1. INTRODUCTION
Nowadays, warranty and its related subjects have become an increasingly important part of aftersale service in the marketing of many products ranging from consumer durables to industrial and commercial products. When a consumer purchases a product, the manufacturer assures the satisfactory performance of the product under predetermined conditions by providing the base warranty (BW) at no cost to the consumer. In contrast to the base warranty, an extended warranty (EW) provides an opportunity for consumers to extend their warranty coverage beyond the base warranty period by purchasing extended warranty contracts. From the consumers’ point of view, extended warranty provides “peace of mind” by protecting them against the product’s failures and its increasing costs. EW providers have also found offering the EW contracts as a good source of revenue. For example, Apple and Dell reported the revenue of offering EW contracts as 43 and 26 percent of their net income, respectively (Extended Warranty Profits, 2004). The EW provider in the automobile industry also may reach profits up to 100 percent by offering EW contracts to consumers (Moore, 2002).
The possibility of offering EW contracts for a wide range of products, besides its considerable profit margins, has made the EW market interesting and attracted many investors’ attention. The academic works on EW contracts have also received increasing attention, especially over the last two decades. Understanding the consumers’ perceptions about EW contracts, studying the effects of consumers’ attitudes on the demand for EW contracts, as well as analyzing other marketingrelated factors have constituted a considerable volume of EW research subject (see for example (Moore, 2002; Stafford, 2006; Albaum and Wiley, 2001; Loveland, 2010). However, the research on EW contracts has not been limited to the qualitative and market study research and mathematical modeling, analyzing, and optimizing EW contracts are also investigated by researchers.
Assuming minimal repairs at failures, Kumar and Chattopadhyay (2004) derived minimum beneficial duration of an EW contract from consumer’s perspective and its corresponding expected cost of EW servicing. Jack and Murthy (2007) also used Stackelberg game model to maximize manufacturer’s expected profit per unit of time and minimize consumer’s expected cost per unit of time. To do this, under minimal repair assumption, the time of EW purchase and its length are assumed to be consumer’s decision variables and the manufacturer determines profit per unit of time on the EW sale and the expected cost of each repair paid by the consumer beyond the warranty coverage. Considering minimal repairs at minor failures and replacement at catastrophic failures, Wu and Longhurst (2011) minimized the expected lifecycle cost of a product by deriving optimal length of EW contracts and optimal opportunitybased age replacement policy. Using the game theory approach, Lam and Lam (2001) also derived the optimal policies of the EW provider and consumer assuming perfect repair at the failures.
In the EW literature, modeling preventive maintenance and its effect on the reliability of product have been investigated by Bouguerra et al. (2012) and Chang and Lin (2012). In the research conducted by Bouguerra et al. (2012), assuming a minimal repair at failures and periodic imperfect preventive repairs, the expected cost of EW servicing incurred by both the consumer and EW providers has been derived. Then, the maximum and minimum acceptable prices of the EW contract from the consumer’s and EW provider’s point of views, respectively, are determined. Chang and Lin (2012) also maximized the expected total profit of selling EW contracts by deriving optimal number and controlledlimit of PMs during the base warranty period as well as the optimal number and degree of PMs during the EW period.
In the reviewed work, a time (age) variable has been used to define warranty coverage region and the type of EW is assumed to be onedimensional. However, for the products sold with twodimensional warranty such as automobiles, the warranty coverage is defined by two variables, namely time (age) and usage (mileage). Considering onedimensional and twodimensional warranty coverage regions and following three corrective repair options of 1) minimal repair at failures, 2) imperfect repair combined with minimal repair, and 3) perfect repair combined with minimal repair, Su and Shen (2012) maximized the profit of EW provider. To do so, the optimal type of EW coverage region (onedimensional or twodimensional coverage regions) as well as the optimal corrective repair option was determined. Majid et al. (2012) also optimized the twodimensional EW contracts and the servicing cost when failures were minimally repaired and the preventive maintenance were carried out on the product. In the mostrecent research, Shahanaghi et al. (2013) proposed a mixed integer nonlinear optimization model to minimize the expected cost of EW servicing from EW provider’s point of view. To do so, assuming a minimal repair at failures during the base and EW periods and a periodic imperfect preventive maintenance with the adjustable degree of repairs during the EW period, they determined the optimal number and degree of preventive repairs. Besides, they provided some guidelines to help the EW providers obtain isowarranty cost curves as well as design a wide range of flexible twodimensional EW contracts and their corresponding optimal maintenance strategies. In the most recent research, considering twodimensional extended warranty contracts Huang et al. (2017) modeled the effect of preventive maintenance and minimized the expected servicing cost of the dealer over warranty period. To do this, regarding customers’ usage rate they were categorized into three classes and for each class of customers a customized preventive maintenance schedule was proposed. From the customer’s perspective; Park et al. (2018) also derived the optimal periodic preventive maintenance policy after the expiration of a twodimensional warranty. The objective function assumed to be minimizing the expected cost rate of the customer. Recently Su and Zhao (2017) also classified and reviewed reported research on extended warranty literature and proposed some future research directions.
As one can expect the EW providers offer EW contracts to make a profit and applying proper upgrade and preventive maintenance strategies may increase the EW providers’ profit by reducing the cost of EW servicing. However, the profit margin of an EW contract as well as the demand for this contract also directly depends on the price of EW contracts. Therefore, to maximize the EW providers’ profit, considering the demand for EW contracts, the optimal price of EW contracts needs to be determined. In this paper, for a first time in the extended warranty literature, an integrated model considering an upgrade action at the beginning of the extended warranty period, periodic preventive maintenance over warranty is derived. Then regarding customers’ demands for extended warranty contracts, the effect of extended warranty coverage and its price on the profitability of the dealer are investigated. In the proposed model, the upgrade improvement level, the numbers of preventive repairs as well as the price of the extended warranty contracts are determined to maximize the expected profit of the dealer. The contribution of the paper can be highlighted from both practical and academic aspects. Form the practical point of view, considering more influential factors which affect the dealer’s expected servicing costs and in turn his/her expected profits may help to a more realistic modeling of the situation and consequently to make more comprehensive and effective decisions. Such an integrated model can be of great interest to academicians. Since, it helps to study the effect of an upgrade, preventive and corrective repairs, warranty coverage, price as well as more importantly their interrelations on the dealers’ profit. Although introducing more variables to a mathematical model may lead to increase in model complexity and in turn affects its ease of use, but at the same time, it helps to capture dynamic and interrelated nature of such decisions.
The rest of this paper is organized as follows. In the next section, the mathematical optimization model of an EW contract and its components is presented. The time to the first failure of product in terms of age and usage, the estimated expected cost of an upgrade, PMs and failure rectification actions, as well as the demand function of EW contracts are derived in Section 2. In Section 3, a realworld failure history of a commercial vehicle is adopted from Shahanaghi et al. (2013) to produce an illustrative numerical example. Using this example, the effects of considering upgrade and PM actions on the EW provider’s profit is studied. Then, assuming four different EW servicing scenarios, the optimal servicing scenario is determined. Finally, in the last section, the conclusion and future research directions are presented.
2. FORMULATION OF THE PROPOSED MODEL
In this section, a mathematical optimization model and its related mathematical models are presented. To do so, it is assumed that a product is sold with a twodimensional base warranty and W_{0} and U_{0} are warranty length and total usage limits, respectively. The product is repairable with increasing failure rate over both base and EW periods. When a consumer purchases the product, his/her usage rate over the base and EW period is constant; but, different consumers may have different random usage rates. By purchasing the product, the consumer has also an opportunity to purchase an EW contract with length W_{1} and total usage limitations U_{1} before the expiration of the base warranty period to cover servicing costs during the EW period.
Moreover, all the failures during the base and EW periods are also assumed to be minimally repaired by the EW provider at no cost to the consumer. Besides, the EW provider performs an upgrade action at the beginning of the EW period to improve the reliability of the product by choosing proper upgrade improvement level as a his/her decision variable. A periodic imperfect preventive maintenance with a predetermined degree of repair is carried out to slow down the degradation process of the product during the EW period, and the number of PMs is decision variable. The time to repair, including corrective and preventive repairs, as well as the time to perform an upgrade action is also assumed to be negligible with respect to the time to failure.
In order to derive these mathematical models, in the next subsection, we derive the time to the first failure of the product in terms of product’s age and total usage.
2.1. Model Notations
The following notation is used to present the proposed mathematical model:

