Topological Data Analysis with Spherical Fuzzy Soft AHPTOPSIS for Environmental Mitigation System
Abstract
:1. Introduction
 The notion of a spherical fuzzy soft set (SFSS) is a combination of an SFS and a SS. An SFSS is a new approach for computational intelligence, data analysis, and fuzzy modeling.
 We define some new operations on SFSSs for the construction of SFSStopology. The idea of spherical fuzzy soft set topology (SFSStopology) is defined with the help of null SFSS, absolute SFSS, SFSSextended union, and SFSSrestricted intersection.
 Novel conceptualizations of SFSStopology are explored, such as, SFSSopen set, SFSSclosed set, SFSSinterior, SFSSclosure, SFSSbase and SFSSsubbase. These notions are illustrated with some numerical examples.
 The concepts of spherical fuzzy soft set separation axioms are proposed and related results are explored.
 We developed an extended choice value method (CVM) and the AHPTOPSIS for SSFSs, respectively.
 The suggested methods are efficient tools for MCDGDM of an environmental mitigation system. An application is designed to identify the ability of the suggested approach to focus the crises addressed by the environmental mitigation system, as well as to signify the validity of numerous major findings through case studies. It is presented in order to justify our technique and demonstrate its applicability and effectiveness.
 The efficiency of suggested methods is demonstrated by a comparative analysis and sensitivity analysis.
2. Preliminaries
 If $s\left({N}_{i}\right)<s\left({N}_{j}\right)$ then ${N}_{i}$ precedes ${N}_{j}$ i.e., ${N}_{i}\prec {N}_{j}$;
 If $s\left({N}_{i}\right)>s\left({N}_{j}\right)$ then ${N}_{i}$ succeeds ${N}_{j}$ i.e., ${N}_{i}\succ {N}_{j}$;
 If $s\left({N}_{i}\right)=s\left({N}_{j}\right)$ then ${N}_{i}\sim {N}_{j}$.
 If $s\left({N}_{i}\right)$ and $s\left({N}_{j}\right)$ coincide and $a\left({N}_{i}\right)$ exceeds $a\left({N}_{j}\right)$ then ${N}_{i}\succ {N}_{j}$;
 If both $s\left({N}_{i}\right),s\left({N}_{j}\right)$ and $a\left({N}_{i}\right),a\left({N}_{j}\right)$ coincide then ${N}_{i}\sim {N}_{j}$
 Inclusion: If ${\mu}_{T}\left(\kappa \right)\le {\mu}_{J}\left(\kappa \right),{\gamma}_{T}\left(\kappa \right)\le {\gamma}_{J}\left(\kappa \right),{\eta}_{T}\left(\kappa \right)\ge {\eta}_{J}\left(\kappa \right)$;
 Equality: If $T\subseteq J$ and $J\subseteq T$ then $T=J$;
 Union: $\left\{(\kappa ,max\{{\mu}_{T}\left(\kappa \right),{\mu}_{J}\left(\kappa \right)\},min\{{\gamma}_{T}\left(\kappa \right),{\gamma}_{J}\left(\kappa \right)\},min\{{\eta}_{T}\left(\kappa \right),{\eta}_{J}\left(\kappa \right)\})\kappa \in X\right\}$;
 Intersection: $\left\{(\kappa ,min\{{\mu}_{T}\left(\kappa \right),{\mu}_{J}\left(\kappa \right)\},min\{{\gamma}_{T}\left(\kappa \right),{\gamma}_{J}\left(\kappa \right)\},max\{{\eta}_{T}\left(\kappa \right),{\eta}_{J}\left(\kappa \right)\right\}\left)\right\kappa \in X\}$;
 Complement: $\left\{(\kappa ,{\eta}_{T}\left(\kappa \right),{\gamma}_{T}\left(\kappa \right),{\mu}_{T}\left(\kappa \right))\right\kappa \in X\}$.
 Inclusion: If ${\mu}_{T}\left(\kappa \right)\le {\mu}_{J}\left(\kappa \right),{\gamma}_{T}\left(\kappa \right)\ge {\gamma}_{J}\left(\kappa \right),{\eta}_{T}\left(\kappa \right)\ge {\eta}_{J}\left(\kappa \right)$.
 Intersection: $\left\{(\kappa ,min\{{\mu}_{T}\left(\kappa \right),{\mu}_{J}\left(\kappa \right)\},max\{{\gamma}_{T}\left(\kappa \right),{\gamma}_{J}\left(\kappa \right)\},max\{{\eta}_{T}\left(\kappa \right),{\eta}_{J}\left(\kappa \right)\})\right\kappa \in X\}$.
${S}_{A}$  ${e}_{1}$  ${e}_{2}$  ⋯  ${e}_{n}$ 
${\kappa}_{1}$  $({\mu}_{11},{\gamma}_{11},{\eta}_{11})$  $({\mu}_{12},{\gamma}_{12},{\eta}_{12})$  ⋯  $({\mu}_{1n},{\gamma}_{1n},{\eta}_{1n})$ 
${\kappa}_{2}$  $({\mu}_{21},{\gamma}_{21},{\eta}_{21})$  $({\mu}_{22},{\gamma}_{22},{\eta}_{22})$  ⋯  $({\mu}_{2n},{\gamma}_{2n},{\eta}_{2n})$ 
⋮  ⋮  ⋮  ⋱  ⋮ 
${\kappa}_{m}$  $({\mu}_{m1},{\gamma}_{m1},{\eta}_{m1})$  $({\mu}_{m2},{\gamma}_{m2},{\eta}_{m2})$  ⋯  $({\mu}_{mn},{\gamma}_{mn},{\eta}_{mn})$ 
 (i)
 ${T}_{1}\subseteq {T}_{2}$, and
 (ii)
 ${S}^{\left(1\right)}\left(e\right)$ is SFSSsubset of ${S}^{\left(2\right)}\left(e\right)$ for all $e\in {T}_{1}$.
3. Spherical Fuzzy Soft Set Topology
 (i)
 ${\varphi}_{E},{\stackrel{\u02d8}{X}}_{E}\phantom{\rule{3.33333pt}{0ex}}\tilde{\in}\stackrel{~}{\xf8}$;
 (ii)
 ${S}_{A},{S}_{B}\tilde{\in}\stackrel{~}{\xf8}$ then ${S}_{A}\tilde{\cap}{S}_{B}\phantom{\rule{3.33333pt}{0ex}}\tilde{\in}\stackrel{~}{\xf8}$;
 (iii)
 If ${S}_{{T}_{i}}\tilde{\in}\stackrel{~}{\xf8},\forall \phantom{\rule{3.33333pt}{0ex}}i\in I$, then ${\tilde{\cup}}_{i\in I}\phantom{\rule{3.33333pt}{0ex}}{S}_{{T}_{i}}\phantom{\rule{3.33333pt}{0ex}}\tilde{\in}\stackrel{~}{\xf8}$.
 (1)
 SFSS interior:The interior ${S}_{A}^{\circ}$ of ${S}_{A}$ is the SFSS extended union of all SFSSopen subsets of ${S}_{A}$. Note that ${S}_{A}^{\circ}$ is the largest SFSSopen subset of ${S}_{A}$.
 (2)
 SFSS closure:The closure $\overline{{S}_{A}}$ of ${S}_{A}$ is the SFSS restricted intersection of all SFSSclosed supersets of ${S}_{A}$. Note that $\overline{{S}_{A}}$ is the smallest SFSSclosed superset of ${S}_{A}$.
 (3)
 SFSS frontier:The boundary or frontier $Fr\left({S}_{A}\right)$ of ${S}_{A}$ is defined as$$Fr\left({S}_{A}\right)=\overline{{S}_{A}}\phantom{\rule{3.33333pt}{0ex}}\tilde{\cap}\phantom{\rule{3.33333pt}{0ex}}\overline{{S}_{A}^{c}}$$
 (4)
 SFSS exterior:The exterior $Ext\left({S}_{A}\right)$ of ${S}_{A}$ is defined as$$Ext\left({S}_{A}\right)={\left({S}_{A}^{c}\right)}^{\circ}$$
 $int\left({\varphi}_{E}\right)={\varphi}_{E}$ and $int\left({\stackrel{\u02d8}{X}}_{E}\right)={\stackrel{\u02d8}{X}}_{E}$.
 $int\left({S}_{A}\right)\subseteq {S}_{A}$.
 A is an SFSS open set ⇔${S}_{A}=int\left({S}_{A}\right)$.
 $int\left(int\left({S}_{A}\right)\right)=int\left({S}_{A}\right).$
 ${S}_{A}\subseteq {S}_{B}\Rightarrow int\left({S}_{A}\right)\subseteq int\left({S}_{B}\right).$
 $int\left({S}_{A}\right)\cup int\left({S}_{B}\right)\subseteq int({S}_{A}\cup {S}_{B}).$
 $int({S}_{A}\cap {S}_{B})=int\left({S}_{A}\right)\cap int\left({S}_{B}\right)$.
 This is obvious by Definition 24.
 This is obvious by Definition 24.
 If ${S}_{A}$ is an SFSS open set in X, then ${S}_{A}$ is itself an SFSS open set in X which contains ${S}_{A}$. Therefore, ${S}_{A}$ itself is the largest SFSS open set contained in ${S}_{A}$ and $int\left({S}_{A}\right)={S}_{A}$. Conversely, suppose that $int\left({S}_{A}\right)={S}_{A}$. Since $int\left({S}_{A}\right)$ is always SFSS open, ${S}_{A}$ must be SFSS open.
 Let $int\left({S}_{A}\right)={S}_{B}$. Then, $int\left({S}_{B}\right)={S}_{B}$ from (3) and then, $int\left(int\left({S}_{A}\right)\right)=int\left({S}_{A}\right)$.
 Consider ${S}_{A}\subseteq {S}_{B}$ as $int\left({S}_{A}\right)\subseteq {S}_{A}\subseteq {S}_{B};\phantom{\rule{3.33333pt}{0ex}}int\left({S}_{A}\right)$ is an SFSS open subset of ${S}_{B}$, so, by the definition, we have that $int\left({S}_{A}\right)\subseteq int\left({S}_{B}\right)$.
 It is clear that ${S}_{A}\subseteq {S}_{A}\cup {S}_{B}$ and ${S}_{B}\subseteq {S}_{A}\cup {S}_{B}$. Thus, $int\left({S}_{A}\right)\subseteq int({S}_{A}\cup {S}_{B})$ and $int\left({S}_{B}\right)\subseteq int({S}_{A}\cup {S}_{B})$. Therefore, we have that $int\left({S}_{A}\right)\cup int\left({S}_{B}\right)\subseteq int({S}_{A}\cup {S}_{B})$, using 5.
 It is known that $int({S}_{A}\cap {S}_{B})\subseteq int\left({S}_{A}\right)$ and $int({S}_{A}\cap {S}_{B})\subseteq int\left({S}_{B}\right)$ by 5. Therefore, that $int({S}_{A}\cap {S}_{B})\subseteq int\left({S}_{A}\right)\cap int\left({S}_{B}\right)$. In addition, from $int\left({S}_{A}\right)\subseteq {S}_{A}$ and $int\left({S}_{B}\right)\subseteq {S}_{B}$, we have $int\left({S}_{A}\right)\cap int\left({S}_{B}\right)\subseteq {S}_{A}\cap {S}_{B}$. These imply that $int({S}_{A}\cap {S}_{B})=int\left({S}_{A}\right)\cap int\left({S}_{B}\right)$
 $cl({\varphi}_{E})={\varphi}_{E}$ and $cl\left({\stackrel{\u02d8}{X}}_{E}\right)={\stackrel{\u02d8}{X}}_{E}$;
 ${S}_{A}\subseteq cl\left({S}_{A}\right)$;
 A is an SFSS closed set ⇔${S}_{A}=cl\left({S}_{A}\right)$;
 $cl\left(cl\left({S}_{A}\right)\right)=cl\left({S}_{A}\right)$;
 ${S}_{A}\subseteq {S}_{B}\Rightarrow cl\left({S}_{A}\right)\subseteq cl\left({S}_{B}\right)$;
 $cl\left({S}_{A}\right)\cup int\left({S}_{B}\right)=cl\left({S}_{A}\right)\cup cl\left({S}_{B}\right)$;
 $cl({S}_{A}\cap {S}_{B})\subseteq cl\left({S}_{A}\right)\cap cl\left({S}_{B}\right)$.
 (1)
 ${\left({S}_{A}^{\circ}\right)}^{c}=\overline{\left({S}_{A}^{c}\right)}$, and
 (2)
 ${\left(\overline{{S}_{A}}\right)}^{c}={\left({S}_{A}^{c}\right)}^{\circ}$.
4. SFSSSeparation Axioms
5. MCDGM by Using SFSS Information
Algorithm 1 Choice value method. 

