Decision-making techniques based on aggregation operators and similarity measures under q-rung orthopair hesitant fuzzy connection numbers and their application

Khawar Hassan; Tanzeela Shaheen; Wajid Ali; Iftikhar Ul Haq; Nadia Bibi; Amal Kumar Adak

doi:10.55976/dma.22024127458-72

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Decision Making and Analysis, Volume 2 Issue 1, 2024 /

Original Research

Decision-making techniques based on aggregation operators and similarity measures under q-rung orthopair hesitant fuzzy connection numbers and their application

Khawar Hassan ¹ , Tanzeela Shaheen ² , Wajid Ali ³ ^* , Iftikhar Ul Haq ⁴ , Nadia Bibi ⁵ , Amal Kumar Adak ⁶

1. Department of Mathematics, Air University, PAF Complex E-9 Islamabad 44230, Pakistan
2. Department of Mathematics, Air University, PAF Complex E-9 Islamabad 44230, Pakistan
3. Department of Mathematics, Air University, PAF Complex E-9 Islamabad 44230, Pakistan
4. Department of Mathematics, Air University, PAF Complex E-9 Islamabad 44230, Pakistan
5. Department of Mathematics, Air University, PAF Complex E-9 Islamabad 44230, Pakistan
6. Department of Mathematics, Ganesh Dutt College, Begusarai, India

Correspondence: wajidali00258@gmail.com

DOI: https://doi.org/10.55976/dma.22024127458-72

Received

07 June 2024
Revised

21 August 2024
Accepted

11 September 2024
Published

29 September 2024

q-rung orthopair fuzzy sets Hesitant fuzzy sets SPA Similarity measure Decision making Optimization Medical diagnosis

Abstract

The concept of q-rung orthopair hesitant fuzzy set represents an advancement and extension of hesitant fuzzy sets, encompassing both fuzzy sets and q-rung orthopair fuzzy sets. q-rung orthopair hesitant fuzzy set characterizes a set of membership and non-membership grades within the interval <a href="#reference_list_-1">[0, 1]</a>, which enhances its adaptability compared to existing methods. This flexibility proves invaluable in providing more insightful data about various objects. The primary objective of this research is to introduce a decision-making technique in the context of q-RHF using the theory of set pair analysis (SPA). q-RHFS effectively handles ambiguous data by incorporating membership and non-membership grades, while the connection number (CN) based on SPA theory manages the intricacies of uncertainty and certainty structures by relying on "identity", "discrepancy" and "contrary" grades. Building on the relationship between q-RHFS and the connection number of set pair analysis, a comprehensive framework known as q-rung hesitant fuzzy connection number set  (qHCNs) is developed. This model not only addresses uncertainty, but also offers valuable insights. Furthermore, this research introduces similarity measures derived from qHCN and examines their advantages through illustrative examples. Additionally, a novel approach to decision modeling utilizing these measures applied to medical diagnosis is also introduced. The application of this established model contributes to an effective approach and demonstrated its soundness and efficiency. In addition, a detailed comparative study is conducted with the existing models and advantages of proposed model. The research concludes with a summary of the authors' findings, highlighting the consistency and effectiveness of their work.

