# NC-VIKOR Based MAGDM Strategy under Neutrosophic Cubic Set Environment.

1. IntroductionSmarandache [1] introduced neutrosophic set (NS) by defining the truth membership function, indeterminacy function and falsity membership function as independent components by extending fuzzy set [2] and intuitionistic fuzzy set [3]. Each of three independent component of NS belons to [[sup.-]0, [1.sup.+]]. Wang et al. [4] introduced single valued neutrosophic set (SVNS) where each of truth, indeterminacy and falsity membership degree belongs to [0, 1]. Many researchers developed and applied the NS and SVNS in various areas of research such as conflict resolution [5], clustering analysis [6-9], decision making [10-39], educational problem [40, 41], image processing [42-45], medical diagnosis [46, 47], social problem [48, 49]. Wang et al. [50] proposed interval neutrosophic set (INS). Ye [51] defined similarity measure of two interval neutrosophic sets and applied it to solve multi criteria decision making (MCDM) problem. By combining SVNS and INS Jun et al. [52], and Ali et al. [53] proposed neutrosophic cubic set (NCS). Thereafter, Zhan et al. [54] presented two weighted average operators on NCSs and applied the operators for MADM problem. Banetjee et al. [55] introduced the grey relational analysis based MADM strategy in NCS environment. Lu and Ye [56] proposed three cosine measures between NCSs and presented MADM strategy in NCS environment. Pramanik et al. [57] defined similarity measure for NCSs and proved its basic properties and presented a new multi criteria group decision making strategy with linguistic variables in NCS environment. Pramanik et al. [58] proposed the score and accuracy functions for NCSs and prove their basic properties. In the same study, Pramanik et al. [59] developed a strategy for ranking of neutrosophic cubic numbers (NCNs) based on the score and accuracy functions. In the same study, Pramanik et al. [58] first developed a TODIM (Tomada de decisao interativa e multicritevio), called the NC-TODIM and presented new NC-TODIM [58] strategy for solving (MAGDM) in NCS environment. Shi and Ye [59] introduced Dombi aggregation operators of NCSs and applied them for MADM problem. Pramanik et al. [60] proposed an extended technique for order preference by similarity to ideal solution (TOPSIS) strategy in NCS environment for solving MADM problem. Ye [61] present operations and aggregation method of neutrosophic cubic numbers for MADM. Pramanik et al. [62] presented some operations and properties of neutrosophic cubic soft set.

Opricovic [63] proposed the VIKOR strategy for a MAGDM problem with conflicting attributes [64-65]. In 2015, Bausys and Zavadskas [66] extended the VIKOR strategy to INS environment and applied it to solve MCDM problem. Further, Hung et al. [67] proposed VIKOR method for interval neutrosophic MAGDM. Pouresmaeil et al. [68] proposed an MAGDM strategy based on TOPSIS and VIKOR in SVNS environment. Liu and Zhang [69] extended VIKOR method in neutrosophic hesitant fuzzy set environment. Hu et al. [70] proposed interval neutrosophic projection based VIKOR method and applied it for doctor selection. Selvakumari et al. [70] proposed VIKOR Method for decision making problem using octagonal neutrosophic soft matrix.

VIKOR strategy in NCS environment is yet to appear in the literature.

Research gap:

MAGDM strategy based on NC-VIKOR. This study answers the following research questions:

i. Is it possible to extend VIKOR strategy in NCS environment?

ii. Is it possible to develop a new MAGDM strategy based on the proposed NC-VIKOR method in NCS environment?

Motivation:

The above-mentioned analysis [64-69] describes the motivation behind proposing a novel NC-VIKOR method based MAGDM strategy under the NCS environment. This study develops a novel NC-VIKOR--based MAGDM strategy that can deal with multiple decision-makers.

The objectives of the paper are:

i. To extend VIKOR strategy in NCS environment.

ii. To define aggregation operator.

iii. To develop a new MAGDM strategy based on proposed NC-VIKOR in NCS environment.

To fill the research gap, we propose NC-VIKOR strategy, which is capable of dealing with MAGDM problem in NCS environment.

The main contributions of this paper are summarized below:

i. We developed a new NC-VIKOR strategy to deal with MAGDM problems in NCS environment.

ii. We introduce a neutrosophic cubic number aggregation operator and prove its basic properties.

iii. In this paper, we develop a new MAGDM strategy based on proposed NC-VIKOR method under NCS environment to solve MAGDM problems.

iv. In this paper, we solve a MAGDM problem based on proposed NC-VIKOR method.

The remainder of this paper is organized as follows: In the section 2, we review some basic concepts and operations related to NS, SVNS, NCS. In Section 3, we develop a novel MAGDM strategy based on NC-VIKOR to solve the MADGM problems with NCS environment. In Section 4, we solve an illustrative numerical example using the proposed NC-VIKOR in NCS environment. Then in Section 5, we present the sensitivity analysis. The conclusions of the whole paper and further direction of research are given in Section 6.

2. Preliminaries

Definition 1. Neutrosophic set

Let X be a space of points (objects) with a generic element in X denoted by x, i.e. x [member of] X. A neutrosophic set [1] A in X is characterized by truth-membership function [t.sub.A](x), indeterminacy-membership function [i.sub.A](x) and falsity-membership function [f.sub.A](x), where [t.sub.A](x), [i.sub.A](x), [f.sub.A](x) are the functions from X to ][sup.-]0, [1.sup.+][ i.e. [t.sub.A], [i.sub.A], [f.sub.A] : X [right arrow] ][sup.-]0, [1.sup.+][ that means [t.sub.A](x), [i.sub.A](x),[f.sub.A](x) are the real standard or nonstandard subset of ][sup.-]0, [1.sup.+][. Neutrosophic set can be expressed as A = {<x, ([t.sub.A](x), [i.sub.A](x), [f.sub.A](x))>: [for all] x [member of] X} and [sup.-]0 <[t.sub.A](x) + [i.sub.A](x) + [f.sub.A](x) [less than or equal to] [3.sup.+].

