Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 7s (2024) 294 https://internationalpubls.com Assessing Healthcare Service Quality in Andhra Pradesh Using Icosagonal Fuzzy Numbers: An Innovative Multi Criteria Decision Making Approach Venkata Prasanna N a , Vinoth S b * a Department of Mathematics & Statistics,Vignan's Foundation for Science, Technology & Research, Vadlamudi, Andhra Pradesh 522213, India. E-mail: prasannakumarnagiri@gmail.com b Department of Mathematics & Statistics,Vignan's Foundation for Science, Technology & Research, Vadlamudi, Andhra Pradesh 522213, India.E-mail: vinomaths6@gmail.com Article History: Received: 01-06-2024 Revised: 03-07-2024 Accepted: 29-07-2024 Abstract: This study uses Icosagonal Fuzzy Numbers (IFNs) to assess the quality of healthcare services in Andhra Pradesh across four zones, we rank characteristics such as appointment scheduling, medical quality, hygiene, lab efficiency, care quality, safety, value for money, dependability, cost-effectiveness, facility quality, and recommendations. Data analysis suggests that hygiene is the most important factor in both zones. A comparison with other fuzzy number approaches (Pentagonal, Hexagonal, Heptagonal, and Hex-Decagonal) reveals that IFNs provide a more nuanced view of service quality. The findings provide insights for enhancing healthcare services in Andhra Pradesh. Keywords: Fuzzy number, icosagonal fuzzy number, defuzzification, mean alpha-cut, hospital quality, different zones in Andhra Pradesh 1. Introduction: Healthcare services are a critical component of modern society, ensuring the well-being and health of populations. These services encompass a wide range of activities, including preventive care, diagnosis, treatment, and rehabilitation. The effectiveness and efficiency of healthcare services can significantly influence the quality of life and overall health outcomes of individuals and communities. In recent years, the application of advanced mathematical and computational techniques has become increasingly important in optimizing healthcare services. One such technique involves the use of fuzzy numbers, which allow for more flexible and realistic modeling of uncertainty and imprecision inherent in healthcare data. This article explores the innovative use of icosagonal fuzzy numbers in the context of healthcare services. Healthcare service quality assessment is critical for improving patient outcomes and optimizing resource allocation. Traditional assessment methods often fall short due to the complexity and inherent uncertainties in healthcare data. Fuzzy set theory, particularly using icosagonal fuzzy numbers, offers a robust framework to address these challenges. Fuzzy numbers, pioneered by L.A. Zadeh [1,17], offer a pragmatic approach to handling imprecise numerical values. Various types of fuzzy sets have been devised to address ambiguity in existing problems, permitting a gradual evaluation of element membership within a set on a scale from 0 to 1. Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 7s (2024) 295 https://internationalpubls.com Interval arithmetic, initially proposed by Dwyer in 1951 and further developed through Zadeh's extension principle and Moore's contributions [19], extends standard arithmetic operations to encompass fuzzy numbers [18]. Triangular and trapezoidal fuzzy numbers have commonly been employed to represent imprecision in real-world scenarios. However, in this study, we advocate the utilization of Icosagonal fuzzy numbers to evaluate healthcare service quality across four zones in Andhra Pradesh. Zadeh [1] pioneered a novel concept known as fuzzy set theory (FST). This foundational theory of uncertainty has been widely and successfully applied across various disciplines. The core concept of fuzzy sets and numbers was formulated by Chang and Zadeh [2]. Subsequently, mathematicians have explored numerous outcomes of this theory [3,4], leading to