A method for joint monitoring and abnormality diagnosis of quality assurance claim quantity and amount

Abstract

The application relates to the technical field of quality management and statistical process control, and provides a joint monitoring and abnormality diagnosis method for quality assurance claim quantity and amount. In each monitoring period, the quality assurance claim quantity of the current period is counted, and the weekly increment of the claim amount is calculated according to the claim quantity as a monitoring statistic; the benchmark parameter is determined based on historical data, the empirical distribution is generated by deducing the theoretical distribution through numerical integration or Monte Carlo simulation in combination with a compound Poisson process, and the dynamic upper control limit is determined according to the significance level; the current increment amount is compared with the control limit, and an alarm is given if the current increment amount exceeds the control limit; candidate variable points are constructed based on the data before the alarm, the abnormal time is determined through maximum likelihood estimation, the candidate models are compared through the Bayesian information criterion, and the optimal model is selected to determine the abnormal type. The joint monitoring and abnormality diagnosis problem under the conditions of highly heterogeneous product base and multiple time variations is solved, dynamic monitoring under the control of the false alarm rate of each monitoring period, abnormal time point estimation and cause identification are realized.

Description

Technical Field

[0001] This invention relates to the field of quality management and statistical process control technology, and to a method for joint monitoring and anomaly diagnosis of the quantity and amount of warranty claims. Background Technology

[0002] Product warranty claims data is a crucial source of information for businesses to obtain information about the on-site reliability of sold products and actual user experience. Changes in data patterns can reflect problems related to product quality and reliability. Monitoring and analyzing warranty claims data helps businesses identify potential problems in product design and manufacturing that were not apparent at an early stage.

[0003] Existing technology proposes a dynamic control chart scheme that enables early detection of reliability issues by monitoring warranty claims throughout the product lifecycle. Instead of pre-setting fixed control limits, it progressively determines these limits by simultaneously considering the randomness of the product sales process and the non-stationary characteristics of product failure modes. In each period, the false alarm rate is controlled at a desired level. Furthermore, a post-signal diagnostic scheme based on maximum likelihood is proposed to help determine the period when problems are most likely to occur. In-depth simulation studies demonstrate that the proposed scheme can detect potential reliability problems in a timely manner and estimate change points with acceptable accuracy.

[0004] However, existing warranty claim monitoring technologies only monitor the number or rate of claims from a single dimension. Problems in the design and manufacturing stages of products often cause not only an abnormal increase in the number of claims, but also changes in the claim amount. These two aspects characterize the potential impact of problems from the two most important aspects: the frequency of occurrence and the severity of the problem. Existing technologies cannot fully identify the compound quality risks of simultaneous anomalies in both quantity and amount. At the same time, the base of insured products has high heterogeneity and multiple time-varying characteristics. Existing control limits are mostly fixed thresholds or approximate settings, lacking dynamic probability control mechanisms. They cannot stably control the false alarm rate in a single period according to a preset significance level, and are prone to missed alarms and false alarms. In addition, there is a lack of effective anomaly diagnosis methods after monitoring alarms. It is impossible to accurately estimate the time of anomaly occurrence, or to distinguish between anomalies with only changes in quantity parameters, only changes in amount parameters, or both parameters changing simultaneously. It is difficult to transform statistical alarms into actionable root cause analysis basis, and cannot provide a systematic solution for early warning and rapid handling of after-sales quality risks. Summary of the Invention

[0005] This invention proposes a joint monitoring and anomaly diagnosis method for the quantity and amount of warranty claims to address the shortcomings of the prior art. This method can identify composite quality risks where the quantity and amount of product warranty claims are simultaneously abnormal. It solves the technical problems of lacking a dynamic probability control mechanism for setting control limits in scenarios where the number of insured products is highly heterogeneous and has multiple time-varying characteristics, making it difficult to stably control the false alarm rate, and lacking effective means to estimate the time of anomaly and determine its causes after an alarm, making it difficult to output operable diagnostic conclusions.

[0006] The technical solution of this invention is: a method for joint monitoring and anomaly diagnosis of the quantity and amount of warranty claims, comprising the following steps: Extract the number of product warranty claims for each monitoring period; when the number of claims is 0, record the total claim amount for that period as 0; when the number of claims is greater than 0, extract the payment amount for each claim and sum them up to obtain the total claim amount for that period. The total claim amount for each monitoring period is used as the monitoring statistic. This statistic is based on the entire life cycle of product production, sales, and warranty. It needs to dynamically filter the set of insured products through a rolling time window and integrate the dual information of claim quantity and claim amount. Based on historical quality assurance data, the claim rate parameters and claim amount distribution parameters under controlled process conditions are determined. Combined with the fitted composite Poisson model, the numerical integration method or Monte Carlo simulation method is used to determine the current dynamic upper control limit according to the preset significance level, so as to control the false alarm rate of each monitoring cycle to be close to the preset level. The current claim amount increment is compared with the dynamic upper control limit. If it does not exceed the limit, the monitoring continues in the next cycle. If it exceeds the limit, an alarm is triggered and the process enters the abnormal diagnosis stage. Based on the monitoring data before the alarm, a set of candidate change points is constructed. The joint log-likelihood values ​​of different candidate change points are compared by the maximum likelihood estimation method to estimate the time of the anomaly. Then, based on the Bayesian information criterion, the fitting comparison of three types of candidate models is performed: only the claim quantity parameter changes, only the claim amount parameter changes, and both parameters change. The optimal model is selected to determine the anomaly type and output the diagnostic conclusion.

