MADM strategy based on some similarity measures in interval bipolar neutrosophic set environment

The paper investigates some similarity measures in interval bipolar neutrosophic environment for multi-attribute decision making problems. At first, we define Hamming and Euclidean distances measures between interval bipolar neutrosophic sets and establish their basic properties. We also propose two similarity measures based on the Hamming and Euclidean distance functions. Using maximum and minimum operators, we define new similarity measures and prove their basic properties. Using the proposed similarity measures, we propose a novel multi attribute decision making strategy in interval bipolar neutrosophic set environment. Lastly, we solve an illustrative example of multi attribute decision making and present comparison analysis to show the feasibility, applicability and effectiveness of the proposed strategy.


Introduction
In order to deal with problems involving indefinite and uncertain information, Zhang [1,2] proposed the bipolar fuzzy sets (BFSs) that utilize the idea of positive and negative preferences of information where the value of membership grade of a component of BFS belongs to [-1, 1].Later, Ezhilmaran and Shankar [3] proposed the bipolar intuitionistic fuzzy sets (BIFSs).Deli et al. [4] defined the bipolar neutrosophic sets (BNSs) as the generalization of fuzzy sets [5], BFSs [1,2], intuitionistic fuzzy sets [6], BIFSs [3] and neutrosophic sets [7], etc.In the same research, Deli et al. [4] presented some operations including the score, accuracy, and certainty functions and two weighted operators to develop a bipolar neutrosophic multi-criteria decision making (MCDM) method.Dey et al. [8] investigated a technique for order preference by similarity to ideal solution (TOPSIS) strategy for multi-attribute decision making (MADM) problems in bipolar neutrosophic setting to obtain the best alternative.
Similarity measure is an important decision making device for many MADM problems such as investment [9,10], pattern recognition [11], supplier selection, construction project selection [12], etc. Deli and Subas [13] and Sahin et al. [14] presented MCDM strategy using correlation coefficient and Jaccard vector similarity measures, respectively.Ulucay et al. [15] proposed four similarity measures namely, Dice, weighted Dice, hybrid and weighted hybrid similarity measure of BNSs and applied them to solve MADM problems.Pramanik et al. [9] presented several projections measures such as bidirectional and hybrid projection measures of BNSs and established their basic properties.In the same research, Pramanik et al. [9], developed three novel algorithms for MADM problems in BNS environment.
Deli et al. [16] and Mahmood et al. [17] grounded the concept of interval bipolar neutrosophic sets (IBNSs) by extending the idea of BNSs [4] and interval neutrosophic sets (INSs) [18].In same paper, Deli et al. [16] defined score, accuracy and certainty functions of IBNS and IBNS weighted average operator and IBNS weighted geometric operator to aggregate the IBNSs and finally, presented a MCDM algorithm.Mahmood et al. [17] defined the union, complement, intersection, and containment of IBNSs and defined some aggregation operators to establish a MADM strategy for IBNSs.In 2018, Pramanik et al. [19] proposed weighted cross entropy measures for bipolar and interval bipolar neutrosophic sets and established their basic properties.Using the weighted cross entropy measures, Pramanik et al. [19] also developed two new MADM strategies.Recently, Pramanik et al. [20] defined correlation coefficient and weighted correlation coefficient measures of IBNSs and established their basic properties and developed a novel MADM strategy with interval bipolar neutrosophic information.

Research gap:
MADM strategy based on some new similarity measures in interval bipolar neutrosophic set environment.

Motivation:
The above-mentioned analysis presents the motivation behind proposing new similarity measure based MADM strategies for MADM problems in IBNS environment.
The objectives of the paper are as follows: 1.To define Hamming distance and Euclidean distance measures and prove their basic properties.
2. To introduce a new similarity measure and prove its basic properties.3. To establish three new MADM strategies based on Hamming distance measure, Euclidean distance measure and a similarity measure using on minimum and maximum operators in IBNS environment.
To deal with the research gap, we propose novel similarity measures for solving MADM problems with interval bipolar neutrosophic information.
The remaining of the article is organized as follows.Section 2 presents some concepts concerned with BFSs, BIFSs, BNSs, and IBNSs.Section 3 introduces new similarity measures based on Hamming and Euclidean distances.Section 3 also introduces new similarity measures between two IBNSs on the basis of maximum and minimum operators and establishes their properties.
Section 4 develops three novel MADM strategies in IBNS environment.Section 5 presents algorithmic representation of the developed three MADM strategies in interval bipolar neutrosophic set environment.Section 6 demonstrates an illustrative example to show the applicability and effectiveness of the proposed strategy and then comparison analysis is presented.Finally, some remarks and future scope of research are provided in the concluding section.

Preliminary
In this section, we recall several basic definitions and properties of BFSs, BIFSs, BNSs, IBNSs respectively, which are useful for the presentation of the paper.

Bipolar fuzzy sets
A BFS [1,2] F is characterized by a positive membership function ) (x

Bipolar intuitionistic fuzzy sets
Assume that X is a non-empty set, then a BIFS [3] G in X is expressed as:

Bipolar neutrosophic set
A BNS [4] H in X is presented as follows: : X  [0, 1] denote the truth, indeterminate, and falsity membership functions respectively of an object x X corresponding to H.
signify the truth, indeterminate, and falsity membership functions of an object x X to some implicit opposite property associated with H.

