Application of Pythagorean Fuzzy  Compact Space to Identify the Preferred City to Live in India

N. B. Gnanachristy; G. K. Revathi

doi:10.5391/IJFIS.2023.23.2.152

Original Article

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International Journal of Fuzzy Logic and Intelligent Systems 2023; 23(2): 152-161

Published online June 25, 2023

https://doi.org/10.5391/IJFIS.2023.23.2.152

Application of Pythagorean Fuzzy Compact Space to Identify the Preferred City to Live in India

N. B. Gnanachristy and G. K. Revathi

Division of Mathematics, Vellore Institute of Technology, Vandalur-Kelambakkam Road, Tamilnadu, India

Correspondence to :
G. K. Revathi (gk_revathi@yahoo.com)

Received: July 19, 2022; Revised: May 3, 2023; Accepted: May 9, 2023

This is an Open Access article distributed under the terms of the Creative Commons Attribution Non-Commercial License (http://creativecommons.org/licenses/by-nc/3.0/) which permits unrestricted noncommercial use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

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Abstract
1. Introduction
2. Preliminaries
3. Pythagorean Fuzzy Compact Space ( )
4. Identification of Preferred Place to Lead a Better Life through Pythagorean Fuzzy Compact Space
5. Comparison of Proposed Study’s Result Using Mamdani Fuzzy Inference System
6. Conclusion
7. Limitation of the Study
Conflict of Interest
Figures
Tables
References
Biographies

The compactness in a Pythagorean fuzzy topological space generalizes the notion of a subset of a closed space. In this study, a Pythagorean fuzzy 𝒢* compact space was defined and its properties were investigated. Further application of the Pythagorean fuzzy 𝒢* compact space is anticipated for examining preferred cities in India using real-time data by considering the temperatures of the cities.

Keywords: Pythagorean fuzzy topology, Pythagorean fuzzy compact space, Pythagorean fuzzy 𝒢* compact space

1. Introduction

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Abstract
1. Introduction
2. Preliminaries
3. Pythagorean Fuzzy Compact Space ( )
4. Identification of Preferred Place to Lead a Better Life through Pythagorean Fuzzy Compact Space
5. Comparison of Proposed Study’s Result Using Mamdani Fuzzy Inference System
6. Conclusion
7. Limitation of the Study
Conflict of Interest
Figures
Tables
References
Biographies

Fuzzy sets are widely employed in various industries, including automobiles, traffic systems, and other processes, where a logic circuit controls gearboxes, anti-skid brakes, and other functions. The concept was introduced by Zadeh [1] in 1965. The type of uncertainty that proceeds for “the class of attractive women,” “the class of tall men,” and other sets cannot be described in conventional mathematical logic languages. The uncertainty that arose in the logic area paved the way for defining membership values. Fuzzy sets were defined using the membership values. Subsequently, Atanassov [2] introduced the non-membership function. An example of the establishment of non-membership is if the box of an apple contains 10 apples, of which John ate seven apples and Caty ate two apples, one of which fell to the floor. Here, we have the logic 710 truth value for John and 310 for Caty; however, it is actually 210. However, the logic states that 310. To break this misconception, non-membership arose. Using both membership and non-membership functions, intuitionistic fuzzy sets (IFSs) were developed, and various properties were studied by Coker [3]. Yager and his colleagues [4–6] established a Pythagorean fuzzy set (PFS) that is an extension of the IFS. He provided an example of membership as 32 and of non-membership as 12. The sum of the two values was greater than 1. These are not available for the IFS. However, they are available for PFS because 322+122≤1 which means PFS is more effective than IFS in decision-making. The computational entitlement of PFS is more factual in analyzing vagueness and uncertainty in many areas, such as decision-making, artificial intelligence, data science, and robotics, because the space of PFS’s membership degree is superior to other fuzzy sets. Therefore, several fuzzy sets were developed and investigated.

1.1 Literature Review of the Study

To view fuzzy sets in a single space, topology concepts were studied and developed by Chang [7], who first introduced fuzzy topology. It was subsequently modified by Lowen [8] and Hutton [9]. Fuzzy topology has had immense applications in recent years. Several topologies [10] developed and studied using different types of fuzzy sets. Classical topology has evolved from classical analysis and has numerous applications in research fields, such as quantum gravity, data mining, machine learning, and data analysis. Likewise, the Pythagorean fuzzy topological space (PFTS) [11] developed using PFS, has many applications in decision-making. Subsequently, Hu [12] discussed product-induced spaces, which are a type of fuzzy topological space (FTS). He demonstrated that every FTS is topologically isomorphic to a definite space of topology, presenting the elementary clues of double points and establishing a fuzzy point neighborhood formation category. Papageorgiou [13] identified a series of fuzzy topological notions, such as fuzzy multifunctions and the fuzzy concept of a neighborhood point, which are helpful in understanding popular fuzzy optimization strategies and games. In his research, he discovered that a shared topological point set can be considered a type of fuzzy topology. The compactness concept in a fuzzy topology was introduced by Chang [7] and later by Gartner et al. [14]. Lowen [8] studied compactness in a fuzzy topology. The concept of compactness is introduced using PFSs. Many authors have studied the application of PFS in decision-making. Akram et al. [15] studied the linguistic Pythagorean fuzzy CRITIC-EDAS method for multiple-attribute group decision-making. They also presented [16] a new outranking method for multicriteria decision-making with complex Pythagorean fuzzy information [17], a multi-criteria decision model using complex Pythagorean fuzzy Yager aggregation operators. The PFS also has a graphical application, and Akram et al. [18] studied the Pythagorean fuzzy graphs (PFGs) application to group decision-making. He also used [19] minimal spanning tree hierarchical clustering algorithm to study similarity measures for the analysis of functional brain networks.

1.2 Motivation and Contribution

Compactness is widespread across numerous streams of mathematics. A novel application of Pythagorean fuzzy compactness in decision-making is interesting. In this paper, some properties of the Pythagorean fuzzy compact space are outlined, and the preferred city in India is identified using the Pythagorean fuzzy compact space by considering temperature as a parameter. The main goal of this study was to provide a topological approach for PFS. In this study, we are motivated to explore the resourcefulness of the compact space in tackling preferred cities to live in India via the maxima method because of its wider scope of applications in real-life problems embedded with imprecision. This study aims to explore the notion of a compact , and its application to real-time temperature-based data.

1.3 Structure of the Paper

The paper is organized as follows: in Section 2, the basic definitions and Pythagorean fuzzy closed set are provided. In Section 3, the Pythagorean fuzzy compact space is defined, and some interesting properties are discussed. In Section 4, an algorithm is proposed and explained in detail for identifying preferable places to lead a better life through a Pythagorean fuzzy compact space. In Section 5, a comparative analysis has been presented between the proposed method and Mamdani fuzzy inference system. Finally, Section 6 presents the conclusions of this study. Section 7 discusses the limitations of this study.