Example 1. Suppose that X = {[x.sub.1], [x.sub.2], [x.sub.3], ..., [x.sub.n]} be the universal set of n points. Let [A.sub.1] be any neutrosophic set in X. Then A expressed as [A.sub.1] = {< [x.sub.1], (0.7, 0.4, 0.3)>: [x.sub.1] [member of] X}.

Definition 2. Single valued neutrosophic set

Let X be a space of points (objects) with a generic element in X denoted by x. A single valued neutrosophic set [4] B in X is expressed as:

B = {< x: ([t.sub.B](x), [i.sub.B](x), [f.sub.B](x)) >: x [member of] X}, where [t.sub.B](x), [i.sub.B](x), [f.sub.B](x) [member of] [0, 1].

For each x [member of] X, [t.sub.B] (x), [i.sub.B] (x), [f.sub.B] (x) [member of] [0, 1] and 0 [less than or equal to] [t.sub.B](x) + [i.sub.B](x) + [f.sub.B](x) [less than or equal to] 3.

Definition 3. Interval neutrosophic set

An interval neutrosophic set [50] [??] of a non empty set H is expreesed by truth-membership function [t.sub.[??]](h) the indeterminacy membership function [i.sub.[??]](h) and falsity membership function [f.sub.[??]](h). For each h [member of] H, [t.sub.[??]](h), [i.sub.[??]](h), [f.sub.[??]](h) [subset or equal to] [0, 1] and [??] defined as follows:

[mathematical expression not reproducible].

Here, we consider [t.sup.-.sub.[??]](h), [.sup.+.sub.[??]](h), [i.sup.- .sub.[??]](h), [i.sup.+.sub.[??]](h), [f.sup.-.sub.[??]](h), [f.sup.+.sub.[??]](h) : H [right arrow] [0, 1] for real applications.

Example 2.

Assume that H = {[h.sub.1], [h.sub.2], [h.sub.3], ..., [h.sub.n]} be a non-empty set. Let [[??].sub.1] be any interval neutrosophic set. Then [[??].sub.1] expressed as [[??].sub.1] = {< [h.sub.1]: [0.30, 0.70], [0.20, 0.45], [0.18, 0.39]: h [member of] H}.

Definition 4. Neutrosophic cubic set

A neutrosophic cubic set [52, 53] in a non-empty set H is defined as N = {< h, [??](h), A(h) >: [for all] h [member of] H}, where [??] and A are the interval neutrosophic set and neutrosophic set in H respectively. Neutrosophic cubic set can be presented as an order pair N = < [??], A >, then we call it as neutrosophic cubic (NC) number.

Example 3.

Suppose that H = {[h.sub.1], [h.sub.2], [h.sub.3], ..., [h.sub.n]} be a non- empty set. Let [N.sub.1] be any NC-number. Then [N.sub.1] can be expressed as [N.sub.1] = {< [h.sub.1]; [0.35, 0.47], [0.20, 0.43], [0.18, 0.42], (0.7, 0.3, 0.5)>: [h.sub.1] [member of] H}.

Some operations of NC-numbers: [52, 53]

i. Union of any two NC-numbers

Let [N.sub.1] = < [[??].sub.1]; [A.sub.1] > and [N.sub.2] = < [[??].sub.2], [A.sub.2] > be any two NC-numbers in a non-empty set H. Then the union of [N.sub.1] and [n.sub.2] denoted by [N.sub.1] [union] [N.sub.2] is defined as follows:

[mathematical expression not reproducible],

where

[mathematical expression not reproducible].

Example 4.

Assume that

[N.sub.1] = < [0.39, 0.47], [0.17, 0.43], [0.18, 0.36], (0.6, 0.3, 0.4)> and [N.sub.2] = < [0.56, 0.70], [0.27, 0.42], [0.15, 0.26], (0.7, 0.3, 0.6)> be two NC-numbers. Then [N.sub.1] [union] [N.sub.2] = < [0.56, 0.7], [0.17, 0.42], [0.15, 0.26], (0.7, 0.3, 0.4)>.

ii. Intersection of any two NC-numbers

Intersection of [N.sub.1] and [N.sub.2] denoted by [N.sub.1] [intersection] [N.sub.2] is defined as follows:

[mathematical expression not reproducible].

Example 5.

Assume that

[N.sub.1] = < [0.45, 0.57], [0.27, 0.33], [0.18, 0.46], (0.7, 0.3, 0.5)> and [N.sub.2] = < [0.67, 0.75], [0.22, 0.44], [0.17, 0.21], (0.8, 0.4, 0.4)> be two NC numbers. Then [N.sub.1] [intersection] [N.sub.2] = < [0.45, 0.57], [0.22, 0.33], [0.18, 0.46], (0.7, 0.3, 0.4)>.

iii. Compliment of a NC-number

Let [N.sub.1] = < [[??].sub.1]1; [A.sub.1] > be a NCS in H. Then compliment of [N.sub.1] = < [[??].sub.1], [A.sub.1] > is denoted by [N.sup.c.sub.1] = {< h, [[??].sup.c.sub.1](h), [A.sup.c.sub.1](h)>: [for all] h [member of] H}.

Here, [mathematical expression not reproducible].

Example 6.

Assume that [N.sub.1] be any NC-number in H in the form: [N.sub.1] = < [.45, .57], [.27, .33], [.18, .46], (.7, .3, .5)>.

Then compliment of [N.sub.1] is obtained as NJ = < [0.18, 0.46], [0.67, 0.73], [0.45, 0.57], (0.5,0.7, 0.7) >.

iv. Containment

Let [mathematical expression not reproducible] be any two NC-numbers in a non-empty set H, then, (i) [N.sub.1] [subset or equal to] [N.sub.2] if and only if

[mathematical expression not reproducible].

Definition 7.

