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BY 4.0 license Open Access Published by De Gruyter Open Access November 2, 2022

Optimal block replacement policy for two-dimensional products considering imperfect maintenance with improved Salp swarm algorithm

  • Enzhi Dong ORCID logo , Zhonghua Cheng ORCID logo , Yue Shuai ORCID logo EMAIL logo and Jianmin Zhao
From the journal Open Physics

Abstract

Human society is entering Industry 4.0. Engineering systems are becoming more complex, which increases the difficulties in maintenance support work. Maintenance plays a very important role in the entire life cycle of a system, and in reality, maintenance is not always perfect, and its maintenance degree is between good as new and bad as old, which should be considered in the maintenance strategy. Under the framework of two-dimensional warranty, this work proposes an optimal two-dimensional block replacement strategy based on the minimum expected warranty cost. Two-dimensional block replacement maintenance is imperfect maintenance. The failure rate reduction method is used to describe the maintenance effect of two-dimensional block replacement. In case analysis, the grid search algorithm, genetic algorithm, particle swarm optimization algorithm, and improved salp swarm algorithm (SSA) are used to find the optimal warranty scheme for the laser module. The improved SSA can converge faster and find a warranty scheme that makes the warranty cost lower. Therefore, managers can use these results to reduce costs and get a win-win extended warranty cost. Rationalization suggestions are put forward for managers to make maintenance decisions through comparative analysis and sensitivity analysis.

Nomenclature

C p

imperfect preventive maintenance cost

C m

corrective maintenance cost.

C Iy

component life cycle cost borne by users with extended warranty service

C In

component life cycle cost borne by users without extended warranty service

G(r)

probability distribution function of utilization rate

M |y

component life cycle cost borne by manufacturer with extended warranty service

M |n

component life cycle cost borne by manufacturer without extended warranty service

r u

upper limit of utilization rate

r l

lower limit of utilization rate

(T 0, U 0)

two-dimensional block replacement interval

[W B, U B]

two-dimensional basic warranty period

[W E, U E]

two-dimensional extended warranty period

[W L, U L]

time limit and usage limit of the product life

ω

repair degree

λ(t|r)

failure rate function of component

1 Introduction

The two-dimensional warranty service strategy generally takes the use time (T) and use degree (U, such as driving mileage, revolutions, etc.) as the constraint boundary for the termination of the warranty period [1]. (Two-dimensional warranty service is adopted for products whose degradation changes with use time and use degree, which can not only take the use of products into account in the decision-making process of warranty service, but also take into account the interests of both the users and manufacturers. With the progress of industrial production and manufacturing technology, the technical complexity of large products (such as automobiles, ships, and aircraft) is gradually increasing, and the types and quantity of products that degraded with the change in use time and use degree are gradually increasing. The demand for two-dimensional warranty service strategy for products is becoming increasingly prominent [2]. Therefore, in order to meet the modeling requirements of two-dimensional warranty service decision-making and provide corresponding data support for the formulation of warranty service terms, this work mainly studies the modeling of products’ two-dimensional warranty service decision-making. Since the two-dimensional warranty strategy is generally used for products whose degradation laws vary with use time and use degree, the scope of the warranty period is also limited by both the use time and use degree [3]. For product warranties, it is in the manufacturer’s interest to use a two-dimensional warranty service strategy for products because it limits the high usage of the product, i.e., the higher the usage, the earlier the warranty period ends.

The block replacement policy refers to the replacement of components in batches at a given time kT (k = 1,2,…). Even if some components are replaced between two replacement intervals, they must be replaced together when the replacement interval T is reached [4]. The advantage of the block replacement policy is that it is easy for the equipment management department to formulate maintenance plans and implement maintenance activities. This strategy is applicable to electronic components and rubber parts which are relatively low in price and a large number of them are used.

1.1 Motivation

With the increasing technical complexity of products, the application of two-dimensional warranty strategy is becoming more and more popular [5]. For example, the basic warranty period of an engineering project emergency vehicle includes two restrictions: calendar time and motorcycle hours. During the two-dimensional warranty period, if preventive maintenance is carried out for the product, the preventive maintenance interval shall also be two-dimensional [6]. At present, it mainly relays on expert experience to determine the two-dimensional preventive maintenance interval for products, which is lack of scientific decision-making basis [7]. The traditional one-dimensional maintenance interval determination method cannot meet the needs of two-dimensional warranty service decision-making, so it is necessary to carry out modeling research on the two-dimensional warranty service with two-dimensional preventive maintenance [8].

