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Open Access

Peer-reviewed

Research Article

Applications of newly defined diamond Pythagorean fuzzy CODAS method via multi-criteria decision-making problems

Abstract

The diverse decision values may fail to capture an accurate perspective when multiple decision-makers are part of the process. To address this challenge, this work introduces the diamond Pythagorean fuzzy set (Dia‑PyFS), an advancement over both the Pythagorean fuzzy (PyFS) set and the interval-valued Pythagorean fuzzy set (IVPyFS). Due to the established extension of intuitionistic fuzzy sets into Pythagorean fuzzy sets, the Dia‑PyFS model, a broader form of the diamond intuitionistic fuzzy set model, demonstrates enhanced performance. We then introduce the Dia‑PyFS as an extension of PyFS. The Diamond Pythagorean fuzzy values that define the elements within Dia‑PyFS may share a common norm. For Dia‑PyFS, we define fundamental algebraic and arithmetic operations, including union, intersection, addition, multiplication, and scalar multiplication, and analyze their primary properties. Additionally, we propose some new Dia‑PyF weighted average and geometric aggregation operators as well as explore their unique properties. We also then propose several algebraic operations between Dia‑PyFVs using general triangular 𝓉-norms and 𝓉-conorms. To transform input values represented by Dia‑PyFs into a single output value, we also introduce specific weighted aggregation operators based on these algebraic methods. Additionally, the Dia‑PyFS framework builds upon the “combinative distance-based assess” (CODAS) methodology, which relies on both Euclidean and Hamming distances. To illustrate the applicability of this new approach, the feasibility and suitability of the Dia‑PyFS set approach for choosing the best options are demonstrated by the summary and comparative analysis of the produced reports.

Citation: Khan MB, Cortez MV, Bin-Mohsin B, Ciurdariu L, Zamri N (2025) Applications of newly defined diamond Pythagorean fuzzy CODAS method via multi-criteria decision-making problems. PLoS One 20(12): e0325018. https://doi.org/10.1371/journal.pone.0325018

Editor: Babek Erdebilli, Ankara Yildirim Beyazit University / Worcester Polytechnic Institute, TÜRKIYE

Received: November 19, 2024; Accepted: June 11, 2025; Published: December 12, 2025

Copyright: © 2025 Khan et al. This is an open access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.

Data Availability: All data is given in the article.

Funding: This project was supported by the Deanship of Scientific Research at King Saud University under the research funding program (ORF-2025-158).

Competing interests: The authors have declared that no competing interests exist.

1. Introduction

The concept of fuzzy sets (FS) was introduced through a function, often referred to as a membership function, which assigns a numerical value between 0 and 1 to represent the degree of an element’s membership. This approach has been applied to manage the uncertainty present in real-world scenarios. Zadeh’s theory of fuzzy sets [1] has proven effective in addressing various types of uncertainty. To better represent uncertainty, many researchers have explored this concept extensively. Later, Atanassov [2] proposed intuitionistic fuzzy sets (IFSs) as an extension of fuzzy sets, incorporating both membership and non-membership degrees. For more information, see [3–10] and the references therein.

Yager [11] introduced Pythagorean fuzzy sets (PyFSs) as an effective tool in multi-criteria decision-making (MCDM) to address a broader array of situations involving uncertainty. This concept extends intuitionistic fuzzy sets (IFS), with membership and non-membership degrees that satisfy the condition that the sum of their squares does not exceed 1. Due to its quadratic structure, a PyFS can widen the variability range for membership and non-membership degrees to the unit circle, thereby enhancing its capacity to represent uncertainty compared to an IFS.

Different types of FSs can be analyzed using points, point pairs, or point triples within a closed interval [12]. This approach requires experts to assign precise numerical values, enhancing the accuracy of the decision-making process. To address the limitations of intuitionistic fuzzy sets (IFS), Atanassov [13] introduced circular intuitionistic fuzzy sets (Cir-IFSs). In Cir-IFSs, the circular form represents the uncertainty within membership and non-membership degrees. Each element's membership and non-membership in a Cir-IFS is depicted using a circular structure, where two non-negative real numbers indicate the circle’s center, with their sum restricted to a maximum of 1. Cir- IFSs facilitate the expression of uncertainty by enabling a flexible adjustment of membership and non-membership degrees. For more information, see [14–23] and the references therein.

Bozyigit et al. [24] recently introduced the concept of circular Pythagorean fuzzy sets (Cir-PyFSs) and circular Pythagorean fuzzy values (Cir-PyFV). This novel approach is a comprehensive extension of Pythagorean fuzzy sets (PyFSs) and circular intuitionistic fuzzy sets (Cir-IFSs). In Cir-PyFS, membership and non-membership degrees are depicted using a circular representation. The circular center consists of non-negative real values uuu and vvv, constrained such that the sum of their squares does not exceed 1. Due to this unique structure, Cir-PyFS effectively models information with circular points defined by a specific center and radius, enhancing its capacity to capture the uncertainty of data. As a result, Cir-PyFS allows experts to evaluate options within a broader and more flexible framework, facilitating the development of more nuanced and complex decisions. For further information, related to IFSs and PyFS, see [25–32] and the references therein.

