Post-Warranty Replacement Models for the Product under a Hybrid Warranty

Abstract

In this article, by considering both a limited number of failure replacements and a limited number of random working cycles as warranty terms, a hybrid warranty (HW) is designed from the manufacturer’s point of view to warrant the product that does successive projects at random working cycles. The warranty cost produced by HW is derived and analyzed. By defining that HW warrants the product, two types of post-warranty replacement models are investigated from the consumer’s point of view to ensure the reliability of the product through HW, i.e., customized post-warranty replacement and uniform post-warranty replacement. Depreciation expense is integrated into each post-warranty replacement. The expected cost rate model is presented for each post-warranty replacement and some special cases are obtained by setting parameters in the expected cost rate. Finally, sensitivities on both HW and post-warranty replacements are analyzed in numerical experiments. It is shown that when a limited number of failure replacements or/and a limited number of random working cycles are introduced to a warranty, the warranty cost can be reduced; and the performance of the uniform post-warranty replacement is superior to the customized post-warranty replacement.

1. Introduction

For manufacturers, offering products (or systems) to warranties can eliminate the consumer’s uncertainty about product quality. For consumers, if a product is purchased under a warranty, the reliability performance of the product can be ensured and maintenance costs during the warranty period (or region) can be shouldered partially or fully by the manufacturer. Therefore, offering each product to a warranty can benefit both manufacturers and consumers.

Recently, warranties have been researched widely from the manufacturer’s (or warrantor’s) point of view. The research on warranties depends on reliability modeling technology. At present, academia and industry are frequently using two types of reliability modeling technology. One type is to model the product lifetime as a distribution function with self-announcing failure, and the other type is to model the product failure (i.e., not self-announcing failure [1]) as a type of degradation process [2,3,4,5,6,7,8,9,10,11,12]. Along this frame, without exception, the research on warranties can be classified into two streams. The first stream is to design degradation-based warranty policies by modeling product failure as a type of degradation process, which has been provided by [13,14,15]. The second stream is to design lifetime-based warranty policies by modeling the product lifetime as a distribution function with self-announcing failure [16,17,18,19,20,21,22,23,24,25,26,27,28,29].

The manufacturer adopts maintenance, replacement, or/and minimal repair to ensure product reliability during the warranty period (or region). However, consumers (i.e., users or warrantees) are concerned about how product reliability during the post-warranty period is sustained. Due to increased maintenance costs, how to model post-warranty maintenance to sustain product reliability has recently received considerable attention. This has also been investigated by following two types of modeling technologies mentioned above. For example, by modeling product failure as a degradation process, condition-based maintenance was used as post-warranty maintenance [15]; by modeling the product lifetime as a distribution function, some post-warranty maintenance policies were modeled [30,31,32,33,34,35,36].

Recently, the development of in situ sensor and measuring technologies has made the product’s health conditions measurable/inspectable. This is an important technological support for designing degradation-based warranty policies. In addition, in situ sensor and measuring technologies can monitor working cycles of the product that performs successive projects at working cycles. From the viewpoint of reliability theory, the product deteriorates with its operating time. In view of this, some random maintenance policies have been investigated to ensure or improve product reliability by modeling working cycles as independent random variables satisfying an identical distribution function [37,38,39,40]. Similarly, for a product that performs successive projects at working cycles, the manufacturer can design a warranty to ensure its reliability performance by monitoring working cycles. The consumer can continue to sustain its reliability by tracking the post-warranty working cycles.

In this article, by introducing random working cycles to a renewing free replacement warranty (RFRW), we design a hybrid warranty (HW) from the manufacturer’s point of view. The requirement of HW is that if the first failure does not occur before the warranty period is reached or before the limited number of random working cycles is completed, whichever occurs earlier, then the product will go through HW at the warranty period or at the completion of the limited random working cycles; if the limited number of failure replacements is reached, then a free repair warranty (FRW) [41,42,43] is triggered to ensure the reliability performance of a new identical product. Defining that HW is used to warrant the product, two types of the consumer’s post-warranty replacements are modeled to sustain the reliability of the product through HW. The first type of the post-warranty replacement is customized according to the difference in HW expiration cases, and the second type of post-warranty replacement is modeled by unifying all cases of HW expiration. For each post-warranty replacement, the expected cost rate model is constructed by integrating a depreciation expense depending on the operating time.

The contribution of this article is listed as follows: (1) a HW is designed to ensure the reliability performance of the product, which does successive projects at working cycles; (2) by defining that HW is extended to post-warranty maintenance, the consumer’s customized post-warranty replacement and the consumer’s uniform post-warranty replacement are modeled to sustain the reliability performance of the product through HW; (3) the performances of the customized and uniformed post-warranty replacements are compared.

This article is organized as follows. In Section 2, we design the manufacturer’s HW and derive and analyze the warranty cost. In Section 3, the expected cost rate models are presented from the consumer’s point of view, by differentiating HW expiration cases. In Section 4, from the consumer’s point of view, a uniform expected cost rate model is presented by unifying all cases of HW expiration. In Section 5, the numerical experiments are used to validate the proposed approaches. Finally, Section 6 draws conclusions.

2. Warranty Design

We assume that the product performs projects and that the random working cycles $Y_i$ ($i=1,2,\dots$) are independent and follow an identical distribution function $G(y)$ with a mean $1/\lambda$. The time $X$ of the first failure for the product follows a general distribution function $F(x)$ with a failure rate function $r(u)$. In addition, assuming that any time to repair or replacement is completely ignored in this article.

2.1. Warranty Definition

Given the warranty period $w$, the number $m$ of random working cycles, and the failure replacement limit $n$, a warranty is described as follows.

(1)

If the first failure occurs before the warranty period $w$ is reached or before the $m\mathrm{th}$ random working cycle is completed, whichever occurs earlier, then the failed product will be replaced with a new identical product, which is called a failure replacement in this article.

(2)

The manufacturer shoulders the replacement cost resulting from the first failure, including but not limited to labor cost and transport cost.

(3)

If the $n\mathrm{th}$ product fails, then the $n\mathrm{th}$ failed product is replaced with a new identical product warranted by the free repair warranty (FRW) [41,42,43] with the time span $w$.

Obviously, the RFRW term (i.e., the first two terms) and FRW term are considered simultaneously; therefore, such a warranty can be called a hybrid warranty (HW).

2.2. Warranty Cost Estimation

Denote $\overline{F}(x)$ by the survival function of the time-to-first-failure $X$, where $\overline{F}(x)=1-F(x)$. When completing the $m\mathrm{th}$ random working cycle, the product’s operating time $S_m$ is a sum of the $m$ random working cycles, i.e., $S_m={\displaystyle \sum _{i=1}^m{Y}_i}$. Let $G^{(m)}(s)$ and ${\overline{G}}^{(m)}(s)$ be the distribution function and survival function of the operating time $S_m$, respectively; then, $G^{(m)}(s)={\displaystyle {\int }_0^s{G}^{(m-1)}(s-u)\mathrm{d}G(u)}$ and ${\overline{G}}^{(m)}(s)=1-G^{(m)}(s)$. According to HW, before the $n\mathrm{th}$ product fails, the product will go through the warranty at the completion of the $m\mathrm{th}$ random working cycle or at the warranty period $w$, whichever occurs earlier. Therefore, the probability $q$ that the product goes through HW before the $n\mathrm{th}$ product fails is given by

$$q=\mathrm{Pr}\{S_m<w,S_m<X\}+\mathrm{Pr}\{w<S_m,w<X\} ={\displaystyle {\int }_0^w\overline{F}(u)\mathrm{d}{G}^{(m)}(u)}+{\overline{G}}^{(m)}(w)\overline{F}(w)=1-{\displaystyle {\int }_0^w{\overline{G}}^{(m)}(u)\mathrm{d}F(u)}$$

(1)

Obviously, the event where the product goes through HW before the $n\mathrm{th}$ product fails and the event where the product does not go through HW before the $n\mathrm{th}$ product fails are mutually exclusive events. Therefore, the probability $p$ that the product does not go through HW before the $n\mathrm{th}$ product fails is given by

$$p=1-q={\displaystyle {\int }_0^w{\overline{G}}^{(m)}(u)\mathrm{d}F(u)}$$

(2)

According to probability theory, the probability that the $k\mathrm{th}$ ($i=1,2,\dots ,n$) product goes through HW is $p^{i-1}q$. Meanwhile, the number of failure replacements is $i-1$. Furthermore, before the $n\mathrm{th}$ product fails, the expected value $E[\kappa ]$ for failure replacement number $i-1$ can be modeled as

$$E[\kappa ]={\displaystyle \sum _{i=1}^n{p}^{i-1}q}(i-1)=\frac{p\left(1-p^{n-1}(nq+p)\right)}{q}$$

(3)

Denote $X_k$ ($k=1,\cdots ,n$) by the lifetime of the $k\mathrm{th}$ product that failed during the warranty region. According to probability theory, the distribution function $H(x)$ of the lifetime $X_k$ can be obtained as

$$H(x)=\mathrm{Pr}\left\{X_k<x|X_k<S_m,X_k<w\right\}=\frac{{\displaystyle {\int }_0^x{\overline{G}}^{(m)}(u)\mathrm{d}F(u)}}{{\displaystyle {\int }_0^w{\overline{G}}^{(m)}(u)\mathrm{d}F(u)}}$$

(4)

where $0<x<w$.

