Case-Based Reasoning System for Aeroengine Fault Diagnosis Enhanced with Attitudinal Choquet Integral

: As the core process of case-based reasoning (CBR), case retrieval is the foundation for CBR success, and the quality of case retrieval depends on the case similarity measure. We improved the CBR system for aeroengine fault diagnosis by embedding the attitudinal Choquet integral (ACI) and 2-order additive measure to consider attribute interactions and decision makers’ attitudes. The enhanced case retrieval method can not only integrate the local similarity, attribute importance


Introduction
When faced with a new problem, it is always helpful to consider the solution to a similar problem from the past. From increasing historical experience, it is possible to identify useful solutions. Case-based reasoning (CBR), a knowledge-based system, is an experience-based method [1]. It solves new problems by retrieving the most similar case from a member of cases [2], which is also the core process of the CBR cycle [3]. CBR is demonstrably suitable for solving problems in fault diagnosis. Yang et al. (2004) added a Petri net to a CBR system to improve the revision function, and successfully applied it to electronic motor diagnosis [4]. Owing to the complexity and uncertainty of the operating environment, Yang et al. (2017) proposed an optimized hybrid model combining CBR and a Bayesian network for embedded software fault diagnosis [5]. A CBR system for the intelligent fault diagnosis of power equipment was proposed by Ma [6].
Decision makers often need to weigh different criteria when making decisions as the factors involved in decision making become more complex. In addition, the importance of different criteria should be considered. Multi-attribute decision making (MADM), or multi-criteria decision making (MCDM), is a decision making methodological framework based on multiple criteria or multiple attributes [7]. MADM provides decision makers with a set of recommendations for alternatives, goals, or solutions [8]. The main task of MADM is to assess a member of the alternatives and then rank them [9,10]. MCDM has been applied in many fields such as supplier [11], material [12], and flotation machine selections [13].

Fault Diagnosis of Aeroengines
As one of the most complicated structures of aircraft, aeroengines work in extremely harsh environments of high temperature, high pressure, and strong vibration for a long time, leading to fatigue, creep, fracture, and many other component failures. Therefore, fault diagnosis of aeroengines is necessary to ensure safe and efficient operation of the engine. There are three widely used methods for aeroengine fault diagnosis: model-based, knowledge-based, and statistical learning methods.
Model-based fault diagnosis needs to analyze the entire structure of mechanical equipment and establish corresponding mathematical or simulation models. Peng et al. (2018) established a simulation model for lubricating oil systems [24]. Using this model, an online fault diagnosis system was built using the parameter trend analysis method. Kim and Mylaraswamy (2006) developed a fault diagnosis and prediction system based on discrete event system modeling and tested the system using actual flight data of startup component failures [25]. They believed that such qualitative modeling methods can be applied to the hierarchical diagnosis of complex large-scale systems because they do not require a detailed model of the system. Wang et al. (2018) established a nonlinear model for the fault diagnosis of a fuel regulator using a particle swarm optimization algorithm combined with a back-propagation neural network for higher diagnosis precision [26]. The nonlinear modeling method is closer to the real operating state of an aeroengine with higher modeling accuracy. This type of method, combined with some conventional fault classification algorithms, can better serve the fault diagnosis of aeroengines.
Knowledge-based fault diagnosis can effectively utilize expert knowledge and experience to make judgments without relying on analytical mathematical models. An expert system is the most classic knowledge-based fault-diagnosis method. Sun et al. (2021) proposed an expert system for aeroengine gas-path fault diagnosis to manage the flight data of various types of engines [27]. Chen, Qu, and Fang (2022) built the first tentative case base in the field of aeroengine fault diagnosis, and developed a CBR system with a highly accurate novel similarity measure for fault diagnosis of aeroengines, where three local similarity measures associated with different attributes were integrated [28].
Statistical learning methods for the fault diagnosis of aeroengines mainly include neural networks, support vector machines, and Bayesian networks. Zhao et al. (2020) used neural network methods such as convolutional neural networks and back-propagation neural networks for aeroengine gas-path fault diagnosis instead of traditional thermodynamic methods [29]. Neural networks can not only obtain higher accuracy of fault diagnosis but also have better adaptability to aeroengine data. Romessis and Mathioudakis (2004) proposed a Bayesian belief network for gas turbine performance fault diagnosis, which can be built from mathematical models rather than hard-to-obtain flight data from malfunctioning operations [30].

