Interval-Valued Neutrosophic Soft Rough Sets
We first defined interval-valued neutrosophic soft rough sets (IVN-soft rough sets for short) which combine interval-valued neutrosophic soft set and rough sets and studied some of its basic properties. This concept is an extension of interval-valued intuitionistic fuzzy soft rough sets (IVIF-soft rough sets).
In 1999, Smarandache introduced the theory of neutrosophic set (NS) , which is the generalization of the classical sets, conventional fuzzy set , intuitionistic fuzzy set , interval-valued fuzzy set , and so on. The concept of neutrosophic set handles indeterminate data whereas fuzzy set theory and intuitionistic fuzzy set theory failed when the relation is indeterminate.
Recently, works on the neutrosophic set theory is progressing rapidly. Bhowmik and Pal [5, 6] defined “intuitionistic neutrosophic set.” Later on Salama and Alblowi  introduced another concept called “generalized neutrosophic set.” Wang et al.  proposed another extension of neutrosophic set which is “single valued neutrosophic sets.” Also, Wang et al.  introduced the notion of interval-valued neutrosophic sets (IVNSs) which is an instance of neutrosophic set. The IVNSs are characterized by an interval membership degree, interval indeterminacy degree, and interval nonmembership degree. Georgiev  explored some properties of the neutrosophic logic and proposed a general simplification of the neutrosophic sets into a subclass of theirs, comprised of elements of . Ye [11, 12] defined similarity measures between interval neutrosophic sets and their multicriteria decision making method. Majumdar and Samant  proposed some types of similarity and entropy of neutrosophic sets. Broumi and Smarandache [14–16] proposed several similarity measures of neutrosophic sets. Chi and Peide  extended TOPSIS to interval neutrosophic sets and so on.
In 1999, a Russian researcher, Molodotsov, proposed a new mathematical tool called “soft set theory” , for dealing with uncertainty and how soft set theory is free from the parameterization inadequacy syndrome of fuzzy set theory, rough set theory, and probability theory. Recently, Deli  introduced the concept of interval-valued neutrosophic soft set as a combination of interval neutrosophic set and soft set. This concept generalizes the concept of the soft set, fuzzy soft set , intuitionistic fuzzy soft set , interval-valued intuitionistic fuzzy soft set , neutrosophic soft set, and intuitionistic neutrosophic soft set .
The rough set theory that was introduced by Pawlak  in 1982, which is a technique for managing the uncertainty and imperfection, can analyze incomplete information effectively. Therefore, many models have been built upon different aspect, that is, universe, relations, object, and operators by many scholars [25–30] such as rough fuzzy sets, fuzzy rough sets, generalized fuzzy rough set, rough intuitionistic fuzzy set, and intuitionistic fuzzy rough sets . It has been successfully applied in many fields such as attribute reduction [32–35], feature selection [36–38], and rule extraction [39–42]. The rough sets theory approximates any subset of objects of the universe by two sets, called the lower and upper approximations. It focuses on the ambiguity caused by the limited discernibility of objects in the universe of discourse.
Moreover, many new rough set models have also been established by combining the Pawlak rough set with other uncertainty theories such as soft set theory. Feng  provided a framework to combine fuzzy sets, rough sets, and soft sets all together, which gives rise to several interesting new concepts such as rough soft sets, soft rough sets, and soft rough fuzzy sets. The combination of hybrid structures of soft sets and rough sets models was also discussed by some researchers [44–46]. Later on, Zhang et al.  proposed the notions of soft rough intuitionistic fuzzy sets and intuitionistic fuzzy soft rough sets, which can be seen as two new generalized soft rough set models, and investigated some properties of soft rough intuitionistic fuzzy sets and intuitionistic fuzzy soft rough sets in detail. Also Saha and Mukherjee  proposed the concept of the notion of soft interval-valued intuitionistic fuzzy rough sets.