[0007] In at least one embodiment of the present invention, the set of products under warranty is selected according to the entire life cycle process of product production, sales and warranty, by a rolling time window, that is, all products under warranty that are before the current monitoring period and whose product age has not exceeded the warranty period.

[0008] In at least one embodiment of the present invention, the numerical integration method derives the probability distribution of the weekly increase in claim amount under controlled conditions and determines the dynamic upper control limit. Specifically, it calculates the current claim rate using product sales data and a claim rate benchmark function, then calculates the theoretical mean and standard deviation of the composite Poisson-exponential sum, constructs an integral grid and calculates the probability density and cumulative distribution function, and finally solves the upper control limit that satisfies the preset significance level through linear interpolation.

[0009] In at least one embodiment of the present invention, the Monte Carlo simulation method simulates the sample distribution characteristics of the weekly increment of claim amount under controlled conditions and determines the dynamic upper control limit. Specifically, it calculates the current claim rate using product sales data and claim rate function, randomly generates multiple claim quantities conforming to the Poisson distribution, generates a single claim amount according to the exponential distribution, and then obtains a total claim amount conforming to the exponential distribution to form a simulation sample. The upper control limit is then determined based on the preset quantile of the sample.

[0010] In at least one embodiment of the present invention, after constructing the candidate change point set, the monitoring data is divided into a change point pre-segment and a change point post-segment according to the candidate change points. The parameters of the change point pre-segment are fixed as the controlled state baseline parameters, and only the parameters of the change point post-segment are subjected to maximum likelihood estimation.

[0011] In at least one embodiment of the present invention, the step of determining the time of anomaly occurrence by the maximum likelihood estimation method specifically involves establishing a corresponding segmented model before and after each candidate variable point, calculating their joint log-likelihood value, and selecting the candidate variable point with the largest joint log-likelihood value as the time of anomaly occurrence.

[0012] In at least one embodiment of the present invention, the fitting of the three candidate models in the anomaly diagnosis stage is based on the joint observation data of the number of claims and the amount of claims before the alarm, and the fitting comparison is performed under the modeling assumption that the statistics of each monitoring period are independent to determine the anomaly type.

[0013] Compared with the prior art, the beneficial effects of the present invention are: This invention proposes a joint monitoring and anomaly diagnosis method for warranty claim quantity and amount. By constructing a dual-dimensional joint monitoring statistic for claim quantity and claim amount, it fully integrates key information on claim frequency and loss severity, enabling more sensitive and comprehensive capture of abnormal changes in product quality and reliability. Employing a dynamic probability control limit based on a composite Poisson process, and relying on numerical integration or Monte Carlo simulation to update the control threshold periodically, it can accurately adapt to the time-varying characteristics of the entire lifecycle, stably control the false alarm rate in each period, and significantly improve the adaptability and accuracy of the monitoring process. After an alarm is triggered, the method accurately locates the time of anomaly occurrence through maximum likelihood estimation. Combining Bayesian information criteria, it optimizes and distinguishes three types of models: those with only quantity parameter changes, those with only amount parameter changes, and those with simultaneous changes in both parameters. This clearly distinguishes the causes of anomalies, transforming simple statistical alarms into interpretable and actionable diagnostic conclusions, effectively shortening the response time for quality problems, and providing a reliable basis for enterprises to quickly conduct root cause analysis and risk management. Ultimately, it forms an integrated after-sales quality risk management solution covering modeling, monitoring, and diagnosis, comprehensively improving the manufacturing enterprise's refined management of warranty claims and early warning capabilities for product reliability. Attached Figure Description

[0014] Figure 1 This invention provides a framework for dynamic monitoring and post-signal diagnosis. Figure 2 This is a diagram illustrating the production scenario of the present invention; Figure 3 This invention relates to the Poisson-index simulation. t = 0 case monitoring results illustration; Figure 4 This is the result of the operation of the monitoring scheme based on the number of claims of this invention; Figure 5 The results are the operational results of the joint monitoring scheme of this invention. Detailed Implementation

[0015] The accompanying drawings described in this invention are merely schematic diagrams of the framework and demonstrations of results in specific scenarios.

[0016] To make the objectives, technical solutions, and advantages of the embodiments of the present invention clearer, the technical solutions of the embodiments of the present invention will be clearly and completely described below with reference to the accompanying drawings. Obviously, the described embodiments are only some, not all, of the embodiments of the present invention. Based on the described embodiments of the present invention, all other embodiments obtained by those skilled in the art without creative effort are within the scope of protection of the present invention.

[0017] Existing technologies primarily focus on changes in claim rates or the number of claims per cycle. However, problems in the design and manufacturing stages often cause not only an abnormal increase in the number of claims but also potential changes in the claim amounts. Both methods characterize the potential impact of problems from the two most important aspects: problem frequency and severity. Within the composite Poisson frequency-severity modeling framework, there is currently a lack of a bivariate joint monitoring and diagnostic solution for the "claim quantity - claim amount" factor.

