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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.

A special class of general inequalities has been identified that provides the opportunity for generating new knowledge that can be used for optimising systems and processes in diverse areas of science and technology. It is demonstrated that inequalities belonging to this class can always be interpreted meaningfully if the variables and separate terms of the inequalities represent additive quantities. The meaningful interpretation of a new algebraic inequality based on the proposed general class of inequalities led to developing a light-weight design for a supporting structure based on cantilever beams, reducing the maximum force upon impact, generating new knowledge about the deflection of elastic elements connected in parallel and series and optimising the allocation of resources to maximise expected benefit. The interpretation of the new inequality yielded that the deflection of elastic elements connected in parallel is at least n^2 times smaller than the deflection of the same elastic elements connected in series, irrespective of the individual stiffness values of the elastic elements. The interpretation of another algebraic inequality from the proposed general class led to a method for decreasing the stiffness of a mechanical assembly by cyclic permutation of the elastic elements building the assembly. The analysis showed that a decrease of stiffness exists only if asymmetry of the stiffness values in the connected elements is present.

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.

The deliberate weaknesses are points of weakness towards which a potential failure is channelled in order to limit the magnitude of the consequences from failure. The paper shows that reducing risk by deliberate weaknesses is a powerful domain-independent method which transcends mechanical engineering and works in various unrelated areas of human activity. A classification has been proposed of categories and classes of deliberate weaknesses reducing risk as well as discussion related to the underlying mechanisms of risk reduction. It is shown that introducing and repositioning existing weaknesses is an effective risk-reduction strategy which transcends engineering and can be applied in many unrelated domains. The paper shows that in the case where the cost of failure of the separate components in a system varies significantly, an approach based on deliberate weaknesses has a significant advantage to the equal-reliability/equal-strength design approach.

This chapter summarizes general guidelines on risk management. The common approach to risk reduction is the domain‐specific approach which relies heavily on root cause analysis and detailed knowledge from the specific domain. The domain‐specific approach to risk reduction created an illusion: that efficient risk reduction can be delivered successfully solely by using methods offered by the specific domain without resorting to general methods for risk reduction. The direct consequence of this illusion is that many industries have been deprived from effective risk‐reducing strategy and reliability improvement solutions. A common approach to reliability improvement is to select a statistical‐based, data‐driven approach. To overcome the major deficiency of the data‐driven approach, the chapter discusses the physics‐of‐failure approach. It also argues that many of the principles for technical risk reduction with general validity are rooted in the reliability and risk theory and cannot possibly be deduced from the general inventive principles formulated in TRIZ.

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 from 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 such 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 totally 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 even mathematics. The domain-independent risk reduction methods presented in this paper promote building products and systems characterised by high-reliability and resilience.

The paper 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 paper demonstrates the application of inequalities to reduce the risk of failure by producing tight uncertainty bounds for properties and risk-critical parameters. Numerous applications of the upper-bound-variance inequality have been demonstrated in bounding uncertainty from multiple sources, among which is the estimation of uncertainty in setting positioning distance and increasing the robustness of electronic devices. The rearrangement inequality has been used to maximise the reliability of components purchased from suppliers. 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 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.

The method of self-reinforcement has been introduced as a domain-independent method for improving reliability and reducing risk. A key feature of self-reinforcement is that increasing the external/internal forces intensifies the system‘s response against these forces. As a result, the driving net force towards precipitating failure is reduced. In many cases, the self-reinforcement mechanisms achieve remarkable reliability increase at no extra cost. Two principal ways of self-reinforcement have been identified: reinforcement by capturing a proportional compensating factor and reinforcement by using feedback loops. Mechanisms of transforming forces and motion into self-reinforcing response have been introduced and demonstrated through appropriate examples. Mechanisms achieving selfreinforcement response by self-aligning, self-anchoring and modified geometry have also been introduced For the first time, the potential of positive feedback loops to achieve self-reinforcement and risk reduction was demonstrated. In this respect, it is shown that self-energizing, fast growth and fast transition provided by positive feedback loops can be used with success for achieving reliability improvement. Finally, a classification was proposed of methods and techniques for reliability improvement and risk reduction based on the method of self-reinforcement.

The paper treats the important problem related to risk controlled by the simultaneous presence of critical events, randomly appearing on a time interval and shows that the expected time fraction of simultaneously present events does not depend on the distribution of events durations. In addition, the paper shows that the probability of simultaneous presence of critical events is practically insensitive to the distribution of the events durations. These counter-intuitive results provide the powerful opportunity to evaluate the risk of overlapping of random events through the mean duration times of the events only, without requiring the distributions of the events durations or their variance. A closed-form expression for the expected fraction of unsatisfied demand for random demands following a homogeneous Poisson process in a time interval is introduced for the first time. In addition, a closed-form expression related to the expected time fraction of unsatisfied demand, for a fixed number of consumers initiating random demands with a specified probability, is also introduced for the first time. The concepts stochastic separation of random events based on the probability of overlapping and the average overlapped fraction are also introduced. Methods for providing stochastic separation and optimal stochastic separation achieving balance between risk and cost of risk reduction are presented.

