Next Article in Journal
Comparing the Normalized Difference Vegetation Index with the Google Street View Measure of Vegetation to Assess Associations between Greenness, Walkability, Recreational Physical Activity, and Health in Ottawa, Canada
Next Article in Special Issue
The Oakville Oil Refinery Closure and Its Influence on Local Hospitalizations: A Natural Experiment on Sulfur Dioxide
Previous Article in Journal
Effect of Bioaugmentation on Biogas Yields and Kinetics in Anaerobic Digestion of Sewage Sludge
Previous Article in Special Issue
Predicting Infectious Disease Using Deep Learning and Big Data
 
 
Font Type:
Arial Georgia Verdana
Font Size:
Aa Aa Aa
Line Spacing:
Column Width:
Background:
Article

A Novel Multiattribute Decision-Making Method Based on Point–Choquet Aggregation Operators and Its Application in Supporting the Hierarchical Medical Treatment System in China

1
School of Economics and Management, Beijing Jiaotong University, Beijing 100044, China
2
School of Mechanical, Electronic and Control Engineering, Beijing Jiaotong University, Beijing 100044, China
*
Authors to whom correspondence should be addressed.
Int. J. Environ. Res. Public Health 2018, 15(8), 1718; https://doi.org/10.3390/ijerph15081718
Submission received: 29 June 2018 / Revised: 27 July 2018 / Accepted: 27 July 2018 / Published: 10 August 2018

Abstract

:
The hierarchical medical treatment system is an efficient way to solve the problem of insufficient and unbalanced medical resources in China. Essentially, classifying the different degrees of diseases according to the doctor’s diagnosis is a key step in pushing forward the hierarchical medical treatment system. This paper proposes a framework to solve the problem where diagnosis values are given as picture fuzzy numbers (PFNs). Point operators can reduce the uncertainty of doctor’s diagnosis and get intensive information in the process of decision making, and the Choquet integral operator can consider correlations among symptoms. In order to take full advantage of these two kinds of operators, in this paper, we firstly define some point operators under the picture fuzzy environment, and further propose a new class of picture fuzzy point–Choquet integral aggregation operators. Moreover, some desirable properties of these operators are also investigated in detail. Then, a novel approach based on these operators for multiattribute decision-making problems in the picture fuzzy context is introduced. Finally, we give an example to illustrate the applicability of the new approach in assisting hierarchical medical treatment system. This is of great significance for integrating the medical resources of the whole society and improving the service efficiency of the medical service system.

1. Introduction

With increasing environmental issues, lung diseases are becoming a serious health problem in China. As the medical facilities in grade III, class A hospitals are much better than those of in other small hospitals, people prefer to go to those relatively high-level hospitals for treatment. As a result, overcrowding in large hospitals is common, far exceeding the coping capacity. At the same time, however, small hospitals or clinics waste medical resources. Under such circumstances, how to better allocate limited medical resources and improve the input and output efficiency of the health care system are new challenges for the medical system in China.
Developing a hierarchical medical treatment system is regarded as key an effective way to solve the problem of insufficient and unbalanced medical resources, in which medical institutions at various levels receive patients according to the degree and urgency of the diseases they have. In such a system, common illnesses are treated at basic clinics, with patients transferred to more specialized facilities if their condition demands it. Serious illnesses should be treated in higher-grade hospitals. At the same time, higher-grade hospitals can also transfer patients down to lower-grade ones as their condition stabilizes. Thus, determining the severity of the illness is a key action in this system. At present, with the increase in the number of patients with lung diseases, establishing an appropriate approach to divide patients under different conditions into different levels of hospitals is an effective way to make full use of limited medical resources and cure more patients with lung diseases. However, the diagnosis for patient’s condition often involves multiple correlative criteria and thus can be described as multiattribute decision making (MADM) problems. This paper proposes a general framework in order to solve the MADM problem, which can be applied in the above scenario.
The essence of MADM is the process of ranking the alternatives and selecting an optimal scheme among a set of alternatives with respect to a list of attribute value. Recently, MADM has received much attention from scholars and has been widely applied to economic management and daily life. For example, Tang et al. [1] proposed an algorithm for group decision making with incomplete hesitant fuzzy linguistic preference relations and applied it to flood disaster risk evaluation. Qi [2] developed two effective multicriteria decision making (MCDM) approaches based on defined prioritized average aggregation operators and applied them to tackle complex emergency response solutions evaluation problems. Lin [3] proposed a linear program and a procedure for solving linguistic MADM problems with risk preferences and incomplete weight information, and further applied it to low-carbon tourism destination selection. Due to the increased complexity of real decision-making problems, we usually have to face the difficulty of representing attribute values appropriately. Chatterjee et al. [4] proposed a novel hybrid method encompassing factor relationship and multi-attributive border approximation area comparison methods for selection and evaluation of non-traditional machining process. Roya et al. [5] proposed a rough group analytic hierarchy process approach to the evaluation supplier criteria in the company for producing metal washers for the automotive industry. Vasiljević et al. [6] developed rough strength relational decision making and trial evaluation laboratory model to analyze the individual priorities of key success factors of hospital’s performance measures. As a generalization of the intuitionistic fuzzy set (IFS) [7], the picture fuzzy set (PFS) introduced by Cuong [8] is a very effective tool to express the complex fuzzy information because it is characterized by three functions expressing the degree of positive, neutral, and negative memberships at the same time. Because of this advantage, the PFS has been widely investigated and quite a few achievements have been made [9,10,11,12,13]. Among them, an important research topic in the research fields of MADM is aggregation operator theory, that can aggregate a collection of individual evaluated values into one. Abbas et al. [14] presented a comprehensive review on aggregation operator theory and decision-making approaches between 1986 and 2017. Among these aggregation operators, traditional aggregation operators, such as arithmetic and geometric operators for the IFS and neutral averaging operators [15] are based on the assumption that the attributes are independent of one another. However, the attributes of the problem are often correlative in the real decision-making process, especially in medical diagnosis. For example, to evaluate patients based on the following symptoms of lung diseases: (vital signs, body temperature, cough and hemoptysis), we want to place more emphasis on hemoptysis than on body temperature. However, on the other hand, we also want to pay more attention to patients who have severe hemoptysis and high body temperature, because hemoptysis and hyperthermia are two classical symptoms of pneumonia. Therefore, we need to find some new ways to deal with these situations where the decision data are correlative. The Choquet integral [16] introduced by Choquet is a useful tool to address the problem. Many scholars have made quite a few achievements in this field and applied the Choquet integral in MADM problems. By using Choquet integral and quasi-arithmetic means, Zhou and Chen [17] proposed a combined continuous quasi-arithmetic Choquet integral operator and a combined continuous generalized Choquet integral operator. In order to globally reflect the interactions between elements, Meng and Zhang [18] further defined the probabilistic generalized semivalue-induced continuous Choquet weighted averaging operator and the induced continuous Choquet geometric mean operator. Xu [19] used the Choquet integral to propose some operators for aggregating intuitionistic fuzzy values with correlative weights and further extended those operators to interval-valued intuitionistic fuzzy sets. Yager [20] proposed an approximation to the Choquet integral criteria aggregation that did not require ordering. By extending Marichal’s concept of entropy for fuzzy measures, Liu et al. [21] proposed a new method for determining fuzzy measures of the Choquet integral. Wen et al. [22] introduced Choquet integral-based linguistic operators under fuzzy heterogeneous environments for supplier selection in supply chain management. Some scholars also extended the Choquet integral to other fuzzy environments, such as in interval intuitionistic fuzzy information [23], the dual hesitant fuzzy environment [24], the interval-valued intuitionistic hesitant fuzzy environment [19] and the Pythagorean fuzzy environment [25]. Point operators are another aggregation tool to reduce the uncertainty of the aggregated arguments and thus obtain intensive information in the process of decision making. Since the point operator was proposed [26], it has been applied to many fields and has attracted increasing attention. Liu and Wang [27] proposed some point operators to translate IFS into another one. Xia and Xu [28] used the point operators to propose some operators for aggregating intuitionistic fuzzy values, and further extended those operators to intuitionistic multiplicative sets [29]. Peng [30,31], and Xing [32] also extended point operators to Pythagorean fuzzy sets, interval-valued Pythagorean fuzzy sets, and dual hesitant fuzzy sets, respectively.
However, the medical diagnosis problem in the real world is complex than many other applications. For instance: (1) We need to exactly express fuzzy information, and picture fuzzy numbers (PFNs) can depict doctors’ diagnoses for patients with respect to the symptoms; (2) We need to consider correlations among symptoms, and then the Choquet integral operator can be utilized to solve this problem; and (3) We need to reduce the uncertainty of doctor’s diagnosis data and get intensive information when diagnosing diseases. We can select point operators to achieve this function by adjusting the degree of doctor’s diagnosis data with some parameters. In order to solve above problems simultaneously, it is necessary to combine point operator with Choquet integral operator under picture fuzzy environment. Thus, the goal of this paper is to establish a new decision-making method that can not only control the certainty of doctor’s diagnosis data, but also deal with these situations where the diagnosis data are correlative. Then we apply new decision-making method to judge patient condition, and patients with different conditions are divided into different levels of hospitals instead of all patients rushing to large hospitals.
The rest of this paper is organized as follows. In the following section, we review some basic concepts related to PFS and the Choquet integral. In Section 3, we define some picture fuzzy point operators. In Section 4, by combining the point operators with Choquet integral operator, we propose the picture fuzzy point–Choquet averaging (PFPCA) operator, the picture fuzzy point–Choquet geometric (PFPCG) operator, the generalized picture fuzzy point–Choquet averaging (GPFPCA) operator and the generalized picture fuzzy point–Choquet geometric (GPFPCG) operator. Some prominent properties and special cases of these proposed operators are also studied. In Section 5, we introduce a novel method for solving MADM with picture fuzzy information based on the proposed operators. In Section 6, we provide an application example about assisting the hierarchical medical system to show the performance of new method.

