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Chapter 6s Statistical Process Control (SPC)

Chapter 6s

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Learning Objectives

When you complete this supplement you should be able to :

1 Explain the purpose of a control chart

2 Build -charts and R-charts

4 List the five steps involved in building control charts

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Statistical Process Control (SPC)

SPC: A process used to monitor standards by taking measurements and corrective action as a product or service is being produced.

The objective of a process control system is to provide a statistical signal when assignable causes of variation are present.

Such a signal can quicken appropriate action to eliminate assignable causes.

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Statistical Process Control (SPC)

Statistical process control (SPC) is a methodology for establishing and maintaining high-quality output, and it is been heavily linked with Six Sigma.

SPC includes a set of tools and principles for:

Determining if a process is stable.

Monitoring a process for possible changes in behavior.

Assessing whether a process is capable of meeting production requirements and customer demands.

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Statistical Process Control (SPC)

Uses statistics and control charts to tell when to take corrective action

Drives process improvement

Four key steps

Measure the process

When a change is indicated, find the assignable cause

Eliminate or incorporate the cause

Restart the revised process

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Inspection

Involves examining items to see if an item is good or defective

Detect a defective product

Does not correct deficiencies in process or product

It is expensive

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When & Where to Inspect

At the supplier’s plant while the supplier is producing

At your facility upon receipt of goods from your supplier

Before costly or irreversible processes

During the step-by-step production process

When production or service is complete

Before delivery to your customer

At the point of customer contact

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Control charts identify variation

Source of variation, many problems can occurred from:

Worker fatigue

Measurement error

Process variability

Tactic to reduce variations:

Robust design

Empowered employees

Quality at source

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Quality at source

The next step in the process is your customer

Ensure perfect product to your customer

Quality at source involves the operator ensuring that the job is done properly. These operators are empowered to self-check their own work.

Employees that deal with a system on a daily basis have a better understanding of the system than anyone else, and they can be very effective at improving the system.

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Quality at source

Poka-yoke is the concept of error-proof devices or techniques designed to pass only acceptable products

You can find a number of everyday examples of Poka-Yoke:

Example: Look at the connector for your computer keyboard or mouse. Its shape prevents it from being connected in the wrong place or turned incorrectly, damaging your computer.

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Type of Variation

Natural or common causes

Special or assignable causes

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Points which might be emphasized include:

- Statistical process control measures the performance of a process, it does not help to identify a particular specimen produced as being “good” or “bad,” in or out of tolerance.

- Statistical process control requires the collection and analysis of data - therefore it is not helpful when total production consists of a small number of units.

- While statistical process control cannot help identify a “good” or “bad” unit, it can enable one to decide whether or not to accept an entire production lot. If a sample of a production lot contains more than a specified number of defective items, statistical process control can give us a basis for rejecting the entire lot. The issue of rejecting a lot which was actually good can be raised here, but is probably better left to later.

1. Natural Variations

Also called common causes

Inherent to the process or random and not controllable

Expected amount of variation

For any distribution there is a measure of central tendency and dispersion (xbar-R Chart)

If the distribution of outputs falls within acceptable limits, the process is said to be "in control"

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2. 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

If present, the process is “out of control”

Eliminate the root causes

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Control charts to monitor processes

To monitor output, we use a control chart

We check things like the mean, range, standard deviation

To monitor a process, we typically use two control charts (x-bar chart & R chart)

Mean (or some other central tendency measure) X-Bar Chart

Variation or dispersion (typically using range or standard deviation) R-Chart

Control Chart is the primary tool of SPC.

