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Robust portfolio selection for individuals: Minimizing the probability of lifetime ruin
Designing prorated lifetime warranty strategy for high-value and durable products under two-dimensional warranty
School of Management, China University of Mining, and Technology, Xuzhou, China |
This paper considers a prorated lifetime warranty strategy for high-value and durable products under two-dimensional warranty. Different from previous studies, the factor of capital time value is introduced into this study for a long-period lifetime warranty coverage. Besides, the mathematical model considers the warranty coverage into three situations and adopts minimal repair and complete/minimal maintenance strategy in the proposed model. To illustrate the proposed model, this paper analyzes the manufacturer cost, consumer cost, additional warranty service price and profit. The main goal of this paper is to provide a comprehensive warranty service to consumers. Through a numerical example, it is found that the proposed lifetime warranty strategy can provide consumers warranty service throughout product lifetime and reduce the expected cost expenditure to consumers. In addition, it can also provide additional profit sources to manufacturers and meet the realization of warranty objectives, while combined maintenance strategy is adopted in lifetime warranty coverage.
References:
[1] |
J. Baik, D. N. P. Murthy and N. Jack,
Two-dimensional failure modeling with minimal repair, Naval Research Logistics, 51 (2014), 345-362.
doi: 10.1002/nav.10120. |
[2] |
R. Banerjee and M. C. Bhattacharjee, Warranty servicing with a Brown-Proschan repair option, Asia-Pacific Journal of Operational Research, 29 (2012), 1240023, 13pp.
doi: 10.1142/S0217595912400234. |
[3] |
J. Baik, D. N. P. Murthy and N. Jack,
Two-dimensional failure modeling with minimal repair, Naval Research Logistics, 51 (2014), 345-362.
doi: 10.1002/nav.10120. |
[4] |
M.-S. Chang, J.-W. Park, Y.-M. Choi, T.-K. Park, B.-O. Choi and C.-J. Shin, Reliability evaluation of scroll compressor for system air conditioner, Journal of Mechanical Science and Technology, 30 (2016), 4459-4463. Google Scholar |
[5] |
W. L. Chang and J.-H. Lin,
Optimal maintenance policy and length of extended warranty within the life cycle of products, Computers and Mathematics with Applications, 63 (2012), 144-150.
doi: 10.1016/j.camwa.2011.11.001. |
[6] |
Y.-H. Chien, The effects of a free-repair warranty on the discrete-time periodic replacement policy, international Journal of Production Economics, 135 (2012), 832-839. Google Scholar |
[7] |
S. Chukova and M. R. Johnston,
Two-dimensional warranty repair strategy based on minimal and complete repairs, Mathematical and Computer Modelling, 44 (2014), 1133-1143.
doi: 10.1016/j.mcm.2006.03.015. |
[8] |
S. K. Gupta, S. De and A. Chatterjee, Warranty forecasting from incomplete two-dimensional warranty data, Reliability Engineering & System Safety, 126 (2014), 1-13. Google Scholar |
[9] |
Z. Hu and X. Du,
Lifetime cost optimization with time-dependent reliability, Engineering Optimization, 46 (2013), 1389-1410.
doi: 10.1080/0305215X.2013.841905. |
[10] |
Y.-S. Huang, E. Chen and J.-W. Ho, Two-dimensional warranty with reliability-based preventive maintenance, IEEE Transactions on Reliability, 62 (2013), 898-907. Google Scholar |
[11] |
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. Google Scholar |
[12] |
Y.-S. Huang, C.-D. Huang and J.-W. Ho,
A customized two-dimensional extended warranty with preventive maintenance, European Journal of Operational Research, 257 (2017), 971-978.
doi: 10.1016/j.ejor.2016.07.034. |
[13] |
B. P. Iskandar and D. N. P. Murthy,
Repair-replace strategies for two-dimensional warranty policies, Mathematical and Computer Modeling, 38 (2003), 1233-1241.
doi: 10.1016/S0895-7177(03)90125-7. |
[14] |
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 (2014), 611-617. Google Scholar |
[15] |
A. Khojandi, L. M. Maillart and O. A. Prokopyev, Optimal planning of life-depleting maintenance activities, IIE Transactions, 46 (2014), 636-652. Google Scholar |
[16] |
Y. Lei, Q. Liu and S. Shum,
Warranty pricing with consumer learning, European Journal of Operational Research, 263 (2017), 596-610.