The relationship between time and the values of products has been discussed in [44,45,46]. In this article, suppose that the product’s depreciation expense is increasing with respect to the operating time $t$, then the depreciation expense $k(t)$ depending on $t$ can be modeled as

$$k(t)=\alpha t^{\beta }$$

(5)

where $\beta >0$ and $\alpha$ ($\alpha >0$) is a depreciation rate satisfying $\alpha =(S_p-v_l)/l^{\beta }$ (here, $S_p$ is the sale price, $l$ is the useful life, and $v_l$ is the salvage at $l$).

For the $k\mathrm{th}$ failed product, its operating time is equal to $X_k$. Therefore, its depreciation expense is $k(X_k)$. Denote $c_R$ by the unit failure replacement cost shoulder for the manufacturer; then, the $k\mathrm{th}$ failed product makes the manufacturer suffer a net cost $c_R-k(X_k)$. When completing the $(i-1)\mathrm{th}$ ($i=1,2,\dots ,n$) failure replacement, the total replacement cost $W{C}_{i-1}$ of the manufacturer can be obtained as

$$W{C}_{i-1}={\displaystyle \sum _{k=0}^{i-1}\left(c_R-k(X_k)\right)}$$

(6)

where $W{C}_0=0$.

$X_k$ ($k=1,\cdots ,n$) are independent and follow the identical distribution function $H(x)$ in Equation (4); thus, the expected value $E\left[W{C}_{i-1}\right]$ of $W{C}_{i-1}$ can be given by

$$E\left[W{C}_{i-1}\right]=E\left[{\displaystyle \sum _{k=0}^{i-1}\left(c_R-k(X_k)\right)}\right] =(i-1){\displaystyle {\int }_0^w\left(c_R-k(x)\right)\mathrm{d}H(x)}=\frac{(i-1){\displaystyle {\int }_0^w\left(c_R-k(x)\right){\overline{G}}^{(m)}(x)\mathrm{d}F(x)}}{{\displaystyle {\int }_0^w{\overline{G}}^{(m)}(u)\mathrm{d}F(u)}}$$

(7)

By HW, for the $n\mathrm{th}$ product failed, it will be replaced with a new identical product warranted by FRW with the time span $w$. In this case, the total replacement cost $W{C}_n$ includes two parts. One part is the replacement cost ${\displaystyle \sum _{k=1}^n\left(c_R-k(X_k)\right)}$ resulting from $n$ failure replacements; the other part is the minimal repair cost $c_m{\displaystyle {\int }_0^{w}r(u)\mathrm{d}u}$ produced by FRW, where $c_m$ is the unit minimal repair cost. By summing, the total cost $W{C}_n$ resulting from $n$ failure replacements can be given by

$$W{C}_n={\displaystyle \sum _{k=0}^n\left(c_R-k(X_k)\right)}+c_m{\displaystyle {\int }_0^{w}r(u)\mathrm{d}u}$$

(8)

As mentioned in Equation (7), the lifetime $X_k$ of Equation (8) follows $H(x)$ in Equation (4), so the expected value $E\left[W{C}_n\right]$ of $W{C}_n$ can be derived as

$$E\left[W{C}_n\right]=E\left[{\displaystyle \sum _{k=0}^n\left(c_R-k(X_k)\right)}\right]+c_m{\displaystyle {\int }_0^{w}r(u)\mathrm{d}u}=\frac{n{\displaystyle {\int }_0^w\left(c_R-k(x)\right){\overline{G}}^{(m)}(x)\mathrm{d}F(x)}}{{\displaystyle {\int }_0^w{\overline{G}}^{(m)}(u)\mathrm{d}F(u)}}+c_m{\displaystyle {\int }_0^{w}r(u)\mathrm{d}u}$$

(9)

As mentioned above, the probability that the $i\mathrm{th}$ product goes through HW is $p^{i-1}q$. In addition, the probability that the $n\mathrm{th}$ failure replacement is triggered is given by $p^n$. According to the calculation method of expectation, the expected value $E[WC]$ of the warranty cost produced by HW can be derived as

$$\begin{array}{ll}E[WC]& ={\displaystyle \sum _{i=1}^n{p}^{i-1}q\cdot E\left[W{C}_{i-1}\right]}+p^n\cdot E\left[W{C}_n\right]= \frac{\left(1-p^n\right){\displaystyle {\int }_0^w\left(c_R-k(x)\right){\overline{G}}^{(m)}(x)\mathrm{d}F(x)}}{q}+p^n\left(c_m{\displaystyle {\int }_0^{w}r(u)\mathrm{d}u}\right)\hfill \\ & =\frac{\xi +p^n\left(q\zeta -\xi \right)}{q}\hfill \end{array}$$

(10)

where $\xi ={\displaystyle {\int }_0^w\left(c_R-k(x)\right){\overline{G}}^{(m)}(x)\mathrm{d}F(x)}$ and $\zeta =c_m{\displaystyle {\int }_0^{w}r(u)\mathrm{d}u}$.

When $n\to \infty$, $E[WC]=\xi /q$; when $n\to 0$, $E[WC]=\zeta$. Furthermore, Proposition 1 can be obtained.

Proposition 1.

(a)

When n is given and m = 0, the warranty cost $E[WC]$ produced by HW satisfies $E[WC] =\left[\varsigma +{[F(w)]}^n\left(\overline{F}(w)c_m{\displaystyle {\int }_0^{w}r(u)\mathrm{d}u}-{\displaystyle {\int }_0^w\left(c_R-k(x)\right)\mathrm{d}F(x)}\right)\right]/\overline{F}(w)$, where $\varsigma ={\displaystyle {\int }_0^w\left(c_R-k(x)\right)\mathrm{d}F(x)}$; this warranty cost is greater than the warranty cost in (10).

(b)

When m is given, the following results can be established.

①

When $q>\xi /\zeta$, the warranty cost $E[WC]$ produced by HW satisfies $\xi /q\le E[WC]\le \zeta$;

②

When $q<\xi /\zeta$, the warranty cost $E[WC]$ produced by HW satisfies $\zeta \le E[WC]\le \xi /q$;

③

When $q=\xi /\zeta$, the warranty cost $E[WC]$ produced by HW satisfies $E[WC]=\zeta$.

Proof .

(a)

$m=0$ means $q=\overline{F}(w)$, $p=F(w)$, and $\underset{m\to 0}{\lim }\xi =\underset{m\to 0}{\lim }{\displaystyle {\int }_0^w\left(c_R-k(x)\right){\overline{G}}^{(m)}(x)\mathrm{d}F(x)}={\displaystyle {\int }_0^w\left(c_R-k(x)\right)\mathrm{d}F(x)}$. Let $\varsigma =\underset{m\to 0}{\lim }\xi ={\displaystyle {\int }_0^w\left(c_R-k(x)\right)\mathrm{d}F(x)}$. By substituting $\varsigma$, $q=\overline{F}(w)$, and $p=F(w)$ into (10), we obtain $E[WC] =\left[\varsigma +{[F(w)]}^n\left(\overline{F}(w)c_m{\displaystyle {\int }_0^{w}r(u)\mathrm{d}u}-{\displaystyle {\int }_0^w\left(c_R-k(x)\right)\mathrm{d}F(x)}\right)\right]/\overline{F}(w)$. As ${\displaystyle {\int }_0^w{\overline{G}}^{(m)}(u)\mathrm{d}F(u)}<F(w)$, $1-{\displaystyle {\int }_0^w{\overline{G}}^{(m)}(u)\mathrm{d}F(u) }>\overline{F}(w)$, and ${\displaystyle {\int }_0^w\left(c_R-k(x)\right){\overline{G}}^{(m)}(x)\mathrm{d}F(x)}<{\displaystyle {\int }_0^w\left(c_R-k(x)\right)\mathrm{d}F(x)}$, the warranty cost in (a) is greater than the warranty cost in Equation (10).