<h5>1. Introduction</h5>
1.1 q-rung orthopair Fuzzy sets 
In 1965, Zadeh introduced a pivotal theory known as the fuzzy set <a href="#reference_list_0">[1]</a>. Fuzzy sets encompass elements from the universal set, each of which is assigned a degree of membership within the range <a href="#reference_list_-1">[0,1]</a>. Numerous scholars have dedicated their efforts to this theory and have applied it in various fields. For instance, Sarfraz <a href="#reference_list_1">[2]</a> harnessed the concept of fuzzy sets in the group decision-making process. Jin et al. <a href="#reference_list_2">[3]</a> extended the scope of fuzzy sets and developed Aczel-Alsina aggregation operators for multi-attribute decision making. Zimmermann <a href="#reference_list_3">[4]</a> also put forth an innovative approach for decision making in fuzzy environments. Alcantund et al. <a href="#reference_list_4">[5]</a> introduced the concept of separable fuzzy soft sets, which enables decision making involving both positive and negative attributes. Ali et al. <a href="#reference_list_5">[6,7]</a> developed a novel framework by building upon existing fuzzy models and employing aggregation operators to amalgamate information for decision systems. Torra <a href="#reference_list_7">[8]</a> introduced the influential concept of generalized fuzzy sets, which he referred to as "hesitant fuzzy sets", wherein membership grades encompass a collection of elements within the range <a href="#reference_list_-1">[0,1]</a>.
Yager <a href="#reference_list_8">[9]</a> introduced an exceptionally robust concept based on Attanssove's intuitionistic sets, which he termed "q-rung orthopair fuzzy sets" (q-ROFSs). In this set, the characterization of membership and non-membership grades is extended in a comprehensive way. Ali <a href="#reference_list_9">[10]</a> provided a new perspective on Yager's q-ROFSs concept. Furthermore, Peng et al. <a href="#reference_list_10">[11]</a> developed a set of information measures built upon q-rung orthopair fuzzy sets. Khan et al. <a href="#reference_list_11">[12]</a> delved deeper into the concept of measures for q-ROFSs. Wang <a href="#reference_list_12">[13]</a> introduced generalized similarity measures utilizing cosine functions for q-ROFSs. Li et al. <a href="#reference_list_13">[14]</a> further explored this idea, focusing on preference relations and their practical applications. Garg et al. <a href="#reference_list_14">[15]</a> made significant contributions by extending the q-ROFS concept, developing powerful aggregation operators and demonstrating their utility in decision-making processes. Shaheen et al. <a href="#reference_list_15">[16]</a> addressed a pivotal question in the literature: the importance of q-ROFSs and provided convincing arguments to support their significance. Oraya et al. <a href="#reference_list_16">[17]</a> showed how this concept can be applied to assess the impact of delays on residential construction projects. Wang et al. <a href="#reference_list_17">[18]</a> further expanded upon Yager's concept by introducing p,q-rung orthopair fuzzy sets and optimized this model. Farid et al. <a href="#reference_list_18">[19]</a> established Aczel-Alsina aggregation operators for multi-criteria decision-making scenarios. Feng <a href="#reference_list_19">[20]</a> explored a probabilistic approach for q-rung sets that helps in the selection of the most appropriate option. Razzaque et al. <a href="#reference_list_20">[21]</a> defined mathematical rings within the q-rung fuzzy environment. Sarkar et al. <a href="#reference_list_21">[22]</a> made a significant contribution by developing a combined framework encompassing q-rung orthopair fuzzy sets and weighted dual hesitant fuzzy sets and applying it to multi-attribute decision-making scenarios. Recently, Jabeen et al. <a href="#reference_list_22">[23]</a> integrated various aggregation operators, including the Bonferroni aggregator and the Aczel-Alsina aggregator, into the q-rung orthopair fuzzy model, exemplifying its practical application.
1.2 Set pair analysis
Zhao et al. <a href="#reference_list_23">[24]</a> introduced the concept of Set Pair Analysis (SPA) as a powerful tool to address data uncertainty. Recognizing the significance of this approach, Decai et al. <a href="#reference_list_24">[25]</a> harnessed the number of connections within Set Pair Analysis to tackle uncertainties by techniques of network planning. Jiang et al. <a href="#reference_list_25">[26]</a> employed this concept in a comprehensive evaluation for urban planning projects. Su et al. <a href="#reference_list_26">[27]</a> further expanded the theory of Set Pair Analysis and applied it to assess the health of urban ecosystems. Ali et al. <a href="#reference_list_27">[28]</a> innovatively combined intuitionistic hesitant fuzzy sets with the connection number from Set Pair Analysis and used this hybrid approach for ranking the best alternatives in multi-criteria decision-making scenarios. Zou et al. <a href="#reference_list_28">[29]</a> utilized the variable fuzzy sets model of Set Pair Analysis and fuzzy AHP to calculate flood risk assessments. Chen et al. <a href="#reference_list_29">[30]</a> focused on models for evaluating water ecological security based on multivariate connection numbers and Markov chains. Xiang et al. <a href="#reference_list_30">[31]</a> conducted