Interval-valued neutrosophic soft sets and its decision making

Abstract

In this paper, the notion of the interval valued neutrosophic soft set (ivn-soft sets) is defined which is a combination of an interval valued neutrosophic set [35] and a soft set [29]. Our ivn-soft sets generalizes the concept of the soft set, fuzzy soft set, interval valued fuzzy soft set, intuitionistic fuzzy soft set, interval valued intuitionistic fuzzy soft set and neutrosophic soft set. Then, we introduce some definitions and operations on ivn-soft sets sets. Some properties of ivn-soft sets which are connected to operations have been established. Also, the aim of this paper is to investigate the decision making based on ivn-soft sets by level soft sets. Therefore, we develop a decision making methods and then give a example to illustrate the developed approach.

1 Introduction

Many fields deal with the uncertain data may not be successfully modeled by the classical mathematics, since concept of uncertainty is too complicate and not clearly defined object. But they can be modeled a number of different approaches including the probability theory, fuzzy set theory [39], intuitionistic fuzzy set [3], rough set theory [32], neutrosophic set theory [33] and some other mathematical tools. These theories have been applied in many real applications to handle uncertainty. In 1999, Molodtsov [29] succesfully proposed a completely new theory so-called soft set theory by using classical sets because its been pointed out that soft sets are not appropriate to deal with uncertain and fuzzy parameters. The theory is a relatively new mathematical model for dealing with uncertainty from a parametrization point of view.

After Molodtsov, there has been a rapid growth of interest in soft sets and their various applications such as; algebraic structures (e.g. [1, 2, 5, 37, 41]), optimization (e.g. [21]), lattice (e.g. [19, 31, 33]), topology (e.g. [8,11, 28, 34]), perron integration, data analysis and operations research (e.g. [29, 30]), game theory (e.g. [14, 29]), clustering (e.g. [4, 27]), medical diagnosis (e.g. [38]), and decision making under uncertainty (e.g. [9, 10, 11, 13, 16, 20, 26]). In recent years, many interesting applications of soft set theory have been expanded by embedding the ideas of fuzzy sets (e.g. [9, 12, 16, 22]), rough sets (e.g. [15]) , intuitionistic fuzzy sets (e.g. [18, 23]), interval valued intuitionistic fuzzy sets (e.g. [17, 40]), neutrosophic sets (e.g. [24, 25]).

Intuitionistic fuzzy sets can only handle incomplete information because the sum of degree true, indeterminacy and false is one in intuitionistic fuzzy sets. But neutrosophic sets can handle the indeterminate information and inconsistent information which exists commonly in belief systems in neutrosophic set since indeterminacy is quantified explicitly and truth-membership, indeterminacy-membership and falsity-membership are independent. It is mentioned in [33, 35]. Therefore, Maji firstly proposed neutrosophic soft sets with operations, which is free of the difficulties mentioned above, in [25]. He also, applied to decision making problems in [24]. After Maji, the studies on the neutrosophic soft set theory have been studied increasingly (e.g. [6, 7]).

From academic point of view, the neutrosophic set and operators need to be specified because is hard to be applied to the real applications. So the concept of interval valued neutrosophic sets [35] which can represent uncertain, imprecise, incomplete and inconsistent information was proposed. In this paper, we first define interval valued neutrosophic soft sets (ivn-soft sets) which is generalizes the concept of the soft set, fuzzy soft set, interval valued fuzzy soft set, intuitionistic fuzzy soft set, interval valued intuitionistic fuzzy soft sets. Then, we introduce some definitions and operations of ivn-soft sets. Some properties of ivn-soft sets which are connected to operations have been established. Also, the aim of this paper is to investigate the decision making based on ivn-soft sets. By means of level soft sets, we develop an adjustable approach to ivn-soft sets based decision making and a examples are provided to illustrate the developed approach.

The relationship among ivn-soft set and other soft sets is illustrated as;

$$\begin{aligned} {\text{Soft}}\;{\text{set}} & \subseteq {\text{Fuzzy}}\;{\text{soft}}\;{\text{set}} \\ & \subseteq {\text{Intuitionistic}}\;{\text{fuzzy}}\;{\text{soft}}\;{\text{set}}\;{\text{(Interval}}\;{\text{valued}}\;{\text{fuzzy}}\;{\text{soft}}\;{\text{set)}} \\ & \subseteq {\text{Interval}}\;{\text{valued}}\;{\text{intuitionistic}}\;{\text{fuzzy}}\;{\text{soft}}\;{\text{set}} \\ & \subseteq {\text{Interval}}\;{\text{valued}}\;{\text{neutrosophic}}\;{\text{soft}}\;{\text{set}} \\ \end{aligned}$$

Therefore, interval valued neutrosophic soft set is a generalization other each the soft sets.

2 Preliminary

In this section, we present the basic definitions of neutrosophic set theory [33], interval valued neutrosophic set theory [35] and soft set theory [29] that are useful for subsequent discussions. More detailed explanations related to this subsection may be found in [6, 7, 12, 17, 24, 25, 33, 35, 36].

Definition 2.1

[33] A neutrosophic set A on the universe of discourse U is defined as

$$A=\big \{\langle x,T_A(x),I_A(x),F_A(x)\rangle : x\in U\big \}$$

where $T_A,I_A,F_A: U \rightarrow ]^-0,1^+[$ and $^-0\le T_A(x)+I_A(x)+F_A(x) \le 3^+$. From philosophical point of view, the neutrosophic set takes the value from real standard or non-standard subsets of $]^-0,1^+[$. But in real life application in scientific and engineering problems it is difficult to use neutrosophic set with value from real standard or non-standard subset of $]^-0,1^+[$. Hence we consider the neutrosophic set which takes the value from the subset of [0, 1].

Here, 1$^+$ = 1 + $\varepsilon$, where 1 is its standard part and $\varepsilon$ its non-standard part. Similarly, $^-0$ = 0 − $\varepsilon$, where 0 is its standard part and $\varepsilon$ its non-standard part.

Definition 2.2

[36] Let U be a space of points (objects), with a generic element in U denoted by u. A single valued neutrosophic sets A in U is characterized by a truth-membership function $T_A$, a indeterminacy-membership function $I_A$ and a falsity-membership function $F_A$. $T_A(u)$; $I_A(u)$ and $F_A(u)$ are real standard or nonstandard subsets of [0, 1]. It can be written as

$$A=\{\langle u,(T_A(u),I_A(u),F_A(u))\rangle:u\in U, \,T_A(u),I_A(u),F_A(u)\in [0,1]\}.$$

There is no restriction on the sum of $T_A(u)$; $I_A(u)$ and $F_A(u)$, so $0\le sup T_A(u) + sup I_A(u) + supF_A(u)\le 3$.

Definition 2.3

[35] Let U be a space of points (objects), with a generic element in U denoted by u. An interval value neutrosophic set (IVN-sets) A in U is characterized by truth-membership function $T_A$, a indeterminacy-membership function $I_A$ and a falsity-membership function $F_A$. For each point $u \in U$; $T_A$, $I_A$ and $F_A \subseteq [0,1]$.

Thus, a IVN-sets over U can be represented by the set of

$$A= \{\langle T_A(u), I_A(u), F_A(u)\rangle /u:u\in U\}$$

Here, $(T_A(u), I_A(u), F_A(u))$ is called interval value neutrosophic number for all $u \in U$ and all interval value neutrosophic numbers over U will be denoted by IVN(U).

Example 2.4

Assume that the universe of discourse $U=\{u_1, u_2\}$ where $u_1$ and characterises the quality, $u_2$ indicates the prices of the objects. It may be further assumed that the values of $u_1$ and $u_2$ are subset of [ 0, 1 ] and they are obtained from a expert person. The expert construct an interval value neutrosophic set the characteristics of the objects according to by truth-membership function $T_A$, a indeterminacy-membership function $I_A$ and a falsity-membership function $F_A$ as follows;

$$A=\{\langle [0.1,1.0], [0.1,0.4], [0.4,0.7]\rangle /u_1,\langle [0.6,0.9], [0.8,1.0], [0.4,0.6] \rangle /u_2\}$$

Definition 2.5

[35] Let A a interval valued neutrosophic sets. Then, for all $u \in U$,

• 1. A is empty, denoted $A=\widetilde{{ \emptyset }}$, is defined by

  $$\widetilde{{ \emptyset }}= \{ \langle [ 0, 0],[ 1, 1],[ 1,1] \rangle/u:u\in U\}$$

• 2. A is universal, denoted $A=\widetilde{{E}}$, is defined by

  $$\widetilde{{E}}= \{\langle[ 1, 1],[ 0, 0],[ 0,0] \rangle/u:u\in U\}$$

• 3. The complement of A is denoted by $\overline{A}$ and is defined by

  $$\begin{aligned} \bar{A} & = \left\{ { \langle [infF_{A} (u),supF_{A} (u)],[1 - supI_{A} (u),{1 - \text{inf I\_A(u)],}}} \right. \\ & \quad \left. {[infT_{A} (u),supT_{A} (u)] \rangle /u:u \in U} \right\} \\ \end{aligned}$$

Definition 2.6

[35] An interval valued neutrosophic set A is contained in the other interval valued neutrosophic set B, $A\widetilde{{ \subseteq }}B$, if and only if

$$\begin{array}{*{20}l} {infT_{A} (u) \le infT_{B} (u)} \hfill & {supI_{A} (u) \ge supI_{B} (u)} \hfill \\ {supI_{A} (u) \ge supI_{B} (u)} \hfill & {infF_{A} (u) \ge infF_{B} (u)} \hfill \\ {infI_{A} (u) \ge infI_{B} (u)} \hfill & {supF_{A} (u) \ge supF_{B} (u)} \hfill \\ \end{array}$$

for all $u \in U$.