[0018] The method proposed in this invention addresses several technical issues in existing technologies. Firstly, existing technologies often focus solely on the number of claims as a monitoring object, neglecting the utilization of claim amount information and the system modeling of the coupling relationship between "frequency and severity." Secondly, in warranty claim monitoring scenarios where the number of insured products is highly heterogeneous and exhibits multiple time-varying characteristics, control limits rarely incorporate dynamic probability control concepts, often relying on fixed thresholds or approximate settings based on the overall sample. This makes it difficult to stably control the false alarm rate within a preset level for each monitoring cycle. Thirdly, the method's ability to estimate the timing of anomalies and determine their causes after an alarm is still insufficient, making it difficult to output actionable diagnostic conclusions.

[0019] Compared to industrial production quality monitoring, warranty claims monitoring suffers from issues such as random sample size variations and heterogeneity of the insured population. Based on the data generation process of warranty claims, and to clearly define the statistical characteristics of warranty claims and establish a scientifically effective statistical model, this invention proposes the following assumptions and settings:

[0020] The production and sales activities of products have discrete periodicity characteristics, and manufacturers in the first... i Expect( i = 1,2,…, p )Production Each unit of product, and this batch of products, will be randomly released and sold within the current period and subsequent periods. In actual operation, time delays between production and sales are quite common. Furthermore, this invention stipulates that the actual warranty will be provided at least in the next period after the sale.

[0021] Assumption 1: This invention assumes that the failure rate is equal to the claim rate, and that the occurrence of product claims follows a claim rate function. The non-homogeneous Poisson process is characterized by a power-law distribution for its claim rate function, i.e.: in For shape parameters, This is the scale parameter.

[0022] Assumption 2: The first i Weekly production and in the first j Weekly sales u Unit 1, up to the 1st kThe cumulative number of warranty claims for the week is denoted as a random variable. Each unit comes with a non-renewable free repair and warranty policy, valid for a period of time. Week. For the period j The sold product units, if within the cycle If a qualified fault occurs within the warranty period, the manufacturer will immediately repair it free of charge (minimum repair). Afterwards, the remaining warranty period will expire. This process can be described as a claim rate. A non-homogeneous Poisson process. Definition The benchmark claim rate for products designed or anticipated by the manufacturer (this parameter is known in advance). ( This represents the actual on-site claim rate of products during the production cycle. That is:

[0023] During the product warranty period, the number of claims often fluctuates over time, therefore the available claim rate is... The model uses a non-homogeneous Poisson process to characterize the differences in failure probability across different time periods. The claim rate function adopts a power-law form because the failure rate of products in reality typically exhibits a "low at first, high later" pattern: new products have a low failure rate in their early stages due to stricter manufacturing processes and quality inspections; as usage time increases, parts gradually wear down, potential defects are gradually exposed, and the claim rate increases over time. The flexibility of the power-law function can well describe this monotonically increasing or non-linearly changing pattern over time, and it is also closer to the failure distribution characteristics in actual after-sales scenarios.

[0024] Assumption 3: The first i Weekly production and in the first j Weekly sales u The unit in the first m The claim amount for each warranty claim is denoted as a random variable. Assume that the claims follow an exponential distribution and that each claim is independent of the others.

[0025] The amount of a single claim is mainly determined by the cost of replacing or repairing the faulty component, typically exhibiting a "long tail" characteristic with most claims being small and a few being large. The exponential distribution can precisely characterize this probability structure. Furthermore, the amount of a single claim is not directly related to other claims; the faulty component, the extent of damage, and the repair plan for each claim are often independent of previous claims, effectively reflecting the cost uncertainty in actual after-sales service.

[0026] Assumption 4: This invention assumes that during the warranty period, the manufacturer only provides the most basic repair services, i.e., only repairing product malfunctions, without replacement, upgrades, or other additional services. All repair services are provided free of charge during the warranty period, and the repair content is limited to fault repair. After the warranty period, any repair services will no longer be provided free of charge by the manufacturer, and consumers will need to bear the relevant costs themselves.

[0027] The various parameters, symbols, and their meanings in the warranty claims monitoring model are as follows: w Warranty period length (in weeks); l pro Production cycle number (weeks); l ext To extend the sales period (number of weeks); l This represents the total number of sales periods (weeks). l = l pro + l ext ；t For warranty time vectors, typically t = 1,2,…, w ; i For production cycle indexing; j Index for sales cycle; p This represents the total number of production periods (weeks). a Product age (time of use); k The current monitoring time period (week); l ( a ) is the failure rate function, which is the product age. a The function; β The shape parameters of the power-law process; or The scaling parameter of a power-law process; v The rate parameter of the exponential distribution, i.e., the mean of the exponential distribution, is 1 / v ; r This represents the relative change in scale parameters after the system goes out of control. t These are the actual variable points in the production process; For the first i Weekly production, the j Weekly sales u Unit 1, up to the 1st k The cumulative number of claims for the week; For the first i Weekly production, the j Weekly sales u Unit 1 m The amount of each claim follows an exponential distribution Exp( v ); D i,j For the firsti Weekly production and in the j Number of units sold per week; S j For the first j Zhou's total claim amount.

