Study on the evaluation model of serialized civil aircraft commonality index based on fuzzy set theory

Abstract

Commonality, a typical commercial feature of serialized civil aircraft study and development, refers to a series of methods of reusing and sharing assets, which were developed based on broad similarity. The common design of serialized civil aircraft is capable of maximally saving R&D, production, operation, and disposal. To maximize the total benefits of manufacturers and operators, the common design of serialized civil aircrafts primarily exploits the commercial experience of serialized products in other fields (e.g., automobiles and mobile phones), whereas a scientific index system and quantitative evaluation model has not been formed. Accordingly, this study proposes a new civil aircraft commonality index evaluation model in accordance with fuzzy set theory and methods. The model follows two branches, i.e., attribute commonality and structural commonality, to develop a multi-level civil aircraft commonality index system. The proposed model can split the commonality into six commonality sub-intervals and build the corresponding standard fuzzy set with the characteristic attribute parameters of the civil aircraft as the elements. Next, based on considerable civil aircraft sample data, a fuzzy test is designed to yield the membership function of the fuzzy set. Thus, a model of evaluating civil aircraft commonality is constructed, taking the characteristic parameters of the civil aircraft to be evaluated as input, and selecting the degree of commonality of each level as output. Lastly, this study employs the evaluation model to evaluate the commonality of Boeing 757-200 with other civil aircrafts. Furthermore, the evaluated results well explain the actual situation, which verifies the effectiveness and practicability of the proposed model.

Keywords

Cost benefit analysis serialized civil aircraft commonality index fuzzy set membership function

Nomenclature

Symbols

The domain of the object

A/B/C

The set of elements in the domain

Membership function

A mapping relationship

Number of boundaries between two sets

P_ξ (x)

The probability density function of the distribution of variable ξ

∅ (x)

Integral of probability density function of standard normal distribution

f (x, y)

The overall importance of factor x than factor y

Weight coefficient vector

Commonality index vector

The absolute value of the difference between the characteristic elements of the aircraft model

{\hat{α}}_{i}

The average of the i-th boundary number

{\hat{σ}}_{i}^{2}

The variance of the i-th boundary number

Abbreviations

Maximum speed

Cruise speed

Empty weight

Maximum take-off weight

Maximum landing weight

Maximum commercial load

TFL

Takeoff field length

LFL

Landing field length

Maximum range

Maximum flying altitude

Takeoff noise

MWS

Main wheel spacing

NMW

Number of main wheels

The number of engines

Engine model

FLW

Fuel load weight

Fuselage length

Fuselage height

Crew number

Wingspan

Wing area

Aspect ratio

Layout style

MPC

Maximum passenger capacity

FRT

Front and rear track

1 Introduction

To produce high-quality and high-versatility products that are demanded in the contemporary market, the strategy of developing serialized products has been primarily adopted by numerous product manufacturing companies [2, 21]. Product series can be also termed as product family, i.e., a set of similar products developed via the product platform. These products exhibit some similar product characteristics, parts and subsystems, whereas differences exist between products to satisfy the individual needs of different customers [7]. Over the past few years, the term commonality has been introduced from product series. In engineering, commonality refers to the situation in which conventional components applied in a product are replaced by newly designed components with the advancing of research and technology. The common component is capable of maintaining all the functions of the component it replaces. For management, when certain inventory units of the manufacturing system are employed for multiple finished products, the common concept is involved [9, 37].

In the late 1970 s, commonality was introduced into the aviation industry by Boeing. Then, Boeing applied the term “Commonality” initially in its product promotion activities, which mainly referred to the similarity of the system and structure of various aircrafts of the 737 series. Furthermore, Boeing stresses that this feature can significantly beneficial to customers’ service and equipment guarantee [1, 32]. In the 1990 s, after the setbacks of the A300, especially the A310, the Airbus A320/330/A340 and other serialized civil aircrafts were launched one after another. To emphasize the important attributes of these serialized products, Airbus adopted the same term “commonality” in its publicity, suggesting that different types of aircraft exhibit a high degree of similarity in cockpit layout and operating rules. Thus, it is ensured that flight crews can easily switch over different aircraft types [31]. Today, the world’s leading civil aircraft manufacturers have introduced the level of commonality into one of the major comprehensive weighing objects of the global civil aircraft optimization design [11, 22].

The difficulty in product series designing refers to the balance between commonality and uniqueness between products, i.e., excessive commonality may cause lack of uniqueness and poor performance of the product, while insufficient commonality may cause increased costs [8 , 20]. Thus, scholars have proposed some common indexes to evaluate the versatility of product series. Collier [5] proposed the earliest commonality measurement index, DCI. DCI exhibits a main advantage that it is easy to calculate, whereas the value range has no fixed boundary, so it is difficult to estimate the increase in commonality during the redesign of a series and comparing different series. Wacker and Trelevan [36] deformed DCI to develop a common index TCCI, which helps compare between series and within series during the product series redesign. It has the defect that it only considers the number of parts, while ignoring other key factors. Siddique, David, and Wang [30] calculated the respective common percentages for components, interfaces between components, as well as assembly. The three percentages were weighted to determine the total measurement value C% of commonality. Such index considers the manufacturing and assembly processes of products, and it can fit different development strategies by weighting factors. It has a defect that this measure only applies to a single platform, instead of the entire product series as a whole for calculation. Kota, Sethuraman, and Miller [13] proposed PCI, a commonality index for production lines. PCI highlights that commonality occurs between products that share the same or various components, instead of unique components. This index formulates a single measurement standard for the entire product series, whereas it superficially understand the commonality of each product in the series. Jiao and Tseng [12] expanded DCI by considering product sales, the number of parts, as well as cost. They stressed that expensive parts in the series have a more significant effect than cheaper parts. Its defect is that the quantity and cost information required for estimation in the index calculation is more sophisticated. Thevenot and Simpson [33, 34] built a common comprehensive index, CMC, which complies with the component, size, shape, material, manufacturing process, assembly, cost of each product, as well as the allowed diversity in the product family; CMC evaluates the product series at a scale of 0 to 1. CMC indexes are evaluated from many aspects; however, an excessive number of data should be analyzed, and sufficient cost data cannot be ensured at the initial stage of design.