doi: 10.1016/j.ejor.2017.06.024. |
[17] |
L.-Z. Lin and H.-R. Yeh, Fuzzy linguistic decision to provide alternatives to market mechanism strategies, Expert Systems with Applications, 37 (2010), 6986-6996. Google Scholar |
[18] |
D. T. Mai, T. Liu, M. D. S. Morris and S. Sun,
Quality coordination with extended warranty for store-brand products, European Journal of Operational Research, 256 (2017), 524-532.
doi: 10.1016/j.ejor.2016.06.042. |
[19] |
S. Mo, J. Zeng and W. Xu, A new warranty policy based on a buyer's preventive maintenance investment, Computers & Industrial Engineering, 111 (2017), 433-444. Google Scholar |
[20] |
D. N. P. Murthy and I. Djamaludin, International journal of production economics, International Journal of Production Economics, 79 (2002), 231-260. Google Scholar |
[21] |
M. Park, K. M. Jung and D. H. Park, Optimal post-warranty maintenance policy with repair time threshold for minimal repair, Reliability Engineering and System Safety, 111 (2013), 147-153. Google Scholar |
[22] |
M. Park, K. M. Jung and D. H. Park,
Optimal maintenance strategy under renewable warranty with repair time threshold, Applied Mathematical Modelling, 43 (2017), 498-508.
doi: 10.1016/j.apm.2016.11.015. |
[23] |
S. H. Park and J. H. Kim, Lifetime estimation of led lamp using gamma process model, Microelectronics Reliability, 57 (2014), 71-78. Google Scholar |
[24] |
A. Rahman and G. Chattopadhyay, Modeling manufacturers' and customers' costs for cost sharing lifetime warranty, Asia-Pacific Journal of Operational Research, 29 (2012), 1250024, 16pp.
doi: 10.1142/S0217595912500248. |
[25] |
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. Google Scholar |
[26] |
L. Shang, S. Si and Z. Cai, Optimal maintenance-replacement policy of products with competing failures after expiry of the warranty, Computers & Industrial Engineering, 98 (2016), 68-77. Google Scholar |
[27] |
S.-H. Sheu and S.-L. Yu,
Warranty strategy accounts for bathtub failure rate and random minimal repair cost, Computers & Mathematics with Applications, 49 (2005), 1233-1242.
doi: 10.1016/j.camwa.2004.06.036. |
[28] |
C. Su and X. Wang, A two-stage preventive maintenance optimization model incorporating two-dimensional extended warranty, Reliability Engineering and System Safety, 155 (2016). 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, International Journal of Production Research, 52 (2014), 6362-6380. Google Scholar |
[30] |
P. Tong, X. Song and L. Zixian, A maintenance strategy for two-dimensional extended warranty based on dynamic usage rate, International Journal of Production Research, 55 (2017), 6362-6380. Google Scholar |
[31] |
S. Varnosafaderani and S. Chukova,
A two-dimensional warranty servicing strategy based on reduction in product failure intensity, Computers & Mathematics with Applications, 63 (2017), 201-213.
doi: 10.1016/j.camwa.2011.11.011. |
[32] |
X. Wang and C. Su, A two-dimensional preventive maintenance strategy for items sold with warranty, International Journal of Production Research, 54 (2016), 5901-5915. Google Scholar |
[33] |
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. Google Scholar |
[34] |
C.-C. Wu, P.-C. Lin and C.-Y. Chou,
Determination of price and warranty length for a normal lifetime distributed product, J. Inf. Optim. Sci., 28 (2007), 335-355.
doi: 10.1080/02522667.2007.10699747. |
[35] |
S. Wu and P. Longhurst, Optimising age-replacement and extended non-renewing warranty policies in lifecycle costing, International Journal of Production Economics, 130 (2011), 262-267. 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. Google Scholar |
show all references
References:
[1] |
J. Baik, D. N. P. Murthy and N. Jack,
Two-dimensional failure modeling with minimal repair, Naval Research Logistics, 51 (2014), 345-362.
doi: 10.1002/nav.10120. |
[2] |
R. Banerjee and M. C. Bhattacharjee, Warranty servicing with a Brown-Proschan repair option, Asia-Pacific Journal of Operational Research, 29 (2012), 1240023, 13pp.
doi: 10.1142/S0217595912400234. |
[3] |
J. Baik, D. N. P. Murthy and N. Jack,
Two-dimensional failure modeling with minimal repair, Naval Research Logistics, 51 (2014), 345-362.
doi: 10.1002/nav.10120. |
[4] |
M.-S. Chang, J.-W. Park, Y.-M. Choi, T.-K. Park, B.-O. Choi and C.-J. Shin, Reliability evaluation of scroll compressor for system air conditioner, Journal of Mechanical Science and Technology, 30 (2016), 4459-4463. Google Scholar |
[5] |
W. L. Chang and J.-H. Lin,
Optimal maintenance policy and length of extended warranty within the life cycle of products, Computers and Mathematics with Applications, 63 (2012), 144-150.
