Interval Complex Neutrosophic Set: Formulation and Applications in Decision-Making

Neutrosophic set is a powerful general formal framework which generalizes the concepts of classic set, fuzzy set, interval-valued fuzzy set, intuitionistic fuzzy set, etc. Recent studies have developed systems with complex fuzzy sets, for better designing and modeling real-life applications. The single-valued complex neutrosophic set, which is an extended form of the single-valued complex fuzzy set and of the single-valued complex intuitionistic fuzzy set, presents difficulties to defining a crisp neutrosophic membership degree as in the single-valued neutrosophic set. Therefore, in this paper we propose a new notion, called interval complex neutrosophic set (ICNS), and examine its characteristics. Firstly, we define several set theoretic operations of ICNS, such as union, intersection and complement, and afterward the operational rules. Next, a decision-making procedure in ICNS and its applications to a green supplier selection are investigated. Numerical examples based on real dataset of Thuan Yen JSC, which is a small-size trading service and transportation company, illustrate the efficiency and the applicability of our approach.


Introduction
Smarandache [12] introduced the Neutrosophic Set (NS) as a generalization of classical set, fuzzy set, and intuitionistic fuzzy set.The neutrosophic set handles indeterminate data, whereas the fuzzy set and the intuitionistic fuzzy set fail to work when the relations are indeterminate.Neutrosophic set has been successfully applied in different fields, including decision-making problems [2, 5-8, 11, 14-16, 19-24, 27, 28].Since the neutrosophic set is difficult to be directly used in real-life applications, Smarandache [12] and Wang et al. [18] proposed the concept of single-valued neutrosophic set and provided its theoretic operations and properties.Nonetheless, in many real-life problems, the degrees of truth, falsehood, and indeterminacy of a certain statement may be suitably presented by interval forms, instead of real numbers [17].To deal with this situation, Wang et al. [17] proposed the concept of Interval Neutrosophic Set (INS), which is characterized by the degrees of truth, falsehood and indeterminacy, whose values are intervals rather than real numbers.Ye [19] presented the Hamming and Euclidean distances between INSs and the similarity measures between INSs based on the distances.Tian et al. [16] developed a multi-criteria decision-making (MCDM) method based on a cross-entropy with INSs [3,10,19,25].
Recent studies in NS and INS have concentrated on developing systems using complex fuzzy sets [9,10,26] for better designing and modeling real-life applications.The functionality of 'complex' is for handling the information of uncertainty and periodicity simultaneously.By adding complex-valued non-membership grade to the definition of complex fuzzy set, Salleh [13] introduced the concept of complex intuitionistic fuzzy set.Ali and Smarandache [1] proposed a complex neutrosophic set (CNS), which is an extension form of complex fuzzy set and of complex intuitionistic fuzzy set.The complex neutrosophic set can handle the redundant nature of uncertainty, incompleteness, indeterminacy, inconsistency, etc., in periodic data.The advantage of CNS over the NS is the fact that, in addition to the membership degree provided by the NS and represented in the CNS by amplitude, the CNS also provides the phase, which is an attribute degree characterizing the amplitude.
Yet, in many real-life applications, it is not easy to find a crisp (exact) neutrosophic membership degree (as in the single-valued neutrosophic set), since we deal with unclear and vague information.To overcome this, we must create a new notion, which uses an interval neutrosophic membership degree.This paper aims to introduce a new concept of Interval-Valued Complex Neutrosophic Set or shortly Interval Complex Neutrosophic Set (ICNS), that is more flexible and adaptable to real-life applications than those of SVCNS and INS, due to the fact that many applications require elements to be represented by a more accurate form, such as in the decision-making problems [4,7,16,17,20,25].For example, in the green supplier selection, the linguistic rating set should be encoded by ICNS rather than by INS or by SVCNS, to reflect the hesitancy and indeterminacy of the decision.This paper is the first attempt to define and use the ICNS in decision-making.The contributions and the tidings of this paper are highlighted as follows: First, we define the Interval Complex Neutrosophic Set (Sect.3.1).Next, we define some set theoretic operations, such as union, intersection and complement (Sect.3.2).Further, we establish the operational rules of ICNS (Sect.3.3).Then, we aggregate ratings of alternatives versus criteria, aggregate the importance weights, aggregate the weighted ratings of alternatives versus criteria, and define a score function to rank the alternatives.Last, a decision-making procedure in ICNS and an application to a green supplier selection are presented (Sects.4,5).
Green supplier selection is a well-known application of decision-making.One of the most important issues in supply chain to make the company operation efficient is the selection of appropriate suppliers.Due to the concerns over the changes in world climate, green supplier selection is considered as a key element for companies to contribute toward the world environment protection, as well as to maintain their competitive advantages in the global market.In order to select the appropriate green supplier, many potential economic and environmental criteria should be taken into consideration in the selection procedure.Therefore, green supplier selection can be regarded as a multi-criteria decision-making (MCDM) problem.However, the majority of criteria is generally evaluated by personal judgement and thus might suffer from subjectivity.In this situation, ICNS can better express this kind of information.
The advantages of the proposal over other possibilities are highlighted as follows: (a) The complex neutrosophic set is a generalization of interval complex fuzzy set, interval complex intuitionistic fuzzy sets, single-valued complex neutrosophic set and so on.For more detail, we refer to Fig. 1 in Sect.3.1.(b) In many real-life applications, it is not easy to find a crisp (exact) neutrosophic membership degree (as in the single-valued neutrosophic set), since we deal with unclear and vague periodic information.To overcome this, the complex interval neutrosophic set is a better representation.(c) In order to select the appropriate green supplier, many potential economic and environmental criteria should be taken into consideration in the selection procedure.Therefore, green supplier selection can be regarded as a multi-criteria decision-making (MCDM) problem.However, the majority of criteria are generally evaluated by personal judgment, and thus, it might suffer from subjectivity.In this  The rest of this paper is organized as follows.Section 2 recalls some basic concepts of neutrosophic set, interval neutrosophic set, complex neutrosophic set, and their operations.Section 3 presents the formulation of the interval complex neutrosophic set and its operations.Section 4 proposes a multi-criteria group decision-making model in ICNS.Section 5 demonstrates a numerical example of the procedure for green supplier selection on a real dataset.Section 6 delineates conclusions and suggests further studies.

