Stepwise type-i interval censored constant correlation competing risks reliability evaluation method

Abstract

The present application relates to product reliability evaluation technical field, specifically step by step I type interval censored constant plus dependent competition failure reliability evaluation method, including the following steps: building modified Weibull distribution competition failure model, designing adaptive step by I type mixed censored test scheme, estimating model parameters based on maximum likelihood estimation method, estimating model parameters based on bayesian analysis method, product reliability analysis;The present application breaks through traditional independent failure assumption, borrows GumbelCopula function and Kendall's accurate characterization two failure mechanism interdependent synergistic effect, avoids risk misjudgment;Based on step by step I type interval censored constant plus life test, accelerates stress shrink period, reduces waste, and records failure data in detail;Two kinds of parameter estimation schemes are provided, confidence interval is built by matching Bootstrap-p method, without large sample hypothesis;Through acceleration equation and least square method parameter conversion, accurate prediction of product actual reliability is provided, which provides basis for whole life cycle management, and has accuracy, efficiency and adaptability.

Description

Technical Field

[0001] This invention relates to the field of product reliability assessment technology, specifically to a stepwise type I interval truncation constant addition dependent competing failure reliability assessment method. Background Technology

[0002] In industrial production and engineering applications, product reliability assessment is a core step in ensuring product quality and reducing failure risks. This is especially true for critical products such as electronic equipment, mechanical components, and aerospace devices, where reliability directly impacts safety and economic efficiency. As product structures become increasingly complex, most product failures are not caused by a single factor, but rather involve multiple competing failure mechanisms working together. This presents a significant challenge to traditional reliability assessment methods.

[0003] Traditional reliability assessments often employ the independent failure assumption, neglecting the synergistic effects and correlations between different failure mechanisms. This leads to discrepancies between the assessment results and the actual failure state of the product, making it difficult to accurately reflect the true reliability level and potentially causing engineering decision-making errors. Furthermore, in life test design, conventional testing methods either require long waiting periods for natural product failure, resulting in lengthy testing cycles and high costs; or, while using accelerated stress to shorten the testing cycle, data recording is often coarse, focusing only on whether the product has failed without distinguishing the effects of different failure mechanisms. This results in low utilization of failure data, loss of critical information, and insufficient support for subsequent model building.

[0004] Regarding parameter estimation and result reliability, existing methods often employ a single estimation strategy, making it difficult to balance assessment accuracy and computational efficiency, and thus failing to adapt to the needs of different engineering scenarios. Furthermore, the construction of parameter confidence intervals often relies on large-sample assumptions, making it difficult to obtain robust parameter confidence ranges when the actual experimental sample size is limited, affecting the persuasiveness and engineering application value of the assessment results. In addition, existing assessment methods often struggle to effectively establish parameter correlations under accelerated stress and normal operating stress, failing to accurately predict product reliability in actual operating environments. This hinders the fulfillment of the needs of product design optimization, quality control, and maintenance strategy development throughout the entire product lifecycle, limiting the supporting role of reliability assessment in product improvement. Therefore, it is necessary to propose a stepwise type I interval truncated constant-addition dependent competing failure reliability assessment method. Summary of the Invention

[0005] To address the problems in the prior art, this invention provides a method for assessing the reliability of progressive type I interval truncation constant addition dependent competing failures.

