Transcript Monte Carlo Simulation using @Risk

Delivering Integrated, Sustainable,
Water Resources Solutions
Monte Carlo Simulation
using @Risk
Robert C. Patev
North Atlantic Division – Regional Technical
Specialist
(978) 318-8394
Delivering Integrated, Sustainable,
Water Resources Solutions
• Topics
– Introduction
– @Risk Basics
– Reliability
– Reporting Guidelines
– @Risk Demonstration
Delivering Integrated, Sustainable,
Water Resources Solutions
• Monte Carlo Simulation
– Types of simulation methods
•
•
•
•
•
Direct – brute force method
Stratified – effort in regions
Latin Hypercube – form of stratified sampling
Importance – selected shift in distributions
Adaptive – form of importance sampling
Delivering Integrated, Sustainable,
Water Resources Solutions
• Introduction to @Risk
– Monte Carlo Simulation (MCS)
– Spreadsheet add-in
• Excel Macros
– User friendly interface
• Easy input
• Many probability distribution functions
• Graphical output
Delivering Integrated, Sustainable,
Water Resources Solutions
• CAVEAT to @Risk
– “Let the engineer beware”
• Not just a “black box” that gives the correct answer
or decision
• Tool to assist in making decisions and arriving at a
solution
• Understand the inputs to your model
• Understand limitations in your spreadsheets
• Cautiously scrutinize and review output (Does it
make sense?)
Delivering Integrated, Sustainable,
Water Resources Solutions
• @Risk Use within the Corps of Engineers
– Reliability Analysis
• Structural
• Geotechnical
– Economic Analysis
– Major Rehabilitation Projects
– System Studies
• ORMSS, GLSLS
Delivering Integrated, Sustainable,
Water Resources Solutions
• @Risk Capabilities
– Easily adds MCS to existing spreadsheet
model
– Fast execution time
– Save MCS results quickly
– User-defined macros
– Complete statistical analysis
• Input
• Output
• Sensitivity
Delivering Integrated, Sustainable,
Water Resources Solutions
• @Risk Basics
– Iterations vs. simulations
• Iteration - an iteration is a single sampling of
random variables
• Simulation - x number of iterations
– Monte Carlo Simulation methods
• Direct sampling
• Latin hypercube sampling
Delivering Integrated, Sustainable,
Water Resources Solutions
Monte Carlo Simulation
using @Risk
1.0
Cumulative Probability
Cumulative Probability
1.0
0
0
Direct Sampling
Latin Hypercube
Delivering Integrated, Sustainable,
Water Resources Solutions
• @Risk Basics
– Random number seed generator
• -1 to 32767 (default = 0)
– Convergence
• Input random variables
• Selected output cells
– User-defined macros
Delivering Integrated, Sustainable,
Water Resources Solutions
• @Risk Basics
– Random Variables
• Numerous discrete/continuous distributions
• Correlation
– Positive/negative
– Examine outputs
• Truncation
– Physical limitations to data
– Examine results
Delivering Integrated, Sustainable,
Water Resources Solutions
• @Risk Basics
Positive
Random Variable B
Random Variable B
Negative
Random Variable A
Random Variable A
Delivering Integrated, Sustainable,
Water Resources Solutions
• @Risk Basics
Truncation
0.4
Area under curve = 1
pdf
0
XL
XU
Delivering Integrated, Sustainable,
Water Resources Solutions
Reliability Using @Risk
• Reliability
R = 1 - P(u)
where, P(u) = Npu / N
Npu = Number of unsatisfactory
performances at limit state < 1.0
N = number of iterations
Delivering Integrated, Sustainable,
Water Resources Solutions
• Random Variables
– Distributions
• Statistical parameters (min/max, mean, std. dev.,
…)
• Distribution types
– Questions - Why use, Where come from, How applied in
model, What other distributions can be used
• Correlation/truncation
– Justification
• Plots of simulated distributions for random
variables and selected “output” cells from
simulation
Delivering Integrated, Sustainable,
Water Resources Solutions
• Sensitivity/Convergence
– Sensitivity
• Identifies the most “critical” variables to the output
• Range: +1 to -1 (closest to (+/-)1, model most
sensitive)
• R-squared method/Rank correlation coefficient
– Convergence
• Limit state functions
• Probability of unsatisfactory performance