Chapter 15 describes the use of the Bayesian network (BN) methodology for reliability assessment and updating of structural and infrastructure systems. A brief review of the BN as a graphical representation of random variables and an efficient framework for encoding their joint distribution and its updating upon observations is presented. D-separation rules describing the flow of information within the network upon observation of random variables are described and methods are presented for discretizing continuous random variables, thus allowing the use of efficient algorithms applicable to BNs with discrete nodes. Efficient BN models for components, systems, random fields, and seismic hazard are developed. For time- or space-variant problems, the dynamic Bayesian network is introduced. This model is used in conjunction with structural reliability methods (FORM, SORM, simulation) to develop enhanced BNs to solve reliability problems for structures under time-varying loads. Detailed examples are presented, including post-earthquake risk assessment of a spatially distributed infrastructure system and reliability assessment of a deteriorating structure under stochastic loads. The chapter concludes with a discussion of the potential of the BN as a tool for near-real-time risk assessment and decision support for constructed facilities, and the need for further research and development to realize this potential.

Chapter 5 presents methods for assessing structural reliability under incomplete probability information, i.e., when complete distributional information on the basic random variables is not available. First, second-moment methods are presented where the available information is limited to the means, variances, and covariances of the basic random variables. These include the mean-centered first-order second-moment (MCFOSM) method, the first-order second-moment (FOSM) method, and the generalized second-moment method. These methods lead to approximate computations of the reliability index as a measure of safety. Lack of invariance of the MCFOSM method relative to the formulation of the limit-state function is demonstrated. The FOSM method requires finding the “design point,” which is the point in a transformed standard outcome space that has minimum distance from the origin. An algorithm for finding this point is presented. Next, methods are presented that incorporate probabilistic information beyond the second moments, including knowledge of higher moments and marginal distributions. Last, a method is presented that employs the upper Chebyshev bound for any given state of probability information. The chapter ends with a discussion of the historical significance of the above methods as well as their shortcomings and argues that they should no longer be used in practice.

This chapter examines two fundamental issues regarding the nature of firms. First, why are they necessary in the business environment? Second, what are their objectives? Addressing these issues involves various aspects of theory which are not always associated with economics: transaction cost theory, property rights theory, motivation theory, information theory and agency theory. Regarding the first issue, the necessity for the existence of firms may appear to be self-evident, but on closer examination we can see that many transactions can be performed between individuals without firms existing at all. The problem is that with complex activities the transaction cost of engagement as individuals can be high, whereas internalizing transactions within firms can reduce this cost. The second issue regarding objectives begins with the concept of profit maximization, and then examines the various assumptions underlying it. Various problem areas related to these assumptions are identified, in particular: the existence of agency problems, the measurement of profit, risk and uncertainty, and multi-product firms. The impact of these problems on firms’ objectives is discussed.

This topic examines the nature of game theory, why it is relevant for managerial decision making, and how it determines decisions. The starting point is an explanation of the nature of game theory in terms of the inter-dependence of decision making, and its wide range of applications in real life. Different types of game and their elements are described. The prisoner’s dilemma illustrates some of the counterintuitive aspects of game theory. Static and dynamic games are analysed, and the different types of equilibrium: dominant strategy equilibrium, iterated dominant strategy equilibrium, Nash equilibrium, subgame perfect Nash equilibrium and mixed strategy equilibrium. Cournot, Bertrand and Stackelberg types of oligopoly and their strategy implications are analysed, and comparisons are drawn between them and with perfect competition and monopoly. Games with uncertain outcomes and repeated games are discussed, along with commitment strategies and credibility. Limitations of standard game theory are discussed, such as the existence of bounded rationality and social preferences. Aspects of behavioural game theory are introduced to account for these factors.

Chapter 7 describes the second-order reliability method (SORM), which employs a second-order approximation of the limit-state surface fitted at the design point in the standard normal space. Three distinct SORM approximations are presented. The classical SORM fits the second-order approximating surface to the principal curvatures of the limit-state surface at the design point. This approach requires computing the Hessian (second-derivative matrix) of the limit-state function at the design point and its eigenvalues as the principal curvatures. The second approach computes the principal curvatures iteratively in the process of finding the design point. This approach requires only first-order derivatives of the limit-state function but repeated solutions of the optimization problem for finding the design point. One advantage is that the principal curvatures are found in decreasing order of magnitude and, hence, the computations can be stopped when the curvature found is sufficiently small. The third approach fits the approximating second-order surface to fitting points in the neighborhood of the design point. This approach also avoids computing the Hessian. Furthermore, it corrects for situations where the curvature is zero but the surface is curved, e.g., when the design point is an inflection point of the surface. Results from the three methods are compared numerically.

In human perception, the role of sparse representation has been studied extensively. As we have alluded to in the Introduction, Chapter 1, investigators in neuroscience have revealed that in both low-level and mid-level human vision, many neurons in the visual pathway are selective for recognizing a variety of specific stimuli, such as color, texture, orientation, scale, and even view-tuned object images [OF97, Ser06]. Considering these neurons to form an overcomplete dictionary of base signal elements at each visual stage, the firing of the neurons with respect to a given input image is typically highly sparse.