MCDM approaches are widely used across various scientific disciplines [33–35]. These approaches are categorized into two main types: human-based methods (such as the best-worst method and the analytic hierarchy process [AHP]) and mathematical methods (including TOPSIS and VIKOR) [36,37]. Mathematical techniques face some limitations: (i) one key aspect in decision-making research is the normalization process, which standardizes different evaluation scales by converting them into dimensionless values. To reach a conclusive result, researchers apply various data normalization methods, such as vector, linear, and linear–max–min normalization. Each normalization technique produces unique scales that influence data behavior, ultimately affecting the final decision [38]. (ii) In the TOPSIS method, the maximum and minimum values are used to define positive and negative ideal solutions. However, certain limitations might affect the general applicability of this approach. For instance, the ideal blood pressure range is typically found between the highest and lowest observed values [39]. (iii) Mathematical methods cannot precisely determine the weights of evaluation criteria. Consequently, an external approach, such as human judgment or prioritization methods, is necessary to assign these weights [40].

Human-based techniques have certain limitations, including: (i) Inconsistencies in factor weighting due to paired comparisons, which is a key issue in human-based methods [41]. For the priorities determined by AHP to be reliable, the pairwise comparison matrices must pass a consistency check. In these matrices, decision-makers assign numerical values to represent paired comparisons based on their expertise and knowledge [42]. However, due to the complexity of the decision problem and possible gaps in experience or knowledge, contradictions can arise within these matrices. When comparing factors, decision-makers must indicate how strongly one factor outweighs another, typically on a scale from 1 to 9, [43]. (ii) The process of subjective comparisons is unusual, making the task mentally demanding. In other words, comparing two unrelated factors poses a challenge as it is not an intuitive process [39]. (iii) Additionally, the method becomes more complex due to the considerable time required for conducting pairwise and reference comparisons across numerous criteria [44].

Moreover, both human and mathematical methods have limitations in managing data ambiguity and imprecision [45,46]. The FWZIC method, developed by [47], has shown superior performance over other human-based MCDM methods due to its unique capability of calculating criteria weights with 0% inconsistency. However, the original FWZIC approach had limitations in addressing the imprecision and uncertainty caused by expert hesitation. FWZIC has since been adapted for various fuzzy contexts—including neutrosophic fuzzy sets, Pythagorean fuzzy sets, cubic PyFS, interval type-2 trapezoidal fuzzy sets, dual-hesitant fuzzy sets, q-rung orthopair fuzzy sets, T-spherical fuzzy sets, and Pythagorean m-polar fuzzy sets—to better handle issues involving ambiguity and imprecision [41–48]. Despite significant efforts, the problem of ambiguity and imprecision in data persists.

An effective solution is essential to comprehensively tackle all the challenges outlined above. This requires the development of a new MCDM mathematical method capable of reducing the number of comparisons, eliminating the need for normalization, and successfully incorporating the concept of an ideal solution. The approach should focus on creating implicit and fair comparisons, avoiding forced or unnatural comparisons, preventing inconsistencies, and minimizing the use of complex mathematical computations. Additionally, the proposed method must account for informational ambiguity. To address these challenges, this study aims to introduce an innovative MCDM technique that leverages the concept of an ideal solution and an opinion matrix. The proposed method will produce rational and coherent decisions, drawing on the insights of the decision-makers.

Recognizing and integrating the diverse needs, interests, and expectations of citizens presents a significant challenge in the execution of SSL projects [49]. Ensuring that the solutions align with the community’s values and objectives is vital, as it directly impacts citizen acceptance. In this context, city planners and decision-makers may find MCDM to be a valuable approach for assessing the relationship between smart cities and public approval. This method can assist in making well-informed decisions during the planning and execution of successful smart city initiatives [40].

Although MCDM methods such as VIKOR (VlseKriterijuska Optimizacija I Komoromisno Resenje) and TOPSIS (Technique for Order of Preference by Similarity to Ideal Solution) [50,51] are effective for ranking alternatives, they are insufficient for determining the exact grade [52,53]. These approaches do not account for the decision threshold or probability, even though they consider the distance between alternatives and ideal solutions [54]. However, three-way decisions (TWDs) could offer a solution to this issue. Additionally, these MCDM methods have limitations, such as the need for normalization, the number of comparisons, inconsistencies, forced comparisons, the extent of mathematical computations, and the presence of informational ambiguity [55]. Recently, Khan et. al [56] introduced the concept of diamond intuitionistic fuzzy sets (Dia‑IFSs). Additionally, some new basic operations and relations are derived then developed Dia‑IFSs via IFS. With the support relation, an illustrated medical diagnosis example is provided as the application of Dia‑IFSs. For further study related to 𝓉-norms and 𝓉-conorms, see [57–63] and the references therein. Al-shami et al., [64] and Al-shami et al., [65] introduced SR‐Fuzzy Sets and (2, 1)-Fuzzy sets as the generalization fuzzy sets. Note that these fuzzy sets applications are more effective than other classical fuzzy sets. Similarly, Thakur et al. [66] another new version of fuzzy sets that give more freedom to decision makers to find the membership and non-membership values. Additionally, the application theses fuzzy sets can found at large scale indifferent fields of our life.