Basic Concepts
Definition 1 [12] Neutrosophic set (NS) Let X be a space of points and let x 2 X.A neutrosophic set S in X is characterized by a truth membership function T S , an indeterminacy membership function I S , and a falsehood membership function F S .T S , I S and F S are real standard or non-standard subsets of 0 À ; 1 þ ½.To use neutrosophic set in some real-life applications, such as engineering and scientific problems, it is necessary to consider the interval 0; 1 ½ instead of 0 À ; 1 þ ½, for technical applications.The neutrosophic set can be represented as: where one has that 0 sup ð Þ 3, and T S , I S and F S are subsets of the unit interval [0, 1].
Definition 2 [9,10] Complex fuzzy set (CFS) A complex fuzzy set S, defined on a universe of discourse X, is characterized by a membership function g S x ð Þ that assigns to any element x 2 X a complex-valued grade of membership in S. The values g S x ð Þ lie within the unit circle in the complex plane, and thus, all forms p S x ð Þ Á e jÁl S ðxÞ where p S x ð Þ and l S x ð Þ are both real-valued and p S x ð Þ 2 0; 1 ½ .The term p S x ð Þ is termed as amplitude term, and e jÁl S ðxÞ is termed as phase term.The complex fuzzy set can be represented as: Definition 3 [13] Complex intuitionistic fuzzy set (CIFS) A complex intuitionistic fuzzy set S, defined on a universe of discourse X, is characterized by a membership function g S x ð Þ and a non-membership function f S x ð Þ, respectively, assigning to an element x 2 X a complex-valued grade to both membership and non-membership in S. The values of g S x ð Þ and f S x ð Þ lie within the unit circle in the complex plane and are of the form g S x ð Þ ¼ p S x ð Þ Á e jÁl S ðxÞ and are all real-valued and p S x ð Þ, r S x ð Þ 2 0; 1 ½ with j ¼ ffiffiffiffiffiffi ffi À1 p .The complex intuitionistic fuzzy set can be represented as: Definition 4 [4] Interval-valued complex fuzzy set (IVCFS) An interval-valued complex fuzzy set A is defined over a universe of discourse X by a membership function In the above equation, C 0;1 ½ is the collection of interval fuzzy sets and R is the set of real numbers.r S x ð Þ is the interval-valued membership function while e jx A x ð Þ is the phase term, with j ¼ ffiffiffiffiffiffi ffi À1 p .
Definition 5 [1] Single-valued complex neutrosophic set (SVCNS) A single-valued complex neutrosophic set S, defined on a universe of discourse X, is expressed by a truth membership function T S ðxÞ, an indeterminacy membership function I S ðxÞ and a falsity membership function F S ðxÞ, assigning a complex-valued grade of T S ðxÞ, I S ðxÞ and F S ðxÞ in S for any x 2 X.The values T S ðxÞ, I S ðxÞ, F S ðxÞ and their sum may all be within the unit circle in the complex plane, and so it is of the following form: T S ðxÞ ¼ p S ðxÞ Á e jl S ðxÞ ; I S ðxÞ ¼ q S ðxÞ Á e jm S ðxÞ and F S ðxÞ ¼ r S ðxÞ Á e jx S ðxÞ ; where p S ðxÞ, q S ðxÞ, r S ðxÞ and l S ðxÞ, m S ðxÞ, x S ðxÞ are, respectively, real values and p S ðxÞ; q S ðxÞ; r S ðxÞ 2 ½0; 1, such that 0 p S ðxÞ þ q S ðxÞ þ r S ðxÞ 3. The single-valued complex neutrosophic set S can be represented in set form as: be a single-valued complex neutrosophic set in X.Then, the complement of a SVCNS S is denoted as S c and is defined by: where Finally, Union of single-valued complex neutrosophic sets