W_{0}, U_{0} : Base warranty’s covered length and total usage

W_{1}, U_{1} : Extended warranty’s covered length and total usage

x : Product age

u : Product’s total usage

X : Time to the first failure of product (random variable)

R : Usage rate (random variable)

r_{D} : Nominal usage rate

$g(r),\hspace{0.17em}G(r)$ : PDF and CDF of the product’s usage rate

M : Number of usage rate scenarios

$[{r}_{i1},\hspace{0.17em}{r}_{i})$ : The ith usage rate subinterval for i =1, 2,…, M

${\overline{r}}_{i}$ : Average usage rate under the ith usage rate scenarios

${f}_{D}(x;{\text{\alpha}}_{0}),\hspace{0.17em}{F}_{D}(x;{\text{\alpha}}_{0})$ : PDF and CDF of the time to the first failure under nominal usage rate where α_{0} is the scale parameter

$f(x;\text{\alpha (}r\text{)}),\hspace{0.17em}\hspace{0.17em}F(x;\text{\alpha}(r))$ : Conditional PDF and CDF of the time to the first failure under usage rate r

$h(x;\text{\alpha (}r\text{)}),\hspace{0.17em}H(x;\text{\alpha}(r))$ : Hazard and cumulative hazard functions associated with $F(x,\hspace{0.17em}\alpha (r))$

c_{m} : Expected cost of minimal repair

c_{PM} : Expected cost of preventive repair

δ_{PM} : Level of applied effort to preventively repair the product

$\mathbf{P}=({p}_{1},\hspace{0.17em}{p}_{2},\hspace{0.17em}\dots ,\hspace{0.17em}{p}_{M})$ : Upgrade improvement level under M usage rate scenarios (decision variable)

