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Optimal maintenance and warranty strategy for second-hand product with periodic preventive maintenance action

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Abstract

This paper proposes an optimal maintenance and warranty policy for the repairable second-hand product incorporating a periodic preventive maintenance strategy to slow down the deterioration of the product and derives an optimal length of non-renewing warranty period. Although a number of maintenance policies for the second-hand product have been discussed in the literature, there exist very few works dealing with the periodic preventive maintenance strategy under non-renewing warranty to obtain an optimal warranty period for the second-hand product. This work considers a non-renewing warranty policy, during which a periodic preventive maintenance is adopted to diminish the degradation process of the product and determines an optimal length of warranty period by optimizing the expected cost rate incurred to the dealer during the maintenance cycle. Since the replacement of the failed second-hand product may not be practical in general, we adopt a concept of full refund instead of replacement in this paper. By pre-setting a repair time threshold, either the refund or minimal repair is provided by the dealer depending on whether the failed second-hand product can be repaired within the repair time threshold or not. The maintenance cycle of the second-hand product starts with the purchase of the product and ends when either the refund occurs within the warranty period or the warranty expires. Given a certain cost structure charged to the dealer, we formulate the expected cost rate during the maintenance cycle of the product and determine an optimal length of warranty period minimizing the objective function. Assuming the power law process for the product’s failures and a Weibull distribution for the repair times, we present a numerical example and investigate the effects of periodic preventive maintenances and other relevant parameters on the optimal solution.

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Abbreviations

PM:

Preventive maintenance

ECR:

Expected cost rate

RFRW:

Repair-full refund warranty

RFRPW:

Repair-full refund preventive maintenance warranty

cdf :

Cumulative distribution function

pdf :

Probability density function

T, Y :

Random variables representing failure time and repair time of the product, respectively

\(F_{pm} \left( t \right),f_{pm} \left( t \right),h_{pm} \left( t \right)\) :

cdf, pdf, Intensity function, respectively, of T after being upgraded at each PM action during the warranty period

\(F_{0} ( \cdot ),f_{0} ( \cdot ),h_{0} ( \cdot )\) :

cdf, pdf and intensity function, respectively, of the failure time for new product without PM actions

\(H_{pm} (a,b):\,\mathop \smallint \limits_{a}^{b} h_{pm} \left( u \right)du\) :

Expected number of failures within an interval \(\left( {a,b} \right)\) when the PM action is adopted.

\(G(y),g(y)\) :

cdf and pdf of the repair time Y, respectively.

x :

Virtual age of the purchasing second-hand product corresponding to the reduced failure rate due to the upgrade prior to its sale.

\(u_{x}\) :

Reduced calendar age that makes the virtual age equal to x

\(\delta\) :

Inter-PM interval between two successive periodic PM actions

\(\omega\) :

Length of warranty period

\(\gamma\) :

Improvement level which measures the PM effect, i.e. \(0 \le {\upgamma } \le 1\).

\(C_{m} ,C_{p}\) :

Random variables representing total minimal repair cost and total PM cost, respectively

\(c_{o} ,c_{u} ,c_{m} ,c_{p} ,c_{f}\) :

Purchasing cost, upgrade cost prior to the purchase, unit minimal repair cost, unit PM cost and unit full refund cost, respectively

\(r_{0}\) :

Repair time threshold

\(t_{f}\) :

Time point at which a full refund is provided

n :

Number of PM actions taken during the warranty period

\(N_{{pm,r_{0} }} (a,b)\) :

Number of failures within an interval \(\left( {a,b} \right)\) when the periodic PM actions are adopted and the repair time threshold is set to \(r_{0}\)

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Acknowledgements

Minjae Park’s work was supported by the National Research Foundation of Korea(NRF) grant funded by the Korea government(MSIT) (NRF-2020R1F1A104823711). Ki Mun Jung’s work was supported by the National Research Foundation of Korea(NRF) grant funded by the Korea government(MSIT) (NRF-2019R1F1A1049570). Dong Ho Park’s research was supported by Basic Science Research Program through the National Research Foundation of Korea(NRF) funded by the Ministry of Science (2020R111A1A0106654211). Minjae Park and Ki Mun Jung contributed equally to this work.

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Authors and Affiliations

  1. College of Business Administration, Hongik University, Seoul, Korea

    Minjae Park

  2. Department of Informational Statistics, Kyungsung University, Busan, Korea

    Ki Mun Jung

  3. Industrial Academic Cooperation Foundation, Hallym University, Chuncheon, Korea

    Dong Ho Park

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  1. Minjae Park
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  2. Ki Mun Jung
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Correspondence to Dong Ho Park.

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Park, M., Jung, K.M. & Park, D.H. Optimal maintenance and warranty strategy for second-hand product with periodic preventive maintenance action. J. Korean Stat. Soc. 50, 773–794 (2021). https://doi.org/10.1007/s42952-021-00127-3

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