VIKOR Method for Interval Neutrosophic Multiple Attribute Group Decision-Making

In this paper, we will extend the VIKOR (VIsekriterijumska optimizacija i KOmpromisno Resenje) method to multiple attribute group decision-making (MAGDM) with interval neutrosophic numbers (INNs). Firstly, the basic concepts of INNs are briefly presented. The method first aggregates all individual decision-makers’ assessment information based on an interval neutrosophic weighted averaging (INWA) operator, and then employs the extended classical VIKOR method to solve MAGDM problems with INNs. The validity and stability of this method are verified by example analysis and sensitivity analysis, and its superiority is illustrated by a comparison with the existing methods.


Introduction
Multiple attribute group decision-making (MAGDM), which has been increasingly investigated and considered by all kinds of researchers and scholars, is one of the most influential parts of decision theory. It aims to provide a comprehensive solution by evaluating and ranking alternatives based on conflicting attributes with respect to decision-makers' (DMs) preferences, and has widely been utilized in engineering, economics, and management. Several traditional MAGDM methods have been developed by scholars in literature, such as the TOPSIS (Technique for Order Preference by Similarity to an Ideal Solution) method [1,2], the VIKOR (VIsekriterijumska optimizacija i KOmpromisno Resenje) method [3][4][5], the PROMETHEE (Preference Ranking Organization Method for Enrichment Evaluations) method [6], the ELECTRE (ELimination Et Choix Traduisant la Realité) method [7], the GRA (Grey Relational Analysis) method [8][9][10], and the MULTIMOORA (Multiobjective Optimization by Ratio Analysis plus Full Multiplicative Form) method [11,12].
Definition 1 [21]. Let X be a space of points (objects), a SVNSs A in X is characterized as following: where the truth-membership function ξ A (x), indeterminacy-membership ψ A (x) and falsity-membership function Definition 2 [22]. Let X be a space of points (objects) with a generic element in fixed set X, denoted by x, where an INS A in X is characterized as follows: where truth-membership function ξ A (x), indeterminacy-membership ψ A (x), and falsity-membership function

An INN can be expressed as
Definition 3 [45].
be an INN, then a score function, SF, is: Definition 4 [45].
be an INN, then an accuracy function, AF A , is defined as: Definition 5 [45].
be two INNs, then: Definition 7 [45]. Let A and B be two INNs, then the normalized Hamming distance between A and B is defined as follows:
Utilize the R k and the interval neutrosophic number weighted averaging (INNWA) operator to get R = r ij m×n .
Step 2. Define the positive ideal solutions R + and negative ideal solutions R − .
For the benefit attribute: For the cost attribute: Step 3. Compute the Γ i and Z i .
where τ j is weight of ϕ j .
Step 4. Compute the Θ i by the following formula: where where θ depicts the decision-making mechanism coefficient. If θ > 0.5, it is for "the maximum group utility"; If θ < 0.5, it is "the minimum regret"; and it is both if θ = 0.5.
Step 5. Rank the alternatives by Θ i , Γ i and Z i according to the selection rule of the traditional VIKOR method.

Numerical Example
In this section, a numerical example is given with INNs. Five possible emerging technology enterprises (ETEs) φ i (i = 1, 2, 3, 4, 5) are selected. Four attributes are selected to evaluate the five possible ETEs: 1 ϕ 1 is the employment creation; 2 ϕ 2 is the development of science and technology;    Then, we use the proposed model to select the best ETE.

Comparative Analysis
In what follows, we compare with the interval neutrosophic number weighted averaging (INNWA) operator and interval neutrosophic number weighted geometric (INNWG) operator [28], INN similarity [33], and INN VIKOR [55]. The results are shown in Table 5.
From the above analysis, it can be seen that the five methods have the same best emerging technology enterprise φ 3 , and the ranking results of Method 1 and Method 2 are slightly different. The proposed INN VIKOR method can reasonably focus a MAGDM problem with INNs. At the same time, compared with Method 5 based on the INN VIKOR method in Reference [55], our proposed method avoids the interval numbers' comparison.

Conclusions
The VIKOR method for a MAGDM presents some conflicting attributes. We extended the VIKOR method to MAGDM with INNs. Firstly, the basic concepts of INNs were briefly presented. The method first aggregates all individual decision-makers' assessment information based on an INNWA operator, and then employs the extended classical VIKOR method for MAGDM problems with INNs. The validity and stability of this method were verified by example analysis and comparative analysis, and its superiority was illustrated by a comparison with the existing methods. In the future, many other methods of INSs need to be explored in for MAGDM, risk analysis, and many other uncertain and fuzzy environments .