Construction method of scrap parts status mapping model based on Dirichlet distribution

Abstract

This invention proposes a method for constructing a damage-quality state mapping model based on the Dirichlet distribution for used parts. Addressing the uncertainty in the quality assessment of used parts, a damage-quality state mapping model (DBMS) based on the Dirichlet distribution is constructed. This model identifies the main failure characteristics through failure behavior analysis of used parts, uses a multinomial distribution to mathematically abstract the damage data, selects the Dirichlet distribution as the prior probability distribution, and updates the posterior distribution parameters using Bayes' theorem, thereby obtaining the expected posterior probability values ​​of the damage data mapped to different quality levels. Furthermore, the Dirichlet-Schwarz evidence theory is introduced to integrate damage information, achieving a comprehensive assessment of the quality status of used parts. To verify the feasibility and effectiveness of the model, a used worm gear is used as a case study, and the results are compared with existing methods. Experimental results show that this model has advantages in prediction accuracy and generalization ability.

Description

Technical Field

[0001] This invention belongs to the technical field of Dirichlet distribution, and particularly relates to a method for constructing a state mapping model of waste parts based on Dirichlet distribution. Background Technology

[0002] The rapid advancement of industrial technology and the continuous expansion of global economic activities have led to an unprecedented scale in the production and use of electromechanical products. Faced with the challenges of limited resources and the urgent need for environmental protection, the recycling and reuse of retired electromechanical products has become a crucial pathway to promoting a circular economy and achieving sustainable development. Retired electromechanical products contain a large number of discarded parts. These parts have undergone prolonged use in various service environments, and their quality often becomes a key factor restricting their remanufacturing or reuse. Accurately and efficiently assessing the quality of discarded parts is of great significance for improving the quality of remanufactured products and achieving the classified utilization and efficient allocation of resources.

[0003] Compared to the quality of manufactured parts, the quality of scrap parts varies significantly due to differences in working conditions, operation time, and wear levels. This uncertainty increases the complexity of remanufacturing process planning, leading to unstable quality in remanufactured products. Many scholars have recognized the necessity of quality assessment during the remanufacturing process of scrap parts. This paper introduces the quality level of parts into the remanufacturing batch scheduling problem, constructing an optimal scheduling model applicable to both deterministic and stochastic quality levels. Addressing the inherent uncertainty in the quality assessment process, the changing trends of the benefits brought by the quality assessment operation are discussed using numerical analysis methods. Therefore, exploring and constructing scientific methods for assessing the quality of scrap parts is crucial for the remanufacturing field. In early research, quality assessment often relied on subjective judgment and experience, lacking systematic assessment methods and quantitative standards. With the development of intelligent technologies, many scholars have begun to explore more scientific and objective quality assessment methods.

[0004] Existing literature indicates that current quality assessment research focuses relatively little on key components of electromechanical equipment. However, data shows that annual losses due to wear and deformation leading to equipment failure and scrap reach as high as 100 billion yuan, and the resulting resource waste has an incalculable impact on the ecological environment. Furthermore, existing research mainly focuses on classifying the quality of scrap parts using methods such as fuzzy sets, rough sets, and random sets, which typically lack flexible adjustment mechanisms and fail to fully utilize damage data to accurately quantify the quality status of in-service electromechanical products and their components. This limits the applicability of such methods in large-scale remanufacturing of scrap parts. Summary of the Invention

[0005] The purpose of this invention is to propose a method for constructing a state mapping model for scrap parts based on the Dirichlet distribution. A Damage-Based Multi-State Mapping Model (DBMS) is proposed for assessing the quality of scrap parts, mapping the damage amount of failure modes to the posterior probability expectation values ​​of different quality levels. First, for scrap parts obtained from the dismantling of retired electromechanical products, quantitative information on the damage amount of failure modes is obtained through intelligent detection methods, constructing a failure sample dataset of scrap parts. Then, the damage amount of failure modes is mathematically abstracted using a multi-distribution model, and the Dirichlet distribution is adopted as the prior probability distribution. Based on this, the prior Dirichlet distribution parameters are determined through non-information priors, and the posterior Dirichlet distribution parameters are updated according to Bayes' theorem, thereby obtaining the posterior probability expectation value. The posterior probability expectation value is introduced into the basic probability quality allocation process, and the DS evidence theory is used to fuse damage information to comprehensively assess the quality status of scrap parts. Finally, the effectiveness of the proposed assessment method is verified through examples.

