Zhang Hongqi Zhao Jiabin Su Chun

(1The 38th Research Institute of CETC, Hefei 230088, China)(2Anhui Technical Standard Innovation Base (Intelligent Design and Munufacturing), Hefei 230088, China)(3School of Mechanical Engineering, Southeast University, Nanjing 211189, China)

Abstract：To increase customers’ satisfaction and promote product’s competitiveness, a customized extended warranty (EW) policy is proposed, where the diversities in both the usage rate and purchase date are considered. The marginal approach is applied to describe the product’s two-dimensional failure in terms of age and usage, respectively. Moreover, minimal repair is adopted to restore the failure, and the virtual age method is applied to depict the effect of preventive maintenance (PM). On this basis, an optimization model is established to minimize the maintenance cost and warranty cost from the manufacturer’s view, and multiple factors are taken into account, including the PM’s intensity and its period, and EW’s interval, etc. A numerical case study is provided to illustrate the effectiveness of the proposed approach. The results show that by considering the product’s usage rate and the purchasing date of EW, the number of failures as well as the cost of maintenance and warranty can be reduced effectively.

Key words：customization; two-dimensional extended warranty; preventive maintenance; usage rate; purchasing date; warranty cost

Warranty is a contractual obligation between the manufacturer and consumers. It can not only protect the consumers against defective items, but can also help the manufacturer to avoid customers’ unreasonable claims[1]. In addition to the basic warranty (BW), the extended warranty (EW) is also widely adopted by the manufacturer to establish a long-term relationship with the customers, and thus to gain the opportunities of product marketing and after-sale service.

Maintenance policy and the corresponding maintenance cost are of great importance for the warranty decision-making. It is reported that the warranty cost accounts for 2% to 10% of products’ operating revenues, and by optimizing the maintenance policy, the warranty cost can be reduced effectively[2]. Chien[3]proposed a periodic replacement maintenance policy for the one-dimensional (1D) warranty, which is characterized by a single variable, e.g. time (age) or usage. A two-dimensional (2D) warranty can take into account the age and usage rate concurrently, and it is more suitable for various industrial products, such as automobiles, CNC machines and engineering machinery, etc. Based on the accelerated failure time model, Shahanaghi et al.[4]proposed a model to optimize the number and degree of preventive repairs so as to minimize the EW provider’s servicing cost. From the manufacturer’s perspective, Wang et al.[5]investigated the periodic imperfect preventive maintenance (PM) for the products covered by a fixed and combined BW and EW region. Su and Shen[6]proposed two types of EW policies from the view of manufacturers, and three kinds of repair options were considered for the failed components. Based on the generalized renewal process and Stackelberg game, Santana et al.[7]designed the price for EW and on-demand maintenance to optimize the manufacturer’s profit.

In the above studies, the proposed warranty policies are rigid, and the customers are provided with the same warranty items. Actually, customers usually have different using habits, diverse operating environments and differentiated affordability, and their requirements for maintenance and warranty are personalized in nature. Moreover, the usage rate will influence product’s failure rate, maintenance policy as well as the warranty cost. Obviously, rigid warranty policies cannot satisfy customers’ personalized needs[8].

Customized warranty aims to consider various factors and meet customers’ diversified demands. Thus, it has attracted much attention in recent years. Based on the attractiveness index, He et al.[9]proposed a demand function to describe the boosted demand of extending the warranty period, and an optimal 2D BW period was obtained by maximising the expected profit. On the basis of the usage rate, Ye et al.[10]grouped the customers and provided them with differentiated EW periods. Based on the dynamic usage rate, Tong et al.[11]proposed a warranty maintenance strategy for 2D EW, a maintenance model was constructed to optimize the maintenance degree and help the service providers reduce the warranty cost. Lam et al.[12]presented an EW model including a free repair period and an EW period, where consumers had the choice to renew or not to renew the warranty period. On this basis, the long-run average cost per unit time was derived, and the optimal policies were obtained for the consumers. Su et al.[13]presented a two-stage PM optimal model and the moment that customers purchased 2D EW was considered. Jack et al.[14]put forward a repair-replace strategy for the products sold with a 2D warranty, and the accelerated failure time (AFT) model was applied to describe the effect of the usage rate on item’s degradation.

