Distance measures between interval complex neutrosophic sets and their applications in multi-criteria group decision making

Abstract: As an extension of neutrosophic set, interval complex neutrosophic set is a new research topic in the field of neutrosophic set theory, which can handle the uncertain, inconsistent and incomplete information in periodic data. Distance measure is an important tool to solve some problems in engineering and science. Hence, this paper presents some interval complex neutrosophic distance measures to deal with multi-criteria group decision-making problems. Firstly, this paper shows the definitions of interval complex neutrosophic set, and especially some novel set theoretic properties. Then, some new distance measures based on Hamming, Euclidean and Hausdorff metrics are proposed. Next, an approach is developed to rank the alternatives in multi-criteria group decision-making problems. Finally, a numerical example is given to demonstrate the practicality and effectiveness of these distance measures.


Introduction
Multi-criteria group decision-making (MCGDM) is the process of ranking a series of alternatives and find the optimal one from them. During the last decades, most decision makers tend to evaluate the rating values of each criterion with crisp number. However, due to the fuzziness of human thinking and the complexity and uncertainty of objective things in real life, the information in MCGDM problem is either vague, imprecise or uncertain [1]. To deal with it, the theory of fuzzy set (FS) [2], intuitionistic fuzzy set (IFS) [3] and interval-valued IFS (IVIFS) [4], which can express the evaluation values more reasonably, were proposed. Based on these theories, some decision making (DM) methodologies have been presented and applied in various disciplines. For instance, Xu [5,6] presented some aggregation operators to aggregation information such as geometric aggregation operator and weighted averaging The purpose of this paper is to construct some interval complex neutrosophic distance measures and apply them into MCGDM problem. The specific arrangements of this article are structured as follows. In section 2, we introduce the concept of ICNS. Section 3 proposes some set theoretic properties of ICNS, such as operational rules, aggregated operator and comparison method. In section 4, we present some distance measures which are satisfied with the axiomatic conditions. A MCGDM approach based on the operator and distance measure is proposed in section 5. In section 6, this paper illustrates the practicality and effectiveness of the proposed approach via a numerical example. In section 7, a conclusion of this paper is given. There is no restriction on the sum of T A (x), I A (x), and F A (x), so 0 − ≤ sup T A (x) + sup

Neutrosophic set
Definition 2.2. [7] The complement of A is denoted by A C and is defined as for every x in X.
for every x in X.
[18] Let X be a space of points(objects) with generic elements in X denoted by x. An ICNS S in X is characterized by a truth-membership function T S (x), an indeterminacy-membership function I S (x), and a falsity-membership function F S (x),which are satisfied the following conditions: where Γ [0,1] is the collection of interval neutrosophic sets and R is the set of real numbers, t S (x) is the interval truth membership function, i S (x) is the interval indeterminate membership function and f S (x) is the interval falsehood membership function, while e jω S (x) , e jψ S (x) and e jφ S (x) are the corresponding interval-valued phase terms, respectively, with j = √ −1. In set theoretic form, an interval complex neutrosophic set can be written as: In (2.1), the amplitude interval-valued terms T S (x), Similarly, for the phase terms: The complement of an ICNS S is denoted by S c and is defined as

Set theoretic properties of interval complex neutrosohic set
Definition 3.1. Let A and B be two ICNSs, then the operational rules of ICNS are defined as follows: (1) The addition of A and B, denoted as A + B, and is defined as: the addition of phase terms is defined below: (2) The product of A and B, denoted as A × B, and is defined as: the product of phase terms is defined below: (3) The scalar multiplication of A is an interval complex neutrosophic set denoted as C = λA, λ > 0, and defined as: The scalar multiplication of phase terms is defined below: The power of A is denoted as D = A λ , λ > 0 and defined as: The power of phase terms is defined below: n) be a collection of interval complex neutrosophic numbers (ICNNs), the interval complex neutrosophic weighted averaging (ICNWA) operator can be defined as follows: ICNWA w (a 1 , a 2 , · · · , a n ) = w 1 a 1 + w 2 a 2 + · · · + w n a n where w = (w 1 , w 2 , · · · , w n ) T is the weight vector of a k , such that 0 < w k < 1, n k=1 w k = 1. Then the ICNWA operator is denoted as following: Especially when the weight vector is w = 1 n , 1 n , · · · , 1 n , the ICNWA operator will reduce to interval complex neutrosophic average (ICNA) operator. Definition 3.3. Let A be an ICNN, then the score function S (A) of A is defined as: Definition 3.4. Let A be an ICNN, then the accuracy function H (A) of A is defined as: Definition 3.5. Let A 1 and A 2 be two ICNNs, and S be the score functions, H be the accuracy
[24] Let X = {x 1 , x 2 , · · · , x n } be the universe of discourse, Φ (X) be the family of ICNSs and R + be the set of non-negative real numbers. A distance measure of interva complex neutrosophic set is a function d : Φ (X) × Φ (X) → R + , which satisfies the following three conditions: for any A, B, C ∈ Φ (X), . Now, we define the some distance measures with interval complex neutrosophic sets as follows: for any two ICNSs A and B, Definition 4.2. The normalized Hamming distance: The normalized Euclidean distance: Definition 4.4. The normalized Hausdorff distance: Definition 4.5. The normalized generalized distance: with λ > 0.
So we can consider d Hm (A, B) as a distance measure. Analogously, the normalized Euclidean distance d E (A, B), normalized Hausdorff distance d Hd (A, B) and normalized generalized distance d G (A, B) be proved as valid distance measures.