To address the challenges associated with the selection and evaluation of electric auto rickshaw frameworks, we propose a novel CODAS method that integrates MCDM within the framework of newly defined diamond-Pythagorean fuzzy sets (Dia‑PyFSs) as well as related aggregation operators. This paper introduce Dia‑PyFSs and the CODAS method, which employs distance measure based on opinion scores in the context of Dia‑PyFSs. The aim of this approach is to compute the Hamming and Euclidean distance formulas for each alternative in decision-making scenarios. Furthermore, decision thresholds for each option are established using Bayesian decision theory within the Dia‑PyF environment. Additionally, the relative assessment matrix method is utilized via assessment scores in the Dia‑PyFS context to prioritize the criteria influencing the selection and assessment of electric auto rickshaw frameworks.

The key benefits of employing MCDM for grading alternatives with the CODAS method are summarized as follows: The proposed approach demonstrates effectiveness in managing information related to dynamic and uncertain scenarios. Moreover, by utilizing MCDM, it can transform conventional ranking outcomes into objective classification results. The method also offers flexibility by adjusting the risk avoidance coefficient, enabling it to dynamically incorporate situational data. Based on these features, the following highlights the main innovations and contributions of this paper:

  • The ideas of Dia-PyFS and Dia-PyFV are introduced in this study.
  • A technique for converting a set of PyFV into Dia-PyFS is obtained, the multi-attribute decision making MCDM can be resolved in this manner.
  • Dia-shape indicate an element's membership or non-membership in a Dia-PyFS. Its structure allows for more sensitive modeling in the continuous environment using MCDM theory.
  • For Dia-PyFS, certain algebraic operations are defined using 𝓉-norms and 𝓉-conorms.
  • Some weighted arithmetic and geometric aggregation operators are supplied with the support of these operations.
  • To support our proposed methodology, this paper includes illustrated MCDM problem via CODAS method.

The structure of the remaining sections is as follows: Section 2 provides a review of the most pertinent literature. Section 3 outlines the core concepts of Dia-PyFSs and Dia-PyFVs. Come properties of Dia-PyFVs are presented and analyzed via basic operations as well as a techniques is obtained to develop Dia-PyFSs via PyFSs in Section 4. Section 5 characterize the basic operation using triangular 𝓉-norms and triangular 𝓉-conorms. In Section 6, some aggregation operators are obtained. Additionally, some exceptional cases are discussed. Additionally, some distance measures are introduced by using Dia-PyFSs. In Section 7, steps of algorithm for MCDM technique is discussed using CODAS method. The proposed methodology is applied to a problem concerning the selection of the optimal solution over electric auto rickshaw framework. Finally, Section 8 offers the conclusion of the paper.

2. Preliminaries

This section explains several fundamental concepts utilized in this study.

Definition 1. ([11]) Let us have a fixed universe E and its sub-set . The set

where is called PyFS and functions indicate the degree of membership (validity, etc.) and non-membership (non-validity, etc.) of element to a fixed set . Now, we can define also function by means of

and it corresponds to degree of indeterminacy (uncertainty, etc.). An PyFV is the pair “” given an element of X. To make things easier to understand, we can write , where , and . The degree of indeterminacy is represented by , subject to the constraints that and .

The definition of the complement of an IFV is as follows:

Definition 2. ([56]) Let us have a fixed universe E and its sub-set . The set

where and is called Dia-IFS and functions indicate the degree of membership (validity, etc.) and non-membership (non-validity, etc.) of element to a fixed set . Now, we can define also function by means of

and it corresponds to degree of indeterminacy (uncertainty, etc.), see Fig 1.

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Fig 1. Geometrical presentation of Dia-PyFS.

https://doi.org/10.1371/journal.pone.0325018.g001

On the other hand can also be defined by using following approach such that.

Let . Then,

Where

Menger's ([58]) notion of probabilistic metric spaces served as the inspiration for Schweizer and Sklar's [41] introduction of the ideas of 𝓉-norm and 𝓉-conorm. In statistics and decision-making, these ideas are essential. The closed unit interval is the basis for the binary operations known as 𝓉-norm and 𝓉-conorms in algebra.

Definition 3. ([59,60]) A mapping :[0,1]×[0,1]→[0,1] that satisfies the following characteristics is called a 𝓉-norm:

Border condition: for all ,

Commutativity: for all ,

Associativity: for all ,

Monotonicity: whenever and for all .

Definition 4. ([59,60]) A mapping :[0,1]×[0,1]→[0,1] that satisfies the following characteristics is called a 𝓉-conorm:

Border condition: for all (border condition),

Commutativity: for all (commutativity),

Associativity: for all (associativity),

Monotonicity: ) whenever and for all (monotonicity).

Definition 5. ([60,61]) A function with that is strictly decreasing and satisfies is referred to as the additive generator of a 𝓉-norm if the relationship holds for all .

The concept of a fuzzy complement is required to determine the additive generator of a dual 𝓉-conorm defined on the interval .

Definition 6. ([3]) A fuzzy complement is a function that meets the following criteria:

  1. (N1) and (boundary conditions),
  2. (N2) whenever for all (monotonicity),
  3. (N3) Continuity,
  4. (N4) for all (involution).

The function given by , where , represents a fuzzy complement. When simplifies to the Pythagorean fuzzy complement .

