Designing prorated lifetime warranty strategy for high-value and durable products under two-dimensional warranty

Tong Peng

doi:10.3934/jimo.2020006

Article Contents

2021, Volume 17, Issue 2: 953-970. Doi: 10.3934/jimo.2020006

This issue Previous Article Robust portfolio selection for individuals: Minimizing the probability of lifetime ruin Next Article An efficient low complexity algorithm for box-constrained weighted maximin dispersion problem

Designing prorated lifetime warranty strategy for high-value and durable products under two-dimensional warranty

Tong Peng

School of Management, China University of Mining, and Technology, Xuzhou, China

Received: August 31, 2018

Revised: June 30, 2019

Early access: January 2020

Published: March 2021

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Abstract

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.

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

Citation:

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Figure 1. Schematic of the combined maintenance model

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Figure 2. Two-dimensional lifetime warranty region

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Table 1. Coordinate values of $ (W_1, U_1), (W_2, U_2 ) $ and $ (W_3, U_3 $)

$ r $	$ W_1 $	$ U_1 $	$ r $	$ W_2 $	$ U_2 $	$ r $	$ W_3 $	$ U_3 $
$ r\ge r_1 $	$ W_r $	$ W_r r $	$ r\le r_2 $	$ W_e $	$ W_e r $	$ r\le r_3 $	$ W_l $	$ W_l r $
$ r\le r_1 $	$ U_r/r $	$ U_r $	$ r>r_2 $	$ U_e/r $	$ U_e $	$ r>r_3 $	$ U_l/r $	$ U_l $

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Table 2. Parameter setting

$ W_t $	$ U_t $	$ r_1 $	$ W_e $	$ U_e $	$ r_2 $	$ W_1 $	$ U_1 $	$ r_3 $	$ C_m $
2	6	3	4	12	3	6	18	3	600
$ C_c $	$ \mathbf{\rho} $	$ \mathbf{\gamma} $	$ \mathbf{\mu} $	$ \mathbf{p}^\prime $	$ \mathbf{\alpha}_\mathbf{0} $	$ \mathbf{\varepsilon} $	$ \mathbf{r}_\mathbf{0} $	$ \mathbf{\tau} $
1200	2	0.1	0.2	0.5	1	2	1	2

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Table 3. Manufacturer cost in base warranty and additional warranty period ($ r = 0.9 $)

Warranty period	$ (0, W_t )\times (0, U_t) $		$ (0, W_e)\times (0, U_e) $		$ (0, W_l)\times(0, U_l) $
Warranty period	Minimal	Complete/minimal	Minimal	Complete/minimal	Minimal	Complete/minimal
$ C_0 $	1388.16	1416.48	4905.06	3541.5	9769.98	6643.14
$ K_r^{'}/K_r^{"}/K_r^{'''} $		0.80		1.67		2.76
$ L_r^{'}/L_r^{"}/L_r^{'''} $		1.48		3.80		5.87

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Table 4. Consumer cost of the product lifetime ($ r = 0.9 $)

Warranty period	$ (0, W_t )\times (0, U_t) $		$ (0, W_e)\times (0, U_e) $		$ (0, W_l)\times(0, U_l) $
Warranty period	Minimal	Complete/minimal	Minimal	Complete/minimal	Minimal	Complete/minimal
$ C_d $	16763.64	14522.16	16763.64	10860.24	15356.88	8411.88
$ C_a $	25194.24	15735.00	22780.2	13077.12	15356.88	8411.88
$ K_r^{'}/K_r^{"}/K_r^{'''} $		0.80		1.67		2.76
$ L_r^{'}/L_r^{"}/L_r^{'''} $		1.48		3.80		5.87

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Table 5. Revised additional warranty service price ($ r = 0.9 $)

Warranty period	$ (0, W_t )\times (0, U_t) $		$ (0, W_e)\times (0, U_e) $		$ (0, W_l)\times(0, U_l) $
Warranty period	Minimal	Complete/minimal	Minimal	Complete/minimal	Minimal	Complete/minimal
$ P $			$ P=P_e $		$ P=0.8P_e+P_a $
$ P $			7033.8	4422.48	15356.88	8411.88
$ K_r^{'}/K_r^{"}/K_r^{'''} $		0.80		1.67		2.76
$ L_r^{'}/L_r^{"}/L_r^{'''} $		1.48		3.80		5.87

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Table 6. Manufacturer additional warranty service profit ($ r = 0.9 $)