Note that an interval valued neutrosophic number $X=(T_X,I_X,F_X)$ is larger than the other interval valued neutrosophic number $Y=(T_Y,I_Y,F_Y)$, denoted $X\widehat{ \le }Y$, if and only if

$$\begin{array}{*{20}l} {infT_{X} \le infT_{Y} } \hfill & {supI_{X} \ge supI_{Y} } \hfill \\ {supT_{X} \le supT_{Y} } \hfill & {supF_{X} \ge supF_{Y} } \hfill \\ {infI_{X} \ge infI_{Y} } \hfill & {supF_{X} \ge supF_{Y} } \hfill \\ \end{array}$$

Definition 2.7

[35] Let A and B be two interval valued neutrosophic sets. Then, for all $u \in U$, $a \in R^+$,

• 1. Intersection of A and B, denoted by $A\widetilde{{ \cap }} B$, is defined by

  $$\begin{aligned} A\widetilde{{ \cap }} B&= \{\langle[ min(inf T_A(u), i nf T_B(u)),min(sup T_A(u), sup T_B(u))],\\ & \quad [ max(inf I_A(u), inf I_B(u)), max(sup IA(x), sup I_B(u)) ],\\ & \quad [ max(inf F_A(u), inf F_B(u)), max(supF_A(u),sup F_B(u))] \rangle/u:u\in U\} \end{aligned}$$

• 2. Union of A and B, denoted by $A\widetilde{{ \cup }} B$, is defined by

  $$\begin{aligned} A\widetilde{{ \cup }} B&= \{ \langle[ max(inf T_A(u), inf T_B(u)),max(sup T_A(u), sup T_B(u))],\\ & \quad [min(inf I_A(u), inf I_B(u)), min(sup I_A(u), sup I_B(u))],\\ & \quad [min(inf F_A(u), inf F_B(u)), min(supF_A(u), sup F_B(u))] \rangle/u:u\in U\} \end{aligned}$$

• 3. Difference of A and B, denoted by $A\widetilde{{ \setminus }} B$, is defined by

  $$\begin{aligned} A\widetilde{{ \setminus }} B&= \{ \langle[ min(inf T_A(u), inf F_B(u)), min(sup T_A(u), sup F_B( x))],\\ & \quad [max(inf I_A(u), 1- sup I_B(u)),max(sup I_A(u), 1- inf I_B(u))],\\ & \quad [max(inf F_A(u), inf T_B(u)), max(supF_A(u), sup T_B(u))] \rangle/u:u\in U\} \end{aligned}$$

• 4. Addition of A and B, denoted by $A\widetilde{{ + }} B$, is defined by

  $$\begin{aligned} A\widetilde{{ + }} B&= \{\langle[ min(inf T_A(u) + inf T_B(u), 1), min(sup T_A(u) + sup T_B(u), 1)],\\ & \quad [min(inf I_A(u) + inf I_B(u), 1), min(sup I_A(u) + sup I_B(u), 1)],\\ & \quad [min(inf F_A(u) + inf F_B(u), 1), min(supF_A(u) + supF_B(u), 1)] \rangle/u:u\in U\}\end{aligned}$$

• 5. Scalar multiplication of A, denoted by $A\widetilde{{ . }} a$, is defined by

  $$\begin{aligned} A\widetilde{{ . }}a&=\{ \langle[ min(inf T_A(u).a, 1), min(sup T_A(u). a, 1)],\\ & \quad [min(infI_A(u).a, 1), min(sup I_A(u).a, 1)],\\ & \quad [min(inf F_A(u).a, 1),min(sup F_A(u) . a, 1)] \rangle/u:u\in U\} \end{aligned}$$

• 6. Scalar division of A, denoted by $A\widetilde{{ / }}a$, is defined by

  $$\begin{aligned} A\widetilde{{ / }} a&= \{ \langle[ min(inf T_A(u)/a, 1), min(sup T_A(u)/ a, 1)],\\ & \quad [min(infI_A(u)/a, 1), min(sup I_A(u)/a, 1)],\\ & \quad [min(inf F_A(u)/a, 1),min(sup F_A(u) / a, 1)] \rangle/u:u\in U\}\end{aligned}$$

• 7. Truth-Favorite of A, denoted by $\widetilde{{ \triangle }} A$, is defined by

  $$\begin{aligned} \widetilde{{ \triangle }} A&= \{ \langle[ min(inf T_A(u) + inf I_A(u), 1), min(sup T_A(u) + sup I_A(u), 1)],[0, 0],\\ & \quad [inf F_A(u) , supF_A(u)] \rangle/u:u\in U\} \end{aligned}$$

• 8. False-Favorite of A, denoted by $\widetilde{{ \nabla }} A$, is defined by

  $$\begin{aligned} \widetilde{{ \nabla }} A &= \{ \langle[ inf T_A(u) ,sup T_A(u) ],[0, 0],\\ & \quad [min(inf F_A(u) + inf I_A(u), 1),min(supF_A(u) + sup I_A(u), 1)] \rangle/u:u\in U\} \end{aligned}$$

Definition 2.8

[29] Let U be an initial universe, P(U) be the power set of U, E be a set of all parameters and $X\subseteq E$. Then a soft set $F_X$ over U is a set defined by a function representing a mapping

$$f_X: E\rightarrow P(U)\, such \,that \,f_X(x)=\emptyset \,\,if\,\, x\notin X$$

Here, $f_X$ is called approximate function of the soft set $F_X$, and the value $f_X(x)$ is a set called x-element of the soft set for all $x \in E$. It is worth noting that the set is worth noting that the sets $f_X(x)$ may be arbitrary. Some of them may be empty, some may have nonempty intersection. Thus, a soft set over U can be represented by the set of ordered pairs

$$F_X= \{(x, f_X(x)): x\in E, f_X(x)\in P(U)\}$$

Example 2.9

Suppose that $U=\{u_1,u_2,u_3,u_4,u_5,u_6\}$ is the universe contains six house under consideration in a real agent and $E=\{x_1=cheap, x_2=beatiful, x_3=green surroundings, x_4=costly, x_5= large\}$.

If a customer to select a house from the real agent then, he/she can construct a soft set $F_X$ that describes the characteristic of houses according to own requests. Assume that $f_X(x_1)= \{u_1,u_2\}$, $f_X(x_2)= \{u_1 \}$, $f_X(x_3)= \emptyset$, $f_X(x_4)= U$, $\{u_1,u_2,u_3,u_4,u_5\}$ then the soft-set $F_X$ is written by

$$F_X=\{(x_1, \{u_1,u_2\}),(x_2, \{u_1,u_4,u_5,u_6\}), (x_4,U), (x_5,\{u_1,u_2,u_3,u_4,u_5\})\}$$

The tabular representation of the soft set $F_X$ is as follow (Table 1):

Table 1 The tabular representation of the soft set $F_X$

Definition 2.10

[26] Let $U=\{u_1,u_2,\ldots,u_k\}$ be an initial universe of objects, $E=\{x_1, x_2,\ldots ,x_m\}$ be a set of parameters and $F_X$ be a soft set over U. For any $x_j \in E$, $f_X(x_j)$ is a subset of U. Then, the choice value of an object $u_i\in U$ is $c_i$, given by $c_i = \sum _j u_{ij}$, where $u_{ij}$ are the entries in the table of the reduct-soft-set. That is,

$$u_{{ij}} = \left\{ {\begin{array}{*{20}l} {1,} \hfill & {u_{i} \in f_{X} (x_{j} )} \hfill \\ {0,} \hfill & {u_{i} \notin f_{X} (x_{j} )} \hfill \\ \end{array} } \right.$$

Example 2.11

Consider the above Example 2.9. Clearly,

$$\begin{aligned} c_{1} & = \sum\limits_{{j = 1}}^{5} {u_{{1j}} } = 4, \\ c_{3} & = c_{6} = \sum\limits_{{j = 1}}^{5} {u_{{3j}} } = \sum\limits_{{j = 1}}^{5} {u_{{6j}} } = 2, \\ c_{2} & = c_{4} = c_{5} = \sum\limits_{{j = 1}}^{5} {u_{{2j}} } = \sum\limits_{{j = 1}}^{5} {u_{{4j}} } = \sum\limits_{{j = 1}}^{5} {u_{{5j}} } = 3 \\ \end{aligned}$$

.

