# American Institute of Mathematical Sciences

March  2021, 17(2): 953-970. doi: 10.3934/jimo.2020006

## 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

Received  September 2018 Revised  July 2019 Published  January 2020

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.

Citation: Tong Peng. Designing prorated lifetime warranty strategy for high-value and durable products under two-dimensional warranty. Journal of Industrial & Management Optimization, 2021, 17 (2) : 953-970. doi: 10.3934/jimo.2020006
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Schematic of the combined maintenance model
Coordinate values of $(W_1, U_1), (W_2, U_2 )$ and $(W_3, U_3$)
 $r$ $W_1$ $U_1$ $r$ $W_2$ $U_2$ $r$ $W_3$ $U_3$ $r\ge r_1$ $W_r$ $W_r r$ $r\le r_2$ $W_e$ $W_e r$ $r\le r_3$ $W_l$ $W_l r$ $r\le r_1$ $U_r/r$ $U_r$ $r>r_2$ $U_e/r$ $U_e$ $r>r_3$ $U_l/r$ $U_l$
 $r$ $W_1$ $U_1$ $r$ $W_2$ $U_2$ $r$ $W_3$ $U_3$ $r\ge r_1$ $W_r$ $W_r r$ $r\le r_2$ $W_e$ $W_e r$ $r\le r_3$ $W_l$ $W_l r$ $r\le r_1$ $U_r/r$ $U_r$ $r>r_2$ $U_e/r$ $U_e$ $r>r_3$ $U_l/r$ $U_l$
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
 $W_t$ $U_t$ $r_1$ $W_e$ $U_e$ $r_2$ $W_1$ $U_1$ $r_3$ $C_m$ 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
Manufacturer cost in base warranty and additional warranty period ($r = 0.9$)
 Warranty period $(0, W_t )\times (0, U_t)$ $(0, W_e)\times (0, U_e)$ $(0, W_l)\times(0, U_l)$ 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
 Warranty period $(0, W_t )\times (0, U_t)$ $(0, W_e)\times (0, U_e)$ $(0, W_l)\times(0, U_l)$ 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
Consumer cost of the product lifetime ($r = 0.9$)
 Warranty period $(0, W_t )\times (0, U_t)$ $(0, W_e)\times (0, U_e)$ $(0, W_l)\times(0, U_l)$ 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
 Warranty period $(0, W_t )\times (0, U_t)$ $(0, W_e)\times (0, U_e)$ $(0, W_l)\times(0, U_l)$ 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
Revised additional warranty service price ($r = 0.9$)
 Warranty period $(0, W_t )\times (0, U_t)$ $(0, W_e)\times (0, U_e)$ $(0, W_l)\times(0, U_l)$ Minimal Complete/minimal Minimal Complete/minimal Minimal Complete/minimal $P$ $P=P_e$ $P=0.8P_e+P_a$ 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
 Warranty period $(0, W_t )\times (0, U_t)$ $(0, W_e)\times (0, U_e)$ $(0, W_l)\times(0, U_l)$ Minimal Complete/minimal Minimal Complete/minimal Minimal Complete/minimal $P$ $P=P_e$ $P=0.8P_e+P_a$ 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
Manufacturer additional warranty service profit ($r = 0.9$)
 Warranty period $(0, W_t )\times (0, U_t)$ $(0, W_e)\times (0, U_e)$ $(0, W_l)\times(0, U_l)$ 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
 Warranty period $(0, W_t )\times (0, U_t)$ $(0, W_e)\times (0, U_e)$ $(0, W_l)\times(0, U_l)$ 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
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
 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
Revised manufacturer cost in the base warranty and additional warranty period ($r = 0.9$)
 Warranty period $(0, W_t )\times (0, U_t)$ $(0, W_e)\times (0, U_e)$ $(0, W_l)\times(0, U_l)$ 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
 Warranty period $(0, W_t )\times (0, U_t)$ $(0, W_e)\times (0, U_e)$ $(0, W_l)\times(0, U_l)$ 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
Revised consumer cost of the product lifetime ($r = 0.9$)
 Warranty period $(0, W_t )\times (0, U_t)$ $(0, W_e)\times (0, U_e)$ $(0, W_l)\times(0, U_l)$ 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
 Warranty period $(0, W_t )\times (0, U_t)$ $(0, W_e)\times (0, U_e)$ $(0, W_l)\times(0, U_l)$ 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
Revised additional warranty service price ($r = 0.9$)
 Warranty period $(0, W_t )\times (0, U_t)$ $(0, W_e)\times (0, U_e)$ $(0, W_l)\times(0, U_l)$ Minimal Complete/minimal Minimal Complete/minimal Minimal Complete/minimal $P$ $P=P_e$ $P=0.8P_e+P_a$ 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
 Warranty period $(0, W_t )\times (0, U_t)$ $(0, W_e)\times (0, U_e)$ $(0, W_l)\times(0, U_l)$ Minimal Complete/minimal Minimal Complete/minimal Minimal Complete/minimal $P$ $P=P_e$ $P=0.8P_e+P_a$ 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
Revised manufacturers' additional warranty service profit ($r = 0.9$)
 Warranty period $(0, W_t )\times (0, U_t)$ $(0, W_e)\times (0, U_e)$ $(0, W_l)\times(0, U_l)$ 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
 Warranty period $(0, W_t )\times (0, U_t)$ $(0, W_e)\times (0, U_e)$ $(0, W_l)\times(0, U_l)$ 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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