(b)

When $q>\xi /\zeta$, the warranty cost $E[WC]$ in Equation (10) is decreasing with respect to $n$. Therefore, we obtain $\underset{n\to \infty }{\lim }E[WC]=\underset{n\to \infty }{\lim }\left(\xi +p^n\left(q\zeta -\xi \right)\right)/q=\xi /q$ and $\underset{n\to 0}{\lim }E[WC]=\underset{n\to 0}{\lim }\left(\xi +p^n\left(q\zeta -\xi \right)\right)/q=\zeta$. In addition, the equality $\xi /q\le E[WC]\le \zeta$ holds. When $q<\xi /\zeta$, the warranty cost $E[WC]$ in Equation (10) is increasing with respect to $n$. Therefore, we obtain $\underset{n\to \infty }{\lim }E[WC]=\underset{n\to \infty }{\lim }\left(\xi +p^n\left(q\zeta -\xi \right)\right)/q=\xi /q$ and $\underset{n\to 0}{\lim }E[WC]=\underset{n\to 0}{\lim }\xi +p^n\left(q\zeta -\xi \right)/q=\zeta$. Furthermore, the equality $\zeta \le E[WC]\le \xi /q$ holds. When $q=\xi /\zeta$, the factor $q\zeta -\xi$ in Equation (10) equates to zero, i.e., $q\zeta -\xi =0$. Therefore, we can obtain the equality $E[WC] =\xi /q$.  □

This proposition indicates that the warranty cost produced by HW can be controlled by specifying the warranty limit $m$ or/and the failure replacement limit $n$.

3. Customized Post-Warranty Replacement Models

According to HW, the product goes through HW at the completion of the $m\mathrm{th}$ random working cycle, at the warranty period $w$, or at the time span $w$, whichever occurs earlier. This means that HW expiration can be classified into three cases. This section customizes the post-warranty replacement model related to each of the three HW expiration cases, under the condition that the product through HW will be replaced at the replacement time $T$ or at failure, whichever takes place earlier.

To model post-warranty replacements conveniently, based on the definition of the life cycle [35], we customize the post-warranty replacement model (i.e., obtain the expected cost rate) of each HW expiration case, as shown below.

3.1. Post-Warranty Replacement Model 1

This section customizes the post-warranty replacement model related to the first HW expiration case, i.e., the case at which the product goes through HW at the completion of the $m\mathrm{th}$ random working cycle. Based on the life cycle definition, we derive the expected cost rate according to the renewal rewarded theorem [47], and seek the optimum replacement time $T^{*}$.

3.1.1. Life Cycle Length Derivation

When the $i\mathrm{th}$ ($i=1,2,\dots ,n$) product goes through HW, the failure replacement number is $i-1$. When the $i\mathrm{th}$ ($i=1,2,\dots ,n$) product goes through HW, the total warranty service period can be obtained as ${\displaystyle \sum _{k=0}^{i-1}{X}_k}$, where $X_0=0$. As the number $i-1$ of failure replacements is a geometric random variable satisfying $p^{i-1}q$, the expected value $E\left[W\right]$ of ${\displaystyle \sum _{k=0}^{i-1}{X}_k}$ can be obtained as

$$E\left[W\right]=E\left[{\displaystyle \sum _{i=1}^n{p}^{i-1}q\left({\displaystyle \sum _{k=0}^{i-1}{X}_k}\right)}\right]=\frac{p\left(1-p^{n-1}(nq+p)\right)}{q}\cdot E\left[X_k\right]=\frac{\left(1-p^{n-1}(nq+p)\right){\displaystyle {\int }_0^{w}x{\overline{G}}^{(m)}(x)\mathrm{d}F(x)}}{q}$$

(11)

where $X_k$ follows $H(x)$ in Equation (4).

For the product through HW at the completion of the $m\mathrm{th}$ random working cycle, its operating time is equal to $S_m$. Therefore, the distribution function $H_{S_m}(s)$ of $S_m$ can be given by

$$H_{S_m}(s)=\mathrm{Pr}\left\{S_m<s|S_m<w,S_m<X\right\}=\frac{{\displaystyle {\int }_0^s\overline{F}(u)\mathrm{d}{G}^{(m)}(u)}}{{\displaystyle {\int }_0^w\overline{F}(u)\mathrm{d}{G}^{(m)}(u)}}$$

(12)

where $0<s<w$.

The expected value $E[S_m]$ of $S_m$ can be derived as

$$E[S_m]={\displaystyle {\int }_0^{w}s\mathrm{d}}{H}_{S_m}(s)=\frac{{\displaystyle {\int }_0^{w}s\overline{F}(s)\mathrm{d}{G}^{(m)}(s)}}{{\displaystyle {\int }_0^w\overline{F}(u)\mathrm{d}{G}^{(m)}(u)}}$$

(13)

According to reliability theory, when the product through HW at the completion of the $m\mathrm{th}$ random working cycle is replaced at the replacement time $T$ or at failure, whichever occurs first, the expected value of the post-warranty period can be derived as ${\displaystyle {\int }_0^w\left({\displaystyle {\int }_0^T\overline{F}(x|s)\mathrm{d}x}\right)\mathrm{d}{H}_{S_m}(s)}$, where $\overline{F}(x|•)=e^{-{\displaystyle {\int }_0^{x}r(•+u)\mathrm{d}u}}$. Based on the life cycle definition, by summing, the expected length $E[L_1]$ of the life cycle can be derived as

$$\begin{array}{ll}E[L_1]& =E\left[W\right]+E\left[S_m\right]+{\displaystyle {\int }_0^w\left({\displaystyle {\int }_0^T\overline{F}(x|s)\mathrm{d}x}\right)\mathrm{d}{H}_{S_m}(s)}\hfill \\ & =\frac{\left(1-p^{n-1}(nq+p)\right){\displaystyle {\int }_0^{w}x{\overline{G}}^{(m)}(x)\mathrm{d}F(x)}}{q}+\frac{{\displaystyle {\int }_0^w\left(s+{\displaystyle {\int }_0^T\overline{F}(x|s)\mathrm{d}x}\right)\overline{F}(s)\mathrm{d}{G}^{(m)}(s)}}{{\displaystyle {\int }_0^w\overline{F}(u)\mathrm{d}{G}^{(m)}(u)}}\hfill \end{array}$$

(14)

3.1.2. Life Cycle Cost Computation

Suppose that each failure makes consumer suffer a failure cost c_f. After the $(i-1)\mathrm{th}$ failure replacement completes, the total failure cost $W{S}_1$ of the consumer can be computed as

$$W{S}_1=\left(i-1\right)c_f$$

(15)

Let $c_F$ be the unit replacement cost resulting from each failure and $c_P$ ($c_R<c_P<c_F$) be the unit replacement cost resulting from the replacement at T, then the replacement cost $C_1(T)$ during the post-warranty period can be expressed as $C_1(T|S_m)=F(T|S_m)(c_F-k(S_m+x))+\overline{F}(T|S_m)\left(c_P-k(S_m+T)\right)$, where $F(T|•)+\overline{F}(T|•)=1$. Furthermore, during the post-warranty period, the expected value $E\left[C_1(T)\right]$ of $C_1(T|S_m)$ is computed as

$$\begin{array}{cc}E\left[C_1(T)\right]& =E\left[C_1(T|S_m)\right]\hfill \\ & =E\left[F(T|S_m)(c_F-k(S_m+x))+\overline{F}(T|S_m)\left(c_P-k(S_m+T)\right)\right]\hfill \\ & =(c_F-c_P){\displaystyle {\int }_0^{w}F(T|s)\mathrm{d}{H}_{S_m}(s)}+c_P-{\displaystyle {\int }_0^w\left({\displaystyle {\int }_0^T\overline{F}(x|s)\mathrm{d}k(s+x)}\right)\mathrm{d}{H}_{S_m}(s)}\hfill \\ & =\frac{(c_F-c_P){\displaystyle {\int }_0^w\left(F(T|s)\right)\overline{F}(s)\mathrm{d}{G}^{(m)}(s)}-{\displaystyle {\int }_0^w\left({\displaystyle {\int }_0^T\overline{F}(x|s)\mathrm{d}k(s+x)}\right)\overline{F}(s)\mathrm{d}{G}^{(m)}(s)}}{{\displaystyle {\int }_0^w\overline{F}(u)\mathrm{d}{G}^{(m)}(u)}}+c_P\hfill \end{array}$$