a comprehensive literature survey spanning from 1989 to 2020 to investigate the development, applications and challenges of Set Pair Analysis in environmental sciences. Zhao et al. <a href="#reference_list_31">[32]</a> explored a novel application of Set Pair Analysis, especially in the identification of mine water sources using the AHP-entropy method. Shi et al. <a href="#reference_list_32">[33]</a> applied this concept to pattern recognition. Pan et al. <a href="#reference_list_33">[34]</a> developed a methodology to evaluate coordination in water resources management and to effectively address conflicts. Ma et al. <a href="#reference_list_34">[35]</a> proposed a model grounded in Set Pair Analysis for comprehensive evaluation of information security in electric power. Yang et al. <a href="#reference_list_35">[36]</a> enhanced the Set Pair Analysis model for urban water security assessment. Chen et al. <a href="#reference_list_36">[37]</a> introduced a model for selecting the optimal evaluation schemes based on the framework of Set Pair Analysis. Finally, Zhou et al. <a href="#reference_list_37">[38]</a> extended the use of Set Pair Analysis in the coupling model for health evaluations of the Huangchuan River.
1.3 Similarity measures
Similarity measures serve the purpose of quantifying the extent of similarity or likeness between two sets, facilitating an evaluation of how closely or comparably these sets align with respect to the attributes of their constituent elements. The choice of a particular similarity measure hinges upon the specific context and the inherent characteristics of the sets to be compared. Diverse similarity measures can be found in the literature, each with its unique applications and constraints. Kausar et al. <a href="#reference_list_38">[39]</a> delved into cosine similarity measures and applied them to enhance the sustainability of solid waste management within the framework of cubic m-polar fuzz sets. Ozhu et al. <a href="#reference_list_39">[40]</a> extended the notion of the dice similarity measure to single-valued neutrosophic type-2 hesitant fuzzy information and employed it in multi-attribute decision-making scenarios. Garg <a href="#reference_list_40">[41]</a> also delved into dice similarity measures and generalized them with dummy mean operators for application in TOPSIS models. Mehmood et al. <a href="#reference_list_41">[42-43]</a> developed cosine similarity measures and aggregation operators for IHFSs and trigonometric similarity measures tailored to bipolar complex fuzzy soft sets and harnessed their approach in pattern recognition and medical diagnosis. Ma et al. <a href="#reference_list_43">[44]</a> adopted similarity measures for extracting potential routes in urban customized bus systems, relying on vehicle trajectory clustering. Jia et al. <a href="#reference_list_44">[45]</a> explored the relationship between Pythagorean fuzzy sets and similarity measures in their research work. Jin et al. <a href="#reference_list_45">[46]</a> extended the concept of picture fuzzy distance and similarity measures within the framework of complete lattices and presented their practical applications. Kirisci et al. <a href="#reference_list_46">[47]</a> introduced innovative cosine similarity measures and distance metrics for the extension of fuzzy sets and  explored preference techniques. Saqlain et al. <a href="#reference_list_47">[48]</a> introduced a set of similarity and distance measures for intuitionistic fuzzy hypersoft sets and presented practical applications. Ali et al. <a href="#reference_list_48">[49]</a> developed the concept of vector similarity measures for dual hesitant fuzzy linguistic term sets and explored their applications. Recently, Rahim et al. <a href="#reference_list_49">[50]</a> enhanced the TOPSIS method with an improved cosine similarity measure tailored to Fermatean fuzzy sets. Ganie et al. <a href="#reference_list_50">[51]</a> ventured into the realm of q-rung orthopair fuzzy sets and explored novel similarity measures and entropy metrics as well as their practical applications. Zhang et al. <a href="#reference_list_51">[52-53]</a> applied the concept of exponential similarity measures within the domain of cubic sets with confidence neutrosophic number, particularly in fuzzy multi-valued scenarios, to develop a group decision-making model.
1.4 Motivation 
Based on an extensive literature review and recognizing the significance of fuzzy sets, SPA, and similarity measures in various domains, we have drawn inspiration to establish a relationship and amalgamate these frameworks to address issues related to uncertainty more comprehensively. It is evident that the amalgamation of these frameworks has the potential to address challenges in a broader range of scenarios where individual structures may fall short. The contributions of this research are elaborated as follows:
<ul>
<li>In this research, we introduced the concept of hesitant fuzzy setsof q-rung orthopair by merging q-rung orthopair fuzzy sets with hesitant fuzzy sets. Hesitant fuzzy sets of q-rung orthopair represent a more general and effective tool for handling ambiguity and uncertainty.</li>
<li>We further enhanced the framework by integratingthe concept of connection number from Set Pair Analysis into the q-rung orthopair hesitant fuzzy environment. This approach led to the development of a new theory known as q-rung orthopair hesitant fuzzy connection number (qHCN) .</li>
