Two-dimensional extended warranty strategy including maintenance level and purchase time: A win-win perspective

Journal Pre-proofs Two-dimensional extended warranty strategy including maintenance level and purchase time: a win-win perspective Zhen He, Dongfan Wang, Shuguang He, Yiwen Zhang, Anshu Dai PII: DOI: Reference:

S0360-8352(20)30028-0 https://doi.org/10.1016/j.cie.2020.106294 CAIE 106294

To appear in:

Computers & Industrial Engineering

Received Date: Revised Date: Accepted Date:

4 July 2019 12 January 2020 13 January 2020

Please cite this article as: He, Z., Wang, D., He, S., Zhang, Y., Dai, A., Two-dimensional extended warranty strategy including maintenance level and purchase time: a win-win perspective, Computers & Industrial Engineering (2020), doi: https://doi.org/10.1016/j.cie.2020.106294

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Two-dimensional extended warranty strategy including maintenance level and purchase time: a win-win perspective Zhen He1 , Dongfan Wang1 , Shuguang He1∗, Yiwen Zhang1 , Anshu Dai2

Acknowledgement This work was supported by the National Natural Science Foundation of China (NSFC) [Grant number 71661147003], [Grant number 71872123], [Grant number 71532008], [Grant number 71802145], and [Grant number 71902180], and [Grant number 71902139].

1

Department of Industrial Engineering, Tianjin University, Tianjin, China 300072 Co-Innovation Center for Computable Modeling in Management Science, Tianjin University of Finance and Economics, Tianjin, China 300072 ∗ Corresponding to : Shuguang He, College of Management and Economics, Tianjin University, Tianjin, 300072 China. ∗ Email: [email protected] 2

1

Two-dimensional extended warranty strategy including maintenance level and purchase time: a win-win perspective

Abstract In practice, consumers need to decide whether and when to purchase extended warranty (EW), and there are many factors that influence this decision, such as product reliability and preventive maintenance (PM) conditions, etc. Under the consideration of consumer PM options, purchase time of EW and consumer usage rate diversity, this research first determines the product failure processes based on Generalised Polya Process (GPP) failure mode. Moreover, considering the impact of consumers purchasing decision of EW on warranty costs, we establish warranty cost models based on the product failure processes. Finally, given the warranty cost model, win-win EW price decision models are proposed in the product life cycle. A real case from a leading automobile manufacturer of China is presented to illustrate the application of the proposed model. The isoline of win-win regions and win-win EW intervals are obtained. Also, two management recommendations are given to help manufacturers make relevant decisions. Keywords: two-dimensional extended warranty, GPP repair, purchase time, preventive maintenance, win−win price interval

Preprint submitted to Computers & Industrial Engineering

January 10, 2020

1. Introduction Existing extended warranty research and market cases show that extended warranty plays an increasingly important role for both manufacturers and consumers (Murthy and Djamaludin (2002)). On one hand, extended warranty provides retail5

ers with as much as a 44-77% profit margin for some major appliance store (Maronick (2007)). On the other hand, a significant number of consumers purchase extended warranty against product failure (Lutz and Padmanabhan (1998)). In general, an extended warranty is a paid service which is sold to consumer, and provides necessary free corrective maintenance (CM) when products failed during the extended warran-

10

ty period (Chu and Chintagunta (2009)). Although the extended warranty imposes additional costs on consumers, more and more consumers tend to purchase extended warranty because it can reduce the repair costs of failed products. At the same time, manufacturers can also strengthen their connections with consumers and increase their own profits through extended warranty (Huang et al. (2017)). Therefore, the

15

study of extended warranty is particularly important. Since the purchase of EW for consumer and the provision of EW for manufacturer require additional costs (Lam and Lam (2001); Zhang et al. (2019)), EW pricing has become an urgent issue. However, few research have focus on the this field (Tong et al. (2014); Hartman and Laksana (2009)). According to literature research, the

20

fundamental factor affecting EW price is the cost of EW, and the cost of extended warranty will change with different CM (or repair) modes (ie, the failure mode of the product) (Murthy, D.N.P. et al. (2006); Zhao and Xie (2017)). Generally speaking, the repair methods are divided into five types (Pham and Wang (1996)), which are perfect repair (Park and Pham (2016)), minimal repair (Park et al. (2013)), imper-

25

fect repair (Chukova et al. (2004); Nguyen et al. (2017)), worse repair (Cha (2014);

2

Lee and Cha (2016)), and worst repair. Different repair hypotheses have a significant impact on the estimation of the EW cost. Jack and Murthy (2001) analysed the warranty costs and examined optimal EW pricing strategy for manufacturers based on minimal repair. Su and Shen (2012) established warranty cost and profit models of 30

one-dimensional (1-D) and two-dimensional (2-D) EW considering imperfect repair combined with minimal repair. Zhao and Xie (2017) predicted warranty costs based on imperfect repairs. To our knowledge, previous scholars have proposed numerous studies on the perfect, minimal, and imperfect repair wherein the assumptions required are too strong and unrealistic in a sense. For example, the failure pro-

35

cess under minimal repair assumption follows the nonhomogeneous Poisson process (NHPP)- that is, the system state after repair is restored to the state before failure. It seems to reflect an unreasonable fact that the future failure process in this case does not depend on the failure history. In view of this consideration, Lee and Cha (2016) proposed an optimal maintenance strategy through Generalised Polya Process

40

(GPP) repair mode based on the 1-dimensional (1-D) case, which solved the gap to some extent. However, in practice, most of the pricing issues for extended warranties are based on minimum repairs (not specifically considering the failure mode of the product), and the pricing of extended warranties requires consideration of more 2-D cases. Therefore, this research proposes an EW price decision model based on GPP

45

failure mode and 2-D EW. Naturally, the free repair service during the extended warranty period also imposes additional costs to the manufacturer. An effective way to reduce warranty cost is to incorporate an appropriate preventive maintenance (PM) plan into warranty (Shahanaghi et al. (2013)). PM operations are planned to both improve product

50

reliability and extend the life of the product (Wu and Zuo (2010)). Manufacturers may benefit from proposing PM if the reduction of warranty cost exceeds the addi3

tional cost incurred by the PM program (Kim et al. (2004)). To reduce the warranty cost, optimisation of the degree and frequency of PM has been the focus in recent literature (Wang et al. (2015); Huang et al. (2015); Ben Mabrouk et al. (2018)); Su 55

and Wang (2016) optimised imperfect PM policies by considering the moment when customers purchase EW. Therefore, PM policies are one of the factors that affect the cost of extended warranty, which naturally affect the price of extended warranty. However, these studies above only optimised the extended warranty costs based on PM from the perspective of either manufacturer or consumer. Actually, according

60

to the Consumer Electronics Association, only 20% of those who purchase extended warranty electronic devices have used the service (Maronick (2007)). At the same time, frequent technological innovations and product prices decline will also affect the sales of extended warranty (Gallego et al. (2014)). It can be seen that it is particularly important to propose an extended warranty price that attracts consumers.

