Triangular dense fuzzy sets and new defuzzification methods

Abstract

This article deals with a new novel defuzzification method for the dense fuzzy sets. In our study, we first define the dense fuzzy set for triangular fuzzy numbers. Then new defuzzification methods have been formulated with crisp convergence tests. Cauchy sequence has been utilized for better illustrations. We show the usefulness and the global justification of newly introduced methods. Applications of new methods in decision making, psychological test, military selection, cryptography, photography, crime research, filtering of noisy environment and risk analysis have been given as its key indexes.

Keywords

Dense fuzzy set triangular fuzzy number defuzzification method

1 Introduction

The concept of fuzzy set was first introduced by Zadeh [21], later it has been applied by Bellman and Zadeh [33] in decision making problems. After that, several researchers were being engaged to characterize the actual nature of the fuzzy sets [Dubois and Prade [6], Kaufmann and Gupta [3], Báez-Sáncheza et al. [1], Beg and Ashraf [15], Ban and Coroianu [2], Deli and Broumi [18]. The concept of eigen fuzzy number sets was developed nicely by Goetschel and Voxman [36]. Baruah [10 –14] discussed the set superimposition and according to the basis of fuzziness he used reference function to characterize the suitable fuzzy membership function of the fuzzy sets itself. Chutia et al. [32] developed an alternative method of finding the membership of a fuzzy number, by the same time also Mahanta et al. [39] were able to construct the structure of fuzzy arithmetic without the help of α –cuts. Roychoudhury and Pedrycz [43] gave the concept of fuzzy complement functional but the Cauchy problem under fuzzy control developed by Bobylev [48, 49]. Buckley [19] have generalized and extended the fuzzy sets to practical applications.

Diamond [26, 27] gave a new structure of k type fuzzy numbers and finally found star shaped fuzzy sets. Diamond and Kloeden [24, 25] had characterized on compact fuzzy sets and parameterized the fuzzy sets into single valued mappings. Fuzzy mappings and fixed point theorem studied by Heilpern [40]. The chaotic iterations of fuzzy sets studied by and interrelations with natural numbers (Peano theorem) and the links with differential equations had been studied by Kloeden [28 –30] respectively.

However, in the process of developing the defuzzification methods, specially on ranking fuzzy numbers Yager [38] has kept a remarkable contribution in the subject itself. Since then, numerous researchers have begun to study over the ranking methods and finally invented numerous formulae over the subject. Researchers like Chu and Tsao [45], Ramli and Mohamad [23], Allahviranloo and Saneifard [44], Ezzati et al. [35], Deng [9] etc. have used the methods for ranking fuzzy numbers based on center of gravity. Cheng [5] discussed a new approach for ranking fuzzy numbers by distance method. Buckley and Chanas [20] have studied a fast method of ranking alternatives using fuzzy numbers, Ezatti and Saneifard [34] developed a method of continuous weighted quasi-arithmetic means. Based on deviation degree, the extensive works over L-R fuzzy numbers made by Wang et al. [50], Kumar et al. [4], Hajjari and Abbasbandy [46], Xu et al. [31] etc. have kept a new milestone in the subject. Yu et al. [47] introduced fuzzy ranking generalizing fuzzy numbers in fuzzy decision making based on the left and right transfer coefficients and areas. De and Sana [41] have used centre of gravity method for a pentagonal fuzzy number in inventory decision making problem.

Beg and Rasid [16], Zhang et al. [8] and Deli [17] studied a new method for ranking fuzzy numbers and its application to group decision making problems. Recently, Rezvani [42] developed a method for ranking generalized exponential trapezoidal fuzzy numbers based on variance.

In our present study, we have introduced dense fuzzy set and its corresponding new defuzzification technique. Moreover, its applicability in multidisciplinary subjects beyond basic sciences and the science itself within has been explained extensively. For defuzzifications, we extend the one of the existing methods. In all the cases the crisp convergences have been tested in a nice way.

2 Triangular dense fuzzy set (TDFS)

In this section we introduce different definitions of dense fuzzy set and triangular dense fuzzy set with their graphic representations and examples for subsequent use.