2. Preliminaries

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Abstract
1. Introduction
2. Preliminaries
3. Pythagorean Fuzzy Compact Space ( )
4. Identification of Preferred Place to Lead a Better Life through Pythagorean Fuzzy Compact Space
5. Comparison of Proposed Study’s Result Using Mamdani Fuzzy Inference System
6. Conclusion
7. Limitation of the Study
Conflict of Interest
Figures
Tables
References
Biographies

In this section, abbreviations and their expansions are introduced, and basic concepts are discussed.

MV (membership value), NMV (non-membership value), PFS (Pythagorean fuzzy set), PFTS (Pythagorean fuzzy topological space), PFOS (Pythagorean fuzzy open set), PFCS (Pythagorean fuzzy closed set), PFI (Pythagorean fuzzy interiors), PFC (Pythagorean fuzzy closure), (Pythagorean Fuzzy open set), (Pythagorean Fuzzy closed set), (Pythagorean Fuzzy open set), (Pythagorean Fuzzy closed set), (Pythagorean Fuzzy interior), (Pythagorean Fuzzy closure), (Pythagorean Fuzzy open cover), PFOC (Pythagorean Fuzzy opencover), PFCo (Pythagorean Fuzzy Compact), (Pythagorean Fuzzy compact space), FIP (Finite intersection properties).

Definition 2.1 [11]

A PFS R of X≠0 is a pair (μ_R, ν_R) where μ_R and ν_R are fuzzy sets of X such that μ_R²(x) + ν_R²(x) = r_R²(x) for any x∈X where the fuzzy set r_R is the strength of commitment at a point.

Definition 2.2 [11]

Let τ be a family of PFS of X≠Ø. If

(i) 0_X, 1_X ∈ τ.
(ii) A_ii ∈ I ⊂ τ, we have ∪ A_i_{_i_∈_I}A_i ∈ τ where I is an arbitrary index set.
(iii) A₁, A₂ ∈ τ, we have A₁ ∩ A₂ ∈ τ, where 0_X = (0, 1) and 1_X = (1, 0), then τ is called a PFT on X.

Definition 2.3 [11]

Let S = (μ_S, ν_S) and R = (μ_R, ν_R) be the two PFS of the set X. Then,

(i) R∪S = (max(μ_R, μ_S), min(ν_R, ν_S)).
(ii) R∩S = (min(μ_R, μ_S), max(ν_R, ν_S)).
(iii) R_c = (ν_R, μ_R).
(iv) R ⊂ S or S ⊃ R if μ_R ≤ μ_S and ν_R ≥ ν_S.

Example 2.4

R = {(0.4, 0.5), (0.2, 0.5))} and S = {(0.6, 0.3), (0.3, 0.4)} be the two PFSs onX = {a, b}, then R∪S = max{(0.4, 0.5), (0.2, 0.5)}∪{(0.6, 0.3), (0.3, 0.4)} = {(0.6, 0.4), (0.3, 0.4)}, R∩S = min{(0.4, 0.5), (0.2, 0.5)}∩{(0.6, 0.3), (0.3, 0.4)} = {(0.4, 0.5), (0.2, 0.5)}, R_c = {(0.5, 0.4), (0.5, 0.2)}, R ⊂ S = {(0.4, 0.5), (0.2, 0.5))} ⊂ {(0.6, 0.3), (0.3, 0.4)}, μ_R ≤ μ_S = 0.4 ≤ 0.6, 0.2 ≤ 0.3, ν_R ≥ ν_S = 0.5 ≥ 0.3, 0.5 ≥ 0.4.

Definition 2.5 [7]

Let X be a PFTS and R = (μ_R, ν_R) be the PFS in X. Subsequently, PFI and PFC are defined as

(i) int(R)=∪{G∣G is a PFOS in X and G⊆R},
(ii) cl(R)=∩{K∣K is a PFCS in X and R⊆K}.

Definition 2.6 [11]

A PFS R is called a Pythagorean fuzzy regular open set if and only if R = int(cl(R)). A PFS S is called a Pythagorean fuzzy regular closed if and only if S = cl(int(S)).

Theorem 2.7 [11]

Let f: X→Y be a function and X and Y≠Ø. Then we have

(i) f⁻¹[B^c] = f⁻¹[B]^c for any PFS B of Y.
(ii) f[f⁻¹[B]] ⊂ B for any PFS B of Y.
(iii) if B₁ ⊂ B₂ then f⁻¹[B₁] ⊂ f⁻¹[B₂] where B₁and B₂ are PFS of Y.
(iv) if A₁ ⊂ A₂ then f[A₁] ⊂ f[A₂] where A₁and A₂ are PFS of X.
(v) f[A]₂ ⊂ f[A₂] for any PFS A of X.
(vi) A ⊂ f⁻¹[f[A]] for any PFS A of X.

Definition 2.8 [7]

The intersection of the elements of each finite subfamily of is nonempty if and only if the family of the FSs has an FIP.

Definition 2.9 [20]

The PFS P of a PFTS X is called a if whenever where is a PFOS and or 1_X. The counterpart to the is the .

Notation

indicates the assortment of all in X.

Definition 2.10 [20]

The PFS P of a PFTS X is called a if whenever where is a PFOS and or1_X. The counterpart to the is the .

Definition 2.11 [20]

The collection , 1_X is in X. The counterpart of is .

Notation

indicates the assortment of all in X.

Definition 2.12 [20]

Let X be a PFTS and W = (μ_W, ν_W) be the PFS in X. Then, and of W are defined as

(i) intG*(W)=∪{R∣isaPFOSinX andR⊆W},
(ii) clG*(W)=∩{S∣SisaPFCSinX andW⊆S}.

Definition 2.13 [20]

Let ℵ: ℵ:X→Y be a mapping from PFTS X to PFTSY.

(i) if ℵ⁻¹(Q) is (resp., ) in X
(ii) for each PFOS (resp., PFCS) Q in Y, then ℵ is called Pythagorean fuzzy continuous ( ),
(iii) if ℵ⁻¹(Q) isPFOS (resp., PFCS) in X for each PFOS (resp., PFCS) Q in Y, then ℵ is called Pythagorean fuzzy continuous (PFC).

3. Pythagorean Fuzzy Compact Space ( )

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Abstract
1. Introduction
2. Preliminaries
3. Pythagorean Fuzzy Compact Space ( )
4. Identification of Preferred Place to Lead a Better Life through Pythagorean Fuzzy Compact Space
5. Comparison of Proposed Study’s Result Using Mamdani Fuzzy Inference System
6. Conclusion
7. Limitation of the Study
Conflict of Interest
Figures
Tables
References
Biographies

This section describes the properties of the Pythagorean fuzzy compact space.

