Design Decisions under Uncertainty with Limited Information Structures and Infrastructures Book Series, Vol. 7

Today's business environment involves design decisions with significant uncertainty. To succeed, decision-makers should replace deterministic methods with a risk-based approach that accounts for the decision maker’s risk tolerance. In many problems, it is impractical to collect data because rare or one-time events are involved. Therefore, we need a methodology to model uncertainty and make choices when we have limited information. This methodology must use all available information and rely only on assumptions that are supported by evidence. This book explains theories and tools to represent uncertainty using both data and expert judgment. It teaches the reader how to make design or business decisions when there is limited information with these tools. Readers will learn a structured, risk-based approach, which is based on common sense principles, for design and business decisions. These decisions are consistent with the decision-maker’s risk attitude. The book is exceptionally suited as educational material because it uses everyday language and real-life examples to elucidate concepts. It demonstrates how these concepts touch our lives through many practical examples, questions and exercises. These are designed to help students learn that first they should understand a problem and then establish a strategy for solving it, instead of using trial-and-error approaches. This volume is intended for undergraduate and graduate courses in mechanical, civil, industrial, aerospace, and ocean engineering and for researchers and professionals in these disciplines. It will also benefit managers and students in business administration who want to make good decisions with limited information.

1. Design Decision under Uncertainty1.1 Decision under Uncertainty1.1.1 Good versus bad decisions1.1.2 Elements of a Decision1.1.3 Limited Information1.2 The Role of Decision Analysis in Engineering Design1.2.1 Sequential decisions in product development1.2.2 Challenges in design decision making under uncertainty and scope of this book1.3 Outline of this Book1.4 ConclusionQuestionsReferences 2. Overview of Theories of Uncertainty and Tools for Modeling Uncertainty2.1 Introduction: Management of Uncertainty in Design2.2 Theories of Uncertainty2.2.1 Intervals2.2.2 Convex Sets2.2.3 Objective Probability2.2.4 Subjective Probability2.2.5 Imprecise Probability2.2.6 Dempster-Shafer Evidence Theory2.3 ConclusionReferences 3. Objective ProbabilityOverview of this chapter3.1 Probability and Random Variables for Modeling Uncertainty3.1.1 Fundamentals of Objective Probability3.1.1.1 Definition of probability3.1.1.2 Axioms of probability3.1.1.3 Conditional probability3.1.1.4 Combined experimentsQuestions and Exercises3.1.2 Random variables3.1.2.1 Discrete random variables3.1.2.2 Continuous random variables3.1.2.3 Conditional Probability Distribution and Density FunctionsQuestions and exercises3.1.3 Multiple random variables3.1.3.1 Discrete random variables3.1.3.2 Continuous random variablesQuestions and exercises3.2 Common probabilistic models3.2.1 Distributions of a single random variable3.2.1.1 Discrete variables3.2.1.2 Continuous variables3.2.2 Joint normal distributionSummary of section 3.2Questions and Exercises3.3 Probability calculations3.3.1 Probability distributions of a function of one random variable3.3.1.1 Probability distribution3.3.1.2 Probability density function3.3.1.3 Mean value and standard deviation of a function of one variable3.3.2 Distribution of functions of multiple random variables3.3.1.1 One function of two variables3.3.2.2 Two functions of two random variables3.3.2.3 The method of auxiliary variables3.3.2.4 Mean value and standard deviation of a function of many variables3.3.2.5 Calculations involving normal random variablesQuestions and Exercises3.4 Concluding RemarksReferencesAppendix 4. Statistical Inference – Constructing Probabilistic Models from Observations4.1 Introduction4.1.1 Objective, scope and summary of this chapter4.2 Estimating mean values of random variables and probabilities of events4.2.1 Sample mean4.2.2 Sample variance4.2.3 Covariance and Correlation4.2.5 Confidence Interval for Variance4.2.6 Probability of an Event4.2.7 How to get the maximum return from your budget for data collection4.3 Statistical hypothesis testing4.4 Selecting input probability distributions4.4.1 Step 1: Select families of probability distributions4.4.2 Step 2: Estimate the distribution parameters4.4.3 Step 3: Assess fit of selected distributions to observed data4.5 Modeling dependent variables4.5.1 Overview of methods for modeling dependence4.5.2 Copulas for modeling dependence4.6 ConclusionReferences 5. Probabilistic Analysis of Dynamic Systems 6. Subjective (Bayesian) Probability6.1 Definition of Subjective Probability6.1.1 Overview6.1.2 Axiomatic definition of probability6.1.3 Conditional probability6.1.4 Principle of insufficient reasonQuestions and problems6.2 Eliciting Expert’s Judgments in Order to Construct Models of Uncertaint6.2.1 Elicitation process6.2.2 Eliciting probabilities6.2.3 Estimation of probabilities of rare events6.2.4 Eliciting probability distributions6.2.5 Representing uncertainty about an elicited distribution by a second-order probabilistic modelQuestions and Problems6.3 Bayesian Analysis6.3.1 Motivation6.3.2 How to update a probability distribution using observations or expert judgment6.3.3 Accounting for imprecision by using probability boundsQuestions and Problems6.4 Heuristics and biases in probability judgmentsQuestions6.5 Concluding remarksReferences 7. Decision Analysis7.1 Introduction 7.1.1 Examples of Decision Problems 7.1.2 Elements of Decision Problems and Terminology 7.1.4 Steps of the Decision Process 7.1.5 Outline of this chapter Questions and Problems7.2 Framing and Structuring Decisions 7.2.1 Define and frame a decision7.2.2 Structure a Decision ProblemQuestions and problems7.3 Solving Decision Problems7.3.1 Backward induction (or folding back the decision tree) Exercises and problems7.4 Performing Sensitivity Analysis7.4.1 Introduction7.4.2 Sensitivity to the definition, framing and structure of the problem7.4.3 One-way sensitivity analysis7.4.4 Two-way sensitivity analysis7.4.5 Sensitivity of the selection of the optimum option to imprecision in probabilitiesQuestions and problems7.5 Modeling Preferences7.5.1 Motivation7.5.2 Simple criteria for decision making7.5.3 Utility7.5.4 Axioms of utilityQuestions and problems7.6 ConclusionReferences 8. Multiattribute Considerations in Design8.1 Tradeoff between attributes8.1.1 Range of negotiability8.1.2 Value functions vs. Utility functionsQuestion and Problems: 8.2 Different multiattribute formulations8.2.1 Single-Attribute based formulations8.2.1.1 Independence conditions8.2.1.2 The additive form of multiattribute utility function8.2.1.3 The multi-linear form of multiattribute utility function8.2.2 Assessing the scaling constants8.2.3 Value function based formulation8.2.4 Attribute dominance utility and multiattribute utility copulas8.3 Solving decision problems under uncertainty using multiattribute utility analysis8.4 ConclusionsQuestions and ProblemsReferences

Zissimos P. Mourelatos, Efstratios Nikolaidis, Vijitashwa Pandey