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Solutions to SPC Practice Problems 1. The overall average on a process you are attempting to monitor is 50 units. The process standard deviation is know n to be 1.72. Determine the upper and low er control limits for a mean chart, if you choose to use a sample size of 5. Set z = 3. The control limits are calculated as follow s: z

n

50 3

1.72 5

50 2.308 Or (47.69, 52.31) 2. Food Storage Technologies produces refrigeration units for food producers and retail food establishments. The overall average temperature that these units maintain is 46 Fahrenheit. The average range is 2 Fahrenheit. Samples of 6 are taken to monitor the production process. Determine the upper and low er control limits for both a mean chart and a range chart for these refrigeration units. Since the standard deviation is not know n, w e w ill use the given information about the average range of the samples, together w ith the follow ing table1: Sampl e Si ze, n M ean Factor, A 2 Upper Range, D 4 2 1.880 3.268 3 1.023 2.574 4 0.729 2.282 5 0.577 2.115 6 0.483 2.004 7 0.419 1.924 8 0.373 1.864 9 0.337 1.816 10 0.308 1.777 12 0.266 1.716 Factors for Computing 3 Sigma Control Chart Limits

Low er Range, D 3 0 0 0 0 0 0.076 0.136 0.184 0.223 0.284

1 Table S6.1 from H eizer and Render, p. 204 (Factors for Computing 3 Sigma Control Chart Limits) Similar to Exhibit TN 7.7, p. 310, in Chase, Jacobs, A quilano.

For the mean chart: 46 0.483

A2 R

46 0.966 Or (45.03, 46.97) For the range chart: D3 R

LCL

0

0 D4 R

UCL

2.004

4.008 3. Sampling 4 pieces of precision-cut w ire (to be used in computer assembly) every hour for the past 24 hours has produced the follow ing results: H our 1 2 3 4 5 6 7 8 9 10 11 12

X (inches)

3.25 3.10 3.22 3.39 3.07 2.86 3.05 2.65 3.02 2.85 2.83 2.97

R (inches) 0.71 1.18 1.43 1.26 1.17 0.32 0.53 1.13 0.71 1.33 1.17 0.40

H our 13 14 15 16 17 18 19 20 21 22 23 24

X (inches)

3.11 2.83 3.12 2.84 2.86 2.74 3.41 2.89 2.65 3.28 2.94 2.64

R (inches) 0.85 1.31 1.06 0.50 1.43 1.29 1.61 1.09 1.08 0.46 1.58 0.97

Develop appropriate control charts and determine w hether there is any cause for concern in the cutting process. Plot the information and look for patterns. First, w e’ll compute the upper and low er limits for both an X chart and an R chart. For the mean chart, note that X 2.982 and R 1.024 : A2 R

2.982 0.729

024

2.982 0.746 Or (2.236, 3.728)

2 Operations

Prof. Juran

For the range chart: D3 R

LCL

0

024

0 UCL

D4 R

2.282

024

2.336 N ow , w e plot the charts: Mean Chart

R Chart

4.0

2.5

3.8 2.0

Sample Range (Inches)

Sample Mean (Inches)

3.5 3.3 3.0 2.8 2.5

1.5

1.0

0.5 2.3 2.0

0.0 1

2

3

4

5

6

7

8

9

10 11 12 13 14 15 16 17 18 19 20 21 22 23 24

1

Hour

2

3

4

5

6

7

8

9

10 11 12 13 14 15 16 17 18 19 20 21 22 23 24

Hour

Based on our rules for interpreting control charts, this process can be assumed to be in control.

3 Operations

Prof. Juran

4. Small boxes of N utraFlakes cereal are labeled “ net w eight 10 ounces.” Each hour, random samples of size n = 4 boxes are w eighed to check process control. Five hours of observations yielded the follow ing data: Ti me 9 a.m. 10 a.m. 11 a.m. 12 a.m. 1 p.m.

Box 1 9.8 10.1 9.9 9.7 9.7

Wei ghts Box 2 Box 3 10.4 9.9 10.2 9.9 10.5 10.3 9.8 10.3 10.1 9.9

Box 4 10.3 9.8 10.1 10.2 9.9

a. Using these data, construct limits for X and R charts. 10.04 and R 0.52 . For the mean chart:

We note that X

X

A2 R

10.04 0.729 0 52

10.04 0.38 Or (9.66, 10.42) For the range chart:

LCL

D3 R

0 0 520

0 UCL

D4 R

2.282 0 520

1.187 b. Is the process in control? Mean Chart

R Chart

10.50

1.4

1.2

Sample Range (Inches)

Sample Mean (Inches)

10.25

10.00

9.75

1.0

0.8

0.6

0.4

0.2

9.50

0.0 1

2

3

4

5

1

Hour

2

3

4

5

Hour

The process appears to be in control.