<li>The paper also extended the concept of Jaccard similarity measures and dice similarity measures.We introduced weighted Jaccard similarity measures and weighted dice similarity measures while discussing their key properties.</li>
<li>These innovative concepts and measures are applied to solve real-life problems, primarily focusing on assessing the degree of similarity. A detailed flowchart of the entire algorithm is presented, and practical applications demonstratethe effectiveness of the models we have</li>
</ul>
The remainder of the paper is structured as follows: Section 2 provides the foundational definitions and ideas that underpin our proposed approach. In Section 3, we delved into the development of the concept of q-rung orthopair hesitant fuzzy connection numbers and outline fundamental operations related to the model. Section 4 delves into the similarity measures for the proposed model and elucidate their properties. Section 5 and Section 6 encompasses the application of the developed algorithm to counsel students based on the evaluation of their study subjects. Finally, Section 7 concludes the paper with the authors' findings and recommendations.
<h5>2. Preliminaries</h5>
In this section, a brief overview of the existing fuzzy models such as IHFSs, q-ROFSs, and Set Pair Analysis theory with some basic operations are discussed.
Definition 1: <a href="#reference_list_41">[42]</a> An IHFS H on x is denoted by two mappings π and ω. It is indicated by following expression: 
H = {(k, π (k), ω (k)) | k ϵ x}
π (k) and ω (k) are sets of certain values in <a href="#reference_list_-1">[0, 1]</a>, representing the grades for membership and non-membership of the element k ϵ x with the condition such that 0 ≤ max (π (k)) + max (ω (k)) ≤ 1.
Definition 2: <a href="#reference_list_41">[42]</a> Score and accuracy function for IHFS H = (πH , ωH ) are defined as:
<img src="https://file.luminescience.cn/66f6506f63ca21727418479" alt="" width="300" height="199" />
Definition 3: <a href="#reference_list_41">[42]</a> For two IHFSs H1 and H2, H1 > H2 where " > " refers preferred to if any of the given condition fulfilled,
 S (H1) > S (H2) ; S (H1) = S (H2) and T (H1) > T (H2);
Definition 4: <a href="#reference_list_8">[9]</a> For a universal set x, A q-rung orthopair fuzzy set (q-ROFS) H over x is formulated as below. H = (k, πH (k), ωH (k) : k ϵ x ) where π : x → <a href="#reference_list_-1">[0, 1]</a> and ω: x → <a href="#reference_list_-1">[0, 1]</a> are respectively the functions granted to assign the degrees of membership and non-membership such that 0 ≤ (πH (k))B + (ωH (k))B ≤ 1, B ≥ 1 Definition 5: The score function to rank the q-ROFN H = （ π，ω）is defined as V (H) = (πB - ωB) Also, an accuracy function is defined as L (H) = (πB + ωB) It is evident that -1 ≤ V (H) ≤ 1 and 0 ≤ L (H) ≤ 1. Definition 6: <a href="#reference_list_23">[24]</a> A connection number between two sets P and Q is defined and represented as, <img src="https://file.luminescience.cn/66f652d41be2f1727419092" alt="" width="200" height="88" />
Where U is the "total number of features" where L and M respectively denotes the "identity" and "contrary" features. M = U-L-N is the "discrepancy" feature of sets P and Q. It is also signified in the following equations E = L/U , F = M/U and G = N/U , which denote the identity, discrepancy and contrary degree, respectively. Clearly, 0 ≤ E, F, G ≤ 1 and E + F +G =1 . Also, i ϵ <a href="#reference_list_-2">[-1,1]</a> is coefficient of "discrepancy degree" and j is the coefficient of "contrary degree" and j = -1.
 Definition 7: <a href="#reference_list_27">[28]</a> The intuitionistic hesitant fuzzy CN set (IHFCNS) corresponding to IHFE H = {(k, π (k), ω (k)) | k ϵ x} is described as ZH = {(k, EH (k) + FH (k) i) + GH (k) j | k ϵ x}
<img src="https://file.luminescience.cn/66f654112df9d1727419409" alt="" width="350" height="117" />
Where al ϵ πH (k) and bl ϵ ωH (k) signified the "identity", "discrepancy" and "contrary" degrees, respectively.
<h5>3. A novel proposed model q-rung orthopair hesitant fuzzy connection number</h5>
In this section, we proposed a novel generalized structure by adding three different frameworks, q-rung orthopair fuzzy sets, hesitant fuzzy sets and the concept of connection number of set pair analysis. This attractive and powerful model is known as connection number of q-rung orthopair hesitant fuzzy. Moreover, some basic operations were also discussed.
Definition 8: An q-RHFS H on x is denoted by two mappings π and ω. It is indicated by following equation: H = {(k, πH (k), ωH (k))| k ϵ x } πH (k) and ωH (k) are sets of certain values in <a href="#reference_list_-1">[0, 1]</a>, representing the membership grades and non-membership grades of the element k ϵ x with the condition such that
<img src="https://file.luminescience.cn/66f6552f5cbd21727419695" alt="" width="350" height="43" />
Definition 9: The q-rung orthopair hesitant fuzzy CN set (qHCNS) corresponding to q-RHFN H = {(k, πH (k), ωH (k))| k ϵ x } is described as ZH = {(k, EH (k) + FH (k) i) + GH (k) j : k ϵ x}  Where,
<img src="https://file.luminescience.cn/66f656d9b66e61727420121" alt="" width="550" height="119" />
Where al ϵ πH (k), bl ϵ ωH (k) and B ≥ 1 signified the "identity", "discrepancy" and "contrary" degrees, respectively.