Chapter 11 addresses time- and/or space-variant structural reliability problems. It begins with a description of problem types as encroaching or outcrossing, subject to the type of dependence on the time or space variable. A brief review of essentials from the random process theory is presented, including second-moment characterization of the process in terms of mean and auto-covariance functions and the power spectral density. Special attention is given to Gaussian and Poisson processes as building blocks for stochastic load modeling. Bounds to the failure probability are developed in terms of mean crossing rates or using a series system representation through parameter discretization. A Poisson-based approximation for rare failure events is also presented. Next, the Poisson process is used to build idealized stochastic load models that describe macro-level load changes or intermittent occurrences with random magnitudes and durations. The chapter concludes with the development of the load-coincidence method for combination of stochastic loads. The probability distribution of the maximum combined load effect is derived and used to estimate the failure probability.

Chapter 9 describes simulation or sampling methods for reliability assessment. The chapter begins by describing methods for generation of pseudorandom numbers for prescribed univariate or multivariate distributions. Next, the ordinary Monte Carlo simulation (MCS) method is described. It is shown that for small failure probabilities, which is the case in most structural reliability problems, the number of samples required by MCS for a given level of accuracy is inversely proportional to the failure probability. Thus, MCS is computationally demanding for structural reliability problems. Various methods to reduce the computational demand of MCS are introduced. These include the use of antithetic variates and importance sampling. For the latter, sampling around design points and sampling in half-space are presented, the latter for a special class of problems. Other efficient sampling methods described include directional sampling, orthogonal-plane sampling, and subset simulation. For each case, expressions are derived for a measure of accuracy of the estimated failure probability. Methods are also presented for computing parameter sensitivities by sampling. Finally, a method is presented for evaluating certain multifold integrals by sampling. This method is useful in Bayesian updating, as described in Chapter 10.

This topic examines the nature of factors that affect what people buy and how much. These factors can be categorised into controllable and uncontrollable by managers. The first category relates to internal factors to the firm and involves the marketing mix. The second category relates to external factors in the business environment. A mathematical framework of analysis is required to quantify the effects of the different variables. This involves the use of demand functions or equations, which are often in a linear or power form. The linear form entails the coefficients of explanatory variables representing the marginal effects of those variables. The power form entails the coefficients or powers of the variables representing elasticities. There is a discussion of the factors determining various elasticities and their interpretation. The importance of elasticities in economic analysis is explained, in terms of managerial decision making and forecasting. The focus is on the application of concepts in demand theory to real-life situations, and the performance of the necessary calculations to make decisions and forecasts. Many solved problems are presented as an aid to this process.

Chapter 2 provides a review of probability theory, focusing on the topics that are essential for the remainder of the book. Included are the elements of set theory, the axioms and basic rules of probability theory, the concept of a random variable, discrete and continuous random variables, univariate and multivariate probability distributions, reliability and hazard functions, expectation and statistical moments, distributions and moments of functions of random variables, and extreme-value distributions. Appendix A presents commonly used probability distribution models with their properties, for easy reference. Thorough mastery of the material in this chapter is essential for understanding the remainder of the book.

This topic examines the nature of market structure, its characteristics and implications for managerial strategy and industrial structure, conduct and performance. The starting point is a discussion of markets and the characteristics that are relevant in terms of determining structure and strategy, in particular pricing. These characteristics relate to number of sellers, type of product, barriers to entry, pricing power and the importance of non-price competition. The four main types of structure, perfect competition, monopoly, monopolistic competition and oligopoly, are described in terms of these characteristics, and in each case both a graphical and algebraic analysis of the equilibrium is presented for both short-run and long-run situations. The different types of structure are compared in terms of price, output, profit, efficiency and welfare. The relationships between structure, conduct and performance are discussed in the light of recent empirical studies, along with some implications for government policy which are examined further in Chapter 14. Case studies illustrate the application of theoretical concepts in practice, particularly in the presence of Covid-19.

The previous three chapters were designed to help you understand the meaning and the method of the Laplace transform and its relation to the Fourier transform (), to show the Laplace transform of a few basic functions (), and to demonstrate some of the properties that make the Laplace transform useful (). In this chapter, you will see how to use the Laplace transform to solve problems in five different topics in physics and engineering. Those problems involve differential equations, so the first section of this chapter () provides an introduction to the application of the Laplace transform to ordinary and partial differential equations. Once you have an understanding of the general concept of solving a differential equation by applying an integral transform, you can work through specific applications including mechanical oscillations (), electrical circuits (), heat flow (), waves (), and transmission lines (). Each of these applications has been chosen to illustrate a different aspect of using the Laplace transform to solve differential equations, so you may find them useful even if you have little interest in the specific subject matter. And as in every chapter, the final section () of this chapter has a set of problems you can use to check your understanding of the concepts and mathematical techniques presented in this chapter.