Proposed Intuitionistic Fuzzy Entropy Measure Along With Novel Multicriteria Sorting Techniques

Recently, the major environmental change and a pandemic called COVID-19 have heavily impacted the economy, business, and health of each country. Moreover, the climatic changes and COVID-19 are calamities to human life. In other words, these two aspects threaten the existence of humans and the sustenance of the overall development of a country. These two factors particularly influence the tourism sector, so a strategy balancing environmental quality and dealing with the ill effects of COVID-19 is formulated to uplift the economic sectors. Atannasov’s intuitionistic fuzzy domain is used to model the environmental quality and COVID-19 due to the involvement of hesitancy and uncertainty. The precise measurement of the imprecision in the information is obtained with the help of entropy measure. The paper analyzes the two aspects using a novel entropy measure based on multiple criteria sorting (MCS). Here, the two MCS problems are solved with the help of two proposed techniques: TOPSIS-GREY-sort and ENTROPY-TOPSIS-GREY-sort. A case study showing the impact of COVID-19 in the Philippines and the environmental quality of Tehran (the capital city of Iran) are considered to validate the functioning of the proposed techniques. We use “A novel sorting method TOPSIS-SORT: an application for Tehran environmental quality evaluation (2016), Ekonomica a management,” and “Current Issues in Tourism 25.2(2022): 168–178, Taylor and Francis” for the comparative analysis.