For satellite, ship, aircraft and other high investment products, cost control is a very important issue [6, 19]. Commonality is one of the most important factors that affect the R&D and manufacturing costs of products, so there is a research topic to quantify the influence of commonality on the R&D and manufacturing costs of products [44]. In order to establish a more refined product commonality cost evaluation model, a comprehensive and in-depth commonality evaluation index is the premise work. The existing commonality evaluation indexes are used to quantify the degree of commonality from the perspective of the number of common components and derivative products among series of products, which can provide a measure for decision-makers to compare the overall commonality of product design schemes. However, due to too few parameter inputs, there are not enough to be the basis for the research on the impact of quantitative commonality on product development cost.

Commonality is a measure to measure the similarity of product structure, components and interfaces. Commonality index is not an absolute division of 0 and 1, but a fuzzy index with transition interval. The common index is not an absolute division of 0 and 1, but a fuzzy index with a transition interval. Take the commonality of a certain interface of two products as an example, one is connected with M16 hexagon bolts, and the other is connected with M18 hexagon bolts. The commonality of the two product interfaces cannot be regarded as 0, because they use the same connection method, nor can it be classified as 1, because they use bolts of different specifications. So their commonality should be in the transitional range of 0-1. Therefore, this manuscript proposes a commonality index evaluation model of civil aircraft based on fuzzy mathematics method, which can evaluate the commonality between aircraft models more scientifically, objectively and precisely, completing the premise research of quantifying the influence of civil aircraft commonality on development cost. The model takes the difference of characteristic attribute parameters of two civil aircraft to be evaluated as the elements of fuzzy set, establishes six standard common sub interval fuzzy sets for each element, designs fuzzy test to obtain the distribution of fuzzy set boundary, and improves the three-division method into six-division method to obtain the membership function of standard fuzzy set. A complete set of evaluation and calculation model of civil aircraft commonality index is established.

Compared with other evaluation index methods, the evaluation model of civil aircraft commonality index established in this paper has the following advantages: the model establishes the index system according to the hierarchical structure of the product, so that the evaluation of the index comprehensively covers most of the characteristics of the product. The evaluation result is the degree of belonging to a certain commonality sub interval and the establishment of the model membership function is based on a large number of similar product data, which is more objective and accurate. With the input of multi characteristic parameters and the output of multi-level common indicators, it can provide comprehensive and in-depth technical reference for decision makers, and lay a more reliable index evaluation foundation for quantifying the impact of commonality on development cost.

2 The basic mathematical theory of the model

2.1 Fuzzy set theory

In order to solve the problem of uncertainty in life and production, Zadeh [41, 42] put forward the theory and method of fuzzy mathematics. Since half a century, this theory and method has attracted extensive attention in various fields and made great achievements [3, 18].

For an ordinary set A, the element u is either u ∈ A or u ∉ A, and one of these two should exist. Thus, the relationship between elements and sets can be represented by two numbers, 0 and 1. For fuzzy set A, there is a function μ_A (x) , μ_A ∈ [0, 1]), which indicates that the degree of all elements belongs to fuzzy set A. In theory, the membership function has a one-to-one correspondence with fuzzy set. Ordinary sets have intersection, union and complement operations to process the transitions and relationships between sets. Besides, fuzzy sets have a set of theories and methods of the mentioned operations [38].

This study sets universe U≠ ∅, A,B,C are three fuzzy sets on the universe. so the operations of intersection, union and complement of fuzzy set are elucidated below:

If ∀x ∈ U, and μ_A (x) ∈ μ_B (x), A ∈ B is marked.

If ∀x ∈ U, and μ_A (x) = μ_B (x), A = B is marked.

If ∀x ∈ U, and μ_C (x) = μ_A (x) ∨ μ_B (x), C = A ∪ B is marked.

If ∀x ∈ U, and μ_C (x) = μ_A (x) ∧ μ_B (x), C = A ∩ B is marked.

If ∀x ∈ U, and μ_B (x) =1 - μ_A (x), $B = \bar{A}$ is marked, and A is termed as the complement of B.

The membership functions of A ∪ B, A ∩ B, $\bar{A}$ are respectively denoted as μ_A∪B, μ_A∩B, $μ_{\bar{A}}$ . According to the mentioned definition, the following formulas are written: $μ_{A \cup B} (x) = μ_{A} (x) \lor μ_{B} (x)$ (1) $μ_{A \cap B} (x) = μ_{A} (x) \land μ_{B} (x)$ (2) $μ_{\bar{A}} (x) = 1 - μ_{A} (x)$ (3)

The function diagram of membership function is shown in Fig. 1. The symbols ∨, ∧ used in the formula are termed as Zadeh operators in fuzzy mathematics. The calculation formula of the symbols is expressed as follows: $μ_{A} (x) \lor μ_{B} (x) = max (μ_{A} (x), μ_{B} (x))$ (4) $μ_{A} (x) \land μ_{B} (x) = min (μ_{A} (x), μ_{B} (x))$ (5)

Fig. 1

Operation diagram of membership function of fuzzy set.

Moreover, fuzzy sets are demonstrated to exhibit the following properties of ordinary sets, i.e., idempotent law, commutative law, associative law, absorption law, distribution law, restoration law, zero-unity law, and duality law [27].

2.2 Method of obtaining membership function

Membership degree is the core idea of fuzzy mathematics [17]. In fact, the root of membership lies in the intermediary transition between objective differences. Therefore, the membership function cannot be set subjectively, but should follow the objective law and establish the practical membership function according to the scientific method [29].

At present, there are several methods to establish membership function, including intuitionistic method [38], reasoning method [10], F-statistic method [16], three-division method [15] and binary comparison sorting method [4]. Each method has its own advantages, disadvantages and applicable conditions [39]. According to the characteristics of civil aircraft parameters. this study adopts the three-division method, and the application of this method is introduced below:

It is assumed that if the membership function of the three fuzzy sets A₁, A₂, A₃ is required to be established on the field U, n fuzzy experiments should be performed first. Each fuzzy experiment determines the division of U, and each division determines a pair of numbers (ξ, η):

ξ: the dividing line between set A₁ and A₂;

η: the dividing line between set A₂and A₃;

If (ξ, η) is yielded, the mapping relationship e is also clarified, thus clarifying the fuzzy concept. Whereas the interval of A₁, A₂, A₃ is a random interval, thus, ξ and η are random variables. This study considers that they exhibit a normal distribution: $ξ N (α_{1}, σ_{1}^{2}) η N (α_{2}, σ_{2}^{2})$ (6)

The number pairs (ξ, η) determine the mapping: $e (ξ, η) : U \to {A_{1}, A_{2}, A_{3}}$ (7)

