A Powerful Domain-Independent Method for Risk and Uncertainty Reduction Based on Algebraic Inequalities
Reliability and risk assessments in the case of uncertainty related to the values of risk-critical parameters still present a great challenge to reliability engineering, risk management and decision making. In many cases, the actual values of the risk-critical parameters (e.g. material properties, dimensions, loads, magnitudes of the consequences of failure) are unknown or are associated with large uncertainty. However, almost all existing reliability analysis tools require reliability data which are unavailable at the design stage, which makes it difficult to compare the performance of different design solutions.
The aim of this book is to introduce a new, powerful domain-independent method for improving reliability and reducing uncertainty and risk, based on algebraic inequalities.
This aim will be achieved by demonstrating the capabilities of algebraic inequalities for (i) revealing the inherent reliability of systems and processes and ranking them in terms of reliability and risk in the absence of knowledge related to the reliabilities of their building parts; (ii) reducing aleatory and epistemic uncertainty; (iii) obtaining tight upper and lower bounds of risk-critical parameters and properties; (iv) supporting risk-critical decisions and (v) optimisation to maximise reliability and achieve robust designs.
Another important objective of this book is to demonstrate the benefits from combining the domain-independent method for risk reduction based on advanced algebraic inequalities and domain-specific knowledge in order to achieve effective uncertainty and risk reduction in diverse application domains. In this respect, the book demonstrates simple and effective solutions in such mature fields as electronics, stress analysis, mechanical design, manufacturing, economics and management. Most of these solutions have never been suggested in standard textbooks and research papers, which demonstrates that the lack of knowledge of the domain-independent method of inequalities for reducing risk and uncertainty made these simple solutions invisible to domain experts.
A formidable advantage of algebraic inequalities is that they do not require knowledge related to the distributions of the variables entering the inequalities. This makes the method of algebraic inequalities ideal for handling deep uncertainty associated with components, properties and control parameters and for ranking designs in the absence of reliability data related to the separate components. The method of algebraic inequalities does not rely on reliability data or detailed knowledge of physical mechanisms underlying possible failure modes. This is why the method is appropriate for new designs, with no failure history and with unknown failure mechanisms. By proving algebraic inequalities related to the reliabilities of competing systems/processes, the proposed method has the potential to reveal their intrinsic reliability and rank them in terms of reliability in the absence of knowledge related to the reliabilities of their building parts.
Suppose that two different system configurations are built by using the same set of n components with performance characteristics (e.g. reliabilities) xl,x2,...,.r„ that are unknown. Let the performance of the first configuration be given by the function /(xb...,x„) while the performance of the second configuration is given by g(xb...,x„). If inequality of the type
could be proved, this would mean that the performance (e.g. reliability) of the first configuration is intrinsically superior to the performance of the second configuration. Then, the first system configuration can be selected and the risk of failure reduced in the absence of knowledge related to the reliabilities of the parts building the systems. The possibility of revealing the intrinsic reliability of competing systems/processes and making a correct ranking under a deep uncertainty related the reliabilities of their components is a formidable advantage of algebraic inequalities.
Similarly, if a particular order among the reliabilities of the separate components is present, the topology of the two competing systems may be the same but for a particular permutation (arrangement) of the components, the system reliability associated with that particular permutation may be superior to any other permutation.
In both cases, the algebraic inequalities reveal the intrinsic reliability of a particular topology or arrangement.
Another formidable advantage of the algebraic inequalities is their capacity to reduce aleatory and epistemic uncertainty and produce tight upper and lower bounds related to uncertain reliability-critical design parameters such as material properties, electrical parameters, dimensions, loads and component reliabilities. As a result, by establishing tight bounds related to properties and parameters, the method of algebraic inequalities can be applied to improve the robustness of designs, by complying them with the worst possible variation of the design parameters. As a result, a number of failure modes can be avoided.
Yet another advantage of the method based on algebraic inequalities consists of its suitability for minimising deviations of key parameters from their required values. As a result, the proposed method can be applied to reduce the sensitivity of components and systems to variation of dimensions and other parameters thereby enhancing robustness and performance. Algebraic inequalities can also be used for maximising the reliability of systems and improving the robustness of manufacturing processes. These applications are demonstrated in the book with using algebraic inequalities to maximise the reliability of parallel-series systems and to find the optimal design parameters that guarantee the smallest deviation of a risk-critical parameter. Algebraic inequalities, can also be used for maximising the performance of engineering systems and processes and this application is demonstrated with maximising the power obtained from a voltage source.
