Volume 20 Issue 5 - November 4, 2011 PDF
A Study of Two-Dimensional Warranty Policies with Preventive Maintenance
Yeu-Shiang Huang* and Chia Yen
Department of Industrial Management Science, College of Management, National Cheng Kung University
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Product deterioration is a major issue of concern in the context of reliability analysis. In dealing with the problem of product deterioration for reliability analysis, it may not be satisfactory to solely consider the effects of time or age. In fact, usage would be another essential factor that accounts for the deterioration. A two-dimensional warranty with considerations for both time and usage for deteriorating products would be more practically accepted and merits both manufacturers and customers. In this paper, the concept of two-dimensional warranty was integrated with the deliberation of preventive maintenance in order to determine the optimal warranty policy which maximizes the manufacturers’ profits. This proposed approach provides manufacturers with guidelines with which to offer customers alternative warranty choices with either a limited time period or a limited usage degree.

Suppose that a nonhomogeneous Poisson process (NHPP) can be used to model the deterioration process of the product, where the deterioration depends on both T(t) and X(t), where T(t) denotes the cumulative operating time and X(t) denotes usage. However, for simplicity, we assume that the product age is equivalent to its cumulative operating time, i.e., T(t)=t. Here the concepts of minimal repair, which indicates that the product is as good as old after a repair, and negligible repair time, which denotes that all breakdowns can be repaired in a relatively short period of time, are assumed in order to have a deteriorating NHPP. If r is defined as the ratio of cumulative usage to time, i.e., , then the intensity function of the NHPP can be represented by λ(t|r)=φ(X(t),t). An appropriate warranty policy with warranty limits K and L along the two dimensions T(t) and t, respectively, needs to be derived to maximize the manufacturer’s overall profits. Let Ω be the set of such policies, i.e., Ω={K,L}. When the product reaches a total age K or usage L, the warranty terminates. Figure 1 shows a two-dimensional warranty policy.
Figure 1 Two-Dimensional Warranty Policy

As can be seen in Figure 1, the warranty terminates when either the age limit K or usage limit L has been reached. There are two possible conditions for warranty termination: (1) when 0<r , warranty terminates at time K; (2) when <r<∞, the warranty terminates at time S which corresponds to the time when the usage limit L is reached. Note that denotes the average ratio of usage to age, which can be obtained from past experience and warranty records. Suppose that, for the literature, we have λ(t|r)θ01r2T(t)+θ3X(t), where θ0, θ1, θ2, and θ3 are deterministic coefficients. Accordingly, when 0<r, the warranty terminates at time K, and the expected warranty cost is given by:

where Cr denotes the repair cost when a failure occurs, and is the expected number of failures within the time period [0, K] assuming a NHPP. When <r<∞, the warranty terminates at time S, S=, and the expected warranty cost is given by:

where is the expected number of failures within the time period [0, S].

For a policy Ω={K,L}, the total expected warranty cost is the sum of the above two costs from equations (2.4) and (2.6), i.e., the cost for the all possible cases of 0<r<∞, which is given by C(Ω)=ECr1+ECr2. However, r differs for each individual customer, and is usually treated as a random variable that has a general distribution G(r), and can be determined by investigating the customer’s warranty records. Therefore, the total expected warranty cost is given by .

Therefore, with appropriate assessments of recovery status of a PM activity and price and warranty elasticities, the optimal warranty policy can be obtained by considering the potential revenue as well as the expected cost derived previously.

The automobile industry has been a popular testing ground for two-dimensional warranty policies. Suppose that the price elasticity and the warranty elasticity are found to be η=-2.2 and ζ=1.85, respectively, from market research conducted by an automobile manufacturer. We assume further that the annual sales of cars is expected to be q=2,000,000. Since the demand is a function of the warranty policy, the problem of interest for the manufacturer would be to find an optimal two-dimensional warranty policy that can maximize the manufacturer’s profit under such an expected sales volume. In our example, we set ru=2.24 and rl=0.16 when r is uniformly distributed to give a uniform distribution with mean and variance being 1.2 and 0.36, respectively. This parameter setting would be realistic in practice, since in general, the annual millage driven by drivers with normal usage condition would range from 8,000 to 15,000 miles per year.  In addition, when r is gamma or lognormally distributed, we let α=4.00, β=3.33, μ=0.07, and σ=0.47 to ensure that these two cases have the same mean values and variances as those in the uniform case. The failure intensity λ(t|r)=0.1+0.2r+(0.3+0.3r)t was adopted, and the time reduction factor, δ(m)=(1+m)e-m, is 0.66 at our selected level of PM m=1.2. In addition, the periodic PM interval is assumed to be three months, i.e., τjj-1=0.25. In order to derive the optimal two-dimensional warranty policy and obtain an estimate of the corresponding expected annual profit, the fixed cost, repair cost, and production cost were set as F=150,000,000; Cr=140; and Cm=15,000; respectively. Table 1 shows the results for the uniform, gamma, and lognormal cases.
Table 1 Results for Uniform and Gamma Cases

Note that the numerical optimal solutions for the warranty limits for the uniform, gamma, and lognormal cases are (7.04 years or 84,483 miles), (2.46 years or 29,525 miles), and (3.63 years or 43,567 miles), respectively. However, to conform to the standards of the automobile industry, the values of the optimal warranty policy for the two dimensions are set to multiples of half a year and 5,000 miles, respectively. In addition, the optimal prices and the annual profits for the uniform, gamma, and lognormal cases can be obtained as ($23,231, $15,337,528,768), ($23,201, $15,317,312,945), and ($23,241, $15,344,272,703), respectively. Since they are fairly similar, the managers may set the price to $23,000 and expect to have an approximate annual profit of $15,000,000,000.

As shown in Table 1, the optimal policies that maximize the manufacturer’s profits are significantly different for the uniform case and the other two distributions, yet the unit price and total profits are similar for all three. This can be explained by the fact that the gamma and the lognormal distributions are more skewed to the right than the uniform distribution is. Figure 2 shows the probability density functions of uniform (2.24,0.16), gamma (4.00,3.33), and lognormal (0.07,0.47) for varying values of r.
Figure 2 PDFs of Uniform (2.24,0.16), Gamma (4.00,3.33), and Lognormal (0.07,0.47)

As can be seen in Figure 2, the long right tails of the gamma and the lognormal distributions show that about one-fifth of the customers have an r ratio greater than the upper limit of the uniform distribution. High-mileage drivers substantially elevate the warranty cost because they increase the number of cars that require maintenance within the warranty term. Therefore, to remain profitable, manufacturers have to offer more restrictive warranty policies if they believe the r values of their customers follow either a gamma or lognormal distribution. As for the difference between the gamma and the lognormal distributions, it can be observed that the lognormal distribution has more mass between the upper and lower limits of the uniform distribution. This means that the warranty policy derived from a group of customers with a lognormal distribution will be slightly more attractive than that derived from those with a gamma distribution.

This paper deals with a two-dimensional warranty problem by considering both the age and usage of deteriorating products. The successive breakdown process of the deteriorating product is assumed to behave as NHPP, and PM is included when determining the optimal warranty policy that maximizes the manufacturer’s profits. The proposed approach provides manufacturers with guidelines on how to offer customers two-dimensional warranty programs with proper time and usage limits.
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