[0006] To achieve the above objectives, this invention provides a method for constructing a state mapping model for scrap parts based on Dirichlet distribution, the method comprising:

[0007] Key features of scrapped parts are integrated into a failure dataset, and a failure feature matrix is ​​constructed based on the dataset; wherein, the key features of scrapped parts include wear, fracture, deformation and quality grade labels;

[0008] n instances are randomly selected from the quality grade labels as training data. Each failure mode contains n values. The data are divided into intervals to identify the specific distribution of damage under different failure modes.

[0009] Based on the identified specific distribution, the parameter a = (a1, a2, ..., a2) is introduced. k The Dirichlet distribution of the component damage data is used to construct a prior probability model. The probability vector θ of the prior probability model is regarded as a random variable under the prior baseline a, i.e., p(θ) ~ Dir(a).

[0010] Construct a probabilistic model under given failure mode f and quality level h, denoted as DPMS. fh The H probability models are integrated to obtain the combined probability model under failure mode f, denoted as DPMS. fTo form a comprehensive probabilistic description of failure mode s, all combined probability models are integrated. For each failure mode f, its corresponding damage data is used as input parameters and processed by the corresponding DBMS model to obtain the expected posterior probability values ​​of a given failure mode f assigned to different quality levels. Finally, the sum T of the expected posterior probabilities of all quality levels under failure mode f is calculated. f According to T f Different basic probability allocation strategies are adopted for different value ranges.

[0011] Furthermore, the feature matrix is ​​represented as follows:

[0012]

[0013] Among them, w ij w represents the wear amount of the i-th discarded component in the j-th failure mode. 11 w represents the wear amount of the first discarded component in the first failure mode. 12 w represents the wear amount of the first discarded component in the second failure mode. 1j w represents the wear amount of the first discarded component in the j-th failure mode. 21 w represents the wear amount of the second discarded component in the first failure mode. 22 w represents the wear amount of the second discarded component in the second failure mode. 21 w represents the wear amount of the second discarded component in the first failure mode. 2j w represents the wear amount of the first discarded component in the j-th failure mode. i1 w represents the wear amount of the i-th scrapped component in the first failure mode. i2 This represents the wear amount of the i-th discarded component in the second failure mode.

[0014] Furthermore, the damage is divided into intervals to identify the specific distribution of damage under different failure modes. Specifically, the Freedman-Diaconi rule is used for adaptive interval division, determining an appropriate number of intervals K for each failure mode f, thereby discretizing the continuous damage data into a finite number of interval categories, as shown below:

[0015]

[0016] Where n represents the number of instances, IQR represents the interquartile range, and f represents each failure mode.

[0017] Furthermore, when discarded parts fail, the damage amount of a specific failure mode will inevitably fall within a certain interval. By statistically analyzing the frequency of events, the probability distribution of each interval category can be estimated:

[0018] Let n represent the frequency of falling within the k-th interval. k The probability associated with it is θ. i The damage data in this set can be formally represented as (n1, n2, ..., n k )~Mult(θ1,θ2,...,θ k ), and satisfy

[0019] Furthermore, the method also includes performing a Jeffreys non-information prior before performing the prior Dirichlet, which is constructed based on the Fisher information matrix, as shown below:

[0020]

[0021] Where p(·) represents the probability distribution function, and k represents the number of intervals divided;

[0022] According to Bayesian nonparametric statistical methods, when the prior distribution of the multiple probability parameters of the specified waste parts is a Dirichlet distribution, the probability density function simultaneously possesses the following properties:

[0023]

[0024] Among them, a i The parameters represent the Dirichlet distribution.

[0025] Furthermore, when Jeffreys' non-informative prior is used, the parameter 'a' of the Dirichlet distribution can be uniquely determined, and a1 = a2 = ... = a k =1 / 2. After determining the prior Dirichlet distribution parameters, based on the property that the Bayesian posterior distribution is proportional to the product of the likelihood function and the prior distribution, we can obtain...

[0026] P(θ1,θ2,...,θ k |n1,n2,...,n k )∝

[0027] Dir(1 / 2+n1,1 / 2+n2,...,1 / 2+n k )

[0028] Where Dir(·) represents the Dirichlet distribution;

[0029] That is, when the observed multivariate distribution data of damage to scrap parts is n=(n1,n2,...,n k When ), we deduce that its posterior distribution is updated with parameters a = (1 / 2 + n1, 1 / 2 + n2, ..., 1 / 2 + n).k If the Dirichlet distribution is such that the expected posterior probability of the Dirichlet distribution is calculated by the following formula:

[0030]

[0031] in, Let represent the expected posterior probability of the i-th interval.

