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The paper explores the probabilistic interpretations of algebraic inequalities and presents several findings. First, the inequality of the additive ratios can be used to increase the probability of an event occurring within a set of mutually exclusive and exhaustive events. The interpretation of this inequality produced a counter‐intuitive result, that for suppliers delivering the same quantity of reliable products alongside unknown numbers of unreliable products, the probability of purchasing a reliable product from a randomly selected supplier is higher than the probability of purchasing a reliable product from the market formed by all suppliers. Next, the paper discusses how averaging the reliabilities of components from different varieties can lead to a significant overestimation of the calculated system reliability, as demonstrated by another algebraic inequality interpretation. Finally, the paper derives tight bounds for the reliability of demand in a load‐strength interference model by interpreting the Hayashi's inequality. Notably, these bounds do not depend on the shape of the load distribution and only require the strength distribution to be known in relatively small vicinities of the lower and upper bound of the load. [ABSTRACT FROM AUTHOR]

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]