Complex vague soft sets and its distance measures

Abstract

In this paper, we propose the concept of complex vague soft sets which are vague soft sets defined in a complex setting. Based on this new concept we define some concepts related to this notion as well as some basic operations namely the complement, union, intersection, AND and OR. The basic properties and relevant laws pertaining to this concept such as the De Morgan’s laws are also verified. We introduce the axiomatic definition of the distance function between two complex vague soft sets and subsequently define several distance measures between complex vague soft sets. Finally some of the algebraic properties of these distance measures are verified.

Keywords

Complex vague soft set complex fuzzy set complex intuitionistic fuzzy set vague soft set fuzzy set

1 Introduction

Fuzzy set theory and complex numbers were first combined to establish the notion of fuzzy complex numbers by Buckley in 1989 [1]. Subsequently, the operations of differentiation and integration on complex fuzzy numbers as well as solving fuzzy linear and quadratic equations involving fuzzy complex numbers were studied from 1990 onwards (see [2 –5]). All this subsequently led to the introduction of fuzzy complex analysis by various researchers (see [6, 7]).

The set of complex numbers has the capability to solve many problems that traditionally cannot be solved using the set of real numbers, such as the evaluation of the improper integrals that represent the electrical resistance in the field of engineering. Thus the application of soft sets, fuzzy sets and the hybrid structures of these sets to complex numbers is an essential step to incorporate the advantages of complex numbers to the notion of soft sets, fuzzy sets and its generalizations. Complex numbers can also be employed in representing information that happens periodically because of the existence of the phase terms and amplitude terms. Ramot et al. [8], Tamir and Kandel [9] and Alkouri and Salleh [10] have illustrated how some information have same values with different meanings at different times can be presented in the form of complex fuzzy sets and complex intuitionistic fuzzy sets.

In 2002, Ramot et al. [8] introduced a new concept called complex fuzzy sets which is an extension to the notion of fuzzy sets. Ramot’s complex fuzzy sets is characterized by a membership function whose range is not limited to [0, 1] but extends to the unit circle in the complex plane, which includes the amplitude and phase terms while the range of Buckley’s complex fuzzy numbers are within the closed interval [0, 1] . In 2011, Tamer et al. [9] introduced a new interpretation of complex fuzzy membership grade, where the complex fuzzy membership grade can be represented in polar form and Cartesian form with fuzzy components for both the amplitude and phase terms. Many researchers have since studied this new form of complex fuzzy sets and developed various concepts related to this notion (see [11, 12]). The membership function for a complex fuzzy set A denoted by μ_A (x) = r_A (x) . e^-iω_A(x) is a complex-valued function that assigns a grade of membership to any element x in the universe of discourse and r_A (x) and ω_A (x) are called the amplitude term and phase term respectively and are both real-valued, with r_A (x) ∈ [0, 1] . The key feature of complex fuzzy sets is the presence of the phase term and its membership function whose values are extended to the unit circle in the complex plane [8]. These features gives complex fuzzy sets wave-like properties that could be used to describe constructive and destructive interference depending on the phase value of an element. It is also very useful in accurately describing problem parameters that involve periodicity and varies with time.

This paper aims to introduce the notion of complex vague soft sets by extending the range of the truth-membership function and the false- membership function from a subinterval of [0, 1] to the unit circle in the complex plane. Thus the concept of complex vague soft sets introduced here has the added advantage of having the characteristics of vague soft sets and the features of complex numbers such as the presence of the amplitude and phase terms and as such, can be used to represent vague and uncertain information that happens periodically and has different meanings at different points in time.

2 Preliminaries

In this section, we recapitulate the concept of soft sets and its different hybrid structures such as vague soft sets, and complex fuzzy sets which are relevant to this paper.

Definition 2.1. [13] A fuzzy setA in a universe of discourse U is characterized by a membership function μ_A (x) that takes on values in the interval [0, 1] where A ⊂ U and μ_A : x → [0, 1] , ∀ x ∈ U .

Definition 2.2. [14] Let X be a space of points (objects) with element of X denoted by x. A vague setV in X is characterized by a truth-membership function t_V : X → [0, 1] and a false-membership function f_V : X → [0, 1]. The value t_V (x) is a lower bound on the grade of membership of x derived from the evidence for x and f_V (x) is a lower bound on the negation of x derived from the evidence against x. The values t_V (x) and f_V (x) both associate a real number in the interval [0, 1] with each point in X, where 0 ≤ t_V (x) + f_V (x) ≤ 1 . This approach bounds the grade of membership of x to a closed subinterval [t_V (x) , 1 - f_V (x)] of [0, 1] .

When X is continuous, the vague set V can be written as: $V = \int [t_{V} (x), 1 - f_{V} (x)] / x, where x \in X .$

When X is discrete, the vague set V can be written as: $V = \sum_{i = 1}^{n} [t_{V} (x_{i}), 1 - f_{V} (x_{i})] / x_{i}, where x_{i} \in X .$

Definition 2.3. [14] Let A = {〈u, [t_A (x) , 1 - f_A (x)] 〉 : u ∈ U} and B = {〈u, [t_B (x) , 1 - f_B (x)] 〉 : u ∈ U} be two vague sets over a universe U . Then we have the following basic operations on vague sets A and B .

(i) The complement of A denoted by A^c is defined by: $t_{A^{c}} (x) = f_{A} (x)$ and $1 - f_{A^{c}} (x) = 1 - t_{A} (x) .$

(ii) The union (intersection) of A and B is a vague set C, denoted as C = A ∪ (∩) B, with lower and upper boundary of the truth-membership function given by: $t_{C} (x) = max (min) (t_{A} (x), t_{B} (x))$ and $\begin{matrix} 1 - f_{C} (x) & = & max (min) (1 - f_{A} (x), 1 - f_{B} (x)) \\ = & 1 - min (max) (f_{A}, f_{B}) . \end{matrix}$

Now we shall recall the definition of soft set. For this, let U be a universe and A be a set of parameters. The set of all subsets of U is denoted by P (U) .

Definition 2.4. [15] A pair (F, A) is called a soft set over U, where F is a mapping given by F : A → P (U). In other words, a soft set over U is a parameterized family of subsets of the universe U. For ɛ ∈ A, F (ɛ) may be considered as the set of ɛ-elements of the soft set (F, A) or as the ɛ-approximate elements of the soft set.

Definition 2.5. [16] Let U be an initial universal set and E be a set of parameters. Let F (U) denote the set of all fuzzy sets of U and A ⊂ E . A pair (F, A) is called a fuzzy soft set over U, where F is a mapping given by F : A → F (U) .

The notion of vague soft sets was introduced by Xu et al. [17] as a generalization of the notion of soft sets. It is an improvement to soft sets and provides a means to deal with the vagueness of the problems involving complex data with a high level of uncertainty and imprecision.

Definition 2.6. [17] A pair $(\hat{F}, A)$ is called a vague soft set over U, where $\hat{F}$ is a mapping given by $\hat{F} : A \to V (U)$ and V (U) is the set of all vague sets of U . In other words, a vague soft set over U is a parameterized family of vague set of the universe U . For all a ∈ A, $μ_{{\hat{F}}_{a}} : U \to [0, 1] \times [0, 1]$ is regarded as the set ɛ-approximate elements of the vague soft set $(\hat{F}, A) .$ Therefore a vague soft set is a collection of approximations of the following form: $\begin{matrix} (\hat{F}, A) & = & {\hat{F} (a_{i}) : i = 1, 2, 3, \dots} \\ = & {\frac{[t_{{\hat{F}}_{(a_{i})}} (x_{i}), 1 - f_{{\hat{F}}_{(a_{i})}} (x_{i})]}{x_{i}} \\ : i = 1, 2, 3, \dots} \end{matrix}$ for all a_i ∈ A and for all x_i ∈ U .

