Design of differentiated warranty coverage that considers usage rate and service option of consumers under 2D warranty policy

Peng Tong; Xiaogang Ma

doi:10.3934/jimo.2020035

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doi: 10.3934/jimo.2020035

Design of differentiated warranty coverage that considers usage rate and service option of consumers under 2D warranty policy

Peng Tong ^1,, and Xiaogang Ma ^2,,

1.	School of Management, China University of Mining and Technology, Jiangsu, China
2.	School of Management, Wuhan Textile University, Hubei, China

^* Corresponding authors: flyingmantong@163.com; xgma@wtu.edu.cn

Received June 2019 Revised September 2019 Published February 2020

Fund Project: The first author is supported by NSF grant the National Natural Science Foundation of China (No. 71701200); the Postdoctoral Fund of China (No. 2016M590525); the Postdoctoral Fund of Jiangsu (No. 1601246C)

Warranty service providers usually provide homogeneous warranty service to improve consumer satisfaction and market share. Considering the difference of consumers, some scholars have carried out studies on maintenance strategies, service pricing, payment method, claim behaviour and warranty cost analysis in recent years. However, few scholars have focused on the differentiated coverage of warranty service that considers usage rate and service option of consumers. On the basis of previous classification criteria on usage rate, this paper divides consumers into heavy, medium and light usage rate groups with clear boundaries. To avoid discrimination in warranty service, this study divides 2D warranty coverage into disjoint sub-regions and adopts different maintenance modes in each sub-region. By formulating and calculating warranty cost model under warranty cost constraints, we can obtain the maximum warranty coverage under usage rate $ r $. Therefore, differentiated warranty scope for consumers in the three groups can be proposed, whilst consumers can choose the most suitable warranty service according to their usage rate. Evidently, the proposed warranty strategy can provide flexible warranty service for consumers, meet the requirements of the warranty cost constraints of warranty service providers and enable enterprises to occupy a favourable position in the market competition.

Keywords: Usage rate, warranty coverage, modelling, cost analysis, 2D warranty.

Mathematics Subject Classification: Primary: 90B50; Secondary: 90B25.

Citation: Peng Tong, Xiaogang Ma. Design of differentiated warranty coverage that considers usage rate and service option of consumers under 2D warranty policy. Journal of Industrial & Management Optimization, doi: 10.3934/jimo.2020035

References:

[1]	A. Akbarov and S. Wu, Forecasting warranty claims considering dynamic over-dispersion, Int. J. Prod. Econ., 139 (2012), 615-622. doi: 10.1016/j.ijpe.2012.06.001. Google Scholar
[2]	J. Baik, D. N. P. Murthy and N. Jack, Two-dimensional failure modeling with minimal repair, Naval Res. Logist., 51 (2004), 345-362. doi: 10.1002/nav.10120. Google Scholar
[3]	W. L. Chang and J.-H. Lin, Optimal maintenance policy and length of extended warranty within the life cycle of products, Comput. Math. Appl., 63 (2012), 144-150. doi: 10.1016/j.camwa.2011.11.001. Google Scholar
[4]	S. Chukova and M. R. Johnston, Two-dimensional warranty repair strategy based on minimal and complete repairs, Math. Comput. Modelling, 44 (2006), 1133-1143. doi: 10.1016/j.mcm.2006.03.015. Google Scholar
[5]	G. Gallego, R. Wang, M. Hu, J. Ward and J. L. Beltran, No claim? Your gain: Design of residual value extended warranties under risk aversion and strategic claim behavior, Manufacturing Service Oper. Management, 17 (2015), 87-100. doi: 10.1287/msom.2014.0501. Google Scholar
[6]	J. C. Hartman and K. Laksana, Designing and pricing menus of extended warranty contracts, Naval Res. Logist., 56 (2009), 199-214. doi: 10.1002/nav.20333. Google Scholar
[7]	Y.-S. Huang, W.-Y. Gau and J.-W. Ho, Cost analysis of two-dimensional warranty for products with periodic preventive maintenance, Reliability Engineering System Safety, 134 (2015), 51-58. doi: 10.1016/j.ress.2014.10.014. Google Scholar
[8]	Y.-S. Huang, C.-D. Huang and J.-W. Ho, A customized two-dimensional extended warranty with preventive maintenance, European J. Oper. Res., 257 (2017), 971-978. doi: 10.1016/j.ejor.2016.07.034. Google Scholar
[9]	B. P. Iskandar and D. N. P. Murthy, Repair-replace strategies for two-dimensional warranty policies, Math. Comput. Modelling, 38 (2003), 1233-1241. doi: 10.1016/S0895-7177(03)90125-7. Google Scholar
[10]	B. P. Iskandar, D. N. P. Murthy and N. Jack, A new repair-replace strategy for items sold with a two-dimensional warranty, Comput. Oper. Res., 32 (2005), 669–682. doi: 10.1016/j.cor.2003.08.011. Google Scholar
[11]	N. Jack, B. P. Iskandar and D. N. P. Murthy, A repair-replace strategy based on usage rate for items sold with a two-dimensional warranty, Reliability Engineering System Safety, 94 (2009), 611-617. doi: 10.1016/j.ress.2008.06.019. Google Scholar
[12]	N. Jack and V. D. D. Schouten, Optimal repair-replace strategies for a warranted product, Int. J. Production Economics, 67 (2000), 95-100. doi: 10.1016/S0925-5273(00)00012-8. Google Scholar
[13]	Z.-L. Lin and Y.-S. Huang, Nonperiodic preventive maintenance for repairable systems, Naval Res. Logist., 57 (2010), 615-625. doi: 10.1002/nav.20418. Google Scholar
[14]	B. Liu, J. Wu and M. Xie, Cost analysis for multi-component system with failure interaction under renewing free-replacement warranty, European J. Oper. Res., 243 (2015), 874-882. doi: 10.1016/j.ejor.2015.01.030. Google Scholar
[15]	D. T. Mai, T. Liu, M. D. S. Morris and S. Sun, Quality coordination with extended warranty for store-brand products, European J. Oper. Res., 256 (2017), 524-532. doi: 10.1016/j.ejor.2016.06.042. Google Scholar
[16]	M. D. C. Moura, J. M. Santana, E. L. Droguett, I. D. Lins and B. N. Guedes, Analysis of extended warranties for medical equipment: A Stackelberg game model using priority queues, Reliability Engineering System Safety, 168 (2017), 338-354. doi: 10.1016/j.ress.2017.05.040. Google Scholar
[17]	D. G. Nguyen and D. N. P. Murthy, An optimal policy for servicing warranty, J. Oper. Res. Soc., 37 (1986), 1081-1088. doi: 10.1057/jors.1986.185. Google Scholar
[18]	D. G. Nguyen and D. N. P. Murthy, Optimal replace repair strategy for servicing products sold with warranty, European J. Oper. Res., 39 (1989), 206-212. doi: 10.1016/0377-2217(89)90193-8. Google Scholar
[19]	M. Park, K. M. Jung and D. H. Park, Optimal warranty policies considering repair service and replacement service under the manufacturer's perspective, Ann. Oper. Res., 244 (2016), 117-132. doi: 10.1007/s10479-014-1740-1. Google Scholar