65

Consumers will assess whether it is worthwhile to purchase an EW at a certain price, and the manufacturer also needs to control the EW price to make it profitable. Therefore, it is necessary to study the extended warranty from a win-win perspective. Bouguerra et al. (2012) proposed a win-win decision-making model for 1-D EW considering different PM strategies. The win-win decision model was extended by

70

Mo et al. (2017) allowing the consumers to invest in the PM costs. These scholars considered the win-win factor to study the issue of extended warranty and PM, and established a win-win bridge between manufacturers and consumers. But, in fact, many factors affect the EW price and win-win strategy, such as the period during which the consumer chooses to do PM (Bouguerra et al. (2012)), the time when the

75

consumer purchases the EW (Su and Wang (2016)), warranty period (Faridimehr and Niaki (2012)), consumer usage rate (Ye and Murthy (2016)) and other consumer behaviours (Tong et al. (2014); Jung et al. (2015); Tong et al. (2004); Wu (2011); 4

Wang et al. (2017)). Motivated by this, our research considers the purchase time of extended warranty, maintenance (CM and PM) policies, and consumer usage rate 80

to establishes the decision model of extended warranty price from the perspective of win-win. The main contributions are as follows: under the consideration of consumer PM options, purchase time of EW and consumer usage rate diversity, this research first determines the product failure processes based on GPP failure mode to describe the

85

repairing cost in the 2-D warranty. Moreover, considering the impact of consumers purchasing decision of EW on warranty costs, we establish warranty cost models based on the product failure processes. Finally, given the warranty cost model above, win-win EW price decision models are proposed in the product life cycle, making both manufacturers and consumers profitable. A real case from a leading automobile

90

manufacturer of China is presented to illustrate the application of the proposed model. The isoline of win-win regions and win-win EW intervals under different purchase time, PM policy and usage rate based on GPP failure mode are obtained to help manufacturers make relevant decisions. The remaining research is organised as follows. In Section 2, we introduce the

95

modelling framework, including assumptions and notations of our model, failure mechanism of the product, imperfect PM actions, and purchase moments of EW. In Section 3, we present the failure rate of the corresponding PM policies. We then introduce the win-win interval considering consumer usage rate under different policies in Section 4. A real case from a leading automobile manufacturer of China

100

to illustrate the applicability of our model is presented in Section 5. Finally, the conclusions are given in Section 6.

5

2. Modelling framework The products considered in this research are repairable and sold with the nonrenewal free repair warranty (FRW), which means that all faults that occur within 105

the basic warranty (BW) region ΩB (WB , UB ) are borne by the manufacturer and the buyer is not required to pay any fees. When the product age exceeds the limit of WB or the usage exceeds the limit of UB , whichever occurs first, manufacturers no longer provide free repair service. At this time, in order to make up for the loss caused by the maintenance of the product after the basic warranty expiry, the consumer

110

will consider purchasing the extended warranty at an additional cost to extend the warranty coverage by ΩE (WE , UE ). WE and UE are the age and usage limits of the extended warranty region ΩE (WE , UE ). Manufacturers need to make EW price profitable, while consumers also need to decide whether to purchase an EW with a certain price. This situation creates a

115

decision difficult to make for manufacturers and consumers. Is there a customised win-win range of EW price that can both cover the manufacturers EW service cost and benefit consumers? In the following sections, the win-win EW price range considering maintenance (CM and PM) level and purchase time will be discussed. 2.1. Assumptions and Notations

120

In this section, the model assumptions and notations used in the research are presented. 2.1.1. Model assumptions The following assumptions are made before developing the mathematical model.

125

(1) Each failure of products affects the performance of the product and results in a warranty claim. All warranty claims are valid. 6

(2) The failure products are repairable and degraded with age and usage. That is, if there is no preventive maintenance intervention, its failure rate will increase with age and/or usage. 130

(3) Both the time to repair the failed product and execute the PM is small enough, so they can be assumed to be negligible. (4) EW is solely provided and served by the manufacturer. Consumers are free to choose whether or not to purchase an EW. (5) The time to purchase EW is divided into two categories: the moment at the time of the product sale and the moment at the end of the BW.

135

(6) We assume that rB =

UB WB

UE =W =U L. E

(7) Consumers are completely rational people, that is, they show complete rationality from the experiments and only accept extended warranty prices that benefit them. 140

2.1.2. Model notations The notations used in this research are given as follows. WB , UB

the age limit and usage limit of the BW region [0, WB ) × [0, UB )

WE , UE

the age limit and usage limit of the EW region

L, U

life cycle and limit usage of the product

t, u

actual age and usage of the product

r, rB

usage rate and nominal usage rate of the product

XB , X

the time to the first failure of the product under nominal usage rate rB and the given usage rate r

G(r), g(r)

distribution function and density function of consumer usage rate

λt|r

the failure rate under GPP repair mode with a given usage rate r

α

degree of repair

7

T, θ, cmc

PM intervals, PM level and cost of each PM action

riy (t), rin (t)

the failure rate at time t with/without purchasing EW under PM policy i, i = I, II, III − 1, III − 2, IV − 1, IV − 2

τj , τjB , τjE

the time to perform PM if the PM actions begin at the beginning of BW period/at the end of BW period/at the end of EW period

ρ

proportion of PM costs incurred by the manufacturer

crc , crm

cost of each GPP repair action borne by consumers and manufacturers

Cin , Ciy

the total maintenance cost borne by the consumer under PM policy i when he does not/do purchase EW

Min , Miy

the total warranty cost for the manufacturer under PM policy i if the consumer does not/do purchase EW

0

ni ,ni(i=1,··· ,6)

number of PM actions of low-usage customers and high-usage customers in each interval as shown in Table 1.

CiM , MiM

the maximum EW price for the consumer/the minimum EW price for the manufacturer under PM policy i

Table 1: Number of preventative maintenance actions in each interval Number

PM Interval

Value

Number

n1

(0, WB ]

n1 = b WTB c

n1

n2

(WB , WB + WE ]

n3

(WB , L]

n3 =

n2 = b WTE c    L−WB − 1 L−WB ∈ N T T  b L−WB c T

n4

(WB + WE , L]

n5

(0, WB + WE ]

n6

(0, L]

n4 =

L−WB T

PM Interval

Value

0

(0, UrBB ]

n1 = b TUrBB c

0

E ( UrBB , UBr+U ] B

0

( UrBB , rUB , ]

n2 n3

∈ /N

   L−WB −WE − 1 T

L−WB −WE T

 b L−WB −WE c T

L−WB −WE T

∈N

0

n4 ∈ /N

E n5 = b WB +W c T   L − 1 L ∈ N T T n6 =  L b L c /N T T ∈

E ( UBr+U , rUB ] B

0

E (0, UBr+U ] B

0

(0, rUB ]

n5 n6

0

0

0

n3 =

0

n4 =

n2 = b TUrEE c    U −UB − 1 U −UB ∈ N T rB T rB

 b U −UB c T rB   U −U B −UE  −1

U −UB T rB

T rB

U −UB −UE T rB

∈N

 b U −UB −UE c T rB

U −UB −UE T rB

∈ /N

∈ /N

0

0

n6 =

E n = b UBT+U rB ] 5  U  U −1 T rB T rB ∈ N

 b U c T rB

U T rB

∈ /N

2.2. Failure mechanism of the product This research considers the two-dimensional warranty problem of age and usage. In order to establish the failure process of a product in terms of both age and us8

age, three main methods are used in the literature: bi-variate, composite scale, and marginal methods (Jack et al. (2009)). The marginal method, as the most used method, assumes that the usage rate of a given customer during the warranty period is constant but varies randomly from customer to customer. Therefore, given a constant usage rate r, the usage of the product is a linear function of its corresponding age-that is, u = rt. Accelerated failure time (AFT) model is a well-known marginal method that is widely used. This research uses the AFT model (Shahanaghi et al. (2013)) to study the impact of consumer usage rate on the time that the product first fails. Let XB and X be the time to the first failure of the product under nominal usage rate rB and the given usage rate r, respectively. Assume that XB , with the distribution function FB (x; αB , βB ), is the time of the first failure under the nominal usage rate rB . According to the AFT model, the distribution function of the time to the first failure X with a given usage rate r is: r F (x; α(r), βB ) = FB (x; ( B )γ αB , βB ) r

(1)

where αB and βB are the scale and shape parameters of XB and γ(≥ 1) is the AFT factor. Then, the corresponding the density function, cumulative hazard function, and failure rate function are f (x; α(r), βB ), Λ(x; α(r), βB ) and λ(x; α(r), βB ), respectively, where λ(x; α(r), βB ) =

f (x; α(r), βB ) . 1 − F (x; α(r), βB )

(2)

Assume that the distribution function and density function are given by G(r) = P (R ≤ r)

and g(r), (rmax ≤ r ≤ rmin ), where rmax and rmin are the lower limit and

upper limit of usage rate, respectively. Besides, we divide customers into two categories: low-usage customers and high-usage customers. The usage rates of low-usage customers and high-usage customers are rL and rH , where rmin ≤ rL < rmid and rmid ≤ rH < rmax .

