Department of Mathematics
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Item Application of Ranking Heptagonal Neutrosophic Fuzzy Numbers to a Tansportation Problem(Library Information Services, COMSATS University Islamabad, Lahore Campus, 2024) Mudasir Nazar, Umer Farooq; CIIT/FA22-BSM-031/LHR, CIIT/FA22-BSM-59/LHR; Dr. Madiha Khalid; LHR TP 9924This thesis introduces the concept of neutrosophic heptagonal numbers and presents a novel approach for their ranking. Neutrosophic heptagonal numbers extend the realm of neutrosophic numbers to a seven-dimensional space, offering a more comprehensive representation of uncertainty in various real-world appli- cations. The developed ranking method provides a structured framework to assess and order these complex numbers, facilitating decision-making processes in uncertain environments. To demonstrate the practical utility of this approach, we apply it to solve a transportation problem framed within the context of a cost matrix. By lever- aging the ranking methodology on the cost matrix, we effectively address the uncertainties inherent in transportation planning, optimizing routes, and mini- mizing costs. Through computational experiments, we validate the effectiveness and efficiency of the proposed method in handling real-world scenarios charac- terized by intricate uncertainties. This research contributes to advancing the understanding and application of neutrosophic heptagonal numbers in decision sciences, offering a valuable tool for analyzing complex systems where uncertainty plays a significant roleItem Group Decision-Making Based on Linguistic Intuitionistic Fuzzy Numbers and Yager Weighted Aggregation Operator(Library Information Services, COMSATS University Islamabad, Lahore Campus, 2025) Aqsa; FA23-RMT-004; Dr. Atiq-ur-Rehman; LHR TP 9752We put forward two novel group decision making methods that utilize the propose Lin guistic Intuitionistic Fuzzy Yager Weighted Arithmetic Aggregation Operator for linguistic intuitionistic fuzzy numbers. The first proposed GDM addresses the situation where the weights of experts and attributes are fully understood; the second GDM approach that is being proposed takes into account the situation in which the weights of experts and the weights of attributes are entirely unknown. First, we propose new operational laws for LIFNs that are based on Yagers’ norm, specifically the addition operation and scalar mul tiplication operations for LIFNs can overcome the limitations of the current addition and scalar multiplication operations of LIFNs. After that, we put forward the LIFYWA AO of LIFN, relying on the proposed addition and scalar multiplication of LIFNs. We also demonstrate several characteristics of the suggested LIFYWA AO. Ultimately, we put for wardtwonewGDMmethodsofLIFNsbasedontheproposedLIFYWAAO.Thesuggested GDMmethodscanovercomethelimitations of current GDM approaches, which are unable to differentiate between the ranking orders of alternatives in certain cases.Item Application of Ranking Hexaagonal Neutrosophic Fuzzy Numbers to an Assignment Problem(Library Information Services, COMSATS University Islamabad, Lahore Campus, 2025) Usama Talib (FA20-BSM-044) : Zafar Iftikhar (FA20-BSM-054); Dr. Madiha Khalid; LHR TP 9920This thesis introduces the concept of neutrosophic hexagonal numbers and presents a novel approach for their ranking. Neutrosophic hexagonal numbers extend the realm of neutrosophic numbers to a seven-dimensional space, offering a more comprehensive representation of uncertainty in various real-world appli cations. The developed ranking method provides a structured framework to assess and order these complex numbers, facilitating decision-making processes in uncertain environments. To demonstrate the practical utility of this approach, we apply it to solve a transportation problem framed within the context of a cost matrix. By lever aging the ranking methodology on the cost matrix, we effectively address the uncertainties inherent in transportation planning, optimizing routes, and mini mizing costs. Through computational experiments, we validate the effectiveness and efficiency of the proposed method in handling real-world scenarios charac terized by intricate uncertainties. This research contributes to advancing the understanding and application of neutrosophic hexagonal numbers in decision sciences, offering a valuable tool for analyzing complex systems where uncertainty plays a significant roleItem Intelligent Solution for Transportation Problems Using Complete Ordering of Single Valued Neutrosophic Numbers(Library Information Services, COMSATS University Islamabad, Lahore Campus, 2025) Muhammad Anees (FA20-BSM-068) : Muhammad Waqar Khizer (FA20-BSM-069); Dr. Madiha Qayyum; LHR TP 9909TheTransportation Problem (TP) is extensively studied in operations research due to its nu merous real-world applications. However, real-world scenarios often involve uncertainty, making it difficult to determine costs, supply, and demand accurately. Fuzzy Numbers are commonly used to address this uncertainty but have limitations. These limitations arise from the inherent uncertainties in human thought patterns, which vary due to different val ues assigned to the three basic degrees of human thought—membership, non-membership, and indeterminacy—termed by Florentin. To better describe uncertainty in real-life situa tions, modern science uses Single Valued Neutrosophic Numbers (SVNS), which assign all three degree functions to any value, providing a more comprehensive mathematical model. Despite their extensive applicability in many Multi-Criteria Decision Making (MCDM) sit uations, ranking SVNS’s remain a significant challenge. Therefore, in this thesis we have f irstly defined a comprehensive ranking model for Single Valued Trapezoidal Neutrosophic Numbers (Normalized/Non-Normalized/ Uniform and Non Uniform Spread) based on four ranking functions/measures involving three existential levels namely- the Basic member ship (involving only membership function)-the Possible membership (a simple compliment of non- membershipfunction)–the Biggest membership (a combination of membership and indeterminacy function) and last but most important a Parametric combination of Possible membership and the Biggest membership. Next, we have employed the last ranking func tion– the parametric combination of Possible membership and the Biggest membership to ix build a Generalized Single Valued Trapezoidal Transportation Problem. In particular, we have utilized generalized trapezoidal data (normailized,/non-normalized/ with Uniform and Non Uniform Spread) to study three different types of transportation problems involving Crip and Single Valued Neutrosophic (triangular and trapezoidal) data for supply, demand, and cell costs. The thesis extends the principles of Least Cost Method (LCM), the North West Corner Method (NWCM), Vogel’s Approximation Method (VAM), and the Modified Distribution Method (MODI) to solve the Transportation Problem within a Single valued Neutrosophic environment. In this thesis, all the above modified solution models for TP incorporates generalized single valued trapezoidal neutrosophic data in different scenarios. Thus they provides the decision maker a flexible intelligence based environment which is capable of accommodating all different types of human mind behaviors (including the three basic behaviors of being optimist, pessimist and neutral ) due to the presence of logical op erations of fuzzy conjunctions and disjunctions that can be modelled by any pair of fuzzy t-norms and their conforms.