# Interval-Valued Complex Fuzzy Geometric Aggregation Operators and Their Application to Decision Making

Authors: Songsong Dai; Lvqing Bi; Bo Hu
Journal: Mathematical Problems in Engineering
 (2020)

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## Abstract

This paper investigates the geometric aggregation operators for aggregating the interval-valued complex fuzzy sets (IVCFSs) whose membership grades are a special set of complex numbers. We develop some geometric aggregation operators under the interval-valued complex fuzzy environment, namely, interval-valued complex fuzzy geometric (IVCFG), interval-valued complex fuzzy weighted geometric (IVCFWG), and interval-valued complex fuzzy ordered weighted geometric (IVCFOWG) operators. Then, we investigate the rotational and reflectional invariances of these operators. Further, a decision-making approach based on these operators is presented under the interval-valued complex fuzzy environment and an example is illustrated to demonstrate the efficiency of the proposed approach.


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## Body

## 1. Introduction

The aggregation operator is a powerful method for decision making, pattern recognition, and cluster analysis. In the past decades, in both theoretical and applied studies, aggregation operators have attained great advances. Many types of aggregation operators have been proposed under different environments, such as fuzzy environment [1–4], intuitionistic fuzzy environment [5–9], interval-valued intuitionistic fuzzy environment [10–14], Pythagorean fuzzy environment [15–17], neutrosophic fuzzy environment [18–20], and hesitant fuzzy environment [21–25].In the above fuzzy environments, membership degrees are the subsets of real numbers. As a generalization of traditional fuzzy set [26], Ramot et al. [27] introduced the concept of complex fuzzy set (CFS), which is characterized by a complex-valued membership function. In many practical situations, complex fuzzy sets are useful [28–40]. Moreover, many researchers extended the concept of CFS to interval-valued complex fuzzy set (IVCFS) [41, 42] and complex intuitionistic fuzzy set (CIFS) [43]. Therefore, many researchers discussed how to aggregate CFSs. Ramot et al [32] introduced the concept of complex fuzzy aggregation. Ma et al. [38] proposed a product-sum aggregation operator under complex fuzzy environment. Bi et al. [39, 40] proposed several aggregation operators under complex fuzzy environment. Garg and Rani [44, 45] investigated the aggregation operators under complex intuitionistic fuzzy environment.However, we still have the key question: why complex fuzzy aggregation? Moreover, why complex fuzzy sets? As mentioned in [46], from mathematical and practical viewpoints, complex fuzzy sets are natural and useful. But complex fuzzy sets (CFSs) remain a puzzle from the intuitive viewpoint. Fuzzy sets and other extensions give intuitively clear way to describe how humans deal with different types of uncertainty. So before we start to examine the information aggregation issue under interval-valued complex fuzzy environment, we first discuss some phenomena which maybe ignored in real life. When we ask the way, two persons may give the answers “it is about 1 km away,” and then we think that 1 km is a reasonable result. However, if their answers are not exclusively same about direction, as shown in Figure 1, 0.95 km also is a reasonable result since C=A+B/2.Figure 1
A target location example.How does this phenomenon affect human decision making? For example, there are two hospitalsH1 and H2; which one is the nearest hospital? Then, we get data about distance and direction from strangers. It is a very interesting case; two strangers both agree that hospital H1 is nearer than hospital H2, but after data aggregation, the result that H2 is nearer than hospital H1 is also reasonable, as shown in Figure 2. The order only relying on the distance is reasonable since we want to go to the nearest hospital. The method based on complex fuzzy aggregation is reasonable since it is a center-based method.Figure 2
A decision-making example.Complex fuzzy aggregation operator can perfectly describe above phenomenon in human decision making since it does not satisfy the property of amplitude monotonicity [40]. Monotonicity is a basic property, which holds in many types of aggregation operators under different fuzzy environments [1–25]. Complex fuzzy aggregation operator as a nonmonotone average is very natural and gives an intuitive way to describe how humans deal with such type of uncertainty.In this paper, we focus on the aggregation operator under interval-valued complex fuzzy environment. IVCFSs also are very appropriate for some applications in real life. For example, when we get lost, we often ask strangers for directions in our daily life. Then, we get some answers, such as “0.5-0.6 km, east” and “0.5–0.7 km, northeast.” These answers can be represented in terms of IVCFSs. We present the theory of the weighted geometric aggregation operators among the IVCFSs. First we review necessary concepts and some basic properties related to this paper inSection 2. In Section 3, we present an interval-valued complex fuzzy weighted geometric (IVCFWG) operator on CFSs. In Section 4, we present an interval-valued complex fuzzy ordered weighted geometric (IVCFOWG) operator. In Section 5, we present a decision-making approach based on the proposed operator under IVCFS environment. Finally, conclusions are given in Section 6.