In the existing literature, the block replacement policy is usually assumed to be a perfect maintenance [9]. This assumption makes the establishment and solution of the model more convenient. However, in fact, the block replacement policy often only replaces individual components in the system. The replacement is difficult to restore the system as new, but is often between “good as new” and “bad as old,” that is, imperfect maintenance [10]. Therefore, it is more practical to regard block replacement policy as imperfect maintenance.

At present, with the progress of manufacturing technology, the reliability of products is higher and higher, and the probability of failure within the basic warranty period is lower. Therefore, users are more and more inclined to purchase extended warranty services. The extended warranty service not only meets the user’s after-sales service requirements for the products after the basic warranty period, but also provides a new profit source for the manufacturer. This work explores the modeling method of effective integration of basic warranty and extended warranty, and strives to obtain a warranty scheme that satisfies both the manufacturer and the user.

1.2 Contribution

This work studies the two-dimensional non-renewal basic warranty cost model and extended warranty cost model when two-dimensional block replacement maintenance is imperfect maintenance. The failure rate reduction method is used to describe the maintenance effect of two-dimensional block replacement, and the optimal two-dimensional block replacement interval is obtained by optimizing the warranty service cost model. This study proposes a discrete utilization rate based salp swarm algorithm (SSA) to solve the problem. This study also introduces the extended warranty mechanism and calculates the extended warranty cost scheme acceptable to manufacturers and users.

The remainder of this article is organized into six sections. Section 2 provides a literature review of related studies. Section 3 presents the description of the model based on reasonable assumptions. Section 4 introduces the model constructing process. Section 5 introduces the improved SSA for solving the model. Section 6 presents a real case scenario to illustrate the applicability of our model. Section 7 presents the conclusion.

2 Related works

Warranty and maintenance are not the same concept. Maintenance is the specific performance of the implementation of warranty work, and the effect of warranty is finally reflected through the effect of maintenance [11]. The specific relationship between the two is: maintenance is the specific work of the implementation of warranty, warranty decision-making includes the selection of warranty mode and maintenance strategy, and the core work of warranty decision-making is the combination of warranty mode and maintenance strategy, so as to achieve the optimal effect of warranty work. This section mainly combs the existing research from two aspects: warranty mode and maintenance strategy.

2.1 Warranty mode

According to the dimensions contained in the warranty period, warranty can be divided into one-dimensional warranty, two-dimensional warranty, and multi-dimensional warranty. Due to the complexity of multi-dimensional warranty modeling and less application at present, there is less research on it. The research on one-dimensional warranty and two-dimensional warranty is the mainstream at present, and two-dimensional warranty is the forefront of warranty theory research. One-dimensional warranty means that the warranty period is determined based on a single variable, usually calendar time or use degree. The two-dimensional warranty policy is that the warranty period is determined by two variables, usually calendar time and use degree, as shown in Figure 1.

Figure 1 
                  Warranty diagram. (a) One-dimensional warranty and (b) two-dimensional warranty.
Figure 1

Warranty diagram. (a) One-dimensional warranty and (b) two-dimensional warranty.

For one-dimensional warranty, Vahdani et al. [12] established the renewal warranty service model of multi-stage degraded repairable products, considering the minimum maintenance with non-negligible maintenance time and the replacement maintenance with negligible maintenance time. Xie et al. [13] established an overall profit evaluation model considering the units within and outside the warranty period. Aggrawal et al. [14] established a warranty service price model considering the product sales cycle, in which the unit failure obeys the exponential distribution. González-Prida et al. [15] used generalized renewal process and inhomogeneous Poisson process to study the optimization decision-making problem of warranty service period. Zhu and Xiang [16] adopted the condition based maintenance strategy. For the multi-component system with two stages, they used the multi-stage random integer model to select the components to be maintained within the limited maintenance time, so as to minimize the total maintenance cost and ensure the reliability of the system. Considering the dependence between components, Safaei et al. [17] proposed the optimal age replacement strategy for parallel systems and series systems.

In reality, the two-dimensional warranty strategy is most widely used in automobile products warranty. In the process of two-dimensional warranty service decision modeling, the first step is to determine the two-dimensional warranty period range [18]. Figure 2 lists several two-dimensional warranty service period ranges, of which the rectangular form is a common case. According to the characteristics of product warranty service and the interests of both users and manufacturers, the rectangular warranty period is suitable for most products. The scope of the two-dimensional warranty service period considered in this study is in the rectangular form.