Let [mathematical expression not reproducible] be any two NC-numbers, then distance [58] between them is defined by D ([N.sub.1], [N.sub.2]) =

[mathematical expression not reproducible] (1)

Definition 2.14: Procedure of normalization

In general, benefit type attributes and cost type attributes can exist simultaneously in MAGDM problem. Therefore the decision matrix must be normalized. Let [a.sub.ij] be a NC-numbers to express the rating value of i-th alternative with respect to j-th attribute ([[PSI].sub.j]). When attribute [[PSI].sub.j] [member of] C or [[PSI].sub.j] [member of] G (where C and G be the set of cost type attribute and set of benefit type attributes respectively) The normalized values for cost type attribute and benefit type attribute are calculated by using the following expression (2).

[mathematical expression not reproducible] (2)

Where, [a.sub.ij] is the performance rating of i th alternative for attribute [[PSI].sub.j] and max [a.sub.j] is the maximum performance rating among alternatives for attribute [[PSI].sub.j].

VIKOR strategy

The VIKOR strategy is an MCDM or multi-criteria decision analysis strategy to deal with multi-criteria optimization problem. This strategy focuses on ranking and selecting the best alternatives from a set of feasible alternatives in the presence of conflicting criteria for a decision problem. The compromise solution [63, 64] reflects a feasible solution that is the closest to the ideal, and a compromise means an agreement established by mutual concessions. The [L.sub.p]-metric is used to develop the stategy [65]. The VIKOR strategy is developed using the following form of [L.sub.p]-metric

[mathematical expression not reproducible].

In the VIKOR strategy, [L.sub.1i] (as [S.sub.i]) and [L.sub.[infinity]i], i (as [R.sub.i]) are utilized to formulate ranking measure. The solution obtained by min Si reflects the maximum group utility ("majority" rule), and the solution obtained by min [R.sub.i] indicates the minimum individual regret of the "opponent".

Suppose that each alternative is evaluated by each criterion function, the compromise ranking is prepated by comparing the measure of closeness to the ideal alternative. The m alternatives are denoted as [A.sub.1], [A.sub.2], [A.sub.3], ..., [A.sub.m]. For the alternative Ai, the rating of the j th aspect is denoted by [[OMEGA].sub.ij], i.e. [[OMEGA].sub.ij] is the value of j th criterion function for the alternative [A.sub.1]; n is the number of criteria.

The compromise ranking algorithm of the VIKOR strategy is presented using the following steps:

Step 1: Determine the best [[OMEGA].sup.+.sub.j] and the worst [[OMEGA].sup.-.sub.j] values of all criterion functions j =1, 2, ..., n. If the j-th function represents a benefit then:

[mathematical expression not reproducible]

Step 2: Compute the values [S.sub.i] and [R.sub.i] ; i = 1, 2, ..., m, by these relations:

[mathematical expression not reproducible],

Here, [w.sub.j] is the weight of the criterion that expressss its relative importance.

Step 3: Compute the values [Q.sub.i]: i = 1, 2, ..., m, using the following relation:

[mathematical expression not reproducible]

Here, v represents "the decision making mechanism coefficient" (or "the maximum group utility"). Here we consider v = 0.5.

Step 4: Preference ranikng order of the the alternatives is done by sorting the values of S, R and Q in decreasing order.

3. VIKOR strategy for solving MAGDM problem in NCS environment

In this section, we propose a MAGDM strategy in NCS environment. Assume that [mathematical expression not reproducible] be a set of r alternatives and [mathematical expression not reproducible] be a set of s attributes. Assume that [mathematical expression not reproducible] be the weight vector of the attributes, where [w.sub.k] [greater than or equal to] 0 and [s.summation over (k=1)] [w.sub.k] = 1. Assume that E ={[E.sub.1], [E.sub.2], [E.sub.3], ..., [E.sub.M]} be the set of M decision makers and [mathematical expression not reproducible] be the set of weight vector of decision makers, where [[zeta].sub.p] [greater than or equal to] 0 and [M.summation over p=1 [[zeta].sub.P] = 1.

The proposed MAGDM strategy consists of the following steps:

Step: 1. Construction of the decision matrix

Let [DM.sup.p] = [([a.sup.p.sub.ij]).sub.rxs] (P = 1, 2, 3, ..., t) be the p-th decision matrix, where information about the alternative [[PHI].sub.i] provided by the decision maker or expert [E.sub.p] with respect to attribute [[PSI].sub.j] (j = 1, 2, 3, ..., s). The p-th decision matrix denoted by [DM.sup.p] (See Equation (3)) is constructed as follows:

[mathematical expression not reproducible] (3)

Here p = 1, 2, 3, ..., M; i = 1, 2, 3, ..., r; j = 1, 2, 3, ..., s.

Step: 2. Normalization of the decision matrix

In decision making situation, cost type attributes and benefit type attributes play an important role to select the best alternative. Cost type attributes and benefit type attributes may exist simultaneously, so the decision matrices need to be normalized. We use Equation (2) for normalizing the cost type attributes and benefit type attributes. After normalization, the normalized decision matrix (Equation (3)) is represented as follows (see Equation 4):

[mathematical expression not reproducible] (4)

Here, p = 1, 2, 3, ..., M; i = 1, 2, 3, ..., r; j = 1, 2, 3, ..., s.

Step: 3. Aggregated decision matrix

For obtaining group decision, we aggregate all the individual decision matrices ([DM.sup.p], p = 1, 2, ..., M) to an aggregated decision matrix (DM) using the neutrosophic cubic numbers weighted aggregation (NCNWA) operator as follows:

[mathematical expression not reproducible] (5)

The NCNWA operator satisfies the following properties:

1. Idempotency

2. Monotoncity

3. Boundedness

Property: 1. Idempotency

If all [a.sup.1.sub.ij], [a.sup.2.sub.ij], ..., [a.sup.M.sub.ij] = a are equal, then [a.sub.ij] = [NCNWA.sub.[zeta]] ([a.sup.1.sub.ij], [a.sup.2.sub.ij], ..., [a.sup.M.sub.ij]) = a

Proof:

Since [a.sup.1.sub.ij] = [a.sup.2.sub.ij] = ... = [a.sup.M.sub.ij] = a, based on the Equation (5), we get

[mathematical expression not reproducible].

