Significance of Frank Aggregation Operators for Multi-Attribute Decision Making in the q-Rung Picture Fuzzy Context PDF

SIGNIFICANCE OF FRANK AGGREGATION OPERATORS FOR MULTI ATTRIBUTE DECISION MAKING IN THE q-RUNG PICTURE FUZZY CONTEXT A THESIS submitted by CHITRA R RA2133001011039 In Partial Fulfilment of the...

SIGNIFICANCE OF FRANK AGGREGATION OPERATORS FOR MULTI ATTRIBUTE DECISION MAKING IN THE q-RUNG PICTURE FUZZY CONTEXT A THESIS submitted by CHITRA R RA2133001011039 In Partial Fulfilment of the Requirements for the Degree of DOCTOR OF PHILOSOPHY DEPARTMENT OF MATHEMATICS FACULTY OF SCIENCE AND HUMANITIES SRM INSTITUTE OF SCIENCE AND TECHNOLOGY KATTANKULATHUR - 603 203 SEPTEMBER 2024  DECLARATION BY THE SCHOLAR I, CHITRA R, hereby declare that the work which is being presented in the thesis entitled, “SIGNIFICANCE OF FRANK AGGREGATION OPERATORS FOR MULTI ATTRIBUTE DECISION MAKING IN THE q -RUNG PICTURE FUZZY CONTEXT” in partial fulfillment of the requirements for the award of the Degree of Doctor of Philosophy is my own work carried out by me under the supervision of Dr. K. PRABAKARAN (Research Supervisor) during the period 2021 to 2024 in the Department of Mathematics, Faculty of Science and Humanities, SRM Institute of Science and Technology, Kattankulathur. The matter presented in this Ph.D. thesis has not been submitted elsewhere for the award of any other degree/diploma. I declare that I have faithfully acknowledged, given credit to and referred to the research workers wherever their works have been cited in the text and the body of the thesis. I further certify that I have not willfully lifted up some other’s work, para, text, data, results, etc., reported in the journals, books, magazines, reports, dissertations, theses, etc., or available at web-sites and have not included them in this Ph.D. thesis and cited as my own work. Place: Kattankulathur (CHITRA R) Date: CERTIFICATE FROM THE SUPERVISOR This is to certify that the above statement made by the candidate is correct to the best of my/our knowledge. Certified that a check for plagiarism has been made on the software available in the University and the contents of thesis have been found free from plagiarism within permissible limits. Dr. K. PRABAKARAN (Research Supervisor)  CERTIFICATE FOR COMPLETION OF MINIMUM REQUIREMENTS AS PER Ph.D. REGULATIONS This is to certify that Ms. Chitra R (RA2133001011039) Department of Mathematics, Faculty of Science and Humanities, of SRM Institute of Science and Technology, a bonafide research scholar registered for the Ph.D. Degree under the supervision of Dr. K. Prabakaran, has satisfactorily completed all the requirements for the submission of thesis for the award of Ph.D. degree as per provisions of the respective Ph.D. Regulations - 2017. She has successfully completed the course work and well- defended the Research Plan Proposal Seminar and the Pre-Submission Seminar presented before the DRCC. She also published research papers in refereed journals of reputed, approved by the University and presented two research papers in conference/seminars. Place: Kattankulathur Dr. K. PRABAKARAN Date: (Research Supervisor) Place: Kattankulathur Dr. Ritesh Kumar Dubey Date: Research Associate Professor & Head Department of Mathematics DRCC Chairperson  COPYRIGHT TRANSFER CERTIFICATE Title of the Thesis : Significance of Frank Aggregation Operators for Multi Attribute Decision Making in the q-Rung Picture Fuzzy Context Name of the Research Scholar : Chitra R Registration Number : RA2133001011039 Department : Mathematics Faculty : Science and Humanities COPYRIGHT TRANSFER The undersigned hereby assigns to the SRM Institute of Science and Technology, all rights under copyright that may exist in and for the above thesis submitted for the award of the Ph.D. degree. Place: Kattankulathur CHITRA R Date: (Research Scholar) [Note: However, the author may reproduce or authorize others to reproduce material extracted verbatim from the thesis or derivative of the thesis for authors personal use, provided that the source and the University’s copyright notice are adequately acknowledged. Also, refer Clause No.: 16 (2006 Regulations), Clause No.: 22 (2014 Regulations) and Clause No. 20 (2017 Regulations)].  ACKNOWLEDGEMENT I would like to begin by expressing my profound gratitude to the Almighty for his blessings upon me and providing the strength and determination necessary to complete this research successfully. This thesis would not have been possible without the invaluable support of many remarkable individuals, each of whom I wish to acknowledge. I wish to express my deep and heartfelt gratitude to my research supervisor, Dr. K. Prabakaran, Assistant Professor, Department of Mathematics, SRM Institute of Science and Technology, for his exceptional guidance and support. His inspiration, empathy, vision, and motivation deeply influenced me, equipping me with the skills needed to complete this work. I am profoundly grateful for the opportunity he provided. I also extend my sincere thanks to the HOD of Mathematics, Dr. Ritesh Kumar Dubey, and all faculty members of the Department of Mathematics for their support and cooperation. I would like to extend my profound gratitude to my Doctoral Committee members Dr. R. Uthayakumar, Professor and Head, Department of Mathematics, The Gandhigram Rural Institute, Gandhigram and Dr. C. Vijayalakshmi, Associate Professor, Department of Statistics and Applied Mathematics, Central University of Tamil Nadu, Thiruvarur for their unwavering support, technical suggestions and constructive comments that enormously helped me in improving my research work. I wish to express my gratitude to my expert committee member Dr. V. Vetrivel, Professor and Head, Department of Mathematics, Indian Institute of Technology, Madras for his valuable suggestions upon my research work. I extend my special gratitude to Dr. T. R. Paarivendhar, Chancellor and Ex Member of Parliment in Perambalur Constituency, Dr. Ravi Pachamuthu, Pro Chancellor Administration and Dr. P. Sathyanarayanan, Pro Chancellor v  vi Academics of SRM Institute of Science and Technology. I profusely thank Dr. C. Muthamizhchelvan, Vice Chancellor, Dr. S. Ponnusamy, Registrar and Dr. A. Duraisamy, Dean Science and Humanities of SRM Institute of Science and Technology for granting me permission to carry out my research in this institution. I would like to acknowledge and offer my special thanks to Dr. K. Gunasekaran, Controller of Examinations, Dr. T.V. Gopal, Dean CET and Dr. D. John Thiruvadigal, Chairperson School of Applied Sciences of SRM Institute of Science and Technology for their guidance and support. I am also grateful to Dr. B. Neppolian, Dean Research of SRM Institute of Science and Technology for his kind cooperation and constant encouragement. I extend my thanks to Dr. P. Rajendran, Librarian, SRM Institute of Science and Technology for providing necessary resources pretaining to my research work. I thank Dr. A. Govindarajan and Dr. B. Vennila for their support at the initial stage of my research. I would also like to express my gratitude to Dr. V. Subburayan for his humble regard during my research period. I extend my sincere thanks to Dr. D. Prakash for his expert guidance in technical tools such as LaTeX, which greatly contributed to the completion of my research work. My heartfelt gratitude goes to Dr. Narasu Sivakumar and Dr. Santhosh Kumar for their caring support and advice. I would also like to recognise the love, support and assistance from my co-scholar, Ms. Subadhra Srinivas. I would also like to express my gratitude and appreciation to the entire SRM Institute of Science and Technology management for proving us with a friendly environment and vibrant staff that make this writing a success. Last but not the least, my heartfelt gratitude to my beloved parents, Mr. S. Rameshbabu and Mrs. S. Rani, my sister Ms. R. Devipriya and my brother Mr. R. Goutham for their love, cooperation, moral support and constant encouragement. Thank you all so much. Chitra R  TABLE OF CONTENTS ABSTRACT......................................... xi LIST OF TABLES...................................... xii LIST OF FIGURES..................................... xiv LIST OF SYMBOLS..................................... xv 1 INTRODUCTION 2 1.1 Background and Motivation.............................. 