Figure 2 
                  Two-dimensional warranty service period. (a) Rectangular, (b) L–shaped, (c) stepped type and (d) triangular.
Figure 2

Two-dimensional warranty service period. (a) Rectangular, (b) L–shaped, (c) stepped type and (d) triangular.

For two-dimensional warranty, by optimizing the two-dimensional warranty service cost model, Banerjee and Bhattacharjee [19] studied the decision-making problem of minimum maintenance or replacement for the first failure in the two-dimensional warranty period. Huang et al. [20] established the Bayesian decision-making model of periodic preventive maintenance of units whose degradation process obeys inhomogeneous Poisson process under the proportional cost sharing warranty strategy, and established the optimization model of periodic preventive maintenance in two-dimensional warranty service of repairable units by using binary joint distribution. Taleizadeh and Mokhtarzadeh [21] used the value risk method to formulate pricing scheme and two-dimensional warranty scheme for products sold online and offline. Lin and Chen [22] analyzed the two-dimensional warranty claim data. Breaking the assumption that the utilization rate is a linear function of age, they mined a more accurate failure law by analyzing the time and mileage data at the time of failure. Song [23] proposed a two-dimensional preventive maintenance and replacement strategy. Under this strategy, preventive maintenance actions are arranged according to age or use degree. Each impact before the n-th impact will lead to product failure or increase in product failure rate. If the product has withstood (n − 1)th shock, replace it with a new product at the nth shock. From the perspective of the manufacturer, the average warranty cost in the whole warranty period is obtained by using the renewal theory. According to the literature review, one-dimensional warranty and two-dimensional warranty mostly concentrate in the basic warranty stage, and there is less research on extended warranty.

2.2 Maintenance strategy

The maintenance strategy adopted during the warranty period will have a great impact on the warranty service cost and products’ performance during the warranty period. Maintenance strategies can be divided into two categories: corrective maintenance strategy and preventive maintenance strategy. Common preventive maintenance strategies include functional check strategy and replacement strategy.

In engineering practice, the failure mode of many products will show some signs in the process of functional degradation to indicate that the failure is about to occur or is occurring. If this sign is found through functional check, preventive measures can be taken in time to avoid the functional failure of the product [24,25]. Delay time is generally used to describe such degradation process [26], and its basic idea is to divide the formation of products failure into two stages: the formation stage of potential failure and the formation stage of functional failure [27]. The duration of these two stages is called initial defect time u and failure delay time h [28]. When the products undergo functional check at the interval of cycle T, t i represents the i-th check point (i = 1,2,3…). There are two situations of product maintenance, as shown in Figure 3. Figure 3(a) shows the failure maintenance after the function failure occurs between the two function check, and Figure 3(b) shows the potential failure maintenance after the potential failure is detected at the function check point.

Figure 3 
                  Two situations in function check. (a) Function failure is found and (b) potential failure is found.
Figure 3

Two situations in function check. (a) Function failure is found and (b) potential failure is found.

It is assumed that the probability density function of potential initial defect time u is g(u) and the cumulative distribution function is G(u). The probability density function of failure delay time h is f(h) and the distribution function is F(h). Then, the specific value of failure renewal probability before time t i is g(u)duF(t i u), and the specific value of check renewal probability at time t i is g(u)du[1 − F(t i u)]. Then, the probability P f (t i − 1,t i ) of failure maintenance during (t i − 1,t i ) is

P f ( t i 1 , t i ) = t i 1 t i g ( u ) F ( t i u ) d u .

The probability P m (t i−1 ,t i ) of potential failure maintenance at time t i is:

P m ( t i 1 , t i ) = t i 1 t i g ( u ) [ 1 F ( t i u ) ] d u .

Periodic replacement strategy mainly includes block replacement strategy and age replacement strategy. At present, the commonly used periodic replacement strategy is one-dimensional periodic replacement strategy. However, with the increasing technical complexity and advanced performance of products, the failure law of many products is affected by many factors (products running time, service time, driving mileage, etc.). When repairing such products, if only one-dimensional periodic replacement interval about time is given, the effect of other influencing factors on failure law will be ignored, resulting in untimely maintenance.

