Optimal preventive maintenance for equipment with two quality states and general failure time distributions

Abstract

We present an economic model for the optimization of preventive maintenance in a production process with two quality states. The equipment starts its operation in the in-control state but it may shift to the out-of-control state before failure or scheduled preventive maintenance. The time of shift and the time of failure are generally distributed random variables. The two states are characterized by different failure rates and revenues. We first derive the structure of the optimal maintenance policy, which is defined by two critical values of the equipment age that determine when to perform preventive maintenance depending on the actual (observable) state of the process. We then provide properties of the optimal solution and show how to determine the optimal values of the two critical maintenance times accurately and efficiently. The proposed model and, in particular, the behavior of the optimal solution as the model parameters and the shift and failure time distributions change are illustrated through numerical examples.

Introduction

It is commonly agreed nowadays that preventive maintenance policies can be very successful in improving equipment reliability while minimizing maintenance-related costs. During the past decades a plethora of preventive maintenance models have been developed and contributed to a good understanding of the properties and effectiveness of preventive maintenance policies under various conditions.

The purpose of this paper is to investigate the structure and properties of optimal preventive maintenance policies in production processes (equipment) that are subject to both quality deterioration and failure. More specifically, we develop, analyze and optimize an economic model for preventive maintenance in the following setting:

• There are two possible quality states during operation (an in-control state and an out-of-control state).

• The failure rate depends on both the quality state of the process and the equipment age.

• The time to shift to the out-of-control state and the times to failure are generally distributed continuous random variables.

• Operation in either one of the quality states generates income, which is higher in the in-control state.

• Downtimes are not negligible.

Production processes, commercial equipment and even consumer products with the above operation and failure characteristics are commonly encountered in practice. For example, consider the production process of forged spare parts, where forging presses are used to form specific shapes on metal sheets. A misplacement or inadequate heating of the metal may cause a slight tilt in the positioning of the hydraulic piston; the press will still be working but the output quality will be significantly lower and the probability of a piston failure due to its improper operation will increase. Another example is the operation of a commercial cooler and freezer. A typical problem is an impeller malfunction; the fan loses its balancing and the result is poor cooling, higher electricity consumption, and higher proneness to failure for the compressor. Note that the same type of malfunction with similar consequences is also common in the operation of personal computers and other electronic devices.

When a production process is characterized by multiple operating states and a failure state, the most important and distinctive feature of the associated model is the definition of the various states and the assumed transition mechanism between them. In order to clearly explain the contribution of this paper and its positioning with respect to existing publications we provide below a brief literature review of related research with emphasis on the aspect of the process deterioration mechanism.

The earlier works on preventive maintenance models for production processes with more than one operating states and a failure state are by Derman, 1962, Derman, 1963, who studies a process that deteriorates moving through a finite number of states according to a Markov chain. Derman assumes that the state of the process is known with accuracy at all discrete points in time and shows that the optimal replacement policy is a control limit policy; that is, the equipment should be replaced as soon as it is observed to operate in a state worse than some critical state. The same result is also obtained by Kolesar (1966) for a similar model but with a more general cost function. More specifically, Kolesar introduces an additional “occupancy” cost associated with being in each operating state.

Another approach to model the process deterioration mechanism has been suggested by Kao (1973). Kao uses a discrete time finite-state semi-Markov process to formulate the problem. This approach can account both for changes in the process state and for the ageing process of the equipment. Thus, Kao examines state-dependent policies as well as state-age-dependent policies and proves that under reasonable conditions the optimal policy is of the control limit type. More recent examples of maintenance models for semi-Markovian deteriorating systems can be found in So, 1992, Lam and Yeh, 1994, Yeh, 1997.

The process operating states can alternatively be expressed by the magnitude of the cumulative damage or wear of the equipment. The process is assumed to be subject to exterior shocks that damage or cause wear to the equipment, thereby increasing its probability of failure. Shock models have been introduced by Taylor (1975) and they have been studied by several researchers, including Feldman, 1976, Bergman, 1978, Gottlieb, 1982, Aven and Gaarder, 1987, Murthy and Iskadar, 1991a, Murthy and Iskadar, 1991b, Qian et al., 2003, Sheu and Chien, 2004. An extensive literature review of shock models up to 1989 is provided in the survey by Valdez-Flores and Feldman (1989). Shock models differ in the assumed stochastic process that governs the time distribution between two successive shocks (state transitions). Shocks may occur according to a Poisson process (e.g., Taylor, 1975), according to a semi-Markov process (e.g., Gottlieb, 1982) or arbitrarily (e.g., Feldman, 1976, Bergman, 1978).

A usual assumption in shock models is that the equipment can only fail at shock times with a generally different failure probability that depends on the accumulated damage or wear. However, Bergman, 1978, Murthy and Iskadar, 1991a, Murthy and Iskadar, 1991b do not restrict failures to shock times but allow them to occur at any point in time. In Bergman, 1978, Murthy and Iskadar, 1991a, Murthy and Iskadar, 1991b assume that the shock process is stationary Poisson but the failure rate, in addition to experiencing a jump whenever a shock occurs, also increases continuously with age while operating in any particular state.

