1. Introduction
The core competitiveness of products depends on product price, product functionality, after-sales service, etc. Among them, after-sales service is not only a powerful guarantee of consumer rights and interests but also a key factor in improving brand image. As the main term of after-sales service, the warranty policy is an important factor that affects product sales and brand image improvement.
Due to the above important values, warranties have been studied extensively from the point of view of warrantors (or manufacturers). According to the technologies of reliability modeling, warranties are divided into the lifetime-based warranty, the on-condition warranty, and the performance-based warranty. Usually, the lifetime-based warranty is designed based on a distribution function of the product’s lifetime. This kind of warranty includes two-dimensional warranties in [
1,
2,
3,
4,
5], one-dimensional warranties in [
6,
7,
8,
9,
10,
11,
12,
13], and three-dimensional warranties in [
14], wherein the product’s lifetime is subject to a distribution function. The on-condition warranty is designed based on a stochastic degradation process. For example, Ref. [
15] designed an on-condition-based renewable free replacement warranty for the product subject to an inverse Gaussian process; Ref. [
16] devised an on-condition warranty by assuming that product deterioration is modeled by a Gamma process; Ref. [
17] designed an optimal on-condition warranty for the product subject to a Wiener process. The performance-based warranty is designed based on a mixture of stochastic degradation processes and distribution functions. For example, Ref. [
18] modeled a performance-based warranty by combining a stochastic degradation process and a distribution function.
Although each of the above types of warranty can guarantee product reliability, this kind of guarantee is limited to the warranty period rather than the whole life cycle of the product. Therefore, consumers must focus on how to guarantee post-warranty reliability. In view of this, a variety of maintenance models were proposed from the point of view of consumers to sustain the post-warranty reliability and lower maintenance costs after the warranty expiration. The related maintenance models are classified as either the on-condition maintenance model or the lifetime-based maintenance model. Previously, Reference [
15] proposed the on-condition maintenance model to sustain the post-warranty reliability, wherein an inverse Gaussian process characterizes the product’s degradation. References [
19,
20,
21] designed and optimized the lifetime-based maintenance models sustaining the post-warranty reliability by letting the product’s lifetime obey a distribution function.
From the viewpoint of an engineering practice, the development and application of the monitoring system (MS) makes it possible to diagnose the health condition of products. In addition, MS is able to monitor the job cycle of the product that works for tasks. For example, in China, sharing bikes (SB) are helping the construction of green and low-carbon cities. By means of MS, SB providers (or users) can monitor the real-time usage of each SB. From reliability theory, the job time that can be estimated by job cycles significantly affects the deterioration of the product working for tasks. Considering this fact [
22,
23,
24,
25] integrated the working cycle (i.e., job cycle) into maintenance theory and studied some random maintenance models that are used for ensuring the product reliability.
Similarly, by designing the limited job cycles a warranty constraint, some novel warranty models can be devised for ensuring product reliability from the point of view of the warrantor; by designing the limited job cycles a replacement limit, some random maintenance models to sustain the post-warranty reliability are investigated from the point of view of the consumer. By designing the limited job cycles a warranty constraint [
26], earlier proposed the warrantor’s two-dimensional free repair warranty models so as to ensure the product reliability, alike earlier investigated consumers’ periodic replacement models, which are used for sustaining the post-warranty reliability.
The two-dimensional free repair warranty first (2DFRWF) in [
26] satisfies that (1) minimal repair is used for removing all product failures before the accomplishment of the
job cycle or before the warranty period w, whichever comes first; (2) warrantors bear the costs of removing all failures. This warranty classifies consumers into two populations. The first population is people whose warranty expiry occurs before the warranty period
at the accomplishment of the
job cycle. The second population is people whose warranty expires before the accomplishment of the
job cycle at the warranty period
. The warranty service period (i.e., a sum of
job cycles) of the former is lower than that (i.e., the length
of the warranty period) of the latter. The occurrence of this case may make the former perceive that they are not treated as equal as the latter. If this perception appears in the first population, then some consumers from the first population will complain to the consumers’ union (CU), and thus the brand image of warrantors is inevitably damaged. If the refund in [
27,
28] is used as compensation for consumers from the first population, then the complaint no longer exists and the brand image is not damaged. By reviewing the literature on warranties, however, it is found that the two-dimensional free repair warranty considering a refund has not been studied to eliminate the consumer’s complaint and ensure the brand image.
