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.

中文翻译：

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关于使用代数不等式提高可靠性和降低风险的两种基本方法

本文介绍了两种通过使用非平凡的代数不等式来提高可靠性和降低风险的基本方法：（a）通过证明从真实系统或过程得出或推测的不等式；（b）对与之相关的现有非平凡的抽象不等式进行有意义的解释真实的系统或过程。代数不等式的强大优势在于，它们在不掌握任何有关控制变量变化的知识的情况下，能够产生与可靠性至关重要的设计参数有关的紧密边界。第一种方法的有效性已通过与在高度不确定性下进行决策相关的示例以及与在其可靠性未知的组件上建立的排名系统相关的示例得到了证明。为了演示第二种方法，对于不等式，已经创建了有意义的解释，这是柯西-舒瓦兹不等式的特例。通过改变变量的解释，对于串联和并联布置的弹性元件，电阻器和电容器，具有相同的不等式。该论文还表明，对超加和不加不等式的有意义的解释可以成功地用于优化各种系统和过程。对超加和不加不等式的有意义的解释已用于最大化指定总位移下的存储弹性应变能，并优化投资收益。最后，对代数不等式的有意义的解释已用于减少不确定性和关于顺序随机事件的大小排名的错误预测的风险。