[0006] The technical solution adopted by this invention to solve its technical problem is: a stepwise type I interval truncation constant addition dependent competing failure reliability assessment method, comprising the following steps: S1. Determine experimental design parameters: Set A stress level, including the normal stress level. and Each accelerating stress level, at each stress level Below, at the initial moment Investment Each product, set chronological time series ( (Expected test time) and phased removal plan To ensure the relocation plan If the number is positive, random samples are randomly selected from the remaining unexpired products using probability sampling. Take =The total number of remaining non-expired products, and let ( Stress level (Total number of failures caused by failure mechanism l) S2. Conduct stepwise type I interval truncation constant-life tests: in each observation interval At the end, record the number of product failures caused by failure mechanism 1 and failure mechanism 2 within the interval. , Randomly select from the remaining unexpired products One removal test, the test until Stop at the specified time, remove all remaining non-failed products, and obtain dependent competition failure data; define characteristic functions. (Product in range) Failure caused by internal failure mechanism (or) (Otherwise), the failure data includes The time interval of failure and the corresponding number of failures ; S3. Construct a dependency competition failure model: 1) Assume that there are two failure mechanisms for the product at each stress level, and the interdependence between the failure mechanisms remains unchanged; 2) Survival time of each failure mechanism , Obtain the parameter as , The exponential distribution; 3) The GumbelCopula function is used to characterize the dependency relationship between the two failure mechanisms. The product reliability function is: ,in For dependent parameters, when The two failure mechanisms are independent of each other; 4) Set the average product lifespan under each failure mechanism. With acceleration stress level Satisfies the acceleration equation: in For the parameter to be estimated, Regarding stress level known functions ; S4. Parameter Estimation: Based on the failure data obtained in step 2, the model parameters are solved using the maximum likelihood estimation method or the midpoint approximation method. and dependent parameters : 1) Maximum Likelihood Estimation: Construct the likelihood function for all products, take its logarithm to obtain the log-likelihood function, and then apply the maximum likelihood function to each product. , and Take the first-order partial derivatives and set them equal to zero to form a system of equations. Use Newton's iteration method to solve the system of equations to obtain the numerical solution of the parameters. 2) Midpoint approximation method: assuming an interval The failure time within the interval is the midpoint. After constructing the simplified likelihood function and taking its logarithm, the solution is obtained. The explicit estimation is then used to solve for the dependent parameters using the fixed-point iteration method. The estimated value; S5. Constructing parameter confidence intervals: The Bootstrap-p method is used to construct confidence levels of... The parameter confidence interval is determined by the following process: 1) Calculate the initial maximum likelihood estimate of parameters based on the original failure data. ; 2) Based on the initial estimates, simulate and generate new stepwise type I interval truncated dependent competition failure data, and calculate the maximum likelihood estimates of the parameters of the new data; 3) Repeat step 5.2 for a total of Second-rate( ),get Group parameter estimates and sort them in ascending order; 4) Determine the confidence interval for each parameter by taking quantiles; S6. Product Reliability Assessment: Based on the acceleration equation, the least squares method is used to obtain... The estimated value is then used to obtain the normal stress level. Failure rate parameter estimates By combining the reliability function constructed using the GumbelCopula function and leveraging the invariance of maximum likelihood estimation, the reliability function at any time under normal stress level can be obtained. Product reliability estimates: ,in, These are estimated values ​​for the dependent parameters.

[0007] Specifically, the GumbelCopula function mentioned in step 3 belongs to the Archimedes Copula family, and its expression is: in, For dependent parameters, , These are the marginal distribution function values ​​or marginal survival function values ​​for the two failure mechanisms, respectively.

[0008] Specifically, the distribution function of the exponential distribution mentioned in step 3 is: ; The reliability function is: .

[0009] Specifically, the dependency relationship described in step 3 can be achieved through... Measurement, and With GumbelCopula parameters The relationship is: .

[0010] Specifically, as described in step 5 The range of values ​​is .

[0011] The beneficial effects of this invention: The stepwise type I interval truncation constant addition dependent competing failure reliability assessment method described in this invention: (i) Accurately characterizing the interdependence of failure mechanisms and improving assessment accuracy: Breaking through the limitations of the traditional independent failure assumption, the Gumbel Copula function is used to quantify the correlation strength between two failure mechanisms, combined with Kendall's τ Measuring interdependencies accurately reflects the synergistic effects between failure mechanisms, solving the problem of misjudging product failure risks by independent failure models, and making reliability assessment results more consistent with the actual working state of the product.

[0012] (ii) Optimize test design and data utilization, taking into account both efficiency and information content: By using a stepwise type I interval truncated constant life test, accelerated stress is set to shorten the test cycle, and unfailed products are gradually removed to reduce resource waste. At the same time, key data such as failure mechanism, failure time interval and number of failures are recorded in detail to obtain sufficient effective information within a limited cost and time, and to avoid data waste and loss of key information.