In addition, it is supposed that the distribution of usage rate G(r) 9

is known for the manufacturer, either through past history of similar products or through customer surveys (Su and Wang (2016)). Thus the rmin and rmax are able to min be known by the same approach. The rmid is given by rmid = rmin + rmax −r . Then, 2

as described by Shahanaghi et al. (2013), the corresponding average usage rates of rL

and rH are Z rmid rL =

rmin

Z rmax sg(s) sg(s) ds, rH = ds rmid − rmin rmid rmax − rmid

(3)

We consider a practical situation where failure of components in the product may lead to a more severe working environment by increasing pressure, temperature, etc., 145

causing instantaneous negative impact to adjacent non-failed components. Such a failure of components will eventually lead to product degradation, which will increase the level of product failure rate. Therefore, the repair actions in this research are considered to be GPP repair (a kind of worse repair) (Lee and Cha (2016)) instead of perfect repair, minimal repair, and imperfect repair. In order to understand the

150

GPP repair, we firstly introduce GPP process as follows. Defination 1.Generalized Polya Process (GPP) A counting process N (t), t ≥ 0 is called Generalized Polya Process (GPP) with the set of parameters (λ(t), α, β), α ≥ 0, β > 0, if 1. N (0) = 0;

155




2. λt = αN (t−) + β λ(t), where λt is the stochastic intensity of an orderly counting process N (t), t ≥ 0. The λt is defined as follows: P r[N (t, t + 4t = 1|Ht− )] E[N (t, t + 4|Ht− )] = lim 4t 4t 4t→0 4t→0

λt = lim

where Ht− ≡ N (u), 0 ≤ u < t is the set of all point events in [0, t). 10

(4)

As defined by Lee and Cha (2016), a repair type is called the ’GPP repair’ if {N (t), t ≥ 0}

is the GPP with the parameter set (λ(t), α, 1). Therefore, the correspond


ing failure intensity under GPP repair mode is given by λt = αN (t−)+1 λ(t). Further, given a usage rate r, the corresponding failure intensity under GPP repair mode is
λt|r = αN (t−) + 1 λ(t|r)

(5)

where r refers to rL or rH (i.e., the value of r depends on whether he/she is a low-usage customer or a high-usage customer) and λ(t|r) refers to λ(x; α(r), βB ), which is given by Formula (2). The parameter α determines the degree of repair with the constraint 160

of α ≥ 0. α = 0 corresponds to the minimal repair and α > 0 implies that the repair is worse repair. Furthermore, an increase of α means that the corresponding repair becomes worse. N (t−) is the total number of failures during time interval (0, t) with its baseline failure rate λ(t). GPP repair assumes that the reliability of the product after a repair is related to the number of previous repairs and the initial reliability

165

of the system, relaxing the assumption of independent increment of NHPP. In summary, given a usage rate r, the distribution function of X (the time to first failure of the product) is derived by the AFT model for the first step. Second, we can obtain the conditional baseline failure rate λ(t|r) by Formula (2). Third, considering the effect of the product failure history on the conditional baseline failure rate λ(t|r),

170

the GPP process is used to obtain the actual conditional failure rate λt|r that varies with both time and the number of failures to describe the worse repairs.


Given usage rate r, the expected number of failures E N (t|r) during the interval (0, t)

introduced by Lee and Cha (2016) is: 

1 E N (t|r) = exp αΛ (t|r) − 1 α

11

(6)

2.3. The imperfect PM actions The system comprises periodic PM and the time interval for PM is T . In addition, the failure process during the (i + 1)th PM interval [iT, (i + 1)T ) follows the GPP repair with the set of parameters (λ(vi+1 (T ) + t), α, 1), 0 < t ≤ T, i = 1, 2, ... where v0 (T ) = 0, vi−1 (T ) < vi (T )

and vi (T ) is non-decreasing in T . vi+1 is the starting intrinsic age

of the GPP just after the (i + 1)th PM, which is proposed by Lee and Cha (2016). Lee and Cha (2016)’s definition of vi+1 is similar to Kijima (1989)’s definition of virtual age. In this research, the effect of PM is that it slows down the degradation of the product so that the starting intrinsic age of the GPP is effectively reduced. The reduction in the starting intrinsic age depends on the PM level θ used. Given the PM level θ, the form of the starting intrinsic age vi+1 (T ) is given by: vi+1 (T ) = vi (T ) + θT, i = 1, 2, ...

(7)

where θ ∈ [0, 1] and v0 (T ) ≡ 0. More specifically, if θ = 0, then vi+1 (T ) = vi (T ), i = 1, 2, ..., which means the product is restored to the state at the beginning of time T (i.e., 175

the PM is perfect maintenance); if θ = 1, then vi+1 (T ) = vi (T ) + T, i = 1, 2, ..., which means the product is restored to the as-bad-as-old state (i.e., the PM is minimal maintenance); while this research considers a general case of 0 < θ < 1, which means the product is partially restored (i.e., the PM is imperfect maintenance). Then, this implies that the baseline failure rate function just after the (i + 1)th PM is given by

180

λ(vi+1 (T ) + t).

Similar to Mo et al. (2017), consumers and manufacturers share the cost of PM during the warranty period in this research. The first PM cost is borne by the manufacturer, while the remaining is borne by the consumer during the BW. Except for the first PM fee borne by the manufacturer, the remaining PM costs are shared by the manufacturer and consumer with the ratio of ρ and 1 − ρ(0 ≤ ρ ≤ 1) during the 12

EW period, if the consumer buys EW. The PM fee per time is a linear function of the PM effort θ, given by: cmc = a + b(1 − θ)

(8)

where b is the marginal PM cost and a is a constant with a, b ≥ 0. The time to repair the failed product and execute PM is negligible. Considering the information asymmetry between consumers and manufacturers, the repair cost per time for consumers is crc , while the repair cost per time for manufacturers is crm , 185

where crc ≥ crm . Considering the impact of PM on EW costs, we divide the PM into the following six categories according to the different PM period selected by the consumer: • Policy I : no PM action during the product lifecycle L. • Policy II : do periodic PM during the product lifecycle L.

190

• Policy III : do periodic PM during the post warranty period. – Policy III -1: do periodic PM after the end of the basic warranty period. – Policy III -2: do periodic PM after the end of the extended warranty period. • Policy IV : do periodic PM during the warranty period.