Definition 2.7. [17] Let $(\hat{F}, A)$ and $(\hat{G}, B)$ be two vague soft sets over a common universe U . Then the basic set theoretic operations between $(\hat{F}, A)$ and $(\hat{G}, B)$ are as given below:

(i) The complement of $(\hat{F}, A),$ denoted by ${(\hat{F}, A)}^{c}$ is defined by ${(\hat{F}, A)}^{c} = ({\hat{F}}^{c}, \neg A)$ where ${\hat{F}}^{c} : \neg A \to V (U)$ and $t_{{\hat{F}}^{c} (α)} (x) = f_{\hat{F} (\neg α)} (x), 1 - f_{{\hat{F}}^{c} (α)} (x) = 1 - t_{\hat{F} (\neg α)} (x), \forall α \in \neg A, x \in U .$

(ii) The union (intersection) of $(\hat{F}, A)$ and $(\hat{G}, B)$ is a vague soft set denoted by $(\hat{H}, C) = (\hat{F}, A) \tilde{\cup} (\tilde{\cap}) (\hat{G}, B),$ where C = A ∪ B and ∀e ∈ C, $t_{{\hat{H}}_{c}} (x) = {\begin{matrix} t_{{\hat{F}}_{c}} (x), & e \in A - B, \\ t_{{\hat{G}}_{c}} (x), & e \in B - A, \\ max (min) (t_{{\hat{F}}_{c}} (x), t_{{\hat{G}}_{c}} (x)), & e \in B \cap A \end{matrix}$ and $\begin{matrix} 1 - f_{{\hat{H}}_{c}} (x) \\ = {\begin{matrix} 1 - f_{{\hat{F}}_{c}} (x), & e \in A - B, \\ 1 - f_{{\hat{G}}_{c}} (x), & e \in B - A, \\ max (min) (1 - f_{{\hat{F}}_{c}} (x), & e \in B \cap A, \\ 1 - f_{{\hat{G}}_{c}} (x)), \end{matrix} \end{matrix}$ where $t_{{\hat{H}}_{c}}$ and $f_{{\hat{H}}_{c}}$ are the respective truth and false membership functions of ${\hat{H}}_{c} .$

The novelty of the concept of complex fuzzy sets introduced by Ramot et al. [8] lies in the range of values in its membership function which is not limited to the interval [1], but extended to the unit circle in the complex plane. Some of the basic concepts pertaining to complex fuzzy sets are as follows:

Definition 2.8. [8] A complex fuzzy setA defined on a universe of discourse U is characterized by a membership function μ_A (x) , that assigns to any element x ∈ U, a complex-valued grade of membership in A . By definition, the values that μ_A (x) may receive all lie within the unit circle in the complex plane and are of the form μ_A (x) = r_A (x) e^iω_A(x), where $i = \sqrt{- 1},$ each of r_A (x) and ω_A (x) are both real-valued and r_A (x) ∈ [0, 1] . The complex fuzzy set A may be represented as the set of ordered pairs: $\begin{matrix} A & = & {(x, μ_{A} (x)) : x \in U} \\ = & {(x, r_{A} (x) e^{i ω_{A} (x)}) : x \in U} . \end{matrix}$

Definition 2.9. [8] Let A and B be two complex fuzzy sets on U and μ_A (x) = r_A (x) e^iω_A(x) and μ_B (x) = r_B (x) e^iω_B(x) be the membership functions of A and B respectively. Then we have the following basic set theoretic operations on complex fuzzy sets A and B .

(i) The complement of A, denoted by $\bar{A},$ is defined as follows: $\begin{matrix} \bar{A} & = & {(x, μ_{\bar{A}} (x)) : x \in U} \\ = & {(x, r_{\bar{A}} (x) e^{i ω_{\bar{A}} (x)}) : x \in U}, \end{matrix}$ where $r_{\bar{A}} (x) = (1 - r_{A} (x))$ and $ω_{\bar{A}} (x) = 2 π - ω_{A} (x) .$

(ii) The union of A and B, denoted by A ∪ B is defined as follows: $\begin{matrix} A \cup B & = & {(x, μ_{A \cup B} (x)) : x \in U} \\ = & {(x, r_{A \cup B} (x) e^{i ω_{A \overset{B}{\cup}} (x)}) : x \in U}, \end{matrix}$ where r_A∪B (x) = max{ r_A (x) , r_B (x) } and ω_A∪B (x) = max(ω_A (x) , ω_B (x)) .

(iii) The intersection of A and B, denoted by A ∩ B is defined as follows: $\begin{matrix} A \cap B & = & {(x, μ_{A \cap B} (x)) : x \in U} \\ = & {(x, r_{A \cap B} (x) e^{i ω_{A \cap B} (x)}) : x \in U}, \end{matrix}$ where r_A∩B (x) = min{ r_A (x) , r_B (x) } and ω_A∩B (x) = min(ω_A (x) , ω_B (x)) .

3 Motivation for and advantages of complex vague soft sets

Complex fuzzy sets introduced by Ramot et al. [8] is an extension to Zadeh’s fuzzy sets and was made possible through the addition of a phase term which describes the changes that occurs in the phase of the elements with respect to time. The theory behind the development of complex fuzzy sets is that in many instances it is often necessary to add a second dimension to the expression of the membership value of an element or object. This is especially true in situations where the elements of the set varies with time. Examples of these include meteorological activities such as sunspot-cyles that describes the number of sunspots observed on the surface of the sun and rainfall patterns in a region which measures the frequency and amount of rainfall recorded in a region, with both these situations involving data that varies from day to day and/or month to month and is periodic in nature. Other similar situations are economic-related activities such as the fluctuations of stock prices which happens on a daily and/or hourly basis and the effects of certain financial factors on the economy of a country or a region.

The inherent difficulties and limitations of complex fuzzy sets are due to the nature of fuzzy sets which strictly requires a single value to be assigned for the degree of membership of an element, which is impractical in practice as it is often impossible to be decided with absolute certainty. Furthermore, this goes against the fundamental characteristic of the membership function which is extremely individualistic, subjective and humanistic in nature and is often dictated by differences in perception, upbringing as well as educational and geographical differences, all of which may cause different individuals to assign different values for the degree of membership of the same element. Alkouri and Salleh [10] generalized complex fuzzy sets by defining it using intuitionistic fuzzy sets (IFSs) instead of fuzzy sets, in a bid to make it more practical, particularly in the context of assigning membership values to the elements. However, despite the many advantages of complex fuzzy sets and by extension, complex intuitionistic fuzzy sets, both have one major drawback, which is the lack of an adequate parameterization tool to facilitate the representation of the parameters in a comprehensive manner.

In view of all these, in this paper, we employ the use soft sets [15] and vague sets introduced by Gau and Buehrer [14]. Soft sets are constructed based on the theory of adequate parameterization. It enables a more comprehensive description of the parameters as there are no restrictions on the approximate description and we can use any medium to describe the parameters such as words, sentences, real numbers, functions and mappings, among others. Vague sets on the other hand, are an extension to fuzzy sets and are regarded as a special case of context-dependent fuzzy sets. In [14], the authors asserted that the downside of using a single value for the grade of membership in fuzzy set theory is that the evidence for and against an element are mixed together, without any indication of how much there is of each type of evidence and thus, this single value does not tell us anything about the accuracy of the membership value. On the other hand, vague sets provides us with an interval-based membership which clearly separates the evidence for and against an element. As such, besides having an estimate of the likelihood of an element belonging to a particular set, we also have a lower and upper bound on this likelihood. Although vague sets are intrinsically related to (IFSs), the interval-based membership values provided by vague sets indicates the possible existence of several data values and this “region of hesitation” better corresponds to the intuition of representing vague data, compared to (IFSs). This simple and subtle but meaningful difference between vague sets and IFSs enables vague sets to better capture the vagueness and uncertainties of the data which is prevalent in most real-life situations. We refer the readers to [18] for more information on the advantages of vague sets over IFSs.

The comparisons, advantages and drawbacks of the various hybrid structures of fuzzy sets, soft sets and complex fuzzy sets discussed above, served as the motivation for this paper. The main objective here is to derive an improved hybrid model of complex fuzzy sets that possesses the ability to: 1) represent two-dimensional information; 2) be more expressive in capturing the vagueness of the data; 3) provide a more adequate parameterization tool that can represent the problem parameters in a more comprehensive manner. The complex vague soft set model introduced in this paper possesses all of the abilities mentioned above as it embodies all of the features of complex fuzzy sets with the added advantage of vague sets and the adequate parameterization characteristic of soft sets. The definition, properties and distance measures between complex vague soft sets will be introduced and dealt with in Sections 4 and 5 respectively.

4 Complex vague soft sets and related concepts

In this section, the definition of complex vague soft sets is established by extending the range of truth-membership functions and false- membership functions of the corresponding vague soft set from the original interval of [1] to a unit circle in the complex plane. This concept is a generalization of the notion of complex fuzzy sets introduced in [8]. Several basic set theoretic operations pertaining to the concept of complex vague soft sets, namely the complement, union, intersection, AND and OR are introduced. Lastly, some of the basic properties of complex vague soft sets and its operations are studied and discussed.