[0028] Combination Figure 1 As shown, a joint monitoring and anomaly diagnosis method for the quantity and amount of warranty claims includes the following steps: Extract the number of product warranty claims for each monitoring period; when the number of claims is 0, record the total claim amount for that period as 0; when the number of claims is greater than 0, extract the payment amount for each claim and sum them up to obtain the total claim amount for that period. The total claim amount for each monitoring period is used as the monitoring statistic. This statistic is based on the entire life cycle of product production, sales, and warranty. It needs to dynamically filter the set of insured products through a rolling time window and integrate the dual information of claim quantity and claim amount. Based on historical quality assurance data, the claim rate parameters and claim amount distribution parameters under controlled process conditions are determined. Combined with the fitted composite Poisson model, the numerical integration method or Monte Carlo simulation method is used to determine the current dynamic upper control limit according to the preset significance level, so as to control the false alarm rate of each monitoring cycle to be close to the preset level. The current claim amount increment is compared with the dynamic upper control limit. If it does not exceed the limit, the monitoring continues in the next cycle. If it exceeds the limit, an alarm is triggered and the process enters the abnormal diagnosis stage. Based on the monitoring data before the alarm, a set of candidate change points is constructed. The joint log-likelihood values ​​of different candidate change points are compared by the maximum likelihood estimation method to estimate the time of the anomaly. Then, based on the Bayesian information criterion, the fitting comparison of three types of candidate models is performed: only the claim quantity parameter changes, only the claim amount parameter changes, and both parameters change. The optimal model is selected to determine the anomaly type and output the diagnostic conclusion.

[0029] Specifically, in step one above, the data acquisition phase involves the following process: During each monitoring cycle... k Initially, the actual number of claims within that period is extracted from the company's after-sales warranty information system. .like If no claims occurred during that period, the total claim amount for that period is recorded as 0, and monitoring can continue for subsequent periods; if Then, the actual compensation amount / repair cost for each claim within that period can be retrieved. And sum them up to get the total amount for the period:

[0030] in Follows an exponential distribution. Therefore, it satisfies This process ensures that the monitoring statistics can simultaneously cover dynamic changes in both quantity and amount.

[0031] Specifically, in step two above, the process of constructing monitoring statistics is as follows: As the main monitoring statistic.

[0032] As a weekly increment for claim amounts, its core function lies in dynamically filtering the "set of insured products" through a rolling time window, accurately adapting to the entire lifecycle of a product from "sales to warranty to warranty expiration," specifically implemented through a four-stage segmented logic: Early stage of production ( ): Filter sales week (Age ≤ Warranty Period) w For products with a cumulative weekly increase in claims (the weekly difference in cumulative value), add up the total amount of new claims this week. Mid-production stage → End of sales stage ):pass Lock "Recent" w For insured products with "weekly sales", calculate the weekly increase and adapt to dynamic screening as sales scale expands; Warranty extension period ( Sales have been terminated; only included in For the remaining insured products, the additional amount will be extracted; Full coverage period ( ): All products are overdue, and the increment is 0.

[0033] This statistic integrates both the number of claims and the claim amount. An abnormal increase in either dimension will drive a significant change in the increase in claim amount. Compared to traditional monitoring solutions that rely on only a single-dimensional indicator, it can more comprehensively capture anomalies in warranty risks. Dynamic monitoring of this statistic can effectively achieve early warning of quality risks.

[0034] Specifically, in step three above, the design and calculation steps for the dynamic probability control limit are as follows: Because warranty claims processes often exhibit non-stationary characteristics that change over time, using fixed control limits can easily lead to underreporting or false reporting. Therefore, this invention introduces a Dynamic Probability Control Limit (DPCL) mechanism:

[0035] ① Parameters under controlled conditions are estimated from historical data. ; ②In the k At the start of the cycle, according to And the composite Poisson distribution, derived using numerical integration methods. Theoretical distribution function ; ③ By significance level Determine the upper control limit UCL k ,satisfy: in H 0 indicates that the process is under control.

[0036] Under controlled assumptions H At level 0, each monitoring cycle corresponds to a non-overlapping time window. Since claim arrival and individual payouts can be considered independent across different cycles, the [missing information - likely a specific timeframe or timeframe] is determined by the [missing information - likely a specific timeframe or timeframe]. k The statistics obtained from each period Other cycles ( They are independent of each other. Based on this independence, the control limit can be set by setting the false alarm rate for a single cycle. To determine.

[0037] This invention explicitly defines the monitoring target as a one-sided inspection scenario that only detects an "unexpected increase" in the number and amount of claims. Considering that manufacturers are primarily concerned with increasing rather than decreasing risk levels, a one-sided Shewhart control chart is used, with only the upper control limit set for targeted monitoring. This control limit will change periodically. k It is dynamically updated as the system progresses, ensuring a consistent level of risk control throughout each cycle, and is adaptive.

[0038] For the special case of a composite Poisson process where the number of claims follows a non-homogeneous Poisson process and the amount of a single claim follows an exponential distribution, two different control limit calculation methods—Monte Carlo simulation and numerical integration—were employed. These two methods have different applicable scopes and implementation principles, complementing each other to achieve dynamic monitoring of the claims process. Tables 1 and 2 show their calculation processes:

[0039] Method A: Numerical Integration Table 1. Numerical Integration Calculation Control Limit Algorithm Method B: Monte Carlo Simulation Method Table 2 Monte Carlo Simulation Control Limit Algorithm Specifically, in step four above, the threshold comparison and alarm determination steps are as follows: When new monitoring statistics After the calculation is completed, with UCL k Comparison: like This indicates that the claims process in this cycle has not deviated significantly from the controlled state, and the system continues to acquire and monitor data in the next cycle. like If the cycle is deemed to have a potential anomaly, an alarm signal is triggered, and the process immediately proceeds to the diagnostic phase.