doi: 10.1016/j.camwa.2011.11.001. |
[6] |
Y.-H. Chien, The effects of a free-repair warranty on the discrete-time periodic replacement policy, international Journal of Production Economics, 135 (2012), 832-839. Google Scholar |
[7] |
S. Chukova and M. R. Johnston,
Two-dimensional warranty repair strategy based on minimal and complete repairs, Mathematical and Computer Modelling, 44 (2014), 1133-1143.
doi: 10.1016/j.mcm.2006.03.015. |
[8] |
S. K. Gupta, S. De and A. Chatterjee, Warranty forecasting from incomplete two-dimensional warranty data, Reliability Engineering & System Safety, 126 (2014), 1-13. Google Scholar |
[9] |
Z. Hu and X. Du,
Lifetime cost optimization with time-dependent reliability, Engineering Optimization, 46 (2013), 1389-1410.
doi: 10.1080/0305215X.2013.841905. |
[10] |
Y.-S. Huang, E. Chen and J.-W. Ho, Two-dimensional warranty with reliability-based preventive maintenance, IEEE Transactions on Reliability, 62 (2013), 898-907. Google Scholar |
[11] |
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. Google Scholar |
[12] |
Y.-S. Huang, C.-D. Huang and J.-W. Ho,
A customized two-dimensional extended warranty with preventive maintenance, European Journal of Operational Research, 257 (2017), 971-978.
doi: 10.1016/j.ejor.2016.07.034. |
[13] |
B. P. Iskandar and D. N. P. Murthy,
Repair-replace strategies for two-dimensional warranty policies, Mathematical and Computer Modeling, 38 (2003), 1233-1241.
doi: 10.1016/S0895-7177(03)90125-7. |
[14] |
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 (2014), 611-617. Google Scholar |
[15] |
A. Khojandi, L. M. Maillart and O. A. Prokopyev, Optimal planning of life-depleting maintenance activities, IIE Transactions, 46 (2014), 636-652. Google Scholar |
[16] |
Y. Lei, Q. Liu and S. Shum,
Warranty pricing with consumer learning, European Journal of Operational Research, 263 (2017), 596-610.
doi: 10.1016/j.ejor.2017.06.024. |
[17] |
L.-Z. Lin and H.-R. Yeh, Fuzzy linguistic decision to provide alternatives to market mechanism strategies, Expert Systems with Applications, 37 (2010), 6986-6996. Google Scholar |
[18] |
D. T. Mai, T. Liu, M. D. S. Morris and S. Sun,
Quality coordination with extended warranty for store-brand products, European Journal of Operational Research, 256 (2017), 524-532.
doi: 10.1016/j.ejor.2016.06.042. |
[19] |
S. Mo, J. Zeng and W. Xu, A new warranty policy based on a buyer's preventive maintenance investment, Computers & Industrial Engineering, 111 (2017), 433-444. Google Scholar |
[20] |
D. N. P. Murthy and I. Djamaludin, International journal of production economics, International Journal of Production Economics, 79 (2002), 231-260. Google Scholar |
[21] |
M. Park, K. M. Jung and D. H. Park, Optimal post-warranty maintenance policy with repair time threshold for minimal repair, Reliability Engineering and System Safety, 111 (2013), 147-153. Google Scholar |
[22] |
M. Park, K. M. Jung and D. H. Park,
Optimal maintenance strategy under renewable warranty with repair time threshold, Applied Mathematical Modelling, 43 (2017), 498-508.
doi: 10.1016/j.apm.2016.11.015. |
[23] |
S. H. Park and J. H. Kim, Lifetime estimation of led lamp using gamma process model, Microelectronics Reliability, 57 (2014), 71-78. Google Scholar |
[24] |
A. Rahman and G. Chattopadhyay, Modeling manufacturers' and customers' costs for cost sharing lifetime warranty, Asia-Pacific Journal of Operational Research, 29 (2012), 1250024, 16pp.
doi: 10.1142/S0217595912500248. |
[25] |
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. Google Scholar |
[26] |
L. Shang, S. Si and Z. Cai, Optimal maintenance-replacement policy of products with competing failures after expiry of the warranty, Computers & Industrial Engineering, 98 (2016), 68-77. Google Scholar |
[27] |
S.-H. Sheu and S.-L. Yu,
Warranty strategy accounts for bathtub failure rate and random minimal repair cost, Computers & Mathematics with Applications, 49 (2005), 1233-1242.