N : Number of planned periodic preventive repairs during the EW period (decision variable)

Price : Price of EW contract (decision variable)
2.2. Modeling the Time to the First Failure of Product
The failure process of a product in terms of its age and usage can be modeled in different ways. In the twodimensional warranty literature, there are three main approaches to model the failure process of the product. In the first approach which is called “bivariate approach,” the time to the first failure of product is modeled using the bivariate distributions. In the “composite scale approach” as a second twodimensional failure modeling approach, product’s age and usage are combined to a new combinatorial variable and the failure process of product is modeled in terms of this variable. In the third and most widely used approach, the usage (age) of the product is modeled as a function of age (usage) and, by substituting usage (age) with this function, the twodimensional failure model is reduced to the onedimensional failure model. This approach is called “marginal approach” and accelerated failure time (AFT), and proportional hazard (PH) models are two wellknown variations of this approach. For more information about these approaches and the related research, please refer to (Jack et al., 2009) .
In this paper, we use marginal approach because from modeling aspect, this approach is somehow easy to model and analyze. Besides, from practical point of view, one can easily drive the usage rate distribution form the field data. To do this we apply AFT model to investigate the effect of consumer’s usage rate on the distribution of the time to the first failure of a product. To do so, let X_{D} denote the time to the first failure of the product under nominal usage rate r_{D} and ${F}_{{X}_{D}}(x,{\alpha}_{0})$ is its distribution function with scale parameter α_{0} . Then, for a given product with usage rate r, the distribution function of the time to the first failure will be:
where γ ≥ 1 is the parameter of the AFT model. As can be seen in (1), based on the AFT model, the conditional distribution function of the time to the first failure of product under usage rate r will be the same as its distribution function with nominal usage rate, in which the scale parameter is $\alpha \left(r\right)={\alpha}_{0}{\left(\frac{{r}_{D}}{r}\right)}^{\gamma}$ instead. Assuming the usage rate of the product as a continues random variable R, regarding usage rate values, infinite number of distribution functions of the time to the first failures will be obtained. In order to facilitate the computations, we derive limited usage rate scenarios by converting continues random variable R into a discreet one. To do so, let g(r), G(r) be the PDF and CDF of the usage rate random variable R with lower and upper bounds R_{L} and , R_{U} respectively. Then, assuming M usage rate scenarios, the average usage rate and the probability of occurrences of each usage rate scenario will be computed by the following two equations:(2)(3)
where $[{r}_{i1},\hspace{0.17em}{r}_{i}),\hspace{0.17em}i=1,\hspace{0.17em}\dots ,\hspace{0.17em}M$ is the ith usage rate subinterval and ${r}_{i}={R}_{L}+i\times \frac{{R}_{U}{R}_{L}}{M}$ for $i=0,\hspace{0.17em}\dots ,\hspace{0.17em}M.$.
Therefore, under the ith usage rate scenario, the expected time of base warranty expiration is ${x}_{BW/{\overline{r}}_{i}}=\text{min}\{{W}_{0},\hspace{0.17em}\frac{{U}_{0}}{{\overline{r}}_{i}}\}$ and the expected total usage of the product at the time base warranty expiration will be ${u}_{BW/{\overline{r}}_{i}}=\text{min}\{{U}_{0},\hspace{0.17em}{\overline{r}}_{i}\hspace{0.17em}{W}_{0}\}.$ For a special case where the EW period begins right after the expiration of base warranty period, the EW’s expiration time will also be ${x}_{EW/{\overline{r}}_{i}}=\text{min}\{{x}_{BW/{\overline{r}}_{i}}+{W}_{1},\hspace{0.17em}{x}_{BW/{\overline{r}}_{i}}+\frac{{U}_{1}}{{\overline{r}}_{i}}\}.$ The expected total usage of the product at the time of EW expiration will be ${u}_{EW/{\overline{r}}_{i}}=\text{min}\{{u}_{BW/{\overline{r}}_{i}}+{U}_{1},\hspace{0.17em}{u}_{BW/{\overline{r}}_{i}}+{\overline{r}}_{i}\hspace{0.17em}{W}_{1}\}.$ This is the same approach which was used by Shahanaghi et al. (2013).
2.3. Modeling Upgrade and its Corresponding Cost
When a consumer purchases an EW contract, the EW provider has an option to improve the reliability of product at the beginning of the EW period by performing an upgrade action. In general, this action is the process of inspecting and repairing or replacing the failed or degraded components with new or younger ones to improve the reliability of the product. From the mathematical modeling point of view, an upgrade action can be interpreted as an imperfect repair. Therefore, the proposed imperfect repair approaches could be used to model an upgrade action and its effect. For more information about the imperfect repair modeling approaches and its possible effects on the reliability of product, please refer to (Wang and Pham, 2006).
