Simulation Modeling and Analysis Session 12 Comparing Alternative System Designs

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 (i th design ) = T ei Number of replications (i th design ) = R i Average response time for replication r (i th design = Y ri Averages and standard deviations over all replications, Y 1 * =  Y ri / R i and Y 2 *, 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 (Y 1 * - Y 2 *) ± t  /2, s.e.(Y 1 * - Y 2 *)

Independent Sampling with Equal Variances Different and independent random number streams are used to simulate the two designs. Var(Y i *) = var(Y ri )/R i =  i 2 /R i Var(Y 1 * - Y 2 *) = var(Y 1 *) + var(Y 2 *) =  1 2 /R 1 +  2 2 /R 2 = V IND Assume the run lengths can be adjusted to produce  1 2 ~  2 2

Independent Sampling with Equal Variances -contd Then Y 1 * - Y 2 * is a point estimate of  1 -  2 S i 2 =  (Y ri - Y i *) 2 /(R i - 1) S p 2 = [(R 1 -1) S (R 2 -1) S 2 2 ]/(R 1 +R 2 -2) s.e.(Y 1 *-Y 2 *) = S p (1/R 1 + 1/R 2 ) 1/2 = R 1 + R 2 -2

Independent Sampling with Unequal Variances s.e.(Y 1 *-Y 2 *) = (S 1 2 /R1 + S 2 2 /R 2 ) 1/2 = (S 1 2 /R 1 + S 2 2 /R 2 ) 2 /M where M = (S 1 2 /R 1 ) 2 /(R 1 -1) + (S 2 2 /R 2 ) 2 /(R 2 -1) Here R 1 and R 2 must be > 6

Correlated Sampling Correlated sampling induces positive correlation between Y r1 and Y r2 and reduces the variance in the point estimator of Y 1 *-Y 2 * Same random number streams used for both systems for each replication r (R 1 = R 2 = R) Estimates Y r1 and Y r2 are correlated but Y r1 and Y s2 (r n.e. s) are mutually independent.

Recall: Covariance var(Y 1 * - Y 2 *) = var(Y 1 * ) + var(Y 2 * ) - 2 cov(Y 1 *, Y 2 * ) =  1 2 /R +  2 2 /R - 2  12  1  2 /R = V CORR = V IND - 2  12  1  2 /R Recall: definition of covariance cov(X1,X2) = E(X1 X2) -  1  2 = = corr(X1 X2)  1  2 = =  1  2

Correlated Sampling -contd Let D r =Y r1 - Y r2 D* = (1/R)  D r = Y 1 * - Y 2 * S D 2 = (1/(R - 1))  (D r - D*) 2 Standard error for the 100(1-  )% confidence interval s.e.(D*) = s.e.(Y 1 * - Y 2 * ) = S D /  R (Y 1 * - Y 2 *) ± t  /2, S D /  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 A n = interarrival time for vehicles n,n+1 S n (1) = brake inspection time for vehicle n in model 1 S n (2) = headlight inspection time for vehicle n in model 1 S n (3) = steering inspection time for vehicle n in model 1 Select R = 10, Total_time = 16 hrs

Example: Auto inspection Independent runs <  1 -  2 < 7.3 Correlated runs <  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  in the confidence interval, based on results obtained using Ro replications R = (t  /2,Ro  S D /  

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  i  j  

Bonferroni Method for Selecting the Best -contd 1.- Specify ,  and R Make R 0 replications for each of the K systems 3.- For each system i calculate Y i * 4.- For each pair of systems i and j calculate S ij 2 and select the largest S max Calculate R = max{R 0, t 2 S max 2 /  2 } 6.- Make R-R0 additional replications for each of the K systems 7.- Calculate overall means Y i ** = (1/R)  Y ri 8.-Select system with largest Y i ** 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 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 Statistical model Y rj =  +  j +  rj where Y rj = observation r for level j  = mean overall effect  j = effect due to level j  rj = random variation in observation r at level j R j = number of observations for level j

Single Factor, Randomized Designs -contd Fixed effects model – levels of factors fixed by analyst –  rj normally distributed –Null hypothesis H 0 :  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 Levels-replications matrix Compute level means (over replications) Y. i * and grand mean Y..* Variation of the response w.r.t. Y..* Y rj - Y..* = (Y. j * - Y..*) + (Y rj - Y. j *) Squaring and summing over all r and j SS TOT = SS TREAT + SS E

ANOVA Test -contd Mean square MS E = SS E /(R-k) is unbiased estimator of var(Y). I.e. E(MS E ) =  2 Mean square MS TREAT = SS TREAT /(k-1) is also unbiased estimator of var(Y). Test statistic F = MS TREAT / MS E If H 0 is true F has an F distribution with k-1 and R-k d.o.f. Find critical value of the statistic F 1-  Reject H 0 if F > F 1- 

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

Linear Regression 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 Observations (data points) (x i,Y i ) i=1,2,..,n Sum of squares of the deviations  i 2 L =   i 2 =  [ Y i -  0 ’ -  1 (x i - x*)] 2 Minimizing w.r.t  0 ’ and  1 find  0 ’* =  Y i /n  1 * =  Y i (x i - x*)/  (x i - x*) 2  0 * =  0 ’* -  1 * x*

Significance Testing Null Hypothesis H 0 :  1 = 0 Statistic (n-2 d.o.f) t 0 =  1 * /  (MS E /S xx ) where MS E =  (Y i - Y pi )/(n-2) S xx =  x i 2 - (  x i ) 2 /n H 0 is rejected if |t 0 | > t  /2,n-2

Multiple Regression Models Y =  0 +  1 x 1 +  2 x  m x m +  Y =  0 +  1 x +  2 x 2 +  Y =  0 +  1 x 1 +  2 x 2 +  3 x 1 x 2 + 