*Article* **Low-Cost Single-Frequency DGNSS/DBA Combined Positioning Research and Performance Evaluation**

**Shengliang Wang 1,2, Xianshu Dong 1,\*, Genyou Liu 2, Ming Gao 2,3, Wenhao Zhao 2,3, Dong Lv 2,3 and Shilong Cao 2,3**

**Abstract:** In recent years, low-cost single-frequency GNSS receivers have been widely used in many fields such as mass navigation and deformation monitoring; however, due to the poor signal quality of low-cost patch antennae, it is difficult for carrier phase real-time kinematic (RTK) technology to fix the integer ambiguity. Differential GNSS (DGNSS) positioning with pseudorange can effectively meet the high robustness and reliability requirements for the submeter to the meter level positioning accuracy of UVA/vehicle/aerospace users. To improve the DGNSS positioning accuracy and reliability of low-cost single-frequency GNSS receivers in complex environments, we propose a differential barometric altimetry (DBA)-assisted DGNSS positioning algorithm, which solves the DGNSS observation equations jointly and rigorously with the Earth ellipsoidal constraint equations constructed by the DBA altitude. The DBA altitude accuracy at different baseline lengths was evaluated in detail, and the DGNSS positioning performance of the single-frequency low-cost u-blox receiver NEO-M8T with a patch antenna and DGNSS/DBA combined positioning performance with the BMP280 barometer was analyzed by several sets of static and dynamic experiments under different environments. The results show that the single-frequency NEO-M8T receiver with patch antenna DGNSS positioning accuracy is submeter level in the static environment and drops to meter level in the dynamic environment. GPS+BDS dual system has higher positioning accuracy than single GPS or single BDS. DGNSS/DBA combination has higher positioning accuracy than DGNSS, especially the root mean square error (RMSE) can be improved by 30% to 80% in the U direction and slightly improved in the N and E directions. This study can provide an effective solution reference for various applications of low-cost sensor fusion positioning in the mass consumer market.

**Keywords:** differential GNSS; DBA; low-cost; combined positioning


**Meet the editor**

Prof. Assoc. Jolanta Iwin-Garzyńska—is the head of the Department of Corporate Finance and Taxation at the Institute of Accounting at the Faculty of Economics and Management, University of Szczecin. She is the author of over 150 books in the field of corporate finance and taxation. Her research deals with the problem of the impact of taxes on the finances of enterprises (potential of

capital, capital structure, cost of capital, profitability of investment, value of assets, etc.). She is a tax adviser (No. entry 10261). She is a practitioner in the field of finance and taxation in commercial companies. In 2007–2010, she became the president of the National Examination Commission for Tax Consulting at the Ministry of Finance. In 2014, she became a member of the Commission.

Contents

**Preface VII**

Jolanta Iwin-Garzyńska

Katarina Teplicka

Ko Hyejin

Deniz Aytaç

**The Case of Nigeria 61**

Zara Ghodsi and Allan Webster

**Associated Distortions 23**

Chapter 1 **Base of Corporate Income Tax and the EU Concept 3**

Chapter 2 **International Aspects of Corporate Income Taxes and**

**Section 2 Taxation, Economic Growth and Welfare State 59**

Ojijo Odhiambo and Oluwatosin Olushola

**Study of the Impact of Tax Equity1 97**

Chapter 3 **Taxes and Their Impact on the Business Sector in Slovakia 41**

Chapter 4 **Taxation and Economic Growth in a Resource-Rich Country:**

Chapter 5 **UK Taxes and Tax Revenues: Composition and Trends 83**

Chapter 6 **How Does a Welfare State achieves Fiscal Sustainability? A**

Chapter 7 **Optimal Taxation of Consumption in the Scope of Changing Elasticities of Demand: Re-reading Ramsey 123**

Liucija Birskyte and Gintare Giriuniene

**Section 1 Corporate Tax 1**


**Author details**

Swapan Kumar Chatterjee1 \*, Snigdha Saha1 and Shahin Muhammed T.K.2

1 Molecular Pharma Pvt. Ltd, Kolkata, West Bengal, India

2 College of Pharmaceutical Sciences, Government Medical College, Kannur, Kerala, India

\*Address all correspondence to<EMAIL_ADDRESS>
© 2023 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

*COVID-19 and Its Impact on Onset and Progression of Parkinson's and Cognitive Dysfunction DOI: http://dx.doi.org/10.5772/intechopen.105667*


**2. Literature Review**

Bitcoin is a digital currency and payment system created by Satoshi Nakamoto [1]. As the first decentralized digital cryptocurrency, bitcoin's price and popularity have risen rapidly since its introduction in 2009. With the growing popularity of bitcoin worldwide, the economic and financial properties of bitcoin have begun to attract the attention of researchers [2–4]. The relevant literature focuses on whether bitcoin is a currency or an asset, what kind of asset bitcoin is, and what kind of risk–return properties bitcoin has.

The earlier literature focused on whether bitcoin was a currency or an asset. Undeniably, there are some commonalities between bitcoin and currency, but from the perspective of monetary function, bitcoin can only be used as a medium of exchange and not as a unit of account or a storage of value [5]; therefore, bitcoin does not have a complete form of currency. Glaser et al. [6] focused on the liquidity of bitcoin when it functions as a medium of exchange, and argued that the convertibility between bitcoin and traditional currencies gives bitcoin liquidity, but the limited supply of bitcoin limits its liquidity. Böhme et al. [7] argued that the liquidity of bitcoin can be significantly weakened due to the frequent delays in bitcoin transactions. However, because of the anonymity of user identities, the bitcoin protocol does not restrict international transfer operations to countries that are on watch lists or embargoed, which provides bitcoin with higher flexibility and liquidity than deposit currencies in supporting international transfers [7]. In terms of the attitudes of bitcoin holders, since most bitcoin users view bitcoin as an investment tool rather than a transactional payment tool [6], the market value of bitcoin is much higher than the size of the economic transactions it facilitates [5], making bitcoin more of a speculative investment tool than a currency. Luther and Salter [8] examined bitcoin's ability to replace traditional currencies based on the increase in bitcoin app downloads after the Cyprus bailout announcement and found that the rise in bitcoin app downloads was insignificant, suggesting that bitcoin is not replacing the currencies of those countries whose domestic banks are in trouble.