Definition 2.12

[13] Let $F_X$ and $F_Y$ be two soft sets. Then,

• 1. Complement of $F_X$ is denoted by $F_X^{\tilde{c}}$. Its approximate function $f_{X^c}(x)= U\setminus f_X(x)\quad \mathrm{for}\,\mathrm{all }\, x \in E$

• 2. Union of $F_X$ and $F_Y$ is denoted by $F_X \tilde{\cup } F_Y$. Its approximate function $f_{X\tilde{\cup } F_Y}$ is defined by

  $$f_{X \tilde{\cup } Y}(x)= f_X(x)\cup f_Y(x)\quad \mathrm{for}\,\mathrm{all }\, x \in E.$$

• 3. Intersection of $F_X$ and $F_Y$ is denoted by $F_X\tilde{\cap }F_Y$. Its approximate function $f_{X \widetilde{{ \cap }} Y}$ is defined by

  $$f_{X \tilde{\cap } Y}(x)= f_X(x)\cap f_Y(x)\quad \mathrm{for}\,\mathrm{all }\, x \in E.$$

3 Interval-valued neutrosophic soft sets

In this section, we give interval valued neutrosophic soft sets (ivn-soft sets) which is a combination of an interval valued neutrosophic sets [35] and a soft sets [29]. Then, we introduce some definitions and operations of ivn-soft sets sets. Some properties of ivn-soft sets which are connected to operations have been established. Some of it is quoted from [6, 7, 12, 13, 17, 18, 24, 25, 35].

Definition 3.1

Let U be an initial universe set, IVN( U ) denotes the set of all interval valued neutrosophic sets of U and E be a set of parameters that are describe the elements of U. An interval valued neutrosophic soft sets(ivn-soft sets) over U is a set defined by a set valued function $\Upsilon _K$ representing a mapping

$$\upsilon _K: E\rightarrow IVN(U)$$

It can be written a set of ordered pairs

$$\Upsilon _K= \{(x, \upsilon _K(x)): x\in E\}$$

Here, $\upsilon _K$, which is interval valued neutrosophic sets, is called approximate function of the ivn-soft sets $\Upsilon _K$ and $\upsilon _K(x)$ is called x-approximate value of $x \in E$. The subscript K in the $\upsilon _K$ indicates that $\upsilon _K$ is the approximate function of $\Upsilon _K$.

Generally, $\upsilon _K$, $\upsilon _L$, $\upsilon _M$,… will be used as an approximate functions of $\Upsilon _K$, $\Upsilon _L$, $\Upsilon _M$,…, respectively.

Note that the sets of all ivn-soft sets over U will be denoted by IVNS(U) .

Now let us give the following example for ivn-soft sets.

Example 3.2

Let $U=\{u_1,u_2\}$ be set of houses under consideration and E is a set of parameters which is a neutrosophic word. Consider $E=\{x_1=cheap, x_2=beatiful, x_3=green surroundings, x_4=costly, x_5= large\}$. In this case, we give an (ivn-soft sets) $\Upsilon _K$ over U as;

$$\begin{aligned} \Upsilon _K &=\{(x_1,\{\langle [0.6,0.8], [0.8,0.9], [0.1,0.5] \rangle /u_1,\langle [0.5,0.8], [0.2,0.9], [0.1,0.7] \rangle /u_2 \}),\\ & \quad (x_2,\{\langle [0.1,0.4], [0.5,0.8], [0.3,0.7] \rangle /u_1,\langle [0.1,0.9], [0.6,0.9], [0.2,0.3] \rangle /u_2 \}),\\ & \quad (x_3,\{\langle [0.2,0.9], [0.1,0.5], [0.7,0.8] \rangle /u_1,\langle [0.4,0.9], [0.1,0.6], [0.5,0.7] \rangle /u_2\}),\\ & \quad (x_4,\{\langle [0.6,0.9], [0.6,0.9], [0.6,0.9] \rangle /u_1,\langle [0.5,0.9], [0.6,0.8], [0.1,0.8] \rangle /u_2\}),\\ & \quad (x_5,\{\langle [0.0,0.9], [1.0,1.0], [1.0,1.1] \rangle /u_1,\langle [0.0,0.9], [0.8,1.0], [0.2,0.5] \rangle /u_2\})\} \end{aligned}$$

The tabular representation of the ivn-soft set $\Upsilon _K$ is as follow (Table 2):

Table 2 The tabular representation of the ivn-soft set $\Upsilon _K$

Definition 3.3

Let $\Upsilon _K \in IVNS(U)$. If $\upsilon _K(x)=\widetilde{{ \emptyset }}$ for all $x\in E$, then $\Upsilon _K$ is called an empty ivn-soft set, denoted by $\Upsilon _{{\widehat{{ \emptyset }}}}$.

Definition 3.4

Let $\Upsilon _K \in IVNS(U)$. If $\upsilon _K(x)=\widetilde{{E}}$ for all $x\in E$, then $\Upsilon _K$ is called a universal ivn-soft set, denoted by $\Upsilon _{{\hat{E}}}$.

Example 3.5

Assume that $U=\{u_1, u_2\}$ is a universal set and $E=\{x_1, x_2, x_3,$ $x_4,x_5\}$ is a set of all parameters. Consider the tabular representation of the $\Upsilon _{{\widehat{{ \emptyset }}}}$ is as follows (Table 3);

Table 3 The tabular representation of the ivn-soft set $\Upsilon _{\widehat{{ \emptyset }}}$

The tabular representation of the $\Upsilon _{\hat{E}}$ is as follows (Table 4);

Table 4 The tabular representation of the ivn-soft set $\Upsilon _{\hat{E}}$

Definition 3.6

Let $\Upsilon _K, \Upsilon _L\in IVNS(U)$. Then, $\Upsilon _K$ is an ivn-soft subset of $\Upsilon _L$, denoted by $\Upsilon _K \widehat{{ \subseteq }} \Upsilon _L$, if $\upsilon _K(x)\widetilde{{ \subseteq }} \upsilon _L(x)$ for all $x \in E$.

Example 3.7

Assume that $U=\{u_1, u_2\}$ is a universal set and $E=\{x_1, x_2, x_3,$ $x_4,x_5\}$ is a set of all parameters. Consider the tabular representation of the $\Upsilon _K$ is as follows (Table 5);

Table 5 The tabular representation of the ivn-soft set $\Upsilon _{K}$

The tabular representation of the $\Upsilon _L$ is as follows (Table 6);

Table 6 The tabular representation of the ivn-soft set $\Upsilon _{L}$

Clearly, by Definition 3.6, we have $\Upsilon _K \widehat{{ \subseteq }} \Upsilon _L$.

Remark 3.8

$\Upsilon _K \widehat{{ \subseteq }} \Upsilon _L$ does not imply that every element of $\Upsilon _K$ is an element of $\Upsilon _L$ as in the definition of the classical subset.

Proposition 3.9

Let $\Upsilon _K, \Upsilon _L,\Upsilon _M \in IVNS(U)$. Then,

• 1. $\Upsilon _{K} \widehat{{ \subseteq }}\Upsilon _{{\hat{E}}}$

• 2. $\Upsilon _{{\widehat{{ \emptyset }}}} \widehat{{ \subseteq }}\Upsilon _{K}$

• 3. $\Upsilon _{K} \widehat{{ \subseteq }}\Upsilon _{K}$

• 4. $\Upsilon _{K} \widehat{{ \subseteq }}\Upsilon _{L}$ and $\Upsilon _{L} \widehat{{ \subseteq }}\Upsilon _{M} \Rightarrow \Upsilon _{K} \widehat{{ \subseteq }}\Upsilon _{M}$

Proof

They can be proved easily by using the approximate function of the ivn-soft sets. $\square$

Definition 3.10

Let $\Upsilon _K, \Upsilon _L \in IVNS(U)$. Then, $\Upsilon _K$ and $\Upsilon _L$ are ivn-soft equal, written as $\Upsilon _K = \Upsilon _L$, if and only if $\upsilon _K(x)= \upsilon _L(x)$ for all $x \in E$.