The two-dimensional age replacement strategy means that if the product does not fail during use, it will be replaced regularly according to the specified two-dimensional age (calendar time and use degree). If a failure occurs within the specified time (two-dimensional age), the failure product shall be repaired, and the age shall be counted again after the repair. Compared with the block replacement strategy, the age replacement strategy has certain flexibility. The two-dimensional age replacement process is shown in Figure 4. For more research on two-dimensional age replacement, please refer to refs [29,30].

Figure 4 
                  Two-dimensional age replacement strategy.
Figure 4

Two-dimensional age replacement strategy.

The two-dimensional block replacement strategy is to replace the product regularly according to the time and use degree after the product is put into use. The two-dimensional block replacement interval can be expressed as (T 0, U 0), in which the time interval is T 0 and the use degree interval is U 0. If any one of the time or use degree of the product reaches the threshold of a given interval, the product shall be replaced. Even if the product is replaced due to functional failure between two regular replacements, it needs to be replaced at the scheduled regular replacement time. If the univariate method [31] is selected to describe the two-dimensional failure law of the product, the two-dimensional block replacement process is shown in Figure 5. Ke and Yao [9] considered three different decision criteria and studied the block replacement policy under uncertain environment on the premise that the component life is a random variable. Schouten et al. [32] studied the optimal block replacement policy of wind turbine system components under the condition of time-varying cost. Zhang et al. [33] considered the duration of the task and gave the optimal plan for block replacement from the perspective of cost and maintainability. Azevedo et al. [34] assumed that the product adopts a corrective replacement strategy and an imperfect maintenance strategy when critical failures and non-critical failures occur, respectively, and used a multi-objective genetic algorithm (GA) to obtain a block replacement plan for products. For more research on two-dimensional block replacement, please refer to ref. [35].

Figure 5 
                  Two-dimensional periodic replacement strategy.
Figure 5

Two-dimensional periodic replacement strategy.

As shown in Figure 5, the shape parameter r 0 of the two-dimensional block replacement area is U 0/T 0, when r > r 0, U 0 is the interval of block replacement, and when r <r 0, T 0 is the interval of block replacement. However, periodic replacement strategy mainly concentrates in the basic warranty stage, and lacks extended warranty research.

3 Model assumptions

The time limit and usage limit of the two-dimensional basic warranty period are W B and U B, and the time limit and usage limit of the two-dimensional extended warranty period are W E and U E. Let is Ω1 = [0, W B) × [0, U B)], Ω2 = [0, W E) × [0, U E)]. There is a linear relationship between time and usage. The relationship between products usage u and time t is u = rt. r is a random variable and is different for different users, g(r) and G(r) are the probability density function and probability distribution function of r, respectively. The product life is two-dimensional, and W L and U L are the time limit and usage limit of the product life, respectively.

Assume that the two-dimensional block replacement maintenance is imperfect maintenance, and the interval is (T 0,  U 0). T 0 is the time interval for two-dimensional block replacement, U 0 is the usage interval for two-dimensional block replacement, and imperfect preventive maintenance is performed regardless of which limit comes first. ω is the repair degree of imperfect preventive maintenance, when ω = 1, imperfect preventive maintenance becomes perfect maintenance, and when ω = 0, imperfect preventive maintenance becomes minimum maintenance. Maintenance time can be ignored due as it is small relative to preventive maintenance intervals and products life.

C p is the cost of imperfect preventive maintenance, and C m is the cost of corrective maintenance. Let n k (k = 1,2,3…) be the number of imperfect preventive maintenance at different stages of the product life cycle, and the specific values are shown in Table 1. The cost of basic warranty shall be borne by the manufacturer, and the cost of extended warranty service shall be borne by the user. Under different usage rates r, there are two implementation situations for two-dimensional imperfect preventive maintenance, as shown in Figure 6.

Table 1

Number of preventive maintenances over each interval in product life

Number Stage Value
n 1 [0, W B) W B / T 0
n 2 [0, W B) W B r / T 0
n 3 [0, U B) U B / U 0
n 4 [0, U B) U B / T 0 r
Figure 6 
               Two-dimensional imperfect preventive maintenance. (a) r ≤ r
                  0 and (b) r > r
                  0.
Figure 6

Two-dimensional imperfect preventive maintenance. (a) rr 0 and (b) r > r 0.

Case 1

When rr 0, the implementation time of imperfect preventive maintenance is T j = jT 0 (j = 1,2,3…), as shown in Figure 6(a).

Case 2

When r > r 0, the implementation time of imperfect preventive maintenance is T j = jU 0/r (j = 1,2,3…), as shown in Figure 6(b).