Nonadditive Measure
Mathematically, a measure is a function that assigns a number to a subset of a set. This number can be compared with the size, volume, probability, etc. The characteristic of classical measures is additivity. For example, the probability of several mutually exclusive events is the sum of the probability of each event. However, in certain cases, the additive measure does not satisfy this requirement. For example, the work efficiency of two people is not simply the sum of their work efficiency. Similarly, in CBR, interactions may exist between attributes, indicating that attributes are not always independent of each other. Therefore, the combined weight of two interactive attributes is not always the sum of the weights of the two attributes.
In response to the above violation of additivity, Choquet (1954) [31] and Sugeno (1974) [32] proposed nonadditive measures, namely fuzzy measures, which were, respectively, the Choquet and Sugeno integrals. These two nonadditive measures have a wide range of applications, particularly in MCDM situations [33]. The MCDM with fuzzy measures can not only fully consider the relative weight of the decision criteria, but also flexibly describe the interaction between criteria [34]. However, with an increase in parameters, the complexity of the nonadditive measure increases exponentially, which is one of the defects of the fuzzy measure. For example, when the cardinality of set X is n, the values of n parameters must be determined for additive measures; for a nonadditive measure, we must compute the values of all subsets of X with 2 n parameters. To address the complexity of the model, many other nonadditive measures have been proposed, such as λ-addition fuzzy measures [35], in which only n − 1 parameters are required. However, λ-addition fuzzy measures also have some disadvantages. For nonadditive measures, there are three types of interactions: independent, redundant, and complementary. In the λ-addition fuzzy measure model, only one interaction exists [36]. Grabisch (1997) [37] proposed an additive discrete fuzzy measure of order k. The definition of the k-order additive measure is based on the pseudo-Boolean function [38], which is an approximate representation of the k-order nonadditive measure. In addition, the Mobius transformation is used to describe k-order additive measures [39]. The k-order additive measures can not only deal with the complexity of nonadditive measures, but also represent three different types of interactions using Shapley indices as the development of the Shapley value [40]. Many scholars have improved and developed k-order additive measures and applied them in various fields. For example, Honda et al. (2022) developed a k-order additive measure for a nondiscrete case [41]. The 2-order additive measure can solve the contradiction between complexity and precision in a wide range of applications [42]. Zhang et al. (2021) proposed a 2-order additive fuzzy measure based on intuitionistic fuzzy sets to quantitatively evaluate the interactions between attributes [43]. Li et al. (2016) proposed a simulation credibility group evaluation method using 2-order additive fuzzy measure to train a traction-drive simulation system [44]. The 2-order additive measure is only concerned with the importance of each attribute and the interaction between two attributes, which is a perfect compromise between the performance and computational complexity of the nonadditive measure, and has been widely used in MADM methods [45].

Preliminaries
In this section, we provide three definitions that constitute the basis for subsequent discussion. Definition 1. Nonadditive measure [32].
If X is a finite nonempty set, then a set function μ: Г(X) → [0, 1] can be defined as a nonadditive measure or fuzzy measure if the set function satisfies the following conditions: where μ is a regular fuzzy measure on X.
where λ indicates the decision maker's attitude satisfying (0, ) λ ∈ ∞ and 1 λ ≠ . A higher λ indicates a more optimistic decision maker. Typically, the value of λ is set to 10 n [19]. μ refers to the importance of individual attributes and their interactions.

CBR Framework
When a new case appears, a CBR system begins with case retrieval. First, key attributes are extracted to represent the case. Generally, attribute data can be divided into several types. In this study, we propose different similarity measures for three types of attribute data. In addition, for an accurate CBR system, the weights of individual attributes and the interaction between attributes should be considered. Many previous studies calculated the weight of attributes using an average operator or weighted average operator, which may lead to a significant difference between the calculated weights and the actual weights. However, the attribute weights are not the same in most cases. In some cases, attributes are not independent of each other, and the interaction between attributes is too important to ignore. Therefore, the joint weights of multiple attributes are not always the sum of the weights of each attribute. We adopt a 2-order additive measure to elaborate on all interactions between attributes. Furthermore, considering the influence of the decision maker's attitude, the ACI was used to obtain the global similarity between the two cases. The framework of the proposed CBR system is shown in Figure 1 and elaborated below.