2. Preliminaries

In the section, we briefly review some basic notions including PFS and the Choquet integral.

2.1. Picture Fuzzy Sets

Definition 1
[8]. Let X be an ordinary fixed set; then a picture fuzzy set P defined on X is given by
P = { x , μ p ( x ) , η p ( x ) , v p ( x ) | x X } ,
where μ p ( x ) is the positive degree of x X , and η p ( x ) and v p ( x ) are the neutral degree and negative degree, respectively, satisfying
μ p ( x ) + η p ( x ) + v p ( x ) 1 .
The uncertainty associated with PFS π P ( x ) = 1 μ p ( x ) η p ( x ) v p ( x ) is also defined. In the case η p ( x ) = 0 , PFS is reduced to the IFS, and when both μ p ( x ) , v p ( x ) = 0 , PFS is reduced to the fuzzy set.
For simplicity, we use the pair ( μ ( x ) , η ( x ) , v ( x ) ) to denote a general PFN that can be denoted by p = ( μ , η , v ) .
Given three PFNs p = ( μ , η , v ) , p 1 = ( μ 1 , η 1 , v 1 ) , p 1 = ( μ 2 , η 2 , v 2 ) , Cuong [8] defined the operations of intersection, union, complement and inclusion for them, which can be described as below:
p 1 p 2 = ( m i n ( μ 1 , μ 2 ) , m a x ( η 1 , η 2 ) , m a x ( v 1 , v 2 ) ) ,
p 1 p 2 = ( m a x ( μ 1 , μ 2 ) , m i n ( η 1 , η 2 ) , m i n ( v 1 , v 2 ) ) ,
p c = ( v , η , μ ) ,
p 1 p 2 ,   if   μ 1 μ 2 , η 1 η 2   and   v 1 v 2 .
Wei [9] further defines some operational laws for PFNs as shown below:
p 1 p 2 = ( ( μ 1 + μ 2 μ 1 μ 2 ) , η 1 η 2 , v 1 v 2 ) ,
p 1 p 2 = ( μ 1 μ 2 , η 1 + η 2 η 1 η 2 , v 1 + v 2 v 1 v 2 ) ,
λ p = ( 1 ( 1 μ ) λ , η λ , v λ ) ,
p λ = ( μ λ , 1 ( 1 η ) λ , 1 ( 1 v ) λ ) .
Definition 2
[13]. For two PFNs p 1 = ( μ 1 , η 1 , v 1 ) , p 2 = ( μ 2 , η 2 , v 2 ) , their relations are defined as follows:
p 1 p 2   iff   x X , μ 1 μ 2 , v 1 v 2 ,
p 1 = p 2   iff   x X , μ 1 = μ 2 , v 1 = v 2 .
In order to rank the PFNs, Garg [13] gave the score function and accuracy function of PFNs.
Definition 3
[13]. Suppose that p = ( μ , η , v ) is a PFN; then the score function of p is shown as follows:
S p = μ p v p .
Definition 4
[13]. Suppose that p = ( μ , η , v ) is a PFN; then the accuracy function of p is shown as follows:
H p = μ p + η p + v p .
Based on the score and accuracy function of PFN, Garg further defines the following ranking rules to compare two PFNs.
Definition 5.
For two PFNs:
if   S p 1 > S p 2 ,   then   p 1 > p 2 ,
if   S p 1 = S p 2 ,   then
if   H p 1 > H p 2 ,   t h e n   p 1 > p 2 ,
if   H p 1 = H p 2 ,   t h e n   p 1 = p 2 .

2.2. Choquet Integral Operator

The fuzzy measure can be used to define a weight on each combination of criteria in the Choquet integral model. In this subsection, we introduce the definitions of fuzzy measure and Choquet integral.
Definition 6
[33]. A fuzzy measure on X is a set function ρ : Γ ( x ) [ 0 , 1 ] , with the following conditions:
(1) 
ρ ( ϕ ) = 0 , ρ ( X ) = 1 (boundary conditions),
(2) 
A , B X and A B , then ρ ( A ) ρ ( B ) (monotonicity).
However, we generally need to determine 2 n 2 values for n criteria, which is quite complex, and thus it is not easy to give such fuzzy measure according to Definition 6. Therefore, the following σ -fuzzy measure ρ is further defined:
ρ ( A B )   =   ρ ( A ) + ρ ( B ) + σ ρ ( A ) ρ ( B ) ,
where A B = ϕ , and the parameter σ [ 1 , + ) denotes the interaction between attributes. In Equation (17):
(1)
If σ = 0, then σ -fuzzy measure ρ reduces to ρ ( A B ) = ρ ( A ) + ρ ( B ) , A B = ϕ , which is defined as an additive measure.
In this situation, if all the elements in X are independent, we get
ρ ( A ) = x i A ρ ( x i ) .
(2)
If all the elements in X are finite, then
ρ ( A ) = ρ ( i = 1 n x i ) = { 1 σ [ i = 1 n ( 1 + σ ρ ( x i ) ) 1 ] , σ 0 x i A ρ ( x i ) , σ = 0 ,
where x i x j = Φ , for i , j = 1 , 2 n , and i j .
(3)
If ρ 0 , then σ -fuzzy measure ρ reduces to ρ ( A B ) ρ ( A ) + ρ ( B ) , which is defined as a super-additive measure.
(4)
If 1 ρ 0 , then σ -fuzzy measure ρ reduces to ρ ( A B ) ρ ( A ) + ρ ( B ) , which is defined as a sub-additive measure.
When using a fuzzy measure to model the importance of decision criteria set S, a well-known aggregation function is the Choquet integral [16].
Definition 7.
Let f be a positive real-valued function on X and ρ be a fuzzy measure on X . The discrete Choquet integral of f with respect to ρ is defined as
( C ) f d ρ = i = 1 n [ ρ ( A σ ( i ) ) ρ ( A σ ( i 1 ) ) ] f σ ( i ) ,
where σ ( i ) denotes a permutation of ( 1 , 2 n ) such that f σ ( 1 ) f σ ( 2 ) f σ ( n ) , and A σ ( 0 ) = ϕ , A σ ( i ) = { x σ ( 1 ) , x σ ( i ) } .

