Optimization model for performance-based warranty decision of degraded systems based on improved sparrow search algorithm

doi:10.23919/JSEE.2025.000135

Journal of Systems Engineering and Electronics ›› 2025, Vol. 36 ›› Issue (5): 1259-1280.doi: 10.23919/JSEE.2025.000135

• SYSTEMS ENGINEERING • Previous Articles

Optimization model for performance-based warranty decision of degraded systems based on improved sparrow search algorithm

Enzhi DONG(), Zhonghua CHENG(), Zichang LIU(), Xi ZHU(), Rongcai WANG(), Yongsheng BAI()

Shijiazhuang Campus of Army Engineering University of PLA, Shijiazhuang 050003, China

Received:2023-12-05 Online:2025-10-18 Published:2025-10-24
Contact: Zhonghua CHENG E-mail:ez_dong@aeu.edu.cn;a15032073178@sina.com;zc_liu1997@aeu.edu.cn;zxtalk@126.com;wrcpromising@163.com;13832106356@163.com
About author:
DONG Enzhi was born in 1997. He received his M.S. degree from the Shijiazhuang Campus of the Army Engineering University of PLA, Hebei, China, in 2022. He is currently pursuing his Ph.D. degree with the Shijiazhuang Campus of the Army Engineering University of PLA, Hebei, China. His research interests include equipment management, equipment support and predictive maintenance. E-mail: ez_dong@aeu.edu.cn

CHENG Zhonghua was born in 1972. He received his Ph.D. degree from the Ordnance Engineering College, Shijiazhuang, in 2005. He is currently a professor at the Shijiazhuang Campus of the Army Engineering University of PLA. His research interests include management science and engineering, maintenance engineering, and reliability engineering. E-mail: a15032073178@sina.com

LIU Zichang was born in 1997. He received his M.S. degree from the School of Electronic Information Engineering at Hebei University, Baoding, China, in 2022. he is current pursuing his Ph.D. degree with the Shijiazhuang Campus of the Army Engineering University of PLA. His research interests include equipment management, equipment support and predictive maintenance. E-mail: zc_liu1997@aeu.edu.cn

ZHU Xi was born in 1997. He received his M.S. degree from the Shijiazhuang Campus of the Army Engineering University of PLA, Hebei, China, in 2021. He is currently pursuing his Ph.D. degree with the Shijiazhuang Campus of the Army Engineering University of PLA, Hebei, China. His research interests include equipment management and material reserve. E-mail: zxtalk@126.com

WANG Rongcai was born in 1996. He received his M.S. degree from the Shijiazhuang Campus of the Army Engineering University of PLA, Hebei, China, in 2021. He is currently pursuing his Ph.D. degree with the Shijiazhuang campus of the Army Engineering University of PLA, Hebei, China. His research interests include equipment management and health management. E-mail: wrcpromising@163.com

BAI Yongsheng was born in 1982. He received his Ph.D. degree from the Ordnance Engineering College, Shijiazhuang in 2010. He is currently a lecturer at the Shijiazhuang Campus of the Army Engineering University of PLA. His research interests include equipment maintenance, equipment support theory and application. E-mail: 13832106356@163.com
Supported by:
This work was supported by the National Natural Science Foundation of China (71871219).

Abstract

Abstract:

Performance-based warranties (PBWs) are widely used in industry and manufacturing. Given that PBW can impose financial burdens on manufacturers, rational maintenance decisions are essential for expanding profit margins. This paper proposes an optimization model for PBW decisions for systems affected by Gamma degradation processes, incorporating periodic inspection. A system performance degradation model is established. Preventive maintenance probability and corrective renewal probability models are developed to calculate expected warranty costs and system availability. A benefits function, which includes incentives, is constructed to optimize the initial and subsequent inspection intervals and preventive maintenance thresholds, thereby maximizing warranty profit. An improved sparrow search algorithm is developed to optimize the model, with a case study on large steam turbine rotor shafts. The results suggest the optimal PBW strategy involves an initial inspection interval of approximately 20 months, with subsequent intervals of about four months, and a preventive maintenance threshold of approximately 37.39 mm wear. When compared to common cost-minimization-based condition maintenance strategies and PBW strategies that do not differentiate between initial and subsequent inspection intervals, the proposed PBW strategy increases the manufacturer’s profit by 1% and 18%, respectively. Sensitivity analyses provide managerial recommendations for PBW implementation. The PBW strategy proposed in this study significantly increases manufacturers’ profits by optimizing inspection intervals and preventive maintenance thresholds, and manufacturers should focus on technological improvement in preventive maintenance and cost control to further enhance earnings.

