Abstract
This paper addresses a finite-horizon profit maximization three-machine replacement problem. More precisely, a model is formulated allowing for preventive maintenance to slow down machine quality and profit reduction caused by obsolescence, to determine the timing of replacing an existing machine by another available machine with improved technology. This decision is considered under uncertainty regarding the introduction time of a machine with a not-yetachieved technology. Given an exponential probability distribution function of the introduction time, the optimality of a bang-bang nonincreasing preventive maintenance control is shown.
Moreover, subproblems maximizing the expected discounted profit are analyzed. Closed-form solutions are provided to compare machines of different technologies and to derive an analytical sensitivity analysis concerned with many issues related to the problem. The results are not necessarily intuitive and simple. For example, different relationships between the planning horizon and the preventive maintenance switching time are presented for the three-machine problem versus the single-machine problem.
The focus of this paper is on the formulation and the analytical analysis of the problem rather than on its computational aspects.
Similar content being viewed by others
References
Chand, S., andSethi, S. P.,Planning Horizon Procedures for Machine Replacement Models with Several Possible Replacement Alternatives, Naval Research Logistics Quarterly, Vol. 29, pp. 483–493, 1982.
Goldstein, T., Ladany, S. P., andMehrez, A.,A Discounted Machine Replacement Model with an Expected Future Technological Breakthrough, Naval Research Logistics Quarterly, Vol. 35, pp. 209–220, 1988.
Goldstein, T., Ladany, S. P., andMehrez, A.,A Dual Machine Replacement Model: A Note on Planning Horizon Procedures for Machine Replacements, Operations Research, Vol. 34, pp. 938–941, 1986.
Sethi, S. P., andThompson, G. L.,Optimal Control Theory, Applications to Management Science, Martinus Nijhoff Publishing Company, Amsterdam, Holland, 1981.
Sethi, S. P., andMorton, T. E.,A Mixed Optimization Technique for the Generalized Machine Replacement Problem, Naval Research Logistics Quarterly, Vol. 19, pp. 471–481, 1972.
Sethi, S. P., andThompson, G. L.,Christmas Toy Manufacturer's Problem: An Application of the Stochastic Maximum Principle, Opsearch, Vol. 14, pp. 161–173, 1977.
Wagner, H. M., andWhitin, T. M.,Dynamic Version of Economic Lot Size Model, Management Science, Vol. 5, pp. 89–96, 1958.
Kamien, N. L., andSchwartz, M. I.,Optimal Maintenance and Sale Age for a Machine Subject to Failure, Management Science, Vol. 17, pp. B495-B504, 1971.
Virtanen, I.,Optimal Maintenance Policy and Planned Sale Date for a Machine Subject to Deterioration and Random Failure, European Journal of Operations Research, Vol. 9, pp. 33–40, 1982.
Gaimon, C., andThompson, G. L.,A Real-Time Solution for Preventive and Repair Maintenance, Optimal Control Application and Methods, Vol. 10, pp. 211–228, 1989.
Blaguiere, A.,Necessary and Sufficient Condition for Optimal Strategies in Impulsive Control, Differential Games and Control Theory, Edited by P. T. Liu and J. G. Sutin, Marcel Dekker, New York, New York, Vol. 3 1979.
Näslund, B.,Simultaneous Determination of Optimal Repair Policy and Service Life, Swedish Journal of Economics, Vol. 68, pp. 63–73, 1966.
Thompson, G. L.,Optimal Maintenance Policy and Sale Date of a Machine, Management Science, Vol. 14, pp. 543–550, 1968.
Pierskalla, W. P., andVoelker, J. A.,A Survey of Maintenance Models: The Control and Surveillance of Deteriorating Systems, Naval Research Logistics Quarterly, Vol. 23, pp. 353–388, 1976.
Leung, L. C., andTanchoco, J. M. A.,Multiple Machine Replacement within an Integrated System Framework, Engineering Economist, Vol. 32, pp. 89–114, 1987.
Ireland, N., andStoneman, P.,Technological Diffusion Expectations and Welfare, Oxford Economic Papers, Vol. 38, pp. 283–304, 1986.
Mehrez, A.,A Note on the Analysis of The Expected Value of Perfect Information with Respect to a Class of R&D Projects, European Journal of Operations Research, Vol. 19, pp. 217–221, 1985.
Larréché, J. C., andSrinivasan, B.,Stratport: A Model for the Evaluation and Formulation of Business Portfolio Strategies, Management Science, Vol. 28, pp. 979–1001, 1982.
Author information
Authors and Affiliations
Additional information
Communicated by M. D. Intriligator
Rights and permissions
About this article
Cite this article
Mahrez, A., Berman, N. Maintenance optimal control three-machine replacement model under technological breakthrough expectations. J Optim Theory Appl 81, 591–618 (1994). https://doi.org/10.1007/BF02193102
Issue Date:
DOI: https://doi.org/10.1007/BF02193102