A condition-based maintenance strategy for heterogeneous populations☆
Introduction
To maintain competitive advantage, manufacturers endeavor as much as possible to produce reliable products. However, during the manufacturing process, some unavoidable manufacturing defects could be introduced, e.g. defects in the raw material, leading to a heterogeneous population of products. The heterogeneous population contains a small proportion of weak/defective products. Compared with the normal products, the weak products have a shorter mean lifetime and are prone to giving rise to early in-use failures. The early in-use failures will cause substantial costs and sometimes are hazardous. In fact, it is not uncommon to observe a heterogeneous population with two sub-populations: a weak sub-population and a normal sub-population. For example, it is widely believed that integrated circuits consist of a small proportion of weak items which have much shorter mean lifetime. The GaAs laser data set, provided by Meeker and Escobar (1998), is a typical sample consisting of a sub-group of normal devices and a sub-group of weak devices; see Tsai, Tseng, and Balakrishnan (2011). Scarf and Cavalcante (2012) considered component heterogeneity when developing an age-based maintenance policy for a single-component system; they assumed that the population of components comprises a mixture of the weak and the strong. Berrade, Scarf, Cavalcante, and Dwight (2013) developed a maintenance policy involving periodic inspections, in which inspections are subject to error; they assumed that the time a system spends in the defective state is a random variable from a mixture distribution. Recent research on heterogeneous data can be found in Cha and Finkelstein, 2013, Erisoglu et al., 2013, Kazmi et al., 2013, among others.
A common practice to tackle a heterogeneous population is to screen out the weak products by means of a burn-in test. Burn-in is an engineering procedure implemented at the end of the manufacturing process. In a burn-in test, all the products are subjected to harsh electrical and thermal conditions that emulate the field operational conditions. At the end of the burn-in test, only the functioning products will be shipped to customers. There has been a bulk of research on developing economical burn-in tests; see Cha and Finkelstein, 2010, Pearn et al., 2013, Post and Bhattacharyya, 2012, Yuan and Kuo, 2010, among others. With the rapid development of modern manufacturing technology, even a weak product is able to operate for a fairly long period of time under aggravated operational conditions. Weeding out weak products via traditional age-based burn-in approach is therefore ineffectual. When there is a performance characteristic whose evolution is closely associated with the lifetime of the product, the condition-based burn-in approach is an attractive alternative (Xiang et al., 2013, Ye et al., 2012). Because a weak product most often deteriorates faster than a normal product, a condition-based burn-in test exercises all the products for a certain period of time and scraps the products with deterioration levels higher than a pre-specified cut-off level. After the burn-in test, all the products with deterioration levels lower than the cut-off level are shipped to customers. In field operation, preventive repairs are often scheduled to further improve reliability and reduce operational costs. See Ahmad and Kamaruddin (2012) for an overview of two maintenance techniques widely adopted by industrial engineers: time-based maintenance and condition-based maintenance. Nonetheless, it is well known that burn-in is costly, including such as burn-in set-up cost, burn-in operational cost and repair/scrap cost from a burn-in failure. Scilicet, by adopting the burn-in procedure, we make a trade-off between early in-use failures and a reduced yield due to the burn-in costs. It is worth noting that nowadays even a weak product is able to operate for a long term. A more judgmatic approach to tackling a heterogeneous population is to directly put all the products into field operation and replace the weak products before they fail. By virtue of appropriately scheduled inspections, early in-use failures can be mitigated, and the failure cost can be countervailed by the long-time operating income. Even if a burn-in test is able to identify most of the defective products, it will cause damage to the normal products, shortening the mean lifetime of the normal products (Cha and Finkelstein, 2011, Ye et al., 2011).
To make full use of weak products and avoid impairing normal products, we develop the inspection–replacement policy for heterogeneous populations with the assumption that a suitable performance characteristic of the products is available. The inspection–replacement policy directly puts all the products in a heterogeneous population into field use without burn-in. The physical degradation of each product will be measured at time . If the degradation of a product exceeds a critical threshold, then it will be treated like a weak product and replaced at time b. If the degradation does not exceed the critical threshold, then it will be treated like a normal product and will be preventively replaced at time , if it survives to time R. A schematic diagram of the inspection–replacement policy is shown in Fig. 1. The purpose of the inspection is to identify and replace weak products, whereas the preventive replacement is to replace aged normal products. If the inspection and the critical threshold are well determined, most of the weak products will be identified by the inspection. It is not difficult to see that the role of the inspection is analogous to a burn-in test except that the inspection is carried out in the field operation. However, there will be some figures of merit by replacing the burn-in approach with the inspection–replacement policy. First of all, the inspection–replacement policy does not need to burn-in products, and therefore saves burn-in costs and avoids impairing normal products. Secondly, the inspection–replacement policy makes full use of weak products, and therefore is expected to yield more profits. Last but not least, the preventive-replacement time R for each product is dynamically determined, depending on the physical degradation of the product obtained by the inspection. The on-line updating technique is expected to return more cost-efficient maintenance policy.
The rest of the paper is organized as follows. Section 2 details the inspection–replacement policy. The long-run cost rate function is derived. A joint burn-in and age-based-replacement policy serving as a benchmark is introduced. Section 3 draws the inspection–replacement policy on two predominant stochastic processes: the gamma process and the Wiener process. Section 4 presents numerical examples to elaborate the methodology developed in Sections 2 Maintenance strategy & cost rate function, 3 Applied to two specific processes. The GaAs laser data set is analyzed. Contradistinctive analysis and sensitivity analysis are performed by using numerical simulations. Section 5 concludes the paper.
Section snippets
Maintenance strategy & cost rate function
The product concerned stands for a component which is randomly drawn from a heterogeneous population. When put into operation, the component undergoes deterioration with the degradation process denoted by , assuming as with the convention. A component is considered to have failed if its deterioration level reaches a given failure threshold, denoted by , which is a known constant. Define T to be a random variable representing the first hitting time of the degradation process
Applied to two specific processes
In this section we draw the advanced inspection–replacement policy on two predominant stochastic processes: the gamma process and the Wiener process. The gamma process is the limit of a shot-noise process with exponential decay, and the wear is non-decreasing. The Wiener process is an almost surely continuous martingale. The gamma process and the Wiener process can be visualized in different contexts and have been studied rather extensively; see Balka et al., 2009, van Noortwijk, 2009 for
Application: GaAs lasers
As a laser device ages, its light output decreases. Increasing a laser device’s operating current may maintain its light output nearly constant. Therefore, the operating current usually serves as a performance characteristic for laser devices. A laser device is considered to have failed if its operating current is too high and crosses a threshold. Fig. 3 shows the percent increase in operating current over time for a sample of fifteen GaAs laser devices (Meeker & Escobar, 1998). The GaAs laser
Conclusions
This study recommended an inspection–replacement policy for heterogeneous populations. The inspection–replacement policy possesses three main advantages. First, the inspection–replacement policy saves the burn-in costs. Second, the inspection–replacement policy makes full use of weak components. Third, the inspection–replacement policy takes into account the reliability criterion. Both sensitivity analysis and contradistinctive analysis were performed to demonstrate the feasibility and
Acknowledgement
Ye is supported by the National Research Foundation Singapore under its Campus for Research Excellence and Technological Enterprise (CREATE).
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This manuscript was processed by Area Editor H.-S. Jacob Tsao.