Instantiation: $e (ξ, η) (x) = {\begin{matrix} A_{1} (x) x ⩽ ξ \\ A_{2} (x) ξ < x ⩽ η \\ A_{3} (x) x > η \end{matrix}$ (8)

The probability P (x ⩽ ξ) refers to the probability that the random variable ξ falls in the interval [x, b). If x increases, [x, b) is smaller, and the probability of falling in the interval [x, b) decreases. Thus, this characteristic of probability P (x ⩽ ξ) is identical to fuzzy set A₁ (x), so: $A_{1} (x) = P (x ⩽ ξ) = \int_{x}^{+ \infty} P_{ξ} (x) dx$ (9) $A_{3} (x) = P (x > η) = \int_{- \infty}^{x} P_{η} (x) dx$ (10) $A_{2} (x) = 1 - A_{1} (x) - A_{3} (x)$ (11)

Where P_ξ (x) and P_η (x) denote the probability densities of random variables ξ and η, respectively.

It is determined by the probability method: $A_{1} (x) = 1 - \emptyset (\frac{x - α_{1}}{σ_{1}})$ (12) $A_{3} (x) = \emptyset (\frac{x - α_{2}}{σ_{2}})$ (13) $A_{2} (x) = \emptyset (\frac{x - α_{1}}{σ_{1}}) - \emptyset (\frac{x - α_{2}}{σ_{2}})$ (14)

Where: $\emptyset (x) = \int_{- \infty}^{x} \frac{1}{\sqrt{2 π}} e^{- \frac{t^{2}}{2}} dt$ (15)

The three membership function graphs are shown in Fig. 2. The above refers to the method to determine the membership function with the three-division method.

Fig. 2

Example diagram of membership function and dividing line.

2.3 Analytic hierarchy process

In the evaluation of an object, it is often evaluated from multiple factors. When the reference data is insufficient, the weight of each factor is difficult to give in the whole [35]. For this type of problem, the American Operations Research in the 1910s Professor Saaty [23 –25] established the Analytic Hierarchy Process, through which the judgment matrix is built by comparing all factors in pairs to calculate the weight vector.

When drawing a pairwise comparison, qualitative languages (e.g., “equal important”, “slightly important”, “obviously important” and “very important”) are generally used for description [26, 28]. The analytic hierarchy process quantifies it and introduces f (x, y) to express the importance of factor x than factor y. If f (x, y) > 1, it means that factor x is more important than factor y; if f (x, y) < 1, factor y is more important than factor x; if f (x, y) = 1, factor x is as important as factor y. Moreover, it is stipulated that f (x, y) = 1/f (y, x) [40]. The specific value of f (x, y) is generally determined by the following method in Table 1 [45].

Table 1
Factor comparison scale table [45]

Factors x and y compared Explanation f (x, y) f (y, x)

x is as important as y x, y have the same contribution to the overall goal 1 1

x is slightly more important than y The contribution of x is slightly greater than y, but not obvious 3 1/3

x is more important than y The contribution of x is significantly greater than y 5 1/5

x is extremely more important than y The contribution of x is significantly greater than y, but there is no overwhelming advantage 7 1/7

x is completely more important than y The contribution of x is overwhelmingly greater than y 9 1/9

The comparison of the importance of x and y is between the above two adjacent judgments Compromise between two adjacent judgments 2,4,6,8 1/2,1/4,1/6,1/8

Factors x and y compared	Explanation	f (x, y)	f (y, x)
x is as important as y	x, y have the same contribution to the overall goal	1	1
x is slightly more important than y	The contribution of x is slightly greater than y, but not obvious	3	1/3
x is more important than y	The contribution of x is significantly greater than y	5	1/5
x is extremely more important than y	The contribution of x is significantly greater than y, but there is no overwhelming advantage	7	1/7
x is completely more important than y	The contribution of x is overwhelmingly greater than y	9	1/9
The comparison of the importance of x and y is between the above two adjacent judgments	Compromise between two adjacent judgments	2,4,6,8	1/2,1/4,1/6,1/8

For a certain practical problem, X = [x₁, x₂ ⋯ x_n] is created as the set of all factors. Fill in the matrix A = (a_ij) _n×n by pairwise comparison, where a_ij = f (x_i, x_j), and A is termed as the judgment matrix. Subsequently, the following steps are performed to calculate the weight vector:

For a certain judgment matrix A, any initial vector $W^{0} = {(w_{1}^{0}, w_{2}^{0}, \dots w_{n}^{0})}^{T}$ is taken, where $w_{i}^{0} \in [0, 1]$ and $max {w_{i}^{0}} = 1$ .

k-1 iterations are performed according to the following formula to determine the k-1 th approximate value of the first eigenvalue λ₁ of A and the normalized eigenvector W corresponding to λ₁ the k-1th normalized approximate value $W^{k - 1} = {(w_{1}^{k - 1}, w_{2}^{k - 1}, \dots w_{n}^{k - 1})}^{T}$ . Subsequently, it yields W^k* = AW^k-1, $W^{k} = W^{k *} / w_{\max}^{k *}$ . It can be verified that $lim_{k \to \infty} W^{k} = \tilde{W}$ , where $\tilde{W}$ denotes the actual weight vector. $W^{i + 1 *} = {AW}^{i}$ (16) $W^{i + 1} = \frac{W^{i + 1 *}}{λ_{\max}^{i + 1}} = \frac{W^{i + 1 *}}{w_{\max}^{i + 1 *}}$ (17)

Given the accuracy requirement ɛ (ɛ > 0) in advance, when $\max | w_{i}^{k} - w_{i}^{k - 1} | < ɛ$ , $W^{k} = {(w_{1}^{k}, w_{2}^{k}, \dots w_{n}^{k})}^{T}$ is taken as the calculation result; otherwise, go to step 2.

However, the final calculation result may not be ensured to comply with the actual situation since human judgment may fall into the contradiction of A > B, B > C, C > A; thus, the consistency of the judgment matrix should be verified. The formula is expressed as follows: $CI = \frac{λ_{1} - n}{n - 1} (n > 1)$ (18) $CR = \frac{CI}{RI}$ (19)

Where CI denotes the consistency index, indicating the degree of A deviation from consistency. The smaller the CI value, the smaller the degree of consistency deviation the judgment matrix will have. Moreover, Satty also introduces the random consistency index RI, which represents the average value of the CI value of considerable randomly generated n-dimensional judgment matrices. This gives the consistency ratio, CR. When CR < 0.1, the judgment matrix is considered to satisfy the consistency requirement.