The optimal preventive maintenance policy in most shock models is shown to be of the control limit type in one or two dimensions. In other words, the maintenance policy is either state-dependent or state-age-dependent. In the former case the equipment is preventively maintained as soon as it reaches or exceeds some threshold of the accumulated damage (state). In the latter case, which is common when the semi-Markovian assumption for the shock process is adopted, the equipment is preventively maintained as soon as the time since the last shock reaches some critical value, which is a function of the accumulated damage.

Apart from the maintenance literature, deterioration of the process condition is also a standard feature of the statistical process control models in the quality field. The process is assumed to operate in the “good” quality state (in-control) until it shifts to an inferior quality state (out-of-control) as a consequence of the occurrence of some assignable causes. The time until the transition to an out-of-control state (quality shift) is usually assumed to be exponentially distributed (Poisson process) but other distributions have been considered as well (Banerjee and Rahim, 1988). In a model of a process with non-exponential transition times Rahim and Banerjee (1993) introduce the use of preventive maintenance actions to protect the equipment against quality shifts. Their work has been further extended by Rahim and Ben-Daya (1998), who develop an integrated model dealing with the determination of the optimal production quantity as well. These quality control models, however, typically assume that the process is not subject to failures that would result in stoppage of operation. The possibility of equipment failure in the quality control context has been investigated by Tagaras (1988) and by Makis and Fung, 1995, Makis and Fung, 1998.

In particular, Tagaras (1988) uses a Markovian approach similar to that of Derman, 1962, Derman, 1963 to describe the evolution of production processes characterized by several quality states (a single in-control state and multiple out-of-control states) and a single failure state. He simultaneously considers quality control and maintenance procedures and derives not only the optimal preventive maintenance time but also the optimal quality control parameters that minimize the expected total cost. Makis and Fung, 1995, Makis and Fung, 1998 determine the optimal lot size and the optimal number of periodic inspections in a production process subject both to random failures and to shifts to an inferior quality state. In the 1998 paper the maintenance policy consists solely of corrective actions, while the 1995 paper accounts for preventive maintenance (replacement) actions too. In the latter case the preventive replacement time is also optimized. In both papers Makis and Fung, 1995, Makis and Fung, 1998 assume that the time to quality deterioration is exponentially distributed while the time to failure is generally distributed but with the same distribution regardless of the quality state of the process.

This paper shares common features with previous work cited above, but it encompasses and examines in a unifying model some of the characteristics that have been previously studied in isolation. More specifically, the similarities and differences from those publications that are closer to our work are as follows:

• The production process is characterized by two quality states (an in-control and an out-of-control state) and a failure state as in Makis and Fung, 1995, Makis and Fung, 1998. However, we use a general distribution for the time of transition to the out-of-control state and we allow for different failure time distributions in each quality state. On the other hand, Makis and Fung consider lot sizing and inspection decisions.

• The time to quality shift (state transition) is arbitrarily distributed as in some shock models but, unlike most shock models, we allow failures to occur at any point in time. In that sense, our model resembles the one suggested by Bergman (1978), but the two models differ in the assumed failure rate pattern. The failure rate in the model of Bergman (1978) remains constant as long as the equipment operates within the same state, implying an exponential distribution for the time to failure, while we make no assumption about the failure rate in each state other than it is non-decreasing in time.

• The failure mechanism adopted in our model is similar to that used by Murthy and Iskadar, 1991a, Murthy and Iskadar, 1991b; the failure rate increases not only whenever a state transition occurs, but also with the equipment age. However, we use a general distribution for the transition times to the out-of-control state in contrast to the exponential distribution for the transition times used by Murthy and Iskadar, 1991a, Murthy and Iskadar, 1991b. In addition, Murthy and Iskadar, 1991a, Murthy and Iskadar, 1991b restrict their analysis to two fixed types of preventive maintenance policies, while this paper contains structural results and properties of the optimal solution.

Furthermore, in contrast to Bergman, 1978, Murthy and Iskadar, 1991a, Murthy and Iskadar, 1991b, we allow non-negligible downtimes and a more general function that permits different operating revenues for each state. On the other hand, though, the models of Bergman, 1978, Murthy and Iskadar, 1991a, Murthy and Iskadar, 1991b consider more than two operating states.

In short, the most important contribution of this paper is that it does not require any of the distributions of the times to quality shift and failure to be exponential, thus significantly enhancing the domain of its applicability.

Our analysis indicates that the optimal preventive maintenance (PM) policy is a two-dimensional control limit policy using both state and age information. That is, the optimal PM policy is characterized by a critical equipment age for each quality state. It is important to note that this type of policy is different from the usual state-age-dependent policies because the latter depend on the current state of the process and the sojourn time in that state rather than the actual total age of the equipment.

The following section describes in more detail the problem and introduces the necessary notation. Section 3 describes the structure of the optimal policy and presents the development of the mathematical model. Some useful properties of the optimal solution are outlined in Section 4. Section 5 provides a numerical example for illustration purposes. Section 6 contains a numerical investigation of the effect of shift and failure time distributions on the optimal solution. In the last section the basic results are summarized and some future research directions are suggested.