2 1.2 Exploring fuzzy sets and their extensions...................... 3 1.3 Navigating essentials of t-norm, t-conorm and Aggregation Operations...... 5 1.3.1 Frank Norm................................... 6 1.3.2 Aggregation Operations............................ 6 1.4 Multi Attribute Decision Making........................... 7 1.5 Literature Review................................... 8 1.6 Overview of the Thesis................................. 11 2 EMPLOYING q̆ -RUNG PICTURE FUZZY FRANK AGGREGATION OPERATORS FOR DECISION-MAKING 14 2.1 Introduction....................................... 14 2.2 Key Concepts...................................... 15 vii  viii 2.3 Results and Discussion................................. 16 2.3.1 MADM Method in the q̆ -rung Picture Fuzzy Framework......... 22 2.3.2 Practical Instance............................... 23 2.3.3 Examining the Influence of Parameters m and q̆ on Decision Outcomes. 24 2.3.4 Comparative Analysis............................. 27 2.4 Conclusion....................................... 28 3 ORDERING q̆ -RUNG PICTURE FUZZY NUMBERS BY POSSIBLE GRADING TECHNIQUE AND ITS UTILIZATION IN DECISION- MAKING PROBLEM 30 3.1 Introduction....................................... 30 3.2 Key Concepts...................................... 31 3.2.1 Possible Grading Technique.......................... 32 3.3 Results and Discussion................................. 32 3.3.1 Mathematical approach to solve MADM using Possible Grading Technique 39 3.3.2 Practical Instance............................... 40 3.3.3 Examining the influence of parameters q̆ and m on decision outcomes. 44 3.3.4 Comparative Analysis............................. 47 3.4 Conclusion....................................... 47 4 EXPLORING GROUP DECISION-MAKING WITH q̆ -RUNG PICTURE FUZZY FRANK BONFERRONI OPERATORS 49  ix 4.1 Introduction....................................... 49 4.2 Key Concepts...................................... 50 4.3 Results and Discussion................................. 51 4.3.1 Framework for Group Decision-Making using Proposed Operator..... 61 4.3.2 Practical Instance............................... 62 4.3.3 Parameters impact on the Decision-Making Outcomes........... 64 4.3.4 Comparison of Proposed and Current Operators.............. 66 4.4 Conclusion....................................... 67 5 INCORPORATING q -RUNG PICTURE FUZZY FRANK PRIORITIZED WEIGHTED AGGREGATORS WITH MULTIMOORA STRATEGY FOR DECISION MAKING 69 5.1 Introduction....................................... 69 5.2 Key Concepts...................................... 70 5.3 Results and Discussions................................ 72 5.3.1 Mathematical Formulation of Integrated MULTIMOORA approach based on Proposed Aggregators........................... 78 5.3.2 Practical Instance............................... 81 5.3.3 Impact of the Parameters m and q̆ on MULTIMOORA q̆ -RPFS.... 86 5.3.4 Comparative examination........................... 87 5.4 Conclusion....................................... 89  x 6 HANDLING DECISION-MAKING ISSUE BASED ON HESITANT FRANK AGGREGATORS IN THE LINGUISTIC q̆ -RUNG PICTURE FUZZY CONTEXT 91 6.1 Introduction....................................... 91 6.2 Key Concepts...................................... 91 6.2.1 Set Operations of H q̆ -RPFS......................... 92 6.2.2 Frank Operations of hesitant q̆ -rung Picture Fuzzy Numbers....... 94 6.3 Results and Discussion................................. 96 6.3.1 MCGDM approach with Hesitant q̆ -rung picture fuzzy data....... 100 6.3.2 Practical Instance............................... 101 6.3.3 The Influence of Frank Parameter on Hesitant q̆ -RPFS.......... 106 6.3.4 Comparative Analysis............................. 107 6.4 Conclusion....................................... 108 7 SUMMARY AND FUTURE WORK 110 7.1 Summary of the Thesis................................. 110 7.2 Future Work...................................... 111 List of Publications.................................... 112 REFERENCES 114 Bibliography......................................... 114  ABSTRACT In the face of growing complexity and uncertainty, fuzzy multi-attribute decision making (MADM) has emerged as an indispensable framework for evaluating alternatives based on multiple criteria. Initiated by Lotfi Zadeh’s fuzzy sets, this field has evolved with the introduction of q̆ -rung picture fuzzy sets ( q̆ -RPFS), which more effectively capture and represent uncertainty compared to traditional fuzzy sets. The focus of this research lies in utilizing Frank t -norm and t -conorm within fuzzy set theory for aggregating q̆ -rung picture fuzzy information, adapting them suitably across different scenarios of MADM. Aggregation operators (aggregators) based on the Frank norm are developed within the q̆ -rung picture fuzzy framework. An essential aspect of decision- making involves ordering and ranking q̆ -RPF data, and a possible grading technique is presented to address this need. Unlike traditional aggregators, the Bonferroni mean (BM) accommodates interdependence among attributes, facilitating the development of q̆ -RPF Frank BM aggregators. In group decision-making, the varied perspectives of different experts necessitate prioritized weighted aggregators (PWA) in the q̆ -RPF context, emphasizing critical attributes and decision-makers. The integration of the MULTIMOORA method with PWA ensures that collective expert preferences leads to more balanced and acceptable outcomes. To handle scenarios where experts exhibit uncertainty or hesitation about certain attributes, hesitant aggregators are developed to capture a range of possible values. Hesitant q̆ -rung picture fuzzy aggregators based on Frank norm operations offer a more comprehensive framework for expressing uncertainty across multiple expert opinions. A comparative analysis, along with numerical illustrations, demonstrates how the q̆ -rung picture fuzzy Frank aggregators lead to more nuanced and reliable decision-making outcomes when compared to conventional methods. Additionally, sensitivity analysis is conducted to ensure the robustness, reliability, and validity of the proposed decision-making framework. xi  LIST OF TABLES TABLE NO. TITLE PAGE NO 2.1 Over all aggregated values of the Locations.................. 24 2.2 Score values of Locations and their ranking.................. 24 2.3 Impact of the variable m upon decision outcomes.............. 26 2.4 Impact of the variable q̆ upon decision outcomes.............. 27 2.5 Comparison of Proposed operators with existing operators......... 28 3.1 Aggregated values for each Marketplace.................... 43 3.2 Ranking Marketplaces based on scores.................... 44 3.3 Influence of a variable q̆ upon decision outcomes.............. 45 3.4 Influence of a variable m upon decision outcomes.............. 