6. AHPTOPSIS Approach for Environmental Mitigation with SFSTopology
 Case studyEnvironmental degradation in Pakistan comprises air quality, water contamination, traffic noise, global warming, chemical abuse, desertification, natural catastrophes, dunes, and storms. A worldwide environmentalperformance index (EPI) has previously labelled Pakistan’s air quality as deplorable. Global warming poses a serious threat to the lives of the citizens in the state. Emissions of greenhouse gases, increasing urbanization, and deforestation all play a role in the current state of affairs. Climate change is running amok in lowincome countries such as Pakistan. It is not alone in being helpless, as advanced economies—most notably China and the United States—postpone lowering emissions. Global warming will have a significant impact on Pakistan, as well as the Maldives and many other island nations. In contrast to many other countries that have addressed the issue of global emissions at the UN, Pakistan is doing little to safeguard its future. Regular agricultural cycles have helped Pakistan’s economy weather several crises. However, if the IPCC Article is accurate, the country will be underwater by 2050. Already, Pakistan is struggling with a slew of environmental challenges. Many lives have been lost due to weather extremes, which also have a major impact on crop cycles and harvests. Floods decimated Pakistan’s two main cities this year: Karachi and Islamabad. Due to landslides, Pakistan’s commercial lifeline with China, the 806kilometer Karakoram Highway, was shut down many times for multiple days. There was considerable deforestation in the northern part of Kohistan and the southern part of Jaglot, which led to the deadly landslides. The logging mafias are swiftly clearing oldgrowth forests north of Shimshal and east of the Skardu Valley, virtually insuring future environmental catastrophes. The state government appears utterly unconcerned about the looming crisis. Not much effort has been put towards meeting its goal of producing 60% of its power from renewable sources by 2030. More than 60% of the country’s electricity is generated from fossil fuels at the moment. Figure 3 shows the key environmental issues in Pakistan.
 Economic ramifications of environmental devastationAgriculture and fishing employ more than twothirds of the workforce in Pakistan and produce over a quarter of the country’s total output. Increased use of finite natural resources is necessary for economic growth. Oddly, the very thing that is enabling this country to develop also constitutes a danger to its longterm safety and stability. A total of 70 percent of Pakistan’s population lives in rural areas and suffers from high poverty levels, according to the World Bank. To make money, these people rely on utilization and conservation, which they tend to misuse. This leads to greater ecological damage, which in turn, enhances impoverishment. This has culminated in a “vicious downward spiral of impoverishment and environmental degradation,” as stated by the World Bank. Pollutionrelated wellness factors influence both urban and rural dwellers, as per a 2013 World Bank evaluation. Air quality is the state’s most critical environmental challenge. Not only do these global impacts harm Pakistanis, but they often put the country’s business in jeopardy. In the article, growing industrialization, globalization, and vehicular use are anticipated to aggravate the situation.
 Water pollutionPakistan is rated as a waterstressed country by the World Economic forum. The Kabul River flows from Afghanistan into Pakistan; whereas the Indus, Jhelum, Chenab, Ravi, and Sutlej Rivers flow from India into Pakistan. Under the Indus Waters Treaty of 1960, water from the Ravi and Sutlej waters is redirected upstream to India for household consumption. The Indus (main stem), Jhelum, and Chenab rivers supply water to the agricultural lands of Punjab and Sindh, though not to the remainder of the region. Pakistan’s economy and the welfare of millions of Pakistanis are strongly effected by resource depletion. With the Law of the Sea Convention and canal diversion, Pakistan’s rivers have fewer diluting flows. The size of the economy, as well as a lack of water treatment, have caused a spike in water pollution. To provide water to people, dumped raw sewage is drained into rivers and the ocean, and unsanitary pipes are used. Water contamination makes it increasingly challenging to acquire safe drinking water and elevates the likelihood of developing an illness carried through raw sewage. There are many ailments that may be largely attributed to filthy water in Pakistan, because of this. Indeed, 45 percent of infant deaths and 60 percent of aquatic infections are caused by diarrhoea.
 Noise pollutionSome of Pakistan’s urbanized areas are plagued by a substantial amount of noise pollution. Noise pollution is generally triggered by traffic, including vehicles, automobiles, lorries, wheelers, and water tankers. An analysis found that Karachi’s main route seemed to have an average noise level of 90 dB and may reach as high as 110 dB. As a matter of fact, this surpasses the 70 dB limit set by the “International Organization for Standardization (ISO)”. According to the studies, the Environmental Quality Agency’s ambient noise standard in Pakistan is 85 decibels (dB). This threshold of noise pollution might have an influence on both auditory and quasi abilities. There seems to be a diversity of nonauditory clinical depression, notably insomnia, hearing and myocardial sickness, neuroendocrine sensitivity to loudness, and mental disorders. There are only a few, inconsistent noise regulations and policies in place. There is no culpability and the municipal and regional environmental conservation agencies are unable to intervene due to various statutory limitations and a loss of specific norms and regulations, which hinders them from doing so.
 Air pollutionWellbeing has been shown to be disproportionately affected by air pollution. For Pakistanis who habitually inhale dirty air, nanoparticle matter variations are a big concern. Respiratory difficulties have been associated to SPM in Pakistan’s largest cities, according to the research. Sustainable fuels such as liquefied petroleum gas (LPG) and improved transportation construction and sustainability can significantly minimize urban air pollution in Pakistan. The government can also adopt mitigation policies to reduce emissions. Pollution levels are increasing in Pakistan’s metro areas. Karachi’s urban air pollution is one of the worst in the world, having a devastating impact on both human development and health. Unsustainable energy use combined with the increased utilisation of automobiles, unauthorized corporate emissions, and debris and polymer combustion have all contributed to urban air pollution. According to the Sindh Environmental Conservation Department, urban air quality is approximately four times that proposed by the World Health Organization. These contaminants contribute to “respiratory disorders, impaired vision, vegetation degradation, and crop production”. Economic production leads to air pollution. An unavoidable byproduct and insufficient air pollution legislation have led to cities’ poor air quality. Abid Omar founded the Pakistan ambient air initiative in 2018, to evaluate the country’s major cities’ air quality. In Pakistan, the US State Department has established three elevated airquality monitoring units. To overcome such environmental issues, various aspects should be taken under view and numerous efforts are required in order to obtain environmentally friendly conditions. Some are listed below:
 The establishment of a large tree plantation.
 Going paperless has the potential to significantly reduce the rate of deforestation on Earth.
 The number of dieselpowered automobiles that pollute the atmosphere should be reduced.
 An effective system for treating and managing sewage should be put in place.
 The practice of living a waterconserving lifestyle should be encouraged.
One of the most crucial objectives is to achieve an appropriate and effective level of environmental remediation and protection. Policymakers and decision makers must acknowledge that a sustainable plan for solving global crises must be a consistent effort comes from a long approach that combines all stakeholders. Inevitably, the project’s performance is determined by organizational commitment and dedication at every phase of the process, in addition to endorsement of adequate systems and guidelines at all levels. A method to tackle the ecological disaster has several merits, some of which are listed below. Recovering a susceptible and priceless expedient.
 Increasing the efficacy of currently available systems.
 Exploiting infrastructure’s massive financial assets.
 Extending the systems’ average life duration.
 Increasing revenue from environmental mitigation services.
 Energydemand reduction.
 Decrease in the service’s carbon footprint.
In order to locate and accentuate the finest solution to environmental issues, a thorough structure of tactics is developed in this study. To be sustainable, the plan chosen must be in harmony with the ecological sector’s integrated approach. Rather than a laborious process of making recommendations that account for the fact that many particular objectives and opportunities exist in the market, a wellorganized approach that can be articulated promptly and succinctly must account for concerns of various individuals and those of the constitutionally sound authorities. Decisions are being made by legislators and selection analysts who are wellversed in the process. A review of the literature on environmental strategic planning undertaken with professionals and authorities, as well as information concerning the region of convenience’s domestic life, resulted in the improvement of these measures. Climateremediation approaches were applied in the environmental distribution network. When a longterm ecological safeguard system exists, clear provisions are often in place. Appraisal attributes are used to assess the effectiveness of each methodology. To choose the optimal method, first, the critical nature of grading parameters should be understood. Attributes are given in Table 5 as the strategies to overcome environmental crises, and judgement criteria are given in Table 6.The linguistic terms for judging alternatives are listed in Table 7.
 Intermediate values for two consecutive linguistic terms will be $\left(I{V}_{1}\right),\left(I{V}_{2}\right),\left(I{V}_{3}\right),$$\left(I{V}_{4}\right)$ as 2, 4, 6, 8, respectively
 Values for inverse composition for each linguistic term will be the reciprocal of its score index.
Algorithm 2 AHPTOPSIS. 