Definition 4.1. Let U be an initial universe, E be a set of parameters under consideration and A ⊂ E . Let P (U) denote the complex vague power set of U. A pair (F, A) is called a complex vague soft set (CVSS) over U, where F is a mapping given by F : A → P (U) such that $\begin{matrix} F (x_{j}) & = & {(x, [r_{t_{F_{a}}} (x), 1 - k_{t_{F_{a}}} (x)] . \\ e^{i [w_{t_{F_{a}}}^{r} (x), 2 π - w_{f_{F_{a}}}^{k} (x)]}) : x \in U} \end{matrix}$ where j = 1, 2, 3, … is the number of parameters,[r_{t
_{F
_a}} (x) , 1 - k_{t
_{F
_a}} (x)] are real-valued in the closedinterval [0, 1], the phase terms $[w_{t_{F_{a}}}^{r} (x), 2 π - w_{f_{F_{a}}}^{k} (x)]$ are real-valued in the interval (0, 2π] , 0 ≤ r_{t
_{F
_a}} (x) + k_{F
_a} (x) ≤ 1 and $i = \sqrt{- 1} .$

The complex truth-membership function t_{F
_a} (x) is defined as: $\begin{matrix} t_{F_{a}} (x) : U \to {a : a \in ℂ, | a | \leq 1}, \\ t_{F_{a}} (x) = r_{t_{F_{a}}} (x) . e^{{iw}_{t_{F_{a}}}^{r} (x)} \end{matrix}$ and the complex false-membership function f_{F
_a} (x) is defined as: $\begin{matrix} f_{F_{a}} (x) : U \to {a : a \in ℂ, | a | \leq 1}, \\ 1 - f_{F_{a}} (x) = (1 - k_{f_{F_{a}}} (x)) . e^{i (2 π - w_{f_{F_{a}}}^{k} (x))}, \end{matrix}$ where t_{F
_a} (x) is the lower bound of the complex grade of membership of x derived from the evidence for x and f_{F
_a} (x) is the lower bound on the negation of x derived from the evidence against x . By definition, the values t_{F
_a} (x) and f_{F
_a} (x) and their sum all lie within a unit circle in the complex plane. As such, a complex vague soft set over a universe U is a parameterized family of complex vague sets of the universe U .

Next, we present an example of a CVSS and the possible interpretations of the upper and lower bounds of the membership functions of the elements in the corresponding CVSS.

Example 4.2. Consider a universal set U, where U ={ x₁ = unemploymentrate, x₂ = inflationrate, x₃ = exchangerate, x₄ = GDP, x₅ = globaloil prices } isa set of financial indicators or indexes that are used to describe the Malaysian economy. Let A = {a₁ = noinfluence, a₂ = minimalinfluence, a₃ = averageinfluence, a₄ = greatinfluence} be a set of parameters that describe the degree of influence of the said financial indicators on the Malaysian economy and suppose that the interaction between this set of parameters and the Malaysian economy are measured in the limited time frame of twelve months i.e. within one year. A CVSS (F, A) for this example will be of the form $\begin{matrix} F (x_{j}) & = & {(x, [r_{t_{F_{a}}} (x), 1 - k_{t_{F_{a}}} (x)] . \\ e^{i [w_{t_{F_{a}}}^{r} (x), 2 π - w_{f_{F_{a}}}^{k} (x)]}) \\ : x \in U, j = 1, 2, 3, \dots} . \end{matrix}$

In the context of this example, the lower and upper bounds of the amplitude terms are a measure of the degree of influence of a financial indicator on the Malaysian economy, while the phase term represents the “phase” of influence, in this case the time lag that characterizes the influence of an economic index, i.e. the time it takes before the influence of (a particular occurrence related to) a financial indicator is evident in the Malaysian economy. These values can be determined or calculated through data obtained from various agencies such as the Central Bank of Malaysia, Kuala Lumpur Stock Exchange (KLSE) and the International Monetary Fund (IMF) or experts such as economists and financial analysts.

Following in this direction, we provide some examples of scenarios that could possibly occur in this context. For example, the statement “Global oil prices have a very large influence on the Malaysian economy and this influence is evident in the Malaysian economy within three to five months”, can be represented by the expression: $F (a_{4}) = (x_{5}, [0.8, 0.9] e^{i [\frac{3}{12} (2 π), \frac{5}{12} (2 π)]})$ which can be simplified to $F (a_{4}) = (x_{5}, [0.8, 0.9] e^{i [\frac{π}{2}, \frac{5 π}{6}]}) .$ The interval [0.8, 0.9] for the amplitude term signifies the very large degree of influence of global oil prices on the Malaysian economy while the interval $[\frac{π}{2}, \frac{5 π}{6}]$ corresponds to the time lag of three to five months (since the influence is measured within a time frame of 12 months) for the influence of global oil prices on the Malaysian economy to become evident. This is reasonable as 30% to 40% of the revenue of the Malaysian government comes from oil and gas. As such, any change in global oil prices would greatly affect (positively or adversely depending on the increase or decrease of global oil prices) the Malaysian economy and judging from the situation in Malaysia in recent months, it is safe to say that it takes between three and five months for any change in global oil prices to become evident in the Malaysian economy. Similarly, the statement “The exchange rate has a very large influence on the Malaysian economy and this is evident within two to three months can be written as: $F (a_{1}) = (x_{3}, [0, 0.05] e^{i [\frac{2}{12} (2 π), \frac{3}{12} (2 π)]})$ and can be simplified to $F (a_{1}) = (x_{3}, [0, 0.05] e^{i [\frac{π}{3}, \frac{π}{2}]}) .$ This statement is reasonable as the degree of membership of exchange rate with respect to the attribute “no influence” is zero or a very low value close to zero.

Example 4.2 illustrates how to employ both components of the amplitude term and the phase term to convey a vague soft set.

Comparison with existing methods in literature

In Example 4.2, we discussed the effects of some of the main financial indicators on the Malaysian economy and as with other economic-related activities where timing is of utmost importance, we discuss the time it takes for these indicators to have an impact on the Malaysian economy. We have proven that both these information can be represented in an accurate and comprehensive manner by using the CVSS model as expounded above. Besides the CVSS model, there are a whole array of hybrid models of fuzzy sets and soft sets. However, the models which are the closest in structure to the CVSS model would be complex fuzzy sets and vague soft sets. As such, we present a comparison between these two models and our CVSS model.

Besides the CVSS model, there are several other methods of incorporating the additional information pertaining to the phase or time lag that elapses before the influence of these indicators becomes evident in the Malaysian economy with the information on the degree of influence of the indicator on the Malaysian economy. One such method would be through the use of vague soft sets to represent these information. In this case, we would have to define two vague soft sets with parameters such as “no influence”, “minimal influence”, “average influence” and “great influence” for the first set and “1–3 months”, “4–6 months” and “6–12 months” for the second set, so that the degree of influence of the indicators on the Malaysian economy and the time it takes for these effects to make an impact on the economy is characterized by its truth and false membership grades in two separate vague soft sets. Although this can be done, it would be highly ambiguous as it would be rather difficult to link the information in both the vague soft sets accurately to see the complete picture, besides causing both aspects of the information to lose its significance or meaning. Insistence on finding the link between these information would necessitate finding the relations between these two vague soft sets, which would in turn require performing tedious composition operations between these two sets. This would result in additional computations which does not in any way increase the accuracy of the representation of the information.

Another alternative to using the vague soft set model is to determine the truth and false membership grades of an indicator by taking into account both the degree of influence of an indicator on the Malaysian economy and the phase or time it takes for these influence to become evident in the economy and combine these values to obtain the truth and false membership values. However, if this were the case, an indicator which has a very large influence on the Malaysian economy but takes only a short period of time for the effect to become evident on the economy would only receive a rather small grade of membership in the vague soft set and vice-versa. It is obvious that both of these methods are inaccurate and can prove to very misleading to the people who will be using these information, especially for the purpose of research and development.