Definition 7. ([63]) Let be a 𝓉-norm and be a 𝓉-conorm on the interval [0, 1]. If and , then and are referred to as dual with respect to the fuzzy complement N.

Remark 1. Let represent a 𝓉-norm on the interval [0,1]. The corresponding dual 𝓉-conorm with regard to the intuitionistic fuzzy complement N is defined as follows:

It is important to mention that qualifies as an Archimedean 𝓉-norm if and only if for all , while is classified as an Archimedean 𝓉-conorm if and only if [60,61]. Klement et al. [62] demonstrated that continuous Archimedean 𝓉-norms can be represented through their additive generators, as established in the following theorem.

Theorem 1 ([62]). Let represent a 𝓉-norm on [0, 1]. The following statements are equivalent:

  1. (i) is a continuous Archimedean 𝓉-norm.
  2. (ii) possesses a continuous additive generator, meaning there exists a continuous, strictly decreasing function with such that for all .

3. Diamond Pythagorean fuzzy sets

This section covers some fundamental properties of the diamond Pythagorean fuzzy set sets utilized in this work.

Definition 8. Let us have a fixed universe E and its sub-set . The set

where and is called Dia‑PyFS and functions indicate the degree of membership (validity, etc.) and non-membership (non-validity, etc.) of element to a fixed set . Now, we can define also function by means of

and it corresponds to degree of indeterminacy (uncertainty, etc.), see Figs 1 and 2.

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Fig 2. Comparison between Dia-IFS and Dia-PyFS.

https://doi.org/10.1371/journal.pone.0325018.g002

On the other hand can also be defined by using following approach such that.

Let . Then,

Where

Note that, if we want to cover the diamond-Pythagorean fuzzy interpretation triangle, then , see Fig 3.

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Fig 3. Triangular coverage of different values of Dia-PyFS.

https://doi.org/10.1371/journal.pone.0325018.g003

Here is the restriction of Definition 8; however, in this case, the Pythagorean fuzzy interpretation triangle cannot be fully covered.

Definition 9. Let us have a fixed universe E and its sub-set . The set

where and with is called Dia-PyFS and functions indicate the degree of membership (validity, etc.) and non-membership (non-validity, etc.) of element to a fixed set . Now, we can define also function by means of

and it corresponds to degree of indeterminacy (uncertainty, etc.), see again Figs 1–2 and 4.

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Fig 4. Triangular coverage of different values of Dia-PyFS.

https://doi.org/10.1371/journal.pone.0325018.g004

3.1. Development of diamond Pythagorean fuzzy sets via Pythagorean fuzzy sets

In this section, we will discuss the procedure of calculating the Dia‑PyFS in order to convert PyFS to Dia‑PyFS.

Assume that there are Pythagorean fuzzy pairs in an PyFS with the following shapes: , where m is a numerical value of an PyFS that contains ni, the number of Pythagorean fuzzy pairs with .

The following formula can be used to determine the “arithmetic average” of diamond Pythagorean fuzzy pairs:

The of has the maximum Euclidean distance value.

thus, PyFS is being changed into Dia‑PyFS.

Definition 10. A diamond Pythagorean fuzzy set is a collection of

where represent the diamond Pythagorean fuzzy value with conditions;

(i) , (ii) , (ii) .

For the sake of simplicity, the set of diamond Pythagorean fuzzy value (Dia‑PyFVs).

In order to facilitate collective decision-making, we now create a mechanism for converting collections of PyFSs into a Dia‑PyFVs.

Proposition 1. Let a set of PyFVs be denoted as . Then

is a Dia‑PyFV with

and

Proof. Since and , then we have

Furthermore, it is obvious that . Note that, for Definition 9, we have

Example 1. The following sets of PyFSs are represented as:

and

With the help of Proposition 1, we find the corresponding Dia‑PyFSs, we have

3.2. Distance measures for Dia‑PyFSs via PyFSs

The next outcomes are introduced for representing different distances over Dia‑PyFSs via Subsection 3 approach.

Definition 11. Let d be a cardinality of E. Then normalized Euclidean distance for two Dia‑PyFSs and is defined as

where q = 1, 2. If q = 1 and q = 2, then distance is known as Hamming distance and Euclidean distance for Dia‑PyFSs, respectively.

Definition 12. Let d be a cardinality of E. Then normalized Euclidean distance for two Dia‑PyFSs and is defined as

where q = 1, 2. If q = 1 and q = 2, then distance is known as Szmidt and Kacprzyk’s form of Hamming distance and, Szmidt and Kacprzyk’s form of Euclidean distance for Dia‑PyFSs, respectively.

4. Diamond Pythagorean fuzzy set related basic operations and relations via max and min approach

The analogs of complement, union, and intersection operations in set theory are defined as follows. Their concepts are inspired by similar definitions within the foundational Dia‑PyFS framework.

For the sake of easy understanding, we will take the following three Dia‑PyFSs over fixed universe :

4.1. Operations

Here is some basic operations between two Dia‑PyFSs and are the following:

Definition 13. Let and be two Dia‑PyFSs. Then,

  1. i. ,
  2. ii. ,
  3. iii. ,
  4. iv. ,
  5. v. ,
  6. vi. ,
  7. vii. ,
  8. viii. ,
  9. ix.
  10. x. ; ,
  11. xi.

