Neutrosophic cubic Heronian mean operators with applications in multiple attribute group decision-making using cosine similarity functions

This article introduces the concept of Heronian mean operators, geometric Heronian mean operators, neutrosophic cubic number–improved generalized weighted Heronian mean operators, neutrosophic cubic number–improved generalized weighted geometric Heronian mean operators. These operators actually generalize the operators of fuzzy sets, cubic sets, and neutrosophic sets. We investigate the average weighted operator on neutrosophic cubic sets and weighted geometric operator on neutrosophic cubic sets to aggregate the neutrosophic cubic information. After this, based on average weighted and geometric weighted and cosine similarity function in neutrosophic cubic sets, we developed a multiple attribute group decision-making method. Finally, we give a mathematical example to illustrate the usefulness and application of the proposed method.


Introduction
The multi-attribute decision-making (MADM) or multi-attribute group decision-making (MAGDM) widely existed in the field of management, military, economy, and engineering techniques [1][2][3] to get an accurate evaluation information in the premises of decision makers (DMs) to make feasible and rational decision.There is a variety of limitations in real-world problems such as uncertainty and complexity of the decisionmaking environment, too much abundant data and inconsistent and indeterminate with respect to fuzzy information.To process this kind of information, in 1965 Zadeh 4 first introduced the fuzzy set (FS) theory.After that Atanassov proposed the intuitionistic fuzzy set (IFS). 5,6In IFS, Atassanav added a non-membership function to decrease the shortcomings in which the FS has only the membership function whereas the IFS is composed of the truth-membership function and falsity-membership function and satisfies the conditions A Tru (u), A Fal (x) 2 ½0, 1 and 0 ł A Tru (u) + A Fal (x) ł 1.
Moreover, in 1998 Smarandache 7 defined the neutrosophic set (NS).In NS, Samarndache added indeterminacy-membership function, that is, NS is characterized by truthmembership A Tru (u), indeterminacy-membership A Ind (u), and falsity-membership A Fal (u).Moreover, the NS is the generalization of FS and IFSs.][10] Further, Jun et al., proposed the concept of neutrosophic cubic set (NCS) by adding truth-membership A Tru (u), indeterminacy-membership A Ind (u), and falsitymembership A Fal (u) in the form of interval NS and truth-membership A Tru (u), indeterminacy-membership A Ind (u) and falsity-membership A Fal (u) in the form of NS. 11 Al-Omeri and Smarandache 12 introduce the idea of neutrosphic sets via neutrosophic topological spaces (NTs), and some other types of NSs such as neutrosophic open sets, neutrosophic continuity, and their application in geographical information system.NCS is the generalization of FS, cubic set, and NS.Many researchers used NCSs in different directions such as, [13][14][15][16][17][18] to have more applications.So many others discussed different aspects of NCS environment on MADM like, Peng et al., 19 Zhang et al., 20 Ye, 21,22 Shi and Ye, 23 Lu and Ye, 24 Pramanik et al., [25][26][27][28] GRA 29 and Dalapati and Pramanik, 30 Liu and Wang 31 proposed the aggregation operator and applied in MAGDM problems.NS theory has various applications in numerous fields such as data record, control theory, problems and decision-making theory.Xu and Yazer 32 and Xu 33 proposed some arithmetic aggregation operators and geometric aggregation operators for intuitionistic fuzzy information and these operators did not consider the correlations of aggregated arguments.After that, in 2007 Beliakov et al. 34 proposed the Heronian mean (HM) operators, which are an important aggregated arguments and possess the characteristic of correlation of aggregation operators.HM operators can deal with the interactions among the attribute values and neutrosophic cubic numbers (NCNs) can easily express the incomplete, indeterminate and inconsistent information.Liu (The research note of HM operators.Shandong University of Finance and Economics, 2012, personal communication) in 2012 extended HM operator to the generalized HM operator. 35Yu and Wu 36 studied the intervalvalued intuitionistic fuzzy information aggregation operators and their applications in decision-making.Further work to aggregate the interval-valued intuitionistic fuzzy information Liu 37 proposed some operators such as generalized interval-valued intuitionistic fuzzy Heronian mean (GIIFHM) operator, generalized interval-valued intuitionistic fuzzy weighted Heronian mean (GIIFWHM) operator, an interval-valued intuitionistic uncertain linguistic weighted geometric average (IVIULWGA) operator, an interval-valued intuitionistic uncertain linguistic ordered weighted geometric (IVIULOWG) operators and also developed the idea of interval-valued intuitionistic uncertain linguistic variables, decision-making problems and their operational laws.Yu 38 proposed the idea of decision-making problems under intuitionistic fuzzy environment and introduced some aggregation operators, such as the intuitionistic fuzzy geometric Heronian mean (IFGHM) operators and the intuitionistic fuzzy geometric weighed Heronian mean (IFGWHM) operators and their properties.Liu et al., 39 proposed the aggregation operator and applied in MAGDM problems.We extend the idea of Li et al., 40 provided in Liu et al. 39 Therefore, in this article, we will extend neutrosophic numbers (NNs) to NCNs, and propose some HM operators for NCNs, including the improved generalized weighted geometric Heronian mean (IGWGHM) operators which can satisfy some properties, such as reducibility, idempotency, monotonicity and boundedness.At the end, these properties are applied to multiattribute group decision-making problem (Figure 1).