(16)

As mentioned above, the number $i-1$ ($i=1,2,\dots ,n$) of failure replacements is a geometric random variable satisfying $p^{i-1}q$. Based on the life cycle definition, by summing and taking expectation, the expected value $E[C(L_1)]$ of the life cycle cost is computed as

$$\begin{array}{ll}E[C(L_1)] & ={\displaystyle \sum _{i=1}^n{p}^{i-1}q\cdot }E\left[W{S}_1\right]+E\left[C_1(T)\right]\hfill \\ & = \frac{p\left(1-q^{n-1}\left(nq+p\right)\right)c_f}{q}+\frac{(c_F-c_P){\displaystyle {\int }_0^w\left(F(T|s)\right)\overline{F}(s)\mathrm{d}{G}^{(m)}(s)}-{\displaystyle {\int }_0^w\left({\displaystyle {\int }_0^T\overline{F}(x|s)\mathrm{d}k(s+x)}\right)\overline{F}(s)\mathrm{d}{G}^{(m)}(s)}}{{\displaystyle {\int }_0^w\overline{F}(u)\mathrm{d}{G}^{(m)}(u)}}+c_P\hfill \end{array}$$

(17)

3.1.3. Cost Rate Modeling

Let $A_1=p\left(1-q^{n-1}\left(nq+p\right)\right)c_f/q+c_P$ and $B_1=\left(1-q^{n-1}\left(nq+p\right)\right){\displaystyle {\int }_0^{w}x{\overline{G}}^{(m)}(x)\mathrm{d}F(x)}/q$; the expected cost rate $C{R}_1(T)$ can be modeled as

$$C{R}_1(T)=\frac{E[C(L_1)] }{E[L_1]}=\frac{ A_1{\displaystyle {\int }_0^w\overline{F}(u)\mathrm{d}{G}^{(m)}(u)}+(c_F-c_P){\displaystyle {\int }_0^w\left(F(T|s)\right)\overline{F}(s)\mathrm{d}{G}^{(m)}(s)}-{\displaystyle {\int }_0^w\left({\displaystyle {\int }_0^T\overline{F}(x|s)\mathrm{d}k(s+x)}\right)\overline{F}(s)\mathrm{d}{G}^{(m)}(s)} }{B_1{\displaystyle {\int }_0^w\overline{F}(u)\mathrm{d}{G}^{(m)}(u)}+{\displaystyle {\int }_0^w\left(s+{\displaystyle {\int }_0^T\overline{F}(x|s)\mathrm{d}x}\right)\overline{F}(s)\mathrm{d}{G}^{(m)}(s)}}$$

(18)

3.1.4. Optimization

The consumer’s goal is to seek an optimum post-warranty replacement policy by optimizing the expected cost rate $C{R}_1(T)$, namely, seek an optimum replacement time $T^{*}$ by means of $C{R}_1(T^{*})=\inf \{T\ge 0\}C{R}_1(T)\}$. Let $F_d(T)$ be the first-order derivative of the expected cost rate $C{R}_1(T)$; then, $F_d(T)$ is obtained as

$$\begin{array}{ll}{F}_{d_1}(T)=& \frac{\left((c_F-c_P){\displaystyle {\int }_0^{w}f(T|s)\overline{F}(s)\mathrm{d}{G}^{(m)}(s)}-{\displaystyle {\int }_0^w\overline{F}(T|s)k^{\prime }(s+T)\overline{F}(s)\mathrm{d}{G}^{(m)}(s)}\right)\times \left(B_1{\displaystyle {\int }_0^w\overline{F}(u)\mathrm{d}{G}^{(m)}(u)}+{\displaystyle {\int }_0^w\left(s+{\displaystyle {\int }_0^T\overline{F}(x|s)\mathrm{d}x}\right)\overline{F}(s)\mathrm{d}{G}^{(m)}(s)}\right)}{{\left(B_1{\displaystyle {\int }_0^w\overline{F}(u)\mathrm{d}{G}^{(m)}(u)}+{\displaystyle {\int }_0^w\left(s+{\displaystyle {\int }_0^T\overline{F}(x|s)\mathrm{d}x}\right)\overline{F}(s)\mathrm{d}{G}^{(m)}(s)}\right)}^2}-\\ & \frac{\left({\displaystyle {\int }_0^w\overline{F}(T|s)\overline{F}(s)\mathrm{d}{G}^{(m)}(s)}\right)\times \left(A_1{\displaystyle {\int }_0^w\overline{F}(u)\mathrm{d}{G}^{(m)}(u)}+(c_F-c_P){\displaystyle {\int }_0^w\left(F(T|s)\right)\overline{F}(s)\mathrm{d}{G}^{(m)}(s)}-{\displaystyle {\int }_0^w\left({\displaystyle {\int }_0^T\overline{F}(x|s)\mathrm{d}k(s+x)}\right)\overline{F}(s)\mathrm{d}{G}^{(m)}(s)}\right)}{{\left(B_1{\displaystyle {\int }_0^w\overline{F}(u)\mathrm{d}{G}^{(m)}(u)}+{\displaystyle {\int }_0^w\left(s+{\displaystyle {\int }_0^T\overline{F}(x|s)\mathrm{d}x}\right)\overline{F}(s)\mathrm{d}{G}^{(m)}(s)}\right)}^2}\end{array}$$

(19)

where $f(T|s)=\mathrm{d}F(T|s)/\mathrm{d}T$ and $k^{\prime }(T+x)=\mathrm{d}w(T+x)/\mathrm{d}T$.

Let $F_m(T)$ be the numerator of $F_d(T)$; then,

$$\begin{array}{cc}\hfill F_m(T)=& \left((c_F-c_P){\displaystyle {\int }_0^{w}f(T|s)\overline{F}(s)\mathrm{d}{G}^{(m)}(s)}-{\displaystyle {\int }_0^w\overline{F}(T|s)k^{\prime }(T+s)\overline{F}(s)\mathrm{d}{G}^{(m)}(s)}\right)\times \hfill \\ & \left(B_1{\displaystyle {\int }_0^w\overline{F}(u)\mathrm{d}{G}^{(m)}(u)}+{\displaystyle {\int }_0^w\left(s+{\displaystyle {\int }_0^T\overline{F}(x|s)\mathrm{d}x}\right)\overline{F}(s)\mathrm{d}{G}^{(m)}(s)}\right)-\left({\displaystyle {\int }_0^w\overline{F}(T|s)\overline{F}(s)\mathrm{d}{G}^{(m)}(s)}\right)\times \hfill \\ & \left(A_1{\displaystyle {\int }_0^w\overline{F}(u)\mathrm{d}{G}^{(m)}(u)}+(c_F-c_P){\displaystyle {\int }_0^{w}F(T|s)\overline{F}(s)\mathrm{d}{G}^{(m)}(s)}-{\displaystyle {\int }_0^w\left({\displaystyle {\int }_0^T\overline{F}(x|s)\mathrm{d}k(s+x)}\right)\overline{F}(s)\mathrm{d}{G}^{(m)}(s)}\right)\hfill \end{array}$$

(20)

Therefore, we can summarize the optimum replacement time $T^{*}$ that minimizes the expected cost rate $C{R}_1(T)$, as follows.

Theorem 1.

The following cases are obtained.

(1)

If $F_m(T)$ is nondecreasing (nonincreasing) and $F_m(0)>0$ or $F_m(\infty )<0$ (or ($F_m(\infty )>0$ or $F_m(0)<0$)), then $T^{*}=0$ or $T^{*}=\infty$ is the optimum solution of the expected cost rate $C{R}_1(T)$.

