We first propose the normal Pythagorean neutrosophic set (NPNS) in this paper, which synthesizes the distribution of the incompleteness, indeterminacy, and inconsistency of the Pythagorean neutrosophic set (PNS) and normal fuzzy number. We also define some properties of NPNS. For solving the decision-making problem of the non-strictly independent and interacting attributes, two kinds of NPNS Choquet integral operators are proposed. First, the NPNS Choquet integral average (NPNSCIA) operator and the NPNS Choquet integral geometric (NPNSCIG) operator are proposed. Then, their calculating formulas are derived, their properties are discussed, and an approach for solving the interacting multi-attribute decision making based on the NPNS is studied. Finally, the two kinds of operators are applied to validate the stability of the new method.
As an important branch of modern decision theory, MADM is widely used in many fields such as economy, management, military, and engineering. One of the core problems of MADM is how to give the attribute values under each attribute. Since Zadeh [
Ideally, the constructed decision indicator system should have the conditions of completeness, representativeness, and independence. However, in many practical cases, these attributes are usually not independent but correlated. In order to solve the MADM problem of attribute correlation, Marichal [
We organize the paper as follows.
For convenience, a normal Pythagorean neutrosophic element (NPNE) is denoted as
and its accuracy functions are
If
If
For
In which
In this section, we define two normal Pythagorean neutrosophic set based Choquet integral operators, one is the normal Pythagorean neutrosophic set Choquet integral averaging (NPNSCIA) operator, and the other is the normal Pythagorean neutrosophic set Choquet integral geometric (NPNSCIG) operator. Meantime, we discuss some properties of them.
NPNSCIA is called the NPNS Choquet integral averaging operator, in which
According to some relevant operation rules of NPNE, we can get the form of the NPNSCIA operator shown in Theorem 1.
In which
Now, we proof
Proof:
When n = 1, we can easily get
When n = 2, we get:
Making a hypothesis, when
Then
This proves Theorem 1.
Proof:
Since
for
It is easy to prove it according to Definition 11, here we omit.
Proof:
Since
Since
Then, with
Theorem 4 has been proved.
Proof:
To Theorem 2,
To Theorems 3–4,
Then
NPNSCIG is called the NPNS Choquet integral geometric operator, in which
According to some relevant operation rules of NPNSs, we can get the form of the NPNSCIG operator shown in Theorem 6.
It is easy to prove Theorems 6–10.
The multi-attribute decision problem with decision information of NPNS is described as follows: There are m schemes
Step 1: Build up
Step 2: Calculate the fuzzy measure of attribute sets;
Step 3: Calculate by using the NPNSCIA or NPNSCIG operator;
Step 4: Calculate each scheme's score value;
Step 5: Choose the best scheme.
The company plans to choose one of the four suppliers
Step 1: In
Step 2: Using
Step 3: Using
Step 4: Using the NPNSCIA operator, we calculate each supplier's comprehensive value.
Step 5: Each supplier's score value can be gotten by using
Step 6: According to the value
While using NPNSCIG operator:
Step 1’–3’: Just as Steps 1–3;
Step 4’: Using the NPNSCIG operator, we calculate each supplier's comprehensive value.
Step 5’: Each supplier's score value can be gotten by using
Step 6’: According to the value
Compared with the literatures [
In this paper, NPNS and Choquet integral operators are combined to define the NPNSCIA operator and NPNSCIG operator, which can consider the incidence relation between indicators. It is proved that they have power equality, displacement invariance, ordered monotonicity, and boundedness. A nonlinear programming model is established and the FM of indicators and indicator sets is solved objectively. In the NPNS environment, by using the defined operators and the established model, the problem of related MADM with attribute weight information unknown or partially attribute weight information unknown is solved effectively. Finally, the case proves that this method is easy and reasonable. This study extended the Choquet integral to NPNS, making the Choquet integral better applied and developed in the related MADM problems.