Optimum Post-Warranty Maintenance Policies for Products with Random Working Cycles

Abstract

The working cycle of the products can be supervised by sensors and other measuring technologies. This fact means that by supervising the working cycle, the manufacturer can devise a warranty policy, and by continuing to supervise the post-warranty working cycle, the consumer can model the post-warranty maintenance. However, in the literature, there is no associated work. Integrating a renewing free-replacement warranty (RFRW) and the number of working cycles, this paper proposes a two-dimensional renewing free-replacement warranty policy, which can be applied to warrant the product and analyze the related warranty cost. By extending the warranty policy to the post-warranty maintenance model, we investigate two kinds of post-warranty maintenance models, including the uniform post-warranty maintenance model and the customized post-warranty maintenance model. For each post-warranty maintenance model, we provide an algorithm to seek the optimum solution. Finally, we provide some numerical experiments to demonstrate the model. The numerical results show that for the produced warranty cost, the traditional RFRW is higher than the proposed warranty policy, and the customized policy is inferior to the uniform policy.

1. Introduction

In our daily life, almost all of us depend on various services and products [1,2,3,4]. We expect that our cars, electrical appliances, mobile phones, computers and so on to function for a rather long time. As a contract, a warranty can benefit both consumers and manufacturers. From the viewpoint of the consumer, when purchasing the product with a warranty, the reliability of the product can be guaranteed, and during the warranty period, the manufacturer can shoulder the repair cost. From the viewpoint of the manufacturer, offering products with a warranty can remove consumer’s doubt regarding product quality and can improve consumer’s reorganization.

Owing to the importance of warranties, warranty policies have been investigated in industry and academia. The literature concerning warranty policies can be divided into two research directions, depending on reliability technology. The first direction aims to devise warranty policies by assuming that the product lifetime with self-announcing failure has a probability distribution function, i.e., devising warranty policies based on the distribution. They are shown in Qiao et al. [5], Luo and Wu [6], Zheng et al. [7], Zhu et al. [8], Liu et al. [9], He et al. [10], Ye and Murthy [11], Xie et al. [12], Tong and Liu [13], Su and Wang [14] and Hooti et al. [15]. The second direction focuses on devising warranty policies by simulating product failure as degradation failure, which is found in Sánchez-Silva and Klutke [16], Zhang et al. [17], Cha et al. [18], Shang et al. [19] and Zhang et al. [20].

From the warranty contract, during the warranty period, the manufacturer is responsible for maintaining product reliability through replacement, maintenance or other methods, while during the post-warranty period, consumers concentrate on how to maintain reliability. Because of increased maintenance costs, the problem of devising a post-warranty maintenance policy has received considerable attention. This problem has been explored in Park et al. [21,22], Liu et al. [23,24], Zhao and Nakagawa [25] Sheu et al. [26] and Shang et al. [19].

Measuring tools, such as advanced sensors, can supervise the working cycle of the product, which can successively perform projects at random working cycles and deteriorate in operating time from the reliability theory. Considering this fact, some researchers have investigated maintenance policies by simulating the working cycle as an independent and identically distributed random variable sequence. Related works have appeared in Zhao et al. [27], Sheu et al. [28], Nakagawa et al. [29,30].

For a product working successively at a random working cycle, by supervising the working cycle, the manufacturer can devise a warranty policy to ensure product reliability. After stopping the warranty, the consumer can keep on supporting the product reliability by tracking the working cycle. However, a warranty policy that can guarantee product reliability has not been developed from the viewpoint of the manufacturer. In addition, from the consumer perspective, post-warranty maintenance models have rarely been explored.

After integrating the number of working cycles of the product and an RFRW policy, from the manufacturer’s perspective, this paper proposes a two-dimensional renewing free-replacement warranty policy. Under the warranty policy, when a failure does not appear before the number of working cycles reaches a predesignated value or before a warranty period limit, whichever appears first, then the product will be covered by a warranty. A maintenance policy is expressed by integrating a classic periodic replacement policy and preventive maintenance, which can help the consumer bear the reliability of the product. Based on the differences in warranty limits, post-warranty policies are customized, and the related cost rates are developed. Unifying each warranty limit, the uniform post-warranty policy is modeled, and the related cost rate is established.

The contribution of this article can be summarized in the following three ways: (a) by integrating the number of working cycles and an RFRW policy, we propose a two-dimensional renewing free-replacement warranty policy, which can guarantee the reliability of the product that executes some projects at a random working cycle; (b) based on the view of the consumer, we extend the warranty policy to the post-warranty maintenance policy; and (c) by reflecting the difference in the warranty limit, we customize the post-warranty maintenance policy, which is different from the conventional literature, where the post-warranty maintenance policy is implicitly uniform.

The remainder of this paper is structured as follows. Section 2 proposes a two-dimensional warranty policy and analyzes the related warranty cost for the manufacturer. In Section 3, by differentiating the warranty limit, the cost rate models for consumers are established. Unifying all warranty limits, Section 4 derives a uniform cost rate based on the view of the consumer. Section 5 presents an algorithm to look for optimum solution values. Section 6 provides a numerical experiment to validate the proposed approach, and sensitivity analysis is executed. Finally, some conclusions are presented in Section 7.

2. Warranty Modeling for Products

We assume that a product works successively, and the working cycle of the $i\mathrm{th}$ ($i=1,2,\dots$) project $Y_i$ has an identical distribution function G(y) = Pr{Y_i < y} possessing a mean $1/\lambda$. The product deteriorates in its operating time, and the time-to-first-failure $X$ has a general distribution function F(x) = Pr{X < x}. Moreover, we assume that the time for preventive maintenance, downtime caused by each minimal repair and replacement time can be neglected completely.

2.1. Two-Dimensional Warranty Proposal

The number for the random working cycle is $m$, which is a pre-specified value and is greater than zero. The warranty period is $w$. The two-dimensional warranty is described below.

(1)

Once product failure occurs, it will be replaced by an identical new product possessing the same warranty when the failure appears before the number of random working cycles satisfies a pre-assigned value $m$, or before the warranty period $w$, whichever appears first;

(2)

the failure replacement enacted by the manufacturer will be completed during the warranty region $(0,m]\times (0,w]$ when the failure does not appear;

(3)

The manufacturer undertakes the whole replacement cost caused by product failure during the warranty region $(0,m]\times (0,w]$, including labor cost, production cost and transport cost.

