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The paper introduces a powerful method for developing lightweight designs and enhancing the load-bearing capacity of common structures. The method, referred to as the 'method of aggregation', has been derived from reverse engineering of sub-additive and super-additive algebraic inequalities. The essence of the proposed method is consolidating multiple elements loaded in bending into a reduced number of elements with larger cross sections but a smaller total volume of material. This procedure yields a huge reduction in material usage and is the first major contribution of the paper. For instance, when aggregating eight load-carrying beams into two beams supporting the same total load, the material reduction was more than 1.58 times. The second major contribution of the paper is in demonstrating that consolidating multiple elements loaded in bending into a reduced number of elements with larger cross sections but the same total volume of material leads to a big increase in the load-bearing capacity of the structure. For instance, when aggregating eight cantilevered or simply supported beams into two beams with the same volume of material, the load-bearing capacity until a specified tensile stress increased twice. At the same time, the load-bearing capacity until a specified deflection increased four times. [ABSTRACT FROM AUTHOR]

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]

Problems with the current methods for reliability improvement are discussed as well as two generic methods for reliability improvement. The paper argues that reliability improvement is underpinned by common principles that provide key input to the design process. The domain-independent methods change the way engineers and scientists approach reliability improvement. The presented generic methods encourage simple low-cost solutions as opposed to some traditional high-cost solutions based on introducing redundancy, condition monitoring, reinforcement and use of expensive materials. The domain-independent methods allow engineers and scientists in a particular domain to access excellent solutions and practices for eliminating failure modes in other domains. In this way, the constant 'reinventing of the wheel' is avoided. As part of the presented approach, a generic method for increasing reliability by increasing the level of balancing and by substitution have been presented. In addition, a new classification of techniques related to increasing the level of balancing has been introduced and discussed for the first time. The paper also proves rigorously 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 maximised by selecting the components from a randomly selected batch. [ABSTRACT FROM AUTHOR]

A method for optimising the design of systems and processes has been introduced that consists of interpreting the left- and the right-hand side of a correct algebraic inequality as the outputs of two alternative design configurations delivering the same required function. In this way, on the basis of an algebraic inequality, the superiority of one of the configurations is established. The proposed method opens wide opportunities for enhancing the performance of systems and processes and is very useful for design in general. The method has been demonstrated on systems and processes from diverse application domains. The meaningful interpretation of an algebraic inequality based on a single-variable sub-additive function led to developing a light-weight design for a supporting structure based on cantilever beams. The interpretation of a new algebraic inequality based on a multivariable sub-additive function led to a method for increasing the kinetic energy absorbing capacity during inelastic impact. The interpretation of a new inequality has been used for maximising the mass of deposited substance during electrolysis. [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]

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]

We propose a framework for analysis and optimization of repairable flow networks by (i) stating and proving the maximum flow minimum flow path resistance theorem for networks with merging flows (ii) a discrete-event solver for determining the variation of the output flow from repairable flow networks with complex topology (iii) a procedure for determining the threshold flow rate reliability for repairable networks with complex topology (iv) a method for topology optimization of repairable flow networks and (v) an efficient algorithm for maximizing the flow in non-reconfigurable flow networks with merging flows. Maximizing the flow in a static flow network does not necessarily guarantee that the flow in the corresponding non-reconfigurable repairable network will be maximized. In this respect, we introduce a new concept related to repairable flow networks: ‘a specific resistance of a flow path’ which is essentially the average percentage of losses from component failures for a flow path from the source to the sink. A very efficient algorithm based on adjacency arrays has also been proposed for determining all minimal flow paths in a network with complex topology and cycles. We formulate and prove a fundamental theorem about non-reconfigurable repairable flow networks with merging flows. The flow in a repairable flow network with merging flows can be maximized by preferentially saturating directed flow paths from the sources to the sink, characterized by the largest average availability. The procedure starts with the flow path with the largest average availability (the smallest specific resistance), and continues by saturating the unsaturated directed flow path with the largest average availability until no more flow paths can be saturated. A discrete-event solver for reconfigurable repairable flow networks with complex topology has also been constructed. The proposed discrete-event solver maximizes the flow rate in the network upon each component failure and return from repair. By maximizing the flow rate upon each component failure and return from repair, the discrete-event solver ensures a larger total output flow during a specified time interval. The designed simulation procedure for determining the threshold flow rate reliability is particularly useful for comparing flow network topologies, and selecting the topology characterized by the largest threshold flow rate reliability. It is also very useful in deciding whether the resources allocated for purchasing extra redundancy are justified. Finally, we propose a new optimization method for determining the network topology combining a maximum output flow rate attained within a specified budget for building the network. The optimization method is based on a branch and bound algorithm combined with pruning the full-complexity network as a way of exploring the possible repairable networks embedded in the full-complexity network. [ABSTRACT FROM AUTHOR]

The exact upper bound of the variance of properties from multiple sources is attained from sampling not more than two sources. This paper discusses important applications of this result referred to as variance upper bound theorem. A new conservative, non-parametric estimate has been proposed for the capability index of a process whose output combines contributions from multiple sources of variation. A new method for assessing and increasing the robustness of processes, operations and products where the mean value can be easily adjusted or is not critical has been presented, based on the variance upper bound theorem. We show that the worst-case variation of a property from multiple sources, obtained by using the variance upper bound theorem, can be used as a basis for developing robust engineering designs and products. If a design is capable of accommodating the worst-case variation of the reliability-critical parameters, it will also be capable of accommodating the variation of the reliability-critical parameters from any combination of sources of variation and mixing proportions. In this respect, a new algorithm for virtual testing based on the variance upper bound theorem has been proposed for determining the probability of a faulty assembly from multiple sources. For sources of variation that can be removed, the robustness can be improved further, by removing the source that yields the largest decrease in the variance upper hound. Consequently, the correspondent algorithm is also presented. A number of engineering applications have been discussed where the variance upper bound theorem can be used to assess and increase the robustness of mechanical and electrical components, manufacturing processes and operations. [ABSTRACT FROM AUTHOR]