Definition 3.1

Let X be the PFTS. A collection P=(μPi,νPi) of subsets of X is said to be cover of X if the union of the element of P is equal to 1_X. i.e., ∪Pi=1X.

Definition 3.2

Let X be the PFTS. The subcover of a cover, P≠0X is a subcollection of P which is a cover.

Definition 3.3

Let X be the PFTS. A cover of P of a PFTS X is said to be a Pythagorean fuzzy open cover if the elements of P are PFOSs of X.

Definition 3.4

Let X be the PFTS. Then, X is said to be a Pythagorean fuzzy -compact space if and only if each of X has a finite open subcover.

Example 3.5

Let X=a be a PFTS with topology τ = {(0, 1), (0.1, 0.9), (0.2, 0.9), (0.4, 0.8), (0.6, 0.7), (0.7, 0.6), (1, 0)}. Let P={(0.4,0.8),(0.6,0.7),(0.7,0.6),(1,0)} be the cover of X and the subcollection {(0.6, 0.7), (0.7, 0.6), (1, 0)} is the subcover, which is a Pythagorean fuzzy open cover. Then, X is a Pythagorean fuzzy compact space.

Proposition 3.6

Every of X is .

Proof

Let T denote a of X and let {Mi∣iɛI}=M be a of X . As R^c is a of X and M={Mi∣iɛI}∪Tc is of X . Because X is , there exists a finite subset I₀ of I such that ∪{M_i|iεI}∪T^c = I_X. Hence T is a .

Proposition 3.7

Every image of is a .

Proof

Let ℵ:X→Y be the from X onto Y. Let {M_i | i ∈ I} be a of Y, and let {ℵ⁻¹(M_i) | i ∈ I} be a of X. Because X is , it has a finite Pythagorean fuzzy subcover say {ℵ⁻1M₁ … ℵ⁻1M_n}. Because ℵ on M₁ … M_n is a of Y,Y is a .

Proposition 3.8

A PFTS X is a PFCo if and only if for every collection {M_i | i ∈ I} of PFCS of X with FIP S ∩M_i ≠ 0_X.

Proof

Let {M_i | i ∈ I} be a collection of PFCS in X with FIP. Assume that ∩M_i = 0_X. Then ∪M_i^c = 1_X. Because X is a PFCo, hence there exists i₁, i₂ … i_n such that ∪M_ij^c = 1_X. ∩M_ij = 0_X which is a contradiction. Hence, ∩M_ij^c ≠ 0_X. Conversely, let {M_i | i ∈ I} be the PFOC of X. Suppose that for every finite i₁, i₂ … i_n we have ∪ M_ij^c ≠ 1_X and ∩M_ij^c ≠ 0_X. Hence, Mjc≠0X is FIP. The hypothesis Mic≠0X implies that ∪ M_ij^c ≠ 1_X. This contradicts {M_i | i ∈ I} is a PFOC of X. Thus, X is PFCo.

4. Identification of Preferred Place to Lead a Better Life through Pythagorean Fuzzy Compact Space

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Abstract
1. Introduction
2. Preliminaries
3. Pythagorean Fuzzy Compact Space ( )
4. Identification of Preferred Place to Lead a Better Life through Pythagorean Fuzzy Compact Space
5. Comparison of Proposed Study’s Result Using Mamdani Fuzzy Inference System
6. Conclusion
7. Limitation of the Study
Conflict of Interest
Figures
Tables
References
Biographies

This section describes the application of a Pythagorean fuzzy compact space to identify the preferred place to live in India using the temperature.

Temperature is the degree of warmth or coldness of the body or environment. The temperature and humidity of the environment can easily affect body temperature. The Earth’s climate depends on temperature. If temperatures are low, then the climate of that place will be colder, whereas if the temperature of a place is higher, the climate will be warmer. Every day is influenced by the temperature, from what to wear, what to cook, where to go, and so on. Without knowing the temperature, we can never go out. Therefore, living at a particular temperature is considered a significant parameter. The temperatures of various cities were considered by assigning membership and non-membership values to the temperature values from 1995 to 2018 using the Pythagorean fuzzy trapezoidal membership function. Pythagorean fuzzy closed sets are obtained. The cover and subcover are derived from the given data and can frame a Pythagorean fuzzy compact.

4.1 Algorithm

Step 1: The temperature of the four cities Chennai, Delhi, Kolkata and Mumbai has been collected from the year 1995–2018 (From January to December) (Figure 1).

Step 2: PFSs were framed by using Pythagorean fuzzy trapezoidal membership function for each year by assigning (μ_PF₍_x₎, ν_PF₍_x₎) to the corresponding temperature value.

Step 3: The PFS obtained satisfies the condition that is provided in the definition of and the graph of the PFSs are drawn using MATLAB (Figure 2).

Step 4: cover of the PFTS is obtained.

Step 5: Pythagorean fuzzy compact space is obtained.

Step 6: Using the mean of maxima method preference is given for the value obtained pMOM=Σj=1MPjm2M+Σj=1KPjk2K, where Pm={P∣μPm=maxi=1MμPi} and Pk={P∣νPm=maxi=1KνPi}.

Step 7: Preferences have been assigned based on the values obtained through this decision-making method.

4.2 Detailed Study of the Proposed Algorithm to Find an Optimal City

Step 1: The data were collected from online sources [21]. Temperature data for the four major cities of Chennai, Delhi, Kolkata, and Mumbai were collected between 1995 and 2018 (from January to December). The average of the 12 months was then computed for each year. Average temperatures in Chennai, Delhi, Kolkata, and Mumbai were determined.

In Table 1, columns 1, 4, 7 and 10 represent the average temperature of the cities Chennai, Delhi, Mumbai and Kolkata respectively. Columns 2, 3, 5, 6, 8, 9, 11, and 12 show the membership and non-membership values for Chennai, Delhi, Mumbai, and Kolkata respectively.

Step 2: PFSs were obtained using the Pythagorean fuzzy trapezoidal function by assigning p₁ = 78.9, p₂ = 81, p₃ = 83, p₄ = 83.5

μPF(x)={0,x≤p1,x-p1p2-p1,p1<x≤p2,1,p2≤x≤p3,p4-xp4-p3,p3≤x<p4,0,x≥p4,νPF(x)={0,x≤p1,1-x-p1p2-p1,p1<x≤p2,1,p2≤x≤p3,1-p4-xp4-p3,p3≤x<p4,0,x≥p4,

where μ_PF₍_x₎ the MV and ν_PF₍_x₎ the NMV. A Pythagorean fuzzy set graph was created using MATLAB (version R2021a), and each point on the graph represents the PFS and the related membership and non-membership values of that temperature.

