New Distance Measures between the Interval-Valued Complex Fuzzy Sets with Applications to Decision-Making

As a generalization of complex fuzzy set (CFS), interval-valued complex fuzzy set (IVCFS) is a new research topic in the field of CFS theory, which can handle two different information features with the uncertainty. Distance is an important tool in the field of IVCFS theory. To enhance the applicability of IVCFS, this paper presents some new interval-valued complex fuzzy distances based on traditional Hamming and Euclidean distances of complex numbers. Furthermore, we elucidate the geometric properties of these distances. Finally, these distances are used to deal with decision-making problem in the IVCFS environment.

is is inconsistent with our vision. e main reason is that distances in [12,16,17] are combining the difference between the amplitude terms and the difference between the phase terms of CFSs. is method ignored the circular structure of CFS. Two CFSs around the center can arbitrarily approach to each other, but their phase terms are completely opposite with the biggest difference. is causes the result, which is not consistent with our intuition. In this environment, using traditional distance between complex number is a more reasonable selection for us to measure the difference between CFSs.
Greenfield et al. [20,21] introduced the IVCFS theory. In real life, when we get some answers such as "0.5 km-0.6 km, east" and "0.5 km-0.7 km, northwest" about the targets, we can represent these answers in terms of IVCFSs. en, we may ask the simple question: what is the distance between "0.5 km-0.6 km, east" and "0.5 km-0.7 km, northwest" (see Figure 2)? Dai et al. [22] proposed some distance measures between IVCFSs. When IVCFSs are reduced to CFSs, this inevitably leads to get the same result in the above instance of Figure 1. erefore, distances in [22] cannot overcome the above drawback of distances of CFSs and are not suitable for IVCFSs in some cases.
e main contribution to this article is summarized as follows: (1) Some new distances for IVCFSs are constructed. ey can overcome the above drawback of distances in IVCFSs. ese distances also are new measures for CFS.
(2) Rotational invariance and reflectional invariance of these distances are studied, and a comparative analysis is provided. (3) ese distances are used in target selection problem when IVCFSs express the locative information of targets. e purpose of this paper is to construct some distances between IVCFSs and apply them into decision-making problem. is article is structured as follows. In Section 2, we introduce the concept of IVCFS. In Section 3, we present some distances for IVCFSs. In Section 4, the rotational invariance and reflectional invariance of our proposed distances are studied. In Section 5, these distances are applied to solve a decision-making problem in IVCFSs information. In Section 6, a conclusion is given.

Preliminaries
In this paper, our discussion is based on IVCFSs. We first recall some basic concepts [1,20,21,[23][24][25][26][27]. Let Let S be a fixed universe, then the following holds: (1) A mapping A: S ⟶ [0, 1] is called a FS on S. For any s ∈ S, its membership degree μ A (s) is For convenience, a value a ∈ I [0,1] · D is called an interval-valued complex fuzzy value (IVCFV), denoted by a � [p a , p a ] · e jq a .
For clarity, we list the membership functions for FS and its generalizations:

Distances between IVCFSs
Definition 1 (see [22]). A function d: (IVCF(S)× IVCF(S)) ⟶ R + ∪ 0 { } is called a distance measure between IVCFSs if it satisfies the following: for any P, Q, R ∈ IVCF(S), Dai et al. [22] defined the following distances in IVCFSs case as follows: for any P, Q ∈ IVCF(S), where S � s 1 , s 2 , . . . , s n , However, these distances are not suitable for localization problem; for example, let μ A (s) ≡ 0.01 · e j0.25π and μ B (s) ≡ 0.01 · e j1.25π ; they are very close, as shown in Figure In order to overcome the abovementioned shortcoming, we introduce some new distances for IVCFSs. Let u � [p 1 , p 1 ] · e jq 1 and v � [p 2 , p 2 ] · e jq 2 be two IVCFVs, and we consider the following distances between u and v: where |a − b| 1 and |a − b| 2 represent traditional Hamming and Euclidean distances of complex numbers a, b ∈ C, respectively.
Based on the above formulas, we define some distances of IVCFSs, for any P, Q ∈ IVCF(S), where S � s 1 , s 2 , . . . , s n , we have the following.
(i) e Hamming distance: (ii) e normalized Hamming distance: (iii) e Euclidean distance: Mathematical Problems in Engineering (iv) e normalized Euclidean distance: Proof. It is easy from max(p 1 , · p Q (s) · e jq Q (s) − p P (s) ·e jq P (s)

