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MD 021 - Operations Management

Statistical Process Control Outline

• Definition of statistical process control (SPC) • Process variation • Characteristics of control charts • Types of control charts • Choosing an SPC method

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Definition of Statistical Process Control

Statistical process control (SPC) is the application of statistical techniques to determine whether the output of a process conforms to the product or service design. SPC is implemented via control charts that are used to monitor the output of the process and indicate the presence of problems requiring further action.

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Process Variation

All processes exhibit variation. For example, a machine that fills cereal boxes will not put exactly the same amount in each box. If we were to collect data over time on the amount of cereal in each box, a plot of the data would be described as a distribution. A distribution is characterized by its: • Mean

• Spread • Range • Standard deviation

• Shape • Symmetric • Skewed

Cereal Production

0

100

200

300

400

500

600

700

9.7 9.75 9.8 9.85 9.9 9.95 10 10.1 10.1 10.2 10.2 10.3 10.3

Average weight (oz.)

No.

of b

oxes

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Quality Measurements Control charts can be used to monitor processes where output is measured as either variables or attributes. Variables measures - Characteristics of a product or service that can be measured on a continuous scale. Examples include length, width, and time. Attributes measures - Characteristics of a product or service that can be quickly counted for acceptable quality. Examples include the number of defects in a product or service.

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Sources of Process Variation

Sources of process variation can be categorized as: • Common causes - Random, natural, or unavoidable, sources of variation

within a process. A process with only common causes of variation is stable (i.e. the mean and spread do not change over time). Such a process is said to be “in a state of statistical control” or “in-control”.

• Assignable causes - Any cause of variation that can be identified and

eliminated, originating from outside the normal process.

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Characteristics of Control Charts

A control chart is a time-ordered diagram to monitor a quality characteristic, consisting of: • A nominal value, or center line – The average of several past samples • Two control limits used to judge whether action is required, an upper

control limit (UCL) and a lower control limit (LCL) • Data points, each consisting of the average measurement calculated from a

sample taken from the process, ordered over time. By the Central Limit Theorem, regardless of the distribution of the underlying individual measurements, the distribution of the sample means will follow a normal distribution. The control limits are set based on the sampling distribution of the quality measurement.

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Purpose of Control Charts

Control charts are intended to reflect only common causes of variation in order to detect assignable causes of variation. Question: If, at the time we are constructing a control chart, there are assignable causes of variation in the process, how can we construct a meaningful control chart? Answer: By carefully choosing our sample size so that only common causes are found within a sample. The control limits are set to: • Usually detect when the process has gone out of control (narrow control limits work

better) • Usually not overreact to random variation (wider control limits work better)

The control limits are set to strike a balance between these competing priorities.

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Control Charts for Variables Standard Deviation of the Process, σ , Known

Control charts for variables (with the standard deviation of the process, σ , known ) monitor the mean, X , of the process distribution. The control limits are: XzXUCL σ+= XzXLCL σ−=

where: X = center line of the chart and the average of several past sample means, z is the standard normal deviate (number of standard deviations from the average), nX /σσ = and is the standard deviation of the distribution of sample means, and n is the sample size

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Example: An automatic filling machine is used to fill 1-liter bottles of cola. The machine’s output is approximately normal with a mean of 1.0 liter and a standard deviation of 0.01 liter. Output is monitored using means of samples of 25 observations. a. Determine upper and lower control limits that will include roughly 97 percent of the sample means when the process is in control.

x = 1.0 liter, σ = .01 liter, n = 25

nzx σ

±:limits Control , [z = 2.17 for 97%]

9957.

2501.17.20.1 is LCL

0043.12501.17.20.1 is UCL

=−

=+

b. Given the sample means: 1.005, 1.001, 0.998, 1.002, 0.995 and 0.999, is the process in control?

1.006

1.0043

1.002

1.000

.998

.9957

.994

UCL

LCL

out

out

Mean (Liters)

• •

• •

*

*

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Control Charts for Variables Standard Deviation of the Process, σ , Unknown

Control charts for variables monitor the mean ( X chart) and variability (R chart) of the process distribution. R chart: To calculate the range of the data, subtract the smallest from the largest measurement in the sample.

