Recursive Aggregation and Its Fusion Process for Intuitionistic Fuzzy Numbers Based on Non-Additive Measure

: In this paper, the recursive aggregation of OWA operators for intuitionistic fuzzy numbers (IFN) based on a non-additive measure (NAM) with σ − λ rules is constructed and investigated in light of the σ − λ rules of a non-additive measure (NAM). Additionally, an integrator is designed by drawing on the genetic algorithm and the process of calculation is elaborated by an example.


Introduction
Since the concept of ordered weighted averaging (OWA) operators was initiated by Yager in 1988 [1], it has received wide application in various domains, including decision expert systems, market surveys, neural networks, analysis, fuzzy logic control, etc. [2][3][4]. In 2005, Yager further put forward the recursive forms of OWA operators [5]. Its fundamental basic idea is to directly derive the aggregated result of n data by using the aggregated result of n − 1 data while keeping the orness level unchanged. However, the attribute indexes are mostly related with each other in the process of deriving aggregation results of n data using the existing aggregation results of n − 1 data. The main reason lies in the fact that the existing recursive forms of OWA operators are established on the base of the classical probability measure and Lebesgue integral integration operator [6]. That is to say, the weight vector is not able to be measured independently, and it may also not meet the countable additivity of the classical probability measure. Fortunately, fuzzy measure, defined by Sugeno in 1974, can be utilized to depict the correlations of attribute indexes [7][8][9][10]. Meanwhile, aggregation functions and aggregation operators are investigated by many researchers recently [11][12][13][14]. Because of the fuzziness and uncertainty of actual decision issues, the evaluation values involved in the decision process are not always expressed as crisp numbers. However, Intuitionistic fuzzy number (IFN) is a critical tool for settling imprecise information [15,16] and could offer the membership degree and the nonmembership degree simultaneously. Thus, IFN performs more flexibly and efficiently than a traditional fuzzy set in addressing uncertainty. In this work, recursive aggregation of OWA operators for IFN based on a non-additive measure (NAM) with σ − λ rules is put forward and researched.
The rest of this work is organized as follows. In Section 2, we review the non-additive measure (NAM) with σ − λ rules and IFN and propose an OWA operator for IFN based on a NAM with σ − λ rules. In Section 3, we derive the recursive forms of OWA operators for IFN according to NAM with σ − λ rules while keeping the orness grade unchanged. In Section 4, the procedure for integrator design in recursive aggregation is designed, and the process of calculation is demonstrated by an example.

Preliminaries
Definition 1 ([7-10]). Let X be a nonempty set and A a σ− algebra on the X. A set function A fuzzy measure µ is called a Sugeno measure if µ satisfies σ − λ rules, briefly denoted as g λ . The fuzzy measure shown in this paper is a Sugeno measure.
By Definitions 5 and 6, we can easily obtain the result below.
Proof. By Definitions 5 and 6, we easily obtain the result below. As It follows that

Theorem 2.
When λ = 0, andβ i is an IFN, then the OWA operator for a series of INFs based on a NAM with σ − λ rules would degenerate to the classic OWA operator form for a series of IFNs. In fact, based on countable additivity, we havẽ

Corollary 1.
When λ = 0, andβ i is a special IFN, namely, real number, i = 1, 2, 3, · · · then the OWA operator for a series of IFN based on a NAM with σ − λ rules degenerates to the classic OWA operator in Reference [2].
The measure of orness involved with an OWA operatorF n of dimension n for IFNs based on a NAM with σ − λ rules can be defined by The measure of andness associated with the OWA operatorF n of dimension n for IFNs based on a NAM with σ − λ rules can be further defined by F n is an OWA operator for IFNs based on a NAM with σ − λ rules, then (1) The measure of orness associated with an OWA operatorF n of dimension n for IFNs based on a NAM with σ − λ rules is defined as (2) The measure of andness associated with the OWA operatorF n of dimension n for IFN based on a NAM with σ − λ rules is defined as Proof.

Remark 1.
When λ = 0, the measure of orness associated with an OWA operatorF n of dimension n for IFNs based on a NAM with σ − λ rules degenerates to the classic case [5].
Proof. The simplest aggregation is for two elements, as Let us now consider the aggregationF 3 . In this case, This leads to the system of independent equations The solution is More generally, in the case of n arguments, we obtain the system of n + 1 independent equations The proof is complete.

Calculation of NAM and Fusion Process Design Based on Recursive Aggregation
Theorem 6. Letα i = (µα i , vα i )(i = 1, 2) be two intuitionistic fuzzy numbers. The distance measure of IFNsα 1 andα 2 , referring to [17], is defined by An OWA operator for IFNs based on a NAM with σ − λ rules is a multiple input and single output model. By solving the model, we can obtain necessary data.
Step 4: Utilize RA for IFNs based on a NAM with σ − λ rules proposed in this work to obtain g (k+1) λ (A j ) with the condition of increasing the attribute index x k+1 .
Step 5: When a new object i gives values to the k + 1 attributes, we can utilize the presented OWA operator to obtainF k(i) . Then, utilize RF for IFNs based on a NAM with σ − λ rules to directly obtain aggregation resultsF k+1(i) of k + 1 attributes from aggregation resultsF k(i) of k attributes.
Step 6: Similarly, when adding, in turn, the attribute index to evaluation, we can always utilize the old aggregation resultsF n−1(i) of n − 1 attributes to directly derive the final aggregation resultsF n(i) of n attributes.

Example 1.
Online shopping is prevalent in e-commercial area. Thus, making a relatively accurate and reasonable evaluation for online shopping is very useful. An online shop's management randomly chooses five customers for a satisfaction evaluation of online shop. The evaluation value is denoted asã ij , andã ij is an IFN. The attributes are x 1 : logistics, x 2 : service attitude, x 3 : price, x 4 : product quality in satisfaction evaluation, and the results are shown in Table 2. When customers further consider "after-sale service" or "payment security", the online shop management want to obtain a new evaluation result.
Step 1: Collecting the evaluation values of the customers for a good in Table 2.
The IFNs in Table 2 shows that the evaluation values of the customers for the attributes of the good. That is to say, the satisfaction evaluation and dissatisfaction evaluation of the customer for the attribute of a good.
Step 1: Build an evaluation function file in Matlab software, save it as fit.m file and store it in the appropriate directory.
Step 2: Genetic algorithm toolbox is executed in the command window of Matlab R2014a to enter the GUI interface of the genetic algorithm and set relevant parameters in the corresponding column. Fitness function is @fit, variables numbers are five. The parameters of the genetic algorithm used in this paper are shown in Table 4. Stopping Generation is 149, and we use the default values for others. The GUI running interface of the genetic algorithm toolbox is shown in Figure 1.
Step 3: After setting the parameters, click "Start" in the GUI running interface of the genetic algorithm toolbox. λ and g (4) λ ({x j })(j = 1, 2, 3, 4) are shown in Table 5, and the operation result diagram is shown in Figure 2.

Conclusions
In this paper, we propose an OWA operator for IFN based on a NAM with σ − λ rules and derive the Recursive Forms of OWA operators for IFN according to NAM with σ − λ rules while keeping the orness grade unchanged. Furthermore, the procedure for integrator design in recursive aggregation is designed and the process of calculation is demonstrated by an example.