Zadeh [
1] started the theory of fuzzy set and since then it has been a significant tool in learning logical subjects. It is applied in many fields, see [
2]. There are numbers of over simplifications/generalization of Zadeh’s fuzzy set idea to interval-valued fuzzy notion [
3], intuitionistic fuzzy set [
4], L-fuzzy notion [
5], probabilistic fuzzy notion [
6] and many others. Zhang [
7,
8], provided the generality of fuzzy sets as bipolar fuzzy sets. The extensions of fuzzy sets with membership grades from
are the bipolar fuzzy sets. The membership grade
of a section directs in bipolar fuzzy set that the section fairly fulfils the couched stand-property, the membership grade
of a section shows that the section fairly fulfils the matter and the membership grade 0 of a section resources that the section is unrelated to the parallel property. While bipolar fuzzy sets and intuitionistic fuzzy sets aspect parallel to one another, they are really distinct sets (see [
3]). When we calculate the place of an objective in a universe, positive material conveyed for a collection of thinkable spaces and negative material conveyed for a collection of difficult spaces [
9]. Naveed et al. [
10,
11,
12], discussed theoretical aspects of bipolar fuzzy sets in detail. Smarandache [
13], gave the notion of neutrosophic sets as a generalization of intutionistic fuzzy sets. The applications of Neutrosophic set theory are found in many fields (see
http://fs.gallup.unm.edu/neutrosophy.htm). Recently Zhang et al. [
14], Majumdar et al. [
15], Liu et al. [
16,
17], Peng et al. [
18] and Sahin et al. [
19] have discussed various uses of neutrosophic set theory in deciding problems. Now a days, neutrosophic sets are very actively used in applications and MCGM problems. Bausys and Juodagalviene [
20], Qun et al. [
21], Zavadskas et al. [
22], Chan and Tan [
23], Hong and Choi [
24], Zhan et al. [
25] studied the applications of neutrosophic cubic sets in multi-criteria decision making in different directions. Anyhow, these approaches use the maximum, minimum operations to workout the aggregation procedure. This leads to subsequent loss of data and, therefore, inaccurate last results. How ever this restriction can be dealt by using famous weighted averaging (WA) operator [
26] and the ordered weighted averaging (OWA) operator [
27]. Medina and Ojeda-Aciego [
28], gave t-notion lattice as a set of triples related to graded tabular information explained in a non-commutative fuzzy logic. Medina et al. [
28] introduces a new frame work for the symbolic representation of informations which is called to as signatures and given a very useful technique in fuzzy modelling. In [
29], Nowaková et al., studied a novel technique for fuzzy medical image retrieval (FMIR) by vector quantization (VQ) with fuzzy signatures in conjunction with fuzzy S-trees. In [
30] Kumar et al., discussed data clustering technique, Fuzzy C-Mean algorithem and moreover Artificial Bee Colony (ABC) algorithm. In [
31] Scellato et al.,discuss the rush of vehicles in urban street networks. Recently Gulistan et al. [
32], combined neutrosophic cubic sets and graphs and gave the concept of neutrosophic cubic graphs with practical life applications in different areas. For more application of neutrosophic sets, we refer the reader to [
33,
34,
35,
36,
37]. Since, the models presented in literature have different limitations in different situations. We mainly concern with the following tools:
We try to blend these two concepts together and try to develop a more powerful tool in the form of neutrosophic bipolar fuzzy sets. In this work we initiate the study of neutrosophic bipolar fuzzy sets which are the generalization of bipolar fuzzy sets and neutrosophic sets. After introducing the definition we give some basic operations, properties and applications of neutrosophic bipolar fuzzy sets. And the rest of the paper is structured as follows;
Section 2 provides basic material from the existing literature to understand our proposal.
Section 3 consists of the basic notion and properties of neutrosophic bipolar fuzzy set.
Section 4 gives the role of weighted aggregation operator in terms of neutrosophic bipolar fuzzy sets. We define neutrosophic bipolar fuzzy weighted averaging operator
and neutrosophic bipolar fuzzy ordered weighted averaging
operators.
Section 5 includes different kinds of similarity measures. In
Section 6, an algorithm for the multiple attribute decision making problems under the neutrosophic bipolar fuzzy environment by using the different kinds of similarity measures of neutrosophic bipolar fuzzy sets and neutrosophic bipolar fuzzy weighted/fuzzy ordered weighted aggregation operators is proposed. In
Section 7, we provide a daily life example related with HOPE foundation, which shows the applicability of the algorithm provided in
Section 6. In
Section 8, we provide a comparison with the previous existing methods. In
Section 9, we discuss conclusion and some future research directions.