4.2. Relations

The relations over Dia‑PyFSs are firstly proposed as follows:

Definition 14. Let and be two Dia‑PyFSs. Then, for all , we have

  • iff
  • iff
  • iff
  • iff
  • iff .

From Definitions 13 and 14, we have concluded the following results:

Proposition 2. Let , and be two Dia‑PyFSs. Then, following properties holds such that

  • , and ,
  • , and ,

Proof. The proof follows similar steps as those used in the operations of PyFSs, and as such, is excluded for brevity.

Proposition 3. Let , and be three Dia‑PyFSs. Then, following properties holds such that

  1. 1) ,and ,
  2. 2) , and ,
  3. 3) ,and ,
  4. 4) , and ,
  5. 5) and ,
  6. 6) and ,
  7. 7) and ,
  8. 8) and ,
  9. 9) and ,
  10. 10) and

Proof. The proof follows a process similar to that used in the operations of PyFSs and is therefore omitted for the sake of brevity.

To keep things concise, it is noted that for the upcoming results, .

5. Diamond Pythagorean fuzzy set related operations

The following definitions pertain to arithmetic operations for Dia‑PyFSs via 𝓉-norm and 𝓉-conorm, including addition, multiplication, and exponentiation:

Definition 15. Let and be two Dia‑PyFSs. Assume that, in the context of diamond Pythagorean fuzzy complement, with as the norm or conorm, and as the dual 𝓉-norm and 𝓉-conorm, respectively. The general algebraic operations among Dia‑PyFSs are defined as follows:

  1. 1) ,
  2. 2) .

It is clear that the operations presented in Definition 15 are based on those outlined in Definition 13, with particular selections for , , and .

Proposition 4. Let and be two Dia‑PyFSs. Assume that, in the context of Pythagorean fuzzy complement, with as the norm or conorm, and as the dual 𝓉-norm and 𝓉-conorm, respectively. Then and are also Dia‑PyFS.

Proof. Sine S is a dual 𝓉-conorm corresponding to Pythagorean fuzzy complement N, then . We know that and is nondecreasing, then we have

Moreover, since the domain of ϙ is the closed unit interval, we conclude that is a Dia‑PyFS. It can also be shown that is a Dia‑PyFV.

As a result, algebraic operations on Dia‑PyFVs can be defined using the additive generators of strict Archimedean 𝓉-norms and 𝓉-conorms.

Definition 16. Let , and assume that and are two Dia‑PyFVs. Suppose that the additive generator of a continuous Archimedean 𝓉-norm is , and the additive generator of a continuous Archimedean 𝓉-norm or 𝓉-conorm is , with . The following definitions describe algebraic operations for Dia‑PyFV:

  1. i.
  2. ii.
  3. iii.
  4. iv.

The following statement verifies that Dia‑PyFVs are also multiplication by constant and power of Dia‑PyFVs.

Proposition 5. Let , and assume that and are two Dia‑PyFVs. Suppose that the additive generator of a continuous Archimedean 𝓉-norm is , and the additive generator of a continuous Archimedean 𝓉-norm or 𝓉-conorm is , with . Then , , and .

Proof. The Proposition 5 makes it obvious that and are . It is well known that and . Now, and are non-decreasing, then

Moreover, since the domain of ϙ is the closed unit interval, we conclude that is a Dia‑PyFS. It can also be shown that is a Dia‑PyFV.

Example 2. Assume that characterized by , and . The algebraic operators are then obtained

  1. a)
  2. b) ,
  3. c)
  4. d)
  5. e)
  6. f)
  7. g)
  8. h)

The following theorem presents some fundamental properties of algebraic operations.

Theorem 2. Let , and are three Dia‑PyFVs with . Suppose that the additive generator of a continuous Archimedean 𝓉-norm is , and the additive generator of a continuous Archimedean 𝓉-norm or 𝓉-conorm is , with . Then, followings hold such that

  1. 1) ,
  2. 2) ,
  3. 3) ,
  4. 4) ,
  5. 5) ,
  6. 6) ,
  7. 7)
  8. 8)

Proof. Obviously, (1), (2), (3) and (4) hold. For (5), we have

For (6), we have

For (7), we have

For (8), we have

6. Aggregation operators via Dia‑PyNs

In data consolidation, especially in decision-making scenarios, the goal of aggregation is to generate a data summary before arriving at a final decision. This section discusses the Dia‑PyF-weighted averaging and geometric aggregation operators. It is important to note that Dia‑PyFVs on E are denoted by Dia‑PyFV(E). Additionally, we calculate the Euclidean distance measure between Dia‑PyFVs.

6.1. Diamond Pythagorean fuzzy weighted averaging aggregation operators

Definition 17. Let be the set of Dia‑PyFVs. Suppose that the additive generator of a continuous Archimedean 𝓉-norm is , and the additive generator of a continuous Archimedean 𝓉-norm or 𝓉-conorm is , with . Then, Dia‑PyFWAA operator with mapping is computed as follows

with weight vector with and .

Theorem 3. Let be the set of Dia‑PyFVs. Suppose that the additive generator of a continuous Archimedean 𝓉-norm is , and the additive generator of a continuous Archimedean 𝓉-norm or 𝓉-conorm is , with . If Dia‑PyFWAA operator is defined with the help of this transformation , then is Dia‑PyFV and we have

with weight vector with and .