195

– Policy IV -1: do periodic PM during the basic warranty period. – Policy IV -2: do periodic PM during the basic and extended warranty period. Since there is no difference in estimating the price of EW under the conditions of Policy I and III -2, we omit the discussion of Policy III -2. 13

200

2.4. Different purchase time of EW From the customer’s point of view, the time to purchase an EW (e.g., an EW of automobile) can be divided into two types (Su and Wang (2016)): • Case 1: Purchase an EW at the time of the product sale. • Case 2: Purchase an EW at the end of the BW period. As we can see in Figure 1 (a) and Figure 1 (b), when EW is purchased at the time of the product sale, the entire warranty region is the rectangle [0, WB +WE ]×[0, UB +UE ]. Thus, the EW region is the rotated L-shaped area (the light gray area plus dark gray area), which is given by [0, WB + WE ] × [0, UB + UE ] minus [0, WB ] × [0, UB ]. Considering (a)

(b)

u

u rB = rE

rH

UB +UE

rB = rE

UB +UE

rL

UE

UE

UB

UB

rL *WB

UB

UB WE

WB

0

WB + WE t

WB

WE

{ {

{ {

WB

0

U B rH

WB + WE t

WB

(c)

(d)

u

u rH

rB

rL * WB +U E

UE WB rL *WB

{

UB + UE

rL

UE

UB

UB

rB

WE UB

WE

WB

WB

{

{

WB

0

{

rE = rB

{

rE = rB

{

205

0

WB + WE t

U B rH

U B rH + WE

t

Figure 1: The extended warranty region for customers with Case 1 (a),(b) and Case 2 (c),(d) 14

210

the difference in customer usage, the EW region of the low-usage customer group is [WB , WB +WE ]×[UB +UE −rL WB ]

(the dark area in (a) of Figure 1), and the EW region

of the high-usage customer group is [ UrHB , WB + WE − UrHB ] × [UB , UB + UE ] (the dark area in (b) of Figure 1). While, when EW is purchased at the end of the BW period, the age limit and usage limit of EW contract are measured from a specific point where 215

the BW expires. Thus, as described in (c) and (d) of Figure 1, the EW region is shifted with respect to the BW region. The EW region of the low-usage customer group is [WB , WB + WE ] × [rL WB , rL WB + UE ] (the dark area in (c) of Figure 1), and the EW region of the high-usage customer group is [ UrHB , UrHB + WE ] × [UB , UB + UE ] (the dark area in (d) of Figure 1). PM planning and corresponding cost sharing

Consumersÿchoice Product sale

Do continuous PM during the BW period and EW period. · The manufacturer shares part of the EW PM Costs.

· PM Policy Ⅱ

Buy EW?

Yes

Case 1

Calculate win-win interval of EW price

No

BW expiry ·

End

No

Buy EW?

Yes

Case 2

·

Plan separately for PM in BW period and EW period. The PM costs during the EW period is borne by the consumer.

Figure 2: Win-win framework of EW based on PM policy II and the moments of customers buying EW

15

It should be noted that the difference purchase time of EW will not only make the 220

scope of the EW different, but also affect the PM scheme. If the consumer buys EW at the end of the BW period, the manufacturer has to executive PM separately during the BW and EW periods. Meanwhile, if consumer purchases EW at the be-

225

ginning of the BW period, the manufacturer can create a continuous PM plan during the BW and EW period, which seemingly projects a higher reliability. Therefore, different purchase time of EW will affect the warranty cost for both consumers and manufacturers, thus affecting the estimation of the win-win interval of EW. Take PM Policy II as an example. Figure 2 is the win-win framework of EW

230

based on PM policy II and the moments of customers buying EW. As shown in Figure 2, the time to purchase EW affects the optimization of PM and whether to the purchase EW or not affects the allocation of the PM costs between manufacturers and consumers. Therefore, the win-win range of the EW price is affected by different purchase time of EW and PM policies.

235

3. Analysis of failure rate Since product failure rate is different due to different PM policies and purchase time of EW, we will discuss the failure rate of products under different conditions. Let riy (t) represent the failure rate at time t when purchasing EW under PM Policy i; while let rin (t) be the failure rate at time t without purchasing EW under PM Policy

240

i,

where i = I, II, III − 1, III − 2, IV − 1, IV − 2 if there is no other explanation in this

research. The time to perform PM actions is denoted as τj , j = 0, 1, 2, ..., with τ0 = 0, if the PM program begins at the beginning of the BW period. The PM instants is denoted as τjB , j = 0, 1, 2, ..., with τ0B = WB , if the PM program begins at the end of the BW 245

period. And the PM instants is denoted as τjE , j = 0, 1, 2, ..., with τ0E = WB + WE , if 16

PM program begins at the end of the EW period. We denote the PM level is θ. The

Failure rate r(t)

failure rate under different conditions is given as follows.

No PM

0 WB

WE

L

t

Figure 3: Rate of failure occurrences for the considered policies In order to show the differences between the above policies more clearly, Figure 3 shows the failure rate under different policies. 250

3.1. The failure rate of Policy I The situation in Policy I is same to the case where the PM level θ = 0. Therefore, the failure rate of Policy I is:

rIy (t|r) = rIn (t|r) = λ(t|r)

17

(9)

3.2. The failure rate of Policy II If the consumer does not buy EW, the failure rate of Policy II is:

C1 rIIy (t|r) =

         

λ vj (T ) + s|r




τj ≤ j ∗T +s < τj +1, j = 1, 2, ..., n5 −1

λ vn5 (T ) + s|r




τn5 ≤ n5 ∗ T + s < WE


   λ vn5+j (T )+WB +WE −τn5 +s|r    
  λ v n5+n4 (T )+WB +WE −τn5 +s|r

E , j = 0, ..., n −1 τjE ≤ WE +j ∗T +s < τj+1 4

τnE4 ≤ WE +n4 ∗T +s < L

(10) If the consumer buys EW, there are two cases due to the time to purchase EW: 255

Case 1 and Case 2. •

Case 1:

The failure rate of Case 1 in Policy II is:

C1 rIIy (t|r) =

                  

λ vj (T ) + s|r




τj ≤ j ∗T +s < τj +1, j = 1, 2, ..., n5 −1

λ vn5 (T ) + s|r




τn5 ≤ n5 ∗ T + s < WE


λ vn5+j (T )+WB +WE −τn5 +s|r
λ vn5+n4 (T )+WB +WE −τn5 +s|r

E , j = 0, ..., n −1 τjE ≤ WE +j ∗T +s < τj+1 4

τnE4 ≤ WE +n4 ∗T +s < L

(11) •

Case 2:

Failure rate of Case 2 in Policy II is: 
  λ vj (T ) + s|r    
   λ vn1 (T ) + s|r     
  λ v (T ) + W − τ + s|r n n +j B 1 1 C2 rIIy (t|r)=
   λ vn1 +n2 (T )+WB − τn1 +s|r    
   λ vn1 +n2 +j (T )+WE+2WB −τnB2−τn +s|r   1  
  λ v B n1+n2+n4(T )+WE+2WB −τn2−τn +s|r 1

τj ≤ j ∗ T + s < τj + 1, j = 1, 2, ..., n1 −1 τn1 ≤ n1 ∗ T + s < WB B , j = 0, ..., n −1 τjB ≤ WB +j ∗T +s < τj+1 2

τnB2 ≤ WB + n2 ∗ T + s < WE E , j = 0, ..., n −1 τjE ≤ WE +j ∗ T +s < τj+1 4

τnE4 ≤ WE + n4 ∗ T + s < L

(12) 18

3.3. The failure rate of Policy III − 1 If the consumer does not buy EW, the failure rate of Policy III − 1 is:

rIII−1n (t|r) =

            

λ(t|r)
λ WB + vj (T ) + s|r
λ WB + vn3 (T ) + s|r

0 ≤ t < WB B , j = 0, ..., n − 1 τjB ≤ WB + j ∗ T + s < τj+1 3

τnB3 ≤ WB + n3 ∗ T + s < L

(13) If the consumer buys EW, the failure rate of Policy III − 1 is:    λ(t|r)     
  λ WB + vj (T ) + s|r rIII−1y (t|r) =
   λ 2WB +WE−τnB2+vn2+j (T )+s|r    
  λ 2W +W −τ B +v n2+n4 (T )+s|r B E n2