The complex fuzzy set model may be able to overcome the problem afflicting vague soft sets. However, there are two major problems with this model; 1) The lack of an adequate parameterization tool which would enable the problem parameters to be defined more comprehensively, unlike the CVSS model which has the added advantage of soft sets. 2) The single-valued grade of membership that is assigned to the elements lack the accuracy of the interval-based grade of membership provided by the CVSS model. As explained previously, it is extremely difficult to assign a single value to describe the degree of influence and the time lag of an indicator as this value is normally obtained from different sources who may have differing opinions. For example, the IMF may opine that global oil prices have a very large influence on the Malaysian economy and it takes only 1-2 months for the change in oil prices to affect the Malaysian economy and thus may choose to assign a value of 0.95 for the amplitude term and $\frac{π}{8}$ for the phase term. The Central Bank of Malaysia on the other hand, may opine that although global oil prices have a substantial effect on the Malaysian economy, it usually takes anywhere from 3–5 months for this effect to become evident in the economy. As such, the central bank may choose to assign values of 0.8 and $\frac{π}{3}$ for the amplitude term and phase term respectively of this element. The CVSS model overcomes this problem by providing an interval based membership value for the amplitude and phase terms of the elements. These features succinctly describe the limitations of using the complex fuzzy set structure to model situations which contain vague, uncertain and subjective data.

From now on, let U be a universe of discourse and (F, A) , (G, B) and (H, C) be three CVSSs over the universe U, which are as defined below: $\begin{matrix} (F, A) & = & {(x, [r_{t_{F_{a}}} (x), 1 - k_{t_{F_{a}}} (x)] . \\ e^{i [w_{t_{F_{a}}}^{r} (x), 2 π - w_{f_{F_{a}}}^{k} (x)]}) : x \in U}, \end{matrix}$

$\begin{matrix} (G, B) & = & {(x, [r_{t_{G_{b}}} (x), 1 - k_{t_{G_{b}}} (x)] . \\ e^{i [w_{t_{G_{b}}}^{r} (x), 2 π - w_{f_{G_{b}}}^{k} (x)]}) : x \in U}, \end{matrix}$ $\begin{matrix} (H, C) & = & {(x, [r_{t_{H_{c}}} (x), 1 - k_{t_{H_{c}}} (x)] . \\ e^{i [w_{t_{H_{c}}}^{r} (x), 2 π - w_{f_{H_{c}}}^{k} (x)]}) : x \in U} . \end{matrix}$

Definition 4.3. Let (F, A) and (G, B) be complex vague soft sets over U . Then (F, A) is said to be a complex vague soft subset of (G, B) if and only if the following conditions are satisfied for all x ∈ U :

r_{t
_{F
_a}} (x) ≤ r_{t
_{G
_b}} (x) and k_{f
_{G
_b}} (x) ≤ k_{f
_{F
_a}} (x) for the amplitude terms;

$w_{t_{F_{a}}}^{r} (x) \leq w_{t_{G_{b}}}^{r} (x)$ and $w_{f_{G_{b}}}^{k} (x) \leq w_{f_{F_{a}}}^{k} (x)$ for the phase terms.

This relationship is denoted as (F, A) ⊂ (G, B) .

Definition 4.4. Let (F, A) and (G, B) be complex vague soft sets over U . Then (F, A) is said to be complex vague soft equal to (G, B) if and only if the following conditions are satisfied for all x ∈ U :

r_{t
_{F
_a}} (x) = r_{t
_{G
_b}} (x) and k_{f
_{F
_a}} (x) = k_{f
_{G
_b}} (x) for the amplitude terms;

$w_{t_{F_{a}}}^{r} (x) = w_{t_{G_{b}}}^{r} (x)$ and $w_{f_{F_{a}}}^{k} (x) = w_{f_{G_{b}}}^{k} (x)$ for the phase term.

This relationship is denoted as (F, A) ≡ (G, B) .

Definition 4.5. Let (F, A) be a complex vague soft set over U . Then:

(F, A) is said to be a null complex vague soft set, denoted by (F, A) _∅ if r_{t
_{F
_a}} (x) = 0, k_{f
_{F
_a}} (x) = 1 and $w_{t_{F_{a}}}^{r} (x) = 0 π, w_{f_{F_{a}}}^{k} (x) = 2 π$ .

(F, A) is said to be a absolute complex vague soft set, denoted by (F, A) _abs if r_{t
_{F
_a}} (x) = 1, k_{f
_{F
_a}} (x) = 0 and $w_{t_{F_{a}}}^{r} (x) = 2 π, w_{f_{F_{a}}}^{k} (x) = 0 π$ .

5 Set theoretic operations on CVSSs

In this section, we introduce some basic operations pertaining to CVSSs and study the important characteristics of the operations of CVSSs such as the commutative, associative and distributive laws as well as illustrate the relationship between the different operations in the form of the relevant De Morgan’s laws.

Definition 5.1. Let (F, A) be a complex vague soft set over U . Then the complement of (F, A) , denoted by (F, A) ^c is defined as (F, A) ^c = (F^c, ¬ A) and defined as $\begin{matrix} {(F, A)}^{c} & = & {(x, [r_{t_{F_{a}}^{c}} (x), 1 - k_{f_{F_{a}}^{c}} (x)] . \\ e^{i [w_{t_{F_{a}}^{c}}^{r} (x), 2 π - w_{f_{F_{a}}^{c}}^{k} (x)]}) : x \in U}, \end{matrix}$ where the complement of the amplitude terms are $r_{t_{F_{a}^{c}}} (x) = k_{f_{F_{a}}} (x)$ and $1 - k_{f_{F_{a}^{c}}} (x) = 1 - r_{t_{F_{a}}} (x)$ and the complement of the phase terms are $w_{t_{F_{a}}^{c}}^{r} (x) = w_{f_{F_{a}}}^{k} (x)$ and $2 π - w_{f_{F_{a}}^{c}}^{k} (x) = 2 π - w_{t_{F_{a}}}^{r} (x) .$

Example 5.2. Consider the expression $F (a_{4}) = (x_{5}, [0.8, 0.9] e^{i [\frac{π}{2}, \frac{5 π}{6}]})$ given in Example 4.2. By using the vague soft complement for both the amplitude and phase terms, we obtain the complement of the expression given by $F (a_{4}) = (x_{5}, [0.1, 0.2] e^{i [\frac{7 π}{6}, \frac{3 π}{2}]}) .$

Proposition 5.3.Let (F, A) be a complex vague soft set over a universeU. Then ((F, A) ^c)) ^c = (F, A) .

Proof. The complement of (F, A) , denoted by (F, A) ^c = (F^c, ¬ A) is defined as: $\begin{matrix} {(F, A)}^{c} & = & {(x, [r_{t_{F_{a}}^{c}} (x), 1 - k_{f_{F_{a}}^{c}} (x)] . \\ e^{i [w_{t_{F_{a}}^{c}}^{r} (x), 2 π - w_{f_{F_{a}}^{c}}^{k} (x)]}) : x \in U} \\ = & {(x, [k_{f_{F_{a}}} (x), 1 - r_{t_{F_{a}}} (x)] . \\ e^{i [w_{f_{F_{a}}}^{k} (x), 2 π - w_{t_{F_{a}}}^{r} (x)]}) : x \in U} . \end{matrix}$

Now let (F, A) ^c = (G, B) = (F^c, ¬ A) . Then we obtain the following: $\begin{matrix} {(G, B)}^{c} & = & {(x, [r_{t_{G_{b}}^{c}} (x), 1 - k_{f_{G_{b}}^{c}} (x)] . \\ e^{i [w_{t_{G_{b}}^{c}}^{r} (x), 2 π - w_{f_{G_{b}}^{c}}^{k} (x)]}) : x \in U} \\ = & {(x, [k_{f_{F_{a}}}^{c} (x), 1 - r_{t_{F_{a}}}^{c} (x)] . \\ e^{i [w_{f_{F_{a}}}^{k} c (x), 2 π - w_{t_{F_{a}}}^{r} c (x)]}) : x \in U} \\ = & {(x, [r_{t_{F_{a}}} (x), 1 - k_{f_{F_{a}}} (x)] . \\ e^{i [w_{t_{F_{a}}}^{r} (x), 2 π - w_{f_{F_{a}}}^{k} (x)]}) : x \in U} \\ = & (F, A) . \end{matrix}$

This completes the proof.

