Viliam Makis

Optimal Replacement In The Proportional Hazards Model

AbstractIn this paper, we examine a replacement problem for a system subject to stochastic deterioration. Upon failure the system must be replaced by a new one and a failure cost is incurred. If the systemis replaced before failure a smaller cost is incurred. The failure of the system depends both on its age and also on values of a diagnostic stochastic process observable at discrete points of time. Cox’s proportional hazards model is used to describe the failure rate of the system. We consider the problem of specifying a replacement rule which minimizes the long-run expected average cost per unit time. The form of the optimal replacement policy is found and an algorithm based on a recursive computational procedure is presented which can be used to obtain the optimal policy and the optimal expected average cost.

A Control-Limit Policy And Software For Condition-Based Maintenance Optimization

Abstract The focus of the paper is the optimization of condition-based maintenance decisions within the contexts of physical asset management. In particular, the analysis of a preventive replacement policy of the control-limit type for a deteriorating system subject to inspections at discrete points of time is presented. Cox’s PHM with a Weibull baseline hazard function and time dependent stochastic covariates is used to describe the failure rate of the system. The methods of estimating model parameters and the calculation of the optimal policy are given. The structure of the decision-making software EXAKT is presented. Experience with collecting, preprocessing and using real oil and vibration data is reported.

Optimal component replacement decisions using vibration monitoring and the proportional-hazards model

This paper describes a case study in which the Weibull proportional-hazards model is used to determine the optimal replacement policy for a critical item which is subject to vibration monitoring. Such an approach has been used to date in the context of monitoring through oil debris analysis, and this approach is extended in this paper to the vibration monitoring context. The Weibull proportional-hazards model is reviewed along with the software EXAKT used for optimization. In particular the case considers condition-based maintenance for circulating pumps in a coal wash plant that is part of the SASOL petrochemical company. The condition-based maintenance policy recommended in this study is based on histories collected over a period of 2 years, and is compared with current practice. The policy is validated using data that arose from subsequent operation of the plant.

Optimal replacement under partial observations

In this paper, we present a framework for the condition-based maintenance optimization. A technical system which can be in one of N operational states or in a failure state is considered. The system state is not observable, except the failure state. The information that is stochastically related to the system state is obtained through condition monitoring at equidistant inspection times. The system can be replaced at any time; a preventive replacement is less costly than failure replacement. The objective is to find a replacement policy minimizing the long run expected average cost per unit time. The replacement problem is formulated as an optimal stopping problem with partial information and transformed to a problem with complete information by applying the projection theorem to a smooth semimartingale process in the objective function. The dynamic equation is derived and analyzed in the piecewise deterministic Markov process stopping framework. The contraction property is shown and an algorithm for the calculation of the value function is presented, illustrated by an example.

Optimal replacement policy and the structure of software for condition‐based maintenance

States that the concept of condition‐based maintenance (CBM) has been widely accepted in practice since it enables maintenance decisions to be made based on the current state of equipment. Existing CBM methods, however, mainly rely on the inspector’s experience to interpret data on the state of equipment, and this interpretation is not always reliable. Aims to present a preventive maintenance policy based on inspections and a proportional hazards modelling approach with time‐dependent covariates to analyse failure‐time data statistically. Presents the structure of the software, currently under develop‐ ment and supported by the CBM Project Consortium.

A note on optimal replacement policy under general repair

Abstract A replacement model with general repair which brings the state of the system to a certain better state is considered. The form of the optimal replacement policy is found and the expected average cost per unit time is derived which can be minimized to obtain the optimal replacement time. Numerical examples are provided.

Multivariate Bayesian Control Chart

A multivariate Bayesian control chart for monitoring process mean under the assumption that the vector of process observations follows a multivariate normal distribution is considered. Traditional control charts such as Hotellings T2, EWMA, and CUSUM charts have been applied to control industrial processes characterized by several measurable variables. It is well known that these traditional, non-Bayesian process control techniques are not optimal, but very few results regarding the structure of the Bayesian control policy have been reported in the literature, all dealing with the univariate, finite-horizon case. In this paper, we formulate the multivariate Bayesian process control problem in the optimal stopping framework. The objective is to find a stopping rule under partial observations, minimizing the long-run expected average cost per unit time for a given sample size and sampling interval. Under standard operating and cost assumptions, it is proved that a control limit policy is optimal, and an algorithm is presented to find the optimal control limit and the minimum average cost.

Optimal lot sizing and inspection policy for an EMQ model with imperfect inspections

An EMQ model with a production process subject to random deterioration is considered. The process can be monitored through inspections, and both the lot size and the inspection schedule are subject to control. The “in-control” periods are assumed to be generally distributed and the inspections are imperfect, i.e., the true state of the process is not necessarily revealed through an inspection. The objective is the joint determination of the lot size and the inspection schedule, minimizing the long-run expected average cost per unit time. Both discrete and continuous cases are examined. A dynamic programming formulation is considered in the case where the inspections can be performed only at discrete times, which is typical for the parts industry. In the continuous case, an optimum inspection schedule is obtained for a given production time and given number of inspections by solving a nonlinear programming problem. A two-dimensional search procedure can be used to find the optimal policy. In the exponential case, the structure of the optimal inspection policy is established using Lagranges method, and it is shown that the optimal inspection times can be found by solving a nonlinear equation. Numerical studies indicate that the optimal policy performs much better than the optimal policy with periodic inspections considered previously in the literature. The case of perfect inspections is discussed, and an extension of the results obtained previously in the literature is presented.

Recursive filters for a partially observable system subject to random failure

We consider a failure-prone system which operates in continuous time and is subject to condition monitoring at discrete time epochs. It is assumed that the state of the system evolves as a continuous-time Markov process with a finite state space. The observation process is stochastically related to the state process which is unobservable, except for the failure state. Combining the failure information and the information obtained from condition monitoring, and using the change of measure approach, we derive a general recursive filter, and, as special cases, we obtain recursive formulae for the state estimation and other quantities of interest. Up-dated parameter estimates are obtained using the EM algorithm. Some practical prediction problems are discussed and an illustrative example is given using a real dataset.

A decision optimization model for condition‐based maintenance

Notes earlier work which commented on the formation of a research group to develop condition‐based maintenance (CBM) decision models and associated software. This paper provides an update on the research direction that has been taken since 1995. In particular, the structure of software for CBM decision making is highlighted, along with possible future research directions.