In addition, random periodic replacement first (RPRF) to sustain the post-warranty reliability in [
26] requires that consumers replace the product through warranty at the replacement time
or the accomplishment of a job cycle, which comes first. Under these requirements, the post-warranty period produced by the replacement at the accomplishment of a job cycle is obviously lower than the post-warranty period produced by the replacement at
. Once the former type of replacement occurs, most of the remaining life for the product through warranty is easily wasted rather than serving consumers. If the replacement at the accomplishment of a job cycle is expanded to the replacement at the accomplishment of the
job cycle, most of the remaining life for the product through warranty unquestionably serves consumers as much as possible, rather than wasting largely. However, random periodic replacement policies that incorporate the replacement at the accomplishment of the
job cycle have not been studied to guarantee the post-warranty reliability.
In this study, by integrating a refund into the first type of warranty (i.e., 2DFRWF) in [
26], a warrantor’s warranty model is devised to eliminate consumer complaints and ensure the brand image. The devised warranty is a two-dimensional free repair warranty with a refund (2DFRW-R) requiring the following: ① if the
job cycle is accomplished before the warranty period
, then the warrantor will provide a refund for making the consumers perceive themselves to be treated fairly, while the warranty service expires at the accomplishment of the
job cycle; ② if the warranty period
will be reached before the
job cycle is accomplished, then the consumer no longer obtains a refund from the warrantor and while the warranty service expires at the warranty period
. In addition, for largely using the remaining life of the product through warranty, we incorporate
job cycles into periodic replacement and investigate a bivariate random periodic replacement (BRPR) requiring that the product through 2DFRW-R be replaced when the
job cycle is accomplished or when the replacement time
is reached, whichever comes earlier. We construct the expected cost rate of BRPR, analyze the existence and uniqueness of the optimal BRPR, and explore the characteristics of the proposed models in numerical examples.
The study’s contributions are highlighted as having two key aspects: (1) by integrating both job cycles and a refund into classic free repair warranty model, a warrantor’s warranty model with fairness characteristics is devised to guarantee the product reliability; (2) a consumer’s BRPR model is studied for sustaining the post-warranty reliability and largely using the remaining life of the product through warranty.
The structure of the present study is listed as follows. 2DFRW-R is defined and the corresponding warranty cost is modeled in
Section 2.
Section 3, from the point of view of the consumer, presents the definition of BRPR and models the cost rate. In
Section 4, the characteristics of 2DFRW-R and BRPR are analyzed in numerical examples. The final conclusion is offered in
Section 5.
3. Bivariate Random Periodic Replacement Model of Consumers
In the case that RPRF in [
26] is used for sustaining the post-warranty reliability, the product with higher job frequency and shorter job cycles is very easily replaced before the replacement time
at the accomplishment of a job cycle. The occurrence of this case wastes the remaining life of the product through the warranty rather than largely serving the consumers. If the replacement before the replacement time
at the accomplishment of a job cycle is expanded to a replacement before the replacement time
at the accomplishment of the
job cycle, the remaining life of the product through warranty can serve the consumer as much as possible, rather than wasting largely.
In view of this, by expanding the accomplishment of a job cycle to the accomplishment of the
(
) job cycle, this section models a bivariate random periodic replacement (BRPR), which is used for sustaining the post-warranty reliability of the product that is warranted by 2DFRW-R and largely using the remaining life of the product through 2DFRW-R. Such a BRPR requires ① the product through 2DFRW-R is preventively replaced at the accomplishment of the
job cycle or at the replacement time
, whichever comes earlier; ② the product through 2DFRW-R undergoes minimal repair at failure before replacement. When
, such a BRPR is reduced to a univariate replacement, which is represented as RPRF in [
26].
To model BRPR conveniently, it is defined that the product’s life cycle is a span that begins from the accomplishment of a new product installation and ends the replacement occurrence at the cost of the consumer, which is similar to [
19,
20,
21]. Clearly, this type of life cycle includes the warranty service period of 2DFRW-R and the post-warranty period of BRPR.