46 3.5 Comparing the proposed operators with current operators......... 47 4.1 Ranking outcomes for fixed m = 2 , u = v = 1................ 64 4.2 Ranking outcomes for fixed q̆ = 3 , u = v = 2................. 65 4.3 Ranking outcomes for fixed q̆ = 2 , m = 2................... 66 xii  xiii 4.4 Comparison of proposed Frank Bonferroni operator and available operators. 66 5.1 Ranking MULTIMOORA outcomes according to Dominance theory.... 86 5.2 Impact of various parameters on MULTIMOORA q̆ -RPFNs........ 87 5.3 Analysis in comparison to other methods.................. 88 6.1 Linguistic variables with associated q̆ -RPF numbers............ 102 6.2 Evaluation of candidates via Linguistic q̆ -RPF Numbers.......... 103 6.3 Candidates ranking for varied m values with fixed q̆............. 107 6.4 Comparison of the available aggregators with the proposed aggregators.. 108  LIST OF FIGURES FIGURE NO. TITLE PAGE NO 2.1 Sensitivity analysis of m using proposed Frank aggregators......... 25 2.2 Sensitivity analysis of q̆ using proposed Frank aggregators......... 25 3.1 Sensitivity analysis of q̆ using ordered Frank aggregators.......... 45 3.2 Sensitivity analysis of m using ordered Frank aggregators......... 46 4.1 Sensitivity analysis of q̆ and m using Bonferroni Frank aggregators.... 65 5.1 Contrasting ranks of the proposed approach with existing techniques.... 88 6.1 Sensitivity analysis of m using proposed hesitant Frank aggregators.... 106 xiv  LIST OF SYMBOLS Y Universal set J˘ Fuzzy set I˘ Intuitionistic fuzzy set P̆ Picture fuzzy set Ŏp q̆-rung orthopair fuzzy set τϱ q̆-rung picture fuzzy set τj q̆-rung picture fuzzy number q̆ The exponent that defines the level of fuzziness Ψ(y) Membership value of y ∈ Y Φ(y) Non-Membership value of y ∈ Y Θ(y) Degree of Neutral Membership value y ∈ Y χ(y) Degree of Refusal membership value of y ∈ Y F(a, b) Frank t-norm operator on the real numbers a, b ∈ [0, 1] ′ F (a, b) Frank t-conorm operator on the real numbers a, b ∈ [0, 1] κ Aggregation function Nn The set of natural numbers indexed by n G Set of Alternatives R Set of Attributes xv  xvi ω̆ Set of Weight vectors B Class of Benefit type Attributes C Class of Cost type Attributes M Decision-maker Dm Decision matrix from decision-maker M b Dm Decision matrix from the bth decision maker Mb δrc Entries of decision matrix Dm b δrc Entries of the bth decision matrix Dm b c δrc Complement of entries in the decision matrix ′ Dm Normalized decision matrix ′ δrc Entries of Normalized decision matrix ′ Dπ(m) Ordered Normalized decision matrix δr Combined preference value of an alternative Gr s Number of alternatives k Number of attributes and associated weight vectors v Number of decision-makers Sτ1 Score function of a q̆-rung picture fuzzy number τ A1τ Accuracy function of a q̆-rung picture fuzzy number τ ⊕ Sum operator ⊗ Product operator  xvii λ̆ Positive real number m Frank parameter τz Set of q̆-rung picture fuzzy numbers τπ(z) Set of ordered q̆-rung picture fuzzy numbers ω̆z Set of Weight vectors corresponding to τz τt Aggregation of t q̆-rung picture fuzzy numbers τ− Minimum of q̆-rung picture fuzzy numbers τ+ Maximum of q̆-rung picture fuzzy numbers P ∗ (τ1 ≥ τ2 ) Possible grading measure that τ1 ≥ τ2 P∗ Possible grading matrix p∗xy Entries of the Possible grading matrix r̆x Crisp value corresponding to τx u, v Parameters of Bonferroni mean operator BM Bonferroni Mean ω̆p(z) Set of Prioritized Weight vectors corresponding to Rz Bsk Prioritized weight matrix corresponding to the matrix Dm Bbsk b Prioritized Weight matrix corresponding to the matrix Dm MM Multimoora MULTIMOORA Multi-Objective Optimization by Ratio Analysis with Full Multiplicative Form  xviii RS Ratio System method RP Reference point method MF Multiplicative Form method ξrRS Combined preference value of an rth alternative in terms of RS ξ˘rRS Score value of an rth alternative in terms of RS ξ¯rRS Normalized value of an rth alternative in terms of RS ξrMF Combined preference value of an rth alternative in terms of MF ξ˘rMF Score value of an rth alternative in terms of MF ξ¯rMF Normalized value of an rth alternative in terms of MF e∗c Reference point with respect to the cth attribute dHamming (τ1 , τ2 ) Hamming distance between τ1 and τ2 drc Chebyshev distance for each alternative with respect to e∗c ξrRP Maximum Chebyshev distance ξ¯rRP Normalized value of an rth alternative in terms of RP H̆f Hesitant Fuzzy set h̆f (y) Membership value of y ∈ Y in the H̆f QH̆ Hesitant q̆-rung picture fuzzy set h̆(y) Hesitant q̆-rung picture fuzzy element of y ∈ Y in the QH̆ lh̆ length of Hesitant q̆-rung picture fuzzy element h̆z Set of Hesitant q̆-rung picture fuzzy numbers  xix Sh̆4 Score value of a Hesitant q̆-rung picture fuzzy number h̆ A2h̆ Accuracy value of a Hesitant q̆-rung picture fuzzy number h̆ h̆+ , h̆− Upper and lower bounds of h̆ SF DW A Spherical Fuzzy Dombi Weighted Average P F DW A Picture Fuzzy Dombi Weighted Average SF W A Spherical Fuzzy Weighted Average SF W G Spherical Fuzzy Weighted Geometric q-RPFEWA q-Rung Picture Fuzzy Einstein Weighted Average PFWA Picture Fuzzy Weighted Average P F HA Picture Fuzzy Hybrid Average PFHWA Picture Fuzzy Hamacher Weighted Average PFDWHM Picture Fuzzy Dombi Weighted Heronian Mean q-RPFDWHM q-Rung Picture Fuzzy Dombi Weighted Hamy Mean q-RPFDWDHM q-Rung Picture Fuzzy Dombi Weighted Dual Hamy Mean SFPWA Spherical Fuzzy Prioritized Weighted Average SFPWG Spherical Fuzzy Prioritized Weighted Geometric HTSDFWA Hesitant T-Spherical Dombi Fuzzy Weighted Average HTSDFWG Hesitant T-Spherical Dombi Fuzzy Weighted Geometric CHAPTER 1  CHAPTER 1 INTRODUCTION 1.1 Background and Motivation In contemporary decision-making scenarios, the complexity and multifaceted nature of real-world problems necessitate robust and sophisticated methods. Multi-attribute decision making stands as a critical framework in this domain, enabling decision-makers to evaluate and prioritize options based on multiple criteria. Traditional MADM techniques, while effective in many situations, often fall short when dealing with the inherent uncertainty and vagueness present in real-world data. The concept of fuzzy sets, revolutionized decision-making by providing a mathematical means to handle uncertainty. Over the years, various extensions of fuzzy sets have been developed to better capture and represent the nuances of real-world problems. Among these, q̆ -rung picture fuzzy sets have emerged as a significant advancement, providing an enhanced framework for expressing uncertainty and vagueness by allowing more degrees of freedom compared to traditional fuzzy sets. On the other hand, Frank aggregators, have exhibited considerable potential owing to their flexibility and capacity to model a wide range of information aggregation. Despite these advancements, the application of q̆ -rung picture fuzzy sets and Frank aggregators in MADM under uncertain environments remains under explored. This thesis aims to address this gap by developing a novel approach that leverages q̆ -rung picture fuzzy Frank aggregators to enhance decision-making accuracy and reliability in uncertain settings. The motivation for this research stems from the growing need to improve decision-making processes in various fields such as finance, engineering, economics, business and human resource management, where uncertainty and multiple attributes are prevalent. By providing a more nuanced and flexible framework for handling MADM, this study seeks to contribute to the theoretical development of Frank aggregators in the q̆ -RPF context and offer practical solutions for complex decision-making problems. 