Property: 3. Monotonicity

Assume that {[a.sup.1.sub.ij], [a.sup.2.sub.ij], ..., [a.sup.M.sub.ij]} and {[a.sup.*1.sub.ij], [a.sup.*2.sub.ij], ..., [a.sup.*M.sub.ij]} be any two set of collections of M NC-numbers with the condition [a.sup.p.sub.ij] [less than or equal to] [a.sup.p*.sub.ij] (p = 1, 2, ..., M), then

[NCNWA.sub.[zeta]] ([a.sup.1.sub.ij], [a.sup.2.sub.ij], ..., [a.sup.M.sub.ij]) [less than or equal to] [NCNWA.sub.[zeta]] ([a.sup.*1.sub.ij], [a.sup.*2.sub.ij], ..., [a.sup.*M.sub.ij]).

Proof:

From the given condition [t.sup.-(p).sub.ij] [less than or equal to] [t.sup.- *(p).sub.ij], we have

[mathematical expression not reproducible].

From the given condition [t.sup.+(p).sub.ij]) [less than or equal to] [t.sup.+*(p).sub.ij], we have

[mathematical expression not reproducible].

From the given condition [mathematical expression not reproducible], we have

[mathematical expression not reproducible].

From the given condition [mathematical expression not reproducible], we have

[mathematical expression not reproducible].

From the given condition [mathematical expression not reproducible], we have

[mathematical expression not reproducible].

From the given condition [mathematical expression not reproducible], we have

[mathematical expression not reproducible].

From the given condition [mathematical expression not reproducible], we have

[mathematical expression not reproducible].

From the given condition [mathematical expression not reproducible], we have

[mathematical expression not reproducible].

From the given condition [t.sup.(p).sub.ij] [less than or equal to] [t.sup.*(p).sub.ij], we have

[mathematical expression not reproducible].

From the above relations, we obtain

[mathematical expression not reproducible].

Property: 2. Boundedness

Let {[a.sup.1.sub.ij], [a.sup.2.sub.ij], ..., [a.sup.1.sub.ij]} be any collection of M NC-numbers.

If

[mathematical expression not reproducible].

Then, [a.sup.-] [less than or equal to] [NCNW.sub.[zeta]] ([a.sup.1.sub.ij] [a.sup.2.sub.ij] ... [a.sup.M.sub.ij]) [less than or equal to] [a.sup.+].

Proof:

From Property 1 and Property 2, we obtain

[mathematical expression not reproducible].

So, we have

[a.sup.-] [less than or equal to] [NCNWA.sub.[zeta]] ([a.sup.1.sub.ij], [a.sup.2.sub.ij], ..., [a.sup.M.sub.ij]) [less than or equal to] [a.sup.+].

Therefore, the aggregated decision matrix is defined as follows:

[mathematical expression not reproducible] (6)

Here, i = 1, 2, 3, ..., r; j = 1, 2, 3, ... s; p =1, 2, ..., M.

Step: 4. Define the positive ideal solution and negative ideal solution

[mathematical expression not reproducible] (7)

[mathematical expression not reproducible] (8)

Step: 5. Compute and [[GAMMA].sub.i] [Z.sub.i]

[[GAMMA].sub.i] and [Z.sub.i] represent the average and worst group scores for the alternative [A.sub.i] respectively with the relations

[mathematical expression not reproducible] (9)

[mathematical expression not reproducible] (10)

Here, [w.sub.j] is the weight of [[PSI].sub.j].

The smaller values of [[GAMMA].sub.i] and [Z.sub.i] correspond to the better average and worse group scores for alternative [A.sub.i], respectively.

Step: 6. Calculate the values of [[phi].sub.i](i = 1, 2, 3, r)

[mathematical expression not reproducible] (11)

Here, [mathematical expression not reproducible] (12)

and [gamma] depicts the decision making mechanism coefficient. If [gamma] > 0.5, it is for "the maximum group utility"; If [gamma] < 0.5, it is "the minimum regret"; and it is both if [gamma] = 0.5.

Step: 7. Rank the priority of alternatives

Rank the alternatives by [[phi].sub.i], [[GAMMA].sub.i] and [Z.sub.i] according to the rule of traditional VIKOR strategy. The smaller value reflects the better alternative.

4. Illustrative example

To demonstrate the feasibility, applicability and effectiveness of the proposed strategy, we solve a MAGDM problem adapted from [51]. We assume that an investment company wants to invest a sum of money in the best option. The investment company forms a decision making board involving of three members ([E.sub.1], [E.sub.2], [E.sub.3]) who evaluate the four alternatives to invest money. The alternatives are Car company ([[PHI].sub.1]), Food company ([[PHI].sub.2]), Computer company ([[PHI].sub.3]) and Arms company ([[PHI].sub.4]). Decision makers take decision to evaluate alternatives based on the attributes namely, risk factor ([[PSI].sub.1]), growth factor ([[PSI].sub.2]), environment impact ([[PSI].sub.3]). We consider three criteria as benefit type based o n Pramanik et al. [58]. Assume that the weight vector of attributes is W = [(0.36, 0.37, 0.27).sup.T] and weight vector of decision makers or experts is [zeta] = [(0.26, 0.40, 0.34).sup.T]. Now, we apply the proposed MAGDM strategy using the following steps.

Step: 1. Construction of the decision matrix

We construct the decision matrices as follows:

Decision matrix for [DM.sup.1] in NCN form

[mathematical expression not reproducible] (13)

Decision matrix for [DM.sup.2] in NCN form

[mathematical expression not reproducible] (14)

Decision matrix for [DM.sup.3] in NC-number form

[mathematical expression not reproducible] (15)

Step: 2. Normalization of the decision matrix

Since all the criteria are considered as benefit type, we do not need to normalize the decision matrices ([DM.sup.1], [DM.sup.2], [DM.sup.3]).