## 2. Preliminaries

In this paper, our discussion is based on interval-valued complex fuzzy set theory. Some basic concepts are recalled below, whereas for other concepts, refer to reports from Ref. [27, 28, 41, 42, 47].LetD be the set of complex numbers on complex unit disk, i.e., D=a∈ℂ|a≤1. Let U be a fixed universe, and a mapping A:U⟶D is called a complex fuzzy set on U.LetI0,1 be the set of all closed subintervals of 0,1, i.e., I0,1=a,b|0≤a≤b≤1. Let D˙ be the boundary set of D i.e., D˙=a∈ℂ|a=1. A mapping A:U⟶I0,1⋅D˙ is called an IVCFS on U. For any x∈U, its membership degree μAx is(1)rAx¯,rAx¯⋅ejνAx,where j=−1, I0,1⋅D˙ is the dot product set of I0,1 and D˙, rAx¯,rAx¯∈I0,1 is the interval-valued amplitude part, and νAx∈ℝ is the phase part.For convenience, we only consider the values onI0,1⋅D˙, which are called interval-valued complex fuzzy values (IVCFVs). Let a=ra¯,ra¯⋅ejνa be an IVCFV, where the interval-valued amplitude part is ra¯,ra¯∈I0,1 and the phase part is νa∈ℝ. The modulus of a is the interval-valued amplitude part ra¯,ra¯, denoted by a. An order of IVCFVs is defined by the interval-valued amplitude part, i.e.,(2)a≤bifra¯≤rb¯andra¯≤rb¯.Leta=ra¯,ra¯⋅ejνa and b=rb¯,rb¯⋅ejνb be two IVCFVs and let the parameters be λ>0 and θ∈ℝ; four operators of IVCFVs include multiplication, power, rotation, and reflection which are defined as follows.(i)
Multiplication of two IVCFVsa,b∈I0,1⋅D˙:(3)a⊗b=ra¯⋅rb¯,ra¯⋅rb¯⋅ejνa+νb.(ii)
Power of an IVCFVa∈I0,1⋅D˙:(4)aλ=ra¯λ,ra¯λ⋅ejλνa.(iii)
Rotation of an IVCFVa∈I0,1⋅D˙:(5)Rotθa=ra¯,ra¯⋅ejνa+θ.(iv)
Reflection of an IVCFVa∈I0,1⋅D˙:(6)Refa=ra¯,ra¯⋅e−jνa.Whena,b∈D, ⊗ is a complex intersection defined by Ramot et al. [27]. When a,b∈0,1, ⊗ is a t-norm [48].Theorem 1.
Suppose thata,b,c are three IVCFVs, and the parameters are λ1>0,λ2>0, and θ1,θ2∈ℝ. Then, we have(1)
a⊗b=b⊗a(2)
a⊗b⊗c=a⊗b⊗c(3)
a⊗bλ1=aλ1⊗bλ1(4)
aλ1+λ2=aλ1⊗aλ2(5)
RefRefa=a(6)
Rotθ1Rotθ2a=Rotθ1+θ2a(7)
Refaλ1=Refaλ1(8)
Refa⊗b=Refa⊗RefbProof.
Leta=ra¯,ra¯⋅ejνa, b=rb¯,rb¯⋅ejνb, and c=rc¯,rc¯⋅ejνc.(1)
By the definition of multiplication of two IVCFVs, we have(7)a⊗b=ra¯⋅rb¯,ra¯⋅rb¯⋅ejνa+νb=rb¯⋅ra¯,rb¯⋅ra¯⋅ejνb+νa=b⊗a.(2)
And(8)a⊗b⊗c=ra¯⋅rb¯,ra¯⋅rb¯⋅ejνa+νb⊗rc¯,rc¯⋅ejνc=ra¯,ra¯⋅ejνa⊗rb¯⋅rc¯,rb¯⋅rc¯⋅ejνb+νc=a⊗b⊗c.(3)
By the definition of power of two IVCFVs, we have(9)a⊗bλ1=ra¯⋅rb¯,ra¯⋅rb¯⋅ejνa+νbλ1=ra¯⋅rb¯λ1,ra¯⋅rb¯λ1⋅ejλ1νa+νb=ra¯λ1⋅rb¯λ1,ra¯λ1⋅rb¯λ1⋅ejλ1νa+λ1νb=aλ1⊗bλ1.(4)