After determining that bitcoin is more of an asset, scholars began comparing bitcoin to traditional assets in an attempt to generalize which asset, or class of assets, bitcoin is more comparable. Bitcoin is often analogized to gold in the literature, and is even referred to as digital gold or the new gold [9]. The similarities between bitcoin and gold are that both have a much higher market value than their intrinsic value, and both derive their market value from scarcity of supply and high mining costs; both have no national attributes, and their supply is not controlled by the government; gold was used as a medium of exchange during the gold standard period but was abandoned later due to lack of liquidity, and bitcoin is likely to face similar problems in the future as the scale of bitcoin users expands. However, there are also differences between bitcoin and gold; for example, people use gold mainly because of its function as a store of value, while the instability of bitcoin prices makes it difficult for it to perform value storage effectively [10]. Klein et al. [11] compared the return volatility of bitcoin and gold and their respective correlations with other asset prices and found that while both prices respond asymmetrically to market shocks, their respective correlations with other asset prices differ significantly, especially during market downturns. Shahzad et al. [12] compared the safehaven, hedging and diversification properties of bitcoin and gold for the G7 stock markets and found that gold outperforms bitcoin in terms of safe-haven and hedging effectiveness

and can provide higher conditional diversification benefits for stock investments than bitcoin, while Thampanya et al. [13] found that neither bitcoin nor gold is a good hedge for the Thailand stock market. Furthermore, Whelan [14] drew an analogy between bitcoin and the U.S. dollar, arguing that both are used as a medium of exchange, but the difference is that the dollar is backed by the government, whereas bitcoin is a private currency issued by the private sector, and, thus, the supply and governance mechanisms for both assets are different. Dyhrberg [10] examined whether bitcoin is more comparative to gold or the U.S. dollar, and found that the behavioral characteristics of bitcoin prices have both dollar- and gold-like components because bitcoin's property falls between a pure medium of exchange and a pure store of value. As such, bitcoin can be classified as an asset that is between the U.S. dollar and gold, and can be used as an effective tool for portfolio management.

More recent studies have begun to focus on the risk–return properties of bitcoin and its risk diversification, hedging and safe haven properties for traditional assets. Although bitcoin has a higher volatility than traditional assets [15], the inclusion of bitcoin in a portfolio may still improve the portfolio's risk–return performance [16–19]. Eisl et al. [20] used a CVaR approach to find that although the inclusion of bitcoin raises the conditional VaR of the portfolio, this additional risk is fully compensated by high returns, which ultimately improves the risk–return ratio. Dyhrberg [21] used daily frequency data to test the hedging effect of bitcoin on U.K. equities and the USD exchange rate and found that bitcoin can be used as a hedge for the FTSE index as well as the USD/EUR and USD/GBP exchange rates over the 2010–2015 period. Yang et al. [22] used a time-frequency domain approach to find that bitcoin has the capability to hedge against the currency market in the long term. Using a daily frequency sample over 2011–2015, Bouri et al. [23] found that bitcoin prices are negatively correlated with the Nikkei 225, MSCI Pacific and commodity indices and, therefore, have the ability to hedge against price fluctuations in these assets. Chan et al. [24] found that bitcoin can be used as an effective hedge for the U.S., European, Canadian, Japanese and Chinese stock markets in the monthly frequency dimension over the period 2010–2017. Wang et al. [25] found that cryptocurrencies are a safe haven rather than a hedge for most international stock indices, and the safe-haven properties are more pronounced in developed markets or markets with larger market capitalizations and higher liquidity. Shahzad et al. [26] showed that the safe-haven role of bitcoin for Chinese, U.S. and other developed and developing stock markets is time-varying and varies across different stock markets. Urquhart and Zhang [27] used high-frequency data to examine bitcoin's hedging and safe-haven capabilities for foreign exchange and found that bitcoin can be a hedge for the CHF, EUR and GBP, a diversifier for the AUD, CAD and JPY, and a safe haven for the CAD, CHF and GBP in times of extreme market turmoil. Wang et al. [28] examined the mean and volatility spillovers between bitcoin and six major financial assets in China and found that bitcoin can hedge China's stock, bond, and monetary markets and can serve as a safe haven for China's monetary market. Smales [29] argued that bitcoin's high volatility, low liquidity and high transaction costs (in terms of time and fees) compared to other assets preclude it from being considered a safe haven until its market matures. Kwapie ´n et al. [3] analyzed detrended cross-correlations between cryptocurrency markets and some traditional markets (including stock, commodity and forex markets) and found that the levels of cross-correlations become higher in turbulent periods.

After the outbreak of the COVID-19 pandemic, the price dynamics and portfolio performance of cryptocurrencies during the pandemic attracted widespread attention [30]. W ˛atorek et al. [4] pointed out that the COVID-19 pandemic has had a significant impact on the cryptocurrency market, transforming cryptocurrencies from a hedging instrument for investors fleeing traditional markets into a part of the global market which is closely linked to traditional financial instruments including currency, stock and commodity. Using a network connectedness model, Balcilar et al. [31] found increasing risk spillovers between cryptocurrencies and global emerging stock markets following the COVID-19 pandemic outbreak, and that cryptocurrencies cannot serve as a diversifier for emerging stock markets in both the short and long term. Caferra and Vidal-Tomas [32] used the wavelet coherence

approach and Markov switching autoregressive model to find that cryptocurrencies have some hedging properties against stock markets in response to shocks from the COVID-19 pandemic. Using the COVID-19 outbreak as a quasi-experiment, Grobys [33] used a difference-in-differences approach to test bitcoin's performance in hedging U.S. stock market risk and found that bitcoin performs poorly in hedging the tail risk of the U.S. market. Conlon and McGee [2] evaluated the safe-haven capability of bitcoin against traditional assets during the sharp decline in global financial markets following the outbreak of COVID-19, and found that bitcoin is not a safe haven for the S&P 500 and that including bitcoin in an equity portfolio at this time would substantially increase the portfolio's downward risk exposure. Conlon et al. [34] further found that bitcoin is not a safe haven for most international stock markets except for China's CSI 300 index. Wen et al. [35] used timevarying parameter vector autoregression to find that bitcoin is not a safe haven for crude oil and stocks during the COVID-19 pandemic. Dutta et al. [36] also found that bitcoin is only a diversifier rather than a safe haven for crude oil during the COVID-19 pandemic.

Although there has been a rich literature on the economic and financial properties of bitcoin, several gaps still remain. First, the sample of relevant studies does not cover a wide enough range of asset classes, and, therefore, they fail to provide a comprehensive dissection of the linkage between bitcoin and various major global assets and the diversification, hedging and safe-haven properties of bitcoin for these assets. Most of the existing studies focus on analyzing the linkage between bitcoin and a specific class of financial assets and bitcoin's diversification, hedging and safe-haven properties for that class of assets, including stocks [24–26,31–34], commodities [36], currencies [22,27], etc. Although the sample selected by Wang et al. [28] covers multiple asset classes, these assets are all Chinese assets and are not globally representative. Second, most of the existing studies have examined the risk–return characteristics of bitcoin and its correlation with other assets in a frequency-specific sample, while few have conducted comparative analyses at different time frequencies. The high short-term volatility and speculative nature of the bitcoin market is likely to impair the short-term correlation between bitcoin and other assets, resulting in the linkage between bitcoin and other assets exhibiting very different characteristics in different time frequency dimensions, and may even lead to changes in bitcoin's risk diversification, hedging and safe-haven properties in different time frequency dimensions.