 Step 1
 Let $A=\{{A}_{1},\xb7\xb7\xb7,{A}_{7}\}$ be the collection of alternatives and $C=\{{\stackrel{\xb4}{C}}_{1},\xb7\xb7\xb7,{\stackrel{\xb4}{C}}_{6}\}$ be the collection of evaluation criteria as given in Table 3 and Table 4, respectively. We will use the set $D=\{{D}_{1},\xb7\xb7\xb7,{D}_{4}\}$ to refer to a group of policymakers / decision makers who have been asked to score each approach on the basis of how well it meets each of the evaluation criteria in terms of SFNs.
 Step 2
 Step 3
 Then, we obtained normalized pairwise comparison matrix, which is given in Table 10.
 Step 4
 Therefore, the required criteria weights are calculated, as shown in Table 11.
 Step 5
 Criteria weights are consistent, as they fulfil the requirement that $CR<0.10$.
 Step 6
 The evaluations of decision makers in terms of decision matrices ${D}_{1}$, ${D}_{2}$, ${D}_{3}$, and ${D}_{4}$ are expressed in Table 12, Table 13, Table 14, and Table 15, respectively. The rows represent the alternatives and the columns represent the parameters in these matrices. Then, the collection of decision matrices forms an SFSStopology.As a result, we arrive at an aggregated decision matrix that looks like Table 16, computed by using $\frac{{D}_{1}+{D}_{2}+{D}_{3}+{D}_{4}}{4}$
 Step 7
 Then, we calculated the weighted SFSS decision matrix, given in Table 17.
 Step 8
 Then, we obtained the SFSSvalued positive ideal solution (SFSVPIS) and SFSSvalued negative ideal solution (SFSVNIS).$SFSVPIS=\{{\ddot{\wp}}_{1}^{+},{\ddot{\wp}}_{2}^{+},{\ddot{\wp}}_{3}^{+},{\ddot{\wp}}_{4}^{+},{\ddot{\wp}}_{5}^{+},{\ddot{\wp}}_{6}^{+}\}\phantom{\rule{0ex}{0ex}}=\left\{(0.2044,0.0168,0.1332),(0.4081,0.0196,0.0651),(0.1381,0.0121,0.0788),\phantom{\rule{0ex}{0ex}}(0.0421,0.0023,0.0291),(0.0348,0.0045,0.0226),(0.0566,0.0060,0.0319)\right\}$.$SFSVNIS=\{{\ddot{\wp}}_{1}^{},{\ddot{\wp}}_{2}^{},{\ddot{\wp}}_{3}^{},{\ddot{\wp}}_{4}^{}\phantom{\rule{0.166667em}{0ex}}{\ddot{\wp}}_{5}^{},{\ddot{\wp}}_{6}^{}\}\phantom{\rule{0ex}{0ex}}=\left\{(0.1323,0.0486,0.1734),(0.0626,0.0326,0.0860),(0.0861,0.0368,0.1118),\phantom{\rule{0ex}{0ex}}(0.0289,0.0142,0.0362),(0.0242,0.0097,0.0287),(0.0400,0.0281,0.0444)\right\}$.
 Step 9
 Step 10
 Each alternative was compared to the ideal solution in Table 20, in order to compute its closeness coefficient.
 Step 11
 The preference order of the alternatives, therefore, is$${\ddot{v}}_{2}\succ {\ddot{v}}_{1}\succ {\ddot{v}}_{7}\succ {\ddot{v}}_{4}\succ {\ddot{v}}_{5}\succ {\ddot{v}}_{3}\succ {\ddot{v}}_{6}$$
7. Conclusions
Author Contributions
Funding
Institutional Review Board Statement
Informed Consent Statement
Data Availability Statement
Conflicts of Interest
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Fuzzy Models  $\mathit{\mu}$  $\mathit{\gamma}$  $\mathit{\eta}$  Constraints 

