#### Abstract

A cubic bipolar fuzzy set (CBFS) is a robust paradigm to express bipolarity and vagueness in terms of bipolar fuzzy numbers and interval-valued bipolar fuzzy numbers. The abstraction of similarity measures (SMs) has a large number of applications in various fields. Therefore, in this study, taking the advantage of CBFSs, three cosine similarity measures for CBFSs are proposed successively by using cosine of the angle between two vectors, new distance measures, and cosine function. Some key properties of these similarity measures (SMs) are explored. Based on suggested SMs, the problem of bacteria recognition is analyzed and an important application is provided to exhibit the efficiency of proposed SMs for CBF information. Moreover, the TOPSIS approach based on cosine SMs is developed for multicriteria group decision-making (MCGDM) problems. An illustrative example about the selection of sustainable plastic recycling process is presented to discuss the efficiency of the suggested MCGDM technique.

#### 1. Introduction

Fuzzy set (FS) theory [1] by using the concept of membership function (MF) is a robust approach for modeling uncertainty. A membership function is the generalization of characteristic function in the crisp set theory. An interval-valued fuzzy set (IVFS) [2] is the generalization of FS that assigns an interval of membership grades to the elements in the universe. The idea of orthopair has been extended to the ordered pair of membership grade (MG) and nonmembership grade (NMG) in the studies of intuitionistic fuzzy sets (IFSs) [3], Pythagorean fuzzy sets (PFSs) [4, 5], and q-rung orthopair fuzzy sets (q-ROPFSs) [6]. The values of MG and NMG are the elements of , i.e., any real number between 0 and 1. A number is called an intuitionistic fuzzy number (IFN) if , a Pythagorean fuzzy number (PFN) if , and a *q*-rung orthopair fuzzy number (q-ROFN) if , .

In many real-life problems, the indeterminacy is an essential factor to express expert opinion of the decision makers (DMs). To express such information, the idea of ordered triples with three components (MG, indeterminacy, and NMG) of neutrosophic set (NS) [7] and single-valued neutrosophic set (SVNS) [8] has been focused by many researchers. The concepts of spherical fuzzy sets [9–11] and picture fuzzy sets [12, 13] are strong models to deal with uncertain real-life problems with three components.

Zhang [14, 15] proposed the notion of bipolar fuzzy set (BFS) and bipolar (crisp) set to deal with bipolarity and fuzziness. Lee [16] proposed some results for bipolar-valued fuzzy sets. Deli et al. [17] studied bipolar neutrosophic set (BNS) and proposed novel features of BNSs with application towards MCDM. Wei et al. [18] studied interval-valued bipolar fuzzy set (IVBFS) for uncertainty and bipolarity and positive and negative intervals based MCDM approach.

A hybrid concept of cubic set (CS) has been studied by Jun et al. [19]. He proposed novel concepts of internal (external) cubic sets of -intersection, -union, -intersection, and -union. The degree of similarity between two objects can be determined by the notion of similarity measure (SM). Ye [20] introduced cosine similarity measures for IFSs. Wei and Wei [21] defined 10 different kinds of similarity measures for medical science and pattern recognition using PFS information using hesitation, MG and NMG, cosine function, and distance measures. Ulucay et al. [22] proposed new SMs for bipolar neutrosophic sets (BNSs) like hybrid vector SMs, Dice SMs, weighted Dice SMs, and weighted hybrid vector SMs. A comparative analysis for different values of the operational parameter is developed to express the validity of suggested SMs. Abdel-Basset et al. [23] investigated medical diagnosis of bipolar disorders by using BNSs-based SMs. They developed new MADM methods based on SMs and their weighted versions and illustrated them with some numerical examples. Tu et al. [24] suggested Dice SMs and Jaccard and cotangent SMs for neutrosophic cubic sets (NCSs) and applied them in MCDM. Lu and Ye [25] defined cosine SMs for NCSs by using cosine functions, distance, and cosine angle of two vectors. They investigated certain properties and propositions of proposed SMs. Peng et al. [26, 27] studied information measures for PFSs and q-ROFSs with corresponding applications in MCDM. Naeem et al. [28] investigated new SMs for PFS information for the analysis of psychological disorder under uncertainty. Hussian and Yang [29] introduced Pythagorean fuzzy Hausdorff metric-based new distance and similarity measures and TOPSIS approach for MCDM.

TOPSIS is a well-known MCDM approach which was first introduced by Hwang and Yoon [30]. Zhang and Xu [31] initiated the Pythagorean fuzzy TOPSIS technique by defining a distance measure. Rani et al. [32] established the TOPIS method based on SMs for the PF environment and applied it for project delivery system selection. Akram et al. [33] developed bipolar fuzzy TOPSIS and utilized it in medical diagnosis. Garg and Arora [34] introduced the IFSS-TOPSIS method by using the correlation coefficient for solving MCDM problems. Garg and Kaur [35] developed TOPSIS based on cubic intuitionistic fuzzy (CIFS) information. They proposed a nonlinear-programming-based MCDM approach to deal with cubic intuitionistic fuzzy (CIFS) uncertain information. Riaz and Tehrim [36–38] initiated the novel hybrid model, namely, cubic bipolar fuzzy set (CBFS), by incorporating the features of BFS and IVBFS. They suggested some AOs named as CBF weighted averaging (geometric) AOs with R (P) orders for external (internal) CBF information.

