Answer:
a) Reliability of the Robot = 0.7876
b1) Component 1: 0.8034
Component 2: 0.8270
Component 3: 0.8349
Component 4: 0.8664
b2) Component 4 should get the backup in order to achieve the highest reliability.
c) Component 4 should get the backup with a reliability of 0.92, to obtain the highest overall reliability i.e. 0.8681.
Step-by-step explanation:
<u>Component Reliabilities:</u>
Component 1 (R1) : 0.98
Component 2 (R2) : 0.95
Component 3 (R3) : 0.94
Component 4 (R4) : 0.90
a) Reliability of the robot can be calculated by considering the reliabilities of all the components which are used to design the robot.
Reliability of the Robot = R1 x R2 x R3 x R4
= 0.98 x 0.95 x 0.94 x 0.90
Reliability of the Robot = 0.787626 ≅ 0.7876
b1) Since only one backup can be added at a time and the reliability of that backup component is the same as the original one, we will consider the backups of each of the components one by one:
<u>Reliability of the Robot with backup of component 1</u> can be computed by first finding out the chance of failure of the component along with its backup:
Chance of failure = 1 - reliability of component 1
= 1 - 0.98
= 0.02
Chance of failure of component 1 along with its backup = 0.02 x 0.02 = 0.0004
So, the reliability of component 1 and its backup (R1B) = 1 - 0.0004 = 0.9996
Reliability of the Robot = R1B x R2 x R3 x R4
= 0.9996 x 0.95 x 0.94 x 0.90
Reliability of the Robot = 0.8034
<u>Similarly, to find out the reliability of component 2:</u>
Chance of failure of component 2 = 1 - 0.95 = 0.05
Chance of failure of component 2 and its backup = 0.05 x 0.05 = 0.0025
Reliability of component 2 and its backup (R2B) = 1 - 0.0025 = 0.9975
Reliability of the Robot = R1 x R2B x R3 x R4
= 0.98 x 0.9975 x 0.94 x 0.90
Reliability of the Robot = 0.8270
<u>Reliability of the Robot with backup of component 3 can be computed as:</u>
Chance of failure of component 3 = 1 - 0.94 = 0.06
Chance of failure of component 3 and its backup = 0.06 x 0.06 = 0.0036
Reliability of component 3 and its backup (R3B) = 1 - 0.0036 = 0.9964
Reliability of the Robot = R1 x R2 x R3B x R4
= 0.98 x 0.95 x 0.9964 x 0.90
Reliability of the Robot = 0.8349
<u>Reliability of the Robot with backup of component 4 can be computed as:</u>
Chance of failure of component 4 = 1 - 0.90 = 0.10
Chance of failure of component 4 and its backup = 0.10 x 0.10 = 0.01
Reliability of component 4 and its backup (R4B) = 1 - 0.01 = 0.99
Reliability of the Robot = R1 x R2 x R3 x R4B
= 0.98 x 0.95 x 0.94 x 0.99
Reliability of the Robot = 0.8664
b2) According to the calculated values, the <u>highest reliability can be achieved by adding a backup of component 4 with a value of 0.8664</u>. So, <u>Component 4 should get the backup in order to achieve the highest reliability.</u>
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c) 0.92 reliability means the chance of failure = 1 - 0.92 = 0.08
We know the chances of failure of each of the individual components. The <u>chances of failure</u> of the components along with the backup can be computed as:
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
So, the <u>reliability for each of the component & its backup</u> is:
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
<u>The reliability of the robot with backups</u> for each of the components can be computed as:
Reliability with Component 1 Backup = R1BB x R2 x R3 x R4
= 0.9984 x 0.95 x 0.94 x 0.90
Reliability with Component 1 Backup = 0.8024
Reliability with Component 2 Backup = R1 x R2BB x R3 x R4
= 0.98 x 0.9960 x 0.94 x 0.90
Reliability with Component 2 Backup = 0.8258
Reliability with Component 3 Backup = R1 x R2 x R3BB x R4
= 0.98 x 0.95 x 0.9952 x 0.90
Reliability with Component 3 Backup = 0.8339
Reliability with Component 4 Backup = R1 x R2 x R3 x R4BB
= 0.98 x 0.95 x 0.94 x 0.9920
Reliability with Component 4 Backup = 0.8681
<u>Component 4 should get the backup with a reliability of 0.92, to obtain the highest overall reliability i.e. 0.8681. </u>