Definition 5.4. Let (F, A) and (G, B) be complex vague soft sets over U . Then the union of two complex vague soft sets (F, A) and (G, B) is a complex vague soft set (H, C) defined as $(F, A) \overset{ˇ}{\cup} (G, B) = (H, C),$ where C = A ∪ B and for all c ∈ C, x ∈ U : t_H(c) (x) $\begin{matrix} t_{H (c)} (x) \\ = {\begin{matrix} r_{t_{F (c)}} (x) . e^{i (w_{t_{F (c)}}^{r} (x))} & c \in A - B, \\ r_{t_{G (c)}} (x) . e^{i (w_{t_{G (c)}}^{r} (x))} & c \in B - A, \\ \begin{matrix} [max (r_{t_{F (c)}} (x), r_{t_{G (c)}} (x)) . \\ e^{i (max (w_{t_{F (c)}}^{r} (x)), (w_{t_{G (c)}}^{r} (x)))}] \end{matrix} & c \in A \cap B, \end{matrix} \end{matrix}$ and $\begin{matrix} 1 - f_{H (c)} (x) \\ = {\begin{matrix} 1 - k_{f_{F (c)}} (x) . e^{i (2 π - w_{f_{F (c)}}^{k} (x))} & c \in A - B, \\ 1 - k_{f_{G (c)}} (x) . e^{i (2 π - w_{f_{G (c)}}^{k} (x))} & c \in B - A, \\ \begin{matrix} [max (1 - k_{f_{F (c)}} (x), 1 - k_{f_{G (c)}} (x)) . \\ e^{i (max (2 π - w_{f_{F (c)}}^{k} (x), 2 π - w_{f_{G (c)}}^{k} (x)))}] \end{matrix} & c \in A \cap B, \end{matrix} \end{matrix}$ where t_H(c) (x) and f_H(c) (x) are the truth membership function and false membership function of H_c (x) respectively. This relationship can be written as $H_{c} (x) = F_{c} (x) \overset{ˇ}{\cap} G_{c} (x) .$

Definition 5.5. Let (F, A) and (G, B) be complex vague soft sets over U . Then the intersection of two complex vague soft sets (F, A) and (G, B) is a complex vague soft set (H, C) defined as $(F, A) \overset{ˇ}{\cap} (G, B) = (H, C),$ where C = A ∪ B and for all c ∈ C, x ∈ U, $\begin{matrix} t_{H (c)} (x) \\ = {\begin{matrix} r_{t_{F (c)}} (x) . e^{i (w_{t_{F (c)}}^{r} (x))} & c \in A - B, \\ r_{t_{G (c)}} (x) . e^{i (w_{t_{G (c)}}^{r} (x))} & c \in B - A, \\ \begin{matrix} [min (r_{t_{F (c)}} (x), r_{t_{G (c)}} (x)) . \\ e^{i (max (w_{t_{F (c)}}^{r} (x)), (w_{t_{G (c)}}^{r} (x)))}] \end{matrix} & c \in A \cap B, \end{matrix} \end{matrix}$ and $\begin{matrix} 1 - f_{H (c)} (x) \\ = {\begin{matrix} 1 - k_{f_{F (c)}} (x) . e^{i (2 π - w_{f_{F (c)}}^{k} (x))} & c \in A - B, \\ 1 - k_{f_{G (c)}} (x) . e^{i (2 π - w_{f_{G (c)}}^{k} (x))} & c \in B - A, \\ \begin{matrix} [min (1 - k_{f_{F (c)}} (x), 1 - k_{f_{G (c)}} (x)) . \\ e^{i (min (2 π - w_{f_{F (c)}}^{k} (x), 2 π - w_{f_{G (c)}}^{k} (x)))}] \end{matrix} & c \in A \cap B, \end{matrix} \end{matrix}$ where t_H(c) (x) and f_H(c) (x) are the truth membership function and false membership function of H_c (x) respectively. This relationship can be written as $H_{c} (x) = F_{c} (x) \overset{ˇ}{\cap} G_{c} (x) .$

Proposition 5.6.Let (F, A) , (G, B) and (H, C) be complex vague soft sets over a universeU . Then the following commutative and associative properties hold true:

$(F, A) \overset{ˇ}{\cup} (G, B) = (G, B) \overset{ˇ}{\cup} (F, A)$

$(F, A) \overset{ˇ}{\cap} (G, B) = (G, B) \overset{ˇ}{\cap} (F, A)$

$(F, A) \overset{ˇ}{\cup} ((G, B) \overset{ˇ}{\cup} (H, C)) = ((F, A) \overset{ˇ}{\cup} (G, B)) \overset{ˇ}{\cup} (H, C)$

$(F, A) \overset{ˇ}{\cap} ((G, B) \overset{ˇ}{\cap} (H, C)) = ((F, A) \overset{ˇ}{\cap} (G, B)) \overset{ˇ}{\cap} (H, C)$

Proof. (a) & (b) The proof is straightforward by Definitions 5.4 and 5.5.

$(F, A) \overset{ˇ}{\cup} (G, B) = (S, D),$ where D = A ∪ B,

$(G, B) \overset{ˇ}{\cup} (H, C) = (T, E),$ where T = B ∪ C,

$((F, A) \overset{ˇ}{\cup} (G, B)) \overset{ˇ}{\cup} (H, C) = (P, K),$ where K = A ∪ B ∪ C,

$(F, A) \overset{ˇ}{\cup} ((G, B) \overset{ˇ}{\cup} (H, C)) = (Q, L),$ where L = A ∪ B ∪ C .

There are several cases to be considered, with the main ones being e ∈ C and e ∉ C .

(i) Let e ∈ C . Four cases to be considered here.

Case 1: If e ∈ A and e ∈ B and e ∈ C, then: $\begin{matrix} P_{e} (x) & = & S_{e} (x) \cup H_{e} (x) = (F_{e} (x) \cup G_{e} (x)) \cup H_{e} (x) \\ = & F_{e} (x) \cup G_{e} (x) \cup H_{e} (x) \end{matrix}$ and $\begin{matrix} Q_{e} (x) & = & F_{e} (x) \cup T_{e} (x) = F_{e} (x) \cup (G_{e} (x) \cup H_{e} (x)) \\ = & F_{e} (x) \cup G_{e} (x) \cup H_{e} (x) . \end{matrix}$

Thus P_e (x) = Q_e (x) .

Case 2: If e ∈ A and e ∉ B and e ∈ C, then P_e (x) = S_e (x) ∪ H_e (x) = F_e (x) ∪ H_e (x) = F_e (x) ∪ T_e (x) = Q_e (x) .

Case 3: If e ∉ A and e ∈ B and e ∈ C, then P_e (x) = S_e (x) ∪ H_e (x) = G_e (x) ∪ H_e (x) = T_e (x) = Q_e (x) .

Case 4: If e ∉ A and e ∉ B and e ∈ C, then P_e (x) = H_e (x) = T_e (x) = Q_e (x) .

(ii) Let e ∉ C . Four cases to be considered here.

Case 5: If e ∈ A and e ∈ B and e ∉ C, then: $\begin{matrix} Q_{e} (x) = F_{e} (x) \cup T_{e} (x) = F_{e} (x) \cup G_{e} (x) \\ = S_{e} (x) = P_{e} (x) . \end{matrix}$

Case 6: If e ∈ A and e ∉ B and e ∉ C, then P_e (x) = S_e (x) = F_e (x) = Q_e (x) .

Case 7: If e ∉ A and e ∈ B and e ∉ C, then Q_e (x) = T_e (x) = G_e (x) = S_e (x) = P_e (x) .

Case 8: Let e ∉ A and e ∉ B and e ∉ C . This is not a possible situation as e must belong to at least one of the three sets A, B or C and therefore it need not be taken into consideration.

This completes the proof.

(d) The proof is similar to the proof of part (c).

Proposition 5.7.Let (F, A) and (G, B) be two complex vague soft sets over a universeU . Then the following De Morgans’s laws hold true:

${((F, A) \overset{ˇ}{\cup} (G, B))}^{c} = {(F, A)}^{c} \overset{ˇ}{cup} {(G, B)}^{c}$

${((F, A) \overset{ˇ}{\cap} (G, B))}^{c} = {(F, A)}^{c} \overset{ˇ}{cup} {(G, B)}^{c}$