[0040] This process forms a closed-loop monitoring mechanism based on "data acquisition, control limit calculation, statistical update, control limit comparison, alarm judgment, and anomaly diagnosis," enabling real-time capture of changes in product reliability.

[0041] Specifically, in step five above, the specific steps for variable point location and anomaly detection are as follows: Based on the joint monitoring solution, an analysis framework centered on change point location and anomaly determination was constructed to address the problem of anomaly diagnosis after control chart alarms.

[0042] Specifically, the diagnostic process first segments the data prior to the alarm on the time axis, considering each candidate segment as a potential change point. Maximum likelihood estimation is used to compare the joint log-likelihood values ​​across different segments to determine the most probable change point. Then, given the change point location, the Bayesian Information Criterion (BIC) is used to fit and compare three types of candidate models (changes only in the claim quantity parameter, changes only in the amount parameter, and changes in both claim quantity and amount parameters simultaneously). The model with the smallest BIC is selected as the optimal explanation to determine the cause of the anomaly. The entire process can be described as a closed-loop flow of "alarm signal, data backtracking, change point estimation, model fitting, BIC comparison, and anomaly determination." This flow is shown in Table 3.

[0043] Table 3 Algorithm for Change Point Identification and Anomaly Detection Through the above steps, this invention aims to form an integrated "modeling-monitoring-diagnosis" methodology framework for warranty claim data, providing a reference for the quantitative monitoring and early warning analysis of after-sales quality risks for manufacturing enterprises.

[0044] Verification of the technical effects of this invention: To verify the effectiveness of the proposed quantity-amount joint monitoring scheme, this invention refers to the simulation settings of Li et al. (2020) and constructs a simulation scenario consistent with the production-sales-quality claim chain. Based on the setting of the claim quantity process, a claim amount process generation mechanism is further introduced so that the simulation data contains both frequency and severity information to support the comparative evaluation of single-dimensional anomalies and joint anomalies.

[0045] Li et al. (2020) Specifically: Li C, Wang X, Li L, et al. On dynamically monitoring aggregate warranty claims for early detection of reliability problems[J]. IISE Transactions, 2020, 52(5): 568-587.

[0046] Suppose a manufacturer has been producing and selling a certain type of product for a long time, and all products sold come with a term... w = 52 weeks of free basic warranty service. The company continuously collects production data, sales data, and warranty claim data, and hopes to achieve early identification of potential reliability issues in the design and manufacturing stages by dynamically monitoring the number and amount of warranty claims.

[0047] First, a simulation scenario that closely resembles actual business operations needs to be constructed to generate a production-sales matrix for simulation. Consider the following representative production scenario: Let the... Weekly output was Then there is

[0048] in This represents the random fluctuations in weekly output, providing a simplified characterization of actual production disturbances. The resulting production trajectory exhibits a typical pattern of first rising, then stabilizing, and then gradually declining.

[0049] Further assuming that all products produced in each production cycle will be in the following 10 Sales are to be completed within 30 weeks, with specific sales dates randomly distributed within this period. The total production cycle is set as follows: p = 130 weeks; after production is completed, unsold inventory requires a maximum of an additional 26 weeks to be sold, therefore the product lifecycle length is taken as... l = 156 weeks. It should be emphasized that the above production and sales settings are only used to construct simulation scenarios. In actual applications, the production and sales matrix should be obtained directly from the company's actual production and sales records on a rolling basis over the period.

[0050] Secondly, a model for warranty claim rates needs to be established. This invention uses a power-law based non-homogeneous Poisson process (NHPP) to describe the unit product claim rate because this model has good adaptability and flexibility in characterizing the evolution of various types of warranty claim rates over time. In this simulation scenario, the baseline claim rate function is denoted as... Its power-law process parameters are set as follows: and This corresponds to the typical degradation behavior of products (such as mechanical products) whose claim rates monotonically increase with usage time.

[0051] Regarding the distribution setting of the claim amount process, the simulation considers using an exponential distribution, and sets the expected value of the claim amount under controlled conditions to 1, that is, sets the amount parameter for the exponential distribution. .

[0052] Due to uncertainties such as fluctuations in materials and processes, supply chain disruptions, and differences in usage environments, the number and amount of claims may exhibit abnormal changes beyond historical norms. This invention uses this scenario as a typical reliability problem for simulation analysis. The simulation pre-sets the timing and magnitude of changes in claim parameters to generate data, but treats these as unknown parameters for inference during model construction and testing. The specific design considers the following three aspects:

[0053] (1) Uncertainty at the moment of change Let the point in time at which the structural changes occur in the claim quantity or amount parameters be... t To reflect the randomness of the timing of reliability issues in practice, rather than using a fixed, known constant, we set: t = 0

[0054] This means that the reliability issues had already occurred before mass production.

[0055] (2) Different abnormal change scenarios Considering that actual warranty risks may manifest as either an abnormal increase in the number of claims or a change in the structure of claim amounts, this invention sets out three scenarios: "change in quantity parameter," "change in amount parameter," and "simultaneous change in both quantity and amount parameters." To facilitate integrated analysis, a common parameter is introduced in the simultaneous change scenario. r Simultaneously depict the intensity of changes in both quantity and monetary amounts.