Local Similarity Measure
In general, there are various data types in CBR systems. In our case base, the data attributes are divided into three types: categorical, numerical, and semantic. Categorical attributes include the aeroengine model, aeroengine category, aeroengine operation state, thrust performance, temperature performance, rotational performance, aeroengine shutdown, and other anomalies. Numerical attributes contain flight height and speed. The semantic attributes contain the fault part and mode in aeroengines. The similarity calculation methods for the three types of data attributes can be found in our previous study [28].
(1) Categorical Attributes. In our CBR system, the similarity of categorical attributes is calculated using Equation (3).
where nc j a is the value of attribute j in new case nc, and i c j a is the value of attribute j in case ci.
where I is the total number of cases.
where J is the total number of attributes.
The similarity between numerical attributes is defined as the normalized distance between two attribute values, calculated using Equations (4) and (5). (3) Semantic Attributes.
The semantic similarity based on tree is used to measure the semantic attribute similarity of the aeroengine fault part and fault mode, which defines the association of the fault mode with the fault part. It consists of two elements: tree structure of the fault part and semantic diagram of the fault mode. Figure 2 shows a schematic diagram of the fault part tree structure of an aeroengine, wherein parts are differentiated and structured to calculate their similarities under the same or different fault modes. Further, a fault mode semantic diagram is developed, as shown in Figure 3. Finally, the association of fault mode with fault part is defined by combining the fault part tree structure and the fault mode semantic diagram, as shown in Figure 4.   The measure of semantic similarity is based on the following three constructs.
(i) In the semantic graph, two nodes with shorter distance are more similar than two nodes with longer distance. The shortest path between nodes ni and nj is denoted as Dist (ni, nj). For example, Dist ("Fracture (Compressor Blade)", "Wear (Compressor Blade)") = 4 implies that the shortest path between Fracture (Compressor Blade) and Wear (Compressor Blade) is 4. (ii) The nearest shared parent node nk of nodes ni and nj is represented by Nspn (ni, nj).
For example, Nspn ("Crack (Compressor Vane)," "Crack (Compressor Blade)") = Compressor, which means compressor is the nearest shared parent node of Crack (Compressor Vane) and Crack (Compressor Blade). (iii) The distance from a node to the root node is defined as the depth of the node, denoted as Depth(nk). For example, Depth (Compressor Vane) = 2 indicates that the depth of node Compressor Vane from root node Aeroengine is 2.
Then, the semantic similarity is defined as a function of the node location in the taxonomy, which is calculated using Equation (6).

A 2-Order Additive Measure Method
The main purpose of the proposed method is to determine the range of interaction by combining the weight of each attribute based on the BWM with the degree of interaction between attributes. The maximum entropy principle is then employed to build an optimization model, and finally, the 2-order additive measure is obtained.
First, we present the theoretical basis of the 2-order additive measure. For , a unique 2-order additive measure can be determined when the interaction between attributes satisfies the following conditions [46]: The calculation of the 2-order additive measure comprises the following five progressive operations.
Step 1. Choose key attributes. Decision makers must choose key attributes to represent a case. The key attribute set is denoted as Step 2. Select the best and the worst attributes among all attributes. The best attribute b a usually refers to the most important or considerable attribute, whereas the worst attribute w a refers to the least important attribute. Notably, if there is more than one best/worst attribute, any can be selected.
Step 3. Compare b a to other attributes using the numbers 1 through 9. The decision maker needs to compare the preference of each attribute (except the best one) over b a in turn. The importance scale is presented in Table 1. The importance ratio vector is denoted as , where bj r denotes the importance ratio of b a to j a . Step 4. Compare other attributes to w a using numbers 1 through 9. Similar to step 3, the preference of other attributes (except the worst one) for w a is compared in turn. The importance ratio vector is denoted as where jw r is the importance ratio of j a to w a .
Step 5. Compute the optimal weight of each attribute. Steps 3 and 4 obtain the importance ratio vectors b R and w R , which can be regarded as the ratios of weights. Therefore, the importance ratio of b a to j a can be expressed as Under ideal conditions, the minimum value of the objective function δ should be 0.
Generally,  (7) is to search for optimal weights to minimize max , (2) Specify attribute interaction intensity.
According to the above theoretical basis, the interaction jk I between j a and k a should satisfy the following inequalities: The minimum value of the interaction is defined as (3) Determine attribute interaction based on the maximum entropy principle.
Applying the maximum entropy principle, the value of interaction ij I can be obtained by solving the following nonlinear programming problem.
(4) Determine value of the Mobius representation.
According to the corresponding relationship between the Mobius representation and jk I [48], the value of Mobius can be calculated using Equation (11). , are computed using Equation (12).