3 Calculation method of civil aircraft commonality index model

3.1 Civil aircraft commonality index system

Any two products often have similarities and their own characteristics, as do two civil aircraft [33]. We cannot directly point out whether these two civil aircraft are common. Instead, a scientific and reasonable calculation method should be used to calculate the degree of commonality between the two civil aircrafts, which is the meaning of the civil aircraft commonality index model.

To find the position of the evaluation object in Fig. 3 more accurately, the “common-unique” interval is divided into 6 sub-intervals, i.e., “Completely Common–Extremely Common”, “Extremely Common–Highly Common”, “Highly Common–Generally Common”, “ Generally Common–Low Common”, “Low Common–Extremely Non-common”, “Extremely Non-common–Completely Unique”, as shown in Fig. 4. Thus, which range the evaluation target is in should be determined, as well as where it is in the interval.

Fig. 3

Commonality interval graph.

Fig. 4

Common sub-interval division diagram.

Before dividing the evaluation object into the above subintervals, it is necessary to define the boundaries of adjacent intervals. But because the common subinterval is also the concept of fuzziness, there is no clear boundary. Therefore, we regard each common subinterval as a fuzzy set, and take the membership degree of the evaluation object belonging to each subinterval as the evaluation result.

According to the definition of commonality, the degree of commonality between the two civil aircrafts is regarded as the degree of similarity between the two civil aircrafts for parts, processes, technology, interfaces and infrastructure. Accordingly, the absolute value of the numerical difference describing the characteristic attributes of the civil aircrafts can be adopted as an explicit expression of the commonality. Then the membership function of civil aircraft attribute parameters is established, and the membership of six common sub intervals of the evaluation object is calculated as the result of common evaluation.

Indeed, many parameters can describe the characteristic attributes of the aircraft., which include cruise speed, maximum weight, take-off field length and other parameters that express the overall characteristic attributes of the civil aircraft, as well as wingspan, wing chord ratio, sweep angle, root tip ratio and other characteristic attribute parameters that describe the aircraft subsystem. Thus, this study builds a multi-level commonality index system of “overall-system-subsystem-component”. The degree of commonality of each level consists of the degree of commonality and structural commonality that describe the characteristics of the level. The structural commonality covers the commonality of the characteristic attributes and the structural commonality of the substructures. The system framework diagram is illustrated in Fig. 5.

Fig. 5

Commonality index system structure diagram.

To calculate a complete commonality index system for an evaluation object, the parameters of each level feature attribute should be collected, and the fuzzy set membership function of each parameter should be established. Obviously, the amount of data and calculation required are extremely huge. Accordingly, the level of the commonality system can be selected according to the actual data collected. For instance, the five overall attribute parameters of the evaluation object, [X1, X2, X3, X4, X5], are only collected, all of which can be quantified. The first-level commonality index can be selected, and the calculation formula for the commonality index of the evaluation object X{x₁, x₂, x₃, x₄, x₅} is expressed as: $C = w_{1} C_{1} + w_{2} C_{2} + w_{3} C_{3} + w_{4} C_{4} + w_{5} C_{5}$ (20) $C_{i} = [c_{i}^{1}, c_{i}^{2}, c_{i}^{3}, c_{i}^{4}, c_{i}^{5}, c_{i}^{6}]$ (21) $c_{i}^{j} = μ_{i}^{j} (x_{i})$ (22)

Where C denotes the commonality index vector, the elements of which represent the membership value of the 6 commonality subintervals; W = [w₁, w₂, w₃, w₄, w₅] is the weight vector, calculated by the analytic hierarchy process; $c_{i}^{j}$ is the degree to which the i-th characteristic attribute parameter belongs to the j sub-interval; $μ_{i}^{j}$ is the membership function of the i-th characteristic attribute parameter in the j interval.

If 3 overall attribute parameters of the evaluation object are collected, 2 systems are overall included, and each subsystem has 3 attribute parameters. The second-level commonality index is selected. The calculation formula for the common index of the evaluation object X{x₁, x₂, x₃, [x_1,2, x_1,2, x_1,3] , [x_2,1, x_2,2, x_2,3]} is written as:

$\begin{matrix} C = w_{1} C_{1} + w_{2} C_{2} + w_{3} C_{3} + \\ w_{4} (w_{4, 1} C_{4, 1} + w_{4, 2} C_{4, 2} + w_{4, 3} C_{4, 3}) \\ + w_{5} (w_{5, 1} C_{5, 1} + w_{5, 2} C_{5, 2} + w_{5, 3} C_{5, 3}) \end{matrix}$ (23) $C_{i, j} = [c_{i, j}^{1}, c_{i, j}^{2}, c_{i, j}^{3}, c_{i, j}^{4}, c_{i, j}^{5}, c_{i, j}^{6}]$ (24) $c_{i, j}^{k} = μ_{i, j}^{k} (x_{i, j})$ (25)

Where $c_{i, j}^{k}$ denotes the degree to which the j-th characteristic attribute parameter of subsystem i belongs to the j interval; $μ_{i, j}^{k}$ represents the membership function of the j-th characteristic parameter of subsystem i.

3.2 Obtaining method of membership function for commonality subintervals of attribute parameter

Figure 5 shows that the calculation of the commonality index system is achieved using the commonality index of each level of attribute parameters. Before calculating the commonality index of an attribute parameter of a certain level of an evaluation object, the membership function of the six commonality fuzzy sets (A₁, A₂, A₃, A₄, A₅, A₆) of the attribute parameter should be set, including “extremely common fuzzy set”, “highly common fuzzy set”, “moderate to high common fuzzy set”, “moderate to low common fuzzy set”, “low common fuzzy set”, and “extremely non-common fuzzy set”. Thus, the six-division method is adopted to solve the problem in this study, which is expanded from the three-division method in section 2.2.

