A new interval constructed belief rule base with rule reliability

The combination rule explosion problem of belief rule base (BRB) is a difficult problem to solve in complex systems and has attracted wide attention. A new interval-constructed belief rule base with rule reliability (IBRB-r) is proposed to solve the problem of combination rule explosion in belief rule base. This model not only proposes a new interval rule construction method, but also designs a new interval rule inference process with rule reliability. This approach can not only clearly indicate the contribution degree of each rule to the model, but also solve the problem of combination rule explosion. This is because combining rules in interval addition form avoids the exponential growth in the number of rules caused by combining rules in Cartesian product form. Therefore, IBRB-r is more suitable for complex system modeling. In the case study section, the structural safety assessment of liquid launch vehicle is introduced to conduct a concrete example analysis. Experimental results show that the proposed model achieves over 95% accuracy under the liquid rocket dataset and has relatively higher accuracy under other datasets as well.


Introduction
The belief rule base (BRB) method based on evidential reasoning was developed based on decision theory, fuzzy theory, the traditional IF-THEN rule base and D-S evidence theory [1]. It is a method driven by mixed data and knowledge that can deal with uncertain information and has good processing performance for small sample data [2]. At present, BRB has been widely used in fault diagnosis [3], medical diagnosis [4], risk assessment [5] and other fields.
Most of the current methods in practical engineering are based on data-driven approaches, such as BPNN, ELM and RBF. For example, Kashem et al. [6] combined principal component analysis and BPNN and applied them to a face recognition system. This method uses PCA to perform dimensionality reduction operations on facial images and the BPNN method for facial recognition. The experimental part proved that this model has an acceptance rate of more than 90% and low time complexity. Ghose et al. [7] proposed a model combining BPNN (back propagation neural network, BPNN) and RBFN (radial basis function network, RBFN) applied to groundwater level detection, which can simulate the fluctuation of groundwater level depth. Bardhan et al. [8] proposed an ELM adaptive neural swarm intelligence method that can be applied to the California bearing ratio in geotechnical engineering. The experimental results showed that the proposed method could predict the ground vibration values more accurately. Benaissa et al. [9] proposed the RBF-based amplitude saturation controller method and designed a neural network-based monitor controller that can be used to control the maglev vehicle monitor. Simulation results verified the effectiveness and robustness of the method. The data-driven approach refers to the databased forecasting, evaluation, scheduling, monitoring, diagnosis, decision-making and optimization of engineering systems using a large amount of sample data to achieve the desired functions [10]. Data-driven modeling is based on sample data, and the quality and quantity of data directly affect the actual effect of modeling [11].
In practical engineering applications, the safety assessment of complex systems relying only on a subjective or objective judgment can no longer meet the needs of the actual system. The expert knowledge base in BRB can be wellinvolved in the safety assessment of complex systems with certain validity and accuracy. However, the assessment model based on BRBs still has the following two problems. On the one hand, due to the complex system situation, when the assessment indexes and referential values are too many, the rules of the safety assessment model based on BRB may cause the phenomenon of "combination explosion." When there are too many attributes or referential points, the traditional BRB adopts the Cartesian product form to establish the belief table, which easily produces the phenomenon of "combination explosion." This will increase the algorithm's complexity and reduce the model's processing performance. On the other hand, the traditional BRB does not consider rule reliability and cannot indicate the contribution of each rule to the assessment model. In practical engineering applications, rules are only partially reliable due to the limitations of environmental noise and conditions. This will directly reduce the model's accuracy and indirectly lead to difficulty in the reduction rules, which could be more conducive to human judgment.
For the rule combination explosion problem in BRB, there are two solutions. First, rule reduction methods are used to reduce rules. In this respect, domestic and foreign scholars have performed much research. Wu et al. [12] introduced information entropy and K-prototypes to remove redundant rules based on fuzzy rough set theory. Yang et al. [13] proposed a rule reduction method based on data envelopment analysis for a belief rule base. Ben Li et al. [14] used Petri nets to solve the problem of combinatorial rule explosion in complex systems. Zhang et al. [15] used the density-based spatial clustering with noise (DBSCAN) algorithm to reduce belief rules, and a new training method based on parameter learning was proposed. Chang et al. [16] introduced gray target (GT), multi-dimensional scale (MDS), isometric mapping (ISOMAP), principal component analysis (PCA) and other feature extraction methods to carry out rule reduction to screen important attributes. Second, the original BRB is replaced by a hierarchical BRB model. Hierarchical BRB can split a multi-attribute dataset, selecting two attributes simultaneously and progressing. This method can effectively reduce the number of attributes to solve the problem of combination rule explosion to a certain extent [17].
The above two methods have been proven effective in reducing rules, but many things could be improved. Using the rule reduction method to reduce rules can easily lead to the loss of precision of BRB, an increase in model complexity and the influence of model representation ability [18]. Specifically, (1) the model cannot guarantee high accuracy after the reduction rule, such as GT and MDS. (2) The accuracy after the reduction rule is ideal. Nevertheless, the algorithm's complexity needs to be lowered to be realized, such as ISOMAP, PCA and DBSCAN. (3) Belief rules based on feature extraction cannot guarantee the integrity and consistency of rule reduction at the same time, such as PCA. With hierarchical BRB modeling, there are two problems: (1) The large structure of the hierarchical BRB model network easily causes the problem of high model complexity. (2) The initial model is constructed hierarchically, and the middle layer cannot be trained. As a result, the results of the middle layer cannot be determined, and the uncertainty of the model increases.
Although these two methods can effectively reduce rules, they need to establish the BRB model first and then reduce rules. In essence, establishing the original model first and then reducing the rules still needs to reduce the rules effectively. In addition, the rule reduction and hierarchical BRB methods do not consider rule reliability and cannot indicate the contribution degree of each rule to the model.
Since the traditional BRB does not consider reliability, the current research only considers attribute reliability and weight. Feng et al. [19] introduced premise attribute reliability into the belief rule base and proposed a BRB model with attribute reliability. However, the model without considering the reliability of rules has some disadvantages. On the one hand, unreliable redundant rules cannot be removed, 1 3 increasing the difficulty of rule reduction and resulting in many rules. On the other hand, the algorithm complexity increases, and the model performance is reduced.
From the above discussion, it is clear that the traditional BRB still has some shortcomings, such as the combinatorial rule explosion problem and the lack of reliability of the rules. Therefore, a new interval-constructed belief rule base with rule reliability (IBRB-r) is proposed. On the one hand, the IBRB-r model has abandoned how the BRB builds the belief table in the form of carte accumulation but is based on adding the interval. This rule combination greatly reduces the number of rules and the complexity of the model. The initial model is built in the interval form, the original BRB model is established, and the explosive problem of the combination rule is solved fundamentally. The IBRB-r model also introduces rule reliability. It uses the experience of the expert to evaluate the knowledge of the rules, which forms the reliability of the rules and the removal of the rules [20]. It is found that the IBRB-r model retains the advantages of the traditional BRB. At the same time, the combination rule problem of BRB is solved without destroying the model structure, and the reliability of the rule is fully considered, which is more suitable for engineering applications [21].
The main contribution points of this paper are as follows. First, a new IBRB-r model for interval form construction and combination rules is proposed. This model can solve the combinatorial rule explosion problem caused by the traditional BRB when the number of premise attributes and referential points increases. Second, the ER rule algorithm is introduced in the model inference process, and the rule reliability is considered. Rule reliability reflects the ability of a source of evidence to provide a correct assessment or solution to a given problem, adding to the objectivity of the rule [22].
The framework of this paper is organized as follows. The first part summarizes the shortcomings of the traditional BRB. To address these shortcomings, the IBRB-r model is proposed. The second part briefly introduces BRB and discusses the challenges of establishing the IBRB-r model. The third part elaborates on the overall structure of the IBRB-r model from three aspects: modeling, reasoning and optimization. The fourth part provides case studies to demonstrate the effectiveness and accuracy of the IBRB-r model. The fifth part summarizes the thesis and the prospect of future work.