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.

The paper introduces the method of separation for improving reliability and reducing technical risk and provides insight into the various mechanisms through which the method of separation attains this goal. A comprehensive classification of techniques for improving reliability and reducing risk, based on the method of separation has been proposed for the first time. From this classification, three principal categories of separation techniques have been identified: (i) assuring distinct functions/properties/behaviour for distinct components or parts (ii) assuring distinct properties/behaviour at distinct time, value of a parameter, conditions or scale and (iii) distancing risk-critical factors. The concept ‘stochastic separation’ of random events and methods for providing a stochastic separation have been introduced. It is shown that separation of properties is an efficient technique for compensating the drawbacks associated with a selection based on homogeneous properties. It is also demonstrated that the method of deliberate weak links and the method of segmentation can be considered as a special case of the method of separation. Finally, the paper demonstrates that the traditional reliability measure ‘safety margin’ is misleading and should not be used as a measure of the relative separation between load and strength.

Repairable flow networks are a new area of research, which analyzes the repair and flow disruption caused by failures of components in static flow networks. This book addresses a gap in current network research by developing the theory, algorithms and applications related to repairable flow networks and networks with disturbed flows. The theoretical results presented in the book lay the foundations of a new generation of ultra-fast algorithms for optimizing the flow in networks after failures or congestion, and the high computational speed creates the powerful possibility of optimal control of very large and complex networks in real time. Furthermore, the possibility for re-optimizing the network flows in real time increases significantly the yield from real production networks and reduces to a minimum the flow disruption caused by failures. The potential application of repairable flow networks reaches across many large and complex systems, including active power networks, telecommunication networks, oil and gas production networks, transportation networks, water supply networks, emergency evacuation networks, and supply networks. The book reveals a fundamental flaw in classical algorithms for maximising the throughput flow in networks, published since the creation of the theory of flow networks in 1956. Despite the years of intensive research, the classical algorithms for maximising the throughput flow leave highly undesirable directed loops of flow in the optimised networks. These flow loops are associated with wastage of energy and resources and increased levels of congestion in the optimised networks.Includes theory and practical examples to build a deep understanding of the issuesWritten by the leading scholar and researcher in this emerging fieldFeatures powerful software tools for analysis, optimization and control of repairable flow networks

The paper raises awareness of the presence of closed parasitic flow loops in the solutions of published algorithm for maximising the throughput flow in networks. If the rooted commodity is interchangeable commodity, a closed parasitic loop can effectively be present even if the routed commodity does not physically travel along a closed loop. The closed parasitic flow loops are highly undesirable loops of flow, which effectively never leave the network. Parasitic flow loops increase the cost of transportation of the flow unnecessarily, consume residual capacity from the edges of the network, increase the likelihood of deterioration of perishable products, increase congestion and energy wastage. Accordingly, the paper presents a theoretical framework related to parasitic flow loops in networks. By using the presented framework, it is demonstrated that the probability of existence of closed and dominated flow loops in networks is surprisingly high. The paper also demonstrates that the successive shortest path strategy for minimising the total length of transportation routes from multiple interchangeable origins to multiple destinations fails to minimise the total length of the routes. It is demonstrated that even in a network with multiple origins and a single destination, the successive shortest path strategy could still fail to minimise the total length of the routes. By using the developed theoretical framework, it is shown that a minimum total length of the transportation routes in a network with multiple interchangeable origins, is attained if and only if no closed parasitic flow loops and dominated flow loops exist in the network. Accordingly, an algorithm for minimising the total length of the transportation routes by eliminating all dominated parasitic flow loops is proposed.

The paper introduces new domain-independent methods for improving reliability and reducing risk based on algebraic inequalities and chain-rule segmentation. Two major advantages of algebraic inequalities for reducing risk have been demonstrated: (i) ranking risky prospects in the absence of any knowledge related to the individual building parts and (ii) reducing the variability of a risk-critical critical output parameter. The paper demonstrates a highly counter-intuitive result derived by using inequalities: if no information about the component reliability characterising the individual 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 paper also demonstrates the benefits from combining domain-independent methods and domain-specific knowledge for achieving risk reduction in several unrelated domains: decision-making, manufacturing, strength of components and kinematic analysis of complex mechanisms. In this respect, the paper introduces the chain rule segmentation method and applies it to reduce the risk of computational errors in kinematic analysis of complex mechanisms. The paper also demonstrates that combining the domain-independent method of segmentation and domain-specific knowledge in stress analysis leads to a significant reduction of the internal stresses and reduction of the risk of overstress failure.