1. A Different Type of Monte Carlo Simulation Jake Blanchard Spring 2010 Uncertainty Analysis for Engineers1

2. Introduction We can use Monte Carlo simulation to study the reliability of a complex system Assign a failure probability (p) to each component Sample a uniform number between 0 and 1 If this number is less than p, assume the component failed Use logic to handle redundancy, etc. Uncertainty Analysis for Engineers2

3. Example Consider a 3-stage process, as diagrammed below Our goal is to find the probability of success of the entire operation, assuming all individual probabilities are independent Branches represent parallel redundancy, so success in a stage requires success of, for example, A or B Uncertainty Analysis for Engineers3 A B C F E D Pr(C)=0.95 Pr(B)=0.8 Pr(A)=0.9 Pr(D)=0.9 Pr(F)=0.5 Pr(E)=0.9

4. Solution Pr(S)=Pr(I)Pr(II)Pr(III) where I, II, and III represent the three stages of the system Pr(I)=Pr(A)+Pr(B)-Pr(AB)=0.9+0.8-0.9*0.8=0.98 (success requires A or B to succeed) Pr(II)=Pr(C)=0.95 Pr(III)=Pr(D)+Pr(E)+Pr(F)-Pr(DE)-Pr(EF)- Pr(DF)+Pr(ABC)=0.9+0.9+0.5-0.9*0.9-0.9*0.5- 0.9*0.5+0.9*0.9*0.5=0.995 So, Pr(S)=0.98*0.95*0.995=0.926 Uncertainty Analysis for Engineers4

5. Script pa=0.9; pb=0.8; pc=0.95; pd=0.9; pe=0.9; pf=0.5; n=10000000 ag=rand(n,1)<pa; bg=rand(n,1)<pb; cg=rand(n,1)<pc; dg=rand(n,1)<pd; eg=rand(n,1)<pe; fg=rand(n,1)<pf; Ig=ag|bg; IIg=cg; IIIg=dg|eg|fg; allgood=mean(Ig&IIg&IIIg) Uncertainty Analysis for Engineers5

6. Another Example What if 2 events are not independent? Consider the system below, where event G is in both stages Uncertainty Analysis for Engineers6 G H J G Pr(G)=0.9 Pr(H)=0.8 Pr(J)=0.7

7. Solution Pr(S)=Pr(G)+Pr(HJ)-Pr(G)*Pr(HJ) So, Pr(S)=0.9+0.8*0.7-0.9*0.8*0.7 Pr(S)=0.956 Uncertainty Analysis for Engineers7

8. Script clear all pg=0.9; ph=0.8; pj=0.7; n=10000000 gg=rand(n,1)<pg; hg=rand(n,1)<ph; jg=rand(n,1)<pj; Ig=gg|hg; IIg=gg|jg; allgood=mean(Ig&IIg) Uncertainty Analysis for Engineers8