Risk and Uncertainty Reduction by Using Algebraic Inequalities

Cover
Half Title
Title Page
Copyright Page
Dedication
Table of Contents
Preface
Author
Chapter 1: Fundamental Concepts Related to Risk and Uncertainty Reduction by Using Algebraic Inequalities
	1 1 Domain-Independent Approach to Risk Reduction
	1 2 A Powerful Domain-Independent Method for Risk and Uncertainty Reduction Based on Algebraic Inequalities
		1 2 1 Classification of Techniques Based on Algebraic Inequalities for Risk and Uncertainty Reduction
	1 3 Risk and Uncertainty
Chapter 2: Properties of Algebraic Inequalities: Standard Algebraic Inequalities
	2 1 Basic Rules Related to Algebraic Inequalities
	2 2 Basic Properties of Inequalities
	2 3 One-Dimensional Triangle Inequality
	2 4 The Quadratic Inequality
	2 5 Jensen Inequality
	2 6 Root-Mean Square–Arithmetic Mean–Geometric Mean–Harmonic Mean (RMS-AM-GM-HM) Inequality
	2 7 Weighted Arithmetic Mean–Geometric Mean (AM-GM) Inequality
	2 8 Hölder Inequality
	2 9 Cauchy-Schwarz Inequality
	2 10 Rearrangement Inequality
	2 11 Chebyshev Sum Inequality
	2 12 Muirhead Inequality
	2 13 Markov Inequality
	2 14 Chebyshev Inequality
	2 15 Minkowski Inequality
Chapter 3: Basic Techniques for Proving Algebraic Inequalities
	3 1 The Need for Proving Algebraic Inequalities
	3 2 Proving Inequalities by a Direct Algebraic Manipulation and Analysis
	3 3 Proving Inequalities by Presenting Them as a Sum of Non-Negative Terms
	3 4 Proving Inequalities by Proving Simpler Intermediate Inequalities
	3 5 Proving Inequalities by Substitution
	3 6 Proving Inequalities by Exploiting the Symmetry
	3 7 Proving Inequalities by Exploiting Homogeneity
	3 8 Proving Inequalities by a Mathematical Induction
	3 9 Proving Inequalities by Using the Properties of Convex/Concave Functions
		3 9 1 Jensen Inequality
	3 10 Proving Inequalities by Using the Properties of Sub-Additive and Super-Additive Functions
	3 11 Proving Inequalities by Transforming Them to Known Inequalities
		3 11 1 Proving Inequalities by Transforming Them to an Already Proved Inequality
		3 11 2 Proving Inequalities by Transforming Them to the Hölder Inequality
		3 11 3 An Alternative Proof of the Cauchy-Schwarz Inequality by Reducing It to a Standard Inequality
		3 11 4 An Alternative Proof of the GM-HM Inequality by Reducing It to the AM-GM Inequality
		3 11 5 Proving Inequalities by Transforming Them to the Cauchy-Schwarz Inequality
	3 12 Proving Inequalities by Segmentation
		3 12 1 Determining Bounds by Segmentation
	3 13 Proving Algebraic Inequalities by Combining Several Techniques
	3 14 Using Derivatives to Prove Inequalities
Chapter 4: Using Optimisation Methods for Determining Tight Upper and Lower Bounds: Testing a Conjectured Inequality by a Simulation – Exercises
	4 1 Using Constrained Optimisation for Determining Tight Upper Bounds
	4 2 Tight Bounds for Multivariable Functions Whose Partial Derivatives Do Not Change Sign in a Specified Domain
	4 3 Conventions Adopted in Presenting the Simulation Algorithms
	4 4 Testing a Conjectured Algebraic Inequality by a Monte Carlo Simulation
	4 5 Exercises
	4 6 Solutions to the Exercises
Chapter 5: Ranking the Reliabilities of Systems and Processes by Using Inequalities
	5 1 Improving Reliability and Reducing Risk by Proving an Abstract Inequality Derived from the Real Physical System or Process
	5 2 Using Algebraic Inequalities for Ranking Systems Whose Component Reliabilities Are Unknown
		5 2 1 Reliability of Systems with Components Logically Arranged in Series and Parallel
	5 3 Using Inequalities to Rank Systems with the Same Topology and Different Component Arrangements
	5 4 Using Inequalities to Rank Systems with Different Topologies Built with the Same Types of Components
Chapter 6: Using Inequalities for Reducing Epistemic Uncertainty and Ranking Decision Alternatives
	6 1 Selection from Sources with Unknown Proportions of High-Reliability Components
	6 2 Monte Carlo Simulations
	6 3 Extending the Results by Using the Muirhead Inequality
Chapter 7: Creating a Meaningful Interpretation of Existing Abstract Inequalities and Linking It to Real Applications
	7 1 Meaningful Interpretations of Abstract Algebraic Inequalities with Applications to Real Physical Systems
		7 1 1 Applications Related to Robust and Safe Design
	7 2 Avoiding Underestimation of the Risk and Overestimation of Average Profit by a Meaningful Interpretation of the Chebyshev Sum Inequality
	7 3 A Meaningful Interpretation of an Abstract Algebraic Inequality with an Application for Selecting Components of the Same Variety
	7 4 Maximising the Chances of a Beneficial Random Selection by a Meaningful Interpretation of a General Inequality
	7 5 The Principle of Non-Contradiction
Chapter 8: Optimisation by Using Inequalities
	8 1 Using Inequalities for Minimising the Deviation of Reliability-Critical Parameters
	8 2 Minimising the Deviation of the Volume of Manufactured Workpieces with Cylindrical Shapes
	8 3 Minimising the Deviation of the Volume of Manufactured Workpieces in the Shape of a Rectangular Prism
	8 4 Minimising the Deviation of the Resonant Frequency from the Required Level for Parallel Resonant LC-Circuits
	8 5 Maximising Reliability by Using the Rearrangement Inequality
		8 5 1 Using the Rearrangement Inequality to Maximise the Reliability of Parallel-Series Systems
		8 5 2 Using the Rearrangement Inequality for Optimal Condition Monitoring
	8 6 Using the Rearrangement Inequality to Minimise the Risk of a Faulty Assembly
Chapter 9: Determining Tight Bounds for the Uncertainty in Risk-Critical Parameters and Properties by Using Inequalities
	9 1 Upper-Bound Variance Inequality for Properties from Different Sources
	9 2 Identifying the Source Whose Removal Causes the Largest Reduction of the Worst-Case Variation
	9 3 Increasing the Robustness of Electronic Products by Using the Variance Upper-Bound Inequality
	9 4 Determining Tight Bounds for the Fraction of Items with a Particular Property
	9 5 Using the Properties of Convex Functions for Determining the Upper Bound of the Equivalent Resistance
	9 6 Determining a Tight Upper Bound for the Risk of a Faulty Assembly by Using the Chebyshev Inequality
	9 7 Deriving a Tight Upper Bound for the Risk of a Faulty Assembly by Using the Chebyshev Inequality and Jensen Inequality
Chapter 10: Using Algebraic Inequalities to Support Risk-Critical Reasoning
	10 1 Using the Inequality of the Negatively Correlated Events to Support Risk-Critical Reasoning
	10 2 Avoiding Risk Underestimation by Using the Jensen Inequality
		10 2 1 Avoiding the Risk of Overestimating Profit
		10 2 2 Avoiding the Risk of Underestimating the Cost of Failure
		10 2 3 A Conservative Estimate of System Reliability by Using the Jensen Inequality
	10 3 Reducing Uncertainty and Risk Associated with the Prediction of Magnitude Rankings
References
Index