[0013] (iii) Provides flexible and reliable parameter solution schemes to enhance the credibility of results: It covers the maximum likelihood estimation method (high accuracy to adapt to strict evaluation scenarios) and the midpoint approximation estimation method (efficiently adapt to rapid evaluation needs), which can flexibly adapt to different application scenarios; at the same time, it adopts the Bootstrap-p method to construct parameter confidence intervals, without relying on large sample assumptions, and can obtain a robust parameter confidence range even with a limited sample size, reducing decision risk.

[0014] (iv) Achieve accurate reliability prediction under normal stress and support full life cycle management: Through acceleration equations and least squares method, accurately calculate failure rate parameters under normal stress level, and combine model derivation to obtain product reliability at any time in actual use environment, providing scientific basis for product design optimization, quality control, maintenance strategy formulation and other full life cycle management links, helping to reduce failure risk and enhance product competitiveness. Attached Figure Description

[0015] The present invention will be further described below with reference to the accompanying drawings and embodiments.

[0016] Figure 1 This is a flowchart illustrating the reliability assessment method for the stepwise type I interval truncation constant addition dependent competing failure provided by the present invention. Detailed Implementation

[0017] To make the technical means, creative features, objectives and effects of this invention easier to understand, the invention will be further described below in conjunction with specific embodiments.