4 Model construction

4.1 Basic warranty cost model

The failure rate function of the component is:

(1) λ ( t | r ) = θ 0 + θ 1 r + θ 2 t 2 + θ 3 r t 2 .

Failure rate fallback method is adopted to describe the effect of imperfect preventive maintenance [175]. That is, after an imperfect preventive maintenance, the failure rate of components becomes

(2) λ t + = ( 1 ω ) λ t ,

where λ t is the component failure rate before the imperfect preventive maintenance is performed and λ t + is the component failure rate after the imperfect preventive maintenance is performed. Then, there are two kinds of failure rates of components between the n-th preventive maintenance and the (n + 1)-th preventive maintenance.

Case 1

When rr 0, the implementation time of imperfect preventive maintenance is T j = j T 0 , and the failure rate of components is

(3) λ ( n T 0 ) + = λ ( t | r ) ω j = 0 n 1 ( 1 ω ) j λ ( ( n j ) T 0 | r ) ( j = 1 , 2 , 3 , .. . ) .

Case 2

When r > r 0, the implementation time of imperfect preventive maintenance is U j = j U 0 ( T j = j U 0 / r ) , and the failure rate of components is

(4) λ ( n T 0 ) + = λ ( t | r ) ω j = 0 n 1 ( 1 ω ) j λ ( ( n j ) U 0 / r | r ) ( j = 1 , 2 , 3 , .. . ) .

Proof

Taking case 1 as an example, the above conclusion is proved by induction. During the first preventive maintenance interval [0, T 0], the failure rate of components is λ ( t | r ) , which conforms to the formula of case 1. Assuming that the formula of case 1 is correct when the number of imperfect preventive maintenance is less than or equal to n, that is, after the n-th preventive maintenance, the failure rate of components is

(5) λ ( n T 0 ) + = λ ( t | r ) ω j = 0 n 1 ( 1 ω ) j λ ( ( n j ) T 0 | r ) .

Then, the failure rate of components before (n + 1)-th preventive maintenance is

(6) λ [ ( n + 1 ) T 0 ] = λ ( ( n + 1 ) T 0 | r ) ω j = 0 n 1 ( 1 ω ) j λ ( ( n j ) T 0 | r ) .

According to formula 2, the failure rate of components after (n + 1)-th preventive maintenance is

(7) λ [ ( n + 1 ) T 0 ] + = ( 1 ω ) λ [ ( n + 1 ) T 0 ] Q = λ ( ( n + 1 ) T 0 | r ) ω λ ( ( n + 1 ) T 0 | r ) + ω j = 0 n 1 ( 1 ω ) j + 1 λ ( ( n j ) T 0 | r ) = λ ( ( n + 1 ) T 0 | r ) ω j = 0 n ( 1 ω ) j λ ( ( n + 1 j ) T 0 | r ) .

The formula proving the method of case 2 is similar to that of case 1.

First, the basic warranty cost model is established. According to the quantity relationship between r 0 and r B , there are two cases in the basic warranty: r 0 r B and r 0 > r B . When r 0 r B , according to different r, there are three cases: 0 < r r 0 , r 0 < r r B , and r > r B . When 0 < r r 0 , the basic warranty cost is

(8) E 1 ( C ) = n 1 C p + C m i = 0 n 1 1 i T 0 ( i + 1 ) T 0 λ ( t | r ) ω j = 0 i 1 ( 1 ω ) j λ ( ( i j ) T 0 | r ) d t + n 1 T 0 W B λ ( t | r ) ω j = 0 n 1 1 ( 1 ω ) j λ ( ( n 1 j ) T 0 | r ) d t .

When r 0 < r r B , the basic warranty cost is

(9) E 2 ( C ) = n 2 C p + C m i = 0 n 2 1 i U 0 / r ( i + 1 ) U 0 / r λ ( t | r ) ω j = 0 i 1 ( 1 ω ) j λ ( ( i j ) U 0 / r | r ) d t + n 2 U 0 / r W B λ ( t | r ) ω j = 0 n 2 1 ( 1 ω ) j λ ( ( n 2 j ) U 0 / r | r ) d t .

When r > r B , the basic warranty cost is

(10) E 3 ( C ) = n 3 C p + C m i = 0 n 3 1 i U 0 / r ( i + 1 ) U 0 / r λ ( t | r ) ω j = 0 i 1 ( 1 ω ) j λ ( ( i j ) U 0 / r | r ) d t + n 3 U 0 / r U B / r λ ( t | r ) ω j = 0 n 3 1 ( 1 ω ) j λ ( ( n 3 j ) U 0 / r | r ) d t .