More recently, Broumi et al.  combined neutrosophic sets with rough sets in a new hybrid mathematical structure called “rough neutrosophic sets” handling incomplete and indeterminate information. The concept of rough neutrosophic sets generalizes fuzzy rough sets and intuitionistic fuzzy rough sets. Also Salama and Broumi  studied roughness of neutrosophic sets. Based on the equivalence relation on the universe of discourse, Mukherjee et al.  introduced soft lower and upper approximation of interval-valued intuitionistic fuzzy set in Pawlak’s approximation space. Motivated by the idea of interval-valued intuitionistic fuzzy soft rough sets introduced in , we extend the interval intuitionistic fuzzy lower and upper approximations to the case of an interval-valued neutrosophic set. The concept of an interval-valued neutrosophic soft rough set is introduced by coupling both the interval-valued neutrosophic soft sets and rough sets.
The paper is structured as follows. In Section 2, we first recall the necessary background on rough sets and interval neutrosophic soft set. Section 3 reviews various proposals for the definition of interval neutrosophic soft rough sets and examines their respective properties. Section 4 presents a multiciteria group decision making scheme under interval-valued neutrosophic soft rough set. Section 5 presents an application of multiciteria group decision making scheme regarding the candidate selection problem. Finally we conclude the paper.
Throughout this paper, let be a universal set and let be the set of all possible parameters under consideration with respect to ; usually parameters are attributes, characteristics, or properties of objects in . We now recall some basic notions of soft sets, interval neutrosophic sets, neutrosophic soft set, interval neutrosophic soft set, rough set, and rough neutrosophic sets. For more details the reader may refer to [1, 9, 18, 19, 24, 49].
Definition 1. Let be a universe of discourse; then the neutrosophic set is an object having the form , where the functions , , and : define, respectively, the degree of membership, the degree of indeterminacy, and the degree of nonmembership of the element to the set with the condition From philosophical point of view, the neutrosophic set takes the value from real standard or nonstandard subsets of . So instead of we need to take the interval for technical applications because will be difficult to apply in the real applications such as in scientific and engineering problems.
Definition 2 (see ). Let be a space of points (objects) with generic elements in denoted by . An interval-valued neutrosophic set (for short IVNS) in is characterized by truth-membership function , indeterminacy-membership function , and falsity-membership function . For each point in , one has that , , and .
For two IVNSs, And , , ; the two relations are defined as follows:(1) if and only if , , , , and ;(2) if and only if , , and for any .The complement of is denoted by and is defined by
As an illustration, let us consider the following example.
Example 3. Assume that the universe of discourse , where characterizes the capability, characterizes the trustworthiness, and indicates the prices of the objects. It may be further assumed that the values of , , and are in and they are obtained from some questionnaires of some experts. The experts may impose their opinion in three components, namely, the degree of goodness, the degree of indeterminacy, and the degree of poorness to explain the characteristics of the objects. Suppose is an interval-valued neutrosophic set (IVNS) of , such that , , , where the degree of goodness of capability is , degree of indeterminacy of capability is , degree of falsity of capability is , and so forth.
Definition 4 (see ). Let be an initial universe set and let be a set of parameters. Let denote the power set of . Consider a nonempty set , . A pair is called a soft set over , where is a mapping given by .
As an illustration, let us consider the following example.
Example 5. Suppose that is the set of houses under consideration; say . Let be the set of some attributes of such houses; say , where stand for the attributes “beautiful,” “costly,” “green surroundings,” and “moderate,” respectively.
In this case, to define a soft set means to point out expensive houses, beautiful houses, and so on. For example, the soft set that describes the “attractiveness of the houses” in the opinion of a buyer, say Thomas, may be defined as follows:
Definition 6 (see ). Let be an initial universe set and let be a set of parameters. Let IVNS denote the set of all interval neutrosophic subsets of . The collection is termed to be the soft interval neutrosophic set over , where is a mapping given by .
The interval neutrosophic soft set defined over a universe is denoted by INSS.
Here,(1) is an ivn-soft subset of , denoted by , if for all ;(2) is an ivn-soft equal to , denoted by , if for all ;(3)the complement of is denoted by and is defined by ;(4)the union of and is denoted by , if for all ;(5)the intersection of and is denoted by , if for all .