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Samples

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

(a) Samples of the product, say five boxes of cereal taken off the filling machine line, vary from each other in weight

Frequency

Weight

Each of these represents one sample of five boxes of cereal

Figure S6.1

FYI

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Samples

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

(b) After enough samples are taken from a stable process, they form a pattern called a distribution

The solid line represents the distribution

Frequency

Weight

Figure S6.1

FYI

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Samples

(c) There are many types of distributions, including the normal (bell-shaped) distribution, but distributions do differ in terms of central tendency (mean), standard deviation or variance, and shape

Weight

Central tendency

Weight

Variation

Weight

Shape

Frequency

Figure S6.1

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

FYI

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Samples

To 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

Weight

Time

Frequency

Prediction

Figure S6.1

FYI

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Samples

To 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

Weight

Time

Frequency

Prediction

Figure S6.1

FYI

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Control Charts

A control chart is a statistical tool used to distinguish between variation in a process resulting from common causes and variation resulting from special causes. It presents a graphic display of process stability or instability over time

Every process has variation. Some variation may be the result of causes which are not normally present in the process. This could be special cause variation.

Some variation is simply the result of numerous, ever-present differences in the process. This is common cause variation. Control Charts differentiate between these two types of variation.

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Students should understand both the concepts of natural and assignable variation, and the nature of the efforts required to deal with them.

Control Charts

One goal of using a Control Chart is to achieve and maintain process stability. And that by:

Determining if a process is stable.

Monitoring a process for possible changes in behavior.

Separating common and special causes of variation

UCL

LCL

Target

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Students should understand both the concepts of natural and assignable variation, and the nature of the efforts required to deal with them.

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

FYI

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This slide helps introduce different process outputs.

It can also be used to illustrate natural and assignable variation.

Control Charts for Variables (x-bar & R chart)

Characteristics that can take any real value

May be in whole or in fractional numbers

Continuous random variables (i.e. the variable can be measured on a continuous scale (e.g. height, weight, length, time etc.)

X- Bar chart tracks changes in the central tendency “Indicates how the average or mean changes over time”

R-chart indicates a gain or loss of dispersion “Indicates how the range of the subgroups changes over time.”

X-Bar and R-Charts are typically used when the subgroup size (n) lies between 2 and 10

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Once the categories are outlined, students may be asked to provide examples of items for which variable or attribute inspection might be appropriate. They might also be asked to provide examples of products for which both characteristics might be important at different stages of the production process.

Control Charts for Variables

Characteristics that can take any real value

May be in whole or in fractional numbers

Continuous random variables

x-chart tracks changes in the central tendency

R-chart indicates a gain or loss of dispersion

These two charts must be used together

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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

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

The mean of the sampling distribution will be the same as the population mean m

FYI

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This slide introduces the difference between “natural” and “assignable” causes.

The next several slides expand the discussion and introduce some of the statistical issues.

Population and Sampling Distributions

Population distributions

Beta

Normal

Uniform

Distribution of sample means

Figure S6.3

99.73% of all

fall within ±

95.45% fall within ±

| | | | | | |

Standard deviation of the sample means

Mean of sample means =

FYI

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Sampling Distribution

= m

(mean)

Sampling distribution of means

Process distribution of means

Figure S6.4

FYI

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It may be useful to spend some time explicitly discussing the difference between the sampling distribution of the means and the mean of the process population.

Sampling Distribution

Mean of process

n = 100

n = 25

Figure S6.4

n = 50

As the sample size increases,

the sampling distribution narrows

FYI

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It may be useful to spend some time explicitly discussing the difference between the sampling distribution of the means and the mean of the process population.

Setting Chart Limits

For x-Charts when we know s

Where = mean of the sample means or a target value set for the process

z = number of normal standard deviations

= standard deviation of the sample means

s = population (process) standard deviation

n = sample size

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Standard deviation calculation

https://www.spcforexcel.com/knowledge/control-chart-basics/estimated-standard-deviation-and-control-charts

The standard deviation is a little more difficult to understand – and to complicate things, there are multiple ways that it can be determined – each giving a different answer. If you are interested to learn more about the standard deviation calculation, please follow the below links.