3.1. Total Cost during the Life Cycle
As mentioned above, it is assumed that the job cycle is independent and obeys the identical distribution that has no memory. Such an assumption shows that the task’s remaining accomplishment time and each job cycle during the post-warranty period are still independent and obey the identical distribution , which has no memory. When the product through 2DFRW-R is preventively replaced at the accomplishment of the job cycle, the job time during the post-warranty period is , where . The respective probabilities that the product through 2DFRW-R is preventively replaced before the replacement time at the accomplishment of the job cycle or before the accomplishment of the job cycle at the replacement time are and .
Denote
by the unit failure cost of each failure occurrence. When 2DFRW-R expires at the accomplishment of the
job cycle and when the product through 2DFRW-R is preventively replaced before the replacement time
at the accomplishment of the
job cycle, the total cost
of BRPR can be computed as
where
;
is the total cost resulting from each failure; and
is a failure rate function at
.
When 2DFRW-R expires at the accomplishment of the
job cycle and when the product through 2DFRW-R is preventively replaced before the accomplishment of the
job cycle at the replacement time
, the total cost
of BRPR is computed as
Under the case where 2DFRW-R expires at the accomplishment of the
job cycle, the total cost
of BRPR is derived as
where the distribution function
of
satisfies
(
);
represents the unit replacement cost, where
.
When 2DFRW-R expires at the warranty period
and when the product through 2DFRW-R is preventively replaced before the replacement time
at the accomplishment of the
job cycle, the total cost
of BRPR is computed as
where
.
When 2DFRW-R expiry occurs at the warranty period
and when the product through 2DFRW-R is preventively replaced before the accomplishment of the
job cycle at the replacement time
, the total cost
of BRPR is computed as
Under the case in which 2DFRW-R expiry occurs at the warranty period
, the total cost
of BRPR is derived as
Under the case where 2DFRW-R warrants the product, the expected total cost
of BRPR is calculated as
As mentioned above, the life cycle is a sum of the warranty service period of 2DFRW-R and the post-warranty period of BRPR. This indicates that the total cost during the life cycle is composed of the total failure cost of 2DFRW-R, the expectation
of the refund related to 2DFRW-R, and the expected total cost of BRPR. By replacing
in
of (5) with
, the total failure cost
of 2DFRW-R is computed as
; the expected value
of the refund related to 2DFRW-R has been offered in (7); and the expected total cost
of BRPR has been provided in (15). By algebraic manipulation, the total cost
during the life cycle is calculated as
3.2. Length of the Life Cycle
2DFRW-R expires at the accomplishment of the
job cycle or at the warranty period
. The respective probabilities are
and
, and the respective warranty service periods are
(
) and
. Hence, the expected length
of the warranty service period of 2DFRW-R is given by
For the product that goes through 2DFRW-R, the probabilities that it is preventively replaced before the replacement time
at the accomplishment of the
job cycle or before the accomplishment of the
job cycle at the replacement time
are represented by
and
, and the respective post-warranty service periods are
(
) and
. Hence, the expected length
of the post-warranty service period produced by BRPR is given by
Under the definition of the life cycle, the length
of the life cycle is expressed as
3.3. The Cost Rate Model
Let
. By the renewal rewarded theorem [
31], the expected cost rate
is given by
where
and
are presented in (19) and (16), respectively.
3.4. Special Models
When
, the cost rate model in (20) is reduced to
As discussed above, reduces 2DFRW-R to FRW. Therefore, the above model belongs to an expected cost rate, wherein FRW is used for warranting the product and BRPR is used to sustain the post-warranty reliability.
When
, the cost rate model in (20) is reduced to
which belongs to an expected cost rate, wherein 2DFRW-R is used for warranting the product and RPRF in [
26] is used for sustaining the post-warranty reliability.
When
and
, the cost rate model in (20) is reduced to
which belongs to an expected cost rate, wherein FRW is used for warranting the product and RPRF in [
26] is used for sustaining the post-warranty reliability.
When
, the cost rate model in (20) is reduced to
makes
. Similar to the case of
,
indicates that the replacement at accomplishment of the of the
job cycle is removed and BRPR is translated into classic periodic replacement in [
32,
33,
34]. Therefore, the model in (24) belongs to an expected cost rate, wherein 2DFRW-R is used for warranting the product and classic periodic replacement is used for sustaining the post-warranty reliability.
