There are three components, A, B, and C, and they are represented by different b

Question

There are three components, A, B, and C, and they are represented by different blocks in the following two reliability block diagrams. Both reliability block diagrams use the same component twice. Let the reliabilities of the components be denoted by RA, RB, and RC. Is there a difference in reliability between the two configurations when the failures or success of all the components are independent of each other? Which system configuration or reliability block diagram has higher reliability? Explain your answer.

Explanation / Answer

Step 1:
To find the reliabilities of Configurations 1 & 2:

Configuration: 1

In top chain, A, B, and C are series.

So,

Reliability of top chain = RA RB RC

In bottom chain, A,B and C are in series.

So,

Reliability of bottom chain = RA RB RC.

Top chain & bottom chains are in parallel.

So,

Reliability of Configuration 1 is:

R1 = 1 - (1 - RA RB RC)2                        (1)

Configuration 2:

A and A are in parallel.

So,

Reliability of AA is:
RAA = 1 - (1-RA)2

B and B are in parallel.

So,

Reliability of BB is:
RBB = 1 - (1-RB)2

C and C are in parallel.

So,

reliability of CC is:
RCC = 1 - (1-RC)2

AA, BB and CC are in series.

So, reliability of Configuration 2 is:
R2 = [1-(1-RA)2] X [1 - (1 - RB)2] X [1- (1 - RC)2]                      (2)

Step 2:

To find out which is greater: R1 or R2:
Let

the reliabilities of A, B and C be 0.9.

Substituting in (1), we get:

R1 = 1 - (1 - 0.9 X 0.9 X 0.9)2 = 0.9266

Substituting in (2), we get:

R2 = [1-0.12]3 = 0.9703

It is seen that R2 > R1.

So,

Configuration 2 has higher reliability.

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