FYI

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Setting Control Limits

Also, the population (process) standard deviation (s) is known to be 1 ounce. The 9 boxes selected in hours 2 through 12 are not shown here, but here Average weight () are shown in the next table:

i.e. z = 3

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Setting Control Limits

Randomly select and weigh nine (n = 9) boxes each hour

WEIGHT OF SAMPLE		WEIGHT OF SAMPLE		WEIGHT OF SAMPLE
HOUR	(AVG. OF 9 BOXES)	HOUR	(AVG. OF 9 BOXES)	HOUR	(AVG. OF 9 BOXES)
1	16.1	5	16.5	9	16.3
2	16.8	6	16.4	10	14.8
3	15.5	7	15.2	11	14.2
4	16.5	8	16.4	12	17.3

Average weight in the first sample (hour 1)

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Setting Control Limits

Average mean of 12 samples

Number

of samples = 12

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Setting Control Limits

Average mean of 12 samples

Number

of samples = 12

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17 = UCL

15 = LCL

16 = Mean

Sample number

| | | | | | | | | | | |

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

Setting Control Limits

Control Chart for samples of 9 boxes

Variation due to assignable causes

Variation due to natural causes

Out of control

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Setting Chart Limits

For x-Charts when we don't know s

where average range of the samples

A2 = control chart factor found in Table S6.1

= mean of the sample means

Ri = range for sample i

k = total number of samples

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Control Chart Factors

TABLE S6.1	Factors for Computing Control Chart Limits (3 sigma)
SAMPLE SIZE, n		MEAN FACTOR, A2	UPPER RANGE, D4	LOWER RANGE, 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

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Setting Control Limits

Process average = 12 ounces

Average range = .25 ounces

Sample size = 5

UCL = 12.144

Mean = 12

LCL = 11.856

From Table S6.1

Super Cola example

labeled as "net weight 12 ounces"

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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

where

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Restaurant Control Limits

For salmon fillets at Darden Restaurants

Sample Mean

x Bar Chart

UCL = 11.524

= 10.959

LCL = 10.394

| | | | | | | | |

1 3 5 7 9 11 13 15 17

11.5 –

11.0 –

10.5 –

Sample Range

Range Chart

UCL = 0.6943

= 0.2125

LCL = 0

| | | | | | | | |

1 3 5 7 9 11 13 15 17

0.8 –

0.4 –

0.0 –

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Setting Control Limits

Average range = 8 minutes

Sample size = 4

From Table S6.1 D4 = 2.282, D3 = 0

UCL = 18.256

Mean = 8

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

FYI

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Mean and Range Charts

R-chart

(R-chart detects increase in dispersion)

UCL

LCL

(b)

These sampling distributions result in the charts below

(Sampling mean is constant, but dispersion is increasing)

x-chart

(x-chart indicates no change in central tendency)

UCL

LCL

Figure S6.5

FYI

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Steps In Building Control Charts

Collect 20 to 25 samples, often of n = 4 or n = 5 observations each, from a stable process, and compute the mean and range of each

Compute the overall means ( and ), set appropriate control limits, usually at the 99.73% level, and calculate the preliminary upper and lower control limits

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Steps In Creating Control Charts

Graph the sample means and ranges on their respective control charts and determine whether they fall outside the acceptable limits

Investigate points or patterns that indicate the process is out of control – try to assign causes for the variation, address the causes, and then resume the process

Collect additional samples and, if necessary, revalidate the control limits using the new data

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Setting Other Control Limits

TABLE S6.2	Common z Values
DESIRED CONTROL LIMIT (%)		Z-VALUE (STANDARD DEVIATION REQUIRED FOR DESIRED LEVEL OF CONFIDENCE)
90.0		1.65
95.0		1.96
95.45		2.00
99.0		2.58
99.73		3.00

FYI

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Control Charts for Attributes

For variables that are categorical

Defective/nondefective, good/bad, yes/no, acceptable/unacceptable

Measurement is typically counting defectives

Charts may measure

Percent defective (p-chart)

Number of defects (c-chart)