Example 7. Let be the set of houses under consideration and is the set of parameters (or qualities). Each parameter is an interval neutrosophic word or sentence involving interval neutrosophic words. Consider . In this case, to define an interval neutrosophic soft set means to point out beautiful houses, costly houses, and so on. Suppose that there are four houses in the universe given by and the set of parameters , where each is a specific criterion for houses: stands for “beautiful,” stands for “costly,” stands for “moderate.”
Definition 8 (see ). Let be an equivalence relation on the universal set . Then the pair is called a Pawlak approximation space. An equivalence class of containing will be denoted by . Now for , the lower and upper approximation of with respect to are denoted by, respectively, and and are defined by Now if , then is called definable; otherwise is called a rough set.
Definition 9 (see ). Let be a nonnull set and let be an equivalence relation on . Let be neutrosophic set in with the membership function , indeterminacy function , and nonmembership function . Then the lower and upper rough approximations of in are denoted by and and, respectively, defined as follows: where It is easy to observe that and are two neutrosophic sets in ; thus NS mapping are, respectively, referred to as the upper and lower rough NS approximation operators, and the pair is called the rough neutrosophic set.
3. Interval Neutrosophic Soft Rough Set
This section is an attempt to extend the concept of an interval-valued intuitionistic fuzzy soft rough set  to the case of an interval-valued neutrosophic rough set.
Definition 10. Let be full soft set over and let the pair be the soft approximation space. Then for an interval-valued neutrosophic set , the lower and upper soft rough approximations of with respect to are denoted by and , respectively, which are interval-valued neutrosophic sets in given by The operators and are called the lower and upper soft rough approximation operators on interval-valued neutrosophic sets. If , then is said to be soft interval-valued neutrosophic definable; otherwise it is called an interval-valued neutrosophic soft rough set.
Theorem 11. Let be a full soft set over and let be the soft approximation space. Then for one has(i), , , , , ;(ii), , , , , .
Proof. (i) Let and . Then for , we have and hence . Consequently, And so Similarly, it can be shown that Thus, we get In a similar manner it can be shown that In a similar manner it can be shown that From (14), (15), and (16) we observe that Now we prove that Let us suppose that such that . Then , , and hence Consequently, Similarly, it can be shown that Thus we get In a similar manner it can be shown that In a similar manner it can be shown that From (22), (23), and (24) we observe that From (17) and (25), we have (ii) Proof is similar to that in (i).
Theorem 12. Let be a full soft set over and let be the soft approximation space. Then for , one has(1);(2);(3);(4);(5);(6);(7);(8).
Proof. (1)–(4) are straightforward.
(5) We have Now Since we have Consequently, Similarly we can get Again since we have Consequently, Similarly we can get Again since we have Consequently, Similarly we can get Using, (31)–(40) we get from (28)
(6) Proof is similar to (5).
(7) We have Now Since we have Consequently, Similarly we can get Again since we have Consequently, Similarly we can get Again since we have Consequently, Similarly we can get Using (46)–(55), we get from (43) (8) Proof is similar to (7).
4. A Multicriteria Group Decision Making Problem
Soft sets, fuzzy soft sets, intuitionistic fuzzy soft sets, and neutrosophic soft sets have been applied by many authors in solving decision making problems. In this section, we illustrate the use of soft sets and, neutrosophic soft sets, interval-valued neutrosophic soft sets, rough sets, interval-valued neutrosophic soft rough sets, and related notions in object evaluation and group decision making.
Let be a set of objects and let be a set of parameters and and let be a full soft set over . Let us assume that we have an expert group consisting of n specialists to evaluate the objects in . Each specialist will examine all the objects in and will point out his/her evaluation result. Let denote the primary evaluation result of the specialist . It is easy to see that the primary evaluation result of the whole expert group can be represented as an interval-valued neutrosophic evaluation soft set over , where is given by , for . Now we consider the soft rough interval-valued neutrosophic approximations of the specialist ’s primary evaluation result with respect to the soft approximation space . Then we obtain two other interval-valued neutrosophic soft sets and over , where is given by and is given by , for . Here can be considered as the evaluation result for the whole expert group with “low confidence,” can be considered as the evaluation result for the whole expert group with “high confidence,” and can be considered as the evaluation result for the whole expert group with “middle confidence.” Let us define two interval-valued neutrosophic sets and by