[0018] like Figure 1 As shown, the present invention provides the following technical solution: Example: A stepwise type I interval truncation constant addition dependent competing failure reliability assessment method, including the following steps: S1. Determine experimental design parameters: Set A stress level, including the normal stress level. and Each accelerating stress level, at each stress level Below, at the initial moment Investment Each product, set chronological time series ( (Expected test time) and phased removal plan To ensure the relocation plan If the number is positive, random samples are randomly selected from the remaining unexpired products using probability sampling. Take =The total number of remaining non-expired products, and let ( Stress level (Total number of failures caused by failure mechanism l) S2. Conduct stepwise type I interval truncation constant-life tests: in each observation interval At the end, record the number of product failures caused by failure mechanism 1 and failure mechanism 2 within the interval. Randomly select from the remaining unexpired products One removal test, the test until Stop at the specified time, remove all remaining non-failed products, and obtain dependent competition failure data; define characteristic functions. (Product in range) Failure caused by internal failure mechanism (or) (Otherwise), the failure data includes The time interval of failure and the corresponding number of failures ; S3. Construct a dependency competition failure model: 1) Assume that there are two failure mechanisms for the product at each stress level, and the interdependence between the failure mechanisms remains unchanged; 2) Survival time of each failure mechanism , Obtain the parameter as , The exponential distribution; 3) The GumbelCopula function is used to characterize the dependency relationship between the two failure mechanisms. The product reliability function is: ,in For dependent parameters, when The two failure mechanisms are independent of each other; 4) Set the average product lifespan under each failure mechanism. With acceleration stress level Satisfies the acceleration equation: in For the parameter to be estimated, Regarding stress level known functions ; S4. Parameter Estimation: Based on the failure data obtained in step 2, the model parameters are solved using the maximum likelihood estimation method or the midpoint approximation method. and dependent parameters : 1) Maximum Likelihood Estimation: Construct the likelihood function for all products, take its logarithm to obtain the log-likelihood function, and then apply the maximum likelihood function to each product. , and Take the first-order partial derivatives and set them equal to zero to form a system of equations. Use Newton's iteration method to solve the system of equations to obtain the numerical solution of the parameters. 2) Midpoint approximation method: assuming an interval The failure time within the interval is the midpoint. After constructing the simplified likelihood function and taking its logarithm, the solution is obtained. The explicit estimation is then used to solve for the dependent parameters using the fixed-point iteration method. The estimated value; S5. Constructing parameter confidence intervals: The Bootstrap-p method is used to construct confidence levels of... The parameter confidence interval is determined by the following process: 1) Calculate the initial maximum likelihood estimate of parameters based on the original failure data. ; 2) Based on the initial estimates, simulate and generate new stepwise type I interval truncated dependent competition failure data, and calculate the maximum likelihood estimates of the parameters of the new data; 3) Repeat step 5.2 for a total of Second-rate( ),get Group parameter estimates and sort them in ascending order; 4) Determine the confidence interval for each parameter by taking quantiles; S6. Product Reliability Assessment: Based on the acceleration equation, the least squares method is used to obtain... The estimated value is then used to obtain the normal stress level. Failure rate parameter estimates By combining the reliability function constructed using the GumbelCopula function and leveraging the invariance of maximum likelihood estimation, the reliability function at any time under normal stress level can be obtained. Product reliability estimates: ,in, These are estimates of the dependency parameters; The GumbelCopula function mentioned in step 3 belongs to the Archimedes Copula family, and its expression is: in, For dependent parameters, , These are the marginal distribution function values ​​or marginal survival function values ​​for the two failure mechanisms, respectively. The distribution function of the exponential distribution mentioned in step 3 is: ; The reliability function is: ; The dependency relationship described in step 3 can be achieved through... Measurement, and With GumbelCopula parameters The relationship is: ; Among them, the steps described in step 5 The range of values ​​is ; When using it, the following steps are included: Step 1: Experimental Preparation and Parameter Design 1) Clearly define the basic information related to the product and identify the two core failure mechanisms that may exist in actual use (such as component wear, circuit overload, etc.) to ensure that subsequent tests focus on the key failure causes; 2) Setting the stress level for accelerated life testing: (Total planning) One stress level, including one normal stress level. (Simulated stress in the actual use environment of the product) and An accelerated stress level (such as a temperature or voltage higher than normal operating conditions, used to shorten the test cycle); 3) Determine the test configuration for each stress level: at each stress level Below, initial moment Investment A complete product; setting chronological time series (in (The total expected trial time is preset), and a phased removal plan is formulated simultaneously. (each) Indicates the observation interval (Number of non-failed products that need to be removed from the test after the test). 4) Ensure the feasibility of the relocation plan: Use probability sampling to select from the remaining unfailed products at the end of each observation period. One, guarantee It is a positive number; at the end of the experiment ,Pick This refers to the total number of remaining non-failed products at this point, and satisfies... , Stress level (Total number of failures caused by failure mechanism l) The second step is to conduct a stepwise type I truncation constant-life test. 1) According to the designed stress level and product quantity, start the test of each stress group simultaneously, monitor the stress state in real time during the test process, and ensure that the stress is stable at the set value; 2) Observe and record in chronological order: within each observation interval At the end, accurately count the number of product failures caused by failure mechanism 1 and failure mechanism 2 within this interval, and record them as follows: , ( Corresponding stress level number, (corresponding failure mechanism number) 3) Perform a phased removal operation: From the remaining unexpired products after the current interval ends, randomly select products according to the preset removal plan. The product removal test was conducted to avoid interference with the data from subsequent continuous testing of these products.

[0019] 4) Trial Termination and Data Processing: When the experiment has reached the expected total time... At this point, stop testing of all stress groups and remove all remaining unfailed products; define the indicator function. (Indicates the product is in the range) Failure caused by internal failure mechanism (or) (This indicates that the product did not fail due to this failure mechanism within this range), and finally, complete dependency-competition failure data is compiled, which includes... The time interval of failure and the corresponding number of failures ; Step 3: Constructing the Dependency Competition Failure Model 1) Clarify the model basis based on experimental assumptions: Determine that the interdependent structure between the two failure mechanisms of the product remains unchanged under each stress level, and that the survival time of each failure mechanism is constant. , Obtain the parameter as , It follows an exponential distribution, and its distribution function is: The reliability function is ; 2) Introducing the GumbelCopula function to characterize the dependency relationship: The GumbelCopula function from the Archimedes Copula family is used to connect two failure mechanisms, and its expression is: , , These are the marginal distribution function values ​​or marginal survival function values ​​for the two failure mechanisms, respectively. For dependent parameters, (When the two failure mechanisms are independent), the reliability function of the product can be obtained. ; 3) Establish acceleration equations: Set the average product lifespan under each failure mechanism. With acceleration stress level Satisfying the acceleration equation , For the parameter to be estimated, Regarding stress level Known functions, such as the temperature-related functions in the Arrhenius equation. ; Step 4: Model parameter estimation (a) Maximum likelihood estimation method (suitable for scenarios requiring high estimation accuracy) 1) Based on the processed failure data, construct the likelihood function for all products. This function comprehensively reflects the failure situation at each stress level and within each observation interval, as well as the status of non-failed products. 2) Take the logarithm of the likelihood function to transform it into a log-likelihood function, which simplifies the subsequent differentiation calculation; 3) For the parameters respectively , and dependent parameters Find the first-order partial derivatives and set each partial derivative to zero to form a system of equations; 4) Solve the above system of equations using the Newton-Raphson iterative method to obtain numerical solutions for each parameter.