Step: 3. Aggregated decision matrix

Using equation eq. (5), the aggregated decision matrix of (13, 14, 15) is presented below:

[mathematical expression not reproducible] (16)

Step: 4. Define the positive ideal solution and negative ideal solution

The positive ideal solution [mathematical expression not reproducible]

and the negative ideal solution

[mathematical expression not reproducible]

Step: 5. Compute and [[GAMMA].sub.i] [Z.sub.i]

Using Equation (9) and Equation (10), we obtain

[mathematical expression not reproducible].

And

[mathematical expression not reproducible].

Step: 6. Calculate the values of [[phi].sub.i]

Using Equations (11), (12) and [gamma] = 0.5, we obtain

[mathematical expression not reproducible]

Step: 7. Rank the priority of alternatives

The preference order of the alternatives based on the traditional rules of the VIKOR startegy is [[PHI].sub.2] > [[PHI].sub.1] > [[PHI].sub.3] > [[PHI].sub.4].

5. The influence of parameter [gamma]

Table 1 shows how the ranking order of alternatives ([[PHI].sub.i]) changes with the change of the value of [gamma]

Figure 2 represents the graphical representation of alternatives ([A.sub.i]) versus [[phi].sub.i] (i = 1, 2, 3, 4) for different values of [gamma].

6. Conclusions

In this paper, we have extended the traditional VIKOR strategy to NC-VIKOR. We introduced neutrosophic cubic numbers weighted aggregation (NCNWA) operator and applied it to aggregate the individual opinion to group opinion prove its three properties. We develpoed a novel NC-VIKOR based MAGDM strategy in neutrosophic cubic set environment. Finally, we solve a MAGDM problem to show the feasibility, applicability and efficiency of the proposed MAGDM strategy. We present a sensitivity analysis to show the impact of different values of the decision making mechanism coefficient on ranking order of the alternatives. The proposed NC-VIKOR based MAGDM strategy can be employed to solve a variety of problems such as logistics center selection [28, 74], teacher selection [75], renewable energy selection [70], fault diagnosis [71], brick selection [76, 77], weaver selection [78], etc.

References

[1] F. Smarandache. A unifying field of logics. Neutrosophy: neutrosophic probability, set and logic, American Research Press, Rehoboth, (1998).

[2] L. A. Zadeh. Fuzzy sets. Information and Control, 8 (3) (1965), 338-353.

[3] K. T. Atanassov. Intuitionistic fuzzy sets. Fuzzy Sets and Systems, 20 (1986), 87-96.

[4] H. Wang, F. Smarandache, Y. Zhang, and R. Sunderraman. Single valued neutrosophic sets. Multispace and Multi-structure, 4 (2010), 410-413.

[5] S. Pramanik, and T. K. Roy. Neutrosophic game theoretic approach to Indo-Pak conflict over Jammu-Kashmir. Neutrosophic Sets and Systems, 2 (2014), 82-101.

[6] J. Ye. Single valued neutrosophic minimum spanning tree and its clustering method. Journal of Intelligent Systems, 23 (2014), 311-324.

[7] J. Ye. Clustering methods using distance-based similarity measures of single-valued neutrosophic sets. Journal of Intelligent Systems, 23(2014), 379-389.

[8] Y. Guo, and A. Sengur. NCM: Neutrosophic c-means clustering algorithm. Pattern Recognition, 48 (8) (2015), 2710-2724.

[9] R. Sahin, Neutrosophic hierarchical clustering algoritms. Neutrosophic Sets and Systems, 2 (2014), 18-24.

[10] P. Biswas, S. Pramanik, and B. C. Giri. Entropy based grey relational analysis method for multi-attribute decision making under single valued neutrosophic assessments. Neutrosophic Sets and Systems, 2 (2014), 102-110.

[11] P. Biswas, S. Pramanik, and B. C. Giri. A new methodology for neutrosophic multi-attribute decision making with unknown weight information. Neutrosophic Sets and Systems, 3 (2014), 42-52.

[12] P. Biswas, S. Pramanik, and B. C. Giri. TOPSIS method for multi-attribute group decision-making under single valued neutrosophic environment. Neural Computing and Applications, (2015). doi: 10.1007/s00521-015-1891-2.

[13] P. Biswas, S. Pramanik, and B. C. Giri. Aggregation of triangular fuzzy neutrosophic set information and its application to multi-attribute decision making. Neutrosophic Sets and Systems, 12 (2016), 20-40.

[14] P. Biswas, S. Pramanik, and B. C. Giri. Value and ambiguity index based ranking method of single-valued trapezoidal neutrosophic numbers and its application to multi-attribute decision making. Neutrosophic Sets and Systems 12 (2016), 127-138.

[15] P. Biswas, S. Pramanik, and B. C. Giri. Multi-attribute group decision making based on expected value of neutrosophic trapezoidal numbers. New Trends in Neutrosophic Theory and Applications-Vol-II. Pons Editions, Brussells (2017). In Press.

[16] P. Biswas, S. Pramanik, and B. C. Giri. Non-linear programming approach for single-valued neutrosophic TOPSIS method. New Mathematics and Natural Computation, (2017). In Press.

[17] I. Deli, and Y. Subas. A ranking method of single valued neutrosophic numbers and its applications to multi-attribute decision making problems. International Journal of Machine Learning and Cybernetics, (2016). doi:10.1007/s13042016-0505-3.

[18] P. Ji, J. Q. Wang, and H. Y. Zhang. Frank prioritized Bonferroni mean operator with single-valued neutrosophic sets and its application in selecting third-party logistics providers. Neural Computing and Applications, (2016). doi:10.1007/s00521-016-2660-6.

[19] A. Kharal. A neutrosophic multi-criteria decision making method. New Mathematics and Natural Computation, 10 (2014), 143-162.

[20] R. X. Liang, J. Q. Wang, and L. Li. Multi-criteria group decision making method based on interdependent inputs of single valued trapezoidal neutrosophic information. Neural Computing and Applications, (2016). doi:10.1007/s00521-016-2672-2.