To establish the membership function of 6 fuzzy sets on the universe U, n fuzzy tests are performed. In each fuzzy test, the division of U is implemented. Each division obtains a group of bounds numbers (η₁, η₂, η₃, η₄, η₅):

η₁: the boundary between extremely common fuzzy set and highly common fuzzy set

η₂: the boundary between highly common fuzzy set and moderate to high common fuzzy set

η₃: the boundary between moderate to high common fuzzy set and moderate to low common fuzzy set

η₄: the boundary between moderate to low common fuzzy set and low common fuzzy set

η₅: the boundary between low common fuzzy set and extremely non-common fuzzy set

The method of each fuzzy test is elucidated below. Six civil aircrafts are randomly selected from the civil aircraft database, and 15 groups of objects to be evaluated can be obtained by pairwise combination. The absolute value θ of the difference between the characteristic elements of each group of objects expresses the commonality of the characteristic. Under the closer to zero difference, it is suggested that the two civil aircrafts are more similar in this element. When the difference is farther from zero, the two civil aircrafts are indicated to be less similar in this element. The absolute value of the difference between the 15 groups of characteristic elements is sorted from small to large θ₁ ⩽ θ₂ ⩽ ⋯ ⩽ θ₁₄ ⩽ θ₁₅, and the boundary numbers obtained through this fuzzy test is defined as: $η_{1} = θ_{1}, η_{2} = θ_{3}, η_{3} = θ_{6}, η_{4} = θ_{10}, η_{5} = θ_{15}$ (26)

The reason why the value interval of the boundary number increases successively is that the θ value exhibits unilateral finiteness, 0⩽ θ < + ∞. If the values are evenly spaced, σ_{η
₁} << σ_{η
₂} << σ_{η
₃} << σ_{η
₄} << σ_{η
₅} is obtained in the final test result, revealing that the variance is significantly different. This study proves that this influence can be effectively reduced by the method of gradually increasing the interval.

Obviously, each experiment may obtain 5 different interval boundary numbers, which theoretically belong to a normal distribution. Thus, the parameter estimation method is adopted to determine the actual parameter value of the normal distribution of each boundary number. ${\hat{α}}_{i} = \frac{1}{n} \sum_{k = 1}^{k = n} η_{i}^{k}$ (27) ${\hat{σ}}_{i}^{2} = \frac{1}{n} \sum_{k = 1}^{k = n} η_{i}^{k 2} - {(\frac{1}{n} \sum_{k = 1}^{k = n} η_{i}^{k})}^{2}$ (28)

Where $η_{i}^{k}$ denotes the value of η_i obtained from the k-th fuzzy test; ${\hat{α}}_{i}$ and ${\hat{σ}}_{i}^{2}$ are the parameter estimates of the normal distribution of η_i.

By considerable fuzzy experiments and the calculation of Equation 27 and 28, the distribution of the boundary numbers can be obtained by replacing the real value with the estimated value:

$\begin{matrix} η_{1} : N ({\hat{α}}_{1}, {\hat{σ}}_{1}^{2}), η_{2} : N ({\hat{α}}_{2}, {\hat{σ}}_{2}^{2}), \\ η_{3} : N ({\hat{α}}_{3}, {\hat{σ}}_{3}^{2}), η_{4} : N ({\hat{α}}_{4}, {\hat{σ}}_{4}^{2}), η_{5} : N ({\hat{α}}_{5}, {\hat{σ}}_{5}^{2}) \end{matrix}$ (29)

The mapping relationship is expressed as: $e_{(η_{1}, η_{2}, η_{3}, η_{4}, η_{5})} (x) = {\begin{matrix} \begin{matrix} A_{1}, x ⩽ η_{1} \\ A_{2}, η_{1} < x ⩽ η_{2} \\ A_{3}, η_{2} < x ⩽ η_{3} \end{matrix} \\ \begin{matrix} A_{4}, η_{3} < x ⩽ η_{4} \\ A_{5}, η_{4} < x ⩽ η_{5} \\ A_{6}, η_{5} < x \end{matrix} \end{matrix}$ (30)

The membership functions of A₁, A₂, A₃, A₄, A₅, A₆ obtained by the fuzzy test are written as follows: $μ_{A_{1}} (x) = \int_{x}^{+ \infty} P_{η_{1}} (u) du = 1 - ϕ (\frac{x - {\hat{α}}_{1}}{{\hat{σ}}_{1}})$ (31) $\begin{matrix} μ_{A_{2}} (x) & = \int_{- \infty}^{x} P_{η_{1}} (u) du - \int_{- \infty}^{x} P_{η_{2}} (u) du \\ = ϕ (\frac{x - {\hat{α}}_{1}}{{\hat{σ}}_{1}}) - ϕ (\frac{x - {\hat{α}}_{2}}{{\hat{σ}}_{2}}) \end{matrix}$ (32) $\begin{matrix} μ_{A_{3}} (x) & = \int_{- \infty}^{x} P_{η_{2}} (u) du - \int_{- \infty}^{x} P_{η_{3}} (u) du \\ = ϕ (\frac{x - {\hat{α}}_{2}}{{\hat{σ}}_{2}}) - ϕ (\frac{x - {\hat{α}}_{3}}{{\hat{σ}}_{3}}) \end{matrix}$ (33) $\begin{matrix} μ_{A_{4}} (x) & = \int_{- \infty}^{x} P_{η_{3}} (u) du - \int_{- \infty}^{x} P_{η_{4}} (u) du \\ = ϕ (\frac{x - {\hat{α}}_{3}}{{\hat{σ}}_{3}}) - ϕ (\frac{x - {\hat{α}}_{4}}{{\hat{σ}}_{4}}) \end{matrix}$ (34) $\begin{matrix} μ_{A_{5}} (x) & = \int_{- \infty}^{x} P_{η_{4}} (u) du - \int_{- \infty}^{x} P_{η_{5}} (u) du \\ = ϕ (\frac{x - {\hat{α}}_{4}}{{\hat{σ}}_{4}}) - ϕ (\frac{x - {\hat{α}}_{5}}{{\hat{σ}}_{5}}) \end{matrix}$ (35) $μ_{A_{6}} (x) = \int_{- \infty}^{x} P_{η_{1}} (u) du = ϕ (\frac{x - {\hat{α}}_{5}}{{\hat{σ}}_{5}})$ (36)

Where $ϕ (x) = \int_{- \infty}^{x} \frac{1}{\sqrt{2 π}} e^{- \frac{t^{2}}{2}} dt$ .

According to the mentioned method, the membership functions of 6 standard fuzzy sets of attribute elements at all levels can be established and the function graph is similar to Fig. 6.

Fig. 6

Schematic diagram of standard fuzzy set membership function.