I. INTRODUCTION
In a decision-making problem, the solution is to provide a technique that selects the best alternative among the given alternatives. Generally, the multiple criteria of the alternatives are analyzed in a decision-making problem.
Here, most decision-makers (DMs) employ a ranking of the alternatives to select the best alternative. The multiple criteria decision-making (MCDM) problems appear in the field of business, medical, and public health sectors. So it has been an active area of research during the last couple of decades The associate editor coordinating the review of this manuscript and approving it for publication was Muhammad Ali Babar .
(see: [1], [2], [3], [4], [5]). Various methods are proposed to solve the MCDM problems (see: [6], [7], [8]). Here, the novel extensions of the Technique for Order Preference by Similarity to Ideal Solution (TOPSIS) and Grey Relational Analysis (GRA) are proposed in the Atanassov intuitionistic fuzzy set (AIFS) environment to solve the MCDM problems. TOPSIS [9], and GRA [10] possess advantages over other MCDM methods due to: (i) computational simplicity, (ii) rationality, (iii) visualization, and (iv) a scalar value that accounts for best and worst alternatives. Moreover, TOPSIS and GRA do not demand any parameter from the DMs during the computation, so the decision-making becomes easy. The AIFS [11] characterizes a criterion based on the degree of belongingness, non-belongingness, and hesitancy. The limitations of the fuzzy set (FS) [12] are dealt with in the AIFS due to the incorporation of an additional degree [13]. A measure-theoretic study of the AIFS is done in [14] and [15]. Some other well-known generalizations of FS are type-2 fuzzy set [16], vague set [17], intervalvalued fuzzy set [18], etc. An interrelationship between interval-valued intuitionistic fuzzy set (IVIFS) and AIFS is recently discussed in [19].
Entropy measures the uncertainty within the information. The amalgamation of entropy and AIFS helps in better decision-making. The notion of fuzzy entropy was given by Zadeh [12], and it is further studied by Kaufman and Magens [20], Higashi and Klir [21], and Szmidt and Kacprzyk [22]. Joshi and Kumar [23], [24] introduced an (R, S) norm-based fuzzy information measure and entropy measure. Parkash et al. [25] proposed two new measures of weighted fuzzy entropy to study the maximum weighted fuzzy entropy principle. Bustince and Burillo [26] gave the notion of intuitionistic fuzzy entropy. Szmidt and Kacprzyk [22] proposed a non-probabilistic type of intuitionistic fuzzy entropy. Vlachos and Sergiadis [27] generalized De Luca and Termini's [28] notion of non-probabilistic entropy to AIFSs. Zhang and Jiang [29] suggested a similarity between AIFSs based upon intuitionistic fuzzy entropy. Chen and Li [30] proposed various entropies on AIFSs. In the same paper, they subdivided the attribute weights into two parts: subjective and objective weights. Subjective weights are determined as per the preference of DMs. The methods such as the AHP method [31], weighted least squares method [32], and Delphi method [33] usually use subjective weights. If subjective weights are not provided, the objective weights are determined using the entropy method, multi-objective programming [34], principle element analysis [35], etc. The entropy method is a commonly used approach that solves MCDM problems with partially known weight or unknown weight information. Zhang and Jiang [29] proposed several entropy methods with unknown weight information of attributes to solve MCDM problems.
The GRA technique computes similarities between the ideal option and given options, and here the increasing order of similarities is used for arranging options in the order of preferences. Moreover, realistic and effective processing of the inadequate information is carried out in GRA [36], [37]. The fuzzy variants of GRA are proposed (see: [38], [39], [40]) to deal with the imprecise information. AIFS is an application-rich and highly studied extension of the fuzzy set (see: [41], [42], [43]). In literature, we come across the proposals of AIFS-based variants of GRA (see: [44], [45], [46]). The reopening of the tourism industry and environmental quality improvisation are the two MCDM problems addressed in the paper. The MCDM method is the umbrella under which choosing, sorting, and ranking techniques are introduced, whereas Multiple criteria sorting (MCS) is a specific name allocated to sorting techniques [47].
The decision-making techniques employed to solve the tourism industry and environmental problems belong to MCS. A finite and pre-defined ordered set called a decision class is assigned to evaluate the multiple criteria in the MCS technique. This technique fuses the decisions and opinions of the DMs. The GRA motivated us to propose an IFGRA-sort [48]. Some interesting sorting techniques are VIKORsort [49], Flow-sort [50], AHP-sort [51], AHP-sort II [52], ELECTRE-TRI [53] and TOPSIS-sort [54], [55], IFGRA-sort [48].
TOPSIS selects an alternative out of the given alternatives nearest to the positive ideal solution and farthest to the negative ideal solution. The collection of best values of criteria is referred to as a positive-ideal solution, and the worst values across all criteria are collected in a set called a negative-ideal solution. It is claimed that the alternative chosen using TOPSIS is the best one. FS efficiently manages the uncertainties associated with the information. The fuzzy versions of the TOPSIS solving uncertainty possessing MCDM problems are introduced in (see: [56], [57]). In [58], a fuzzy TOPSIS involving the MCDM problem is remodeled in a classical domain. In classical TOPSIS, both the assessments and criteria weights of DMs are assigned specific numerical values. This is seldom the case in real-world situations because the criteria weight selection involves uncertainty, and the hundred percent precision in the DM perception is rarely achieved. We model the criteria weight selection and DM perception in the Atannasov intuitionistic fuzzy domain due to the presence of uncertainty and hesitancy in DM perceptions. The AIFS-based TOPSIS technique solves supplier selection problems [59], data envelopment analysis [60], multi-person MCDM problem [61], group decision-making problem [62], entropy-based public blockchain problem [63]. In [64], a survey of the research work related to the TOPSIS is given.
The notable contributions of the paper are as follows: • A novel entropy measure compatible with the AIFS environment is proposed.
• The MCS explores either subjective or objective weights, so we proposed a subjective weight-dependent technique called IFGRA-sort in [48]. Here, we have introduced the TOPSIS-GREY-sort technique, which uses entropy-driven objective weight. The proposed entropy measure introduces ENTROPY-TOPSIS-GREY-sort. The proposed entropy measure is used to calculate the objective weights in ENTROPY-TOPSIS-GREY-sort. A convex combination of subjective and objective weights has been used in the proposed ENTROPY-TOPSIS-GREY-sort to analyze the effect of both subjective and objective weights upon decision-making. VOLUME 11, 2023 • We have successfully solved the problem of the COVID-19 pandemic concerning domestic tourism in the Philippines. Further, a novel methodology to improvise the environmental quality of Tehran has been suggested in the AIFS environment. The paper is divided into six sections: Section II recalls the basic concepts. Section III mathematically proposes an entropy measure within the AIFS environment. In section IV, two MCS techniques, TOPSIS-GREY-sort and ENTROPY-TOPSIS-GREY-sort, are introduced. The proposed techniques are validated over case studies of domestic tourism in the Philippines and the environmental quality of Tehran in section V. Finally, the conclusion and future work is stated in section VI.

