James A. Reneke

A review of different perspectives on uncertainty and risk and an alternative modeling paradigm

Abstract The literature in economics, finance, operations research, engineering and in general mathematics is first reviewed on the subject of defining uncertainty and risk. The review goes back to 1901. Different perspectives on uncertainty and risk are examined and a new paradigm to model uncertainty and risk is proposed using relevant ideas from this study. This new paradigm is used to represent, aggregate and propagate uncertainty and interpret the resulting variability in a challenge problem developed by Oberkampf et al. [2004, Challenge problems: uncertainty in system response given uncertain parameters. Reliab Eng Syst Safety 2004; 85(1): 11–9]. The challenge problem is further extended into a decision problem that is treated within a multicriteria decision making framework to illustrate how the new paradigm yields optimal decisions under uncertainty. The accompanying risk is defined as the probability of an unsatisfactory system response quantified by a random function of the uncertainty.

Reliable Design Optimization Under Aleatory and Epistemic Uncertainties

In the engineering design community, decision making methodologies to select the “best” design from among feasible designs is one of the most critical part of the design process. As the design models become increasingly realistic, the decision making methodology becomes increasingly complex. That is, because of the realistic design models, more and more decisions are made under uncertain environments without making any unrealistic assumptions. A decision maker is usually forced to work with uncertainties of which some stochastic information is known (aleatory) or no information is known (epistemic). In this paper, we discuss both forms of uncertainties and their modeling methodologies. We also define risk as a random function of these uncertainties and propose a risk quantification technique. Existing methods to handle aleatory uncertainties are discussed and an alternative search based decision making methodology is proposed to handle epistemic uncertainties. We illustrate our decision making methodology using the side-impact crashworthiness problem presented by Gu, et.al. [1]. In addition to the aleatory uncertainties considered by these researchers, we model a couple of non-design variables as epistemic uncertainties in our decision problem. Lack of information of these epistemic uncertainties increases the complexity of the side-impact crashworthiness problem significantly. However, the proposed methodology helps to identify a robust design with respect to epistemic uncertainty.Copyright

Affordable Upgrades of Complex Systems: A Multilevel, Performance-Based Approach

A modeling and methodological approach to complex system decision making is proposed. A system is modeled as a multilevel network whose components interact and decisions on affordable upgrades of the components are to be made under uncertainty. The system is studied within a framework of overall performance analysis in a range of exogenous environments and in the presence of random inputs. The methodology makes use of stochastic analysis and multiple-criteria decision analysis. An illustrative example of upgrading an idealized industrial production system with complete computations is included.

Stochastic differential equations in mathematical demography: a review

The use of stochastic differential equations in mathematical demography is reviewed. Such equations have been introduced in stochastic versions of the exponential and logistic growth models. Their recent use in the study of stochastic models of mortality is also examined.

A Diffusion Model of Forest Succession

Based on a tree by tree replacement mechanism, a diffusion model of forest stand canopy composition is formulated and analyzed. The model is used to explore composition dichotomies by estimating coefficients from forest stand data and interpreting the results in terms of mechanisms for succession. The model yields a concrete characterization of the succession phenomenon known as the climax state.

Design Under Uncertainty: Balancing Expected Performance and Risk

The problem of quantifying uncertainty in the design process is approached indirectly. Nonquantifiable variability resulting from lack of knowledge is treated as epistemic uncertainty and quantifiable variability caused by random influences is treated as aleatory uncertainty. The emphasis in this approach is on the effects of epistemic uncertainty, left unquantified, on design performance. Performance is treated as a random function of the epistemic uncertainties that are considered as independent variables, and a design decision is based on the mean and variance of design performance. Since the mean and variance are functions of the uncertainties, multicriteria decision methods are employed to determine the preferred design. The methodology is illustrated on a three-spring model with stochastic forcing and two uncertain damping coefficients. Based on the example, the concept of balancing expected performance and risk is explored in an engineering context. Risk is quantified using aleatory uncertainty for fixed values of epistemic uncertainty. The study shows the unique features of this approach in which risk-based design decisions are made under both aleatory and epistemic uncertainties without assuming a distribution for epistemic uncertainty.

Linear feedback control for a class of nonlinear systems

The authors discuss a class of nonlinear vector systems that admit a Lyapunov function of the form V(x)=(x- alpha )/sup T/ rho (x- alpha ). Examples of this class include Lorenzs system, the starting point for the modern study of systems with chaotic attractors, and Euler systems arising in the study of rotational motion of rigid and linked systems such as satellites and robots. In the context of Lyapunovs second method, sufficient conditions that produce either of the following two types of global behavior are discussed: the origin is a global asymptotic stable point or the system is point dissipative. Associated with each set of conditions is a linear algebra problem relating to controllability.<<ETX>>

Stieltjes Integral Equations in Partially Ordered Sets

Abstract Stieltjes integral equations are considered in partially ordered sets of real valued functions, and connections are made using integral inequalities to the existence and uniqueness problems of hereditary systems which are not Lipschitz.

Multidisciplinary design optimisation under uncertainty: an information model approach

The design of a multi–purpose vehicle capable of performing diverse missions in diverse environmental conditions requires a multi–disciplinary approach. Uncertain missions implied by a multi–purpose design goal and uncertain environmental conditions resulting from multiple operating theatres involve tradeoffs in vehicle performance. An information model approach to handling tradeoffs is presented. The design process is conceptualized as proceeding in stages. At each stage, the design problem is decomposed from the top down into design levels and interacting components having uncertain elements on each level. The components may require different knowledge bases and models with different mathematical structures, time and size scales, calling for higher–level coordination. Component performances are modelled as random functions of uncertainties considered as deterministic variables. Information models are developed making use of second–order statistics of the random performance functions and an algebra of their reduced–order representations. Decision–making proceeds from the bottom up. Higher–level design decisions, the result of tradeoffs between alternative component designs, are based on the information models of the component performance functions. Preferred overall designs are determined within a finite set of feasible designs by means of multi–criteria optimisation methods without using mathematical programming. The methodology is illustrated by a simplified two–component vehicle design problem.

Decision making using a multi-criteria approach in a wholesale electrical power market

In this paper, we formulate a multi-criteria decision problem using a reproducing kernel Hilbert space approach to stochastic processes applied to a wholesale electrical power market. Our approach provides an alternative to stochastic, robust optimization commonly used in recent years in that we consider a range of uncertainty rather than just the most likely case or worst case scenario. In particular, we model operation of the electricity market of the ISO New England, an independent Regional Transmission Organization serving the New England area, using hourly zonal data for the last five years. Alternative investments in production or infrastructure must be evaluated in an environment of risk and Knightian uncertainty. Assuming that performance is modeled as a random function of deterministic but uncertain demand allows risk to be measured by value at risk curves over a range of uncertain demand. The preferred investment is determined by a preference rule defined on the set of value at risk curves, a standard multi-criteria approach.