[0032] Furthermore, the basic probability allocation strategy includes:

[0033] A. Supplementary Allocation: When T f When <1, 1-T f Assigned to the identification frame Θ, it represents a measure of global uncertainty, reflecting the uncertainty that cannot be accurately assigned to a specific quality level due to insufficient information on the failure mode f;

[0034] B. Complete allocation: When T f When θ = 1, the constraints of the basic probability function are naturally satisfied. f This is directly used as the effective basic probability allocation result, meaning that all probabilities are fully allocated to each quality level.

[0035] C. Normalization adjustment: When T f When the value is greater than 1, it indicates that the expected probability values ​​assigned to multiple quality levels are relatively high. In order to maintain the rationality of the probability distribution, the expected posterior probability values ​​need to be normalized.

[0036] The beneficial technical effects of the present invention are at least as follows:

[0037] This paper proposes a Damage-Based Multi-State Mapping Model (DBMS) for the quality assessment of scrap parts, mapping the damage amount of failure modes to the posterior probability expectation values ​​of different quality levels. First, for scrap parts obtained from the dismantling of retired electromechanical products, quantitative information on the damage amount of failure modes is obtained through intelligent detection methods, constructing a dataset of scrap part failure samples. Then, the damage amount of failure modes is mathematically abstracted using a multi-distribution model, and a Dirichlet distribution is adopted as the prior probability distribution. Based on this, the prior Dirichlet distribution parameters are determined through non-information priors, and the posterior Dirichlet distribution parameters are updated according to Bayes' theorem, thus obtaining the posterior probability expectation value. The posterior probability expectation value is introduced into the basic probability quality allocation process, and DS evidence theory is used to fuse damage information to comprehensively assess the quality status of scrap parts. Finally, the effectiveness of the proposed assessment method is verified through examples. Attached Figure Description

[0038] The present invention will be further described with reference to the accompanying drawings, but the embodiments in the drawings do not constitute any limitation on the present invention. For those skilled in the art, other drawings can be obtained based on the following drawings without creative effort.

[0039] Figure 1 This is a flowchart of the waste parts quality assessment method of the present invention.

[0040] Figure 2 This is a schematic diagram showing the number of intervals for different failure modes within each quality level according to embodiments of the present invention. Detailed Implementation

[0041] Embodiments of the present invention are described in detail below. Examples of these embodiments are shown in the accompanying drawings, wherein the same or similar reference numerals denote the same or similar elements or elements having the same or similar functions throughout. The embodiments described below with reference to the accompanying drawings are exemplary and are only used to explain the present invention, and should not be construed as limiting the present invention.

[0042] Example 1

[0043] DS evidence theory is a mathematical framework for handling uncertain information. It integrates evidence from different information sources through combination rules, making it applicable to decision-making and reasoning in various uncertain environments and providing strong support for problem-solving in multiple fields. The identification frame Θ is the foundation of DS evidence theory; it represents the complete set of all possible outcomes of a problem, with a finite number of mutually exclusive elements. The power set 2 of the identification frame... Θ It contains all subsets of Θ, each subset corresponding to a proposition. The Basic Probability Assignment (BPA) is a mapping function that assigns probabilities to different propositions within the power set; its mathematical expression is m:2. Θ →[0,1], and satisfy the following conditions

[0044]

[0045] In the formula, the basic probability mass m(A) quantifies the degree to which evidence supports the truth of proposition A. When m(A) > 0, proposition A is called a focal element. The Dempster evidence theory integrates uncertainty information from different evidence sources through the Dempster composition rule, and its mathematical expression is as follows:

[0046]

[0047] Among them, A and B i For propositions in the power set 2Θ, m i (B i Proposition B iThe basic probability mass under the i-th source of evidence, where K is the normalization coefficient, is calculated using the following formula:

[0048]

[0049] Furthermore, the Dirichlet distribution is a widely used multivariate probability distribution whose parameters are determined by a set of positive real vectors α = (α1, α2, ..., α...). k Composed of ) and α i >0. The probability density function of the Dirichlet distribution is defined as:

[0050]

[0051] Where θ=(θ1,θ2,…,θ k ) is a multivariate random variable, θ i Let represent the probability value of a random variable in the i-th category, and satisfy . B(α) is a multivariate Beta function, defined as follows:

[0052]

[0053] A key characteristic of the Dirichlet distribution is its conjugate relationship with the multinomial distribution, often used as the conjugate prior distribution of the multinomial distribution. In Bayesian statistics, if one distribution is the conjugate prior of another, then given data, the posterior distribution will be of the same type as the prior distribution. Specifically, when a random variable X follows a multinomial distribution Mult(n,θ), where n is the number of trials and θ is a set of probability parameters, the likelihood function of the parameter θ for a given sample data D is expressed as...