So, when r 0 r B , the basic warranty cost of the components is

(11) E ( C B 1 ) = 0 r 0 E 1 ( C ) g ( r ) d r + r 0 r B E 2 ( C ) g ( r ) d r + r B E 3 ( C ) g ( r ) d r .

When r 0 > r B , according to different r, there are three cases: 0 < r r B , r B < r r 0 , and r > r 0 . When 0 < r r B , the basic warranty cost is

(12) E 4 ( C ) = n 1 C p + C m i = 0 n 1 1 i T 0 ( i + 1 ) T 0 λ ( t | r ) ω j = 0 i 1 ( 1 ω ) j λ ( ( i j ) T 0 | r ) d t + n 1 T 0 W B λ ( t | r ) ω j = 0 n 1 1 ( 1 ω ) j λ ( ( n 1 j ) T 0 | r ) d t .

When r B < r r 0 , the basic warranty cost is

(13) E 5 ( C ) = n 4 C p + C m i = 0 n 4 1 i T 0 ( i + 1 ) T 0 λ ( t | r ) ω j = 0 i 1 ( 1 ω ) j λ ( ( i j ) T 0 | r ) d t + n 4 T 0 U B / r λ ( t | r ) ω j = 0 n 4 1 ( 1 ω ) j λ ( ( n 4 j ) T 0 | r ) d t .

When r > r 0 , the basic warranty cost is

(14) E 6 ( C ) = n 3 C p + C m i = 0 n 3 1 i U 0 / r ( i + 1 ) U 0 / r λ ( t | r ) ω j = 0 i 1 ( 1 ω ) j λ ( ( i j ) U 0 / r | r ) d t + n 3 U 0 / r U B / r λ ( t | r ) ω j = 0 n 3 1 ( 1 ω ) j λ ( ( n 3 j ) U 0 / r | r ) d t .

So, when r 0 > r B , the basic warranty cost of the components is

(15) E ( C B 2 ) = r L r B E 4 ( C ) g ( r ) d r + r B r 0 E 5 ( C ) g ( r ) d r + r 0 r u E 6 ( C ) g ( r ) d r .

To sum up, the basic warranty cost under different conditions is

(16) E ( C B ) = r l r 0 E 1 ( C ) g ( r ) d r + r 0 r B E 2 ( C ) g ( r ) d r + r B r u E 3 ( C ) g ( r ) d r r 0 r B r l r B E 4 ( C ) g ( r ) d r + r B r 0 E 5 ( C ) g ( r ) d r + r 0 r u E 6 ( C ) g ( r ) d r r 0 > r B .

4.2 Extended warranty cost scheme

Next from the perspective of the manufacturer and the user, the component life cycle cost borne by all parties is established to determine the win-win extended warranty cost. It is considered to carry out two-dimensional imperfect preventive maintenance only in the basic warranty period, and only corrective maintenance in other life stages. From the perspective of users, assuming that C Iy (C In) is the component life cycle cost borne by users when they choose extended warranty service (do not choose extended warranty service), there is an upper limit on the cost E ( C E ) of extended warranty service to satisfy users, that is

(17) C I y + E ( C E ) C I n .

In calculating C In, there are two cases: r L > r B and r L r B . For case r L > r B , when r 0 r L > r B , if the user will not choose extended warranty service, the full life cycle cost borne by the user is

(18) C I n = C m r l r u U B / r U L / r λ ( t | r ) ω j = 0 n 3 1 ( 1 ω ) j λ ( ( n 3 j ) U 0 / r | r ) d t g ( r ) d r + r L r 0 U B / r U L / r λ ( t | r ) ω j = 0 n 4 1 ( 1 ω ) j λ ( ( n 4 j ) T 0 | r ) d t g ( r ) d r + r B r L U B / r T L λ ( t | r ) ω j = 0 n 4 1 ( 1 ω ) j λ ( ( n 4 j ) T 0 | r ) d t g ( r ) d r + r l r B W B T L λ ( t | r ) ω j = 0 n 1 1 ( 1 ω ) j λ ( ( n 1 j ) T 0 | r ) d t g ( r ) d r .