Up to now, there are still some problems with the customized warranty policies. For example, it is usually assumed that customers buy EW only when the product is sold or when the BW is expired. In fact, customers may buy EW at any time. Moreover, the proposed PM policies are usually of periodicity and iso-strength. Considering the above shortcomings, this study puts forward some improvements on the customized warranty. The main contributions include: 1) Customers can purchase EW at any time during the product’s useful life; 2) A non-equal periodic PM policy is proposed, and a more flexible warranty can be obtained; 3) The corresponding maintenance method, warranty period and warranty cost can be customized.

1 Models of Failure and Maintenance

1.1 Failure modeling

Up to now, three approaches have been proposed for the failure modeling of 2D warranty, i.e., the bivariate method, composite scale method and marginal method[1,14]. Among them, the marginal method defines the failure strength as a function of product’s age and usage rate, and it considers the influence of age and usage on the times of failures simultaneously. In this study,T(t) andU(t) denote the product’s cumulative use time and usage at timet, respectively, and it is assumed that the repair time can be ignored. Thus,T(t)=tand the product’s cumulative usage isU(t)=rt.

Letλ(t|r) denote the failure strength function of the product. The mathematical expressions ofλ(t|r) can be obtained by data-fitting. For details, one can refer to Refs.[5,15-16]. Therefore, the failure strength function model is

λ(t|r)=θ0+θ1r+θ2T(t)+θ3U(t)=θ0+θ1r+(θ2+θ3r)t

(1)

The extreme value of Eq.(1) is close to 0; that is, the inherent reliability of a new product isλ(0|r)=θ0+θ1r. Considering that the product’s inherent reliability is independent of its usage rate,θ1=0. In addition, the derived function of Eq.(1) isθ2+θ3r, which means that the change of failure strength is linearly related to the usage rate, regardless of the using time. This study considers the impact of age, usage as well as their interaction on the failure rate. Assuming that the product’s inherent reliability isθ0, we can obtain the improved two-dimensional warranty failure model as

λ(t|r)=θ0+θ1T(t)+θ2U(t)+θ3T(t)U(t)=

θ0+θ1t+θ2rt+θ3rt2

(2)

whereθiare the constants with positive value,i=0, 1, 2, and 3, respectively.

1.2 A novel non-equal strength PM model

Up to now, three types of maintenance policies are widely applied in the warranty terms[6,17]: 1) Policy 1: Minimal repair is adopted for all the failures. 2) Policy 2: Minimal repair is adopted in BW, and minimal repair as well as periodic PM are used in EW. 3) Policy 3: Minimal repair is adopted before the customers purchase EW; and after that, minimal repair and periodic PM are adopted until the end of EW.

With the increase in cumulative usage and using time, the product’s failure rate will usually tend to increase. To make the maintenance policy more flexible, we propose a new non-equal strength periodic PM, and it is called Policy 4. Except that the maintenance strength during each period may be different, the other assumptions are the same as Policy 3.

The inspection period of PM isP(ΔT, ΔU), where ΔTand ΔUare the step sizes of the time and usage, respectively. In other words, when time reaches ΔTor the usage reaches ΔU, whichever comes first, one time of PM will be adopted. Thus,P=min(ΔT, ΔU/r). It includes three types of cases, as shown in Fig.1. The number of PMs during the warranty period isn=fix(W/P), where fix is a rounding down function. In the 2D warranty, the actual inspection time nodes are different from the change of usage rates. When the usage rate is equal to the designed usage rater0, i.e.r0=W/U, the inspection period for PM isP(ΔT, ΔU) and the warranty period will be ended atW. When the usage rater1>r0andr2

Fig.1 PM period under different usage rates

The virtual age method is used to describe the effect of PM. The virtual age factor isδ(m)=(1+m)e-m, where 0≤m≤M. Particularly, whenm=0, the effect of PM is ignored; and whenm=M(Mis infinite), the effect of PM is the same as that of the replacement. For an incomplete PM, 0

During the warranty period, the maintenance policy and failure rate’s variation tendency are shown in Fig.2, wheretis the product’s using time, andλis the product’s failure strength. At the time ofT+P, the manufacturer implements the first PM with the maintenance strength ofm1. The failure rate is reduced fromλ2to the failure rate, where the product’s virtual age isδ(m1)(T+P), and 0<δ(m1)<1. By analogy, after a fixed period, a PM will be performed, and the maintenance strength in each period is irrelevant. Specially,δ(mn-1)=0.