Key words: performance-based warranty, gamma process, periodic inspection, intelligent optimization algorithm

Enzhi DONG, Zhonghua CHENG, Zichang LIU, Xi ZHU, Rongcai WANG, Yongsheng BAI. Optimization model for performance-based warranty decision of degraded systems based on improved sparrow search algorithm[J]. Journal of Systems Engineering and Electronics, 2025, 36(5): 1259-1280.

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Notation	Description
W	The performance-based warranty period
$ {b_p} $	The preventive maintenance threshold
$ {b_f} $	The failure threshold
$ Y(t) $	The system degradation process over time
C_p	The cost of preventive maintenance
C_f	The cost of corrective maintenance
T_p	The duration of preventive maintenance
T_f	The duration of the corrective maintenance
C_i	The inspection cost
T_i	The time spent on the inspection
T₁	The first inspection interval
T	The subsequent inspection interval
T_max	The inspection interval time limit
T_work	The time of operation under system warranty
T_halt	The downtime during the system warranty period
E_1,k(T)_halt	The expected downtime if preventive replacement is implemented at the kth inspection
E_2,k(T)_halt	The expected downtime if preventive replacement is implemented at the kth inspection
E_1,k(C)	The expected warranty cost if preventive replacement is implemented at the kth inspection
E_2,k(C)	The expected warranty cost if preventive replacement is implemented at the kth inspection
E(T_halt)	The expected downtime during the warranty period
EC	The expected warranty cost during the warranty period
ER	The total benefits of the performance-based warranty for the manufacturer
A_min	The minimum system availability acceptable to the user
S_p	The product price
Q(W, b_p, S_p)	The product demand function
EP	The warranty profit for manufacturers
L	The total profit on sales of the manufacturer’s products
L_s	The total profit for the manufacturer
prob{·}	The probability of an event occurring

Number	Test function	Optimal value	Definition domain	Dimension
1	$ {F_1}\left( x \right) = \displaystyle\sum\limits_{i = 1}^n { - {x_i}\sin {\sqrt {\left\| {{x_i}} \right\|} }} $	−12569.5	[−500,500]	30
2	$ {F_2}\left( x \right) = - \displaystyle\sum\limits_{i = 1}^5 {{{\left[ {\left( {x - {a_i}} \right){{\left( {x - {a_i}} \right)}^{{T}}} + {c_i}} \right]}^{ - 1}}} $	−10.1532	[0,10]	4
3	$ {F_3}\left( x \right) = - \displaystyle\sum\limits_{i = 1}^7 {{{\left[ {\left( {x - {a_i}} \right){{\left( {x - {a_i}} \right)}^{{T}}} + {c_i}} \right]}^{ - 1}}} $	−10.4029	[0,10]	4
4	$ {F_4}(x) = - \displaystyle\sum\limits_{i = 1}^{10} {{{\left[ {\left( {x - {a_i}} \right){{\left( {x - {a_i}} \right)}^{{T}}} + {c_i}} \right]}^{ - 1}}} $	−10.5364	[0,10]	4

Warranty policy	T₁/Month	T/Month	b_p/mm	EP/10⁴CNY	Q	$ S_p^ * /10^4{\mathrm{CNY}}$	$L_s/10^4{\mathrm{CNY}} $	${\mathrm{EC}}/10^4{\mathrm{CNY}} $
PBW	20.16	4.21	37.39	4.61	46544	1405.21	3.760×10⁷	2.23
Policy 1	19.36	3.63	37.88	4.57	46546	1405.22	3.758×10⁷	2.21
Policy 2	11.13	11.13	26.76	3.90	46505	1406	3.754×10⁷	2.56