(2)

If $F_m(0)<0$ and $F_m(\infty )>0$, then there exists at least a finite optimum solution T* minimizing $C{R}_1(T)$.

(3)

If the condition of (2) holds and $F_m(T)$ is strictly monotonous, then there exists a finite and unique optimum solution $T^{*}$ satisfying $F_m(T^{*})=0$.

This theorem presents the existence and uniqueness of the optimum replacement time $T^{*}$ by discussing the first-order derivative. The existence and uniqueness of the optimum solution has been extensively studied; interested readers can consult [33,34,35,37,38,39,48].

3.2. Post-Warranty Replacement Model 2

This section customizes the post-warranty replacement model related to the second HW expiration case, i.e., the case where the product goes through HW at the warranty period $w$. Similar to Section 3.2, we model the expected cost rate by means of the renewal rewarded theorem.

3.2.1. Life Cycle Length Derivation

For the product through HW at the warranty period $w$, its warranty service period is equal to $w$. When it is replaced at the replacement time $T$ or at failure, whichever occurs first, the expected value of the post-warranty period can be derived as ${\displaystyle {\int }_0^T\overline{F}(x|w)\mathrm{d}x}$, where $\overline{F}(x|w)=e^{-{\displaystyle {\int }_0^x\lambda (w+u)\mathrm{d}u}}$. Similar to Equation (14), by summing, the expected length $E[L_2]$ of the life cycle can be derived as

$$E[L_2]=E\left[W\right]+w+{\displaystyle {\int }_0^T\overline{F}(x|w)\mathrm{d}x}=\frac{\left(1-p^{n-1}\left(nq+p\right)\right){\displaystyle {\int }_0^{w}x{\overline{G}}^{(m)}(x)\mathrm{d}F(x)}}{q}+w+{\displaystyle {\int }_0^T\overline{F}(x|w)\mathrm{d}x}$$

(21)

3.2.2. Life Cycle Cost Computation

Similar to the derivation of Equation (16), the expected value $E\left[C_2(T)\right]$ of the replacement cost is expressed as

$$E\left[C_2(T)\right]=F(T|w)(c_F-c_P)+c_P-{\displaystyle {\int }_0^T\overline{F}(x|w)\mathrm{d}k(w+x)}$$

(22)

According to Equation (17), the expected total cost resulting from $n$ failure replacements is ${\displaystyle \sum _{i=1}^n{p}^{i-1}q\cdot }E\left[W{S}_1\right]$. Based on the life cycle definition, by summing, the expected value $E[C(L_2)]$ of the life cycle cost is computed as

$$\begin{array}{cc}\hfill E[C(L_2)] & ={\displaystyle \sum _{i=1}^n{p}^{i-1}q\cdot }E\left[W{S}_1\right]+E\left[C_2(T)\right]\hfill \\ & =\frac{p\left(1-p^{n-1}\left(nq+p\right)\right)c_f}{q}+F(T|w)(c_F-c_P)+c_P -{\displaystyle {\int }_0^T\overline{F}(x|w)\mathrm{d}k(w+x)}\hfill \end{array}$$

(23)

3.2.3. Cost Rate Modeling

Letting $B_2=\left(1-p^{n-1}\left(nq+p\right)\right){\displaystyle {\int }_0^{w}x{\overline{G}}^{(m)}(x)\mathrm{d}F(x)}/q+w$, the expected cost rate $C{R}_2(T)$ can be modeled as

$$C{R}_2(T)=\frac{E[C(L_2)] }{E[L_2]}=\frac{ A_1+F(T|w)(c_F-c_P) -{\displaystyle {\int }_0^T\overline{F}(x|w)\mathrm{d}k(w+x)} }{B_2+{\displaystyle {\int }_0^T\overline{F}(x|w)\mathrm{d}x}}$$

(24)

The consumer’s goal is still to seek an optimum post-warranty replacement policy by optimizing the expected cost rate $C{R}_2(T)$. Similar to the derivative principle of Theorem 1, the existence and uniqueness of the optimum replacement time $T^{*}$ can be obtained by discussing the first-order derivative of the expected cost rate $C{R}_2(T)$. Herein, we no longer present them.

3.3. Post-Warranty Replacement Model 3

The post-warranty replacement model related to the third HW expiration case at which the product goes through HW at the time span $w$ is customized in this subsection. Next, we model the expected cost rate.

3.3.1. Life Cycle Length Derivation

Until the $n\mathrm{th}$ product fails, the total warranty service period can be obtained as ${\displaystyle \sum _{k=1}^n{X}_k}$. For the product through HW at the time span $w$, its warranty service period is equal to the time span $w$. In addition, the expected value of the post-warranty period can be similarly calculated as ${\displaystyle {\int }_0^T\overline{F}(x|w)\mathrm{d}x}$. By summing and taking expectation, the expected length $E[L_3]$ of the life cycle can be derived as

$$E[L_3]=E\left[{\displaystyle \sum _{k=1}^n{X}_k}\right]+w+{\displaystyle {\int }_0^T\overline{F}(x|w)\mathrm{d}x}=\frac{n{\displaystyle {\int }_0^{w}x{\overline{G}}^{(m)}(x)\mathrm{d}F(x)}}{{\displaystyle {\int }_0^w{\overline{G}}^{(m)}(u)\mathrm{d}F(u)}}+w+{\displaystyle {\int }_0^T\overline{F}(x|w)\mathrm{d}x}$$

(25)

where the lifetime $X_k$ follows the distribution function $H(x)$ in Equation (4).

3.3.2. Life Cycle Cost Computation

The total failure cost $W{S}_3$ until the $n\mathrm{th}$ failure replacement completion can be obtained is $W{S}_3={\displaystyle \sum _{i=0}^n{c}_f}+c_f{\displaystyle {\int }_0^{w}r(u)\mathrm{d}u}$. Therefore, the total failure cost $W{S}_u$ until RFW expiration can be computed as

$$E\left[W{S}_3\right]=n{c}_f+c_f{\displaystyle {\int }_0^{w}r(u)\mathrm{d}u}$$

(26)

Similar to the derivation of Equation (16), the expected value $E\left[C_3(T)\right]$ of the replacement cost is computed as

$$E\left[C_3(T)\right]=F(T|w)(c_F-c_P)+c_P-{\displaystyle {\int }_0^T\overline{F}(x|w)\mathrm{d}k(w+x)}$$

(27)

Based on the life cycle definition, by summing Equations (24) and (25), the expected value $E[C(L_3)]$ of the life cycle cost be computed as

$$E[C(L_3)]=E\left[W{S}_3\right]+E\left[C_3(T)\right] =n{c}_f+c_f{\displaystyle {\int }_0^{w}r(u)\mathrm{d}u}+F(T|w)(c_F-c_P)+c_P-{\displaystyle {\int }_0^T\overline{F}(x|w)\mathrm{d}k(w+x)}$$

(28)

3.3.3. Cost Rate Modeling

By the renewal rewarded theorem, the expected cost rate $C{R}_3(T)$ can be modeled as

$$C{R}_3(T) =\frac{E[C(L_3)]}{E[L_3]}=\frac{n{c}_f+c_f{\displaystyle {\int }_0^{w}r(u)\mathrm{d}u}+F(T|w)(c_F-c_P)+c_P-{\displaystyle {\int }_0^T\overline{F}(x|w)\mathrm{d}k(w+x)}}{n{\displaystyle {\int }_0^{w}x{\overline{G}}^{(m)}(x)\mathrm{d}F(x)}/{\displaystyle {\int }_0^w{\overline{G}}^{(m)}(u)\mathrm{d}F(u)}+w+{\displaystyle {\int }_0^T\overline{F}(x|w)\mathrm{d}x} }$$

(29)

Herein, the existence and uniqueness of the optimum replacement time $T^{*}$ are no longer summarized, because it is similar to Theorem 1.

4. Uniformed Post-Warranty Replacement Model

Section 3 has customized three post-warranty replacement models by differentiating HW expiration cases. In this section, a uniform post-warranty replacement model will be derived by simultaneously considering all cases of HW expiration. Next, we derive the expected cost rate.