Proof. (a) Suppose that $(F, A) \overset{ˇ}{\cup} (G, B) = (H, K) .$ Then ${((F, A) \overset{ˇ}{\cup} (G, B))}^{c} = {(H, K)}^{c}$ and it follows that ∀x ∈ U : $\begin{matrix} {(H, K)}^{c} \\ = {(x, (\begin{matrix} [r_{t_{H_{K}}^{c}} (x), 1 - k_{f_{H_{K}}^{c}} (x)] . \\ e^{i [w_{t_{H_{K}}^{c}}^{r} (x), 2 π - w_{f_{H_{K}}^{c}}^{k} (x)]} \end{matrix}))} \\ = {(x, (\begin{matrix} [k_{f_{H_{K}}} (x), 1 - r_{t_{H_{K}}} (x)] . \\ e^{i [w_{f_{H_{K}}}^{k} (x), 2 π - w_{t_{H_{K}}}^{r} (x)]} \end{matrix}))} \\ = {(x, {([max (r_{t_{F (c)}} (x), r_{t_{G (c)}} (x)) . e^{i (max (w_{t_{F (c)}}^{r} (x)), (w_{t_{G (c)}}^{r} (x)))}])}^{c} \\ . {([max \begin{matrix} (1 - k_{f_{F (c)}} (x), 1 - k_{f_{G (c)}} (x)) . \\ e^{i (max (2 π - w_{f_{F (c)}}^{k} (x), 2 π - w_{f_{G (c)}}^{k} (x)))} \end{matrix}])}^{c})} \\ = {(x, [min (r_{t_{F (c)}} (x), r_{t_{G (c)}} (x)) . e^{i (min (w_{t_{F (c)}}^{r} (x)), (w_{t_{G (c)}}^{r} (x)))}] . \\ [min \begin{matrix} (1 - k_{f_{F (c)}} (x), 1 - k_{f_{G (c)}} (x)) . \\ e^{i (min (2 π - w_{f_{F (c)}}^{k} (x), 2 π - w_{f_{G (c)}}^{k} (x)))} \end{matrix}])} \\ = {(\begin{matrix} x, [r_{t_{F_{a}}} (x), 1 - k_{f_{F_{a}}} (x)] . \\ e^{i [w_{t_{F_{a}}}^{r} (x), 2 π - w_{f_{F_{a}}}^{k} (x)]} \end{matrix})} \overset{ˇ}{\cap} \\ {(\begin{matrix} x, [r_{t_{G_{b}}} (x), 1 - k_{f_{G_{b}}} (x)] . \\ e^{i [w_{t_{G_{b}}}^{r} (x), 2 π - w_{f_{G_{b}}}^{k} (x)]} \end{matrix})} \\ = {(F, A)}^{c} \overset{ˇ}{\cap} {(G, B)}^{c} . \end{matrix}$

Thus, it follows that ${(H, K)}^{c} = ((F, A) \overset{ˇ}{\cup} (G, B))^{c} = {(F, A)}^{c} \overset{ˇ}{\cap} {(G, B)}^{c}$ and this completes the proof.

(b) The proof is similar to the proof of part (a).

Definition 5.8. The restricted union of two complex vague soft sets (F, A) and (G, B) over a universe U, denoted by $(F, A) {\overset{ˇ}{\cup}}_{R} (G, B) = (H, C),$ where C = A ∩ B and for all c ∈ C, x ∈ U, $\begin{matrix} t_{H (c)} (x) & = & [max (r_{t_{F (c)}} (x), r_{t_{G (c)}} (x)) \\ . e^{i (max (w_{t_{F (c)}}^{r} (x)), (w_{t_{G (c)}}^{r} (x)))}] \end{matrix}$ and $\begin{matrix} 1 - f_{H (c)} (x) \\ = [\begin{matrix} max (1 - k_{f_{F (c)}} (x), 1 - k_{f_{G (c)}} (x)) . \\ e^{i (max (2 π - w_{f_{F (c)}}^{k} (x), 2 π - w_{f_{G (c)}}^{k} (x)))} \end{matrix}] \end{matrix}$

where t_H(c) (x) and f_H(c) (x) are the truth membership function and false membership function of H_c (x) respectively. This relationship can be written as H_c (x) = F_c (x) ∪ G_c (x) .

Definition 5.9. The restricted intersection of two complex vague soft sets (F, A) and (G, B) over a universe U, denoted by $(F, A) {\overset{ˇ}{\cap}}_{R} (G, B) = (H, C),$ where C = A ∩ B and for all c ∈ C, x ∈ U, $\begin{matrix} t_{H (c)} (x) & = & [min (r_{t_{F (c)}} (x), r_{t_{G (c)}} (x)) \\ . e^{i (min (w_{t_{F (c)}}^{r} (x)), (w_{t_{G (c)}}^{r} (x)))}] \end{matrix}$ and $\begin{matrix} 1 - f_{H (c)} (x) \\ = [\begin{matrix} min (1 - k_{f_{F (c)}} (x), 1 - k_{f_{G (c)}} (x)) . \\ e^{i (min (2 π - w_{f_{F (c)}}^{k} (x), 2 π - w_{f_{G (c)}}^{k} (x)))} \end{matrix}] \end{matrix}$ where t_H(c) (x) and f_H(c) (x) are the truth membership function and false membership function of H_c (x) respectively. This relationship can be written as H_c (x) = F_c (x) ∩ G_c (x) .

Definition 5.10. Let (F, A)and (G, B) be two complex vague soft sets over U. Then (F, A) OR (G, B) , denoted by $(F, A) \overset{ˇ}{\lor} (G, B) = (H, A \times B),$ is a complex vague soft set with the lower bound and the upper bound of H_(α,β) (x) defined by $\begin{matrix} t_{H_{(α, β)}} (x) & = & [max (r_{t_{F (α)}} (x), r_{t_{G (β)}} (x)) \\ . e^{i (max (w_{t_{F (α)}}^{r} (x)), (w_{t_{G (β)}}^{r} (x)))}] \end{matrix}$ and

$\begin{matrix} 1 - f_{H (c)} (x) \\ = [\begin{matrix} max (1 - k_{f_{F (α)}} (x), 1 - k_{f_{G (β)}} (x)) . \\ e^{i (max (2 π - w_{f_{F (α)}}^{k} (x), 2 π - w_{f_{G (β)}}^{k} (x)))} \end{matrix}] \end{matrix}$

for all (α, β) ∈ A × B and x ∈ U . This relationship can be written as H_(α,β) (x) = F_α (x) ∪ G_β (x) .

Definition 5.11. Let (F, A) and (G, B) be two complex vague soft sets over U . Then (F, A) AND (G, B) , denoted by $(F, A) \overset{ˇ}{\land} (G, B) = (H, A \times B),$ is a complex vague soft set with the lower bound and the upper bound of H_(α,β) (x) defined by

$\begin{matrix} t_{H_{(α, β)}} (x) & = & [min (r_{t_{F (α)}} (x), r_{t_{G (β)}} (x)) \\ . e^{i (min (w_{t_{F (α)}}^{r} (x)), (w_{t_{G (β)}}^{r} (x)))}] \end{matrix}$ and $\begin{matrix} 1 - f_{H (c)} (x) \\ = [\begin{matrix} min (1 - k_{f_{F (α)}} (x), 1 - k_{f_{G (β)}} (x)) . \\ e^{i (min (2 π - w_{f_{F (α)}}^{k} (x), 2 π - w_{f_{G (β)}}^{k} (x)))} \end{matrix}] \end{matrix}$ for all (α, β) ∈ A × B and x ∈ U . This relationship can be written as H_(α,β) (x) = F_α (x) ∩ G_β (x) .

Proposition 5.12.Let (F, A) , (G, B) and (H, C) be complex vague soft sets over a universeU . Then the following commutative and associative laws of theANDandORoperations hold true:

$(F, A) \overset{ˇ}{\lor} (G, B) = (G, B) \overset{ˇ}{\lor} (F, A)$

$(F, A) \overset{ˇ}{\land} (G, B) = (G, B) \overset{ˇ}{\land} (F, A)$

$((F, A) \overset{ˇ}{\lor} (G, B)) \overset{ˇ}{\lor} (H, C) = (F, A) \overset{ˇ}{\lor} ((G, B) \overset{ˇ}{\lor} (H, C))$

$((F, A) \overset{ˇ}{\land} (G, B)) \overset{ˇ}{\land} (H, C) = (F, A) \overset{ˇ}{\land} ((G, B) \overset{ˇ}{\land} (H, C))$

Proof. The proofs follow directly from the commutative and associative properties of the max and min operators.

Proposition 5.13.Let (F, A) and (G, B) be two complex vague soft sets over a universeU . Then the following De Morgan’s laws hold true:

${((F, A) \overset{ˇ}{\lor} (G, B))}^{c} = {(F, A)}^{c} \overset{ˇ}{\land} {(G, B)}^{c}$

${((F, A) \overset{ˇ}{\land} (G, B))}^{c} = {(F, A)}^{c} \overset{ˇ}{\lor} {(G, B)}^{c}$

Proof. (a) & (b) The proof is similar to that of Proposition 5.7.