In this paper, we use probabilistic (p, q) rule approach to model an upgrade action at the beginning of the EW period and its effect on the reliability of the product. According to this approach, when a product is upgraded, the product state will be “as good as a new” state with probability p and “as bad as an old” one with probability q = (1 − p), where $p\in [0,\hspace{0.17em}1].$. The parameter p is called “Brown and Proschan’s parameter” and is interpreted as the level of upgrading improvement (Shafiee et al., 2011). Therefore, if x is the age of the product right before an upgrade action, then, U, a reduction in its age after an upgrade action is a random variable which follows from the Bernoulli distribution with parameter p as follows:(4)
where the expected value of reduction in the age of product is E[U] = px.
Let X_{i} be conditional time to the first failure under the ith usage rate scenario and $f(x;\text{\alpha (}{\overline{r}}_{i}\text{)})$ and $F(x;\text{\alpha}({\overline{r}}_{i}))$ be conditional PDF and CDF of X_{i}, respectively. Then, assuming minimal upgrade at the beginning of the EW period, the distribution of the time to the first failure, i.e. ${F}_{m}(x;\text{\alpha (}{\overline{r}}_{i}\text{)}),$ during the EW period will be:(5)
and its density function is:
Based on the probabilistic (p, q) rule approach and assuming negligible time to perform an upgrade action, the corresponding failure rate after an upgrade is a discrete mixture of F(t) and F_{m}(t) governed by the discrete distribution $\{p,\hspace{0.17em}(1p)\}.$
Then, by replacing Eqs. (6) and (7) in the above equations and simplification, the product’s hazard rate function will be:
wher $R(x;\text{\alpha (}{\overline{r}}_{i}\text{)})=1F(x;\text{\alpha (}{\overline{r}}_{i}\text{)})\hspace{0.17em}\text{and}\hspace{0.17em}f(x;\text{\alpha (}{\overline{r}}_{i}\text{)})=\frac{dF(x,\text{\alpha (}{\overline{r}}_{i}\text{)})}{dx}$ are reliability function and PDF of X_{i}, respectively. In Eq. (8), x is the time passed from the beginning of the EW period and ${x}_{BW/{\overline{r}}_{i}}$ is the estimated age of the product at the time of upgrade action. Then, the cumulative hazard rate of product after an upgrade action is ${H}_{u}(x;\text{\alpha (}{\overline{r}}_{i}\text{)},\hspace{0.17em}{p}_{i})={\displaystyle {\int}_{0}^{x}{h}_{u}(t;\text{\alpha (}{\overline{r}}_{i}\text{)},\hspace{0.17em}{p}_{i})dt}.$ This process is the same approach which was used by Shafiee et al. (2011).
For a given product with the average usage rate ${\overline{r}}_{i}$ and considering an upgrade action with an improvement level , p_{i} the cost of the upgrade action can be modeled as a nonlinear function of the product’s age and usage at the time of upgrade action (or equivalently at the time of base warranty expiration). It should be recalled that the EW coverage is assumed to begin right after the expiration of the base warranty period. Therefore, the cost of the upgrade action will be:
where c_{0} is the fixed cost of an upgrade action and parameters c, ψ, ω, and ξ are positive real values that can be estimated using the upgrade historical data. ${x}_{BW/{\overline{r}}_{i}}=\frac{{U}_{0}}{{\overline{r}}_{i}}\hspace{0.17em}\text{and}\hspace{0.17em}{u}_{BW/{\overline{r}}_{i}}={\overline{r}}_{i}\times {W}_{0}$ are also the estimated age and usage of the product at the time of upgrade action. It is worth mentioning that the proposed cost function in Eq. (9) is an extension of onedimensional upgrade cost model introduced by Chattopadhyay and Murthy (2004) to the twodimensional case where the total usage of product at the time of upgrade is also incorporated.
Finally, considering M possible usage rate scenarios and their corresponding probabilities ${\overline{P}}_{{s}_{i}}\text{for}\hspace{0.17em}i=1,\hspace{0.17em}2,\hspace{0.17em}\mathrm{...},\hspace{0.17em}M,$ the expected cost of upgrade action, i.e. $E[{C}_{u}(\mathbf{P};\hspace{0.17em}\mathbf{X},\hspace{0.17em}\mathbf{U})]$, will be:(10)
where $X=({\overline{x}}_{1},\hspace{0.17em}{\overline{x}}_{2},\hspace{0.17em}\dots ,\hspace{0.17em}{\overline{x}}_{M}),\hspace{0.17em}U=({\overline{u}}_{1},\hspace{0.17em}{\overline{u}}_{2},\hspace{0.17em}\dots ,\hspace{0.17em}{\overline{u}}_{M})$ are vectors of estimated ages and usages, respectively, and $\mathbf{P}=({p}_{1},\hspace{0.17em}{p}_{2},\dots ,\hspace{0.17em}{p}_{M})$ is to upgrade improvement level under M usage rate scenarios. It is worth noting that P is a decision vector and needs to be optimally determined to maximize the EW providers’ profit.
2.4. Periodic Preventive Maintenance Mathematical Model and its Corresponding Cost