The method of algebraic inequalities reduces risk at no extra cost or at a low cost, unlike many other methods (e.g. ‘introducing redundancy’, ‘selecting better materials’, ‘strengthening weak links’ and ‘condition monitoring’).
The method of algebraic inequalities is domain-independent because of the domain-independent nature of mathematics. Consequently, the method is demonstrated in such diverse domains as mechanical design, electronic circuits design, project management, economics, decision-making under uncertainty, manufacturing and quality control.
Finally, another formidable advantage of algebraic inequalities is that they admit a meaningful interpretation in terms of uncertainty, reliability and risk which can be attached to a real system or process and used to obtain new physical properties and bounds. To the best of our knowledge, creating meaningful interpretation for existing non-trivial abstract inequalities and attaching it to a real system or process has not yet been explored in the reliability and risk literature. Covering this gap constitutes an important objective of this book.
Algebraic inequalities have been used extensively in mathematics. For a long time, simple inequalities are being used to express error bounds in approximations and constraints in linear programming models (Figure 1.1).
The properties of a number of useful non-trivial algebraic inequalities, such as the arithmetic mean - geometric mean (AM-GM) inequality, Cauchy-Schwarz
FIGURE 1.1 Simple inequality constraints used in linear programming.
inequality, the rearrangement inequality, the Chebyshev inequalities, Jensen inequality, Muirhead inequality, Holder inequality, and so on, have also been well documented (Bechenbach and Bellman, 1961; Kazarinoff, 1961; Engel, 1998; Hardy et al„ 1999; Steele, 2004; Pachpatte, 2005; Sedrakyan and Sedrakyan, 2010; Lugo, 2013).
In reliability and risk research, inequalities have been used exclusively as a mathematical tool for reliability and risk evaluation and for characterisation of reliability functions (Berg and Kesten, 1985; Makri and Psillakis, 1996; Ebeling, 1997; Xie and Lai. 1998; Dohmen, 2006; Hill et ah, 2013; Kundu and Ghosh, 2017). It is important to guarantee that the reliability of a system meets certain minimal expectations and inequalities have been used (Ebeling, 1997) for obtaining lower and upper bounds on the system reliability by using minimal cut sets and minimal path sets. Xie and Lai (1998), for example, used simple conditional inequalities to obtain more accurate approximations for system reliability, instead of the usual minimal cut and minimal cut bounds. By using improved Bonferroni inequalities, the lower and upper bounds of system reliability were derived by Makri and Psillakis (1996). Inequality-based reliability estimates for complex systems have also been proposed by Hill et ah (2013).
In the reliability and risk research, algebraic inequalities have also been used to express relationships between random variables and their transformations to generate insight into the structure of reliability distributions. Simple inequalities with relation to reliability prediction have been used by Berg and Kesten (1985); inequalities involving expectations have been used by Kundu and Ghosh (2017) for characterising some well-known reliability distributions. Well-known inequalities about a random variable X with unknown probability distribution are the Chebyshev inequality and Markov inequality (DeGroot, 1989). These inequalities are related to the probability that the distance of a random value from a specified number or from its mean will be larger than a specified quantity. An inequality related to the probability that the distances between the locations of a fixed number of uniformly distributed points on a segment will be greater than a specified quantity has been proved in Todinov (2002a).
Trivial inequalities, obtained from solving with respect to one of the variables, have been used for specifying the upper bound of the lineal density of Poisson- distributed flaws to guarantee a probability of clustering below a maximum acceptable level (Todinov, 2005, 2006a, 2016).
In reliability and risk research, inequalities have been used exclusively as a tool for reliability and risk evaluation and for characterisation of reliability functions. However, these reliability-related applications of inequalities are very limited and oriented towards measuring the reliability performance of the systems, instead of providing direct input to the design process by reducing uncertainty and improving the reliability of components, systems and processes.
Despite that standard reliability textbooks (Ramakumar, 1993; Lewis, 1996; O’Connor, 2002; Dhillon, 2017; Modarres et al., 2017) do allocate some space on reliability improvement methods such as introducing redundancy, derating, eliminating common cause and condition monitoring, there is a clear lack of discussion related to reducing uncertainty and risk by using algebraic inequalities.
To the best of our knowledge of existing approaches involving inequalities in reliability and risk, the method of algebraic inequalities has not yet been used as a domain-independent uncertainty and risk reduction method. Covering this gap in the available literature is one of the objectives of this book.