FYI

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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

where

FYI

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Instructors may wish to point out the calculation of the standard deviation reflects the binomial distribution of the population

p-Chart for Data Entry

SAMPLE NUMBER	NUMBER OF ERRORS	FRACTION DEFECTIVE	SAMPLE NUMBER	NUMBER OF ERRORS		FRACTION DEFECTIVE
1	6	.06	11		6	.06
2	5	.05	12		1	.01
3	0	.00	13		8	.08
4	1	.01	14		7	.07
5	4	.04	15		5	.05
6	2	.02	16		4	.04
7	5	.05	17		11	.11
8	3	.03	18		3	.03
9	3	.03	19		0	.00
10	2	.02	20		4	.04
					80

FYI

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p-Chart for Data Entry

SAMPLE NUMBER	NUMBER OF ERRORS	FRACTION DEFECTIVE	SAMPLE NUMBER	NUMBER OF ERRORS		FRACTION DEFECTIVE
1	6	.06	11		6	.06
2	5	.05	12		1	.01
3	0	.00	13		8	.08
4	1	.01	14		7	.07
5	4	.04	15		5	.05
6	2	.02	16		4	.04
7	5	.05	17		11	.11
8	3	.03	18		3	.03
9	3	.03	19		0	.00
10	2	.02	20		4	.04
					80

(because we cannot have a negative percent defective)

FYI

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.11 –

.10 –

.09 –

.08 –

.07 –

.06 –

.05 –

.04 –

.03 –

.02 –

.01 –

.00 –

Sample number

Fraction defective

| | | | | | | | | |

2 4 6 8 10 12 14 16 18 20

p-Chart for Data Entry

UCLp = 0.10

LCLp = 0.00

p = 0.04

FYI

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.11 –

.10 –

.09 –

.08 –

.07 –

.06 –

.05 –

.04 –

.03 –

.02 –

.01 –

.00 –

Sample number

Fraction defective

| | | | | | | | | |

2 4 6 8 10 12 14 16 18 20

p-Chart for Data Entry

UCLp = 0.10

LCLp = 0.00

p = 0.04

Possible assignable causes present

FYI

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Control Limits for c-Charts

Population will be a Poisson distribution, but applying the central limit theorem allows us to assume a normal distribution for the sample statistics

FYI

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Instructors may wish to point out the calculation of the standard deviation reflects the Poisson distribution of the population where the standard deviation equals the square root of the mean

c-Chart for Cab Company

Day

Number defective

14 –

12 –

10 –

8 –

6 –

4 –

2 –

0 –

UCLc = 13.35

LCLc = 0

c = 6

Cannot be a

negative number

FYI

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Select points in the processes that need SPC

Determine the appropriate charting technique

Set clear and specific SPC policies and procedures

Managerial Issues and Control Charts

Three major management decisions:

FYI

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Which Control Chart to Use

TABLE S6.3	Helping You Decide Which Control Chart to Use
VARIABLE DATA USING AN x-CHART AND R-CHART
Observations are variables Collect 20 – 25 samples of n = 4, or n = 5, or more, each from a stable process and compute the mean for the x-chart and range for the R-chart Track samples of n observations

FYI

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Which Control Chart to Use

TABLE S6.3	Helping You Decide Which Control Chart to Use
ATTRIBUTE DATA USING A P-CHART
Observations are attributes that can be categorized as good or bad (or pass–fail, or functional–broken), that is, in two states We deal with fraction, proportion, or percent defectives There are several samples, with many observations in each
ATTRIBUTE DATA USING A C-CHART
Observations are attributes whose defects per unit of output can be counted We deal with the number counted, which is a small part of the possible occurrences Defects may be: number of blemishes on a desk; flaws in a bolt of cloth; crimes in a year; broken seats in a stadium; typos in a chapter of this text; or complaints in a day

FYI

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Patterns in Control Charts

Normal behavior. Process is "in control."

Upper control limit

Target

Lower control limit

Figure S6.7

FYI

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Ask the students to imagine a product, and consider what problem might cause each of the graph configurations illustrated.

Patterns in Control Charts

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

Upper control limit

Target

Lower control limit

Figure S6.7

FYI

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Ask the students to imagine a product, and consider what problem might cause each of the graph configurations illustrated.

Patterns in Control Charts

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

Upper control limit

Target

Lower control limit

Figure S6.7

FYI

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Ask the students to imagine a product, and consider what problem might cause each of the graph configurations illustrated.