And(10)aλ1+λ2=ra¯λ1+λ2,ra¯λ1+λ2⋅ejλ1+λ2νa=ra¯λ1⋅ra¯λ1,ra¯λ1⋅ra¯λ2⋅ejλ1νa+λ2νa=aλ1+λ2.(5)
By the definition of reflection of two IVCFVs, we have(11)RefRefa=Refra¯,ra¯⋅e−jνa=ra¯,ra¯⋅e−−jνa=ra¯,ra¯⋅ejνa=a.(6)
By the definition of rotation of two IVCFVs, we have(12)Rotθ1Rotθ2a=Rotθ1ra¯,ra¯⋅ejνa+θ2=ra¯,ra¯⋅ejνa+θ2+θ1=Rotθ1+θ2a.(7)
For any two IVCFVsa,b and real value λ>0, we have(13)Refaλ=ra¯,ra¯⋅e−jνaλ=ra¯λ,ra¯λ⋅ej−λνa=Refaλ.(8)
For any two IVCFVsa,b, we have(14)Refa⊗Refb=ra¯,ra¯⋅e−jνa⊗rb¯,rb¯⋅e−jνb=ra¯⋅rb¯,ra¯⋅rb¯⋅ej−νa−νb=ra¯⋅rb¯,ra¯⋅rb¯⋅e−jνa+νb=Refa⊗b.
Moreover, the multiplication and power operators have the following properties.Theorem 2.
Suppose thata=ra¯,ra¯⋅ejνa, b=rb¯,rb¯⋅ejνb, c=rc¯,rc¯⋅ejνc, and d=rd¯,rd¯⋅ejνd are four IVCFVs; the parameters are λ1>0,λ2>0, and θ∈ℝ. Then we have(1)
(Amplitude monotonicity) Ifa≤b,c≤d, then a⊗c≤b⊗d(2)
(Amplitude boundedness)ra¯∧rb¯,ra¯∧rb¯≤a⊗b≤1(3)
Ifλ1≤λ2, then aλ2≤aλ1(4)
Ifa≤b, then aλ1≤aλ1, Refa≤Refb, Rotθa≤RotθaProof.
Since the amplitude terms of IVCFVs are interval values, we can obtain the above results from the properties of multiplication operator of interval values.
Rotational invariance and reflectional invariance are two important geometric properties of complex fuzzy operators [28, 29, 39]. They show that a complex fuzzy operator is invariant under a rotation and a reflection, respectively. We define the following two similar properties for interval-valued complex fuzzy operators as follows.(i)
An interval-valued complex fuzzy operatorv:I0,1⋅D˙n⟶I0,1⋅D˙ is rotationally invariant if and only if, for any θ,(15)vRotθa1,…,Rotθan=Rotθva1,…,an.(ii)
An interval-valued complex fuzzy operatorv:I0,1⋅D˙n⟶I0,1⋅D˙ is reflectionally invariant if and only if(16)vRefa1,…,Refan=Refva1,…,an.
Then, for the multiplication operator of IVCFVs, we have the following results.Theorem 3.
The multiplication operator of IVCFVs is reflectionally invariant.Proof.
Trivial from Theorem1 (16).Theorem 4.
The multiplication operator of IVCFVs is not rotationally invariant.Proof.
For any two IVCFVsa=ra¯,ra¯⋅ejνa, b=rb¯,rb¯⋅ejνb and real value θ, we have(17)Rotθa⊗Rotθb=ra¯,ra¯⋅ejνa+θ⊗rb¯,rb¯⋅ejνb+θ=ra¯⋅rb¯,ra¯⋅rb¯⋅ejνa+νb+2θ,Rotθa⊗b=ra¯⋅rb¯,ra¯⋅rb¯⋅ejνa+νb+θ.
Sinceνa+νb+2θ≠νa+νb+θ, the multiplication operator is not rotationally invariant.