Obviously, the warranty includes two types of warranty limits and a renewing free-replacement warranty term. Therefore, it is called a two-dimensional renewing free-replacement warranty (2DRFRW) policy.

2.2. Warranty Cost Modeling

The operating time $S_m$ for the product is the sum of the $m$ random working cycles, i.e., $S_m={\displaystyle \sum _{i=1}^m{Y}_i}$, which has a distribution function $G^{(m)}(s)$ ($G^{(m)}(s)={\displaystyle {\int }_0^s{G}^{(m-1)}(s-u)\mathrm{d}G(u)}$) and a survival function ${\overline{G}}^{(m)}(s)$ (${\overline{G}}^{(m)}(s)=1-G^{(m)}(s)$). Based on the 2DRFRW, when the failure does not appear during the warranty region, then the product undergoes a warranty. Therefore, the product will undergo a warranty at the warranty limit $w$ or at the warranty limit $m$, whichever appears first. Then, the probability that the product undergoes a warranty is

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

(1)

where ${\displaystyle {\int }_0^w\overline{F}(u)\mathrm{d}{G}^{(m)}(u)}=\mathrm{Pr}\{S_m<w,S_m<X\}$, ${\overline{G}}^{(m)}(w)\overline{F}(w)=\mathrm{Pr}\{S_m>w,X>w\}$ and $1-F(\cdot )=\overline{F}(\cdot )$.

During the warranty region, product failure can be classified into two cases. The first case is that the failure appears before the warranty limit $m$. Another case is that the failure appears before the warranty limit $w$. Thus, the probability that the product fails during the warranty region can be computed as

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

(2)

Note that the second equality in Equation (2) can be established by means of the total probability formula in probability theory.

Applying probability theory, the probability that until the $i\mathrm{th}$ ($i=1,2,\dots$) product undergoes a warranty is $p^{i-1}q$. Based on this fact, the number of failure replacements is $i-1$. Moreover, the expected number $E[\kappa ]$ of failure replacements can be given as

$$E[\kappa ]=\frac{p}{q}=\frac{{\displaystyle {\int }_0^w{\overline{G}}^{(m)}(u)\mathrm{d}F(u)}}{1-{\displaystyle {\int }_0^w{\overline{G}}^{(m)}(u)\mathrm{d}F(u)}}={\displaystyle \sum _{i=1}^{\infty }{p}^{i-1}q}(i-1)$$

(3)

According to the traditional renewing (renewable) free-replacement (repair) warranty (RFRW) policy (see Wang et al. [31], Lee et al. [32] and Marshall et al. [33]), the expected number $E[\kappa ]$ of failure replacements is

$$E[\kappa ]=F(w)/\overline{F}(w)$$

(4)

Since $F(w)={\displaystyle {\int }_0^w\mathrm{d}F(u)}>{\displaystyle {\int }_0^w{\overline{G}}^{(m)}(u)\mathrm{d}F(u)}$ and $\overline{F}(w)=1-{\displaystyle {\int }_0^w\mathrm{d}F(u)}<1-{\displaystyle {\int }_0^w{\overline{G}}^{(m)}(u)\mathrm{d}F(u)}$, the inequality $F(w)/\overline{F}(w)>{\displaystyle {\int }_0^w{\overline{G}}^{(m)}(u)\mathrm{d}F(u)}/\left(1-{\displaystyle {\int }_0^w{\overline{G}}^{(m)}(u)\mathrm{d}F(u)}\right)$ holds. This inequality indicates that the replacement frequency caused by the traditional RFRW is larger than the replacement frequency caused by the proposed 2DRFRW. Therefore, the proposed 2DRFRW policy can produce a smaller warranty cost compared with the traditional RFRW.

3. Customized Post-Warranty Maintenance Policy

For a product to undergo a warranty at the warranty limit $m$ or $w$, it has worked with $S_m$ or $w$. When a maintenance policy is applied to preserve the post-warranty reliability of the product at the warranty limit $m$, this maintenance policy can produce a maintenance schedule of its own. A similar situation exists for a product undergoing a warranty at the warranty limit $w$. Because the history working times are different, i.e., $S_m\ne w$, their respective maintenance schedules are two distinct schedules, which are reflected exactly in the difference of the optimum value of decision variables. This fact indicates that we can customize post-warranty maintenance policy to preserve post-warranty reliability.

For each of the two cases (or two warranty limits), the related post-warranty maintenance policy will be customized from the viewpoint of the consumer based on a maintenance policy, which is formulated by integrating preventive maintenance (PM) at warranty expiration into a conventional periodic replacement policy (see Zhao et al. [34], Toledo et al. [35] and Chien [36]). We refer to the integrated maintenance policy as the proposed maintenance policy. Moreover, we define the life cycle of a product as an interval at the consumer’s expense, from its installation time to its replacement time, which is similar to Shang et al. [37] and Liu et al. [24]. According to this definition, we can derive the expected cost rate associated with each post-warranty maintenance policy.

3.1. Post-Warranty Maintenance Policy 1

For the case where the product undergoes a warranty at the warranty limit $m$, the post-warranty maintenance policy will be customized from the standpoint of the consumer based on the proposed maintenance policy.

3.1.1. Life Cycle Cost Modeling

When the product undergoes a warranty at the warranty limit $m$, then the operating time of the product is equal to $S_m$. In this case, the distribution function $H_{S_m}(s)$ of the operating time $S_m$ can be computed as

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

(5)

where $w>s>0$.