Patterns in Control Charts

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

Upper control limit

Target

Lower control limit

Figure S6.7

FYI

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Ask the students to imagine a product, and consider what problem might cause each of the graph configurations illustrated.

Patterns in Control Charts

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

Upper control limit

Target

Lower control limit

Figure S6.7

FYI

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Ask the students to imagine a product, and consider what problem might cause each of the graph configurations illustrated.

Patterns in Control Charts

Erratic behavior. Investigate.

Upper control limit

Target

Lower control limit

Figure S6.7

FYI

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Ask the students to imagine a product, and consider what problem might cause each of the graph configurations illustrated.

Patterns in Control Charts

Run test

Identify abnormalities in a process

Runs of 5 or 6 points above or below the target or centerline suggest assignable causes may be present

Process may not be in statistical control

There are a variety of run tests

FYI

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Process Capability

The natural variation of a process should be small enough to produce products that meet the standards required

A process in statistical control does not necessarily meet the design specifications

Process capability is a measure of the relationship between the natural variation of the process and the design specifications

FYI

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Process Capability Ratio

Cp =

Upper Specification – Lower Specification

A capable process must have a Cp of at least 1.0

Does not look at how well the process is centered in the specification range

Often a target value of Cp = 1.33 is used to allow for off-center processes

Six Sigma quality requires a Cp = 2.0

FYI

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Process Capability Ratio

Cp =

Upper Specification - Lower Specification

Insurance claims process

Process mean x = 210.0 minutes

Process standard deviation s = .516 minutes

Design specification = 210 ± 3 minutes

FYI

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Process Capability Ratio

Cp =

Upper Specification - Lower Specification

Insurance claims process

Process mean x = 210.0 minutes

Process standard deviation s = .516 minutes

Design specification = 210 ± 3 minutes

= = 1.938

213 – 207

6(.516)

FYI

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Process Capability Ratio

Cp =

Upper Specification - Lower Specification

Insurance claims process

Process mean x = 210.0 minutes

Process standard deviation s = .516 minutes

Design specification = 210 ± 3 minutes

= = 1.938

213 – 207

6(.516)

Process is capable

FYI

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Process Capability Index

A capable process must have a Cpk of at least 1.0

A capable process is not necessarily in the center of the specification, but it falls within the specification limit at both extremes

Cpk = minimum of , ,

Upper Specification – x Limit

Lower x – Specification Limit

FYI

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Process Capability Index

New Cutting Machine

New process mean x = .250 inches

Process standard deviation s = .0005 inches

Upper Specification Limit = .251 inches

Lower Specification Limit = .249 inches

FYI

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Process Capability Index

New Cutting Machine

New process mean x = .250 inches

Process standard deviation s = .0005 inches

Upper Specification Limit = .251 inches

Lower Specification Limit = .249 inches

Cpk = minimum of ,

(.251) - .250

(3).0005

FYI

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Process Capability Index

New Cutting Machine

New process mean x = .250 inches

Process standard deviation s = .0005 inches

Upper Specification Limit = .251 inches

Lower Specification Limit = .249 inches

Cpk = minimum of ,

(.251) - .250

(3).0005

.250 - (.249)

(3).0005

FYI

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Process Capability Index

New Cutting Machine

New process mean x = .250 inches

Process standard deviation s = .0005 inches

Upper Specification Limit = .251 inches

Lower Specification Limit = .249 inches

Cpk = = 0.67

.001

.0015

New machine is NOT capable

Cpk = minimum of ,

(.251) - .250

(3).0005

.250 - (.249)

(3).0005

Both calculations result in

FYI

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Lower specification limit

Upper specification limit

Interpreting Cpk

Cpk = negative number

Cpk = zero

Cpk = between 0 and 1

Cpk = 1

Cpk > 1

Figure S6.8

FYI

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Acceptance Sampling

Form of quality testing used for incoming materials or finished goods

Take samples at random from a lot (shipment) of items

Inspect each of the items in the sample

Decide whether to reject the whole lot based on the inspection results

Only screens lots; does not drive quality improvement efforts

FYI

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Here again it is useful to stress that acceptance sampling relates to the aggregate, not the individual unit. You might also discuss the decision as to whether one should take only a single sample, or whether multiple samples are required.