Proof. By hypothesis, is a Dia‑PyFV. By utilizing mathematical induction, it can be seen that the second part is also true. If n = 2, then we have

Let us temporarily assume that the following expression hold such that

We now have

That concludes the proof.

Corollary 1. Assume that characterized by, and . The algebraic Dia‑PyFWAA operators are then obtained such that

and

6.2. Diamond Pythagorean fuzzy weighted geometric aggregation operators

Definition 18. Let be the set of Dia‑PyFVs. Suppose that the additive generator of a continuous Archimedean 𝓉-norm is , and the additive generator of a continuous Archimedean 𝓉-norm or 𝓉-conorm is , with . Then, diamond Pythagorean fuzzy weighted geometric aggregation (Dia‑PyFWGA) operator with mapping is computed as follows

with weight vector with and .

Theorem 4. Let be the set of Dia‑PyFVs. Suppose that the additive generator of a continuous Archimedean 𝓉-norm is , and the additive generator of a continuous Archimedean 𝓉-norm or 𝓉-conorm is , with . If Dia‑PyFWGA operator is defined with the help of this transformation , then is Dia‑PyFV and we have

with weight vector with and .

Proof. By using same arguments like Theorem 4, it can be proven.

Corollary 2. Assume that characterized by, and σ(𝓉) = -log (1–𝓉2). The algebraic Dia‑PyFWGA operators are then obtained such that

and

6.3. Distance measures via Dia‑PyFSs

The next outcomes are introduced for representing different distances over Dia‑PyFSs via Subsection 3 approach.

Definition 19. Let d be a cardinality of E. Then normalized Euclidean distance for two Dia‑PyFSs and is defined as

where q = 1, 2. If q = 1 and q = 2, then distance and are known as Hamming distance and Euclidean distance for Dia‑PyFSs, respectively.

Definition 20. Let d be a cardinality of E. Then normalized Euclidean distance for two Dia‑PyFSs and is defined as

where q = 1, 2. If q = 1 and q = 2, then distance and are known as Szmidt and Kacprzyk’s form of Hamming distance and, Szmidt and Kacprzyk’s form of Euclidean distance for Dia‑PyFSs, respectively.

7. Applications of Dia‑PyFSs in MCDM scenarios via CODAS technique

Input:

Step 1: A team of decision-makers (DMrs) should be formed using Dia‑PyF data in the Dia‑PyFS format for an appropriate number of alternatives and attributes, denoted as lj (∈ ℕ). Here, = {, , ..., , } represents a group of experts. The preferences of each experts are evaluated using Dia‑PyFVs. Thus, the decision data, expressed as Dia‑PyFVs, are arranged in the decision matrix (M), including , , and , along with the corresponding weighting vector .

Step 2: Standardization of Dia‑PyF data inputs:

To achieve optimal and precise results, it is essential to normalize the input data before proceeding with further calculations. As a result, the Dia‑PyF analysis can be standardized by

In this case, since the input data for all attributes is identical, there is no need to normalize the data. All alternatives and criteria in our specific problem are of the same nature.

Step 3: Using the Proposition 1, we find Dia‑PyF decision matrix (M) based on the decision data provided in matrix Mk (k = 1, 2, 3) in order to determine the alternatives’ by with help of Proposition 1 overall preference values, Ąi.

Step 4: To complete the data, experts first assign weights to each criterion. This ensures that the final decision reflects the collective input of all health experts.

Step 5: To calculate the weighted Dia‑PyF decision matrix, we use following equation for the weight vector (j = 1, 2, 3, 4, 5) such that:

Step 6: Determine the solution that identify the diamond Pythagorean fuzzy ideal solution (FIS) by selecting the maximum value for each criterion across all alternative

Step 7: For each alternative, compute the Dia‑PyF -Hamming and Dia‑PyF -Euclidean distance from the FIS.

And

Or

and

where .

Step 8: Construct the relative assessment matrix (RAM) using the calculated values of and from the previous step as follows:

,

Where Λik represents a threshold function used to determine whether the Euclidean distances of two alternatives are equivalent. It is defined as follows:

Here, γ represents a threshold function used to determine whether the Euclidean distances of two alternatives are equivalent.

Step 9: The attribute with the highest score receives the top rank via RAM and should be selected as the final choice such that:

7.1. Selection of electric auto rickshaw via Dia‑PyFSs

This section presents a practical case study focused on evaluating and selecting a location for an electric auto rickshaw shredding facility in the Asian countries. The developed Dia‑PyF CODAS method is applied to determine the optimal facility location from a set of four possible alternatives, considering a range of conflicting quantitative and qualitative evaluation criteria.