II. BASIC CONCEPTS
This section has three subsections. The basic definitions helpful in understanding the paper are provided in section II-A. The two algorithms, called Intuitionistic FUZZY TOPSIS and Intuitionistic fuzzy GREY RELATIONAL ANALYSIS, are given in section II-B and II-C, respectively. [11]): Atanassov intuitionistic fuzzy set (AIFS) Q over the universe of discourseX is of the form:

Definition 1 (Atanassov Intuitionistic Fuzzy Set
The membership function, µ Q :X → [0, 1] and nonmembership function, ν Q :X → [0, 1] are defined in such a manner that: calculates the hesitation degree whereas µ Q (x) and ν Q (x) assigns a membership degree and non-membership degree to x ∈X , respectively. Here, the ordered pairx = (µ, ν) is used to represent an AIFS.

B. INTUITIONISTIC FUZZY TOPSIS
Let Q = {Q 1 , Q 2 , . . . Q p } be the set of the alternatives and C = {C 1 ,C 2 , . . .C q } be the set of the criteria. The steps implementing Intuitionistic fuzzy TOPSIS method [59], [69] are given as follows: Step1: The weights are assigned to G numbers of DMs based on their opinions. First grade the k th DM with an AIFN D k = [µ k , ν k , π k ] and his opinion is given a weightage λ k as follows: Step2: Evaluate criteria-dependent intuitionistic fuzzy decision matrices corresponding to the DMs and then aggregate them. Let the intuitionistic fuzzy decision matrix corresponding to k th DM be . . q) are aggregated using the IFWA operator defined in [65] as follows: Step3: Calculation of the criteria weights, W = [w 1 , w 2 , w 3 , . . . .w q ], w n = (µ n , ν n , π n )(n = 1, 2, . . . , q).
It is not necessary for all the criteria to be equally important in decision-making. So, a criteria weight matrix is calculated in which the perspectives of DMs regarding the criteria are merged. Here, the k th DM rates criterion x n with the help of an AIFN w The IFWA operator combines the weights of entire criteria, and we obtain the criteria weight w n of x n as follows: Step4: Aggregation of criteria weights and intuitionistic fuzzy decision matrices.
The criteria weights W and intuitionistic fuzzy decision matrix H are aggregated using the binary operation ⊕ given in [70]. Hence, Then, the aggregated weighted intuitionistic fuzzy decision matrix H is given as: Step5: Evaluation of intuitionistic fuzzy positive-ideal solution (Q + ) and intuitionistic fuzzy negative-ideal solution (Q − ).
Let us denote the benefit and cost criteria with J 1 and J 2 , respectively. The following equations calculate Q + and Q − : Step6: Evaluation of the separation measures. Q + and Q − based separation measures S + and S − for each alternative are calculated using the normalized euclidean distance measure [67] as follows: Step7: Evaluate relative closeness coefficient (RC m ) between intuitionistic ideal solution (Q * ) and alternative Q m as follows: Step8: Select the best alternative, the alternative getting maximum (RC m ) value.

C. INTUITIONISTIC FUZZY GREY RELATIONAL ANALYSIS
Let Q = {Q 1 , Q 2 , · · · , Q p } be the set of alternatives with attribute setC = {C 1 ,C 2 , · · · ,C q }. Moreover, a set of weight vectors W = {w 1 , w 2 , · · · , w q } T is assigned to attribute setC such that w n ∈ [0, 1], q n=1 w n = 1. The steps implementing intuitionistic fuzzy GRA [71] method are as follows: Step1: Standard decision matrix. Let the standard decision matrix H = [h mn ] p ×q is given in the form h mn = µ mn , ν mn .
Step2: Evaluation of the positive ideal solution and negative ideal solution.
The standard decision matrix based intuitionistic fuzzy positive ideal solution (PIS) Q + and intuitionistic fuzzy negative ideal solution (NIS) Q − is calculated as follows: Step3: Calculation of the grey relational coefficient (GRC). The PIS, NIS, and an alternative are used in the distance measure (see: Eq.(1)), which results in a positive GRC (ξ + mn ) and negative GRC (ξ − mn ) as follows: Here, we have an identification coefficient η ∈ [0, 1], its commonly used value is η = 0.5.
Step4: Compute PIS based grey relational grade (GRG) and NIS based GRG corresponding to an alternative.
PIS based GRG (ξ + m ) and NIS based GRG (ξ − m ) of each criterion are calculated as follows: We have assigned a weight vector W over the attributeC n such that 0 ≤ w n ≤ 1 and q n=1 w n = 1.
Step5: Determination of the relative closeness coefficient. For alternative Q m , the relative closeness coefficient (RC m ) is given in terms of ξ + m and ξ − m as follows: Step6: Rank the alternatives. Ranking of alternatives is based on the descending values of RC m , that is, the most important alternative is assigned the maximum value of RC m .