Definition 2.1. Consider the fuzzy set $\tilde{A}$ whose components are sequence of functions generating from the mapping of natural numbers with a crisp number x. Now if all the components converge to the crisp number x as n→ ∞ then the fuzzy sets under considerations are called dense fuzzy set (DFS) (See Fig. 1).

Definition 2.2. Let a fuzzy number $\tilde{A} = 〈 a_{1}, a_{2}, a_{3} 〉$ with a₁ = a₂f_n and a₃ = a₂g_n, where f_n and g_n are the sequence of functions. If f_n and g_n are both converge to 1 as n→ ∞ then the fuzzy set $\tilde{A} = 〈 a_{1}, a_{2}, a_{3} 〉$ converges to a crisp singleton {a₂}. Then we call the fuzzy set $\tilde{A} = 〈 a_{1}, a_{2}, a_{3} 〉$ as a Triangular dense fuzzy set (TDFS).

Definition 2.3.Alternative definition of TDFS. Let, $\tilde{B} = 〈 f_{n}, g_{n}, h_{n} 〉$ be three sequences of the elements of a triangular fuzzy set, where f_n and g_n are the sequence of functions. Now, if f_n → a₂, g_n → a₂ and h_n → a₂ holds for n→ ∞ then the fuzzy set $\tilde{B}$ converges to a crisp singleton {a₂}. Then we call the fuzzy set $\tilde{B} = 〈 f_{n}, g_{n}, h_{n} 〉$ as a Triangular dense fuzzy set.

Definition 2.4.Definition of TDFS based on Cartesian product of two sets. Let $\tilde{A}$ be the fuzzy number whose components are the elements of $R \times N, R$ being the set of real numbers and N being the set of natural numbers with the membership grade satisfying the functional relation $μ : R \times N \to [0, 1]$ . Now as n→ ∞ if μ (x, n) → 1 for some $x \in R and n \in N$ then we call the set $\tilde{A}$ as dense fuzzy set. If $\tilde{A}$ is triangular then it is called TDFS. Now, if for some n in N, μ (x, n) attains the highest membership degree 1 then the set itself is called “Normalized Triangular Dense Fuzzy Set” or NTDFS (See Fig. 2).

Definition 2.5.Definition of TDFS based on non-membership function. Let $\tilde{A}$ be the fuzzy number whose components are the elements of $R \times N$ whose non-membership grade satisfying the functional relation $γ : R \times N \to [0, 1]$ . Now as n→ ∞ if γ (x, n) → 0 for some $x \in R and n \in N$ then we call the set $\tilde{A}$ as dense fuzzy set. If we consider the fuzzy number $\tilde{A}$ of the form $\tilde{A} = 〈 a_{1}, a_{2}, a_{3} 〉$ then we call it “Triangular Dense Fuzzy Set”. Now, if for n = 0, γ (x, n) attains the highest membership degree 1 then we can express this fuzzy number as “Normalized Triangular Dense Fuzzy Set” or NTDFS (See Fig. 3).

Example 2.6. As per definitions (1–4) let us assume the TDFS as follows $\begin{matrix} \tilde{A} = 〈 a_{2} (1 - \frac{ρ}{1 + n}), a_{2}, a_{2} (1 + \frac{σ}{1 + n}) 〉, \\ for 0 < ρ, σ < 1 \end{matrix}$ (1)

The memberships function for 0 ≤ n is defined as follows: $\begin{matrix} μ (x, n) = \\ {\begin{matrix} 0 if x < a_{2} (1 - \frac{ρ}{1 + n}) and x > a_{2} (1 + \frac{σ}{1 + n}) \\ {\frac{x - a_{2} (1 - \frac{ρ}{1 + n})}{\frac{ρ a_{2}}{1 + n}}} if a_{2} (1 - \frac{ρ}{1 + n}) \leq x \leq a_{2} \\ {\frac{a_{2} (1 + \frac{σ}{1 + n}) - x}{\frac{σ a_{2}}{1 + n}}} if a_{2} \leq x \leq a_{2} (1 + \frac{σ}{1 + n}) \end{matrix} \end{matrix}$ (2)

