1. Delivering Integrated, Sustainable, Water Resources Solutions Monte Carlo Simulation using @Risk Robert C. Patev North Atlantic Division – Regional Technical Specialist (978) 318-8394

2. Delivering Integrated, Sustainable, Water Resources Solutions Topics –Introduction –@Risk Basics –Reliability –Reporting Guidelines –@Risk Demonstration

3. 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

4. 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

5. 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?)

6. 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

7. 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

8. 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

9. Delivering Integrated, Sustainable, Water Resources Solutions Direct Sampling Cumulative Probability 01.0 Latin Hypercube Cumulative Probability 01.0 Monte Carlo Simulation using @Risk

10. 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

11. 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

12. Delivering Integrated, Sustainable, Water Resources Solutions @Risk Basics Random Variable A Random Variable B Negative Random Variable A Random Variable B Positive

13. Delivering Integrated, Sustainable, Water Resources Solutions @Risk BasicsTruncation0 0.4 pdf XL XU Area under curve = 1

14. Delivering Integrated, Sustainable, Water Resources Solutions Reliability R = 1 - P(u) where, P(u) = N pu / N N pu = Number of unsatisfactory performances at limit state < 1.0 N = number of iterations Reliability Using @Risk

15. 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

16. 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