Fig.2 Failure rate’s change tendency with PM policy

The duration of a PM isW-T, then the times of PM during the warranty period isn=fix((W-T)/P). After thej-th PM, the product’s virtual age isvj(j=1, 2, …,n-1,n), then we have

(3)

For the minimal repair, the average cost isCm, and the cost of thej-th PM isCP(mj). During thej-th PM period, the total warranty cost isCj, which includes the cost for the minimal repair and PM. Therefore, we can obtain the warranty cost optimization model for each PM period.

(4)

Let the total warranty cost during the warranty period beC. We can obtain the warranty cost optimization model during the entire warranty period as follows:

(5)

2 Modeling of Warranty Cost

In this study, two factors are considered to analyze the warranty cost, i.e., the product’s usage rate and the EW’s purchasing date. It is assumed that for a particular customer, the usage rate is subject to a random distributionG(r), andg(r) is the density distribution function. Based on the usage rate, a novel method is proposed to divide the customers into different groups, as shown in Fig.3. In Fig.3,rlandrhrepresent the lowest usage rate and the highest usage rate, respectively,rl≤r≤rh;kpoints of the usage rate are taken equally spaced during the warranty period, i.e.rl,r2, …,rh; and the customers are divided into group 1 to groupk-1.

Fig.3 Customer classification based on usage rate

According to the time that the customer purchases the EW, two cases are considered.

Case1Customer purchases EW before the expiration of BW, i.e. 0≤T≤Wband 0≤U≤Ub. In this case, the manufacturer provides customers with the same rectangular EW period, and determines the PM policy for the remaining BW and EW period according to the EW’s purchasing date, as shown in Fig.4.

Fig.4 Purchasing EW before BW’s expiration

Case2Customer purchases EW at some point after the BW is expired, i.e.T>WborU>Ub. In this case, the manufacturer provides the customer with a customized EW periodΩ′e=(W′e,U′e) as well as the maintenance polices according to the purchasing date and usage rate, as shown in Fig.5.

Fig.5 Purchasing EW after BW expired

(6)

Considering the difference in customer’s usage rate,P1andmjare the variables to be optimized, and in the total warranty period, the expected warranty cost optimization model is

0≤mj≤M

(7)

(8)

(9)

(10)

Considering the customers’ usage rate as well as the optimizing variables ofP2andmj, we obtain the expected cost optimization model for the manufacturer as follows:

0≤mj≤M

(11)

(12)

Considering that the EW period will continue to decrease with the using time, when the using time reaches a certain threshold value, the manufacturer cannot make a profit or will even lose money by providing the warranty. Therefore, the manufacturer should set a boundary for the EW. By using Eq.(9), the deadlineTdfor EW’s purchasing date can be obtained.

3 Case Study

In this section, a numerical example is provided to illustrate the effectiveness of the proposed policies. Both the BW and EW periods are (3 years, 6×104km). The parameters in the failure strength function are set as follows:θ0=0.05,θ1=0.7,θ2=0.5, andθ3=0.1. It is supposed that the customer’s usage rate is subject to a uniform distribution,rl=0.5×104km/year,rh=3.5×104km/year, andr0=2×104km/year. In total, there are 6 levels of maintenance strength, i.e.,m=0, 1, 2, 3, 4, 5 andM, and the corresponding PM costs areCp= 0, 20, 50, 90, 150, 240, and 3 000 dollar, respectively. The average minimal repair cost isCm= 250 dollar. In addition, we discretize the PM period intoP(ΔT, ΔU) and ΔTand ΔUare the integers in units of month and 103km, respectively.

According to Eq.(6), we can obtain the PM periods (i.e. ΔTand ΔU), PM strength (m) and the corresponding total warranty cost (C) whenT=0. When the usage rate isr0, the comparison of different maintenance policies (presented in Section 2.2) is shown in Tab.1. Among them, Policy 4 has the lowest total warranty cost of 2 041.28 dollar. It indicates that compared with the other three policies, the proposed maintenance policy can achieve better results. The corresponding PM period is (4 months, 7×103km). Furthermore, the strength of the first two PM periods is 0; and for the other PM periods, the strength of 2 is adopted.

Tab.1 Comparison of total warranty cost under different policies

Considering the maintenance policies and purchasing date of EW, the minimum warranty cost is shown in Fig.6. It can be found that Policy 1 and Policy 2 have nothing to do with the purchasing date, and the warranty cost of Policy 2 is lower than that of Policy 1. The total warranty cost of Policy 3 and Policy 4 will increase with the delay of purchasing date. At the end of BW (i.e. the 36 months), the warranty cost for Policy 3 is 9 891.10 dollar, which is the same as that of Policy 2. Furthermore, by adopting Policy 4, the minimum warranty cost is 2 030.49 dollar at 3 and 7 months, respectively. Obviously, regardless of the purchasing date of EW, the proposed maintenance policy can obtain the lowest warranty cost for the manufacturer.