Global Similarity
The global similarity between cases is the weighted sum of similarity obtained by integrating the local similarity of different attributes calculated using ACI, as shown in Equation (13).  In the study of Chen et al. (2022) [28], a prototype CBR system for fault diagnosis of aeroengines has been established. However, the system does not consider the weight of attributes, interaction between attributes, and the attitude of decision makers in determining the similarity measure. To address the above defects, this study introduces the 2-order additive measure, an approximation of the nonadditive measure, and ACI to improve the case retrieval method. In particular, the preliminaries presented in Section 3 constitute the theoretical basis of nonadditive measures. The CBR system enhanced with the improved retrieval method can diagnose aeroengine faults more efficiently.

Experimental Study and Discussion
This section illustrates the proposed retrieval method using an experimental study. In the experiment, there are four cases of aeroengine fault diagnosis in the case base, as listed in Table 2. The case base is stored in ACCESS software, in which each case is represented by five attributes: the aeroengine model, aeroengine category, aeroengine operation state, other anomalies, and aeroengine fault part and fault mode, denoted by a1, a2, a3, a4, and a5, respectively. Here, a1, a2, a3, a4 are categorical attributes, while a5 is a semantic attribute. Note that in this experiment, we do not cover numerical attributes that are relatively simple and easy to handle. New case information is presented in Table 3. In CBR, the existing cases in the case base must be sorted; therefore, the most similar case is selected to solve the new case.

Nonadditive Measure Calculation
First, BWM is used to calculate the initial attribute weights. Attribute set A = {a1, a2, a3, a4, a5}. According to the decision maker, a5 is the best attribute (ab) and a2 is the worst attribute (aw). The importance ratios of ab to the other attributes are listed in Table 4, and the importance ratios of other attributes to aw are listed in Table 5.  The optimal attribute weights can then be obtained by solving the following optimization problem:  ω ω ω ω ω > > > > . The calculation result is consistent with the decision maker's attitude.
After obtaining the initial attribute weights, the interactions between attributes must be considered. According to Equation (9), we can get X12 = min (2I1/4, 2I2/4) = min (0.0316, 0.0233) = 0.0233. Here, In refers to ωn. In this experiment study, the attribute number n = 5. Similarly, all values for Xjk are calculated and the results are listed in Table 6.

Xjk
Value Second, the interaction interval is divided into subintervals to specify the intensity of the attribute interaction. In the experiment, t = 5. Therefore, the interval [ , ] jk jk X X − is divided into five sub-intervals: Here, we use К1 to К5 to denote the five subintervals. The decision maker then determines the interaction between any two attributes as a negative/independent/positive relationship. According to the decision maker, the interaction between a1 and a2 is quite negative, thus, the interaction interval between a1 and a2 belongs to К1. Similarly, a1 and a3 , a2  and a3 belong to К2; a2 and a4, a2 and a5 belong to К3; a1 and a4, a1 and a5, a3 and a4, a3 and a5 belong to К4; a4 and a5 belongs to К5. Therefore, 12 12  . Similarly, we can obtain all the interaction intervals for the possible attribute pairs, as listed in Table 7.
By using MATLAB programming, we obtain all the values of Ijk, as listed in Table 8.

Ijk
Value Fourth, the values of Mobius representation can be obtained using Equation (11), as listed in Table 9. For example,  Finally, the 2-order additive measures of all subsets of A are calculated using Equation (12). The results are listed in Table 10. For example,