4 Establishment and application of civil aircraft commonality index model

4.1 Civil aircraft commonality index model establishment

The performance data and structural parameters of 97 aircraft types are collected from the “World Civil Aircraft Manual” [43], including the full range of civil aircraft types and some derivative types of two mainstream civil aircraft manufacturers, i.e., Boeing and Airbus. Moreover, the civil aircraft models of non-mainstream civil aircraft companies (e.g., MD (McDonnell Douglas), IL (Ilyshin), AN (Andonov) and CRJ (Bombardier)), and some niche civil aircraft models (e.g., Xinzhou, Concord, ARJ, Fokker, Trident civil aircraft) are incorporated into the sample library. To be specific, most of the branch line and main line civil aircraft models are involved, which lays a solid data foundation for the commonality index model.

This study takes the establishment of a two-layers civil aircraft commonality index system as an example to express the modeling method of the civil aircraft commonality index system. The parameters describing the attributes of the aircraft are selected, including maximum speed (MS), cruise speed (CS), empty weight (EW), maximum take-off weight (TW), Maximum landing weight (LW), maximum commercial load (CL), takeoff field length (TFL), landing field length (LFL), maximum range (MR), maximum flying altitude (FA), and takeoff noise (TN). The aircraft’s subsystems consist of power system, fuselage system, wing system and undercarriage system. The attribute parameters describing the power system include maximum thrust of a single engine (MT), the number of engines (NE), the engine model (NM), and the fuel load weight (FLW). The attribute parameters describing the fuselage system include fuselage length (FL), fuselage height (FH), maximum passenger capacity (MPC), and crew number (NC). The attribute parameters describing the wing system include wingspan (WS), wing area (WA) and aspect ratio (AR). The attribute parameters describing the undercarriage include layout style (LS), main wheel spacing (MWS), front and rear track (FRT), as well as number of main wheels (NMW). The index system diagram is given in Fig. 7.

Fig. 7

Framework diagram of the two-layers commonality index system for civil aircraft.

All parameters can be quantified in Fig. 7, except for the engine model and undercarriage layout. The quantifiable parameters are standardized according to Equation 37, and then the absolute value of the standardized parameter value difference between the two civil aircraft models acts as the model enter. $y_{i} = \frac{x_{i} - min {x_{i}, 1 ⩽ i ⩽ n}}{max {x_{i}, 1 ⩽ i ⩽ n} - min {x_{i}, 1 ⩽ i ⩽ n}}$ (37)

Where y_i denotes standardized dimensionless data; x_i represents original data.

For the engine model, the data is processed according to Equation 38. $y = 0.5 * C + 0.3 * T + 0.2 * N$ (38)

Where y denotes the quantified dimensionless data. When two engines belong to the same company, C = 0; otherwise, C = 1. When two engines belong to the same serial number, T = 0; otherwise, T = 1. When two engines are of the same model, N = 0; otherwise, N = 1.

For the undercarriage layout, the data are processed according to Equation 39. $y = 0.6 * A + 0.6 * B + 0.3 * C$ (39)

Where y denotes the quantified dimensionless data. When the undercarriage of the two civil aircraft models refer to multi-support point type and the rear three point type, respectively, A = 1; otherwise, A = 0. When the undercarriage of the two civil aircraft models refer to front three point type and rear three point type, respectively, B = 1; otherwise, B = 0. When the undercarriage of the two civil aircraft models are front three point type and multi-support point type, respectively, C = 1; otherwise, C = 0.

12 aviation industry experts and professors were invited to fill in the judgment matrix according to the contribution of each attribute parameters to the commonality of the upper level structure (Table 3). Subsequently, the weight coefficient of each attribute parameter is obtained according to the calculation method in Section 2.3, and the results are listed in Table 4.

Table 2

Summary table of RI value of n order judgment matrix

Matrix order	3	4	5	6	7	8	9
RI	0.52	0.89	1.12	1.26	1.36	1.41	1.46
Matrix order	10	11	12	13	14	15
RI	1.49	1.52	1.54	1.56	1.58	1.59

Table 3

Judgment matrix of the contribution degree of civil aircraft performance parameters

	MS	CS	EW	TW	LW	CL	TFL	LFL	MR	FA	TN	Calculation results
MS	1	0.66	0.5	1	1	3	4	4	2	4	4	0.14
CS	1.52	1	0.66	1	1	4	5	5	3	2	3	0.15
EW	2	1.52	1	2	2	3	5	5	3	2	4	0.19
TW	1	1	0.5	1	1	2	2	2	2	1.5	3	0.11
LW	1	1	0.5	1	1	2	1	1	1	1.5	3	0.09
CL	0.33	0.25	0.33	0.5	0.5	1	1	1	0.8	0.5	1	0.04
TFL	0.25	0.2	0.2	0.5	1	1	1	1	0.6	0.5	1	0.04
LFL	0.25	0.2	0.2	0.5	1	1	1	1	0.6	0.5	2	0.05
MR	0.5	0.33	0.33	0.5	1	1.25	1.67	1.67	1	2	1	0.07
FA	0.25	0.5	0.5	0.67	0.67	2	2	2	0.5	1	2	0.07
TN	0.25	0.33	0.25	0.33	0.33	1	1	0.5	1	0.5	1	0.04

Table 4

The weight coefficient of each parameter in the commonality index system

Civil aircraft commonality (1)	Attribute commonality (0.4)	MS (0.14), CS (0.15), EW (0.19), TW (0.11), LW (0.09), CL (0.04), TFL (0.04), LFL (0.05), MR (0.07), FA (0.07), TN (0.04)
	Structural commonality (0.6)	Power system (0.4)	MT (0.18), NE (0.11), NT (0.58), FLW (0.13)
		Fuselage system (0.2)	FL (0.22), FH (0.21), MPC (0.47), NC (0.10)
		Wing system (0.26)	WS (0.24), WA (0.32), AR (0.44)
		Undercarriage system (0.14)	LS (0.39), MWS (0.26), FRT (0.23), NMW (0.12)

Samples are randomly selected from the data of 97 civil aircraft models, which is repeated for do 5000 times for fuzzy tests. 6 civil aircraft models are randomly selected each time, and 15 sets of comparative data are obtained by pairwise combination. According to the method in section 3.2, the average value and variance of the boundary numbers of 5 fuzzy sets are determined to estimate the parameters of the normal distribution (Table 5).