Fig.6 Warranty cost for different purchases time and maintenance policies

According to Policy 4, the optimized maintenance policies for different purchasing dates are shown in Tab.2. The PM period is rounded to (4 months, 7×103km). Additionally, the PM strength will change with the purchasing date. For example, during the first seven months, the PM strength is 2/0. It means that only the last time of PM is 0 (the last time of PM is canceled), and all the remaining maintenance strength is 2. The meaning of other PM’s strength is as follows: 3/2/0 means that the strength of the first PM is 3, the strength of the last PM is 0, and the remaining’s is 2; 3/2 means that the PM strength of the first PM is 3, and the PM strength of the remaining month is 2.

Tab.2 Optimized maintenance strategies for different purchasing dates

According to the proposed method in Section 2, customers are divided into six groups:r1=0.75×104km/year,r2=1.25×104km/year,r3=1.75×104km/year,r4=2.25×104km/year,r5=2.75×104km/ year, andr6=3.25×104km/year. Based on Eqs.(7) and (8), we can obtain the average warranty cost, and the comparison of the two situations is shown in Fig.7. For the customers with a high usage rate, the BW will end earlier; while for the divided customers’ groups, the total warranty cost can be reduced in stages. The expected total warranty cost is 2 252.77 and 2 342.16 dollar for the divided and undivided usage rate strategies, respectively. Obviously, for the manufacturer it is beneficial to divide the customers into different groups based on their usage rate.

Fig.7 Comparison of warranty cost of different customer groups

Fig.8 Warranty cost changing with purchasing date for different groups

Similarly, by minimizing the maintenance cost, we can obtain the maintenance policy for each group, as shown in Tab.3. From Tab.3, when the usage rate is low, the corresponding PM period is long, and the start time for PM will be postponed. Additionally, the PM strength may be different during each period. For instance, the PM strength for Groups 1, 2 and 4 is 2/0. It means that the strength for the last time of PM is 0 and the strength for the rest is 2. The PM strength of Group 3 are 2, 2, 2, 1, 2, 2, 2, 1, 2, 2, 2, 1, 2, 2, 2, 1, respectively.

Tab.3 Optimized maintenance policy for each customer group

When BW is expired, we can obtain the EW periods for different purchasing times accordingly, as shown in Tab.4. It shows that the warranty period tends to decrease with the increase in the purchasing time, which is provided by the manufacturer. With the constraint ofW′e=2, the manufacturer stops selling the EW. In the 84th month, the EW period is (24 months, 4×104km). Therefore, the deadline for purchasing EW is set to beTd= 84 months in this case, and the optimized PM period is 4 months. At the end of BW, the PM strength is 3/2, during the EW, the PM strength becomes 4/2. It means that the PM strength for the first time is 4, and it is 2 for the rest of the time. In addition, we can also obtain the warranty cost under different maintenance policies. After optimization, the total warranty cost fluctuates around 3 700 dollar.

Tab.4 Maintenance strategies under different purchasing times

Similarly, if a customer purchases EW after the BW expired, the manufacturer can also provide him/her with a customized maintenance policy. According to Eq.(9), we can obtain the optimized maintenance policy as well as the warranty cost for the customers with any usage rate. The results are similar to Fig.8 and Tab.3. Here, we do not discuss the details furthermore.

4 Conclusions

1) By combining minimal repair and PM, better maintenance effect can be obtained compared with applying only minimal repair. Furthermore, there is an appropriate time point during the BW period to update the BW maintenance policy and start the PM.

2) For the customers, the warranty cost increases if they postpone purchasing EW. Meanwhile, the corresponding warranty period is also shortened. Therefore, the manufacturer should adopt some preferential measures to encourage customers to purchase EW as early as possible.

3) Besides the purchasing date, the customer’s usage rate should be considered in a customized warranty policy. It can help the manufacturer to reduce the expected number of failures and the warranty cost during the warranty period.

4) In addition to the usage rate, in the future study, the following factors can also be integrated into the customized warranty policy, including the product’s operating environment, customer’s risk preference and their learning ability, etc.