Ali et al. [39] proposed Einstein geometric aggregation operators using novel complex interval-valued Pythagorean fuzzy sets. Alosta et al. [40] developed a new AHP-RAFSI approach for resolving a location selection problem. Hashemkhani Zolfani et al. [41] introduced a VIKOR- and TOPSIS-focused reanalysis of the MADM methods based on logarithmic normalization. Ramakrishnan and Chakraborty [42] proposed a cloud TOPSIS model for green supplier selection. Dobrosavljevic and Urosevic [43] suggested analysis of business process management defining and structuring activities. Yorulmaz et al. [44] proposed a robust Mahalanobis distance-based TOPSIS to evaluate the economic development of provinces. Petrovic and Kankaras [45] developed a hybridized IT2FS-DEMATEL-AHP-TOPSIS multicriteria decision-making approach as a case study of selection and evaluation of criteria for determination of air traffic control radar position. Badi and Pamucar [46] introduced a supplier selection method for steel-making companies by using combined Grey-MARCOS. Riaz et al. [47] proposed essential characteristics for soft multiset topology and robust MCDM applications.

The advantages and objectives of this manuscript are as follows: (1) To deal with vagueness and bipolarity with cubic bipolar fuzzy sets (CBFSs) which are a superior model to existing bipolar fuzzy models. (2) To define cosine SMs between CBFSs based on cosine of the angle between two vectors, new distance measures, and cosine function. Moreover, their weighted extensions are also introduced. (3) To apply these similarity measures to bacteria recognition problem. (4) To propose the TOPSIS approach based on cosine SMs to deal with the plastic recycling method selection problem.

The arrangement of this manuscript is planned as follows: In Section 2, we discuss some rudimentary concepts of bipolarity and fuzziness. In Section 3, we define cosine SMs, weighted cosine SMs, and related propositions. In Section 4, we establish an algorithm to handle pattern recognition problems under the CBF environment and a complex pattern recognition problem is presented to exhibit the efficiency of proposed algorithm. In Section 5, we introduce TOPSIS approach based on cosine SMs and an application concerning the selection of most sustainable plastic recycling process is discussed. Finally, we assess the validity and usefulness of our suggested technique by comparing it with some existing methodologies. Section 6 is designed for concluding remarks to express advantages and objectives of this manuscript.

#### 2. Preliminaries

Some rudimentary concepts can be reviewed to understand the necessary fundamentals related to this manuscript (see [1, 2, 14, 18, 19, 36–38]).

*Definition 1 (see [37]). *A cubic bipolar fuzzy set (CBFS) on the universe of discourse can be defined aswhere is an IVBFS and is a BFS on . Thus, CBFS can also be written aswhere and represent the interval-valued positive and negative MGs, respectively, and and represent the single-valued positive and negative MGs, respectively, of an object .

##### 2.1. Operations on CBFSs

*Definition 2 (see [37]). *Let and be two CBFSs on and . Then, the operations on these CBFSs under -order are given as follows:(i)(ii)(iii)(iv)(v)(vi)(vii) if and , and ,

*Definition 3 (see [37]). *Let and be two CBFSs on and . Then, we have the following:(i)(ii)(iii)(iv)(v)(vi)(vii) if and , and ,

#### 3. Cosine Similarity Measures for CBFSs

In this section, we define three cosine SMs for CBFSs. We examine their properties and give examples for better understanding. Later on, the weighted versions of these similarity measures will also be presented.

*Definition 4. *Let be a finite universe of discourse.

Let and be two CBFSs on ; then, cosine SM based on the cosine of the angle between two vectors is given by

Theorem 1. *For two CBFSs and , the cosine SM proposed in equation (3) possesses the following properties:*(i)*(ii)**(iii)** if *

*Proof. *(i)It is obvious that . We only have to show that . The Cauchy–Schwarz inequality further implies that Utilizing the above inequality, we have Hence, .(ii)It is obvious.(iii)If , then , , , and , for all . Thus, .

*Example 1. *Let and be two CBFSs on . Then, by utilizing equation (3), we calculate the cosine similarity measure between and as

*Definition 5. *Let be a finite universe of discourse, and let and be two CBFSs on ; then, cosine SM based on distance is given by

Theorem 2. *For two CBFSs and , the cosine similarity measure proposed in equation (7) possesses the following properties:*(i)*(ii)**(iii)** iff *

*Proof. *(i)We know that , , , , , and , for all . By combining all these inequalities, we get Hence, .(ii)It is obvious.(iii)If , then , , , and , for all . Therefore,So, .

Conversely, consider . Since , we infer thatThis gives , , , and , for all . Hence, .

*Example 2. *Consider the CBFSs and from the previous example. By utilizing equation (7), another cosine similarity measure between and can be computed as

*Definition 6. *Let and be two CBFSs on ; then, cosine similarity measure based on cosine function is defined as

Theorem 3. *For two CBFSs and , the cosine similarity measure proposed in equation (12) satisfies the following conditions:*(i)*(ii)**(iii)** if *

*Proof. *(i)Let , , , and . We see that , . On substituting the values of , and , we get .(ii)Since cosine is an even function, .(iii)If , then , , , and , for all . Hence, one can easily infer that .

*Example 3. *Consider the CBFSs and from Example 1. Then, by utilizing equation (12), we calculate cosine SM between and given by