Planning and implementing preventive maintenance during an EW period may help the EW provider control the degradation process of product and its corresponding cost. In this paper, we assume that, during the EW time span W_{1}, regardless of consumers’ usage rate, one single timebased periodic preventive maintenance with the predetermined degree of repairs is planned and the number of PMs during the EW period is the EW provider’s decision variable. Then, conditioning on product usage rate, we estimate the number of performed PMs on the product up to its EW expiration times. In order to do so, first, we model the effect of preventive repairs on the state of the product using the virtual age type I model introduced by Kijima et al. (1988).
Let N be the number of planned PMs during the EW time span W_{1} and ${T}_{PM}=\frac{{W}_{1}}{N+1}$ be the time between two successive PMs. If v_{k}_{1} is the virtual age of the product after the (k −1) th preventive repair, then based on Kijima’s model, the virtual age of the product after the kth preventive repair will be:(11)
Where $0\le {\delta}_{PM}\le 1$ and it is the level of the applied effort to preventively repair the product. Recall that it is assumed to be predetermined and constant over the EW period. As can be seen, based on this approach, ${\delta}_{PM}=1$ means that the kth preventive repair completely removes the effect of the product aging between the times of the (k  1)th and k th preventive repairs. ${\delta}_{PM}=0$ also means that the preventive repair is minimal and has no significant effect on the state of the product. Other values of δ_{PM} show that preventive repairs can partially remove product’s aging effect between the times of two successive PMs.
Let a periodic preventive maintenance with N planned preventive repairs are conducted by the EW provider to control the degradation process of the product during the time span W_{1}. As one can expect, regarding consumers’ usage rate and in turn the warranty coverage length, the product experiences the different number of PMs up to their EW expiration times, see Figure 1.
Therefore, assuming N planned PMs for the given product with a given average usage rates ${\overline{r}}_{i}$, the number of performed PMs will be:(12)
As a result, the expected number of performed PMs during the extended warranty period will be:(13)
where ${\overline{P}}_{{s}_{i}}$ is the probability of occurrences of the ith usage rate scenario. Finally, assuming c_{PM} as an expected cost of each PM action, the expected cost of PM actions during the EW period will be $E[{C}_{PM}(N)]={c}_{PM}\times E[{N}_{PM}(N)].$ It is worth noting that the number of planned PMs (N) is the EW provider’s decision variable.
2.5. Modeling Subsequent Failures and Driving Expected Number of Failures During EW
In this paper, it is assumed that, when a product fails during the EW period, the EW provider minimally repairs the product at no cost to the consumer in a negligible time. Besides, periodic preventive maintenance with imperfect preventive repairs during the EW period is performed to slow down the degradation process of the product. In such a situation, for each PM time interval, the number of failures up to a time between two successive PMs constitutes a nonhomogeneous Poisson process (NHPP). Therefore, the intensity function of NHPP will be the same as the hazard function of the time to the first failure of product (Baik et al., 2004) and can be used to estimate the expected number of failures during successive PMs.
As stated before, conditioning on the consumers’ usage rate, for a product with a given average usage rate ${\overline{r}}_{i},$ the number of performed PMs during the EW period will be ${N}_{PM}^{\left(i\right)}(N).$ For this product, the number of failures during the EW period is the sum of the number of failures during the first ${N}_{PM}^{\left(i\right)}(N)$ PM time intervals with equal length T_{PM} and the number of failures during the time of the last PM up to EW expiration time. Consequently, conditioning on the usage rate ${\overline{r}}_{i}$ and considering p_{i} as a corresponding upgrade improvement level as well as , δ_{PM} a predetermined degree of preventive repairs, the expected number of failures during the j th PM for $j=1,\hspace{0.17em}2,\hspace{0.17em}\dots ,\hspace{0.17em}{N}_{PM}^{\left(i\right)}(N)$ will be as follows:(14)
where ${v}_{j1}$ and ${v}_{j1}+{T}_{PM}$ are virtual age of the product after the (j−1) th PM and virtual age of product right before the jth PM action, respectively. ${h}_{u}(x;\text{\alpha (}{\overline{r}}_{i}\text{)},\hspace{0.17em}{p}_{i},\hspace{0.17em}{\delta}_{PM})$ is also the hazard function of the product after an upgrade action, which can be computed using Eq. (8). Then, the virtual age of product right after the last PM (v_{L}) and the remaining covered time of the EW contract ${T}_{L}=\text{min}\{{W}_{1},\hspace{0.17em}\frac{{U}_{1}}{{\overline{r}}_{i}}\}{N}_{PM}^{\left(i\right)}(N)\times {T}_{PM},$ the expected number of failures during the last interval will be:(15)