III. A NEW INTUITIONISTIC FUZZY ENTROPY MEASURE
In this section, a new entropy measure is proposed.

IV. PROPOSED TECHNIQUES A. PROPOSED INTUITIONISTIC FUZZY TOPSIS-GREY SORTING (TOPSIS-GREY-SORT)
The proposed technique (TOPSIS-GREY-sort) is defined for a dataset with p alternatives. The q numbers of criteria are involved to define the alternatives. The aim of the technique is the correct classification of p alternatives in g classes {d 1 , d 2 , . . . , d g }. The TOPSIS-SGREY-sort technique is implemented in thirteen steps as follows: Step1: Construct a decision matrix H k = (h k mn ) p ×q corresponding to k th decision-maker (DM) k = 1, 2, . . . , G. The g classes based information of a criterion forms a limit profile, and it is stored in the collection Y = In Y , the upper limit of the preceding class and lower limit of the succeeding class is same, i.e., y is the lower limit of j th class and y with the help of the pre-defined -aggregation operator.
Step5: Calculate intuitionistic fuzzy entropy in intuitionistic fuzzy decision matrix˜ = (λ mn ) p×q as follows: The E mn denotes entropy value at the mn th position in intuitionistic fuzzy decision matrix. Step6: Normalize the intuitionistic fuzzy entropy values as follows: Step7: Evaluate the objective weight of each criterion.
Here, we mainly employ the entropy weight method to calculate the initial objective weights. The matrixẼ = {e mn , m = 1, 2 · · · p, n = 1, 2 · · · q} is involved in the calculation of objective weight of a criterion: Step8: Derive the weighted matrix out of the intuitionistic fuzzy decision matrix.
The following formula proposed by Atanassov in [11] is used to construct a weighted matrix out of an intuitionistic fuzzy decision matrix (˜ ): where, 0 < W n < 1.
The aggregated weighted intuitionistic fuzzy decision matrix H is shown in the equation at the bottom of the next page, denotes the mn th entry of aggregated weighted intuitionistic fuzzy decision matrix.
Step9: Calculate the intuitionistic fuzzy positive ideal solution (Q + ) and the intuitionistic fuzzy negative ideal solution (Q − ) from H , which is given as: Step10: Determine the grey relational coefficient (GRC). The use of Q + n and Q − n in intuitionistic fuzzy normalized distance measure (1), calculates positive GRC (ξ + mn ) and negative GRC (ξ − mn ) which is as follows: Generally, during computation the identification coefficient η = 0.5, η ∈ [0, 1] is used, and the same is taken here also.
Step11: Assign grey relational grade (GRG) to every alterative with the help of ξ + mn and ξ − mn as follows: here the n th criterion is assigned weight W n (obtained in Step7), and 0 ≤ W n ≤ 1; q n=1 W n = 1.
Step13: Find the deviations of the upper limit profile and lower limit profile of the j th class from Q + and Q − . These deviations are RC x j u n and RC x j l n , respectively. If RC m values of the alternatives satisfy eq. 45, then such alternatives belong to the j th interval class.
Here, Steps 1-4 are the same as proposed in TOPSIS-GREYsort, and from Step 5, a different approach is adopted.
where, E mn represents each entropy value in the intuitionistic fuzzy decision matrix.
Step6: Normalization of intuitionistic fuzzy entropy values by repeating the Step-6 of TOPSIS-GREY-sort.
Step7: To aggregate and then evaluate data, some series of moderate and accurate weights as much as possible is undoubtedly needed. The weights are considered in two aspects. On the one hand, the objective weights are calculated through the entropy weight method such that its influence on data can be felt. On the other hand, the experts give subjective weights to express the decision preference. Finally, the weighted average value of objective and subjective weights is taken.
Step7.1 Evaluate the objective weights of criteria. For objective weight (say W 1 n ) calculation repeat the Step-7 of TOPSIS-GREY.
Step7.3 Combine objective and surjective weights Step8 Repeat the steps 8-13 of TOPSIS-GREY-sort, by using the combined weights obtained in Step-7.3 for GRG calculation in this algorithm, for final sorting.