An electric auto rickshaw, or e-rickshaw, is a three-wheeled electric vehicle well-suited for short-range transportation within urban and suburban areas. It presents an environmentally friendly alternative to traditional gasoline or diesel-powered rickshaws by minimizing both air pollution and noise in crowded cities. Powered by rechargeable batteries-typically lithium-ion or lead-acid—these vehicles generally have a range of 50–120 kilometers on a full charge, depending on the model and battery size. Choosing an electric auto-rickshaw offers several environmental and economic advantages. As they produce zero tailpipe emissions, electric rickshaws help improve air quality, especially in high-density areas. They are also quieter than their fuel-powered counterparts, making them a less intrusive option for urban transportation. With fewer moving components, electric auto-rickshaws incur lower maintenance expenses over time, and the cost of recharging the battery is often significantly lower than refueling with gasoline or diesel. Electric auto-rickshaws are generally designed for low to moderate speed, typically achieving speeds between 25 and 35 km/h, which is adequate for short, frequent city trips. With seating for two to four passengers, depending on the model, they are efficient for local commuting. When selecting an electric auto-rickshaw, potential buyers should consider charging infrastructure, as it affects convenience and operational range. In some areas, dedicated charging stations are available, while in others, e-rickshaws can be recharged at home overnight or utilize battery-swapping systems at specific locations. Opting for an electric auto-rickshaw provides several benefits, including reduced environmental impact, minimal noise, and cost savings for both drivers and riders. Many governments support electric vehicle use through incentives, subsidies, or tax breaks, promoting them as part of clean energy and sustainable mobility initiatives. However, there are factors to consider before selecting an electric auto-rickshaw. Limited battery capacity and charging durations may constrain travel range, and charging facilities may not be evenly distributed across regions. Additionally, these vehicles are usually lightweight and best suited to paved roads, which can limit their performance on rough surfaces or with heavier loads. Electric auto-rickshaws have gained significant traction in South Asian nations like Pakistan, Sri lanka, Bangladesh, Nepal, and India, where they serve as an affordable and eco-friendly transportation option. In densely populated urban centers, electric rickshaws meet the demand for economical, low-emission transit, making them an effective solution for sustainable urban transportation.

Four three-wheeled electric auto-rickshaws (Ą1, Ą2, Ą3, and Ą4) were evaluated by three experts (DM1, DM2, and DM3) based on five criteria (, , , , and ) such that:

Standard Passenger Electric Auto-rickshaws : These are the most common types of electric auto-rickshaws, designed for short-distance travel in urban and semi-urban areas. They typically carry 2–4 passengers and have a compact body that allows them to navigate through narrow lanes and crowded streets. These vehicles have an open body design, with no doors or windows, offering a more affordable option. They are commonly used in areas with warm climates and are suitable for short-distance travel in cities and towns.

Deluxe or Premium Passenger Electric Auto-rickshaws : These vehicles are designed to provide more comfort and amenities, such as cushioned seating, better suspension, air conditioning, and sometimes entertainment systems. They may also have more spacious interiors and are often used for premium services or tourist transport. These auto-rickshaws feature closed cabins, offering more protection from weather elements such as rain or extreme heat. The enclosed body provides a more comfortable ride, making them suitable for urban areas with variable weather conditions.

Convertible Passenger Electric Auto-rickshaws : These vehicles allow for flexibility in usage. The rear section can be adjusted or converted to accommodate more passengers or cargo, depending on demand. This type is useful for operators who wish to switch between passenger transport and cargo delivery without requiring multiple vehicles.

Solar-Powered Passenger Electric Auto-rickshaws : These models incorporate solar panels on the roof, which help recharge the battery during the day, extending the vehicle’s range. Solar-powered e-rickshaws are particularly useful in regions with abundant sunlight and can reduce operating costs by using renewable energy.

Following are the criteria that will help to find best alternative such that:

  1. Battery-Powered (): Electric auto-rickshaws are powered by rechargeable batteries, commonly lithium-ion or lead-acid batteries. Battery capacity varies, with a typical range of around 50–120 kilometers per full charge, depending on the model and battery type.
  2. Environmentally Friendly (): E-rickshaws produce zero tailpipe emissions, making them an eco-friendly solution in cities striving to reduce pollution levels. With electricity as their fuel source, they contribute less to air pollution and are quieter than traditional fuel-powered auto-rickshaws.
  3. Speed and Performance (): Typically, e-rickshaws are designed for low-speed travel, with maximum speeds ranging from 25–35 km/h (15–22 mph). They are optimized for short, intra-city routes, carrying 2–4 passengers over relatively short distances.
  4. Charging Infrastructure (): The expansion of charging stations is critical for e-rickshaws to operate efficiently in large numbers. In many regions, charging infrastructure is being developed specifically for these vehicles. Some e-rickshaws can be charged overnight at home, while others may use swappable battery systems at designated stations for convenience.
  5. Load Capacity (): Electric auto-rickshaws are generally designed for lightweight transportation and may not be as robust as traditional models when carrying heavier loads or navigating rough terrains. The suggested algorithm's geometrical interpretation is presented in the form of Fig 5.
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Fig 5. Flow chart of the proposed algorithm.

https://doi.org/10.1371/journal.pone.0325018.g005

7.2. Using Algorithm 1 to solve

Step 1: We apply this algorithm to process the input data, where three medical professionals are tasked with evaluating four emergency options, Ąi (i = 1, 2, 3, 4), in relation to five criteria (i = 1, 2, 3, 4, 5). For the construction of the decision matrices for the three experts, DMζ (ζ = 1, 2, 3), and the criteria , the PyFVs for Linguistic terms are given in Table 1.

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Table 1. PyFVs for Linguistic terms.

https://doi.org/10.1371/journal.pone.0325018.t001

Step 2: The expert assessments for decision-making have been collected and recorded using PyF phrases, as shown in Tables 2–. The PyFDM, outlined in Tables 5– are derived from Tables 2–. respectively, using the PyFVs defined in Table 1.