2  3 1.2 Exploring fuzzy sets and their extensions Fuzzy set theory, introduced by Lotfi Zadeh in 1965, provides a mathematical framework for dealing with uncertainty and imprecision. Unlike classical set theory, where elements either belong to a set or not, fuzzy sets allow for partial membership, which is characterized by a membership function that assigns each element a value in [0, 1], referred to as the membership value (MV). This flexibility makes fuzzy sets (FS) particularly useful for modeling real-world scenarios where binary classifications are insufficient. Later, Atanassov’s intuitionistic fuzzy sets (IFS) further refined this concept by including a non-membership value (NV). Yager’s development of Pythagorean fuzzy sets slightly relaxed the constraints of IFS. Subsequently, Cuong’s picture fuzzy sets (PFS) extended IFS by incorporating an indeterminacy value (IV), offering a more nuanced representation of uncertainty. Kutlu and Kahraman’s development of spherical fuzzy sets further relaxed the constraints of PFS. Unlike IFS, where the sum of the MV and NV must be less than or equal to 1, Yager’s q̆ -rung orthopair fuzzy set ( q̆ -ROFS) relaxes this constraint by allowing the sum of q̆ th powers of MV and NV to be less than or equal to 1. To reinforce the concept of PFS, q̆ -rung picture fuzzy sets ( q̆ -RPFS) were presented by Li et al. as an extension of PFS, offering a sophisticated framework for capturing and representing uncertainty more effectively and flexibly than traditional fuzzy sets. Unlike PFS, where the sum of the MV, IV and NV must be less than or equal to 1, q̆ -RPFS relaxes this constraint by allowing the sum of q̆ th powers of MV, IV and NV to be less than or equal to 1. To understand these concepts, essential definitions of various types of FS are presented as follows: Definition 1.2.1 (Fuzzy set ). A fuzzy set J˘ within a universe of discourse Y is specified as a set of ordered pairs J˘ = {(y, ΨJ˘(y))|y ∈ Y}. Here, ΨJ˘ denotes the membership function that assigns a membership value (MV) ΨJ˘(y) to each element y , ranging from 0 to 1. Definition 1.2.2 (Intuitionistic fuzzy set ). An intuitionistic fuzzy set I˘ within a universe of discourse Y is specified as I˘ = {(y, ΨI˘(y), ΦI˘(y))|y ∈ Y} where ΨI˘(y) and ΦI˘(y) indicates the MV and NV of y respectively, with the condition that for every  4 y ∈ Y ensuring that 0 ≤ (ΨI˘(y) + ΦI˘(y)) ≤ 1. Also, the indeterminacy value of y ∈ Y to the set I˘ is obtained by, (1 − {ΨI˘(y) + ΦI˘(y)}). Definition 1.2.3 (Picture fuzzy set ). A picture fuzzy set P̆ within a universe of discourse Y is specified as P̆ = {(y, ΨP̆ (y), ΘP̆ (y)), ΨP̆ (y)|y ∈ Y} where ΨP̆ (y) , ΘP̆ (y) and ΦP̆ (y) indicates the MV, IV and NV of y respectively, with the condition that for every y ∈ Y ensuring that (ΨP̆ (y) + ΘP̆ (y) + ΦP̆ (y)) ≤ 1. Also, the refusal value (RV) of y ∈ Y to the set P̆ is obtained by, (1 − {ΨP̆ (y) + ΘP̆ (y) + ΦP̆ (y)}). Definition 1.2.4 ( q̆ -rung orthopair fuzzy set ). Let Y indicate the universal set. The q̆ -rung orthopair fuzzy set Ŏp on Y is elucidated as: Ŏp = {⟨y, Ψŏp (y), Φŏp (y)⟩ : y ∈ Y} , where the MV of y expressed by Ψŏp (y) and the NV of y expressed as Φŏp (y) in a way that meets the requirement, (Ψŏp (y))q̆ + (Φŏp (y))q̆ ≤ 1 , q̆ is a positive integer. Also, the indeterminacy value of y ∈ Y to the set Ŏp is obtained by, (1 − (Ψŏp (y))q̆ − (Φŏp (y))q̆ )1/q̆. Definition 1.2.5 ( q̆ -rung picture fuzzy set ). Let Y indicate the universal set. The q̆ -rung picture fuzzy set τϱ on Y is elucidated as: τϱ = {⟨y, Ψτϱ (y), Θτϱ (y), Φτϱ (y)⟩ : y ∈ Y} , where the MV of y stated as Ψτϱ (y) , the IV of y stated as Θτϱ (y) and the NV of y stated as Φτϱ (y) in a way that meets the requirement, 0 ≤ (Ψτϱ (y))q̆ + (Θτϱ (y))q̆ + (Φτϱ (y))q̆ ≤ 1 , q̆ is a positive integer. Also, the refusal grade χτϱ of y ∈ Y is obtained by, (1 − (Ψτϱ (y))q̆ − (Θτϱ (y))q̆ − (Φτϱ (y))q̆ )1/q̆. For ease, a q̆ -rung picture fuzzy number ( q̆ -RPFN) is indicated as, τj = ⟨Ψτj , Θτj , Φτj ⟩ ∈ τϱ. Definition 1.2.6 (Operational laws ). Let τ, τ1 and τ2 are the three q̆ -RPFNs then their operating principles are given below: 1. τ1 ⊕ τ2 = ⟨(Ψq̆τ1 + Ψq̆τ2 − Ψq̆τ1 Ψq̆τ2 )1/q̆ , Θτ1 Θτ2 , Φτ1 Φτ2 ⟩. 2. τ1 ⊗ τ2 = ⟨Ψτ1 Ψτ2 , (Θq̆τ1 + Θq̆τ2 − Θq̆τ1 Θq̆τ2 )1/q̆ , (Φq̆τ1 + Φq̆τ2 − Φq̆τ1 Φq̆τ2 )1/q̆ ⟩. 3. λ̆τ = ⟨(1 − (1 − Ψq̆τ )λ̆ )1/q̆ , Θλ̆τ , Φλ̆τ ⟩ ( λ̆ ≻ 0 ). 4. τ λ̆ = ⟨Ψλ̆τ , (1 − (1 − Θq̆τ )λ̆ )1/q̆ , (1 − (1 − Φq̆τ )λ̆ )1/q̆ ⟩.  5 1.3 Navigating essentials of t-norm, t-conorm and Aggregation Operations Functions that qualify as fuzzy intersections and fuzzy unions are commonly referred to in the literature as t -norms and t -conorms, respectively. The following definitions related to t -norm and t -conorm are adapted from a book by George J. Klir and Bo Yuan. Definition 1.3.1 ( t -norm). The intersection of two fuzzy sets J˘1 and J˘2 , is defined generally by a mathematical operation that operates on the unit interval; that is, this operation is represented by a function called t-norm of the form If : [0, 1] × [0, 1] → [0, 1]. For each element y in the universal set Y , this function takes as input the pair of membership values of y in sets J˘1 and J˘2 , and outputs the membership value of y in the fuzzy set representing the intersection of J˘1 and J˘2. Therefore, (ψJ˘1 ∩J˘2 )(y) = If [ψJ˘1 (y), ψJ˘2 (y)] for all y ∈ Y. A fuzzy intersection If or t-norm is a binary operation described on the unit interval [0, 1], adhering to the following axioms ∀ p, q, r ∈ [0, 1] : Axiom 1 : If (p, 1) = p (condition at the boundary). Axiom 2 : q ≤ r =⇒ If (p, q) ≤ If (q, r) (monotonic). Axiom 3 : If (p, q) = If (q, p) (commutativity). Axiom 4 : If (p, If (q, r)) = If (If (p, q), r) (associativity). Definition 1.3.2 ( t -conorm). The union of two fuzzy sets J˘1 and J˘2 , is defined generally by a mathematical operation that operates on the unit interval; that is, this operation is represented by a function called t-conorm of the form Uf : [0, 1] × [0, 1] → [0, 1]. For each element y in the universal set Y , this function takes as input the pair of membership values of y in sets J˘1 and J˘2 , and outputs the membership value of y in the fuzzy set representing the union of J˘1 and J˘2. Therefore, (ψJ˘1 ∪J˘2 )(y) = Uf [ψJ˘1 (y), ψJ˘2 (y)] for all y ∈ Y. A fuzzy union Uf or t-conorm is a binary operation described on the unit interval [0, 1], adhering to the following axioms ∀ p, q, r ∈ [0, 1] :  6 Axiom 1 : Uf (p, 0) = p (condition at the boundary). Axiom 2 : q ≤ r =⇒ Uf (p, q) ≤ Uf (q, r) (monotonic). Axiom 3 : Uf (p, q) = Uf (q, p) (commutativity). Axiom 4 : Uf (p, Uf (q, r)) = Uf (Uf (p, q), r) (associativity). 1.3.1 Frank Norm In fuzzy set theory, the Frank norm, named after the mathematician M. J. Frank that is widely studied for its unique properties and applications in fuzzy logic and decision- making systems. The behavior of the Frank norm can be adjusted by the parameter m , influencing the characteristics of the norm. Definition 1.3.3 (Frank norm operations ). If we consider the real numbers a and b within the closed unit interval [0,1], then the Frank t-norm and Frank t-conorm are determined as follows: (ma −1)(mb −1) F(a, b) = logm (1 + m−1 ). ′ (m1−a −1)(m1−b −1) F (a, b) = 1 − logm (1 + m−1 ) where (a, b) ∈ [0, 1] × [0, 1] and m ̸= 1. Both the Frank t -norm and t -conorm (FTT) satisfy important properties such as boundary conditions, idempotency, monotonicity, and commutativity ensuring their applicability in decision-making frameworks. 