Question

A production line has three machines A, B, and C, with reliabilities of .99, .96, and .93, respectively. The machines are arranged so that if one breaks down, the others must shut down. Engineers are weighing two alternative designs for increasing the line’s reliability. Plan 1 involves adding an identical backup line, and plan 2 involves providing a backup for each machine. In either case, three machines (A, B, and C) would be used with reliabilities equal to the original three.

a. Which plan will provide the higher reliability?

b. Explain why the two reliabilities are not the same.

c. What other factors might enter into the decision of which plan to adopt?

Answer 1

a.         Prob(line works) = .99 × .96 × .93 = .8839

#1:  

Prob(line works) + Prob(line fails) × Prob(backup works)

= .8839 + [(1 - .8839) × (.8839)] = .9865

#2:

Prob:

.99 + [(1 - .99) × .99]  = .9999

.96 + [(1 - .96) × .96]  = .9984

.93 + [(1 - .93) × .93]  = .9951

Overall: .9999 × .9984 × .9951 = .9934

Plan 2 is better (.9934 > .9865)

b.

In #1 the system will fail if any one of original and any one of backup machines fail.

In #2 the system will fail if any one of original and its backup machine both fail.

So, overall failing is more likely for #1.