Step 3: In Chennai city, it is possible to get between the PFSs (0.635912, 0.771762) and (0.7568513, 0.651658) and also between the PFSs (0.482005, 0.876168) and (0.635912, 0.771762). Similarly, it is possible to obtain in Kolkata and Mumbai.

Step 4: cover is obtained for the cities Chennai, Kolkata and Mumbai. However, cover is not obtained for Delhi.

Step 5: Therefore, Pythagorean fuzzy compact space is obtained except for Delhi city because cover is not obtained for Delhi.

Step 6: Mean of maxima method for PFS fuzzy-rule-based systems use fuzzification, inference, and composition processes to evaluate linguistic if, then rules. They produce fuzzy results that must be converted into crisp outputs. Defuzzification was used to transform the fuzzy findings into crisp findings. Defuzzification is the process of transforming a fuzzy output into a single crisp value for a given fuzzy set. Many defuzzification techniques are used in decision-making. This method is suitable and effective [18, 22] for this real-time data. To provide preferences for the city based on the temperature mean of maxima method, a PFS was defined and used. The mean of maxima for PFS is given by

pMOM=∑j=1MPjm2M+∑j=1KPjk2K,

where PM={P∣μPm=maxi=1MμPi} and Pk={P∣νPK=maxi=1KνPi}.

Step 7: Based on the above formula, p_MOM is calculated for various cities; for Chennai, the maximum in MV and minimum in NMV are taken, and we obtain 11 values. The average was used for all the values, and we obtained 82.70980. Similarly, for other cities, the preferences have been fixed (Table 2). It was concluded that Chennai is the preferred place to live when using the temperature parameter for all seasons. This interpretation of preferences can also be observed by using the Mamdani fuzzy inference system (FIS). This provides reasonable results with a relatively simple structure. Although the above result is flawless, Mamdani FIS was used to compare the above results for a single value of the parameter temperature. Here, for every temperature value of the cities of Chennai, Delhi, Kolkata and Mumbai, the rule was applied using the Mamdani FIS to determine that Chennai is the preferred place to live in.

5. Comparison of Proposed Study’s Result Using Mamdani Fuzzy Inference System

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Abstract
1. Introduction
2. Preliminaries
3. Pythagorean Fuzzy Compact Space ( )
4. Identification of Preferred Place to Lead a Better Life through Pythagorean Fuzzy Compact Space
5. Comparison of Proposed Study’s Result Using Mamdani Fuzzy Inference System
6. Conclusion
7. Limitation of the Study
Conflict of Interest
Figures
Tables
References
Biographies

In this section, the results obtained through the algorithm developed.

Fuzzy inference is the process of utilizing fuzzy logic to create a map from a given input to an output. Mapping provides a foundation for making decisions and identifying patterns. The Mamdani system is presented to shape the final function locally, that is, specific rules to adjust the input-output connection on explicit portions of the input space without affecting the relation in other regions. This surface viewer from the FIS helps view preferences graphically, making the analysis more accurate and clearer. Here, single input and output were considered (Figure 3).

5.1 Input

Here, we considered the input as the temperature in three categories: cold, moderate, and hot, according to the data available. We generated these categories in the fuzzy trapezoidal membership function in the range of 70°C–90°C.

5.2 Output

Here, the output as a preference in the three categories is not preferable, preferable, and highly preferable. We generated these categories in the fuzzy triangular membership function in the range is 70%-110% (Figure 5, Table 4).

5.3 Rules Applied

The rules applied in Mamdani FIS is

• If the temperature is cold then it is not preferable.
• If the temperature is moderate then it is preferable.
• If the temperature is high then it is highly not preferable.

Based on these rules, we obtained a graph of each temperature value for the corresponding preferences (Figures 6).

5.4 Surface Viewer

This graph provides a brief overview of the temperature and the corresponding preference details (Figure 7).

6. Conclusion

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Abstract
1. Introduction
2. Preliminaries
3. Pythagorean Fuzzy Compact Space ( )
4. Identification of Preferred Place to Lead a Better Life through Pythagorean Fuzzy Compact Space
5. Comparison of Proposed Study’s Result Using Mamdani Fuzzy Inference System
6. Conclusion
7. Limitation of the Study
Conflict of Interest
Figures
Tables
References
Biographies

From these results, it is recognized that many factors are related to temperature, even though urbanization has an impact on temperature [23]. Another significant reason for this increase in temperature is pollution [24]. Chakraborty et al. [24] concluded that urbanization and pollution are the reasons for the increase in temperature. The Air Quality Index (AQI) in Delhi has become poorer [25]. This causes pollution, leading to an increase in temperature. Compared with the data and sources [26] available regarding Delhi’s temperature, it was identified that Delhi is the least preferred place to live in among the cities, and temperature is the basis factor needed for the place to live in. According to the inference of the value computed using p_MOM, Chennai was the first place to reside based on temperature and Delhi was the least favored (Table 5). This could be due to Delhi’s high pollution levels compared with other cities. This could result in an increase in temperature, making the weather hot. The results are verified using Mamdani FIS. Mamdani FIS without changing the functions of the input and output, the Mamdani FIS provides a clear relationship between the functions. The primary goal of analyzing the collected results using FIS was to analyze the study using a topological approach using FIS. This evaluation provides a clear depiction that Delhi is preferred less than the other cities.

7. Limitation of the Study

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Abstract
1. Introduction
2. Preliminaries
3. Pythagorean Fuzzy Compact Space ( )
4. Identification of Preferred Place to Lead a Better Life through Pythagorean Fuzzy Compact Space
5. Comparison of Proposed Study’s Result Using Mamdani Fuzzy Inference System
6. Conclusion
7. Limitation of the Study
Conflict of Interest
Figures
Tables
References
Biographies

Preferences were made based on a single parameter: temperature.

This study can be enhanced by exploring different parameters for different regions, defining the Pythagorean fuzzy relationships between the parameters for different locations, and using a decision-making technique. This can also be replicated in other cities and countries across the globe by selecting different countries.

Conflict of Interest

Go to

Abstract
1. Introduction
2. Preliminaries
3. Pythagorean Fuzzy Compact Space ( )
4. Identification of Preferred Place to Lead a Better Life through Pythagorean Fuzzy Compact Space
5. Comparison of Proposed Study’s Result Using Mamdani Fuzzy Inference System
6. Conclusion
7. Limitation of the Study
Conflict of Interest
Figures
Tables
References
Biographies

No potential conflict of interest relevant to this article was reported.