[20]	M. Park and H. Pham, Cost models for age replacement policies and block replacement policies under warranty, Appl. Math. Model., 40 (2016), 5689-5702. doi: 10.1016/j.apm.2016.01.022. Google Scholar
[21]	X. Qin, Q. Su and S. H. Huang, Extended warranty strategies for online shopping supply chain with competing suppliers considering component reliability, J. Systems Sci. Systems Engineering, 26 (2017), 753-773. doi: 10.1007/s11518-017-5355-3. Google Scholar
[22]	M. Reimann and W. Zhang, Joint optimization of new production, warranty servicing strategy and secondary market supply under consumer returns, Pesquisa Operacional, 33 (2013), 325-342. doi: 10.1590/S0101-74382013000300001. Google Scholar
[23]	M. Shafiee, M. Finkelstein and S. Chukova, Burn-in and imperfect preventive maintenance strategies for warranted products, Proceedings of the Institution of Mechanical Engineers, Part O: Journal of Risk and Reliability, 225 (2011), 211-218. doi: 10.1177/1748006X11398584. Google Scholar
[24]	K. Shahanaghi, R. Noorossana, S. G. Jalali-Naini and M. Heydari, Failure modeling and optimizing preventive maintenance strategy during two-dimensional extended warranty contracts, Engineering Failure Analysis, 28 (2013), 90-102. doi: 10.1016/j.engfailanal.2012.09.006. Google Scholar
[25]	L. Shang, S. Si and Z. Cai, Optimal maintenance-replacement policy of products with competing failures after expiry of the warranty, Comput. Industrial Engineering, 98 (2016), 68-77. doi: 10.1016/j.cie.2016.05.012. Google Scholar
[26]	C. Su and J. Shen, Analysis of extended warranty policies with different repair options, Engineering Failure Analysis, 25 (2012), 49-62. doi: 10.1016/j.engfailanal.2012.04.002. Google Scholar
[27]	C. Su and X. Wang, A two-stage preventive maintenance optimization model incorporating two-dimensional extended warranty, Reliability Engineering System Safety, 155 (2016), 169-178. doi: 10.1016/j.ress.2016.07.004. Google Scholar
[28]	C. Tom and P. Elmira, Maintenance policies with two-dimensional warranty, Reliability Engineering System Safety, 77 (2002), 61–69. Google Scholar
[29]	P. Tong, Z. Liu, F. Men and L. Cao, Designing and pricing of two-dimensional extended warranty contracts based on usage rate, Internat. J. Prod. Res., 52 (2014), 6362-6380. doi: 10.1080/00207543.2014.940073. Google Scholar
[30]	P. Tong, X. Song and L. Zixian, A maintenance strategy for two-dimensional extended warranty based on dynamic usage rate, Internat. J. Prod. Res., 55 (2017), 5743-5759. doi: 10.1080/00207543.2017.1330573. Google Scholar
[31]	H. Vahdani, H. Mahlooji and A. Eshraghnia Jahromi, Warranty servicing for discretely degrading items with non-zero repair time under renewing warranty, Comput. Industrial Engineering, 65 (2013), 176-185. doi: 10.1016/j.cie.2011.08.012. Google Scholar
[32]	S. Varnosafaderani and S. Chukova, A two-dimensional warranty servicing strategy based on reduction in product failure intensity, Comput. Math. Appl., 63 (2012), 201-213. doi: 10.1016/j.camwa.2011.11.011. Google Scholar
[33]	J. Wang, Z. Zhou and H. Peng, Flexible decision models for a two-dimensional warranty policy with periodic preventive maintenance, Reliability Engineering System Safety, 162 (2017), 14-27. doi: 10.1016/j.ress.2017.01.012. Google Scholar
[34]	Y. Wang, Z. Liu and Y. Liu, Optimal preventive maintenance strategy for repairable items under two-dimensional warranty, Reliability Engineering System Safety, 142 (2015), 326-333. doi: 10.1016/j.ress.2015.06.003. Google Scholar
[35]	W. Xie, Optimal pricing and two-dimensional warranty policies for a new product, Internat. J. Prod. Res., 55 (2017), 6857-6870. doi: 10.1080/00207543.2017.1355578. Google Scholar
[36]	Z.-S. Ye and D. N. P. Murthy, Warranty menu design for a two-dimensional warranty, Reliability Engineering System Safety, 155 (2016), 21-29. doi: 10.1016/j.ress.2016.05.013. Google Scholar