4 Operations

Prof. Juran

c. What other steps should the quality department follow at this point? There aren’t really very much data here, nor are the sample sizes very large. We could increase the sample size and continue to monitor the charts over time. 5. You are attempting to develop a quality monitoring system for some parts purchased from Warton & Kotha M anufacturing Co. These parts are either good or defective. You have decided to take a sample of 100 units. Develop a table of the appropriate upper and low er control chart limits for various values of the fraction defective in the sample taken. The values for p in this table should range from 0.02 to 0.10 in increments of 0.02. Develop the upper and low er control limits for a 99.73% confidence interval. H ere is how to calculate the limits for the first value of the proportion defective:

p z

p

p n

0.02 3

0.02 0 98 100

0.02 3 0140

0.02 0.0420 Or (0.0000, 0.0620) (N ote that w e have substituted zero for any negative number. In this case the low er control limit w ould have been –0.022.) H ere is the completed table of control limits: p 0.02 0.04 0.06 0.08 0.10

n = 100 UCL LCL 0.0620 0.0000 0.0988 0.0000 0.1312 0.0000 0.1614 0.0000 0.1900 0.0100

5 Operations

Prof. Juran

6. In the past, the defect rate for your product has been 1.5%. What are the upper and low er control chart limits if you w ish to use a sample size of 500 and z = 3?

p z

p

p n

0.015 3

0.015 985 500

0.015 3 0 0054

0.015 0.0163 Or (0.000, 0.0313) 7. Refer to problem 6 above. If the defect rate w ere 3.5% instead of 1.5%, w hat w ould be the control limits?

p z

p

p n

0.035 3

0.035 0 965 500

0.035 3 0082

0.035 0.0247 Or (0.0103, 0.0597) 8. Refer to problems 6 and 7 above. M anagement w ould like to reduce the sample size to 100 units. If the past defect rate has been 3.5%, w hat w ould happen to the control limits (z = 3)? Should this action be taken? Explain your answ er.

p z

p

p n

0.035 3

0.035 965 100

0.035 3 0 0182

0.035 0.0551 Or (0.0000, 0.0901) This turns out to result in a rather large increase in the w idth of the control range (the upper limit increases by more than 50%). This may have a significant negative effect on quality, by increasing the risk of a Type II error.

6 Operations

Prof. Juran

9. Blackburn, Inc., an equipment manufacturer in N ashville, has submitted a sample cutoff valve to improve your manufacturing process. Your process engineering department has conducted experiments and found that the valve has a mean ( ) of 8.00 and a standard deviation ( ) of 0.04. Your desired performance is defined by the specification limits (7.865, 8.135). What is the Cpk of the Blackburn valve? Cpk

min

SL X X LSL , 3 3

min

.135 8 8 7.865 , 3 * 0.04 3 * 0.04

min

125,1.125

1.125 This process is capable. It is centered betw een the specification limits, and its variability is sufficiently small that it w ill meet customer needs almost all of the time. 10. The manager of the Oat Flakes plant desires a quality specification w ith a mean of 16 ounces, an upper specification limit of 16.5, and a low er specification limit of 15.5. In this example, the process is know n to have a population standard deviation of 1.0 ounces. Using the data below , in w hich w eights are expressed in ounces, determine the standard deviation of the 12 w eights and then determine the Cpk of the process. Wei ght of Sampl e H our A vg. of 9 Boxes 1 16.1 2 16.8 3 15.5 4 16.5

N ote that X

Wei ght of Sampl e H our A vg. of 9 Boxes 5 16.5 6 16.4 7 15.2 8 16.4

Wei ght of Sampl e H our A vg. of 9 Boxes 9 16.3 10 14.8 11 14.2 12 17.3

16.00 . Cpk

min

SL X X LSL , 3 3

min

6.5 6.0 16.0 5.5 , 3* 1 3* 1

min 0 1667,0.1667

0.1667 This process is not capable; it is too variable to meet customer needs.

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Prof. Juran