6 Distance measure between CVSSs

Liu [19] is one of the pioneers in the research pertaining to the development of distance measures and similarity measures of fuzzy sets. In [19], the concept of distance measures between fuzzy sets were introduced and subsequently used to define the similarity measures and entropy of fuzzy sets. In the same spirit, we propose the axiomatic definition of the distance measure between CVSSs. Subsequently, we extend and propose various distance measures between CVSSs, namely the Hamming, Normalized Hamming, Euclidean and Normalized Euclidean distances as well as the fifth distance between CVSSs.

From now on, let CVSS (U) denote the set of all complex vague soft sets in U .

Definition 6.1. A real-valued non-negative function d : CVSS (U) × CVSS (U) → [0, 1] is called a distance measure between CVSSs if it satisfies the following conditions for any (F, A) , (G, B) , (H, C) ∈ CVSS (U) :

d ((F, A) , (G, B)) = d ((G, B) , (F, A))

d ((F, A) , (G, B)) = 0 ⇔ (F, A) ≡ (G, B)

0 ≤ d ((F, A) , (G, B)) ≤ 1

d ((F, A) , (F, A) ^c) = 1 if and only if (F, A) is a complex fuzzy set.

d ((F, A) , (G, B)) ≤ d ((F, A) , (H, C)) + d ((H, C) , (G, B)) (triangle inequality)

Definition 6.2. Let d be a distance measure between CVSSs. The various distances between CVSSs are defined as follows:

The Hamming distance:

$d_{H} ((F, A), (G, B)) = \frac{1}{4} \sum_{j = 1}^{m} \sum_{i = 1}^{m} [\begin{matrix} | r_{t_{F (a_{i})}} (x_{j}) - r_{t_{G (b_{i})}} (x_{j}) | + | k_{f_{G (b_{i})}} (x_{j}) - k_{f_{F (a_{i})}} (x_{j}) | \\ + \frac{1}{2 π} (| w_{t_{F_{(a_{i})}}}^{r} (x_{j}) - w_{t_{G_{(b_{i})}}}^{r} (x_{j}) | + | w_{f_{G_{(b_{i})}}}^{k} (x_{j}) - w_{f_{F_{(a_{i})}}}^{k} (x_{j}) |) \end{matrix}]$

The normalized Hamming distance:

d_{H}^{N} ((F, A), (G, B)) = \frac{d_{H} ((F, A), (G, B))}{mn}

The Euclidean distance:

\begin{matrix} d_{E} ((F, A), (G, B)) \\ = \sqrt{\frac{1}{4} \sum_{j = 1}^{n} \sum_{i = 1}^{m} [\begin{matrix} {(r_{t_{F_{(a_{i})}}} (x_{j}) - r_{t_{G_{(b_{i})}}} (x_{j}))}^{2} + {(k_{f_{G_{(b_{i})}}} (x_{j}) - k_{f_{F_{(a_{i})}}} (x_{j}))}^{2} \\ + \frac{1}{4 π^{2}} ({(w_{t_{F (a_{i})}}^{r} (x_{j}) - w_{t_{G (b_{i})}}^{r} (x_{j}))}^{2} + {(w_{f_{G (b_{i})}}^{k} (x_{j}) - w_{f_{F (a_{i})}}^{k} (x_{j}))}^{2}) \end{matrix}]} \end{matrix}

The normalized Euclidean distance:

d_{E}^{N} ((F, A), (G, B)) = \frac{d_{E} ((F, A), (G, B))}{\sqrt{mn}}

The fifth distance:

$\begin{matrix} d_{F} ((F, A), (G, B)) = \frac{1}{2 mn} \sum_{j = 1}^{n} \sum_{i = 1}^{m} \\ [\begin{matrix} \frac{| r_{t_{F (a_{i})}} (x_{j}) - r_{t_{G (b_{i})}} (x_{j}) | + | k_{f_{G (b_{i})}} (x_{j}) - k_{f_{F (a_{i})}} (x_{j}) |}{4} \\ + \frac{1}{2 π} (\frac{| w_{t_{F_{(a_{i})}}}^{r} (x_{j}) - w_{t_{G_{(b_{i})}}}^{r} (x_{j}) | + | w_{f_{G_{(b_{i})}}}^{k} (x_{j}) - w_{f_{F_{(a_{i})}}}^{k} (x_{j}) |}{4}) \end{matrix} \end{matrix}$ $\begin{matrix} + \frac{max (| r_{t_{F_{(a_{i})}}} (x_{j}) - r_{t_{G_{(b_{i})}}} (x_{j}) |, | k_{f_{G_{(b_{i})}}} (x_{j}) - k_{f_{F_{(a_{i})}}} (x_{j}) |)}{2} \\ + \frac{1}{2 π} (\frac{max (| w_{t_{F (a_{i})}}^{r} (x_{j}) - w_{t_{G (b_{i})}}^{r} (x_{j}) |, | w_{f_{G (b_{i})}}^{k} (x_{j}) - w_{f_{F (a_{i})}}^{k} (x_{j}) |)}{2}) \end{matrix}] .$

Theorem 6.3.All the distance measures given in Definition 6.2 are distance functions between CVSSs.

Proof. It is obvious that all the distance measures given in Definition 6.2 satisfy conditions (i) –(iv) of Definition 6.1. As such, for the sake of brevity, we only prove the triangle inequality condition for the Hamming distance. $\begin{matrix} d_{H} ((F, A), (G, B)) = \frac{1}{4} \sum_{j = 1}^{n} \sum_{i = 1}^{m} [\begin{matrix} | r_{t_{F_{(a_{i})}}} (x_{j}) - r_{t_{G_{(b_{i})}}} (x_{j}) | + | k_{f_{G_{(b_{i})}}} (x_{j}) - k_{f_{F_{(a_{i})}}} (x_{j}) | \\ + \frac{1}{2 π} (| w_{t_{F (a_{i})}}^{r} (x_{j}) - w_{t_{G (b_{i})}}^{r} (x_{j}) | + | w_{f_{G (b_{i})}}^{k} (x_{j}) - w_{f_{F (a_{i})}}^{k} (x_{j}) |) \end{matrix}] \\ = \frac{1}{4} \sum_{j = 1}^{n} \sum_{i = 1}^{m} [\begin{matrix} | r_{t_{F_{(a_{i})}}} (x_{j}) - r_{t_{G_{(b_{i})}}} (x_{j}) \pm r_{t_{H_{(c_{i})}}} (x_{j}) | + | k_{f_{G_{(b_{i})}}} (x_{j}) - k_{f_{F_{(a_{i})}}} (x_{j}) \pm k_{f_{H_{(c_{i})}}} (x_{j}) | \\ + \frac{1}{2 π} (| w_{t_{F (a_{i})}}^{r} (x_{j}) - w_{t_{G (b_{i})}}^{r} (x_{j}) \pm w_{t_{H (c_{i})}}^{r} (x_{j}) | + | w_{f_{G (b_{i})}}^{k} (x_{j}) - w_{f_{F (a_{i})}}^{k} (x_{j}) \pm w_{f_{H (c_{i})}}^{k} (x_{j}) |) \end{matrix}] \\ \leq \frac{1}{4} \sum_{j = 1}^{n} \sum_{i = 1}^{m} [\begin{matrix} \begin{matrix} | r_{t_{F_{(a_{i})}}} (x_{j}) - r_{t_{G_{(b_{i})}}} (x_{j}) | + | r_{t_{G_{(b_{i})}}} (x_{j}) - r_{t_{H_{(c_{i})}}} (x_{j}) | \\ + | k_{f_{G_{(b_{i})}}} (x_{j}) - k_{f_{F_{(a_{i})}}} (x_{j}) | + | k_{f_{F_{(a_{i})}}} (x_{j}) - k_{f_{H_{(c_{i})}}} (x_{j}) | \end{matrix} \\ + \frac{1}{2 π} (\begin{matrix} | w_{t_{F (a_{i})}}^{r} (x_{j}) - w_{t_{G (b_{i})}}^{r} (x_{j}) | + | w_{t_{G (b_{i})}}^{r} (x_{j}) - w_{t_{H (c_{i})}}^{r} (x_{j}) | \\ + | w_{f_{G (b_{i})}}^{k} (x_{j}) - w_{f_{F (a_{i})}}^{k} (x_{j}) | + | w_{f_{F (a_{i})}}^{k} (x_{j}) - w_{f_{H (c_{i})}}^{k} (x_{j}) | \end{matrix}) \end{matrix}] \\ = (\frac{1}{4} \sum_{j = 1}^{n} \sum_{i = 1}^{m} [\begin{matrix} | r_{t_{F_{(a_{i})}}} (x_{j}) - r_{t_{H_{(c_{i})}}} (x_{j}) | + | k_{f_{H_{(c_{i})}}} (x_{j}) - k_{f_{F_{(a_{i})}}} (x_{j}) | \\ + \frac{1}{2 π} (| w_{t_{F (a_{i})}}^{r} (x_{j}) - w_{t_{H (c_{i})}}^{r} (x_{j}) | + | w_{f_{H (c_{i})}}^{k} (x_{j}) - w_{f_{F (a_{i})}}^{k} (x_{j}) |) \end{matrix}]) \\ + (\frac{1}{4} \sum_{j = 1}^{n} \sum_{i = 1}^{m} [\begin{matrix} | r_{t_{H_{(c_{i})}}} (x_{j}) - r_{t_{G_{(b_{i})}}} (x_{j}) | + | k_{f_{G_{(b_{i})}}} (x_{j}) - k_{f_{H_{(c_{i})}}} (x_{j}) | \\ + \frac{1}{2 π} (| w_{t_{H (c_{i})}}^{r} (x_{j}) - w_{t_{G (b_{i})}}^{r} (x_{j}) | + | w_{f_{G (b_{i})}}^{k} (x_{j}) - w_{f_{H (c_{i})}}^{k} (x_{j}) |) \end{matrix}]) \\ = d_{H} ((F, A), (H, C)) + d_{H} ((H, C), (G, B)) . \end{matrix}$