Figures

Go to

Abstract
1. Introduction
2. Preliminaries
3. Pythagorean Fuzzy Compact Space ( )
4. Identification of Preferred Place to Lead a Better Life through Pythagorean Fuzzy Compact Space
5. Comparison of Proposed Study’s Result Using Mamdani Fuzzy Inference System
6. Conclusion
7. Limitation of the Study
Conflict of Interest
Figures
Tables
References
Biographies

Fig. 1.

This diagram depicts the framework of this study.

Fig. 2.

Representation of fuzzified value of Pythagorean fuzzy set using MATLAB.

Fig. 3.

Input and output in Mamdani in FIS.

Fig. 4.

Input in Mamdani FIS.

Fig. 5.

Output in Mamdani FIS.

Fig. 6.

Rule viewer in Mamdani FIS.

Fig. 7.

A graph of each temperature in Mamdani FIS.

Tables

Go to

Abstract
1. Introduction
2. Preliminaries
3. Pythagorean Fuzzy Compact Space ( )
4. Identification of Preferred Place to Lead a Better Life through Pythagorean Fuzzy Compact Space
5. Comparison of Proposed Study’s Result Using Mamdani Fuzzy Inference System
6. Conclusion
7. Limitation of the Study
Conflict of Interest
Figures
Tables
References
Biographies

Table. 1.

Table 1. Average temperature of four cities from the year 1998–2018 with membership values (MV) and non-membership values (NMV) corresponding to each city.

Column 1	Column 2	Column 3	Column 4	Column 5	Column 6	Column 7	Column 8	Column 9	Column 10	Column 11	Column 12
Chennai	MV	NMV	Delhi	MV	NMV	Kolkata	MV	NMV	Mumbai	MV	NMV
82.7156	1	0	76.9983	0	1	79.4627	0.59305	0.80516	81.1046	1	0
82.6576	1	0	75.7541	0	1	79.5226	0.6238	0.7815	81.7451	1	0
83.3838	0.4820	0.8762	74.4573	0	1	78.7301	0	1	81.7858	1	0
84.2123	0	1	76.7767	0	1	79.9644	0.8156	0.5786	82.1979	1	0
83.2978	0.6359	0.7718	77.1373	0	1	79.8025	0.7510	0.6603	81.8214	1	0
82.9820	1	0	76.5954	0	1	79.7008	0.7075	0.7067	81.7104	1	0
83.2123	0.7585	0.6517	76.2107	0	1	78.9021	0.0370	0.9993	81.0630	1	0
83.4660	0.2607	0.9654	77.6841	0	1	80.2411	0.9155	0.4022	82.2751	1	0
83.4921	0.1261	0.9920	75.8663	0	1	79.8266	0.7610	0.6488	81.4370	1	0
82.5467	1	0	77.2787	0	1	80.3885	0.9645	0.2640	80.6027	1	0
82.7329	1	0	76.5595	0	1	79.1142	0.3659	0.9306	81.2625	1	0
82.9647	1	0	77.7984	0	1	79.9589	0.8135	0.5815	81.3005	1	0
82.0337	1	0	77.0748	0	1	79.5756	0.6498	0.7601	82.5531	1	0
82.3557	1	0	76.2656	0	1	79.2724	0.4824	0.8759	82.1156	1	0
84.3293	0	1	77.8814	0	1	80.4962	0.9988	0.0490	83.0447	0.9543	0.2989
82.9195	1	0	78.2364	0	1	80.5627	1	0	82.6871	1	0
82.9373	1	0	76.6962	0	1	79.4178	0.5689	0.8224	82.2847	1	0
83.8896	0	1	76.9049	0	1	80.1792	0.8942	0.4477	81.6964	1	0
82.9622	1	0	76.7625	0	1	79.3855	0.5508	0.8346	81.7392	1	0
83.4827	0.1858	0.9826	77.0011	0	1	79.8811	0.7831	0.6219	82.7939	1	0
84.1790	0	1	77.6071	0	1	80.6566	1	0	83.6986	0	1
84.5000	0	1	80.5456	1	0	81.3205	1	0	82.9213	1	0
84.7586	0	1	78.9932	0.2413	0.9705	79.8584	0.7739	0.6333	83.4044	0.4373	0.8993
84.9737	0	1	78.3545	0	1	79.7614	0.7337	0.6794	83.88	0	1

Table. 2.

Table 2. Calculated p_MOM for four cities.

City	Calculated p_MOM	Preference
Chennai	82.70980	First
Delhi	80.54563	Fourth
Mumbai	80.84659	Third
Kolkata	81.86986	Second

Table. 3.

Table 3. Temperature and its input parameters value (unit: °C).

Temperature	Corresponding parameters in FIS in MATLAB
Cold	77, 78.9, 81, 82
Moderate	80, 81, 82, 83
High	82, 83.5, 84, 84.5

Table. 4.

Table 4. Preference and its output parameter value (unit: %).

Preference	Corresponding Parameters in FIS in MATLAB
Not preferable	70, 80, 90
Preferable	80, 90, 100
Highly not preferable	90, 100, 110

Table. 5.

Table 5. Preference.

City	Preference
Chennai	First
Kolkata	Second
Mumbai	Third
Delhi	Four

References

Go to

Abstract
1. Introduction
2. Preliminaries
3. Pythagorean Fuzzy Compact Space ( )
4. Identification of Preferred Place to Lead a Better Life through Pythagorean Fuzzy Compact Space
5. Comparison of Proposed Study’s Result Using Mamdani Fuzzy Inference System
6. Conclusion
7. Limitation of the Study
Conflict of Interest
Figures
Tables
References
Biographies