1.3.2 Aggregation Operations Aggregation operations on fuzzy sets are operations by which several fuzzy sets are combined in a desirable way to produce a single fuzzy set. Definition 1.3.4 (Aggregation Operator ). Formally, any aggregation operation on n fuzzy sets ( n ≥ 2 ) is defined by a function κ : [0, 1]n → [0, 1]. When applied  7 to fuzzy sets J˘1 , J˘2 ,... , J˘n defined on Y , function κ produces an aggregate fuzzy set J˘ by operating on the membership grades of these sets for each y ∈ Y. Thus, ˘ = κ(J˘1 (y), J˘2 (y),... , J˘n (y)) for each y ∈ Y. To be recognized as a meaningful J(y) aggregation function, κ must meet at least the following axiomatic requirements, which capture the fundamental concept of aggregation. Axiom 1 : κ(0, 0,..., 0) = 0 and κ(1, 1,... , 1) = 1 (condition at the boundary). Axiom 2 : For any pair (x1 , x2 ,... , xn ) and (y1 , y2 ,... , yn ) of n -tuples such that xi , yi ∈ [0, 1] for all i ∈ Nn if xi ≤ yi ∀i ∈ Nn then κ(x1 , x2 ,..., xn ) ≤ κ(y1 , y2 ,..., yn ) that is, κ is monotonic increasing in all its arguments. Axiom 3 : κ is a continuous function. Axiom 4 : κ is an idempotent function that is κ(y, y,... , y) = y ∀y ∈ [0, 1]. 1.4 Multi Attribute Decision Making The decision-making process consists of several stages: recognizing the issues, forming preferences, assessing the options, and selecting the most suitable alternatives. In cases involving a single criterion, decision-making is generally straightforward, as the decision- maker simply selects the option with the highest preference score [11, 12]. However, evaluating alternatives across multiple criteria introduces challenges such as determining attribute weights, managing preference dependencies, and resolving conflicts among attributes, complicating the process. To address these complexities, more advanced approaches are required. Multi-Criteria Decision-Making (MCDM) was developed to handle conflicting attributes in decision-making, facilitating the identification of the optimal alternative. In light of the MADM literature, the following definitions are necessary : Definition 1.4.1 (Alternative). Alternatives refer to the various options or choices that can be evaluated during the process of making a decision. In MADM, these alternatives are assessed based upon specified attributes. Definition 1.4.2 (Attribute). In MADM, attributes are the particular criteria or characteristics employed to assess alternatives. These attributes may be classified as either  8 quantitative or qualitative factors. These attributes can be classified as either beneficial, where a greater amount is regarded as more advantageous for the decision maker, or cost- related, where a lower amount is considered more desirable. Definition 1.4.3 (Decision matrix ). In MADM, a decision matrix is typically denoted as Dm = [δrc ]s×k , as shown in equation (1.1). In this matrix, Gr signifies the alternatives and Rc (1 ≤ r ≤ s, 1 ≤ c ≤ k) refers to the attributes respectively. The term δrc indicates the performance ratings of rth alternative concerning the cth attribute. R1 R2 Rk   G1  δ11 δ12 ··· δ1k     δ21 δ22 ··· δ2k    G2 Dm =   (1.1).. . ...... . ..     Gs δs1 δs2 ··· δsk 1.5 Literature Review Aggregation operators (AOs) play a pivotal role in combining fuzzy evaluations to arrive at a consensus or an optimal decision. In addition to individual decision-making, multi-attribute group decision-making (MAGDM) enhances the process by incorporating diverse perspectives and preferences from multiple decision-makers, leading to a more collaborative and comprehensive approach under conditions of uncertainty. Researchers have extensively explored MADM and MAGDM techniques, establishing aggregation operators based on various norms within different fuzzy frameworks. Xu and Yager presented geometric AOs for IFS, and subsequently, xu proposed AOs for IFS and demonstrated their application in MCDM. Yager introduced the intuitionistic fuzzy (IF) Bonferroni mean operator and suggested generalizations of this operator. Later, Xu and Yager developed IF geometric BM operators and applied them to MCDM issue. Subsequently, Wang and Liu developed AOs for IFS under the Einstein norm, and utilized them in MADM. Xia et al. analyzed about Archimedean norms in the IF environment, developed specific AOs, and devised a MCDM approach, respectively. Yu  9 proposed IF prioritized AOs, that can handle prioritization among criteria, making the method more practical for MAGDM. Das et al. proposed geometric AOs based on IFS and presented an algorithm for applying these operators to MADM. Researchers have been examined grading the various IF numbers using different types of measurements including score, accuracy, and possibility measurements. Xu and Da developed the possibility degree approach for grading the numbers in the form of interval data. The possibility degree measurement for IF numbers was introduced by Wei and Tang [25, 26]. Gao & Dammak et al. provided a description of the possible measurement of interval-valued IFS and how it has been used in MCDM situations. A novel generalized enhanced score function was proposed by Garg to grade the various interval-valued intuitionistic fuzzy numbers. Liu et al. proposed IF Heronian mean operators, while Li et al. introduced IF Hamy mean operators under Dombi norms and employed them to address MAGDM issues. In the Pythagorean fuzzy perspective, Garg [32, 33] proposed and presented AOs under the Einstein norm and utilized them in MADM issues. Rahman et al. introduced a Pythagorean fuzzy weighted averaging AO and applied it to a MAGDM issue. Wu and Wei developed Pythagorean fuzzy AOs based on the Hamacher norm and utilized them in a MADM issue. Yi et al. proposed Pythagorean fuzzy Frank AOs and demonstrated their application to a MADM problem for airline service quality assessment. Shahzadi et al. introduced six Pythagorean fuzzy Yager AOs and applied them to MADM issues. In the framework of q̆ -ROFS, Liu and Wang proposed AOs and methods for MADM. Liu et al. and Wei et al. developed q̆ -ROF Heronian mean operators to employ them in MAGDM and MADM techniques respectively. Jana et al. established q̆ -ROF Dombi AOs and used them to develop a model for solving MADM. Wang et al. proposed q̆ -ROF hamy mean AOs and applied them to an enterprise resource planning system selection. Akram and Shahzadi extended Yager AOs to q̆ -ROFS and developed an algorithm for MADM. Akram et al. proposed AOs that combine q̆ -ROF soft sets with the Yager norm, demonstrating their application in MAGDM. Akram et al. established q̆ -ROF AOs using the Einstein norm, and illustrated their application in MADM. Farid and Riaz presented AOs based upon Aczel-Alsina operations within  10 the q̆ -ROF framework. MADM problems based on PFS and SFS have emerged as an exciting area of study. Wei proposed a picture fuzzy (PF) cross-entropy measure and developed a corresponding MADM procedure. Wang introduced the notions and operating rules of PFS, developed geometric AOs and applied them in MADM. Jin et al. presented Pearson’s PF correlation to address MADM issues. Khan proposed picture fuzzy AOs based upon Einstein norms and illustrated their use in MAGDM. Tian et al. developed PF weighted AOs and applied them to MCDM with incomplete criteria weights. Abdullah and Ashraf proposed a series of picture fuzzy AOs for MAGDM. Qin et al. introduced a MCDM method using PF Archimedean power Maclaurin symmetric mean operators. Qiyas et al. developed Yager AOs for PFS and employed them to the selection of emergency programs. Singh and Ganie introduced a correlation coefficient (CC) for PFS and