[0054]

[0055] Using the Dirichlet distribution as the prior distribution of θ, according to Bayes' theorem, the posterior distribution of θ is proportional to the product of the likelihood function and the prior distribution; that is, the posterior distribution can be expressed as...

[0056]

[0057] This indicates that, due to the conjugate of the Dirichlet distribution and the multinomial distribution, the posterior distribution of θ is still a Dirichlet distribution, with its parameters adjusted to α+n based on the sample data.

[0058] like Figure 1As shown, in the field of electromechanical product remanufacturing research, component failure is generally regarded as a key factor driving product retirement, covering a wide range of situations from complete functional loss to partial functional degradation. During their service life, electromechanical products are affected by multiple uncertain factors such as material properties, maintenance quality, operating environment, and workload, resulting in significant randomness and dispersion in the failure behavior of their internal components. Specifically, even components of the same model may exhibit different quality states due to differences in failure characteristics. Therefore, the characterization and quantification of failure characteristics are fundamental to assessing the quality status of components. By deeply analyzing and measuring the failure modes and degrees of failure in discarded components, a detailed profile of component failure behavior can be constructed.

[0059] The quality assessment process for decommissioned electromechanical products and their used parts is as follows: Figure 1 As shown, firstly, the failure characteristics of scrap parts, including wear, fracture, and deformation, are quantified using intelligent detection methods to construct a dataset of scrap part failure samples. Then, a multinomial distribution is used to mathematically abstract the damage data, and the Dirichlet distribution is selected as the prior probability distribution. Bayes' theorem is used to calculate the posterior probability of failure characteristics belonging to different quality states, and a differentiated adjustment strategy is employed to construct the basic probability assignment. Finally, through evidence fusion reasoning, the quality status of the scrap parts is comprehensively evaluated.

[0060] Wear, fracture, and deformation are the three most common failure characteristics in discarded parts, prevalent not only in various retired electromechanical products but also profoundly affecting their service life and reliability. In remanufacturing practices, the various failure characteristics and damage levels of discarded parts are often described qualitatively or vaguely, leading to subjectivity and uncertainty in quality assessment and increasing the risk of performance degradation after remanufacturing. Therefore, discarded parts obtained from dismantling retired electromechanical products are first appropriately cleaned, and then quantitative information on relevant failure characteristics is obtained through non-destructive testing techniques to assess the quality of the parts.

[0061] Table 1 provides quantitative data on three main failure characteristics of used machine tool drive spindles. The corresponding quantitative ranges are represented by specific numerical ranges. For example, if a drive spindle part exhibits wear failure, and its wear amount w is less than 1.0 mm³, it is considered slight wear, indicating that the damage is within an acceptable low level. When the wear amount w is between 1.0 mm³ and 2.0 mm³, the wear degree is defined as moderate. When the wear amount w exceeds 2.0 mm³, it is considered severe wear. Drive spindle parts with slight and moderate wear can be restored to their original performance through remanufacturing. Drive spindle parts with severe wear are subject to material recycling and will not be further studied.

[0062] Table 1. Quantification of Failure Characteristics of Machine Tool Transmission Spindles

[0063]

[0064]

[0065] Furthermore, in evidence theory, Basic Probability Assignment (BPA) is the process of calculating the basic probability of each piece of evidence within the identification framework Θ, a process accomplished through a basic probability assignment function. In practical applications, determining the basic probability assignment becomes a crucial issue. Traditional BPA methods often rely on expert experience or heuristic rules, making it difficult to guarantee consistency and objectivity in the assignment. Therefore, this paper proposes a basic probability assignment method based on the Dirichlet distribution in evidence theory.

[0066] Based on intelligent detection methods and digital technology, a dataset U of failure samples of scrap parts is systematically constructed. The dataset integrates three key failure characteristics of scrap parts in a structured manner: wear (Fw), fracture (Fc), and deformation (Fd), and attaches a quality grade label (H) to reflect the comprehensive performance status of each scrap part.