3 Defuzzification method based on α-cuts

The left and the right α-cuts of a triangular dense fuzzy number $\tilde{A} = μ (x, n)$ are, $L^{- 1} (α, n) = a_{2} (1 - \frac{ρ}{1 + n} + \frac{ρ α}{1 + n})$

and $R^{- 1} (α, n) = a_{2} (1 + \frac{σ}{1 + n} - \frac{σ α}{1 + n}) .$

So, L^-1 (α, n) + R^-1 (α, n) $\begin{matrix} = a_{2} (2 - \frac{ρ}{1 + n} + \frac{ρ α}{1 + n} + \frac{σ}{1 + n} - \frac{σ α}{1 + n}) \\ = a_{2} {2 + \frac{(ρ - σ) α}{1 + n} + \frac{σ - ρ}{1 + n}} \end{matrix}$

Now, we use the Yager’s [38] method of ranking index as an extension of the new defuzzification method for the TDFS $\tilde{A}$ and it is stated as follows: $I (\tilde{A)} = \frac{1}{2 N} \sum_{n = 0}^{N} \int_{0}^{1} {L^{- 1} (α, n) + R^{- 1} (α, n)} d α$ (3)

Thus, $\begin{matrix} \int_{0}^{1} {L^{- 1} (α, n) + R^{- 1} (α, n)} d α \\ = a_{2} {2 + \frac{σ - ρ}{2 (1 + n)}} . \end{matrix}$

Hence $I (\tilde{A)} = \frac{1}{2 N} \sum_{n = 0}^{N} a_{2} {2 + \frac{σ - ρ}{2 (1 + n)}} = \frac{a_{2}}{2 N} [2 N + \frac{σ - ρ}{2} {\frac{1}{1 + 0} + \frac{1}{1 + 1} + \frac{1}{1 + 2} + \dots \dots . + \frac{1}{1 + N}}]$ (4)

Obviously, as N → ∞, $I (\tilde{A)} \to a_{2}$ .∥Note that, if we take the above formula for continuous case, then above method reduces to the following: $\begin{matrix} I (\tilde{A)} = \frac{1}{2 k} \int_{0}^{k} \int_{0}^{1} {L^{- 1} (α, y) + R^{- 1} (α, y)} d α dy \\ = a_{2} {1 + \frac{(σ - ρ)}{4 k} Log (1 + k)} \end{matrix}$ (5)

4 Defuzzification based on area compensation

The area compensation method for the general triangular fuzzy number is defined as $R_{old} = \frac{\int_{a_{1}}^{a_{2}} x μ (x) dx + \int_{a_{2}}^{a_{3}} x μ (x) dx}{\int_{a_{1}}^{a_{2}} μ (x) dx + \int_{a_{2}}^{a_{3}} μ (x) dx}$ (6)

Now we propose the new defuzzification method and is defined as

$\begin{matrix} R_{new} = \frac{\sum_{n = 1}^{N} [\int_{a_{1}}^{a_{2}} x μ (x, n) dx + \int_{a_{2}}^{a_{3}} x μ (x, n) dx]}{\sum_{n = 0}^{N} [\int_{a_{1}}^{a_{2}} μ (x, n) dx + \int_{a_{2}}^{a_{3}} μ (x, n) dx]} \\ = \frac{P}{Q} \end{matrix}$ (7)

Here,

$\begin{matrix} P = \sum_{n = 0}^{N} [\int_{a_{1}}^{a_{2}} x μ (x, n) dx + \int_{a_{2}}^{a_{3}} x μ (x, n) dx] \\ = \sum_{n = 0}^{N} [\int_{a_{1}}^{a_{2}} \frac{x^{2} - a_{1} x}{a_{2} - a_{1}} dx + \int_{a_{2}}^{a_{3}} \frac{a_{3} x - x^{2}}{a_{3} - a_{2}} dx] by (2) \\ = \sum_{n = 0}^{N} [\frac{(a_{3} - a_{1}) (a_{1} + a_{2} + a_{3})}{6}] \\ = \sum_{n = 0}^{N} \frac{a_{2}^{2}}{6} [\frac{σ^{2} - ρ^{2}}{{(1 + n)}^{2}} + \frac{3 (σ + ρ}{(1 + n)}] \end{matrix}$ (8)

and

$\begin{matrix} Q = \sum_{n = 0}^{N} [\int_{a_{1}}^{a_{2}} μ (x, n) dx + \int_{a_{2}}^{a_{3}} μ (x, n) dx] \\ = \sum_{n = 0}^{N} [\int_{a_{1}}^{a_{2}} \frac{x - a_{1}}{a_{2} - a_{1}} dx + \int_{a_{2}}^{a_{3}} \frac{a_{3} - x}{a_{3} - a_{2}} dx] \\ = \sum_{n = 0}^{N} \frac{a_{2} (σ + ρ)}{2 (1 + n)} by (2) \end{matrix}$ (9)