## 3. Interval-Valued Complex Fuzzy Weighted Geometric Operators

In this section, we introduce the weighted geometric operators in an interval-valued complex fuzzy environment and discuss their fundamental characteristics.Definition 1.
Letaii=1,2,…,n be a collection of IVCFVs; an interval-valued complex fuzzy weighted geometric (IVCFWG) operator is defined as(18)IVCFWGa1,a2,…,an=⊗i=1naiwi,where wi∈0,1 for all i and ∑i=1nwi=1.
Whenwi=1/ni=1,2,…,n, then the IVCFWG operator is denoted by interval-valued complex fuzzy geometric (IVCFA) operator, i.e.,(19)IVCFWGa1,a2,…,an=⊗i=1nai1/n.
Whenai∈I0,1i=1,2,…,n, the IVCFWG operator can reduce to a traditional interval-valued fuzzy weighted geometric operator of values on unit interval I0,1.
Whenai∈0,1i=1,2,…,n, the IVCFWG operator can reduce to a traditional weighted geometric operator [49] of real numbers on unit interval [0, 1].
Whenai∈0,1 and wi=1/ni=1,2,…,n, the CFWP operator can reduce to a traditional geometric mean operator [50] of real numbers on unit interval [0, 1].Theorem 5.
Letaii=1,2,…,n be a collection of IVCFVs; then, the aggregated value IVCFWGa1,a2,…,an is also an IVCFV and(20)IVCFWGa1,a2,…,an=∏i=1nrai¯wi,∏i=1nrai¯wi⋅ej∑i=1nwiνai,where wi∈0,1 for all i and ∑i=1nwi=1.Proof.
It is easy to check that the aggregated value is also an IVCFV. Now, we prove equation (20) by mathematical induction method.
Forn=2, we have a1w1=ra1¯w1,ra1¯w1⋅ejw1νa1, a2w2=ra2¯w2,ra2¯w2⋅ejw2νa2; then,(21)IVCFWGa1,a2=a1w1⊗a2w2=ra1¯w1ra2¯w1,ra1¯w1ra2¯w2⋅ejw1νa1+w2νa2=∏i=12rai¯w1,∏i=12rai¯wi⋅ej∑i=12wiνai.
If equation (20) holds for n=k, i.e.,(22)IVCFWGa1,a2,…,ak=∏i=1krai¯w1,∏i=1krai¯wi⋅ej∑i=1kwiνai,then for n=k+1,(23)IVCFWGa1,a2,…,ak+1=IVCFWGa1,a2,…,ak⊗ak+1wk+1=∏i=1krai¯wi,∏i=1krai¯wi⋅ej∑i=1kwiνai⊗ak+1wk+1=∏i=1krai¯wi⋅rak+1¯wk+1,∏i=1krai¯wi⋅rak+1¯wk+1⋅ej∑i=1kwiνai+wk+1νak+1=∏i=1k+1rai¯wi,∏i=1k+1rai¯wi⋅ej∑i=1k+1wiνai.
Therefore, equation (20) holds for all n.
The working of the proposed IVCFWG operator is explained with a numerical example as follows.Example 1.