[0067] U = [F w ,F c ,F d [H](8)

[0068] Since a single failure characteristic, such as wear, may encompass multiple specific failure modes, its failure data can be represented in the following matrix form:

[0069]

[0070] Among them, w ij This represents the wear amount of the i-th scrap component under the j-th failure mode. Similarly, fracture and deformation can also be described in detail using a similar matrix structure. Based on the scrap component failure sample dataset, n instances are randomly selected from quality level h as training data. Each failure mode contains n values, which are then divided into intervals to identify the specific distribution of damage under different failure modes. The Freedman-Diaconi rule provides an adaptive interval division strategy that comprehensively considers the proportional relationship between the interquartile range (IQR) of the data and the sample size (n), and is applicable to data with different distribution patterns. Using this rule, an appropriate number of intervals K can be determined for each failure mode f, thereby discretizing the continuous damage data into a finite number of interval categories. The calculation expression is as follows:

[0071]

[0072] When used parts fail, the damage amount for a specific failure mode inevitably falls within a certain interval. Therefore, damage data points under different failure modes can be considered as the result of an experiment, and the observations falling within a specific interval correspond to the event frequency of that interval category. Through statistical analysis of the event frequencies, the probability distribution of each interval category can be estimated, and the frequency falling within the k-th interval can be represented as n. i The probability associated with it is θ. i Then, the damage data can be formally represented as, and satisfies the following: Thus, a multivariate distribution model was constructed to describe the damage distribution under specific failure modes of scrap parts.

[0073] Furthermore, in Bayesian nonparametric methods, the Dirichlet distribution is considered the natural conjugate prior of the multinomial distribution. The introduced parameter is a = (a1, a2, ..., a...). k The Dirichlet distribution of the multinomial component damage data is used as a prior probability model. The probability vector θ is regarded as a random variable under the prior baseline a, i.e., p(θ) ~ Dir(a).

[0074] In most practical applications, directly estimating the prior Dirichlet distribution parameter α is often limited by the scarcity or lack of information in the failure data of scrap parts. To reduce the over-reliance on prior knowledge or assumptions, non-information prior inference methods are used to obtain robust and effective prior distributions. Jeffreys prior is a commonly used non-information prior [15-16], which is constructed based on the Fisher information matrix and does not depend on any specific parameter value. According to the mathematical derivation in reference

[17] , in the case of a multivariate distribution of the damage amount of scrap parts failure, Jeffreys prior follows the following relationship

[0075]

[0076] Furthermore, according to Bayesian nonparametric statistical methods, when the prior distribution of the multiple probability parameters of the specified waste parts is a Dirichlet distribution, the probability density function simultaneously possesses the following characteristics.

[0077]

[0078] From formulas (11) and (12), it can be deduced that when Jeffreys' non-information prior is used, the parameter 'a' of the Dirichlet distribution can be uniquely determined, and a1 = a2 = ... = a k = 1 / 2. After determining the prior Dirichlet distribution parameters, based on the property that the Bayesian posterior distribution is proportional to the product of the likelihood function and the prior distribution, we can obtain...

[0079] P(θ1,θ2,...,θ k |n1,n2,...,n k )∝

[0080] Dir(1 / 2+n1,1 / 2+n2,...,1 / 2+n k (13)

[0081] That is, when the observed multinomial distribution data of damage to scrap parts is n=(n1,n2,...,nk), it can be deduced that its posterior distribution is updated with parameters a=(1 / 2+n1,1 / 2+n2,...,1 / 2+n k The Dirichlet distribution of ). As can be seen from reference

[18] , the expected value of the posterior probability of the Dirichlet distribution at this time can be calculated by the following formula.

[0082]

[0083] Furthermore, following the steps described above, a probabilistic model, denoted as DPMBfh, can be constructed under the given failure mode f and quality level h. This model provides the expected posterior probability values ​​for each interval category, and its expression is as follows:

[0084]

[0085] To determine the basic probabilistic quality allocation of failure modes at different quality levels, H probabilistic models are integrated to obtain a combined probabilistic model under failure mode f, denoted as DPMSf, to form a comprehensive probabilistic description of failure mode f.