In reliability engineering, PM cost increases with both the reliability increment and age. The reliability increment is usually simulated by failure rate reduction or/and age reduction. In this article, age reduction is applied as a measure of the reliability increment. For the product through warranty at warranty limit $m$, its age equates to its operating time $S_m$. Denote the function $S_m\left(1-\phi (\mathit{n})\right)$ with the decision variable $\mathit{n}$ ($\mathit{n}=0,1,\dots$) by the reliability increment at $S_m$, where $\phi (\mathit{n})$ is a decreasing function with the maintenance ability level $\mathit{n}$, $1\ge \phi (\mathit{n})\ge 0$, $\phi (0)=1$ and $\phi (\infty )=0$. When the proposed maintenance policy is applied to preserve the post-warranty reliability, then the PM cost at $S_m$ can be modeled as an increasing function as follows:

$$c_h{\left(1-\phi (\mathit{n})\right)}^{\alpha }{\left(S_m\right)}^{\alpha +\beta }=C_{S_m}^{PM}=c_h{\left(\left(1-\phi (\mathit{n})\right)S_m\right)}^{\alpha }{\left(S_m\right)}^{\beta }$$

(6)

where $c_h>0$, $\alpha >0$ and $\beta >0$. Obviously, when $\phi (\infty )=0$, then the PM at $S_m$ translates into perfect PM; when $\phi (0)=1$, then any maintenance (including PM and minimal repair) is not performed.

Since $S_m$ follows $H_{S_m}(s)$ in Equation (5), the expected value $E[C_{S_m}^{PM}]$ of PM cost at $S_m$ can be derived as

$$\frac{c_h{\left(1-\phi (\mathit{n})\right)}^{\alpha }{\displaystyle {\int }_0^w{\left(s\right)}^{\alpha +\beta }\overline{F}(s)\mathrm{d}{G}^{(m)}(s)}}{{\displaystyle {\int }_0^w\overline{F}(u)\mathrm{d}{G}^{(m)}(u)}}=[c_h{\left(1-\phi (\mathit{n})\right)}^{\alpha }{\left(S_m\right)}^{\alpha +\beta }]=E[C_{S_m}^{PM}]$$

(7)

The product undergoing PM at $S_m$ has a failure rate $r(\phi (\mathit{n})S_m+u)$. When the proposed maintenance policy is applied to preserve the reliability of the product after warranty expiration, the product will be replaced if the operating time reaches the replacement time $\mathit{T}$. Therefore, until replacement occurs at $\mathit{T}$, the total minimal repair cost $\mathsf{\Lambda }(\mathit{T}|S_m)$ caused by minimal repair is given by

$$c_m{\displaystyle {\int }_0^{\mathit{T}}r(\phi (\mathit{n})S_m+u)\mathrm{d}u}=\mathsf{\Lambda }(\mathit{T}|S_m)$$

(8)

where $c_m$ is the unit minimal repair cost.

Since $S_m$ follows $H_{S_m}(s)$ in Equation (5), the expected value $\mathsf{\Lambda }(\mathit{T})$ of the total minimal repair cost $\mathsf{\Lambda }(\mathit{T}|S_m)$ can be computed as

$$\frac{c_m{\displaystyle {\int }_0^w\left({\displaystyle {\int }_0^{\mathit{T}}r(\phi (\mathit{n})s+u)\mathrm{d}u}\right)\overline{F}(s)\mathrm{d}{G}^{(m)}(s)}}{{\displaystyle {\int }_0^w\overline{F}(u)\mathrm{d}{G}^{(m)}(u)}}=E\left[\mathsf{\Lambda }(\mathit{T}|S_m)\right]=\mathsf{\Lambda }(\mathit{T})$$

(9)

Denote $c_f$ by the unit failure cost caused by unit failure replacement. By Equation (3), the expected failure cost is calculated as $E[\kappa ]c_f$. Let $c_p$ be the unit replacement cost. By the definition of the life cycle, by summing $E[\kappa ]c_f$, $E[C_{S_m}^{PM}]$, $\mathsf{\Lambda }(\mathit{T})$ and $c_p$, the expected value of the life cycle cost can be obtained as

$$\begin{array}{ll}E\left[C(L_1)\right]& =E[\kappa ]c_f+E\left[C_{S_m}^{PM}\right]+\mathsf{\Lambda }(\mathit{T})+c_p\\ & =c_p+\frac{{\displaystyle {\int }_0^w\left(c_h{\left(1-\phi (\mathit{n})\right)}^{\alpha }{\left(s\right)}^{\alpha +\beta }+c_m{\displaystyle {\int }_0^{\mathit{T}}r(\phi (\mathit{n})s+u)\mathrm{d}u}\right)\overline{F}(s)\mathrm{d}{G}^{(m)}(s)}}{{\displaystyle {\int }_0^w\overline{F}(u)\mathrm{d}{G}^{(m)}(u)}}+\frac{c_f{\displaystyle {\int }_0^w{\overline{G}}^m(u)\mathrm{d}F(u)}}{1-{\displaystyle {\int }_0^w{\overline{G}}^m(u)\mathrm{d}F(u)}}\end{array}$$

(10)

3.1.2. Life Cycle Length Modeling

Let $X_k$ be the lifetime of the $k\mathrm{th}$ failed product during the warranty region; then, according to probability theory, the distribution function $H(x)$ of the lifetime $X_k$ can be represented as

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

(11)

where $0<x<w$.

Until the $i\mathrm{th}$($i=1,2,\dots$) product undergoes a warranty, the manufacturer performs $i-1$ failure replacements. Thus, the total warranty service period until the $i\mathrm{th}$ product undergoes the warranty can be obtained as ${\displaystyle \sum _{k=1}^{i-1}{X}_k}$. Since the number $i-1$ of failure replacements follows a geometric distribution $p^{i-1}q$, the expected value of the warranty service period ${\displaystyle \sum _{k=1}^{i-1}{X}_k}$ can be expressed as

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

(12)

where $q$ and $p$ have been offered in Equations (1) and (2), respectively, and $E\left[X_k\right]={\displaystyle {\int }_0^{w}x\mathrm{d}H(x)}={\displaystyle {\int }_0^{w}x{\overline{G}}^{(m)}(x)\mathrm{d}F(x)}/{\displaystyle {\int }_0^w{\overline{G}}^{(m)}(u)\mathrm{d}F(u)}$.

For the product undergoing a warranty at the warranty limit $m$, its warranty service period equals the operating time $S_m$. Therefore, the expected value of the warranty service period (i.e., $S_m$) can be obtained as

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

(13)

where $H_{S_m}(s)$ has been offered in Equation (5).