Proposition 3.11

Let $\Upsilon _K, \Upsilon _L,\Upsilon _M \in IVNS(U)$. Then,

• 1. $\Upsilon _K = \Upsilon _L$ and $\Upsilon _L =\Upsilon _M \Leftrightarrow \Upsilon _K=\Upsilon _M$

• 2. $\Upsilon _K \widehat{{ \subseteq }} \Upsilon _L$ and $\Upsilon _L \widehat{{ \subseteq }} \Upsilon _K \Leftrightarrow \Upsilon _K =\Upsilon _L$

Proof

The proofs are trivial. $\square$

Definition 3.12

Let $\Upsilon _K \in IVNS(U)$. Then, the complement $\Upsilon _K^{\hat{{c}}}$ of $\Upsilon _K$ is an ivn-soft set such that

$$\upsilon _K^{\hat{{c}}}(x)=\overline{\upsilon _K}(x),\mathrm{for}\,\mathrm{all }\, x \in E.$$

Example 3.13

Consider the above Example 3.7, the complement $\Upsilon _L^{\hat{{c}}}$ of $\Upsilon _L$ can be represented into the following table (Table 7);

Table 7 The tabular representation of the ivn-soft set $\Upsilon _L^{\hat{{c}}}$

Proposition 3.14

Let $\Upsilon _K \in IVNS(U)$. Then,

• 1. $(\Upsilon _K^{\hat{{c}}})^{\hat{{c}}}= \Upsilon _K$

• 2. $\Upsilon _{\widehat{{ \emptyset }}} ^{\hat{{c}}} = \Upsilon _{\hat{E}}$

• 3. $\Upsilon _{\hat{E}} ^{\hat{{c}}} = \Upsilon _{\widehat{{ \emptyset }}}$

Proof

By using the fuzzy approximate functions of the ivn-soft set, the proofs can be straightforward. $\square$

Theorem 3.15

Let $\Upsilon _K \in IVNS(U)$. Then, $\Upsilon _K\widehat{{ \subseteq }}\Upsilon _L \Leftrightarrow \Upsilon _L^{\hat{{c}}}\widehat{{ \subseteq }}\Upsilon _K^{\hat{{c}}}$

Proof

By using the fuzzy approximate functions of the ivn-soft set, the proofs can be straightforward. $\square$

Definition 3.16

Let $\Upsilon _K, \Upsilon _L \in IVNS(U)$. Then, union of $\Upsilon _K$ and $\Upsilon _L$, denoted $\Upsilon _K \widehat{{ \cup }} \Upsilon _L$, is defined by

$$\upsilon _{K \widehat{{ \cup }} L}(x)=\upsilon _{K}(x)\widetilde{{ \cup }}\upsilon _{L}(x)\quad \mathrm{for}\,\mathrm{all }\, x \in E.$$

Example 3.17

Consider the above Example 3.7, the union of $\Upsilon _K$ and $\Upsilon _L$, denoted $\Upsilon _K \widehat{{ \cup }} \Upsilon _L$, can be represented into the following table (Table 8);

Table 8 The tabular representation of the ivn-soft set $\Upsilon _K \widehat{{ \cup }} \Upsilon _L$

Theorem 3.18

Let $\Upsilon _K, \Upsilon _L \in IVNS(U)$. Then, $\Upsilon _K \widehat{{ \cup }} \Upsilon _L$ is the smallest ivn-soft set containing both $\Upsilon _K$ and $\Upsilon _L$.

Proof

The proofs can be easily obtained from Definition 3.16. $\square$

Proposition 3.19

Let $\Upsilon _K, \Upsilon _L,\Upsilon _M \in IVNS(U)$. Then,

• 1. $\Upsilon _K\widehat{{ \cup }} \Upsilon _K = \Upsilon _K$

• 2. $\Upsilon _K \widehat{{ \cup }} \Upsilon _{\widehat{{ \emptyset }}} = \Upsilon _K$

• 3. $\Upsilon _K \widehat{{ \cup }} {\Upsilon _{\hat{E}}} = {\Upsilon _{\hat{E}}}$

• 4. $\Upsilon _K \widehat{{ \cup }} \Upsilon _L= \Upsilon _L\widehat{{ \cup }} \Upsilon _K$

• 5. $(\Upsilon _K \widehat{{ \cup }} \Upsilon _L)\widehat{{ \cup }} \Upsilon _M= \Upsilon _K \widehat{{ \cup }} (\Upsilon _L\widehat{{ \cup }} \Upsilon _M)$

Proof

The proofs can be easily obtained from Definition 3.16. $\square$

Definition 3.20

Let $\Upsilon _K, \Upsilon _L \in IVNS(U)$. Then, intersection of $\Upsilon _K$ and $\Upsilon _L$, denoted $\Upsilon _K \widehat{{ \cap }} \Upsilon _L$, is defined by

$$\upsilon _{K \widehat{{ \cap }} L}(x)=\upsilon _K(x)\widetilde{{ \cap }}\upsilon _{L}(x)\quad \mathrm{for}\,\mathrm{all }\, x \in E.$$

Example 3.21

Consider the above Example 3.7, the intersection of $\Upsilon _K$ and $\Upsilon _L$, denoted $\Upsilon _K \widehat{{ \cap }} \Upsilon _L$, can be represented into the following table (Table 9);

Table 9 The tabular representation of the ivn-soft set $\Upsilon _K \widehat{{ \cap }} \Upsilon _L$

Proposition 3.22

Let $\Upsilon _K, \Upsilon _L \in IVNS(U)$. Then, $\Upsilon _K \widehat{{ \cap }} \Upsilon _L$ is the largest ivn-soft set containing both $\Upsilon _K$ and $\Upsilon _L$.

Proof

The proofs can be easily obtained from Definition 3.20. $\square$

Proposition 3.23

Let $\Upsilon _K, \Upsilon _L,\Upsilon _M \in IVNS(U)$. Then,

• 1. $\Upsilon _K \widehat{{ \cap }} \Upsilon _K = \Upsilon _K$

• 2. $\Upsilon _K \widehat{{ \cap }} \Upsilon _{\widehat{{ \emptyset }}} = \Upsilon _{\widehat{{ \emptyset }}}$

• 3. $\Upsilon _K \widehat{{ \cap }} {\Upsilon _{\hat{E}}} = \Upsilon _K$

• 4. $\Upsilon _K \widehat{{ \cap }} \Upsilon _L= \Upsilon _L\widehat{{ \cap }}\Upsilon _K$

• 5. $(\Upsilon _K \widehat{{ \cap }} \Upsilon _L)\widehat{{ \cap }} \Upsilon _M=\Upsilon _K \widehat{{ \cap }} (\Upsilon _L\widehat{{ \cap }} \Upsilon _M)$

Proof

The proof of the Propositions 1- 5 are obvious. $\square$

Remark 3.24

Let $\Upsilon _K \in IVNS(U)$. If $\Upsilon _K \ne \Upsilon _{\widehat{{ \emptyset }}}$ or $\Upsilon _K \ne \Upsilon _{\hat{E}}$, then $\Upsilon _K \widehat{{ \cup }} \Upsilon _K^{\hat{c}}\ne \Upsilon _{\hat{E}}$ and $\Upsilon _K \widehat{{ \cap }}\Upsilon _K^{\hat{c}}\ne \Upsilon _{\widehat{{ \emptyset }}}$.

Proposition 3.25

Let $\Upsilon _K, \Upsilon _L \in IVNS(U)$. Then, De Morgan’s laws are valid

• 1. $(\Upsilon _K\, \widehat{{ \cup }} \,\Upsilon _L)^{\hat{c}}= \Upsilon _K^{\hat{c}}\, \widehat{{ \cap }} \,\Upsilon _L^{\hat{c}}$

• 2. $(\Upsilon _K \, \widehat{{ \cap }} \, \Upsilon _L)^{\hat{c}} = \Upsilon _K^{\hat{c}} \, \widehat{{ \cup }} \, \Upsilon _L^{\hat{c}}.$

Proof

The proofs can be easily obtained from Definition 3.12, Definition 3.16 and Definition 3.20. $\square$

Proposition 3.26

Let $\Upsilon _K, \Upsilon _L,\Upsilon _M \in IVNS(U)$. Then,

• 1. $\Upsilon _K \widehat{{ \cup }} (\Upsilon _L \widehat{{ \cap }} \Upsilon _M)= (\Upsilon _K \widehat{{ \cup }} \Upsilon _L) \widehat{{ \cap }} (\Upsilon _K \widehat{{ \cup }} \Upsilon _M)$

• 2. $\Upsilon _K \widehat{{ \cap }} (\Upsilon _L \widehat{{ \cup }}\Upsilon _M)= (\Upsilon _K\widehat{{ \cap }} \Upsilon _L)\widehat{{ \cup }} (\Upsilon _K \widehat{{ \cap }} \Upsilon _M)$

• 3. $\Upsilon _K \widehat{{ \cup }} (\Upsilon _K \widehat{{ \cap }} \Upsilon _L)= \Upsilon _K$

• 4. $\Upsilon _K \widehat{{ \cap }} (\Upsilon _K \widehat{{ \cup }}\Upsilon _L)= \Upsilon _K$

Proof

The proofs can be easily obtained from Definition 3.16 and Definition 3.20. $\square$

Definition 3.27

Let $\Upsilon _K, \Upsilon _L \in IVNS(U)$. Then, OR operator of $\Upsilon _K$ and $\Upsilon _L$, denoted $\Upsilon _K \widehat{{ \bigvee }} \Upsilon _L$, is defined by a set valued function $\Upsilon _O$ representing a mapping

$$\upsilon _O: E\times E\rightarrow IVN(U)$$

where

$$\upsilon _O(x,y)=\upsilon _K(x)\widetilde{{ \cup }}\upsilon _{L}(y)\quad \mathrm{for}\,\mathrm{ all }\,(x,y) \in E\times E.$$

