**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*~ *<sup>S</sup>* 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* <sup>þ</sup> ð Þ *<sup>υ</sup>A*ð Þ *<sup>x</sup>* <sup>þ</sup> ð Þ <sup>Π</sup>*A*ð Þ *<sup>x</sup>* ≤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.