significant advancements and diverse applications of FST. As a result, the topic has garnered substantial interest and attention. In our everyday life, we frequently encounter fuzziness in various decision-making scenarios. These situations often arise in contexts such as quality assessment, game theory, and other areas. Assessing the quality of healthcare services is of paramount importance, especially in diverse regions like Andhra Pradesh. With its varying demographics and healthcare infrastructure across different zones, ensuring high-quality healthcare delivery becomes crucial. Effective quality assessment mechanisms not only enhance patient outcomes but also contribute to the overall efficiency and reliability of healthcare systems. In Andhra Pradesh, where access to healthcare services may vary significantly between urban and rural areas, comprehensive quality assessment initiatives can help identify gaps, allocate resources effectively, and improve the overall standard of care. By addressing specific challenges and tailoring strategies to the unique needs of each zone, healthcare providers can strive towards achieving equitable, accessible, and high-quality healthcare services throughout the region. The notion of fuzzy numbers has been described as a fuzzy subset of the real number line by Dubois and Prade [15]. A fuzzy number embodies a quantity whose values are nuanced rather than precise, diverging from the exactness of single-valued numbers [12, 5]. To navigate the nuances of real-life scenarios, numerous scholars have turned to triangular and trapezoidal fuzzy numbers [8, 10, 11, 14]. Moreover, the introduction of hexagonal, octagonal, and decagonal fuzzy numbers has aimed to dispel ambiguity [4, 6, 7, 9]. Many researchers have prioritized addressing uncertain linguistic terms within group decision-making processes [3, 11, 13]. In decision-making scenarios, experts may resort to uncertain linguistic terms to articulate opinions when faced with uncertainty and information gaps. Such terms are frequently employed as inputs in decision analysis endeavors. Traditionally, linguistic values have been represented using fuzzy numbers like triangular and trapezoidal shapes. However, confining membership functions to triangular or trapezoidal forms becomes intricate when vagueness extends across twenty distinct points. Thus, this paper embarks on exploring a novel concept: the Icosagonal Fuzzy Number (IFN), tailored for uncertain linguistic environments. This paper’s work is organised as follows: Section 2 provides a quick overview of the fundamental definitions of fuzzy numbers. Section 3 presents symmetric and asymmetric representations of Icosagonal fuzzy numbers with corresponding α-cuts. Section 4 discusses defuzzification algorithms for linear and Non-linear Icosagonal fuzzy numbers with symmetry. Also, this section proposes a ranking method based on the mean of method. Section 5 includes comparison of proposed Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 7s (2024) 296 https://internationalpubls.com defuzzification approaches for different zones of Andhra Pradesh. The conclusion of this paper is proceeded in Section 6. 2. Mathematical Preliminaries: This section covers essential definitions and notations that will be utilized throughout the paper. Definition 2.1: Let ̆̀ be the universal set, then the fuzzy set denoted by ̃ is the family of ordered pairs ( ̆̀, ̂ ̆̀)), where the first element ̆̀ is the member of universal set ̆̀ and the second element represents the membership value corresponding to the element ̆̀ ∈ ̆̀ in [0, 1] i.e. ̃= {(( ̆̀, ̂ ̆̀))| ̆̀ ∈ ̆̀} and the mapping ̃ : ̆̀ → [0, 1] is the membership function of ̃. Definition 2.2: A fuzzy set ̃ in ̃is called a fuzzy number if it satisfies the following axioms: (1) There exist at least one ̆̀ ∈ ̃ with ̃ ̆̀ = 1. (2) Membership function ̃ ̆̀ is piecewise continuous. (3) ̃ must be a convex fuzzy set. Definition 2.3: The - cut of a fuzzy set ̃ defined on the universal set ̆̀ is a crisp set denoted by ̃ , which contains all those elements of ̆̀, whose membership value is greater than or equal to ∈ [0, 1] in ̃ i.e. ̃ = { ̆̀ ∈ ̆̀| ̃ ̆̀) ≥ } for ∈ [0, 1]. 