Theorem 6.4. All the distance measures given in Definition 6.2 satisfies the following properties:

d ((F, A) , (G, B)) = d ((F, A) ∩ (G, B) , (F, A) ∪ (G, B))

d ((F, A) , (F, A) ∩ (G, B)) = d ((G, B) , (F, A) ∪ (G, B))

d ((F, A) , (F, A) ∪ (G, B)) = d ((G, B) , (F, A) ∩ (G, B))

Proof. For the sake of brevity only the proof for the Hamming distance measure is given here. $\begin{matrix} d_{H} ((F, A) \cap (G, B), (F, A) \cup (G, B)) \\ = \frac{1}{4} \sum_{j = 1}^{n} \sum_{i = 1}^{m} [| min (r_{t_{F_{(a_{i})}}} (x_{j}), r_{t_{G_{(b_{i})}}} (x_{j})) - max (r_{t_{F_{(a_{i})}}} (x_{j}), r_{t_{G_{(b_{i})}}} (x_{j})) | \\ + | min (1 - k_{f_{F (a_{i})}} (x_{j}), 1 - k_{f_{G (b_{i})}} (x_{j})) - max (1 - k_{f_{F (a_{i})}} (x_{j}), 1 - k_{f_{G (b_{i})}} (x_{j})) | \\ + \frac{1}{2 π} (| min (w_{t_{F (a_{i})}}^{r} (x_{j}), w_{t_{G (b_{i})}}^{r} (x_{j})) - max (w_{t_{F (a_{i})}}^{r} (x_{j}), w_{t_{G (b_{i})}}^{r} (x_{j})) | \\ + | \min (2 π - w_{f_{F_{(a_{i})}}}^{k} (x_{j}), 2 π - w_{f_{G_{(b_{i})}}}^{k} (x_{j})) - \max (2 π - w_{f_{F_{(a_{i})}}}^{k} (x_{j}), 2 π - w_{f_{G_{(b_{i})}}}^{k} (x_{j})) |)] \\ = \frac{1}{4} \sum_{j = 1}^{n} \sum_{i = 1}^{m} [| min (r_{t_{F (a_{i})}} (x_{j}), r_{t_{G (b_{i})}} (x_{j})) - max (r_{t_{F (a_{i})}} (x_{j}), r_{t_{G (b_{i})}} (x_{j})) | + | (1 \\ - max (k_{f_{F (a_{i})}} (x_{j}), k_{f_{G (b_{i})}} (x_{j}))) - (1 - min (k_{f_{F (a_{i})}} (x_{j}), k_{f_{G (b_{i})}} (x_{j}))) | \\ + \frac{1}{2 π} (| min (w_{t_{F (a_{i})}}^{r} (x_{j}), w_{t_{G (a_{i})}}^{r} (x_{j})) - max (w_{t_{F (a_{i})}}^{r} (x_{j}), w_{t_{G (b_{i})}}^{r} (x_{j})) | \\ + | (2 π - max (w_{f_{F (a_{i})}}^{k} (x_{j}), w_{f_{G (b_{i})}}^{k} (x_{j}))) - (2 π - min (w_{f_{F (a_{i})}}^{k} (x_{j}), w_{f_{G (b_{i})}}^{k} (x_{j}))) |)] \\ = \frac{1}{4} \sum_{j = 1}^{n} \sum_{i = 1}^{m} [| min (r_{t_{F (a_{i})}} (x_{j}), r_{t_{G (b_{i})}} (x_{j})) - max (r_{t_{F (a_{i})}} (x_{j}), r_{t_{G (b_{i})}} (x_{j})) | + | min (k_{f_{F (a_{i})}} (x_{j}), k_{f_{G (b_{i})}} (x_{j})) \\ - max (k_{f_{F (a_{i})}} (x_{j}), k_{f_{G (b_{i})}} (x_{j})) | \\ + \frac{1}{2 π} (| min (w_{t_{F (a_{i})}}^{r} (x_{j}), w_{t_{G (b_{i})}}^{r} (x_{j})) - max (w_{t_{F (a_{i})}}^{r} (x_{j}), w_{t_{G (b_{i})}}^{r} (x_{j})) | \\ + | min (w_{f_{F (a_{i})}}^{k} (x_{j}), w_{f_{G (b_{i})}}^{k} (x_{j})) - max (w_{f_{F (a_{i})}}^{k} (x_{j}), w_{f_{G (b_{i})}}^{k} (x_{j})) |)] \\ = \frac{1}{4} \sum_{j = 1}^{n} \sum_{i = 1}^{m} [| r_{t_{F (a_{i})}} (x_{j}) - r_{t_{G (b_{i})}} (x_{j}) | + | k_{f_{F (a_{i})}} (x_{j}) - k_{f_{G (b_{i})}} (x_{j}) | \\ + \frac{1}{2 π} (| w_{t_{F (a_{i})}}^{r} (x_{j}) - w_{t_{G (b_{i})}}^{r} (x_{j}) | + | w_{f_{F (a_{i})}}^{k} (x_{j}) - w_{f_{G (b_{i})}}^{k} (x_{j}) |)] \\ = \frac{1}{4} \sum_{j = 1}^{n} \sum_{i = 1}^{m} [| r_{t_{F (a_{i})}} (x_{j}) - r_{t_{G (b_{i})}} (x_{j}) | + | k_{f_{G (b_{i})}} (x_{j}) - k_{f_{F (a_{i})}} (x_{j}) | \\ + \frac{1}{2 π} (| w_{t_{F (a_{i})}}^{r} (x_{j}) - w_{t_{G (b_{i})}}^{r} (x_{j}) | + | w_{f_{G (b_{i})}}^{k} (x_{j}) - w_{f_{F (a_{i})}}^{k} (x_{j}) |)] \\ = d_{H} ((F, A), (G, B)) . \end{matrix}$

(ii) & (iii) The proof is similar to that of part (i).

7 Conclusion

The work in this paper is theoretical in nature. This paper deals with the establishment of the notion of complex vague soft sets which extends the range

the membership function which is characterized by an amplitude term and a phase term to a unit circle in the complex plane and is an improvement to the notion of complex fuzzy sets. The phase term of this concept on the other hand, provides the ability to deal with complex data which consist of two dimensional information, particularly data which involves periodicity, while the amplitude term gives it the ability to consider values of the membership function in more detail compared to classical soft sets, fuzzy sets or its hybrid structures. The basic set theoretic operations pertaining to this new model were introduced and the algebraic properties of these operations were studied. The axiomatic definition of the distance function between CVSSs was proposed, several distance measures between CVSSs were introduced and some fundamental algebraic properties of these distance measures were proposed andverified.

8 Suggestions for future research

In this paper we have introduced the concept of complex vague soft sets, its properties and the distance measure between CVSSs. The distance measures introduced here can be used to develop the similarity measure and subsequently the entropy of CVSSs. These are important characteristics of any type of fuzzy set and has a lot of potential to be applied in problem solving and decision making processes to obtain more accurate solutions.

Footnotes

Acknowledgments

The authors would like to gratefully acknowledge the financial assistance received from the Ministry of Education, Malaysia and UCSI University, Malaysia under Grant no. FRGS/1/2014/ST06/UCSI/03/1.

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