Abstract

As a generalization of fuzzy sets and intuitionistic fuzzy sets, neutrosophic sets have been developed to represent uncertain, imprecise, incomplete, and inconsistent information existing in the real world. And interval neutrosophic sets (INSs) have been proposed exactly to address issues with a set of numbers in the real unit interval, not just a specific number. However, there are fewer reliable operations for INSs, as well as the INS aggregation operators and decision making method. For this purpose, the operations for INSs are defined and a comparison approach is put forward based on the related research of interval valued intuitionistic fuzzy sets (IVIFSs) in this paper. On the basis of the operations and comparison approach, two interval neutrosophic number aggregation operators are developed. Then, a method for multicriteria decision making problems is explored applying the aggregation operators. In addition, an example is provided to illustrate the application of the proposed method.

1. Introduction

Zadeh proposed his remarkable theory of fuzzy sets (FSs in short) in 1965 [1] to encounter different types of uncertainties. Since then, it has been applied successfully in various fields [2]. As the traditional fuzzy set uses one single value to represent the grade of membership of the fuzzy set A defined on a universe, it cannot handle some cases where is hard to be defined by a specific value. So interval valued fuzzy sets (IVFSs) were introduced by Turksen [3]. And to cope with the lack of knowledge of nonmembership degrees, Atanassov introduced intuitionistic fuzzy sets (IFSs in short) [4–7], an extension of Zadeh’s FSs. In addition, Gau and Buehrer [8] defined vague sets. Later on, Bustince pointed out that vague sets and Atanassov’s IFSs are mathematically equivalent objects [9]. As for the present, IFSs have been widely applied in solving multicriteria decision making problems [10–14], neural networks [15, 16], medical diagnosis [17], color region extraction [18, 19], market prediction [20], and so forth.

IFSs took into account the membership degree, nonmembership degree, and degree of hesitation simultaneously. So IFSs are more flexible and practical in addressing the fuzziness and uncertainty than the traditional FSs. Moreover, in some actual cases, the membership degree, nonmembership degree, and hesitation degree of an element in the IFS may not be a specific number. Hence, it was extended to the interval valued intuitionistic fuzzy sets (IVIFSs in brief) [21]. To handle the situations where people are hesitant in expressing their preference over objects in a decision making process, hesitant fuzzy sets (HFSs) were introduced by Torra [22] and Torra and Narukawa [23].

Although the FSs theory has been developed and generalized, it can not deal with all sorts of uncertainties in different real physical problems. Some types of uncertainties such as the indeterminate information and inconsistent information can not be handled. For example [24], when we ask about the opinion of an expert about a certain statement, he or she may say that the possibility that the statement is true is 0.5, that the statement is false is 0.6, and the degree that he or she is not sure is 0.2. This issue is beyond the scope of FSs and IFSs. Therefore, some new theories are required.

Smarandache coined neutrosophic logic and neutrosophic sets (NSs) in 1995 [25, 26]. A NS is a set where each element of the universe has a degree of truth, indeterminacy, and falsity, respectively, and which lies in , the nonstandard unit interval [27]. Obviously, it is the extension to the standard interval in IFSs. And the uncertainty present here, that is, indeterminacy factor, is independent of truth and falsity values while the incorporated uncertainty is dependent on the degrees of belongingness and nonbelongingness in IFSs [28]. And for the aforementioned example, by means of NSs, it can be expressed as .

However, without being specified, it is difficult to apply in the real applications. Hence, the single valued neutrosophic set (SVNS) was proposed, which is an instance of NSs [24, 28]. Furthermore, the information energy of SVNSs, correlation and correlation coefficient of SVNSs, and a decision making method by the use of SVNSs were presented [29]. In addition, Ye also introduced the concept of simplified neutrosophic sets (SNSs), which can be described by three real numbers in the real unit interval , and proposed a multicriteria decision making method using aggregation operators for SNSs [30]. Majumdar and Samant introduced a measure of entropy of a SVNS [28].

In fact, sometimes the degree of truth, falsity, and indeterminacy of a certain statement can not be defined exactly in the real situations but denoted by several possible interval values. So the interval neutrosophic set (INS) was required, similar to IVIFS. Wang et al. proposed the concept of INS and gave the set-theoretic operators of INS [31]. The operations of INS were discussed in [32]; yet the comparison methods were not seen there. Furthermore, Ye defined the Hamming and Euclidean distances between INSs and proposed the similarity measures between INSs based on the relationship between similarity measures and distances [33]. However, in some cases, the INS operations in [31] might be irrational. For instance, the sum of any element and the maximum value should be equal to the maximum one, while it does not hold with the operations in [31]. In addition, to the best of our knowledge, the existing literatures do not put forward the INS aggregation operators and decision making method, which were vitally important for INSs to be utilized in the real situations in scientific and engineering applications. Therefore, the operations and comparison approach between interval neutrosophic numbers (INNs) and the aggregation operators for INSs are defined in this paper to be used. Thus, a multicriteria decision making method is established based on the proposed operators; an illustrative example is given to demonstrate the application of the proposed method.

The rest of the paper is organized as follows. Section 2 briefly introduces interval numbers, properties of t-norm and t-conorm, and concepts and operations of NSs, SNSs, and INSs. And the operations and comparison approach for INSs are defined on the basis of the IVIFS theory in Section 3. The INN aggregation operators are given and a decision making method is developed for INSs by means of the INN aggregation operators in Section 4. In Section 5, an illustrative example is presented to illustrate the proposed method and the comparison analysis and discussion are given. Finally, Section 6 concludes the paper.

2. Preliminaries

In this section, some basic concepts and definitions related to INSs, including interval numbers, t-norm and t-conorm, and the definitions and operations of NSs, SNSs, and INSs are introduced, which will be utilized in the rest of the paper.

