## One of the industrial robots designed by a leading producer of servomechanisms has four major components. Components’ reliabilities are 0.98

One of the industrial robots designed by a leading producer of servomechanisms has four major components. Components’ reliabilities are 0.98, 0.95, 0.94, and 0.90. All of the components must function in order for the robot to operate effectively. a. Compute the reliability of the robot. (Round your answer to 4 decimal places.) Reliability b1. Designers want to improve the reliability by adding a backup component. Due to space limitations, only one backup can be added. The backup for any component will have the same reliability as the unit for which it is the backup. Compute the reliability of the robot. (Round your answers to 4 decimal places.) Reliability Component 1 Component 2 Component 3 Component 4 b2. Which component should get the backup in order to achieve the highest reliability? Component 1 Component 2 Component 3 Component 4 c. If one backup with a reliability of 0.92 can be added to any one of the main components, which component should get it to obtain the highest overall reliability? Component 1 Component 2 Component 3 Component 4

## Answers ( )

Answer:a)

Reliability of the Robot = 0.7876b1) Component 1:

0.8034Component 2:

0.8270Component 3:

0.8349Component 4:

0.8664b2)

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:Component Reliabilities:Component 1 (

R1) : 0.98Component 2 (

R2) : 0.95Component 3 (

R3) : 0.94Component 4 (

R4) : 0.90a)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 ≅ 0.7876b1)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:Reliability of the Robot with backup of component 1can 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 = 0.9996Reliability 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 = 0.8034Similarly, to find out the reliability of component 2: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 = 0.9975Reliability 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 = 0.8270Reliability of the Robot with backup of component 3 can be computed as: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 = 0.9964Reliability 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 = 0.8349Reliability of the Robot with backup of component 4 can be computed as: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 = 0.99Reliability 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 = 0.8664b2)According to the calculated values, thehighest reliability can be achieved by adding a backup of component 4 with a value of 0.8664. So,Component 4 should get the backup in order to achieve the highest reliability.c)0.92 reliability means the chance of failure = 1 – 0.92 =0.08We know the chances of failure of each of the individual components. The

of the components along with the backup can be computed as:chances of failureComponent 1 = 0.02 x 0.08 =

0.0016Component 2 = 0.05 x 0.08 =

0.0040Component 3 = 0.06 x 0.08 =

0.0048Component 4 = 0.10 x 0.08 =

0.0080So, the

is:reliability for each of the component & its backupComponent 1 (

R1BB) = 1 – 0.0016 =0.9984Component 2 (

R2BB) = 1 – 0.0040 =0.9960Component 3 (

R3BB) = 1 – 0.0048 =0.9952Component 4 (

R4BB) = 1 – 0.0080 =0.9920for each of the components can be computed as:The reliability of the robot with backupsReliability 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 = 0.8024Reliability 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 = 0.8258Reliability 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 = 0.8339Reliability 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 = 0.8681Component 4 should get the backup with a reliability of 0.92, to obtain the highest overall reliability i.e. 0.8681.