Modeling Consecutive Failures of Repairable Systems, with Applications in Warranty Cost Analysis

<p>Most engineered systems are inclined to fail sometime during their lifetime. Many of these systems are repairable and not necessarily discarded and replaced upon failure. Unlike replacements, where the failed system is replaced with a new and identical system, not all repairs have an equivalent effect on the working condition of the system. Describing the effect of repairs is a requirement in modeling consecutive failures of a repairable system–at the very least, it is assumed that a repair simply returns the failed system to an operational state without affecting its working condition (i.e. the repair is minimal). Although this assumption simplifies the modeling process, it is not the most accurate description of the effect of repair in real situations. Often, along with returning a failed system to an operational state, repairs can improve the working condition of the system, and thus, increase its reliability which impacts on the rate of future failures of the system. Repair models provide a generalized framework for realistic modeling of consecutive failures of engineered systems, and have broad applications in fields such as system reliability and warranty cost analysis. The overall goal of this research is to advance the state of the art in modeling the effect of general repairs, and hence, failures of repairable systems. Two specific types of system are considered: (i) a system whose working condition initially improves with time or usage, and whose lifetime is modeled as a univariate random variable with a non-monotonic failure rate function; (ii) a system whose working condition deteriorates with age and usage, and whose lifetime is modeled as a bivariate random variable with an increasing failure rate function. Most univariate lifetime distributions used to model system lifetimes are assumed to have increasing failure rate functions. In such cases, modeling the effect of general repairs is straightforward– the effect of a repair can bemodeled as a possible decrease, proportional to the effectiveness of the repair, in the conditional intensity function of the associated failure process. For instance, a general repair can be viewed as the replacement of the failed system with an identical system at a younger age, so that the conditional failure intensity following the repair is lower than the conditional failure intensity prior to the failure. When the failure rate function is initially decreasing, specifically bathtub-shaped, general repair models suggested for systems with increasing failure rate functions can only be applied when initial repairs are assumed to be minimal. In this study, we propose a new approach to modeling the effect of general repairs on systems with a bathtub-shaped failure rate function. The effect of a general repair is characterized as a modification in the conditional intensity function of the corresponding failure process, such that the system following a general repair is at least as reliable as a system that has not failed. We discuss applications of the results in the context of warranty cost analysis and provide numerical illustrations to demonstrate properties of the models. Sometimes the failures of a system may be attributed to changes in more than one measure of its working condition– for instance, the age and some measure of the usage of the system (such as, mileage). Then, the system lifetime is modeled as a bivariate random variable. Most general repair models for systems with bivariate lifetime distributions involve reducing the failure process to a one-dimensional process by, for instance, assuming a relationship between age and usage or by defining a composite scale. Then, univariate repair models are used to describe the effect of repairs. In this study, we propose a new approach to model the effect of general repairs performed on a system whose lifetime is modeled as a bivariate random variable, where the distributions of the bivariate inter-failure lifetimes depend on the effect of all previous repairs and following a general repair, the system is at least as reliable as a system that has not failed. The lifetime of the original system is assumed to have an increasing failure rate (specifically, hazard gradient vector) function. We discuss applications of the associated failure process in the context of two-dimensional warranty cost analysis and provide simulation studies to illustrate the results. This study is primarily theoretical, with most of the results being analytic. However, at times, due to the intractability of some of the mathematical expressions, simulation studies are used to illustrate the properties and applications of the proposed models and results.</p>

Modeling Consecutive Failures of Repairable Systems, with Applications in Warranty Cost Analysis