Preliminaries
In this section, we give some helpful terminologies from the existing literature.

Definition 1 (NS).
Let U be a non-empty set. 7A neutrsophic set in U is a structure of the form A = fu; A Tru (u), A Ind (u), A Fal (u)ju 2 U g, is characterized by a truth-membership Tru, indeterminacy-membership Ind and falsity-membership Fal, where A Tru , A Ind , Definition 2 (NCS).Let X be a non-empty set. 11A NCS over U is defined in the form of a pair O = (Å, L) where

Definition 3 (HM operator
where I = ½0, 1 then the function HM is called Heroinan mean (HM) operator.

Definition 4 (geometric Heronian mean operator).
A GHM operator of dimension n is a mapping GHM : I n !I such that (The research note of HM operators.Shandong University of Finance and Economics, 2012, personal communication) where x, y ø 0 and I = ½0, 1.Then the function GHM x, y is called generalized Heroinan mean (GHM) operator.
It is easy to prove that GHM operator has the following properties: Theorem 3 (boundedness).GHM operator lies between the max and min operators, that is Since the HM and geometric mean (GM) operator only consider the interrelationship of the e input arguments and do not take their own weights into account.In the following, we will introduce another HM operator which is called the weighted generalized Heronian mean (GWHM) operator and shown as follows.
The IGGWHM has the properties, such as reducibility, idempotency, monotonicity, and boundedness (The research note of HM operators.Shandong University of Finance and Economics, 2012, personal communication).
2. When x = 0, then Similarly, IGGWHM 0, y does not have any relationship with y.
Definition 11 (the NCNIGWHM operator).Let x, y ø 0, and Q j = ( Ra j , S b j ) where Raj = f ÃTru (u j ), ÃInd (u j ), ÃFal (u j )g and S b = fA Tru (u j ), A Ind (u j ), A Fal (u j )g (j = 1, 2, . . ., n) be a collection of NCNs with the weight vector W = (w 1 , w 2 , . . ., w n ) T such that w j ø 0 and P n j = 1 w j = 1, then an NCNIGWH operator of dimension n is a mapping NCNIGWH : C n !C, and has where C is the set of all NCNs.Theorem 8. Let x, y ø 0, and Q j = ( Ra j , S b j ) (j = 1, 2, :::, n) be a collection of NCNs with the weight vector W = (w 1 , w 2 , :::, w n ) T such that w j ø 0 and P n j = 1 w j = 1, then the result aggregated from Definition 11 is still an NCN, and even Proof.Since and then Furthermore International Journal of Distributed Sensor Networks which complete the proof of Theorem 8 h Moreover, the NCNIGWHM operator also has the following properties.