3. Theory of Icosagonal Fuzzy Number and Its Different Representations: In this section, we study different types of icosagonal fuzzy numbers. Definition 3.1: A linear icosagonal fuzzy number with symmetry ̃ given by Figure 1 is represented by ̂ ={( ̀ ̀ ̀ ̀ ̀ ̀ ̀ ̀ ̀ ̀ ̀ ̀ ̀ ̀ ̀ ̀ ̀ ̀ ̀ ̀ ); ( ̅̈ , ̅̈ , ̅̈ , ̅̈ ) } and its membership function is defined as follows: Figure 1: Linear Icosagonal fuzzy number with symmetry ̅̈ ̅̈ ̅̈ ̅̈ ̃ ̀̆ Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 7s (2024) 297 https://internationalpubls.com ̃ ̆̀ { ̆̀ ̀ ̅̈ . ̆̀ ̀ ̀ ̀ / ̀ ̆̀ ̀ ̅̈ ̀ ̆̀ ̀ ̅̈ ̅̈ ̅̈ . ̆̀ ̀ ̀ ̀ / ̀ ̆̀ ̀ ̅̈ ̀ ̆̀ ̀ ̅̈ ̅̈ ̅̈ . ̆̀ ̀ ̀ ̀ / ̀ ̆̀ ̀ ̅̈ ̀ ̆̀ ̀ ̅̈ ̅̈ ̅̈ . ̆̀ ̀ ̀ ̀ / ̀ ̆̀ ̀ ̅̈ ̀ ̆̀ ̀ ̅̈ ̅̈ . ̆̀ ̀ ̀ ̀ / ̀ ̆̀ ̀ ̀ ̆̀ ̀ ̅̈ ̅̈ . ̀ ̆̀ ̀ ̀ / ̀ ̆̀ ̀ ̅̈ ̀ ̆̀ ̀ ̅̈ ̅̈ ̅̈ . ̀ ̆̀ ̀ ̀ / ̀ ̆̀ ̀ ̅̈ ̀ ̆̀ ̀ ̅̈ ̅̈ ̅̈ . ̀ ̆̀ ̀ ̀ / ̀ ̆̀ ̀ ̅̈ ̀ ̆̀ ̀ ̅̈ ̅̈ ̅̈ . ̀ ̆̀ ̀ ̀ / ̀ ̆̀ ̀ ̅̈ ̀ ̆̀ ̀ ̅̈ . ̀ ̆̀ ̀ ̀ / ̀ ̆̀ ̀ ̆̀ ̀ } where 0 < ̅̈ ̅̈ ̅̈ ̅̈ 1. Definition 3.2: The - cut of linear icosagonal fuzzy number with symmetry is the collection of all ̆̀ ∈ ̆̀, whose membership function ̃ ̆̀ is greater than or equal to i.e., ̃ , ̆̀ ∈ ̆̀ ̃ ̆̀ - for ∈ [0, 1] as Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 7s (2024) 298 https://internationalpubls.com ̃ { ̃ ̀ . ̅̈ / ̀ ̀ ∈ [ ̅̈ ] ̃ ̀ . ̅̈ ̅̈ ̅̈ / ̀ ̀ ∈ [ ̅̈ ̅̈ ] ̃ ̀ . ̅̈ ̅̈ ̅̈ / ̀ ̀ ∈ [ ̅̈ ̅̈ ] ̃ ̀ . ̅̈ ̅̈ ̅̈ / ̀ ̀ ∈ [ ̅̈ ̅̈ ] ̃ ̀ . ̅̈ ̅̈ / ̀ ̀ ∈ [ ̅̈ ] ̃ ̀ . ̅̈ ̅̈ / ̀ ̀ ∈ [ ̅̈ ] ̃ ̀ . ̅̈ ̅̈ ̅̈ / ̀ ̀ ∈ [ ̅̈ ̅̈ ] ̃ ̀ . ̅̈ ̅̈ ̅̈ / ̀ ̀ ∈ [ ̅̈ ̅̈ ] ̃ ̀ . ̅̈ ̅̈ ̅̈ / ̀ ̀ ∈ [ ̅̈ ̅̈ ] ̃ ̀ . ̅̈ / ̀ ̀ ∈ [ ̅̈ ] } Where ̃ ̃ ̃ ̃ ̃ ̃ ̃ ̃ ̃ ̃ are the increasing and decreasing functions of correspondingly in the respective intervals. Definition 3.3: A linear icosagonal fuzzy number with asymmetry ̃ given by Figure 2 is represented by ̃ = {( ̀ ̀ ̀ ̀ ̀ ̀ ̀ ̀ ̀ ̀ ̀ ̀ ̀ ̀ ̀ ̀ ̀ ̀ ̀ ̀ ); ( ̅̈ , ̅̈ , ̅̈ , ̅̈ , ̈ ̅ , ̈ ̅ , ̈ ̅ , ̈ ̅ } and its membership function is defined as follows: Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 7s (2024) 299 https://internationalpubls.com ̃ ̆̀ { ̆̀ ̀ ̅̈ . ̆̀ ̀ ̀ ̀ / ̀ ̆̀ ̀ ̅̈ ̀ ̆̀ ̀ ̅̈ ̅̈ ̅̈ . ̆̀ ̀ ̀ ̀ / ̀ ̆̀ ̀ ̅̈ ̀ ̆̀ ̀ ̅̈ ̅̈ ̅̈ . ̆̀ ̀ ̀ ̀ / ̀ ̆̀ ̀ ̅̈ ̀ ̆̀ ̀ ̅̈ ̅̈ ̅̈ . ̆̀ ̀ ̀ ̀ / ̀ ̆̀ ̀ ̅̈ ̀ ̆̀ ̀ ̅̈ ̅̈ . ̆̀ ̀ ̀ ̀ / ̀ ̆̀ ̀ ̀ ̆̀ ̀ ̅̈ ̅̈ . ̀ ̆̀ ̀ ̀ / ̀ ̆̀ ̀ ̅̈ ̀ ̆̀ ̀ ̅̈ ̅̈ ̅̈ . ̀ ̆̀ ̀ ̀ / ̀ ̆̀ ̀ ̅̈ ̀ ̆̀ ̀ ̅̈ ̅̈ ̅̈ . ̀ ̆̀ ̀ ̀ / ̀ ̆̀ ̀ ̅̈ ̀ ̆̀ ̀ ̅̈ ̅̈ ̅̈ . ̀ ̆̀ ̀ ̀ / ̀ ̆̀ ̀ ̅̈ ̀ ̆̀ ̀ ̅̈ . ̀ ̆̀ ̀ ̀ / ̀ ̆̀ ̀ ̆̀ ̀ } where 0 < ̅̈ < ̈ ̅ < ̅̈ < ̈ ̅ < ̅̈ < ̈ ̅ ̅̈ < ̈ ̅ < 1. Definition 3.4: The -cut of linear icosagonal fuzzy number with asymmetry is the collection of all ̆̀ ∈ ̆̀, whose membership function ̃ ̆̀ is greater than or equal to i.e. ̃ , ̆̀ ∈ ̆̀ ̃ ̆̀ - for ∈ [0, 1] as Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 7s (2024) 300 https://internationalpubls.com ̃ { ̃ ̀ . ̅̈ / ̀ ̀ ∈ [ ̅̈ ] ̃ ̀ . ̅̈ ̅̈ ̅̈ / ̀ ̀ ∈ [ ̅̈ ̅̈ ] ̃ ̀ . ̅̈ ̅̈ ̅̈ / ̀ ̀ ∈ [ ̅̈ ̅̈ ] ̃ ̀ . ̅̈ ̅̈ ̅̈ / ̀ ̀ ∈ [ ̅̈ ̅̈ ] ̃ ̀ . ̅̈ ̅̈ / ̀ ̀ ∈ [ ̅̈ ] ̃ ̀ ( ̈ ̅ ̈ ̅ ) ̀ ̀ ∈ * ̈ ̅ + ̃ ̀ ( ̈ ̅ ̈ ̅ ̈ ̅ ) ̀ ̀ ∈ * ̈ ̅ ̈ ̅ + ̃ ̀ ( ̈ ̅ ̈ ̅ ̈ ̅ ) ̀ ̀ ∈ * ̈ ̅ ̈ ̅ + ̃ ̀ ( ̈ ̅ ̈ ̅ ̈ ̅ ) ̀ ̀ ∈ * ̈ ̅ ̈ ̅ + ̃ ̀ . ̈ ̅ / ̀ ̀ ∈ * ̈ ̅ + } where ̃ ̃ ̃ ̃ ̃ ̃ ̃ ̃ ̃ ̃ are the increasing and decreasing functions of correspondingly in the respective intervals. Figure 2: Linear Icosagonal fuzzy number with asymmetry Definition 3.5: A non-linear icosagonal fuzzy number with symmetry ̃ given by Figure 3 is represented by ̃ = {( ̀ ̀ ̀ ̀ ̀ ̀ ̀ ̀ ̀ ̀ ̀ ̀ ̀ ̀ ̀ ̀ ̀ ̀ ̀ ̀ ); ( ̃ , ̃ , ̃ ̃ ̃ ; ̃ , ̃ , ̃ , ̃ ̃ ̅̈ ̅̈ ̅̈ ̅̈ ) } and its membership function is defined as follows: ̅̈ ̅̈ ̅̈ ̅̈ ̂ ̀̆ ̈ ̅ ̈ ̅ ̈ ̅ ̈ ̅ Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 7s (2024) 30 https://internationalpubls.com ̃ ̆̀ { ̆̀ ̀ ̅̈ . ̆̀ ̀ ̀ ̀ / ̃ ̀ ̆̀ ̀ ̅̈ ̀ ̆̀ ̀ ̅̈ ̅̈ ̅̈ . ̆̀ ̀ ̀ ̀ / ̃ ̀ ̆̀ ̀ ̅̈ ̀ ̆̀ ̀ ̅̈ ̅̈ ̅̈ . ̆̀ ̀ ̀ ̀ / ̃ ̀ ̆̀ ̀ ̅̈ ̀ ̆̀ ̀ ̅̈ ̅̈ ̅̈ . ̆̀ ̀ ̀ ̀ / ̃ ̀ ̆̀ ̀ ̅̈ ̀ ̆̀ ̀ ̅̈ ̅̈ . ̆̀ ̀ ̀ ̀ / ̃ ̀ ̆̀ ̀ ̀ ̆̀ ̀ ̅̈ ̅̈ . ̀ ̆̀ ̀ ̀ / ̃ ̀ ̆̀ ̀ ̅̈ ̀ ̆̀ ̀ ̅̈ ̃ ̅̈ . ̀ ̆̀ ̀ ̀ / ̃ ̀ ̆̀ ̀ ̅̈ ̀ ̆̀ ̀ ̅̈ ̅̈ ̅̈ . ̀ ̆̀ ̀ ̀ / ̃ ̀ ̆̀ ̀ ̅̈ ̀ ̆̀ ̀ ̅̈ ̅̈ ̅̈ . ̀ ̆̀ ̀ ̀ / ̃ ̀ ̆̀ ̀ ̅̈ ̀ ̆̀ ̀ ̅̈ . ̀ ̆̀ ̀ ̀ / ̃ ̀ ̆̀ ̀ ̆̀ ̀ } where 0 < ̅̈ < ̅̈ < ̅̈ ̅̈ < 1. Definition 3.6: The cut of non-linear icosagonal fuzzy number with symmetry is the collection of all ̆̀ ∈ ̆̀, whose membership function ̃ ̆̀ is greater than or equal to i.e. ̃ , ̆̀ ∈ ̆̀ ̃ ̆̀ - for ∈ [0, 1] as Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 7s (2024) 302 https://internationalpubls.com ̃ { ̃ ̀ . ̅̈ / ̃ ̀ ̀ ∈ [ ̅̈ ] ̃ ̀ . ̅̈ ̅̈ ̅̈ / ̃ ̀ ̀ ∈ [ ̅̈ ̅̈ ] ̃ ̀ . ̅̈ ̅̈ ̅̈ / ̃ ̀ ̀ ∈ [ ̅̈ ̅̈ ] ̃ ́ ̀ . ̅̈ ̅̈ ̅̈ / ̃ ̀ ̀ ∈ [ ̅̈ ̅̈ ] ̃ ́ ̀ . ̅̈ ̅̈ / ̃ ̀ ̀ ∈ [ ̅̈ ] ̃ ̀ . ̅̈ ̅̈ / ̃ ̀ ̀ ∈ [ ̅̈ ] ̃ ̀ . ̅̈ ̅̈ ̅̈ / ̃ ̀ ̀ ∈ [ ̅̈ ̅̈ ] ̃ ̀ . ̅̈ ̅̈ ̅̈ / ̃ ̀ ̀ ∈ [ ̅̈ ̅̈ ] ̃ ̀ . ̅̈ ̅̈ ̅̈ / ̃ ̀ ̀ ∈ [ ̅̈ ̅̈ ] ̃ ̀ . ̅̈ / ̃ ̀ ̀ ∈ [ ̅̈ ] } where ̃ ̃ ̃ ̃ ̃ ̃ ̃ ̃ ̃ ̃ are the increasing and decreasing functions of correspondingly in the respective intervals. Figure 3: Non-Linear Isagonal fuzzy number with symmetry ̅̈ ̅̈ ̅̈ ̅̈ ̃ ̀̆ Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 7s (2024) 303 https://internationalpubls.com 4. Defuzzification: Defuzzification plays a crucial role in converting fuzzy inference results into precise values. To achieve this, fuzzy logic utilizes various techniques such as the centroid method, the greatest of maxima, the smallest of maxima, the mean of maxima, the bisector of area, and the graded mean integral value. In this proposal, we outline four distinct methods for defuzzifying symmetric linear icosagonal fuzzy numbers. These techniques are based on several defuzzification strategies that can also be applied to non-linear symmetric icosagonal fuzzy numbers (NLS), among other scenarios. This approach provides a comprehensive and versatile perspective on the defuzzification process, tailored to the specific characteristics of the fuzzy numbers in question. 4.1. Defuzzification of LS Based on the Approach of Mean of α-cut Method for NLS: In this section, we proposed a method to compute the defuzzification of linear icosagonal fuzzy number with symmetry as follows: Since the left and right -cuts of a non-linear icosagonal fuzzy number with symmetry are given by ̀ . ̅̈ / ̃ ̀ ̀ ∈ [ ̅̈ ] ̀ . ̅̈ / ̃ ̀ ̀ ∈ [ ̅̈ ] ̀ . ̅̈ ̅̈ ̅̈ / ̃ ̀ ̀ ∈ [ ̅̈ ̅̈ ] ̀ . ̅̈ ̅̈ ̅̈ / ̃ ̀ ̀ ∈ [ ̅̈ ̅̈ ] ̀ . ̅̈ ̅̈ ̅̈ / ̃ ̀ ̀ ∈ [ ̅̈ ̅̈ ] ̀ . ̅̈ ̅̈ ̅̈ / ̃ ̀ ̀ ∈ [ ̅̈ ̅̈ ] ̀ . ̅̈ ̅̈ ̅̈ / ̃ ̀ ̀ ∈ [ ̅̈ ̅̈ ] ̀ . ̅̈ ̅̈ ̅̈ / ̃ ̀ ̀ ∈ [ ̅̈ ̅̈ ] Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 7s (2024) 304 https://internationalpubls.com ̀ . ̅̈ ̅̈ / ̃ ̀ ̀ ∈ [ ̅̈ ] ̀ . ̅̈ ̅̈ / ̃ ̀ ̀ ∈ [ ̅̈ ] Then the mean of cut of ̂ is ̃ ∫ . / ̃ ∫ . / ̅̈ ∫ . / ̅̈ ̅̈ ∫ . / ̅̈ ̅̈ ∫ . / ̅̈ ̅̈ ∫ . / ̅̈ ̃ ∫ ( ̀ . ̅̈ / ̃ ̀ ̀ ̀ . ̅̈ / ̃ ̀ ̀ ) ̅̈ ∫ ( ̀ . ̅̈ ̅̈ ̅̈ / ̃ ̀ ̀ ̀ . ̅̈ ̅̈ ̅̈ / ̃ ̀ ̀ ) ̅̈ ̅̈ ∫ ( ̀ . ̅̈ ̅̈ ̅̈ / ̃ ̀ ̀ ̀ . ̅̈ ̅̈ ̅̈ / ̃ ̀ ̀ ) ̅̈ ̅̈ ∫ ( ̀ . ̅̈ ̅̈ ̅̈ / ̃ ̀ ̀ ̀ . ̅̈ ̅̈ ̅̈ / ̃ ̀ ̀ ) ̅̈ ̅̈ ∫ ( ̀ . ̅̈ ̅̈ / ̃ ̀ ̀ ̀ . ̅̈ ̅̈ / ̃ ̀ ̀ ) ̅̈ Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 7s (2024) 305 https://internationalpubls.com ̃ ̅̈ ̀ ̀ ̅̈ ̀ ̀ ( ̃ ) ̅̈ ̀ ̀ ( ̃ ) ̅̈ ̅̈ ̀ ̀ ̅̈ ̅̈ ̀ ̀ ( ̃ ) ̅̈ ̅̈ ̀ ̀ ( ̃ ) ̅̈ ̅̈ ̀ ̀ ̅̈ ̅̈ ̀ ̀ ̃ ̅̈ ̅̈ ̀ ̀ ( ̃ ) ̅̈ ̅̈ ̀ ̀ ̅̈ ̅̈ ̀ ̀ ̃ ̅̈ ̅̈ ̀ ̀ ( ̃ ) ̅̈ ̀ ̀ ̅̈ ̀ ̀ ̃ ̅̈ ̀ ̀ ( ̃ ) If we set all the non-linear parameters as unity i.e., ̃ , ̃ , ̃ ̃ , ̃ ̃ , ̃ , ̃ , ̃ ̃ =1, then the proposed defuzzification technique applies for the linear icosagonal fuzzy number with symmetry as follows; ̃ ̅̈ ̀ ̀ ̀ ̀ ̅̈ ̅̈ ̀ ̀ ̀ ̀ ̅̈ ̅̈ ̀ ̀ ̀ ̀ ̅̈ ̅̈ ̀ ̀ ̀ ̀ ̅̈ ̀ ̀ ̀ ̀ In particular, for ̅̈ = 0.20, ̅̈ = 0.40, ̅̈ = 0.60 and ̅̈ = 0.80 we have ̃ ̀ ̀ ̀ ̀ ̀ ̀ ̀ ̀ ̀ ̀ ̀ ̀ ̀ ̀ ̀ ̀ ̀ ̀ ̀ ̀ Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 7s (2024) 306 https://internationalpubls.com 4.2. Calculation of fuzzified values: In this section, employing a systematic approach