Therefore, for this product, the expected number of failures during its EW period will be:(16)
As a result, considering all the usage rate scenarios and their corresponding probabilities, the expected number of failures over the EW period will be:(17)
Finally, assuming c_{m} as a expected cost of a rectification action, the expected failure cost incurred by the EW provider will be $E[{C}_{F}]={c}_{m}\times E[{N}_{F}].$ It is worth noting that the expected cost of the failure rectification during the EW contract depends on the two following decision variables: the number of planned PMs (N) and upgrade levels (P). So, in the following lines, we use the notation $E[{C}_{F}(P,\hspace{0.17em}N)]$ to show these dependencies.
2.6. Demand for EW Contract
The conducted research on the marketing aspect of EW contracts reveals that many factors may affect the consumer’s demand for EW contracts: type of product, its reliability, product purchase price, repair cost, warranty policy, length and price of warranty, consumer’s risk attitude, loyalty of warranty provider’s brand, marketing strategy, advertisement, etc. However, there are limited works on the mathematical modeling of the demand for EW contracts. For the first time in the EW literature, Bryant and Gerner (1982) studied the effect of product failure and cost of servicing on the demand for EW contracts. Desai and Padmanabhan (2004) derived a linear demand function for the EW contract by considering product’s failure probability and utilities of riskaverse consumers. Lai et al. (2005) also proposed a linear demand function considering the EW’s price and length. Chen et al. (2009) discussed the purchase of EW contracts for electronic products. They examined the impact of product characteristics (hedonic/utilitarian and manufacturer’s warranty), retailer actions (promotions as well as feature advertising), and consumer characteristics (income, gender, and prior usage) on the purchase of EW contracts. Jindal (2011) also considered the relative importance of different preferences to explain consumers’ choice on the purchasing of EW contracts under risk. In the most recent research, Su and Shen (2012) extended the demand function proposed by Glickman and Berger (1976) to obtain the demand for twodimensional EW contracts. However, they ignored the effect of EW’s total usage limitation on the demand for twodimensional contract. In this paper, we simply develop the linear demand function proposed by Li et al. (2008) to predict the demand for twodimensional EW contracts. To this end, we incorporate the effects of EWs both length and total usage limitations by considering the price per unit of the EW contract’s coverage region. Based on this model, the demand for EW linearly decreases when its price increases. Besides, increasing the coverage region of warranty leads to increase in the demand for extended warranty contract in a nonlinear manner. Therefore, we have:(18)
where q is the demand for the new product or, equivalently, the maximum number of consumers who may purchase an EW contract and $a\in [0,\hspace{0.17em}1]$ is sensitivity of consumer about price per unit of the coverage region.
2.7. TwoDimensional EW Mathematical Optimization Model
In this subsection, a mixed integer nonlinear optimization model is presented to maximize the expected profit of providing EW contract from the EW provider’s point of view. In this model, the decision variables are $\mathbf{P}=({p}_{1},\hspace{0.17em}{p}_{2},\hspace{0.17em}\dots ,\hspace{0.17em}{p}_{M}),$ upgrade improvement level, N as the number of planned PMs during the EW period, and the price of the EW contract. Let $E[{C}_{EW}(P,\hspace{0.17em}N)]$ be the expected cost of EW service; then, we have:(19)
where $E[{C}_{u}(P)],\hspace{0.17em}E[{C}_{PM}(N)],\hspace{0.17em}\text{and}\hspace{0.17em}E[{C}_{F}(P,\hspace{0.17em}N)]$ are expected cost of upgrade, preventive maintenance, and failure rectifications during the EW contract, respectively. As a result, twodimensional EW optimization model is formulated as follows:
where T_{c} is consumer’s minimum acceptable time between two successive PMs. This constraint means that consumers do not prefer to bring their products for PMs in any time intervals. The proposed optimization model can be solved using the following three steps:
Let N_{Max} be the maximum number of preventive repairs that can be planned during the EW period while the constraint ${N}_{Max}<\frac{{W}_{1}}{{T}_{c}}$ still holds. Then, for $N=n,\hspace{0.17em}\hspace{0.17em}n=0,\hspace{0.17em}1,\hspace{0.17em}2,\hspace{0.17em}\dots ,\hspace{0.17em}{N}_{Max},$ a given number of planned PMs during the EW period, the following steps are implemented:

Step 1: For N = n, by minimizing the expected cost of EW servicing, i.e. $E[{C}_{EW}(P,\hspace{0.17em}n)],$ the optimal upgrade improvement level ${P}_{n}^{*}=({p}_{1,n}^{*},{p}_{2,n}^{*},\hspace{0.17em}\mathrm{...},\hspace{0.17em}{p}_{M,n}^{*})$ is obtained.

Step 2: For N = n and the corresponding upgrade improvement level ${\mathbf{P}}_{n}^{*},$ the obtained value of ${E}^{*}[{C}_{EW}({P}_{n}^{*},\hspace{0.17em}n)]$ is placed in Eq. (20) and the optimal price of the EW contract $(Pric{e}_{n,{P}_{n}^{*}}^{*})$ is determined to maximize the EW provider’s profit. In this step, the optimal value of the objective function in Eq. (20), i.e. EW provider’s optimal profit, is also computed.

Step 3: By repeating the above two steps for all the feasible number of PMs (N = n, n = 0,1, 2, , ), Max … N the EW provider’s optimal profit for each possible number of PMs is derived. Finally, maximum value between the obtained optimal profit values and its corresponding decisions will be the overall optimal solution of Eq. (20).
Using these steps, the optimal number of planned PMs, optimal upgrade level, as well as the optimal price of the EW contract one can obtain to maximize the expected profit of the EW provider.
In the next subsection, we will derive the mathematical optimization model for a special case where the distributions of the time to the first failure and consumer’s usage rate follow the Weibull distributions.
2.8. Mathematical Optimization Model for a Special Case: Weibull Distribution
Let X be a random variable representing the time to the first failure of product, which follows Weibull distribution with scale parameter α_{0} and shape parameter β. Besides, let R be consumer’s usage rate random variable, which follows Weibull distribution with scale and shape parameters α_{r} and β_{r}, respectively. For the assumed special case, the presented formulas in the previous subsections will be as follows:
The average usage rate and its corresponding probability under the ith usage rate scenario will be:(21)(22)
Besides, for a given product with average usage rate ${\overline{r}}_{i}$ and upgrade improvement level , p_{i} the hazard function of the time to the first failure of the product after an upgrade action will be:(23)
Finally, the mathematical optimization model will be:
As can be seen, for a special case, the derived mathematical optimization model is also a mixed integer nonlinear programming problem and the presented steps in the previous subsection can be used to obtain the optimal solution. It is worth noting that, in Eq. (24), the objective function is concave and the optimal solution of (24) is a globally optimal solution. Please see the appendix for the proof.
3. AN ILLUSTRATIVE NUMERICAL EXAMPLE
In this section, in order to illustrate the proposed model, we use a realworld failure history of a commercial vehicle reported by Shahanaghi et al. (2013) to reproduce a numerical example. The distributions of the time to the first failure and consumer’s usage rate as well as other values of model parameters are presented in Table 1.
Then, through this example, we study the effect of an upgrade and PM actions on the expected cost of EW servicing and, in turn, on the profitability of EW contracts. To do so, we consider the following four EW servicing scenarios:
Scenario 1) The EW provider (insurer) limits his/her servicing strategy to minimal repair at failures and no upgrade and PM actions are performed on the product during the EW period. In the automotive industry, with a slight variation, this scenario is called mechanical breakdown insurance (MBI). In the MBI, the insurer only pays repair bills for the failed products and no further investigations are made on the product.
Scenario 2) The EW provider minimally rectifies all failures, and a periodic preventive maintenance with a predetermined and constant degree of repairs is performed on the product during EW period.
Scenario 3) The EW provider performs an upgrade action at the beginning of the EW period and all failures during the EW period are minimally repaired.
Scenario 4) The EW provider performs an upgrade action at the beginning of the EW period and minimally repairs all failures of product. Besides, a PM with the predetermined degree of repair is carried on the product to control the degradation process of product.
Considering these servicing scenarios and assumed values of Table 1, we apply the model (24) to obtain optimal profit of the EW provider and its corresponding decision variables. The provided results show that, for both scenarios 2 and 4, where PMs are carried out on the product, the EW provider’s profit is maximized by performing preventive repairs every six months. For other possible numbers of PMs, the corresponding optimal profit of the EW provider under scenarios 2 and 4 is depicted in Figures 2 and 3, respectively.
For servicing scenarios where an upgrade action is considered, i.e. scenarios 3 and 4, the optimal upgrade improvement levels regarding usage rate scenarios and the number of performed PMs during the EW period are also presented in Table 2.
As expected, considering different servicing strategies may lead to different servicing costs, prices, and in turn different profits. For the four mentioned servicing strategies, the expected cost of failures, PMs, upgrade, as well as optimal prices and profits are presented in Table 3.
As observed in Table 3, the servicing scenario 4, i.e. performing both upgrade and PM actions during the assumed EW contract, is the most profitable scenario. In order to make a comprehensive comparison between the effects of the four mentioned servicing scenarios, we consider scenarios 1 as the base scenario. Then, we compare the amounts of improvements resulted by performing scenarios 2 to 4 with that of scenario 1. The resulted improvements are presented in terms of rectification cost (CM), expected servicing cost (Total cost), price, and demand for the EW as well as the EW provider’s profit (see Figure 4).
As can be seen in Figure 4, the provided results reveal that considering an upgrade action at the beginning of the EW period may effectively reduce the servicing cost and the price of EW contract as well as increase the demand for EW contracts and, in turn, profitability of offering twodimensional EW contracts. The results also show that the effect of performing preventive maintenance during the EW period may not be as considerable as an upgrade action. However, it should be noted that, in this paper, it is assumed that preventive repairs can only remove the effect of product aging during two successive PMs. This means that, in the best case, where δ_{PM} = 1, performing PMs can only restore the virtual age of the product to its age at the beginning of the EW period.
