Question 17.11: Consider a system, shown in Figure 17.13, with six component......
Consider a system, shown in Figure 17.13, with six components, which has the following reliability block diagram.
The reliabilities of the components are as follows:
R_{1} = 0.95
R_{2}=0.90
R_{3} = 0.80
R_{4} = 0.85
R_{5} = 0.75
R_{6} = 0.90.
(a) Find the exact reliability of the system using the series-parallel model.
(b) Find all the minimum paths and minimum cuts for the above system.
(c) Find the lower bound and the upper bound on the system reliability using the equations for the bounds on system reliability, which uses the minimum paths and minimum cuts.
(a) R_S=R_{1234} \times R_{56} \\ =\left[1-\left(1-R_1 R_2\right)\left(1-R_3\right)\left(1-R_4\right)\right] \times\left[1-\left(1-R_5\right)\left(1-R_6\right)\right] \\ =[1-(1-0.95 \times 0.90)(1-0.80)(1-0.85)] \times[1-(1-0.75)(1-0.90)] \\ =0.970759.
(b) The table below
(c) Using Equation 17.61, we have
R_U = 1-\left(1-R_1 R_2 R_5\right)\left(1-R_1 R_2 R_6\right)\left(1-R_3 R_5\right)\left(1-R_3 R_6\right)\left(1-R_4 R_5\right)\left(1-R_4 R_6\right) \\ =1-(1-0.95 * 0.90 * 0.75)(1-0.95 * 0.90 * 0.90)(1-0.80 * 0.75) \\ *(1-0.80 * 0.90)(1-0.85 * 0.75)(1-0.85 * 0.90) \\ =0.999211and
R_L=\left[1-\left(1-R_5\right)\left(1-R_6\right)\right]\left[1-\left(1-R_1\right)\left(1-R_3\right)\left(1-R_4\right)\right]\left[1-\left(1-R_2\right)\left(1-R_3\right)\left(1-R_4\right)\right] \\ =[1-0.25 * 0.10][1-0.05 * 0.2 * 0.15][1-0.1 * 0.2 * 0.15] \\ =0.970617.
\prod_{k=1}^S P\left[\beta_k(X)=1\right] \leq P[\phi(X)=1] \leq 1-\prod_{j=1}^p\left\{1-P\left[\alpha_j(X)\right]=1\right\} (17.61)
Thus, the reliability bounds are 0.970617 ≤ R_S ≤ 0.999211.
The lower bound is much better because there is less dependency between the minimum cuts (fewer components share different minimum cuts) than for minimum paths (where some components are part of several minimum paths).
| Components for minimal paths | Components for minimal cuts |
| 1, 2, 5 | 5, 6 |
| 1, 2, 6 | 1, 3, 4 |
| 3, 5 | 2, 3, 4 |
| 3, 6 | |
| 4, 5 | |
| 4, 6 |