Based on the proposed maintenance policy, when the operating time of the product undergoing the warranty at $m$ reaches the replacement time $\mathit{T}$, this product will be replaced. In this case, the post-warranty period of the product is equal to $\mathit{T}$. According to the definition of the life cycle, by summing $E\left[S_m\right]$, $E\left[W\right]$ and $\mathit{T}$, the expected length of the life cycle can be computed as

$$E[L_1]=E\left[S_m\right]+E\left[W\right]+\mathit{T}=\frac{{\displaystyle {\int }_0^{w}s\overline{F}(s)\mathrm{d}{G}^{(m)}(s)}}{{\displaystyle {\int }_0^w\overline{F}(u)\mathrm{d}{G}^{(m)}(u)}}+\frac{{\displaystyle {\int }_0^{w}x{\overline{G}}^{(m)}(x)\mathrm{d}F(x)}}{1-{\displaystyle {\int }_0^w{\overline{G}}^{(m)}(u)\mathrm{d}F(u)}}+\mathit{T}$$

(14)

3.1.3. Cost Rate Modeling

The expected length $E[L_1]$ of the life cycle and the expected value $E\left[C(L_1)\right]$ of the life cycle cost are presented in Equations (14) and (10), respectively.

Let

$$B_1={\displaystyle {\int }_0^{w}x{\overline{G}}^m(x)\mathrm{d}F(x)}/\left(1-{\displaystyle {\int }_0^w{\overline{G}}^m(u)\mathrm{d}F(u)}\right)+{\displaystyle {\int }_0^{w}s\overline{F}(s)\mathrm{d}{G}^{(m)}(s)}/{\displaystyle {\int }_0^w\overline{F}(u)\mathrm{d}{G}^{(m)}(u)}$$

$$A_1=c_f{\displaystyle {\int }_0^w{\overline{G}}^m(u)\mathrm{d}F(u)}/\left(1-{\displaystyle {\int }_0^w{\overline{G}}^m(u)\mathrm{d}F(u)}\right)+c_P$$

Applying the renewal rewarded theorem, the expected cost rate $C{R}_1(\mathit{n},\mathit{T})$ can be calculated as

$$C{R}_1(\mathit{n},\mathit{T})=\frac{{\displaystyle {\int }_0^w\left(c_m{\displaystyle {\int }_0^{\mathit{T}}r(\phi (\mathit{n})s+u)\mathrm{d}u}+c_h{\left(1-\phi (\mathit{n})\right)}^{\alpha }{\left(s\right)}^{\alpha +\beta }\right)\overline{F}(s)\mathrm{d}{G}^{(m)}(s)}/{\displaystyle {\int }_0^w\overline{F}(u)\mathrm{d}{G}^{(m)}(u)+A_1}}{\mathit{T}+B_1}$$

(15)

3.2. Post-Warranty Maintenance Policy 2

In this section, the post-warranty maintenance policy will be customized from the standpoint of the consumer based on the proposed maintenance policy, given the case in which the product undergoes a warranty at the warranty limit $w$.

3.2.1. Life Cycle Cost Modeling

When the product undergoes a warranty at the warranty limit $w$, then its age is equal to $w$. When PM is performed at $w$, similar to Equation (6), the related PM cost $E\left[C_w^{PM}\right]$ can be given by

$$c_h{\left(1-\phi (\mathit{n})\right)}^{\alpha }{\left(w\right)}^{\alpha +\beta }=E\left[C_w^{PM}\right]$$

(16)

Until the product undergoing a warranty at $w$ is replaced at $\mathit{T}$, the expected value $\mathsf{\Lambda }(\mathit{T}|w)$ of the total minimal repair cost generated by minimal repair is given by

$$c_m{\displaystyle {\int }_0^{\mathit{T}}r(\phi (\mathit{n})w+u)\mathrm{d}u}=\mathsf{\Lambda }(\mathit{T}|w)$$

(17)

where $r(\phi (\mathit{n})w+u)$ is a failure rate function for the product undergoing PM at $w$.

According to the definition of the life cycle, by summing $E[\kappa ]c_f$, $c_P$, $\mathsf{\Lambda }(\mathit{T}|w)$ and $E\left[C_w^{PM}\right]$, the expected value of the life cycle cost can be expressed by

$$\begin{array}{ll}E\left[C(L_2)\right]& =E\left[C_w^{PM}\right]+\mathsf{\Lambda }(\mathit{T}|w)+c_P+E[\kappa ]c_f\\ & =\frac{c_f{\displaystyle {\int }_0^w{\overline{G}}^m(u)\mathrm{d}F(u)}}{1-{\displaystyle {\int }_0^w{\overline{G}}^m(u)\mathrm{d}F(u)}}+c_m{\displaystyle {\int }_0^{\mathit{T}}r(\phi (\mathit{n})w+u)\mathrm{d}u}+c_P+c_h{\left(1-\phi (\mathit{n})\right)}^{\alpha }{\left(w\right)}^{\alpha +\beta }\end{array}$$

(18)

3.2.2. Life Cycle Length Modeling

For the product undergoing a warranty at $w$, its warranty service period is $w$. Using the definition of the life cycle, by summing $E\left[W\right]$, $\mathit{T}$ and $w$, the expected length $E\left[L_2\right]$ of the life cycle can be given by

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

(19)

3.2.3. Cost Rate Modeling

Let

$$A_2=c_f{\displaystyle {\int }_0^w{\overline{G}}^m(u)\mathrm{d}F(u)}/\left(1-{\displaystyle {\int }_0^w{\overline{G}}^m(u)\mathrm{d}F(u)}\right)+c_P$$

$$B_2={\displaystyle {\int }_0^{w}x{\overline{G}}^m(x)\mathrm{d}F(x)}/\left(1-{\displaystyle {\int }_0^w{\overline{G}}^m(u)\mathrm{d}F(u)}\right)+w$$

According to the renewal rewarded theorem, the expected cost rate $C{R}_2(\mathit{n},\mathit{T})$ can be calculated as

$$C{R}_2(\mathit{n},\mathit{T})=\frac{c_h{\left(1-\phi (\mathit{n})\right)}^{\alpha }{\left(w\right)}^{\alpha +\beta }+A_2+c_m{\displaystyle {\int }_0^{\mathit{T}}r(\phi (\mathit{n})w+u)\mathrm{d}u}}{\mathit{T}+B_2}$$

(20)

4. Uniform Post-Warranty Maintenance Policy

In this section, by simultaneously considering the two cases mentioned above, a uniform post-warranty maintenance policy will be modeled from the standpoint of consumers in terms of the proposed maintenance policy. The cost rate associated with the uniform post-warranty maintenance policy will be developed using the life cycle definition, which is provided in Section 3.