3. Some Point Operations for Picture Fuzzy Numbers and Their Properties

Motivated by the idea of intuitionistic fuzzy point operators [28] and dual hesitant fuzzy point operators [32], we will define a series of picture fuzzy point operations to obtain more intensive information and further analyze some desirable properties of these operations, which are very useful in the remainder of this paper.
Definition 8.
For a PFN p = ( μ , η , v ) , let α , β , γ [ 0 , 1 ] , we define some PF point operators: PFNPFN as follows:
D α , β ( p ) = { μ p + α π p , η p + β π p , v p + ( 1 α β ) π p } ,
F α , β , γ ( p ) = { μ p + α π p , η p + β π p , v p + γ π p } ,
where α + β + γ 1
G α , β , γ ( p ) = { α μ p , β η p , γ v p } .
It is obvious that the above PF point operators transform a PFN into another one. From Equations (21) and (22), we know that D α ( p ) assigns all the uncertainty into the other three parts of a PFS, while F α , β , γ ( p ) only assigns part of the uncertainty. Meanwhile, we can get π D α ( p ) = 1 π p , and π F α , β , γ ( p ) = π p ( 1 α β γ ) , which means that F α , β , γ ( p ) and D α ( p ) can reduce the uncertainty of PFS, and increase the positive degree, neutral degree, and positive degree. Similarly, From Equation (23), we know that G α , β , γ ( p ) can reduce the positive degree, neutral degree, and positive degree, and π G α , β , γ = ( 1 α μ p β η p γ v p ) , which means that G α , β , γ ( p ) increases the uncertainty of PFS.
Then, we discuss some properties of the operator F α , β , γ ( p ) in detail.
Theorem 1.
Let p = ( μ , η , v ) be a PFN and taking α , β , γ [ 0 , 1 ] , then
( F α , β , γ ( p c ) ) c = F γ , β , α ( p ) ,
( G α , β , γ ( p c ) ) c = G γ , β , α ( p ) .
If
α = μ p μ p + η p + v p ,   β = η p μ p + η p + v p   and   γ = v p μ p + η p + v p   then   F α , β , γ ( p ) = ( α , β , γ ) .
Proof. 
We prove the Equation (24) holds, and (25), (26) can be proved analogously.
(1)
From p c = ( v , η , μ ) , we get
( F α , β , γ ( p c ) ) c = ( v p + α π p , η p + β π p , μ p + γ π p ) c = F γ , β , α ( p ) .
(2)
Then
F α , β , γ ( p ) = ( μ p + μ p μ p + η p + v p π p , η p + η p μ p + η p + v p π p , v p + v p μ p + η p + v p π p ) = ( μ p μ p + η p + v p , η p μ p + η p + v p , v p μ p + η p + v p ) = ( α , β , γ ) .
Based on the operations of the PFNs, let D α 0 ( p ) = F α , β , γ 0 ( p ) = G α , β , γ 0 ( p ) = H α , β , γ 0 ( p ) = p ; we then get the following Theorem 2. ☐
Theorem 2.
Let p = ( μ , η , v ) be a PFN and taking α , β , γ [ 0 , 1 ] , and α + β + γ 0 , then
D α n ( γ ) = { μ p + α π p , η p + β π p , v p + ( 1 α β ) π p } ,
F α , β , γ n ( p ) = ( μ p + α π p τ , η p + β π p τ , v p + γ π p τ ) ,
where τ = 1 ( 1 α β γ ) n α + β + γ ,
G α , β , γ n ( p ) = ( μ p α n , η p β n , v p γ n ) .
The proof of this theorem is provided in Appendix A.
In the following, a numeric example is forwarded to illustrate Theorems 1 and 2.
Example 1.
Let p = ( 0.15 ,   0.35 ,   0.25 ) be a PFN, then the point operators of p can be calculated according to Definition 8 (Suppose α = 0.4 , β = 0.3 , γ = 0.2 ). Firstly, we can obtain π p = 1 ( 0.15 + 0.35 + 0.25 ) = 0.25 , and τ = 1 ( 1 α β γ ) n α + β + γ = 1 0.1 n 0.9 , then we have
D α , β ( p ) = { 0.15 + 0.25 α , 0.35 + 0.25 β , 0.25 + 0.25 ( 1 α β ) } = ( 0.25 ,   0.425 ,   0.325 ) ,
F α , β , γ ( p ) = ( 0.15 + 0.25 α , 0.35 + 0.25 β , 0.25 + 0.25 γ ) = ( 0.25 ,   0.425 ,   0.3 ) ,
G α , β , γ ( p ) = ( 0.15 α , 0.35 β , 0.25 γ ) = ( 0.06 ,   0.105 ,   0.05 ) .
Similarly,
D α n ( γ ) = ( 0.25 ,   0.425 ,   0.325 ) ,
F α , β , γ n ( p ) = ( 0.15 + 1 0.1 n 9 , 0.35 + 0.75 × ( 1 0.1 n ) 9 , 0.25 + 5 × ( 1 0.1 n ) 9 ) ,
G α , β , γ n ( p ) = { 0.15 × 0.4 n , 0.35 × 0.3 n , 0.25 × 0.2 n } .
From Theorem 2, we can easily obtain the following properties.
Theorem 3.
Let p = ( μ , η , v ) be a PFS, and n be a positive integer. Taking α , β , γ [ 0 , 1 ] , then
( F α , β , γ n ( p c ) ) c = F γ , β , α n ( p ) ,
( G α , β , γ n ( p c ) ) c = G γ , β , α n ( p ) .
Theorem 4.
Let p = ( μ , η , v ) be a PFS, and n be a positive integer. Taking α , β , γ [ 0 , 1 ] , the relation is defined as A B if and only if μ F α , β , γ n ( p ) μ F α , β , γ n 1 ( p ) , and v F α , β , γ n ( p ) v F α , β , γ n 1 ( p ) , and then
( F α , β , γ n ( p c ) ) c = F γ , β , α n ( p ) ,
π F α , β , γ n π F α , β , γ n 1 .
If   α = μ p μ p + η p + v p , β = η p μ p + η p + v p , γ = v p μ p + η p + v p ,   then   F α , β , γ n ( p ) = F α , β , γ ( γ ) .
Definition 9.
Let α , β , γ [ 0 , 1 ] , and α + β + γ ≤ 1. We define the following limit:
lim n F α , β , γ n ( p ) = lim n { μ F α , β , γ n ( p ) , η F α , β , γ n ( p ) , v F α , β , γ n ( p ) } .
Theorem 5.
Let α , β , γ [ 0 , 1 ] , and α + β + γ 1 ; then we have
lim n F α , β , γ n ( p ) = = D α α + β + γ , β α + β + γ n ( p ) .
Proof of Theorem 5.
According to Theorem 7, we get
lim n μ F ξ ζ ( γ ) n = lim n ( μ p + α π p 1 ( 1 α β γ ) n α + β + γ ) = μ p + α α + β + γ π p ,
lim n η F ξ ζ ( γ ) n = lim n ( η p + β π p 1 ( 1 α β γ ) n α + β + γ ) = η p + β α + β + γ π p ,
lim n v F ξ ζ ( γ ) n = = v p + γ α + β + γ π p .
So we have
lim n F α , β , γ n ( p ) = lim n { μ F α , β , γ n ( p ) , η F α , β , γ n ( p ) , v F α , β , γ n ( p ) } = { μ p + α α + β + γ π p , η p + β α + β + γ π p , v p + γ α + β + γ π p } = D α α + β + γ , β α + β + γ n ( p )
. ☐

4. Picture Fuzzy Point–Choquet Integral Aggregation Operators and Their Properties

In order to get more intensive information from PFS and efficiently deal with correlations among arguments at the same time, we combine picture fuzzy point operators with the Choquet integral operator to propose some new class of aggregation operators for aggregating picture fuzzy information in this section. Some desirable properties of proposed aggregation operators are also discussed in detail.

