Answer:
a) Robot Reliability = 0.7876
b1) Component 1: 0.8034
Component 2: 0.8270
Component 3: 0.8349
Component 4: 0.8664
b2) To maximize overall reliability, Component 4 should be backed up.
c) To achieve the highest reliability of 0.8681, backup for Component 4 with a reliability of 0.92 should be implemented.
Step-by-step explanation:
Component Reliabilities:
Component 1 (R1): 0.98
Component 2 (R2): 0.95
Component 3 (R3): 0.94
Component 4 (R4): 0.90
a) The reliability of the robot can be determined by calculating the reliabilities of the individual components that constitute the robot.
Robot Reliability = R1 x R2 x R3 x R4
= 0.98 x 0.95 x 0.94 x 0.90
Robot Reliability = 0.787626 ≅ 0.7876
b1) As only a single backup can be used at once, and its reliability matches that of the original, we evaluate each component's backup sequentially:
Robot Reliability with Component 1 backup is calculated by first assessing the failure probability of the component plus its backup:
Failure probability = 1 - R1
= 1 - 0.98
= 0.02
Combined failure probability for Component 1 and backup = 0.02 x 0.02 = 0.0004
Thus, reliability of combined Component 1 and backup (R1B) = 1 - 0.0004 = 0.9996
Robot Reliability = R1B x R2 x R3 x R4
= 0.9996 x 0.95 x 0.94 x 0.90
Robot Reliability = 0.8034
To determine reliability of Component 2:
Failure probability for Component 2 = 1 - 0.95 = 0.05
Combined failure probability of Component 2 and backup = 0.05 x 0.05 = 0.0025
Reliability of Component 2 with backup (R2B) = 1 - 0.0025 = 0.9975
Robot Reliability = R1 x R2B x R3 x R4
= 0.98 x 0.9975 x 0.94 x 0.90
Robot Reliability = 0.8270
Robot Reliability with backup of Component 3 calculates as follows:
Failure probability for Component 3 = 1 - 0.94 = 0.06
Combined failure probability of Component 3 and backup = 0.06 x 0.06 = 0.0036
Reliability for Component 3 with backup (R3B) = 1 - 0.0036 = 0.9964
Robot Reliability = R1 x R2 x R3B x R4
= 0.98 x 0.95 x 0.9964 x 0.90
Robot Reliability = 0.8349
Robot Reliability with Component 4 backup calculates as:
Failure probability for Component 4 = 1 - 0.90 = 0.10
Combined failure probability of Component 4 and backup = 0.10 x 0.10 = 0.01
Reliability for Component 4 and backup (R4B) = 1 - 0.01 = 0.99
Robot Reliability = R1 x R2 x R3 x R4B
= 0.98 x 0.95 x 0.94 x 0.99
Robot Reliability = 0.8664
b2) The best reliability is achieved with the backup of Component 4, yielding a value of 0.8664. Thus, Component 4 is the best candidate for backup to optimize reliability.
c) A reliability of 0.92 indicates a failure probability of = 1 - 0.92 = 0.08
We can compute the probability of failure for each component along with its backup:
Component 1 = 0.02 x 0.08 = 0.0016
Component 2 = 0.05 x 0.08 = 0.0040
Component 3 = 0.06 x 0.08 = 0.0048
Component 4 = 0.10 x 0.08 = 0.0080
Thus, the reliabilities for each component and its backup become:
Component 1 (R1BB) = 1 - 0.0016 = 0.9984
Component 2 (R2BB) = 1 - 0.0040 = 0.9960
Component 3 (R3BB) = 1 - 0.0048 = 0.9952
Component 4 (R4BB) = 1 - 0.0080 = 0.9920
Reliability of robot including backups for each of the components can be calculated as:
Reliability with Backup for Component 1 = R1BB x R2 x R3 x R4
= 0.9984 x 0.95 x 0.94 x 0.90
Reliability with Backup for Component 1 = 0.8024
Reliability with Backup for Component 2 = R1 x R2BB x R3 x R4
= 0.98 x 0.9960 x 0.94 x 0.90
Reliability with Backup for Component 2 = 0.8258
Reliability with Backup for Component 3 = R1 x R2 x R3BB x R4
= 0.98 x 0.95 x 0.9952 x 0.90
Reliability with Backup for Component 3 = 0.8339
Reliability with Backup for Component 4 = R1 x R2 x R3 x R4BB
= 0.98 x 0.95 x 0.94 x 0.9920
Reliability with Backup for Component 4 = 0.8681
To maximize overall reliability, Component 4 should be backed up at a reliability of 0.92, achieving an overall reliability of 0.8681.
