1 Introduction
Warranty analysis of a manufactured product is an integral part of statistical quality improvement. Improper warranty analysis may affect the business goal and the perceived quality can be in turmoil. Warranty is defined (see Blischke et al., 2011) as a contractual agreement between manufacturer (or seller) and consumer (or buyer) that is entered into upon sale of a product. This contract defines the compensation available to the buyer if the performance of the product is found to be unsatisfactory. Therefore, by providing warranties, manufacturer ensures the consumer about the product reliability. Hence, a longer warranty period usually ensures the consumer higher reliability of the product. However, if the product reliability is low, but manufacturer offers an unrealistically large warranty period, then it may incur high penalty cost to the manufacturer. Also, if the warranty period is smaller in comparison with the other competitors in the market, then sales volume of the product may decrease. Therefore, finding an appropriate warranty period is an important task for the manufacturer. The usual way to find the warranty length is based on the assessment of product reliability. The reliability assessment is typically done through a lifetesting experiment. In practice, lifetests are conducted under various censoring schemes in order to save time and cost of the experimentation.
In this article, we consider TypeII unified hybrid censoring scheme, abbreviated as TypeII UHCS, (see Balakrishnan et al., 2008) which is the generalization of the generalized TypeI and TypeII hybrid censoring schemes (see Chandrasekar et al., 2004). The TypeII UHCS can be described as follows. The testing starts with units and alongside two integers and two time points are chosen such that and . If the th failure occurs before time , terminate the test at . If the th failure occurs before and th failure occurs between and , terminate the test at th failure time. If the th failure occurs before and th failure occurs after , terminate the test at . If the th failure occurs after and th failure occurs before , terminate the experiment at th failure time. If the th failure occurs after and th failure occurs after , terminate the test at . Finally, if the th failure occurs after time , terminate the experiment at th failure time. The advantage of TypeII UHCS is that it ensures at least failures and the maximum test duration is . A schematic representation of TypeII UHCS is presented in Figure 1.
The most commonly used warranty policies are free replacement warranty (FRW) policy, prorata warranty (PRW) policy and combined FRW/PRW policy (see Murthy and Blischke, 2006; Blischke et al., 2011). An important feature of a warranty policy is that if the product fails during the warranty period, consumer will get full or prorated compensation from the manufacturer. Under the FRW policy, if the product fails during the warranty period, a nonrepairable product is replaced by an identical one free of charge. In case of repairable product, the manufacturer will repair the product free of cost. On the other hand, if the product fails under PRW policy, the manufacturer will provide a prorated compensation to the consumer. Sometimes, a combination of both the policies are also considered which is termed as FRWPRW policy. We consider determination of warranty length for a combined FRW/PRW policy based on data observed under TypeII UHCS. Although there are many works on determination of warranty length for different policies based on complete data (see Menezes and Currim, 1992; DeCroix, 1999; Wu et al., 2006), there are few works under censored data. GutiérrezPulido et al. (2006) determined Bayesian optimal warranty length under prorata warranty policy where they considered twoparameter Weibull distribution as product lifetime. Wu and Huang (2010) investigated a decision problem under combined FRWPRW policy. They used a Bayesian approach to determine the optimal warranty lengths under TypeII progressive censoring scheme for a Rayleigh distribution. Chakrabarty et al. (2019) investigated optimal reliability acceptance sampling plans under TypeI hybrid censoring schemes by taking warranty cost as constraint. Budhiraja and Pradhan (2019) developed optimal reliability acceptance sampling plans under progressive TypeI interval censoring with random removal using a cost model which consists of warranty cost as a component. The aim of this article is twofold. First, a generalized censoring scheme is used to develop the proposed methodologies, therefore, it can be easily extended to the other censoring schemes which are the special cases of TypeII UHCS. Second, this article proposes a nonlinear prorated rebate cost and compared it with linear prorated rebate cost proposed by Wu and Huang (2010). It has been shown that the proposed nonlinear rebate function gives a larger warranty period with maximum profit in comparison with the linear rebate function.
The organization of the paper is as follows. In Section 2, lifetime model and posterior predictive distribution based on the data obtained through TypeII UHCS are discussed. In Section 3, we have derived various nonlinear cost functions such as rebate function, economic benefit function, warranty cost function, dissatisfaction cost function. Using the cost functions, an utility function is constructed in this section which is maximized to compute optimal warranty length. The computational methodology to obtain optimal warranty lengths is discussed in Section 4. A reallife data are analyzed to illustrate the proposed method in Section 5 and, finally, some concluding remarks are made in Section 6.
2 Lifetime model and posterior distribution
Suppose that are the lifetimes of testing units which follow a lognormal distribution LN
. The probability density function (PDF) and the cumulative distribution function (CDF) of LN
are given by(1) 
and