[0086] DBMS f =(DPMS) f1 DBMS f2 DBMS fH (16)

[0087] By analogy, we obtain the combined probability model for all failure modes (DBMS1, DBMS2, ..., DBMS). F When assessing the quality condition of scrap parts, for each failure mode f, its corresponding damage data is used as an input parameter. This data is then processed by the corresponding DBMS model to obtain the expected posterior probability of assigning a given failure mode f to different quality levels, denoted as θ. f =(θ f1 ,θ f2 ,…,θ fHThese expected values ​​directly reflect the probability that the scrapped parts belong to each quality grade under the given failure mode f. To ensure the consistency and completeness of the basic probability allocation, the sum T of the expected posterior probabilities for all quality grades under failure mode f is calculated. f ,Right now According to T f Different basic probability allocation strategies are adopted for different value ranges.

[0088] 1. Replenishment Allocation: When T f When <1, 1-Tf is assigned to the identification frame Θ, representing a measure of global uncertainty, reflecting the uncertainty that cannot be accurately assigned to a specific quality level due to insufficient information on failure mode f.

[0089] 2. Complete allocation: When T f When θ = 1, the constraints of the basic probability function are naturally satisfied. f This is directly used as the effective basic probability allocation result, meaning that all probabilities are fully allocated to each quality level.

[0090] 3. Normalization adjustment: When T f When the value is greater than 1, it indicates that the expected probability values ​​assigned to multiple quality levels are relatively high. In order to maintain the rationality of the probability distribution, the expected posterior probability values ​​need to be normalized.

[0091] Using the above method, a set of basic probability assignments that meet the requirements can be generated for each failure mode f, providing a quantitative and structured probabilistic framework for assessing the quality status of scrap parts.

[0092] Example 2

[0093] To illustrate the calculation process of the proposed BPA generation method and verify its effectiveness in assessing the quality of scrap parts, scrap worm gears were selected as the main research object to verify their application value and accuracy in practical engineering problems. Furthermore, to further verify the universality and robustness of the proposed method, the classic Iris dataset was introduced as an auxiliary verification tool.

[0094] Worm gears are common transmission components in electromechanical equipment. Due to the complexity of their working environment and operating conditions, their failure behaviors are diverse. After in-depth analysis and testing, the main failure characteristics of used worm gears are wear, fracture, and deformation, as shown in Table 2.

[0095] Table 2 Failure Characteristics of Used Worm Gear

[0096]

[0097] After systematic collection and organization, 60 sets of failure sample data U of waste turbine worm gears were obtained. Based on the actual remanufacturing cost, the quality status of the waste turbine worm gears was divided into three levels: good quality (G), medium quality (M), and severely degraded (B), as shown in Table 3.

[0098] Table 3 Dataset of Failure Samples of Scrap Turbine Worms

[0099]

[0100] The Iris dataset is a classic multivariate dataset containing 150 samples, each corresponding to one species of iris, divided into three different species (Setosa, Versicolor, and Virginica). Each sample contains four features: sepal length (SL), sepal width (SW), petal length (PL), and petal width (PW), all of which are continuous variables.

[0101] For the scrap worm gear and Iris dataset, a stratified random sampling method was used to randomly select 80% of the data from each category as the training sample set to build the DPMB model; the remaining 20% ​​of the data was used as the test sample set to calculate the generated basic probability assignment (BPA) to evaluate the model performance.

[0102] In the quality assessment of scrap worm gears, an identification framework Θ = {G, M, B} is constructed, which includes three different quality levels. Using equation (10) and a pre-established training sample set, the number of intervals for different failure modes within each quality level is calculated and determined, as shown in the following results. Figure 2 As shown.

[0103] Based on the number of intervals for different failure modes, the boundary values ​​of each interval are obtained. Then, the frequency of damage amounts for each failure mode in the training sample set is statistically analyzed to identify its distribution within each interval. Furthermore, the expected posterior probability of the failure mode under different quality levels is calculated based on Equation 14. Thus, a corresponding DPMB model is constructed for each failure mode to achieve a probabilistic mapping from damage amount to quality status. Taking failure mode w1 as an example, it is divided into three intervals in different quality levels, and the corresponding interval boundary values ​​and expected posterior probability values ​​are calculated. The results are summarized in Tables 4 and 5.

[0104] Table 4. Interval division of failure mode w1

[0105]

[0106] Table 5 Expected posterior probability of failure mode w1

[0107]

[0108] The dataset of waste turbine worm gear test samples is shown in Table 6. Based on the damage exhibited by each sample under different failure modes, the expected posterior probabilities belonging to different quality levels are matched from the corresponding DPMB models. Furthermore, based on the sum of the calculated posterior probabilities, a differentiated basic probability allocation strategy is adopted to calculate the final basic probability allocation result.