Zadeh, LA (1965). Fuzzy sets. Information and Control. 8, 338-353. https://doi.org/10.1016/S0019-9958(65)90241-X
Atanassov, KT (1999). Intuitionistic Fuzzy Sets. Heidelberg, Germany: Physica-Verlag https://doi.org/10.1007/978-3-7908-1870-3_1
Coker, D (1997). An introduction to intuitionistic fuzzy topological spaces. Fuzzy Sets and Systems. 88, 81-89. https://doi.org/10.1016/S0165-0114(96)00076-0
Yager, RR . Pythagorean fuzzy subsets., Proceedings of 2013 Joint IFSA World Congress and NAFIPS Annual Meeting (IFSA/NAFIPS), 2013, Edmonton, Canada, Array, pp.57-61. https://doi.org/10.1109/IFSA-NAFIPS.2013.6608375
Yager, RR (2014). Pythagorean membership grades in multicriteria decision making. IEEE Transactions on Fuzzy Systems. 22, 958-965. https://doi.org/10.1109/TFUZZ.2013.2278989
Yager, RR, and Abbasov, AM (2013). Pythagorean membership grades, complex numbers, and decision making. International Journal of Intelligent Systems. 28, 436-452.
Chang, CL (1968). Fuzzy topological spaces. Journal of Mathematical Analysis and Applications. 24, 182-190. https://doi.org/10.1016/0022-247X(68)90057-7
Lowen, R (1976). Fuzzy topological spaces and fuzzy compactness. Journal of Mathematical Analysis and Applications. 56, 621-633. https://doi.org/10.1016/0022-247X(76)90029-9
Hutton, B (1980). Products of fuzzy topological spaces. Topology and its Applications. 11, 59-67.
Zahan, I, and Nasrin, R (2021). An introduction to fuzzy topological spaces. Advances in Pure Mathematics. 11, 483-501. https://doi.org/10.4236/apm.2021.115034
Olgun, M, Unver, M, and Yardımcı, S (2019). Pythagorean fuzzy topological spaces. Complex & Intelligent Systems. 5, 177-183. https://doi.org/10.1007/s40747-019-0095-2
Hu, CH (1985). Fuzzy topological spaces. Journal of Mathematical Analysis and Applications. 110, 141-178.
Papageorgiou, NS (1985). Fuzzy topology and fuzzy multifunctions. Journal of Mathematical Analysis and Applications. 109, 397-425. https://doi.org/10.1016/0022-247x(85)90159-3
Gantner, TE, Steinlage, RC, and Warren, RH (1978). Compactness in fuzzy topological spaces. Journal of Mathematical Analysis and Applications. 62, 547-562. https://doi.org/10.1016/0022-247X(78)90148-8
Akram, M, Ramzan, N, and Deveci, M (2023). Linguistic Pythagorean fuzzy CRITIC-EDAS method for multiple-attribute group decision analysis. Engineering Applications of Artificial Intelligence. 119. article no. 105777
Akram, M, Zahid, K, and Alcantud, JCR (2022). A new out-ranking method for multicriteria decision making with complex Pythagorean fuzzy information. Neural Computing and Applications. 34, 8069-8102. https://doi.org/10.1007/s00521-021-06847-1
Akram, M, Peng, X, and Sattar, A (2021). Multi-criteria decision-making model using complex Pythagorean fuzzy Yager aggregation operators. Arabian Journal for Science and Engineering. 46, 1691-1717. https://doi.org/10.1007/s13369-020-04864-1
Akram, M, Habib, A, and Davvaz, B (2019). Direct sum of n Pythagorean fuzzy graphs with application to group decision-making. Journal of Multiple Valued Logic & Soft Computing. 33, 75-115.
Habib, A, Akram, M, and Kahraman, C (2022). Minimum spanning tree hierarchical clustering algorithm: a new Pythagorean fuzzy similarity measure for the analysis of functional brain networks. Expert Systems with Applications. 201. article no. 117016
Gnanachristy, NB, and Revathi, GK (2021). A view on Pythagorean fuzzy contra continuous function. Journal of Physics: Conference Series. 2115. article no. 012041
Saha, R. (2009) . Temperature of different cities of India. Available: https://www.kaggle.com/datasets/riturajsaha/temperature-of-different-cities-of-india
Naaz, S, Alam, A, and Biswas, R (2011). Effect of different defuzzification methods in a fuzzy based load balancing application. International Journal of Computer Science Issues (IJCSI). 8, 261-267.
Dhorde, A, Dhorde, A, and Gadgil, AS (2009). Long-term temperature trends at four largest cities of India during the twentieth century. Journal of Indian Geophysical Union. 13, 85-97.
Chakraborty, SD, Kant, Y, and Bharath, BD (2014). Study of land surface temperature in Delhi city to managing the thermal effect on urban developments. International Journal of Advanced Scientific and Technical Research. 4, 439-450.
Asian News International. (2021) . Air quality in Delhi, Noida slips to ‘very poor’ category. Available: https://www.ndtv.com/delhi-news/air-quality-in-delhi-noida-slips-to-very-poor-category-gurugrams-improves-to-moderate-2649805
Angelov, P (1995). Crispification: defuzzification of intuitionistic fuzzy sets. Busefal. 64, 51-55.

Biographies

Go to

Abstract
1. Introduction
2. Preliminaries
3. Pythagorean Fuzzy Compact Space ( )
4. Identification of Preferred Place to Lead a Better Life through Pythagorean Fuzzy Compact Space
5. Comparison of Proposed Study’s Result Using Mamdani Fuzzy Inference System
6. Conclusion
7. Limitation of the Study
Conflict of Interest
Figures
Tables
References
Biographies

N. B. Gnanachristy is currently doing her research in the Division of Mathematics, the School of Advanced Sciences, Vellore Institute of Technology Chennai, Chennai-127, Tamil Nadu, India in fuzzy topology and its applications. The author has published papers in journals. E-mail: nbgnanam92@gmail.com

G. K. Revathi is currently working as an associate professor in the Division of Mathematics, the School of Advanced Sciences, Vellore Institute of Technology Chennai, Chennai-127, Tamil Nadu, India. Her area of interest are topology and its applications, fuzzy set theory and its application, fuzzy inference theory, mathematical modelling, decision-making, data analysis and machine learning algorithms. E-mail: gk revathi@yahoo.com

Article

Original Article

International Journal of Fuzzy Logic and Intelligent Systems 2023; 23(2): 152-161

Published online June 25, 2023 https://doi.org/10.5391/IJFIS.2023.23.2.152

Application of Pythagorean Fuzzy Compact Space to Identify the Preferred City to Live in India

N. B. Gnanachristy and G. K. Revathi

Division of Mathematics, Vellore Institute of Technology, Vandalur-Kelambakkam Road, Tamilnadu, India

Correspondence to:G. K. Revathi (gk_revathi@yahoo.com)

Received: July 19, 2022; Revised: May 3, 2023; Accepted: May 9, 2023

Abstract

Keywords: Pythagorean fuzzy topology, Pythagorean fuzzy compact space, Pythagorean fuzzy 𝒢,* compact space

1. Introduction

Fuzzy sets are widely employed in various industries, including automobiles, traffic systems, and other processes, where a logic circuit controls gearboxes, anti-skid brakes, and other functions. The concept was introduced by Zadeh [1] in 1965. The type of uncertainty that proceeds for “the class of attractive women,” “the class of tall men,” and other sets cannot be described in conventional mathematical logic languages. The uncertainty that arose in the logic area paved the way for defining membership values. Fuzzy sets were defined using the membership values. Subsequently, Atanassov [2] introduced the non-membership function. An example of the establishment of non-membership is if the box of an apple contains 10 apples, of which John ate seven apples and Caty ate two apples, one of which fell to the floor. Here, we have the logic $\frac{7}{10}$ truth value for John and $\frac{3}{10}$ for Caty; however, it is actually $\frac{2}{10}$ . However, the logic states that $\frac{3}{10}$ . To break this misconception, non-membership arose. Using both membership and non-membership functions, intuitionistic fuzzy sets (IFSs) were developed, and various properties were studied by Coker [3]. Yager and his colleagues [4–6] established a Pythagorean fuzzy set (PFS) that is an extension of the IFS. He provided an example of membership as $\frac{\sqrt{3}}{2}$ and of non-membership as $\frac{1}{2}$ . The sum of the two values was greater than 1. These are not available for the IFS. However, they are available for PFS because ${\frac{\sqrt{3}}{2}}^{2} + {\frac{1}{2}}^{2} \leq 1$ which means PFS is more effective than IFS in decision-making. The computational entitlement of PFS is more factual in analyzing vagueness and uncertainty in many areas, such as decision-making, artificial intelligence, data science, and robotics, because the space of PFS’s membership degree is superior to other fuzzy sets. Therefore, several fuzzy sets were developed and investigated.