Definition 3.28

Let $\Upsilon _K, \Upsilon _L \in IVNS(U)$. Then, AND operator of $\Upsilon _K$ and $\Upsilon _L$, denoted $\Upsilon _K \widehat{{ \bigwedge }} \Upsilon _L$, is defined by a set valued function $\Upsilon _A$ representing a mapping

$$\upsilon _A: E\times E\rightarrow IVN(U)$$

where

$$\upsilon _A(x,y)=\upsilon _K(x)\widetilde{{ \cap }}\upsilon _{L}(y)\quad \mathrm{for}\,\mathrm{ all }\,(x,y) \in E\times E.$$

Proposition 3.29

Let $\Upsilon _K, \Upsilon _L,\Upsilon _M \in IVNS(U)$. Then,

• 1. $(\Upsilon _K \widehat{{ \bigvee }} \Upsilon _L)^{\hat{c}}= \Upsilon _K^{\hat{c}} \widehat{{ \bigwedge }} \Upsilon _L^{\hat{c}}$

• 2. $(\Upsilon _K \widehat{{ \bigwedge }} \Upsilon _L)^{\hat{c}} = \Upsilon _K^{\hat{c}} \widehat{{ \bigvee }} \Upsilon _L^{\hat{c}}.$

• 3. $(\Upsilon _K \widehat{{ \bigvee }} \Upsilon _L)\widehat{{ \bigvee }} \Upsilon _M=\Upsilon _K \widehat{{ \bigvee }} (\Upsilon _L\widehat{{ \bigvee }} \Upsilon _M)$

• 4. $(\Upsilon _K \widehat{{ \bigwedge }} \Upsilon _L)\widehat{{ \bigwedge }} \Upsilon _M=\Upsilon _K \widehat{{ \bigwedge }} (\Upsilon _L\widehat{{ \bigwedge }} \Upsilon _M)$

Proof

The proof of the Propositions 1–4 are obvious. $\square$

Definition 3.30

Let $\Upsilon _K, \Upsilon _L \in IVNS(U)$. Then, difference of $\Upsilon _K$ and $\Upsilon _L$, denoted $\Upsilon _K \widehat{{ \setminus }} \Upsilon _L$, is defined by

$$\begin{aligned} \upsilon _{K \widehat{{ \setminus }} L}(x)=\upsilon _{K}(x)\widetilde{{ \setminus }}\upsilon _{L}(x)\quad \mathrm{for}\,\mathrm{all }\, x \in E. \end{aligned}$$

Definition 3.31

Let $\Upsilon _K, \Upsilon _L \in IVNS(U)$. Then, addition of $\Upsilon _K$ and $\Upsilon _L$, denoted $\Upsilon _K \widehat{{ +}} \Upsilon _L$, is defined by

$$\upsilon _{K \widehat{{ + }} L}(x)=\upsilon _{K}(x)\widetilde{{ + }}\upsilon _{L}(x)\quad \mathrm{for}\,\mathrm{ all }\, x \in E.$$

Proposition 3.32

Let $\Upsilon _K, \Upsilon _L , \Upsilon _M \in IVNS(U)$. Then,

• 1. $\Upsilon _{K}(x)\widehat{{ + }}\Upsilon _{L}(x)\widehat=\Upsilon _{L}(x)\widehat{{ +}}\Upsilon _{K}(x)$

• 2. $(\Upsilon _{K}(x)\widehat{{ + }}\Upsilon _{L}(x))\widehat{{ + }}\Upsilon _{M}(x)=\Upsilon _{K}(x)\widehat{{ + }}(\Upsilon _{L}(x)\widehat{{ + }}\Upsilon _{M}(x))$

Proof

The proofs can be easily obtained from Definition 3.31. $\square$

Definition 3.33

Let $\Upsilon _K \in IVNS(U)$. Then, scalar multiplication of $\Upsilon _K$, denoted $a\widehat{{ \times }}\Upsilon _K$, is defined by

$$a\widehat{{ \times }}\Upsilon _K=a\widetilde{{ . }}\upsilon _K\quad \mathrm{for}\,\mathrm{ all }\, x \in E.$$

Proposition 3.34

Let $\Upsilon _K, \Upsilon _L , \Upsilon _M \in IVNS(U)$. Then,

• 1. $\Upsilon _{K}(x)\widehat{{ \times }}\Upsilon _{L}(x)=\Upsilon _{L}(x)\widehat{{ \times }}\Upsilon _{K}(x)$

• 2. $(\Upsilon _{K}(x)\widehat{{ \times }}\Upsilon _{L}(x))\widehat{{ \times }}\Upsilon _{M}(x)=\Upsilon _{K}(x)\widehat{{ \times }}(\Upsilon _{L}(x)\widehat{{ \times }}\Upsilon _{M}(x))$

Proof

The proofs can be easily obtained from Definition 3.33. $\square$

Definition 3.35

Let $\Upsilon _K \in IVNS(U)$. Then, scalar division of $\Upsilon _K$, denoted $\Upsilon _K\hat{/}a$, is defined by

$$\Upsilon _K\hat{/}a =\Upsilon _K\widetilde{{ / }}a \quad \mathrm{for}\,\mathrm{ all }\, x \in E.$$

Example 3.36

Consider the above Example 3.7, for $a=5$, the scalar division of $\Upsilon _K$, denoted $\Upsilon _K\hat{/}5$, can be represented into the following table (Table 10);

Table 10 The tabular representation of the ivn-soft set $\Upsilon _K\hat{/}5$

Definition 3.37

Let $\Upsilon _K \in IVNS(U)$. Then, truth-Favorite of $\Upsilon _K$, denoted $\widehat{{ \bigtriangleup }} \Upsilon _K$, is defined by

$$\widehat{{ \bigtriangleup }} \Upsilon _K= \widetilde{{ \bigtriangleup }} \upsilon _K\quad \mathrm{for}\,\mathrm{ all }\, x \in E.$$

Example 3.38

Consider the above Example 3.7, the truth-Favorite of $\Upsilon _K$, denoted $\widehat{{ \bigtriangleup }} \Upsilon _K$, can be represented into the following table (Table 11);

Table 11 The tabular representation of the ivn-soft set $\widehat{{ \bigtriangleup }} \Upsilon _K$

Proposition 3.39

Let $\Upsilon _K, \Upsilon _L \in IVNS(U)$. Then,

• 1. $\widehat{{ \bigtriangleup }}\widehat{{ \bigtriangleup }} \Upsilon _K =\widehat{{ \bigtriangleup }}\Upsilon _K$

• 2. $\widehat{{ \bigtriangleup }} (\Upsilon _K \widehat{{ \cup }} \Upsilon _K)\widehat{{ \subseteq }} \widehat{{ \bigtriangleup }} \Upsilon _K \widehat{{ \cup }} \widehat{{ \bigtriangleup }}\Upsilon _K$

• 3. $\widehat{{ \bigtriangleup }} (\Upsilon _K \widehat{{ \cap }} \Upsilon _K)\widehat{{ \subseteq }} \widehat{{ \bigtriangleup }} \Upsilon _K \widehat{{ \cap }} \widehat{{ \bigtriangleup }}\Upsilon _K$

• 4. $\widehat{{ \bigtriangleup }} (\Upsilon _K \widehat{{ + }} \Upsilon _K)=\widehat{{ \bigtriangleup }} \Upsilon _K \widehat{{ + }} \widehat{{ \bigtriangleup }}\Upsilon _K$

Proof

The proofs can be easily obtained from Definition 3.16, Definition 3.20 and Definition 3.37. $\square$

Definition 3.40

Let $\Upsilon _K \in IVNS(U)$. Then, False-Favorite of $\Upsilon _K$, denoted $\widehat{{ \bigtriangledown }} \Upsilon _K$, is defined by

$$\widehat{{ \bigtriangledown }} \Upsilon _K= \widetilde{{ \bigtriangledown }} \upsilon _K\quad \mathrm{for}\,\mathrm{ all }\, x \in E.$$

Example 3.41

Consider the above Example 3.7, the False-Favorite of $\Upsilon _K$, denoted $\widehat{{ \bigtriangledown }} \Upsilon _K$, can be represented into the following table (Table 12);

Table 12 The tabular representation of the ivn-soft set $\widehat{{ \bigtriangledown }} \Upsilon _K$

Proposition 3.42

Let $\Upsilon _K, \Upsilon _L \in IVNS(U)$. Then,

• 1. $\widehat{{ \bigtriangledown }}\widehat{{ \bigtriangledown }} \Upsilon _K \widehat=\widehat{{ \bigtriangledown }}\Upsilon _K$

• 2. $\widehat{{ \bigtriangledown }} (\Upsilon _K \widehat{{ \cup }} \Upsilon _K)\widehat{{ \subseteq }} \widehat{{ \bigtriangledown }} \Upsilon _K \widehat{{ \cup }} \widehat{{ \bigtriangledown }}\Upsilon _K$

• 3. $\widehat{{ \bigtriangledown }} (\Upsilon _K \widehat{{ \cap }} \Upsilon _K)\widehat{{ \subseteq }} \widehat{{ \bigtriangledown }} \Upsilon _K \widehat{{ \cap }} \widehat{{ \bigtriangledown }}\Upsilon _K$

• 4. $\widehat{{ \bigtriangledown }} (\Upsilon _K \widehat{{ + }} \Upsilon _K)=\widehat{{ \bigtriangledown }} \Upsilon _K \widehat{{ + }} \widehat{{ \bigtriangledown }}\Upsilon _K$

Proof

The proof can be easily obtained from Definition 3.16, Definition 3.20 and Definition 3.40. $\square$

Theorem 3.43

Let P be the power set of all ivn-soft sets defined in the universe U. Then $(P, \widehat{{ \cap }}, \widehat{{ \cup }})$ is a distributive lattice.

