Innovative Applications of O.R.An unpunctual preventive maintenance policy under two-dimensional warranty☆
Introduction
In order to retain in today’s highly competitive market, manufacturers are forced not only to provide high quality products on time but also to offer appealing after-sales services for their products. One such kind of after-sales services is product warranty. Nowadays, almost all products are sold with warranty contracts due to competition, industrial obligations, and/or customer requirements (Luo, Wu, 2018, Xie, Ye, 2016). Offering warranty is by no means free from the manufacturers’ perspective. The warranty cost, resulting from the servicing of warranty claims, is a huge financial burden to manufacturers, and can account for as much as 15% of net sales (Murthy & Djamaludin, 2002). In reality, most manufacturers regard warranty as an overhead element, and they are always under pressure to keep warranty expenses staying within limits (Liu, Wu, Xie, 2015, Zhao, He, Xie, 2018). As a result, manufacturers are constantly seeking effective warranty cost reduction strategies. Incorporating appropriate preventive maintenance (PM) programs into warranty policies is one feasible way that offers potential solution to warranty cost reduction (Chien, Zhang, Yin, 2019, Kim, Djamaludin, Murthy, 2004).
Generally, PM activities are performed before an item breaks down and aim to reduce the item’s degradation and its risk of failure (Shafiee & Chukova, 2013). In practice, the PM schedule of warranted items is usually prescribed by the manufacturer, while PM implementation relies upon either maintenance crews to execute the operations onsite (for installed systems, such as household heating systems and heavy machines) or item owners to return their items to authorized maintenance centers. This paper focuses particularly on the latter case, though our model can be easily applied to the former case. In this situation, one common yet challenging issue is maintenance unpunctuality that stems from the separate nature of PM specification and execution (He, Maillart, & Prokopyev, 2017). For instance, in the automobile warranty and maintenance context, it is not uncommon that some owners do not follow the manufacturer-prescribed PM schedule strictly. Possible reasons include (i) scheduled PM instants are at working days, (ii) exact PM instants may be difficult to record precisely, especially for heavy-usage customers, and (iii) some customers might simply miss certain PM instants.
In recent years, a Chinese automaker, SAIC-GM-Wuling Automobile, has adopted an unpunctual PM policy in its warranty and maintenance practice. This policy allows customers to return their vehicles slightly earlier or later than scheduled PM instants in a tolerated range, provided that the vehicles are still under warranty. From the manufacturer’s perspective, customer unpunctuality may deteriorate maintenance effectiveness, which in turn increases warranty servicing cost. Therefore, it is necessary to explore the impacts of customer unpunctuality on the optimization of PM policy and the resultant warranty expenses. This paper aims to serve this necessity, with specific focus on automobiles protected by a two-dimensional (2-D) warranty.
The literature on warranty-oriented PM optimization generally deals with optimal trade-off between the investment in a PM program and the resultant reduction in warranty servicing costs. This topic has received considerable attentions over the last decades; see Shafiee and Chukova (2013) for a comprehensive literature review. Since this paper is interested in optimal unpunctual PM policy under 2D warranty, we confine our attention to the interplay between 2D warranties and PM policies.
A 2D warranty simultaneously employs two variables, usually age and usage, to characterize the warranty policy (Xie, Shen, Zhong, 2017, Ye, Murthy, 2016). A typical example of 2D warranties is automobile warranties, with usage being mileage travelled. In the 2D warranty context, the optimization of imperfect PM policies has been gaining in popularity. Huang and Yen (2009) and Huang, Chen, and Ho (2013) sought to determine optimal 2D warranty terms with consideration of periodic and non-periodic (reliability-based) PM strategies, respectively. Wang and Su (2016) and Su and Wang (2016) proposed a 2D PM policy for items sold with a 2D warranty, under which items should be preventively maintained every K units of age or every L units of usage, whichever comes first. Iskandar and Husniah (2017) studied a 2D lease contract for repairable equipment subject to a periodic imperfect PM policy. Wang, Zhou, and Peng (2017a) developed a game-theoretical decision model for a 2D warranty policy with a periodic PM policy, where customers are entitled to either accept or reject each PM during extended warranty period. Huang, Huang, and Ho (2017) studied a new 2D extended warranty policy in which customized PM schedules are offered to different customer categories. Wang, Liu, Liu, and Li (2017b) examined the worthiness of pre-sale upgrade and post-sale PM policies for second-hand products sold with a 2D warranty. Wang, Li, and Xie (2019) investigated optimal upgrade and PM strategies for industrial equipment under successive usage-based lease contracts with a 2D warranty period. Li, Liu, Wang, and Li (2019) studied optimal burn-in and PM strategies for repairable products sold with a 2D base warranty and an optional extended warranty. Recently, Wang and Xie (2018) reviewed the state of the art on integrated studies of 2D warranties and PM policies.
Nevertheless, the existing studies on warranty-oriented PM optimization implicitly assume that scheduled PM activities during the warranty period are punctually executed. As stated earlier, however, maintenance crews/customers may be unpunctual, i.e., the PM activities may be performed either earlier or later than intended. To the best of our knowledge, there are currently three reported publications related to the optimal planning of unpunctual PM/inspection policies. He et al. (2017) studied optimal age replacement policies with and without minimal repair in anticipation of maintenance unpunctuality. They assumed that the potential unpunctuality of maintenance crew causes actual replacement instants to deviate from planned instants in a probabilistic manner. Zhao, Al-Khalifa, Hamouda, and Nakagawa (2017) also discussed a random age replacement policy in which an item is replaced before failure at a random point of time. In a different problem setting, Scarf, Cavalcante, and Lopes (2019) investigated an inspection-replacement strategy for a critical system (whose failures can be immediately revealed) in which inspection opportunities arise at random.