show all references

References:

[1]	A. Akbarov and S. Wu, Forecasting warranty claims considering dynamic over-dispersion, Int. J. Prod. Econ., 139 (2012), 615-622. doi: 10.1016/j.ijpe.2012.06.001. Google Scholar
[2]	J. Baik, D. N. P. Murthy and N. Jack, Two-dimensional failure modeling with minimal repair, Naval Res. Logist., 51 (2004), 345-362. doi: 10.1002/nav.10120. Google Scholar
[3]	W. L. Chang and J.-H. Lin, Optimal maintenance policy and length of extended warranty within the life cycle of products, Comput. Math. Appl., 63 (2012), 144-150. doi: 10.1016/j.camwa.2011.11.001. Google Scholar
[4]	S. Chukova and M. R. Johnston, Two-dimensional warranty repair strategy based on minimal and complete repairs, Math. Comput. Modelling, 44 (2006), 1133-1143. doi: 10.1016/j.mcm.2006.03.015. Google Scholar
[5]	G. Gallego, R. Wang, M. Hu, J. Ward and J. L. Beltran, No claim? Your gain: Design of residual value extended warranties under risk aversion and strategic claim behavior, Manufacturing Service Oper. Management, 17 (2015), 87-100. doi: 10.1287/msom.2014.0501. Google Scholar
[6]	J. C. Hartman and K. Laksana, Designing and pricing menus of extended warranty contracts, Naval Res. Logist., 56 (2009), 199-214. doi: 10.1002/nav.20333. Google Scholar
[7]	Y.-S. Huang, W.-Y. Gau and J.-W. Ho, Cost analysis of two-dimensional warranty for products with periodic preventive maintenance, Reliability Engineering System Safety, 134 (2015), 51-58. doi: 10.1016/j.ress.2014.10.014. Google Scholar
[8]	Y.-S. Huang, C.-D. Huang and J.-W. Ho, A customized two-dimensional extended warranty with preventive maintenance, European J. Oper. Res., 257 (2017), 971-978. doi: 10.1016/j.ejor.2016.07.034. Google Scholar
[9]	B. P. Iskandar and D. N. P. Murthy, Repair-replace strategies for two-dimensional warranty policies, Math. Comput. Modelling, 38 (2003), 1233-1241. doi: 10.1016/S0895-7177(03)90125-7. Google Scholar
[10]	B. P. Iskandar, D. N. P. Murthy and N. Jack, A new repair-replace strategy for items sold with a two-dimensional warranty, Comput. Oper. Res., 32 (2005), 669–682. doi: 10.1016/j.cor.2003.08.011. Google Scholar
[11]	N. Jack, B. P. Iskandar and D. N. P. Murthy, A repair-replace strategy based on usage rate for items sold with a two-dimensional warranty, Reliability Engineering System Safety, 94 (2009), 611-617. doi: 10.1016/j.ress.2008.06.019. Google Scholar
[12]	N. Jack and V. D. D. Schouten, Optimal repair-replace strategies for a warranted product, Int. J. Production Economics, 67 (2000), 95-100. doi: 10.1016/S0925-5273(00)00012-8. Google Scholar
[13]	Z.-L. Lin and Y.-S. Huang, Nonperiodic preventive maintenance for repairable systems, Naval Res. Logist., 57 (2010), 615-625. doi: 10.1002/nav.20418. Google Scholar
[14]	B. Liu, J. Wu and M. Xie, Cost analysis for multi-component system with failure interaction under renewing free-replacement warranty, European J. Oper. Res., 243 (2015), 874-882. doi: 10.1016/j.ejor.2015.01.030. Google Scholar
[15]	D. T. Mai, T. Liu, M. D. S. Morris and S. Sun, Quality coordination with extended warranty for store-brand products, European J. Oper. Res., 256 (2017), 524-532. doi: 10.1016/j.ejor.2016.06.042. Google Scholar
[16]	M. D. C. Moura, J. M. Santana, E. L. Droguett, I. D. Lins and B. N. Guedes, Analysis of extended warranties for medical equipment: A Stackelberg game model using priority queues, Reliability Engineering System Safety, 168 (2017), 338-354. doi: 10.1016/j.ress.2017.05.040. Google Scholar
[17]	D. G. Nguyen and D. N. P. Murthy, An optimal policy for servicing warranty, J. Oper. Res. Soc., 37 (1986), 1081-1088. doi: 10.1057/jors.1986.185. Google Scholar
[18]	D. G. Nguyen and D. N. P. Murthy, Optimal replace repair strategy for servicing products sold with warranty, European J. Oper. Res., 39 (1989), 206-212. doi: 10.1016/0377-2217(89)90193-8. Google Scholar
[19]	M. Park, K. M. Jung and D. H. Park, Optimal warranty policies considering repair service and replacement service under the manufacturer's perspective, Ann. Oper. Res., 244 (2016), 117-132. doi: 10.1007/s10479-014-1740-1. Google Scholar