We selected the prices of bitcoin and 14 major financial assets covering stock, bond, commodity and currency over the period 2013–2021 as a sample to test the dynamic linkage of bitcoin with each asset and the linkage's variation in different time frequency dimensions, and to further identify the risk diversification, hedging and safe-haven properties of bitcoin for various assets. We first used the ADCC-GARCH approach to quantitatively measure the dynamic correlation between bitcoin and other traditional assets. Based on the estimated dynamic conditional correlation (DCC) series, we further adopted Ratner and Chiu's [37] approach to identify the risk diversification, hedging and safe-haven properties of bitcoin for various types of assets to assess the extent to which bitcoin can be used as a diversifier, hedge or safe haven for those assets. As a result, we show that: (i) bitcoin is positively linked to risk assets, including stocks, bonds and commodities, and negatively linked to the U.S. dollar, which is a typical safe-haven asset, so bitcoin is closer in nature to a risk asset than a safe-haven asset; (ii) the high short-term volatility and speculative nature of the bitcoin market makes its long-term correlation with other asset prices stronger than the short-term correlation; (iii) the positive linkage between the prices of bitcoin and risk assets increases sharply under extreme shocks (e.g., the outbreak of COVID-19 in early 2020); (iv) bitcoin can hedge against the U.S. dollar, and in the long term, bitcoin can hedge against the Chinese stock market and act as a safe haven for the U.S. stock market and crude oil. However, for most other traditional assets, bitcoin is only an effective diversifier.

The contribution of this study is twofold. First, our selected sample covers four major asset classes—stock, bond, commodity and currency—which specifically include 14 major representative global financial assets. Based on this sample, we provide a comprehensive analysis of the linkage between bitcoin and major global assets as well as bitcoin's diversification, hedging and safe haven properties for these assets. Second, we also perform a comparative analysis in different time-frequency dimensions, comparing the dynamic correlation between bitcoin and each asset in daily, weekly and semi-monthly frequency dimensions as well as the differences in bitcoin's risk diversification, hedging and safe-haven capabilities for other assets in different time-frequency dimensions.


4.1.1. Inputs

Carbonaceous material, especially the organic material (dissolved/suspended) present in wastewater, conducts its chemical energy potential. Exceptions are energy in the form of heat (for wastewater above ambient temperature) and the use of wastewater as growth media for species to absorb carbon dioxide using sunlight energy (photosynthesis) because of their inorganic constituents [35].


**2. Biology and ecology of Patellid limpets**

#### **2.1. Taxonomy and distribution**

variable and unpredictable environmental conditions. Therefore, many of their peculiar morphological and biological characteristics can be understood as adaptations to this environment. The biological traits of limpets vary inter- and intraspecifically as a result of genetic differences and environmental influences [1]. Parameters such as growth, reproduction, and mortality are dependent on a complex array of selective forces and are important in understanding the distribution and abundance of a species [2, 3]. Differences in limpet populations from distinct geographic areas are most probably explained by specific environmental and anthropogenic conditions, essentially oligotrophy, sea water temperature, and fishing pressure. Thus, for some of the biological traits, it is expected to find patterns, like temperature which changes somewhat consistently with latitude and has a profound effect on the growth of limpet species, with species inhabiting higher latitudes growing more slowly and achieving larger maximum sizes, therefore having a longer lifespan than limpets from lower

Patellid limpets are also subjected to anthropogenic impacts on the coastal ecosystems such as, pollution, habitat removal, and harvest which in some cases has led to the reduction of abundance or even the disappearance of limpets from large areas. The decline of these species, which may have been further accelerated by the progressive deterioration of the coastline, continues at an alarming rate and many of their stocks are on the verge of disappearance. To avert this situation, regulators have established several measures including the implementation of closed seasons and areas where limpet harvest is interdicted, minimum size of capture, and catch limits. Limpet populations seem to respond, in general, in a positive way to these measures; however, the response is closely linked to the ability of the regulators to enforce said

Another popular strategy adopted in the protection of the rocky shores and limpets is the implementation of marine protected areas (MPAs). The effectiveness of MPAs in protecting exploited populations of limpets and underlying their overall success in increasing density and abundance as well as promoting healthy size composition with impact on the reproductive output of these species is well known. Nonetheless, several limitations are recognized that can negatively affect the protective role of MPAs such as, naturally occurring variations of the species biology and ecology as well as limitations regarding the management of MPAs, for instance, the lack of surveillance and enforcement of protection

The aim of this work is to review a broad spectrum of subjects associated to Patellid limpets' biology such as growth and reproduction, also the consequences of commercial exploitation on the stocks of these species and the effects of marine protected areas in the biology and populational dynamics of these intertidal grazers. The focus is on determining the effect of identified biotic and abiotic factors that influence these biological characteristics and geographical patterns recognized to be closely connected to growth and reproduction, such as latitude. Regarding conservation of Patellidae, the authors aim to elucidate how beneficial MPAs are in their role of protection of exploited populations and in enhancing recruitment

latitudes.

72 Organismal and Molecular Malacology

measures.

regulations.

and yield of adjacent exploited populations.

Patellid limpets are marine gastropod grazers belonging to the family Patellidae Rafinesque, 1815 that comprises the genera *Patella* Linnaeus, 1758, *Cymbula*, H. Adams & A. Adams, 1854, *Helcion* Montfort, 1810, and *Scutellastra* H. Adams & A. Adams, 1854. The worldwide distribution of Patellidae species is anti-tropical with half of the known species restricted to southern Africa and the North-Eastern Atlantic where a high diversity of species is found, while relatively few species are present in the Indian and Pacific Oceans [4, 5]. The Patellidae family is currently represented by, at least, 49 recognized species [6]. The genus *Patella* is comprised of 14 recognized species with a geographical distribution restricted to the North-Eastern Atlantic and the Mediterranean Sea; the genus *Cymbula* includes 10 species found in Southern Africa, South-Eastern Atlantic, and Mediterranean; the genus *Helcion* is represented by four species restricted to Southern Africa, while *Scutellastra* encompasses 21 species with a wide distribution ranging from Southern Africa to the Indo-West and Eastern Pacific [4, 7–10]. Limpets are subject to an array of environmental stresses as a result of their extended vertical distribution, which ranges from the upper to the lower shore levels. Thus, limpets can exhibit varying degrees of structural adaptations since their position relative to the shore influences their exposure to desiccation, hydrodynamic action of the waves, temperature variation, and tidal width [11–14]. This impressive phenotypic plasticity allied to the relatively simple shell geometry, convergent shell shape, and sculpturing results in an unclear Patellid limpet's taxonomy, in such a way that the initial generic names, with broad geographical range, had to be re-evaluated based on superficial similarities [15].