Fuzzy set (FS) [7]  🗸  ×  ×  An FS deals with vagueness 
in terms of $\mu $ with $0\le \mu \le 1$  
Intuitionistic fuzzy set  🗸  ×  🗸  An IFS assigns a pair of PMD and 
(IFS) [8]  NMD with $0\le \mu +\eta \le 1$  
Pythagorean fuzzy set  🗸  ×  🗸  A PFS assigns a pair of PMD and 
(PFS) [9,10]  NMD with $0\le {\mu}^{2}+{\eta}^{2}\le 1$  
qRung orthopair fuzzy set  🗸  ×  🗸  A qROFS assigns a pair of PMD and 
(qROFS) [11]  NMD with $0\le {\mu}^{q}+{\eta}^{q}\le 1,q\ge 1$  
Neutrosophic set  🗸  🗸  🗸  An NS assigns three indexes, truthness T, 
(NS) [12]  indeterminacy I, and falsity F, with  
$T,I,F\in ]{0}^{},{1}^{+}[$, $T+I+F\in [{0}^{},{3}^{+}]$  
Singlevalued neutrosophic  🗸  🗸  🗸  An NS assigns three indexes, truthness T, 
set (SVNS) [13]  indeterminacy I, and falsity F, with  
$T,I,F\in [0,1]$, $T+I+F\in [0,3]$  
Picture fuzzy set  🗸  🗸  🗸  A PFS assigns PMD, ND, and NMD, 
(PFS) [14,15,16]  such that $0\le \mu +\gamma +\eta \le 1$  
Spherical fuzzy set  🗸  🗸  🗸  A PFS assigns PMD, ND, and NMD, 
(SFS) [17,18,19]  such that $0\le {\mu}^{2}+{\gamma}^{2}+{\eta}^{2}\le 1$ 
Models  Researchers  Applications 

