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Weekly Updates1. Introduction
Suppose a product deteriorates both with time and with usage. For an elapsed time t, let T(t) denote the cumulative operating time, and X(t) denote usage degree, where the cumulative operating time is a measure of the total time that the product has actually been operated, and the usage is another measure of the operating time. For example, assume that a person owns an automobile for 2 years, with a cumulative driving time of 800 hours, and an accumulated driving mileage of 18000 miles. In this example, t is 2 years, T(2) is 0.0048 years, and X(2) is 18000 miles. These two time-dependent measures would be the metrics utilized in a two-dimensional warranty problem. An appropriate warranty policy with warranty limits along the two dimensions needs to be derived to maximize the manufacturer's overall profits. A two-dimensional warranty is often represented by a rectangle with age on the horizontal axis and usage on the vertical axis, and we use this construct here. Maintenance and replacement policies can be represented in this two-dimensional space by defining the age-usage regions in which each policy should be used.
The attractiveness of a product is not solely based on its tangible aspects such as price, functionality and aesthetics, but also on its intangible aspects such as company reputation and customer service. In fact, physical features are not always the primary factors that differentiate products. Instead, after-sales service and maintenance play important roles in both safeguarding the rights and interests of consumers and promoting the sales and reputation of product providers. In particular, a good service and maintenance warranty policy signals an image of high-quality products and is thus a powerful marketing weapon. Some manufacturers even provide competitive customer service programs by amalgamating preventive maintenance with their warranty policies. However, maintaining an attractive warranty policy is typically costly, especially for products that deteriorate quickly. The trade-off between high warranty costs and an increased market share is of special importance for managers when making marketing decisions.
Products may be repaired rather than replaced after breakdowns, since they can be restored to fully operate the required functions by methods other than replacing the entire product. However, the successive times between breakdowns are not necessarily identically distributed, as in renewal processes for products that cannot be repaired. More generally, the successive times between breakdowns would become smaller which is an indication of deterioration. In order to model deterioration. Non-homogeneous Poisson Processes (NHPPs) were introduced to make the modeling of time-dependent behavior of repairable products more tractable (Crow, 1974; Thompson, 1981; Hartler, 1989; Rigdon and Basu, 2000). In most models in which NHPPs are used to model the deterioration process, it is assumed: (i) that repair times can be neglected; (ii) that repairs take place instantaneously after failure; and (iii) that the repaired product behaves like a product of the same age that has experienced no failure (i.e., minimal repair). For example, Cox and Lewis (1966) employed a NHPP to model air conditioning equipment in the airplane industry, Ascher and Feingold (1984) stated that a NHPP is plausible in practice for modeling the breakdown process of a repairable system, Deshpande and Singh (1995) considered a replacement policy for repairable systems with minimal repairs by using a NHPP, Huang (2001) employed the Bayesian approach and a NHPP to develop a decision process to determine whether an overhaul decision is optimal for a deteriorating repairable system, and Huang and Zhuo (2004) used a similar procedure to obtain an optimal warranty policy for deteriorating products. Therefore, the use of NHPPs in describing the breakdown process of a deteriorating product is reasonable and justified in this study.
Warranties and Preventive Maintenance (PM) for deteriorating products have been intensively studied for decades. Polatoglu and Sahin (1998) studied two warranty models that consider customers' repurchasing behaviors: Free Replacement Warranty (FRW) and pro-rata warranty. El-Ferik and Ben-Daya (2006) developed a hybrid age-based model for deteriorating equipment with an imperfect PM program, considering both hazard rate and effective age to determine the number of PM actions and the length of PM intervals that minimize the total long-term expected cost per unit time. Maillart (2006) investigated joint optimization of PM and condition monitoring by using Markov decision processes, allowing the decision maker to examine wear-related variables to assess the deterioration of the system. Jung et al. (2000) proposed a feasible PM policy which starts after the warranty term has expired, and derived the optimal number of PM actions and their durations that minimize the long-term maintenance cost per unit time. Kim et al. (2004) used the concept of virtual age, where the effective age is younger than the actual age after performing a PM action, to investigate the expected warranty costs for suppliers and buyers when the entire PM cost is paid for by the buyer.