#### **2.2. Feeding habits and ecological importance**

Limpets are grazing herbivores that feed, by scraping the rocky substrate with the radula, on microbial biofilms which are primarily composed of cyanobacteria and microalgae, including diatoms, spores, and other propagules of macroalgae and invertebrates [16, 17]. Limpets' feeding habits are essential in structuring intertidal communities [16–19] since limpet grazing is a key process in rocky shores involved in determining macroalgal abundance and in modifying ecosystem stability, indirectly enhancing or inhibiting the establishment of other organisms [17]. The decline of population density of limpets might result in an abnormal development of algae diversity as reported by Boaventura et al. [20] or in the occupation of their ecological niche by competing organisms such as barnacles or sea urchins [21–23]. However, the effect of these grazers is not limited to the removal of algae, and very often they can affect other animal species through competitive interactions [24] and by providing secondary habitats for other invertebrates that settle either on top of, or beneath, their shells [19, 25, 26]. Grazers may also affect the rate of succession [27] or cause different assemblages to develop [28]. Thus, limpets are rightfully considered to be keystone species in intertidal communities [29].

#### **2.3. Movement and homing**

Patellid limpets are considered to some extent semi-sessile organisms; nonetheless, they perform small movements in the area surrounding their usual fixation site. This behavior is designated as homing and can often be observed through the scar that remains in the rocky substrate where the limpet settles. Limpet movement patterns and homing behavior have been extensively studied for *Patella vulgata* Linnaeus, 1758 [30], *Patella depressa* Pennant, 1777 [31], *Patella rustica* Linnaeus, 1758 [32], *Patella ferruginea* Gmelin, 1791 [33], *Scutellastra flexuosa* (Quoy & Gaimard, 1834), and *Scutellastra argenvillei* (Krauss, 1848) [34]. This homing behavior has different functions in different species such as avoiding desiccation [35, 36], reducing predation and intraspecific competition [37–40], responding to wave action [41, 42] and defending territory or asserting dominance [43, 44]. The mechanism that is most widely accepted as being responsible for the homing behavior reports limpets following chemical trails, laid down on the outward trip, on their way back to the fixation site [31, 45, 46].

#### **2.4. Growth**

Biological parameters such as growth rate, asymptotic length, and age structure reflect the overall state of health of a population and are commonly used as stock assessment tools of exploited marine organisms. Growth, reproductive strategy, and mortality are dependent on a complex array of selective forces [2] and are important in understanding the distribution and abundance of a species [3]. To determine these parameters, most studies usually resort to the capture-recapture method [22, 47–50] or length-frequency distribution analysis [51–53]. Over the past decades, intensive research has focused on the biology of limpets, due to their diversity and ecological significance; however, there remain gaps in the knowledge concerning these species' age structure and growth patterns.

Patellid limpets, like many marine gastropods, exhibit both intra- and interspecific seasonal variation in growth rates [54]. Although some intraspecific variation may be genetically controlled [55], external factors such as changes in food availability [56, 57], wave action [58–60], and vertical distribution on the shore [61] are thought to influence growth rates. Other factors such as population density, available grazing area, predation, and competition are indicated as influencing growth rates of mollusks supporting the idea that the strategy of diverting energy to reproduction and vice versa, according to the organisms' needs, influences growth rates [24, 49, 62, 63]. It has been suggested that limpets with greater growth rates have smaller lifespan while limpets with slow growth are generally long-lived [46]. As such, rapidly growing limpets are usually associated with early maturation, high mortality, and a short lifespan [46, 64].

Clarke et al. [49] observed a latitudinal cline in annual shell growth of the polar limpet *Nacella concinna* (Strebel, 1908). This latitudinal pattern could nevertheless be masked by inter-annual variability. The authors suggest that the observed variation could be the result of a simultaneous change in both growth rate and the duration of the growth period. This change would result from the shorter duration of the seasonal blooming of epiphytic microalgal and microbial biomass at higher latitudes. Another factor influencing growth rates in *N. concinna* is seawater temperature, with warmer temperatures that last longer producing higher growth rates.

*Scutellastra* and *Cymbula* species that occur at similar latitudes present variations in terms of growth, namely in maximum size and growth rates. When compared to tropical limpets belonging to the genus *Cellana* H. Adams, 1869, limpets from temperate regions are generally larger, with wider lifespan and slower growth rates. Additionally, limpets inhabiting the artic regions such as *N. conccina* achieve larger sizes, even wider lifespans, and slower growth rates. This latitudinal pattern has been usually associated with the latitudinal variation of temperature, photoperiod, and insolation [49]. Even though it is consensual that species from lower latitudes grow more rapidly than species from higher latitudes [49, 52], it is not yet clarified whether physiological constraints, a reduced or prolonged growing season, or combination of both might be the cause of dissimilar growth rates at differing latitudes [49].

Nevertheless, due to Patellids' anti-tropical distribution, growth patterns are difficult to observe, particularly when considering latitude. Within this family, variations in growth are mostly derived from prevalent local environmental factors. Nonetheless, when comparing to other Patellogastropoda, a latitudinal pattern becomes apparent, in which at lower latitudes limpets grow at faster rates and achieve smaller sizes, while at higher latitudes, they grow at slower rates and achieve larger sizes. For instance, for the polar limpet *N. concinna* reported growth rates range between 0.059 and 0.323 year−1, while the highest growth rate is exceptionally high for a limpet inhabiting the polar regions, probably due to specific characteristics of the habitat in Signy Island [49]. The overall growth rates are inferior to those reported for limpets of the genus *Cellana* that inhabit lower latitudes in temperate and tropical regions with growth rates ranging from 0.400 to 1.661 year−1. Patellid limpets exhibit intermediate growth rates ranging from 0.117 year−1 in *Scutellastra choclear* (Born, 1778) and 1.020 year−1 in *Cymbula oculus* (Born, 1778) reflecting their anti-tropical distribution.

However, the nonlinearity of growth of marine organisms renders the direct comparison of growth parameters impossible [65]. As such, determination and comparison of the overall growth performance of different marine species is achieved using the growth performance index (GPI) of Pauly and Munro [66], which relates the asymptotic length and growth rate [66]. Nonetheless, the growth performance index in Patellogastropod limpets exhibits the same pattern as growth rates with decreasing GPI as latitude increases and ranging from 1.942 in *N. concinna* to 3.653 in *Cymbula granatina* (Linnaeus, 1758), suggesting that growth performance of limpets varies with latitude. Within the Patellidae family the variation of GPI is reduced with values ranging from 2.42 for *S. cochlear* to 3.65 for *C. granatina* from South Africa [62], which is in agreement with Sparre et al. [67] who claim that the growth performance index remains relatively constant at similar rates between related taxa. The variability results therefore due to abiotic and biotic factors that different species are subject to, such as greater or lesser extent of hydrodynamics, desiccation, predation, competition, and temperature.