[0109] Table 6 Dataset of Test Samples for Scrap Worm Gear

[0110]

[0111]

[0112] Table 7 shows the basic probability allocation and fusion results of the scrap turbine worm gear sample T1. Each failure mode is considered as a source of evidence, and each value represents the basic probability allocation of the corresponding source of evidence for a specific quality level. According to the Dempster synthesis rule in Equation (2), the basic probability allocations from different sources of evidence are synthesized. The fusion results show that for the scrap turbine worm gear sample T1, the basic probability quality of its quality condition belonging to level B is the highest at 0.9169, while the basic probability quality of its quality condition belonging to levels G and M is relatively low, at 0.0235 and 0.0592 respectively. The probability allocation for uncertain quality states in the entire set Θ is extremely low at 0.0004. This indicates that under the current evaluation framework, based on all evidence information and combined with the maximum membership principle, the quality condition of the test sample T1 is determined to be level B, which is consistent with the quality level results in Table 6.

[0113] Table 7. Basic probability allocation and fusion results of waste turbine worm gear sample T1

[0114]

[0115] To verify the effectiveness and feasibility of the proposed basic probability allocation method, it was compared and analyzed with methods in existing literature. Specifically, experiments were conducted on the scrap worm gear dataset U and the iris dataset Iris, with ten independent experiments performed on each dataset to ensure the reliability of the results. In each experiment, the dataset was randomly divided into a training set and a test set, and the average recognition accuracy was calculated. The results of the ten experiments are summarized in Table 8.

[0116] Table 8 Comparison of Evaluation Results by Different Methods

[0117]

[0118]

[0119] By comparing the performance of different evaluation methods on the scrap turbine worm gear dataset U and the iris dataset Iris, it can be seen that the evaluation method proposed in this paper exhibits high accuracy and stability on both datasets. In particular, on dataset U, its average accuracy is slightly higher than the other two methods, which verifies its effectiveness in uncertainty modeling and reasoning.

[0120] In summary, this paper proposes a damage-quality state mapping model that integrates Dirichlet distribution and evidence theory. It identifies key failure characteristics through component failure behavior analysis, and based on acquired damage quantity data samples, combines multinomial distribution and Dirichlet distribution within a Bayesian framework to model the quality probability of scrap components. This model is then incorporated into evidence theory to construct the basic probabilistic quality for each failure mode. Combination rules are used to fuse different evidence bodies to obtain the final probability output, thus establishing a probabilistic mapping mechanism from damage quantity to quality state, providing decision support for the quality status assessment of scrap components from decommissioned electromechanical products. A scrap worm gear is selected as a case study, supplemented by the classic Iris dataset. The proposed method is compared with two existing methods, validating the feasibility and effectiveness of the model. Experimental results show that the model has advantages in prediction accuracy and generalization ability.

[0121] Given that high-quality damage data for used parts is often difficult to obtain in practical applications, further research could focus on reducing reliance on a single data source by integrating existing data resources such as historical maintenance records and actual usage conditions. Simultaneously, efforts should be made to explore integrating this model into existing remanufacturing quality control processes to achieve broader practical applications.

[0122] Although embodiments of the invention have been shown and described, those skilled in the art will understand that various changes, modifications, substitutions and variations can be made to these embodiments without departing from the principles and spirit of the invention, the scope of which is defined by the claims and their equivalents.