Example 1: Let H1 = {(0.8, 0.3), (0.4, 0.6)} and H2 = { (0.5, 0.3), (0.4, 0.1)} be two q-ROHFNs and for B = 6, we have two qHCNs such that,
ZH1 = {0.0218 + 0.9740i + 0.0041j} ZH2 = {0.0013 + 0.9983i + 0.0003j}
Definition 10: For two qHCNs Z1 (k) = E1 (k) + F1 (k)i + G1 (k)j and Z2 (k) = E2 (k) + F2 (k)i + G2 (k)j, we have Z1 (k) = Z2 (k) ⇔ E1 (k) = E2 (k), F1 (k) = F2 (k), G1 (k) = G2 (k) Z1 (k) ≤ Z2 (k) if E1 (k) ≤ E2 (k), G1 (k) ≥ G2 (k). Z1 (k)C = G1 (k) + F1 (k)i + E1 (k) j is the complement of the qHCN Z1 (k) = E1 (k) + F1 (k)i + G1 (k)j
Definition 11: If Z1 = E1+ F1i +G1j and Z2 = E2 + F2i +G2j are two qHCNs, then
<img src="https://file.luminescience.cn/66f66619de8d01727424025" alt="" width="500" height="87" />
and are also qHCNs.
<h5>4. Power aggregation operators for q-rung orthopair hesitant fuzzy connection number</h5>
In this section, we defined the aggregation operators to calculate qHCNs and developed a series of power aggregation operators for qHCNs such as power average qHCNs aggregation operators (qHCPA), power weighted average qHCN aggregation operators (qHCPWA), power ordered weighted average qHCN aggregation operators (qHCPOW), power geometric qHCN aggregation operators (qHCPG), power weighted geometric qHCN aggregation operators (qHCPWG), and power ordered weighted geometric qHCN aggregation operators (qHCPOWG). Furthermore, the characteristics and properties of these operators were also discussed.
4.1 Power average aggregation operators for q-rung orthopair hesitant fuzzy connection numbers
Definition 12: Suppose that Zi (k) (i ϵ N) is be the collection of q-RHCNs then power average operator will be a mapping qHCPA: Zin → Z and it is defined as
<img src="https://file.luminescience.cn/66f66749d23a21727424329" alt="" width="500" />
wherever, T (Zi) is the Sup of ith the biggest qHCE (Zi) by all the other qHCEs, that is,
<img src="https://file.luminescience.cn/66f667d82acbf1727424472" alt="" width="200" height="61" />
here Sup (Zi, Zj) is the Sup for Zi from Zj, and it is calculated by, Sup (Zi, Zj) = 1- d (Zi, Zj)
The Sup fulfills the given characteristics: Sup (Zi, Zj) ϵ <a href="#reference_list_-1">[0,1]</a> Sup (Zi, Zj) = Sup (Zj, Zi) Sup (Zi, Zj) ≥ Sup (Zs, Zt), if d (Zi- Zj) ≥ d (Zs- Zt) where d is distance. The support (Sup) amount is a similarity indicator.
 Definition 13: Let Zi be a set of qHCEs and w = (w1, w2,..., wn)T is weight vector of Zi, wi > 0 and <img src="https://file.luminescience.cn/66f66e4c460f51727426124" alt="" width="40" height="36" />
wi = 1. The power weight average operator qHCPWA: Zn → Z is defined as
<img src="https://file.luminescience.cn/66f66ed4b54ea1727426260" alt="" width="500" height="263" />
The following properties can be easily justified for the qHCPWA operators.
<img src="https://file.luminescience.cn/66f66f31c87fe1727426353" alt="" width="400" height="155" />
 Definition 14: Let Zi be a group of qHCEs, the power order weight average operator is a mapping qHCPOWA: Zn → Z and is defined as
<img src="https://file.luminescience.cn/66f66fabbbe8b1727426475" alt="" width="550" height="265" />
where σ(1), σ(2),…σ(n), indicates permutation of (1, 2, ... n), in which Zσ(i-1) ≥ Zσ(i), wi (i ϵ N) is group of weights so that
<img src="https://file.luminescience.cn/66f671a59af881727426981" alt="" width="300" height="130" />
and T (Zσ(j)) implies the Sup of jth largest qHCE T (Zσ(j)) by all the other (qHCEs), that is,
<img src="https://file.luminescience.cn/66f6720e1711a1727427086" alt="" width="250" height="86" />
where <img src="https://file.luminescience.cn/66f6732558fc51727427365" alt="" width="180" height="48" />shows the Sup of jth is the biggest qHCE Zσ(j), for the ith largest IHCE Zσ(i).
Some properties of qHCPOWA operator are as follows.
<img src="https://file.luminescience.cn/66f677108c3851727428368" alt="" width="400" height="156" />
Where Z'j is a permutation of Zj.
Definition 15: Suppose Zi is a group of qHCEs, the power hybrid average operator qHCPHA: Zn → Z is defined as
<img src="https://file.luminescience.cn/66f677922f85e1727428498" alt="" width="500" height="293" />
Where Zσ(i) is the ith largest objects in qHCE arguments
<img src="https://file.luminescience.cn/66f678c35043b1727428803" alt="" width="350" height="39" />
and it is the weighting vector of qHCE arguments Zi (i = 1, 2, ..., n), wi ϵ <a href="#reference_list_-1">[0,1]</a> and <img src="https://file.luminescience.cn/66f67b485d4d61727429448" alt="" width="50" height="40" /> wi = 1 and wi is a group such that
<img src="https://file.luminescience.cn/66f67b9b7dc881727429531" alt="" width="300" height="104" />
and <img src="https://file.luminescience.cn/66f67bf34af2e1727429619" alt="" width="60" height="31" />is the Sup of jth biggest qHCEs by all the other (qHCEs), that is,
<img src="https://file.luminescience.cn/66f67c365a9081727429686" alt="" width="250" height="86" />
Where <img src="https://file.luminescience.cn/66f67da97983b1727430057" alt="" width="180" height="49" />shows the Sup of jth biggest qHCE <img src="https://file.luminescience.cn/66f67c94c04a91727429780" alt="" width="45" height="41" />for the ith biggest qHCE <img src="https://file.luminescience.cn/66f8e95fc0c331727588703" alt="" width="48" height="36" />. Specifically, qHCPHA is reduced to qHCPWA operator if
<img src="https://file.luminescience.cn/66f8e9ab26fd31727588779" alt="" width="180" height="49" />