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Table 2. Linguistic decision matrix DM1 via PyFVs.

https://doi.org/10.1371/journal.pone.0325018.t002

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Table 3. Linguistic decision matrix DM2 via PyFVs.

https://doi.org/10.1371/journal.pone.0325018.t003

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Table 4. Linguistic decision matrix DM3 via PyFVs.

https://doi.org/10.1371/journal.pone.0325018.t004

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Table 5. Decision matrix DM1 via PyFVs.

https://doi.org/10.1371/journal.pone.0325018.t005

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Table 6. Decision matrix DM2 via PyFVs.

https://doi.org/10.1371/journal.pone.0325018.t006

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Table 7. Decision matrix DM3 via PyFVs.

https://doi.org/10.1371/journal.pone.0325018.t007

Step 3: Next, we calculate the combined Dia‑PyF information using the Proposition 1, as detailed in Table 8.

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Table 8. Collective decision matrix (DM) via Dia‑PyFVs.

https://doi.org/10.1371/journal.pone.0325018.t008

Step 4: The five weight vectors for the Dia‑PyF data are presented as follows:

The weights are calculated using the formula given in Step 4, resulting in the final weight vector: . This vector clearly satisfies the condition .

Step 5: We determine the Dia‑PyFDM using the weight vector . Consequently, we obtain the weighted Dia‑PyFDM decision matrices, as shown in Table 9.

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Table 9. Weighted decision matrix (DM) via Dia‑PyFVs.

https://doi.org/10.1371/journal.pone.0325018.t009

Step 6: For diamond Pythagorean fuzzy ideal solution, we have

Step 7: Next, we calculate the Hamming and Euclidean distances of each alternative Ąi from FIS. We represent these distances as and , with the results as follows:

and

Step 8: The relative assessment matrix is derived from the above data using equations in previous Step 8. Here, threshold parameter is set at a value of 0.15.

Step 9: The score of four potential options are determine by summing of each of the row such that R1 = −0.079, R2 = −0.309, R3 = 251, and R4 = .320 and then are prioritized according to the decreasing values of their assessment scores. The order of ranking is as follows: Ą4 ≻ Ą3 ≻ Ą1 ≻ Ą2. Thus, based on the diamond Pythagorean fuzzy CODAS approach, “solar-powered passenger electric auto-rickshaws” is identified as the most suitable ride for the people in the Asian country.

As this is pioneering research, there are currently no alternative solutions available for direct comparison using a pure Dia‑PyF framework. Therefore, this section contrasts the proposed methodologies with established MCDM approaches, emphasizing their application in a PyF environment. Since both Dia‑PyF expand upon traditional PyFs, we can obtain a PyF from any Dia‑PyF by setting ℵ = 0 across all elements. Applying this approach to the data from our case study yields results also summarized in Table 10. It is observed that the ranking of alternatives changes and is also influenced by the specific MCDM method used. However, our methodologies offer the advantage of fully utilizing the information provided by experts, as they operate within the broader Dia‑PyF environment. Notably, the Dia‑PyF CODAS method is enhanced through the use of empirically effective Hamming and Euclidean distance measures.

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Table 10. Analyzing the intended work in comparison to the classical work.

https://doi.org/10.1371/journal.pone.0325018.t010

8. Conclusion

This study introduces the Dia‑PyFS, focusing on its diamond structure as a primary feature. The Dia‑PyFS forms a diamond with a norm capped at two and a center defined by validity and non-validity values. By encapsulating data within a diamond structure, the two-dimensional Dia‑PyFS can effectively manage high-level inaccurate data. The properties of algebraic operations within Dia‑PyFSs are explored. Using the PyF t-norm and t-conorm, weighted and geometric operations for Dia‑PyFS are defined within this framework. Additionally, DM via PyFS and DM via Dia‑PyFS are examined to measure the gap between the aggregated value and the optimal solution. The proposed aggregation operators and CODAS methods provide tools for selecting an electric autorickshaw based on multiple criteria. The suggested operations and DMs are guaranteed to be reliable through comparison and visualization.

In PyFS and IVPyFS frameworks, input data consist of a point and an interval along a single dimension, whereas, in the Dia‑PyF framework, the input data form a two-dimensional diamond. This diamond approach in Dia‑PyFSs provides a broader scope for handling uncertainty compared to PyFS and IVPyFS, making it particularly useful for applications requiring high accuracy in managing imprecision. For instance, Dia‑PyFSs offer enhanced effectiveness in fields such as machine learning, deep learning, pattern recognition, and areas like medical diagnostics. Future research will primarily focus on exploring alternative t-norm and t-conorm operators for aggregating Dia‑PyFSs. Furthermore, transforming PyFS clusters into Dia‑PyFSs may allow for the definition and application of diverse distance and similarity metrics between clusters in fields like computer image analysis and medical treatment. Additionally, the diamond structure could be applied to specific values, such as orthopair fuzzy sets, and generalized fuzzy sets, such as quasi rung orthopair fuzzy sets and picture fuzzy sets. Aggregation operations and distance measures of these developed sets will also be utilized in collaborative decision-making processes.

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