Improving reliability and reducing risk by using inequalities

Abstract

This article introduces a powerful domain-independent method for improving reliability and reducing risk based on algebraic inequalities which transcends mechanical engineering and can be applied in many unrelated domains. The article demonstrates the application of inequalities to reduce the risk of failure by producing sharp uncertainty bounds for properties and risk-critical parameters. Applications of the upper-bound-variance inequality have been demonstrated in bounding uncertainty from multiple sources. With the help of the rearrangement inequality, a highly counter-intuitive result has been obtained. If no information about the component reliability characterising the individual, independent suppliers is available, purchasing components from a single supplier or from the smallest possible number of suppliers maximises the probability of a high-reliability assembly. The Cauchy–Schwartz inequality has been applied for determining sharp bounds of mechanical properties and the Chebyshev's inequality for determining a lower bound for the reliability of an assembly. The inequality of the inversely correlated random events has been introduced and applied for ranking risky prospects involving units with unknown probabilities of survival.