Acceptance Sampling

Form of quality testing used for incoming materials or finished goods

Take samples at random from a lot (shipment) of items

Inspect each of the items in the sample

Decide whether to reject the whole lot based on the inspection results

Only screens lots; does not drive quality improvement efforts

Rejected lots can be:

Returned to the supplier

Culled for defectives (100% inspection)

May be re-graded to a lower specification

FYI

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Operating Characteristic Curve

Shows how well a sampling plan discriminates between good and bad lots (shipments)

Shows the relationship between the probability of accepting a lot and its quality level

FYI

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You can use this and the next several slides to begin a discussion of the “quality” of the acceptance sampling plans. You will find additional slides on “consumer’s” and “producer’s” risk to pursue the issue in a more formal manner in subsequent slides.

Return whole shipment

The "Perfect" OC Curve

% Defective in Lot

P(Accept Whole Shipment)

100 –

75 –

50 –

25 –

0 –

| | | | | | | | | | |

0 10 20 30 40 50 60 70 80 90 100

Cut-Off

Keep whole shipment

FYI

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An OC Curve

Probability of Acceptance

Percent defective

| | | | | | | | |

0 1 2 3 4 5 6 7 8

 = 0.05 producer's risk for AQL

 = 0.10

Consumer's risk for LTPD

LTPD

AQL

Bad lots

Indifference zone

Good lots

Figure S6.9

FYI

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AQL and LTPD

Acceptable Quality Level (AQL)

Poorest level of quality we are willing to accept

Lot Tolerance Percent Defective (LTPD)

Quality level we consider bad

Consumer (buyer) does not want to accept lots with more defects than LTPD

FYI

S6 - ‹#›

Once the students understand the definition of these terms, have them consider how one would go about choosing values for AQL and LTPD.

Producer's and Consumer's Risks

Producer's risk ()

Probability of rejecting a good lot

Probability of rejecting a lot when the fraction defective is at or above the AQL

Consumer's risk (b)

Probability of accepting a bad lot

Probability of accepting a lot when fraction defective is below the LTPD

FYI

S6 - ‹#›

This slide introduces the concept of “producer’s” risk and “consumer’s” risk. The following slide explores these concepts graphically.

OC Curves for Different Sampling Plans

n = 50, c = 1

n = 100, c = 2

FYI

S6 - ‹#›

This slide presents the OC curve for two possible acceptance sampling plans.

Average Outgoing Quality

where

Pd = true percent defective of the lot

Pa = probability of accepting the lot

N = number of items in the lot

n = number of items in the sample

AOQ =

(Pd)(Pa)(N – n)

FYI

S6 - ‹#›

It is probably important to stress that AOQ is the average percent defective, not the average percent acceptable.

Average Outgoing Quality

If a sampling plan replaces all defectives

If we know the true incoming percent defective for the lot

We can compute the average outgoing quality (AOQ) in percent defective

The maximum AOQ is the highest percent defective or the lowest average quality and is called the average outgoing quality limit (AOQL)

FYI

S6 - ‹#›

It is probably important to stress that AOQ is the average percent defective, not the average percent acceptable.

Automated Inspection

Modern technologies allow virtually 100% inspection at minimal costs

Not suitable for all situations

FYI

S6 - ‹#›

SPC and Process Variability

(a) Acceptance sampling (Some bad units accepted; the "lot" is good or bad)

(b) Statistical process control (Keep the process "in control")

Lower specification limit

Upper specification limit

Process mean, m

Figure S6.10

FYI

S6 - ‹#›

This may be a good time to stress that an overall goal of statistical process control is to “do it better,” i.e., improve over time.

