Zadeh[1] introduced the concept of fuzzy set theory to address the issue of vagueness, uncertainty and imprecision inherent in real-life decision-making scenarios. Since then, numerous extensions of fuzzy sets have emerged, including IFS[2], PFS[3] and Q-ROFS[4], each serving as a generalization of traditional fuzzy sets. More recently, Dutta and Borah[5] have developed the HyFS which offers significant flexibility and enough space to the diverse need of decision-makers. The HyFS has garnered substantial interest among researchers focused on decision-making amidst the increasing complexity of contemporary challenges. Numerous studies have utilized HyFS to tackle a variety of MCDM problems. Divsalar et al[6] effectively applied HyFS to the sustainable supplier selection problem within the daily industry. Zavadskas et al[7] employed hyperbolic fuzzy data to establish personalized priorities for contract clauses. Additionally, Banik and Dutta[8] employed HyFS to identify crime-prone zones in Dibrugarh city.

The selection of optimal MWTT is crucial for safeguarding public health and protecting the environment. Extensive research has been conducted to identify the most effective strategies for enhancing MWTT, with fuzzy set theory beinga key component of these studies. However, there is currently no published evaluations concerning MWTT assessment using the HyFS preference structure.

The MCDM methods established in fuzzy environment have demonstrated their efficacy in the selection process for MWTT, highlighting the necessity for further exploration and application of HyFS in this essential area. To address the significant challenges associated with MWTT through traditional decision-making methods[9, 10]. Dursun et al [11] have pioneered a MCDM technique within a fuzzy framework for MWT. Their work introduced two robust fuzzy MCDM frameworks, employing fuzzy integral and a hierarchical distance-based technique which represent a sustain-able advancement in this field. Building on Dursun et al[11] insights, Liu et al[12] advanced the determining the optimal selection for MWT. This was followed by the development of an ILT-MULTIOORA technique by Liu et al[13] which uti-lizes interval 2-tuple linguistic variables. Liu et al[14] evaluated the weights of the criteria using the 2-tuple DAMETAL method and applied the fuzzy MULTIMOORA methodology to assess MWTT comprehensively.

Lu et al[15] introduced an interval 2-tuple induced-TOPSIS method and Xiao[16] unvieled a technique for prioritizing MWTT that assigns crisp values to linguistic variables and leverages D-numbers integration for comprehensive assessment. Hinduja and Pandey[17] formulated a hybrid algorithm that enhances the selection for MWTT. Additionally, Li et al[18] contributed to the dialogue by proposing a hybrid MCDM technique that synergizes the DEMATEL and TOPSIS methods within an interval-valued fuzzy environment. Despite these advancements, a notable gap persists in the literature regarding methodologies that seamlessly integrate both objective and subjective weights for MWT attributes. Both weights plays a vital role in minimizing potential losses while enhancing decision accuracy. The MWT selection problem remains largely unexplored. There exists a significant gap in research utilizing an integrated entropy measure, SWARA model combined with COPRAS or ARAS approaches in a hyperbolic fuzzy environment. Therefore, this research proposes an innovative approach to MWTT selection within a hyperbolic fuzzy environment, effective harnessing the strengths of both objective and subjective weights.

By employing MCDM, healthcare facility can adopt efficient solutions that not only comply with regulatory standards, but also advanced sustainability goals. This strategic approach is vital for safeguarding public health and protecting our environment for future generations. One constructive approach to enhancing contemporary MCDM techniques involves carefully establishing appropriate weights for various criteria.

In consideration of the preceding points, the following motivation drive this research :

1. Existing fuzzy set theories often encounter challenges in addressing certain paradoxical situations, creating a significant gap in the handling of uncertainty. The implementation of hyperbolic fuzzy set offers a sophisticated approach that allows for full membership, partial non-membership and their reversals.

2. Entropy measure and score functions plays a pivotal role in determining the weights of criteria and evaluate alternatives within MCDM models. These methodologies can facilitate the development of advanced MCDM techniques such as COPRAS, SAW, ARAS etc. Furthermore, there is no development of entropy measure within a hyperbolic fuzzy environment, highlighting a valuable opportunity for innovation in this area.

3. There is a significant lack of literature addressing the application of hybrid MCDM methods within the hyperbolic fuzzy context.

In consideration of the preceding points, the following are the objectives of this research :

1. Developing a novel score function for HyFS.

2. Developing an entropy measure for HyFS for determining the objective weight of the criteria.

3. A case study to endorse the probable impact of the proposed algorithm in MWTT selection using HyFS.

The structure of this paper is organized as follows :

In section 2, we provide the definitions of existing fuzzy sets, including IFS, PFS, Q-ROFS and HyFS with some basic operations on HyFS. Section 3 presents a novel entropy measure, score function within hyperbolic fuzzy environment and the algorithms for the EM-SWARA-COPRAS and EM-SWARA-ARAS methods under the HyFSs context. Section 4 demonstrates the practical application of the developed methodology through a case study on selecting the most desirable MWTT option in India. The case study illustrates the strength and robustness of the proposed approach. Also, a comparative analysis is presented that underscores its advantages. Finally, section 5 conclude the paper and outlines future research scope.

The entropy measure plays a pivotal role in the development of MCDM techniques, as it is essential for identifying the objective weight of attributes. Despite its importance, the hyperbolic fuzzy framework has lacked a suitable entropy measure until now.

Definition 5. Let, HyFN H = (x, y) such that x, y ∈ [0, 1] and 0 ≤ x.y ≤ 1.

A function E : HyFS(X) → [0, 1] is a HyF entropy measure if (i)E(H) = 0 if H is a crisp set.