V. EXPERIMENTAL VALIDATION OF PROPOSED TECHNIQUES OVER TWO MCDM PROBLEMS
In this section, we solve two types of MCDM problems with the help of the proposed sorting techniques. The impact of COVID-19 on the tourism sector of the Philippines is analyzed, and a strategy to reopen domestic tourism is suggested. Further, the environmental quality of Tehran is studied in the second problem, and a remedy for improvising environmental quality is recommended. Here, Tehran's tourist sites and regions are sorted into different classes as per the exposure to COVID-19 and their environmental quality.
A. DOMESTIC TOURISM PROBLEM OF PHILIPPINES [73] The countries have taken vital initiatives for economic recovery amidst the COVID-19 pandemic. Yet, the tourism industry is fully at the fag-end in the sectors soon to be reopened. Generally, the strategies to restart the tourism industry do not incorporate the perceptions of the tourists. It is clear that there is fear amongst tourists because of the pandemic outbreak, so the strategies to restart the tourism industry should give due importance to the perceptions of the tourists also.
The tourism industry and COVID-19 infection rate are correlated [74], [75], [76]. As per the exposure to COVID-19, the tourist sites of the Philippines [73] are categorized into three classes, namely ''Low exposure interval class (LEIC),'' ''Moderate exposure interval class (MEIC),'' and ''High exposure interval class (HEIC).'' The two sorting techniques, TOPSIS-GREY-sort and ENTROPY-TOPSIS-GREY-sort, are used to study a six criteria-based dataset of twenty identified tourist sites in the central Philippine province to promote domestic tourism (see : Table 3). Initially, the limit profile of each criterion is known, and aggregated decision matrix is provided in Table 1. The detailed plan of the Philippines has a dataset with which the threshold of each class is identified.

1) TOPSIS-GREY
In this technique, the subjective weights of the experts are not used, rather entropy weight determination technique assigns the objective weights.
Step3: Normalize matrix and use Eq.(31) for the normalization process (use the minimized criterion value, as small-scale values are preferable). In Table 2, the normalized matrix = (λ mn ) p×q is given.
Step12: RC m values are calculated using Eq.(44) for each alternative and limit profile and these values are in Table 6.
Step13: Eq.(45) is used to assign alternatives to appropriate classes. The final sorting of tourist sites as per their exposure to COVID-19 is shown in Fig. 2.

2) ENTROPY-TOPSIS-GREY-SORT
In this technique, the subjective weights provided by the experts and the objective weights calculated using the entropy weight determination technique are merged, and their cumulative effects on decision-making are studied. Here Step 1-4 are same as in TOPSIS-GREY-sort, so the same dataset, normalized matrix, same value of β = 0.9 and α = 0.5 and IF aggregated matrix is obtained.  Step5: The intuitionistic fuzzy entropy values (E mn ) inẼ are obtained using˜ = (λ mn ) p×q and using Eq.(23). Each value given inẼ is of the form 1.0e-16.   Step6: The normalized intuitionistic fuzzy entropy values (e mn ) inẼ are obtained using Eq. (33).
Step7: The objective weights of the criteria are calculated usingẼ and the entropy method (see: Eq.(34)) and we obtain weight values as {0.1967, 0.0661, 0.1751, 0.1346, 0.1425, 0.2850}. Subjective weights provided by experts for each criteria is given in Table-  Step9: Use Eq.
Step13: Eq.(45) is used to assign alternatives to appropriate classes. The final sorting of tourist sites as per their exposure to COVID-19 is shown in Fig. 3.