[21] R. X. Liang, J. Q. Wang, and H. Y. Zhang. A multi-criteria decision-making method based on single-valued trapezoidal neutrosophic preference relations with complete weight information. Neural Computing and Applications, (2017). doi: 10.1007/s00521-017-2925-8.

[22] P. Liu, Y. Chu, Y. Li, and Y. Chen. Some generalized neutrosophic number Hamacher aggregation operators and their application to group decision making. International Journal of Fuzzy System, 16 (2) (2014), 242-255.

[23] P. D. Liu, and H. G. Li. Multiple attribute decisionmaking method based on some normal neutrosophic Bonferroni mean operators. Neural Computing and Applications, 28 (2017), 179-194.

[24] P. Liu, and Y. Wang. Multiple attribute decisionmaking method based on single-valued neutrosophic normalized weighted Bonferroni mean. Neural Computing and Applications, 25(7) (2014), 2001-2010.

[25] J. J. Peng, J. Q. Wang, J. Wang, H. Y. Zhang, and X. H. Chen. Simplified neutrosophic sets and their applications in multi-criteria group decision-making problems. International Journal of Systems Science, 47 (10) (2016), 2342-2358.

[26] J. Peng, J. Wang, H. Zhang, and X. Chen. An outranking approach for multi-criteria decision-making problems with simplified neutrosophic sets. Applied Soft Computing, 25, 336-346.

[27] S. Pramanik, D. Banerjee, and B. C. Giri. Multi criteria group decision making model in neutrosophic refined set and its application. Global Journal of Engineering Science and Research Management, 3 (6) (2016), 12-18.

[28] S. Pramanik, S. Dalapati, and T. K. Roy. Logistics center location selection approach based on neutrosophic multi-criteria decision making. In F. Smarandache, & S. Pramanik (Eds), In F. Smarandache, & S. Pramanik (Eds), New trends in neutrosophic theory and applications (pp. 161-174). Brussels: Pons Editions, 2016.

[29] R. Sahin, and M. Karabacak. A multi attribute decision making method based on inclusion measure for interval neutrosophic sets. International Journal of Engineering and Applied Sciences, 2 (2) (2014), 13-15.

[30] R. Sahin, and A. Kucuk. Subsethood measure for single valued neutrosophic sets. Journal of Intelligent and Fuzzy System, (2014). doi:10.3233/IFS-141304.

[31] R. Sahin, and P. Liu. Maximizing deviation method for neutrosophic multiple attribute decision making with incomplete weight information. Neural Computing and Applications, (2015), doi: 10.1007/s00521-015-1995-8.

[32] M. Sodenkamp. Models, methods and applications of group multiple-criteria decision analysis in complex and uncertain systems. Dissertation, University of Paderborn, (2013), Germany.

[33] J. Ye. Multicriteria decision-making method using the correlation coefficient under single-valued neutrosophic environment. International Journal of General Systems, 42 (2013a), 386-394.

[34] J. Ye. Single valued neutrosophic cross-entropy for multi criteria decision making problems. Applied Mathematical Modelling, 38 (3) (2013), 1170-1175.

[35] J. Ye. A multi criteria decision-making method using aggregation operators for simplified neutrosophic sets. Journal of Intelligent and Fuzzy Systems, 26 (2014), 2459-2466.

[36] J. Ye. Trapezoidal neutrosophic set and its application to multiple attribute decision-making. Neural Computing and Applications, 26 (2015), 1157-1166.

[37] J. Ye. Bidirectional projection method for multiple attribute group decision making with neutrosophic number. Neural Computing and Applications, (2015). doi: 10.1007/s00521-015-2123-5.

[38] J. Ye. Projection and bidirectional projection measures of single valued neutrosophic sets and their decision making method for mechanical design scheme. Journal of Experimental and Theoretical Artificial Intelligence, (2016). doi:10.1080/0952813X.2016.1259263.

[39] S. Pramanik, S. Dalapati, S. Alam, F. Smarandache, and T. K. Roy. NS-cross entropy-based MAGDM under single-valued neutrosophic set environment. Information, 9(2) (2018), 37.

[40] K. Mondal, and S. Pramanik. Multi-criteria group decision making approach for teacher recruitment in higher education under simplified Neutrosophic environment. Neutrosophic Sets and Systems, 6 (2014), 28-34.

[41] K. Mondal, and S. Pramanik. Neutrosophic decision making model of school choice. Neutrosophic Sets and Systems, 7 (2015), 62-68.

[42] H. D. Cheng, and Y. Guo. A new neutrosophic approach to image thresholding. New Mathematics and Natural Computation, 4 (2008), 291-308.

[43] Y. Guo, and H. D. Cheng. New neutrosophic approach to image segmentation. Pattern Recognition, 42 (2009), 587-595.

[44] Y. Guo, A. Sengur, and J. Ye. A novel image thresholding algorithm based on neutrosophic similarity score. Measurement, 58 (2014), 175-186.

[45] M. Zhang, L. Zhang, and H. D. Cheng. A neutrosophic approach to image segmentation based on watershed method. Signal Processing, 90 (5) (2010), 1510-1517.

[46] J. Ye. Improved cosine similarity measures of simplified neutrosophic sets for medical diagnoses. Artificial Intelligence in Medicine, 63 (2015), 171-179.

[47] S. Ye, and J. Ye. Dice similarity measure between single valued neutrosophic multi sets and its application in medical diagnosis. Neutrosophic Sets and System, 6 (2014), 49-54.

[48] K. Mondal, and S. Pramanik. A study on problems of Hijras in West Bengal based on neutrosophic cognitive maps. Neutrosophic Sets and Systems, 5(2014), 21-26.

[49] S. Pramanik, and S. Chakrabarti. A study on problems of construction workers in West Bengal based on neutrosophic cognitive maps. International Journal of Innovative Research in Science. Engineering and Technology, 2(11) (2013), 6387-6394.

