A cafeteria serving line has a coffee urn (like a system) from which customers serve them selves. Arrivals at the urn follow a Poisson distribution at the rate of 3 per minute. In serving themselves, customers take about 15 seconds, exponentially distributed. (a) How many customers would you expect to see, on average, at the coffee urn? (b) How long would you expect it to take to get a cup of coffee? (c) What percentage of time is the urn being used?
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Production & Operation Management
Home Assignment 2
Deadline : May 22nd(Mon) 2022, 23:59
1. (10-10) A cafeteria serving line has a coffee urn (like a system) from which
customers serve them selves. Arrivals at the urn follow a Poisson distribution at the
rate of 3 per minute. In serving themselves, customers take about 15 seconds,
exponentially distributed.
(a) How many customers would you expect to see, on average, at the coffee urn?
(b) How long would you expect it to take to get a cup of coffee?
(c) What percentage of time is the urn being used?
(d) What is the probability that 3 or more people are in the cafeteria?
2. (10-9) The Heart Association plans to install a free blood pressure testing booth.
Previous experience indicates that, on average, 10 persons per hour request a test.
Assume arrivals are Poisson distributed from an infinite population. Blood pressure
measurements can be made at a constant time of 5 minutes each.
(a) What average number in line can be expected?
(b) What average number of persons can be expected to be in the system?
(c) On average, how much time will it take to measure a person’s blood pressure,
including waiting time?
(d) On weekends, the arrival rate can be expected to increase to over 12 per hour.
What effect will this have on the number in the waiting line?
3. (10-16) customers enter the camera store at the average rate of 6 per hour. the store
is staffed by 1 employee, who takes an average of 6 minutes to serve each arrival. A
new clerk has been hired for the camera store. who also takes an average of 6
minutes to serve each arrival. How long would a customer expect to spend in the
store now? Assume this is a simple Poisson arrival, exponentially distributed service
time situation (use Exhibit 10.9 in textbook or Page 263 in lecture note).
4. (10-11) An engineering firm retains a technical specialist to assist 4 design engineers
working on a project. On average, each help request for assistance takes the
specialist 1 hour. The engineers require help from the specialist on the average of
once each day. Because each assistance takes about an hour, each engineer can work
for 7 hours, on average, without assistance.
(a) How many engineers, on average, are waiting for the technical specialist for help?
(b) What is the average time an engineer has to wait for the specialist?
(c) What is the probability an engineer will have to wait in line for the specialist?
Please refer to the table below or Exhibit 10.10 in the textbook.
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5. Using Exhibit 10.14 in the textbook (or Page 277 in lecture note), simulate 10 arrivals
and fill out the table below (colored cell).
6. (13-5) Design specifications require that a key dimension on a product measure 100 ± 10
units. A process being considered for producing this product has a standard deviation
of 4 units.
(a) What is process capability here?
(b) Suppose the process average shifts to 92. Calculate the new process capability.
(c) After (b), what percentage of the items produced will be defective?
| Bob | Ray | |||||||||
| item # |
random # |
start time |
performan ce time |
finish time |
wait time |
random # |
start time |
performan ce time |
finish time |
wait time |
| 1 | 10 | 10 | ||||||||
| 2 | 21 | 21 | ||||||||
| 3 | 61 | 61 | ||||||||
| 4 | 28 | 28 | ||||||||
| 5 | 02 | 02 | ||||||||
| 6 | 12 | 12 | ||||||||
| 7 | 53 | 53 | ||||||||
| 8 | 23 | 23 | ||||||||
| 9 | 16 | 16 | ||||||||
| 10 | 83 | 83 |
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7. (13-7) 10 samples of 15 parts each were taken from an
ongoing process to establish a p-chart for control. The
samples and the number of defectives inn each are
shown in the following table. Draw a p-chart for 95%
confidence (1.96 standard deviation).
| Sample | n | # of defects |
| 1 | 15 | 3 |
| 2 | 15 | 1 |
| 3 | 15 | 0 |
| 4 | 15 | 0 |
| 5 | 15 | 0 |
| 6 | 15 | 2 |
| 7 | 15 | 0 |
| 8 | 15 | 3 |
| 9 | 15 | 1 |
| 10 | 15 | 0 |
8. (13-8) A shirt manufacturer buys cloth by the 100-yard roll
from a supplier. For setting up a control chart to manage
the irregularities, the following data were collected from a
sample provided by the supplier. Using the data, draw a
c-chart with z=2.
| Sample | Irregularities |
| 1 | 3 |
| 2 | 5 |
| 3 | 2 |
| 4 | 6 |
| 5 | 5 |
| 6 | 4 |
| 7 | 6 |
| 8 | 3 |
| 9 | 4 |
| 10 | 5 |
9. & 10. (13-13) The following table contains
the measurements of the key length
dimension from a fuel injector. These
samples of size 5 were taken at one-hour
intervals. Draw a three-sigma —chart
and R-chart for the length of fuel injector
(Use Exhibit 13.7 or Page 363 in lecture
note).
| Sample | Observations | ||||
| 1 | 2 | 3 | 4 | 5 | |
| 1 | .486 | .499 | .493 | .511 | .481 |
| 2 | .499 | .506 | .516 | .494 | .529 |
| 3 | .496 | .500 | .515 | .488 | .521 |
| 4 | .495 | .506 | .483 | .487 | .489 |
| 5 | .472 | .502 | .526 | .469 | .481 |
| 6 | .473 | .495 | .507 | .493 | .506 |
| 7 | .495 | .512 | .490 | .471 | .504 |
| 8 | .525 | .501 | .498 | .474 | .485 |
| 9 | .497 | .501 | .517 | .506 | .516 |
| 10 | .495 | .505 | .516 | .511 | .497 |
| 11 | .495 | .482 | .468 | .492 | .492 |
| 12 | .483 | .459 | .526 | .506 | .522 |
| 13 | .521 | .512 | .493 | .525 | .510 |
| 14 | .487 | .521 | .507 | .501 | .500 |
| 15 | .493 | .516 | .499 | .511 | .513 |
| 16 | .473 | .506 | .479 | .480 | .523 |
| 17 | .477 | .485 | .513 | .484 | .496 |
| 18 | .515 | .493 | .493 | .485 | .475 |
| 19 | .511 | .536 | .486 | .497 | .491 |
| 20 | .509 | .490 | .470 | .504 | .512 |
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