Preliminary: BRB and existing problems
In 2006, Yang proposed a belief rule base inference method based on evidential inference rules [20]. BRB is a rule-based expert system that can use a mixture of data and knowledge to drive modeling and establish a nonlinear mapping between input and output. The belief distribution of a BRB can effectively represent the multisource information of uncertainty, including probabilistic uncertainty and fuzzy uncertainty [23]. The BRB has a set of belief rules, and the k − th rule can be expressed as follows: where the k rule of the BRB is denoted as R k . The i − th premises attribute is denoted by x i (i = 1, ..., M) . The set of referential values of M premise attributes in rule k is denoted as A k i (i = 1, ⋅ ⋅ ⋅, M) . The i − th results are denoted as D i (i = 1, ..., N) , and the corresponding belief degree of each result under the k − th rule is denoted as i,k (i = 1, ⋅ ⋅ ⋅, N) . The rule weight of rule k is denoted as k . The total number of rules is denoted L . The meanings of the symbols appearing in the paper are shown in Table 1.
BRB mainly comprises a knowledge base, inference engine and optimization algorithm. The knowledge base of BRB is expert knowledge acquired through longterm and extensive practice, which is the accumulation of professional knowledge. BRB adopts evidential reasoning (ER), and the reasoning process can be traced and explained. The optimization algorithm of BRB uses projection covariance matrix adaptation evolutionary strategies (P-CMA-ES). This chapter will analyze the original BRB model in detail from the perspectives of modeling, reasoning and optimization and point out the existing problems. Figure 1 shows the overall structure of the traditional BRB model, which comprises three parts: knowledge base, inference machine, and optimization algorithm. The BRB knowledge base is represented as a series of belief rules. The initial expert knowledge base is constructed based on expert knowledge generated by long-term practice, which can convert qualitative information into quantitative input. The inference machine uses ER, which has good interpretability and process traceability. The optimization model is designed to optimize the parameters of the initial model constructed from expert knowledge to further improve the model's accuracy.

BRB modeling process: modeling and problems
This section will detail the traditional BRB modeling process and its problems. The following will focus on analyzing the modeling process of traditional BRB from three aspects: problem mechanism analysis, referential point setting and belief table construction.