Proof. For any complex numbers
and 0 ≤ |a − b| 2 ≤ 2, and hence, h(P, Q) ≤ n 1 2 , and q(P, Q) ≤ (1/2n) n 1 2 ≤ 1. In general, we use e (A, B) to measure the distance between two targets A and B. But when targets are in the city, h (A, B) is viewed as the city block distance may be more reasonable. Figure 3 shows an instance of the difference between two distances.
Based on the relations among IVCFS, IVFS, CFS, and FS, we give the comparison of our proposed distances of IVCFSs with IVFS, CFS, and FS. Based on the reduction of IVCFSs, the comparison results are shown in Remarks 1-3. □ Remark 1. When IVCFSs are reduced to CFSs, the abovedefined functions (4)-(7) are distances for CFSs based on traditional Hamming and Euclidean distances of complex numbers defined as follows:

Mathematical Problems in Engineering
Remark 2. When IVCFSs are reduced to IVFSs, the abovedefined functions (4)- (7) are distances for IVFSs based on Hausdorff metric defined as follows: Remark 3. When IVCFSs are reduced to FSs, the abovedefined functions (4)- (7) are distances for FSs as follows:
And the reflection of P, denoted Ref(P), is defined as Ref μ P (s) � p P (s), p P (s) · e j 2π− q P (s) ( ) .
Dai et al. [22] gave the following definitions for distance measures between IVCFSs.
Definition 2 (see [22]). Let d is a distance for IVCFSs, and d is rotationally invariant if for any α and P, Q ∈ IVCF(S).
Definition 3 (see [22]). Let d is a distance for IVCFSs, and d is reflectionally invariant if for any P, Q ∈ IVCF(S).

Theorem 3.
e above-defined distances e, q are reflectionally and rotationally invariant. Proof. It is easy from the facts that traditional Euclidean distance between complex numbers is reflectionally and rotationally invariant.

Theorem 4.
e above-defined distances h, l are reflectionally invariant, but not rotationally invariant.

Proof
(1) It is easy from the fact that | (a + bj) − (c + dj) for any complex numbers a + bj and c + dj. us, Similarly, we can get l( . us, the above-defined distances h, l are not rotationally invariant.

Numerical Example for Decision-Making
In real life, we may get some answers such as "0.5 km-0.6 km, east" and "0.5 km-0.7 km, northwest" about the targets. ese answers can be represented in terms of IVCFSs. Now, we consider a decision-making problem in the environment of IVCFSs. Assume that the ideal target is 1, and there are four alternatives (T 1 , T 2 , T 3 , T 4 ). en, rating values of these alternatives are given by five natives (E 1 , E 2 , E 3 , E 4 ), and then, we try to find the nearest alternative. e corresponding rating values of alternatives given by natives are shown in Table 1. Now, we compute the distance between the ideal target and A i (i � 1, 2, 3, 4) based on the distance functions (5)- (8).
e results are shown in Table 2.
Here, we use the technique for order preference by similarity to an ideal solution (TOPSIS) [28] for decision-making. Based on the TOPSIS method, we try to find the nearest alternative to the ideal target, and thus, the best alternative is the one with the nearest distance to the ideal target.
e results are shown in Table 3, in which T i ≻ T k means T i is nearer than T k for the ideal target. en, as we can see in Table 4, T 2 is the best alternative in this example.

Conclusions
CFS and IVCFSs are used to describe locative information with uncertainty in some real-world applications; for example, when we ask for directions, we may get answers such as "0.5 km-0.6 km, east", "0.8 km, West," and "0.5 km-0.7 km, northwest" about the targets. en, we need to measure the difference between objects and estimate how long it will take to get to the close object. In this case, distances in [12,16,17,22] are not suitable. In this paper, we have presented some new distances for IVCFSs by using traditional Euclidean distance between complex numbers. ey are suitable for measuring the distance between objects. We used these distances to deal with the location decision problem under uncertain situations. ese distance measures include the Hamming distance h, the normalized Hamming distance l, the Euclidean distance e, and the normalized Euclidean distance q. Furthermore, the distances h and l are reflectionally invariant but not rotationally invariant, and distances e and q are both reflectionally and rotationally invariant. Finally, based on these distances, we presented an illustrative example for location decision-making under IVCFS situation.
Note that we give a drawback of distances in [12,16,17,22] from a specific application of IVCFS. Many angles of analysis of distances are needed. In future research, we expect to develop more distances of CFS and its extension from different angles and apply them in different applications, such as engineering, economics, and medicine.

FS:
Fuzzy set CFS: Complex fuzzy set IVFS: Interval-valued fuzzy set IVCFS: Interval-valued complex fuzzy set IVCFV: Interval-valued complex fuzzy value IVCF (S): e set of all IVCFSs of S TOPSIS: Technique for order preference by similarity to an ideal solution.

Data Availability
e data used to support the findings of this study are included in the article.

Conflicts of Interest
e authors declare that there are no conflicts of interest.