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The AutoSimOA Project

AUTOMATING D.E.S OUTPUT ANALYSIS:. The AutoSimOA Project. HOW MANY REPLICATIONS TO RUN. Katy Hoad, Stewart Robinson, Ruth Davies Warwick Business School OR49 Sept 07. A 3 year, EPSRC funded project in collaboration with SIMUL8 Corporation. OUTLINE Introduction Methods Algorithm

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The AutoSimOA Project

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  1. AUTOMATING D.E.S OUTPUT ANALYSIS: The AutoSimOA Project HOW MANY REPLICATIONS TO RUN Katy Hoad, Stewart Robinson, Ruth Davies Warwick Business School OR49 Sept 07 A 3 year, EPSRC funded project in collaboration with SIMUL8 Corporation.

  2. OUTLINE Introduction Methods Algorithm Test Methodology Test Results Extended Algorithm & Results Discussion Summary

  3. Objective To provide an easy to use method, that can be incorporated into existing simulation software, that enables practitioners to obtain results of a specified accuracy from their discrete event simulation model. (Only looking at analysis of a single scenario)

  4. = summary statistic from rep1 N replications = summary statistic from repN Response measure of interest Output data from model Introduction • Initial Setup: • Any warm-up problems already dealt with. • Run length (m) decided upon. • Modeller decided to use multiple replications to obtain better estimate of mean performance. • Multiple replications performed by changing the random number streams used by the model and re-running the simulation.

  5. QUESTION IS… How many replications are needed? • Limiting factors: computing time and expense. If performing N replications achieves a sufficient estimate of mean performance: > N replications: Unnecessary use of computer time and money. < N replications: Inaccurate results → incorrect decisions.

  6. 4 main methods found in the literature for choosing N: • Rule of Thumb • Run at least 3 to 5 replications. • Advantage: Very simple. • Disadvantage: Does not use characteristics of model output. • No measured precision level.

  7. 2. Simple Graphical Method • Plot Cumulative mean -v- number of replications • Visually select point where cumulative mean line becomes “flat”. Use this as N. Advantages: Simple Uses output of interest in decision. Disadvantages: Subjective No measured precision level.

  8. 3.Confidence Interval (with Specified Precision) Method • User decides size of error they can tolerate. • Run increasing numbers of replications, • Construct Confidence Intervals around sequential cumulative mean of output variable until desired precision achieved. Advantages: Relies upon statistical inference to determine number of replications required. Allows the user to tailor accuracy of output results to their particular requirement or purpose for that model and result. Disadvantage:Many simulation users do not have the skills to apply such an approach.

  9. 4.Prediction Formula Method • User decides size of error they can tolerate. • Run a few replications, estimate variance & mean • Use formula to predict N. • Check desired precision achieved – if not amend N and repeat Advantages: Simple. Uses data from model. Provides specified precision. Disadvantage:Can be very inaccurate especially for small number of replications. If variance estimate low underestimate N If variance estimate high overestimate N

  10. Chose to automate: Confidence Interval (with Specified Precision) Method

  11. The replication algorithm interacts with the simulation model sequentially.

  12. We define the precision, dn, as the ½ width of the Confidence Interval expressed as a percentage of the cumulative mean: Where n is the current number of replications carried out, is the student t value for n-1 df and a significance of 1-α, is the cumulative mean, snis the estimate of the standard deviation, calculated using results Xi (i = 1 to n) of the n current replications. ALGORITHM DEFINITIONS

  13. Stopping Criteria • Simplest method: Stop when dn 1st found to be ≤ desired precision, drequired , and recommend that number of replications, Nsol, to the user. • Problem: Data series could prematurely converge, by chance, to incorrect estimate of the mean, with precision drequired , then diverge again. • ‘Look-ahead’ procedure: When dn 1st found to be ≤ drequired, algorithm performs set number of extra replications, to check that precision remains ≤ drequired.

  14. ‘Look-ahead’ procedure kLimit = ‘look ahead’ value. Actual number of replications checked ahead is a function of this user defined value: Function relates ‘look ahead’ period length with current value of n.

  15. Replication Algorithm 95% confidence limits Precision ≤ 5% Cumulative mean, f(kLimit) Nsol + f(kLimit) Nsol

  16. Precision ≤ 5% Precision > 5% Precision ≤ 5% f(kLimit) Nsol + f(kLimit) Nsol Nsol

  17. TESTING METHODOLOGY • 24 artificial data sets created: Left skewed, symmetric, right skewed; Varying values of relative standard deviation (stdev/mean). • Advantage: true mean and variance known. • Artificial data set: 100 sequences of 2000 data values. • 8 real models selected. • Different lengths of ‘look ahead’ period looked at: kLimit values = 0 (i.e. no ‘look ahead’ period), 5, 10, 25. • drequiredvalue kept constant at 5%.

  18. 5 performance measures • Coverage of the true mean • Bias • Absolute Bias • Average Nsol value • Comparison of 4. with Theoretical Nsol value • For real models: ‘true’ mean and st.dev values - estimated from whole sets of output data (3000 to 11000 data points).

  19. Results • Nsol values for individual algorithm runs are very variable. • Average Nsol values for 100 runs per model close to the theoretical values of Nsol. • Normality assumption appears robust. • Using a ‘look ahead’ period improves performance of the algorithm.

  20. Impact of different look ahead periods on performance of algorithm

  21. Eg.s of changes in Nsol & improvement in estimate of true mean

  22. Examples of changes in Nsol & improvement in estimate of true mean

  23. DISCUSSION • kLimit default value set to 5. • Initial number of replications set to 3. • Multiple response variables - Algorithm run with each response - use maximum estimated value for Nsol. • Different scenarios - advisable to repeat algorithm every few scenarios to check that precision has not degraded significantly. • Inclusion into simulation package: Full explanations of algorithm and results.

  24. SUMMARY • Selection and automation of Confidence Interval (with Specified Precision) Method for estimating the number of replications to be run in a simulation. • Algorithm created with ‘look ahead’ period -efficient and performs well on wide selection of artificial and real model output. • ‘Black box’ - fully automated and does not require user intervention.

  25. Thank you for listening. ACKNOWLEDGMENTSThis work is part of the Automating Simulation Output Analysis (AutoSimOA) project (http://www.wbs.ac.uk/go/autosimoa) that is funded by the UK Engineering and Physical Sciences Research Council (EP/D033640/1). The work is being carried out in collaboration with SIMUL8 Corporation, who are also providing sponsorship for the project. Katy Hoad, Stewart Robinson, Ruth Davies Warwick Business School OR49 Sept 07

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