demonstrated its applicability in MADM, pattern recognition and medical diagnosis. Singh et al. generalized CC for PFS, highlighting its applicability in MADM and pattern recognition. In the context of SFS, Kutlu and Kahraman proposed the TOPSIS method for the optimal site selection of electric vehicle charging stations. Elmira et al. developed BM operators for SFS and and applied them to MAGDM. Princy and Mohana presented a MADM method based on SF cross-entropy. Naeem et al. proposed SF rough Hamacher AOs for addressing MADM. Due to their efficiency and flexibility, q̆ -rung picture fuzzy sets have garnered significant attention from researchers in the realm of MADM. Liu et al. developed q̆ -rung picture fuzzy ( q̆ -RPF) Yager AOs and applied them to handle MADM issues. Akram and Shumaiza proposed MCDM methods based upon q̆ -RPF sets using them to select a housing society and an industrial robot. Feng and Guan presented q̆ - RPF Schweizer-Sklar Maclaurin symmetric mean operators and applied to MADM issues. Chitra and Prabakaran , developed a series of q̆ -RPF aggregators under Frank operators to approach MADM issues, such as selecting a location for starting a business venture and selecting Marketplaces for investment respectively.  11 1.6 Overview of the Thesis This research significantly advances fuzzy decision-making by enhancing the application of Frank aggregation operators within the q̆ -rung picture fuzzy context. The approach developed through these proposed operators serves as a valuable tool for improving decision processes and addressing challenges in multi-attribute decision-making. The comprehensive overview of the study is as follows: Chapter 1 analyzes the background and motivations behind this research, providing essential definitions and a literature survey that establishing the relevance of aggregation operators in decision-making and identifying gaps in existing literature to set the stage for subsequent chapters. Chapter 2 introduces q̆ -rung picture fuzzy Frank weighted aggregation operators providing a robust foundational framework for addressing complex decision-making challenges. This chapter establishes their effectiveness across various decision-making processes, offering a reliable approach to tackling these issues. Chapter 3 presents ordered q̆ -rung picture fuzzy Frank weighted operators building on the previous chapter. It also introduces a new technique called possible grading measurement for ordering and ranking q̆ -rung picture fuzzy numbers. The application of these operators, offers a structured approach to solving complex decision-making problems. Chapter 4 proposes novel Frank weighted Bonferroni averaging and geometric operators within the q̆ -rung picture fuzzy framework. These operators are designed to improve the aggregation of opinions in group decision-making by utilizing the flexibility and structure of the Bonferroni mean. These operators ensure that group decisions are more balanced, comprehensive, and reflective of diverse viewpoints, making them particularly useful in handling complex multi-attribute group decision-making scenarios under uncertainty. Chapter 5 integrates the proposed q̆ -rung picture fuzzy Frank prioritized weighted aggregators with the MULTIMOORA strategy, creating a powerful framework for  12 addressing multi-attribute group decision-making problems where the weights of criteria are not explicitly defined. This hybrid model ensures more balanced and comprehensive decision outcomes in uncertain and complex group decision-making scenarios. Chapter 6 delves into the concepts of hesitant and q̆ -rung picture fuzzy sets, highlighting their significance in decision-making contexts. This Chapter develops hesitant q̆ -rung picture fuzzy weighted aggregators based on the Frank norm, which allow for a more nuanced aggregation of information from multiple experts. These operators are then applied to a specific group decision-making problem, demonstrating their effectiveness in capturing the diverse opinions and hesitations of decision-makers. Chapter 7 concludes the research by summarizing the key findings and providing suggestions for future research in this field. CHAPTER 2  CHAPTER 2 EMPLOYING Q̆ -RUNG PICTURE FUZZY FRANK AGGREGATION OPERATORS FOR DECISION-MAKING 2.1 Introduction In the present scientific era, modeling uncertainty in MADM methodologies is crucial for addressing real-world situations. MADM is a regularly employed analytical mechanism whose principal goal is to choose the optimal alternative from a limited set of alternatives based on decision-maker’s preference information. However, the limitations of crisp set theory have been highlighted by the ambiguities and imprecision inherent in human judgments. As a result, Zadeh established the notion of fuzzy set (FS) to accommodate uncertain knowledge, allowing experts to express their degree of satisfaction through membership grades ranging over the closed interval [0, 1]. Over the years, many researchers have extended the idea of FS into various forms of FS. Ultimately, Li et al. presented q̆ -RPFS which are more suitable than a PFS or an SFS as they generalize the limitations of both. With society’s ongoing complexity and quick theoretical advancements, researchers have shown great interest in developing the aggregators to integrate uncertain information across various fuzzy settings to utilize them in MADM issues. Zhang et al. developed aggregators in the intuitionistic fuzzy domain. Seikh and Mandal established aggregators in the picture fuzzy domain. Moreover, q̆ -rung picture fuzzy sets can manage uncertain information more precisely and flexibly due to the parameter q̆. Additionally, FTT operations are well-suited for data aggregation with its operational parameter. In this chapter, q̆ -rung picture fuzzy Frank weighted averaging operator and q̆ -rung picture fuzzy Frank weighted geometric operator are proposed by extending the q̆ - rung orthopair fuzzy Frank arithmetic and geometric aggregators presented by Seikh 14  15 and Mandal. Furthermore, this chapter outlines an approach to address a MADM problem, demonstrating the utility and effectiveness of the proposed operators. 2.2 Key Concepts This section includes primary definitions and essential concepts. Definition 2.2.1 (Score and Accuracy functions ). Let τ = (Ψτ , Θτ , Φτ ) be a q̆ -rung picture fuzzy number. Then the score ( Sτ1 ) and the accuracy ( A1τ ) functions are defined as follows, Sτ1 = (1 + Ψq̆τ − Θq̆τ − Φq̆τ )/2 (2.1) A1τ = Ψq̆τ + Θq̆τ + Φq̆τ (2.2) Let us consider τ1 = (Ψτ1 , Θτ1 , Φτ1 ) and τ2 = (Ψτ2 , Θτ2 , Φτ2 ) to be two q̆ -RPFNs. Then from the equations (2.1) and (2.2), if Sτ11 ≻ Sτ12 then τ1 ≻ τ2. If Sτ11 = Sτ12 then, (i) if A1τ1 ≻ A1τ2 , then τ1 ≻ τ2. (ii) if A1τ1 = A1τ2 , then τ1 = τ2. Definition 2.2.2 ( q̆ -rung picture fuzzy Frank operational rules ). Let us take two positive real numbers to be λ̆ and m with m ̸= 1. Thus the Frank t -norm and Frank t -Conorm operations for q̆ -RPFNs, τ = (Ψτ , Θτ , Φτ ) , τ1 = (Ψτ1 , Θτ1 , Φτ1 ) and τ2 = (Ψτ2 , Θτ2 , Φτ2 ) are elucidated as follows: q̆ q̆ q̆ q̆ 1−Ψτ 1−Ψτ Θτ Θ 1 −1)(m 2 −1) 1 −1)(m τ2 −1) ˆ τ1 ⊕ τ2 = {(1 − logm (1 + (m m−1 ))1/q̆ , (logm (1 + (m m−1 ))1/q̆ , q̆ q̆ Φ Φ (m τ1 −1)(m τ2 −1) (logm (1 + m−1 ))1/q̆ }. q̆ q̆ q̆ q̆ Ψτ Ψ 1−Θτ 1−Θτ 1 −1)(m τ2 −1) 1 −1)(m 2 −1) ˆ τ1 ⊗ τ2 = {(logm (1 + (m m−1 ))1/q̆ , (1 − logm (1 + (m m−1 ))1/q̆ , q̆ q̆ 1−Φτ 1−Φτ (m 1 −1)(m 2 −1) (1 − logm (1 + m−1 ))1/q̆ }. q̆ q̆ (m1−Ψτ −1)λ̆ 1/q̆ (mΘτ −1)λ̆ 1/q̆ ˆ λ̆τ = {(1 − logm (1 + (m−1)λ̃−1 )) , (logm (1 + (m−1)λ̃−1 )) , q̆ (mΦτ −1)λ̆ (logm (1 + (m−1)λ̃−1 ))1/q̆ }.  