. be a collection of NCNs, and
min ÃTru (u j ), min ÃInd (u j ), min ÃFal (u j ), min A Tru (u j ), min A Ind (u j ), min A Fal (u j ) À Á Proof.Since Q j ø Q À j , then based on Theorems 10 and 11, we have Like wise, we can get which completes the proof.h We will discuss some special cases of the NCNIGWHM with respect to parameters x and y, as follows: 2. When y = 0, then we have

NCNIGWGHM operator
Definition 12. Let x, y ø 0, and Q j = ( Ra j , S b j ) where Raj = f ÃTru (u j ), ÃInd (u j ), A ^Fal (u j )g and S b = fA Tru (u j ), A Ind (u j ), A Fal (u j )g (j = 1, 2, . . ., n) be a collection of NCNs with the weight vector W = (w 1 , w 2 , . . ., w n ) T such that w j ø 0 and P n j = 1 w j = 1, then an NCNIGWGHM operator of dimension n is a mapping NCNIGWGHM : C n !C, and has where C is the set of all NCNs.
Theorem 12. Let x, y ø 0, and Q j = ( Ra j , S b j ) where Raj = f ÃTru (u j ), ÃInd (u j ), A ^Fal (u j )g and S b = fA Tru (u j ), A Ind (u j ), A Fal (u j )g (j = 1, 2, . . ., n) be a collection of NCNs with the weight vector W = (w 1 , w 2 , . . ., w n ) T such that w j ø 0 and P n j = 1 w j = 1, then the aggregated value by equation ( 23) can be expressed as Similar, the proofs of Theorem 8 and Theorem 12 are omitted.Moreover, similar to the proofs of Theorems 9-11, it is easy to prove that the NCNIGWGHM operator also has the following properties.

When
The approach to multiple attribute group decision-making with NCNs In this section, we shall introduce the approach to multiple attribute group decision-making with the help of the NCNs.We apply NCN-improved generalized weighted Heronian mean operator to deal with the attribute group decision-making problems under the NCNs environment with an illustrated example.

Algorithm
Step 1.The DMs take their analysis of each alternatives based on each criteria.The performance of each alternatives H i with respect to each crteria G j .
Step 2. Calculate the NCNIGWHM operator (h k i1 , h k i2 , . . ., h k in ) to obtain the collective evaluation value of alternatives H i with respect to each criteria G j .
Step 3. Calculate the cosine similarity using Definition 10 in article. 24tep 4. Rank all the alternatives.

Numerical example
This section introduces an illustrative example to show the application of the above MAGDM method based on NCN.An investment company intends to choose one product to invest its money from four alternatives H i (i = 1, 2, 3, 4): Where H 1 = medicine company, H 2 = textile company, H 3 = mobile company, and H 4 = car company.The weights of the indicators are w = (0:5, 0:3, 0:1, 0:1).Three criteria have been evaluated and they are shown as follows: G 1 = Tax Rate, G 2 = Demand/Supply and G 3 = Wages.In order to get a most suitable choice we will use the above-mentioned algorithm as follows: International Journal of Distributed Sensor Networks Step 2. Calculate the NCNIGWHM operator by formula (15) to obtain the collective evaluation value (h k i1 , h k i2 , . . ., h k in ) of alternatives H i with respect to each criterion G j and w = (0:5, 0:3, 0:1, 0:1), we can get  Step 4. Rank all the alternatives, we get the sequence of candidates as follows: h 1 1 h 2 1 h 3 1 h 4 shown in Figure 3.

Conclusion
In this article, we have discussed the idea of NCNs and different operators such as HM, GHM, weighted Heronian mean, generalized Heronian mean, and generalized weighted geometric mean operators.We applied HM to the NCSs.The NCS can be defined as the three elements such as truth, indeterminate, and incomplete information.The Heronian mean can represent the relationship of the aggregated values and MADM method.Finally, a numerical example is given to verify the proposed method.

Declaration of conflicting interests
The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Figure 1 .
Figure 1.A flowchart of NCNs based on MAGDM problem.

Step 1 .
Let H = fH 1 , H 2 , H 3 , H 4 g be a set of alternatives and G = fG 1 , G 2 , G 3 g be the set of criteria.Let D be set of decision matrix.The decision matrix evaluates each alternative based on given criteria.

Step 3 .Figure 2 .
Figure 2. Line chart of alternatives versus score values of alternatives.

Figure 3 .
Figure 3. Graphical representation of the ranking values of alternatives.
).A HM operator of dimension n is a mapping HM : I n !I such that (The research note of HM operators.Shandong University of