Chapter 4 Homework Problems
COB 300C – Mike Busing
*******CAUTION – SOME SOLUTIONS IN BACK OF TEXTBOOK ARE INCORRECT. THE FOLLOWING SOLUTIONS ARE CORRECT.************
6. One of the industrial robots designed by a leading producer of servomechanisms has four major components. Components’ reliabilities are .98, .95, .94, and .90. All of the components must function in order for the robot to operate effectively.
a. Compute the reliability of the robot.
The key here is that ALL FOUR COMPONENTS must work in order for the robot to operate effectively. The four component reliabilities are INDEPENDENT EVENTS (one working or failing does not affect the probability that another will work or fail). This being the case, we know from simple probability theory that the probability of success (robot operating effectively) is the product of the four component reliabilities.
.98 X .95 X .94 X .90 = .7876
b. 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. Which component should get the backup in order to achieve the highest reliability?
Intuitively, we might think that a process is only as good as its weakest component. WE’RE EXACTLY RIGHT! That being the case, the component with reliability of .90 should be given the backup (let’s call it component #4). This is known as REDUNDANCY. We know that the probability of component #4 working and the probability of the backup for component #4 working are independent events. Additionally, the probabilities of the five components (4 main + 1 backup) being reliable are also independent events. Again, by employing simple probability theory we can find the probability that the robot operates effectively.
REVISED COMPONENT 4 RELIABILITY:
[# 4 functions] [# 4 fails & backup for #4 functions]
.90 + (1 - .90) X (.90) = .99
Therefore, with the backup in place, component 4 (with backup) has a reliability of .99
The new ROBOT RELIABILITY IS:
.98 X .95 X .94 X .99 = .8664
- If one backup with a reliability of .92 can be added to any one of the main components, which component should get it to obtain the highest overall reliability?
Again, the correct approach is to try to increase the weak link as much as possible.
REVISED COMPONENT 4 RELIABILITY:
[# 4 functions] [# 4 fails & backup for #4 functions]
.90 + (1 - .90) X (.92) = .9920
Therefore, with the backup in place, component 4 (with backup) has a reliability of .9920
The new ROBOT RELIABILITY IS:
.98 X .95 X .94 X .9920 = .8681
16. An office manager has received a report from a consultant that includes a section on equipment replacement. The report indicates that scanners have a service life that is normally distributed with a mean of 41 months and a standard deviation of 4 months. On the basis of this information, determine the percentage of scanners that can be expected to fail in the following time periods:
a. Before 38 months of service.
-.75 0
z-scale
38 41
months
You can employ an excel function (NORMSDIST) to find the probability corresponding to the calculated z value OR simply use Appendix B (Caution! Appendix B has errors).
b. Between 40 and 45 months of service.
-0.25 0 1.00
z-scale
40 41 45
months
To obtain the probability identified above, we subtract the probability corresponding to z=-.25 from the probability corresponding to z=1.00 à Probability = .841345-.401294 = .4401
c. Within - 2 months of mean life.
-0. 5 0
z-scale
39 41
months
To obtain the probability identified above, we subtract the probability corresponding to z = -.5 from the probability corresponding to z =0.000 à Probability = .5000-.3085 = .1915
21. A manager must decide between two machines. The manager will take into account each machine’s operating costs and initial costs, and its breakdown and repair times. Machine A has a projected average operating time of 142 hours and a projected average repair time of seven hours. Projected times for machine B are an average operating time of 65 hours and a repair time of two hours. What are the projected availabilities of each machine?
Machine A
Machine B