B. RESULT DISCUSSION AND COMPARATIVE ANALYSIS
• In comparison to [73], the results given here are of generalized nature, and the tourist sites are categorized as a 'low exposure interval class (LEIC)', a 'moderate exposure interval class (MEIC)', and the 'high exposure   13,13], and HEIC = [7,7]. These specific valued results are depicted in Fig. 3), the figure can be analyzed as, on average, COVID-19 highly impacts seven tourist sites, and thirteen are moderately affected. So, to resume the tourism industry for economic purposes, authorities have excellent insight into decision-making and resource allocation for these tourist sites. In different political boundaries, the standard COVID-19 guidelines and public health measures are followed differently. Hence, we suggest that authorities first consider reopening domestic tourism in place of international tourism. The authorities find it easy to monitor, supervise, and control domestic tourism, motivating the governments to reopen international tourism.
• It is not always the case that subjective weights (weights allocated by DMs) are known. In such a case, objective weights are calculated using a data-driven weight determination scheme. Here, we have carried out the individual study of objective and subjective weights, and their cumulative effect is also analyzed. TOPSIS-GREYsort uses objective weights (calculated by entropy method) which is (0.1543, 0.1995, 0.1614, 0.1763, 0.1740, 0.1345). Moreover, the convex combination of subjective weight (see:    C. ENVIRONMENTAL QUALITY PROBLEM OF TEHRAN [54] Tehran is the capital city of Iran, situated in the northern part of Iran and on the southern slopes of the Alborz Mountains. It is the biggest city in Iran and is divided into 22 districts. It is the most important city in Iran. It hosts many national and international events and is the land of ministries, government institutions, and private organizations. The better air, rail, and road connectivities make it a transport hub and an economic driver across the country. Almost one-fourth of the urban population of Iran lives in Tehran, which has caused a significant expansion and escalation of pressures on the environment (Research and Planning Center of Tehran, 2012). The high-altitude mountains surround the city in three directions, blocking the air passage. As a result, Tehran faces varying environmental problems and issues, such as air pollution mitigation and management, waste management, water and wastewater management, urban safety, and greening. In other words, the geographical conditions of Tehran have a cascading effect on its environment.
The environmental quality problem of Tehran studied in [54] is dealt with in this paper also. Districts of Tehran are classified into five classes according to their environmental conditions: high suitable interval class (HSIC), suitable interval class (SIC), moderately suitable interval class (MSIC), low suitable interval class (LSIC), and very low suitable interval class (VLSIC). The study involves 22 identified districts of Tehran for environmental quality under five criteria, 'urban green space per capita (C 1 )', 'waste and soil pollution management (C 2 )', ' quality of environmental management of water and waste water projects (C 3 )', 'air pollution management (C 4 )' and 'ratio of incidents to the station (C 5 )' (see: Table 9). The limit profiles of criteria are initially known. The basis of this criteria was the annual report of the Tehran municipality so that the quality of the environment could be assessed accordingly. For the threshold values of each class, data from the environmental plan, detailed plan, the comprehensive goal of greenery, plan of environment, plan of water and wastewater, and the experts opinion were used. So, a proper study and analysis help us to recognize the districts having poor environmental quality. These districts must be prioritized to establish a better environment and, thus, better living conditions. The proposed sorting techniques are implemented in the upcoming subsections. Moreover, the geographic location of the city makes the air pollution in Tehran even worse. The sorting techniques TOPSIS-GRAsort and ENTROPY-TOPSIS-GREY-sort are implemented here, categorizing the districts as per their environmental quality. The aggregated decision matrix, along with the limit profile is calculated (see: Table 10).

1) TOPSIS-GREY-SORT
In this technique, the subjective weights of the experts are not used, rather entropy weight determination technique assigns the objective weights.
Step1: Construct the decision matrix H k = (h k mn ) p ×q using the data provided by Tehran environmental management. Here h k mn is the assessment of k th decision-maker k = 1, 2, . . . , G regarding n th criterion (n = 1, 2, . . . , q) of the m th district (m = 1, 2, . . . , p ). The expert group provides limit profiles of 5 classes (g = 5), namely; HSIC, SIC, MSIC, LSIC, and VLSIC. The limit profiles of each class are shown in Table 9. The well-known aggregation function called weighted mean aggregates H k , 1 ≤ k ≤ G and it results an aggregated decision matrix H = (h mn ) p ×q , where Step2: The aggregated decision matrix = (H , X ) = (χ mn ) p×q takes into account the role of both limit profile and H = (h mn ) p ×q during decision making (see : Table 10).
Step3: Normalize matrix and use Eq. (31) for the normalization process (use the minimized criterion value, as small-scale values are preferable). In Table 11, the normalized matrix = (λ mn ) p×q is given.
Step12: RC m values are calculated using Eq.(44) for each alternative and limit profile, and these values are in Table 14.
Step13: Eq.(45) is used to assign alternatives to appropriate classes. The final sorting of tourist sites as per their exposure to COVID-19 is shown in Fig. 6.