Thus,

$\begin{matrix} R_{new} = \frac{P}{Q} = \frac{\sum_{n = 0}^{N} \frac{a_{2}^{2}}{6} [\frac{σ^{2} - ρ^{2}}{{(1 + n)}^{2}} + \frac{3 (σ + ρ}{(1 + n)}]}{\sum_{n = 0}^{N} \frac{a_{2} (σ + ρ)}{2 (1 + n)}} \\ = \frac{a_{2}}{3} \frac{\sum_{n = 0}^{N} [\frac{σ - ρ}{{(1 + n)}^{2}} + \frac{3}{(1 + n)}]}{\sum_{n = 0}^{N} \frac{1}{(1 + n)}} \end{matrix}$ $\begin{matrix} = \frac{a_{2}}{3} \frac{[\begin{matrix} (σ - ρ) {1 + \frac{1}{2^{2}} + \frac{1}{3^{2}} + \frac{1}{4^{2}} + \dots + \frac{1}{N^{2}} + \frac{1}{{(1 + N)}^{2}}} \\ + 3 {1 + \frac{1}{2} + \frac{1}{3} + \frac{1}{4} + \dots \dots \dots \dots \dots + \frac{1}{N} + \frac{1}{(1 + N)}} \end{matrix}]}{[1 + \frac{1}{2} + \frac{1}{3} + \frac{1}{4} + \dots \dots \dots \dots \dots + \frac{1}{N} + \frac{1}{(1 + N)}]} \\ = a_{2} [1 + \frac{(σ - ρ)}{3} \frac{{1 + \frac{1}{2^{2}} + \frac{1}{3^{2}} + \frac{1}{4^{2}} + + \frac{1}{N^{2}} + \frac{1}{{(1 + N)}^{2}}}}{{1 + \frac{1}{2} + \frac{1}{3} + \frac{1}{4} + \dots \dots \dots \dots \dots . + \frac{1}{N} + \frac{1}{(1 + N)}}}] \end{matrix}$ (10)

Now, the term $\frac{{1 + \frac{1}{2^{2}} + \frac{1}{3^{2}} + \frac{1}{4^{2}} + + \frac{1}{N^{2}} + \frac{1}{{(1 + N)}^{2}}}}{{1 + \frac{1}{2} + \frac{1}{3} + \frac{1}{4} + \dots \dots \dots \dots \dots + \frac{1}{N} + \frac{1}{(1 + N)}}} \to 0 as N \to \infty$

Because we have, $\begin{matrix} lim_{N \to \infty} \frac{{1 + \frac{1}{2^{2}} + \frac{1}{3^{2}} + \frac{1}{4^{2}} + + \frac{1}{N^{2}} + \frac{1}{{(1 + N)}^{2}}}}{{1 + \frac{1}{2} + \frac{1}{3} + \frac{1}{4} + \dots \dots \dots \dots \dots + \frac{1}{N} + \frac{1}{(1 + N)}}} \\ = lim_{k \to 0} k \times \frac{\lim_{k \to 0} \sum_{r = 1}^{N + 1} \frac{1}{{(rk)}^{2}}}{\lim_{k \to 0} \sum_{r = 1}^{N + 1} (\frac{1}{rk})} \\ = lim_{k \to 0} k \times \frac{\lim_{k \to 0} \sum_{r = 1}^{N + 1} \frac{1}{{(rk)}^{2}}}{\lim_{k \to 0} \sum_{r = 1}^{N + 1} (\frac{1}{rk})} \end{matrix}$

Hence, R_new → a₂.