<p>Most engineered systems are inclined to fail sometime during their lifetime. Many of these systems are repairable and not necessarily discarded and replaced upon failure. Unlike replacements, where the failed system is replaced with a new and identical system, not all repairs have an equivalent effect on the working condition of the system. Describing the effect of repairs is a requirement in modeling consecutive failures of a repairable system–at the very least, it is assumed that a repair simply returns the failed system to an operational state without affecting its working condition (i.e. the repair is minimal). Although this assumption simplifies the modeling process, it is not the most accurate description of the effect of repair in real situations. Often, along with returning a failed system to an operational state, repairs can improve the working condition of the system, and thus, increase its reliability which impacts on the rate of future failures of the system. Repair models provide a generalized framework for realistic modeling of consecutive failures of engineered systems, and have broad applications in fields such as system reliability and warranty cost analysis. The overall goal of this research is to advance the state of the art in modeling the effect of general repairs, and hence, failures of repairable systems. Two specific types of system are considered: (i) a system whose working condition initially improves with time or usage, and whose lifetime is modeled as a univariate random variable with a non-monotonic failure rate function; (ii) a system whose working condition deteriorates with age and usage, and whose lifetime is modeled as a bivariate random variable with an increasing failure rate function. Most univariate lifetime distributions used to model system lifetimes are assumed to have increasing failure rate functions. In such cases, modeling the effect of general repairs is straightforward– the effect of a repair can bemodeled as a possible decrease, proportional to the effectiveness of the repair, in the conditional intensity function of the associated failure process. For instance, a general repair can be viewed as the replacement of the failed system with an identical system at a younger age, so that the conditional failure intensity following the repair is lower than the conditional failure intensity prior to the failure. When the failure rate function is initially decreasing, specifically bathtub-shaped, general repair models suggested for systems with increasing failure rate functions can only be applied when initial repairs are assumed to be minimal. In this study, we propose a new approach to modeling the effect of general repairs on systems with a bathtub-shaped failure rate function. The effect of a general repair is characterized as a modification in the conditional intensity function of the corresponding failure process, such that the system following a general repair is at least as reliable as a system that has not failed. We discuss applications of the results in the context of warranty cost analysis and provide numerical illustrations to demonstrate properties of the models. Sometimes the failures of a system may be attributed to changes in more than one measure of its working condition– for instance, the age and some measure of the usage of the system (such as, mileage). Then, the system lifetime is modeled as a bivariate random variable. Most general repair models for systems with bivariate lifetime distributions involve reducing the failure process to a one-dimensional process by, for instance, assuming a relationship between age and usage or by defining a composite scale. Then, univariate repair models are used to describe the effect of repairs. In this study, we propose a new approach to model the effect of general repairs performed on a system whose lifetime is modeled as a bivariate random variable, where the distributions of the bivariate inter-failure lifetimes depend on the effect of all previous repairs and following a general repair, the system is at least as reliable as a system that has not failed. The lifetime of the original system is assumed to have an increasing failure rate (specifically, hazard gradient vector) function. We discuss applications of the associated failure process in the context of two-dimensional warranty cost analysis and provide simulation studies to illustrate the results. This study is primarily theoretical, with most of the results being analytic. However, at times, due to the intractability of some of the mathematical expressions, simulation studies are used to illustrate the properties and applications of the proposed models and results.</p>

Availability Optimization of Multicomponent Products with Economic Dependence under Two-Dimensional Warranty

For multicomponent products, the maintenance of every component separately will increase the downtime and reduce the availability of products during the warranty period. To solve this problem, the economic dependence between the components is considered in this paper. Firstly, a single-component two-dimensional (2D) preventive maintenance (PM) availability model is established, and the simulated annealing algorithm is adopted to calculate the optimal 2D PM interval to achieve the maximal availability of any single component. Then, to ensure that the warranty cost of each component does not exceed the budget, the PM benchmark interval is introduced, and the PM work is optimized following the method of grouping maintenance. Based on this, the 2D preventive grouping maintenance availability model of multicomponent products is established. Finally, an example is given to verify the proposed method, and the results indicate that the proposed method increases the availability of multicomponent products during the 2D warranty period.

