Journal of Quality Engineering and Management

Determining the optimal time policy model for the products with multiple failure states by considering a mixture distribution

Document Type : Original Article

Authors

Masoud Amini 1
Mohammad Saber Fallahnezhad 1
Mohammad Saleh Owlia 1
Mohammadali Vahdat 1
Shahaboddin Kharazmi 2

1 Department of Industries, Faculty of Industrial Engineering, Yazd University, Yazd, Iran.

2 Department of Mechanics, Faculty of Mechanical Engineering, Semnan University, Semnan, Iran.

https://doi.org/10.48313/jqem.2025.522094.1522
Abstract
Purpose: This study aims to optimize warranty periods for complex products by examining the role of warranties in customer retention and cost management. The proposed model uses a mixed statistical distribution to simultaneously model minor and major failures, seeking to minimize the product's life-cycle cost while maintaining customer satisfaction.
Methodology: The mathematical model defines life-cycle costs, establishes an objective function to minimize total costs, and determines the optimal warranty period. A numerical example and sensitivity analysis are used for validation, and the model is solved using Maple 2024.
Findings: The optimal warranty period was identified as 3.5-4.5 time units, and the model achieved a 23% cost reduction compared to conventional methods. Sensitivity analysis showed that changes in failure probability and failure rate directly affect the optimal warranty length.
Originality/Value: Using a mixed statistical distribution to model different failure types simultaneously offers an innovative, more realistic approach. This model provides a practical tool for adjusting warranty policies and reducing life-cycle costs, with potential for further development by incorporating dependent failures and real-world data.

Keywords

Warranty
Mixture distribution
Life cycle
Cost rate
[1]     Shafiee, M., & Chukova, S. (2013). Maintenance models in warranty: A literature review. European journal of operational research, 229(3), 561–572. https://doi.org/10.1016/j.ejor.2013.01.017
[2]      Blischke, W. R., &  Murthy, D. N. P. (1992). Product warranty management — I: A taxonomy for warranty policies. European journal of operational research, 62(2), 127-148 https://doi.org/10.1016/0377-2217(92)90242-2
[3]     Majid, H. A., Kasim, N. H., Jamahir, N. I., & Samah, A. A. (2012). Soft computing methods in warranty problems: Review and recent applications (2003-2012). International journal of computer science issues (IJCSI), 9(4), 190. https://B2n.ir/nt1501
[4]     Yang, D., He, Z., & He, S. (2016). Warranty claims forecasting based on a general imperfect repair model considering usage rate. Reliability engineering & system safety, 145, 147–154. https://doi.org/10.1016/j.ress.2015.09.012
[5]     Murthy, D. N. P., & Djamaludin, I. (2002). New product warranty: A literature review. International journal of production economics, 79(3), 231–260. https://doi.org/10.1016/S0925-5273(02)00153-6
[6]     Stamenković, D., Popović, V., Spasojević-Brkić, V., & Radivojević, J. (2011). Combination free replacement and pro-rata warranty policy optimization model. Istrazivanja I projektovanja za privredu, 9(4), 457–464. http://dx.doi.org/10.5937/jaes9-1202
[7]     Wu, S. (2013). A review on coarse warranty data and analysis. Reliability engineering & system safety, 114, 1–11. http://dx.doi.org/10.5937/jaes9-1202
[8]     Park, M., Jung, K. M., & Park, D. H. (2017). Optimal maintenance strategy under renewable warranty with repair time threshold. Applied mathematical modelling, 43, 498–508. https://doi.org/10.1016/j.apm.2016.11.015
[9]     Bai, J., & Pham, H. (2006). Cost analysis on renewable full-service warranties for multi-component systems. European journal of operational research, 168(2), 492–508. https://doi.org/10.1016/j.ejor.2004.03.034
[10]   Cai, K., He, S., & He, Z. (2020). Information sharing under different warranty policies with cost sharing in supply chains. International transactions in operational research, 27(3), 1550–1572. https://doi.org/10.1111/itor.12597
[11]    Zhang, Y. L., & Wang, G. J. (2019). A geometric process warranty model using a combination policy. Communications in statistics - theory and methods, 48(6), 1493–1505. https://doi.org/10.1080/03610926.2018.1433853
 [12]     Chang, C. C. (2021). Optimal preventive replacement policy for operating products with renewing free-replacement warranty. Communications in statistics - theory and methods, 50(18), 4255–4270. https://doi.org/10.1080/03610926.2020.1713371
 [13] Zhang, Y. L., & Wang, G. J. (2011). An extended replacement policy for a deteriorating system with multi-failure modes. Applied mathematics and computation, 218(5), 1820–1830. https://doi.org/10.1016/j.amc.2011.06.066
[14]   Barlow, R., & Hunter, L. (1960). Optimum preventive maintenance policies. Operations research, 8(1), 90–100. https://doi.org/10.1287/opre.8.1.90
[15]   Yeh, R. H., Chen, M. Y., & Lin, C. Y. (2007). Optimal periodic replacement policy for repairable products under free-repair warranty. European journal of operational research, 176(3), 1678–1686. https://doi.org/10.1016/j.ejor.2005.10.047
[16]   Wang, L., Pei, Z., Zhu, H., & Liu, B. (2018). Optimising extended warranty policies following the two-dimensional warranty with repair time threshold. Eksploatacja i niezawodność – maintenance and reliability, 20(4), 523–530. https://doi.org/10.17531/ein.2018.4.1
[17]   Murthy, D. N. P., & Djamaludin, I. (2002). New product warranty: A literature review. International journal of production economics, 79(3), 231–260. https://doi.org/10.1016/S0925-5273(02)00153-6
[18]   Liu, P., & Wang, G. (2023). Optimal preventive maintenance policies for products with multiple failure modes after geometric warranty expiry. Communications in statistics - theory and methods, 52(24), 8794–8813. https://doi.org/10.1080/03610926.2022.2076115
[19]    Block, H. W., Borges, W. S., & Savits, T. H. (1985). Age-dependent minimal repair. Journal of applied probability, 22(2), 370–385. https://B2n.ir/nt1501
Journal of Quality Engineering and Management
Volume 15, Issue 2
Summer 2025
Pages 137-147

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