Table 5

Normal distribution parameter table of each characteristic attribute parameter

Variance	Mean	Variance	Mean	Variance	Mean	Variance	Mean	Variance	Mean	Minimum value	Maximum value	Parameter
$η_{5}, σ_{5}^{2}$	η₅, μ₅	$η_{4}, σ_{4}^{2}$	η₄, μ₄	$η_{3}, σ_{3}^{2}$	η₃, μ₃	$η_{2}, σ_{2}^{2}$	η₂, μ₂	$η_{1}, σ_{1}^{2}$	η₁, μ₁
0.074	0.3916	0.0137	0.1584	0.0019	0.0757	0.0007	0.0346	0.0001	0.0052	405	1127	MS
0.0992	0.4667	0.0278	0.1893	0.0031	0.0807	0.0008	0.0343	0.0001	0.0053	380	1041.25	CS
0.0388	0.4739	0.0137	0.2467	0.0043	0.1195	0.0013	0.0523	0.0002	0.0122	5.735	275.3	EW
0.0371	0.4672	0.0133	0.2456	0.0044	0.1209	0.0012	0.052	0.0002	0.0127	9.5	427	LW
0.0457	0.4695	0.0141	0.2365	0.0045	0.1141	0.0013	0.0488	0.0002	0.0119	9.5	584	TW
0.0435	0.3794	0.0085	0.1848	0.0025	0.0934	0.0008	0.0417	0.0002	0.0087	2267	151000	CL
0.0395	0.5199	0.011	0.2731	0.0044	0.1462	0.0015	0.0666	0.0003	0.0171	350	3410	TFL
0.0352	0.3905	0.0087	0.1917	0.0029	0.1001	0.0008	0.0452	0.0002	0.0113	270	3000	LFL
0.0336	0.6604	0.018	0.3692	0.0058	0.1925	0.0023	0.0875	0.0006	0.0221	1750	15791	MR
0.0468	0.3449	0.0108	0.1438	0.0026	0.0672	0.0006	0.0282	0.0001	0.0044	4200	18288	FA
0.0435	0.4138	0.0106	0.2022	0.0031	0.1022	0.0011	0.0437	0.0002	0.0081	79.7	119.5	TN
0.0338	0.672	0.0303	0.3947	0.0082	0.1676	0.0026	0.0693	0.0004	0.0134	41	343	MT
0.0798	0.8652	0.1664	0.5132	0.0475	0.1274	0	0	0	0	2	4	NE
0.0029	0.9994	0.0034	0.9933	0.0253	0.9505	0.0946	0.7533	0.0791	0.2931	/	/	NM
0.045	0.466	0.0156	0.2377	0.0049	0.1092	0.0013	0.0414	0.0001	0.0076	3069	355850	FLW
0.0323	0.5933	0.013	0.3204	0.0051	0.1751	0.0024	0.0832	0.0005	0.0185	15.67	74.8	FL
0.0256	0.5363	0.0111	0.3032	0.0049	0.157	0.002	0.064	0.0003	0.0118	5.05	24.1	FH
0.0315	0.4029	0.0095	0.2069	0.0027	0.1011	0.0008	0.0462	0.0002	0.0112	26	840	MPC
0.0541	0.3780	0.0279	0.2156	0.0056	0.0179	0	0	0	0	2	5	NC
0.0362	0.5463	0.0144	0.2931	0.0059	0.153	0.002	0.0613	0.0003	0.0105	21.21	79.8	WS
0.0461	0.4762	0.0136	0.2568	0.0056	0.1231	0.0014	0.0451	0.0001	0.0074	74.98	831	WA
0.0308	0.3148	0.0053	0.1595	0.0016	0.0818	0.0007	0.0361	0.0001	0.0058	1.7	11.4	AR
0.0228	0.1932	0.0123	0.0702	0.0021	0.0111	0	0	0	0	/	/	LS
0.0608	0.6438	0.0379	0.431	0.0415	0.1052	0	0	0	0	2	6	NMW
0.0379	0.558	0.0125	0.2889	0.0055	0.1593	0.0023	0.0745	0.0004	0.0163	6.345	33.58	FRT
0.0226	0.6003	0.0173	0.3728	0.007	0.1582	0.0024	0.062	0.0003	0.008	3.17	14.33	MWS

According to the method in Section 3.2, the membership functions of the six standard fuzzy sets (A₁, A₂, A₃, A₄, A₅, A₆) of each attribute parameter in the commonality index system are established respectively.

Taking empty weight, a parameter describing the civil aircraft characteristic attribute, as an example, through 5000 fuzzy tests the distribution of the 5 boundary numbers is: $\begin{matrix} η_{1} \sim N (0.0112, 0.0002) η_{2} \sim N (0.0523, 0.0013) \\ η_{3} \sim N (0.1195, 0.0043) η_{4} \sim N (0.2467, 0.0137) \\ η_{5} \sim N (0.4739, 0.0388) \end{matrix}$ (40)

The function graph of the boundary numbers are shown in Fig. 8. According to formula 31-36, the membership function of the six common sub-interval fuzzy sets with empty weight parameters is: $μ_{A_{1}} (x) = 1 - ϕ (\frac{x - 0.0112}{\sqrt{0.0002}})$ (41) $μ_{A_{2}} (x) = = ϕ (\frac{x - 0.0112}{\sqrt{0.0002}}) - ϕ (\frac{x - 0.0523}{\sqrt{0.0013}})$ (42) $μ_{A_{3}} (x) = ϕ (\frac{x - 0.0523}{\sqrt{0.0013}}) - ϕ (\frac{x - 0.1195}{\sqrt{0.0043}})$ (43) $μ_{A_{4}} (x) = ϕ (\frac{x - 0.1195}{\sqrt{0.0043}}) - ϕ (\frac{x - 0.2467}{\sqrt{0.0137}})$ (44) $μ_{A_{5}} (x) = = ϕ (\frac{x - 0.2467}{\sqrt{0.0137}}) - ϕ (\frac{x - 0.4739}{0.0388})$ (45) $μ_{A_{6}} (x) = ϕ (\frac{x - 0.4739}{0.0388})$ (46)

Where $ϕ (x) = \int_{- \infty}^{x} \frac{1}{\sqrt{2 π}} e^{- \frac{t^{2}}{2}} dt$ . The membership function graph of six common subintervals of the parameter empty weight is shown in Fig. 9.

Fig. 8

Distribution chart of boundary number of commonality subintervals of empty weight.

Fig. 9

The membership function graph of sub-interval standard fuzzy sets for the commonality of empty weight.