Data Mining in Automotive Warranty Analysis

<p>This thesis is about data mining in automotive warranty analysis, with an emphasis on modeling the mean cumulative warranty cost or number of claims (per vehicle). In our study, we deal with a type of truncation that is typical for automotive warranty data, where the warranty coverage and the resulting warranty data are limited by age and mileage. Age, as a function of time, is known for all sold vehicles at all time. However, mileage is only observed for a vehicle with at least one claim and only at the time of the claim. To deal with this problem of incomplete mileage information, we consider a linear approach and a piece-wise linear approach within a nonparametric framework. We explore the univariate case, as well as the bivariate case. For the univariate case, we evaluate the mean cumulative warranty cost and its standard error as a function of age, a function of mileage, and a function of actual (calendar) time. For the bivariate case, we evaluate the mean cumulative warranty cost as a function of age and mileage. The effect of reporting delay of claim and several methods for making prediction are also considered. Throughout this thesis, we illustrate the ideas using examples based on real data.</p>

Data Mining in Automotive Warranty Analysis

<p>This thesis is about data mining in automotive warranty analysis, with an emphasis on modeling the mean cumulative warranty cost or number of claims (per vehicle). In our study, we deal with a type of truncation that is typical for automotive warranty data, where the warranty coverage and the resulting warranty data are limited by age and mileage. Age, as a function of time, is known for all sold vehicles at all time. However, mileage is only observed for a vehicle with at least one claim and only at the time of the claim. To deal with this problem of incomplete mileage information, we consider a linear approach and a piece-wise linear approach within a nonparametric framework. We explore the univariate case, as well as the bivariate case. For the univariate case, we evaluate the mean cumulative warranty cost and its standard error as a function of age, a function of mileage, and a function of actual (calendar) time. For the bivariate case, we evaluate the mean cumulative warranty cost as a function of age and mileage. The effect of reporting delay of claim and several methods for making prediction are also considered. Throughout this thesis, we illustrate the ideas using examples based on real data.</p>

Reliability-Based Opportunistic Maintenance Modeling for Multi-component Systems with Economic Dependence under Base Warranty

Maintenance usually plays a key role in controlling a multi-component production system within normal operations. Furthermore, the failure of components in the production system will also cause large economic losses for users due to the shutdown. Meanwhile, manufacturers of the production system will be confronted with the challenges of the warranty cost. Therefore, it is of great significance to optimize the maintenance strategy to reduce the downtime and warranty cost of the system. Opportunistic maintenance (OM) is a quite important solution to reduce the maintenance cost and improve the system performance. This paper studies the OM problem for multi-component systems with economic dependence under base warranty (BW). The irregular imperfect preventive maintenance (PM) is performed to reduce the failure rate of components at a certain PM reliability threshold. Moreover, the OM optimization model is developed to minimize the maintenance cost under the optimal OM reliability threshold of each component. A simulated annealing (SA) algorithm is proposed to determine the optimal maintenance cost of the system and the optimal OM threshold under BW. Finally, a numerical example of a belt conveyor drive device in a port is introduced to demonstrate the feasibility and advantages of the proposed model in maintenance cost optimization.

A two-echelon supply chain model with price and warranty dependent demand and pro-rata warranty policy under cost sharing contract

In this article, a two-echelon supply chain model with a single-vendor a single-buyer is considered. The vendor's production process is imperfect and the market demand is assumed to be dependent on the buyer's selling price and warranty period. The vendor consents to return a definite portion of the buyer's purchase value, if any product is found defective within the length of warranty. The refund value or the warranty cost is considered as a function of the warranty period and the buyer's selling price of the item. This warranty cost is assumed to be fully borne by the vendor in the first model (Model I) while in the second model (Model II), it is assumed that the buyer agrees to bear a portion of the warranty cost. The proposed models are solved under decentralized scenario. We also derive and optimize the average total profit of the supply chain in order to obtain the optimal decisions of the centralized model. We consider a Stackelberg game between the vendor and the buyer in the decentralized scenario, where the vendor is assumed to be the leader and the buyer as the pursuer. Through numerical study, it is observed that, with respect to all the key decisions of the models, Model II provides better outcomes than Model I. Sensitivity analysis is also carried out to examine the impacts of changes of parameter-values on the optimum decisions.