# Control charts, PERT, Forecasting, Economic Order Quantity

Problem 1. The perfect Circle Company manufactures bushings. Once each hour a sample of 125 finished bushings is drawn from the output; each bushing is examined by a technician. Those which fail are classified as defective; the rest are satisfactory. Here are data on ten consecutive samples taken in one week:

Sample no. 1 2 3 4 5 6 7 8 9 10

Defective 15 13 16 11 13 14 20 25 30 45

a. What type of control chart should be used here?

b. What is the centerline of the chart?

c. What is the lower and upper control limits (LCL and UCL) based on 99.7% confidence level?

d. What statistic should be plotted on the control chart?

e. Draw the control chart and plot using Excel.

f. Is this system under control?

g. What should the quality control engineer do?

Problem 2. A project consists of 8 activities, lettered A through H below. For each activity, the preceding activity is given, and a probabilistic estimate of the time required to complete it. Times are in days.

Preceding Optimistic Most Likely Pessimistic

Activity Activity Time Time Time

A -- 2 days 4 days 6 days

B A 3 6 9

C A 2 5 11

D -- 2 10 12

E C, 4 8 15

F B,E 2 4 12

G D 3 4 11

H F,G 1 1 1

a. Determine the expected time for each activity assuming it satisfies the beta distribution.

b. Determine the variance for each activity assuming beta distribution.

c. Draw the PERT network for this project, with the activities on the arrows.

d. List all possible paths through the network (from beginning to end).

e. What is the expected duration of each path?

f. What is the variance of each path?

g. What is the critical path(s) based on the expected durations?

h. What is the probability to finish the project in 17 days?

Problem 3. National Mixer, Inc., sells can openers. Monthly sales for a seven-month period were as follows:

Sales

Month (000 units)

Feb. . . . . . . . . . 19

Mar. . . . . . . . . . 18

Apr. . . . . . . . . 15

May . . . . . . . . . 20

Jun. . . . . . . . . . 18

Jul. . . . . . . . . . . 22

Aug. . . . . . . . . . 20

a. Plot the monthly data using Excel.

b. Forecast September sales volume using each of the following:

(1) A linear trend equation.

(2) A five-month moving average.

(3) Exponential smoothing with a smoothing constant equal to .20, assuming a February forecast of 19(000).

(4) The naïve approach.

(5) A weighted average using .60 for August, .30 for July, and .10 for June.

Problem 4. The manager of a car wash received a revised price list from the vendor who supplies soap, and the promise of a shorter lead time for deliveries. Formerly the lead time was four days, but now the vendor promises a 25% reduction in the lead time. Annual usage of soap is 4,500 gallons. The car wash is open 360 days a year. Assume that daily usage follows the normal distribution, and that it has a standard deviation of 2 gallons per day. The ordering cost is $30 and annual carrying cost is 25% of the purchasing price. The revised price list (cost per gallon) is shown in the table.

Quantity Unit Price

1-499 $2.00

500-999 1.70

1000+ 1.62

a. What order quantity is optimal?

b. What ROP is appropriate if the acceptable risk of a stockout is 1.5 percent?

I have attached the problem in the word doc.

#### Solution Summary

Answers 4 questions dealing with Control charts, PERT network, Forecasting (using linear trend equation, moving average, Exponential smoothing, optimal order quantity