doi: 10.1016/j.camwa.2004.06.036. |
[28] |
C. Su and X. Wang, A two-stage preventive maintenance optimization model incorporating two-dimensional extended warranty, Reliability Engineering and System Safety, 155 (2016). 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, International Journal of Production Research, 52 (2014), 6362-6380. Google Scholar |
[30] |
P. Tong, X. Song and L. Zixian, A maintenance strategy for two-dimensional extended warranty based on dynamic usage rate, International Journal of Production Research, 55 (2017), 6362-6380. Google Scholar |
[31] |
S. Varnosafaderani and S. Chukova,
A two-dimensional warranty servicing strategy based on reduction in product failure intensity, Computers & Mathematics with Applications, 63 (2017), 201-213.
doi: 10.1016/j.camwa.2011.11.011. |
[32] |
X. Wang and C. Su, A two-dimensional preventive maintenance strategy for items sold with warranty, International Journal of Production Research, 54 (2016), 5901-5915. Google Scholar |
[33] |
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. Google Scholar |
[34] |
C.-C. Wu, P.-C. Lin and C.-Y. Chou,
Determination of price and warranty length for a normal lifetime distributed product, J. Inf. Optim. Sci., 28 (2007), 335-355.
doi: 10.1080/02522667.2007.10699747. |
[35] |
S. Wu and P. Longhurst, Optimising age-replacement and extended non-renewing warranty policies in lifecycle costing, International Journal of Production Economics, 130 (2011), 262-267. 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. Google Scholar |


2 | 6 | 3 | 4 | 12 | 3 | 6 | 18 | 3 | 600 |
1200 | 2 | 0.1 | 0.2 | 0.5 | 1 | 2 | 1 | 2 |
2 | 6 | 3 | 4 | 12 | 3 | 6 | 18 | 3 | 600 |
1200 | 2 | 0.1 | 0.2 | 0.5 | 1 | 2 | 1 | 2 |
Warranty period | ||||||
Minimal | Complete/minimal | Minimal | Complete/minimal | Minimal | Complete/minimal | |
1388.16 | 1416.48 | 4905.06 | 3541.5 | 9769.98 | 6643.14 | |
0.80 | 1.67 | 2.76 | ||||
1.48 | 3.80 | 5.87 |
Warranty period | ||||||
Minimal | Complete/minimal | Minimal | Complete/minimal | Minimal | Complete/minimal | |
1388.16 | 1416.48 | 4905.06 | 3541.5 | 9769.98 | 6643.14 | |
0.80 | 1.67 | 2.76 | ||||
1.48 | 3.80 | 5.87 |
Warranty period | ||||||
Minimal | Complete/minimal | Minimal | Complete/minimal | Minimal | Complete/minimal | |
16763.64 | 14522.16 | 16763.64 | 10860.24 | 15356.88 | 8411.88 | |
25194.24 | 15735.00 | 22780.2 | 13077.12 | 15356.88 | 8411.88 | |
0.80 | 1.67 | 2.76 | ||||
1.48 | 3.80 | 5.87 |
Warranty period | ||||||
Minimal | Complete/minimal | Minimal | Complete/minimal | Minimal | Complete/minimal | |
16763.64 | 14522.16 | 16763.64 | 10860.24 | 15356.88 | 8411.88 | |
25194.24 | 15735.00 | 22780.2 | 13077.12 | 15356.88 | 8411.88 | |
0.80 | 1.67 | 2.76 | ||||
1.48 | 3.80 | 5.87 |
Warranty period | ||||||
Minimal | Complete/minimal | Minimal | Complete/minimal | Minimal | Complete/minimal | |
7033.8 | 4422.48 | 15356.88 | 8411.88 | |||
0.80 | 1.67 | 2.76 | ||||
1.48 | 3.80 | 5.87 |
Warranty period | ||||||
Minimal | Complete/minimal | Minimal | Complete/minimal | Minimal | Complete/minimal | |
7033.8 | 4422.48 | 15356.88 | 8411.88 | |||
0.80 | 1.67 | 2.76 | ||||
1.48 | 3.80 | 5.87 |
Warranty period | ||||||
Minimal | Complete/minimal | Minimal | Complete/minimal | Minimal | Complete/minimal | |
0 | 0 | 2128.74 | 880.5 | 5586.9 | 1768.74 | |