respectively, where and denote unknown parameters of the distribution. Here, is the CDF of standard normal distribution. The lognormal distribution is quite popular in reliability studies because of the flexibility of its shape (see Johnson et al., 1994). Suppose represent corresponding ordered lifetimes. Let and represent the number of failures and the duration of the lifetesting, respectively, under a TypeII UHCS. Therefore, represents a TypeII UHCS data defined as
Based on the data obtained by a TypeII UHCS, the likelihood function is given by
(2) 
where, , and are the observed values of , and , respectively. It is assumed that the joint prior distribution of
follows a normalgamma distribution with probability density function
(3)  
where, and . The hyper parameters and reflect prior knowledge about unknown parameters of interest, where and . The posterior distribution is given as
The posterior predictive distribution, which represents the current beliefs of the decision taker about the failure time, is given as
Now we are presenting the expression of the Fisher information about which will be used in the next section for applying MetropolisHastings (MH) algorithm to compute optimal warranty length. The Fisher information under TypeII UHCS is given by (see Sen et al., 2020)
where , , and represent the Fisher information about under TypeI censoring, TypeII censoring and TypeI hybrid censoring schemes, respectively. The expressions of each of them are given as follows
(4)  
(5)  
(6) 
where is the hazard function of , is the density of ,
is the vector
and is defined as the matrix for . It may be noted that the expressions of the Fisher information about under TypeI hybrid censored data with schemes and are similar as presented in equation 6.3 Cost functions
FRWPRW policy can be viewed as choosing two positive time points and such that , in which FRW policy is applicable in period and PRW policy is applicable in period . Different choices of and raise to FRW or PRW policies as subcase of FRWPRW policy. For instance, when , it reduces to FRW policy and, when , it reduces to PRW policy. Wu and Huang (2010) considered a linear prorata rebate function which is the function of the remaining time of the warranty period. Assuming be the sales price of a certain product, Wu and Huang (2010) defined the cost of reimbursing an item, which is linear in nature under FRWPRW policy, as
A pictorial diagram of reimbursing an item under FRWPRW policy can be visualized in Figure 2.
In this article, we propose a nonlinear rebate function under FRWPRW policy, which is defined as
where the parameter () controls the nonlinearity of the prorated rebate function. It is noted that if increases, then the prorated rebate function looks like linear. A graphical representation of this can be seen in Figure 3. Interpretation of Figure 3 is described in Section 5. To compute the optimal warranty length of the products, it is required to construct an utility function, which will be optimized. In this article, three cost functions such as economic benefit function, warranty cost function and dissatisfaction cost function are considered which were proposed by GutiérrezPulido et al. (2006). In the subsequent sections, we have discussed those cost functions.
3.1 Economic benefit function
By providing a suitable warranty period, the manufacturer may expect an increase in sales volume of the products which results in economic benefit. Therefore, the economic benefit function is considered as the monotone increasing function of the average of twostage warranty lengths. But, it may be noted that if the manufacturer gives a larger warranty period than their competitor, then the consumer might consolidate certain doubts about the product. As a consequence of that, considering a bounded benefit function should be a realistic choice for the manufacturer. Therefore, an economic benefit function, denoted as , is considered here as (see Wu and Huang, 2010)
where is the manufacturers profit for one product, is the potential number of products to be sold with this warranty policy and is the parameter to control the speed of increment in benefit. can be uniquely determined from the ratio of two special quantities in the combined FRW/PRW policy. Assuming is the standard market warranty under FRW policy, let us consider the following ratio
The ratio indicates whether the percentage of benefit increases if the manufacturer changes warranty policy from FRW to PRW. Note that cease to zero when any of the warranty lengths goes to infinity. It interprets the fact that it is unrealistic to expect economic benefit if the warranty length is too long. Let . Then, is a strictly monotone increasing function, which can take any value between 0.5 and 1. Note that and . Therefore, can be determined uniquely by solving the equation for given , where .
3.2 Warranty cost function
Warranty cost is the direct cost to the manufacturer for reimbursing the products which fail during warranty period. Let us denote warranty cost by . It is defined as
where is defined in Section 4.1. To find the expected number of failures during the warranty period, we consider the probabilities of failure before time period , between time period and and after time period . Thus, the warranty cost function can be formulated as
where denotes the indicator function and represents the posterior predictive cumulative distribution function.
3.3 Dissatisfaction cost function
We consider another cost function which is the manufacturer’s indirect cost to the product. This is called as dissatisfaction cost or penalty cost. Typically, the consumers have certain expectation about the product lifetime. Suppose that the consumer’s expected lifetime of the product is , which can be considered as the mean, median or percentile of the posterior predictive distribution. Now, if the product fails during the twostage warranty period or if it fails immediately after the expiration of combined warranty period, then the consumers have certain dissatisfaction about the product. This can indirectly affect on company’s reputation to the buyer and also it can reduce the future sale volumes of the product. Therefore, we split the total dissatisfaction cost into three time intervals as follows.
 Case I:

Item fails in the time period ,
 Case II:

Item fails in the time period ,
 Case III:

Item fails in the time period .
Note that we assume the consumer’s expected lifetime greater than second stage warranty period . Customers often seek for higher reliable products and, thus, this assumption is quite practical in nature.
The dissatisfaction cost in Case I (that is, the product fails in the FRW policy) is defined as
Now let us formulate the dissatisfaction cost in case II (that is, the product fails in the PRW policy). We propose a dissatisfaction cost, which is decreased nonlinearly with the remaining time of the PRW period. At , per unit cost of dissatisfaction is and, at , per unit cost of dissatisfaction is where with . Therefore, the dissatisfaction cost in Case II is defined as
Finally, suppose that the product fails during immediate expiration of warranty and before the consumer’s expected lifetime of the product. Therefore, being an unsatisfied customer, it incurred some dissatisfaction cost. We propose that the dissatisfaction cost decreases nonlinearly with remaining time of consumer’s expected lifetime of the product and the dissatisfaction reaches to zero at . Hence, the dissatisfaction cost in Case III is defined as
Therefore, the total dissatisfaction cost is given by summing the above three costs and defined as
A pictorial diagram of the dissatisfaction cost structure can be visualized in Figure 4.
3.4 Utility function
Utilizing the three proposed cost functions in Sections 4.2, 4.3 and 4.4, we define an utility function as (see GutiérrezPulido et al., 2006)
It is note that the time to failure is the random quantity in utility function. Therefore, the expected value of the utility function is given by
(7)  
where
4 Optimal warranty length
In this section, we proposed a method to compute optimal warranty length. In order to compute the optimal warranty length, that is, the optimal values of and , the expected utility function given in equation 7 is maximized with respect to and . Therefore, the optimal warranty is the solution of the following optimization problem
(8) 
where is the set of all positive real numbers. In general, the optimization problem in (8
) does not have a closed form analytical solution. Nevertheless, in this article, we propose to use MetropolisHasting (MH) algorithm to compute the Bayes estimates of (
7) and, then, using that estimate in (8), we compute the optimal solution. The procedure of MH algorithm suggests that the samples from a posterior distribution can be generated using some proposal density. Commonly, a symmetric type proposal density such as can be taken into consideration. Here, we consider a bivariate normal , proposal density where denotes the inverse of the information matrix. Since we are generating samples from a bivariate normal distribution, few negative observations for may appear, which is not acceptable. In this regard, we propose the following steps of the MH algorithm to draw samples from the corresponding posterior density. Step 1:

Set initial value of as
 Step 2:

For repeat the following steps

[label=()]

Set

Generate a new candidate parameter value from

Set

Calculate

Update with probability ; otherwise set

The above procedure will generate observations of . Some initial observations of size , say, are discarded as burnin observations and the remaining observations can be used to compute the Bayes estimate of in (7). Subsequently, the corresponding Bayes estimate can be computed as
Finally, the optimal warranty period in (8) is computed by solving the optimization problem
This is a nonlinear optimization problem with two real continuous decision variables. NewtonRaphson method can be used to solve this problem.
5 Numerical illustration with reallife data analysis
In this section, a reallife data set is considered for illustrative purpose. The data set is taken from Proschan (1963). The data represent the intervals of successive failure times in hours of the air conditioning system of Boeing 7912 jet airplane. The corresponding ordered times are listed as 1, 3, 5, 7, 11, 11, 11, 12, 14, 14, 14, 16, 16, 20, 21, 23, 42, 47, 52, 62, 71, 71, 87, 90, 95, 120, 120, 225, 246, 261. GutiérrezPulido et al. (2005) fitted the data with lognormal distribution and calculated the hyper parameter values as and . By using the censoring scheme , we have generated TypeII UHCS data as 1, 3, 5, 7, 11, 11, 11, 12, 14, 14, 14, 16, 16, 20, 21, 23, 42, 47, 52, 62, 71, 71, 87, 90, 95. Suppose that the sales price of this product is and the production cost of the product is . Therefore, the profit per unit product is . The manufacturer gives a standard warranty, which is the
th quantile of the posterior predictive distribution under the FRW policy i.e. standard warranty is
hours. Since, we are considering combined FRW/PRW policy, manufacturer is interested to change the warranty policy from FRW to PRW and assumed that the percentage of benefit remains to be i.e., . In this case, the unique solution to the equation is . Since the customer dissatisfaction indirectly affect on the image of the company, so, the product sales may be reduced. Therefore, we assume that proportions of customer dissatisfaction are = (0.09, 0.04). The consumer will satisfy with the product if its lifetime reaches customer’s expectation over the product lifetime. Here, we assume customer’s expected lifetime of the product is , which is the median of the posterior predictive distribution. We observed that for linear rebate function, the optimal warranty length under combined FRW/PRW policy is and the maximum value is $ 175.281M. Using the proposed nonlinear rebate function defined in Section 4.1, the optimal warranty lengths under the combined FRW/PRW policy are computed as
For , and optimal value is $ 183.622M,

For, , and optimal value is $ 180.159M

For , and optimal value is $ 178.761M,

For , and optimal value is $ 177.013M,

For , and optimal value is $ 175.679M,

For , and optimal value is $ 175.384M.
It is observed that the optimal warranty lengths under nonlinear rebate function are wider than that of linear rebate function. Also, optimal warranty length and optimal value both decrease with increasing . For and , the corresponding nonlinear rebate functions are plotted in Figure 3. From Figure 3, it is observed that when increases, the nonlinear prorated rebate function looks like a linear prorated rebate function.
6 Conclusion
In our study, We have considered lognormal as the lifetime distribution of the product, however, the proposed methodologies can be extended easily to other lifetime distributions. This article also proposes a nonlinear prorated rebate cost and compared it with linear prorated rebate cost proposed by Wu and Huang (2010). For nonlinear prorated rebate cost, a larger warranty period with maximum profit is obtained in comparison with the linear rebate cost. This is the prime advantage of choosing a nonlinear prorated rebate cost function.
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