and qHCPHA is reduced to IHCPOWA operator if 
<img src="https://file.luminescience.cn/66f8e9db23cc81727588827" alt="" width="170" height="51" />
4.2 Power geometric aggregation operators for q-rung orthopair hesitant fuzzy connection numbers 
Definition 16: Suppose Zi is family of qHCEs, power geometric operator qHCPG: Zn→Z is defined as
<img src="https://file.luminescience.cn/66f8ea99a6e351727589017" alt="" width="500" height="253" />
where
<img src="https://file.luminescience.cn/66f8eac3aef701727589059" alt="" width="200" height="61" />
Definition 17: Let Zi be a group of qHCEs then power weight geometric operator qHCPWG: Zn →Z is defined as
<img src="https://file.luminescience.cn/66f8eb1d1c0c61727589149" alt="" width="520" height="239" />
The following characteristics can be easily proved for qHCPWG operator.
<img src="https://file.luminescience.cn/66f8eb4a573c91727589194" alt="" width="400" height="166" />
 Definition 18: Let Zi is a group of qHCEs, the power order weight geometric operator of dimension 'n', qHCPOWG: Zn →Z is defined as
<img src="https://file.luminescience.cn/66f8ebb9ec0511727589305" alt="" width="550" height="253" />
where σ(1), σ(2), …σ(n) indicates permutation of (1,2, …, ), where Zσ (i-1) ≥ Zσ(i), wi (i = 1, 2,...n) is a group of weights so that,
<img src="https://file.luminescience.cn/66f8ec4078fb31727589440" alt="" width="300" height="131" />
and T (Zσ(i)) implies the Sup of ith biggest qHCE Zσ(i) by all the other qHCEs, that is,
<img src="https://file.luminescience.cn/66f8ece302ce01727589603" alt="" width="250" height="85" />
where <img src="https://file.luminescience.cn/66f8ed3071c371727589680" alt="" width="200" height="50" />shows the Sup of ith is the biggest qHCE Zσ(i), for the jth largest qHCE Zσ(j).
Some properties of qHCPOWG operator are as follows,
<img src="https://file.luminescience.cn/66f8ed9a34ac71727589786" alt="" width="450" height="265" />
 Definition 19: Suppose Zi is a group of qHCEs, the power hybrid geometric operator of objects 'n' qHCPHG: Zn → Z is defined as
<img src="https://file.luminescience.cn/66f8ee14bca111727589908" alt="" width="600" height="264" />
where Zσ(i) is the ith largest object in qHCEs arguments
<img src="https://file.luminescience.cn/66f8ee70244eb1727590000" alt="" width="300" height="70" />
and it is the weighting vector of IHCE influences Zi as well as wi is a group where,
<img src="https://file.luminescience.cn/66f8eeb71d0071727590071" alt="" width="320" height="143" />
and <img src="https://file.luminescience.cn/66f8ef3dcfb7c1727590205" alt="" width="70" height="38" />is the Sup of jth biggest qHCEs <img src="https://file.luminescience.cn/66f8ef5bc6f391727590235" alt="" width="48" height="36" />by all the other (qHCEs), that is,
<img src="https://file.luminescience.cn/66f8ef96d667e1727590294" alt="" width="300" height="103" />
where <img src="https://file.luminescience.cn/66f8efcab38a31727590346" alt="" width="180" height="49" />shows the Sup of jth largest qHCE <img src="https://file.luminescience.cn/66f8f0f124b611727590641" alt="" width="45" height="39" />, for the ith biggest qHCE <img src="https://file.luminescience.cn/66f8f03c8d9ca1727590460" alt="" width="52" height="37" /> . Specifically, qHCPHG is reduced to qHCPWG operator if
<img src="https://file.luminescience.cn/66f8e9ab26fd31727588779" alt="" width="180" height="49" />
and qHCPHG is reduced to operator if
<img src="https://file.luminescience.cn/66f8e9db23cc81727588827" alt="" width="170" height="51" />
Theorem 1: The qHCPA operator aggregates the "n" and again generates qHCE. Proof: The proof is straightforward. Remarks: Similarly, qHCPWA, qHCPOWA, qHCPHA, qHCPG, qHCPWG, qHCPOWG and qHCPHG also generates qHCE.
<h5>5. Some novel similarity measures for q-rung orthopair hesitant fuzzy connection numbers (qHCNs)</h5>
This section described the generalized form of similarity measures (SM) for qHCNs. Some characteristics are aslo explained with Jaccard SMs and Dice SMs. Similarity measures are vital tool for the similarity degree among sets. Definition 20: <a href="#reference_list_41">[42]</a> A real-valued function s: x×x → <a href="#reference_list_-1">[0,1]</a> is called the similarity degree, if it follows the given properties for R, S, T ϵ x,
<img src="https://file.luminescience.cn/66f8f26c18d9b1727591020" alt="" width="250" height="146" />
We straightforwardly developed Jaccard similarity measure and Dice similarity measure. In this part, we described SMs and weighted SMs (WSM) between the q-RHCNs based on the set theoretic approach for a collection of two q-RHCNs Z1 and Z2.
Definition 21: For two qHCNs Z1 and Z2, their Jaccard SMs are defined as:
<img src="https://file.luminescience.cn/66f8f2e773f411727591143" alt="" width="600" height="135" />
The JSMs for qHCN satisfy the following properties of SM: 0 ≤ Jac (Z1, Z2) ≤ 1 Jac (Z1, Z2) = Jac (Z2, Z1) Jac (Z1, Z2) =1 ⇔ Z1 = Z2
Definition 22: For two qHCNs Z1 and Z2, their weighted Jaccard JSM (WJSM) are defined as
<img src="https://file.luminescience.cn/66f8f3d2ba7671727591378" alt="" width="600" height="134" />
The WJSM for qHCN satisfies the following properties of SM: 0 ≤ Jacw(Z1, Z2) ≤ 1 Jacw(Z1, Z2) = Jacw(Z2, Z1) Jacw(Z1, Z2) = 1⇔ Z1 = Z2
When we supposed the weight vector is <img src="https://file.luminescience.cn/66f8e9ab26fd31727588779" alt="" width="180" height="49" />
 