4.1. Picture Fuzzy Point–Choquet Averaging Operator

Definition 10.
Let Ω be the set of all PFNs, and p i = ( μ i , η i , v i ) ( i = 1 , 2 , , m ) be a collection of PFNs, taking α i , β i , γ i [ 0 , 1 ] . Then we define the series of PFPCA operators): Ω m Ω , if
F ( C 1 ) p d ρ = P F P C A D α , β n ( p 1 , p 2 p n ) = i = 1 m ( ρ ( A σ ( i ) ) ρ ( A σ ( i 1 ) ) ) D α σ ( i ) , β σ ( i ) n ( p σ ( i ) ) ,
F ( C 2 ) p d ρ = P F P C A F α , β , γ n ( p 1 , p 2 p n ) = i = 1 m ( ρ ( A σ ( i ) ) ρ ( A σ ( i 1 ) ) ) F α σ ( i ) , β σ ( i ) , γ σ ( i ) n ( p σ ( i ) ) ,
F ( C 3 ) p d ρ = P F P C A G α , β , γ n ( p 1 , p 2 p n ) = i = 1 m ( ρ ( A σ ( i ) ) ρ ( A σ ( i 1 ) ) ) G α σ ( i ) , β σ ( i ) , γ σ ( i ) n ( p σ ( i ) ) ,
where σ ( i ) denotes a permutation of ( 1 , 2 m ) such that p σ ( 1 ) p σ ( 2 ) p σ ( m ) , and G σ ( i ) is the attribute corresponding to p σ ( i ) .
By operational laws defined in Section 2.1, we can obtain the following theorem.
Theorem 6.
Let p i = ( μ i , η i , v i ) ( i = 1 , 2 , , m ) be a collection of PFNs, and σ ( i ) be a permutation of ( 1 , 2 m ) such that p σ ( 1 ) p σ ( 2 ) p σ ( m ) , G σ ( i ) is the attribute corresponding to p σ ( i ) , and A σ ( 0 ) = ϕ A σ ( i ) = { G σ ( 1 ) , G σ ( i ) } , taking ω ˜ i = ρ ( A σ ( i ) ) ρ ( A σ ( i 1 ) ) , α i , β i , γ i [ 0 , 1 ] , α i + β i + γ i 1 . Then, the aggregated values by the series of PFPCA operators are also PFNs:
P F P C A D α , β n ( p 1 , p 2 p n ) = { 1 i = 1 m ( 1 ( μ p σ ( i ) + α i π p σ ( i ) ) ) ω ˜ i , 1 i = 1 m ( 1 ( η p σ ( i ) + β i π p σ ( i ) ) ) ω ˜ i i = 1 m ( v p σ ( i ) + ( 1 α i β i ) π p σ ( i ) ) ω ˜ i } ,
P F P C A F α , β , γ n ( p 1 , p 2 , , p m ) = ( ( 1 i = 1 m ( 1 μ F α i , β i , γ i n ( p σ ( i ) ) ) ω ˜ i ) , i = 1 m η F α i , β i , γ i n ( p σ ( i ) ) ω ˜ i , i = 1 m v F α i , β i , γ i n ( p σ ( i ) ) ω ˜ i )
where
μ F α i , β i , γ i n ( p i ) = μ p σ ( i ) + α i π p σ ( i ) 1 ( 1 α i β i γ i ) n α i + β i + γ i ,
η F α i , β i , γ i n ( p i ) = η p σ ( i ) + β i π p σ ( i ) 1 ( 1 α i β i γ i ) n α i + β i + γ i ,
v F α i , β i , γ i n ( p σ ( i ) ) = v p σ ( i ) + γ i π p σ ( i ) 1 ( 1 α i β i γ i ) n α i + β i + γ i ,
P F P C A G α , β , γ n ( p 1 , p 2 , , p m ) = ( ( 1 i = 1 m ( 1 μ p σ ( i ) α i n ) ω ˜ i ) , i = 1 m ( η p σ ( i ) β i n ) ω ˜ i , i = 1 m ( v p σ ( i ) γ i n ) ω ˜ i ) .
The proof of this theorem is provided in Appendix B.
In the following, a numeric example is forwarded to illustrate Theorem 6.
Example 2.
Let p 1 = ( 0.25 ,   0.35 ,   0.15 ) , p 2 = ( 0.42 ,   0.18 ,   0.37 ) , p 3 = ( 0.34 ,   0.27 ,   0.16 ) be PFN. Then we aggregate the three PFNs by the following steps:
Step 1. Identify the fuzzy measure of the n attributes of G according to Equations (17) and (19). Suppose that the fuzzy measures of attributes of G are given as follows:
ρ ( G 1 ) = 0.38 , ρ ( G 2 ) = 0.27 , ρ ( G 3 ) = 0.36 .
Firstly, according to Equation (19), the value of σ is obtained: σ = 0.029 , and then the fuzzy measures of attribute sets of G = { G 1 , G 2 , G 3 , G 4 } can be calculated by Equation (13), shown as follows:
ρ ( G 1 , G 2 ) = 0.65 , ρ ( G 1 , G 3 ) = 0.74 ,   ρ ( G 2 , G 3 ) = 0.63 ,   ρ ( G 1 , G 2 , G 3 ) = 1 .
Step 2. By score functions, we rearrange the three PFNs in descending order, shown as follows:
s ( p 1 ) = 0.1 , s ( p 2 ) = 0.05 , s ( p 3 ) = 0.18 ,
p σ ( 1 ) = ( 0.34 ,   0.27 ,   0.16 ) , p σ ( 2 ) = ( 0.25 ,   0.35 ,   0.15 ) , p σ ( 3 ) = ( 0.42 ,   0.18 ,   0.37 ) .
Then we can get
A σ ( 1 ) = { G 3 } , A 1 σ ( 2 ) = { G 1 , G 3 } , A 1 σ ( 4 ) = { G 1 , G 2 , G 3 } ,
ρ A σ ( 1 ) = ρ G 3 = 0.36 , ρ A σ ( 2 ) ρ A σ ( 1 ) = ρ G 1 G 3 ρ G 3 = 0.38 , ρ A σ ( 3 ) ρ A σ ( 2 ) = ρ G 2 G 3 G 4 ρ G 1 G 3 = 0.26 .
.
Step 3. Calculate the point operators of pi according to Definition 8 (Suppose α = 0.3 , β = 0.4 , γ = 0.1 , n = 3 ). Firstly, we can obtain π p σ ( 1 ) = 1 ( 0.25 + 0.35 + 0.15 ) = 0.25 , π p σ ( 2 ) = 1 ( 0.42 + 0.18 + 0.37 ) = 0.03 , π p σ ( 3 ) = 1 ( 0.34 + 0.27 + 0.16 ) = 0.23 . Then we have
F α , β , γ 3 ( p σ ( 1 ) ) = ( 0.25 + 0.25 × 1 0.2 3 0.8 , 0.35 + 0.25 × 1 0.2 3 0.8 , 0.15 + 0.25 × 1 0.2 3 0.8 ) = ( 0.56 , 0.66 , 0.46 ) ,
F α , β , γ 3 ( p σ ( 2 ) ) = ( 0.42 + 0.03 × 1 0.2 3 0.8 , 0.18 + 0.03 × 1 0.2 3 0.8 , 0.37 + 0.03 × 1 0.2 3 0.8 ) = ( 0.46 , 0.22 , 0.41 ) , F α , β , γ 3 ( p σ ( 3 ) ) = ( 0.34 + 0.23 × 1 0.2 3 0.8 , 0.27 + 0.23 × 1 0.2 3 0.8 , 0.16 + 0.23 × 1 0.2 3 0.8 ) = ( 0.63 , 0.56 , 0.45 )