0.80 | 1.67 | 2.76 | ||||
1.48 | 3.80 | 5.87 |
Warranty period | ||||||
Minimal | Complete/minimal | Minimal | Complete/minimal | Minimal | Complete/minimal | |
0 | 0 | 2128.74 | 880.5 | 5586.9 | 1768.74 | |
0.80 | 1.67 | 2.76 | ||||
1.48 | 3.80 | 5.87 |
Warranty period | |||||||||||||||
Complete/minimal | Complete/minimal | Complete/minimal | |||||||||||||
Base warranty | After base warranty | Base warranty | After base warranty | Base warr anty | After base warranty | ||||||||||
Subinter val | |||||||||||||||
Cost of subinterval | 239.52 | 958.98 | 217.98 | 565 2.36 | 886 9.8 | 988.02 | 1554.48 | 1997.52 | 0.48 | 885 6.12 | 138 8.16 | 181 3.06 | 6579.46 | 19. 36 | 1.55 328 e-5 |
15522.16 | 10854.12 | 8411.88 | |||||||||||||
0.80 | 0.80 | 0.80 | 0.80 | 1.67 | 1.67 | 1.67 | 1.67 | 2.76 | 2.76 | 2.76 | 2.76 | ||||
1.48 | 1.48 | 1.48 | 1.48 | 3.80 | 3.80 | 3.80 | 3.08 | 5.87 | 5.87 | 5.87 | 5.87 |
Warranty period | |||||||||||||||
Complete/minimal | Complete/minimal | Complete/minimal | |||||||||||||
Base warranty | After base warranty | Base warranty | After base warranty | Base warr anty | After base warranty | ||||||||||
Subinter val | |||||||||||||||
Cost of subinterval | 239.52 | 958.98 | 217.98 | 565 2.36 | 886 9.8 | 988.02 | 1554.48 | 1997.52 | 0.48 | 885 6.12 | 138 8.16 | 181 3.06 | 6579.46 | 19. 36 | 1.55 328 e-5 |
15522.16 | 10854.12 | 8411.88 | |||||||||||||
0.80 | 0.80 | 0.80 | 0.80 | 1.67 | 1.67 | 1.67 | 1.67 | 2.76 | 2.76 | 2.76 | 2.76 | ||||
1.48 | 1.48 | 1.48 | 1.48 | 3.80 | 3.80 | 3.80 | 3.08 | 5.87 | 5.87 | 5.87 | 5.87 |
Warranty period | ||||||
Minimal | Complete/minimal | Minimal | Complete/minimal | Minimal | Complete/minimal | |
1388.16 | 1388.16 | 4905.06 | 3599.4 | 9769.98 | 6643.14 | |
2 | 2 | 2.76 | ||||
2 | 3.80 | 5.87 |
Warranty period | ||||||
Minimal | Complete/minimal | Minimal | Complete/minimal | Minimal | Complete/minimal | |
1388.16 | 1388.16 | 4905.06 | 3599.4 | 9769.98 | 6643.14 | |
2 | 2 | 2.76 | ||||
2 | 3.80 | 5.87 |
Warranty period | ||||||
Minimal | Complete/minimal | Minimal | Complete/minimal | Minimal | Complete/minimal | |
16763.64 | 16763.64 | 16763.64 | 10860.24 | 15356.88 | 8411.88 | |
25194.24 | 25194.24 | 22780.2 | 13077.12 | 15356.88 | 8411.88 |
Warranty period | ||||||
Minimal | Complete/minimal | Minimal | Complete/minimal | Minimal | Complete/minimal | |
16763.64 | 16763.64 | 16763.64 | 10860.24 | 15356.88 | 8411.88 | |
25194.24 | 25194.24 | 22780.2 | 13077.12 | 15356.88 | 8411.88 |
Warranty period | ||||||
Minimal | Complete/minimal | Minimal | Complete/minimal | Minimal | Complete/minimal | |
7033.8 | 4422.48 | 15356.88 | 8411.88 | |||
2 | 2 | 2.76 | ||||
2 | 3.80 | 5.87 |
Warranty period | ||||||
Minimal | Complete/minimal | Minimal | Complete/minimal | Minimal | Complete/minimal | |
7033.8 | 4422.48 | 15356.88 | 8411.88 | |||
2 | 2 | 2.76 | ||||
2 | 3.80 | 5.87 |
Warranty period | ||||||
Minimal | Complete/minimal | Minimal | Complete/minimal | Minimal | Complete/minimal | |
0 | 0 | 2128.74 | 823.08 | 5586.9 | 1768.74 | |
2 | 2 | 2.76 | ||||
2 | 3.80 | 5.87 |
Warranty period | ||||||
Minimal | Complete/minimal | Minimal | Complete/minimal | Minimal | Complete/minimal | |
0 | 0 | 2128.74 | 823.08 | 5586.9 | 1768.74 | |
2 | 2 | 2.76 | ||||
2 | 3.80 | 5.87 |
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