1.1 Literature Review of the Study

1.2 Motivation and Contribution

1.3 Structure of the Paper

2. Preliminaries

In this section, abbreviations and their expansions are introduced, and basic concepts are discussed.

Definition 2.1 [11]

A PFS R of $X \neq 0$ is a pair (μ_R, ν_R) where μ_R and ν_R are fuzzy sets of $X$ such that μ_R²(x) + ν_R²(x) = r_R²(x) for any $x \in X$ where the fuzzy set r_R is the strength of commitment at a point.

Definition 2.2 [11]

Let τ be a family of PFS of $X \neq Ø$ . If

(i) 0_X, 1_X ∈ τ.
(ii) A_ii ∈ I ⊂ τ, we have ∪ A_i_{_i_∈_I}A_i ∈ τ where I is an arbitrary index set.
(iii) A₁, A₂ ∈ τ, we have A₁ ∩ A₂ ∈ τ, where 0_X = (0, 1) and 1_X = (1, 0), then τ is called a PFT on X.

Definition 2.3 [11]

Let S = (μ_S, ν_S) and R = (μ_R, ν_R) be the two PFS of the set $X$ . Then,

(i) R∪S = (max(μ_R, μ_S), min(ν_R, ν_S)).
(ii) R∩S = (min(μ_R, μ_S), max(ν_R, ν_S)).
(iii) R_c = (ν_R, μ_R).
(iv) R ⊂ S or S ⊃ R if μ_R ≤ μ_S and ν_R ≥ ν_S.

Example 2.4

Definition 2.5 [7]

Let $X$ be a PFTS and R = (μ_R, ν_R) be the PFS in $X$ . Subsequently, PFI and PFC are defined as

(i) $i n t (R) = \cup {G ∣ G is a PFOS in X and G \subseteq R}$ ,
(ii) $c l (R) = \cap {K ∣ K is a PFCS in X and R \subseteq K}$ .

Definition 2.6 [11]

A PFS R is called a Pythagorean fuzzy regular open set if and only if R = int(cl(R)). A PFS S is called a Pythagorean fuzzy regular closed if and only if S = cl(int(S)).

Theorem 2.7 [11]

Let f: $X \to Y$ be a function and $X$ and $Y \neq Ø$ . Then we have

(i) f⁻¹[B^c] = f⁻¹[B]^c for any PFS B of $Y$ .
(ii) f[f⁻¹[B]] ⊂ B for any PFS B of $Y$ .
(iii) if B₁ ⊂ B₂ then f⁻¹[B₁] ⊂ f⁻¹[B₂] where B₁and B₂ are PFS of $Y$ .
(iv) if A₁ ⊂ A₂ then f[A₁] ⊂ f[A₂] where A₁and A₂ are PFS of $X$ .
(v) f[A]₂ ⊂ f[A₂] for any PFS A of $X$ .
(vi) A ⊂ f⁻¹[f[A]] for any PFS A of $X$ .

Definition 2.8 [7]

The intersection of the elements of each finite subfamily of is nonempty if and only if the family of the FSs has an FIP.

Definition 2.9 [20]

The PFS P of a PFTS $X$ is called a if whenever where is a PFOS and or 1_X. The counterpart to the is the .

Notation

indicates the assortment of all in $X$ .

Definition 2.10 [20]

The PFS P of a PFTS $X$ is called a if whenever where is a PFOS and or1_X. The counterpart to the is the .

Definition 2.11 [20]

The collection , 1_X is in $X$ . The counterpart of is .

Notation

indicates the assortment of all in $X$ .

Definition 2.12 [20]

Let $X$ be a PFTS and W = (μ_W, ν_W) be the PFS in $X$ . Then, and of W are defined as

(i) ${i n t}_{G^{*}} (W) = \cup {R ∣ i s a P F O S i n X a n d R \subseteq W}$ ,
(ii) ${c l}_{G^{*}} (W) = \cap {S ∣ S i s a P F C S i n X a n d W \subseteq S}$ .

Definition 2.13 [20]

Let ℵ: $ℵ : X \to Y$ be a mapping from PFTS $X$ to $PFTS Y$ .

(i) if ℵ⁻¹(Q) is (resp., ) in $X$
(ii) for each PFOS (resp., PFCS) Q in $Y$ , then ℵ is called Pythagorean fuzzy continuous ( ),
(iii) if ℵ⁻¹(Q) isPFOS (resp., PFCS) in $X$ for each PFOS (resp., PFCS) Q in $Y$ , then ℵ is called Pythagorean fuzzy continuous (PFC).

3. Pythagorean Fuzzy Compact Space ( )

This section describes the properties of the Pythagorean fuzzy compact space.

Definition 3.1

Let $X$ be the PFTS. A collection $P = (μ_{P_{i}}, ν_{P_{i}})$ of subsets of $X$ is said to be cover of $X$ if the union of the element of $P$ is equal to 1_X. i.e., $\cup P_{i} = 1_{X}$ .

Definition 3.2

Let $X$ be the PFTS. The subcover of a cover, $P \neq 0_{X}$ is a subcollection of $P$ which is a cover.

Definition 3.3

Let $X$ be the PFTS. A cover of $P$ of a PFTS $X$ is said to be a Pythagorean fuzzy open cover if the elements of $P$ are PFOSs of $X$ .

Definition 3.4

Let $X$ be the PFTS. Then, $X$ is said to be a Pythagorean fuzzy -compact space if and only if each of $X$ has a finite open subcover.

Example 3.5

Let $X = a$ be a PFTS with topology τ = {(0, 1), (0.1, 0.9), (0.2, 0.9), (0.4, 0.8), (0.6, 0.7), (0.7, 0.6), (1, 0)}. Let $P = {(0.4, 0.8), (0.6, 0.7), (0.7, 0.6), (1, 0)}$ be the cover of $X$ and the subcollection {(0.6, 0.7), (0.7, 0.6), (1, 0)} is the subcover, which is a Pythagorean fuzzy open cover. Then, $X$ is a Pythagorean fuzzy compact space.