Let
A and B be two SVCNSs in X. Then: where ; ; where _ and ^denote the max and min operators, respectively.To calculate the phase terms e jÁl A[B ðxÞ , e jÁm A[B ðxÞ and e jÁx A[B ðxÞ , we refer to [1].

Definition 8 [1] Intersection of single-valued complex neutrosophic sets
Let A and B be two SVCNSs in X. Then: where ðxÞ ; where _ and ^denote the max and min operators, respectively.To calculate the phase terms e jÁl A[B ðxÞ , e jÁm A[B ðxÞ and e jÁx A[B ðxÞ , we refer to [1].
3 Interval Complex Neutrosophic Set with Set Theoretic Properties

Interval Complex Neutrosophic Set
Before we present the definition, let us consider an example below to see the advantages of the new notion ICNS.
Example 1 Suppose we have a car component factory.Each worker from this factory receives 10 car components per day to polish.
• NS The best worker, John, successfully polishes 9 car components, 1 car component is not finished, and he wrecks 0 car component.Then, John's neutrosophic work is (0.9, 0.1, 0.0).The worst worker, George, successfully polishes 6, not finishing 2, and wrecking 2. Thus, George's neutrosophic work is (0.6, 0.2, 0.2).• INS The factory needs to have one worker coming in the weekend, to work for a day in order to finish a required order from a customer.Since the factory management cannot impose the weekend overtime to workers, the manager asks for a volunteer.How many car components are to be polished during the weekend?Since the manager does not know which worker (W) will volunteer, he estimates that the work to be done in a weekend day will be: W([0.6, 0.9], [0.1, 0.2], [0.0, 0.2]), i.e., an interval for each T, I, F, respectively, between the minimum and maximum values of all workers.• CNS The factory's quality control unit argues that although many workers correctly/successfully polish their car components, some of the workers do a work of a better quality than the others.Going back to John and George, the factory's quality control unit measures the work quality of each of them and finds out that: John's work is (0.9 9 e 0.6 , 0.1 9 e 0.4 , 0.0 9 e 0.0 ), and George's work is (0.6 9 e 0.7 , 0.2 9 e 0.5 , 0.2 9 e 0.1 ).Thus, although John polishes successfully 9 car components, more than George's 6 successfully polished car components, the quality of John's work (0.6, 0.4, 0.0) is less than the quality of George's work (0.7, 0.5, 0.1).
It is clear from the above example that the amplitude and phase (attribute) of CNS should be represented by intervals, which better catch the unsure values of the membership.Let us come back to Example 1, where the factory needs to have one worker coming in the weekend to work for a day, in order to finish a certain order from a customer.Again, the manager asks for a volunteer worker W. We find out that the number of car components that will be done over one weekend day is W([0.6,0.9], [0.1, 0.2], [0.0, 0.2]), which are actually the amplitudes for T, I, F. But what will be their quality?Indeed, their quality will be W([0.6,0.9] 9 e [0.6, [0.0, 0.2] 9 e [0.0, 0.1] ), by taking the [min, max] for each corresponding phases for T, I, F, respectively, for all workers.Therefore, we should propose a new notion for such the cases of decision-making problems.

