1st Edition

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

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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 Uncertainty**1.1 Decision under Uncertainty

1.1.1 Good versus bad decisions

1.1.2 Elements of a Decision

1.1.3 Limited Information

1.2 The Role of Decision Analysis in Engineering Design

1.2.1 Sequential decisions in product development

1.2.2 Challenges in design decision making under uncertainty and scope of this book

1.3 Outline of this Book

1.4 Conclusion

Questions

References

**2. Overview of Theories of Uncertainty and Tools for Modeling Uncertainty**

2.1 Introduction: Management of Uncertainty in Design

2.2 Theories of Uncertainty

2.2.1 Intervals

2.2.2 Convex Sets

2.2.3 Objective Probability

2.2.4 Subjective Probability

2.2.5 Imprecise Probability

2.2.6 Dempster-Shafer Evidence Theory

2.3 Conclusion

References

**3. Objective Probability**Overview of this chapter

3.1 Probability and Random Variables for Modeling Uncertainty

3.1.1 Fundamentals of Objective Probability

3.1.1.1 Definition of probability

3.1.1.2 Axioms of probability

3.1.1.3 Conditional probability

3.1.1.4 Combined experiments

Questions and Exercises

3.1.2 Random variables

3.1.2.1 Discrete random variables

3.1.2.2 Continuous random variables

3.1.2.3 Conditional Probability Distribution and Density Functions

Questions and exercises

3.1.3 Multiple random variables

3.1.3.1 Discrete random variables

3.1.3.2 Continuous random variables

Questions and exercises

3.2 Common probabilistic models

3.2.1 Distributions of a single random variable

3.2.1.1 Discrete variables

3.2.1.2 Continuous variables

3.2.2 Joint normal distribution

Summary of section 3.2

Questions and Exercises

3.3 Probability calculations

3.3.1 Probability distributions of a function of one random variable

3.3.1.1 Probability distribution

3.3.1.2 Probability density function

3.3.1.3 Mean value and standard deviation of a function of one variable

3.3.2 Distribution of functions of multiple random variables

3.3.1.1 One function of two variables

3.3.2.2 Two functions of two random variables

3.3.2.3 The method of auxiliary variables

3.3.2.4 Mean value and standard deviation of a function of many variables

3.3.2.5 Calculations involving normal random variables

Questions and Exercises

3.4 Concluding Remarks

References

Appendix

**4. Statistical Inference – Constructing Probabilistic Models from Observations**4.1 Introduction

4.1.1 Objective, scope and summary of this chapter

4.2 Estimating mean values of random variables and probabilities of events

4.2.1 Sample mean

4.2.2 Sample variance

4.2.3 Covariance and Correlation

4.2.5 Confidence Interval for Variance

4.2.6 Probability of an Event

4.2.7 How to get the maximum return from your budget for data collection

4.3 Statistical hypothesis testing

4.4 Selecting input probability distributions

4.4.1 Step 1: Select families of probability distributions

4.4.2 Step 2: Estimate the distribution parameters

4.4.3 Step 3: Assess fit of selected distributions to observed data

4.5 Modeling dependent variables

4.5.1 Overview of methods for modeling dependence

4.5.2 Copulas for modeling dependence

4.6 Conclusion

References

**5. Probabilistic Analysis of Dynamic Systems**

**6. Subjective (Bayesian) Probability**6.1 Definition of Subjective Probability

6.1.1 Overview

6.1.2 Axiomatic definition of probability

6.1.3 Conditional probability

6.1.4 Principle of insufficient reason

Questions and problems

6.2 Eliciting Expert’s Judgments in Order to Construct Models of Uncertaint

6.2.1 Elicitation process

6.2.2 Eliciting probabilities

6.2.3 Estimation of probabilities of rare events

6.2.4 Eliciting probability distributions

6.2.5 Representing uncertainty about an elicited distribution by a second-order probabilistic model

Questions and Problems

6.3 Bayesian Analysis

6.3.1 Motivation

6.3.2 How to update a probability distribution using observations or expert judgment

6.3.3 Accounting for imprecision by using probability bounds

Questions and Problems

6.4 Heuristics and biases in probability judgments

Questions

6.5 Concluding remarks

References

**7. Decision Analysis**7.1 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 Problems

7.2 Framing and Structuring Decisions

7.2.1 Define and frame a decision

7.2.2 Structure a Decision Problem

Questions and problems

7.3 Solving Decision Problems

7.3.1 Backward induction (or folding back the decision tree)

Exercises and problems

7.4 Performing Sensitivity Analysis

7.4.1 Introduction

7.4.2 Sensitivity to the definition, framing and structure of the problem

7.4.3 One-way sensitivity analysis

7.4.4 Two-way sensitivity analysis

7.4.5 Sensitivity of the selection of the optimum option to imprecision in probabilities

Questions and problems

7.5 Modeling Preferences

7.5.1 Motivation

7.5.2 Simple criteria for decision making

7.5.3 Utility

7.5.4 Axioms of utility

Questions and problems

7.6 Conclusion

References

**8. Multiattribute Considerations in Design**8.1 Tradeoff between attributes

8.1.1 Range of negotiability

8.1.2 Value functions vs. Utility functions

Question and Problems:

8.2 Different multiattribute formulations

8.2.1 Single-Attribute based formulations

8.2.1.1 Independence conditions

8.2.1.2 The additive form of multiattribute utility function

8.2.1.3 The multi-linear form of multiattribute utility function

8.2.2 Assessing the scaling constants

8.2.3 Value function based formulation

8.2.4 Attribute dominance utility and multiattribute utility copulas

8.3 Solving decision problems under uncertainty using multiattribute utility analysis

8.4 Conclusions

Questions and Problems

References

### Biography

Zissimos P. Mourelatos, Efstratios Nikolaidis, Vijitashwa Pandey