[50] H. Wang, F. Smarandache, Y. Q. Zhang, and R. Sunderraman. Interval Neutrosophic Sets and Logic: Theory and Applications in Computing; Hexis: Phoenix, AZ, USA, 2005.

[51] J. Ye. Similarity measures between interval neutrosophic sets and their applications in multi criteria decisionmaking. Journal of Intelligent and Fuzzy Systems, 26 (2014), 165-172.

[52] Y. B. Jun, F. Smarandache, and C. S. Kim. Neutrosophic cubic sets. New Mathematics and Natural Computation, 13 (1) (2017), 41-54.

[53] M. Ali, I. Deli, and F. Smarandache. The theory of neutrosophic cubic sets and their applications in pattern recognition. Journal of Intelligent & Fuzzy Systems, 30 (4) (2016), 1957-1963.

[54] J. Zhan, M. Khan, M. Gulistan, and A. Ali. Applications of neutrosophic cubic sets in multi-criteria decision making. International Journal for Uncertainty Quantification, 7(5) (2017), 377-394.

[55] D. Banerjee, B. C. Giri, S. Pramanik, and F. Smarandache. GRA for multi attribute decision making in neutrosophic cubic set environment. Neutrosophic Sets and Systems, 15 (2017), 60-69.

[56] Z. Lu, and J. Ye. Cosine measures of neutrosophic cubic sets for multiple attribute decision-making. Symmetry 9 (7) (2017), 121.

[57] S. Pramanik, S. Dalapati, S. Alam, and T. K. Roy, and F. Smarandache. Neutrosophic cubic MCGDM method based on similarity measure. Neutrosophic Sets and Systems, 16, (2017), 44-56.

[58] S. Pramanik, S. Dalapati, S. Alam, and T. K. Roy. NC-TODIM-Based MAGDM under a Neutrosophic Cubic Set Environment. Information 8 (4) (2017), 149.

[59] L. Shi, and J. Ye. Dombi aggregation operators of neutrosophic cubic sets for multiple attribute decisionmaking. Algorithms, 11 (3), 29, (2018). doi:10.3390/a11030029

[60] S. Pramanik, P. P. Dey, B. C. Gin, and F. Smarandache. An extended TOPSIS for multi-attribute decision making problems with neutrosophic cubic information. Neutrosophic Sets and Systems, 17 (2017), 20-28.

[61] S. Pramanik, S. Dalapati, S. Alam, and T. K. Roy. Some operations and properties of neutrosophic cubic soft set. Global Journal of Research and Review, 4 (2) (2017).

[62] J. Ye. Operations and aggregation method of neutrosophic cubic numbers for multiple attribute decision-making. https://books.google.co.in/books?id=GhRPDwAAQBA J.

[63] S. Opricovic. Multicriteria Optimization of Civil Engineering Systems; Faculty of Civil Engineering: Belgrade, Serbia, 1998.

[64] S. Opricovic, G. H. Tzeng. Compromise solution by MCDM methods: A comparative analysis of VIKOR and TOPSIS. European journal of operational research, 156 (2004), 445-455.

[65] S. Opricovic, G. H. Tzeng. Extended VIKOR method in comparison with outranking methods. European journal of operational research, 178 (2) (2007), 514-529.

[66] R. Bausys, and E. K. Zavadskas. Multicriteria decision making approach by VIKOR under interval neutrosophic set environment. Economic Computation and Economic Cybernetics Studies and Research, 4 (2015), 33-48.

[67] Y. Huang, G. Wei, and C. Wei. VIKOR method for interval neutrosophic multiple attribute group decisionmaking. Information, 8 (2017), 144. doi:10.3390/info8040144.

[68] H. Pouresmaeil, E. Shivanian, E. Khorram, and H. S. Fathabadi. An extended method using TOPSIS and VIKOR for multiple attribute decision making with multiple decision makers and single valued neutrosophic numbers. Advances and Applications in Statistics, 50(4) (2017), 261.

[69] P. Liu, L. Zhang. The extended VIKOR method for multiple criteria decision making problem based on neutrosophic hesitant fuzzy set, (2014).

[70] J. Hu, L. Pan, and X. Chen. An interval neutrosophic projection-based VIKOR method for selecting doctors. Cognitive Computation, 9(6) (2017), 801-816.

[71] K. Selvakumari, and M. A. Priyadharshini. VIKOR method for decision making problem using octagonal neutrosophic soft matrix. International Journal of Latest Engineering Research and Applications, 2(7) (2017), 41-45.

[72] J. R. San Cristobal. Multi-criteria decision-making in the selection of a renewable energy project in Spain: The VIKOR method. Renewable Energy, 36 (2011), 498-502.

[73] J. Ye. Fault diagnoses of steam turbine using the exponential similarity measure of neutrosophic numbers. Journal of Intelligent and Fuzzy System 30 (2016), 1927-1934.

[74] S. Pramanik, S, Dalapati, and T. K. Roy. Neutrosophic multi-attribute group decision making strategy for logistic center location selection. In F. Smarandache, M. A. Basset & V. Chang (Eds.), Neutrosophic Operational Research, Vol. III. Pons Asbl, Brussels, 2018, 13-32.

[75] S. Pramanik, and D. Mukhopadhyaya. Grey relational analysis based intuitionistic fuzzy multi-criteria group decision-making approach for teacher selection in higher education. International Journal of Computer Applications, 34 (2011), 21-29.

[76] K. Mondal, and S. Pramanik. Intuitionistic fuzzy multicriteria group decision making approach to quality-brick selection problem Journal of Applied Quantitative Methods, 9(2) (2014), 35-50.

[77] K. Mondal, and S. Pramanik. Neutrosophic decision making model for clay-brick selection in construction field based on grey relational analysis. Neutrosophic Sets and Systems, 9 (2015), 64-71.

[78] P. P. Dey, S. Pramanik, and B. C. Giri. Multi-criteria group decision making in intuitionistic fuzzy environment based on grey relational analysis for weaver selection in khadi institution. Journal of Applied and Quantitative Methods, 10 (4) (2015), 1-14.