with linguistic variables crafted to encapsulate and scrutinize the complexity within the collected responses, we precisely computed the fuzzified values using linear icosagonal fuzzy numbers. This involved a detailed procedure where the gathered data was rigorously examined using a series of linguistic factors designed to uncover and capture the intricate nuances embedded in the provided responses. Linguistic variables Icosagonal Fuzzy Number (IFN) 1. Very dissatisfied (1.0, 2.5, 4.0, 5.5, 7.0, 8.5, 10.0, 11.5, 13.0, 14.5, 16.0, 17.5, 19.0, 20.5, 22.0, 23.5, 25.0, 26.5, 28.0, 29.5) 2. Dissatisfied (2.5, 4.0, 5.5, 7.0, 8.5, 10.0, 11.5, 13.0, 14.5, 16.0, 17.5, 19.0, 20.5, 22.0, 23.5, 25.0, 26.5, 28.0, 29.5, 31.0) 3. Neutral (4.0, 5.5, 7.0, 8.5, 10.0, 11.5, 13.0, 14.5, 16.0, 17.5, 19.0, 20.5, 22.0, 23.5, 25.0, 26.5, 28.0, 29.5, 31.0, 32.5) 4. Satisfied (5.5, 7.0, 8.5, 10.0, 11.5, 13.0, 14.5, 16.0, 17.5, 19.0, 20.5, 22.0, 23.5, 25.0, 26.5, 28.0, 29.5, 31.0, 32.5, 34.0) 5. Very Satisfied (7.0, 8.5, 10.0, 11.5, 13.0, 14.5, 16.0, 17.5, 19.0, 20.5, 22.0, 23.5, 25.0, 26.5, 28.0, 29.5, 31.0, 32.5, 34.0, 35.5) 5. Comparative Study of proposed different zones in Andhra Pradesh: In this section, we compared all the proposed different zones in Andhra Pradesh using defuzzification technique. The composition of zones and districts in each zone of Andhra Pradesh is as hereunder: Zone-I: Srikakulam, Vizianagaram and Visakhapatnam Zone-II: East Godavari, West Godavari and Krishna Zone-III: Guntur, Prakasam and Nellore Zone-IV: Chittor, Kadapa, Anantapur and Kurnool ZONE – I S. No Criteria Defuzzified value Rank (Icosagonal FN) 1 Booking an appointment 19.8269 2 2 Quality of medicals 19.5962 4 3 Hygiene 19.9038 1 4 Time taken by labs 19.0962 8 5 Care provided by medical professionals & nurses 19.6346 3 6 Safety measures 19.4423 6 7 Value for money 19.2885 7 Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 7s (2024) 307 https://internationalpubls.com 8 Reliability and recovery 19.5962 4 9 Cost effectiveness 18.9808 9 10 Quality of food, lift facility & security 18.5962 10 11 Recommendation 16.9038 11 Table 1: Computation for ranking of various parameters (health services) in Zone-I using defuzzified values. ZONE – I S . n o Criteria Penta - gonal D.Val ue Ra nk (P) Hexago nal D.Valu e Ra nk (H) Hept a- gonal D.Val ue Rank (Hep ta) Hex- Decago nal D.Valu e Rank (Hex. D) Icosago nal D. Value Ra nk (I) 1 Booking an appointment 16.15 38 2 14.8782 2 13.10 26 2 14.189 1 2 19.8269 2 2 Quality of medicals 15.69 23 4 14.4936 4 12.79 49 4 13.996 8 4 19.5962 4 3 Hygiene 16.30 77 1 15.0064 1 13.20 51 1 14.253 2 1 19.9038 1 4 Time taken by labs 14.69 23 8 13.6603 8 12.12 82 8 13.580 1 8 19.0962 8 5 Care provided by medical professional s & nurses 15.76 92 3 14.5577 3 12.84 62 3 14.028 8 3 19.6346 3 6 Safety measures 15.38 46 6 14.2372 6 12.58 97 6 13.868 6 6 19.4423 6 7 Value for money 15.07 69 7 13.9808 7 12.38 46 7 13.740 4 7 19.2885 7 8 Reliability and recovery 15.69 23 4 14.4936 4 12.79 49 4 13.996 8 4 19.5962 4 9 Cost effectivenes s 14.46 15 9 13.4679 9 11.97 44 9 13.484 0 9 18.9808 9 1 0 Quality of food, lift facility & security 13.69 23 10 12.8269 10 11.46 15 10 13.163 5 10 18.5962 10 1 1 Recommend ation 10.30 77 11 10.0064 11 9.205 1 11 11.753 2 11 16.9038 11 Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 7s (2024) 308 https://internationalpubls.com Table 2: Defuzzified Values and Rankings of Healthcare Service Quality Criteria. 