Similarity Calculation by ACI
The local similarity between ci and nc can be obtained as shown in Table 11. Recall that a1, a2, a3, and a4 are categorical attributes and a5 is a semantic attribute. Hence, the similarity measures a1, a2, a3, and a4 are calculated using Equation (3), and the similarity measures of a5 are calculated using Equation (6).
After obtaining the nonadditive measures and local similarity, the ACI is used to aggregate them and calculate the global similarity.
Using ACI, the similarity between c1 and nc is calculated from Equation (2), as follows:  Table 10. In the experiment, the similarity between c1, c2, c3, and c4 and nc are 0.7709, 0.9455, 0.9373, and 0.2752, and the corresponding λ values are set to 10 2 , 10 6 , 10 4 and 10 respectively. The value of λ is determined by the decision maker, which varies with the decision maker's different attitudes towards ci and nc. Notably, the decision maker is most optimistic about c2 and most pessimistic about c4; therefore, we set a larger λ for c2 and a smaller λ for c4. Table 12 shows the global similarity between cn and nc under different aggregation functions, including ACI, AO (average operator), and OWA (ordered weighted averaging), where the attribute weights in OWA are determined by the BWM and calculated by Model (14).  Table 12, the preference ranking of solutions to the new coming case by ACI and OWA is c2 ≻ c3 ≻ c1 ≻≻ c4. However, AO produces a preference ranking c3≻ c2 ≻ c1 ≻≻ c4, which is inconsistent with the preference derived from the decision maker's attitude. This is because the attribute importance varies significantly, and the AO does not consider this. Both ACI and OWA consider the attribute importance, thus producing reasonable ranking results. In particular, the preference ranking from ACI is closer to the decision maker's cognition. However, regarding the magnitude of global similarity given by ACI, c2 (with a similarity of 0.9455) has only a 0.87% advantage over c3 (with a similarity of 0.9373). Thus, when solving nc, c3, in addition to the first choice c2, is worth considering. Perhaps the combination of c2 and c3 will produce a better solution for nc. This is because ACI considers the decision maker's attitude in addition to the interaction between attributes, thus helping the decision maker to make higher-quality decisions. Notably, decision makers must possess professional knowledge and decision making ability when determining the value of λ in ACI. In CBR embedded with ACI, correct judgment of λ can yield a more accurate case retrieval result, whereas wrong judgment may lead to an unreasonable result.

Sorting Results and Comparison
CBR is an experience-based approach to knowledge-intensive problem solving. The case base of the CBR system contains cases that are considered to have been successfully solved. However, in case representation, one type of empirical knowledge, namely the attitude or preference of decision makers, is difficult to describe and model. The key to the success of CBR is to retrieve the most appropriate cases to solve a new coming case. Therefore, case retrieval is the core process of CBR cycle. Chen et al. (2022) [28] adopted the AO method to calculate the global similarity and obtained high retrieval accuracy. In this study, we synthesize other important but often ignored factors in similarity calculation, including attribute weight, interaction between attributes, and attitude of decision makers, to further improve the accuracy of case retrieval while inheriting the reliability and validity of the method proposed in [28]. Note that in the above experiment, the preference ranking obtained by using AO violates the preference associated with the decision maker's attitude. The improved retrieval method actually adjusts and optimizes the ranking result from AO, producing a more comprehensive and reasonable ranking. It is this difference that indicates the superiority of the improved retrieval method. At the same time, in practice, we may continuously verify and improve the validity of the improved retrieval methods while updating and enriching the case base. For a new coming case, the CBR system first recommends several cases and ranks them from most similar to least similar. Then, without showing the ranking result of the system, the decision maker is required to rank the cases recommended by the system according to his expertise and attitude. If the two rankings are consistent, it indicates that the improved retrieval method is reliable and valid. Otherwise, we need to examine the cause of the inconsistency and determine whether to further improve the retrieval method.

Conclusions
In this study, an improved case-retrieval method for a CBR system for aeroengine fault diagnosis is developed. To overcome the limitation of assuming that the attributes of the case description are independent of each other, we consider the interaction between attributes. A simple and practical approximation of a nonadditive measure, that is, the 2order additive measure, is introduced into the aggregation of attribute weights, which combines the weights of each attribute with the interaction between attributes to determine the nature and intensity of the interaction. The calculation procedure of the 2-order additive measures for all subsets of the attribute set is presented. The ACI can be used to determine the global similarity between two cases by synthesizing the local similarity, attribute weight, and attitude characteristics of the decision maker. Through an experimental study in the field of aeroengine fault diagnosis, along with a comparison analysis, the application of the proposed method is demonstrated, and its effectiveness is verified. The results show that the method is feasible in CBR for aeroengine fault diagnosis and can improve the accuracy of case retrieval.
In complex knowledge-intensive fields such as aeroengineering, intricate relationships among attributes with positive or negative interactions often exist. In addition, when determining case similarity in a CBR system, the decision maker's attitude is an influencing factor that cannot be ignored. The ACI incorporating a nonadditive measure can appropriately quantify attribute interactions and embed the decision maker's attitude. The CBR system enhanced with ACI improves the case retrieval accuracy by providing highquality recommendations.
There is room for improvement in our work. We will collect new aeroengine fault diagnosis cases to refine and enrich potentially critical attributes. With the updating and expansion of the case base, we will further verify the validity of the proposed retrieval method and continue to improve it through practical application. For example, when determining attribute weight, the judgment information about attribute importance generally has fuzzy characteristics. Therefore, it is worth considering the adoption of fuzzy set theory to determine attribute weight. In addition, utility theory can be introduced within the framework of ACI to consider the preference of decision makers in the future.