1. ETM 607 – Input Modeling General Idea of Input Modeling Data Collection Identifying Distributions Parameter estimation Goodness of Fit tests Selecting input models without data Input modeling software (lab).

2. Model Inputs can be constants, but are typically random variables. Input models can have a significant impact on model outputs and conclusions one draws from models. Outputs are a result of simulation model. Inputs are typically collected and modeled by observing the real system or talking with “experts”. See Jerry Mobley example. ETM 607 – Input Modeling Model Inputs Model Logic Model Outputs

3. General Idea Observe process and collect data (or discuss with expert if data process or data not available). Identify a probability distribution associated with collected data. Usually involves creating a histogram and matching to known pdf’s or using input modeling software. Choose parameters for the selected distribution (e.g. mean, standard deviation, min, max, etc…) Parameters are an output from input modeling software. Evaluate for goodness of fit. ETM 607 – Input Modeling

4. How do you collect data? Historical records Stop watch and hours of observation Standards Expert advice Others? ETM 607 – Input Modeling

5. How do you collect data? Examples from my recent experience YKK MF1 Spooling Line – extensive data collection performed by intern the previous summer, and “expert” knowledge ETM 607 – Input Modeling Value (min.) / Distribution Comments newCSI Inspection Process Number of heads6newCSI 1 - newCSI 6 Number of operators1CSI Operator 1 Setup Time~Triangular(1.1,1.3,1.5)Time includes operator travel time to machine. Process Time57.454400 (yards) x.914 (meters/yards) / 70 (meters/min) Move completed zundo Time~Triangular(1.0,1.5,2.0) Time includes operator travel time to zundo. Tape Stop frequency (per shift)~Exp(400 yards) Does not require operator to fix. Only stops if tape is running. Tape Stop duration~Uniform(.6,.75)Between 36 and 45 seconds Downtime frequency~Exp(1400 yards) Includes all minor machine stops. Only stops if machine (tape) is running. Downtime duration~Triangular(1.0,1.5,2.5)Time includes operator travel time to machine. Other CSI operator duties – frequency9 / hour Includes all other operator duties. This frequency and duration can be manipulated to obtain desired operator idle time. Other CSI operator duties – duration~Triangular(1.0,1.5,3.0) OtherZundos are always available to be processed at each CSI head (process never starved). oldCSI Inspection Process Number of heads10oldCSI 1 - oldCSI 10 Number of operators1CSI Operator 2 Setup Time~Triangular(1.1,1.3,1.5) Time includes operator travel time to machine. Process Time50.274400 (yards) x.914 (meters/yards) / 80 (meters/min) Move completed zundo Time~Triangular(1.0,1.5,2.0) Time includes operator travel time to zundo. Tape Stop frequency~Exp(400 yards)Requires operator to fix. Assumed same number of tape stops as for newCSI machines. Tape Stop duration~Uniform(1.5,2.5) Time includes operator travel time to machine. Report showed average min to restart as 2.03 Downtime frequency~Exp(1400 yards) Includes all minor machine stops. Used same values as newCSI. Downtime duration~Triangular(1.1,1.2,1.5) Time includes operator travel time to machine. Used same values as newCSI.

6. How do you collect data? Examples from my recent experience Bassett Furniture – Door Assembly Department: Used extensive stopwatch time studies (I was being paid by the hour). C130 PDM/HVM: Historical data and “expert” opinion. ETM 607 – Input Modeling

7. Creating Histograms: How do you create a histogram from collected data? 1.Divide range of data into intervals (number of intervals rule of thumb: square root of sample size). 2.Label horizontal axis to conform to the intervals selected. 3.Count frequency of observations within each interval. 4.Label the vertical axis so that the total occurrence can be plotted for each interval. 5.Plot the frequencies on the vertical axis. ETM 607 – Fitting Distribution to Collected Data

8. Ink-Blot Test Suppose you collected and plotted the following. What distribution would you select? ETM 607 – Fitting Distribution to Collected Data A - D - B -C - E -

9. Ink-Blot Test Plots generated using Arena’s Input Analyzer. ETM 607 – Input Modeling A - Triangular D - Uniform B - ExponentialC - Normal E - Lognormal

10. Ink-Blot Test How would you model the following data? Insert figure 9.1 ETM 607 – Input Modeling

11. Which family of distributions fits your data (e.g. exponential, uniform, lognormal, beta, uniform, normal, etc…)? Ink-blot method using histogram of collected data. Good discussion on page 346-347 if histogram not available. ETM 607 – Selecting the Family of Distributions

12. Once a distribution is determined, how do you obtain the proper parameters? if grouped into k 2 Important Parameters frequency distributions Sample Mean Sample Variance ETM 607 – Parameter Estimation

13. Suggested Estimators for Frequently Used Distributions Insert Table 9.3 ETM 607 – Parameter Estimation

14. How well does the observed data fit the distributional form you select? Interesting comment: “If very little data are available, then a goodness-of-fit test is unlikely to reject any candidate distribution; but if a lot of data are available, then a goodness-of-fir test will likely reject all candidate distributions.” ETM 607 – Goodness of Fit Tests

15. Chi-Square Test Already reviewed when evaluating Random Numbers. In general compares distributional histogram to observed histogram. Compare this sample statistic to critical values found in Table A.6. Where k is the number of cells, s is the number parameters in the hypothesized distribution. H 0 : The random variable, X, conforms to the distributional assumption with the parameters(s) given by the parameter estimate(s). H 1 : The random variable X does not conform. ETM 607 – Goodness of Fit Tests

16. Chi-Square Test - cont Finding E i : When testing the Random number, R, we used where n is the number of observations and k is the number of cells. In general, where n is the number of observations and p i is the theoretical, hypothesized probability associated with the i th interval. ETM 607 – Goodness of Fit Tests

17. Chi-Square Test – cont Finding E i : For discrete distribution, simply substitute the value for x i into the pmf and calculate. For continuous distributions, for the cell/interval given by [a i-1, a i ]. ETM 607 – Goodness of Fit Tests

18. Chi-Square Test – cont Finding E i : Alternatively, one could require a series of p i values (that sum to 1.0) and determine the endpoints of each interval [a i, a i-1 ]. See example 9.18. ETM 607 – Goodness of Fit Tests

19. How do you develop input models when no data is available? Processes in design phase Data expensive to collect Rare occurrences Triangular distribution (min, most likely, max) better than Uniform (min, max). Obtain experimental results by making multiple runs with different input models to observe the sensitivity. Normal distribution often satisfies customer. ETM 607 – Models when No Data is Available

20. Lab Utilize Arena Input Analyzer ETM 607 – Input Modeling