Leta1=0.6,0.7⋅ej0.1π, a2=0.3,0.5⋅ej0.2π, a3=0.6,0.8⋅ej0.7π, and a4=0.4,0.7⋅ej0.4π be four IVCFVs and w=0.1,0.2,0.3,0.4T be the associated weight vector. Then, by using equation (20),(24)∏i=14rai¯w1=0.60.1⋅0.30.2⋅0.60.3⋅0.40.4=0.4441,∏i=14rai¯wi=0.70.1⋅0.50.2⋅0.80.3⋅0.70.4=0.6812,∑i=14wiνai=0.1⋅0.1π+0.2⋅0.2π+0.3⋅0.7π+0.4⋅0.4π=0.42π,we obtain IVCFWGa1,a2,a3,a4=0.4441,0.6812⋅ej0.42π.
Based on Theorem5, the proposed IVCFWG operator satisfies the following properties.Theorem 6.
Letaii=1,2,…,n and bii=1,2,…,n be two collections of IVCFVs, the weights be wi∈0,1i=1,2,…,n, and ∑i=1nwi=1. Then, we have the following properties:(1)
(Idempotency) Ifai=ai=1,2,…,n, then(25)IVCFWGa1,a2,…,an=a.(2)
(Amplitude monotonicity) Ifai≤bii=1,2,…,n, then(26)IVCFWGa1,a2,…,an≤IVCFWGb1,b2,…,bn.(3)
(Amplitude boundedness)(27)r1≤IVCFWGa1,a2,…,an≤r2,where r1,r2 are two interval values:(28)r1=minirai¯,minirai¯,r2=maxirai¯,maxirai¯.Proof.
(1)
Letai=a=ra¯,ra¯, i=1,2,…,n; then,(29)IVCFWGa1,a2,…,an=∏i=1nra¯w1,∏i=1nra¯wi⋅ej∑i=1nwiνa=ra¯∑i=1nwi,ra¯∑i=1nwi⋅ejνa=ra¯,ra¯⋅ejνa=a.(2)
The property of the amplitude monotonicity is trivial from Theorem2 (4).(3)
The property of the amplitude boundedness is easily obtained from idempotency and amplitude monotonicity.
Note that idempotency is concerned with both the phase part and amplitude part of IVCFVs. Amplitude boundedness and amplitude monotonicity are only concerned with the amplitude part of IVCFVs.Theorem 7.
The IVCFWG operator is reflectionally invariant.Proof.
For any collection of IVCFVsaii=1,2,…,n, from equation (20), we have(30)IVCFWGRefa1,Refa2,…,Refan=∏i=1nrai¯wi,∏i=1nrai¯wi⋅ej∑i=1n−wiνai=∏i=1nrai¯wi,∏i=1nrai¯wi⋅e−j∑i=1nwiνai,RefIVCFWGa1,a2,…,an=Ref⊗i=1naiwi=∏i=1nrai¯wi,∏i=1nrai¯wi⋅e−j∑i=1nwiνai.
Then,IVCFWGRefa1,Refa2,…,Refan=Ref⊗i=1nwiai is reflectionally invariant.Theorem 8.
The IVCFWG operator is rotationally invariant.Proof.
For any collection of IVCFVsaii=1,2,…,n and any θ∈ℝ, we have(31)IVCFWGRotθa1,Rotθa2,…,Rotθan=∏i=1nrai¯wi,∏i=1nrai¯wi⋅ej∑i=1nwiνai+θ=∏i=1nrai¯wi,∏i=1nrai¯wi⋅ej∑i=1nwiνai+∑i=1nwiθ=∏i=1nrai¯wi,∏i=1nrai¯wi⋅ej∑i=1nwiνai+θ=RotθIVCFWGa1,a2,…,an.
Then, the IVCFWG operator is rotationally invariant.