Problem mechanism analysis
The first step in modeling should be identifying the problem the model intends to solve. It is important to identify the factors that influence the problem and determine (1) with rule weight k and attribute weight 1,k , 2,k , .., M,k k ∈ {1, 2, ..., L},

Set referential values and referential points
Referential points should be selected where the attributes have typical significance, usually in most datasets. The referential point contains upper and lower bounds and can represent the range of data values. The number of referential points should be defined according to the actual problem. The more general referential points there are, the higher the model accuracy, but the model complexity will also increase. For example, if one premise attribute has five referential points and another premise attribute has six referential points, 30 belief rules need to be established. However, if you have five premise attributes and 20 referential points for each premise The i − th parameter vector in the vector of the s + 1 generation The three assessment grades of liquid launch vehicle structure safety, represent normal, medium and low, respectively  Expected utility attribute, you need to establish 20 5 rules. With an increasing number of premise attributes and referential values, the number of rules increases exponentially, and the established rules are prone to the problem of combination rule explosion.

Build a belief table
After setting the premise attributes and the result referential points and referential values, the belief table needs to be established. According to the corresponding belief rules, the belief table is constructed. The following is a brief example of how traditional BRB builds belief tables. For example, suppose that each prerequisite attribute has three referential levels, A, B, C and I, J, and K, while the result has four referential levels L, M, O, H. In this case, nine belief rules need to be constructed, and the corresponding belief table is shown in Table 2.
The above traditional BRB constructs belief tables in the form of Cartesian products. In the above example, the number of attributes is small, and the number of referential values and referential points is small, so the combination of rules of the belief table does not produce a combination explosion. However, when the number of premise attributes, referential values, and referential points increase, it is easy to produce the problem of combination rule explosion. This is mainly due to how the rules are combined because the number of rules that build belief tables as Cartesian products increase exponentially. The traditional BRB is only suitable for simple systems with a small number of prerequisite attributes and referential values. With a large number of prerequisite attributes and referential values triggering an explosion of combinatorial rules, BRB is no longer applicable.

The reasoning process of BRB: reasoning and problems
As the inference engine of BRB, ERs are a multi-attribute decision-making method formed based on decision theory and D-S evidence theory. Its belief framework has good performance in describing uncertain problems, so it is chosen by BRB for model reasoning. The inference of BRB is mainly divided into four parts: calculation

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A new interval constructed belief rule base with rule… of rule fitness, calculation of rule activation weight, rule synthesis by the ER analytic algorithm, and calculation of expected utility value. First, the rule fitness is calculated. This step completes the input data transformation according to the premise attributes' different properties. The equivalent transformation based on utility can fully retain the features of the original data and is suitable for the input data transformation of the BRB reasoning process.
Then, the rule activation degree is calculated. BRB combines rules in the form of Cartesian products, and each combination rule has a different practical meaning. In practical models, not every rule is equally important. The activation degree of each rule is different.
Then, the ER analytic approach is used for rule synthesis. In 2007, Yang proposed the ER analytical method [24]. In model reasoning, the ER analytical method is widely used. Finally, the model output results are calculated based on utility theory. The BRB model uses ER reasoning. ER inference integrates multiple sources of information and can show better processing ability for uncertain information. Moreover, the process of ER inference is traceable and explainable, which is obvious in the circuit. However, traditional BRB reasoning needs to consider rule reliability, which impacts model integrity and representation ability. Therefore, it is necessary to introduce rule reliability into the new model to enhance its integrity.

BRB optimization process: the parameter training model
The initial parameters of BRB are given by experts, including the belief, attribute weight and rule weight. Based on long-term practice, experts can give a general distribution in line with the trend of the actual system, but there may be better solutions to the model. Therefore, it is necessary to design an optimization model to correct the initial distribution given by expert knowledge so that BRB can achieve the optimal effect.
First, there are clear optimization objectives. The mean square error (MSE) is an important indicator to measure the effectiveness of a model [27]. The smaller the numerical gap between the model output and the label value is, the better the model optimization effect. The objective function of BRB optimization can be expressed as: where the optimized parameters include the rule weight, attribute weight and belief. The MSE is calculated as follows: where T train is the number of training samples and Z * and Z are the label value and the output value of the BRB model, respectively. Based on the above analysis and discussion, the complete optimization objective function of the BRB can be expressed as: Then, an appropriate optimization algorithm is selected to construct the optimization model. The optimization model continuously modifies the parameters by calculating the MSE value of the label and the BRB model output value. This process is shown in Fig. 2. As can be seen from Fig. 2, after the data samples are input to the initial BRB, ER inference is required. The initial BRB model constructed from expert knowledge is obtained after ER inference. However, the initial BRB model needs to be more accurate due to the system's complexity and the limitations of expert knowledge. For this reason, the optimization algorithm further optimizes the parameters of the initial BRB model by learning and training the input parameters of the real system to improve the accuracy of the BRB.