Proof

The proof can be easily obtained by showing properties; idempotency, commutativity, associativity and distributivity. $\square$

4 ivn-soft set based decision making

In this section, we present an adjustable approach to ivn-soft set based decision making problems by extending the approach to interval-valued intuitionistic fuzzy soft set based decision making [40]. Some of it is quoted from [18, 26, 35, 40].

Definition 4.1

Let $\Upsilon _K \in IVNS(U)$. Then a relation form of $\Upsilon _K$ is defined by

$$R_{{\Upsilon _{K} }} = \{ (r_{{\Upsilon _{K} }} (x,u)/(x,u)):r_{{\Upsilon _{K} }} (x,u) \in IVN(U),x \in E,u \in U\}$$

where $r_{\Upsilon _K}:E\times U \rightarrow IVN(U)\,\,and\,\,r_{\Upsilon _K}(x,u)=\upsilon _{K(x)}(u)$ for all $x\in E$ and $u\in U$.

That is, $r_{\Upsilon _K}(x,u)=\upsilon _{K(x)}(u)$ is characterized by truth-membership function $T_K$, a indeterminacy-membership function $I_K$ and a falsity-membership function $F_K$. For each point $x\in E$ and $u\in U$; $T_K$, $I_K$ and $F_K \subseteq [0,1]$.

Example 4.2

Consider the above Example 3.7, then, $r_{\Upsilon _K}(x,u)=\upsilon _{K(x)}(u)$ can be given as follows

• $\upsilon _{K(x_1)}(u_1)= \langle [0.6,0.8], [0.8,0.9], [0.1,0.5]\rangle$,

• $\upsilon _{K(x_1)}(u_2)= \langle [0.5,0.8], [0.2,0.9], [0.1,0.7] \rangle$,

• $\upsilon _{K(x_2)}(u_1)= \langle [0.1,0.4], [0.5,0.8], [0.3,0.7] \rangle$,

• $\upsilon _{K(x_2)}(u_1)= \langle [0.1,0.9], [0.6,0.9], [0.2,0.3] \rangle$,

• $\upsilon _{K(x_3)}(u_1)= \langle [0.2,0.9], [0.1,0.5], [0.7,0.8] \rangle$,

• $\upsilon _{K(x_3)}(u_2)= \langle [0.4,0.9], [0.1,0.6], [0.5,0.7] \rangle$,

• $\upsilon _{K(x_4)}(u_1)= \langle [0.6,0.9], [0.6,0.9], [0.6,0.9] \rangle$,

• $\upsilon _{K(x_4)}(u_2)= \langle [0.5,0.9], [0.6,0.8], [0.1,0.8] \rangle$,

• $\upsilon _{K(x_5)}(u_1)= \langle [0.0,0.9], [1.0,1.0], [1.0,1.0]\rangle$,

• $\upsilon _{K(x_5)}(u_2)= \langle [0.0,0.9], [0.8,1.0], [0.2,0.5] \rangle$.

Zhang et al.[40] introduced level-soft set and different thresholds on different parameters in interval-valued intuitionistic fuzzy soft sets. Taking inspiration these definitions we give level-soft set and different thresholds on different parameters in ivn-soft sets.

Definition 4.3

Let $\Upsilon _K \in IVNS(U)$. For $\alpha , \beta , \gamma \subseteq [0,1]$, the $(\alpha , \beta , \gamma )$-level soft set of $\Upsilon _K$ is a crisp soft set, denoted $(\Upsilon _{K};{\langle\alpha , \beta , \gamma \rangle })$, defined by

$$(\Upsilon _{K};{\langle\alpha , \beta , \gamma \rangle })= \{(x_i,\{u_{ij}:u_{ij}\in U, \mu (u_{ij})=1\}) : x_i\in E\}$$

where,

$$\mu (u_{{ij}} ) = \left\{ {\begin{array}{*{20}l} {1,} \hfill & {(\alpha ,\beta ,\gamma )\hat{ \le }\upsilon _{{K(x_{i} )}} (u_{j} )} \hfill \\ {0,} \hfill & {{\text{others}}} \hfill \\ \end{array} } \right.$$

for all $u_j\in U$.

Obviously, the definition is an extension of level soft sets of interval-valued intuitionistic fuzzy soft sets [40].

Remark 4.4

In Definition 4.3, $\alpha =(\alpha _1,\alpha _2) \subseteq [0,1]$ can be viewed as a given least threshold on degrees of truth-membership, $\beta =( \beta _1, \beta _2) \subseteq [0,1]$ can be viewed as a given greatest threshold on degrees of indeterminacy-membership and $\gamma =( \gamma _1, \gamma _2) \subseteq [0,1]$ can be viewed as a given greatest threshold on degrees of falsity-membership. If $(\alpha , \beta , \gamma ) \widehat{{ \le }}\upsilon _{K(x_i)}(u)$, it shows that the degree of the truth-membership of u with respect to the parameter $x_i$ is not less than $\alpha$, the degree of the indeterminacy-membership of u with respect to the parameter $x_i$ is not more than $\gamma$ and the degree of the falsity-membership of u with respect to the parameter $x_i$ is not more than $\beta$. In practical applications of inv-soft sets, the thresholds $\alpha$, $\beta$, $\gamma$ are pre-established by decision makers and reflect decision makers’ requirements on “truth-membership levels”, “indeterminacy-membership levels” and “falsity-membership levels”, respectively.

Example 4.5

Consider the above Example 3.7.

Clearly the ( [0.3, 0.4], [0.3, 0.5], [0.1, 0.2])-level soft set of $\Upsilon _K$ as follows

$$(\Upsilon _{K};{\langle [0.3,0.4], [0.3,0.5], [0.1,0.2]\rangle })=\{(x_1,\{u_1\}),(x_4,\{u_1,u_2\})\}$$

Note 4.6

In some practical applications the thresholds $\alpha , \beta , \gamma$ decision makers need to impose different thresholds on different parameters. To cope with such problems, we replace a constant value the thresholds by a function as the thresholds on truth-membership values, indeterminacy-membership values and falsity-membership values, respectively.

Theorem 4.7

Let $\Upsilon _K,\Upsilon _L \in IVNS(U)$. Then,

• 1. $(\Upsilon _{K};\langle\alpha _1, \beta _1, \gamma _1\rangle)$ and $(\Upsilon _{K};\langle\alpha _2, \beta _2, \gamma _2\rangle)$ are $\langle\alpha _1, \beta _1, \gamma _1\rangle$-level soft set and $\langle\alpha _2, \beta _2, \gamma _2\rangle$-level soft set of $\Upsilon _K$, respectively. If $\langle\alpha _2, \beta _2, \gamma _2\rangle\widehat{{ \le }} \langle\alpha _1, \beta _1, \gamma _1\rangle$, then we have $(\Upsilon _{K};\langle\alpha _1, \beta _1, \gamma _1\rangle)\tilde{\subseteq } (\Upsilon _{K};\langle\alpha _2, \beta _2, \gamma _2\rangle)$.

• 2. $(\Upsilon _{K};\langle\alpha , \beta , \gamma \rangle)$ and $(\Upsilon _{L};\langle\alpha , \beta , \gamma \rangle)$ are $\langle\alpha , \beta , \gamma$-level soft set $\Upsilon _K$ and $\Upsilon _L$, respectively. If $\Upsilon _K\widehat{{ \subseteq }} \Upsilon _L$, then we have $(\Upsilon _{K};\langle\alpha , \beta , \gamma \rangle)\tilde{\subseteq } (\Upsilon _{L};\langle\alpha , \beta , \gamma \rangle)$.

Proof

The proof of the theorems are obvious. $\square$

Definition 4.8

Let $\Upsilon _K \in IVNS(U)$. Let an interval-valued neutrosophic set $\langle\alpha , \beta , \gamma \rangle_{{\Upsilon _K}}:E \rightarrow IVN(U)$ in U which is called a threshold interval-valued neutrosophic set. The level soft set of $\Upsilon _K$ with respect to $\langle\alpha , \beta , \gamma \rangle_{{\Upsilon _K}}$ is a crisp soft set, denoted by $(\Upsilon _{K};\langle\alpha , \beta , \gamma \rangle_{{\Upsilon _K}})$, defined by;

$$(\Upsilon _{K};{\langle\alpha , \beta , \gamma \rangle }_{{\Upsilon _K}})= \{(x_i,\{u_{ij}:u_{ij}\in U, \mu (u_{ij})=1\}): x_i\in E\}$$

where,

$$\mu (u_{{ij}} ) = \left\{ {\begin{array}{*{20}l} {1,} \hfill & {\langle \alpha ,\beta ,\gamma \rangle _{{\Upsilon _{K} }} (x_{i} )\widehat{{ \le }}\upsilon _{{K(x_{i} )}} (u_{j} )} \hfill \\ {0,} \hfill & {others} \hfill \\ \end{array} } \right.$$

for all $u_j\in U$.