$ \beta $	PBW			Policy 1			Policy 2
$ \beta $	T₁/Month	T/Month	b_p/mm	T₁/Month	T/Month	b_p/mm	T₁/Month	T/Month	b_p/mm
0.8	10.00	12.54	29.87	14.74	4.99	38.63	11.12	11.12	31.44
0.9	18.35	5.03	37.87	17.61	4.38	38.36	11.62	11.62	28.86
1	21.35	4.46	37.40	20.49	3.85	37.88	11.78	11.78	26.76
1.1	24.25	4.10	36.45	23.32	3.53	36.95	11.61	11.61	25.27
1.2	27.36	1.50	12.29	26.11	3.35	35.66	9.70	9.70	26.78

$ \beta $	PBW			Policy 1			Policy 2
$ \beta $	T₁/Month	T/Month	b_p/mm	T₁/Month	T/Month	b_p/mm	T₁/Month	T/Month	b_p/mm
0.8	10.00	11.21	30.62	13.92	4.72	38.63	10.50	10.50	31.44
0.9	17.33	4.75	37.86	16.63	4.13	38.36	10.98	10.98	28.86
1	20.16	4.21	37.39	19.36	3.63	37.88	11.13	11.13	26.76
1.1	22.91	3.88	36.45	22.04	3.32	36.99	10.96	10.96	25.27
1.2	25.84	1.50	13.13	25.05	1.50	18.33	6.52	6.52	31.40

Optimization model for performance-based warranty decision of degraded systems based on improved sparrow search algorithm

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[1]	Shuai ZHAO, Viliam MAKIS, Shaowei CHEN, Yong LI. Health evaluation method for degrading systems subject to dependent competing risks [J]. Journal of Systems Engineering and Electronics, 2018, 29(2): 436-444.
[2]	Zhijun Cheng, Zheng Yang, and Bo Guo. Optimal opportunistic maintenance model of multi-unit systems [J]. Journal of Systems Engineering and Electronics, 2013, 24(5): 811-817.
[3]	Lin Tan, Chiming Guo, Zhijun Cheng, and Bo Guo. Optimal maintenance decisions for gamma deteriorating systems [J]. Journal of Systems Engineering and Electronics, 2011, 22(5): 788-793.
[4]	Lin Tan, Zhijun Cheng, Bo Guo, and Shiyu Gong. Condition-based maintenance policy for gamma deteriorating systems [J]. Journal of Systems Engineering and Electronics, 2010, 21(1): 57-61.

$ \beta $	PBW			Policy 1			Policy 2
$ \beta $	T₁/Month	T/Month	b_p/mm	T₁/Month	T/Month	b_p/mm	T₁/Month	T/Month	b_p/mm
0.8	13.62	5.37	37.68	13.19	4.47	38.63	9.95	9.95	31.44
0.9	16.41	4.50	37.87	15.76	3.91	38.37	10.40	10.40	28.86
1	19.11	3.99	37.40	18.34	3.44	37.88	10.54	10.54	26.76
1.1	22.27	1.50	17.16	20.88	3.14	36.99	10.38	10.38	25.27
1.2	24.48	1.50	10.32	23.73	1.50	10.18	6.18	6.18	31.40

p	PBW			Policy 1			Policy 2
p	T₁/Month	T/Month	b_p/mm	T₁/Month	T/month	b_p/mm	T₁/Month	T/Month	b_p/mm
0.6	36.20	5.10	40.35	30.25	5.01	39.06	20.32	17.32	35.23
0.7	34.30	4.62	39.16	27.46	4.72	39.63	16.24	15.14	34.01
0.8	25.60	4.48	38.52	25.69	4.13	38.56	13.25	12.35	33.15
0.9	20.16	4.21	37.39	19.36	3.63	37.88	11.13	11.13	26.76