Chapter 13

quality control and improvement

13-1.A production process is monitored by tracking various numeric parameters pertaining to the items produced, the tolerances desired, etc. Quality control/improvement is a means of checking and maintaining standards in the process, with the help of sampling methods and statistical indicators.

13-2.Variance, particularly the effort to minimize it for the production parameters of interest.

13-3.1) Natural, random variation;2) variation due to assignable causes.

13-4.A control chart is a time plot of a statistic, such as a sample mean, range, etc., with a centerline and upper and lower control limits. The plots are used to identify any potential problems with a production process.

13-5.A center line representing the desired value of a production parameter is given, as well as upper and lower limits for acceptable variation away from the center value. The sampled parameters are then charted, typically in order corresponding to the chronology of the process, and certain anomalous patterns can then be discerned to determine whether the process in “in control” or not, and if not, the direction in which to make corrections.

13-6.The variable being charted must be assumed to have an at least approximately normal distribution. Variation of the random, natural kind cannot be singled out by means of quality control charts.

13-7.A method for determining whether a particular lot or group of items is acceptable. It usually involves making an assumption of a probability that any one item is acceptable, then checking a sample of outcomes for whether the proportion of acceptable items in the sample falls within a certain range in a binomial distribution corresponding to the assumed probability.

13-8.Various experiments in the design and production methods used are made to determine which factors most affect the parameters of interest. ANOVA methods are often used to make this determination. By adopting modified methods in the full-scale production process, the overall quality control results may be improved.

13-9.

a) 77.62%

Type of Error / Freq. / cum %
Omissions / 164 / 47.67%
Quantity Entry / 103 / 77.62%
Part Number / 45 / 90.70%
Date / 24 / 97.67%
Withdraw/Deposit / 8 / 100.00%

b) first 3 causes: Omissions, Quantity Entry Errors and Part Number Errors. (90.70%)

13-10.

Type of Defect / Freq.
oil leaks / 106
cracked blocks / 62
cylinders / 29
radiators / 17
ignition / 10

13-11.

Pareto Diagram
Type of Defect / Freq.
chlorofluorocarbons / 61
air toxins / 30
wastes / 8
other / 1


13-12.Both the average number of bugs (i.e., the “process mean”) and the variance in number of defects
over all projects are steadily decreasing over time.

13-13. Since the sample mean is a random variable which tends toward a normal distribution, we expect it to cluster around the process mean over repeated observations. Thus a time plot of sample means of aggregated observations is a way to see whether this clustering behavior is in fact taking place, so we will be able to detect any major shift in the process mean (the process then being out of control). The chart is constructed by means of estimates of the parameters of interest: process mean and process standard deviation.

13-14.

The waiting time process is in control.

n / 4 / x-bar-bar / 5.575
UCL / 9.256
LCL / 1.894

13-15.Random sampling, so that the observations are independent.

13-16.

Process is not in control:

n / 3 / x-bar-bar / 124.2
UCL / 130.6
LCL / 117.7

13-17.Process is in control

n / 5 / x-bar-bar / 5.9
UCL / 7.054
LCL / 4.746

13-18.We wish to monitor and control the process variance.

13-19.An R-chart is easier to calculate by hand, but an s-chart gives information more directly related to the actual process standard deviation.

13-20.Sample range and sample standard deviation, as random variables, actually have skewness (are not symmetrically distributed about their means).

13-21.Process is in control

R-bar / 5.05 / s-bar / 2.272
UCL / 11.52 / UCL / 5.148
LCL / 0 / LCL / 0

13-22.Out of control: the sixth sample

R-bar / 6.3 / s-bar / 3.313
UCL / 16.22 / UCL / 8.509
LCL / 0 / LCL / 0

13-23.Process in control

R-bar / 2 / s-bar / 0.899
UCL / 4.23 / UCL / 1.879
LCL / 0 / LCL / 0

13-24.Process is in control

n / 5 / x-bar-bar / 1.261 / R-bar / 0.075 / s-bar / 0.031
UCL / 1.305 / UCL / 0.16 / UCL / 0.066
LCL / 1.218 / LCL / 0 / LCL / 0

