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

ANSWER :

a)

Plan 1 :

Reliability of machines A, B, C in series = 0.99*0.96*0.93 = 0.883872 = 0.8839

Reliability of back up machines = 0.8839 (being identical to A, B, C)

Failure possibility of original set up = Failure of back up system = (1 - 0.8839)

Reliability of the Plan 1 (with back up set) 

= 1 - Failure of  both main and back up sets

= 1 - (1 - 0.8839)(1 - 0.8839)

= 0.9865 

Plan 2 :

Each machine has a back up individually :

Reliability of machine A with back up = 1 - (1 - 0.99)(1- 0.99) = 0.9999

Reliability of machine B with back up = 1 - (1 - 0.96)(1- 0.96) = 0.9984

Reliability of machine C with back up = 1 - (1 - 0.93)(1- 0.93) = 0.9951

Reliability of Plan 2  = 0.9999 * 0.9984 * 0.9951 = 0.9934 .

Hence, Plan 2 has higher reliability of 0.9934 (ANSWER)

b) 

In Plan 1, any one of the 3 machines failure in both the main line and back up line will 

cause stoppage of the system. So, failure probability will be higher and reliability lower.

In Plan 2, individual machine A or B or C and their corresponding back up failure

will cause the system to fail. This lowers overall probability of failure and reliability 

Improves.

c)

Other factors that may affect the decision may be :

Flow of material : 

Flow of material in Plan 2 will vary from case to case of failure, while in Plan 1

Flow is linear and simple either through main line or through back up line. 

Space requirement , costs etc. are not much different. 

Answer 2

Following is the diagram of Plan 1 and Plan 2 designs for reliability

Plan 1 – Identical Backup line

Plan 2 – Backup plan

a)

The reliability of Plan 1 is shown as follows:

P (line work) + P (line fail) × P (backup work)

the reliability of plan 2 is shown as follow

P (line work) + P (line fail) × P (backup work)

Thus, 0.9934 > 0.9865

Since the reliability of Plan 2 is greater than Plan 1. Therefore, Plan 2 is better

b)

The reliability of both the plan is not same because in Plan 1, the system will fail if any one original and backup fails. In plan 2, the system will fail only if component and backup fails

c)

The factors that might enter into the decision are as follows:

1. Distinguish between space for line and individual backups

2. When needed, backups can easily shift

3. There might be cost differences

Answer 3

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