Note that, if we take, the above sum as for continuous form then our formula reduces to $R_{new}^{'} = \frac{\int_{y = 0}^{k} [\int_{a_{1}}^{a_{2}} x μ (x, y) dx + \int_{a_{2}}^{a_{3}} x μ (x, y) dx] dy}{\int_{y = 0}^{k} [\int_{a_{1}}^{a_{2}} μ (x, n) dx + \int_{a_{2}}^{a_{3}} μ (x, n) dx] dy} = \frac{P^{'}}{Q^{'}}$ (11)

Where, on simplifications, $P^{'} = \frac{a_{2}^{2}}{6} (σ + ρ) [\frac{k (σ - ρ)}{1 + k} + 3 Log (1 + k)] .$

And $Q^{'} = \frac{a_{2}}{2} (σ + ρ) Log (1 + k) .$

Thus, $\begin{matrix} R_{new}^{'} = \frac{P^{'}}{Q^{'}} = \frac{a_{2}}{3} [3 + (σ - ρ) \frac{k}{(1 + k) Log (1 + k)}] \\ = a_{2} [1 + \frac{(σ - ρ)}{3} \frac{k}{(1 + k) Log (1 + k)}] \end{matrix}$ (12)

Now, $lim_{k \to \infty} \frac{k}{(1 + k) Log (1 + k)} (\frac{\infty}{\infty} form) \to 0,$

on L’Hospital rule.

Hence, $R_{new}^{'} \to a_{2}$ .

5 Defuzzification based on graded mean integration method

We have, the defuzzification formula defined as

$\begin{matrix} P_{old} = \frac{\frac{1}{2} \int_{0}^{1} (L_{A}^{- 1} (α) + R_{A}^{- 1} (α)) d α}{\int_{0}^{1} α d α} \\ = \frac{(a_{1} + 4 a_{2} + a_{3})}{6} \\ P_{new} = \frac{\sum_{n = 0}^{N} \frac{1}{2} \int_{0}^{1} (L_{A}^{- 1} (α, n) + R_{A}^{- 1} (α, n)) d α}{N \int_{0}^{1} α d α} \\ = \frac{\frac{1}{4} \sum_{n = 0}^{N} a_{2} {2 + \frac{σ - ρ}{2 (1 + n)}}}{\frac{N}{2}} \\ = a_{2} [1 + \frac{(σ - ρ)}{4 N} {1 + \frac{1}{2} \\ + \frac{1}{3} + \dots \dots \dots + \frac{1}{N} + \frac{1}{1 + N}}] \end{matrix}$ (13)

Obviously, as N→ ∞, P_new → a₂. From Cauchy sequence, we have, $\begin{matrix} lim_{N \to \infty} \frac{1}{N} {1 + \frac{1}{2} + \frac{1}{3} + \dots \dots \dots + \frac{1}{N} + \frac{1}{1 + N}} \\ = lim_{N \to \infty} \frac{1}{N^{2}} {N + \frac{N}{2} + \frac{N}{3} + \dots \dots \dots + \frac{N}{N} + \frac{N}{1 + N}} \end{matrix}$

and $\begin{matrix} lim_{k \to 0} k \times \lim_{k \to 0} k \sum_{r = 1}^{N + 1} \frac{1}{rk} = lim_{k \to 0} k \times \lim_{ɛ \to 0} \int_{ɛ}^{1} \frac{dx}{x} \\ = 0 \times \lim_{ɛ \to 0} log (\frac{1}{ɛ}) = 0 . \end{matrix}$

6 Implication of dense fuzzy sets

The fuzzy sets were first developed to capture the decision under linguistic ambiguity in its semantic meaning which has been using day by day in practice. Moreover, as per existing knowledge concerned, at the very beginning of a competitive sphere or any kind of psychological testing and diagnosis the ultimate vagueness assumes high. Therefore, whenever we began to talk with and experiencing with the same subject the ambiguity began to decrease from the process. By this way, individual as well as organizational behaviour have been changed. Such kind of changes cannot be ignored and cannot be avoided by means of any costs rather it has been recognised. Finally, no ambiguity will be viewed after careful nourishment of the subject itself. Hence, the dense fuzzy set may be applied for the following subjects.

