This paper investigates the optimal threshold value of failure rates for leased products with a Weibull lifetime distribution. Within a lease period, any product failure is rectified by minimal repairs and the lessor may incur a penalty when the time required to perform a minimal repair exceeds a reasonable time limit. To reduce product failures, additional preventive maintenance actions are carried out when the failure rate reaches a threshold value. Under this maintenance scheme, a mathematical model of the expected total cost is established. Based on the model, the optimal threshold value and the corresponding maintenance degrees are derived such that the expected total cost is minimized. The structural properties of the optimal policy are investigated and an efficient algorithm is provided to search for the optimal policy. Finally, numerical examples are given to illustrate the features of the optimal policy.
Due to the increase in complexity of products and rapid advances in technological innovation, there is a trend to lease a product rather than own a product. For a leased product, performing maintenance actions usually requires expensive equipment and professional technicians, which is not economical for the lessee (the one leasing the product). Therefore, the maintenance of the product is usually specified in a lease contract to ensure that the product could fulfill its intended purpose [1]. In this paper, we propose a maintenance scheme, in which preventive maintenance actions are taken when the failure rate of the leased product reaches a certain threshold value since this maintenance scheme can be easily specified in a lease contract in practice.
Maintenance actions usually can be classified into two major categories: (i) Corrective Maintenance (CM) and (ii) Preventive Maintenance (PM). CM actions are used to rectify a failed product back to its operational state, and PM actions are performed to improve the operational state of the product to avoid failures. For repairable products, various maintenance policies have been extensively discussed in the literature [2], [3], [4] and [5]. In practice, minimal repair is the most commonly used CM action to restore a failed product [6] and [7]. After minimal repair, the product is operational but the failure rate of the product remains unchanged. Various issues associated with minimal repair can be found in the literature [6], [7], [8] and [9].
To reduce the number of failures and possible penalties within the lease period, PM actions are usually taken by the lessor. Many PM policies have been proposed and studied under various situations, such as finite or infinite horizon [2] and [3], and perfect or imperfect maintenance [4], [10] and [11]. Nakagawa [6] proposed an imperfect PM for repairable products in which the degree of PM is described by the reduction in age of the product. Since then, the age-reduction method has been widely adopted in the research of imperfect maintenance policies.
Another method for describing the degree of a preventive maintenance is the failure-rate reduction method (FRRM). In 1993, Chan and Shaw [12] proposed two methods of failure-rate reduction: (i) failure rate with fixed reduction; that is, the failure rate is reduced by a fixed quantity after each PM action; and (ii) failure rate with proportional reduction; that is, the failure rate is reduced by an amount proportional to the current failure rate. In this paper, we will adopt the fixed failure-rate-reduction method to describe the degree of PM and derive the optimal PM policy for a leased product, since this method can be clearly specified in a lease contract.
The remainder of this paper is organized as follows. The mathematical model is developed in Section 2 for the case when the failure density is Weibull. The properties of the optimal PM policy are investigated in Section 3. In Section 4, the impact of providing preventive maintenance within a lease period is illustrated through numerical examples. Finally, some conclusions are drawn in the last section.
In this paper, we investigate the optimal threshold value of the failure rate before PM actions need to be performed within the lease period. We found that the optimal threshold value of the failure rate is equal to the degree of PM. This means that each PM action should reduce the failure rate to zero if it is worthwhile to perform a PM action. Furthermore, we show that there exists a unique optimal PM policy within the lease period such that the expected total cost is minimized. An efficient algorithm is also provided to search for the optimal policy. From numerical examples, we found that the cost reduction is significant for a long lease period if PM actions are performed.
The uniqueness property of the optimal PM policy obtained in this paper is for the case when the life-time distribution of the leased product is Weibull. However, for the case when the life-time distribution is a general distribution other than Weibull, the uniqueness property might not exist or the conditions for the existence of the optimal policy might be very complicated. Furthermore, some generalizations such as nonlinear maintenance cost, time-dependent penalty cost, or various penalty schemes, are extended issues for future study in this area.