(ii)E(H) attains its unique maximum when x = y = (iii)E(H) = E(H )

1 1 1 1

(iv)E(H1) ≤ E(H2) if H1 is crisper than H2, i.e., x1 ≤ x2 ≤ , y1 ≤ y2 ≤ or x1 ≥ x2 ≥ , y1 ≥ y2 ≥ where

2 2 2 2

H1 = (x1, y1) and H2 = (x2, y2) A novel HyF entropy measure is proposed as given below :

(2x + 2y − 2) ENP(H) = 1 − . (1) xy + 1 Theorem 1. The function EN P (Λ) is an HyF entropy measure.

We propose a novel score function with the following definitions :

Definition 6. Let, H = (x, y) be a HyFN, then the score function SNP of H is defined as xy + x + 1 SNP(H) = . (2) 2(y + 1) where SN P (H) ∈ [0, 1]

Extremely good (EG) (1,0) Very good (VG) (0.95,0.05) Good (G) (0.75,0.20) Medium good (MG) (0.65,0.30) Fair (F) (0.55,0.40) Medium poor (MP) (0.40,0.50) Poor (P) (0.30,0.60) Very poor (VP) (0.20,0.70) Extremely poor (EP) (0.10,0.80)

Extremely significant (1,0) Very significant (0.90,0.05) Significant (0.70,0.20) Moderate (0.60,0.30) Insignificant (0.40,0.50) Very insignificant (0.30,0.80) Extremely insignificant (0.10,0.80) C r i t e r i a

C1 C2 C3 C4 C5 C6 C7 C8 C9 C10

A1 E1 G P P VG F F F VG G VG E2 F MG F G MG P G P VP G E3 MP G P P G P VG F MG VG

E4 F F P P P VG MG F G MG A2 E1 VG VP F G F MP VG G G VG E2 P G F G MG P G F MP MG E3 F MG P MP G MG G MG F G E4 F MG VP P MP G VG F VG P A3 E1 G VP F G F MP VG G G VG E2 P G G G P F F F P VG E3 F G P P G MP F F F G E4 P VP F VP MG G G G MG MP A4 E1 V MP MG G MG MG P VG MG F E2 P MG F MG MG G MG G P MG E3 F F P P VG MP F G F MG E4 F P VP P MP G G VG D MP A5 E1 VG G F VG F MG F VG MG G E2 VG VG MG F MG G MG MG G G E3 P P G P G MP F G P MP E4 P P MG MG MP F F G F P

The weights of the DEs are calculated with the results presented in Table 3. For each alternative based on the criteria, a HyF decision matrix is developed by the panel of experts as illustrated in Table 4. The AHyF-DM is then constructed using DE’s weight. The integrated weights for the criteria are determined by combining objective and subjective weights. The objective weights are derived using the Entropy Measure. The entropy values are calculated by employing equation(1) and zj kj pj Sj C5 0.8474 - 1 1 0.1264 C4 0.8354 0.0120 1.0120 0.9881 0.1249 C7 0.6898 0.1456 1.1456 0.9763 0.1234 C8 0.6584 0.0314 1.0314 0.9466 0.1197 C6 0.5197 0.1387 1.1387 0.8313 0.1051 C1 0.2643 0.2554 1.2554 0.6622 0.0837 C3 0.2486 0.0157 1.0157 0.6520 0.0825 C2 0.2336 0.0150 1.0150 0.6424 0.0812 C9 0.1898 0.0438 1.0438 0.6154 0.0778 C10 0.1566 0.0332 1.0332 0.5956 0.0753

The SWARA model is employed to determine the subjective weights of the criteria. The subjective weight of the criteria are determined using SWARA model is presented in the last column of Table 7. Finally, the integrated/final weights of the criteria are determined. The final weights of the criteria are presented as follows :

wj = {0.1094, 0.1195, 0.1275, 0.1121, 0.0977, 0.1043, 0.0865, 0.0749, 0.0956, 0.0725}

Initially, the COPRAS method is applied to assess the alternative for MWT methods in India. The relative grade and utility grade are presented in Table 8. The results in Table 8 distinctly indicates that the ranking order for MWTT is A1 > A2 > A4 > A3 > A5. Alternative A1 emerges as the best option for MWTT.

Moreover, we strengthen our analysis by applying the ARAS method within hyperbolic fuzzy framework. Table 12 shows that the alternatives are ranked as A1 > A2 > A4 > A3 > A5. Thus, A1 is the best option for MWTT.

Comparative Analysis. In this section, we present comparison of our proposed technique against existing methodologies, demonstrating the efficacy and advantage of our developed approach for MWTT selection.

In comparison with other developed methodologies for MWTT selection, our approach demonstrates the follow-ing strengths :

1. Methodologies[12, 28, 46, 47, 49, 50] focuses solely on objective weights, disregarding subjective evaluations while methodologies[14, 16, 19, 22, 48, 51] emphasizes subjective weights, disregarding objective weights. This imbalance results in a flawed evaluation, allowing decision-makers to harness valuable human insight while maintaining rigorous analytical standards.

2. In methodologies[14, 16, 28, 47, 49, 50] weights assigned to experts are assumed randomly rather than meticulously computed, resulting in potentially biased and unrealistic outcomes. In critical fields like healthcare and medical waste management such oversights can have profound consequences. Our proposed methodology methodically computes expert weights, ensuring accuracy and reliability.

3. Methodologies[16, 20, 51] neglects the critical nature of attributes which can adversely affect the ranking of alternatives. Our proposed methodology rigorously accounts for these attributes, enhancing the credibility of the results.

4. Additionally, the findings by methodologies[21, 22] which positioned alternative A5 as the second-best option for MWTT, are equally questionable. This option is characterized by harmful emissions, being costly and sustainable energy consumption, all factors that undermining it viability.