4.1. Life Cycle Cost Modeling

As mentioned above, the case in which the product is undergoing a warranty can be divided into two kinds of cases. The first case is that the product undergoes a warranty at the warranty limit $m$. The second case is that the product undergoes a warranty at the warranty limit $w$.

The probability $Q_1$ that the first case appears can be computed as $Q_1={\displaystyle \sum _{i=1}^{\infty }{p}^{i-1}{q}_1}$, where $q_1$ $q_1=\mathrm{Pr}\{S_m<w,S_m<X\}={\displaystyle {\int }_0^w\overline{F}(u)\mathrm{d}{G}^{(m)}(u)}$. The probability $Q_2$ that the second case appears can be computed as $Q_2={\displaystyle \sum _{i=1}^{\infty }{p}^{i-1}{q}_2}$, where $q_2=\mathrm{Pr}\{w<S_m,w<X\}={\overline{G}}^{(m)}(w)\overline{F}(w)$.

In addition, during the post-warranty period $\mathit{T}$, the expected costs associated with the two kinds of cases are $E\left[C_{S_m}^{PM}\right]+E\left[\mathsf{\Lambda }(\mathit{T}|S_m)\right]+c_p$ and $E\left[C_w^{PM}\right]+\mathsf{\Lambda }(\mathit{T}|w)+c_p$. By definition of the life cycle in Section 3, the expected value $E\left[C(L_u)\right]$ of the life cycle cost can be given by

$$\begin{array}{l}E\left[C(L_u)\right]=Q_1\left(E\left[C_{S_m}^{PM}\right]+E\left[\mathsf{\Lambda }(\mathit{T}|S_m)\right]+c_p\right)+E[\kappa ]c_f+Q_2\left(E\left[C_w^{PM}\right]+\mathsf{\Lambda }(\mathit{T}|w)+c_p\right)\\ =\left[q_2\left(E\left[C_w^{PM}\right]+\mathsf{\Lambda }(\mathit{T}|w)\right)+q_1\left(E\left[C_{S_m}^{PM}\right]+E\left[\mathsf{\Lambda }(\mathit{T})\right]\right)\right]\left({\displaystyle \sum _{i=1}^{\infty }{p}^{i-1}}\right)+c_p+E[\kappa ]c_f=\\ c_p+\frac{c_m{\displaystyle {\int }_0^{\mathit{T}}r(u)\mathrm{d}u}+c_f{\displaystyle {\int }_0^w{\overline{G}}^m(u)\mathrm{d}F(u)}+{\displaystyle {\int }_0^w{\overline{G}}^{(m)}(s)\mathrm{d}\left(c_m{\displaystyle {\int }_0^{\mathit{T}}r(\phi (\mathit{n})s+u)\mathrm{d}u+c_h{\left(1-\phi (\mathit{n})\right)}^{\alpha }{\left(s\right)}^{\alpha +\beta }}\right)\overline{F}(s)}}{1-{\displaystyle {\int }_0^w{\overline{G}}^m(u)\mathrm{d}F(u)}}\end{array}$$

(21)

where $E[\kappa ]$ was provided in Equation (3), $E[C_{S_m}^{PM}]$ was provided in Equation (7), $E[\mathsf{\Lambda }(\mathit{T}|S_m)]$ was presented in Equation (9), $E[C_w^{PM}]$ was given by Equation (16), and $\mathsf{\Lambda }(\mathit{T}|w)$ was expressed by Equation (17).

4.2. Life Cycle Length Modeling

For a product undergoing a warranty at $m$ or at $w$, whichever occurs first, its expected operating times associated with two kinds of cases are $E[S_m]+\mathit{T}$ and $w+\mathit{T}$. Similar to the derivation principle of Equation (21), by the definition of the life cycle in Section 3, the expected value $E[L_u]$ of the life cycle length can be given by

$$E[L_u]=\mathit{T}+\frac{{\displaystyle {\int }_0^w{\overline{G}}^{(m)}(u)\overline{F}(u)\mathrm{d}u}}{1-{\displaystyle {\int }_0^w{\overline{G}}^{(m)}(u)\mathrm{d}F(u)}}=E[W]+Q_1\left(E[S_m]+\mathit{T}\right)+Q_2\left(w+\mathit{T}\right)$$

(22)

where $E[W]$ was offered in Equation (12), and $E[S_m]$ was provided in Equation (13).

4.3. Cost Rate Modeling

The expected value $E[C(L_u)]$ of the life cycle cost and the expected value $E[L_u]$ of the life cycle length were given, respectively, by Equations (21) and (22). Let $A_3=c_f{\displaystyle {\int }_0^w{\overline{G}}^m(u)\mathrm{d}F(u)}/\left(1-{\displaystyle {\int }_0^w{\overline{G}}^m(u)\mathrm{d}F(u)}\right)+c_p$; then, the expected cost rate $C{R}_u(\mathit{n},\mathit{T})$ can be given by

$$\begin{array}{l}C{R}_u(\mathit{n},\mathit{T})=\\ \frac{c_m{\displaystyle {\int }_0^{\mathit{T}}r(u)\mathrm{d}u}+A_3\left(1-{\displaystyle {\int }_0^w{\overline{G}}^m(u)\mathrm{d}F(u)}\right)+{\displaystyle {\int }_0^w{\overline{G}}^{(m)}(s)\mathrm{d}\left(c_m{\displaystyle {\int }_0^{\mathit{T}}r(\phi (\mathit{n})s+u)\mathrm{d}u}+c_h{\left(1-\phi (\mathit{n})\right)}^{\alpha }{\left(s\right)}^{\alpha +\beta }\right)\overline{F}(s)}}{\mathit{T}\left(1-{\displaystyle {\int }_0^w{\overline{G}}^{(m)}(u)\mathrm{d}F(u)}\right)+{\displaystyle {\int }_0^w{\overline{G}}^{(m)}(u)\overline{F}(u)\mathrm{d}u}}\end{array}$$

(23)

4.4. Special Cases

Obviously, in Equation (23), the expected cost rate $C{R}_u(\mathit{n},\mathit{T})$ is a generalized model. Hence, if some parameters are assigned special values, the generalized model can be reduced as some special models.