In any kind of decision making process

Example: Suppose, in a newly open inventory system, the demand rate for a particular commodity is totally unknown due to the non experience and lack of knowledge over public needs. But as the day passes the fact will be understood by the decision maker and finally be settled in a finite demand for that particular item.

Let initially the demand of an item assumes a fuzzy number $\begin{matrix} \tilde{A} = 〈 a_{2} (1 - \frac{ρ}{1 + n}), a_{2}, a_{2} (1 + \frac{σ}{1 + n}) 〉, \\ for 0 < ρ, σ < 1 for n = 0 . \end{matrix}$ But as the days going on, after n days later the demand rate appears as above for n ≠ 0.

We see that, as $n \to \infty, \tilde{A} \to a_{2}$ (a finite value).

Psychological testing/ military selection

Example: Let us consider we are interested to perform a best psychological test among a group of researcher/youth to know whether all of them are fulfilling the basic merit or not. To do this, we make a questionnaire and apply over each of them. We observe that, as soon as the questionnaire has been distributed among the researchers/youth they began to answer. But, it is viewed that, initially they make few wrong responses and if we put the time duration quite more, then the chances of getting right response become high. Now this kind of chances can be put in the following fuzzy information $\begin{matrix} \tilde{A} = 〈 (1 - \frac{ρ}{t}), 1, (1 + \frac{σ}{t}) 〉, \\ for 0 < ρ, σ < 1 and t > 0 \end{matrix}$

Note that, as $t \to \infty, \tilde{A} \to 1$ .

Measurement of perfection and accuracy of any kind of computation

Example: We know, for any kind of measurement the errors made by the experimenter follows the Gaussian curve. Unlike confidence interval is statistics, the fuzzy confidence interval (towards certainties) will be near the most affordable fuzzy interval measure $\begin{matrix} \tilde{A} = 〈 a (1 - \frac{σ}{n}), (1 + \frac{σ}{n}) a 〉, \\ for 0 < σ, n \in N \end{matrix}$

Cryptography

Example: The basic insights behind the cryptography are the process of encoding and decoding of a particular message. With the help of dense fuzzy set, we may use several membership functions taking n as binary digits in the associated membership function and reset the decoding message as its defuzzified value and vice versa.

The respective fuzzy set may be of the form $\begin{matrix} \tilde{A} = 〈 a (1 - \frac{σ}{n}), (1 + \frac{ρ - σ}{2 n}) a, (1 + \frac{ρ}{n}) a 〉, \\ for 0 < σ, ρ < 1 and n \in N \end{matrix}$

$\begin{matrix} \tilde{B} = 〈 b (1 - \frac{σ}{n}), (1 + \frac{ρ - σ}{2 n}) b, b, (1 + \frac{ρ}{n}) b 〉 \\ for 0 < σ, ρ < 1 and n \in N \end{matrix}$

Analysing the fuzzy differential and integral calculus

Example: In dense fuzzy set we see, the fuzzy numbers around x can be stated as the sequence of functions of natural numbers. From the Definition 3, we see, for $\tilde{B} = 〈 f_{n}, g_{n} 〉, h_{n}, lim_{n \to \infty} f_{n} = lim_{n \to \infty} g_{n} = lim_{n \to \infty} h_{n}$

Therefore, by this study we may incorporate the following integral relation

$\int_{a}^{b} f (x) dx = lim_{n \to \infty} \frac{1}{n} \sum_{r = 1}^{n} f (a + \frac{r}{n})$ from the sum of the sequences {f_n (x) } _n.

Photography

Example: Taking several membership functions, also using complex numbers in developing the membership functions different images can be performed. The closure values of n giving the darkness of the image and brightness correspond to the amplitude of the complex fuzzy membership function. It may be the following complex function Z (x, n) = f_n (x) + ig_n (x), where f_n and g_n are dense sequences.