Step 4. Utilize the P F P C A F α , β , γ n ( p 1 , p 2 , p 3 ) operator to aggregate the three PFNs and get the aggregated p as follows:
p = i = 1 m ( ρ ( A σ ( i ) ) ρ ( A σ ( i 1 ) ) ) F α i , β i , γ i n ( p σ ( i ) ) = ( ( 1 i = 1 m ( 1 μ F α i , β i , γ i n ( p σ ( i ) ) ) ρ ( A σ ( i ) ) ρ ( A σ ( i 1 ) ) ) , i = 1 m ( η F α i , β i , γ i n ( p σ ( i ) ) ) ρ ( A σ ( i ) ) ρ ( A σ ( i 1 ) ) , i = 1 m ( v F α i , β i , γ i n ( p σ ( i ) ) ) ρ ( A σ ( i ) ) ρ ( A σ ( i 1 ) ) ) = ( ( 1 ( 1 0.56 ) 0.36 × ( 1 0.46 ) 0.38 × ( 1 0.63 ) 0.26 ) , 0.66 0.36 × 0.22 0.38 × 0.56 0.26 , 0.46 0.36 × 0.41 0.38 × 0.45 0.26 ) .
Example 2 gives a detailed portrait of the P F P C A F α , β , γ n ( p 1 , p 2 , p 3 ) operator. It should be pointed out that the P F P C A F α , β , γ n ( p 1 , p 2 , p 3 ) operator includes a reorder step and it is similar to the famous ordered weighted averaging (OWA) operator. In the following, we discuss some properties of the above PFPCA operators.
Theorem 7.
Let p i = ( μ i , η i , v i ) ( i = 1 , 2 , , m ) be a collection of PFNs. Taking k 0 , then
P F P C A D α , β n ( k p 1 , k p 2 , , k p m ) = k P F P C A D α , β n ( p 1 , p 2 , , p m ) ,
P F P C A F α , β , γ n ( k p 1 , k p 2 , , k p m ) = k P F P C A F α , β , γ n ( p 1 , p 2 , , p m ) ,
P F P C A G α , β , γ n ( k p 1 , k p 2 , , k p m ) = k P F P C A G α , β , γ n ( p 1 , p 2 , , p m ) .
Proof. 
We prove the Equation (50) holds for all m, and the others can be proved analogously.
By the operational law in Section 2.2, we have
k p i = ( 1 ( 1 μ i ) k , η i k , v i k )  
and
P F P C A F α , β , γ n ( k p 1 , k p 2 , , k p m ) = ( 1 i = 1 m ( 1 μ F α i , β i , γ i n ( p σ ( i ) ) ) k ω ˜ i , i = 1 m η F α i , β i , γ i n ( p i ) k ω ˜ i , i = 1 m v F α i , β i , γ i n ( p σ ( i ) ) k ω ˜ i ) ,
and hence
k P F P C A F α , β , γ n ( p 1 , p 2 , , p m ) = k ( 1 i = 1 m ( 1 μ F α i , β i , γ i n ( p σ ( i ) ) ) ω ˜ i , i = 1 m η F α i , β i , γ i n ( p σ ( i ) ) k ω ˜ i , i = 1 m v F α i , β i , γ i n ( p σ ( i ) ) ω ˜ i ) ( 1 i = 1 m ( 1 μ F α i , β i , γ i n ( p σ ( i ) ) ) k ω ˜ i , i = 1 m η F α i , β i , γ i n ( p σ ( i ) ) k ω ˜ i , i = 1 m v F α i , β i , γ i n ( p σ ( i ) ) k ω ˜ i ) = P F P C A F α , β , γ n ( k p 1 , k p 2 , , k p m ) .
Therefore, Equation (50) holds, which completes the proof. ☐
Theorem 8.
Let pi and qi be two collections of PFNs, then
P F P C A D α , β n ( p 1 q 1 , p 2 q 2 , , p n q m ) = P F P C A D α , β n ( p 1 , p 2 , , p m ) P F P C A D α , β n ( q 1 , q 2 , , q m )
P F P C A F α , β , γ n ( p 1 q 1 , p 2 q 2 , , p n q m ) = P F P C A F α , β , γ n ( p 1 , p 2 , , p m ) P F P C A F α , β , γ n ( q 1 , q 2 , , q m )
P F P C A G α , β , γ n ( p 1 q 1 , p 2 q 2 , , p n q m ) = P F P C A G α , β , γ n ( p 1 , p 2 , , p m ) P F P C A G α , β , γ n ( q 1 , q 2 , , q m )
Proof. 
We prove the Equation (53) holds for all m, and the others can be proved analogously.
By the operational law in Section 2.2, we have
p i q i = ( μ p i + μ q i μ p i μ q i , η p i η q i , v p i v q i ) ,
P F P C A F α , β , γ n ( p 1 q 1 , p 2 q 2 , , p n q m ) = ( ( 1 i = 1 m ( 1 μ F α i , β i , γ i n ( p σ ( i ) ) ) ω ˜ i ( 1 μ F α i , β i , γ i n ( q σ ( i ) ) ) ω ˜ i ) , i = 1 m η F α i , β i , γ i n ( p σ ( i ) ) ω ˜ i η F α i , β i , γ i n ( q σ ( i ) ) ω ˜ i , i = 1 m v F α i , β i , γ i n ( p σ ( i ) ) ω ˜ i v F α i , β i , γ i n ( q σ ( i ) ) ω ˜ i ) ,
P F P C A F α , β , γ n ( p 1 , p 2 , , p m ) P F P C A F α , β , γ n ( q 1 , q 2 , , q m ) = ( 1 i = 1 m ( 1 μ F α i , β i , γ i n ( p σ ( i ) ) ) ω ˜ i ( 1 μ F α i , β i , γ i n ( q σ ( i ) ) ) ω ˜ i , i = 1 m η F α i , β i , γ i n ( p σ ( i ) ) ω ˜ i η F α i , β i , γ i n ( q σ ( i ) ) ω ˜ i , i = 1 m ν F α i , β i , γ i n ( p σ ( i ) ) ω ˜ i ν F α i , β i , γ i n ( q σ ( i ) ) ω ˜ i ) = P F P C A F α , β , γ n ( p 1 , p 2 , , p m ) P F P C A F α , β , γ n ( q 1 , q 2 , , q m ) .
Therefore, Equation (53) holds, which completes the proof. ☐
Theorem 9.
(Idempotency). If p i = ( μ i , η i , v i ) are equal, i.e., p i = p = ( μ , η , v ) for all i, then
P F P C A D α , β n ( p 1 , p 2 , , p m ) = D α , β n ,
P F P C A F α , β , γ n ( p 1 , p 2 , , p m ) = F α , β , γ n ,
P F P C A G α , β , γ n ( p 1 , p 2 , , p m ) = G α , β , γ n .
Proof. 
We prove the Equation (56) holds for all m, and the others can be proved analogously.
Since p i = p = ( μ , η , v ) for all i, then