0 ≤ t < WB B , j = 0, ..., n −1 τjB ≤ WB + j ∗ T + s < τj+1 2 E ,j = 0, ..., n −1 τjE ≤ WB +WE +j ∗T +s < τj+1 4

τnE4 ≤ WB + WE + n4 ∗ T + s < L

(14) 3.4. The failure rate of Policy IV − 1 and Policy IV − 2 260

In this section, the failure rates of Policy IV − 1 and Policy IV − 2 are established. 3.4.1. Policy IV − 1 As for Policy IV − 1, whether purchase EW or not has no effect on preventive maintenance programs. Therefore, the failure rate in the condition of purchasing EW is the same as the one of not purchasing EW, which is given by:

rIV −1n (t|r) = rIV −1y (t|r) =

  

λ vj (T ) + s|r




τj ≤ j ∗ T + s < τj+1 , j = 0, ..., n1 − 1

  λ vn (T ) + s|r 1




τn1 ≤ n1 ∗ T + s < L

(15)

19

3.4.2. Policy IV − 2 If the consumer does not buy EW, the failure rate of Policy IV − 2 is:

rIV −2n (t|r) =

         

λ vj (T ) + s|r




τj ≤ j ∗T + s < τj+1 ,

λ vn1 (T ) + s|r




τn1 ≤ n1 ∗ T + s < WB




B , j = 0, ..., n −1 τjB ≤ WB +j ∗T +s < τj+1 2

   λ vn1 +j (T )+WB −τn1 +s|r    
  λ v n1 +n2 (T )+WB −τn1 +s|r

j = 0, ..., n1 − 1

τnB2 ≤ WB + n2 ∗ T + s < L

(16) If the consumer buys EW, there are two cases due to the time to purchase EW: Case 1 and Case 2. 265

•

Case 1:

The failure rate of Case 1 in Policy IV − 2 is: C1 rIV −2y (t|r)

•

=

  

λ vj (T ) + s|r




  λ vn (T ) + s|r
5

τj ≤ j ∗ T + s < τj+1 , j = 0, ..., n5 − 1

(17)

τn5 ≤ n5 ∗ T + s < L

Case 2:

Since the PM programs in the condition of purchasing EW at the end of the BW period is the same as the condition of not purchasing EW. Therefore, the failure rate of Case 2 in Policy IV − 2 is: C2 rIV −2y (t|r) = rIV −2n (t|r)

(18)

4. Model analysis Since the win-win interval of EW price is different due to different PM policies and purchase time of EW, we will establish and discuss the win-win interval under 270

different conditions in this section. Cin is the cost borne by the consumer when they do not purchase the EW in the product life cycle under PM Policy i. Ciy is the total 20

cost borne by the consumer when purchasing EW under PM Policy i in the product life cycle, except for the fee of EW contract. CW s the price of EW. It is acceptable for the consumer only when CW + Ciy ≤ Cin . Therefore, the maximum EW price CiM 275

that the consumer can accept is CiM = Cin − Ciy , i.e., CW ≤ CiM . Min

is the warranty cost for the manufacturer during the BW period if the con-

sumer does not purchase EW under PM Policy i. Miy is the warranty cost for the manufacturer during BW and EW periods under the PM Policy i if the consumer purchases EW. MiM is the minimum EW price acceptable by the manufacturer under 280

PM Policy i. It is profitable for the manufacturer only when Miy − CW ≤ Min . Therefore, the lowest EW price MiM that the manufacturer can accept is MiM = Miy − Min i.e., CW ≥ MiM . Our goal is to find a win-win interval [MiM , CiM ] based on different maintenance levels (CM and PM), EW purchase time, and usage rates.

285

4.1. The win-win interval of EW price for Policy I The maximum EW price that consumers can afford under Policy I is: CIM = CIn − CIy =

Z rmid Z WB +WE 
crc  exp α rIn (t|r)dt − 1 dG(r) rmin α WB

(19)

The minimum EW price that manufacturer can accept under Policy I is: MIM = MIy − MIn =

Z rmid Z WB +WE 
crm  exp α rIn (t|r)dt − 1 dG(r) α rmin WB

(20)

Therefore, the win-win EW price interval for manufacturers and low-usage consumers under Policy I is [MIM , CIM ] 4.2. The win-win interval of EW price for Policy II •

Case 1: 21

The maximum EW price that consumers can afford of Case 1 under Policy II is: C1 C1 C1 CIIM =CIn − CIy  n5 −1  Z rmid Z τB Z L  j+1
 
crc X = exp α rIIn (t|r)dt −1 + exp α rIIn (t|r)dt −1 rmin α τjB τnB5 j=0 Z τE Z L nX 4 −1     j+1

C1 C1 − exp α rIIy (t|r)dt − 1 − exp α rIIy (t|r)dt − 1 dG(r) j=0

τjE

τnE4

+ cmc n6 − n1 − (n5 − n1 − 1) (1 − ρ) − n4




(21)

The minimum EW price that manufacturer can accept of Case 1 under Policy II is: C1 C1 C1 MIIM =MIy − MIn

=

 n5 −1  Z rmid Z τj+1 Z WE
 
 crm X C1 C1 exp α rIIy (t|r)dt −1 + exp α rIIy (t|r)dt −1 α τj τn 5 rmin j=0 Z τj+1 Z WB nX 1 −1    

− exp α rIIn (t|r)dt − 1 − exp α rIIn (t|r)dt − 1 dG(r) j=0

τj

τn 1


+ cmc 1 + (n5 − n1 − 1) ρ 290

(22)

Therefore, the win-win EW price interval for manufacturers and low-usage conC1 , C C1 ]. sumers of Case 1 under Policy II is [MIIM IIM

•

Case 2:

The maximum EW price that consumers can afford of Case 2 under Policy II is: C2 C2 C2 CIIM =CIn − CIy

 n3 −1  Z rmid Z τB Z L j+1
 
 crc X = exp α rIIn (t|r)dt −1 + exp α rIIn (t|r)dt −1 rmin α τjB τnB3 j=0 Z τE Z L nX 4 −1     j+1

C2 C2 − exp α rIIy (t|r)dt − 1 − exp α rIIy (t|r)dt − 1 dG(r) j=0

τjE

+ cmc n2 − (n2 − 1) (1 − ρ)

τnE4




(23) 22

The minimum EW price that manufacturer can accept of Case 2 under Policy II is: C2 C2 C2 MIIM =MIy − MIn  n1 −1  Z rmid Z τj+1 Z WB
 
 crm X C2 C2 = exp α rIIy (t|r)dt −1 + exp α rIIy (t|r)dt −1 α rmin τj τn1 j=0 Z τB Z WE nX 2 −1     j+1

C2 C2 + exp α rIIy (t|r)dt − 1 − exp α rIIy (t|r)dt − 1 dG(r) τjB

j=0

−

nX 1 −1 

τnB2

Z τj+1 exp α τj

j=0

+ cmc 1 + (n2 − 1) ρ

Z WB   

rIIn (t|r)dt − 1 − exp α rIIn (t|r)dt − 1 dG(r) τn 1




(24)

Therefore, the win-win EW price interval for manufacturers and low-usage conC2 , C C2 ]. sumers of Case 2 under Policy II is [MIIM IIM

295

4.3. The win-win interval of EW price for Policy III − 1 The maximum EW price that consumers can afford under Policy III − 1 is: CIII−1M =CIII−1n − CIII−1y Z rmid nX Z τB Z L −1 j+1
 
 crc 3  exp α rIII−1n(t|r)dt −1 + exp α rIII−1n(t|r)dt −1 = rmin α τjB τnB j=0

−

nX 4 −1

Z τE j+1

exp α

j=0

τjE

3


  rIII−1y (t|r)dt −1 − exp α

+ cmc n3 − (n2 − n1 ) (1 − ρ) − n4

Z L τnE4


 rIII−1y (t|r)dt −1 dG(r)




