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New results related to maximizing the reliability of common systems with interchangeable redundancies at a component level have been obtained by using the method of algebraic inequalities. It is shown that for systems with independently working components with interchangeable redundancies, the system reliability corresponding to a symmetric arrangement of the redundant components is always inferior to the system reliability corresponding to an asymmetric arrangement of the redundant components, irrespective of the probabilities of failure of the different types of components. It is also shown that for series–parallel systems, the system reliability is maximized by arranging the main components in ascending order of their probabilities of failure, whereas the redundant components are arranged in descending order of their probabilities of failure. Finally, this article derives rigorously the highly counterintuitive result that if two components must be selected from n batches containing reliable and faulty components with unknown proportions, the likelihood that both components will be reliable is maximized by selecting both components from a randomly selected batch.

The reverse engineering of a valid algebraic inequality often leads to a projection of a novel physical reality characterized by a distinct signature: the algebraic inequality itself. This paper uses reverse engineering of valid algebraic inequalities for generating new knowledge and substantially improving the reliability of common series-parallel systems. Our study emphasizes that in the case of series-parallel systems with interchangeable redundant components, the asymmetric arrangement of components always leads to higher system reliability than a symmetric arrangement. This finding remains valid, irrespective of the particular reliabilities characterizing the components. Next, the paper presents novel system reliability inequalities whose reverse engineering enabled significant enhancement of the reliability of series-parallel systems with asymmetric arrangements of redundant components, without knowledge of the individual component reliabilities. Lastly, the paper presents a new technique for validating complex algebraic inequalities associated with series-parallel systems. This technique relies on permutation of variable values and the method of segmentation.

—The paper examines the profound impact on the forecasted system reliability when one assumes average reliabilities on demand for components of various kinds but of the same type. In this paper, we use reverse engineering of a novel algebraic inequality to demonstrate that the prevalent practice of using average reliability on demand for components of the same type but different varieties to calculate system reliability on demand is fundamentally flawed. This approach can introduce significant errors due to the innate variability of components within a given type. Additionally, the paper illustrates the optimization of engineering processes using reverse engineering of sub-additive algebraic inequalities based on concave power laws. Employing reverse engineering on these sub-additive inequalities has paved the way for strategies that enhance the performance of diverse industrial processes. The primary advantage of these subadditive inequalities lies in their simplicity, rendering them particularly suitable for reverse engineering.

The paper reveals that a prediction of system reliability on demand based on average reliabilities on demand of components is a fundamentally flawed approach. A physical interpretation of algebraic inequalities demonstrated that assuming average component reliabilities on demand entails an overestimation of the system reliability on demand for systems with components logically arranged in series and series-parallel and underestimation of the reliability on demand for systems with components logically arranged in parallel. The key reason for these discrepancies is the variability of components from the same type. Techniques for countering variability by promoting asymmetric response through inversion have also been introduced. The paper demonstrates that variability during assembly operations can affect negatively the reliability of mechanical systems. Accordingly, techniques for reducing the variability of stresses during assembly operations have been discussed. Finally, the paper provides a discussion related to the reasons for the relatively slow adoption of domain-independent methods for improving reliability despite their numerous advantages.

The paper demonstrates a new domain-independent method for improving reliability and reducing risk including two fundamental approaches: the forward approach, based on deriving algebraic inequalities from real systems and processes; and the inverse approach, based on deriving new knowledge by meaningful interpretation of existing correct algebraic inequalities. The forward approach has been used to prove the domain-independent principle of the well-ordered systems which are characterised by the smallest possible risk of failure. The inverse approach has been used to generate new knowledge related to the relationship of the equivalent elastic constants of elements arranged in series and parallel and the upper and lower bounds of the percentage of faulty components in pooled batches of components with unknown sizes.

The paper introduces a method for increasing the impact of additive quantities by meaningful interpretation of multivariate sub-additive and super-additive functions. The paper demonstrates that the segmentation of additive quantities through sub-additive and super-additive functions can be used to generate new knowledge and optimise systems and processes and the presented algebraic inequalities are applicable to any area of science and technology. The meaningful interpretation of the modified Cauchy-Schwarz inequality, led to a method for increasing of the power output from a voltage source and to a method for increasing the capacity for absorbing strain energy of loaded mechanical components. It was found that the existence of asymmetry is essential to increasing the strain energy absorbing capacity and the power output. Loaded elements experiencing the same displacement do not yield an increase of the absorbed strain energy. Similarly, loaded resistances experiencing the same current do not yield an increase of the power output. Finally, the meaningful interpretation of an algebraic inequality in terms of potential energy, resulted in a general necessary condition for minimising the sum of powers of distances to a fixed number of points in space.

The paper demonstrates for the first time uncertainty reduction and attaining superior performance through segmentation based on algebraic inequalities. Meaningful interpretation of algebraic inequalities has been used for generating new knowledge in unrelated application domains. Thus, the method of segmentation through an abstract inequality led to a new theorem related to electrical circuits. The power output from a source with particular voltage, on elements connected in series, is smaller than the total power output from the segmented sources applied to the individual elements. Segmentation attained through the same abstract inequality led to another new theorem related to electrical capacitors. The energy stored by a charge of given size on a single capacitor is smaller than the total energy stored in multiple capacitors with the same equivalent capacity, by segmenting the initial charge over the separate capacitors. Finally, inequalities based on sub-additive and superadditive functions have been introduced for reducing uncertainty and obtaining superior performance by a segmentation or aggregation of controlling factors. By a meaningful interpretation of sub-additive and super-additive inequalities, superior performance has been achieved for processes described by a powerlaw dependence.

The paper discusses applications of the domain-independent method of algebraic inequalities, for reducing uncertainty and risk. Algebraic inequalities have been used for revealing the intrinsic reliability of competing systems and ranking the systems in terms of reliability in the absence of knowledge related to the reliabilities of their components. An algebraic inequality has also been used to establish the principle of the well-ordered parallel-series systems which, in turn, has been applied to maximize the reliability of common parallel-series systems. The paper introduces linking an abstract inequality to a real process by a meaningful interpretation of the variables entering the inequality and its left- and right-hand parts. The meaningful interpretation of a simple algebraic inequality led to a counterintuitive result. If two varieties of items are present in a large batch, the probability of selecting randomly two items of different variety is smaller than the probability of selecting randomly two items of the same variety.

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.