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Simulation Modeling and AnalysisPowerPoint Presentation

Simulation Modeling and Analysis

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Outline

- Comparing Two Designs
- Comparing Several Designs
- Statistical Models
- Metamodeling

Comparing two designs

- Let the average measures of performance for designs 1 and 2 be 1 and 2.
- Goal of the comparison: Find point and interval estimates for 1 - 2

Example

- Auto inspection system design
- Arrivals: E(6.316) min
- Service:
- Brake check N(6.5,0.5) min
- Headlight check N(6,0.5) min
- Steering check N(5.5,0.5) min

- Two alternatives:
- Same service person does all checks
- A service person is devoted to each check

Comparing Two Designs -contd

- Run length (ith design ) = Tei
- Number of replications (ith design ) = Ri
- Average response time for replication r (ith design = Yri
- Averages and standard deviations over all replications, Y1* = S Yri / Ri and Y2* , are unbiased estimators of 1 and 2.

Possible outcomes

- Confidence interval for 1 - 2 well to the left of zero. I.e. most likely 1 < 2.
- Confidence interval for 1 - 2 well to the right of zero. I.e. most likely 1 > 2.
- Confidence interval for 1 - 2 contains zero. I.e. most likely 1 ~ 2.
- Confidence interval
(Y1* - Y2*) ± t /2, s.e.(Y1* - Y2*)

Independent Sampling with Equal Variances

- Different and independent random number streams are used to simulate the two designs.
Var(Yi*) = var(Yri)/Ri = i2/Ri

Var(Y1* - Y2*) = var(Y1*) + var(Y2*)

= 12/R1 + 22/R2= VIND

- Assume the run lengths can be adjusted to produce 12 ~22

Independent Sampling with Equal Variances -contd

- Then Y1* - Y2* is a point estimate of 1 - 2
Si2 = (Yri - Yi*)2/(Ri - 1)

Sp2 = [(R1-1) S12 + (R2-1) S22]/(R1+R2-2)

s.e.(Y1*-Y2*) = Sp (1/R1 + 1/R2)1/2

= R1 + R2 -2

Independent Sampling with Unequal Variances

s.e.(Y1*-Y2*) = (S12/R1 + S22/R2)1/2

= (S12/R1 + S22/R2)2/M

where

M = (S12/R1)2/(R1-1) + (S22/R2)2/(R2-1)

- Here R1 and R2 must be > 6

Correlated Sampling

- Correlated sampling induces positive correlation between Yr1 and Yr2 and reduces the variance in the point estimator of Y1*-Y2*
- Same random number streams used for both systems for each replication r (R1 = R2 = R)
- Estimates Yr1 and Yr2 are correlated but Yr1 and Ys2 (r n.e. s) are mutually independent.

Recall: Covariance

var(Y1* - Y2*) = var(Y1* ) + var(Y2* ) -

2 cov(Y1* , Y2* ) =

= 12/R + 22/R - 2 121 2/R = VCORR

= VIND - 2 121 2/R

Recall: definition of covariance

cov(X1,X2) = E(X1 X2) - m1 m2 =

= corr(X1 X2) s1 s2 =

= r s1 s2

Correlated Sampling -contd

- Let Dr =Yr1 - Yr2
D* = (1/R) Dr = Y1* - Y2*

SD2 = (1/(R - 1)) (Dr - D*)2

- Standard error for the 100(1- )% confidence interval
s.e.(D*) = s.e.(Y1* - Y2* ) = SD/ R

(Y1* - Y2*) ± t /2, SD/ R

Correlated Sampling -contd

- Random Number Synchronization Guides
- Dedicate a r.n. stream for a specific purpose and use as many streams as needed. Assign independent seeds to each stream at the beginning of each run.
- For cyclic task subsystems assign a r.n. stream.
- If synchronization is not possible for a subsystem use an independent stream.

Example: Auto inspection

An = interarrival time for vehicles n,n+1

Sn(1) = brake inspection time for vehicle n in model 1

Sn(2) = headlight inspection time for vehicle n in model 1

Sn(3) = steering inspection time for vehicle n in model 1

- Select R = 10, Total_time = 16 hrs

Example: Auto inspection

- Independent runs
-18.1 < 1-2 < 7.3

- Correlated runs
-12.3 < 1-2 < 8.5

- Synchronized runs
-0.5 < 1-2 < 1.3

Confidence Intervals with Specified Precision

- Here the problem is to determine the number of replications R required to achieve a desired level of precision e in the confidence interval, based on results obtained using Ro replications
R = (t /2,Ro-1 SD/e)2