[0056] (3) Description of the increase in claim parameters Assuming that after a reliability issue occurs, the occurrence of claims still follows a power-law function, the shape parameter of the process remains constant, while its scale parameter and monetary parameter are adjusted from the original level by the same proportional coefficient, that is: , in r The relative increase in the claim parameter. Scale parameter. Decreasing this parameter will increase the expected number of failures (or claims) per unit time, and the monetary value will also increase. Decreasing the value of a claim increases the expected amount per claim, and both changes reflect a deterioration in product reliability. The simulation selects:

[0057] These represent three levels of reliability degradation: slight, moderate, and significant.

[0058] Regarding the monitoring strategy, the maximum expected false alarm rate is set to... Compared with the classic Shewhart control chart The design is consistent with the requirements to ensure a low false alarm probability when the process is under control. This part of the research will evaluate the performance of the proposed joint monitoring and diagnostic scheme based on Monte Carlo simulation methods.

[0059] Based on the above parameter settings, this invention conducts simulation studies (10,000 random runs for each single case). The monitoring results in this section are presented in a "mean + standard deviation" format. The values ​​without parentheses represent the average probability of "at least one signal occurring" in the corresponding scenario obtained from multiple Monte Carlo simulations, while the values ​​in parentheses represent the standard deviation of these probabilities across 10,000 simulations. The following table shows the monitoring and diagnostic results for different scenarios.

[0060] Table 4 Poisson Index (Simulation) t =0 case monitoring results From the perspective of the changing scenarios, under this change point setting, the highest detection probability corresponds to "both quantity and amount parameters changing". r Taking 0.5 as an example, after the point change ( t +1, t [+11] Within the time window, the detection probabilities for cases where only the quantity parameter changes and cases where only the amount parameter changes are 0.4597 and 0.1319, respectively, while the detection probability for cases where both quantity and amount change simultaneously reaches 0.8796; in ( t +1, t Within [+21], the detection probability of simultaneous changes in both parameters jumps directly to 1.0000, significantly higher than 0.6791 for the case of only quantitative parameter change and 0.2110 for the case of only monetary parameter change. This indicates that when both claim frequency and claim amount undergo structural changes simultaneously, the response of the joint statistic is significant, and the control chart can achieve a high detection rate within a short time window.

[0061] From the magnitude of variation rFrom the perspective of detection probability, r The increase was significant, with the most pronounced increase occurring in the initial time window after the change point. In the case where both quantity and amount parameters change, ( t +1, t The detection probability within +11] is determined by r =0.1 when 0.0468, increased to r =0.25 when 0.1560, and in r When the value is 0.5, it jumps to 0.8796; within a longer window (τ+1, τ+21], it rises from 0.2224 to 0.9642, and finally reaches 1.0000. Figure 3 The curves in the text show the same trend: r The larger the value, the faster the curve rises from near zero to near 1 after the change point, indicating that the joint monitoring statistic can better reflect the impact of the variation amplitude and has strong sensitivity to moderate and severe reliability degradation, while a longer observation period is required under slight degradation.

[0062] The diagnostic results table still uses the "mean + standard deviation" format, where... The delay between the moment the control chart sends a signal and the actual change point (signal delay). The error in the change point estimation obtained based on the maximum likelihood diagnostic scheme is... This indicates the baseline estimation error obtained under ideal conditions (assuming the claim parameters after the change point are known), used for comparison with the actual diagnostic plan; flag1 " "Change", flag2 v "Change" and "flag3" v "Change" indicates that the diagnostic plan will determine the change point as "only the claim quantity parameter". "Change", "Claim Amount Parameter Only" v "Change" and " and v The ratio of "simultaneous change" is used to characterize the diagnostic scheme of the present invention's ability to distinguish different types of change points.

[0063] Table 5 Poisson Index ( t =0) Diagnostic results for each situation Overall, with the magnitude of variation r Increasing from 0.1 to 0.5, the average signal delay All delays were significantly shortened, with the average delay for changes in only the quantity parameter decreasing from approximately 30 weeks to 11 weeks, and for changes in both parameters decreasing from approximately 26 weeks to 9 weeks. Furthermore, the delay was consistently the smallest when both quantity and amount changed simultaneously, while it was relatively larger when only the amount parameter changed. This indicates that the more severe the degradation and the more comprehensive the parameter information involved, the more timely the diagnostic initiation. The corresponding change point estimation error... and ideal reference error The absolute values ​​are all around 1, but the error range varies. r The fact that the variation amplitude increases but converges indicates that the change point localization results become more stable as the variation amplitude increases.

[0064] Based on the anomaly detection results, the recognition accuracy is higher in scenarios involving single-parameter changes. However, when the actual situation is "only the claim quantity parameter..." "Change", flag1 The proportion of "change" has remained relatively stable at over 80%, while flag2 " v "Change" and "flag3" v The probability of misjudging "change" remained at a low level; when the actual situation was "only the monetary parameter" v When the condition changes, flag2 consistently achieves a correct judgment rate of 70% across all scenarios. The 80% accuracy rate indicates that the diagnostic protocol still has a strong ability to distinguish between purely monetary abnormalities. In contrast, the most difficult situation to identify is when both the quantity and amount change simultaneously, with an accuracy rate of approximately 50%. 70%, and in r When the value is 0.1, the proportion of misjudgments as "only quantitative parameters have changed" is the highest, reflecting that when the magnitude is small, the quantitative dimension plays a stronger dominant role in the diagnostic statistics.

[0065] In summary, the Poisson-exponential joint monitoring and diagnostic scheme performs well in terms of detection speed, change point location accuracy, and anomaly detection capability, and has high diagnostic reliability, especially for changes in single parameters such as quantity or amount.