## 4. Interval-Valued Complex Fuzzy Ordered Weighted Geometric Operators

Based on the order of IVCFVs defined by equation (2) and the ordered weighted averaging (OWA) operator introduced by Yager [1], we define an interval-valued complex fuzzy ordered weighted geometric (IVCFOWG) operator as follows.Definition 2.
Letaii=1,2,…,n be a collection of CFVs; then, an IVCFOWG operator is defined as(32)IVCFOWGa1,a2,…,an=⊗i=1naσiwi,where wi∈0,1i=1,2,…,n and ∑i=1nwi=1, σ1,σ2,…,σn is a permutation of 1,2,…,n such that aσi−1≥aσi for all i.
Especially, whenwi=1/ni=1,2,…,n, the IVCFOWG operator is reduced to the IVCFWG operator.
Similar to the IVCFWG operator, the IVCFOWG operator has the following properties.Theorem 9.
Letaii=1,2,…,n be a collection of IVCFVs; then, the aggregated value IVCFOWGa1,a2,…,an is also an IVCFV and(33)IVCFOWGa1,a2,…,an=∏i=1nraσi¯wi,∏i=1nraσi¯wi⋅ej∑i=1i=1wiaσi,where wi∈0,1 for all i and ∑i=1nwi=1, σ1,σ2,…,σn is a permutation of 1,2,…,n such that aσi−1≥aσi for all i.Proof.
The proof is similar to Theorem5.
The working of the proposed IVCFOWG operator is explained with a numerical example as follows.Example 2.
Leta1=0.6,0.7⋅ej0.1π, a2=0.3,0.5⋅ej0.2π, a3=0.6,0.8⋅ej0.7π, and a4=0.4,0.7⋅ej0.4π be four IVCFVs and w=0.1,0.2,0.3,0.4T be the associated weight vector. By the order from equation (2), we have a3≥a1≥a4≥a2. Then, by using equation (33),(34)∏i=14raσi¯wi=0.60.1⋅0.60.2⋅0.40.3⋅0.30.4=0.4026,∏i=14raσi¯wi=0.80.1⋅0.80.2⋅0.70.3⋅0.50.4=0.6369,∑i=14wiνai=0.1⋅0.7π+0.2⋅0.1π+0.3⋅0.4π+0.4⋅0.2π=0.29π,we obtain IVCFOWGa1,a2,a3,a4=0.4026,0.6369⋅ej0.29π. It is different from IVCFWGa1,a2,a3,a4=0.4441,0.6812⋅ej0.42π (see Example 1).Theorem 10.
Letaii=1,2,…,n be a collection of CFVs, CFOWP weights be wi∈0,1i=1,2,…,n, and ∑i=1nwi=1. Then, we have the following properties.(1)
(Idempotency) Ifai=ai=1,2,…,n, then(35)IVCFOWGa1,a2,…,an=a.(2)
(Amplitude monotonicity) Ifai≤bii=1,2,…,n, then(36)IVCFWGa1,a2,…,an≤IVCFWGb1,b2,…,bn.(3)
(Amplitude boundedness)(37)r1≤IVCFOWGa1,a2,…,an≤r2.where r1,r2 are two interval values:(38)r1=minirai¯,minirai¯,r2=maxirai¯,maxirai¯.Proof.
The proof is similar to Theorem6.Theorem 11.
The IVCFOWG operator is reflectionally invariant and rotationally invariant.Proof.
The proof is similar to Theorems7 and 8.
Note that the operator⊗ does not have the property of rotational invariance, but the IVCFWG and IVCFOWG operators defined based on ⊗ operator have the property of rotational invariance.
Here, we investigate the aggregation operators based on special class of the IVCFVs, which belong to subsets of the upper-right quadrant of the complex unit disk. LetD1=ejv|v∈0,π/2; we consider the aggregation operator on I0,1⋅D1.
Let us consider the closeness of IVCFVs onI0,1⋅D1 under the IVCFWG and IVCFOWG operations. For the IVCFWG operator, we have the following result.Theorem 12.
Letz1,z2,…,zn∈I0,1⋅D1. Then, the aggregated value(39)IVCFWGz1,z2,…,zn∈I0,1⋅D1.Proof.
DenotingIVCFWGz1,z2,…,zn=t¯,t¯⋅ejν, from Theorem 5, t¯,t¯ is an interval value. Since ν=∑i=1nwi⋅νzi is a weighted arithmetic aggregation operator of real numbers on 0,π/2, then we have ν∈0,π/2. Thus, IVCFWGz1,z2,…,zn∈I0,1⋅D1.
Similar to the above theorem, we have the following.Theorem 13.
Letz1,z2,…,zn∈I0,1⋅D1. Then, the aggregated value(40)IVCFOWGz1,z2,…,zn∈I0,1⋅D1.The above theorems show us that the IVCFWG and the IVCFOWG operators are close under values onI0,1⋅D1.Consider other quadrants of the complex unit disk. Let(41)Dk=z=ejθ|θ∈k−1π2,kπ2,for k=1 to 4.Now, we discuss the closeness of the IVCFWG and the IVCFOWG operators on values of other quadrants of the complex unit disk. Plainly, we have the following.Theorem 14.
For anyk∈1,2,3,4, if z1,z2,…,zn∈I0,1⋅Dk, then we have(42)IVCFWGz1,z2,…,zn∈I0,1⋅Dk,IVCFOWGz1,z2,…,zn∈I0,1⋅Dk.Proof.
Similar to Theorem12.


