in Fig. 12.

[See PDF for image]

Fig. 12

Influence of the parameters $\mathit{f}$ and θ on the decision results

Although the parameters influence the numerical scores of the alternatives, sensitivity analysis confirms that their variations do not affect the relative rankings. This demonstrates the robustness of the proposed decision-making model, as minor adjustments to $\theta$ and $f$ do not alter the decision-making process. The stability in rankings can be attributed to the smoothing effect of the fractional Diophantine fuzzy Hamacher aggregation operators, which help mitigate extreme fluctuations. Additionally, the use of the softplus activation function in the output layer plays a crucial role in ensuring smooth and stable score calculations. Unlike traditional activation functions, the softplus function provides a continuous, differentiable, and non-linear transformation that prevents abrupt changes in output values. This characteristic helps regulate the model’s output, reducing the sensitivity of numerical scores to small parameter variations while preserving meaningful differences between alternatives. Together, these features enhance the model’s reliability and enable decision-makers to confidently use the model for consistent ranking outcomes, even under uncertain or variable parameter conditions.

Advantages and limitations

In this section, we discuss the advantages and disadvantages of the proposed decision-making model. We use a fractional Diophantine fuzzy neural network to extract expert information that is based on fractional Diophantine fuzzy information and Hamacher aggregation operators. Some of the benefits and drawbacks of our suggested decision-making model are covered below:

Advantages:

1. Fractional Diophantine fuzzy neural networks are a versatile tool for solving real-life issues since they can be applied to a variety of domains, including multi-criteria decision-making, production, regression, pattern recognition, and classification.

2. We present an innovative framework called the fractional Diophantine fuzzy neural network, based on the fractional Diophantine fuzzy Hamacher aggregation operator, to resolve MCGDM problems in an enhanced way. It is an AI technique that provides more accurate results based on expert input compared to traditional MCGDM techniques.

3. Fractional Diophantine fuzzy neural networks are able to handle complex input data and uncertain expert knowledge efficiently.

4. FDFS accommodates a large amount of fractional data and gives decision-making experts greater flexibility in dealing with membership degrees (MG), non-membership degrees (NMG), and reference parameters (RPs). The limitations of traditional fuzzy models are mitigated by FDFS, which expand the decision-making space and enable more accurate and flexible communication among experts.

5. Like other neural network-based models, the proposed approach is capable of high adaptability and pattern recognition; however, it can also be susceptible to overfitting, especially when trained on small or imbalanced datasets. This limitation is proactively addressed through a sensitivity analysis of key parameters ($\theta$ and $f$), which improves model robustness and helps mitigate overfitting. Future extensions may include regularization strategies to further enhance generalizability.

6. The model produces expert-driven results in less time due to fewer computational steps, which makes it promising for time-sensitive applications. While real-time decision-making in high-frequency environments may require further optimization, this model offers a solid foundation for future real-time enhancements.

7. In contrast to recent machine learning-based hospital selection frameworks, which often rely solely on data-driven models and may suffer from a lack of interpretability or limited ability to incorporate linguistic uncertainty, the proposed FDFNN model offers a hybrid approach. By combining fractional Diophantine fuzzy sets with neural networks, our method not only handles complex, imprecise, and fractional decision data but also retains transparency in the decision-making process. This dual capability allows for more nuanced and context-sensitive evaluations—something that purely data-driven ML models often struggle with, particularly in high-stakes emergency environments where expert knowledge and dynamic criteria must be considered simultaneously.

Limitations:

1. The proposed work is very effective for handling decision-making problems, but in most real-life situations, the decision-making problems involve three-dimensional or time-dependent data that cannot be handled by the current model architecture.

2. Fractional Diophantine neural networks are not able to process large-scale datasets efficiently, which are typically encountered in big data applications. Therefore, scalability to large hospital networks or industrial-scale decision-making remains a challenge.

3. To implement fractional Diophantine neural networks effectively, the complete input signal from all experts is necessary. Missing or incomplete data can negatively impact the decision output, especially in real-time or dynamic environments.

4. Although the model is capable of modeling uncertainty through fractional fuzzy logic, it is sensitive to noisy data. Its robustness may decline if the input contains a high level of noise or inconsistencies.

Conclusion and future directions

The fractional Diophantine fuzzy set (FDFS) is a novel concept introduced in this study to increase its flexibility and usefulness in real-world situations. Since fractional Diophantine fuzzy sets can handle large amounts of fractional data and give decision-making experts more flexibility in terms of MD, NMD, and RPs, they are an extension of the existing frictional fuzzy set and nonlinear Diophantine fuzzy sets. The limitations of conventional fuzzy models are so minimized by FDFS, which broadens the scope of decision-making and allows experts to share information more accurately. In information science and decision-making models, fractional Diophantine fuzzy information is a more useful way to describe uncertainty. Following that, we established novel Hamacher operating rules for FDFNs, as well as score and accuracy functions for ranking any FDFN, and discussed their desirable properties, respectively. Next, we developed a distance equation to calculate the distance between any two FDFNs. We next examine the desirable qualities of these innovative operational rules and construct a set of Hamacher weighted aggregation operators to aggregate any FDFNs. The three levels of our suggested decision-making model—the input layer, the hidden layer, and the output layer—are described in detail. Furthermore, these innovative aggregation operators and the proposed decision-making model are applied for selecting hospital in emergency situations in Peshawar. The decision-making process becomes complex due to unknown weight vectors. We use fractional Diophantine fuzzy distance measures to locate unknown weight vectors. Additionally, a comparative analysis has been performed with existing MCGDM techniques in order to verify and illustrate the efficacy of the proposed decision-making methodology. Furthermore, we provide the impact of using different values of the parameters $\mathrm{\theta }$ and $f$ to check his influence on the decision results of the fractional Diophantine fuzzy neural network. Finally, we discuss the advantages and disadvantages of the proposed decision-making model.

Future directions

Although the suggested approach is highly good at managing decision-making difficulties, most real-world scenarios include three-dimensional data that FDFS is unable to manage. We will expand the suggested model to various fuzzy extensions, including spherical fuzzy sets (SFSs), complex spherical fuzzy sets (CSFSs), etc., and various decision-making techniques in order to address this limitation in the future.

Acknowledgements

This work was partly supported by Yunnan Fundamental Research Projects (No. 202401AT070479), and Yunnan Provincial Xingdian Talent Support Program.

Author contributions

Methodology, S.A. and N.A.; Formal analysis, M.B.; Data curation, M.N.; Writing—original draft, S.A. and N.A. All authors have read and agreed to the published version of the manuscript.

Data availability

No datasets were generated or analysed during the current study.

Declarations

Ethics approval and consent to participate

This paper does not contain any studies with animals performed by any of the authors.

Competing interests

The authors declare no competing interests.

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

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