When
and
, the cost rate model in (20) is reduced to
which belongs to an expected cost rate, wherein FRW is used for warranting the product and classic periodic replacement is used for sustaining the post-warranty reliability.
3.5. Optimizing
In this subsection, we seek an optimal BRPR, i.e., seek an optimal solution combination by minimizing in (20). Other cost rate models can be similarly sought.
Considering that and are undefined and nonspecific, to seek optimal analytical solutions is intractable. A two-stage optimization method is used to summarize the existence and uniqueness of the optimal BRPR, as shown below.
Differentiating
with respect to
for any
, the derivative
of
is presented by
Denote
by the numerator of
; then,
Obviously, .
Let
be zero; then, the following expression can be obtained:
Obviously, the left of (26) equates to the model in (20).
Let equate to the right of (26), i.e., . By minimizing in (20), the existence and uniqueness of the optimal solution are summarized as the theorem below.
Theorem 1. For any , the results below can be obtained.
- ①
When, we have
The optimal solution satisfyingexists uniquely and the optimal expected cost ratesatisfies, ifis strictly increasing toalong with the increase of;
the optimal solutionsatisfiesand the optimal expected cost ratesatisfies, ifis strictly decreasing to along with the increase of;
at least an optimal solution that satisfies exists and the optimal expected cost rate satisfies , if is nonmonotonic along with the increase of and if there is more than one root for the equation .
- ②
When, we have
the optimal solutionsatisfying exists uniquely and the optimal expected cost ratesatisfies, ifis strictly increasing to along with the increase of;
at least an optimal solution that satisfiesexists and the optimal expected cost rateequates to, ifis nonmonotonic along with the increase of and if,
an optimal solutionsatisfying exists uniquely and the optimal expected cost rateequates to, ifis strictly decreasing toalong with the increase of.
For a fixed
, by minimizing
in (20), the existence and uniqueness of the optimal solution
are summarized based on the inequality
. The inequality
is equivalent to the inequality
Denote
by the left of the above inequality; then, we obtain
Furthermore, by minimizing in (20), the existence and uniqueness of the optimal solution are summarized as follows.
Theorem 2. For any, we can obtain the results below.
An optimal solution that satisfies simultaneously andexists uniquely if increases strictly to along with the increase of .
Such a way is one of methods to seek optimal policies of the expected cost rate models in
Section 3.4, and here no longer provide them.
4. Numerical Examples
For convenience, the first failure time of the hydropower dam inspection equipment is assumed to obey a distribution function , wherein , and ; assume that each job cycle of the hydropower dam inspection equipment is independent and obeys the identical distribution function , where ; assume that and that other parameters are provided when needed.
4.1. Characteristic Analysis of 2DFRW-R
To show how
,
, and
affect 2DFRW-R, we plot
Figure 1 by letting
,
,
,
,
,
,
and
. As shown in
Figure 1A (where
), for a fixed
, the increases in the warranty limit
decrease the warranty cost of 2DFRW-R to the warranty cost of FRW, which is obtained by computing
. In addition,
Figure 1A indicates that the warranty cost of 2DFRW-R increases with the increase in
for a fixed
. The above phenomena signal that the warranty cost of 2DFRW-R is lowered by limiting the size of
and/or by shortening the warranty period
. Third,
Figure 1A shows that the warranty cost of 2DFRW-R exceeds the warranty cost of 2DFRWF in [
26] when the warranty limit
is smaller.
Figure 1B (where
) shows that when
is smaller, the warranty cost of 2DFRW-R increases with the increase in
. This means that the warranty cost of 2DFRW-R can rapidly decline to a lower value by lowering
when all job cycles are smaller (i.e., the case in which
is greater, similarly hereinafter). In addition,
Figure 1B shows that no matter how
changes, the warranty cost of 2DFRW-R gradually decreases to the warranty cost of FRW with the increase in
.
4.2. Characteristic Analysis of BRPR
To explore the effect of
and
on the optimal BRPR, we make
Figure 2 by letting
,
,
,
,
,
,
,
and
.
Figure 2A (where
and
) shows that with the increase in the repair cost
, the minimum expected cost rate
increases while the optimal replacement time
decreases. This indicates that the lower repair cost can reduce the minimum expected cost rate and lengthen the optimal post-warranty service period.