Crime research / Behavioural study

Example: Here the ‘truth’ and ‘untruth’ of a particular event can be measured by the degree of fuzziness over number of observations or frequencies of interviews. For instance, at the time of interview, we put the values of ‘truth’ as {x}, then the fuzzy information behind this fact is given by $\begin{matrix} \tilde{C} = 〈 x_{n} (1 - \frac{σ}{n}), x, (1 + \frac{ρ}{n}) x_{n} 〉 \\ for 0 < σ, ρ < 1 and n \in N, x_{n} \in R \end{matrix}$

i) Truth value converges in a harmonic region (Shown in Fig. 1)

ii) Truth value deviates with frequencies of interviews (Shown in Fig. 4)

The concerned fuzzy set is given by $\begin{matrix} \tilde{M} = 〈 x, x, (1 + \frac{ρ}{n}) x_{n} 〉 \\ for 0 < ρ < 1 and n \in N, x_{n} \in R \end{matrix}$

iii) Truth value converges with frequencies of interviews (Shown in Fig. 5)

In this case, the fuzzy set can be stated as follows: $\begin{matrix} \tilde{M} = 〈 x_{n} (1 - \frac{σ}{n}), x, x 〉 \\ for 0 < σ < 1 and n \in N, x_{n} \in R \end{matrix}$

iv) Truth value never be found (Shown in Fig. 6)

In this case, the subject has decided not to speak truth all the time.

The corresponding fuzzy set is given by $\begin{matrix} \tilde{M} = 〈 x_{n} (- 1 + \frac{σ}{n}), x, (1 - \frac{ρ}{n}) x_{n} 〉 \\ for 0 < σ, ρ < 1 and n \in N, x_{n} \in R \end{matrix}$

v) Truth value may come within a small sphere of divergence (Shown in Fig. 7)

In this case the corresponding fuzzy set may be the interval valued DFS and it can be defined as follows: $\begin{matrix} \tilde{M} = 〈 P_{n} (x), x, Q_{n} (x) 〉 \\ for 0 < σ, ρ < 1 and n \in N, x_{n} \in R \end{matrix}$

Where, $P_{n} (x) = 〈 x (1 - \frac{2 ρ}{n}), (1 - \frac{ρ}{n}) x 〉$ and $Q_{n} (x) = 〈 x (1 + \frac{σ}{n}), (1 + \frac{2 σ}{n}) x 〉 .$

Filtering of noisy environment or digital image

In this area, specially on Gaussian noise Kaur and Singh [7] removal, a verification and cross verification can be done by developing the filter mechanism under dense fuzzy rule. This can only be possible by taking the dense fuzzy set like $\begin{matrix} \tilde{P_{m, n} (x)} = 〈 x (1 - \frac{ρ}{m}), x, (1 + \frac{σ}{n}) x 〉 \\ for 0 < σ, ρ < 1 and m, n \in N, x \in R \end{matrix}$

Here m, n can be resets by the experimenter alone. Then taking mean, median, mode, quartiles etc. of the proposed fuzzy noise level or fuzzy digital images an experimenter can easily reach to his specified goal.

Vulnerability/ risk assessment of the subjects under disaster management

Any kind of disaster whether it is natural (earth quake, Tsunami, Strom etc) or manmade (flood, accidents, military attacks etc.) must be harmful to any kind of subjects like loss of property, concrete like structure constructions even for human health. The present situations may be evaluated and this may be posing by means of fuzzy sets for n = 0, then its distortions may be observed after the occurrences of the several frequencies of disasters n = 1, 2. 3 ... .. Here, all types of dense fuzzy sets (wherever necessary) can be utilised to predict the final vulnerability or risk. The existing research may be viewed in Takacs [22] and Mishra et al. [37].

7 Conclusion

Here we have developed a new fuzzy set named dense fuzzy set. In this study, we have also extended few existing defuzzification methods by considering the sequence of functions for the case of triangular fuzzy number. In real life problems, the basic aim of fuzzy sets is to measure the precision and truth value of a particular event. To get this, several efforts can be taken care of. It may come in a rapid way (cases of rapid convergence of the sequence), it may come slowly (cases of harmonic sequence) or it never be achieved (cases of fully divergent sequences). Each case has been explained with their respective graphical view. We plan to further study in future on applications of our method in fuzzy differential and integral calculus. In addition to applications in several existing fields we also expect our research work will have wide applications in new multidisciplinary areas of research of environmental hazards like, risk assessment of the subjects under disaster management, filtering of pollution levels, etc.

Footnotes

Acknowledgments

Authors are grateful to the editor and the referees for a very careful reading and their valuable comments and suggestions.

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