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Uncertainty: Overview00:59

Uncertainty: Overview

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In analytical chemistry, we often perform repetitive measurements to detect and minimize inaccuracies caused by both determinate and indeterminate errors. Despite the cares we take, the presence of random errors means that repeated measurements almost never have exactly the same magnitude. The collective difference between these measurements - observed values - and the estimated or expected value is called uncertainty. Uncertainty is conventionally written after the estimated or expected value.
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Propagation of Uncertainty from Random Error00:59

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An experiment often consists of more than a single step. In this case, measurements at each step give rise to uncertainty. Because the measurements occur in successive steps, the uncertainty in one step necessarily contributes to that in the subsequent step. As we perform statistical analysis on these types of experiments, we must learn to account for the propagation of uncertainty from one step to the next. The propagation of uncertainty depends on the type of arithmetic operation performed on...
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Mechanistic Models: Compartment Models in Algorithms for Numerical Problem Solving01:29

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Mechanistic models play a crucial role in algorithms for numerical problem-solving, particularly in nonlinear mixed effects modeling (NMEM). These models aim to minimize specific objective functions by evaluating various parameter estimates, leading to the development of systematic algorithms. In some cases, linearization techniques approximate the model using linear equations.
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Statically indeterminate problems are those where statics alone can not determine the internal forces or reactions. Consider a structure comprising two cylindrical rods made of steel and brass. These rods are joined at point B and restrained by rigid supports at points A and C. Now, the reactions at points A and C and the deflection at point B are to be determined. This rod structure is classified as statically indeterminate as the structure has more supports than are necessary for maintaining...
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Uncertainty: Confidence Intervals00:54

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The confidence interval is the range of values around the mean that contains the true mean. It is expressed as a probability percentage. The interpretation of a 95% confidence interval, for instance, is that the statistician is 95% confident that the true mean falls within the interval. The upper and lower limits of this range are known as confidence limits. The confidence limits for the true mean are estimated from the sample's mean, the standard deviation, and the statistical factor...
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Propagation of Uncertainty from Systematic Error01:10

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The atomic mass of an element varies due to the relative ratio of its isotopes. A sample's relative proportion of oxygen isotopes influences its average atomic mass. For instance, if we were to measure the atomic mass of oxygen from a sample, the mass would be a weighted average of the isotopic masses of oxygen in that sample. Since a single sample is not likely to perfectly reflect the true atomic mass of oxygen for all the molecules of oxygen on Earth, the mass we obtain from this...
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Recoverable robust single machine scheduling with polyhedral uncertainty.

Matthew Bold1, Marc Goerigk2

  • 1STOR-i Centre for Doctoral Training, Lancaster University, Lancaster, UK.

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Summary
This summary is machine-generated.

This study addresses recoverable robust scheduling under uncertainty, developing new models to minimize total flow time. The research introduces novel formulations for single-machine scheduling problems with adjustable job orderings.

Keywords:
Budgeted uncertaintyPolyhedral uncertaintyRecoverable robustnessRobust optimizationScheduling

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Area of Science:

  • Operations Research
  • Combinatorial Optimization
  • Scheduling Theory

Background:

  • Single-machine scheduling problems are crucial in operations research.
  • Uncertainty in job processing times complicates scheduling decisions.
  • Recoverable robust optimization offers a framework to handle such uncertainties by allowing post-realization adjustments.

Purpose of the Study:

  • To develop and analyze recoverable robust formulations for a single-machine scheduling problem.
  • To minimize total flow time under polyhedral uncertainty.
  • To investigate efficient solution approaches for the scheduling problem and its subproblems.

Main Methods:

  • Formulation of the scheduling problem within a recoverable robust optimization framework.
  • Analysis of the incremental subproblem using max-weight matching theory.
  • Derivation of compact mathematical formulations (matching-based and assignment-based).

Main Results:

  • A general result for max-weight matching problems with specific edge weights is proven.
  • Two novel compact formulations (matching-based and assignment-based) are derived for the recoverable robust scheduling problem.
  • Computational experiments compare the performance of the proposed models.

Conclusions:

  • The proposed matching-based and assignment-based formulations provide efficient ways to solve the recoverable robust single-machine scheduling problem.
  • The study contributes to the field of robust optimization and scheduling theory.
  • The findings offer practical insights for decision-makers facing uncertain processing times.