Figure 2B (where
and
) indicates that both the minimum expected cost rate
and the optimal replacement time
increase when
is increasing. This means that the greater replacement cost
is unable to reduce the minimum expected cost rate, but it is able to extend the optimal post-warranty service period.
To explore the effect of
and
on the optimal BRPR, we make
Table 1 by letting
,
,
,
,
,
,
,
,
and
.
Table 1 shows that for a given
, with the increase in the warranty limit
, the minimum expected cost rate
increases while the optimal replacement time
decreases. Such a phenomenon indicates that a smaller
can reduce the minimum expected cost rate and lengthen the optimal post-warranty service period. As shown in
Table 1, for a given
, with the increase in
, the minimum expected cost rate
decreases while the optimal replacement time
increases. From the whole perspective, this means that when all job cycles are smaller, the minimum expected cost rate is lowered and the optimal post-warranty service period is extended.
4.3. Comparison
To use largely the remaining life of the product through 2DFRW-R, in this study, we have planned that BRPR sustains the post-warranty reliability. If
, BRPR is reduced to RPRF in [
27], i.e., the post-warranty reliability is sustained by any of the above random periodic replacements. Random periodic replacement with the best performance is an ideal replacement policy. This means that an ideal maintenance policy is chosen by ranking the performance of the above random periodic replacements.
In addition, 2DFRWF in [
26] and 2DFRW-R in this study are able to warrant the product. Under these cases, how the above warranties affect the optimal BRPR is an interesting focus.
In this subsection, from the numerical perspective, we provide consumers with a guide for selecting an ideal maintenance policy and explore how the above warranties affect the optimal BRPR.
To compare the performance of BRPR and RPRF and explore the effect of warranties on the optimal BRPR, we take 2DFRW-R and FRW as examples and make
Figure 3, where
,
,
,
,
,
,
,
,
and
.
As indicated in
Figure 3A (The symbol (*) denotes the minimum point), where
, the minimum expected cost rate of RPRF exceeds the minimum expected cost rate of BRPR, and the optimal replacement time of RPRF also exceeds the optimal replacement time of BRPR. Because the dimensions are not the same, the above changes mean that the performance of BRPR and RPRF cannot be ascertained in
Figure 3A.
Figure 3B shows that the minimum expected cost rate of BRPR under RFW exceeds the minimum expected cost rate of BRPR under 2DFRW-R, and the optimal replacement time of BRPR under RFW also exceeds the optimal replacement time of BRPR under 2DFRW-R. Similarly, the dimensions are not the same, thus these changes still cannot indicate how RFW or 2DFRW-R affect the optimal BRPR.
Next, we rank random periodic replacements by means of the numerical method below. We take BRPR and RPRF as examples.
Let and be the optimal values of the total costs during the life cycle, which are related to BRPR and RPRF, respectively. Furthermore, denote and by the optimal values for the lengths of the life cycles related to both. Third, let and be the cycle lengths related to both, under the condition at which the total costs of two random periodic replacements are multiplicative. Then, the numerical method below is presented.
Step 1:Letting and ;
Step 2:BRPR is an ideal maintenance policysustaining the post-warranty reliability if; RPRF is an ideal maintenance policysustaining the post-warranty reliability if; every of them is able to sustain the post-warranty reliability whose key cause is that their performance is equivalent if.
By means of the numerical method, we next determine the rank of random periodic replacements under 2DFRW-R.
Let
,
,
,
,
,
,
,
,
and
; we make
Table 2. As shown in
Table 2, the cycle length
is longer than the cycle length
, i.e.,
, when
is the same. This means that BRPR’s performance is better than RPRF’s performance, i.e., compared with RPRF, BRPR in this study can largely use the remaining life of the product through 2DFRW-R.
Next, we make the symbol assumption similar to
Table 2 and use the numerical method to explore how 2DFRW-R and FRW affect the optimal BRPR. We make
Table 3 by letting
,
,
,
,
,
,
,
,
and
. As indicated in
Table 3, the cycle length
is longer than the cycle length
, i.e.,
, when
is the same. This means that the performance of BRPR under 2DFRW-R is better than that of BRPR under FRW. In other words, compared with FRW, 2DFRW-R is able to lengthen the optimal post-warranty service period and lower the minimum expected cost rate.