Question
According to the above analysis of the traditional BRB, it can be found that the traditional BRB still has many things that could be improved. These deficiencies greatly degrade the model's performance and must be addressed properly. It can be summarized as follows: 1. Rule combination explosion in the BRB modeling process The combination explosion of rules is a thorny issue for the BRB. Based on the analysis of the traditional BRB belief table construction method in Sect. 2.1, it can be found that combining rules in the form of Cartesian products easily produces a combination explosion. As the number of premise attributes and referential Traditional BRB reasoning uses the ER analytic approach, but the ER analytic approach does not consider rule reliability. This results in an incomplete model that indicates how much each rule contributes to the model. 3. BRB is not suitable for engineering applications with many attributes and referential values. Traditional BRB modeling methods are not suitable for engineering applications. The complex conditions of data noise, a large number of premise attributes, and many referential values and referential points in engineering applications cause this.
The above analysis proposes a new interval-constructed belief rule base with rule reliability (IBRB-r). In this model, belief tables are constructed as intervals, and rule reliability is considered. This not only solves the problem of combination rule explosion in traditional BRB modeling, but also solves the problem that rule reliability is not considered in traditional BRB reasoning, which can be fully applied to engineering practice.

Model description based on IBRB-r
Similar to a traditional BRB, the IBRB-r model is also composed of a series of belief rules. The difference is that the new model has a great change in the modeling method, and rule reliability is introduced into the reasoning method. Assuming that the premise attributes are independent of each other, the IBRB-r model can be described by the IF-THEN statement as where the M premise property is denoted as x i (i = 1, ..., M) . The referential interval of the M premise attribute can be denoted as [a i , b i ] , where i = 1, ..., M . The rule reliability of rule k is denoted as k . The rule weight of rule k is denoted as k .

Problem formulation
Section 2 proposes the IBRB-r model based on the need for more traditional BRB analysis. IBRB-r can perfectly solve the problem that the traditional BRB model easily causes the explosion of combination rules and does not consider the reliability of rules. This section will propose how to construct the IBRB-r model from the following three perspectives and formulate the problem.
Problem 1: How to reasonably design the modeling process of the IBRB-r model. According to the analysis in Sect. 2, there are problems in the combination of rules in the modeling process of traditional BRB, which easily causes the explosion of combined rules. Therefore, it is necessary to propose a new rule combination method to reasonably design the modeling process of the new model. The set of parameters after reasonable modeling can be described as follows: where L represents the number of rules. 1 , ..., L represents the initial belief given by the expert knowledge. 1 , ..., L represents the initial rule reliability. 1 , ..., L represents the initial rule weight. w 0 represents the parameter set of belief degrees, rule weight, and rule reliability.
Problem 2: How to reasonably design the reasoning process of the IBRB-r model. The traditional BRB does not consider rule reliability, so it is necessary to add rule reliability in the reasoning process to reasonably design the reasoning process of the IBRB-r model. The reasoning process of the new model can be formulated as follows: where Z represents the expected utility value of the IBRB-r model and represents the parameter set in the model inference process. x represents the set of prerequisite attributes of the model. inference (⋅) represents the formal description function of the inference process.
Question 3: How to reasonably optimize the parameters of the IBRB-r model. It is also a problem to select an appropriate optimization algorithm and optimize the parameters of IBRB-r. The optimization process of the new model can be described as follows: where w best is the optimal parameter set of the IBRB-r model. is a set of parameters in the optimization model. optimize (⋅) is a formal description function of the optimization process.

IBRB-r modeling process: a new way to set referential intervals and a new way to build belief tables
The modeling method of the IBRB-r model has been greatly changed from that of the traditional BRB. This section will detail the IBRB-r model's modeling process in detail from problem mechanism analysis, referential point setting and belief table construction.

Mechanism analysis of the problem
Regarding mechanism analysis, IBRB-r is the same as a traditional BRB. They all need to first identify the main influencing factors and possible outcomes of the problem, that is, the premise properties and possible outcomes. After that, data samples of premise attributes and possible outcomes are fed into the model for modeling, reasoning, and optimization.

Setting the new referential interval
Unlike the traditional BRB, IBRB-r sets, the referential values and referential points of the premise attributes in referential levels. IBRB-r replaces the referential value of the premise attribute with the referential interval and the referential point with the referential level. When the sample data of the premise attribute fall into one of the referential intervals, the corresponding belief rule will be activated. The activated rules are then involved in model inference and optimization.
IBRB-r's new referential interval setting is suitable for engineering applications. In practical engineering applications, monitoring data are easily affected by noise, and data samples have great uncertainty. This method of representing data referential points in the form of intervals allows for a better description of model uncertainty to be applied in engineering practice.
For example, suppose two premise attributes each have three referential levels, and the result has four referential levels. The following is an example of setting the new referential level and interval. The referential level and referential interval settings of the two premise attributes are shown in Tables 3 and 4, respectively. The referential levels and referential intervals of the results are shown in Table 5.