In general, based on the provided results for the four mentioned servicing scenarios, it can be concluded that making an investment to upgrade the product at the beginning of the EW period and, at the same time, considering preventive maintenance may significantly improve the profitability of the offering twodimensional EW contracts. Moreover, it helps the EW provider improve the reliability of products and offer EW contracts at lower prices, which are of great interest for consumers. In order to study the effect of EW coverage region and the level of preventive maintenance on the profitability of the EW provider, we run a sensitivity analysis on these parameters. The result is presented and discussed in the next subsection.
3.1. Sensitivity Analysis
As presented in the numerical example, the EW length and usage are assumed to be constant, i.e. W_{1} = 2 (Years) and ${U}_{1}=13.5\times ({10}^{4}Km).$ However, as expected, any changes in the EW coverage region may affect the demand for EW contracts, its servicing cost, its price, and finally EW provider’s profit. In the following lines for different values of extended warranty length and usage limitations, we conduct sensitivity analyses. The result for EW length is shown in Table 4.
As demonstrated in Table 4, with increasing the length of EW contracts, the number of PMs, expected servicing cost, price, demand for EW, and in turn the profit of EW provider are increased. However, a conflict may be raised that, with increasing the price, one can expect that the demand for EW contracts will be decreased, while the provided result shows increase in demand?! The answer relies on the fact that the price of the EW contract is not the only influential factor and the length of EW contracts also contributes to the demand (see Eq. 18 ). Since the growth rate of EW length is bigger than the growth rate of price, one can expect that the fraction $\frac{Price}{{W}_{1\times}{U}_{1}}$ begins to decrease and, in turn, the demand for EW contracts begin to increase. For example, when W_{1} increases from 4 to 5, the growth rate will be 25%, while at the same time the growth rate of price will be 4%. It should be noted that the presented result in Table 4 shows that increasing the length of EW up to the product’s end of life will improve the profitability of the EW provider; however, more investigation reveals that, if the length of the EW contract increases beyond the eight years, the profit of EW contract begins to decrease. This situation relies on the fact that, when a product ages, the expected number of failures, its corresponding cost, and in turn the price of the EW contracts will dramatically increase in a way that the growth rate of the price will be bigger than the growth rate of EW length. Therefore, the demand for EW contract and, as a result, the profit of EW provider starts to decrease.
The sensitivity analysis of EW covered usage is also carried out, the result of which is shown in Table 5.
According to Table 5, with increasing the covered usage of the product under EW contract, the number of PMs, expected servicing cost, price of EW contract, demand, as well as the profit of EW provider are increased. The amount of these increases is considerable and shows that, by adding the warranted usage, one can expect that the profit of EW provider dramatically increases. It should be noted that the behavior of the EW provider’s profit changes regarding the EW usage is similar to its behavior with respect to EW length. In other words, if the EW usage is bigger than 85×(10^{4} Km), then the profit of EW provider will be decreased and almost the same justification is beyond this behavior.
As seen in Table 6, with increasing the level of preventive maintenance, the number of planned PMs increases, while the expected servicing cost and price decrease and, at the same time, the demand for EW contract as well as EW provider’s profit increases. To explain this behavior, it is worth noting that, in the sensitivity analysis of , δ_{PM} it is assumed that when the level of the preventive maintenance increases, the cost of each preventive repair remains constant, i.e. c_{PM}=5(Monetary unit). c_{PM} = Monetary unit Therefore, increasing the effectiveness of preventive repair without increasing its cost will causes an increase in the number of PMs and, in turn, this increase will more effectively slowdown the degradation process of product and its failure rate. As a result, decreasing the expected cost of servicing and price and increasing the demand and profit of EW provider are expected.
4. CONCLUSION AND FUTURE RESEARCH DIRECTIONS
Proposing extended warranty contracts for products such as automobiles is a good opportunity for manufacturers, dealers, and insurers as well as other independent companies to make a profit. It also provides consumers with “peace of mind.” However, applying improper servicing strategies may lead to thin profit margin or even loss for EW providers. From the consumer’s perspective, it also may lead to poor quality of services or disruption in the servicing of products. Therefore, from the managerial aspect, modeling the product failure process under different servicing strategies and driving optimal service pattern as well as proper price of EW contracts will help the EW provider to offer highquality services at affordable price to the customer, and at the same time, to maximize his/her profit that is of great interest.
To the best knowledge of the authors, for the first time in the literature, in this paper, we modeled and studied the effect of an upgrade action at the beginning of the EW span, a periodic preventive maintenance during EW, and the price of EW contracts on provider’s profit in an integrated framework. To do so, a mixed integer nonlinear mathematical model was proposed to determine upgrade improvement level, number of preventive repairs, and optimal price of EW contract by maximizing EW provider’s profit. Then, through a numerical example, the effect of four different servicing scenarios on the profit and price of EW contracts was studied.
The provided results revealed that making an investment to upgrade products at the beginning of the EW period as well as planning and implementing preventive maintenance during EW period may improve the profitability of the EW contracts up to 206 percent in comparison with “minimal repair at failures” servicing strategy. Besides, results showed that this servicing strategy could not only significantly improve the profitability of offering EW contracts, but also helps the EW provider provide highquality services by effectively improving the reliability of products. Besides, it could help the EW provider offer EW contracts with lower prices (up to 16%), which is of great interest to consumers.