[20]	M. Park and H. Pham, Cost models for age replacement policies and block replacement policies under warranty, Appl. Math. Model., 40 (2016), 5689-5702. doi: 10.1016/j.apm.2016.01.022. Google Scholar
[21]	X. Qin, Q. Su and S. H. Huang, Extended warranty strategies for online shopping supply chain with competing suppliers considering component reliability, J. Systems Sci. Systems Engineering, 26 (2017), 753-773. doi: 10.1007/s11518-017-5355-3. Google Scholar
[22]	M. Reimann and W. Zhang, Joint optimization of new production, warranty servicing strategy and secondary market supply under consumer returns, Pesquisa Operacional, 33 (2013), 325-342. doi: 10.1590/S0101-74382013000300001. Google Scholar
[23]	M. Shafiee, M. Finkelstein and S. Chukova, Burn-in and imperfect preventive maintenance strategies for warranted products, Proceedings of the Institution of Mechanical Engineers, Part O: Journal of Risk and Reliability, 225 (2011), 211-218. doi: 10.1177/1748006X11398584. Google Scholar
[24]	K. Shahanaghi, R. Noorossana, S. G. Jalali-Naini and M. Heydari, Failure modeling and optimizing preventive maintenance strategy during two-dimensional extended warranty contracts, Engineering Failure Analysis, 28 (2013), 90-102. doi: 10.1016/j.engfailanal.2012.09.006. Google Scholar
[25]	L. Shang, S. Si and Z. Cai, Optimal maintenance-replacement policy of products with competing failures after expiry of the warranty, Comput. Industrial Engineering, 98 (2016), 68-77. doi: 10.1016/j.cie.2016.05.012. Google Scholar
[26]	C. Su and J. Shen, Analysis of extended warranty policies with different repair options, Engineering Failure Analysis, 25 (2012), 49-62. doi: 10.1016/j.engfailanal.2012.04.002. Google Scholar
[27]	C. Su and X. Wang, A two-stage preventive maintenance optimization model incorporating two-dimensional extended warranty, Reliability Engineering System Safety, 155 (2016), 169-178. doi: 10.1016/j.ress.2016.07.004. Google Scholar
[28]	C. Tom and P. Elmira, Maintenance policies with two-dimensional warranty, Reliability Engineering System Safety, 77 (2002), 61–69. Google Scholar
[29]	P. Tong, Z. Liu, F. Men and L. Cao, Designing and pricing of two-dimensional extended warranty contracts based on usage rate, Internat. J. Prod. Res., 52 (2014), 6362-6380. doi: 10.1080/00207543.2014.940073. Google Scholar
[30]	P. Tong, X. Song and L. Zixian, A maintenance strategy for two-dimensional extended warranty based on dynamic usage rate, Internat. J. Prod. Res., 55 (2017), 5743-5759. doi: 10.1080/00207543.2017.1330573. Google Scholar
[31]	H. Vahdani, H. Mahlooji and A. Eshraghnia Jahromi, Warranty servicing for discretely degrading items with non-zero repair time under renewing warranty, Comput. Industrial Engineering, 65 (2013), 176-185. doi: 10.1016/j.cie.2011.08.012. Google Scholar
[32]	S. Varnosafaderani and S. Chukova, A two-dimensional warranty servicing strategy based on reduction in product failure intensity, Comput. Math. Appl., 63 (2012), 201-213. doi: 10.1016/j.camwa.2011.11.011. Google Scholar
[33]	J. Wang, Z. Zhou and H. Peng, Flexible decision models for a two-dimensional warranty policy with periodic preventive maintenance, Reliability Engineering System Safety, 162 (2017), 14-27. doi: 10.1016/j.ress.2017.01.012. Google Scholar
[34]	Y. Wang, Z. Liu and Y. Liu, Optimal preventive maintenance strategy for repairable items under two-dimensional warranty, Reliability Engineering System Safety, 142 (2015), 326-333. doi: 10.1016/j.ress.2015.06.003. Google Scholar
[35]	W. Xie, Optimal pricing and two-dimensional warranty policies for a new product, Internat. J. Prod. Res., 55 (2017), 6857-6870. doi: 10.1080/00207543.2017.1355578. Google Scholar
[36]	Z.-S. Ye and D. N. P. Murthy, Warranty menu design for a two-dimensional warranty, Reliability Engineering System Safety, 155 (2016), 21-29. doi: 10.1016/j.ress.2016.05.013. Google Scholar