Case A: if $m=0$, the model can be rewritten as

$$C{R}_u(\mathit{n},\mathit{T})=\frac{c_m{\displaystyle {\int }_0^{\mathit{T}}r(\phi (\mathit{n})w+u)\mathrm{d}u}+c_p{c}_{f}F(w)/\overline{F}(w)+c_h{\left(1-\phi (\mathit{n})\right)}^{\alpha }{\left(w\right)}^{\alpha +\beta }}{\mathit{T}+{\displaystyle {\int }_0^w\overline{F}(u)\mathrm{d}u}/\overline{F}(w)}$$

(24)

If $m=0$, then the sum $S_m$ equals 0 and ${\overline{G}}^{(m)}(u)$ equals 1, i.e., $\underset{m\to 0}{\lim }{S}_m=0$ and $\underset{m\to 0}{\lim }{\overline{G}}^{(m)}(u)=1$. This fact indicates that the warranty limit $m$ can be ignored and that the proposed 2DRFRW can be translated into the traditional RFRW.

Therefore, this model is the expected cost rate at which the conventional RFRW is applied as a unique warranty policy, and the proposed maintenance policy is applied as the post-warranty maintenance policy.

Case B: if $\mathit{n}=0$, the model above can be rewritten as

$$C{R}_u(\mathit{n},\mathit{T})=\frac{{\displaystyle {\int }_0^w{\overline{G}}^{(m)}(s)\mathrm{d}\left(c_m{\displaystyle {\int }_0^{\mathit{T}}r(s+u)\mathrm{d}u}\right)\overline{F}(s)}+A_3\left(1-{\displaystyle {\int }_0^w{\overline{G}}^m(u)\mathrm{d}F(u)}\right)+c_m{\displaystyle {\int }_0^{\mathit{T}}r(u)\mathrm{d}u}}{\mathit{T}\left(1-{\displaystyle {\int }_0^w{\overline{G}}^{(m)}(u)\mathrm{d}F(u)}\right)+{\displaystyle {\int }_0^w{\overline{G}}^{(m)}(u)\overline{F}(u)\mathrm{d}u}}$$

(25)

where $A_3=c_f{\displaystyle {\int }_0^w{\overline{G}}^m(u)\mathrm{d}F(u)}/\left(1-{\displaystyle {\int }_0^w{\overline{G}}^m(u)\mathrm{d}F(u)}\right)+c_p$.

When $\mathit{n}=0$, we have $\phi (0)=1$. This fact indicates that PMs at $S_m$ and $w$ were ignored. Therefore, this model is the expected cost rate where a conventional periodic replacement policy is applied as the post-warranty maintenance policy.

Case C: if $m=\mathit{n}=0$, then the model can be rewritten as

$$C{R}_u(\mathit{n},\mathit{T})=\frac{c_{f}F(w)/\overline{F}(w)+c_p+c_m{\displaystyle {\int }_0^{\mathit{T}}r(w+u)\mathrm{d}u}}{\mathit{T}+{\displaystyle {\int }_0^w\overline{F}(u)\mathrm{d}u}/\overline{F}(w)}$$

(26)

This model is the expected cost rate, where the conventional RFRW is applied to warrant the product, and a conventional periodic replacement policy is applied to preserve the post-warranty reliability of the product.

5. Optimizing

Decision variables, including the maintenance ability level $\mathit{n}$ and the replacement time $\mathit{T}$ in the three cost rate models mentioned above, are identical. Since the expression of $F(x)$ is nonspecific and undefined, it is difficult to obtain the analytically optimum solutions. However, the existence and uniqueness of optimum solutions can be concluded by discussing the first-order derivative with respect to the decision variables of the cost rate. A similar process has been presented and extensively discussed in detail in the literature (see Park et al. [22], Zhao et al. [27] and Sheu et al. [28]). Considering this fact, the existence and uniqueness of optimum solutions are not summarized. In contrast, a numerical expression of optimum solutions can be obtained by applying the following algorithm when all parameters except the decision variables are given.

Let $i=0$, $\mathsf{\Upsilon }$ and $\mathit{T}$ be two empty sets, then,

Step 1: Given $i$, calculate the optimum solution ${\mathit{T}}_i^{*}$ and the optimum value $C{R}_j(i,{\mathit{T}}_i^{*})$ ($j=1,2,u$) by optimizing model $C{R}_j(i,\mathit{T})$, and then regard ${\mathit{T}}_i^{*}$ as the $i\mathrm{th}$ element of the set $\mathit{T}$ and $C{R}_j(i,{\mathit{T}}_i^{*})$ as the $i\mathrm{th}$ element of the set $\mathsf{\Upsilon }$;

Step 2: Let $i=i+1$, then repeat Step 1 until i = n;

Step 3: Seeking the minimum value ${\mathit{T}}_i^{*}$ in the set $\mathit{T}$ and seeking the maximum value $C{R}_j(i,{\mathit{T}}_i^{*})$ in the set $\mathsf{\Upsilon }$, then the optimum replacement time ${\mathit{T}}_{}^{*}={\mathit{T}}_i^{*}$ and the optimum maintenance ability level ${\mathit{n}}^{*}=i$.

6. Numerical Experiments

The intelligent mobile robot is applied to check the hidden trouble of the high-voltage electric equipment. The manufacturer and consumer can detect the state of robots by using advanced measuring technology, such as turn off, turn on, operating time and failure time. The robot is powered on when it is applied and is powered off when the use is completed. The time span between power on and off is a working cycle that can be described by a random variable.