5.1. Comparative Analysis of Icosagonal Fuzzy Number Rankings with Proposed and Existing Rankings ZONE – I Defuzzified values S. No Criteria Penta- gonal (R1) Hexagonal (R2) Hepta- gonal (R3) Hex- Decagonal (R4) Icosagonal (R5) Relation between R1, R2, R3, R4 & R5 1 Booking an appointment 16.1538 14.8782 13.1026 14.1891 19.8269 R3 < R4 < R2 < R1 < R5 2 Quality of medicals 15.6923 14.4936 12.7949 13.9968 19.5962 R3 < R4 < R2 < R1 < R5 3 Hygiene 16.3077 15.0064 13.2051 14.2532 19.9038 R3 < R4 < R2 < R1 < R5 4 Time taken by labs 14.6923 13.6603 12.1282 13.5801 19.0962 R3 < R4 < R2 < R1 < R5 5 Care provided by medical professionals & nurses 15.7692 14.5577 12.8462 14.0288 19.6346 R3 < R4 < R2 < R1 < R5 6 Safety measures 15.3846 14.2372 12.5897 13.8686 19.4423 R3 < R4 < R2 < R1 < R5 7 Value for money 15.0769 13.9808 12.3846 13.7404 19.2885 R3 < R4 < R2 < R1 < R5 8 Reliability and recovery 15.6923 14.4936 12.7949 13.9968 19.5962 R3 < R4 < R2 < R1 < R5 9 Cost effectiveness 14.4615 13.4679 11.9744 13.4840 18.9808 R3 < R2 < R4 < R1 < R5 10 Quality of food, lift facility & security 13.6923 12.8269 11.4615 13.1635 18.5962 R3 < R2 < R4 < R1 < R5 11 Recommendation 10.3077 10.0064 9.2051 11.7532 16.9038 R3 < R2 < Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 7s (2024) 309 https://internationalpubls.com R1 < R4 < R5 Table 3: Order of a Icositetragon Fuzzy Number with the proposal and existing rankings *R1, R2, R3, R4, R5 denote the ranking of Pentagonal, Hexagonal, Heptagonal, Hex Decagonal, and Icosagonal Fuzzy Numbers respectively. ZONE – II S. No Criteria Defuzzified value Rank (Icosagonal FN) 1 Booking an appointment 19.69 5 2 Quality of medicals 20.02 1 3 Hygiene 19.63 6 4 Time taken by labs 19.51 7 5 Care provided by medical professionals & nurses 19.72 3 6 Safety measures 19.81 2 7 Value for money 19.12 10 8 Reliability and recovery 19.72 3 9 Cost effectiveness 19.18 8 10 Quality of food, lift facility & security 19.15 9 11 Recommendation 16.72 11 Table 4: Computation for ranking of various parameters (health services) in Zone-II using defuzzified values. ZONE – II S . n o Criteria Penta - gonal D.Val ue Ra nk (P) Hexago nal D.Valu e Ra nk (H) Hept a- gonal D.Val ue Rank (Hep ta) Hex- Decago nal D.Valu e Rank (Hex. D) Icosago nal D. Value Ra nk (I) 1 Booking an appointment 15.88 5 14.65 5 12.92 5 14.08 5 19.69 5 2 Quality of medicals 16.54 1 15.20 1 13.36 1 14.35 1 20.02 1 3 Hygiene 15.76 6 14.55 6 12.84 6 14.03 6 19.63 6 4 Time taken by labs 15.52 7 14.35 7 12.68 7 13.93 7 19.51 7 5 Care provided by medical professional s & nurses 15.94 3 14.70 3 12.96 3 14.10 3 19.72 3 Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 7s (2024) 3 0 https://internationalpubls.com 6 Safety measures 16.12 2 14.85 2 13.08 2 14.18 2 19.81 2 7 Value for money 14.74 10 13.70 10 12.16 10 13.60 10 19.12 10 8 Reliability and recovery 15.94 3 14.70 3 12.96 3 14.10 3 19.72 3 9 Cost effectivenes s 14.86 8 13.80 8 12.24 8 13.65 8 19.18 8 1 0 Quality of food, lift facility & security 14.80 9 13.75 9 12.2 9 13.63 9 19.15 9 1 1 Recommend ation 9.94 11 9.70 11 8.96 11 11.60 11 16.72 11 Table 5: Defuzzified Values and Rankings of Healthcare Service Quality Criteria. 5.2. Comparative Analysis of Icosagonal Fuzzy Number Rankings with Proposed and Existing Rankings ZONE – II Defuzzified values S. No Criteria Penta- gonal (R1) Hexagonal (R2) Hepta- gonal (R3) Hex- Decagonal (R4) Icosagonal (R5) Relation between R1, R2, R3, R4 & R5 1 Booking an appointment 15.88 14.65 12.92 14.08 19.69 R3 < R4 < R2 < R1 < R5 2 Quality of medicals 16.54 15.20 13.36 14.35 20.02 R3 < R4 < R2 < R1 < R5 3 Hygiene 15.76 14.55 12.84 14.03 19.63 R3 < R4 < R2 < R1 < R5 4 Time taken by labs 15.52 14.35 12.68 13.93 19.51 R3 < R4 < R2 < R1 < R5 5 Care provided by medical professionals & nurses 15.94 14.70 12.96 14.10 19.72 R3 < R4 < R2 < R1 < R5 6 Safety measures 16.12 14.85 13.08 14.18 19.81 R3 < R4 < R2 < R1 < R5 7 Value for money 14.74 13.70 12.16 13.60 19.12 R3 < R4 < R2 < R1 < R5 8 Reliability and recovery 15.94 14.70 12.96 14.10 19.72 R3 < R4 < R2 < R1 < R5 9 Cost effectiveness 14.86 13.80 12.24 13.65 19.18 R3 < R4 < R2 < R1 < R5 10 Quality of food, lift facility & security 14.80 13.75 12.2 13.63 19.15 R3 < R4 < R2 < R1 < R5 Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 7s (2024) 3 https://internationalpubls.com 11 Recommendation 9.94 9.70 8.96 11.60 16.72 R3 < R2 < R1 < R4 < R5 Table 6: Order of a Icositetragon