Figure 1. Termination point of 2D warranty service

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Figure 2. Schematic of the maintenance strategy under 2D warranty

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Figure 3. Trend diagram of $ W_r-U_r $

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Figure 4. Diagram of the differentiated warranty service strategy

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Figure 5. Curve of $ W_r-U_r $ ($ ε = 0.9 $)

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Figure 6. Curve of $ W_r-U_r $ ($ \varepsilon = 1.1 $)

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Table 1. The interval of usage rate intensity

Usage intensity	Low limit of interval	Upper limit of interval
Light	$ r_{l1} $	$ r_{l2} $
Medium	$ r_{l2} $	$ r_{h1} $
Heavy	$ r_{h1} $	$ r_{h2} $

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Usage intensity	Low limit of interval	Upper limit of interval
Light	$ r_{l1} $	$ r_{l2} $
Medium	$ r_{l2} $	$ r_{h1} $
Heavy	$ r_{h1} $	$ r_{h2} $

Table 2. The interval of usage rate intensity

Failure	$ C_{mi}(Yuan) $	$ C_{ci} (Yuan) $	$ f_i(w,r_d) $	$ \lambda_{0} $	$ k $
A31	1000	5000	$ 6.32E-04 w^{1.06}{e^{-({w}/{4.03})}}^{2.06} $	4.03	2.06
A18	3200	6400	$ 2.20E-02 w^{0.59}{e^{-({w}/{3.22})}}^{1.59} $	3.22	1.59
A88	800	4800	$ 2.53E-03 w^{0.45}{e^{-({w}/{2.48})}}^{1.45} $	2.48	1.45
A20	2600	7800	$ 5.67E-02 w^{0.49}{e^{-({w}/{2.90})}}^{1.49} $	2.90	1.49
A30	4500	9000	$ 4.78E-03 w^{0.62}{e^{-({w}/{3.40})}}^{1.62} $	3.40	1.62
A10	3000	12000	$ 5.82E-02 w^{0.51}{e^{-({w}/{2.94})}}^{1.51} $	2.94	1.51
A16	2900	8700	$ 9.24E-03 w^{0.42}{e^{-({w}/{2.76})}}^{1.42} $	2.76	1.42
A50	3500	10500	$ 6.98E-02 w^{0.85}{e^{-({w}/{3.28})}}^{1.85} $	3.28	1.85
A15	1700	3400	$ 1.36E-02 w^{0.51}{e^{-({w}/{3.14})}}^{1.51} $	3.14	1.51
A40	2800	5600	$ 1.33E-01 w^{0.49}{e^{-({w}/{2.69})}}^{1.49} $	2.69	1.49
A17	4600	9200	$ 1.89E-02 w^{0.55}{e^{-({w}/{3.22})}}^{1.55} $	3.22	1.55