CrispTOPSIS  Hwang and Yoon [50]  The fighter aircraft problem 
FuzzyTOPSIS  Chen [51]  Selection of a systemanalysis engineer 
BFTOPSIS  Akram et al. [52]  Skin disorder diagnosis 
FSSTOPSIS  Eraslan and Karaaslan [53]  Selection of a house 
IFSSTOPSIS  Garg and Arora [54]  Supplierselection problem 
SFTOPSIS  Kahraman et al. [55]  Selection of a hospital location 
PmpFTOPSIS  Naeem et al. [56]  Selection of an advertisement mode 
HFSTOPSIS  Senvar et al. [57]  Hospitalsite selection 
PFTOPSIS  Zhang and Xu [58]  MCDM based on PFSs to examine efficiency among domestic airlines 
TOPSIS  Kahraman et al. [59]  Ranking of alternatives for location problem in supplychain management 
${\mathit{S}}_{\mathit{E}}$  ${\mathit{p}}_{1}$  ${\mathit{p}}_{2}$  ${\mathit{p}}_{3}$  ${\mathit{p}}_{4}$  ${\mathit{p}}_{5}$  ${\mathit{p}}_{6}$ 

${x}_{1}$  (0.469, 0.131, 0.630)  (0.589, 0.128, 0.338)  (0.811, 0.008, 0.213)  (0.638, 0.213, 0.419)  (0.291, 0.316, 0.362)  (0.429, 0.214, 0.586) 
${x}_{2}$  (0.234, 0.346, 0.189)  (0.000, 0.000, 1.000)  (0.783, 0.132, 0.189)  (0.789, 0.102, 0.289)  (0.278, 0.118, 0.346)  (0.000, 0.000, 1.000) 
${x}_{3}$  (0.271, 0.213, 0.348)  (0.769, 0.139, 0.169)  (0.000, 0.000, 1.000)  (0.532, 0.243, 0.411)  (0.291, 0.381, 0.293)  (0.781, 0.131, 0.639) 
${x}_{4}$  (0.795, 0.142, 0.231)  (0.249, 0.321, 0.256)  (0.330, 0.142, 0.479)  (0.359, 0.134, 0.651)  (0.594, 0.287, 0.367)  (0.801, 0.095, 0.121) 
${x}_{5}$  (0.256, 0.389, 0.180)  (0.393, 0.102, 0.597)  (0.435, 0.134, 0.596)  (0.795, 0.112, 0.280)  (0.000, 0.000, 1.000)  (0.286, 0.327, 0.179) 
${x}_{6}$  (0.692, 0.134, 0.128)  (0.643, 0.260, 0.189)  (0.000, 0.000, 1.000)  (0.279, 0.321, 0.340)  (0.788, 0.103, 0.211)  (0.327, 0.256, 0.441) 
${x}_{7}$  (0.297, 0.216, 0.310)  (0.781, 0.118, 0.171)  (0.181, 0.310, 0.490)  (0.497, 0.115, 0.324)  (0.237, 0.310, 0.212)  (0.505, 0.123, 0.486) 
X  $\mathit{S}={\mathit{\mu}}^{2}{\mathit{\eta}}^{2}$  Ranking 

${x}_{1}$  $0.1322$  2 
${x}_{2}$  $0.0436$  7 
${x}_{3}$  $0.2080$  6 
${x}_{4}$  $0.0548$  3 
${x}_{5}$  $0.2010$  5 
${x}_{6}$  $0.1749$  1 
${x}_{7}$  $0.0367$  4 
Code  Strategies 

${A}_{1}$  Forest conservation 
${A}_{2}$  Disaster mitigation 
${A}_{3}$  Environmental legislation 
${A}_{4}$  Eliminate the use of fossilfuel vehicles 
${A}_{5}$  Eliminate singleuse plastics 
${A}_{6}$  Agriculture that is sustainable 
${A}_{7}$  Mitigation of environmental aspects of aviation 
Code  Strategies  Explanation 

${\stackrel{\xb4}{C}}_{1}$  Cost figure  Expenditure associated with the implementation of the criteria 
${\stackrel{\xb4}{C}}_{2}$  Benefit period  Calculation of the effective life span of the criteria 
${\stackrel{\xb4}{C}}_{3}$  Energy Saved  For a solution to be viable, it must be able to cut energy consumption and globalwarming emissions. 
${\stackrel{\xb4}{C}}_{4}$  Supply reliability  The criteria may be preferable if it is capable of saving a longterm service and easing supply constraints. 
${\stackrel{\xb4}{C}}_{5}$  Flexibility  The criteria should be tailored to meet diverse needs and uncertainties in order to be more flexible. 
${\stackrel{\xb4}{C}}_{6}$  Social acceptance  If the criteria has ability to be accepted by the localities. 
Linguistic Terms  Score Index 

Equal importance (EI)  1 
Moderate importance (MI)  3 
Strong important (SI)  5 
Verystrong importance (VI)  7 
Extreme importance (EXI)  9 
${\stackrel{\xb4}{\mathit{C}}}_{1}$  ${\stackrel{\xb4}{\mathit{C}}}_{2}$  ${\stackrel{\xb4}{\mathit{C}}}_{3}$  ${\stackrel{\xb4}{\mathit{C}}}_{4}$  ${\stackrel{\xb4}{\mathit{C}}}_{4}$  ${\stackrel{\xb4}{\mathit{C}}}_{6}$  

${\stackrel{\xb4}{C}}_{1}$  $EI$  $I{V}_{2}$  $VI$  $MI$  $SI$  $I{V}_{1}$ 
${\stackrel{\xb4}{C}}_{2}$  $I{V}_{2}$  $EI$  $I{V}_{1}$  $VI$  $I{V}_{2}$  $SI$ 
${\stackrel{\xb4}{C}}_{3}$  $VI$  $I{V}_{1}$  $EI$  $MI$  $I{V}_{3}$  $EXI$ 
${\stackrel{\xb4}{C}}_{4}$  $MI$  $VI$  $MI$  $EI$  $SI$  $VI$ 
${\stackrel{\xb4}{C}}_{5}$  $SI$  $I{V}_{2}$  $I{V}_{2}$  $SI$  $EI$  $I{V}_{2}$ 
${\stackrel{\xb4}{C}}_{6}$  $I{V}_{1}$  $SI$  $EXI$  $VI$  $I{V}_{2}$  $EI$ 
${\stackrel{\xb4}{\mathit{C}}}_{1}$  ${\stackrel{\xb4}{\mathit{C}}}_{2}$  ${\stackrel{\xb4}{\mathit{C}}}_{3}$  ${\stackrel{\xb4}{\mathit{C}}}_{4}$  ${\stackrel{\xb4}{\mathit{C}}}_{4}$  ${\stackrel{\xb4}{\mathit{C}}}_{6}$  