#### **2.5. Reproduction**

**2.3. Movement and homing**

74 Organismal and Molecular Malacology

**2.4. Growth**

Patellid limpets are considered to some extent semi-sessile organisms; nonetheless, they perform small movements in the area surrounding their usual fixation site. This behavior is designated as homing and can often be observed through the scar that remains in the rocky substrate where the limpet settles. Limpet movement patterns and homing behavior have been extensively studied for *Patella vulgata* Linnaeus, 1758 [30], *Patella depressa* Pennant, 1777 [31], *Patella rustica* Linnaeus, 1758 [32], *Patella ferruginea* Gmelin, 1791 [33], *Scutellastra flexuosa* (Quoy & Gaimard, 1834), and *Scutellastra argenvillei* (Krauss, 1848) [34]. This homing behavior has different functions in different species such as avoiding desiccation [35, 36], reducing predation and intraspecific competition [37–40], responding to wave action [41, 42] and defending territory or asserting dominance [43, 44]. The mechanism that is most widely accepted as being responsible for the homing behavior reports limpets following chemical trails, laid

Biological parameters such as growth rate, asymptotic length, and age structure reflect the overall state of health of a population and are commonly used as stock assessment tools of exploited marine organisms. Growth, reproductive strategy, and mortality are dependent on a complex array of selective forces [2] and are important in understanding the distribution and abundance of a species [3]. To determine these parameters, most studies usually resort to the capture-recapture method [22, 47–50] or length-frequency distribution analysis [51–53]. Over the past decades, intensive research has focused on the biology of limpets, due to their diversity and ecological significance; however, there remain gaps in the knowledge concerning

Patellid limpets, like many marine gastropods, exhibit both intra- and interspecific seasonal variation in growth rates [54]. Although some intraspecific variation may be genetically controlled [55], external factors such as changes in food availability [56, 57], wave action [58–60], and vertical distribution on the shore [61] are thought to influence growth rates. Other factors such as population density, available grazing area, predation, and competition are indicated as influencing growth rates of mollusks supporting the idea that the strategy of diverting energy to reproduction and vice versa, according to the organisms' needs, influences growth rates [24, 49, 62, 63]. It has been suggested that limpets with greater growth rates have smaller lifespan while limpets with slow growth are generally long-lived [46]. As such, rapidly growing limpets are usually associated with early maturation, high mortality, and a short lifespan [46, 64].

Clarke et al. [49] observed a latitudinal cline in annual shell growth of the polar limpet *Nacella concinna* (Strebel, 1908). This latitudinal pattern could nevertheless be masked by inter-annual variability. The authors suggest that the observed variation could be the result of a simultaneous change in both growth rate and the duration of the growth period. This change would result from the shorter duration of the seasonal blooming of epiphytic microalgal and microbial biomass at higher latitudes. Another factor influencing growth rates in *N. concinna* is seawater temperature, with warmer temperatures that last longer producing higher growth rates.

down on the outward trip, on their way back to the fixation site [31, 45, 46].

these species' age structure and growth patterns.

Patellid limpets have a simple reproductive system, consisting of a simple gonad inserted in the visceral mass and a reduced gonoduct leading to the right nephridium [68, 69]. These species are not externally sexually dimorphic, and sex determination is only possible through macroscopic observation of the gonads. Spawning results in the release of oocytes and sperm directly in the ocean where fecundation occurs. According to Orton et al. [68], spawning is stimulated by environmental triggers, such as high wind speed and wave action. An increase in phytoplankton concentration may also stimulate spawning as suggested by Underwood [24] who observed that gastropod species with planktotrophic larvae spawn when phytoplankton concentration is high.

Most limpet species have a reproductive cycle with a gonadal development stage culminating in a spawning period followed by a resting phase. The spawning period varies inter- and intraspecifically; it may also vary from year to year and is supposed to be triggered by temperature variations, increased wave action, and onshore winds [70]. In regions with higher temperatures, spawning occurs in a short period contrary to what happens in regions with colder waters, where the development of the gonads requires a longer time period [71]. *P. vulgata* is believed to be a winter breeder, with spawning occurring from October to March; however, in colder localities, sexual maturation occurs earlier [68]. On the other hand, in south-west England, *P. depressa* is considered a summer breeder [72] with spawning occurring between late July and early September and without a resting phase unlike *P. vulgata*. The same authors suggested that an increase in temperature associated with wave action stimulates spawning in this species. *Patella ulyssiponensis* Gmelin, 1791 has a spawning period that lasts from October to December, being also considered a winter breeder in south-west England [59, 68, 73]. Orton et al. [68] and Orton and Southward [72] suggested that although the development of the gonad in *P. vulgata* and *P. depressa*, respectively, is well related with temperature, the act of spawning is triggered by violent onshore storms. Thompson [59] also found *P. ulyssiponensis* spawning during the autumn storms. Hence, it seems likely that spawning cannot take place until a population is sufficiently mature, but after that stage is reached, the first strong windstorm will trigger spawning [59]. Another factor that potentially affects the timing of spawning in limpets is food availability; Underwood [24] reported that species with planktotrophic larval stage time spawning with periods when phytoplankton concentrations are high. One such case is that of the closely related species of *P. ulyssiponensis* from the Portuguese mainland and *Patella aspera* Röding, 1798 from Madeira Island. *P. ulyssiponensis* is reported to be a summer breeder while *P. aspera* was reported to be a winter breeder with spawning occurring when the phytoplankton concentration is higher (P. Henriques, pers. comm.). Similarly, it has been reported that in limpets with restricted geographic distribution, the reproductive cycle is influenced by geographic locality, namely in the timing of gametogenesis and spawning [62, 74]. For limpets with broader geographic distribution, it is possible that the reproductive cycle is adjusted to regional environmental conditions [74].

Limpets, like many sessile or sedentary marine invertebrates, have life cycles that include a prolonged pelagic larval phase that can last up to 2 weeks as reported by Hawkins et al. [75] for *Patella* species. Veliger larvae remain in the water column as plankton until eventually fixating in the rocky substrate on the inferior level of the coast. As the juveniles grow, they begin a slow vertical migration, colonizing different levels of the rocky shores [76], leading to variability in patterns of recruitment [77]. Moreover, larvae in the water column are subject to processes of physical transport that can disperse them from the site of reproduction [78]. Thus, the number of recruits on a specific location may be independent of the local larvae production [16, 79] and influenced by current regimes. Nonetheless, limpet populations cannot be considered fully open or fully closed, since some local larval retention is likely to occur despite larval dispersal [80, 81].

directly in the ocean where fecundation occurs. According to Orton et al. [68], spawning is stimulated by environmental triggers, such as high wind speed and wave action. An increase in phytoplankton concentration may also stimulate spawning as suggested by Underwood [24] who observed that gastropod species with planktotrophic larvae spawn when phyto-