(25)

The minimum EW price that manufacturer can accept under Policy III − 1 is: MIII−1M =MIII−1y − MIII−1n  n2 −1  Z rmid Z τB j+1
 crm X = exp α rIII−1y (t|r)dt −1 α rmin τjB j=0

23

Z WE  
+ exp α rIII−1y (t|r)dt −1 dG(r) τnB2


+ cmc 1 + (n2 − 1) ρ

(26)

Therefore, the win-win EW price interval for manufacturers and low-usage consumers under Policy III − 1 is [MIII−1M , CIII−1M ]. 4.4. The win-win interval of EW price for Policy IV − 1 and Policy IV − 2 In this section, the win-win interval of EW price considering Policy IV − 1 and 300

Policy IV − 2 are established. 4.4.1. The win-win interval of EW price for policy IV − 1 The maximum EW price that consumers can afford under Policy IV − 1 is: CIV −1M = CIV −1n − CIV −1y =

Z rmid Z WB +WE 
crc  exp α rIV −1n (t|r)dt − 1 dG(r) rmin α WB

(27)

The minimum EW price that manufacturer can accept under Policy IV − 1 is: MIV −1M = MIV −1y − MIV −1n =

Z rmid Z WB +WE 
crm  exp α rIV −1n (t|r)dt − 1 dG(r) α rmin WB

(28)

Therefore, the win-win EW price interval for manufacturers and low-usage consumers under Policy IV − 1 is [MIV −1M , CIV −1M ]. 4.4.2. The win-win interval of EW price for policy IV − 2 •

305

Case 1:

The maximum EW price that consumers can afford of Case 1 under Policy IV − 2 is: C1 C1 C1 CIV −2M =CIV −2n − CIV −2y Z rmid nX Z τB Z L −1  j+1
 
crc 2  = exp α rIV −2n(t|r)dt −1 + exp α rIV −2n(t|r)dt −1 rmin α τjB τnB j=0

2

24

 Z L 

C1 − exp α rIV (t|r)dt −1 dG(r)+cmc n5 −n1 − (n5 −n1 −1) (1−ρ) −2y

(29)

WE

For the manufacturer, the situation of Case 1 under Policy IV − 2 is the same as the situation of Case 1 under Policy II (Simulation results and theoretical formulas also prove this). Therefore, the minimum EW price that the manufacturer can accept of Case 1 under Policy IV-2 is: C1 C1 MIV −2M = MIIM

(30)

Therefore, the win-win EW price interval for manufacturers and low-usage conC1 C1 sumers of Case 1 under Policy IV − 2 is [MIV −2M , CIV −2M ].

•

Case 2:

The maximum EW price that consumers can afford of Case 2 under Policy IV − 2 is: C2 C2 C2 CIV −2M =CIV −2n − CIV −2y

 n2 −1  Z rmid Z τB j+1
 crc X = exp α rIV −2n (t|r)dt −1 rmin α τjB j=0 Z WE  
+ exp α rIV −2n (t|r)dt −1 dG(r)

(31)

τnB2

+ cmc n2 − (n2 − 1) (1 − ρ)




For the manufacturer, the situation of Case 2 under Policy IV − 2 is the same as the situation of Case 2 under Policy II (Simulation results and theoretical formulas also prove this). Therefore, the minimum EW price that the manufacturer can accept of Case 2 under Policy IV − 2 is: C2 C2 MIV −2M = MIIM .

(32)

Therefore, the win-win EW price interval for manufacturers and low-usage con310

C2 C2 sumers of Case 2 under Policy IV − 2 is [MIV −2M , CIV −2M ].

25

Because of the similarity of the win-win interval modelling method for high usage rate and low usage rate, this research omits the modelling of the win-win interval of EW price for high-usage manufacturers and high-usage customers. 5. Case study 315

In this section, the failure history of the automobile produced in China is used to illustrate application of the proposed method. The warranty data we collected includes the time to the first failure (day) and the total usage (kilometer) at the fai− 14

Total usage (10000Km)

12

10

8

6

4

2

0 0

0.5

1

1.5

2

Time to the first failure (Year)

Figure 4: Scatter plot of the time to the first failures and total usages for the automobiles lure times of 1,087 automobiles sold in 2010 during the basic warranty period. The scatter plot of the time to the first failure and the total usage of these vehicles is 320

shown in Figure 4. In order to estimate the distribution of the time to the first failure and the distribution of the consumer usage rate, we plotted their histograms and estimated the parameters of their distribution. As shown in Figure 5, the time to the first failure

26

is subject to the Weibull distribution with the scale parameter αB = 0.35 and the shape 325

parameter βB = 1.18. And, the usage rate follows a Log-normal distribution with the mean µ = 1.05 and the standard deviation σ = 0.89. Both of the two distributions pass the hypothesis test. For the automobiles above, the basic warranty expires when the age of the product exceeds 3 (years) from the time to purchase or its total usage exceeds 6 × 104 (km)

330

(kilometer), whichever occurs first. Similarly, we consider the extended warranty is the rectangular with WB = 3 (years) and UB = 6 × 104 (km). And, we assume length of the life cycle and limit usage of the automobile are L = 9 (years) and U = 1.8 × 105 (km), respectively. (a)

(b)

120

200 180

100

160 140

Frequency

Frequency

80

60 40

120 100 80 60 40

20

20 0 0

0.5

1

1.5

2

2.5

0 0

3

Time to the first failure (Year)

10

20

30

40

50

Usage rate (10000Km/Year)

Figure 5: Histogram of the time to the first failure and the total usage rate for the automobiles We consider the AFT factor γ = 1.05, rmin = 0 (km/year) and rmax = 4.8 × 105 (k335

m/year). Moreover, the PM interval T = 0.17 (years) and cmc = 30 + 10(1 − θ). The proportion of the preventive maintenance cost per time undertaken by the manufacturer is ρ = 0.2. Simultaneously, the repair cost per time for the manufacturer and the repair fee per time for the consumer is crm = 50($) and crc = 100($).

27

We can get the win-win EW prices with certain PM level and repair degree for the 340

considered policies. Taking Policy III − 1 for example, when the PM level θ = 0.6 and repair degree α = 0.058, the win-win interval for the EW price is obtained between 2065 ($) ((b) of Figure 6) and 4018 ($) ((b) of Figure 7). The best compromise would be the intermediate price for this interval. Therefore, the best EW price for Policy III − 1

345

is 3041.5 ($).