[0020] (ii) Midpoint approximation method (suitable for scenarios where computational efficiency is a priority) 1) Simplified failure time assumption: assuming the product's failure time is within the observation period. The failure time within the interval occurs at the midpoint of the interval, i.e. ; 2) Construct a simplified likelihood function based on the midpoint assumption, take its logarithm, and then solve for it directly. The explicit estimate.

[0021] 3) Solve for dependent parameters using the fixed-point iteration method. First give initial value Substitute into the iterative formula to calculate the subsequent iteration values. Repeat the iteration until , To preset a very small positive number, it is usually taken as ),at this time That is The estimated value ; Step 5: Constructing Parameter Confidence Intervals (Bootstrap-p Method) 1) Based on the original failure data, the initial maximum likelihood estimates of the parameters are obtained using the parameter estimation method described above. .

[0022] 2) Based on the initial estimate, simulate and generate a new set of stepwise type I interval truncated dependent competition failure data, and calculate the maximum likelihood estimate of the parameters corresponding to the new data according to the same parameter estimation method.

[0023] 3) Repeat step 5.2 for a total of Second-rate( ), The larger the confidence interval, the more reliable it is. Group parameter estimates.

[0024] 4) The group parameter estimates are sorted in ascending order, and the confidence level is determined by taking the quantiles. (Commonly used) (i.e., 95% confidence level) is the parameter confidence interval, for example... The confidence interval is , The same applies to the confidence intervals; Step 6: Product Reliability Assessment 1) Solve for the failure rate parameters under normal stress levels: based on the acceleration equation Using the least squares method An estimate was made, and the following results were obtained. Normal stress level Substituting into the acceleration equation, the estimated failure rate parameters under normal stress are obtained. ; 2) Calculate product reliability under normal stress: A reliability function is constructed using the GumbelCopula function, and the invariance of maximum likelihood estimation is utilized to... , , Substitute into the formula This allows us to obtain the result at any time under normal stress levels. The product reliability estimate is used to complete the product reliability assessment; Step 7: Result Verification and Output 1) Verify the reasonableness of parameter estimation: through Measuring the dependency between failure mechanisms Determine the dependency parameters based on engineering experience. Does the estimated value match the actual failure characteristics of the product? 2) Output evaluation report: Clearly record the test design parameters, failure data details, model parameter estimates and confidence intervals, and product reliability at different times under normal stress, providing a basis for product reliability improvement and life prediction.

[0025] The foregoing has shown and described the basic principles, main features, and advantages of the present invention. Those skilled in the art should understand that the present invention is not limited to the above embodiments. The embodiments and descriptions in the specification are merely illustrative of the principles of the invention. Various changes and modifications can be made to the invention without departing from its spirit and scope, and all such changes and modifications fall within the scope of protection claimed by the present invention. The scope of protection of the present invention is defined by the appended claims and their equivalents.