Most limpet species have a reproductive cycle with a gonadal development stage culminating in a spawning period followed by a resting phase. The spawning period varies inter- and intraspecifically; it may also vary from year to year and is supposed to be triggered by temperature variations, increased wave action, and onshore winds [70]. In regions with higher temperatures, spawning occurs in a short period contrary to what happens in regions with colder waters, where the development of the gonads requires a longer time period [71]. *P. vulgata* is believed to be a winter breeder, with spawning occurring from October to March; however, in colder localities, sexual maturation occurs earlier [68]. On the other hand, in south-west England, *P. depressa* is considered a summer breeder [72] with spawning occurring between late July and early September and without a resting phase unlike *P. vulgata*. The same authors suggested that an increase in temperature associated with wave action stimulates spawning in this species. *Patella ulyssiponensis* Gmelin, 1791 has a spawning period that lasts from October to December, being also considered a winter breeder in south-west England [59, 68, 73]. Orton et al. [68] and Orton and Southward [72] suggested that although the development of the gonad in *P. vulgata* and *P. depressa*, respectively, is well related with temperature, the act of spawning is triggered by violent onshore storms. Thompson [59] also found *P. ulyssiponensis* spawning during the autumn storms. Hence, it seems likely that spawning cannot take place until a population is sufficiently mature, but after that stage is reached, the first strong windstorm will trigger spawning [59]. Another factor that potentially affects the timing of spawning in limpets is food availability; Underwood [24] reported that species with planktotrophic larval stage time spawning with periods when phytoplankton concentrations are high. One such case is that of the closely related species of *P. ulyssiponensis* from the Portuguese mainland and *Patella aspera* Röding, 1798 from Madeira Island. *P. ulyssiponensis* is reported to be a summer breeder while *P. aspera* was reported to be a winter breeder with spawning occurring when the phytoplankton concentration is higher (P. Henriques, pers. comm.). Similarly, it has been reported that in limpets with restricted geographic distribution, the reproductive cycle is influenced by geographic locality, namely in the timing of gametogenesis and spawning [62, 74]. For limpets with broader geographic distribution, it is possible that the reproductive

plankton concentration is high.

76 Organismal and Molecular Malacology

cycle is adjusted to regional environmental conditions [74].

Limpets, like many sessile or sedentary marine invertebrates, have life cycles that include a prolonged pelagic larval phase that can last up to 2 weeks as reported by Hawkins et al. [75] for *Patella* species. Veliger larvae remain in the water column as plankton until eventually fixating in the rocky substrate on the inferior level of the coast. As the juveniles grow, they begin a slow vertical migration, colonizing different levels of the rocky shores [76], leading to variability in patterns of recruitment [77]. Moreover, larvae in the water column are subject to processes of physical transport that can disperse them from the site of reproduction [78]. Thus, the number of recruits on a specific location may be independent of the local larvae production [16, 79] and influenced by current regimes. Nonetheless, limpet populations cannot be Orton [82] suggested the existence of the phenomenon of protandrous hermaphroditism in limpets of the genus *Patella* based on sexual dimorphism in size-frequency of *P. vulgata*; subsequently Thompson [59], Branch [46], and Le Quesne [83] observed that some individuals reach maturity as males and become females in the more advanced stages of their life cycle. This phenomenon of sequential hermaphroditism is also suggested to occur in species of the genera *Cymbula* [46], *Helcion* [74], and *Scutellastra* [62, 84, 85]. Not all male limpets change sex, since a considerable proportion of males can be found in the larger size-groups, these individuals might eventually change sex or remain as males if the signals that lead to sex change are not present [86]. Also, some limpet species are sequential hermaphrodites in which the sex change can be reverted as reported for *P. ferruginea* by Guallart et al. [87].

Sex change in limpet species is thought to be genetically controlled. However, high variability in the timing or on the limpet size at which the change occurs suggests that environmental factors may influence the process. Species such as *C. oculus* have a relatively fixed timing of sex change [88], while in other species, the sex change occurs at sizes that are highly variable. These differences in size and age at which the sex change occurs are often mediated by environmental factors [46, 89–92]. For instance, sex change in mollusks can be delayed in populations where large females are present [89, 90]. Additionally, in populations subjected to higher mortality rates or slower growth rates, sex change seems to occur earlier [93]. Also, it has been reported that social control of sex change occurs in Patellogastropod limpets [91, 92]. In this case, several possible cues for sex change have been suggested such as, contact frequency between individuals, available movement area, food availability, growth rate, pheromonal information, and communication by mucus traces left by individuals during foraging excursions [91].

Hermaphroditism is an evolutionarily advantageous strategy for species with low population densities or low motility such as limpets, since under such conditions, hermaphroditism is supposed to increase the likelihood of successful fertilization [87]. Reproductive success in broadcast spawners, such as limpets, is correlated to the quantity of gametes released into the water column. It is believed that larger limpets produce more gametes than smaller individuals. Additionally, sex change in protandrous hermaphrodite species results in an increase of female individuals in the larger size classes. Thus, the sex distribution through sizes in protandrous hermaphrodite limpets makes these species extremely vulnerable to harvest [33], since the depletion of larger and more fecund individuals and females in a higher percentage may potentially alter the sex ratio and reduce the reproductive output of populations [86].


**5. Conclusions**

−517 to −605 MPa, which is improved by 17%. The max CRS varies from −684 to −794 MPa and the increase rate is about 16%. It is mainly due to the improved shot velocity after increasing SP intensity, which can cause more severe plastic deformation on surface, the deeper surface deformation layer and the larger CRS. Comparing the results by simulation and experiment, it reveals that the variation trends of residual stress by simulation are similar with the results from experiment.

**Figure 16.** Depth distribution of CRS on the matrix and composite under three different SP intensities [36], (a) Ti-6Al-4V;

**Figure 15.** (a) Schematic figure of residual stress measurement coordinate; (b) photo of the residual stress measurement

**Figure 16(b)** shows the experimental results of residual stress distribution on the composite 8% (TiB+TiC)/Ti-6Al-4V. From the figure, the depths of surface deformation layer are 200, 250, and 300 μm corresponding to 0.15, 0.30, and 0.45 mmA, which are shallower than the matrix's. The difference is resulted from the existence of reinforcements' resistance to the deformation of surface. Moreover, SP intensity has direct relation to the shot velocity. The larger impact velocity, the higher impact kinetic energy, and the depths of surface deformation layer are deeper.

**4.3. Residual stress distribution of 8% (TiB+TiC)/Ti-6Al-4V**

42 Finite Element Method - Simulation, Numerical Analysis and Solution Techniques

using X-ray stress analyzer.

(b) 8% (TiB+TiC)/Ti-6Al-4V.

In order to study the effect of various parameters on the residual stress distribution after SP, LS/DYNA analysis module in ANSYS is utilized to establish the finite element model for Ti-6Al-4V and (TiB+TiC)/Ti-6Al-4V, and both 3D homogeneous and inhomogeneous models are set up. The influence of coverage rate, shot radius, and shot velocity on residual stress distribution is studied using the multi-layer shot balls to simulate the actual SP process. The main results are concluded as the following:

**5.** The results obtained from the inhomogeneous SP model reveal that the compressive and tensile residual stresses are introduced in (TiB+TiC)/Ti-6Al-4V. The max CRS and tensile residual stress are −1511 and +1155 MPa, respectively. CRS appear in matrix, but the tensile residual stresses gets generated in the reinforcements, which reveals the higher yield strength of reinforcements. This stress distribution indicates the effect of reinforcements, keeping the adverse tensile stresses in reinforcements and retarding the damage to matrix.