16 q̆ q̆ (mΨτ −1)λ̆ 1/q̆ (m1−Θτ −1)λ̆ 1/q̆ ˆ τ λ̆ = {(logm (1 + (m−1)λ̃−1 )) , (1 − logm (1 + (m−1)λ̃−1 )) , q̆ (m1−Φτ −1)λ̆ (1 − logm (1 + (m−1)λ̃−1 ))1/q̆ }. Since Frank norm operations have yet to be explored in the q̆ -rung picture fuzzy environment, q̆ -rung picture fuzzy averaging and geometric accumulation operators have been proposed and derived. Definition 2.2.3 (q̆ -rung picture fuzzy Frank weighted averaging ( q̆-RPFFWA ) operator). Let τz = (Ψτz , Θτz , Φτz ) (z = 1, 2,... , t) be a number of q̆ -RPFNs with their Pt corresponding weight vectors ω̆z (z = 1, 2,... , t) , satisfies ω̆z = 1, ω̆z ∈ [0, 1]. Then z=1 the operator q̆-RPFFWA : τ t → τ is defined as, t L q̆-RPFFWA(τ1 , τ2 ,... , τt ) = ω̆z τz. z=1 Definition 2.2.4 (q̆ -rung picture fuzzy Frank weighted geometric ( q̆-RPFFWG ) operator). Let τz = (Ψτz , Θτz , Φτz ) (z = 1, 2,... , t) be a number of q̆ -RPFNs with their Pt corresponding weight vectors ω̆z (z = 1, 2,... , t) , satisfies ω̆z = 1, ω̆z ∈ [0, 1]. Then z=1 the operator q̆-RPFFWG : τ t → τ is defined as, t (τz )ω̆z. N q̆-RPFFWG(τ1 , τ2 ,... , τt ) = z=1 2.3 Results and Discussion This section presents the theorems related to the proposed operators and their properties. Theorem 2.3.1. If τz = (Ψτz , Θτz , Φτz ) (z = 1, 2,... , t) represents an ensemble of q̆ - RPFNs then their accumulated value using the q̆ -RPFFWA operator is again a q̆ -RPFN. t M q̆-RPFFWA(τ1 , τ2 ,... , τt ) = ω̆z τz z=1  t t  Y q̆ Y q̆ 1−Ψ ω̆ 1/q̆ Θ ω̆ 1/q̆ (1 − logm (1 + (m τz − 1) )) , (logm (1 + (m τz − 1) )) ,   z z        z=1 z=1 = t (2.3) Φq̆τz  Y  − 1)ω̆z ))1/q̆.      (logm (1 + (m   z=1  17 Proof. The theorem can be proved inductively as follows: For t = 2 , by definition 2.2.2, we get 2 L q̆-RPFFWA(τ1 , τ2 ) = ω̆z τz = ω̆1 τ1 ⊕ ω̆2 τ2 z=1 q̆ q̆ q̆ 1−Ψ τ1 −1)ω̆1 Θ Φ (m (m τ1 −1)ω̆1 1/q̆ (m τ1 −1)ω̆1 1/q̆ = {(1 − logm (1 + (m−1)ω̆1 −1 ))1/q̆ , (logm (1 + (m−1)ω̆1 −1 )) , (logm (1 + (m−1)ω̆1 −1 )) } q̆ q̆ q̆ 1−Ψ τ2 −1)ω̆2 Θ Φ (m (m τ2 −1)ω̆2 1/q̆ (m τ2 −1)ω̆2 1/q̆ ))1/q̆ , (logm (1 L {(1 − logm (1 + (m−1)ω̆2 −1 + (m−1)ω̆2 −1 )) , (logm (1 + (m−1)ω̆2 −1 )) } 2 q̆ 2 q̆ (m1−Ψτz − 1)ω̆z ))1/q̆ , (logm (1 + (mΘτz − 1)ω̆z ))1/q̆ , Q Q = {(1 − logm (1 + z=1 z=1 2 2 Φq̆τz − 1)ω̆z ))1/q̆ } since Q P (logm (1 + (m ω̆z = 1. z=1 z=1 It turns out to be correct for t = 2. If the consequence is correct for t = p , then p L q̆-RPFFWA(τ1 , τ2 ,..., τp ) = ω̆z τz z=1 p q̆ p q̆ (m1−Ψτz − 1)ω̆z ))1/q̆ , (logm (1 + (mΘτz − 1)ω̆z ))1/q̆ , Q Q = {(1 − logm (1 + z=1 z=1 p Φq̆τz ω̆z 1/q̆ Q (logm (1 + (m − 1) )) }. z=1 Now for t = p + 1 , p+1 L p L L q̆-RPFFWA(τ1 , τ2 ,..., τp , τp+1 ) = ω̆z τz = ω̆z τz ω̆p+1 τp+1 z=1 z=1   p q̆ 1/q̆   p q̆ 1/q̆ (m1−Ψτz −1)ω̆z (mΘτz −1)ω̆z Q Q = 1 − logm 1 + z=1 p P  , logm 1 + z=1 p P  , ω̆z −1 ω̆z −1 (m−1)z=1 (m−1)z=1   p q̆ 1/q̆ (mΦτz −1)ω̆z Q logm 1 + z=1 p P . ω̆z −1 (m−1)z=1   q̆ 1−Ψτ 1/q̆   Θ q̆ 1/q̆ L (m p+1 −1)ω̆p+1 (m τp+1 −1)ω̆p+1 1 − logm 1 + (m−1)ω̆p+1 −1 , logm 1 + (m−1)ω̆p+1 −1 ,   Φ q̆ 1/q̆ (m τp+1 −1)ω̆p+1 logm 1 + (m−1)ω̆p+1 −1.   p+1 1/q̆   p+1 1/q̆ 1−Ψq̆τz Q ω̆z Q Θq̆τ ω̆ = 1 − logm 1 + (m − 1) , logm 1 + (m z − 1) z , z=1 z=1  18   p+1 1/q̆ p+1 Q Φq̆τ ω̆ P logm 1 + (m z − 1) z since ( ω̆z = 1). z=1 z=1 Thus, assuming the assertion holds for t = p it will also hold for p + 1. Therefore, by the principle of mathematical induction, the given assertion is valid for all positive integers t. Theorem 2.3.2 (Idempotent). If all the q̆ -rung picture fuzzy numbers τz = (Ψτz , Θτz , Φτz ) (z = 1, 2,... , t) are identical, i.e., τz = τ , ∀z (z = 1, 2,... , t) , then q̆-RPFFWA(τ, τ,... , τ ) = τ. Proof. If τz = τ , ∀z then we get: t q̆ (m1−Ψτz − 1)ω̆z ))1/q̆ , Q q - RP F F W A(τ1 , τ2 ,..., τt ) = {(1 − logm (1 + z=1 t q̆ t q̆ (mΘτz − 1)ω̆z ))1/q̆ , (logm (1 + (mΦτz − 1)ω̆z ))1/q̆ } Q Q (logm (1 + z=1 z=1 t q̆ t q̆ (m1−Ψτ − 1)ω̆z ))1/q̆ , (logm (1 + (mΘτ − 1)ω̆z ))1/q̆ , Q Q = {(1 − logm (1 + z=1 z=1 t q̆ (mΦτ − 1)ω̆z ))1/q̆ } Q (logm (1 + z=1 t P t P ω̆z ω̆z 1−Ψq̆τ 1/q̆ Θq̆τ = {(1 − logm (1 + (m − 1)z=1 )) , (logm (1 + (m − 1)z=1 ))1/q̆ , t P ω̆z Φq̆τ (logm (1 + (m − 1)z=1 ))1/q̆ } q̆ q̆ q̆ = {(1 − logm (1 + (m1−Ψτ − 1)))1/q̆ , (logm (1 + (mΘτ − 1)))1/q̆ , (logm (1 + (mΦτ − 1)))1/q̆ }
= (Ψq̆τ )1/q̆ , (Θq̆τ )1/q̆ , (Φq̆τ )1/q̆ = {Ψτ , Θτ , Φτ } = τ. Hence the proof. Theorem 2.3.3 (Bounded). Let τz = (Ψτz , Θτz , Φτz ) (z = 1, 2,... , t) be a collection of q̆ -rung picture fuzzy numbers. Suppose τ − = min{τ1 , τ2 ,... , τt } and τ + = max{τ1 , τ2 ,... , τt } then τ − ≤ q̆-RPFFWA(τ1 , τ2 ,... , τt ) ≤ τ +.  19 Proof. Let τ − = (Ψ− , Θ− , Φ− ) and τ + = (Ψ+ , Θ+ , Φ+ ). Thus, Ψ− = minz {Ψτz } , Θ− = maxz {Θτz } , Φ− = maxz {Φτz } and Ψ+ = maxz {Ψτz } , Θ+ = minz {Θτz } , Φ+ = minz {Φτz }. + )q̆ q̆ − )q̆ Therefore, (m1−(Ψ − 1)ω̆z ≤ (m1−Ψτz − 1)ω̆z ≤ (m1−(Ψ − 1)ω̆z t t q̆ + )q̆ (m1−(Ψ − 1)ω̆z ) ≤ logm (1 + (m1−Ψτz − 1)ω̆z ) Q Q Subsequently, logm (1 + z=1 z=1 t 1−(Ψ− )q̆ − 1)ω̆z ) Q ≤ logm (1 + (m z=1 t t q̆ − )q̆ (m1−(Ψ −1)ω̆z ))1/q̆ ≤ (1−logm (1+ (m1−Ψτz −1)ω̆z ))1/q̆ Q Q It follows that: (1−logm (1+ z=1 z=1 t 1−(Ψ+ )q̆ − 1)ω̆z ))1/q̆. Q ≤ (1 − logm (1 + (m z=1 Similarly, t t q̆ + )q̆ (m(Θ − 1)ω̆z ))1/q̆ ≤ (logm (1 + (mΘτz − 1)ω̆z ))1/q̆ Q Q (logm (1 + z=1 z=1 t (Θ− )q̆ − 1)ω̆z ))1/q̆ and Q ≤ (1ogm (1 + (m z=1 t t q̆ + )q̆ (m(Φ − 1)ω̆z ))1/q̆ ≤ (logm (1 + (mΦτz − 1)ω̆z ))1/q̆ Q Q (logm (1 + z=1 z=1 t (Φ− )q̆ − 1)ω̆z ))1/q̆. Q ≤ (1ogm (1 + (m z=1 Thus, τ − ≤ q̆-RPFFWA(τ1 , τ2 ,..., τt ) ≤ τ +. Theorem 2.3.4 (Monotone). Let τz∗ = (Ψτz∗ , Θτz∗ , Φτz∗ ) (z = 1, 2,... , t) and τz = (Ψτz , Θτz , Φτz ) (z = 1, 2,... , t) be two groups of q̆ -rung picture fuzzy numbers. Suppose that Ψτz ≤ Ψτz∗ , Θτz ≥ Θτz∗ and Φτz ≥ Φτz∗ , ∀z. Then, q̆-RPFFWA(τ1 , τ2 ,... , τt ) ≤ q̆-RPFFWA(τ1∗ , τ2∗ ,... , τt∗ ). Proof. If Ψτz ≤ Ψτz∗ , Θτz ≥ Θτz∗ and Φτz ≥ Φτz∗ , ∀(z = 1, 2,..., t) , then q̆ 1−Ψq̆τ ∗ (m1−Ψτz − 1)ω̆z ≥ (m z − 1)ω̆z t t q̆ 1−Ψq̆τ ∗ (m1−Ψτz − 1)ω̆z ) ≥ logm (1 + − 1)ω̆z ) Q Q Subsequently, logm (1 + (m z z=1 z=1  20 t t q̆ 1−Ψq̆τ ∗ (m1−Ψτz − 1)ω̆z ))1/q̆ ≤ (1 − logm (1 + − 1)ω̆z ))1/q̆. Q Q It follows that: (1 − logm (1 + (m z z=1 z=1 Similarly, it can be shown that, t t q̆ Θq̆τ ∗ (mΘτz − 1)ω̆z ))1/q̆ ≥ (logm (1 + − 1)ω̆z ))1/q̆ and Q Q (logm (1 + (m z z=1 z=1 t t Φq̆τz Φq̆τ ∗ − 1)ω̆z ))1/q̆ ≥ (logm (1 + − 1)ω̆z ))1/q̆. Q Q (logm (1 + (m (m z z=1 z=1 Therefore, t q̆ t q̆ (m1−Ψτz − 1)ω̆z ))1/q̆ )q̆ − ((logm (1 + (mΘτz − 1)ω̆z ))1/q̆ )q̆ − Q Q ((1 − logm (1 + z=1 z=1 t Φq̆τz − 1)ω̆z ))1/q̆ )q̆ Q ((logm (1 + (m z=1 t t 1−Ψq̆τ ∗ Θq̆τ ∗ − 1)ω̆z ))1/q̆ )q̆ − ((logm (1 + − 1)ω̆z ))1/q̆ )q̆ − Q Q ≤ ((1 − logm (1 + (m z (m z z=1 z=1 t Q Φq̆τ ∗ ω̆z 1/q̆ q̆ ((logm (1 + (m z − 1) )) ). z=1 Let us denote τ = q̆-RPFFWA(τ1 , τ2 ,..., τt ) and τ ∗ = q̆-RPFFWA(τ1∗ , τ2∗ ,...