Fuzzy Number with the proposal and existing rankings *R1, R2, R3, R4, R5 denote the ranking of Pentagonal, Hexagonal, Heptagonal, Hex Decagonal, and Icosagonal Fuzzy Numbers respectively ZONE – III S. No Criteria Defuzzified value Rank (Icosagonal FN) 1 Booking an appointment 19.6636 5 2 Quality of medicals 19.8364 2 3 Hygiene 19.7107 3 4 Time taken by labs 19.2866 7 5 Care provided by medical professionals & nurses 19.6322 6 6 Safety measures 19.8835 1 7 Value for money 18.9725 10 8 Reliability and recovery 19.6872 4 9 Cost effectiveness 19.0982 9 10 Quality of food, lift facility & security 19.1139 8 11 Recommendation 16.5301 11 Table 7: Computation for ranking of various parameters (health services) in Zone-III using defuzzified values. ZONE – III S . n o Criteria Penta - gonal D.Val ue Ra nk (P) Hexago nal D.Valu e Ra nk (H) Hept a- gonal D.Val ue Rank (Hep ta) Hex- Decago nal D.Valu e Rank (Hex. D) Icosago nal D. Value Ra nk (I) 1 Booking an appointment 15.82 72 5 14.6060 5 12.88 48 5 14.053 0 5 19.6636 5 2 Quality of medicals 16.17 28 2 14.8940 2 13.11 52 2 14.197 0 2 19.8364 2 3 Hygiene 15.92 15 3 14.6846 3 12.94 76 3 14.092 3 3 19.7107 3 4 Time taken by labs 15.07 33 7 13.9777 7 12.38 22 7 13.738 9 7 19.2866 7 5 Care provided by medical professional 15.76 44 6 14.5537 6 12.84 29 6 14.026 8 6 19.6322 6 Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 7s (2024) 3 2 https://internationalpubls.com s & nurses 6 Safety measures 16.26 70 1 14.9725 1 13.17 80 1 14.236 3 1 19.8835 1 7 Value for money 14.44 50 10 13.4542 10 11.96 34 10 13.477 1 10 18.9725 10 8 Reliability and recovery 15.87 43 4 14.6453 4 12.91 62 4 14.072 6 4 19.6872 4 9 Cost effectivenes s 14.69 63 9 13.6636 9 12.13 09 9 13.581 8 9 19.0982 9 1 0 Quality of food, lift facility & security 14.72 77 8 13.6898 8 12.15 18 8 13.594 9 8 19.1139 8 1 1 Recommend ation 9.560 2 11 9.3835 11 8.706 8 11 11.441 8 11 16.5301 11 Table 8: Defuzzified Values and Rankings of Healthcare Service Quality Criteria. 5.3. Comparative Analysis of Icosagonal Fuzzy Number Rankings with Proposed and Existing Rankings ZONE – III Defuzzified values S. No Criteria Penta- gonal (R1) Hexagonal (R2) Hepta- gonal (R3) Hex- Decagonal (R4) Icosagonal (R5) Relation between R1, R2, R3, R4 & R5 1 Booking an appointment 15.8272 14.6060 12.8848 14.0530 19.6636 R3 < R4 < R2 < R1 < R5 2 Quality of medicals 16.1728 14.8940 13.1152 14.1970 19.8364 R3 < R4 < R2 < R1 < R5 3 Hygiene 15.9215 14.6846 12.9476 14.0923 19.7107 R3 < R4 < R2 < R1 < R5 4 Time taken by labs 15.0733 13.9777 12.3822 13.7389 19.2866 R3 < R4 < R2 < R1 < Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 7s (2024) 3 3 https://internationalpubls.com R5 5 Care provided by medical professionals & nurses 15.7644 14.5537 12.8429 14.0268 19.6322 R3 < R4 < R2 < R1 < R5 6 Safety measures 16.2670 14.9725 13.1780 14.2363 19.8835 R3 < R4 < R2 < R1 < R5 7 Value for money 14.4450 13.4542 11.9634 13.4771 18.9725 R3 < R2 < R4 < R1 < R5 8 Reliability and recovery 15.8743 14.6453 12.9162 14.0726 19.6872 R3 < R4 < R2 < R1 < R5 9 Cost effectiveness 14.6963 13.6636 12.1309 13.5818 19.0982 R3 < R4 < R2 < R1 < R5 10 Quality of food, lift facility & security 14.7277 13.6898 12.1518 13.5949 19.1139 R3 < R4 < R2 < R1 < R5 11 Recommendation 9.5602 9.3835 8.7068 11.4418 16.5301 R3 < R2 < R1 < R4 < R5 Table 9: Order of a Icositetragon Fuzzy Number with the proposal and existing rankings *R1, R2, R3, R4, R5 denote the ranking of Pentagonal, Hexagonal, Heptagonal, Hex Decagonal, and Icosagonal Fuzzy Numbers respectively. ZONE – IV S. No Criteria Defuzzified value Rank (Icosagonal FN) 1 Booking an appointment 19.9579 2 2 Quality of medicals 19.9134 3 3 Hygiene 19.8540 4 4 Time taken by labs 19.5866 7 5 Care provided by medical professionals & nurses 19.7500 5 6 Safety measures 20.0025 1 7 Value for money 19.2153 8 8 Reliability and recovery 19.6609 6 9 Cost effectiveness 18.8441 10 10 Quality of food, lift facility & security 19.1262 9 11 Recommendation 16.4233 11 Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 7s (2024) 3 4 https://internationalpubls.com Table 10: Computation for ranking of various parameters (health services) in Zone-IV using defuzzified values. ZONE – IV S . n o Criteria Penta - gonal D.Val ue Ra nk (P) Hexago nal D.Valu e Ra nk (H) Hepta - gonal D.Val ue Rank (Hept a) Hex- Decago nal D.Valu e Rank (Hex. D) Icosago nal D. Value Ra nk (I) 1 Booking an appointment 16.39 39 2 15.0965 2 13.27 72 2 14.2983 2 19.9579 2 2 Quality of medicals 16.32 67 3 15.0223 3 13.21 78 3 14.2611 3 19.9134 3 3 Hygiene 16.20 79 4 14.9233 4 13.13 86 4 14.2116 4 19.8540 4 4 Time taken by labs 15.67 33 7 14.4777 7 12.78 22 7 13.9889 7 19.5866 7 5 Care provided by medical professionals & nurses 16.00 00 5 14.7500 5 13.00 00 5 14.1250 5 19.7500 5 6 Safety measures 16.50 50 1 15.1708 1 13.33 66 1 14.3354 1 20.0025 1 7 Value for money 14.93 07 8 13.8589 8 12.28 71 8 13.6795 8 19.2153 8 8 Reliability and recovery 15.82 18 6 14.6015 6 12.88 12 6 14.0507 6 19.6609 6 9 Cost effectiveness 14.18 81 10 13.2401 10 11.79 21 10 13.3700 10 18.8441 10 1 0 Quality of food, lift facility & security 14.75 25 9 13.7104 9 12.16 83 9 13.6052 9 19.1262 9 1 1 Recommend ation 9.346 5 11 9.2054 11 8.564 4 11 11.3527 11 16.4233 11 Table 11: Defuzzified Values and Rankings of Healthcare Service Quality Criteria. 5.4. Comparative Analysis of Icosagonal Fuzzy Number Rankings with Proposed and Existing Rankings ZONE – IV Defuzzified values Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 7s (2024) 3 5 https://internationalpubls.com S. No Criteria Penta- gonal (R1) Hexagonal (R2) Hepta- gonal (R3) Hex- Decagonal (R4) Icosagonal (R5) Relation between R1, R2, R3, R4 & R5 1 Booking an appointment 16.3939 15.0965 13.2772 14.2983 19.9579 R3 < R4 < R2 < R1 < R5 2 Quality of medicals 16.3267 15.0223 13.2178 14.2611 19.9134 R3 < R4 < R2 < R1 < R5 3 Hygiene 16.2079 14.9233 13.1386 14.2116 19.8540 R3 < R4 < R2 < R1 < R5 4 Time taken by labs 15.6733 14.4777 12.7822 13.9889 19.5866 R3 < R4 < R2 < R1 < R5 5 Care provided by medical professionals & nurses 16.0000 14.7500 13.0000 14.1250 19.7500 R3 < R4 < R2 < R1 < R5 6 Safety measures 16.5050 15.1708 13.3366 14.3354 20.0025 R3 < R4 < R2 < R1 < R5 7 Value for money 14.9307 13.8589 12.2871 13.6795 19.2153 R3 < R4 < R2 < R1 < R5 8 Reliability and recovery 15.8218 14.6015 12.8812 14.0507 19.6609 R3 < R4 < R2 < R1 < R5 9 Cost effectiveness 14.1881 13.2401 11.7921 13.3700 18.8441 R3 < R2 < R4 < R1 < R5 10 Quality of food, lift facility & security 14.7525 13.7104 12.1683 13.6052 19.1262 R3 < R4 < R2 < R1 < R5 11 Recommendation 9.3465 9.2054 8.5644 11.3527 16.4233 R3 < R2 < R1 < R4 < R5 Table 12: Order of a Icositetragon Fuzzy Number with the proposal and existing rankings *R1, R2, R3, R4, R5 denote the ranking of Pentagonal, Hexagonal, Heptagonal, Hex Decagonal, and Icosagonal Fuzzy Numbers respectively. Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 7s (2024) 3 6 https://internationalpubls.com 6. Conclusion: This paper presents the idea of Icosagonal Fuzzy Numbers (IFNs) and how defuzzification methods use them. We have shown the efficacy of IFNs in handling complexity in linguistic data by contrasting defuzzification techniques among various zones in Andhra Pradesh. Compared to conventional shapes like trapezoidal or triangular fuzzy numbers, the suggested solutions offer a more nuanced depiction of fuzzy numbers. Our results demonstrate the benefits of applying IFNs to healthcare service quality assessment, providing more accurate and thorough assessments. This study adds to the body of knowledge in fuzzy set theory by creating new directions for investigation and real-world applications in situations involving decision-making when ambiguity and uncertainty are common. References: [1] Zadeh LA. Fuzzy sets. Inf Control. 1965;8(3):338–353. 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