Proposition 3.6

Every of $X$ is .

Proof

Let T denote a of $X$ and let ${M_{i} ∣ i ɛ I} = M$ be a of $X$ . As R^c is a of $X$ and $M = {M_{i} ∣ i ɛ I} \cup T^{c}$ is of $X$ . Because $X$ is , there exists a finite subset I₀ of I such that ∪{M_i|iεI}∪T^c = I_X. Hence T is a .

Proposition 3.7

Every image of is a .

Proof

Let $ℵ : X \to Y$ be the from $X$ onto $Y$ . Let {M_i | i ∈ I} be a of $Y$ , and let {ℵ⁻¹(M_i) | i ∈ I} be a of $X$ . Because $X$ is , it has a finite Pythagorean fuzzy subcover say {ℵ⁻1M₁ … ℵ⁻1M_n}. Because ℵ on M₁ … M_n is a of $Y, Y$ is a .

Proposition 3.8

A PFTS $X$ is a PFCo if and only if for every collection {M_i | i ∈ I} of PFCS of $X$ with FIP S ∩M_i ≠ 0_X.

Proof

Let {M_i | i ∈ I} be a collection of PFCS in $X$ with FIP. Assume that ∩M_i = 0_X. Then ∪M_i^c = 1_X. Because $X$ is a PFCo, hence there exists i₁, i₂ … i_n such that ∪M_ij^c = 1_X. ∩M_ij = 0_X which is a contradiction. Hence, ∩M_ij^c ≠ 0_X. Conversely, let {M_i | i ∈ I} be the PFOC of $X$ . Suppose that for every finite i₁, i₂ … i_n we have ∪ M_ij^c ≠ 1_X and ∩M_ij^c ≠ 0_X. Hence, $M_{j}^{c} \neq 0_{X}$ is FIP. The hypothesis $M_{i}^{c} \neq 0_{X}$ implies that ∪ M_ij^c ≠ 1_X. This contradicts {M_i | i ∈ I} is a PFOC of $X$ . Thus, $X$ is PFCo.

4. Identification of Preferred Place to Lead a Better Life through Pythagorean Fuzzy Compact Space

This section describes the application of a Pythagorean fuzzy compact space to identify the preferred place to live in India using the temperature.

4.1 Algorithm

Step 1: The temperature of the four cities Chennai, Delhi, Kolkata and Mumbai has been collected from the year 1995–2018 (From January to December) (Figure 1).

Step 2: PFSs were framed by using Pythagorean fuzzy trapezoidal membership function for each year by assigning (μ_PF₍_x₎, ν_PF₍_x₎) to the corresponding temperature value.

Step 3: The PFS obtained satisfies the condition that is provided in the definition of and the graph of the PFSs are drawn using MATLAB (Figure 2).

Step 4: cover of the PFTS is obtained.

Step 5: Pythagorean fuzzy compact space is obtained.

Step 6: Using the mean of maxima method preference is given for the value obtained $p_{M O M} = \frac{Σ_{j = 1}^{M} P_{j}^{m}}{2 M} + \frac{Σ_{j = 1}^{K} P_{j}^{k}}{2 K}$ , where $P^{m} = {P ∣ μ_{P^{m}} = {m a x}_{i = 1}^{M} μ_{P^{i}}}$ and $P^{k} = {P ∣ ν_{P^{m}} = {m a x}_{i = 1}^{K} ν_{P^{i}}}$ .

Step 7: Preferences have been assigned based on the values obtained through this decision-making method.

4.2 Detailed Study of the Proposed Algorithm to Find an Optimal City

Step 2: PFSs were obtained using the Pythagorean fuzzy trapezoidal function by assigning p₁ = 78.9, p₂ = 81, p₃ = 83, p₄ = 83.5

\begin{array}{l} μ_{P F (x)} = {\begin{array}{l} 0, & x \leq p_{1}, \\ \sqrt{\frac{x - p_{1}}{p_{2} - p_{1}}}, & p_{1} < x \leq p_{2}, \\ 1, & p_{2} \leq x \leq p_{3}, \\ \sqrt{\frac{p_{4} - x}{p_{4} - p_{3}}}, & p_{3} \leq x < p_{4}, \\ 0, & x \geq p_{4}, \end{array} \\ ν_{P F (x)} = {\begin{array}{l} 0, & x \leq p_{1}, \\ 1 - \sqrt{\frac{x - p_{1}}{p_{2} - p_{1}}}, & p_{1} < x \leq p_{2}, \\ 1, & p_{2} \leq x \leq p_{3}, \\ 1 - \sqrt{\frac{p_{4} - x}{p_{4} - p_{3}}}, & p_{3} \leq x < p_{4}, \\ 0, & x \geq p_{4}, \end{array} \end{array}

Step 4: cover is obtained for the cities Chennai, Kolkata and Mumbai. However, cover is not obtained for Delhi.

Step 5: Therefore, Pythagorean fuzzy compact space is obtained except for Delhi city because cover is not obtained for Delhi.

p_{M O M} = \frac{\sum_{j = 1}^{M} P_{j}^{m}}{2 M} + \frac{\sum_{j = 1}^{K} P_{j}^{k}}{2 K},

where $P^{M} = {P ∣ μ_{P^{m}} = {m a x}_{i = 1}^{M} μ_{P^{i}}}$ and $P^{k} = {P ∣ ν_{P^{K}} = {m a x}_{i = 1}^{K} ν_{P^{i}}}$ .

5. Comparison of Proposed Study’s Result Using Mamdani Fuzzy Inference System

In this section, the results obtained through the algorithm developed.

5.1 Input

5.2 Output

5.3 Rules Applied

The rules applied in Mamdani FIS is

• If the temperature is cold then it is not preferable.
• If the temperature is moderate then it is preferable.
• If the temperature is high then it is highly not preferable.

Based on these rules, we obtained a graph of each temperature value for the corresponding preferences (Figures 6).

5.4 Surface Viewer

This graph provides a brief overview of the temperature and the corresponding preference details (Figure 7).

6. Conclusion

7. Limitation of the Study

Preferences were made based on a single parameter: temperature.

Abstract
1. Introduction
2. Preliminaries
3. Pythagorean Fuzzy Compact Space ( )
4. Identification of Preferred Place to Lead a Better Life through Pythagorean Fuzzy Compact Space
5. Comparison of Proposed Study’s Result Using Mamdani Fuzzy Inference System
6. Conclusion
7. Limitation of the Study

Fig 1.

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