Applications of inequalities have also been considered in physics (Rastegin, 2012) and engineering (Cloud et ah, 1998; Samuel and Weir, 1999). However, in the mechanical engineering design literature (Cloud et ah, 1998; French, 1999; Thompson, 1999; Samuel and Weir, 1999; Collins, 2003; Norton, 2006; Pahl et ah, 2007; Childs, 2014; Budinas and Nisbett, 2015; Gullo and Dixon. 2018; Mott et ah, 2018) there is also a lack of discussion on the use of complex algebraic inequalities to improve reliability and reduce risk. In engineering design, the application of inequalities is mainly confined to inequalities linking design variables required to satisfy various design constraints in order to guarantee that the design will perform its required functions (Samuel and Weir, 1999).
Recently, work related to applying advanced algebraic inequalities for uncertainty and risk reduction was published in Todinov (2019c), where a highly counter-intuitive result in decision making under deep uncertainty was obtained by using the Muirhead’s inequality. In Todinov (2019d), an advanced inequality known as ‘upper-bound variance theorem’ has been applied to create a sharp upper bound for the variance of properties from multiple sources. This book extends this work by introducing various applications of the new domain-independent method based on algebraic inequalities for improving reliability and reducing risk and uncertainty.
Classification of Techniques Based on Algebraic Inequalities for Risk and Uncertainty Reduction
The book introduces an important method based on algebraic inequalities for reliability improvement and uncertainty and risk reduction. The method consists of creating relevant meaning for the variables entering the algebraic inequalities and providing a meaningful interpretation of the different parts of the inequality. As a result, the abstract inequality is linked with a real physical system or process.
By using this method, various meaningful interpretations have been made of the same abstract inequality. It has been shown that the equivalent resistance of n resistors arranged in parallel is at least n times smaller than the average resistance of the same resistors, irrespective of the individual values of the resistors. This upper bound is much stronger than the well-known upper bound: the equivalent resistance of n resistors arranged in parallel is smaller than the least resistance. Similar interpretation has been given to the equivalent elastic constant of n elastic elements arranged in series and parallel and to the capacity n capacitors arranged in series and in parallel.
The meaningful interpretation of an abstract algebraic inequality led to a new theorem related to electrical circuits. The power output from a voltage source, on elements arranged in series, is smaller than the total power output from the segmented source where each voltage segment is applied to a separate element.
Finally, an important principle related to algebraic inequalities has been formulated: If a correct algebraic inequality permits a meaningful interpretation related to a real process/experiment, the realisation of the process/experiment must yield results that do not contradict the abstract inequality.
As a domain-independent method, algebraic inequalities reduce risk by (i) revealing the intrinsic reliability of systems and processes and ranking them in terms of reliability and risk; (ii) reducing epistemic uncertainty; (iii) reducing aleatory uncertainty; (iv) maximising system reliability and system performance; (v) minimising the risk of faulty assembly; (vi) providing tight upper and lower bounds for the variation of risk-critical parameters; (vii) providing support for risk-critical decisions and (viii) creating the basis for robust designs and processes.
A classification has been presented in Figure 1.2, which includes the different ways through which algebraic inequalities reduce uncertainty and risk.
Despite the existence of a substantial amount of literature on algebraic inequalities and a significant amount of solved examples (Bechenbach and Bellman, 1961; Kazarinoff, 1961; Engel, 1998; Hardy et al., 1999; Steele, 2004; Pachpatte, 2005; Sedrakyan and Sedrakyan, 2010), there is absence of a systematic exposition of the different techniques through which an advanced algebraic equality can be tested and proved. A good awareness of the most important tools through which an algebraic inequality can be tested and proved significantly increases the benefit from using inequalities in reliability improvement and risk reduction and enhances the power of researchers in conjecturing and proving inequalities.
A number of powerful methods and techniques for testing and proving inequalities have not been discussed in the literature. To the best of our knowledge the method of Monte Carlo simulation for testing a conjectured algebraic inequality has not been discussed in the literature. In some cases, discussion regarding a particular method or technique does exist, but it is insufficient or inadequate. Filling this gap in the current literature determined another objective of the book: to introduce systematically various important techniques through which an inequality can be conjectured and proved.
FIGURE 1.2 Using inequalities for improving reliability and reducing risk and uncertainty.
The application of algebraic inequalities has been demonstrated in various unrelated domains of human activity: reliability engineering, manufacturing, decision making, mechanical engineering design, electric engineering design, project management, economics and business planning, which shows that the method of algebraic inequalities is indeed a domain-independent method for improving reliability and reducing risk and uncertainty.