Comparing Several System Designs

- Consider K alternative designs
- Performance measure i
- Procedures
- Fixed sample size
- Sequential sampling (multistage)

Comparing Several System Designs -contd

- Possible Goals
- Estimation of each i
- Comparing i to a control 1
- All possible comparisons
- Selection of the best i

Bonferroni Method for Multiple Comparisons

- Consider C confidence intervals 1-i
- Overall error probability E = j
- Probability all statements are true (the parameter is contained inside all C.I.’s)
P 1 - E

- Probability one or more statements are false
P E

Example: Auto inspection (contd)

- Alternative designs for addition of one holding space
- Parallel stations
- No space between stations in series
- One space between brake and headlight inspection
- One space between headlight and steering inspection

Bonferroni Method for Selecting the Best

- System with maximum expected performance is to be selected.
- System with maximum performance and maximum distance to the second best is to be selected.
i - max j ij

Bonferroni Method for Selecting the Best -contd

1.- Specify , and R0

2.- Make R0 replications for each of the K systems

3.- For each system i calculate Yi*

4.- For each pair of systems i and j calculate Sij2 and select the largest Smax2

5.-Calculate R = max{R0, t2 Smax2 / 2}

6.- Make R-R0 additional replications for each of the K systems

7.- Calculate overall means Yi** = (1/R) Yri

8.-Select system with largest Yi** as the best

Statistical Models to Estimate the Effect of Design Alternatives

- Statistical Design of Experiments
- Set of principles to evaluate and maximize the information gained from an experiment.

- Factors (Qualitative and Quantitative), Levels and Treatments
- Decision or Policy Variables.

Single Factor, Randomized Designs Alternatives

- Single Factor Experiment
- Single decision factor D ( k levels)
- Response variable Y
- Effect of level j of factor D, j

- Completely Randomized Design
- Different r.n. streams used for each replication at any level and for all levels.

Single Factor, Randomized Designs -contd Alternatives

- Statistical model
Yrj = + j + rj

where

Yrj = observation r for level j

= mean overall effect

j = effect due to level j

rj = random variation in observation r at level j

Rj= number of observations for level j

Single Factor, Randomized Designs -contd Alternatives

- Fixed effects model
- levels of factors fixed by analyst
- rj normally distributed
- Null hypothesis H0: j = 0 for all j=1,2,..,k
- Statistical test: ANOVA (F-statistic)

- Random effects model
- levels chosen at random
- j normally distributed

ANOVA Test Alternatives

- Levels-replications matrix
- Compute level means (over replications) Y.i* and grand mean Y..*
- Variation of the response w.r.t. Y..*
Yrj - Y..* = (Y.j* - Y..*) + (Yrj - Y.j*)

- Squaring and summing over all r and j
SSTOT = SSTREAT + SSE

ANOVA Test -contd Alternatives

- Mean square MSE = SSE/(R-k) is unbiased estimator of var(Y). I.e. E(MSE) = 2
- Mean square MSTREAT = SSTREAT/(k-1) is also unbiased estimator of var(Y).
- Test statistic
F = MSTREAT / MSE

- If H0 is true F has an F distribution with k-1 and R-k d.o.f.
- Find critical value of the statistic F1-
- Reject H0 if F > F1-

Metamodeling Alternatives

- Independent (design) variables xi, i=1,2,..,k
- Output response (random) variable Y
- Metamodel
- A simplified approximation to the actual relationship between the xi and Y
- Regression analysis (least squares)
- Normal equations

Linear Regression Alternatives

- One independent variable x and one dependent variable Y
- For a linear relationship
E(Y:x) = 0 + 1 x

- Simple Linear Regression Model
Y = 0 + 1 x +

Linear Regression -contd Alternatives

- Observations (data points)
(xi,Yi) i=1,2,..,n

- Sum of squares of the deviations i2
L =i2 = [ Yi - 0’ - 1(xi - x*)]2

- Minimizing w.r.t 0’ and 1 find
0’* = Yi /n

1*= Yi (xi - x*)/ (xi - x*)2

0* = 0’* - 1*x*

Significance Testing Alternatives

- Null Hypothesis H0: 1 = 0
- Statistic (n-2 d.o.f)
t0 = 1*/(MSE/Sxx)

where

MSE = (Yi - Ypi)/(n-2)

Sxx = xi2 - ( xi )2/n

- H0 is rejected if |t0| > t/2,n-2

Multiple Regression Alternatives

- Models
Y = 0 + 1 x1 + 2 x2 + ... + m xm +

Y = 0 + 1 x+ 2 x2 +

Y = 0 + 1 x1 + 2 x2 + 3 x1 x2 +

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