${\stackrel{\xb4}{C}}_{1}$  $1.000$  $4.000$  $7.000$  $3.000$  $5.000$  $2.000$ 
${\stackrel{\xb4}{C}}_{2}$  $0.250$  $1.000$  $0.500$  $7.000$  $4.000$  $5.000$ 
${\stackrel{\xb4}{C}}_{3}$  $0.143$  $2.000$  $1.000$  $3.000$  $6.000$  $9.000$ 
${\stackrel{\xb4}{C}}_{4}$  $0.333$  $0.143$  $0.333$  $1.000$  $5.000$  $0.143$ 
${\stackrel{\xb4}{C}}_{5}$  $0.200$  $0.250$  $0.167$  $0.200$  $1.000$  $4.000$ 
${\stackrel{\xb4}{C}}_{6}$  $0.500$  $0.200$  $0.111$  $7.000$  $0.250$  $1.000$ 
${\stackrel{\xb4}{\mathit{C}}}_{1}$  ${\stackrel{\xb4}{\mathit{C}}}_{2}$  ${\stackrel{\xb4}{\mathit{C}}}_{3}$  ${\stackrel{\xb4}{\mathit{C}}}_{4}$  ${\stackrel{\xb4}{\mathit{C}}}_{4}$  ${\stackrel{\xb4}{\mathit{C}}}_{6}$  

${\stackrel{\xb4}{C}}_{1}$  $0.243$  $0.527$  $0.768$  $0.142$  $0.235$  $0.095$ 
${\stackrel{\xb4}{C}}_{2}$  $0.061$  $0.132$  $0.055$  $0.330$  $0.188$  $0.236$ 
${\stackrel{\xb4}{C}}_{3}$  $0.035$  $0.263$  $0.110$  $0.142$  $0.282$  $0.426$ 
${\stackrel{\xb4}{C}}_{4}$  $0.081$  $0.019$  $0.037$  $0.047$  $0.235$  $0.007$ 
${\stackrel{\xb4}{C}}_{5}$  $0.049$  $0.033$  $0.018$  $0.009$  $0.047$  $0.189$ 
${\stackrel{\xb4}{C}}_{6}$  $0.122$  $0.026$  $0.012$  $0.330$  $0.012$  $0.047$ 
${\stackrel{\xb4}{C}}_{1}$  $0.335$ 
${\stackrel{\xb4}{C}}_{2}$  $0.167$ 
${\stackrel{\xb4}{C}}_{3}$  $0.210$ 
${\stackrel{\xb4}{C}}_{4}$  $0.071$ 
${\stackrel{\xb4}{C}}_{5}$  $0.058$ 
${\stackrel{\xb4}{C}}_{6}$  $0.092$ 
${\stackrel{\xb4}{\mathit{C}}}_{1}$  ${\stackrel{\xb4}{\mathit{C}}}_{2}$  ${\stackrel{\xb4}{\mathit{C}}}_{3}$  ${\stackrel{\xb4}{\mathit{C}}}_{4}$  ${\stackrel{\xb4}{\mathit{C}}}_{4}$  ${\stackrel{\xb4}{\mathit{C}}}_{6}$  

${A}_{1}$  $(0.82,0.03,0.21)$  $(0.19,0.35,0.28)$  $(0.79,0.16,0.31)$  $(0.68,0.21,0.40)$  $(0.24,0.39,0.41)$  $(0.65,0.13,0.37)$ 
${A}_{2}$  $(0.16,0.07,0.78)$  $(0.41,0.23,0.56)$  $(0.34,0.19,0.59)$  $(0.42,0.32,0.38)$  $(0.65,0.17,0.39)$  $(0.47,0.21,0.51)$ 
${A}_{3}$  $(0.52,0.28,0.37)$  $(0.45,0.31,0.36)$  $(0.17,0.38,0.41)$  $(0.34,0.47,0.29)$  $(0.46,0.11,0.52)$  $(0.62,0.18,0.25)$ 
${A}_{4}$  $(0.47,0.32,0.38)$  $(0.27,0.47,0.38)$  $(0.80,0.04,0.36)$  $(0.35,0.41,0.28)$  $(0.71,0.13,0.29)$  $(0.56,0.29,0.34)$ 
${A}_{5}$  $(0.73,0.18,0.26)$  $(0.19,0.37,0.49)$  $(0.47,0.21,0.52)$  $(0.61,0.24,0.38)$  $(0.52,0.32,0.47)$  $(0.43,0.24,0.26)$ 
${A}_{6}$  $(0.43,0.21,0.35)$  $(0.27,0.36,0.53)$  $(0.61,0.28,0.31)$  $(0.71,0.09,0.25)$  $(0.15,0.39,0.53)$  $(0.67,0.21,0.19)$ 
${A}_{7}$  $(0.83,0.04,0.19)$  $(0.38,0.27,0.57)$  $(0.46,0.19,0.52)$  $(0.75,0.18,0.26)$  $(0.59,0.28,0.37)$  $(0.37,0.28,0.49)$ 
${\stackrel{\xb4}{\mathit{C}}}_{1}$  ${\stackrel{\xb4}{\mathit{C}}}_{2}$  ${\stackrel{\xb4}{\mathit{C}}}_{3}$  ${\stackrel{\xb4}{\mathit{C}}}_{4}$  ${\stackrel{\xb4}{\mathit{C}}}_{4}$  ${\stackrel{\xb4}{\mathit{C}}}_{6}$  

${A}_{1}$  $(0.00,0.00,1.00)$  $(0.00,0.00,1.00)$  $(0.00,0.00,1.00)$  $(0.00,0.00,1.00)$  $(0.00,0.00,1.00)$  $(0.00,0.00,1.00)$ 
${A}_{2}$  $(0.00,0.00,1.00)$  $(0.00,0.00,1.00)$  $(0.00,0.00,1.00)$  $(0.00,0.00,1.00)$  $(0.00,0.00,1.00)$  $(0.00,0.00,1.00)$ 
${A}_{3}$  $(0.00,0.00,1.00)$  $(0.00,0.00,1.00)$  $(0.00,0.00,1.00)$  $(0.00,0.00,1.00)$  $(0.00,0.00,1.00)$  $(0.00,0.00,1.00)$ 
${A}_{4}$  $(0.00,0.00,1.00)$  $(0.00,0.00,1.00)$  $(0.00,0.00,1.00)$  $(0.00,0.00,1.00)$  $(0.00,0.00,1.00)$  $(0.00,0.00,1.00)$ 
${A}_{5}$  $(0.00,0.00,1.00)$  $(0.00,0.00,1.00)$  $(0.00,0.00,1.00)$  $(0.00,0.00,1.00)$  $(0.00,0.00,1.00)$  $(0.00,0.00,1.00)$ 
${A}_{6}$  $(0.00,0.00,1.00)$  $(0.00,0.00,1.00)$  $(0.00,0.00,1.00)$  $(0.00,0.00,1.00)$  $(0.00,0.00,1.00)$  $(0.00,0.00,1.00)$ 
${A}_{7}$  $(0.00,0.00,1.00)$  $(0.00,0.00,1.00)$  $(0.00,0.00,1.00)$  $(0.00,0.00,1.00)$  $(0.00,0.00,1.00)$  $(0.00,0.00,1.00)$ 
${\stackrel{\xb4}{\mathit{C}}}_{1}$  ${\stackrel{\xb4}{\mathit{C}}}_{2}$  ${\stackrel{\xb4}{\mathit{C}}}_{3}$  ${\stackrel{\xb4}{\mathit{C}}}_{4}$  ${\stackrel{\xb4}{\mathit{C}}}_{4}$  ${\stackrel{\xb4}{\mathit{C}}}_{6}$  

