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S6S6 Statistical Process Control

Statistical Process Control

PowerPoint presentation to accompany Heizer and Render Operations Management, 10e Principles of Operations Management, 8e

PowerPoint slides by Jeff Heyl

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Statistical Process Control

The objective of a process control system is to provide a statistical

signal when assignable causes of variation are present

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Variability is inherent in every process Natural or common

causes Special or assignable causes

Provides a statistical signal when assignable causes are present

Detect and eliminate assignable causes of variation

Statistical Process Control (SPC)

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Natural Variations Also called common causes Affect virtually all production processes Expected amount of variation Output measures follow a probability

distribution For any distribution there is a measure

of central tendency and dispersion If the distribution of outputs falls within

acceptable limits, the process is said to be “in control”

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Assignable Variations

Also called special causes of variation Generally this is some change in the process

Variations that can be traced to a specific reason

The objective is to discover when assignable causes are present Eliminate the bad causes Incorporate the good causes

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SamplesTo measure the process, we take samples and analyze the sample statistics following these steps

(d) If only natural causes of variation are present, the output of a process forms a distribution that is stable over time and is predictable

WeightTimeF

req

uen

cy Prediction

Figure S6.1

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SamplesTo measure the process, we take samples and analyze the sample statistics following these steps

(e) If assignable causes are present, the process output is not stable over time and is not predicable

WeightTimeF

req

uen

cy Prediction

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Figure S6.1

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Control Charts

Constructed from historical data, the purpose of control charts is to help distinguish between natural variations and variations due to assignable causes

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Process Control

Figure S6.2

Frequency

(weight, length, speed, etc.)Size

Lower control limit Upper control limit

(a) In statistical control and capable of producing within control limits

(b) In statistical control but not capable of producing within control limits

(c) Out of control

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Types of Data

Characteristics that can take any real value

May be in whole or in fractional numbers

Continuous random variables

Variables Attributes Defect-related

characteristics Classify products

as either good or bad or count defects

Categorical or discrete random variables

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Central Limit Theorem

Regardless of the distribution of the population, the distribution of sample means drawn from the population will tend to follow a normal curve

1. The mean of the sampling distribution (x) will be the same as the population mean m

x = m

s n

sx =

2. The standard deviation of the sampling distribution (sx) will equal the population standard deviation (s) divided by the square root of the sample size, n

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Control Charts for Variables

For variables that have continuous dimensions Weight, speed, length,

strength, etc.

x-charts are to control the central tendency of the process

R-charts are to control the dispersion of the process

These two charts must be used together

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Setting Chart LimitsFor x-Charts when we know s

Upper control limit (UCL) = x + zsx

Lower control limit (LCL) = x - zsx

where x = mean of the sample means or a target value set for the processz = number of normal standard deviations

sx = standard deviation of the sample means

= s/ ns = population standard deviationn = sample size

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Setting Control LimitsHour 1

Sample Weight ofNumber Oat Flakes

1 172 133 164 185 176 167 158 179 16

Mean 16.1s = 1

Hour Mean Hour Mean1 16.1 7 15.22 16.8 8 16.43 15.5 9 16.34 16.5 10 14.85 16.5 11 14.26 16.4 12 17.3