When r L > r 0 r B , if the user will not choose extended warranty service, the full life cycle cost borne by the user is

(19) C I n = C m r L r u U B / r U L / r λ ( t | r ) ω j = 0 n 3 1 ( 1 ω ) j λ ( ( n 3 j ) U 0 / r | r ) d t g ( r ) d r + r 0 r L U B / r T L λ ( t | r ) ω j = 0 n 3 1 ( 1 ω ) j λ ( ( n 3 j ) U 0 / r | r ) d t g ( r ) d r + r B r 0 U B / r T L λ ( t | r ) ω j = 0 n 4 1 ( 1 ω ) j λ ( ( n 4 j ) T 0 | r ) d t g ( r ) d r + r l r B W B T L λ ( t | r ) ω j = 0 n 1 1 ( 1 ω ) j λ ( ( n 1 j ) T 0 | r ) d t g ( r ) d r .

When r L > r B > r 0 , if the user will not choose extended warranty service, the full life cycle cost borne by the user is

(20) C I n = C m r L r u U B / r U L / r λ ( t | r ) ω j = 0 n 3 1 ( 1 ω ) j λ ( ( n 3 j ) U 0 / r | r ) d t g ( r ) d r + r B r L U B / r T L λ ( t | r ) ω j = 0 n 3 1 ( 1 ω ) j λ ( ( n 3 j ) U 0 / r | r ) d t g ( r ) d r + r 0 r B W B T L λ ( t | r ) ω j = 0 n 2 1 ( 1 ω ) j λ ( ( n 2 j ) U 0 / r | r ) d t g ( r ) d r + r l r 0 W B T L λ ( t | r ) ω j = 0 n 1 1 ( 1 ω ) j λ ( ( n 1 j ) T 0 | r ) d t g ( r ) d r .

For case r L r B

(21) C In = C m r 0 r u U B / r U L / r λ ( t | r ) ω j = 0 n 3 1 ( 1 ω ) j λ ( ( n 3 j ) U 0 / r | r ) d t g ( r ) d r + r B r 0 U B / r U L / r λ ( t | r ) ω j = 0 n 4 1 ( 1 ω ) j λ ( ( n 4 j ) T 0 | r ) d t g ( r ) d r + r L r B W B U L / r ( λ ( t | r ) ω j = 0 n 1 1 ( 1 ω ) j λ ( ( n 1 j ) T 0 | r ) ) d t g ( r ) d r + r l r L W B T L ( λ ( t | r ) ω j = 0 n 1 1 ( 1 ω ) j λ ( ( n 1 j ) T 0 | r ) ) d t g ( r ) d r r L r B < r C m r B r u U B / r U L / r ( λ ( t | r ) ω j = 0 n 3 1 ( 1 ω ) j λ ( ( n 3 j ) U 0 / r | r ) ) d t g ( r ) d r + r 0 r B W B U L / r ( λ ( t | r ) ω j = 0 n 2 1 ( 1 ω ) j λ ( ( n 2 j ) U 0 / r | r ) ) d t g ( r ) d r + r L r 0 W B U L / r ( λ ( t | r ) ω j = 0 n 1 1 ( 1 ω ) j λ ( ( n 1 j ) T 0 | r ) ) d t g ( r ) d r + r l r L W B T L ( λ ( t | r ) ω j = 0 n 1 1 ( 1 ω ) j λ ( ( n 1 j ) T 0 | r ) ) d t g ( r ) d r r L r 0 r B C m r B r u U B / r U L / r ( λ ( t | r ) ω j = 0 n 3 1 ( 1 ω ) j λ ( ( n 3 j ) U 0 / r | r ) ) d t g ( r ) d r + r L r B W B U L / r ( λ ( t | r ) ω j = 0 n 2 1 ( 1 ω ) j λ ( ( n 2 j ) U 0 / r | r ) ) d t g ( r ) d r + r 0 r L W B T L ( λ ( t | r ) ω j = 0 n 2 1 ( 1 ω ) j λ ( ( n 2 j ) T 0 | r ) ) d t g ( r ) d r + r l r 0 W B T L ( λ ( t | r ) ω j = 0 n 1 1 ( 1 ω ) j λ ( ( n 1 j ) T 0 | r ) ) d t g ( r ) d r r 0 < r L r B .