## 5. An Approach to Decision Making with the IVCFWG Operator

In this section, we present an approach using the IVCFWG operator to a decision making with interval-valued complex fuzzy information.We consider a target selection application of CFSs. Assume that our position is fixed, and then we can measure the distance and angle of the possible alternatives by using a position sensor and an angular sensor (or other attributes from expert opinions). Assume that we get a measurementd±ϵ,q. Here, we use an interval value d¯,d¯ to represent d±ϵ by setting d¯=d−ϵ and d¯=d+ϵ. To improve the target location accuracy, we repeatedly measure the alternatives. Then, the target is selected in the following approach according to aggregation theory.Letxii=1,2,…,n be the possible alternatives and ekk=1,2,…,m be the experts. Then, the decision maker provides a decision matrix A=aikn×m, where aik is an interval-valued complex fuzzy value given by the expert ek for alternative xi. Further, assume that w=w1,w2,…,wnT is the weight vector of the different experts such that wi≥0i=1,2,…,n and ∑i=1nwi=1. The process can be summarized as follows:Step 1. Transform the decision matrix A into the normalized interval-valued complex fuzzy decision matrix C=cikn×m using cik=aik/d where d=maxikaik.Step 2. Aggregate all the interval-valued complex fuzzy values aikk=1,2,…,m and get the overall interval-valued complex fuzzy value bi corresponding to the alternative xi by the IVCFWG operator, bi=IVCFWGai1,ai2,…,ain.Step 3. Rank the overall IVCFVs bii=1,2,…,n using (2).Next, we give an example to illustrate the above approach.Example 3.
Suppose we want to go to the nearest bank, then we often use GPS navigation system or ask strangers for directions.
Suppose that there are three alternativesxii=1,2,3 and three experts (GPS or stranger) eii=1,2,3 with the weight vector 0.4,0.3,0.3T. The three experts evaluate the three alternatives under the IVCFS environment, and their corresponding rating values are summarized in the decision matrix C=aik3×3 represented in Table 1, where cik=rik⋅ejνikπ is an interval-valued complex fuzzy value, rik¯,rik¯ represents the distance between rik¯ km and rik¯ km, and νik represents the direction for the alternative xi.Step 1. The values cik do not need normalization.Step 2. Aggregate the interval-valued complex fuzzy values bi of the alternatives xi by using the IVCFWG operator (see equation (20)):(43)∏i=13ra1i¯w1=0.190.4⋅0.110.3⋅0.160.3=0.1532,∏i=13ra1i¯wi=0.250.4⋅0.230.3⋅0.230.3=0.2378,∑i=13wiνa1i=0.4⋅0.11+0.3⋅0.13+0.3⋅0.11⋅π=0.116π,and we obtainb1=IVCFWGa11,a12,a13=0.1532,0.2378⋅ej0.116π.