Both He et al. (2017) and Zhao et al. (2017) focus on optimal preventive replacement problems over an infinite planning horizon, where optimal replacement thresholds are determined by minimizing long-run expected cost rates. Similarly, Scarf et al. (2019) deal with the long-run cost rate induced by inspections and subsequent replacements. The long-run cost rate is formulated as the ratio of expected maintenance cost per renewal cycle to expected renewal cycle length, according to the renewal reward theorem. In real applications, however, maintenance planning horizons are mostly finite, e.g., the limited warranty period for Ford vehicles is 3 years or 36000 miles, whichever comes first. In this scenario, the optimization problem is then to identify an optimal number of PM activities to minimize the expected total maintenance cost over the finite horizon of interest (e.g., warranty period). On the other hand, a PM activity is usually imperfect in the sense that the item’s reliability status, after PM, is improved but not as good as new. Our work distinguishes itself from He et al. (2017), Zhao et al. (2017), and Scarf et al. (2019) by well incorporating the two aspects—finite planning horizon and imperfect PM effect.
This paper makes an early attempt to study an unpunctual imperfect PM policy for repairable items sold with a 2D warranty. In our setting, the potential unpunctuality of customers returning their items for PM is what causes actual PM instants differing from scheduled instants. The unpunctual PM policy is developed upon the 2D PM policy in Wang and Su (2016) and Su and Wang (2016). The unique characteristic of this policy is that it allows customers to slightly advance or postpone planned (originally periodical) PM interventions in a tolerated range. Specifically, under the unpunctual policy, the jth PM action should be performed at age jK ± ΔK or at usage jL ± ΔL, whichever occurs first. In particular, K and L are scheduled age- and usage-based PM intervals, and ± ΔK and ± ΔL are tolerated unpunctual ranges in age and usage dimensions, respectively. The unpunctual policy gives customers a flexibility, to some extent, of planning PM activities according to their personal schedules, instead of following the manufacturer-prescribed PM schedule strictly.
Since the proposed unpunctual PM policy tolerates a degree of customer unpunctuality in PM execution, two natural questions arise: What are the impacts of customer unpunctuality on the optimization of PM policy and the resulting warranty cost? How to design (or modify) a new (or an existing) PM policy in anticipation of customer unpunctuality? The primary purpose of this paper is to answer these questions. To this end, we first identify the optimal PM policy (i.e., number and degree of PM activities) and the corresponding warranty cost that anticipate customer unpunctuality, and then compare them with those of the punctual policy. The remainder of the article is organized as follows. Section 2 develops two optimization models to determine optimal PM decisions for both punctual and unpunctual PM policies. Essentially, the unpunctual model extends its punctual counterpart by assuming that customer unpunctuality is governed by a specific probability distribution. Section 3 deals with the optimization and comparison of the two PM policies. Because it is difficult to carry out the analysis in a general setting, we first consider two special forms—linear and quadratic—of the failure rate function, and then extend the analysis to a general failure rate form. The corresponding results under two possible unpunctuality distributions—uniform and triangular—are also discussed and compared. Section 4 presents numerical studies to demonstrate the unpunctual PM policy and then answer the two questions raised above. Finally, Section 5 concludes this paper and suggests some future research topics.
Section snippets
Two-dimensional warranty and failure modeling
Consider that a repairable item is sold with a 2D free repair warranty policy which is characterized by a rectangular region as shown in Fig. 1. Under this policy, any eligible failures within the warranty period will be rectified by the manufacturer at no cost to the customer. The warranty contract terminates when either the item age reaches its limit W or the cumulative usage exceeds its limit U, whichever comes first. That is, if the usage rate is low (
Model analysis
In this section, the effects of customer unpunctuality on the optimal PM policy and the total warranty cost are explored by comparing optimal punctual and unpunctual policies. It is well known that the Weibull failure rate in Eq. (2) bears different shapes according to the value of shape parameter β. For the sake of mathematical tractability, we first consider two special cases in which and respectively. The two cases represent linear and quadratic approximations of a general failure
Numerical studies
In this section, numerical studies are presented to demonstrate the unpunctual PM policy. Section 4.1 describes parameter settings used in the studies. Sections 4.2 and 4.3 then present numerical results for the punctual and unpunctual PM policies, respectively. Finally, Section 4.4 discusses observations from the numerical studies and their managerial implications, and also answers the two questions raised earlier.
Concluding remarks
This paper investigates an unpunctual PM policy for repairable items sold with a 2D warranty, under which the jth PM activity should be performed at age jK ± ΔK or at usage jL ± ΔL, whichever occurs first. This policy offers customers a large degree of flexibility and convenience by tolerating their unpunctual PM behaviors to some extent. The expected total warranty costs for the unpunctual PM policy and its punctual counterpart are formulated under the assumption that customer unpunctuality is
Acknowledgements
This work was supported by the National Natural Science Foundation of China (grant numbers 71601166, 71532008) and also by the Research Grants Council of Hong Kong under Theme-based Research Fund (grant number T32-101/15-R) and General Research Fund (grant number CityU 11203815, CityU 11203519). The authors are indebted to the Editor and the four anonymous referees for their constructive comments and suggestions, which led to significant improvements of this paper.
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A preliminary version of this paper, Wang, Li, and Xie (2018), has appeared in the Proceedings of the 10th IMA International Conference on Modelling in Industrial Maintenance and Reliability, 2018.