n = 9

LCLx = x - zsx = 16 - 3(1/3) = 15 ozs

For 99.73% control limits, z = 3

UCLx = x + zsx = 16 + 3(1/3) = 17 ozs

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17 = UCL

15 = LCL

16 = Mean

Setting Control Limits

Control Chart for sample of 9 boxes

Sample number

| | | | | | | | | | | |1 2 3 4 5 6 7 8 9 10 11 12

Variation due to assignable

causes

Variation due to assignable

causes

Variation due to natural causes

Out of control

Out of control

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Setting Chart Limits

For x-Charts when we don’t know s

Lower control limit (LCL) = x - A2R

Upper control limit (UCL) = x + A2R

where R = average range of the samples

A2 = control chart factor found in Table S6.1 x = mean of the sample means

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Control Chart Factors

Table S6.1

Sample Size Mean Factor Upper Range Lower Range n A2 D4 D3

2 1.880 3.268 0

3 1.023 2.574 0

4 .729 2.282 0

5 .577 2.115 0

6 .483 2.004 0

7 .419 1.924 0.076

8 .373 1.864 0.136

9 .337 1.816 0.184

10 .308 1.777 0.223

12 .266 1.716 0.284

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Setting Control Limits

Process average x = 12 ouncesAverage range R = .25Sample size n = 5

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Setting Control Limits

UCLx = x + A2R= 12 + (.577)(.25)= 12 + .144= 12.144 ounces

Process average x = 12 ouncesAverage range R = .25Sample size n = 5

From Table S6.1

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Setting Control Limits

UCLx = x + A2R= 12 + (.577)(.25)= 12 + .144= 12.144 ounces

LCLx = x - A2R= 12 - .144= 11.857 ounces

Process average x = 12 ouncesAverage range R = .25Sample size n = 5

UCL = 12.144

Mean = 12

LCL = 11.857

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R – Chart

Type of variables control chart Shows sample ranges over time

Difference between smallest and largest values in sample

Monitors process variability Independent from process mean

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Setting Chart Limits

For R-Charts

Lower control limit (LCLR) = D3R

Upper control limit (UCLR) = D4R

whereR = average range of the samples

D3 and D4 = control chart factors from Table S6.1

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Setting Control Limits

UCLR = D4R= (2.115)(5.3)= 11.2 pounds

LCLR = D3R= (0)(5.3)= 0 pounds

Average range R = 5.3 poundsSample size n = 5From Table S6.1 D4 = 2.115, D3 = 0

UCL = 11.2

Mean = 5.3

LCL = 0

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Mean and Range Charts

(a)

These sampling distributions result in the charts below

(Sampling mean is shifting upward but range is consistent)

R-chart(R-chart does not detect change in mean)

UCL

LCL

Figure S6.5

x-chart(x-chart detects shift in central tendency)

UCL

LCL

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Mean and Range Charts

R-chart(R-chart detects increase in dispersion)

UCL

LCL

Figure S6.5

(b)

These sampling distributions result in the charts below

(Sampling mean is constant but dispersion is increasing)

x-chart(x-chart does not detect the increase in dispersion)

UCL

LCL

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Control Charts for Attributes

For variables that are categorical Good/bad, yes/no,

acceptable/unacceptable

Measurement is typically counting defectives

Charts may measure Percent defective (p-chart) Number of defects (c-chart)

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Control Limits for p-Charts

Population will be a binomial distribution, but applying the Central Limit Theorem

allows us to assume a normal distribution for the sample statistics

UCLp = p + zsp^

LCLp = p - zsp^

where p = mean fraction defective in the samplez = number of standard deviationssp = standard deviation of the sampling distribution

n = sample size

^

p(1 - p)n

sp =^

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p-Chart for Data EntrySample Number Fraction Sample Number FractionNumber of Errors Defective Number of Errors Defective

1 6 .06 11 6 .062 5 .05 12 1 .013 0 .00 13 8 .084 1 .01 14 7 .075 4 .04 15 5 .056 2 .02 16 4 .047 5 .05 17 11 .118 3 .03 18 3 .039 3 .03 19 0 .00

10 2 .02 20 4 .04Total = 80

(.04)(1 - .04)

100sp = = .02^

p = = .0480

(100)(20)

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Sample number

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p-Chart for Data Entry

UCLp = p + zsp = .04 + 3(.02) = .10^

LCLp = p - zsp = .04 - 3(.02) = 0^

UCLp = 0.10

LCLp = 0.00

p = 0.04

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.11 –

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Sample number

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2 4 6 8 10 12 14 16 18 20

p-Chart for Data Entry

UCLp = p + zsp = .04 + 3(.02) = .10^

LCLp = p - zsp = .04 - 3(.02) = 0^

UCLp = 0.10

LCLp = 0.00

p = 0.04

Possible assignable

causes present

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Patterns in Control Charts

Normal behavior. Process is “in control.”

Upper control limit

Target

Lower control limit

Figure S6.7

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Upper control limit

Target

Lower control limit

Patterns in Control Charts

One plot out above (or below). Investigate for cause. Process is “out of control.”

Figure S6.7

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Upper control limit

Target

Lower control limit

Patterns in Control Charts

Trends in either direction, 5 plots. Investigate for cause of progressive change.

Figure S6.7

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Upper control limit

Target

Lower control limit

Patterns in Control Charts

Two plots very near lower (or upper) control. Investigate for cause.

Figure S6.7

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Upper control limit

Target

Lower control limit

Patterns in Control Charts

Run of 5 above (or below) central line. Investigate for cause.

Figure S6.7

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Upper control limit

Target

Lower control limit

Patterns in Control Charts

Erratic behavior. Investigate.

Figure S6.7