(44)∏i=13ra2i¯w1=0.120.4⋅0.180.3⋅0.180.3=0.1531,∏i=13ra2i¯wi=0.230.4⋅0.240.3⋅0.20.3=0.2234,∑i=13wiνa2i=0.4⋅0.52+0.3⋅0.54+0.3⋅0.52⋅π=0.526π,and we obtainb2=IVCFWGa11,a12,a13=0.1531,0.2234⋅ej0.126π.(45)∏i=13ra3i¯w1=0.10.4⋅0.190.3⋅0.170.3=0.1422,∏i=13ra3i¯wi=0.160.4⋅0.230.3⋅0.190.3=0.1878,∑i=13wiνa3i=0.4⋅1.11+0.3⋅1.12+0.3⋅1.14⋅π=1.122π,and we obtainb3=IVCFWGa11,a12,a13=0.1422,0.1878⋅ej0.122π.Step 3. Rank the interval-valued complex fuzzy values bii=1,2,3: b1>b2>b3, which shows that the alternative x3 is the optimal choice, i.e., the nearest bank.
An IVCFS is a generalization of complex fuzzy sers (CFS), interval-valued fuzzy sets (IVFS), and fuzzy sets (FS). IVCFS can handle two-dimensional information in a single set. Then, IVCFS contains much more information than CFS, IVFS, and FS.
It is revealed from the present study that the aggregation operators under IVFSs and FSs [5, 6] are the special cases of the proposed aggregation operators. Thus, the proposed aggregation operators can be equivalently utilized to solve the DM problem under these existing environments by setting phase term to be zero while the existing operators [5, 6] are unable to solve the problems under the IVCFSs environment considered in the present paper. Thus, interval-valued complex fuzzy aggregation operators are more general than some (interval-valued) fuzzy aggregation operators.Table 1
Decision matrix.
e1e2e3x10.19,0.25⋅ej0.11π0.11,0.23⋅ej0.13π0.16,0.23⋅ej0.11πx20.12,0.23⋅ej0.52π0.18,0.24⋅ej0.54π0.18,0.20⋅ej0.52πx30.10,0.16⋅ej1.11π0.19,0.23⋅ej1.12π0.17,0.19⋅ej1.14π


## 6. Conclusion

As mentioned in [50], the basic feature of aggregation operators is their monotonicity property. However, aggregation operators under the complex fuzzy environment [40] are not monotone. Interestingly, such aggregation operators can explain some phenomena in our real life.In this paper, we discussed the geometric aggregation operators under the interval-valued complex fuzzy environment. Two interval-valued complex fuzzy aggregation operators, the IVCFWG and the IVCFOWG operators, are developed and their properties are studied. It is also interesting to note that both the IVCFWG and the IVCFOWG operators are rotationally invariant and reflectionally invariant. Further, based on the proposed operators, we presented a decision-making approach under interval-valued complex fuzzy information. An illustrative example is given for illustrating the proposed approach.It was observed that the aggregation operator based on nonadditive integrals (Choquet [51] and Sugeno integrals [52]) is one of the hot topics in this field [49]. As future work, we can consider the complex fuzzy aggregation operators based on complex integrals.



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