P F P C A F α , β , γ n ( p 1 , p 2 , , p m ) = ( ( 1 i = 1 m ( 1 μ F α i , β i , γ i n ( p σ ( i ) ) ) ω ˜ i ) , i = 1 m η F α i , β i , γ i n ( p σ ( i ) ) ω ˜ i , i = 1 m v F α i , β i , γ i n ( p σ ( i ) ) ω ˜ i ) = ( 1 ( 1 μ F α i , β i , γ i n ( p ) ) i = 1 m ω i , ( η F α i , β i , γ i n ( p ) ) i = 1 m ω i , ( v F α i , β i , γ i n ( p ) ) i = 1 m ω i ) = ( 1 ( 1 μ F α i , β i , γ i n ( p ) ) , η F α i , β i , γ i n ( p ) , v F α i , β i , γ i n ( p ) ) = ( μ F α i , β i , γ i n ( p ) , η F α i , β i , γ i n ( p ) , v F α i , β i , γ i n ( p ) ) = F α , β , γ n .
 ☐
Theorem 10.
(Monotonicity) Let p i = ( μ p i , η p i , v p i ) and q i = ( μ q i , η q i , v q i ) ( i = 1 , 2 , , m ) be two collections of PFN. If μ p i μ q i , η p i η q i and v p i v q i holds for all i ( i = 1 , 2 , , m ) , then
P F P C A D α , β n ( p 1 , p 2 , , p m ) P F P C A D α , β n ( q 1 , q 2 , , q m ) ,
P F P C A F α , β , γ n ( p 1 , p 2 , , p m ) P F P C A F α , β , γ n ( q 1 , q 2 , , q m ) ,
P F P C A G α , β , γ n ( p 1 , p 2 , , p m ) P F P C A G α , β , γ n ( q 1 , q 2 , , q m ) .
Proof. 
We prove the Equation (59) holds for all m, and the others can be proved analogously.
By Theorem 6, we get
P F P C A F α , β , γ n ( p 1 , p 2 , , p m ) = ( ( 1 i = 1 m ( 1 μ F α i , β i , γ i n ( p σ ( i ) ) ) ω ˜ i ) , i = 1 m η F α i , β i , γ i n ( p σ ( i ) ) ω ˜ i , i = 1 m v F α i , β i , γ i n ( p σ ( i ) ) ω ˜ i ) , P F P C A F α , β , γ n ( q 1 , q 2 , , q m ) = ( ( 1 i = 1 m ( 1 μ F α i , β i , γ i n ( q σ ( i ) ) ) ω ˜ i ) , i = 1 m η F α i , β i , γ i n ( q σ ( i ) ) ω ˜ i , i = 1 m v F α i , β i , γ i n ( q σ ( i ) ) ω ˜ i ) .
Since μ p i μ q i and v p i v q i , we can get
( 1 i = 1 m ( 1 μ F α i , β i , γ i n ( p σ ( i ) ) ) ω ˜ i ) ( 1 i = 1 m ( 1 μ F α i , β i , γ i n ( q σ ( i ) ) ) ω ˜ i )  
and
i = 1 m η F α i , β i , γ i n ( p σ ( i ) ) ω ˜ i i = 1 m η F α i , β i , γ i n ( q σ ( i ) ) ω ˜ i , i = 1 m v F α i , β i , γ i n ( p σ ( i ) ) ω ˜ i i = 1 m v F α i , β i , γ i n ( q σ ( i ) ) ω ˜ i .
By Definition 6, we get P F P C A F α , β , γ n ( p 1 , p 2 , , p m ) P F P C A F α , β , γ n ( q 1 , q 2 , , q m ) . ☐
Theorem 11.
(Boundedness) Let p i = ( μ i , η i , v i ) ( i = 1 , 2 , , m ) be a collection of PFNs, then
d D α , β n P F P C A D α , β n d D α , β n + ,
d F α , β , γ n P F P C A F α , β , γ n ( p 1 , p 2 , , p m ) d F α , β , γ n + ,
d G α , β , γ n P F P C A G α , β , γ n ( p 1 , p 2 , , p m ) d G α , β , γ n + ,
where d Δ + = ( m a x i ( μ Δ ) , m i n i ( v Δ ) ) and d Δ = ( m i n i ( μ Δ ) , m a x i ( v Δ ) ) and Δ denotes D α , β n , F α , β , γ n , G α , β , γ n .
Proof. 
We prove the Equation (62) holds for all m, and the others can be proved analogously.
From Theorem 6, we can get
P F P C A F α , β , γ n ( p 1 , p 2 , , p m ) = ( ( 1 i = 1 m ( 1 μ F α i , β i , γ i n ( p σ ( i ) ) ) ω ˜ i ) , i = 1 m η F α i , β i , γ i n ( p σ ( i ) ) ω ˜ i , i = 1 m v F α i , β i , γ i n ( p σ ( i ) ) ω ˜ i ) .
By the definition of d F ξ , ζ n + , d F ξ , ζ n we can get
1 i = 1 m ( 1 m i n ( μ F α i , β i , γ i n ( p σ ( i ) ) ) ) ω ˜ i 1 i = 1 m ( 1 μ F α i , β i , γ i n ( p σ ( i ) ) ) ω ˜ i 1 i = 1 m ( 1 m a x ( μ F α i , β i , γ i n ( p σ ( i ) ) ) ) ω ˜ i F P C A F α , β , γ n ( p 1 , p 2 , , p m ) P F P C A F α , β , γ n ( p 1 , p 2 , , p m ) F P C A F α , β , γ n ( p 1 + , p 2 + , , p m + ) .
By Definition 7, we get d F α , β , γ n P F P C A F α , β , γ n ( p 1 , p 2 , , p m ) d F α , β , γ n + . ☐
By giving different values of the parameters, we get the following special cases.
Theorem 12.
Let p i = ( μ i , η i , v i ) ( i = 1 , 2 , , m ) be a collection of PFNs, then
(1) 
If ω i = ω ˜ i = ρ ( A σ ( i ) ) ρ ( A σ ( i 1 ) ) , then the series of PFPCA operators are all reduced to the series of picture fuzzy point averaging (PFPWA) operators. In particular, if m i = 1 m , ( i = 1 , 2 , , m ) , then PFPCA operators is reduced to a picture fuzzy averaging (PFA) operator, which is defined as:
P F A = ( 1 ( i = 1 m ( 1 μ i 2 ) ) 1 / m , ( i = 1 m η i ) 1 / m , ( i = 1 m v i ) 1 / m ) .  
(2) 
If n = 0 , ω ˜ i = ρ ( A σ ( i ) ) ρ ( A σ ( i 1 ) ) , and ρ ( A ) = i = 1 | A | ω ˜ i for all A X , where | A | is the number of the elements in set A, ω ˜ = ( ω ˜ 1 , ω ˜ 2 , ω ˜ m ) T , ω ˜ i [ 0 , 1 ] , i = 1 m ω ˜ i = 1 , then the PFPCA operator is reduced to a picture fuzzy order-weighted averaging (PFOWA) operator defined by Garg [13].
(3) 
If n = 0 , ω ˜ i = ρ ( A σ ( i ) ) ρ ( A σ ( i 1 ) ) , then the series of PFPCA operators are all reduced to the series of picture fuzzy weighted averaging (PFWA) operators defined by Garg [13].