**References**

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Finite Element Dynamic Analysis on Residual Stress Distribution of Titanium Alloy and Titanium…

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[2] Schulze V. Characteristics of surface layers produced by shot peening. In: Proceeding of the Eighth International Conference on Shot Peening ICSP-8 in Garmisch-Partenkirchen

[3] Kim K-H, Kim Y-C, Jeon E-C, Kwon D. Evaluation of indentation tensile properties of Ti alloys by considering plastic constraint effect. Materials Science and Engineering A.

[4] Benedetti M, Fontanari V, Monelli B. Numerical simulation of residual stress relaxation in shot peened high-strength aluminum alloys under reverse bending fatigue. Journal of

[5] Kim T, Lee JH, Lee H, Cheong S-k. An area-average approach to peening residual stress under multi-impacts using a three-dimensional symmetry-cell finite element model

[6] Prasannavenkatesan R, Zhang J, McDowell DL, Olson GB, Jou H-J. 3D modeling of subsurface fatigue crack nucleation potency of primary inclusions in heat treated and shot peened martensitic gear steels. International Journal of Fatigue. 2009;**31**:1176-1189

[7] Guagliano M. Relating Almen intensity to residual stresses induced by shot peening: A numerical approach. Journal of Materials Processing Technology. 2001;**110**:277-286

[8] Rouhaud E, Ouakka A, Ould C, Chaboche J, Francois M. Finite elements model of shot peening, effects of constitutive laws of the material, Proceedings ICSP-9, Paris, France.

[9] Ould C, Rouhaud E, François M, Chaboche JL. A kinematic hardening finite elements model to evaluate residual stresses in shot-peened parts, local measurements by X-ray

[10] Baragetti S, Guagliano M, Vergani L. A numerical procedure for shot peening optimisation by means of non-dimensional factors. International Journal of Materials and

[11] Boyce B, Chen X, Hutchinson J, Ritchie R. The residual stress state due to a spherical

[12] Evans R. Shot peening process: Modelling, verification, and optimisation. Materials

[13] Levers A.A. Prior, finite element analysis of shot peening. Journal of Materials Processing

diffraction. In: Mater. Sci. Forum Trans Tech Publ. 2006. pp. 161-166

hard-body impact. Mechanics of Materials. 2001;**33**:441-454

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with plastic shots. Materials & Design. 2010;**31**:50-59

**6.** The experimental results from XRD method are shown that the surface CRS increased from −545 to −724 MPa and the max CRS varies from −655 to −819 MPa. The ranges of residual stress distribution in experiments are in good agreement with the simulated results by 3D finite element dynamic analysis.

From all results and discussion, using 3D finite element dynamic analysis to simulate the residual stress distribution of titanium matrix composite is reasonable, especially for the stress distribution in and around the reinforcements.


**Abstract**

Fuzzy sets have been extensively researched and results have been developed based on the extensions of fuzzy sets. In this chapter, fuzzy sets and its extensions are discussed. Z-numbers along with weighted sum product assessment method is used to obtain a feasible solution to the location selection problem for installation of smog towers in a densely populated locality. The degrees of freedom namely degree of membership, degree of non-membership and the degree of hesitancy have been expressed as Zadeh's Z-number with probability quotient for the degrees. Further, ranking of the alternatives based on Z-numbers and WASPAS to allocate smog towers to residential areas stricken by air pollution.

**Keywords:** Z-numbers, WASPAS, fuzzy set, Generalized fuzzy set, smog tower, air pollution

### **1. Introduction**

Mathematics is a study of quantity, structure, space and change. Patterns are observed to understand the structures and reasoning is provided to real time phenomena. Mathematics can be subdivided into arithmetic, algebra, geometry and analysis. Further, there are subdivisions linking the core of mathematics to other fields like logic, set theory, empirical mathematics and more recently to the rigorous study of uncertainty, imprecision and vagueness. Set theory is a branch of mathematical logic that studies 'sets', a collection of well-defined objects. An object under consideration either 'belongs to' a set or 'does not belong' to a set. Thus classical set theory could answer membership of an element in terms of 0's and 1's. This binary logic could not translate the imprecision prevailing in the real world. The need to bridge the precise classical mathematics with the imprecise world gave birth to the concept of fuzzy sets. These sets were introduced independently by Zadeh [1] and Dieter Klaua in 1965 as an extension of classical set theory. In contrast to binary terms, fuzzy set theory allowed gradual assessment of the membership of elements in a set described by a membership function valued in the interval [0,1]. Zadeh went on to propose new operations in fuzzy logic and proved that fuzzy logic was a generalization of classical Boolean logic. He also proposed the concept of fuzzy numbers which were special case of fuzzy sets. The mathematical operations were also defined and thus fuzzy sets paved the way for many extensions, whose edifice stands strongly on the mathematics concept.

### **1.1 The extensions**

Interval-valued fuzzy sets (IVFS) were introduced as an extension to fuzzy sets in which the membership degrees are represented by an interval value reflecting the uncertainty in assigning membership degrees. Larger the interval, more uncertainty is seen in assigning membership degrees.

Intuitionistic Fuzzy sets (IFS) is also an extension of fuzzy set introduced by Atanassov [2]. The addition of 'degree of non-membership' of an element improved the efficiency of modeling uncertainty.

Interval-Valued Intuitionistic Fuzzy Set (IVIFS):

Atanassov [3] combined IVFS and IFS to form IVIFS where in are all intervals in [0,1].

Neutrosophic Sets:

Having defined fuzzy sets, IVFS, IFS and IVIFS, researchers still could not handle uncertainty efficiently. The question of, "what if I had a neutral opinion?" had to be answered. Thus, Smarandache broke free the inter-dependencies of all three membership functions. Thus neutrosophic sets were defined

$$\begin{aligned} A &= \{ (\mathfrak{x}, \mu\_A(\mathfrak{x}), \nu\_A(\mathfrak{x}))/\mathfrak{x} \in X \} \\ \mu\_A(\mathfrak{x}): X &\to [0, 1], \nu\_A(\mathfrak{x}): X \to [0, 1], \Gamma \end{aligned}$$

such that,

$$\mathbf{0} \le \mu\_A(\mathbf{x}) + \nu\_A(\mathbf{x}) + \Pi\_A(\mathbf{x}) \le \mathbf{3}.$$

Neutrosophic sets, thus, generalized all the sets with classic set theory as the foundation.

In a nutshell, the holy trinity were introduced as a trigger for astounding research all over the world.