To explain the warranty and the approach proposed in this paper, we assume that the lifetime of the robot follows a two-parameter Weibull function $F(u)$ with the failure rate function $r(u)=a{(u)}^b$, where $a>0$ and $b>0$, and we assume that the working cycle follows an exponential distribution function $G(u)$ with a failure rate $\lambda$, i.e., $G(u)=1-\exp (-\lambda u)$. Some constant parameters are presented in

Table 1. Parameter values.

${\mathit{c}}_{\mathit{m}}$	${\mathit{c}}_{\mathit{f}}$	${\mathit{c}}_{\mathit{h}}$	${\mathit{c}}_{\mathit{R}}$	${\mathit{c}}_{\mathit{p}}$	$\mathit{\alpha }$	$\mathit{\beta }$
0.3	0.2	1	10	15	1	1

, while other parameters not mentioned in Table 1 are provided when needed.

For convenience, in this article, the 2DRFRW is expressed as PW. From the standpoint of reliability engineering, it is impossible that the product is “as good as new” after maintenance. This fact means that the maintenance ability is restricted, i.e., the value of maintenance ability $\mathit{n}$ is not boundless. We use $\phi (\mathit{n})=(\mathit{n}+1)e^{-\mathit{n}}$ to simulate the reliability alteration caused by PM, where $\mathit{n}=0,2,\cdots ,5$ represents the maintenance ability level. PM is not performed when $\mathit{n}=0$, and the maximum value of the level is obtained when $\mathit{n}=5$.

6.1. Warranty Cost Sensitivity Analysis

Let $w=2$, $\lambda =0.5$, $a=1.2$ and $b=1$; Figure 1 is plotted. From Figure 1, when the warranty limit $m$ increases, the warranty cost (green line) generated by PW first increases and then tends to the warranty cost (red line) generated by RFRW. The corresponding warranty service period $S_m$ (where $S_m={\displaystyle \sum _{i=1}^m{Y}_i}$) increases and gradually tends to the warranty period $w$. When $S_m=w$, i.e., the warranty service period equates to the warranty period, then PW translates into RFRW. Hence, when the warranty limit $m$ increases, the warranty cost generated by PW first closes and then equates the warranty cost generated by RFRW.

To illustrate the effect of the working cycle on the warranty cost, Figure 2 is plotted, where $m=1$, $w=2$, $a=1.2$ and $b=1$. From Figure 2, the warranty cost (green line) associated with PW first increases and then tends to the warranty cost (red line) associated with RFRW when $\lambda$ decreases. When the failure rate $\lambda$ decreases, then the mean value $E[Y_i]$ ($=1/\lambda$) for the working cycle $Y_i$ increases. This variation tendency concluded that the mean value $E[Y_i]$ first closes and then equates the warranty period $w$. Hence, when the rate $\lambda$ decreases, the warranty cost generated by PW first closes and then equates the warranty cost generated by RFRW. This fact means that from the overall standpoint, when the random working cycle is greater, the warranty cost generated by PW is increasing.

Synthesizing Figure 1 and Figure 2, it can be concluded that when the random working cycle is determined by the projects, the manufacturer can cut the warranty cost by identifying the number $m$ of working cycles.

6.2. Post-Warranty Maintenance Sensitivity Analysis

To illustrate the post-warranty maintenance policies, we refer to two types of post-warranty maintenance as Policy 1 (in Section 3.1) and Policy 2 (Section 3.2) and refer to the uniform post-warranty maintenance policy as Policy 3 (in Section 4).

6.2.1. Policy 1 Sensitivity Analysis

To display the effect of the ability level $\mathit{n}$ on the optimum cost rate $C{R}_1({\mathit{n}}^{*},{\mathit{T}}^{*})$ and the optimum replacement time ${\mathit{T}}^{*}$, Figure 3 is plotted, where $m=2$, $\lambda =0.5$, $w=2$, $a=1.2$ and $b=1$. From Figure 3, the optimum replacement time ${\mathit{T}}^{*}$ exists and is increasing in $\mathit{n}$. In addition, using the algorithm presented in Section 5, the optimum level ${\mathit{n}}^{*}$ equates to the maximum value $5$, i.e., maintenance ability level ${\mathit{n}}^{*}=5$. Finally, at the warranty limit $S_m$, compared with the case ignoring PM (i.e., $\mathit{n}=0$), PM (i.e., $\mathit{n}\ne 0$) can not only cut the optimum cost rate $C{R}_1({\mathit{n}}^{*},{\mathit{T}}^{*})$ but also expand the optimum replacement time ${\mathit{T}}^{*}$. This fact means that compared with the traditional periodic replacement policy, the maintenance policy is superior.

If the random working cycle follows an exponential distribution function $G(u)=1-\exp (-\lambda u)$, then the mean value for the working cycle is $1/\lambda$. This fact indicates that the mean value is a decreased function with respect to $\lambda$. That is, the working cycle can be explained by $\lambda$ indirectly. By considering this relationship, Figure 4 ($m=2$, $w=2$, $a=1.2$ and $b=1$) associated with $\lambda$ illustrates the effect of the working cycle on the optimum cost rate $C{R}_1({\mathit{n}}^{*},{\mathit{T}}^{*})$ and the optimum replacement time ${\mathit{T}}^{*}$.

From Figure 5, the optimum cost rate $C{R}_2({\mathit{n}}^{*},{\mathit{T}}^{*})$ and the optimum replacement time ${\mathit{T}}^{*}$ exist. In addition, applying the algorithm provided in Section 5, the optimum level ${\mathit{n}}^{*}$ exists and equates $5$. Finally, at the warranty limit $w$, compared with the case ignoring PM (i.e., $\mathit{n}=0$), PM (i.e., $\mathit{n}\ne 0$) can cut the optimum cost rate $C{R}_2({\mathit{n}}^{*},{\mathit{T}}^{*})$ and expand the optimum replacement time ${\mathit{T}}^{*}$. This fact again means that compared with the conventional periodic replacement policy, the proposed maintenance policy is superior.

To illustrate the effect of the working cycle on the optimum cost rate $C{R}_2({\mathit{n}}^{*},{\mathit{T}}^{*})$ and the optimum replacement time ${\mathit{T}}^{*}$, Figure 6 is plotted, where $m=2$, $w=2$, $a=1.2$ and $b=1$.