New belief table construction method
The IBRB-r model proposes a new way of constructing a belief table and rule combination. Its rules are combined in additive form, not in Cartesian product form. This combination of rules results in a dramatic change in the construction of belief tables Table 3 Referential levels and referential intervals of prerequisite attribute 1

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and has the following advantages. On the one hand, the combination of rules avoids the exponential growth of the number of rules and can effectively reduce the number of rules. This kind of rule combination perfectly solves the problem of combination rule explosion. On the other hand, constructing a belief table in addition to form is more suitable for engineering practice. Due to a large number of premise attributes, referential points and referential values in engineering practice, it is easy to cause large and complex problems in the model. Using an addition form to construct a belief table can not only solve the problem of combination rule explosion, but also simplify the model and reduce its complexity. The following is an example of the new belief table construction. If the value of each attribute is divided into different non-overlapping interval ranges, each interval corresponds to a rule. If you have two premise properties, one premise property has three referential intervals, and another premise property has four referential intervals, then the number of rules is 3 + 4 = 7. However, the traditional BRB requires 3*4 = 12 rules. In this case, the number of rules in the IBRBr model is reduced by 42% compared with the traditional BRB. Assuming one premise property has six referential intervals and another premise property has ten referential intervals, then the number of rules is 6 + 10 = 16. However, the traditional BRB requires 6*10 = 60 rules. In this case, the number of rules in the IBRB-r model is reduced by 73% compared with the traditional BRB. Given that one premise property has 30 referential intervals and another premise property has 30 referential intervals, the number of rules is 30 + 30 = 60. However, the traditional BRB requires 30*30 = 900 rules. It can be seen that in this case, the number of rules of the IBRB-r model is reduced by 93% compared with the traditional BRB.
As shown in Tables 2, 3, 4, assume that the referential points for one of the prerequisite attributes are A, B and C, and the referential points for the other prerequisite attribute are I, J and K. Figures 3, 4 show a comparison of the two ways of combining the rules.
The form of the BRB combination rule requires that the referential points between different attributes are combined in the form of a Cartesian product. As shown in Fig. 3, the three referential points A, B, and C for one of the attributes and the three referential points I, J, and K for the other attribute are combined As seen from Fig. 4, IBRB-r combines rules in the form of interval addition. Setting the three referential intervals of one attribute as A, B and C and the three referential intervals of another attribute as I, J and K, we can obtain 3 + 3 = 6 interval rules. These six rules are A, B, C, I, J, and K. This is different from the way the rules of BRB are combined. It should be noted that IBRB-r is in the form of "or" when combining rules, but it is in the form of "and" when the rules are activated. In other words, although each interval of each attribute is used as a rule when constructing interval rules, each attribute is activated as a rule in the inference process of IBRB-r.
As a result, the new belief table construction table is shown in Table 6. Compared with the traditional BRB belief table construction method, the belief table constructed by the IBRB-r model greatly reduces the difficulty of rule construction. In addition, the way IBRB-r constructs belief tables completely solves the problem of exploding combination rules.

New rule activation mode
After completing the modeling process, the traditional BRB activates the rule when the matching degrees of the prerequisite attributes are not 0. However, the IBRB-r IBRB-r rule combination mode A B C I J K Fig. 4 Rule construction of IBRB-r   table shown in Table 5, when premise one attribute falls in the interval [a 1 , b 1 ] and another premise attribute falls in the interval [c 2 , d 2 ] , rules 1 and 5 are activated accordingly.

Newly introduced ER rule
The original ER parsing algorithm determines that the evidence is completely reliable, considering only the weight of the evidence. However, evidence is obtained in different ways, making the obtained evidence only partially reliable [25]. Therefore, the introduction of evidence reliability is necessary. The weight of evidence is different from reliability, which reflects the ability of the source of evidence to provide a solution to the problem to be solved and is an inherent evidentiary property that is an objective concept [25]. ER rules can distinguish between the reliability and weight of evidence and incorporate them into the belief distribution of inferences and evidence [25]. The inference process of IBRB-r introduces ER rules, which further considers the rule reliability compared with the ER analytic approach. Here, the rules of IBRB-r are equivalent to the evidence in the ER rules. Different ways of obtaining evidence are different, and it is easy to be disturbed by the environment in obtaining evidence. As a result, evidence may not be completely reliable, so the ER rule introduces evidence reliability, that is, the regular reliability of the IBRB-r model. Different from the ER analytic approach, the reasoning process of the ER rule algorithm is mainly shown in Fig. 5.
The rules in the IBRB-r model modeling process are used as evidence in the ER rules. Mark the independent evidence in section L as e i (i = 1, ..., L) . The identification framework is denoted as Θ , which consists of N assessment level D n (n = 1, ..., N) . This can be represented as Θ = {D 1 , ..., D N } . A piece of evidence can then be represented as the following belief distribution: where n,i is expressed as the belief degree that the assessment scheme is evaluated as D n under the evidence e i . Θ,i is expressed as global ignorance, that is, the belief of the i − th attribute concerning the identification framework Θ. Suppose that the weight of evidence is i (i = 1, ..., L) , and i ∈ [0, 1] . The evidence reliability is i (i = 1, ..., L) , and i ∈ [0, 1] [0,1]. Then, the belief distribution of evidence mixed weighting with reliability can be expressed as: where the power set is denoted by (Θ) . The mixed probability mass of the i − th attribute in hierarchy D n is denoted as m n,i , which can be obtained by the following formula: where the normalized coefficient is denoted by c w,i = 1 (1 + i − i ) , which satisfies ∑ N n=1m n,i +m (Θ),i = 1 . The joint support of any two pieces of evidence n,e(2) is calculated as follows: Then, the joint support of L independent evidence n,e(L) can be generalized to be computed in the following way:  , D n ⊆ Θ, D n ≠ ∅ where k = 3, 4, ..., L n, e(k) is the belief degree of the former k attributes with respect to the level D n after fusion, and m n, e(1) = m n,1 , m (Θ),e(1) = m (Θ),1 . Through the above formula calculation, the comprehensive assessment results can be obtained as follows: The utility at level D n is denoted as u(D n ) . The final output is Z , the expected utility. The final expected utility is calculated as follows:

IBRB-r parameter optimization process: P-CMA-ES algorithm
Sections 3.1 and 3.2 establish the initial interval rule model from expert knowledge, but influenced by the complexity of the system and the limitation of expert knowledge, the initial interval rule model constructed by expert knowledge may need to be more accurate and reliable. In this way, the initial interval rule model constructed based on expert knowledge cannot accurately reflect the actual prediction results. Therefore, the monitored or collected data samples should be fully utilized. The structure and parameters of the optimization model are adapted to the actual design of the model to build a new optimization model.
The parameters to be optimized in the optimization model of IBRB-r include belief, rule reliability and rule weights. The model parameters of IBRB-r need to have practical physical meaning, so the parameters to be optimized should be trained under constraints with practical meaning [26]. According to the practical meaning, the constraints on the parameters can be set as follows: For the BRB parameter optimization problem, the main optimization methods include sequential quadratic programming (SQP) and constrained particle swarm optimization (PSO), P-CMA-ES and the differential evolution algorithm (DE). Zhou et al. [28] used PSO, SQP, P-CMA-ES and other algorithms to optimize the hidden power set BRB and found that the model using the P-CMA-ES algorithm had higher accuracy. Cao et al. [29] compared the optimization effects of DE and P-CMA-ES on interpretable BRBs and found that P-CMA-ES has high accuracy and interpretability.
Due to the superiority of P-CMA-ES in the BRB optimization algorithm, the P-CMA-ES algorithm was used to optimize the initial parameters in subsequent experiments IBRB-r and BRB. The parameters to be optimized for the BRB and IBRB-r optimization models are different. The parameters to be optimized for BRB include belief degree, attribute weights and rule weights, while the parameters to be optimized for IBRB-r include belief degree, rule reliability and rule weights [26]. Unlike the objective function of BRB, the construction of the IBRB-r model needs to optimize the objective function as follows explicitly: where the MSE is calculated as follows: The P-CMA-ES algorithm's optimization process is divided into six parts: parameter initialization, sampling, projection, updating the next generation means, updating the covariance matrix and recursive execution [30].
First, the parameters are initialized. Initialize the parameters of the BRB that need to be optimized, including the belief degree, rule weights and attribute weights. The initialization parameter can be expressed as: Then, the sample. The parameters of each generation are obtained through a sampling operation, which can be expressed as follows: where s+1 i is used to represent the solution i in generation s + 1 . w s is used to represent the mean of the s generation search distribution. s is used to represent the step size of generation s . C s is used to represent the s − th covariance matrix. N(⋅) is used to represent the normal distribution. is used to represent the number of offspring.
Then, the projection. To satisfy the constraints, the solution is projected onto the feasible hyperplane, which can be expressed as: (21) min MSE ( , , ) st.  Then, the mean of the next-generation search distribution is updated [31]. Denote the weight coefficient as h i and the offspring population size as . The i − th solution of solutions of the s + 1 generation search distribution is denoted as s i∶ . The operation to update the mean can be expressed as: Then, the covariance matrix is updated [32]. Update the covariance matrix using the following equation: where the generation step s is denoted as s . The learning rate is denoted as e 1 and e 2 . The evolutionary path of generation s + 1 is denoted as P s+1 e . The offspring population of generation s is denoted as s . The i − th parameter vector in the vector of the s + 1 generation is denoted as K s+1 i: . After the modeling, inference and optimization of the IBRB-r model, the complete IBRB-r model can be obtained. Compared with the traditional BRB model, the IBRB-r model has greatly changed the modeling and reasoning processes. Among them, IBRB-r designed a new referential interval and a new belief table construction method in the modeling process, and a new rule activation method and a new ER rule were designed in the reasoning process. The specific IBRB-r model structure is shown in Fig. 6.

Case study
This experiment takes the result safety assessment of a liquid launch vehicle as the main case to prove the effectiveness and accuracy of the IBRB-r model. At the same time, the effectiveness and accuracy of the IBRB-r model under different datasets are compared. The data "Rocket" for the structural safety of the liquid launch vehicle was collected from a laboratory platform. The monitoring indicators of the experimental platform of a liquid launch vehicle include shaking, inclining, ambient temperature, ambient humidity, etc. [33]. Only shaking and inclining are used to evaluate the structural safety of the liquid launch vehicle. This is because the ambient temperature and humidity are unchanged during the experiment, so the influence of these two factors is not considered temporarily. There are 515 liquid launch vehicle monitoring data points, 445 experimental training samples and 70 test samples. The other datasets of the experimental control group are from the public UCI platform dataset.