13-25.Process is out of control on the 9th sample (R and s charts)

n / 5 / x-bar-bar / 12.62 / R-bar / 0.184 / s-bar / 0.07
UCL / 12.73 / UCL / 0.388 / UCL / 0.147
LCL / 12.51 / LCL / 0 / LCL / 0

removing the 9th sample:

n / 5 / x-bar-bar / 12.63 / R-bar / 0.154 / s-bar / 0.06
UCL / 12.72 / UCL / 0.326 / UCL / 0.126
LCL / 12.54 / LCL / 0 / LCL / 0

13-26.Process is in control

5 / x-bar-bar / 2 / R-bar / 0.025 / s-bar / 0.011
UCL / 2.014 / UCL / 0.054 / UCL / 0.022
LCL / 1.985 / LCL / 0 / LCL / 0

13-27.All points are well within the p-chart limits; process is in control.

13-28.From a p-chart: = 0.0333UCL = 0.1317

The 12th sample exceeds the UCL.

p Chart / batteries
n / 30
1 / 2 / 3 / 4 / 5 / 6 / 7 / 8 / 9 / 10 / 11 / 12 / 13 / 14 / 15
x / 1 / 1 / 0 / 0 / 1 / 2 / 0 / 1 / 0 / 0 / 2 / 5 / 0 / 1
p / 0.03 / 0.03 / 0 / 0 / 0.03 / 0.07 / 0 / 0.03 / 0 / 0 / 0.07 / 0.17 / 0 / 0.03
p-bar / 0.03
UCL / 0.13
LCL / 0

13-29.All points are well within the p-chart limits; process is in control.

13-30.It may be very hard to detect any defective items at all.

13-31.The tenth sample barely exceeds the UCL = 8.953; otherwise in control.

c Chart / Imperfections per yard
1 / 2 / 3 / 4 / 5 / 6 / 7 / 8 / 9 / 10 / 11 / 12 / 13 / 14 / 15 / 16 / 17
c / 5 / 3 / 4 / 8 / 2 / 3 / 1 / 2 / 5 / 9 / 2 / 2 / 2 / 3 / 4 / 2 / 1
c-bar / 3.4
UCL / 9
LCL / 0

13-32.The 10th and 12th observation exceeds the UCL = 34.33.

13-33.All points within c-chart limits; process is in control.

13-34.The number of defectives is assumed to follow a Poisson distribution.

13-35.Answer will vary.

13-36.Using a c chart, we find the 10th observation exceeds the UCL = 27.01.

13-37.The twentieth observation far exceeds the UCL = 8.92/100; also the last nine observations are all on one side of the center line = 3.45/100.

13-38.R Chart for tile: Process is in control.

R-bar / 0.623
UCL / 1.247
LCL / 0

13-39.X-bar Chart for tile: Process is in control.

x-bar-bar / 2.432
UCL / 2.733
LCL / 2.131

13-40.s Chart for tile: Process is in control.

s-bar / 0.254
UCL / 0.501
LCL / 0.008

13-41.

X-bar, R and s Charts
Groups
1 / 2 / 3 / 4 / 5 / 6 / 7 / 8 / 9 / 10 / 11 / 12 / 13 / 14 / 15
577 / 579 / 576 / 579 / 577 / 579 / 579 / 577 / 577 / 584
577 / 580 / 580 / 580 / 580 / 576 / 578 / 579 / 579 / 580
579 / 578 / 580 / 580 / 578 / 578 / 580 / 578 / 579 / 582
580 / 580 / 579 / 578 / 580 / 577 / 578 / 579 / 577 / 579
578 / 580 / 576 / 577 / 578 / 578 / 577 / 580 / 576 / 580
578.2 / 579.4 / 578.2 / 578.8 / 578.6 / 577.6 / 578.4 / 578.6 / 577.6 / 581
3 / 2 / 4 / 3 / 3 / 3 / 3 / 3 / 3 / 5
1.304 / 0.894 / 2.049 / 1.304 / 1.342 / 1.14 / 1.14 / 1.14 / 1.342 / 2
n / 5 / x-bar-bar / 578.6 / R-bar / 3.2 / s-bar / 1.366
UCL / 580.5 / UCL / 6.768 / UCL / 2.853
LCL / 576.8 / LCL / 0 / LCL / 0

The X-bar chart shows the process out of control, and the R chart and s chart show a potential upward trend.

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