Warranty period	$ (0, W_t )\times (0, U_t) $		$ (0, W_e)\times (0, U_e) $		$ (0, W_l)\times(0, U_l) $
Warranty period	Minimal	Complete/minimal	Minimal	Complete/minimal	Minimal	Complete/minimal
$ \pi $	0	0	2128.74	880.5	5586.9	1768.74
$ K_r^{'}/K_r^{"}/K_r^{'''} $		0.80		1.67		2.76
$ L_r^{'}/L_r^{"}/L_r^{'''} $		1.48		3.80		5.87

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Table 7. Base warranty cost and consumer cost of the product lifetime ($ r = 0.9 $)

Warranty period	$ (0, W_t )\times (0, U_t) $					$ (0, W_e)\times (0, U_e) $					$ (0, W_l)\times(0, U_l) $
	Complete/minimal					Complete/minimal					Complete/minimal
	Base warranty			After base warranty		Base warranty		After base warranty			Base warr anty	After base warranty
Subinter val	$ [0, 0.8) $	$ [0.8, 1.48) $	$ [1.48, 2) $	$ [2, 4) $	$ [4, 6] $	$ [0, 1.67) $	$ [1.67, 2) $	$ [2, 3.8) $	$ [3.8, 4) $	$ [4, 6] $	$ [0, 2) $	$ [2, 2.76) $	$ [2.76, 4) $	$ [4, 5.78) $	$ [5.78, 6] $
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
$ C_d $				15522.16				10854.12				8411.88
$ K_r^{'}/K_r^{"}/K_r^{'''} $	0.80	0.80	0.80	0.80		1.67	1.67	1.67	1.67		2.76	2.76	2.76		2.76
$ L_r^{'}/L_r^{"}/L_r^{'''} $	1.48	1.48	1.48	1.48		3.80	3.80	3.80	3.08		5.87	5.87	5.87		5.87

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Table 8. Revised manufacturer cost in the base warranty and additional warranty period ($ r = 0.9 $)

Warranty period	$ (0, W_t )\times (0, U_t) $		$ (0, W_e)\times (0, U_e) $		$ (0, W_l)\times(0, U_l) $
Warranty period	Minimal	Complete/minimal	Minimal	Complete/minimal	Minimal	Complete/minimal
$ C_0 $	1388.16	1388.16	4905.06	3599.4	9769.98	6643.14
$ K_r^{'}/K_r^{"}/K_r^{'''} $		2		2		2.76
$ L_r^{'}/L_r^{"}/L_r^{'''} $		2		3.80		5.87

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Table 9. Revised consumer cost of the product lifetime ($ r = 0.9 $)

Warranty period	$ (0, W_t )\times (0, U_t) $		$ (0, W_e)\times (0, U_e) $		$ (0, W_l)\times(0, U_l) $
Warranty period	Minimal	Complete/minimal	Minimal	Complete/minimal	Minimal	Complete/minimal
$ C_d $	16763.64	16763.64	16763.64	10860.24	15356.88	8411.88
$ C_a $	25194.24	25194.24	22780.2	13077.12	15356.88	8411.88

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Table 10. Revised additional warranty service price ($ r = 0.9 $)

Warranty period	$ (0, W_t )\times (0, U_t) $		$ (0, W_e)\times (0, U_e) $		$ (0, W_l)\times(0, U_l) $
Warranty period	Minimal	Complete/minimal	Minimal	Complete/minimal	Minimal	Complete/minimal
$ P $			$ P=P_e $		$ P=0.8P_e+P_a $
$ P $			7033.8	4422.48	15356.88	8411.88
$ K_r^{'}/K_r^{"}/K_r^{'''} $		2		2		2.76
$ L_r^{'}/L_r^{"}/L_r^{'''} $		2		3.80		5.87

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Table 11. Revised manufacturers' additional warranty service profit ($ r = 0.9 $)

Warranty period	$ (0, W_t )\times (0, U_t) $		$ (0, W_e)\times (0, U_e) $		$ (0, W_l)\times(0, U_l) $
Warranty period	Minimal	Complete/minimal	Minimal	Complete/minimal	Minimal	Complete/minimal
$ \pi $	0	0	2128.74	823.08	5586.9	1768.74
$ K_r^{'}/K_r^{"}/K_r^{'''} $		2		2		2.76
$ L_r^{'}/L_r^{"}/L_r^{'''} $		2		3.80		5.87

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