When calculating C Iy, there are six cases, including r L r E r B , r E < r L < r B , r E < r B < r L , r B < r E < r L , r B < r L < r E , and r L < r B < r E . In order to reduce the number of discussions, let r E = r B . When r L r E = r B ,

(22) C I y = C m r 0 r u U E / r U L / r λ ( t | r ) ω j = 0 n 3 1 ( 1 ω ) j λ ( ( n 3 j ) U 0 / r | r ) d t g ( r ) d r + r B r 0 U E / r U L / r λ ( t | r ) ω j = 0 n 4 1 ( 1 ω ) j λ ( ( n 4 j ) T 0 | r ) d t g ( r ) d r + r L r B W E U L / r ( λ ( t | r ) ω j = 0 n 1 1 ( 1 ω ) j λ ( ( n 1 j ) T 0 | r ) ) d t g ( r ) d r + r l r L W E T L ( λ ( t | r ) ω j = 0 n 1 1 ( 1 ω ) j λ ( ( n 1 j ) T 0 | r ) ) d t g ( r ) d r r L r B < r 0 C m r B r u U E / r U L / r ( λ ( t | r ) ω j = 0 n 3 1 ( 1 ω ) j λ ( ( n 3 j ) U 0 / r | r ) ) d t g ( r ) d r + r 0 r B W E U L / r ( λ ( t | r ) ω j = 0 n 2 1 ( 1 ω ) j λ ( ( n 2 j ) U 0 / r | r ) ) d t g ( r ) d r + r L r 0 W E U L / r ( λ ( t | r ) ω j = 0 n 1 1 ( 1 ω ) j λ ( ( n 1 j ) T 0 | r ) ) d t g ( r ) d r + r l r L W E T L ( λ ( t | r ) ω j = 0 n 1 1 ( 1 ω ) j λ ( ( n 1 j ) T 0 | r ) ) d t g ( r ) d r r L r 0 r B C m r B r u U E / r U L / r ( λ ( t | r ) ω j = 0 n 3 1 ( 1 ω ) j λ ( ( n 3 j ) U 0 / r | r ) ) d t g ( r ) d r + r L r B W E U L / r ( λ ( t | r ) ω j = 0 n 2 1 ( 1 ω ) j λ ( ( n 2 j ) U 0 / r | r ) ) d t g ( r ) d r + r 0 r L W E T L ( λ ( t | r ) ω j = 0 n 2 1 ( 1 ω ) j λ ( ( n 2 j ) T 0 | r ) ) d t g ( r ) d r + r l r 0 W E T L ( λ ( t | r ) ω j = 0 n 1 1 ( 1 ω ) j λ ( ( n 1 j ) T 0 | r ) ) d t g ( r ) d r r 0 < r L r B .

When r B = r E < r L ,

(23) C I y = C m r 0 r u U E / r U L / r λ ( t | r ) ω j = 0 n 3 1 ( 1 ω ) j λ ( ( n 3 j ) U 0 / r | r ) d t g ( r ) d r + r L r 0 U E / r U L / r λ ( t | r ) ω j = 0 n 4 1 ( 1 ω ) j λ ( ( n 4 j ) T 0 | r ) d t g ( r ) d r + r E r L U E / r T L λ ( t | r ) ω j = 0 n 4 1 ( 1 ω ) j λ ( ( n 4 j ) T 0 | r ) d t g ( r ) d r + r l r B W E T L λ ( t | r ) ω j = 0 n 1 1 ( 1 ω ) j λ ( ( n 1 j ) T 0 | r ) d t g ( r ) d r r 0 r L > r B C m r L r u U E / r U L / r λ ( t | r ) ω j = 0 n 3 1 ( 1 ω ) j λ ( ( n 3 j ) U 0 / r | r ) d t g ( r ) d r + r 0 r L U E / r T L λ ( t | r ) ω j = 0 n 3 1 ( 1 ω ) j λ ( ( n 3 j ) U 0 / r | r ) d t g ( r ) d r + r B r 0 U E / r T L λ ( t | r ) ω j = 0 n 4 1 ( 1 ω ) j λ ( ( n 4 j ) T 0 | r ) d t g ( r ) d r + r l r B W E T L λ ( t | r ) ω j = 0 n 1 1 ( 1 ω ) j λ ( ( n 1 j ) T 0 | r ) d t g ( r ) d r r L > r 0 r B C m r L r u U E / r U L / r λ ( t | r ) ω j = 0 n 3 1 ( 1 ω ) j λ ( ( n 3 j ) U 0 / r | r ) d t g ( r ) d r + r B r L U E / r T L λ ( t | r ) ω j = 0 n 3 1 ( 1 ω ) j λ ( ( n 3 j ) U 0 / r | r ) d t g ( r ) d r + r 0 r