${A}_{1}$  $(0.62,0.21,0.38)$  $(0.39,0.41,0.28)$  $(0.84,0.07,0.19)$  $(0.19,0.37,0.64)$  $(0.48,0.25,0.53)$  $(0.23,0.48,0.51)$ 
${A}_{2}$  $(0.70,0.18,0.27)$  $(0.53,0.24,0.36)$  $(0.63,0.15,0.31)$  $(0.34,0.28,0.59)$  $(0.61,0.14,0.32)$  $(0.63,0.17,0.28)$ 
${A}_{3}$  $(0.53,0.23,0.46)$  $(0.19,0.46,0.51)$  $(0.47,0.32,0.35)$  $(0.56,0.19,0.43)$  $(0.37,0.41,0.46)$  $(0.46,0.21,0.43)$ 
${A}_{4}$  $(0.43,0.26,0.37)$  $(0.25,0.31,0.42)$  $(0.76,0.25,0.24)$  $(0.28,0.39,0.51)$  $(0.21,0.42,0.38)$  $(0.18,0.36,0.59)$ 
${A}_{5}$  $(0.35,0.29,0.44)$  $(0.46,0.39,0.49)$  $(0.39,0.17,0.61)$  $(0.51,0.08,0.48)$  $(0.75,0.19,0.10)$  $(0.31,0.27,0.48)$ 
${A}_{6}$  $(0.15,0.37,0.72)$  $(0.23,0.41,0.53)$  $(0.59,0.14,0.47)$  $(0.61,0.04,0.39)$  $(0.52,0.28,0.39)$  $(0.79,0.05,0.20)$ 
${A}_{7}$  $(0.54,0.16,0.61)$  $(0.17,0.49,0.35)$  $(0.52,0.27,0.41)$  $(0.62,0.19,0.40)$  $(0.81,0.16,0.19)$  $(0.59,0.26,0.38)$ 
${\stackrel{\xb4}{\mathit{C}}}_{1}$  ${\stackrel{\xb4}{\mathit{C}}}_{2}$  ${\stackrel{\xb4}{\mathit{C}}}_{3}$  ${\stackrel{\xb4}{\mathit{C}}}_{4}$  ${\stackrel{\xb4}{\mathit{C}}}_{4}$  ${\stackrel{\xb4}{\mathit{C}}}_{6}$  

${A}_{1}$  $(1.00,0.00,0.00)$  $(1.00,0.00,0.00)$  $(1.00,0.00,0.00)$  $(1.00,0.00,0.00)$  $(1.00,0.00,0.00)$  $(1.00,0.00,0.00)$ 
${A}_{2}$  $(1.00,0.00,0.00)$  $(1.00,0.00,0.00)$  $(1.00,0.00,0.00)$  $(1.00,0.00,0.00)$  $(1.00,0.00,0.00)$  $(1.00,0.00,0.00)$ 
${A}_{3}$  $(1.00,0.00,0.00)$  $(1.00,0.00,0.00)$  $(1.00,0.00,0.00)$  $(1.00,0.00,0.00)$  $(1.00,0.00,0.00)$  $(1.00,0.00,0.00)$ 
${A}_{4}$  $(1.00,0.00,0.00)$  $(1.00,0.00,0.00)$  $(1.00,0.00,0.00)$  $(1.00,0.00,0.00)$  $(1.00,0.00,0.00)$  $(1.00,0.00,0.00)$ 
${A}_{5}$  $(1.00,0.00,0.00)$  $(1.00,0.00,0.00)$  $(1.00,0.00,0.00)$  $(1.00,0.00,0.00)$  $(1.00,0.00,0.00)$  $(1.00,0.00,0.00)$ 
${A}_{6}$  $(1.00,0.00,0.00)$  $(1.00,0.00,0.00)$  $(1.00,0.00,0.00)$  $(1.00,0.00,0.00)$  $(1.00,0.00,0.00)$  $(1.00,0.00,0.00)$ 
${A}_{7}$  $(1.00,0.00,0.00)$  $(1.00,0.00,0.00)$  $(1.00,0.00,0.00)$  $(1.00,0.00,0.00)$  $(1.00,0.00,0.00)$  $(1.00,0.00,0.00)$ 
${\stackrel{\xb4}{\mathit{C}}}_{1}$  ${\stackrel{\xb4}{\mathit{C}}}_{2}$  ${\stackrel{\xb4}{\mathit{C}}}_{3}$  ${\stackrel{\xb4}{\mathit{C}}}_{4}$  ${\stackrel{\xb4}{\mathit{C}}}_{4}$  ${\stackrel{\xb4}{\mathit{C}}}_{6}$  

${A}_{1}$  $(0.6100,0.0600,0.3975)$  $(0.3950,0.1900,0.3900)$  $(0.6575,0.0575,0.3750)$  $(0.4675,0.1450,0.5100)$  $(0.4300,0.1600,0.4850)$  $(0.4700,0.305,0.4700)$ 
${A}_{2}$  $(0.4650,0.0625,0.5125)$  $(0.4850,0.1175,0.4800)$  $(0.4925,0.0850,0.4750)$  $(0.4400,0.1500,0.4925)$  $(0.5650,0.0775,0.4275)$  $(0.5250,0.0950,0.4475)$ 
${A}_{3}$  $(0.5125,0.1275,0.4575)$  $(0.4100,0.1925,0.4675)$  $(0.4100,0.1750,0.4400)$  $(0.4575,0.1300,0.4950)$  $(0.4575,0.1300,0.4950)$  $(0.5200,0.0975,0.4200)$ 
${A}_{4}$  $(0.4750,0.1450,0.4375)$  $(0.3800,0.1950,0.4500)$  $(0.6400,0.0725,0.4000)$  $(0.4075,0.2000,0.4475)$  $(0.4800,0.1375,0.4175)$  $(0.4350,0.1625,0.4825)$ 
${A}_{5}$  $(0.5200,0.1175,0.4250)$  $(0.4125,0.1900,0.4950)$  $(0.4650,0.0950,0.5325)$  $(0.5300,0.0800,0.4650)$  $(0.5675,0.1275,0.3925)$  $(0.4350,0.1275,0.435)$ 
${A}_{6}$  $(0.3950,0.1450,0.5175)$  $(0.3750,0.1925,0.5150)$  $(0.5500,0.1050,0.4450)$  $(0.5800,0.0325,0.4100)$  $(0.4175,0.1675,0.4800)$  $(0.6150,0.0650,0.3475)$ 
${A}_{7}$  $(0.5925,0.0500,0.4500)$  $(0.3875,0.1900,0.4800)$  $(0.4950,0.1150,0.4825)$  $(0.5925,0.0925,0.4150)$  $(0.6000,0.1100,0.3900)$  $(0.4900,0.1350,0.4675)$ 
${\stackrel{\xb4}{\mathit{C}}}_{1}$  ${\stackrel{\xb4}{\mathit{C}}}_{2}$  ${\stackrel{\xb4}{\mathit{C}}}_{3}$  ${\stackrel{\xb4}{\mathit{C}}}_{4}$  ${\stackrel{\xb4}{\mathit{C}}}_{4}$  ${\stackrel{\xb4}{\mathit{C}}}_{6}$  