4.2. Picture Fuzzy Point–Choquet Geometric Operator

Definition 11.
Let Ω be the set of all PFNs, and p i = ( μ i , η i , v i ) ( i = 1 , 2 , , m ) be a collection of PFNs, taking α i , β i , γ i [ 0 , 1 ] . Then we define the series of PFPCG operators: Ω m Ω , if
F ( C 4 ) p d ρ = P F P C G D α , β n ( p 1 , p 2 p m ) = i = 1 m ( ρ ( A σ ( i ) ) ρ ( A σ ( i 1 ) ) ) D α σ ( i ) , β σ ( i ) n ( p σ ( i ) ) ,
F ( C 5 ) p d ρ = P F P C G F α , β , γ n ( p 1 , p 2 p m ) = i = 1 m ( ρ ( A σ ( i ) ) ρ ( A σ ( i 1 ) ) ) F α σ ( i ) , β σ ( i ) , γ σ ( i ) n ( p σ ( i ) ) ,
F ( C 6 ) p d ρ = P F P C G G α , β , γ n ( p 1 , p 2 p m ) = i = 1 m ( ρ ( A σ ( i ) ) ρ ( A σ ( i 1 ) ) ) G α σ ( i ) , β σ ( i ) , γ σ ( i ) n ( p σ ( i ) ) ,
where σ ( i ) denotes a permutation of ( 1 , 2 m ) such that p σ ( 1 ) p σ ( 2 ) p σ ( m ) , and G σ ( i ) is the attribute corresponding to p σ ( i ) , A σ ( i ) = { G σ ( 1 ) , G σ ( i ) } , A σ ( 0 ) = ϕ .
By operational laws defined in Section 2.1, we can obtain the following theorem.
Theorem 13.
Let p i = ( μ i , η i , v i ) ( i = 1 , 2 , , m ) be a collection of PFNs, and σ ( i ) be a permutation of ( 1 , 2 m ) such that p σ ( 1 ) p σ ( 2 ) p σ ( m ) , G σ ( i ) is the attribute corresponding to p σ ( i ) , A σ ( i ) = { G σ ( 1 ) , G σ ( i ) } , A σ ( 0 ) = ϕ . Taking ω ˜ i = ρ ( A σ ( i ) ) ρ ( A σ ( i 1 ) ) , α i , β i , γ i [ 0 , 1 ] , α i + β i + γ i 1 , then the aggregated values by the series of PFPCG operators are also PFNs, and
P F P C G D α , β n ( p 1 , p 2 p m ) = { i = 1 m ( μ p σ ( i ) + ( 1 γ i β i ) π p σ ( i ) ) ω ˜ i , 1 i = 1 m ( 1 ( η p σ ( i ) + β i π p σ ( i ) ) ) ω ˜ i , i = 1 m ( 1 ( v p σ ( i ) + γ i π p σ ( i ) ) ) ω ˜ i }
P F P C G F α , β , γ n ( p 1 , p 2 , , p m ) = ( i = 1 m μ F α i , β i , γ i n ( p σ ( i ) ) ω ˜ i , 1 i = 1 m ( 1 η F α i , β i , γ i n ( p σ ( i ) ) ) ω ˜ i , 1 i = 1 m ( 1 v F α i , β i , γ i n ( p σ ( i ) ) ) ω ˜ i )  
where
μ F α i , β i , γ i n ( p i ) = μ p σ ( i ) + α i π p σ ( i ) 1 ( 1 α i β i γ i ) n α i + β i + γ i ,
η F α i , β i , γ i n ( p i ) = η p σ ( i ) + β i π p σ ( i ) 1 ( 1 α i β i γ i ) n α i + β i + γ i ,
v F α i , β i , γ i n ( p σ ( i ) ) = v p σ ( i ) + γ i π p σ ( i ) 1 ( 1 α i β i γ i ) n α i + β i + γ i
P F P C G G α , β , γ n ( p 1 , p 2 , , p m ) = ( ( 1 i = 1 m ( 1 μ p σ ( i ) α i n ) ω ˜ i ) , i = 1 m ( η p σ ( i ) β i n ) ω ˜ i , i = 1 m ( v p σ ( i ) γ i n ) ω ˜ i )
Theorem 14.
Let p i = ( μ i , η i , v i ) ( i = 1 , 2 , , m ) be a collection of PFNs. Taking k 0 , then
P F P C G D α , β n ( p 1 k , p 2 k , , p m k ) = ( P F P C G D α , β n ( p 1 , p 2 , , p m ) ) k ,
P F P C G F α , β , γ n ( p 1 k , p 2 k , , p m k ) = ( P F P C G F α , β , γ n ( p 1 , p 2 , , p m ) ) k ,
P F P C G G α , β , γ n ( p 1 k , p 2 k , , p m k ) = ( P F P C G G α , β , γ n ( p 1 , p 2 , , p m ) ) k .
Theorem 15.
Let p i = ( μ p i , η p i , v p i ) and q i = ( μ q i , η q i , v q i ) ( i = 1 , 2 , , m ) be two collections of PFNs, then
P F P C G D α , β n ( p 1 q 1 , p 2 q 2 , , p m q m ) = P F P C G D α , β n ( p 1 , p 2 , , p m ) P F P C G D α , β n ( q 1 , q 2 , , q m ) ,
P F P C G F α , β , γ n ( p 1 q 1 , p 2 q 2 , , p m q m ) = P F P C G F α , β , γ n ( p 1 , p 2 , , p m ) P F P C G F α , β , γ n ( q 1 , q 2 , , q m ) ,
P F P C G G α , β , γ n ( p 1 q 1 , p 2 q 2 , , p m q m ) = P F P C G G α , β , γ n ( p 1 , p 2 , , p m ) P F P C G G α , β , γ n ( q 1 , q 2 , , q m ) .
Parallel to Theorems 9–11, the series of PFPCG operators have properties similar to PFPCA operators such as idempotency, monotonicity, and boundedness under some conditions, which are omitted in order to save space.
ω i = ω ˜ i = ρ ( A σ ( i ) ) ρ ( A σ ( i 1 ) ) .  

4.3. Generalized Picture Fuzzy Point–Choquet Averaging Operator

Definition 12.
Let p i = ( μ i , η i , v i ) ( i = 1 , 2 , , m ) be a collection of PFNs, taking α i , β i , γ i [ 0 , 1 ] , λ 0 , and α i + β i + γ i 1 . Then we define a series of GPFPCA operators: Ωm → Ω, if
F ( C 7 ) p d ρ = G P F P C A D α , β n ( p 1 , p 2 p n ) = ( i = 1 m ( ρ ( A σ ( i ) ) ρ ( A σ ( i 1 ) ) ) ( D α σ ( i ) , β σ ( i ) n ( p σ ( i ) ) ) λ ) 1 / λ ,
F ( C 8 ) p d ρ = G P F P C A F α , β , γ n ( p 1 , p 2 p n ) = ( i = 1 m ( ρ ( A σ ( i ) ) ρ ( A σ ( i 1 ) ) ) ( F α σ ( i ) , β σ ( i ) , γ σ ( i ) n ( p σ ( i ) ) ) λ ) 1 / λ ,
F ( C 9 ) p d ρ = G P F P C A G α , β , γ n ( p 1 , p 2 p n ) = ( i = 1 m ( ρ ( A σ ( i ) ) ρ ( A σ ( i 1 ) ) ) ( G α σ ( i ) , β σ ( i ) , γ σ ( i ) n ( p σ ( i ) ) ) λ ) 1 / λ .
where G σ ( i ) is the attribute corresponding to p σ ( i ) , A σ ( i ) = { G σ ( 1 ) , G σ ( i ) } , A σ ( 0 ) = ϕ , and σ ( i ) denotes a permutation of ( 1 , 2 , , m ) such that p σ ( 1 ) p σ ( 2 ) p σ ( m ) .
By operational laws defined in Section 2.1, we can obtain the following theorem.
Theorem 16.
Let p i = ( μ i , η i , v i ) ( i = 1 , 2 , , m ) be a collection of PFNs, and σ ( i ) be a permutation of ( 1 , 2 m ) such that p σ ( 1 ) p σ ( 2 ) p σ ( m ) , G σ ( i ) is the attribute corresponding to p σ ( i ) , A σ ( 0 ) = ϕ , A σ ( i ) = { G σ ( 1 ) , G σ ( i ) } , and taking ω ˜ i = ρ ( A σ ( i ) ) ρ ( A σ ( i 1 ) ) , α i , β i , γ i [ 0 , 1 ] , α i + β i + γ i 1 , then the aggregated values by the series of GPFPCA operators are also PFNs.
(1)
G P F P C A D α , β n ( p 1 , p 2 p n ) = { 1 i = 1 m ( 1 ( μ p σ ( i ) + α i π p σ ( i ) ) λ ) ω ˜ i , 1 ( 1 i = 1 m ( 1 ( 1 η p σ ( i ) ( 1 β i ) π p σ ( i ) ) λ ) ω ˜ j ) 1 / λ ,   1 ( 1 i = 1 m ( 1 ( 1 v p σ ( i ) ( 1 γ i ) π p σ ( i ) ) λ ) ω ˜ j ) 1 / λ } ;
(2)
G P F P C A F α , β , γ n ( p 1 , p 2 , , p m ) = ( ( 1 i = 1 m ( 1 μ F α i , β i , γ i n ( p σ i ) λ ) ω ˜ i ) 1 λ , 1 [ 1 i = 1 m ( 1 ( 1 η F α i , β i , γ i n ( p σ i ) ) λ ) ω ˜ i ] 1 λ , 1 [ 1 i = 1 m ( 1 ( 1 v F α i , β i , γ i n ( p σ i ) ) λ ) ω ˜ i ] 1 λ )
where
μ F α i , β i , γ i n ( p i ) = μ p σ ( i ) + α i π p σ ( i ) 1 ( 1 α i β i γ i ) n α i + β i + γ i ,
η F α i , β i , γ i n ( p i ) = η p σ ( i ) + β i π p σ ( i ) 1 ( 1 α i β i γ i ) n α i + β i + γ i ,
v F α i , β i , γ i n ( p σ ( i ) ) = v p σ ( i ) + γ i π p σ ( i ) 1 ( 1 α i β i γ i ) n α i + β i + γ i ;
(3) G P F P C A F α , β , γ n ( p 1 , p 2 , , p m ) =
( ( 1 i = 1 m ( 1 μ p σ ( i ) λ α i n λ ) ω ˜ i ) 1 λ , 1 [ 1 i = 1 m ( 1 ( 1 η p σ ( i ) β i n ) λ ) ω ˜ i ] 1 λ , 1 [ 1 i = 1 m ( 1 ( 1 v p σ ( i ) γ i n ) λ ) ω ˜ i ] 1 λ ) .
Parallel to Theorems 9–11, the series of GPFPCA operators have properties similar to PFPCA operators such as idempotency, monotonicity, and boundedness under some conditions, which are omitted in order to save space.