σ x = σ

σ x

= µ

x =

2σ x

=σ x = σ

3σ x

+3σ x

+3s

+2σ x

+2s

+1σ x

+1s

−1σ x

-1s

−2σ x

-2s

−3σ x

-3s

x =

Upper control limit (UCL) = x +zσ x

Upper control limit (UCL)=x+zs

Lower control limit (LCL) = x −zσ x

Lower control limit (LCL)=x-zs

=σ / n

=s/n

σ x

x =

= 17+13+16+18+17+16+15+17+16

9 =16.1 ounces

17+13+16+18+17+16+15+17+16

=16.1 ounces

= = Avg of 9 boxes( )

i=1

∑ 12

⎡

⎣

⎢ ⎢ ⎢ ⎢

⎤

⎦

⎥ ⎥ ⎥ ⎥

= =

Avg of 9 boxes

( )

i=1

x =16 ounces n=9 z=3 σ =1 ounce

x=16 ounces

n=9

z=3

s=1 ounce

x =

LCLx = − zσ x =16−3 1 9

⎛

⎝ ⎜

⎞

⎠ ⎟=16−3

1 3 ⎛

⎝ ⎜ ⎞

⎠ ⎟=15 ounces

LCL

= -zs

=16-3

=15 ounces

UCLx = + zσ x =16+3 1 9

⎛

⎝ ⎜

⎞

⎠ ⎟=16+3

1 3 ⎛

⎝ ⎜ ⎞

⎠ ⎟=17 ounces

UCL

= +zs

=16+3

=17 ounces

x =

UCLx = + A2R LCLx = − A2R

UCL

= +A

LCL

= -A

R= R i

i=1

∑ k

i=1

x =

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

UCL

= +A

=12+(.577)(.25)

=12+.144

=12.144 ounces

LCLx = − A2R =12−.144 =11.856 ounces

LCL

= -A

=12-.144

=11.856 ounces

x =

Upper control limit (UCLR) = D4R Lower control limit (LCLR) = D3R

Upper control limit (UCL

) = D

Lower control limit (LCL

) = D

UCL R = upper control chart limit for the range

LCL R = lower control chart limit for the range

D 4 and D

3 = values from Table S6.1

UCL

= upper control chart limit for the range

LCL

= lower control chart limit for the range

and D

= values from Table S6.1

x =

R –

–

UCLR = D4R = (2.282)(8) =18.256 minutes

UCL

=(2.282)(8)

=18.256 minutes

LCL R = D

3 R

=(0)(8) =0 minutes

LCL

=(0)(8)

=0 minutes

x =

UCL p = p+zσ

LCL p = p−zσ

UCL

=p+zs

LCL

=p-zs

σ̂ p =

p 1− p( ) n

p1-p

()

p= mean fraction (percent) defective in the samples z= number of standard deviations

σ p = standard deviation of the sampling distribution

n= number of observations in each sample

p= mean fraction (percent) defective in the samples

z= number of standard deviations

=standard deviation of the sampling distribution

n= number of observations in each sample

σ p is estimated by

is estimated by

p= Total number of errors

Total number of records examined =

80 (100)(20)

=.04

σ̂ p =

(.04)(1−.04) 100

=.02 (rounded up from .0196)

Total number of errors

Total number of records examined

(100)(20)

=.04

(.04)(1-.04)

100

=.02 (rounded up from .0196)

UCL p = p+zσ̂

p =.04+3(.02)=.10

LCL p = p−zσ̂

p =.04−3(.02)=0

UCL

=p+z

=.04+3(.02)=.10

LCL

=p-z

=.04-3(.02)=0

c = mean number of defects per unit

c = standard deviation of defects per unit

c= mean number of defects per unit

c= standard deviation of defects per unit

Control limits (99.73%) =c ±3 c

Control limits (99.73%) =c±3c

c = 54 complaints/9 days = 6 complaints/day

c= 54 complaints/9 days = 6 complaints/day

UCLc =c +3 c

=6+3 6 =13.35

UCL

=c+3c

=6+36

=13.35

LCLc =c −3 c

=6−3 6 =0

LCL

=c-3c

=6-36