2) ENTROPY-TOPSIS-GREY-SORT
In this technique, the subjective weights provided by the experts and the objective weights calculated using the entropy weight determination technique are merged, and their cumulative effects on decision-making are studied. Here Step 1-4 are same, so same dataset, normalized matrix, same value of β = 0.75 and α = 1.5 and IF aggregated matrix are obtained.
Step6: The normalized intuitionistic fuzzy entropy values (e mn ) inẼ are obtained using Eq.(33).    Subjective weights provided by experts for each criterion are given in Table 9. The combined weight W is given as follows: {0.2132, 0.2114, 0.2108, 0.2012, 0.1633}.
Step8: The weighted intuitionistic fuzzy decision matrix is in Table-15.
Step12: Using Eq.(44), the RC m values of each alternative based on their limit profile are calculated. These values are given in Table 16.
Step13: Eq.(45) is used to assign alternatives to appropriate classes. The final sorting of tourist sites as per their exposure to COVID-19 is shown in Fig. 7.

D. RESULT DISCUSSION AND COMPARATIVE ANALYSIS
• In comparison to [54], the results given here are of generalized nature, and twenty-two districts of Tehran VOLUME 11, 2023  Fig. 7), the figure can be explained that five districts have HSIC environment, only one district has a SIC environment, fifteen districts have a MSIC environment, whereas one district has a LSIC environment and none of the districts have a VLSIC environment. So, to improve the environmental quality of the entire Tehran, authorities need to focus on only one district of the LSIC environment (as per the ENTROPY-TOPSIS-GREY-sort ). The results of TOPSIS-GREY-sort and ENTROPY-TOPSIS-GREY-sort are similar to those obtained in [54]. We suggest the maximum number of districts belonging to the LSIC environment is three (as per ENTROPY-TOPSIS-GREY-sort) and two (as per TOPSIS-GREY-sort), and nearly the results are identical. Thus three districts need to be looked upon to improve the environmental quality of Tehran.
• It is not always the case that subjective weights (weights allocated by DMs) are known. In such a case, objective weights are calculated using a data-driven weight determination scheme. Here, we have carried out the individual study of objective and subjective weights, and both weights' cumulative effect is also analyzed. TOPSIS-GREY-sort uses objective weights (calculated by entropy method) which is (0.1013, 0.3557, 0.1588, 0.2750, 0.1093). Moreover, the convex combination of subjective weight (see:

VI. CONCLUSION
In the AIFS environment, the hesitancy and uncertainty involved have been suitably dealt with due to the use of the MCS method. Here ratings at the attributes and alternatives level have been expressed in linguistic terms, and it is characterized by a generalized fuzzy set called AIFS. The domestic tourism problem concerning the reopening of the tourist sites in the Philippines amidst COVID-19 and the environmental quality of Tehran has been satisfactorily solved, which demonstrates the efficacy of our proposed novel sorting technique. Further, it is worth noting that the proposed methods and entropy measures have a generic structure; hence, they can also be applied to other MCS application domains. Some of the contributions and insights of proposed MCS techniques are as follows: (1) the proposed entropy measure in the AIFS environment models the uncertainty involved in MCS problems in a more generalized manner without much increasing the computational cost, (2) it profitably sorts tourist sites into three classes (i.e., LEIC, MEIC, and HEIC) and sorts districts of Tehran into five classes (i.e., HSIC, SIC, MSIC, LSIC, and VLSIC). It remains efficient with the increase in the number of classes by adding the limit profiles, and (3) the proposed MCS techniques offer considerable potential for other domain applications. The case study results provide a vision to stakeholders in the tourism and public health sectors in appropriately designing and implementing measures as the industry is slowly recovering amidst the pandemic and implementing a strategy in different districts of Tehran to improve their environmental quality as per their sorting into other classes. Because of the variation of the tuning parameters instead of crisp values, we are obtaining ranges through the proposed sorting techniques, which in general, helps the stakeholders and authorities. A brief study of subjective and objective weights is also carried out in this work. This work offers an AIFS extension of the TOPSIS-Sort. In a similar manner, type-2 fuzzy sets (T2FS) and IVIFS-based models of TOPSIS-sort can be proposed.