Obviously, the definition is an extension of level soft sets of interval-valued intuitionistic fuzzy soft sets [40].

Remark 4.9

In Definition 4.8, $\alpha =(\alpha _1,\alpha _2) \subseteq [0,1]$ can be viewed as a given least threshold on degrees of truth-membership, $\beta =( \beta _1, \beta _2) \subseteq [0,1]$ can be viewed as a given greatest threshold on degrees of indeterminacy-membership and $\gamma =( \gamma _1, \gamma _2) \subseteq [0,1]$ can be viewed as a given greatest threshold on degrees of falsity-membership of u with respect to the parameter x.

If ${\langle\alpha , \beta , \gamma \rangle }_{{\Upsilon _K}}(x_i) \widehat{{ \le }}\upsilon _{K(x_i)}(u)$ it shows that the degree of the truth-membership of u with respect to the parameter $x_i$ is not less than $\alpha$, the degree of the indeterminacy-membership of u with respect to the parameter $x_i$ is not more than $\gamma$ and the degree of the falsity-membership of u with respect to the parameter $x_i$ is not more than $\beta$.

Definition 4.10

Let $\Upsilon _K \in IVNS(U)$. Based on $\Upsilon _K$, we can define an interval-valued neutrosophic set $\langle\alpha , \beta , \gamma \rangle^{avg}_{{\Upsilon _K}}:E \rightarrow IVN(U)$ by

$$\langle\alpha , \beta , \gamma \rangle^{avg}_{{\Upsilon _K}}(x_i)=\sum _{u\in U}\upsilon _{K(x_i)}(u)/{|U|}$$

for all $x\in E$.

The interval-valued neutrosophic set $\langle\alpha , \beta , \gamma \rangle^{avg}_{{\Upsilon _K}}$ is called the avg-threshold of the ivn-soft set ${\Upsilon _K}$. In the following discussions, the avg-level decision rule will mean using the avg-threshold and considering the avg-level soft set in ivn-soft sets based decision making.

Let us reconsider the ivn-soft set ${\Upsilon _K}$ in Example 3.7. The avg-threshold $\langle\alpha , \beta , \gamma \rangle^{avg}_{{\Upsilon _K}}$ of ${\Upsilon _K}$ is an interval-valued neutrosophic set and can be calculated as follows:

$$\langle\alpha , \beta , \gamma \rangle^{avg}_{{\Upsilon _K}}(x_1)=\sum _{i=1}^2\upsilon _{K(x_1)}(u_i)/{|U|}=\langle [0.55,0.8], [0.5,0.9], [0.1,0.6] \rangle$$

$$\langle\alpha , \beta , \gamma \rangle^{avg}_{{\Upsilon _K}}(x_2)=\sum _{i=1}^2\upsilon _{K(x_2)}(u_i)/{|U|}=\langle [0.1,0.65], [0.55,0.85], [0.25,0.5] \rangle$$

$$\langle\alpha , \beta , \gamma \rangle^{avg}_{{\Upsilon _K}}(x_3)=\sum _{i=1}^2\upsilon _{K(x_3)}(u_i)/{|U|}=\langle [0.15,0.9], [0.1,0.55], [0.6,0.75] \rangle$$

$$\langle\alpha , \beta , \gamma \rangle^{avg}_{{\Upsilon _K}}(x_4)=\sum _{i=1}^2\upsilon _{K(x_4)}(u_i)/{|U|}=\langle [0.55,0.9], [0.6,0.85], [0.35,0.85] \rangle$$

$$\langle\alpha , \beta , \gamma \rangle^{avg}_{{\Upsilon _K}}(x_5)=\sum _{i=1}^2\upsilon _{K(x_5)}(u_i)/{|U|}=\langle [0.0,0.9], [0.9,1.0], [0.6,0.75] \rangle$$

Therefore, we have

$$\begin{aligned} \langle\alpha , \beta , \gamma \rangle^{avg}_{{\Upsilon _K}}&= \{\langle [0.55,0.8], [0.5,0.9], [0.1,0.6] \rangle /x_1,\langle [0.1,0.65],[0.55,0.85],\\ & \quad [0.25,0.5] \rangle /x_2,\langle [0.15,0.9], [0.1,0.55], [0.6,0.75] \rangle /x_3,\langle [0.55,0.9],\\ & \quad [0.6,0.85], [0.35,0.85] \rangle /x_4, \langle [0.0,0.9], [0.9,1.0], [0.6,0.75] \rangle /x_5 \} \end{aligned}$$

Example 4.11

Consider the above Example 3.7. Clearly;

$$(\Upsilon _{K};\langle\alpha , \beta , \gamma \rangle^{avg}_{{\Upsilon _K}})=\{(x_5,\{u_2\})\}$$

Definition 4.12

Let $\Upsilon _K \in IVNS(U)$. Based on $\Upsilon _K$, we can define an interval-valued neutrosophic set $\langle\alpha , \beta , \gamma \rangle^{Mmm}_{\Upsilon _K}:A \rightarrow IVN(U)$ by

$$\begin{aligned} \langle \alpha ,\beta ,\gamma \rangle _{{\Upsilon _{K} }}^{{Mmm}} & = \left\{ {\langle [max_{{u \in U}} \{ infT_{{\upsilon _{{K(x_{i} )}} (u)}} \} ,max_{{u \in U}} \{ supT_{{\upsilon _{{K(x_{i} )}} (u)}} \} ],} \right. \\ & \quad [min_{{u \in U}} \{ infI_{{\upsilon _{{K(x_{i} )}} (u)}} \} ,min_{{u \in U}} \{ supI_{{\upsilon _{{K(x_{i} )}} (u)}} \} ], \\ & \quad \left. {[min_{{u \in U}} \{ infF_{{\upsilon _{{K(x_{i} )}} (u)}} \} ,min_{{u \in U}} \{ supF_{{\upsilon _{{K(x_{i} )}} (u)}} \} ]\rangle /x_{i} :x_{i} \in E} \right\} \\ \end{aligned}$$

The interval-valued neutrosophic set $\langle\alpha , \beta , \gamma \rangle^{Mmm}_{\Upsilon _K}$ is called the max-min-min-threshold of the ivn-soft set $\Upsilon _K$. In what follows the Mmm-level decision rule will mean using the max-min-min-threshold and considering the Mmm-level soft set in ivn-soft sets based decision making.

Definition 4.13

Let $\Upsilon _K \in IVNS(U)$. Based on $\Upsilon _K$, we can define an interval-valued neutrosophic set $\langle\alpha , \beta , \gamma \rangle^{mmm}_{\Upsilon _K}:E \rightarrow IVN(U)$ by

$$\begin{aligned} \langle \alpha ,\beta ,\gamma \rangle _{{\Upsilon _{K} }}^{{mmm}} & = \left\{ {\langle [min_{{u \in U}} \{ infT_{{\upsilon _{{K(x_{i} )}} (u)}} \} ,min_{{u \in U}} \{ supT_{{\upsilon _{{K(x_{i} )}} (u)}} \} ],} \right. \\ & \quad [min_{{u \in U}} \{ infI_{{\upsilon _{{K(x_{i} )}} (u)}} \} ,min_{{u \in U}} \{ supI_{{\upsilon _{{K(x_{i} )}} (u)}} \} ], \\ & \quad \left. {[min_{{u \in U}} \{ infF_{{\upsilon _{{K(x_{i} )}} (u)}} \} ,min_{{u \in U}} \{ supF_{{\upsilon _{{K(x_{i} )}} (u)}} \} ]\rangle /x_{i} :x_{i} \in E} \right\} \\ \end{aligned}$$

The interval-valued neutrosophic set $\langle\alpha , \beta , \gamma \rangle^{mmm}_{\Upsilon _K}$ is called the min-min-min-threshold of the ivn-soft set $\Upsilon _K$. In what follows the mmm-level decision rule will mean using the min-min-min-threshold and considering the mmm-level soft set in ivn-soft sets based decision making.

