Richard M. Feldman

A survey of preventive maintenance models for stochastically deteriorating single‐unit systems

A survey of the research done on preventive maintenance is presented. The scope of the present survey is on the research published after the 1976 paper by Pierskalla and Voelker [98]. This article includes optimization models for repair, replacement, and inspection of systems subject to stochastic deterioration. A classification scheme is used that categorizes recent research into inspection models, minimal repair models, shock models, or miscellaneous replacement models.

Optimal replacement policies with minimal repair and age-dependent costs

Abstract In this paper, we study a modified minimal repair/replacement problem that is formulated as a Markov decision process. The operating cost is assumed to be a nondecreasing function of the systems age. The specific maintenance actions for a manufacturing system to be considered are whether to have replacement, minimal repair or keep it operating. It is shown that a control limit policy , or in particular a ( t , T ) policy, is optimal over the space of all possible policies under the discounted cost criterion. A computational algorithm for the optimal ( t , T ) policy is suggested based on the total expected discounted cost.

Foundations of stochastic development

Abstract A consistent mathematical theory of stochastic poikilotherm development has been derived based upon a minimum set of biological assumptions obtained from the literature. In the subsequent analysis, the resulting developmental rate can be justifiably represented as a random variable. Three cases are considered: (1) developmental rates dependent only on temperature, (2) rates dependent on both temperature and age, and (3) rates dependent on a general function of temperature and time. The analysis provides a mathematical foundation for the current practice of superimposing a probability distribution function on a biological time scale to describe the development of individuals from a population.

A stochastic model of a temperature-dependent population☆

Abstract The theoretical basis is developed for a population model which allows the use of constant temperature experimental data in predicting the size of an insect population for any variable temperature environment. The model is based on a stochastic analysis of an insects mortality, development, and reproduction response to temperature. The key concept in the model is the utilization of a physiological time scale. Different temperatures affect the population by increasing an individuals physiological age by differing rates. Conditions for the temperature response properties are given which establish the validity of the model for variable temperature regimes. These conditions refer to the relationship between chronological and physiological age. Reasonable agreement between the model and field populations demonstrates the practicality of this approach.

Optimal Replacement with Semi-Markov Shock Models Using Discounted Costs

Consider a system that is subject to a sequence of randomly occurring shocks; each shock causes some damage of random magnitude to the system. Any of the shocks might cause the system to fail, and the probability of such a failure is a function of the sum of the magnitudes of damage caused by all previous shocks. When the system fails, it must be immediately replaced and a failure cost is incurred. If the system is replaced before failure, a smaller replacement cost is incurred, and that cost may depend upon the state of the system at replacement time. The purpose of this paper is to determine the optimal replacement policy among the class of policies that replace at shock times. The main assumption is that the cumulative damage process is a semi-Markov process. The cost criterion is to minimize the discounted cost of replacement. See Feldman [Feldman, R. M. 1976. Optimal replacement with semi-Markov shock models. J. Appl. Probab.13 108--117; Feldman, R. M. 1977. The maintenance of systems governed by semi-Markov shock models. Proceedings of the Conference on the Theory and Applications of Reliability with Emphasis on Bayesian and Nonparametric Methods. Edited by C. P. Tsokos. Academic Press.] for the average cost criterion case. The general approach will be to apply optimal stopping theory to the replacement problem.

A mathematical model of field population dynamics of the southern pine beetle, dendroctonus frontalis

Abstract The southern pine beetle, Dendroctonus frontalis Zimmermann, is a major insect pest of pines in the southern U.S.A. A population dynamic model of D. frontalis within an infestation was developed. The model is based on the integration of several component submodels that describe individual life stage processes of the beetle. The inclusion of variable forest stand density and microclimate conditions results in a model response which closely follows actual field responses. A sensitivity analysis was performed to investigate effects of weather conditions and the importance of the spatial patterns within the forest.

Optimization of process improvement and inspection location for serial manufacturing

SUMMARY This paper deals with serial manufacturing systems with imperfect inspection and rework of defective items. An inspector may not correctly identify a defective item; however, nondefective items are always correctly identified. Inspection stations, when they are used, screen all incoming units. Our objective is to develop a method that will determine which process steps to improve, the amount of process improvement, and the placement of inspection stations so that the expected per unit total cost of production is minimized. The dynamics of the process is modelled as a Markov renewal process and a gradient search procedure is used to optimize process improvement.

Stage-structured matrix models for organisms with non-geometric development times

Matrix models have been used to model population growth of organisms for many decades. They are popular because of both their conceptual simplicity and their computational efficiency. For some types of organisms they are relatively accurate in predicting population growth; however, for others the matrix approach does not adequately model growth rate. One of the reasons for the lack of accuracy is that most matrix-based models implicitly assume a specific degree of variability in development times for the organism. Because the variability is implicit, the implied variances are often not verified with experimental data. In this paper, we shall present extensions to the stage-classified matrix models so that organisms with arbitrary means and standard deviations of development times can be modeled.

An Improved Policy Iteration Algorithm for Semi-Markov Maintenance Problems

Abstract This paper uses a general maintenance problem and a new policy iteration algorithm to solve most semi-Markovian maintenance models. This algorithm decomposes the maintenance process into a deteriorating process and a sequence of replacement and repair actions so that the number of states involved in the decision process is less than the state space of the unmaintained process. Consequently, the number of equations to be solved at every iteration step can be substantially reduced. The improved policy iteration algorithm and the conventional policy iteration algorithm are compared in a set of randomly generated semi-Markov maintenance problems.

Within-tree life process models of the southern pine beetle, dendroctonus frontalis

Abstract This paper develops models of gallery construction, emergence and re-emergence for use in a general population dynamics model of the southern pine beetle, Dendroctonuc frontalis Zimmermann. Models of these processes were originally developed from laboratory data, and are extended here to account for fluctuating temperatures and variable attack densities under field conditions. The resulting models were tested using data from three natural populations (infestations) from east Texas. These tests reveal that the laboratory-derived models closely predict the timing and length of gallery, and the timing of emergence and re-emergence in the field.