[0066] Specific embodiments of the present invention: This section focuses on demonstrating the solution's monitoring response capabilities and anomaly detection output under runaway conditions, while also showcasing the invention's technical effectiveness. Therefore, the dataset used is a set of warranty claim data under simulated runaway scenarios. Specifically, the quality anomaly is set to occur at the beginning of the warranty period: setting a variable point. t =0, and let the claim rate function and the claim amount parameter shift simultaneously after the change point, with the shift coefficient taking r=0.3. Under the aforementioned runaway scenario, a warranty claim observation sequence was generated over 207 monitoring periods. A total of 470,711 products were sold throughout the entire period, with 193,133 claims occurring, totaling RMB 276,160.50. In terms of data structure, this chapter uses a three-stage approach—"production-sales-in-warranty"—to characterize the claim process, and organizes product sales according to the production and sales periods. Due to space limitations, only the sales volume of the products corresponding to the first ten production weeks in the first ten sales periods is shown in Table 6.

[0067] Table 6 Sales figures for the first ten weeks Furthermore, the number of claims and the amount of claims for each monitoring period are summarized to form the input sequence for the control chart. Table 7 shows the monitoring period information for the first twenty weeks, including the number of claims and the amount of claims for each period, as well as the dynamic probability control limit (UCL) corresponding to the joint scheme of this invention and the scheme of Li et al. (2020), which is used to intuitively demonstrate the dynamic updating of the control limit as the number of insured products evolves.

[0068] Table 7 Monitoring Period Information for the First Twenty Weeks During the monitoring process, the solution combines real-time production-sales information to obtain the insured scale for each period and updates the current control limit accordingly. Then, it calculates monitoring statistics and makes alarm judgments. When the statistics exceed the limit for the first time and trigger an alarm, the solution takes the first time the limit is exceeded and its neighborhood interval as input, and further calls the anomaly discrimination method constructed in this invention to first estimate the change point of the runaway occurrence, and then performs type discrimination in the candidate runaway mode, thereby further refining the alarm signal into an interpretable diagnostic conclusion.

[0069] Figure 4 and Figure 5 The dynamic control chart monitoring results of the monitoring scheme based on the number of claims by Li et al. (2020) and the combined scheme of "number of claims – amount of claims" of this invention are shown respectively, where the horizontal axis represents the monitoring period. k The blue line represents the corresponding monitoring statistics, and the orange line represents the dynamic probability control limit. The red vertical line indicates the alarm moment when the statistical value first exceeds the control limit. From the perspective of the initial alarm time, under the same out-of-control settings, the monitoring scheme based on the number of claims proposed by Li et al. (2020) showed that... The value is 15, while the joint monitoring scheme of this invention... A value of 10 indicates that the joint scheme can provide an earlier signal of loss of control in this scenario. This result reflects that when degradation affects both the claim rate and the level of single loss, introducing a monetary dimension to construct a joint statistic can more fully characterize the distribution deviation under loss of control conditions, thereby shortening the number of observation periods required from the occurrence of an anomaly to the first alarm.

[0070] Table 6-3 Diagnostic Results of Dataset After the alarm is triggered, this invention further invokes anomaly diagnosis methods to distinguish between three types of changes: "change in quantity parameters only," "change in amount parameters only," and "change in both quantity and amount parameters simultaneously," and provides an estimate of the change point at which the loss of control occurs. Table 6-3 shows the diagnostic output for this dataset: the results classify this dataset as "change in both quantity and amount simultaneously" (corresponding to flag 3), and the change point estimation result is as follows: =0, which is consistent with the preset variable point in this case. t The value of 0 remains consistent. This indicates that, under this case setting, the combined solution can not only trigger monitoring signals earlier but also provide anomaly detection results consistent with the data settings after the alarm. This provides more targeted statistical basis for subsequent after-sales quality issues categorized by "frequency-severity".

[0071] Based on specific implementation examples, the beneficial effects of this invention are as follows: 1. By constructing a joint monitoring statistic for both the quantity and amount of warranty claims, integrating dual information, it overcomes the limitation of existing technologies that only monitor a single quantitative dimension. When both quantity and amount change simultaneously, r =0.5, the detection probability within 11 cycles after the change point reaches 87.96%, which is better than the 45.97% of single-dimensional monitoring, significantly improving the early identification capability of complex reliability problems.

[0072] 2. A dynamic probability control limit mechanism is adopted, which updates the control limit periodically according to the time-varying characteristics of the base number of products under warranty, so as to achieve precise control of the false alarm rate in each monitoring cycle, effectively adapting to the high heterogeneity and non-stationarity of the quality assurance process, and providing two calculation methods: numerical integration and Monte Carlo simulation, which balances accuracy and practicality.

[0073] 3. After the alarm is triggered, the system uses maximum likelihood-based change point estimation and Bayesian information criterion-based model selection to pinpoint the time of loss of control (with estimation error controlled within one cycle) and identify the anomaly type (single parameter change identification accuracy reaches 70%-80% or more). This transforms the statistical alarm signal into interpretable and actionable diagnostic conclusions, providing decision support for the root cause analysis of quality problems.

[0074] 4. Compared with existing technologies that only monitor the number of claims, under the same runaway settings, the signal delay is reduced from 15 cycles to 10 cycles, and alarms are issued 33% earlier. This forms an integrated "modeling-monitoring-diagnosis" framework, providing a systematic solution for the quantitative monitoring and early warning decision-making of after-sales quality risks for manufacturing enterprises.