τt∗ ). Then by definition 2.2.1 , we will have Sτ1 ≤ Sτ1∗. (I) If Sτ1 ≺ Sτ1∗ then τ ≺ τ ∗ q̆-RPFFWA(τ1 , τ2 ,.., τt ) ≺ q̆-RPFFWA(τ1∗ , τ2∗ ,.., τt∗ ). (II) If Sτ1 = Sτ1∗ then, t q̆ t q̆ ((1 − logm (1 + (m1−Ψτz − 1)ω̆z ))1/q̆ )q̆ − ((logm (1 + (mΘτz − 1)ω̆z ))1/q̆ )q̆ Q Q z=1 z=1 t q̆ (mΦτz − 1)ω̆z ))1/q̆ )q̆ Q −((logm (1 + z=1 t t 1−Ψq̆τ ∗ Θq̆τ ∗ − 1)ω̆z ))1/q̆ )q̆ − ((logm (1 + − 1)ω̆z ))1/q̆ )q̆ Q Q = ((1 − logm (1 + (m z (m z z=1 z=1 t Φq̆τ ∗ − 1)ω̆z ))1/q̆ )q̆. Q −((logm (1 + (m z z=1 Thus, from the condition Ψτz ≤ Ψτz∗ , Θτz ≥ Θτz∗ and Φτz ≥ Φτz∗ , ∀z , we have t t q̆ 1−Ψq̆τ ∗ (m1−Ψτz − 1)ω̆z ))1/q̆ )q̆ = ((1 − logm (1 + − 1)ω̆z ))1/q̆ )q̆ , Q Q ((1 − logm (1 + (m z z=1 z=1  21 t t q̆ Θq̆τ ∗ (mΘτz − 1)ω̆z ))1/q̆ )q̆ = ((logm (1 + − 1)ω̆z ))1/q̆ )q̆ and Q Q ((logm (1 + (m z z=1 z=1 t t q̆ Φq̆τ ∗ (mΦτz − 1)ω̆z ))1/q̆ )q̆ = ((logm (1 + − 1)ω̆z ))1/q̆ )q̆. Q Q ((logm (1 + (m z z=1 z=1 Therefore, from the equation (2.2) we have, t q̆ t q̆ A1τ = ((1 − logm (1 + (m1−Ψτz − 1)ω̆z ))1/q̆ )q̆ + ((logm (1 + (mΘτz − 1)ω̆z ))1/q̆ )q̆ + Q Q z=1 z=1 t Φq̆τz − 1)ω̆z ))1/q̆ )q̆ Q ((logm (1 + (m z=1 t t 1−Ψq̆τ ∗ Θq̆τ ∗ − 1)ω̆z ))1/q̆ )q̆ + ((logm (1 + − 1)ω̆z ))1/q̆ )q̆ Q Q = ((1 − logm (1 + (m z (m z z=1 z=1 t Φq̆τ ∗ − 1)ω̆z ))1/q̆ )q̆ = A1τ ∗ Q + ((logm (1 + (m z z=1 ⇒ From (I) & (II) we obtain, q̆-RPFFWA(τ1 , τ2 ,..., τt ) ≤ q̆-RPFFWA(τ1∗ , τ2∗ ,..., τt∗ ). Theorem 2.3.5. If τz = (Ψτz , Θτz , Φτz ) (z = 1, 2,... , t) represents an ensemble of q̆ - RPFNs then their accumulated value using the q̆-RPFFWG operator is again a q̆ -RPFN. t O q̆-RPFFWG(τ1 , τ2 ,... , τt ) = (τz )ω̆z z=1 t t   Y q̆ Y q̆ Ψ ω̆ 1/q̆ 1−Θ ω̆ 1/q̆ (m τz − 1) z )) , (1 − logm (1 + (m τz − 1) z )) ,     (logm (1 +     z=1 z=1 = t (2.4) 1−Φq̆τz  Y   ω̆z 1/q̆     (1 − logm (1 + (m − 1) ))   z=1 Proof. The proof of this theorem follows a similar approach to that of Theorem 2.3.1. Theorem 2.3.6 (Idempotent). If all the q̆ -rung picture fuzzy numbers τz = (Ψτz , Θτz , Φτz ) (z = 1, 2,... , t) are alike, i.e., τz = τ , ∀z (z = 1, 2,... , t) , then q̆-RPFFWG(τ, τ,... , τ ) = τ. Proof. The proof of this theorem follows a similar approach to that of Theorem 2.3.2.  22 Theorem 2.3.7 (Bounded). Let τz = (Ψτz , Θτz , Φτz )(z = 1, 2,... , t) be a collection of q̆ -rung picture fuzzy numbers. Suppose τ − = min{τ1 , τ2 ,... , τt } and τ + = max{τ1 , τ2 ,... , τt } then τ − ≤ q̆-RPFFWG(τ1 , τ2 ,... , τt ) ≤ τ +. Proof. The proof of this theorem follows a similar approach to that of Theorem 2.3.3. Theorem 2.3.8 (Monotone). Let τz∗ = (Ψτz∗ Θτz∗ , Φτz∗ )(z = 1, 2,... , t) and τz = (Ψτz , Θτz , Φτz ) (z = 1, 2,... , t) be two groups of q̆ -rung picture fuzzy numbers. Suppose Ψτz ≤ Ψτz∗ , Θτz ≥ Θτz∗ and Φτz ≥ Φτz∗ , ∀z , then q̆-RPFFWG(τ1 , τ2 ,... , τt ) ≤ q̆-RPFFWG(τ1∗ , τ2∗ ,... , τt∗ ). Proof. The proof of this theorem follows a similar approach to that of Theorem 2.3.4. 2.3.1 MADM Method in the q̆ -rung Picture Fuzzy Framework Consider a collection of ‘ s ’ alternatives (choices) represented by G = {G1 , G2 ,... , Gs } that the decision-maker ( M ) has evaluated using a set of various criteria R = {R1 , R2 ,... , Rk } corresponding to the weight vectors ω̆ = (ω̆1 , ω̆2 ,... , ω̆k ) with ω̆k ≻ 0 Pk and ω̆t = 1. The decision matrix denoted as Dm defined by, Dm = (δrc )s×k = t=1 (Ψrc , Θrc , Φrc )s×k. The steps involved in the decision-making are as follows: Step 1: On the basis of decision-maker’s preference for the alternatives, form the decision matrix Dm using the q̆ -RPF data, where Dm = (δrc )s×k = (Ψrc , Θrc , Φrc )s×k. Step 2: In a given MADM problem, if the attributes are of different types, namely benefit type (B) and cost type (C) , apply the condition given below to transform the ′ ′ ′ ′ ′ given matrix Dm into normalized q̆ -RPF matrix, Dm = (δrc )s×k = (Ψrc , Θrc , Φrc )s×k.   ′ δrc  if c ∈ B δrc = (2.5)  (δrc )c if c ∈ C  where (δrc )c = (Φrc , Θrc , Ψrc ).  23 Step 3: Using the q̆-RPFFWA and q̆-RPFFWG operators, compute the combined value δr with respect to the alternative Gr (r = 1, 2,... , s) by applying equations (2.3) & (2.4), respectively. Step 4: Utilizing equation (2.1), calculate the scores Sδ1r (r = 1, 2,... , s) corresponding to each aggregated value δr (r = 1, 2,... , s). Step 5: Select the most desirable choice Gr (r = 1, 2,... , s) by ranking the choices in decreasing order based on the scores obtained in the previous step. 2.3.2 Practical Instance Consider a business person is looking for a place to start a new venture. Let G1 , G2 , G3 , G4 are probable selection places. The location’s characteristics are R1 : labor availability; R2 : transport facility; R3 : Security; and R4 : raw material availability. The decision-maker has assigned weights to each attribute: ω̆1 = 0.2, ω̆2 = 0.3, ω̆3 = 0.3, ω̆4 = 0.2. The associated decision matrix Dm , provided by decision-maker, is given below: Step 1: The q̆ -rung picture fuzzy decision matrix Dm , which expresses the evaluations of alternatives, is shown below: G1 G2 G3 G4   R1 (0.5, 0.03, 0.4) (0.7, 0.03, 0.1) (0.4, 0.01, 0.4) (0.4, 0.08, 0.4)   R2  (0.4, 0.01, 0.5) (0.6, 0.04, 0.2) (0.4, 0.03, 0.5) (0.3, 0.03, 0.4)      R3  (0.3, 0.05, 0.5) (0.6, 0.06, 0.3) (0.5, 0.07, 0.4) (0.2, 0.01, 0.6)      R4 (0.5, 0.05, 0.3) (0.3, 0.07, 0.4) (0.5, 0.03, 0.4) (0.5, 0.07, 0.4) Step 2: Since the given attributes are of the same type, the normalization process for the matrix can be skipped. Step 3: Compute the combined value δr (when m = 2 and q̆ = 3) with respect to  24 the location Gr (r = 1, 2,... , s) using the q̆-RPFFWA and q̆-RPFFWG operators, as specified in equations (2.3) & (2.4), respectively. Table 2.1 presents the aggregated performance values of each location based upon the proposed operators. Step 4: Using equation (2.1), compute the scores Sδ1r for r = 1, 2, 3, 4. Table 2.2 displays the scores and rankings of the locations respectively. Table 2.1: Over all aggregated values of the Locations Accumulation operators q̆-RPFFWA q̆-RPFFWG δ1 = (0.4272,0.0279,0.4324) δ1 = (0.4018,0.0409,0.4544) δ2 = (0.5928,0.0477,0.2262) δ2 = (0.5416,0.0541,0.2872) δ3 = (0.4561,0.0311,0.4279) δ3 = (0.4475,0.0489,0.4355) δ4 = (0.3659,0.0311,0.4526) δ4 = (0.3123,0.0564,0.4816) Table 2.2: Score values of Locations and their ranking Accumulation Operators S 1δ1 S 1δ2 S 1δ3 S 1δ4 Ranking order q̆-RPFFWA 0.4985 0.5983 0.5082 0.4781 G2 > G3 > G1 > G4 q̆-RPFFWG 0.4855 0.5675 0.5035 0.4593 G2 > G3 > G1 > G4 Step 5: Therefore, it is clear that the second location, G2 is the optimal choice. 2.3.3 Examining the Influence of Parameters m and q̆ on Decision Outcomes A sensitivity analysis is conducted to evaluate the effects of parameters m and q̆ on decision outcomes using the q̆-RPFFWA and q̆-RPFFWG operators. First, for a fixed q̆ value of 3 the analysis explores varying m values ranging from 3 to 10. The scores  25 and rankings of each location are assessed across these m values. Table 2.3 reveals that the ranking consistently follows G2 ≻ G3 ≻ G1 ≻ G4 , indicating that G2 is the preferred location, irrespective of the variations in m. (a) Scores for each Location based on (b) Scores for each Location based on q̆-RPFFWA operator q̆-RPFFWG operator Figure 2.1: Sensitivity analysis of m using proposed Frank aggregators (a) Scores for each Location based on (b) Scores for each Location based on q̆-RPFFWA operator q̆-RPFFWG operator Figure 2.2: Sensitivity analysis of q̆ using proposed Frank aggregators  26 Conversely, for a fixed m value of 3, the analysis examines q̆ values of 4, 5, 7, 9, 12, 15, and 20. Table 2.4 illustrates that, even with varying q̆ values, G2 consistently emerges as the optimal location. Figures 2.1 and 2.2 visually represent the scores of each location for these specified parameter settings. Table 2.3: Impact of the variable m upon decision outcomes m Aggregators S 1δ1 S 1δ2 S 1δ3 S 1δ4 Ranking order OC 3 q̆-RPFFWA 0.4984 0.5979 0.5082 0.4779 G2 ≻ G3 ≻ G1 ≻ G4 G2 q̆-RPFFWG 0.4856 0.5682 0.5035 0.4595 G2 ≻ G3 ≻ G1 ≻ G4 G2 4 q̆-RPFFWA 0.4983 0.5976 0.5081 0.4777 G2 ≻ G3 ≻ G1 ≻ G4 G2 q̆-RPFFWG 0.4857 0.5688 0.5036 0.4597 G2 ≻ G3 ≻ G1 ≻ G4 G2 5 q̆-RPFFWA 0.4982 0.5975 0.5081 0.4776 G2 ≻ G3 ≻ G1 ≻ G4 G2 q̆-RPFFWG 0.4858 0.5692 0.5036 0.4598 G2 ≻ G3 ≻ G1 ≻ G4 G2 6 q̆-RPFFWA 0.5140 0.5996 0.5233 0.4958 G2 ≻ G3 ≻ G1 ≻ G4 G2 q̆-RPFFWG 0.4858 0.5695 0.5036 0.4599 G2 ≻ G3 ≻ G1 ≻ G4 G2 7 q̆-RPFFWA 0.4981 0.5972 0.5081 0.4774 G2 ≻ G3 ≻ G1 ≻ G4 G2 q̆-RPFFWG 0.4859 0.5697 0.5036 0.4600 G2 ≻ G3 ≻ G1 ≻ G4 G2 8 q̆-RPFFWA 0.4980 0.5971 0.5080 0.4774 G2 ≻ G3 ≻ G1 ≻ G4 G2 q̆-RPFF