**1. Introduction**

Multiple-criteria decision making (MCDM) is a branch of decision making theory where the aim of an individual is to select the most acceptable alternatives among the feasible ones under some criteria. This criteria dependence in decision can be found in real life on a regular basis. While handling some real life decision making problems, a decision maker often faces trouble due to the presence of several kinds of uncertainties in the data, which is very natural. An attempt was made for the first time by Zadeh [1] by introducing a novel concept of fuzzy set theory. Immediately after that, several improvisations of fuzzy sets were made and implemented in the decision making process. For instance, rough sets by Pawlak [2], intuitionistic fuzzy sets [3], interval valued intuitionistic fuzzy sets [4] by Atanassov, soft sets by Molodtsov [5], etc. Unlike the classical logic, a fuzzy set associates a degree of membership value to every element of the universe of discourse, which can range from 0 to 1, whereas an intuitionistic fuzzy set associate a degree of membership *µ* ∈ [0, 1] and a degree of non-membership *ν* ∈ [0, 1], where 0 ≤ *µ* + *ν* ≤ 1. The margin of indeterminacy or hesitation *π* is defined as *π* = 1 − *µ* − *ν*. Smarandache in [6,7] proposed neutrosophic sets. In neutrosophic sets, the indeterminacy membership function walks along independently of the truth membership or of the falsity membership. Neutrosophic theory has been widely explored by researchers (see [8–13]) for application purpose in handling real life situations involving uncertainty. Although the hesitation margin of neutrosophic theory is independent of the truth or falsity membership, looks more general than intuitionistic fuzzy sets yet. Recently, in [14] Atanassov et al. studied the relations between inconsistent intuitionistic fuzzy sets [15], picture fuzzy sets [16], neutrosophic sets [7] and intuitionistic fuzzy sets [3]; however, it remains in doubt that whether the indeterminacy associated to a particular

element occurs due to the belongingness of the element or the non-belongingness. This has been pointed out by Chattejee et al. [17] while introducing a more general structure of neutrosophic set viz. quadripartitioned single valued neutrosophic set (QSVNS). The idea of QSVNS is actually stretched from Smarandache, s four numerical-valued neutrosophic logic [18] and Belnap, s four valued logic [19], where the indeterminacy is divided into two parts, namely, "unknown" i.e., neither true nor false and "contradiction" i.e., both true and false. In the context of neutrosophic study however, the QSVNS looks quite logical. Also in their study, Chatterjee et al. [17] analyzed a real life example for a better understanding of a QSVNS environment and showed that such situations occur very naturally. They have also solved a decision making problem pertaining to pattern recognition showing the application capability of QSVNS.

Bipolarity often reflects the tendency of the human mind in reasoning to make a decision on the basis of +ve and -ve information. Lee [20,21] introduced bipolar fuzzy set as an extension of fuzzy set. Bipolar fuzzy model with some hybrid structures were studied in [22–25] shows the flexibility of bipolar fuzzy sets for solving decision making problems. Bipolar fuzzy set and neutrosophic set have been put together for a more general framework viz. bipolar neutrosophic sets (BNS) by Deli et al. [26]. Sahin et al. [27,28] introduced the Jaccard vector similarity measure, hybrid vector similarity measure, and dice similarity measure with applications to decision making problems. Jamil et al. [29] applied bipolar neutrosophic Hamacher averaging operator in group decision making problems. Bipolar neutrosophic sets help in handling uncertain information in a reliable way. It is an extension of the bipolar fuzzy set and neutrosophic set, which can deal with real life problems involving positive and negative information. Looking at the work of Chatteree et al. [17], unlike neutrosophic set, it is in doubt whether the negative indeterminacy associated to some elements of the universe of discourse is due to the occurrence of the counter-property or the non-occurrence of the counter-property.

To overcome the aforesaid situation we merged the bipolar neutrosophic set and QSVNS to introduce a more general structure, namely, quadripartitioned single valued bipolar neutrosophic sets (QSVBNS). The word "quadripartitioned" refers to four values i.e., in case of QSVBNS the negative indeterminacy of the BNS is divided into two parts alongside truth and falsity membership alike QSVNS.

First, we develop some set theoretic results on QSVBNS and then formulas for similarity measures were framed, and finally a real life problem was dealt with using the MCDM method in this setting. A comparison was made in application of a real life problem, where it is seen that the use of QSVBN system gives a better result compared to fuzzy sets and bipolar fuzzy sets. The paper is organized as follows: Section 2 recalls some preliminaries results. In Section 3, QSVBNS is introduced and some basic operations on QSVBNS are dealt with; an example of QSVBNS is also presented. Several similarity measures between QSVBNS are defined and their properties are studied in Section 4. In Section 5 we give an algorithm based on quadripartitioned bipolar weighted similarity measure to deal with the multi-criteria decision making problem in a QSVBN environment. Based on the given algorithm, a real life problem in decision making is solved in Section 6. A detailed discussion about the obtained result is analyzed in Section 7. Finally, Section 8 concludes the paper.