From Figure 6, the optimum cost rate $C{R}_2({\mathit{n}}^{*},{\mathit{T}}^{*})$ and the optimum replacement time ${\mathit{T}}^{*}$ uniquely exist. Additionally, they increase with respect to the failure rate $\lambda$. These facts indicate how a larger working cycle can reduce the cost rate and replacement time.

6.2.2. Policy 3 Sensitivity Analysis

To display whether the optimum replacement time ${\mathit{T}}^{*}$ and the optimum ability level ${\mathit{n}}^{*}$ exist,

Table 2. Optimum solution and optimum value.

Solution and Value	$\mathit{n}=\mathbf{0}$	$\mathit{n}=\mathbf{1}$	$\mathit{n}=\mathbf{2}$	$\mathit{n}=\mathbf{3}$	$\mathit{n}=\mathbf{4}$	$\mathit{n}=\mathbf{5}$
${\mathit{T}}^{*}$	5.8467	5.8728	5.9054	5.9257	5.9362	5.9412
$C{R}_u({\mathit{T}}^{*})$	4.5655	4.4903	4.3964	4.3374	4.3067	4.2921

is presented, where $m=2$, $w=0.5$, $\lambda =0.5$, $a=1.2$ and $b=1$. From Table 2, the optimum ${\mathit{T}}^{*}$ and ${\mathit{n}}^{*}$ exist, and the maximum value ${\mathit{n}}^{*}$ is $5$. In addition, at the warranty limits, PM (i.e., $\mathit{n}\ne 0$) can cut the optimum cost rate $C{R}_u({\mathit{n}}^{*},{\mathit{T}}^{*})$ and expand the optimum replacement time ${\mathit{T}}^{*}$ compared with the case ignoring PM (i.e., $\mathit{n}=0$), i.e., Case B in Section 4.4. This fact indicates that the maintenance policy is superior to the conventional periodic replacement policy.

6.3. Comparisons

To compare the optimum solution and optimum value, we draw Figure 7, where $m=2$, ${\mathit{n}}^{*}=5$, $\lambda =0.5$, $w=2$, $a=1.1$ and $b=1$.

Based on Figure 7, the optimum time ${\mathit{T}}^{*}$ of Policy 2 is largest; the optimum time ${\mathit{T}}^{*}$ of Policy 1 is smallest; the optimum time ${\mathit{T}}^{*}$ of Policy 3 is moderate; the optimum cost rate of Policy 1 is smallest; the optimum cost rate of Policy 2 is largest; and the optimum cost rate of Policy 3 is moderate. These variations cannot show the superiority between the uniform policy and the customized policies.

To compare the superiority between the customized policies and the uniform policy, some illustrations are given below. The cost rate $C{R}^{*}$ can be calculated according to $C{R}^{*}=\left(Q_1\cdot E[C(L_1)]+Q_2\cdot E[C(L_2)]\right)/\left(Q_1\cdot E[L_1]+Q_2\cdot E[L_2]\right)$, where $E[L_1]$ and $E[C(L_1)]$ are obtained, respectively, by substituting the optimum time ${\mathit{T}}^{*}$ of Policy 1 into Equations (14) and (10); $E[L_2]$ and $E[C(L_2)]$ are obtained, respectively, by substituting the optimum time ${\mathit{T}}^{*}$ of Policy 2 into Equations (19) and (18). The cycle length $L^{*}$ can be represented as $L^{*}=Q_1\cdot E[L_1]+Q_2\cdot [L_2]$. We call the case associated with both $C{R}^{*}$ and $L^{*}$ a comprehensive case, which is similar to the case described in Equation (23), an integrated case composed of Policies 1 and 2. Moreover, $L_u^{*}$ and $L^{**}$ represent the duration lengths associated with Policy 3 and the comprehensive case, respectively, when the cost of the comprehensive case is equal to t the cost of Policy 3. According to these illustrations, we present

Table 3. Comparison.

$\mathit{n}$	The Comprehensive Case		Policy 3		Duration Length
	$\mathit{C}{\mathit{R}}^{*}$	${\mathit{L}}^{*}$	$\mathit{C}{\mathit{R}}_{\mathit{u}}({\mathit{T}}^{*})$	$\mathit{E}[{\mathit{L}}_{\mathit{u}}]$	${\mathit{L}}^{**}$	${\mathit{L}}_{\mathit{u}}^{*}$
0	3.8426	7.1134	2.9210	9.5335	20.7782	36.6334
1	3.7692	7.2070	2.8790	9.5868	20.7490	36.1346
2	3.6761	7.3218	2.8265	9.6529	20.6951	35.4850
3	3.6170	7.3927	2.7935	9.6941	20.6515	35.0636
4	3.6170	7.4293	2.7763	9.7155	20.6260	35.1410
5	3.6170	7.4466	2.7682	9.7257	20.6137	35.1779

, where $m=2$, $w=1$, $\lambda =1$, $a=1$ and $b=1$.

From Table 3, the duration length is smaller than the duration length, i.e., $L_u^{*}>L^{**}$, when the ability level is given. This relationship indicates that at a smaller cost rate, the uniform policy can expand the post-warranty time. That is, compared with customized policies, the performance of the uniform policy is superior.

7. Conclusions

This paper, from the standpoint of the manufacturer, proposes a two-warranty renewal free-replacement policy to appropriately deserve the product, which, at random working cycles, performs projects successively. By comparing it with the conventional warranty policy, the characteristics of the warranty policy are explained. Defining that the product with random working cycles is warranted by the warranty policy, the post-warranty maintenance is simulated from the standpoint of the consumer by differing each warranty limit. Additionally, considering all warranty limits, uniform post-warranty maintenance is simulated from the standpoint of consumers. Numerical experiments are presented to perform sensitivity analysis on the warranty cost, which is produced by all post-warranty maintenance policies and the proposed warranty policy. It is shown that compared with the conventional warranty policy, the proposed warranty policy can yield a smaller warranty cost, and compared with the post-warranty maintenance policies, the uniform policy is superior.

In the future, by integrating imperfect preventive maintenance into the warranty period, we will develop warranty models for products that have random working cycles. In addition, we will consider a research topic in which the warranty policy is devised by supervising downtime between successive projects.