Combination of Qualitative and Quantitative Sources of Knowledge for Risk Assessment in the Framework of Possibility Theory^*

Abstract

This paper focuses on a representation of system reliability in the framework of possibility theory. Particularly, given a (probabilistic) quantitative knowledge pertaining to the time to failure of a system (risk function) and some qualitative knowledge about the degree of pessimism and optimism of the information supplied by the quantitative knowledge, one constructs a possibilistic reliability function of the system. The latter models the possibility of the system to survive up to the current time t. The proposal is motivated by the observation that the system may fail even if it is considered unlikely by the (probabilistic) risk function. Besides, research from cognitive science shows that probability values, particularly when a human factor is involved, tend to be overestimated or underestimated depending on the level of confidence the user is facing. This methodology involves implicitly the combination of the two pieces of knowledge into a more refined knowledge expressed in the possibility framework. Rational assumptions are put forward in order to guide the construction of the possibilistic model. Several ramifications of the proposal, particularly considering special exponential lifetime distributions will be investigated. Particularly, when reasoning in average, or looking for typical elements, new results on (possibilistic) mean time to failure (PMTF) will be pointed out. On the other hand the absence of any qualitative knowledge makes the proposal as a counterpart of the so-called probability–possibility transformations investigated in fuzzy/possibility literature. Comparisons with the preceding will be investigated for special cases of exponential lifetime distributions.

Keywords:

• Risk
• lifetime distribution
• possibility theory
• probability–possibility transformations

Acknowledgements

This work is supported by DIRC inter-disciplinary research project on dependability of human-computer organizations funded by British EPSRC, which is gratefully acknowledged.

Notes

After a period as an operational scientist in industry, Martin Newby joined Bradford University as a lecturer in industrial technology and later as a lecturer in operational research and statistics. From 1988 to 1995 he was Associate Professor of Industrial Engineering at the Technological University Eindhoven, and was also a member of The Frits Philips Institute for Quality Management. Since 1995 he has been with City University as Professor of Statistical Science. He is active in national and international professional bodies in statistics and reliability and is adviser to the Committee on Defense Equipment Reliability and Maintenance in the UK.

Mourad Oussalah obtained a Masters degree in Electronics from the Polytechnic National Institute of Algiers in 1992, and a D.E.A from the University of Paris XII in 1994. In 1998, he obtained a Ph.D. in robotics and data fusion at the University of Evry Val Essonnes, France. He worked as an assistant professor from 1996 to 1998 at Evry University. In October 1998, he joined Katolieke Universiteit Leuven (Belgium) as a research fellow in the project Active Sensing for Intelligent Machines (1999–2004) where he applied new artificial intelligence tools to robotics tasks. In January 2001, he joined CSR as a research fellow to work on software dependability within the DIRC project. Since 2003, he has been a lecturer at the University of Birmingham. His topics of interests include data fusion, reliability analysis, uncertainty handling in the framework of probability, possibility and evidence theories, random sets, and robotics.

^‡A different expression was used in Heilpern (1992), but it can easily be proven that the two expressions are equivalent.

^*This paper is an augmented version of the paper by Oussalah et al. (2001) presented by the authors at the Eusflat 2001 Conference, Leicester, UK.