# Risk and Uncertainty Reduction by Using Algebraic Inequalities

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Risk and Uncertainty Reduction by Using Algebraic Inequalities book

# Risk and Uncertainty Reduction by Using Algebraic Inequalities

DOI link for Risk and Uncertainty Reduction by Using Algebraic Inequalities

Risk and Uncertainty Reduction by Using Algebraic Inequalities book

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This book covers the application of algebraic inequalities for reliability improvement and for uncertainty and risk reduction. It equips readers with powerful domain-independent methods for reducing risk based on algebraic inequalities and demonstrates the significant benefits derived from the application for risk and uncertainty reduction.

Algebraic inequalities:

• Provide a powerful reliability-improvement, risk and uncertainty reduction method that transcends engineering and can be applied in various domains of human activity

• Present an effective tool for dealing with deep uncertainty related to key reliability-critical parameters of systems and processes

• Permit meaningful interpretations which link abstract inequalities with the real world

• Offer a tool for determining tight bounds for the variation of risk-critical parameters and complying the design with these bounds to avoid failure

• Allow optimising designs and processes by minimising the deviation of critical output parameters from their specified values and maximising their performance

This book is primarily for engineering professionals and academic researchers in virtually all existing engineering disciplines.

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

1.3 Risk and uncertainty

2. PROPERTIES OF ALGEBRAIC INEQUALITIES AND 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's inequality

2.6 Root-mean square – Arithmetic mean – Geometric mean – Harmonic mean (RMS-AM-GM-HM) inequality

2.7 Weighted Arithmetic mean-Geometric (AM-GM) mean inequality

2.8 Hölder's inequality

2.9 Cauchy-Schwarz inequality

2.10 Rearrangement inequality

2.11 Chebyshev's sum inequality

2.12 Muirhead's inequality

2.13 Markov's inequality

2.14 Chebyshev's inequality

2.15 Minkowski inequality

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 a 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

3.10 Proving inequalities by using the properties of sub-additive and super-additive functions

3.11 Proving inequalities by transforming them to known inequalities

3.12 Proving inequalities by a segmentation

3.13 Proving algebraic inequalities by combining several techniques

3.14 Using derivatives to prove inequalities

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

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 inequalities for ranking systems whose component reliabilities are unknown

5.3 Using inequalities for ranking systems with the same topology and different components arrangements

5.4 Using inequalities to rank systems with different topologies built with the same type of components

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's inequality

7. CREATING A MEANINGFUL INTERPRETATION OF EXISTING ABSTRACT INEQUALITIES AND LINKING IT TO REAL APPLICATIONS

7.1 Meaningful interpretations of an abstract algebraic inequality with several applications to real physical systems

7.2 Avoiding underestimation of the risk and overestimation of average profit by a meaningful interpretation of the Chebyshev's sum inequality

7.3 A meaningful interpretation of an abstract algebraic inequality with an application to 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

8. INEQUALITIES MINIMISING THE RISK OF A FAULTY ASSEMBLY AND OPERATION

8.1 Using inequalities for minimising the deviation of reliability-critical parameters

8.2 Minimising the deviation of the volume of manufactured cylindrical workpieces with cylindrical shape

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

8.6 Using the rearrangement inequality to minimise the risk of a faulty assembly

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 devices 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 for resistors with uncertain values

9.6 Determining a tight upper bound for the risk of a faulty assembly by using the Chebyshev's inequality

9.7 Deriving a tight upper bound for the risk of a faulty assembly by using the Chebyshev's inequality and Jensen's inequality

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's inequality

10.3 Reducing uncertainty and risk associated with the prediction of the magnitudes ranking related to random outcomes

11. REFERENCES