Summary
The state of a system, which is subject to random failure, is described by a Markov process. Preventive replacements are possible at any stopping time. If the failure rate is not a monotone function, the optimal replacement policy that minimizes long run cost per unit time is in general no control limit rule. Conditions are given, under which the optimal policy can be determined.
Zusammenfassung
Der Zustand eines Systems werde bis zu seinem Ausfall durch einen Markov Prozeß beschrieben. Eine vorsorgliche Wartungserneuerung kann zu einer Stoppzeit durchgeführt werden. Ist die Ausfallrate nicht monoton, so wird die optimale Erneuerungsstrategie nach dem Durchschnittskostenprinzip i. a. keine “control limit rule” sein. Es werden Bedingungen angegeben, unter denen eine optimale Politik bestimmt werden kann.
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References
Abdel-Hameed M (1977) Optimal replacement policies for devices subject to a gamma wear process. In: Tsokos CP, Shimi JN (eds) The theory and applications of reliability. Academic Press, New York, pp 397–412
Barlow R, Proschan F (1967) Mathematical theory of reliability. John Wiley, New York
Bergman B (1978) Optimal replacement under a general failure model. Adv Appl Probab 10:431–451
Bosch K, Jensen U (1983) Instandhaltungsmodelle — Eine Übersicht, Teil 2. OR Spektrum 5:129–148
Cinlar E, Jacod J (1981) Representation of semimartingale Markov processes in terms of Wiener processes and Poisson random measures. In: Cinlar E et al (eds) Seminar on stochastic processes. Birkhäuser, Boston, pp 159–242
Dynkin EB (1965) Markov processes, vol I. Springer, Berlin Göttingen Heidelberg
Feldman RM (1976) Optimal replacement with semi-Markov shock models. J Appl Probab 13:108–117
Feldman RM (1977) Optimal replacement for systems governed by Markov additive shock processes. Ann Probab 5:413–429
Gottlieb G (1982) Optimal replacement for shock models with general failure rate. Oper Res 30:82–92
Nummelin E (1980) A general failure model: optimal replacement with state depencent replacement and failure costs. Math Oper Res 5:381–387
Smith WL (1955) Regenerative stochastic processes. Proc Soc London Ser A 232:6–31
Taylor HM (1975) Optimal replacement under additive damage and other failure models. Naval Res Logist Q 22: 1–18
Yamada K (1980) Explicit formula of optimal replacement under additive shock processes. Stochastic Process Appl 9: 193–208
Zuckerman D (1978) Optimal replacement policy for the case where the damage process is a one-sided Lévy process. Stochastic Process Appl 7:141–152
Zuckerman D (1978) Optimal stopping in a semi-Markov shock model. J Appl Probab 15:629–634
Zuckerman D (1980) Optimal replacement under additive damage and self-restoration. RAIRO 14:115–127
Zuckerman D (1980) Optimal stopping in terminating onesided processes with unbounded reward functions. Z Oper Res Ser A 24:145–153
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Jensen, U. Optimal replacement with non-monotone failure rates. OR Spektrum 6, 53–57 (1984). https://doi.org/10.1007/BF01721253
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DOI: https://doi.org/10.1007/BF01721253