#### **1.2 Recent extensions**

Picture Fuzzy sets (PFS):

These sets were introduced by Cuong [4] to model situations where in human opinions involved refusal towards a particular event. For instance, voting in an election could have four categories of people; people wanting to vote for a particular party, people abstaining from voting, people not wanting to vote for a party and people refusing to vote. Thus degree of refusal membership is given by *ηA*ð Þ *x* , with

$$\begin{aligned} A &= \{ (\mathfrak{x}, \mu\_A(\mathfrak{x}), \nu\_A(\mathfrak{x}))/\mathfrak{x} \in \mathcal{X} \} \\ \mu\_A(\mathfrak{x}): &X \to [0, \mathfrak{1}], \nu\_A(\mathfrak{x}): X \to [0, \mathfrak{1}], \Pi\_A(\mathfrak{x}): X \to [0, \mathfrak{1}] \\ &\mathbf{0} \le \mu\_A(\mathfrak{x}) + \nu\_A(\mathfrak{x}) + \Pi\_A(\mathfrak{x}) < \mathbf{1}. \end{aligned}$$

Pythagorean Fuzzy Sets (PyFS):

These sets were introduced as a generalization for IFS by Yager [5]. The main feature of PyFS is that it is characterized by the degrees in which the sum of the square of each of the parameters equal to 1.

Let X be a universal set. Then a Pythagorean fuzzy set A, which is a set of ordered pairs over X, is defined by the following;

$$A = \{ (\mathfrak{x}, \mu\_A(\mathfrak{x}), \nu\_A(\mathfrak{x}))/\mathfrak{x} \in X \}$$

*Location Selection for Smog Towers Using Zadeh's Z-Numbers Integrated with WASPAS DOI: http://dx.doi.org/10.5772/intechopen.95906*

$$\begin{aligned} \mu\_A(\mathfrak{x}): &X \to [0, 1], \nu\_A(\mathfrak{x}): X \to [0, 1] \\ \mathbf{0} \le (\mu\_A(\mathfrak{x}))^2 + (\nu\_A(\mathfrak{x}))^2 \le \mathbf{1}, \\ \Pi\_A(\mathfrak{x}) = \sqrt{\mathbf{1} - \left( (\mu\_A(\mathfrak{x}))^2 + (\nu\_A(\mathfrak{x}))^2 \right)} \end{aligned}$$

Hesitant Fuzzy Set (HFS):

HFS were introduced by Torra [6]. Hesitant fuzzy sets were defined in terms of a function that returns a set of membership values for each element in the domain.

HFS, to a large extent, were able to model uncertainty, but with in-depth research, a significant drawback appeared, namely, loss of information. To overcome this drawback, Zhu and Xu [7] proposed the concept of Probabilistic Hesitant Fuzzy Set (PHFS) which incorporates distribution information in HFS. PHFS depicts not only the hesitancy of decision-makers when they are irresolute for one thing or the other, but also hesitant distribution information.

Spherical Fuzzy Sets (SFS):

These sets were introduced as an extension to Picture fuzzy sets and Pythagorean fuzzy sets [8] (**Figure 1**).

A spherical fuzzy set *A*~ *S* of the universe of discourse U is given by

$$\mu\_A(\mathfrak{x}): X \to [0, \mathfrak{1}], \nu\_A(\mathfrak{x}): X \to [0, \mathfrak{1}], \Pi\_A(\mathfrak{x}): X \to [0, \mathfrak{1}]$$

With 0 ≤ð Þ *μA*ð Þ *x* þ ð Þ *υA*ð Þ *x* þ ð Þ Π*A*ð Þ *x* ≤1, for any x in the universal set U.

The spherical fuzzy sets extend PFS and PyFS, but however, these sets are nothing but a particular case of Neutrosophic sets.

#### **1.3 Decision-making techniques**

Decision-making is process which involves problem-solving yielding a solution deemed to be "optimal" or satisfactory to an extent. A major part of decisionmaking involves analysis of finite set of alternatives with respect to a given set of criteria. The task involves ranking these alternatives based on feasibility when all

**Figure 1.** *Spherical fuzzy set.*

the criteria are considered simultaneously. This area of decision-making has always attracted researchers and is still a highly debatable concept as there are many such methods which yield different results when applied to the same set of data. Emotion also appears to aid the decision-making process. Decision-making often occurs in uncertainty about whether one's choices will lead to benefit or harm.

Under fuzzy environment:

Decision making under uncertainty means a decision process in which the constraints or goals are fuzzy in nature, but the system need not be fuzzy. As per Bellman and Zadeh [9], fuzzy goals and constraints can be precisely defined as fuzzy sets in the space of alternatives. A fuzzy decision is then viewed as an intersection of the given goals and constraints. Decision-making under uncertainty basically translates to taking decisions in which the goals or constraints are fuzzy in nature. This implies that the constraints consists of alternatives whose boundaries are not sharply defined. An example of a fuzzy goal is "x should be in the vicinity of y", where y is a constant. Here vicinity is a source of fuzziness.

We thus divide the decision-making process into seven steps:

#### **1.4 Zadeh's Z-numbers**

A Z-number is an ordered pair of fuzzy numbers (A,B). Z-number is associated with a real-valued uncertain variable X, with the first component A, playing the role of a fuzzy restriction R(X), on the values of which X can take, written X is A. *R X*ð Þ : *X is A* ! *μA*ð Þ *u* .

Here *μA*ð Þ *u* is the degree to which u satisfies the constraint.

The second component B, is referred to as certainty or reliability or probability or strength of belief related to the component A.

For example, (finding an enclosed space of 900 s.m. in a densely populated area, low, not sure) [10].

#### **1.5 Weighted aggregated sum product assessment (WASPAS)**

WASPAS method was introduced by Zavadskas et al. [11]. This MCDM method is a combination of two simple decision-making techniques; Weighted Sum Model (WSM) and Weighted Product Model (WPM).

The total relative importance is given by

$$\mathbf{Q}\_{i} = \lambda \sum\_{j=1}^{n} \overline{\mathbf{x}}\_{ij} w\_{j} + (\mathbf{1} - \lambda) \prod\_{j=1}^{n} \left( \overline{\mathbf{x}}\_{ij} \right)^{w\_{j}}, \lambda = \mathbf{0}, \mathbf{0}.\mathbf{1}, \mathbf{0}.\mathbf{2}, \dots, \mathbf{1} \tag{1}$$

*Location Selection for Smog Towers Using Zadeh's Z-Numbers Integrated with WASPAS DOI: http://dx.doi.org/10.5772/intechopen.95906*

Here,

*xij* is the performance of ith alternative with respect to the jth criterion. *xij* is the normalized value of *xij* evaluated as follows;

$$\overline{\mathfrak{X}}\_{\vec{\eta}} = \frac{\mathfrak{X}\_{\vec{\eta}}}{\max\_{i} \mathfrak{X}\_{\vec{\eta}}} \text{ for beneficial criteria} \tag{2}$$

$$\mathfrak{X}\_{\vec{\eta}} = \frac{\min\_{i} \mathfrak{x}\_{\vec{\eta}}}{\mathfrak{x}\_{\vec{\eta}}} \text{ for non- beneficial criteria} \tag{3}$$

In the next section a literature review of the various aspects of decision-making using Z-numbers will be discussed.