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Joint Optimization of Preventive Maintenance and Inventory Control – Analytical Approach

This chapter investigates an integrated strategy of inventory control and preventive maintenance for a randomly failing production unit subject to a minimum required availability level. The production unit is submitted to a maintenance action as soon as it reaches a certain age m or at failure, whichever occurs first. A buffer stock, with level h, is built up at time A from the start of a production cycle in order to allow a continuous supply of the subsequent production unit at a constant rate during repair and preventive maintenance actions whose respective durations are random. A mathematical model and a numerical procedure are developed to simultaneously find the optimal values of the three decision variables (m*, h*, A*), which minimize the total average cost per time unit and satisfy the availability constraint.

1.1. Introduction

Production units’ failures are considered as an important source of disturbance and loss of productivity, particularly in a just-in-time manufacturing context. High availability levels must be guaranteed in order to provide the production units with the required effective capacity. Implementing preventive maintenance policies represents the favored means to reach this objective. However, the deployment of maintenance actions often requires the complete stopping of the maintained production units. To minimize the impact of these stops on the production system global performance, the maintenance and production control policies should be considered concurrently.

The simultaneous consideration of maintenance policies and production planning and control has recently become an important research area. Abdelnour et al. [ABD 95] studied the effect of maintenance policies on just-in-time production systems. Some studies have examined the conditions of building buffer stocks to guarantee the continuous supply of the subsequent production unit during the interruptions of service due to repair or preventive maintenance. Van der Duyn Schouten and Vanneste [VAN 95] proposed a preventive maintenance policy for two machines and one buffer between them, based not only on the age of the machine but also on the size of the buffer, which are both used to determine when to perform a preventive maintenance action. Meller and Kim [MEL 96] studied the impact of a preventive maintenance policy on a two-machine system with a fixed-capacity buffer between the machines. They suppose in their model that the machines’ failure rates are constant and that repair time is exponentially distributed and preventive maintenance actions are known at constant duration. Chan [CHA 01] developed a simulation model to evaluate the performance of a production line in the presence of several maintenance strategies. Rempelmeier [REM 01] considered the performance evaluation of a non-homogeneous production system taking into account the maintenance parameters and the quality of the manufactured products. Chelbi and Ait- Kadi [CHE 04] developed an analytical model to determine both the buffer stock size and the preventive maintenance period for an unreliable production unit, which is submitted to regular preventive maintenance of random duration.

Other related works appearing in the literature include [GRO 92a] and [GRO 92b], which simultaneously investigated the lot sizing of the buffer stock under the contrast of the machine breakdowns. In the same context, Cheung and Hausmann [CHE 97] proposed through their study a simultaneous optimization of strategic stock and the policy of maintenance of the age type. Later, Gharbi and Kenne [GHA 00] and Kenne and Gharbi [KEN 01] considered the ordering of the flow of production and the preventive maintenance by using the Markovian model. Recently, Chelbi and Rezg [CHE 06] developed an integrated study of production and inventory applied on a system with random failing subjected to a minimum required availability level.

The present study focuses on a joint policy of maintenance and inventory control for a repairable production unit subject to random failures, which supplies input to a subsequent unit operating according to a just-in-time configuration. The production unit is submitted to a maintenance action as soon as it reaches a certain age m or at failure, whichever occurs first. According to the proposed strategy, a buffer stock h is built-up at the maximum production rate at time A from the start of a production cycle, and not from the beginning of the cycle as is commonly supposed in the related literature. This buffer is developed to hedge against potential future capacity shortage during repair or planned maintenance actions whose respective durations are random. Once the inventory level reaches h, one should produce exactly enough to satisfy the demand.

We develop a mathematical model allowing the simultaneous consideration of the following three decision variables: the threshold inventory level h, the preventive maintenance critical age m and the inventory buildup start time A. The objective is to minimize the total expected cost per time unit including maintenance and inventory related costs, under the constraint of a minimum required stationary availability level of the production unit. We also propose a numerical procedure to generate the optimal values of the decision variables for any given set of input parameters.

The remainder of the chapter is organized as follows: section 1.2 presents the description problem and used notation. section 1.3 is dedicated to the development of the analytical model. section 1.4 presents the optimization phase. A numerical example and sensitivity study are studied in sections 1.5 and 1.6. Some concluding remarks and indications about possible extensions to this work are provided in section 1.7.

1.2. Description problem and notations

We are interested in a production system composed of one randomly failing machine, which supplies a subsequent unit with a single product. The demand rate d is constant. The machine has a maximum production rate Umax with Umax > d. A buffer stock is built up to allow continuous supply during the repair following failures and during the execution of planned preventive maintenance actions (Figure 1.1).

Figure 1.1. The considered production system

Considering A as the time at which the buildup of the buffer stock from the beginning of a production cycle starts, and s(t) as the inventory level at instant t, the production control policy is as follows:

The maintenance policy consists of submitting the machine to a preventive maintenance action as soon as it reaches a certain age m or at failure, whichever occurs first. Each maintenance action restores the machine to a state as good as new.

We defined the accumulation rate as δ, which is equal to the difference between the buffer’s input and output rates.

The inventory level evolution during a production cycle goes through four phases as shown in Figure 1.2.

Figure 1.2. The inventory level evolution during a production cycle

During phase I, production corresponds to demand, i.e. δ = 0. Phase II stands for the buffer buildup period, i.e. δ = Umax –d. Phase III is the saturation period in which s(t) is kept at level h and the production rate is equal to d, i.e. δ = 0 . Phase IV corresponds to a maintenance action during which production is stopped and demand is satisfied from the buffer stock, i.e. δ = –d. A loss is incurred if the maintenance time exceeds h/d.

The production cycle Tcyc is divided into two periods: TBM – the machine work time (case of failure or preventive maintenance) and TTR – the machine down time (case of failure or preventive maintenance).

Our objective is to simultaneously determine the age m at which preventive maintenance must be performed, the size of the buffer stock h and the instant at which its buildup should start, so as to operate at minimum cost per time unit over an infinite horizon and meet the minimum required stationary availability level concurrently.

All costs related to maintenance and inventory are supposed to be known and constant. The following notations will also be used throughout the model:

Cs : holding cost of a product unit during a unit of time;

Cl : loss cost due to an unsatisfied demand of one product unit during a unit of time;

Cpm : preventive maintenance action cost;

Ccm : corrective maintenance action cost;

f(t): probability density function associated with the machine-time-to- failure;

F(t): probability distribution function associated with the machine-time-to-failure;

R(t): reliability function associated with the machine, R(t) = 1 – F(t);

r(t): machine instantaneous failure rate function;

μ: the machine average time to failure;

ξ: the instant of failure of the machine;

gp (t): probability density function associated with the duration of a preventive maintenance action;

gc(t): probability density function associated with the duration of a corrective maintenance...