Claims

1. A stepwise type I interval truncation constant addition dependent competing failure reliability assessment method, characterized in that, Includes the following steps: S1. Determine experimental design parameters: Set A stress level, including the normal stress level. and Each accelerating stress level, at each stress level Below, at the initial moment Investment Each product, set chronological time series ( (Expected test time) and phased removal plan To ensure the relocation plan If the number is positive, random samples are randomly selected from the remaining unexpired products using probability sampling. Take =The total number of remaining non-expired products, and let ( Stress level (Total number of failures caused by failure mechanism l) S2. Conduct stepwise type I interval truncation constant-life tests: in each observation interval At the end, record the number of product failures caused by failure mechanism 1 and failure mechanism 2 within the interval. , Randomly select from the remaining unexpired products One removal test, the test until Stop at the specified time, remove all remaining non-failed products, and obtain dependent competition failure data; define characteristic functions. (Product in range) Failure caused by internal failure mechanism (or) (Otherwise), the failure data includes The time interval of failure and the corresponding number of failures , ; S3. Construct a dependency competition failure model: 1) Assume that there are two failure mechanisms for the product at each stress level, and the interdependence between the failure mechanisms remains unchanged; 2) Survival time of each failure mechanism , Obtain the parameter as , The exponential distribution; 3) The GumbelCopula function is used to characterize the dependency relationship between the two failure mechanisms. The product reliability function is: ,in For dependent parameters, when The two failure mechanisms are independent of each other; 4) Set the average product lifespan under each failure mechanism. With acceleration stress level Satisfies the acceleration equation: in For the parameter to be estimated, Regarding stress level known functions ; S4. Parameter Estimation: Based on the failure data obtained in step 2, the model parameters are solved using the maximum likelihood estimation method or the midpoint approximation method. and dependent parameters : 1) Maximum Likelihood Estimation: Construct the likelihood function for all products, take its logarithm to obtain the log-likelihood function, and then apply the maximum likelihood function to each product. , and Take the first-order partial derivatives and set them equal to zero to form a system of equations. Use Newton's iteration method to solve the system of equations to obtain the numerical solution of the parameters. 2) Midpoint approximation method: assuming an interval The failure time within the interval is the midpoint. After constructing the simplified likelihood function and taking its logarithm, the solution is obtained. The explicit estimation is then used to solve for the dependent parameters using the fixed-point iteration method. The estimated value; S5. Constructing parameter confidence intervals: The Bootstrap-p method is used to construct confidence levels of... The parameter confidence interval is determined by the following process: 1) Calculate the initial maximum likelihood estimate of parameters based on the original failure data. ; 2) Based on the initial estimates, simulate and generate new stepwise type I interval truncated dependent competition failure data, and calculate the maximum likelihood estimates of the parameters of the new data; 3) Repeat step 5.2 for a total of Next, get Group parameter estimates and sort them in ascending order; 4) Determine the confidence interval for each parameter by taking quantiles; S6. Product Reliability Assessment: Based on the acceleration equation, the least squares method is used to obtain... The estimated value is then used to obtain the normal stress level. Failure rate parameter estimates By combining the reliability function constructed using the GumbelCopula function and leveraging the invariance of maximum likelihood estimation, the reliability function at any time under normal stress level can be obtained. Product reliability estimates: ,in, These are estimated values ​​for the dependent parameters.

2. The method for assessing reliability of stepwise type I interval truncation constant addition dependent competing failures according to claim 1, characterized in that: The GumbelCopula function mentioned in step 3 belongs to the Archimedes Copula family, and its expression is: in, For dependent parameters, , These are the marginal distribution function values ​​or marginal survival function values ​​for the two failure mechanisms, respectively.

3. The stepwise type I interval truncation constant addition dependent competing failure reliability assessment method according to claim 1, characterized in that: The distribution function of the exponential distribution mentioned in step 3 is: ； The reliability function is: 。 4. The method for assessing reliability of stepwise type I interval truncation constant addition dependent competing failures according to claim 1, characterized in that: The dependency relationship described in step 3 can be achieved through... Measurement, and With GumbelCopula parameters The relationship is: .

5. The stepwise type I interval truncation constant addition dependent competing failure reliability assessment method according to claim 1, characterized in that: Step 5 The range of values ​​is .

Images