at that point the WJSM would shift into JSM. Otherwise speaking when <img src="https://file.luminescience.cn/66f8f4d5a38f11727591637" alt="" width="80" />  k = 1, 2, 3, ...n then WJSM (Z1, Z2) = JSM (Z1, Z2)
 Definition 23: For two qHCNs Z1 and Z2, their Dice SMs (DSM) are defined as:
<img src="https://file.luminescience.cn/66f8f5b64ea9b1727591862" alt="" width="600" height="92" />
The DSM for qHCN satisfies the following properties of SM: 0 ≤ Dic (Z1, Z2) ≤ 1 Dic (Z1, Z2) = Dic (Z2, Z1) Dic (Z1, Z2) =1 ⇔ Z1 = Z2
Definition 24: For two qHCNs Z1 and Z2, their weighted DSM (WDSM) are defined as:
<img src="https://file.luminescience.cn/66f8f638f26841727591992" alt="" width="600" height="91" />
The WDSM for qHCNs satisfies the following properties of SM: 0 ≤ Dicw (Z1, Z2) ≤ 1 Dicw (Z1, Z2) = Dicw (Z2, Z1) Dicw (Z1, Z2) =1 ⇔ Z1 = Z2
When we assumed the weight vector is <img src="https://file.luminescience.cn/66f8e9ab26fd31727588779" alt="" width="180" height="49" />
At that point the WDSM would change into DSM. Otherwise speaking when <img src="https://file.luminescience.cn/66f8f4d5a38f11727591637" alt="" width="80" />  k = 1, 2, 3, ...n then WDSM (Z1, Z2) =DSM (Z1, Z2).
<h5>6. An algorithm for decision-making techniques based on the structure of q-rung orthopair hesitant fuzzy connection numbers (qHCNs)</h5>
In this section, we employed a well-established approach that relies on similarity measures and qHCNs to make decisions. The information system has been developed with the guidance from experts. Similarity degree plays a crucial role in ranking both known and unknown alternatives. Let us consider a unique set of alternatives denoted as A = A1, A2,..., Am, which necessitates investigation with the support of a set of known alternatives represented by P. Additionally, we had a collection of attributes, denoted as G = {G1, G2, ..., Gn}. To facilitate our analysis, we introduced a weight vector for these attributes, denoted as ω = (ωt) for t ranging from 1 to n, where ωt is non-negative, and the sum of all ωt values equals 1. Therefore, we would apply the Jaccard weighted SM to find the similarity degrees. The algorithm is briefly described step by step in Figure 1.
<img src="https://file.luminescience.cn/66f8f7b161f9e1727592369" alt="" width="500" height="304" />
Step 1: Evaluate the known and unknown information of alternatives/attributes in the form of qHCNs. Step 2: Develop the given data in the form of qHCNs according to the proposed approach in Definition 9. Step 3: Calculate the similarity degrees between the known information and unknown information of alternatives. Step 4: Rank all alternatives based on the aggregated similarity degrees and select the best one.
<h5>7. Applications in decision-making processes</h5>
The purpose of this section is to utilize the above developed similarity measures to resolve a real-life difficulty with q-RHF data.
Example 2: Suppose that a counseling center has five various disciplines, namely History (G1), Mathematics (G2), Biology (G3), Political studies (G4), and Economics (G5) related to career determination in the field of Politics (A1), Pharmacy (A2), Teaching (A3) and Anatomy (A4). Suppose that an unknown student P goes to the counseling center for getting assistance to select his/her appropriate profession. The purpose of the problem is to establish the most projected profession for the student P in the A1, A2 , A3 and A4. To achieve this aim, the following stages of the suggested approach are developed, which are summarized as follows. Step 1: A specialist provides the preference to each discipline connected to the profession, the q-RHFS were summarized in the following Table 1.
<img src="https://file.luminescience.cn/66f8f91fc85b81727592735" alt="" width="780" height="203" />
Step 2: qHCNs related to the preferences of the given data and student are prepared by applying Definition 9, and their results were summarized as Table 2.
<img src="https://file.luminescience.cn/66f8fa0667c5d1727592966" alt="" width="700" height="299" />
 Step 3: By allocating the weight vector w = (0.15, 0.10. 0.20, 0.25, 0.30)T corresponding to G1, G2, G3, G4 and G5 then by using the recommended weighted similarity measure in Definition 14 and respectively, their corresponding measurement values are given S (A1P) = 0.035158, S (A2P) = 0.387027, S (A3P) = 0.190423, S (A4P) = 0.183446. Step 4: From these calculated findings, we calculated the rank order of the alternatives as A2 > A3 > A4 > A1, Where ">" refers to “preferred to”. A2 is the best attribute for the students according to the proposed operator. Example 3: In this section, we illustrated the practical application of our established method in medical diagnosis through an example related to infectious diseases. Considered a scenario where we had a set of patients, denoted as A = {A1, A2, A3}, each of them required a medical diagnosis. We prepared a list of symptoms associated with potential diseases, represented in the form of q-RHF information. These symptoms are denoted as:
s = { Muscle pain (M), Temperature (T), Headache (H), Fever (F)}
Then, assumed we had a diagnosed patient, referred to as P, who presented with a specific set of symptoms. These symptoms served as valuable input data for determining the level of disease severity. By employing our method, we aimed to calculate the degree of similarity between the symptoms exhibited by patient P and those associated with various diseases, ultimately to assist in the diagnosis process. Step 1: The q-RHF information of all patients are given in the Table 3.
<img src="https://file.luminescience.cn/66f8fcd54c7a31727593685" alt="" width="700" height="197" /> Step 2: By definition 9 of qHCN, we received the alternatives values for B = 2 in Table 4.
<img src="https://file.luminescience.cn/66f8fd1b935ab1727593755" alt="" width="600" height="262" /> Step 3: By allocating the weight vector w = (0.5, 0.2. 0.2, 0.1)T corresponding to M, T, H and F then by using the recommended weighted Jaccard similarity measure in definition 14 and respectively, their corresponding measurement values are given. S (A1P) = 0.0107, S (A2P) = 0.0084, S (A3P) = 0.0073 Step 4: From these calculated findings, we calculated the rank order of the alternatives as A1 > A2 > A3 From the above ranking, we could easily diagnose the status of patients by their symptoms. It is confirmed that A1 was more similar to the P who was diagnosed patient.
7.1 Comparative analysis