An approach for MCGDM
In this section, a MCGDM approach is presented by using the operational rules and above-defined distance measures for ICNSs.
Assume that a committee of l decision makers (D h , h = 1, 2, · · · , l) is responsible for evaluating m alternatives (A p , p = 1, 2, · · · , m) under n selection criteria (C q , q = 1, 2, · · · , n), where the performance ratings are ICNNs. The weight vector of the creteria is w q (q = 1, 2, · · · , n) which satisfies 0 < w q < 1 and n q=1 w q = 1. Then the steps of the proposed MAGDM method as follows: Step 1: Aggregate the ratings of alternatives versus criteria. Let x hpq = T L hpq , T U hpq , I L hpq , I U hpq , F L hpq , F U hpq , h = 1, 2, · · · , l; p = 1, 2, · · · , m; q = 1, 2, · · · , n be the performance ratings evaluated by decision maker D h for alternative A p versus criterion C q , then its decision-making matrix can be denoted as M (h) = x hpq m×n . By using the ICNA operator, we can get the aggregating decision making matrix M = x pq m×n = T pq , I pq , F pq m×n with aggregation ratings, where x pq is defined as follows: Step 2: Calculating the weighted decision-making matrix R according to the criteria weights. where r pq = w q * x pq and Step 3: Determination of the positive ideal solution (PIS) and negative ideal solution (NIS).
Step 4: Calculating the distance between each alternative and PIS, NIS.
where d r pq , R + q and d r pq , R − q are defined in section 3.
Step 5: Calculating the closeness coefficients of alternatives.
Step 6: Ranking the alternatives. The larger value of closeness coefficients CC p , the better alternative A p is.

Numerical example
In this section, we applies the proposed MCGDM method for green supplier selection. The managers would like to manage the suppliers effectively, due to an increasing number of them [18]. Data were collected by conducting semi-structured interviews with managers and department heads. Three managers (decision-makers), i.e., D 1 − D 3 , were requested to separately proceed to their own evaluation. Five criteria, namely Price/cost (C 1 ), Quality (C 2 ), Delivery (C 3 Table 3. The Performance Rating From Decision maker D 3 .
Then the complete MCGDM procedure is characterized by the following steps: Step 1: Aggregated performance ratings. According to Eq (5.1), the aggregated ratings of three suppliers versus five criteria from three decision makers are given in Table 4. Step 2: Calculating the weighted decision making matrix R = r pq m×n . According to the attribute weight w = {0.2, 0.3, 0.25, 0.15, 0.1} and Eq (5.2), the weighted decision making matrix is obtained and shown in Table 5. Step 5: Calculating the distance measures between alternatives and PIS, NIS. We compute the distance measures by Eqs (5.5) and (5.6) which are shown in Table 6.  Step 6: Computing the closeness coefficients. According to Eq (5.7), the closeness coefficients are obtained and shown in Table 7. From these results, it is obvious that the ranking of three suppliers is A 3 > A 2 > A 1 , and the optimal supplier is A 3 .
By comparing with the method proposed in [18], the biggest difference is the criteria weights which are the interval complex neutrosophic numbers in [18] whereas real numbers in our paper, but one similarity is that the weights are satisfied that w 2 > w 3 > w 1 > w 4 > w 5 , so the ranking result and optimal supplier are in the same way which can show our approach is practical and effective.

Conclusions
It is obvious that interval complex neutrosophic set is a useful tool for dealing with the uncertain, inconsistent and incomplete information in periodic data. The aim of this paper is to introduce some interval complex neutrosophic distance measures and apply them into MCGDM problems. Hence, based on the Hamming, Euclidean, Hausdorff metrics, we present some distance measures for ICNSs, and an approach is developed to handle the MCGDM problems. At the beginning of this article, we briefly introduce some definitions and set theoretic properties of ICNS. Next, in order to obtain the best alternative(s), we propose an approach based on some distance measures for MCGDM problems. Finally, we illustrate the application of the proposed method thorough a numerical example. From the result we can see the practicality and effectiveness of this method.
As further work, we may develop more information measures and techniques for decision-making problems under interval complex neutrosophic environment and apply them into different fields, such as venture capital, pattern recognition and comprehensive evaluation.