The control limits are:

UCL D R LCL D RR R= =4 3 and where: R = average of several past R values and is the central line of the control chart, and D D3 4, = constants that provide three standard deviation (three-sigma) limits for a given sample size (from Table 10.3 on Page 470).

X chart: The control limits are:

UCL X A R LCL X A RX X= + = −2 2 and where: X = central line of the chart and the average of past sample means, and A2 = constant to provide three sigma limits for the process mean (from Table 10.3 on Page 470).

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Control Chart for Variables Example X , R Charts

Webster Chemical Company produces mastics and caulking for the construction industry. The product is blended in large mixers and then pumped into tubes and capped. Webster is concerned whether the filling process for tubes of caulking is in statistical control. The process should be centered on 8 ounces per tube. Several samples of eight tubes are taken and each tube is weighed in ounces.

Tube number Sample 1 2 3 4 5 6 7 8

1 7.98 8.34 8.02 7.94 8.44 7.68 7.81 8.11 2 8.23 8.12 7.98 8.41 8.31 8.18 7.99 8.06 3 7.89 7.77 7.91 8.04 8.00 7.89 7.93 8.09 4 8.24 8.18 7.83 8.05 7.90 8.16 7.97 8.07 5 7.87 8.13 7.92 7.99 8.10 7.81 8.14 7.88 6 8.13 8.14 8.11 8.13 8.14 8.12 8.13 8.14

a) Assuming that taking only 6 samples is sufficient, use the data in the table to construct three-sigma R-chart and

X -chart control limits. Do these control charts exhibit any potential assignable causes? b) The first and sixth samples appear to be outside the control limits of the R-chart. Webster looks for assignable

causes and quickly notes that the weighing scale was gummed up with caulking. Apparently, a tube was not properly capped. The sticky scale did not correctly read the variation in weights for the sixth sample. Delete that data and recalculate R , UCLR , and LCLR . Are the new charts appropriate now?

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Solution to Control Chart for Variables Example

a) Tube number

Sample 1 2 3 4 5 6 7 8 Avg. Range 1 7.98 8.34 8.02 7.94 8.44 7.68 7.81 8.11 8.040 0.76 2 8.23 8.12 7.98 8.41 8.31 8.18 7.99 8.06 8.160 0.43 3 7.89 7.77 7.91 8.04 8.00 7.89 7.93 8.09 7.940 0.32 4 8.24 8.18 7.83 8.05 7.90 8.16 7.97 8.07 8.050 0.41 5 7.87 8.13 7.92 7.99 8.10 7.81 8.14 7.88 7.980 0.33 6 8.13 8.14 8.11 8.13 8.14 8.12 8.13 8.14 8.130 0.03

8.050 0.38 X = 8.050, R = 0.38, n = 8 From Table 10.3:

909.7)38.0(37.0050.8

191.8)38.0(37.0050.8

053.0)38.0(14.0

707.0)38.0(86.1

2

2

3

4

=−=−=

=+=+=

===

===

RAXLCL

RAXUCL

RDLCL

RDUCL

X

X

R

R

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b) We delete the sixth observation and recalculate the control limits. The ranges, including the range for the first

sample, are now all within the revised control limits, and the process averages also fall inside of the revised control limits.

868.7)45.0(37.0034.8

201.8)45.0(37.0034.8

063.0)45.0(14.0

837.0)45.0(86.1

034.85

980.7050.8940.7160.8040.8

45.05

)33.041.032.043.076.0(

2

2

3

4

=−=−=

=+=+=

===

===

=++++

=

=++++

=

RAXLCL

RAXUCL

RDLCL

RDUCL

X

R

X

X

R

R

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Control Charts for Attributes

p-Chart

A p-chart is a commonly used control chart for attributes, whereby the quality characteristic is counted, rather than measured, and the entire item or service can be declared good or defective. The standard deviation of the proportion defective, p, is: σ p p p n= −( ) /1 , where n = sample size, and p= average of several past p values and central line on the chart. Using the normal approximation to the binomial distribution, which is the actual distribution of p,

UCL p z LCL p zp p p p= + = −σ σ and where z is the normal deviate (number of standard deviations from the average).