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Failure	$ C_{mi}(Yuan) $	$ C_{ci} (Yuan) $	$ f_i(w,r_d) $	$ \lambda_{0} $	$ k $
A31	1000	5000	$ 6.32E-04 w^{1.06}{e^{-({w}/{4.03})}}^{2.06} $	4.03	2.06
A18	3200	6400	$ 2.20E-02 w^{0.59}{e^{-({w}/{3.22})}}^{1.59} $	3.22	1.59
A88	800	4800	$ 2.53E-03 w^{0.45}{e^{-({w}/{2.48})}}^{1.45} $	2.48	1.45
A20	2600	7800	$ 5.67E-02 w^{0.49}{e^{-({w}/{2.90})}}^{1.49} $	2.90	1.49
A30	4500	9000	$ 4.78E-03 w^{0.62}{e^{-({w}/{3.40})}}^{1.62} $	3.40	1.62
A10	3000	12000	$ 5.82E-02 w^{0.51}{e^{-({w}/{2.94})}}^{1.51} $	2.94	1.51
A16	2900	8700	$ 9.24E-03 w^{0.42}{e^{-({w}/{2.76})}}^{1.42} $	2.76	1.42
A50	3500	10500	$ 6.98E-02 w^{0.85}{e^{-({w}/{3.28})}}^{1.85} $	3.28	1.85
A15	1700	3400	$ 1.36E-02 w^{0.51}{e^{-({w}/{3.14})}}^{1.51} $	3.14	1.51
A40	2800	5600	$ 1.33E-01 w^{0.49}{e^{-({w}/{2.69})}}^{1.49} $	2.69	1.49
A17	4600	9200	$ 1.89E-02 w^{0.55}{e^{-({w}/{3.22})}}^{1.55} $	3.22	1.55

Table 3. Age and usage parameters of the 2D warranty coverage ($ \varepsilon = 0.9 $)

$ w_n $	Value	$ u_n $	Value
$ W_l $	3.35	$ U_l $	2.18
$ W_m $	2.44	$ U_m $	4.39
$ W_h $	1.83	$ U_h $	5.49

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$ w_n $	Value	$ u_n $	Value
$ W_l $	3.35	$ U_l $	2.18
$ W_m $	2.44	$ U_m $	4.39
$ W_h $	1.83	$ U_h $	5.49

Table 4. Age and usage parameters of the 2D warranty coverage ($ \varepsilon = 1.1 $)

$ w_n $	Value	$ u_n $	Value
$ W_l $	3.53	$ U_l $	2.29
$ W_m $	2.28	$ U_m $	4.10
$ W_h $	1.50	$ U_h $	4.50

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$ w_n $	Value	$ u_n $	Value
$ W_l $	3.53	$ U_l $	2.29
$ W_m $	2.28	$ U_m $	4.10
$ W_h $	1.50	$ U_h $	4.50

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2019 Impact Factor: 1.366

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2019 Impact Factor: 1.366

American Institute of Mathematical Sciences

Design of differentiated warranty coverage that considers usage rate and service option of consumers under 2D warranty policy

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American Institute of Mathematical Sciences

Design of differentiated warranty coverage that considers usage rate and service option of consumers under 2D warranty policy

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