Set Theoretic Operations of Interval Complex
Neutrosophic Set Definition 10 Let A and B be two interval complex neutrosophic set over X which are defined by T B, and it is defined as: for all x 2 X.The union of the phase terms remains the same as defined for single-valued complex neutrosophic set, with the distinction that instead of subtractions and additions of numbers, we now have subtractions and additions of intervals.The symbols _,^represent max and min operators.

Definition 11 Let
A and B be two interval complex neutrosophic set over X which are defined by T B is denoted as A \ B, and it is defined as: for all x 2 X.Similarly, the intersection of the phase terms remains the same as defined for single-valued complex neutrosophic set, with the distinction that instead of subtractions and additions of numbers we now have subtractions and additions of intervals.The symbols _,^represent max and min operators.

Definition 12 Let
A be an interval complex neutrosophic set over X which is defined by T , and it is defined as: where Proposition 1 Let A, B and C be three interval complex neutrosophic sets over X. Then: (c) The scalar multiplication of A is an interval complex neutrosophic set denoted as C ¼ k A and defined as: The scalar of phase terms is defined below: . ..; m; p ¼ 1; . ..; n; q ¼ 1; . ..; h: Using the operational rules of the IVCNS, the averaged suitability rating x op ¼ ð½T L op ; T U op ; ½I L op ; I U op ; ½F L op ; F U op Þ can be evaluated as: where

Aggregate the Importance Weights
Let w pq ¼ ð½T L pq ; T U pq ; ½I L pq ; I U pq ; ½F L pq ; F U pq Þ be the weight assigned by decision-maker D q to criterion C p ; where ð Þ ; p ¼ 1; . ..; n; q ¼ 1; . ..; h: Using the operational rules of the IVCNS, the average weight w p ¼ ð½T L p ; T U p ; ½I L p ; I U p ; ½F L p ; F U p Þ can be evaluated as: 3 Aggregate the Weighted Ratings of Alternatives Versus Criteria The weighted ratings of alternatives can be developed via the operations of interval complex neutrosophic set as follows:
Table 2 displays the importance weights of the five criteria from the three decision-makers.The aggregated weights of criteria obtained by Eq. ( 4) are shown in the last column of Table 2.

Compute the Total Value of Each Alternative
Table 3 presents the final fuzzy evaluation values of each supplier using Eq. ( 5).It is believed that uncertain, ambiguous, indeterminate, inconsistent and incomplete periodic/redundant information can be dealt better with intervals instead of single values.This paper aimed to propose the interval complex neutrosophic set, which is more adaptable and flexible to real-life problems than other types of fuzzy sets.The definitions of interval complex neutrosophic set, accompanied by the set operations, were defined.The relationship of interval complex neutrosophic set with other existing approaches was presented.
A new decision-making procedure in the interval complex neutrosophic set has been presented and applied to a decision-making problem for the green supplier selection.Comparison between the proposed method and the related methods has been made to demonstrate the advantages and applicability.The results are significant to enrich the knowledge of neutrosophic set in the decision-making applications.
Future work plans to use the decision-making procedure to more complex applications, and to advance the interval complex neutrosophic logic system for forecasting problems.

Fig. 1
Fig. 1 Relationship of complex neutrosophic set with different types of fuzzy sets

Table 1
Aggregated ratings of suppliers versus the criteria

Table 2
The importance and aggregated weights of the criteria

Table 3
The final fuzzy evaluation values of each supplier

Table 4
Modified score function of each alternative

Table 5
The importance and aggregated weights of the criteria

Table 6
Aggregated ratings of suppliers versus the criteria