4.1. Life Cycle Length Derivation

As mentioned above, HW expiration includes three cases, which are that the product goes through HW at the completion of the $m\mathrm{th}$ random working cycle, the product goes through HW at the warranty period $w$, and the new identical product goes through HW at the time span $w$. The probability $Q_1$ the first case occurs is computed as $Q_1={\displaystyle \sum _{i=1}^n{p}^{i-1}{q}_1}$, where $q_1$ ($=\mathrm{Pr}\{S_m<w,S_m<X\}={\displaystyle {\int }_0^w\overline{F}(u)\mathrm{d}{G}^{(m)}(u)}$) is the probability that the product goes through HW at the completion of the $m\mathrm{th}$ random working cycle. The probability $Q_2$ that the second case occurs is computed as $Q_2={\displaystyle \sum _{i=1}^n{p}^{i-1}{q}_2}$, where $q_2$ ($=\mathrm{Pr}\{w<S_m,w<X\}={\overline{G}}^{(m)}(w)\overline{F}(w)$) is the probability that the product goes through HW at the warranty period $w$. The probability $Q_3$ that the third case occurs is computed as $Q_3=p^n$. In addition, the expected values of the life cycles corresponding to three cases are $E[{\displaystyle \sum _{k=0}^{i-1}{X}_k}]+E\left[S_m\right]+E_1\left[T\right]$, $E[{\displaystyle \sum _{k=0}^{i-1}{X}_k}]+w +E_2\left[T\right]$, and $E[{\displaystyle \sum _{k=1}^n{X}_k}]+w+E_3\left[T\right]$, respectively, where $X_0=0$, $i=1,2,\dots ,n$, and $E_j\left[T\right]$ ($j=1,2,3$) is the expected value of the post-warranty period related to the $j\mathrm{th}$ case. Therefore, the expected value $E\left[L_u\right]$ of the life cycle can be obtained as

$$\begin{array}{ll}E\left[L_u\right]& =Q_1\cdot \left(E[{\displaystyle \sum _{k=0}^{i-1}{X}_k}]+E\left[S_m\right]+E_1\left[T\right]\right)+Q_2\cdot \left(E[{\displaystyle \sum _{k=0}^{i-1}{X}_k}]+w +E_2\left[T\right]\right)+Q_3\cdot \left(E[{\displaystyle \sum _{k=1}^n{X}_k}]+w+E_3\left[T\right]\right)\\ & =\frac{p(1-p^n)}{q}\cdot E\left[X_k\right]+\left({\displaystyle \sum _{i=1}^n{p}^{i-1}}\right)\left[q_1\left({\displaystyle {\int }_0^{w}x\mathrm{d}{H}_{S_m}(x)}+E_1\left[T\right]\right)+q_2\left(w +E_2\left[T\right]\right)\right]+p^n\left(w+E_3\left[T\right]\right) \\ & =\frac{(1-p^n)\left[{\displaystyle {\int }_0^w{\overline{G}}^{(m)}(u)\overline{F}(u)\mathrm{du}}+{\displaystyle {\int }_0^w{\overline{G}}^{(m)}(s)\mathrm{d}\left(\overline{F}(s){\displaystyle {\int }_0^T\overline{F}(x|s)\mathrm{d}x}\right)}+{\displaystyle {\int }_0^T\overline{F}(x)\mathrm{d}x}\right]}{q}+ p^n\left(w+{\displaystyle {\int }_0^T\overline{F}(x|w)\mathrm{d}x}\right)\end{array}$$

(30)

where $E_1\left[T\right]={\displaystyle {\int }_0^w\left({\displaystyle {\int }_0^T\overline{F}(x|s)\mathrm{d}x}\right)\overline{F}(s)\mathrm{d}{G}^{(m)}(s)}/{\displaystyle {\int }_0^w\overline{F}(u)\mathrm{d}{G}^{(m)}(u)}$, $E_2\left[T\right]=E_3\left[T\right]={\displaystyle {\int }_0^T\overline{F}(x|w)\mathrm{d}x}$, and $E\left[X_k\right]={\displaystyle {\int }_0^{w}x{\overline{G}}^{(m)}(x)\mathrm{d}F(x)}/{\displaystyle {\int }_0^w{\overline{G}}^{(m)}(u)\mathrm{d}F(u)}$.

4.2. Life Cycle Cost Computation

For a product through HW at one of three HW expiration cases, which are the completion of the $m\mathrm{th}$ random working cycle, the warranty period $w$, and the time span $w$, the total costs related to three HW expiration cases are $(i-1)c_f+E[C_1(T)]$, $(i-1)c_f+E[C_2(T)]$, and $n{c}_f+c_f{\displaystyle {\int }_0^w\lambda (u)\mathrm{d}u}+E[C_3(T)]$, respectively, where $i=1,2,\dots ,n$. Therefore, similar to the derivation principle of Equation (28), the expected value $E[C(L_u)]$ of the life cycle cost is given by

$$\begin{array}{ll}E[C(L_u)]& =Q_1\cdot \left((i-1)c_f+E[C_1(T)]\right)+Q_2\cdot \left((i-1)c_f+E[C_2(T)]\right)+ Q_3\cdot \left(n{c}_f+c_f{\displaystyle {\int }_0^w\lambda (u)\mathrm{d}u}+E[C_3(T)]\right)\\ & =E[\kappa ]c_f+\left({\displaystyle \sum _{i=1}^n{p}^{i-1}{q}_1}\right)\cdot E[C_1(T)]+\left({\displaystyle \sum _{i=1}^n{p}^{i-1}{q}_2}\right)\cdot E[C_2(T)]+p^n\cdot \left(n{c}_f+c_f{\displaystyle {\int }_0^w\lambda (u)\mathrm{d}u}+E[C_3(T)]\right)\\ & =\frac{p\left(1-p^n\right)c_f}{q}+ p^n\left(c_f{\displaystyle {\int }_0^{w}r(u)\mathrm{d}u}+F(T|w)(c_F-c_P)-{\displaystyle {\int }_0^T\overline{F}(x|w)\mathrm{d}k(w+x)}\right)+c_P+\\ & \frac{(1-p^n)\left[\begin{array}{l}(c_F-c_P)\left({\displaystyle {\int }_0^w{\overline{G}}^{(m)}(s)\mathrm{d}\left(F(T|s)\overline{F}(s)\right)}+F(T)\right)-\\ {\displaystyle {\int }_0^w{\overline{G}}^{(m)}(s)\mathrm{d}\left(\overline{F}(s){\displaystyle {\int }_0^T\overline{F}(x|s)\mathrm{d}k(s+x)}\right)}-{\displaystyle {\int }_0^T\overline{F}(x)\mathrm{d}k(x)}\end{array}\right]}{q}\end{array}$$

(31)

where $E[\kappa ]$ has been derived in Equation (3), $E[C_1(T)]$ has been offered in Equation (16), $E[C_2(T)]$ has been provided in Equation (20), and $E[C_3(T)]$ has been expressed in Equation (25).

4.3. Cost Rate Modeling

Letting $A_3=p\left(1-p^n\right)c_f/q+ p^n{c}_f{\displaystyle {\int }_0^{w}r(u)\mathrm{d}u}+c_P$ and $B_3=(1-p^n){\displaystyle {\int }_0^w{\overline{G}}^{(m)}(u)\overline{F}(u)\mathrm{d}u}/q+ p^{n}w$, the expected cost rate $C{R}_u(T)$ can be modeled as

$$C{R}_u(T)=\frac{E[C(L_u)] }{E[L_u]}=\frac{\begin{array}{l}{A}_{3}q+ p^{n}q\left(F(T|w)(c_F-c_P)-{\displaystyle {\int }_0^T\overline{F}(x|w)\mathrm{d}k(w+x)}\right)+\\ (1-p^n)\left[\begin{array}{l}(c_F-c_P)\left({\displaystyle {\int }_0^w{\overline{G}}^{(m)}(s)\mathrm{d}\left(F(T|s)\overline{F}(s)\right)}+F(T)\right)-\\ {\displaystyle {\int }_0^w{\overline{G}}^{(m)}(s)\mathrm{d}\left(\overline{F}(s){\displaystyle {\int }_0^T\overline{F}(x|s)\mathrm{d}k(s+x)}\right)}-{\displaystyle {\int }_0^T\overline{F}(x)\mathrm{d}k(x)}\end{array}\right]\end{array}}{B_{3}q+(1-p^n)\left[{\displaystyle {\int }_0^w{\overline{G}}^{(m)}(s)\mathrm{d}\left(\overline{F}(s){\displaystyle {\int }_0^T\overline{F}(x|s)\mathrm{d}x}\right)}+{\displaystyle {\int }_0^T\overline{F}(x)\mathrm{d}x}\right]+p^{n}q{\displaystyle {\int }_0^T\overline{F}(x|w)\mathrm{d}x}}$$

(32)

The existence and uniqueness of the optimum replacement time $T^{*}$ in the expected cost rate $C{R}_u(T)$ can be summarized, which is similar to Theorem 1.