2.1. Interval Numbers and Their Operations

Interval numbers and their operations are of utmost importance to explore the operations for INSs. So some definitions and operations of interval numbers are given below.

Definition 1 (see [34–37]). Let , and then is called an interval number. In particular, if , then is reduced to a positive interval number.
Consider any two interval numbers and , and then their operations are defined as follows:(1); (2); (3); (4), ;(5).

Definition 2 (see [37]). Let and , and , and then the degree of possibility of is formulated by

Suppose that there are interval numbers and each interval number is compared to all interval numbers by using (1), namely,

Then a complementary matrix can be constructed as follows: where , , .

2.2. t-Norm and t-Conorm

The t-norm and its dual t-conorm play an important role in the construction of operation rules and averaging operators of INSs. Here, some basic concepts are introduced.

Definition 3 (see [38, 39]). A function is called t-norm if it satisfies the following conditions:(1);(2); (3);(4)if , then .

Definition 4 (see [38, 39]). A function is called t-conorm if it satisfies the following conditions:(1);(2);(3);(4)if , then .

Definition 5 (see [38, 39]). A t-norm function is called Archimedean t-norm if it is continuous and for all . An Archimedean t-norm is called strictly Archimedean t-norm if it is strictly increasing in each variable for . A t-conorm function is called Archimedean t-conorm if it is continuous and for all . An Archimedean t-conorm is called strictly Archimedean t-conorm if it is strictly increasing in each variable for .

It is well known [39, 40] that a strict Archimedean t-norm can be expressed via its additive generator as and similarly applied to its dual t-conorm with . We observe that an additive generator of a continuous Archimedean t-norm is a strictly decreasing function .

There are some well-known Archimedean t-conorms and t-norms [41].(1)Let , , , and . Then algebraic t-conorm and t-norm are obtained: (2)Let , , , and . Then Einstein t-conorm and t-norm are obtained: (3)Let , , , and , . Then Hamacher t-conorm and t-norm are obtained:

2.3. Definitions and Operations of NSs and SNSs

Definition 6 (see [31]). Let be a space of points (objects), with a generic element in denoted by . A NS in is characterized by a truth-membership function , an indeterminacy-membership function , and a falsity-membership function .  , , and are real standard or nonstandard subsets of ; that is, , , and . There is no restriction on the sum of , , and , so .

Definition 7 (see [31]). A NS is contained in the other NS , denoted by , if and only if , , , , , and for .

Since it is difficult to apply NSs to practical problems, Ye reduced NSs of nonstandard intervals into a kind of SNSs of standard intervals that will preserve the operations of NSs [30].

Definition 8 (see [30]). Let be a space of points (objects), with a generic element in denoted by . A NS in is characterized by , , and , which are single subintervals/subsets in the real standard ; that is, , , and . Then, a simplification of is denoted by which is called a SNS. It is a subclass of NSs.

The operational relations of SNSs are also defined in [30].

Definition 9 (see [30]). Let and be two SNSs. For any ,(1),  ,(2),(3),(4).

There are some limitations in Definition 9.

In some situations, the operations, such as and , as given in Definition 9, might be irrational. This will be shown in the example below.

For example, let two simplified neutrosophic numbers (SNNs) and . Obviously, is the maximum of SNSs. It is notable that the sum of any number and the maximum number should be equal to the maximum one. However, according to in Definition 9, . Hence, does not hold and so do the other equations in Definition 9. It shows that the operations above are incorrect.

In addition, the similarity measure for SNSs in [30] on the basis of the operations does not satisfy any cases.

For instance, let the alternatives , and the ideal alternative . According to the decision making method based on the cosine similarity measure for SNSs under the simplified neutrosophic environment in [30], we can obtain that , ; that is, the alternative is equal to the alternative . However, for ,   and , it is clear that the alternative is superior to the alternative .

2.4. Definitions and Operations of INSs

Definition 10 (see [31]). Let be a space of points (objects) with generic elements in denoted by . An INS in is characterized by a truth-membership function , an indeterminacy-membership function , and a falsity-membership function . For each point in , we have that , , , and , . We only consider the subunitary interval of . It is the subclass of a NS. Therefore, all INSs are clearly NSs.

Definition 11 (see [31]). An INS is contained in the other INS , , if and only if , , , ,  and ,
for any .

Definition 12 (see [31]). Two INSs and are equal, written as , if and only if and .

Definition 13 (see [31]). The addition of two INSs and is an INS , written as , whose truth-membership, indeterminacy-membership, and falsity-membership functions are related to those of and by , , ,   , , ,
for all in .

As to be known, when , it should satisfy and for B being the minimum value of INSs. And when , as the largest element of INSs, it should satisfy and . Let . That is , , and . According to Definition 13, , , , , , and ; that is, , so that Definition 13 does not hold.

Definition 14 (see [31]). The Cartesian product of two INSs defined on the universe and defined on the universe is an INS , written as , whose truth-membership, indeterminacy-membership, and falsity-membership functions are related to those of and by , , , , , ,
for all in , in .

Being similar to Definition 13, Definition 14 does not hold in some cases. Therefore, new operation rules for INSs should be explored.

3. Operations and Comparison Approach for INSs

3.1. Operations for INSs

Xu defined some operations of interval valued intuitionistic fuzzy numbers [42]. Based on these operations and preliminaries in Section 2, the operations of two INSs can be defined as follows.

Definition 15. Let two INNs , , and . The operations for INNs are defined based on the Archimedean t-conorm and t-norm as below:(1)(2)(3)(4)
Let and be both INNs. If we assign its generator a specific form, specific operations for INSs will be obtained. When , we have(5)(6)(7)(8)