[0075] The above embodiments are merely specific implementations of the present invention, used to illustrate the technical solutions of the present invention, and not to limit them. The protection scope of the present invention is not limited thereto. Although the present invention has been described in detail with reference to the foregoing embodiments, those skilled in the art should understand that any person skilled in the art can still modify or easily conceive of changes to the technical solutions described in the foregoing embodiments within the technical scope disclosed in the present invention, or make equivalent substitutions for some of the technical features; and these modifications, changes, or substitutions do not cause the essence of the corresponding technical solutions to deviate from the spirit and scope of the technical solutions implemented in the present invention, and should all be covered within the protection scope of the present invention.

Claims

1. A method for joint monitoring and anomaly diagnosis of warranty claim quantity and amount, characterized in that, Includes the following steps: The current period's warranty claim quantity is extracted according to the preset monitoring cycle. When the current period's claim quantity is zero, the current period's total claim amount is set to zero. When the current period's claim quantity is greater than zero, all compensation amounts are summed to obtain the current period's total claim amount. Subsequently, the incremental claim amount between adjacent periods is calculated as a monitoring statistic. The set of products within the warranty period in the current period is selected based on the rolling time window, and the two types of monitoring data, namely the number of claims and the claim amount, are integrated. Based on historical warranty claim sample data, the claim rate parameters and claim amount distribution parameters under controlled conditions are obtained by fitting. Combined with the composite Poisson-exponential model, the numerical integration method or Monte Carlo simulation method is used to calculate and update the dynamic upper control limit in real time on a cycle with a preset significance level. Compare the weekly increase in the current claim amount with the corresponding period's dynamic upper control limit. If it does not exceed the dynamic upper control limit, continue monitoring in the next period. If it exceeds the dynamic upper control limit, trigger an abnormal alarm. Historical monitoring data before the alarm node is extracted to construct a candidate change point set. The maximum likelihood estimation method is used to determine the actual time of the anomaly. Based on the Bayesian information criterion, three types of diagnostic models are fitted and compared: only the claim quantity parameter changes, only the claim amount parameter changes, and both the quantity and amount parameters change simultaneously. The optimal fitting model is selected based on the minimum value of the Bayesian information criterion. The anomaly type is determined based on the parameter change type corresponding to the optimal model, and the diagnostic conclusion is output.

2. The method for joint monitoring and anomaly diagnosis of warranty claim quantity and amount as described in claim 1, characterized in that, The numerical integration method is used to derive the probability distribution of the weekly increase in claim amount under controlled conditions and determine the dynamic upper control limit. Specifically, the current claim rate is calculated using product sales data and the claim rate benchmark function. Then, the theoretical mean and standard deviation of the composite Poisson-exponential sum are calculated. An integral grid is constructed and the probability density and cumulative distribution function are calculated. Finally, the upper control limit that meets the preset significance level is solved by linear interpolation.

3. The method for joint monitoring and anomaly diagnosis of warranty claim quantity and amount as described in claim 1, characterized in that, The Monte Carlo simulation method simulates the sample distribution characteristics of the weekly increase in claim amount under controlled conditions and determines the dynamic upper control limit. Specifically, it calculates the current claim rate using product sales data and claim rate function, randomly generates multiple claim quantities that conform to the Poisson distribution, and generates a single claim amount based on the exponential distribution, thereby obtaining a total claim amount that conforms to the exponential distribution to form a simulation sample, and then determines the upper control limit based on the preset quantile of the sample.

4. The method for joint monitoring and anomaly diagnosis of warranty claim quantity and amount as described in claim 1, characterized in that, The rolling time window dynamically filters the set of products under warranty, specifically by selecting all products on sale that are still under warranty and whose product age has not exceeded the warranty period before the current monitoring period, adapting to the time-varying characteristics of the entire product lifecycle of production, sales, and warranty.

5. The method for joint monitoring and anomaly diagnosis of warranty claim quantity and amount as described in claim 1, characterized in that, After constructing the candidate change point set, the monitoring data is divided into the change point pre-change point segment and the change point post-change point segment according to the candidate change points. The parameters of the change point pre-change point segment are fixed as the control state baseline parameters, and only the parameters of the change point post-change point segment are estimated by maximum likelihood.

6. The method for joint monitoring and anomaly diagnosis of warranty claim quantity and amount as described in claim 5, characterized in that, The method of determining the time of anomaly occurrence by means of maximum likelihood estimation is as follows: for each candidate variable point, a corresponding segmented model before and after the variable point is established, and their joint log-likelihood value is calculated. The candidate variable point with the largest joint log-likelihood value is selected as the time of anomaly occurrence.

7. The method for joint monitoring and anomaly diagnosis of warranty claim quantity and amount as described in claim 5, characterized in that, The fitting of the three candidate models in the anomaly diagnosis stage is based on the joint observation data of the number of claims and the amount of claims before the alarm, and the fitting comparison is carried out under the modeling assumption that the statistics of each monitoring period are independent to determine the anomaly type.

8. The method for joint monitoring and anomaly diagnosis of warranty claim quantity and amount as described in claim 1, characterized in that, The diagnostic conclusion of the abnormal type is output in the form of an identifier, which includes three states: only the quantity parameter changes, only the amount parameter changes, and both the quantity and amount parameters change simultaneously.

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