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Optimal preventive maintenance level for a repairable product under warranty subject to multimode failure process

Azmat Ullah (Antai College of Economics and Management, Shanghai Jiao Tong University, Shanghai, China)
Muhammad Ayat (Department of Industrial and Management Engineering, Hanyang University, Ansan, Republic of Korea)
Hakeem Ur Rehman (Institute of Quality and Technology Management, University of the Punjab, Lahore, Pakistan)
Lochan Kumar Batala (School of Economics, Henan University, Kaifeng, China)

International Journal of Quality & Reliability Management

ISSN: 0265-671X

Article publication date: 10 November 2020

Issue publication date: 6 April 2021




The purpose of this paper is to develop a model that determines whether how much effort of preventive maintenance action is worthwhile for the consumer over the post-sale product life cycle of a repairable complex product where the product is under warranty and subject to stochastic multimode failure process, that is, damaging failure and light failure with different probabilities.


The expected life cycle cost is designed for a warranted product from the consumer perspective. The product failure is quantified with failure rate function, which is the number of failures incurred over the product life cycle. The authors consider the failure rate function reduction method in their model where the scale parameter of a failure rate function is maximized by applying the optimal preventive maintenance level. The scale parameter of any failure distribution refers to the meantime to failure (MTTF). The first-order condition is applied with respect to the maintenance level in order to achieve the convexity of the nonlinear function of the expected life cycle cost function.


The authors have found analytically the close form of the preventive maintenance level, which can be used to find the optimal reduced form of the failure rate function of the product and the minimum product expected life cycle cost under the given condition of multimode stochastic failure process. The authors have suggested different maintenance policies to consumers in order to implement the proposed preventive maintenance model under different conditions. A numerical example further illustrated the analytical model by considering the Weibull distribution.

Practical implications

The consumer may use this study in the accurate modeling of the life cycle cost of a product that is under warranty and fails with a multimode failure process. Also, the suggested preventive maintenance approach of this study helps the consumer in making appropriate maintenance decisions such as to minimize the expected life cycle cost of a product.


This study proposes an accurate estimation of a life cycle cost for a product that is under the support of warranty and fails with multimode. Furthermore, for such a kind of product, which is under warranty and fails with multimode, this study suggests a new preventive maintenance approach that assures the minimum expected life cycle cost.



Conflicts of interest: We would like to assure you that the authors have no conflict of interest in the submission of this manuscript, and all authors have approved the manuscript for publication.


Ullah, A., Ayat, M., Rehman, H.U. and Batala, L.K. (2021), "Optimal preventive maintenance level for a repairable product under warranty subject to multimode failure process", International Journal of Quality & Reliability Management, Vol. 38 No. 5, pp. 1193-1214.



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