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The paper introduces two fundamental approaches for reliability improvement and risk reduction by using nontrivial algebraic inequalities: (a) by proving an inequality derived or conjectured from a real system or process and (b) by creating meaningful interpretation of an existing nontrivial abstract inequality relevant to a real system or process. A formidable advantage of the algebraic inequalities can be found in their capacity to produce tight bounds related to reliability‐critical design parameters in the absence of any knowledge about the variation of the controlling variables. The effectiveness of the first approach has been demonstrated by examples related to decision‐making under deep uncertainty and examples related to ranking systems built on components whose reliabilities are unknown. To demonstrate the second approach, meaningful interpretation has been created for an inequality that is a special case of the Cauchy‐Schwarz inequality. By varying the interpretation of the variables, the same inequality holds for elastic elements, resistors, and capacitors arranged in series and parallel. The paper also shows that meaningful interpretation of superadditive and subadditive inequalities can be used with success for optimizing various systems and processes. Meaningful interpretation of superadditive and subadditive inequalities has been used for maximizing the stored elastic strain energy at a specified total displacement and for optimizing the profit from an investment. Finally, meaningful interpretation of an algebraic inequality has been used for reducing uncertainty and the risk of incorrect prediction about the magnitude ranking of sequential random events. [ABSTRACT FROM AUTHOR]

The popular domain-specific approach to risk reduction created the illusion that efficient risk reduction can be delivered successfully solely by using methods offered by the specific domain. As a result, many industries have been deprived of efficient risk reducing strategy and solutions. This paper argues that risk reduction is underlined by domain-independent methods and principles which, combined with knowledge from the specific domain, help to generate effective risk reduction solutions. In this respect, the paper introduces a powerful method for reducing the likelihood of computational errors based on combining the domain-independent method of segmentation and local knowledge of the chain rule for differentiation. The paper also demonstrates that lack of knowledge of domain-independent principles for risk reduction misses opportunities to reduce the risk of failure even in a mature field like stress analysis. The domain-independent methods for risk reduction do not rely on reliability data or knowledge of physical mechanisms underlying possible failure modes and are particularly well suited for developing new designs, with unknown failure mechanisms and failure history. In many cases, the reliability improvement and risk reduction by using the domain-independent methods reduces risk at no extra cost or at a relatively small cost. The presented domain-independent methods work across unrelated domains and this is demonstrated by the supplied examples which range from various areas of engineering and technology, computer science, project management, health risk management, business and mathematics. The domain-independent risk reduction methods presented in this paper promote building products and systems characterised by high-reliability and resilience. [ABSTRACT FROM AUTHOR]

The paper provides for the first time a comprehensive introduction into the mechanisms through which the method of separation achieves risk reduction and into the ways it can be implemented in engineering designs. The concept stochastic separation of critical random events on a time interval, which consists of guaranteeing with a specified probability a specified degree of distancing between the random events, is introduced. Efficient methods for providing stochastic separation by reducing the duration times of overlapping critical random events on a time interval are presented. The paper shows that the probability of overlapping of critical events, randomly appearing on a time interval, is practically insensitive to the distribution of their duration times and to the variance of the duration times as long as the mean of the duration times remains the same. A rigorous proof is presented that this statement is valid even for two random events on a time interval. The paper also provides insight into various mechanisms through which deterministic separation improves reliability and reduces risk. It is demonstrated that the separation on properties is an efficient technique for compensating the drawbacks associated with homogeneous properties. It is demonstrated that improving reliability by including redundancy, improving reliability by segmentation and some of the deliberate weak link techniques and stress limiters techniques for reducing risk are effectively special cases of a deterministic separation. Finally, the paper demonstrates that in a number of cases, the way to extract benefit from the method of separation is to build and analyse a mathematical model based on the method of separation. A comprehensive classification of the discussed methods for stochastic and deterministic separation is also presented. [ABSTRACT FROM AUTHOR]