Bivariate replacement policy for a system subject to shocks

ABSTRACT

A bivariate replacement policy for a two-unit system under the shock-effect is studied. Unit 1 is subject to one of the two types of shocks. A type 1 shock causes a minor failure of unit 1, while a type 2 shock makes a catastrophic failure of unit 1. The probabilities of the shock types depend on the number of shocks since the last replacement. Each minor failure of unit 1 brings some cumulative damage to unit 2. When the cumulative damage of unit 2 reaches L, unit 2 fails, which simultaneously causes the failure of unit 1, resulting in a catastrophic failure of system. Moreover, unit 2 with cumulative damage x has a minor failure with probability $\omega \left(x\right)$ at unit 1 failure instant, and this failure is removed by a minimal repair. A more general replacement policy is considered, in which the system is replaced when the cumulative damage to unit 2 has reached l (≤ L), or the nth type 1 shock occurs, or when type 2 shock occurs or the accumulative damage to unit 2 exceeds L, whichever comes first. The expression for average cost rate is developed and the corresponding optimal policy is determined analytically and numerically.

KEYWORDS:

• Minimal repair
• shock damage
• replacement policy
• optimization

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No potential conflict of interest was reported by the author(s).

Additional information

Notes on contributors

Shey-Huei Sheu

Shey-Huei Sheu is a Chair Professor of the Department of Business Administration at the Asia University. He was a Chair Professor of the Industrial Management Department at the National Taiwan University of Science and Technology. He received his M.Sc. (1979) in Applied Mathematics from the National Tsing Hua University, and his Ph.D. (1987) in Statistics from the University of Kentucky. He has published over one hundred and fifty papers in several prestigious journals in the fields of statistics, reliability, operations research, and systems engineering.

Tzu-Hsin Liu

Tzu-Hsin Liu is presently a postdoctoral researcher at the Department of Finance, Chaoyang University of Science and Technology, Taiwan. She received her Masters and PhD degrees in applied mathematics from National Chung-Hsing University, Taiwan, in 2005 and 2011, respectively. Her current research interests include queueing theory, maintenance policy, and statistics.

Wei-Teng Sheu

Wei-Teng Sheu is currently a PhD student at Department of Business Administration, National Taiwan University of Science and Technology. His research interests include in the areas of quality, forecasting methodology, marketing, and E-Commerce.

Jau-Chuan Ke

Jau-Chuan Ke is a Professor of Applied Statistics at the National Taichung University of Science and Technology, Taiwan. He received his Ph.D. degree in Applied Mathematics from National Chung-Hsing University, Taiwan. He is an editorial board member of Applied Mathematical Modelling, Quality Technology & Quantitative Management and Queueing Models and Service Management. His research interests include queueing theory, reliability, statistics, and decision theory.

Zhe-George Zhang

Zhe George Zhang is a Professor of Operations Research in the Department of Decision Sciences at Western Washington University and an Adjunct Professor of Management Science in the Faculty of Business Administration at Simon Fraser University. He received a B.Sc. degree in Computer Engineering from Nankai University, an MBA degree from Schulich School of Business at York University, and a Ph. D. degree in Operations Research from the University of Waterloo. His research interests include queuing theory and applications, stochastic dynamic programming, probability models in reliability, and supply chain management issues.