In regard to practice, the PM level θ and repair degree α can be assumed from historical data or engineering analysis (Konno (2010); Lee and Cha (2016)). Our research find that the shape of the curve obtained is similar when the preventive maintenance level θ changes in (0, 1) and the repair degree α changes in the subinterval

4

x 10 4 3.8 3.6 3.4 3.2 3 2.8 2.6 2.4 2.2 2 1.8 1.6 1.4 1.2 1 0.8 0.6 0.4 0.2 0 0

(a)

Policy Ⅰ Policy Ⅳ-1

0.02

0.04

0.06

0.08

0.1

The minimum EW price accepted by manufacturers ($)

The minimum EW price accepted by manufacturers ($)

within (0, ∞) (taking the PM degree θ = 0.6 and α change in (0, 0.1) as an example). (b)

5000 4500

Policy ⅡCase 2 (Option Ⅳ-2 Case 2) Policy ⅡCase 1(Option Ⅳ-2 Case 1) Policy Ⅲ-1

4000 3500 3000

(0.058, 2065)

2500 2000 1500 0

0.02

0.04

0.06

0.08

0.1

Repair degree α (PM level θ=0.6)

Repair degree α (PM level θ=0.6)

Figure 6: The effects of repair degree on the minimum EW price accept by manufacturers 350

The EW price threshold for consumers and manufacturers are shown in Figure 6 and Figure 7, which indicate that: (a) The repair degree α affects the EW price acceptable to manufacturers and consumers. The MAEW (the maximum EW price afforded by the consumer) and 28

MIEW (the minimum EW price acceptable to the manufacturer) for Policy I and 355

Policy IV − 1, compared with other four policies, are more significantly impacted by the repair degree. Indeed, with fewer or no PM actions, the expected number of failures increases during the extended warranty period. Therefore, the extended warranty price that makes the manufacturer profitable and covers the cost of the x 10 4 3.8 3.6 3.4 3.2 3 2.8 2.6 2.4 2.2 2 1.8 1.6 1.4 1.2 1 0.8 0.6 0.4 0.2 0 0

4

(a)

(b)

The maximum EW price afforded by consumers ($)

The maximum EW price afforded by consumers ($)

consumer becomes larger. Policy Ⅰ Policy Ⅳ-1 Policy Ⅳ-2 Case 1

0.02

0.04

0.06

0.08

0.1

Repair degree α (PM level θ=0.6)

4200 4100 4000 3900 3800 3700 (0.058,4018) 3600 Policy Ⅲ-1 Policy ⅡCase 1 Policy ⅡCase 2 Policy Ⅳ-2 Case 2

3500 3400 0

0.02

0.04

0.06

0.08

0.1

Repair degree α (PM level θ=0.6)

Figure 7: The effects of repair degree on the maximum EW price afforded by consumers 360

(b) The extended warranty price acceptable to manufacturers and consumers under the condition of Case 1 is always greater than the condition of Case 2 in both Policy II and Policy IV − 2. Indeed, as PM actions are continuously implemented at the connection point between the BW and the EW (Case 1), the additional cost of the continuous PM is greater than the repair cost reduced by PM. Consequently, the EW

365

costs for manufacturers increase. Meanwhile, consumers would be ready to spend more money to purchase extended warranty, given that the continuous PM increase product reliability and reduce the number of failures during the post-warranty period. For Policy II and Policy IV − 2, as shown in Figure 8 - Figure 11, we can see that:

29

370

(a) Although the EW price acceptable to manufacturers and consumers under the condition of Case 1 is greater than the condition of Case 2 in Policy II and Policy IV − 2,

the win-win interval under Case 1 is greater than Case 2 in most situations

(as showed in Figure 8 - Figure 11). Therefore, both manufacturers and consumers can benefit more in Case 1 than that in Case 2, in general. But as shown in (a) of Figure 11, the win-win price range of Case 1 is greater than Case 2 when θ < 0.58; the win-win price range of Case 1 and Case 2 are both 1690 when θ = 0.58; the win-win price range of Case 1 is smaller than Case 2 when θ > 0.58. Therefore, the win-win price range of Case 1 and Case 2 under Policy IV − 2 needs to be analysed specifically

Policy Ⅳ-2 Case 1 Policy Ⅳ-2 Case 2

0.02

0.04

0.06

0.08

0.1

Repair degree α (PM level θ=0.2) (c) 10000 9000 8000 7000 6000 5000 4000 3000 2000 1000 0

Policy Ⅳ-2 Case 1 Policy Ⅳ-2 Case 2

0.02

0.04

0.06

0.08

0.1

Win-win range of EW prices ($)

(a) 10000 9000 8000 7000 6000 5000 4000 3000 2000 1000 0

Win-win range of EW prices ($)

Win-win range of EW prices ($)

when the PM level θ is high and the repair degree α is very small.

Win-win range of EW prices ($)

375

Repair degree α (PM level θ=0.6)

(b) 10000 9000 8000 7000 6000 5000 4000 3000 2000 1000 0

Policy Ⅳ-2 Case 1 Policy Ⅳ-2 Case 2

0.02

0.04

0.06

0.08

0.1

Repair degree α (PM level θ=0.4) (d) 10000 9000 8000 7000 6000 5000 4000 3000 2000 1000 0

Policy Ⅳ-2 Case 1 Policy Ⅳ-2 Case 2

0.02

0.04

0.06

0.08

0.1

Repair degree α (PM level θ=0.8)

Figure 8: The impact of purchase time and maintenance level on the winwin interval of EW for Policy IV-2 30

0.02

0.04

0.06

0.08

0.1

Repair degree α (PM level θ=0.2) (c) 2300 2200 2100 2000 1900 1800 1700 1600 1500 1400 0

Policy Ⅱ Case 1 Policy Ⅱ Case 2

0.02

0.04

0.06

0.08

0.1

Win-win range of EW prices ($)

Policy Ⅱ Case 1 Policy Ⅱ Case 2

Win-win range of EW prices ($)

Win-win range of EW prices ($) Win-win range of EW prices ($)

(a) 2300 2200 2100 2000 1900 1800 1700 1600 1500 1400 0

Repair degree α (PM level θ=0.6)

(b) 2300 2200 2100 2000 1900 1800 1700 1600 1500 1400 0

Policy Ⅱ Case 1 Policy Ⅱ Case 2

0.02

0.04

0.06

0.08

0.1

0.08

0.1

Repair degree α (PM level θ=0.4) (d) 2300 2200 2100 2000 1900 1800 1700 1600 1500 1400 0

Policy Ⅱ Case 1 Policy Ⅱ Case 2

0.02

0.04

0.06

Repair degree α (PM level θ=0.8)

Figure 9: The impact of purchase time and maintenance level on the winwin interval of EW for Policy II 380

(b) As shown in Figure 8 - Figure 9, compared with Case 1 and Case 2 for Policy II , the differences in the curve shape of win-win range between Case 1 and Case 2 for Policy IV − 2 are much larger. Actually, there is no significant difference in the EW costs for manufacturers in the condition of continuous PM actions (Case 1) or discontinuous PM actions (Case 2). Unlike manufacturers, who only need

385

to consider the EW cost during the warranty period, consumers need to consider the maintenance cost beyond the warranty period. Compared with Policy II , no PM is carried out during the post-warranty period for Policy IV − 2, resulting that the number of the expected failures will increase during the post-warranty period. Therefore, discontinuous PM actions will increase more product failures compared

390

with continuous PM actions for Policy IV − 2. Thereby, consumers (Policy IV − 2, 31

Policy Ⅱ Case 1 Policy Ⅱ Case 2

0.2

0.4

0.6

0.8

1

PM level θ (Repair degree α=0.0002) (c) 3000 2750 2500 2250 2000 1750 1500 1250 1000 750 0

Policy Ⅱ Case 1 Policy Ⅱ Case 2

0.2

0.4

0.6

0.8

1

Win-win range of EW prices ($)

(a) 3000 2750 2500 2250 2000 1750 1500 1250 1000 750 0

Win-win range of EW prices ($)

Win-win range of EW prices ($)

Win-win range of EW prices ($)

Case 1) gets reluctant to buy EW in this situation (lower threshold value of EW).