Establishment of a structural safety assessment model of a liquid launch vehicle based on IBRB-r
Based on the IBRB-r model, combined with shaking and inclining indexes and assessment results, the safety assessment rules of a liquid launch vehicle structure can be described as IF-THEN rules as follows: where Shaking and Inclining of the liquid launch vehicle structure are the prerequisite attributes of IBRB-r. The experiment set 20 referential intervals for each attribute. k is the rule reliability of rule k , and k is the rule weight of rule k . D 1 , D 2 and D 3 , as the three assessment grades of liquid launch vehicle structure safety, represent normal, medium and low, respectively. Through the analysis of data samples, the referential levels and referential intervals of Shaking and Inclining are set as shown in Table 7. The larger the sample value of the two attributes, the higher the assessment level. The assessment result's referential level and referential interval are set as shown in Table 8, which reflects the safe failure probability of the liquid launch vehicle result and thus reflects its safe state.
After setting the referential level and interval for shaking and incline, the initial IBRB-r liquid launch vehicle safety assessment model can be obtained. The initial model specifically includes the initial belief degree, rule reliability and rule weight, Then safety state is {(D 1 , 1,k ), (D 2 , 2,k ), (D 3 , 3,k )} with rule reliability k and rule weight k k ∈ {1, 2, ..., L},  as shown in Table 9. Table 9 shows the initial parameters and clearly shows the method of constructing the belief table of the IBRB-r model. When the two prerequisite properties are shaking and inclining, each with 20 referential intervals, the traditional BRB requires 21 referential points. At this time, the traditional BRB has 21*21 = 441 rules, while the IBRB-r model only needs 20 + 20 = 40 rules, which reduces the number of rules by nearly 91%. This greatly reduces the number of rules and makes the model much less complex.

Inference and optimization of the structural safety assessment model of a liquid launch vehicle based on IBRB-r
After model establishment, the output belief degree, rule reliability and rule weight will be obtained after model reasoning and parameter optimization. The parameter values after inference and optimization are shown in Table 10. Referential value 1.0 0.5 0

Curve comparison between the model output value and real value
After modeling, reasoning and optimization of IBRB-r, the expected utility value output by the IBRB-r model can be obtained. The comparison of the expected utility value and result label of the IBRB-r-based liquid launch vehicle structure safety assessment model is shown in Fig. 7. In this experiment, numerical fitting also compares the IBRB-r model with the traditional BRB model. The comparison between the output expected utility value and the real value of the BRB is shown in Fig. 7.
From the comparison of model output values and true values in Fig. 7a, it can be seen that only two points of IBRB-r are misclassified, and the remaining 68 predicted values have a strong fit with the true values. However, with the same number of referential points for comparison, it can be seen from the comparison of model output values and true values in Fig. 7b that the predicted and true values of traditional BRB for this test sample have significantly more error points than IBRB-r, which indicates that IBRB-r has relatively higher accuracy.
In addition, this experiment also combines the IBRB-r model with a backpropagation neural network (BPNN) [34][35][36], an extreme learning machine (ELM)   [37][38][39] and a radial basis function neural network (RBF) [40], and the comparison between the obtained model output value and the real value is shown in Fig. 8. The three methods in Fig. 8 show that the predicted values of the data-driven BPNN, ELM, and RBF-based models all have more than three error points compared to the true values, and the curve fits are significantly weaker than the proposed IBRB-r. As seen from Fig. 8, the comparison curve fitting between the output value of the IBRB-r model and the real value is better than that of BRB, BPNN, ELM and RBF. This shows the superiority of the IBRB-r model in terms of model accuracy.

Comparison of the IBRB-r model before and after improvement
The improved IBRB-r model is compared with the traditional BRB before improvement in the following dimensions: the number of parameters, the number of rules, and the complexity. The experimental results are shown in Table 11. This IBRB-r-based liquid launch vehicle safety assessment model experiment has two prerequisite attributes with 20 referential intervals each. The number of rules based on interval addition to obtain IBRB-r is 20*2 = 40. The BRB set in the same control condition of 20 referential intervals has 21 referential points, so the combination rules based on the Cartesian product form yield 21*21 = 441 rules. The parameter scale of IBRB-r includes the number of belief degrees (40*3 = 120), the number of rule weights (40), the number of rule reliabilities (40), and the total number of parameters to be optimized, 200, obtained by adding the three together. The BRB set for the same 20 referential intervals in the control condition includes the number of belief degrees (441*3 = 1323), the number of rule weights (441), and the number of attribute weights (2), and the total number of parameters to be optimized, 1766, is obtained by adding the three together.
According to Table 11, the IBRB-r model comprehensively outperforms the traditional BRB model regarding the total number of parameters, number of rules, complexity and accuracy. The specific summary is as follows:

Accuracy comparison of different methods
In the structural safety assessment model of liquid launch vehicle structures, the accuracy of different methods under this dataset is compared in addition to the comparison before and after the model improvement. The experimental results are shown in Table 12. As seen from Table 12, the IBRB-r model is superior to the traditional BRB and data-driven BPNN, ELM and RBF methods in terms of accuracy. In the structural safety assessment of liquid rockets, the IBRB-r, BRB, BPNN, ELM, and RBF methods in Table 12 have good accuracy rates of more than 90% in several experiments. However, IBRB-r has fewer rules than the traditional BRB. Compared with the data-driven BPNN, ELM, and RBF, the inference process of IBRB-r possesses interpretability. The proposed model achieves an accuracy of over 95%, and its inference process is interpretable. This is because IBRB-r is based on a semiquantitative modeling approach that combines subjectivity and objectivity, and the inference process of the model possesses interpretability.