However, considering the single effect of either time or age in dealing with the problem of product deterioration for reliability analysis may not be satisfactory because usage is often another essential factor that accounts for deterioration. Blischke and Murthy (1992) noticed this crucial issue and separated warranty policies into two categories, one-dimensional and two-dimensional policies, respectively. Recent years have seen significantly increased attention to two-dimensional warranties in the field of reliability analysis. Murthy et al. (1995) analyzed the manufacturer's expected costs by using the two-dimensional structure of product age and usage degree under the FRW policy. Baik et al. (2003) focused on selecting the optimal repair or replacement policy to minimize the manufacturer's service costs for a two-dimensional warranty problem. Instead of using the conventional two-dimensional structure of time and usage, Majeske (2003) proposed a two-dimensional framework with quality and reliability being the two different dimensions based on an analysis of warranty data from the automobile industry. Iskandar et al. (2005) derived the structure of the optimal repair-and-replacement strategy for items sold with a two-dimensional warranty with pre-specified age and usage limits. Failures are minimally repaired if the age and usage are small, and as the age and/or usage approach the warranty limits. In between, it is optimal to replace, not repair.
The complexity of two-dimensional warranty problems has hindered researchers from further investigating these problems. For this reason, some research has turned to a more restrictive direction in which the two dimensions may have some possible relationship. For example. Gertsbahk and Kordonsky (1998) suggested a method to calculate warranty limits for a two-dimensional warranty by using a linear combination of the scales of the two dimensions, and Iskandar and Murthy (2003) assumed that the two effects of age and usage on deterioration are linearly related and derived the optimal warranty limits for deteriorating products.
Despite the importance of PM and the two-dimensional structure for deteriorating products, integrated discussion of both concerns has received relatively little attention in the literature. A two-dimensional warranty policy that takes both time and usage into account would be more realistic, and its flexibility would allow manufacturers to increase profits by attracting customers with different usage modes (e.g., high usage within a short period or low usage within a longer period). Furthermore, the virtual age (or equivalently, effective age after accounting for the beneficial effects of PM; Kim et al. (2004)) can be used to characterize product reliability when determining a suitable optimal warranty policy. Therefore, in this paper, we seek to optimize a warranty policy with the goal of maximizing the manufacturers' profits by integrating the concept of a two-dimensional warranty and the consideration of PM under assumptions similar to those in the model of Iskandar and Murthy (2003). In particular, they assume that the measures of age and usage are linearly related through a multiplicative factor whose distribution can be estimated. Our proposed approach differs from other methods in two respects: (i) PM is included in the model development; and (ii) instead of having a predetermined warranty term, the method provides manufacturers with guidelines on how to offer customers two-dimensional warranty programs with proper time and usage limits.
This paper is organized as follows: Section 2 explains the concept of a two-dimensional warranty in more detail; Section 3 develops the proposed two-dimensional warranty model with consideration of PM; Section 4 demonstrates the usefulness of the proposed model by testing it on a practical numerical case, with sensitivity analyses carried out to investigate important factors that may affect the optimal decisions; and Section 5 contains the concluding remarks.
The following notation is used throughout this paper:
T(t) = the cumulative operating time at time t;
X(t) = the cumulative usage (according to a metric other than operating time) at time t;
[OMEGA] = the selected decision [OMEGA] = {K, L};
K = the age limit under the warranty;
L = the usage limit under the warranty;
r = the ratio of usage to operating time; i.e., L/K r = L/K;
[r.sub.u] = the upper bound on r for the uniform distribution;
[r.sub.1] = the lower bound on r for the uniform distribution;
[alpha] = the shape factor for the gamma distribution;
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