${A}_{1}$  $(0.2044,0.0201,0.1332)$  $(0.0660,0.0317,0.0651)$  $(0.1381,0.0121,0.0788)$  $(0.0332,0.0103,0.0362)$  $(0.0249,0.0093,0.0281)$  $(0.0433,0.0281,0.0432)$ 
${A}_{2}$  $(0.1558,0.0209,0.1717)$  $(0.4081,0.0196,0.0802)$  $(0.1034,0.0179,0.0998)$  $(0.0312,0.0107,0.0350)$  $(0.0328,0.0045,0.0248)$  $(0.0483,0.0087,0.0412)$ 
${A}_{3}$  $(0.1717,0.0427,0.1533)$  $(0.0684,0.0321,0.0781)$  $(0.0861,0.0368,0.0924)$  $(0.0325,0.0092,0.0351)$  $(0.0266,0.0075,0.0287)$  $(0.0478,0.0090,0.0386)$ 
${A}_{4}$  $(0.1591,0.0486,0.1466)$  $(0.0635,0.0326,0.0752)$  $(0.1344,0.0152,0.0840)$  $(0.0289,0.0142,0.0318)$  $(0.0278,0.0080,0.0242)$  $(0.0400,0.0150,0.0444)$ 
${A}_{5}$  $(0.1742,0.0394,0.1424)$  $(0.0689,0.0317,0.0827)$  $(0.0977,0.0199,0.1118)$  $(0.0376,0.0057,0.0330)$  $(0.0329,0.0074,0.0228)$  $(0.0400,0.0117,0.0400)$ 
${A}_{6}$  $(0.1323,0.0486,0.1734)$  $(0.0626,0.0321,0.0860)$  $(0.1155,0.0221,0.0935)$  $(0.0412,0.0023,0.0291)$  $(0.0242,0.0097,0.0278)$  $(0.0566,0.0060,0.0319)$ 
${A}_{7}$  $(0.1984,0.0168,0.1508)$  $(0.0647,0.0317,0.0802)$  $(0.1040,0.0242,0.1013)$  $(0.0421,0.0066,0.0295)$  $(0.0348,0.0064,0.0226)$  $(0.0451,0.0124,0.0430)$ 
${d}_{1}^{+}={d}^{+}({\ddot{v}}_{1},\mathrm{SFSV}\text{}\mathrm{PIS})$  $0.3440$ 
${d}_{2}^{+}={d}^{+}({\ddot{v}}_{2},\mathrm{SFSV}\text{}\mathrm{PIS})$  $0.0785$ 
${d}_{3}^{+}={d}^{+}({\ddot{v}}_{3},\mathrm{SFSV}\text{}\mathrm{PIS})$  $0.3490$ 
${d}_{4}^{+}={d}^{+}({\ddot{v}}_{4},\mathrm{SFSV}\text{}\mathrm{PIS})$  $0.3510$ 
${d}_{5}^{+}={d}^{+}({\ddot{v}}_{5},\mathrm{SFSV}\text{}\mathrm{PIS})$  $0.3467$ 
${d}_{6}^{+}={d}^{+}({\ddot{v}}_{6},\mathrm{SFSV}\text{}\mathrm{PIS})$  $0.3589$ 
${d}_{7}^{+}={d}^{+}({\ddot{v}}_{7},\mathrm{SFSV}\text{}\mathrm{PIS})$  $0.3475$ 
${d}_{1}^{}={d}^{}({\ddot{v}}_{1},\mathrm{SFSV}\text{}\mathrm{NIS})$  $0.1119$ 
${d}_{2}^{}={d}^{}({\ddot{v}}_{2},\mathrm{SFSV}\text{}\mathrm{NIS})$  $0.3497$ 
${d}_{3}^{}={d}^{}({\ddot{v}}_{3},\mathrm{SFSV}\text{}\mathrm{NIS})$  $0.0545$ 
${d}_{4}^{}={d}^{}({\ddot{v}}_{4},\mathrm{SFSV}\text{}\mathrm{NIS})$  $0.0725$ 
${d}_{5}^{}={d}^{}({\ddot{v}}_{5},\mathrm{SFSV}\text{}\mathrm{NIS})$  $0.0619$ 
${d}_{6}^{}={d}^{}({\ddot{v}}_{6},\mathrm{SFSV}\text{}\mathrm{NIS})$  $0.0518$ 
${d}_{7}^{}={d}^{}({\ddot{v}}_{7},\mathrm{SFSV}\text{}\mathrm{NIS})$  $0.0850$ 
${C}_{1}^{}\left({\ddot{v}}_{1}\right)$  0.2454 
${C}_{2}^{}\left({\ddot{v}}_{2}\right)$  0.8167 
${C}_{3}^{}\left({\ddot{v}}_{3}\right)$  0.1351 
${C}_{4}^{}\left({\ddot{v}}_{4}\right)$  0.1712 
${C}_{5}^{}\left({\ddot{v}}_{5}\right)$  0.1515 
${C}_{6}^{}\left({\ddot{v}}_{6}\right)$  0.1261 
${C}_{7}^{}\left({\ddot{v}}_{7}\right)$  0.1965 
Models  Advantages and Limitations 

Soft set (SS) (Molodtsov [35])  An SS deals with uncertainty in terms of a parameterized collection of the subsets of the universe. 
It can not deal with spherical fuzzy information.  
Spherical fuzzy set (SFS) ([17,18,19])  It deals with spherical fuzzy information in terms of three indexes of PMD, ND, and NMD. 
It can not deal with parameterizations.  
Spherical fuzzy soft set ([82])  A strong hybrid model of SS and SFS to deal with uncertainty in terms of a parameterized collection of spherical fuzzy subsets. 
It defines classes of parameters and their approximate elements. 
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Riaz, M.; Tanveer, S.; Pamucar, D.; Qin, D.S. Topological Data Analysis with Spherical Fuzzy Soft AHPTOPSIS for Environmental Mitigation System. Mathematics 2022, 10, 1826. https://doi.org/10.3390/math10111826
Riaz M, Tanveer S, Pamucar D, Qin DS. Topological Data Analysis with Spherical Fuzzy Soft AHPTOPSIS for Environmental Mitigation System. Mathematics. 2022; 10(11):1826. https://doi.org/10.3390/math10111826
Chicago/Turabian StyleRiaz, Muhammad, Shaista Tanveer, Dragan Pamucar, and DongSheng Qin. 2022. "Topological Data Analysis with Spherical Fuzzy Soft AHPTOPSIS for Environmental Mitigation System" Mathematics 10, no. 11: 1826. https://doi.org/10.3390/math10111826