[79] Abdel-Basset, M., Mohamed, M., Smarandache, F., & Chang, V. (2018). Neutrosophic Association Rule Mining Algorithm for Big Data Analysis. Symmetry, 10(4), 106.

[80] Abdel-Basset, M., & Mohamed, M. (2018). The Role of Single Valued Neutrosophic Sets and Rough Sets in Smart City: Imperfect and Incomplete Information Systems. Measurement. Volume 124, August 2018, Pages 47-55

[81] Abdel-Basset, M., Gunasekaran, M., Mohamed, M., & Smarandache, F. A novel method for solving the fully neutrosophic linear programming problems. Neural Computing and Applications, 1-11.

[82] Abdel-Basset, M., Manogaran, G., Gamal, A., & Smarandache, F. (2018). A hybrid approach of neutrosophic sets and DEMATEL method for developing supplier selection criteria. Design Automation for Embedded Systems, 1-22.

[83] Abdel-Basset. M., Mohamed, M., & Chang, V. (2018). NMCDA: A framework for evaluating cloud computing services. Future Generation Computer Systems, 86, 12-29.

[84] Abdel-Basset, M., Mohamed, M., Zhou, Y., & Hezam, I. (2017). Multi-criteria group decision making based on neutrosophic analytic hierarchy process. Journal of Intelligent & Fuzzy Systems, 33(6), 4055-4066.

[85] Abdel-Basset, M.; Mohamed, M.; Smarandache, F. An Extension of Neutrosophic AHP-SWOT Analysis for Strategic Planning and Decision-Making. Symmetry 2018, 10, 116.

Received: April 30, 2018. Accepted: May 10, 2018.

Surapati Pramanik (1), Shyamal Dalapati (2), Shariful Alam (3), Tapan Kumar Roy (4),

(1) Department of Mathematics, Nandalal Ghosh B.T. College, Panpur, P.O.- Narayanpur, District -North 24 Parganas, Pin code-743126, West Bengal, India. E-mail: sura pati@yahoo.co.in

(2) Department of Mathematics, Indian Institute of Engineering Science and Technology, Shibpur, P.O.-Botanic Garden, Howrah-711103, West Bengal, India. E-mail: dalapati shvamal30@gmail.com

(3) Department of Mathematics, Indian Institute of Engineering Science and Technology, Shibpur, P.O.-Botanic Garden, Howrah-711103, West Bengal, India. E-mail: salam50in@yahoo.co.in

(4) Department of Mathematics, Indian Institute of Engineering Science and Technology, Shibpur, P.O.-Botanic Garden, Howrah-711103, West Bengal,

Caption: Figure. 1 Decision making procedure of proposed MAGDM method

Caption: Fig 2. Graphical representation of ranking of alternatives for different values of [gamma].

Table 1. Values of [[phi].sub.i] (i = 1, 2, 3, 4) and ranking of alternatives for different values of [gamma]. Values of Values of Preference order of [gamma] [[phi].sub.i] alternatives [gamma] = 0.1 [[phi].sub.1] = [[PHI].sub.2] > 0.22, [[phi].sub.2] [[PHI].sub.1] > = 0.04, [[PHI].sub.3] > [[phi].sub.3] = [[PHI].sub.4] 0.62, [[phi].sub.4] = 1 [gamma] = 0.2 [[phi].sub.1] = [[PHI].sub.2] > 0.24, [[phi].sub.2] [[PHI].sub.1] > = 0.08, [[PHI].sub.3] > [[phi].sub.3] = [[PHI].sub.4] 0.55, [[phi].sub.4] = 1 [gamma] = 0.3 [[phi].sub.1] = [[PHI].sub.2] > 0.26, [[phi].sub.2] [[PHI].sub.1] > = 0.12, [[PHI].sub.3] > [[phi].sub.3] = [[PHI].sub.4] 0.48, [[phi].sub.4] = 1 [gamma] = 0.4 [[phi].sub.1] = [[PHI].sub.2] > 0.29, [[phi].sub.2] [[PHI].sub.1] > = 0.16, [[PHI].sub.3] > [[phi].sub.3] = [[PHI].sub.4] 0.41, [[phi].sub.4] = 1 [gamma] = 0.5 [[phi].sub.1] = [[PHI].sub.2] > 0.31, [[phi].sub.2] [[PHI].sub.1] > = 0.2, [[phi].sub.3] [[PHI].sub.3] > = 0.34, [[PHI].sub.4] [[phi].sub.4] = 1 [gamma] = 0.6 [[phi].sub.1] = [[PHI].sub.2] > 0.34, [[phi].sub.2] [[PHI].sub.3] > = 0.24, [[PHI].sub.1] > [[phi].sub.3] = [[PHI].sub.4] 0.28, [[phi].sub.4] = 1 [gamma] = 0.7 [[phi].sub.1] = [[PHI].sub.3] > 0.36, [[phi].sub.2] [[PHI].sub.2] > = 0.28, [[PHI].sub.1] > [[phi].sub.3] = [[PHI].sub.4] 0.21, [[phi].sub.4] = 1 [gamma] = 0.8 [[phi].sub.1] = [[PHI].sub.3] > 0.39, [[phi].sub.2] [[PHI].sub.2] > = 0.32, [[PHI].sub.1] > [[phi].sub.3] = [[PHI].sub.4] 0.14, [[phi].sub.4] = 1 [gamma] = 0.9 [[phi].sub.1] = [[PHI].sub.3] > 0.42, [[phi].sub.2] [[PHI].sub.2] > = 0.36, [[PHI].sub.1] > [[phi].sub.3] = [[PHI].sub.4] 0.07, [[phi].sub.4] = 1

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Author: | Pramanik, Surapati; Dalapati, Shyamal; Alam, Shariful; Roy, Tapan Kumar |
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Publication: | Neutrosophic Sets and Systems |

Article Type: | Report |

Date: | Jun 1, 2018 |

Words: | 6812 |

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