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Control Chart for Attributes Example p-Chart

A sticky scale brings Webster’s attention to whether caulking tubes are being properly capped. If a significant proportion of the tubes aren’t being sealed, Webster is placing their customers in a messy situation. Tubes are packaged in large boxes of 144. Several boxes are inspected and the following numbers of leaking tubes are found:

Sample Tubes Sample Tubes Sample Tubes 1 3 8 6 15 5 2 5 9 4 16 0 3 3 10 9 17 2 4 4 11 2 18 6 5 2 12 6 19 2 6 4 13 5 20 1 7 2 14 1 Total 72

Construct p-chart with three-sigma control limits.

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Solution to Control Chart for Attributes Example

n = 144, p = =72

20 1440 025

( ).

σ

σ

σ

p

p p

p p

p pn

UCL p z

LCL p z

=−

=−

=

= + = + =

= − = − = −

( ) . ( . ) .

. ( . ) .

. ( . ) .

1 0 025 1 0 025144

0 013

0 025 3 0 013 0 064

0 025 3 0 013 0 014 (adjusted to zero)

The highest proportion of defectives occurs in sample #10, but is still within control limits: p10 9 144 0 0625= =/ . . Therefore, the p-chart is appropriate.

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Control Charts for Attributes

c-Chart

A c-chart is another type of control chart for attributes, whereby the quality characteristic is counted as the number of defects/ unit. Using the normal approximation to the Poisson distribution, which is the actual distribution of c,

czcLCLczcUCL cc −=+= and where c is the average number of defects/unit and the center line of the c-chart.

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Control Chart for Attributes Example c-Chart

At Webster Chemical, lumps in the caulking compound could cause difficulties in dispensing a smooth bead from the tube. Testing for the presence of lumps destroys the product, so Webster takes random samples. The following are the results of the study:

Tube # Lumps Tube # Lumps Tube # Lumps 1 6 5 6 9 5 2 5 6 4 10 0 3 0 7 1 11 9 4 4 8 6 12 2

Determine the c-chart with two-sigma upper and lower control limits for this process. Solution:

c

c

UCL c z

LCL c z

c

c c

c c

=+ + + + + + + + + + +

=

= = =

= + = + =

= − = − =

( )

( )( )

( )( )

6 5 0 4 6 4 1 6 5 0 9 212

4

4 2

4 2 2 8

4 2 2 0

σ

σ

σ

The eleventh tube has too many lumps (9), so it may have assignable causes of variations.

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(a) Trend UCL

LCL

(b) Cycles UCL

LCL

(c) Bias UCL

LCL

(d) Mean shift UCL

LCL

(e) Too much dispersion UCL

LCL

Examples of Nonrandom Patterns

Source: Figure 10.11, Page 476, from Stevenson

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(a) Process is capable

Lower specification

Upper specification

Nominal Value

100 12080

Process distribution

(b) Process is not capable

Upper specification

Nominal Value

100 12080

Process distribution

Lower specification

Process Capability

Relationship between Process Distribution and Specifications

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Process Capability Exercise

Webster Chemical’s nominal weight for filling tubes of caulk is 8.00 ounces ± 0.60 ounces. The target process capability ratio is 1.33. The current distribution of the filling process is centered on 8.054 ounces with a standard deviation of 0.192 ounces. Compute the process capability index to assess whether the filling process is capable and set properly. Solution: Process capability ratio:

Cp = =−

=Upper specification - Lower specification

68 6 7 46 0192

10417σ

. .( . )

.

Process capability index:

0.948

948.0)192.0(3

8.054-8.600 ,135.1)192.0(3400.7054.8

3X -ion specificatUpper ,

3ionspecificatLower - X of miminum

=

==−

=σσpkC

The process is not capable of consistently meeting specifications according to the minimum capability level set by Webster.

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Supplementary Material:

Calculating p

nxp i

i =

ip = proportion of defective products

ix = number of the defected products n = sample size m = number of samples

nm

x

nmxxx

mnx

nx

nx

mppp

p

m

ii

m

m

m

⋅=

⋅+++

=++

=+++

=∑=121

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21 )...(...)...(

productsofnumbertotalproductsdefectedofnumbertotalp =