4.4. Special Cases

Case A: When $m=\infty$, the model above is rewritten as

$$C{R}_u(T)=\frac{F(w)\left(1-{[\overline{F}(w)]}^n\right)c_f/\overline{F}(w)+c_P+ {[\overline{F}(w)]}^n{c}_f{\displaystyle {\int }_0^{w}r(u)\mathrm{d}u}+ F(T|w)(c_F-c_P)-{\displaystyle {\int }_0^T\overline{F}(x|w)\mathrm{d}k(w+x)}}{\left(1-{[\overline{F}(w)]}^n\right){\displaystyle {\int }_0^w\overline{F}(x)\mathrm{d}x}/\overline{F}(w)+ {[\overline{F}(w)]}^{n}w+{\displaystyle {\int }_0^T\overline{F}(x|w)\mathrm{d}x}}$$

(33)

$m=\infty$ means ${\overline{G}}^{(m)}(s)=1$. This indicates that the warranty limit $m$ fails and HW is transformed into RFRW with the failure replacement limit $n$. Therefore, this model is an expected cost rate where RFRW with the failure replacement limit $n$ warrants the product, and age replacement sustains reliability.

Case B: When $n=\infty$, the model above is rewritten as

$$C{R}_u(T)=\frac{A_{2}q+(c_F-c_P)\left({\displaystyle {\int }_0^w{\overline{G}}^{(m)}(s)\mathrm{d}\left(F(T|s)\overline{F}(s)\right)}+F(T)\right)-{\displaystyle {\int }_0^w{\overline{G}}^{(m)}(s)\mathrm{d}\left(\overline{F}(s){\displaystyle {\int }_0^T\overline{F}(x|s)\mathrm{d}k(s+x)}\right)}-{\displaystyle {\int }_0^T\overline{F}(x)\mathrm{d}k(x)}}{B_{3}q+{\displaystyle {\int }_0^w{\overline{G}}^{(m)}(s)\mathrm{d}\left(\overline{F}(s){\displaystyle {\int }_0^T\overline{F}(x|s)\mathrm{d}x}\right)}+{\displaystyle {\int }_0^T\overline{F}(x)\mathrm{d}x}}$$

(34)

where $A_2=p{c}_f/q+c_P$ and $B_3={\displaystyle {\int }_0^w{\overline{G}}^{(m)}(x)\overline{F}(x)\mathrm{d}x}/q$.

$n=\infty$ means $p^n=0$, which means that the RFW and the failure replacement limit $n$ are ignored simultaneously and that HW is transformed into a two-dimensional renewing free replacement warranty (2DRFRW), where $m$ and $w$ are two warranty limits. This model is an expected cost rate where 2DRFRW is used as a product warranty and age replacement is used as a policy to sustain reliability.

Case C: When $n=0$ and $m=\infty$, the model above is rewritten as

$$C{R}_u(T)=\frac{c_P+c_f{\displaystyle {\int }_0^{w}r(u)\mathrm{d}u}+ F(T|w)(c_F-c_P)-{\displaystyle {\int }_0^T\overline{F}(x|w)\mathrm{d}k(w+x)}}{ w+{\displaystyle {\int }_0^T\overline{F}(x|w)\mathrm{d}x}}$$

(35)

$n=0$ means $p^n=1$. $m=\infty$ means ${\overline{G}}^{(m)}(s)=1$. These results indicate that the warranty limit $m$ and RFRW with the failure replacement limit $n$ fail simultaneously and that HW is transformed into FRW. This model is an expected cost rate at which FRW is used as a product warranty and age replacement is used as a policy to sustain reliability.

Case D: When $n=\infty$, $m=\infty$, and $k(•)=0$, the model above is rewritten as

$$C{R}_u(T)=\frac{F(w)c_f/\overline{F}(w)+c_P+ F(T|w)(c_F-c_P)}{{\displaystyle {\int }_0^w\overline{F}(x)\mathrm{d}x}/\overline{F}(w)+{\displaystyle {\int }_0^T\overline{F}(x|w)\mathrm{d}x}}$$

(36)

$n=\infty$ and $m=\infty$ indicate that HW is transformed into RFRW [49,50]. Therefore, this model is an expected cost rate where the depreciation expense is ignored, RFRW warrants the product, and age replacement sustains reliability.

Other cases are also obtained by means of a similar method, and herein, we no longer present them.

5. Numerical Experiments

Recently, delivery robots have begun to be used in some factories. In situ sensor and measuring technologies are integrated into this type of delivery robot. By means of in situ sensor and measuring technologies, management can detect the working cycles of each delivery robot, which can help to perform the reliability management during the life cycle for delivery robots.

Assume that the time span between turn on and turn off is a delivery cycle (i.e., a working cycle) of the delivery robot; assume that the time $X$ of the first failure for the delivery robot follows a two-parameter Weibull function $F(x)$ whose failure rate function is given by $\lambda (u)=a{(u)}^b$ where $a,b>0$; assume that all random working cycles follow an identical distribution function $G(y)=1-\exp (-\lambda y)$, where $\lambda$ is a constant failure rate; and assume that some constant parameters are offered in

Table 1. Parameter values.

cm	cf	α	β	ch	cR	cP	λ	a	b
0.1	0.1	0.01	1	1	6	10	0.1	0.1	1

and other parameters (except the decision variable) not to be assigned a value in Table 1 are provided when needed.

5.1. Sensitivity Analysis of HW

Letting $m=3$ and $w=1$, we plot Figure 1. As shown in Figure 1, the warranty cost produced by HW increases to a constant cost that is the warranty cost produced by HW at which the failure replacement limit $n$ is ignored, i.e., the two-dimensional renewing free replacement warranty (2DRFRW) policy mentioned in Case B. As the failure replacement limit $n$ increases, the replacement frequency produced by HW increases to a constant frequency, which is a replacement frequency produced by 2DRFRW. Therefore, given a failure replacement cost $c_R=6$, the warranty cost produced by HW first closes and then equates to the warranty cost produced by 2DRFRW under the case at which the failure replacement limit $n$ increases.

To display how the warranty limit $m$ affects HW, we plot Figure 2, where $n=3$ and $w=1$. As indicated in Figure 2, the warranty cost produced by HW increases to a constant cost, which is the warranty cost resulting from RFRW with the failure replacement limit $n=3$, i.e., the warranty mentioned in Case A, as the warranty limit $m$ increases. When the warranty limit $m$ increases, the corresponding warranty service period $S_m$ ($S_m={\displaystyle \sum _{i=1}^m{Y}_i}$) tends gradually to infinity and ${\overline{G}}^{(m)}(\cdot )$ tends gradually to one. In this case, HW transforms into RFRW with the failure replacement limit $n=3$. Therefore, given $c_R=6$, when the warranty limit $m$ increases, the warranty cost produced by HW first closes and then equates to the warranty cost produced by RFRW with the failure replacement limit $n=3$.

Figure 3 displays how the failure replacement limit $n$ and the warranty limit $m$ affect HW, where $w=1$. As displayed in Figure 3, the warranty cost produced by HW first increases and then tends to a constant cost, which is the warranty cost resulting from RFRW (i.e., the traditional FRW mentioned in Case C), as the failure replacement limit $m$ and the warranty limit $m$ increase. This indicates numerically that the warranty cost produced by HW is lower than the warranty cost produced by RFRW.

Based on the above discussion and analysis, it is easy to conclude that the manufacturer can reduce or control the warranty cost by setting the number $m$ of random working cycles or/and the failure replacement limit $n$ when all random working cycles are uncontrollable, namely all random working cycles are decided by the project performed.

5.2. Sensitivity Analysis of the Post-Warranty Replacements

To illustrate the post-warranty replacements, we refer to three types of customized post-warranty replacements as policy 1 (in Section 3.1), policy 2 (in Section 3.2), and policy 3 (in Section 3.3), and call the uniform post-warranty replacement model policy 4 (in Section 4).