Comparison of different methods in different datasets
In addition to model comparison before and after improvement, this experiment also compares the methods in different datasets. The dataset used from the UCI platform and the dataset information are shown in  Table 14 compares the models in each dataset before and after improvement. As seen from Table 14, the number of rules and parameter size of IBRB-r are substantially reduced, and the accuracy rate is higher than that of the traditional BRB under different datasets. In particular, the number of rules of IBRB-r under the Banana dataset is only equivalent to 0.09% of the traditional BRB, which substantially reduces the number of rules but yields a higher accuracy rate, which proves the superiority of the proposed IBRB-r model. Table 15 shows that the proposed IBRB-r model is universal. With multiple datasets, it outperforms the traditional BRB in many dimensions, such as the number of parameters, the number of rules, the complexity of the model and the accuracy. Compared with the traditional BRB, the number of rules and parameters to be trained and optimized in the IBRB-r model are greatly reduced, which perfectly solves the problem of combination rule explosion in the traditional BRB. The IBRBr model also achieves the same or even higher accuracy than the traditional BRB in a short time. All these results fully illustrate the universality, accuracy and superiority of the IBRB-r model. In addition, the experiment also compares the accuracy of different methods in different datasets, and the results are shown in Table 15.
As seen from Table 14, IBRB-r has fewer rules than the traditional BRB, which can avoid the problem of combinatorial rule explosion caused by too many rules. As seen from Table 15, IBRB-r achieves more than 85% model accuracy under different datasets and has higher accuracy compared to BRB, BPNN, ELM, and RBF methods.

Analysis and summary of experimental results
As an improved model, the IBRB-r model is superior to the traditional BRB model in all aspects while retaining the advantages of the traditional BRB model. According to Tables 11, 12, 13, 14 and 15, compared with the traditional BRB model based on rule modeling, IBRB-r has the following characteristics: 1) the IBRB-r model completely solves the problem of combination rule explosion. When the number of premise attributes and referential points is large, the number of rules in the traditional BRB increases exponentially. This easily leads to the explosion of combination rules. However, IBRB-r constructs belief tables in the form of interval addition, which completely solves the problem of combination rule explosion. 2) IBRB-r introduces ER rules and considers the rules' reliability, making the model more complete.
3) The IBRB-r model uses simpler models to achieve higher model accuracy than the traditional BRB. According to Tables 14 and 15, the IBRB-r model with different datasets takes less time but has higher accuracy. This fully shows that IBRB-r can reduce the complexity of the model and has high precision. 4) The IBRB-r model has universality and is more suitable for engineering applications. Because the traditional BRB can only use a hierarchical method to address multiattribute and multireferential problems, the model network structure is large. As a result, it is difficult to apply it to engineering practice. However, IBRB-r can deal with the problem of multiple attributes and multiple referential points. As shown in Tables 14 and 15, IBRB-r performs better on different datasets and is suitable for practical engineering applications. Compared with the data-driven BPNN, ELM and RBF, IBRB-r has the following characteristics: 1) IBRB-r has low dependency on samples and better processing ability for small sample data. It uses a hybrid data-knowledge-driven approach and can obtain high-precision results even when the sample data size is small. However, the model accuracy of BPNN, ELM and RBF based on the data-driven mode depends on data samples, and the model accuracy is low in the case of small samples. 2) The inference engine of the IBRB-r model is transparent, and the inference process can be traced. However, the data-driven BPNN, ELM and RBF models' internal structure must be visible. 3) IBRB-r can better process qualitative and quantitative information. IBRB-r can fully use expert knowledge for qualitative  analysis and data for quantitative analysis. This semiquantitative modeling method can ensure the high accuracy of the experimental results.

Conclusion
The traditional BRB rule combination easily causes the combination rule explosion, and it is unsuitable to address engineering application problems with multiple attributes and referential points. The IBRB-r model is improved on the traditional BRB to solve this problem. Its main contributions are as follows: (1) IBRB-r innovatively combines rules in the form of interval addition in the modeling process. This method greatly reduces the complexity of the model and completely solves the problem of combination rule explosion.
(2) IBRB-r innovatively introduces ER rules in the reasoning process and considers the reliability of rules.
The improved IBRB-r model can deal with multi-attribute and multi-referential problems well and is suitable for engineering applications. In the experimental part, the effectiveness and accuracy of the IBRB-r model are fully verified by analyzing the safety evaluation case of a liquid rocket body structure. At the same time, the experimental results of IBRB-r and other methods in different datasets are compared, which proves the universality of the IBRB-r model.
In future research, more attention can be paid to the following aspects.