PM level θ (Repair degree α=0.02)

(b) 3000 2750 2500 2250 2000 1750 1500 1250 1000 750 0

Policy Ⅱ Case 1 Policy Ⅱ Case 2

0.2

0.4

0.6

0.8

1

PM level θ (Repair degree α=0.002) (d) 3000 2750 2500 2250 2000 1750 1500 1250 1000 750 0

Policy Ⅱ Case 1 Policy Ⅱ Case 2

0.2

0.4

0.6

0.8

1

PM level θ (Repair degree α=0.2)

Figure 10: The impact of purchase time and maintenance level on the winwin interval of EW for Policy II (c) At the same time, Figure 10 and Figure 11 show that the influence of purchase time of EW on the win-win range for Policy II is not significant. The similarly situation can also be found for Policy IV −2 when the repair degree is small (the prod395

uct reliability is high). However, with the repair degree getting large (the product reliability gets low), the influence of EW purchase time on Policy IV − 2 is obvious. In fact, due to the low reliability of product, the expected number of failures is large. Therefore, manufacturers are disinclined to sell EW (high threshold of EW price). Compared with Policy II (implement PM in the whole life cycle), the number of

400

failures in Policy IV − 2 is more likely to be affected by continuous PM or not. Natu32

rally, when discontinuous PM is implemented, the expected failures increase during the post-warranty period. In other words, consumers need to bear more expenses during the post-warranty period. In view of this, they are not willing to purchase

(0.98,1846) (0.58,1690)

(0.98,1836) (0.58,1690) Policy Ⅳ-2 Case 1 Policy Ⅳ-2 Case 2 0.2

0.4

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Win-win range of EW prices ($)

(a) 2200 2100 2000 1900 1800 1700 1600 1500 1400 1300 1200 1100 0

PM level θ (Repair degree α=0.0002) (c) 2200 2100 2000 1900 1800 1700 1600 1500 1400 1300 1200 1100 0

Policy Ⅳ-2 Case 1 Policy Ⅳ-2 Case 2 0.2

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(b) 2200 2100 2000 1900 1800 1700 1600 1500 1400 1300 1200 1100 0

Win-win range of EW prices ($)

Win-win range of EW prices ($)

Win-win range of EW prices ($)

EW at Case 1 (high threshold value of EW).

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PM level θ (Repair degree α=0.02)

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Policy Ⅳ-2 Case 1 Policy Ⅳ-2 Case 2

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Figure 11: The impact of purchase time and maintenance level on the winwin interval of EW for Polic IV-2 405

Figure 12 and Figure 13 are contour maps of the EW win-win price range for Policy IV − 2. As shown in Figure 12 and Figure 13, for a given EW win-price range 720 ($), all combinations of and on the respective curves can be presented as possible coverage areas, all of which have a win-win price range of 720 ($). The win-win range of Policy IV − 2 Case 1 and Policy IV-2 Case 2 are affected by both repair degree α

410

and preventive maintenance level θ. However, the impact of preventive maintenance 33

level θ on Policy IV − 2 Case 2 is more dominant than that on Policy IV − 2 Case 1. For example, in Figure 13, when θ = 0.1, the win-win interval contour changes little, no matter how α changes. 860

0.015

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Figure 12: Isoline of win-win interval of EW for PM Policy IV-2 Case 1

0.015 1850

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Figure 13: Isoline of win-win interval of EW for PM Policy IV-2 Case 2 Take Policy II as an example. According to the Figure 14, we give the following 415

suggestions for consumers with high and low usage rates: (a) Consumers with low or high usage rate are advised to buy EW in Case 1, because the win-win range of EW in Case 1 is larger than that in Case 2. Thus, the consumer can benefit more in Case 1, compared with Case 2. (b) For the condition of buying EW at the time of Case 1, low-usage customers

420

have a larger win-win range and benefit more than high-usage customers, but this difference will decrease as the PM level increases. 34

(b)

Policy Ⅱ Case 1(low-usage) Policy Ⅱ Case 2(low-usage) Policy Ⅱ Case 1(high-usage) Policy Ⅱ Case 2(high-usage)

Win-win range of EW prices ($)

Win-win range of EW prices ($)

(a) 2300 2200 2100 2000 1900 1800 1700 1600 1500 1400 1300 1200 1100 1000 900 800 700 600 500 400 300 200 100 0 0

0.2

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2300 2200 2100 2000 1900 1800 1700 1600 1500 1400 1300 1200 1100 1000 900 800 700 600 500 400 300 200 100 0 0

Policy Ⅱ Case 1(low-usage) Policy Ⅱ Case 2(low-usage) Policy Ⅱ Case 1(high-usage) Policy Ⅱ Case 2(high-usage)

0.02

0.04

0.06

0.08

0.1

Repair degree α (PM level θ=0.2)

Figure 14: Win-win range of EW price with low-usage and high-usage customers (c) For the condition of buying EW at the time of Case 2, there is no significant difference in the win-win interval between low-usage customers and high-usage customers. However, as the repair degree increases, low-usage customers benefit slightly 425

more than high-usage customers. 6. Conclusion In this research, we established win-win intervals of 2-D EW of consumer durables considering different purchase times of EW, maintenance strategies (CM and PM), and usage rate during the product life cycle. For consumer durables, it is very vital to

430

establish an appropriate customised EW decision model to both benefit consumers and cover the actual warranty cost borne by manufacturers. However, according to the authors’ knowledge, most of those scholars established warranty cost models based on the assumption of minimum repair, without considering the actual failure mode of the product. Actually, when an electrical device fails due to an external 35

435

shock, the component that has not failed will also experience this shock. With the growing trend of system degradation from external shock, the system failure rate increases. Therefore, it is more practical to consider the GPP failure mode. In this point of view, the GPP repair was applied to our win-win decision model, which intends to fill the gap between actual requirement from manufacturers and the

440

existed extended warranty decision-making research. Moreover, we investigated the win-win EW price interval model by taking the purchasing time of EW, different PM polices, and consumer usage rate into account. Finally, a real case from automobile manufacturer in China was presented to illustrate the application of the proposed strategy.

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The results show that the win-win EW interval increases with the growth of repair degree and preventive maintenance level. For Policy II , the win-win EW interval of Case 1 is greater than the condition of Case 2, while the opposite result is obtained for Policy IV − 2 when the PM level is small. Two management recommendations are available for this discovery from a win-win perspective:

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(1) Manufacturers should promote extended warranty when consumers acquire the products, except for certain special condition (e.g., when the preventive maintenance level is small and the product reliability is high for Policy IV − 2). (2) Manufacturers should increase the promotion of EW for products with low reliability and poor preventive maintenance, due to a larger win-win EW interval

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compared with other conditions. Future research may consider more specific issues, such as the win-win extended warranty model for consumers and manufacturers with repair time penalty costs. In addition, the game problem between multi-stake holders based on GPP repair mode and the purchase time of extended warranty is also very interesting. This

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research establishes a win-win EW model under the assumption that the consumer 36

is a completely rational person. However, the customers may tend to show relatively limited rationality from some experiments. Thus it is interesting to consider this effect in the future work. 7. Disclosure statement 465

No potential conflict of interest was reported by the authors. 8. References Bouguerra, S., Chelbi, A., and Rezg, N. (2012). A Decision Model for Adopting an Extended Warranty under Different Maintenance Policies. International Journal of Production Economics, 135(2), 840-849.

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Highlights    

Expanded the application of Generalised Polya Process in warranty decision making Preventive maintenance, purchase time of EW and usage diversity are considered A real case is presented to illustrate the application of the proposed model Give suggestions on extended warranty purchase decisions from win-win perspective

Author Contributions Statement 

Zhen He and Dongfan Wang provided the idea and formulated the aims;



Zhen He, Dongfan Wang and Shuguang He developed the methodology and designed the models;



Yiwen Zhang collected the data;



Dongfan Wang, Shuguang He and Yiwen Zhang analyzed the data and simulated the model;



Shuguang He analyzed the experimental results;



Zhen He and Dongfan Wang wrote the manuscript.



Anshu Dai did the revision of the modeling framework and provided management explanations and applications in the case study section.