Optimum Value and Optimum Solution

To explore whether the optimum solution (i.e., $T^{*}$) and optimum value (i.e., $C{R}_j(T^{*})$, $j=1,2,3,4$) exist uniquely, we plot Figure 4 ($m=2$ and $w=2.2$), where Subplot a/b/c/d indicate policies 1/2/3/4, respectively. Figure 4 shows that the optimum cost rate $C{R}_j(T^{*})$ and the optimum replacement time $T^{*}$ exist uniquely. In addition, the optimum cost rate $C{R}_j(T^{*})$ and the optimum replacement time $T^{*}$ decrease with the increase in the failure replacement limit $n$. These changes show that a greater failure replacement limit $n$ can reduce the cost rate but cannot extend the replacement time.

5.3. Comparison

To illustrate the performances of the post-warranty replacements, we plot

Table 2. Comparison of different policies.

n	Policy 1		Policy 2		Policy 3		Policy 4
	E[L1]	CR1(T*)	E[L2]	CR2(T*)	E[L3]	CR3(T*)	E[L4]	CR4(T*)
1	3.6502	3.1813	4.7858	2.1517	5.6057	1.7051	5.0340	1.9952
2	3.8896	2.9583	4.9554	2.0354	6.5207	1.3952	5.1149	1.9458
3	4.0476	2.8279	5.0690	1.9650	7.5210	1.1757	5.1416	1.9300

, where $m=2$ and $w=2$.

Table 2 shows that when the failure replacement limit $n$ increases, the optimum cost rate $C{R}_j(T^{*})$ of policy $j$ ($j=1,2,3,4$) decreases, whereas the optimum life cycle length $E[L_j]$ increases. In addition, given $n$, the optimum life cycle lengths have the relationship, $E[L_1]<E[L_2]<E[L_4]<E[L_3]$, whereas the optimum cost rates have the relationship.$C{R}_3(T^{*})<C{R}_4(T^{*})<C{R}_2(T^{*})<C{R}_1(T^{*})$. These relationships indicate that the superior sequence among policies $j$ ($j=1,2,3,4$) is $\mathrm{P}\mathrm{o}\mathrm{l}\mathrm{i}\mathrm{c}\mathrm{y} 1<\mathrm{P}\mathrm{o}\mathrm{l}\mathrm{i}\mathrm{c}\mathrm{y} 2<\mathrm{P}\mathrm{o}\mathrm{l}\mathrm{i}\mathrm{c}\mathrm{y} 4<\mathrm{P}\mathrm{o}\mathrm{l}\mathrm{i}\mathrm{c}\mathrm{y} 3$. By the analysis in Section 4, the cost rate model related to policy 4 is a generalized model at which three HW expiration cases are integrated. Therefore, the superior sequence above indicates that policy 4 can sustain the reliability of the product through HW at the warranty limit $m$ or at the warranty period $w$ but cannot sustain the reliability of the product through HW at the time span $w$.

These cannot illustrate the superior sequence between the customized post-warranty replacements and the uniform post-warranty replacement.

To illustrate the superior sequence between the customized post-warranty replacements (i.e., policy 1/2/3) and the uniform post-warranty replacement (i.e., policy 4), we make some illustration, as follows. The cost rate $C{R}_I^{*}$ can be computed as

$$C{R}_I^{*}=\frac{E[\kappa ]c_f+Q_1\cdot E\left[C_1(T)\right]+Q_2\cdot E\left[C_2(T)\right]+Q_3\cdot E\left[C_3(T)\right]}{E[W]+Q_1\cdot \left(E(S_m)+E_1\left[T\right]\right)+Q_2\cdot \left(w+E_2\left[T\right]\right)+Q_3\cdot \left(w+E_3\left[T\right]\right)}$$

(37)

where $Q_1$, $Q_2$, and $Q_3$ are given in the analysis of Equation (28); $E_1\left[T\right]$ and $E\left[C_1(T)\right]$ are obtained by substituting the optimum replacement time $T^{*}$ of policy 1 into ${\displaystyle {\int }_0^w\left({\displaystyle {\int }_0^T\overline{F}(T|s)\mathrm{d}x}\right)\overline{F}(s)\mathrm{d}{G}^{(m)}(s)}/{\displaystyle {\int }_0^w\overline{F}(u)\mathrm{d}{G}^{(m)}(u)}$ and Equation (16); $E_2\left[T\right]$ and $E\left[C_2(T)\right]$ are obtained by substituting the optimum replacement time $T^{*}$ of policy 2 into ${\displaystyle {\int }_0^T\overline{F}(x|w)\mathrm{d}x}$ and Equation (20); $E_3\left[T\right]$ and $E\left[C_3(T)\right]$ are obtained by substituting the optimum replacement time $T^{*}$ of policy 3 into ${\displaystyle {\int }_0^T\overline{F}(x|w)\mathrm{d}x}$ and Equation (25). The total cost $C{L}_I^{*}$ can be given by $C{L}_I^{*}=E[W]+Q_1\cdot \left(E(S_m)+E_1\left[T\right]\right)+Q_2\cdot \left(w+E_2\left[T\right]\right)+Q_3\cdot \left(w+E_3\left[T\right]\right)$. In this article, we call the case related to $C{R}_I^{*}$ the integrated case, which is similar to the case in Equation (30) and is composed of policy 1, policy 2, and policy 3. In addition, $C{L}_I^{*}$ and $C{L}_4^{*}$ represent two total costs (i.e., life cycle costs) related to the integrated case and policy 4, respectively, when the life cycle length related to policy 4 is equal to the life cycle length related to the integrated case.

Based on the above illustrations, we plot

Table 3. Comparison of the Integrated Policy and Policy 4.

$\mathit{n}$	The Integrated Case		Policy 4		Comparison
	$\mathit{E}[{\mathit{L}}^{*}]$	$\mathit{C}{\mathit{R}}^{*}$	$\mathit{E}[{\mathit{L}}_4]$	$\mathit{C}{\mathit{R}}_4({\mathit{T}}^{*})$	$\mathit{C}{\mathit{L}}_{\mathit{I}}^{*}$	$\mathit{C}{\mathit{L}}_4^{*}$
1	6.4653	2.2241	5.0340	1.9952	72.3863	64.9364
2	6.6007	2.1832	5.1149	1.9458	73.7090	65.6939
3	6.6500	2.1719	5.1416	1.9300	74.2608	65.9899

where $m=2$ and $w=2$. Table 3 shows that the total cost $C{L}_I^{*}$ is greater than the total cost $C{L}_4^{*}$, i.e., $C{L}_I^{*}>C{L}_4^{*}$, when the failure replacement limit $n$ is given. This relationship means that uniform post-warranty replacement can extend the replacement time at a lower cost. Namely, the performance of the uniform post-warranty replacement (i.e., policy 4) is superior to the customized post-warranty replacements (i.e., policy 1/2/3), from a whole performance perspective.

6. Conclusions

In this article, a hybrid warranty (HW) was designed from the manufacturer’s point of view by introducing a limited number of random working cycles and a limited number of failure replacements to the renewing free replacement warranty (RFRW). The warranty cost related to HW was obtained and the characteristics of HW were illustrated by discussing warranty terms. From the consumer’s point of view, the customized post-warranty replacements were modeled by distinguishing HW expiration cases; the uniform post-warranty replacement was also modeled by unifying all cases of HW expiration. Each post-warranty replacement requires that the product through HW be replaced at a replacement time or at failure, whichever takes place earlier. The depreciation expense with operating time is incorporated into each post-warranty replacement. By discussing parameters, it is found that some classic cost rate models are special cases of the cost rate model derived in this article. Sensitivity analysis on some key parameters on HW and the post-warranty replacements were performed in numerical experiments. It was shown that when the number of random working cycles or/and the number of failure replacements is designed as a warranty term, the warranty cost can be reduced; and the performance of the uniform post-warranty replacement is superior to the performance of the customized post-warranty replacements from a whole performance perspective.

In this article, we only considered the product whose failure belongs to self-announcing failure while ignoring the defective stage [51,52] and product’s performance. Therefore, some other directions can also be investigated by considering the product’s performance, as shown below.

• By characterizing the product’s performance as a stochastic degradation process, some performance-based warranties and performance-based maintenance models can be defined and investigated.
• By combining the product’s performance and the product’s working cycles, we can define and construct some random warranty models and some random maintenance models.