Theorem 4.14

Let $\Upsilon _K \in IVNS(U)$. Then, $(\Upsilon _{K};\langle\alpha , \beta , \gamma \rangle^{avg}_{{\Upsilon _K}})$, $(\Upsilon _{K};\langle\alpha , \beta , \gamma \rangle^{Mmm}_{{\Upsilon _K}}), (\Upsilon _{K};\langle\alpha , \beta , \gamma \rangle^{mmm}_{{\Upsilon _K}}))$ are the avg-level soft set, Mmm-level soft set, mmm-level soft set of $\Upsilon _K \in IVNS(U)$, respectively. Then,

• 1. $(\Upsilon _{K};\langle\alpha , \beta , \gamma \rangle^{Mmm}_{{\Upsilon _K}})\tilde{\subseteq } (\Upsilon _{K};\langle\alpha , \beta , \gamma \rangle^{avg}_{{\Upsilon _K}})$

• 2. $(\Upsilon _{K};\langle\alpha , \beta , \gamma \rangle^{Mmm}_{{\Upsilon _K}})\tilde{\subseteq } (\Upsilon _{K};\langle\alpha , \beta , \gamma \rangle^{mmm}_{{\Upsilon _K}})$

Proof

The proof of the theorems are obvious. $\square$

Theorem 4.15

Let $\Upsilon _K,\Upsilon _L \in IVNS(U)$. Then,

• 1. Let $\langle\alpha _1, \beta _1, \gamma _1\rangle^i_{\Upsilon _K}$ and $\langle\alpha _2, \beta _2, \gamma _2\rangle^i_{\Upsilon _K}$ for $i\in \{{avg},{Mmm},{mmm}\}$ be two threshold interval-valued neutrosophic sets. Then, $(\Upsilon _{K};\langle\alpha _1, \beta _1, \gamma _1\rangle_{\Upsilon _K})$ and $(\Upsilon _{K};\langle\alpha _2, \beta _2, \gamma _2\rangle_{\Upsilon _K})$ are $\langle\alpha _1, \beta _1, \gamma _1\rangle_{\Upsilon _K}$-level soft set and $\langle\alpha _2, \beta _2, \gamma _2\rangle_{\Upsilon _K}$-level soft set of $\Upsilon _K$, respectively. If $\langle\alpha _2, \beta _2, \gamma _2\rangle_{\Upsilon _K} \widehat{{ \le }} \langle\alpha _1, \beta _1, \gamma _1\rangle_{\Upsilon _K}$, then we have $(\Upsilon _{K};\langle\alpha _1, \beta _1, \gamma _1\rangle_{\Upsilon _K})\tilde{\subseteq } (\Upsilon _{K};\langle\alpha _2, \beta _2, \gamma _2\rangle_{\Upsilon _K})$.

• 2. Let $\langle\alpha , \beta , \gamma \rangle_{\Upsilon _K}$ be a threshold interval-valued neutrosophic sets. Then, $(\Upsilon _{K};\langle\alpha , \beta , \gamma \rangle_{\Upsilon _K})$ and $(\Upsilon _{L};\langle\alpha , \beta , \gamma \rangle_{\Upsilon _K})$ are $\langle\alpha , \beta , \gamma$-level soft set $\Upsilon _K$ and $\Upsilon _L$, respectively. If $\Upsilon _K\widehat{{ \subseteq }} \Upsilon _L$, then we have $(\Upsilon _{K};\langle\alpha , \beta , \gamma \rangle_{\Upsilon _K})\tilde{\subseteq } (\Upsilon _{L};\langle\alpha , \beta , \gamma \rangle_{\Upsilon _K})$.

Proof

The proof of the theorems are obvious. $\square$

Now, we construct an ivn-soft set decision making method by the following algorithm;

Algorithm:

• 1. Input the ivn-soft set $\Upsilon _K$,

• 2. Input a threshold interval-valued neutrosophic set $\langle\alpha , \beta , \gamma \rangle^{avg}_{\Upsilon _K}$ (or $\langle\alpha , \beta , \gamma \rangle^{Mmm}_{\Upsilon _K}, \langle\alpha , \beta , \gamma \rangle^{mmm}_{\Upsilon _K}$) by using avg-level decision rule (or Mmm-level decision rule, mmm-level decision rule) for decision making.

• 3. Compute avg-level soft set $(\Upsilon _{K};\langle\alpha , \beta , \gamma \rangle^{avg}_{{\Upsilon _K}})$ (or Mmm-level soft set ($(\Upsilon _{K};\langle\alpha , \beta , \gamma \rangle^{Mmm}_{{\Upsilon _K}})$, mmm-level soft set $(\Upsilon _{K};\langle\alpha , \beta , \gamma \rangle^{mmm}_{{\Upsilon _K}})))$

• 4. Present the level soft set $(\Upsilon _{K};\langle\alpha , \beta , \gamma \rangle^{avg}_{{\Upsilon _K}})$ (or the level soft set($(\Upsilon _{K};\langle\alpha , \beta , \gamma \rangle^{Mmm}_{{\Upsilon _K}}$, the level soft set $(\Upsilon _{K};\langle\alpha , \beta , \gamma \rangle^{mmm}_{{\Upsilon _K}})))$ in tabular form.

• 5. Compute the choice value $c_i$ of $u_i$ for any $u_i\in U$,

• 6. The optimal decision is to select $u_k$ if $c_k=max_{u_i \in U}c_i.$

Remark 4.16

If k has more than one value then any one of $u_k$ may be chosen.

If there are too many optimal choices in Step 6, we may go back to the second step and change the threshold (or decision rule) such that only one optimal choice remains in the end.

Remark 4.17

The aim of designing the Algorithm is to solve ivn-soft sets based decision making problem by using level soft sets. Level soft sets construct bridges between ivn-soft sets and crisp soft sets. By using level soft sets, we need not treat ivn-soft sets directly but only cope with crisp soft sets derived from them after choosing certain thresholds or decision strategies such as the mid-level or the top?bottom-level decision rules. By the Algorithm, the choice value of an object in a level soft set is in fact the number of fair attributes which belong to that object on the premise that the degree of the truth-membership of u with respect to the parameter x is not less than “truth-membership levels”, the degree of the indeterminacy-membership of u with respect to the parameter x is not more than “indeterminacy-membership levels” and the degree of the falsity-membership of u with respect to the parameter x is not more than “falsity-membership levels”.

Example 4.18

Suppose that a customer to select a house from the real agent. He can construct a ivn-soft set $\Upsilon _K$ that describes the characteristic of houses according to own requests. Assume that $U=\{u_1,u_2,u_3,u_4,u_5,u_6\}$ is the universe contains six house under consideration in an real agent and $E=\{x_1=cheap, x_2=beatiful, x_3=green surroundings, x_4=costly, x_5= large\}$.

Now, we can apply the method as follows:

• 1. Input the ivn-soft set $\Upsilon _K$ as,

  Table 13 The tabular representation of the ivn-soft set $\widehat{{ \bigtriangledown }} \Upsilon _K$

• 2. Input a threshold interval-valued neutrosophic set $\langle\alpha , \beta , \gamma \rangle^{avg}_{\Upsilon _K}$ by using avg-level decision rule for decision making as;

  $$\begin{aligned} \langle \alpha ,\beta ,\gamma \rangle _{{\Upsilon _{K} }}^{{avg}} & = \left\{ {\langle [0.41,0.76],[0.56,0.9],[0.18,0.63]\rangle /x_{1} ,\langle [0.31,0.7],[0.46,0.66],} \right. \\ & \quad [0.31,0.58]\rangle /x_{2} ,\langle [0.41,0.8],[0.21,0.53],[0.61,0.76]\rangle /x_{3} ,\langle [0.45,0.81], \\ & \quad \left. {[0.61,0.86],[0.45,0.86]\rangle /x_{4} ,\langle [0.25,0.65],[0.7,0.9],[0.61,0.76]\rangle /x_{5} {\text{ }}} \right\} \\ \end{aligned}$$

• 3. Compute avg-level soft set $(\Upsilon _{K};\langle\alpha , \beta , \gamma \rangle^{avg}_{{\Upsilon _K}})$ as;

  $$(\Upsilon _{K};\langle\alpha , \beta , \gamma \rangle^{avg}_{{\Upsilon _K}})=\{(x_2, \{u_3\}), (x_3,\{u_4\}), (x_4,\{u_6\}, (x_5,\{u_3\})\}$$

• 4. Present the level soft set $(\Upsilon _{K};\langle\alpha , \beta , \gamma \rangle^{avg}_{{\Upsilon _K}})$ in tabular form as (Table 14);

  Table 14 The tabular representation of the soft set $F_X$

• 5. Compute the choice value $c_i$ of $u_i$ for any $u_i\in U$ as;

  $$c_1=c_2=c_5= \sum _{j=1}^5 u_{1j}= \sum _{j=1}^6 u_{2j}= \sum _{j=1}^5 h_{5j}=0,$$
  $$c_4= c_6= \sum _{j=1}^5 u_{4j}= \sum _{j=1}^6 h_{6j}=1$$
  $$c_3= \sum _{j=1}^5 u_{3j}=2$$

• 6. The optimal decision is to select $u_3$ since $c_3=max_{u_i \in U}c_i.$

Note that this decision making method can be applied for group decision making easily with help of the Definition 3.27 and Definition 3.28.

5 Conclusion

In this paper, the notion of the interval valued neutrosophic soft sets (ivn-soft sets) is defined which is a combination of an interval valued neutrosophic sets[35] and a soft sets[29]. Then, we introduce some definitions and operations of ivn-soft sets sets. Some properties of ivn-soft sets which are connected to operations have been established. Finally, we propose an adjustable approach by using level soft sets and illustrate this method with some concrete examples. This novel proposal proves to be feasible for some decision making problems involving ivn-soft sets. It can be applied to problems of many fields that contain uncertainty such as computer science, game theory, and so on.