Claims

1. A method for constructing a state mapping model for scrap parts based on Dirichlet distribution, characterized in that, The method includes: Key features of scrapped parts are integrated into a failure dataset, and a failure feature matrix is ​​constructed based on the dataset; wherein, the key features of scrapped parts include wear, fracture, deformation and quality grade labels; n instances are randomly selected from the quality grade labels as training data. Each failure mode contains n values. The data are divided into intervals to identify the specific distribution of damage under different failure modes. Based on the identified specific distribution, the following parameters are introduced: a =( a 1, a 2,..., a k The Dirichlet distribution of the component damage data is used to construct a prior probability model. The probability vector θ of the prior probability model is regarded as a random variable under the prior baseline a, i.e. p ( θ )~Dir( a ); Construct a predetermined failure mode f The probabilistic model under quality level h is denoted as The failure modes are obtained by integrating H probability models. f The following combination probability model is denoted as To form a response to failure modes f A comprehensive probabilistic description, integrating all combined probabilistic models, for each failure mode f The corresponding damage data is used as input parameters and processed by the corresponding DBMS model to match the given failure mode. f The expected posterior probabilities are assigned to different quality levels, and finally the failure modes are calculated. f The sum of the expected posterior probabilities of all quality levels ,according to Different basic probability allocation strategies are adopted for different value ranges; The method further includes performing a Jeffreys non-information prior before performing the prior Dirichlet, which is constructed based on the Fisher information matrix, as shown below: ； Where p(•) represents the probability distribution function, and k represents the number of intervals; the frequency of falling within the k-th interval is represented by n. k The probability associated with it is θ. i Then, the damage data of this group is formally represented as ( n 1, n 2,..., n k Mult( θ 1, θ 2,..., θ k ), and satisfy ; According to Bayesian nonparametric statistical methods, when the prior distribution of the multiple probability parameters of the specified waste parts is a Dirichlet distribution, the probability density function simultaneously possesses the following properties: ； in, The parameter represents the Dirichlet distribution, and the prior belief represents the probability of each interval. When Jeffreys' non-informative prior is used, the parameter 'a' of the Dirichlet distribution can be uniquely determined, and a 1= a 2=‧‧‧= a k =1 / 2. After determining the prior Dirichlet distribution parameters, based on the property that the Bayesian posterior distribution is proportional to the product of the likelihood function and the prior distribution, we can obtain... ； Where Dir(•) represents the Dirichlet distribution; That is, when the multivariate distribution data of damage to scrap parts are observed to be n =( n 1, n 2,..., n k When ), it is deduced that its posterior distribution is updated with parameters as follows: a =(1 / 2+ n 1, 1 / 2+ n 2,...,1 / 2+ n k If the Dirichlet distribution is such that the expected posterior probability of the Dirichlet distribution is calculated by the following formula: ； in, Indicates the first i The expected value of the posterior probability for each interval.

2. The method for constructing a state mapping model for scrap parts based on Dirichlet distribution according to claim 1, characterized in that, The feature matrix is ​​represented as follows: ； in, This represents the wear amount of the i-th discarded component in the j-th failure mode. This indicates the wear amount of the first discarded component in the first failure mode. This indicates the amount of wear on the first discarded component in the second failure mode. This represents the wear amount of the first discarded component in the j-th failure mode. This indicates the wear amount of the second discarded component in the first failure mode. This indicates the amount of wear of the second discarded component in the second failure mode. This indicates the wear amount of the second discarded component in the first failure mode. This represents the wear amount of the first discarded component in the j-th failure mode. This represents the wear amount of the i-th discarded component in the first failure mode. This represents the wear amount of the i-th discarded component in the second failure mode.

3. The method for constructing a state mapping model of scrap parts based on Dirichlet distribution according to claim 1, characterized in that, The process involves dividing the data into intervals to identify the specific distribution of damage under different failure modes. Specifically, the Freedman-Diaconi rule is used for adaptive interval division, determining an appropriate number of intervals K for each failure mode f. This discretizes the continuous damage data into a finite number of interval categories, as shown below: ； Where n represents the number of instances, IQR represents the interquartile range, and f represents each failure mode.

4. The method for constructing a state mapping model of scrap parts based on Dirichlet distribution according to claim 3, characterized in that, When used parts fail, the damage amount for a specific failure mode will inevitably fall within a certain range. By statistically analyzing the frequency of events, the probability distribution of each range category can be estimated: The frequency falling within the k-th interval is represented as... n k The probability associated with it is θ i Then, the damage data of this group is formally represented as ( n 1, n 2,..., n k Mult( θ 1, θ 2,..., θ k ), and satisfy .

5. The method for constructing a state mapping model of scrap parts based on Dirichlet distribution according to claim 1, characterized in that, The basic probability allocation strategy includes: A. Surplus Allocation: When When < 1, 1- Assigned to the identification frame Θ, it represents a measure of global uncertainty, reflecting the uncertainty that cannot be accurately assigned to a specific quality level due to insufficient information on the failure mode f; B. Full allocation: when When = 1, the constraints of the basic probability function are naturally satisfied. This is directly used as the effective basic probability allocation result, meaning that all probabilities are fully allocated to each quality level. C. Normalization adjustment: When When the value is greater than 1, it indicates that the expected probability values ​​assigned to multiple quality levels are relatively high. In order to maintain the rationality of the probability distribution, it is necessary to normalize the expected posterior probability values.

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