In this section, we made a comparison between the existing studies and proposed model for the advancement and consistency of our structure. In addition, we conducted an in-depth comparison between the existing studies and our proposed model to highlight its advancements and consistency in structure. Subsequently, we discussed the advantages and limitations of our proposed model in detail. We evaluated the performance of the proposed method by comparing it with current operators, such as power average aggregation and power geometric aggregation, specifically within the context of intuitionistic hesitant fuzzy data, as proposed by Tahir et al. <a href="#reference_list_41">[42]</a>, Garg et al. <a href="#reference_list_40">[41]</a>, and Wajid et al. <a href="#reference_list_27">[28]</a>. For this comparative analysis, we utilized the data from example 2, and the corresponding results were presented in Table 5. This evaluation provided valuable insights into the effectiveness and efficiency of our approach relative to the existing methodologies.
<img src="https://file.luminescience.cn/66f8fddddd0eb1727593949" alt="" width="580" height="269" />
In this analysis, we observed that the methods outlined by Tahir et al. <a href="#reference_list_41">[42]</a> and Wajid et al. <a href="#reference_list_27">[28]</a> were well-suited for an intuitionistic hesitant fuzzy environment. However, our proposed model stood out as it was more robust and versatile. Unlike the previous approaches, our model was capable of handling not only intuitionistic hesitant fuzzy environments but also pythagorean hesitant, fermatean hesitant and q-rung hesitant fuzzy environments. Furthermore, by setting the parameter B = 1 our proposed method successfully reproduced all the results achieved by the approaches defined in <a href="#reference_list_27">[28, 42]</a>. This consistency demonstrated the reliability and broader applicability of our developed model and made it a superior alternative for dealing with various types of hesitant fuzzy environments.
7.2 Advantages and limitation of the proposed approach
Some benefits and limitations of the proposed approach are listed as below:
<ul>
<li>The proposed model serves as a generalized framework that extends both hesitant fuzzy sets and q-Rung orthopair fuzzy sets. By incorporating the connection number from set pair analysis, this model significantly enhances its ability to handle data uncertainty more effectively than existing approaches.</li>
<li>The inclusion of the parameter B in the q-Rung hesitant connection number (q-HCN) model enables it to encompass various fuzzy environments, such as intuitionistic hesitant connection numbers, pythagorean hesitant connection numbers, fermatean hesitant connection numbers and square root fuzzy data.</li>
<li>As demonstrated in Table 5, when the parameter B = 1, the results generated by the proposed model align closely with those of existing models, thereby confirming the correctness and consistency of the proposed approach.</li>
<li>To facilitate data processing, the model included a series of aggregation operators, and several similarity measures had been applied to address real-world problems, which showed the practical applicability of the model in diverse scenarios.</li>
<li>However, the proposed model is limited to handling q-Rung fuzzy data. It did not extend to p,q-Rung orthopair fuzzy sets, where it failed to provide accurate results.</li>
<li>Similarly, the model is not equipped to handle bipolar fuzzy information, thus made it unsuitable for p,q-Bipolar fuzzy data neither.</li>
</ul>
<h5>8. Conclusion</h5>
In this research, we introduced a novel concept: q-rung orthopair hesitant fuzzy sets, which was achieved by amalgamating q-rung orthopair fuzzy sets (q-ROFSs) with hesitant fuzzy sets (HFSs). q-rung orthopair hesitant fuzzy sets, denoted as q-RHFSs, represented a more versatile and potent tool for effectively managing situations characterized by ambiguity and uncertainty. We enriched this framework by integrating the notion of the connection number from set pair analysis into the q-rung orthopair hesitant fuzzy environment, thereby giving rise to an entirely new theoretical construct termed q-rung orthopair hesitant fuzzy connection number (qHCN). Furthermore, this research extended the scope of similarity measures, specifically Jaccard and Dice similarity measures, by introducing weighted versions of these measures. The salient properties of these novel measures had been thoroughly examined. We also presented a novel algorithm designed to calculate the attributes of alternatives by utilizing the weighted similarity measures to discern the characteristics of both known and unknown alternatives. An illustrative medical diagnosis problem had been successfully addressed by using this innovative approach, which highlighted its potential applications in the domains of pattern recognition, artificial intelligence and decision makings. We conducted an in-depth comparative study which incorporated the latest research findings and included a detailed analysis of both the benefits and limitations of the proposed model. This comprehensive evaluation provides a clearer understanding of how our model stands in relation to current advancements and identifies areas where it excels as well as where it may have constraints. Our future endeavors will encompass the application of this approach across various research domains, with a focus on addressing practical challenges. We intend to explore further extensions of fuzzy models <a href="#reference_list_53">[54-57]</a>, including bipolar fuzzy sets, complex fuzzy sets and cubic fuzzy models. Additionally, we will delve into neural network problems and endeavor to devise models rooted in optimization theory as part of our ongoing research agenda <a href="#reference_list_57">[58-59]</a>.
<h5>Conflict of interest</h5>
There is no conflict of interest.
<h5>Authors'contributions</h5>
Khawar Hassan contributed to the design and execution of experiments and research methodologies, Tanzeela Shaheen supervised this research and assisted in data collection and analysis. Wajid Ali focused on the theoretical framework and optimization processes. Iftikhar ul Haq provided critical insights into data interpretation and ensured the accuracy of results, while Nadia Bibi assisted with literature reviews, refining the analysis, and manuscript writing. Amal Kumar Adak contributed through computational analysis, technical support, and the development of algorithms.

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Hassan, K., Shaheen, T., Ali, W., Haq, . I. U., Bibi, N., & Adak, A. K. (2024). Decision-making techniques based on aggregation operators and similarity measures under q-rung orthopair hesitant fuzzy connection numbers and their application. Decision Making and Analysis, 2(1), 58–72. https://doi.org/10.55976/dma.22024127458-72

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Decision-making techniques based on aggregation operators and similarity measures under q-rung orthopair hesitant fuzzy connection numbers and their application

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Decision-making techniques based on aggregation operators and similarity measures under q-rung orthopair hesitant fuzzy connection numbers and their application

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