4.4. Generalized Picture Fuzzy Point–Choquet Geometric Ooperator

Definition 13.
Let p i = ( μ i , η i , v i ) ( i = 1 , 2 , , m ) be a collection of PFNs, taking α i , β i , γ i [ 0 , 1 ] , λ 0 , and α i + β i + γ i 1 . Then we define a series of GPFPCG operators: Ωm → Ω, if
F ( C 10 ) p d ρ = G P F P C G D α , β n ( p 1 , p 2 p m ) = 1 λ i = 1 m ( λ D α σ ( i ) , β σ ( i ) n ( p σ ( i ) ) ) ( ρ ( A σ ( i ) ) ρ ( A σ ( i 1 ) ) ) ,
F ( C 11 ) p d ρ = G P F P C G F α , β , γ n ( p 1 , p 2 p m ) = 1 λ i = 1 m ( λ F α σ ( i ) , β σ ( i ) γ σ ( i ) n ( p σ ( i ) ) ) ( ρ ( A σ ( i ) ) ρ ( A σ ( i 1 ) ) ) ,
F ( C 12 ) p d ρ = G P F P C G G α , β , γ n ( p 1 , p 2 p m ) = 1 λ i = 1 m ( λ G α σ ( i ) , β σ ( i ) γ σ ( i ) n ( p σ ( i ) ) ) ( ρ ( A σ ( i ) ) ρ ( A σ ( i 1 ) ) ) .
Similarly, we can obtain the following theorem:
Theorem 17.
Let p i = ( μ i , η i , v i ) ( i = 1 , 2 , , m ) be a collection of PFNs, and σ ( i ) be a permutation of ( 1 , 2 m ) such that p σ ( 1 ) p σ ( 2 ) p σ ( m ) , G σ ( i ) is the attribute corresponding to p σ ( i ) , A σ ( 0 ) = ϕ , and A σ ( i ) = { G σ ( 1 ) , G σ ( i ) } . Taking ω ˜ i = ρ ( A σ ( i ) ) ρ ( A σ ( i 1 ) ) , α i , β i , γ i [ 0 , 1 ] , α i + β i + γ i 1 , then the aggregated values by the series of GPFPCG operators are also PFNs, and
(1)
G P F P C G D α , β n ( p 1 , p 2 p n ) = { 1 ( 1 i = 1 m ( 1 ( 1 μ p σ ( i ) ( 1 α i ) π p σ ( i ) ) λ ) ω ˜ i ) 1 / λ , 1 i = 1 m ( 1 ( η p σ ( i ) + β i π p σ ( i ) ) λ ) ω ˜ i , 1 i = 1 m ( 1 ( v p σ ( i ) + γ i π p σ ( i ) ) λ ) ω ˜ i } ;
(2)
G P F P C G F α , β , γ n ( p 1 , p 2 , , p m ) = { 1 ( 1 i = 1 m ( 1 ( 1 μ p σ ( i ) ( 1 α σ ( i ) ) π p σ ( i ) ) λ ) ω ˜ i ) 1 / λ , 1 i = 1 m ( 1 ( η p σ ( i ) + β σ ( i ) π p σ ( i ) ) λ ) ω ˜ i , 1 i = 1 m ( 1 ( v p σ ( i ) + γ σ ( i ) π p σ ( i ) ) λ ) ω ˜ i } ,
where
μ F α i , β i , γ i n ( p i ) = μ p σ ( i ) + α i π p σ ( i ) 1 ( 1 α i β i γ i ) n α i + β i + γ i ,
η F α i , β i , γ i n ( p i ) = η p σ ( i ) + β i π p σ ( i ) 1 ( 1 α i β i γ i ) n α i + β i + γ i , v F α i , β i , γ i n ( p σ ( i ) ) = v p σ ( i ) + γ i π p σ ( i ) 1 ( 1 α i β i γ i ) n α i + β i + γ i ;
(3)
G P F P C G G α , β , γ n ( p 1 , p 2 , , p m ) = ( 1 [ 1 i = 1 m ( 1 ( 1 μ p σ ( i ) α i n ) λ ) ω i ] 1 λ , ( 1 i = 1 m ( 1 η p σ ( i ) λ β i n λ ) ω ˜ i ) 1 λ , ( 1 i = 1 m ( 1 v p σ ( i ) λ γ i n λ ) ω ˜ i ) 1 λ ) .
Parallel to Theorems 13–15, the series of GPFPCG operators have properties such as idempotency, monotonicity, and boundedness under some conditions, which are omitted in order to save space.
In fact, the correlations of these proposed aggregation operators can be further studied. Here, we take PFPCAF α , β , γ n as an example.
Theorem 18.
Let p i = ( μ i , η i , v i ) ( i = 1 , 2 , , m ) be a collection of PFNs, Then the operation of complement on them is as follows:
P F P C A F α , β , γ n ( p 1 c , p 2 c , , p m c ) = P F P C A F α , β , γ n ( p 1 , p 2 , , p m ) c ,
P F P C G F α , β , γ n ( p 1 c , p 2 c , , p m c ) = P F P C G F α , β , γ n ( p 1 , p 2 , , p m ) c ,
G P F P C A F α , β , γ n ( p 1 c , p 2 c , , p m c ) = G P F P C G F α , β , γ n ( p 1 , p 2 , , p m ) c ,
G P F P C G F α , β , γ n ( p 1 c , p 2 c , , p m c ) = G P F P C A F α , β , γ n ( p 1 , p 2 , , p m ) c .
By Theorems 3–5, we can easily obtain the following theorems.
Theorem 19.
Let p i = ( μ i , η i , v i ) ( i = 1 , 2 , , m ) be a collection of PFNs, then the operation of the complement to aggregation operators is as follows:
[ P F P C A F α , β , γ n ( p 1 c , p 2 c , , p m c ) ] c = P F P C A F α , β , γ n ( p 1 , p 2 , , p m ) ,
[ P F P C G F α , β , γ n ( p 1 c , p 2 c , , p m c ) ] c = P F P C G F α , β , γ n ( p 1 , p 2 , , p m ) ,
[ G P F P C G F α , β , γ n ( p 1 c , p 2 c , , p m c ) ] c = G P F P C G F α , β , γ n ( p 1 , p 2 , , p m ) ,
[ G P F P C A F α , β , γ n ( p 1 c , p 2 c , , p m c ) ] c = G P F P C A F α , β , γ n ( p 1 , p 2 , , p m ) .
Theorem 20.
Let p i = ( μ i , η i , v i ) ( i = 1 , 2 , , m ) be a collection of PFNs, then
lim n P F P C A F α , β , γ n ( p 1 , p 2 , , p m ) = P F P C A D α α + β + γ , β α + β + γ , n ( p 1 , p 2 , , p m ) ,
lim n P F P C G F α , β , γ n ( p 1 , p 2 , , p m ) = P F P C G D α α + β + γ , β α + β + γ , n ( p 1 , p 2 , , p m ) ,
lim n G P F P C A F α , β , γ n ( p 1 , p 2 , , p m ) = G P F P C A D α α + β + γ , β α + β + γ , n ( p 1 , p 2 , , p m ) ,
lim n G P F P C G F α , β , γ n ( p 1 , p 2 , , p m ) = G P F P C G D α α + β + γ , β α + β + γ , n ( p 1 , p 2 , , p m ) .
Theorem 21.
Let p i = ( μ i , η i , v i ) be a collection of PFNs, If α i = μ p i μ p i + η p i + v p i , β i = η p i μ p i + η p i + v p i , γ i = v p i μ p i +