Interval bipolar neutrosophic set
An IBNS [16,17] K in X is represented as follows: where K is characterized by positive and negative truth- Definition 1. Ref. [16,17]: Consider L and M be two IBNSs in the universe of discourse X.
for all x X. Definition 2. Ref. [16,17]: Suppose L and M are two IBNSs defined over X.Then we say Definition 3. Ref. [18]: The complement of an
Again, we have, It is clear that the bigger the value of SMj (L, M) (j = 1, 2) represents L is more similar to M.
Theorem 2. The interval bipolar neutrosophic similarity measure Q1 (L, M) should satisfy the following properties: for every x X.Using these inequalities, we have the following similarity measures: we can obtain that Q1 (L, N)  Q1 (L, M).
Similarly, we have However, if we consider the importance differences in the independent components (i.e., positive and negative truth-membership ) in a IBNS, we can input the weights of the independent components in Eq. ( 9).Therefore, we define another similarity measure between L and M as follows:  are the weights of the independent components in a IBNS and , Eq (10) reduces to Eq. ( 9).
Furthermore, if we consider the importance dissimilarities in X = {x1, x2, ..., xn}, we require to consider the weight of every point xi (i = 1, 2, ..., n).Therefore, we construct another similarity measure between L and M as given below.

Proof
The properties of the Theorem 3 can be established in the same way as the Theorem 2. If the weight values of the independent elements in a IBNS are   = 0.05,   = 0.05,   = 0.07,

Interval bipolar multi-attribute decision making strategies based on the proposed similarity measures
In this section, we present three novel similarity measures for MADM problem in interval bipolar neutrosophic environment.Consider Y = {Y1, Y2, …, Ym}, (m  2) be a discrete set of m feasible alternatives, Z = {Z1, Z2, …, Zn}, (n  2) be a set of attributes under consideration and wj be the weight vector of the attributes such that 0  wj  1 and   n w 1 j j = 1.Three MADM strategies are presented in compact form as follows.
Step 1.The decision maker assigns the rating of performance value of alternative Yi (i = 1, 2, …, m) with respect to the predefined attribute Zj (j = 1, 2, …, n) in terms of interval bipolar neutrosophic  Step 2. The interval bipolar neutrosophic positive ideal solution (IBN-PIS) can be defined by utilizing a maximum operator for the benefit attribute and a minimum operator for the cost attribute as follows: where H1 and H2 are benefit and cost type attributes, respectively.
Step 3. In the step, we propose three similarity measures between an alternative Yi , i = 1, 2, …, m and the ideal solution Y + as given below.
Step 4. Using the weighted similarity measures SM1 (Yi, Y + ), SM2 (Yi, Y + ) and Q (Yi, Y + ) the ranking order of the alternatives is obtained and the best alternative is selected.The bigger value of SM1 (Yi, Y + ) reflects the better alternative.Similar result holds for SM2 (Yi, Y + ) and Q (Yi, Y + ).

Algorithmic representation of the proposed MADM strategies in interval bipolar neutrosophic environment
We present three MADM strategies in algorithmic form using the following steps: Step 1.The decision maker provides the interval bipolar neutrosophic decision matrix Step 2. IBN-PIS is computed from the specified interval bipolar neutrosophic decision Step 3. The weighted similarity measures SM1 (Yi, Y + ), SM2 (Yi, Y + ) and Q (Yi, Y + ) between each alternative Yi , i = 1, 2, …, m and the ideal solution Y + are computed respectively, using Eqs.( 12), ( 13) and ( 14).ii.Based on SM2 (Yi, Y + ), the highest value of SM2 (Yi, Y + ) reflects that Yi, i = 1, 2, …, m is the best alternative.
iii.Based on Q (Yi, Y + ), the highest value of Q (Yi, Y + ) signifies that Yi, i = 1, 2, …, m is the best alternative.
Step 5. Choose the most desirable alternative.Step 6. Stop.
Representation of the proposed strategy is shown in Figure 1.

Illustrative numerical example
Consider a decision making problem discussed in [17,19] where there are four possible alternatives to invest money namely: a food company (Y1), a car company (Y2), a arm company (Y3), and a computer company (Y4).The investment company must take a decision based on the three predefined attributes namely: growth analysis (Z1), risk analysis (Z2), and environment analysis (Z3) where Z1, Z2 are benefit type and Z3 is cost type attributes [21].Suppose the weight vector of Z1, Z2, and Z3 is given by w = (w1, w2, w3) = (0.35, 0.25, 0.4) [17].Now the decision making steps are shown as given below.
Step 5. Thereby, the computer company (Y4) is the most suitable option to invest money.studied weight cross entropy measure to find the best option.It is to be observed that proposed strategies obtained the same ranking order as the method discussed by Mahmood et al. [17].
However, Pramanik et al. [19] obtained different ranking order.We present the comparison of the results obtained from the different strategies in Table 1.


where xX.A BFS F is expressed as follows:

Definition 5 :
Let L and M be two IBNSs in X = {x1, x2, …, xn}, then we propose two similarity measures between L and M based on the defined Hamming and Euclidean distance measures as given below.

Figure 1 .
Figure 1.Conceptual representation of the proposed strategies