The membership function of the standard fuzzy set of the commonality sub-intervals of other attribute parameters can be obtained in the same manner, as an attempt to establish the two-layers index evaluation model of the commonality of the entire civil aircraft. Subsequently, this model can be used to evaluate the commonality of any two civil aircraft models. The evaluation method is that after obtaining all the attribute parameters of the two models (Table 5), Equation 37–39 are used for standardization. Subsequently, the absolute value of the attribute difference between the two civil aircraft models is substituted into the membership function of the standard fuzzy set of the commonality sub-interval of each attribute parameter to obtain the commonality index vector of each attribute. Lastly, weighted summation is performed according to the coefficient table (Table 4) to determine the commonality index of the evaluation object.

4.2 Calculation example-Boeing 757

The following will take the commonality evaluation of Boeing 757 with other civil aircraft models as an example to demonstrate the application of the commonality evaluation model for civil aircraft. Then analyze whether the evaluation results are in line with the actual model situation to verify the correctness and validity of the model.

The Boeing 757 refers to a dual-engine, medium- and long-range transport aircraft built by Boeing Commercial Airplanes. It can be applied for both longer and shorter routes. It takes up a certain place in the Boeing series, even the global civil aircraft market. Taking the basic model of Boeing 757-200 as an example, and using the commonality index model proposed here, this study analyzes the commonality of Boeing 757-200 with other civil aircraft models, i.e., modified models of the same model, 757-300, models with the same seat in the same series (100–200 seats), 707-320B, 717-200, 727-100, 737-900, models with different seats in the same series, 767-200, 747-400, 777-200, different series of the same seat model, A318-100, A320-200, Douglas transport aircraft DC-8-30, Bombardier Group transport aircraft CRJ1000, different series of different seat models, A380-100.

Figure 10 indicated that whether Boeing 757-200 and Boeing 757-300 have commonality at the overall layer or at the system layer, their membership coefficients that pertain to the extremely common sub-interval are significantly larger than those belonging to other common sub-intervals. Only at the layer of fuselage commonality, the coefficient of the moderately to high common sub-interval is higher than that of other sub-intervals, since the Boeing 757-300 is extended from the Boeing 757-200. Besides, their main system and structure all comply with the identical design. Only the length of the fuselage increases by 7.11m, and the passenger capacity increases by 20%. Accordingly, the two civil aircraft models maintain a high degree of commonality, revealing that the evaluated results of the commonality index model and the actual situation remain consistent.

Fig. 10

Commonality indexes between Boeing 757-200 and Boeing 757-300.

Figure 11 shows that the commonality index of Boeing 757-200 and Airbus A318-100 fall to the moderately to low common sub-interval. Though the two models are typical models of 100-200 seat planes, they exhibit their own characteristics at the structural layer. For instance, the undercarriage of the A318-100 adopts the rear three-point type, while the Boeing 757-200 has the front three-point type. However, the two models pertain to the same seat level, so certain commonalities exist in attribute parameters. Comprehensive consideration of the common level is also consistent with the moderately to low common interval.

Fig. 11

Commonality indexes between Boeing 757-200 and Airbus A318-100.

As indicated from Figs. 12 and 13, besides the extremely high commonality between the Boeing 757-200 and its modified Boeing 757-300, the commonality with other Boeing aircraft of the identical series is roughly at a moderately to high level. This is because there should be some design concept inheritance between the aircraft designs of the identical series. However, each model has different performance biases that cause differences at the structural layer, which leads to a moderately to high common level. Compared with models of different seat class, the Boeing 757-200 has a moderately to low or low level of commonality, and even the commonality with the A380-100 is in extremely non-common interval. Since different seat classes require different performance, different system structures and manufacturers have different design habits and ideas. In brief, the results of such commonality indexes are consistent with the actual situation.

Fig. 12

Comparison of commonality indexes between Boeing 757-200 and the same series of aircraft.

Fig. 13

Comparison of commonality indexes between Boeing 757-200 and different seat class aircraft.

From the analysis of the commonality evaluation results of Boeing 757, it can be seen that the evaluation results of the civil aircraft commonality evaluation model in this paper can explain the commonality problems of the models, which proves that the model can be used for commonality evaluation among civil aircraft models. The evaluation result can objectively and in-depth measure the degree of commonality between models, and can be used as a technical reference for the design of commonality of models and the quantitative study of common benefits.

5 Conclusion

In this study, a novel evaluation model is built for the commonality of civil aircraft. According to the fuzziness of the commonality of civil aircraft, the model adopts the clustering analysis method in fuzzy mathematics. Taking the parameter difference between the characteristic attributes of the two civil aircraft to be evaluated as the elements of the fuzzy set, 6 standard common sub-interval fuzzy sets are established for the respective element. Subsequently, based on the system characteristics of the civil aircraft, a multi-level civil aircraft commonality index system is built with attribute commonality and structural commonality as two branches. The analytic hierarchy process is adopted to assign weight coefficients to each level and its elements. Moreover, this study collects the parameters of the civil aircraft and designs a fuzzy test to obtain the distribution of the fuzzy set boundary number. Next, the three-division method is optimized as the six-division method to establish the membership function of the standard fuzzy set, establishing a complete set of evaluation model of commonality index of civil aircraft. Lastly, this study evaluates the commonality of Boeing 757-100 with many other aircrafts based on the commonality evaluation model of civil aircraft. The evaluated results consistent with the actual situation also prove the effectiveness of the evaluation model built in this study. In the above research, the following conclusions are drawn:

Fuzzy set theory exhibits good applicability and application value to evaluate the commonality between products.

The analysis of the case results shows that there is a high degree of commonality between the basic and derivative models of civil aircraft models. Moreover, the inheritance of design ideas for the same series of civil aircraft models also makes them have high commonality.

As impacted by the large difference in performance requirements of civil aircraft models in different seat class, the commonality between civil aircraft models of different seat class is poor, and even models of the same series may be at a moderately to low level of commonality.

The evaluation model of civil aircraft commonality index proposed in this study can evaluate the degree of commonality between any models, and the evaluated results can be effectively referenced for the decision-making of manufacturers and operators. Besides, it can lay a numerical foundation for building a commonality benefit model for civil aircraft.

Footnotes

Acknowledgments

This research was supported by the National Natural Science Foundation of China (Grant Nos. 11972301, 11972300), and Fundamental Research Funds for the Central Universities of China (Grant No. G2019KY05203).

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