statistical process control

© 2011 Pearson Education, Inc. publishing as Prentice Hall S6 - 1

S6S6 Statistical Process Control

Statistical Process Control

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

PowerPoint slides by Jeff Heyl


OutlineOutline

Statistical Process Control (SPC) Control Charts for Variables

The Central Limit Theorem

Setting Mean Chart Limits (x-Charts)

Setting Range Chart Limits (R-Charts)

Using Mean and Range Charts

Control Charts for Attributes

Managerial Issues and Control Charts


Outline – ContinuedOutline – Continued

Process Capability

Process Capability Ratio (Cp)

Process Capability Index (Cpk )

Acceptance Sampling Operating Characteristic Curve

Average Outgoing Quality


Learning ObjectivesLearning Objectives

When you complete this supplement When you complete this supplement you should be able to:you should be able to:

1. Explain the use of a control chart

2. Explain the role of the central limit theorem in SPC

3. Build x-charts and R-charts

4. List the five steps involved in building control charts


Learning ObjectivesLearning Objectives

When you complete this supplement When you complete this supplement you should be able to:you should be able to:

5. Build p-charts and c-charts

6. Explain process capability and compute Cp and Cpk

7. Explain acceptance sampling

8. Compute the AOQ


Statistical Process ControlStatistical Process Control

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

signal when assignable causes of signal when assignable causes of variation are presentvariation are present


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 Statistical Process Control (SPC)(SPC)


Natural VariationsNatural 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”


Assignable VariationsAssignable 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


SamplesSamplesTo 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

Fre

qu

ency

Weight

#

## #

##

##

#

# # ## # ##

# # ## # ## # ##

Each of these represents one sample of five

boxes of cereal

Figure S6.1



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

The solid line represents the

distribution

Fre

qu

ency

WeightFigure S6.1



(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

Fre

qu

ency

Figure S6.1



(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



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

WeightTimeF

req

uen

cy Prediction

????

???

???

??????

???

Figure S6.1


Control ChartsControl Charts

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


Process ControlProcess 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


Types of DataTypes 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


Central Limit TheoremCentral 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

x =

n

x =

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


Population and Sampling Population and Sampling DistributionsDistributions

Three population distributions

Beta

Normal

Uniform

Distribution of sample means

Standard deviation of the sample means

= x =

n

Mean of sample means = x

| | | | | | |

-3x -2x -1x x +1x +2x +3x

99.73% of all xfall within ± 3x

95.45% fall within ± 2x

Figure S6.3


Sampling DistributionSampling Distribution

x = (mean)

Sampling distribution of means

Process distribution of means

Figure S6.4


Control Charts for VariablesControl 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


Setting Chart LimitsSetting Chart LimitsFor For xx-Charts when we know -Charts when we know

Upper control limit (UCL) = x + zx

Lower control limit (LCL) = x - zx

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

x =standard deviation of the sample means

=/ n =population standard deviationn =sample size


Setting Control LimitsSetting Control LimitsHour 1

Sample Weight ofNumber Oat Flakes

1 172 133 164 185 176 167 158 179 16

Mean 16.1 = 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 - zx = 16 - 3(1/3) = 15 ozs

For 99.73% control limits, z = 3

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


17 = UCL

15 = LCL

16 = Mean

Setting Control LimitsSetting 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


Setting Chart LimitsSetting Chart Limits

For For xx-Charts when we don’t know -Charts when we don’t know

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


Control Chart FactorsControl 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



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



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


From From Table S6.1Table S6.1



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

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


UCL = 12.144

Mean = 12

LCL = 11.857


Restaurant Control LimitsRestaurant Control LimitsFor salmon filets at Darden Restaurants

Sam

ple

Mea

n

x Bar Chart

UCL = 11.524

x – 10.959

LCL – 10.394| | | | | | | | |

1 3 5 7 9 11 13 15 17

11.5 –

11.0 –

10.5 –

Sam

ple

Ran

ge

Range Chart

UCL = 0.6943

R = 0.2125LCL = 0| | | | | | | | |

1 3 5 7 9 11 13 15 17

0.8 –

0.4 –

0.0 –


Restaurant Control LimitsRestaurant Control Limits

SpecificationsLSL 10USL 12

Capability

Mean = 10.959Std.dev = 1.88Cp = 1.77Cpk = 1.7

Capability Histogram

LSL USL

10.2

10.5

10.8

11.1

11.4

11.7

12.0


R – ChartR – 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


Setting Chart LimitsSetting Chart Limits

For For RR-Charts-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



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


Mean and Range ChartsMean 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


Mean and Range ChartsMean 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


Steps In Creating Control Steps In Creating Control ChartsCharts

1. Take samples from the population and compute the appropriate sample statistic

2. Use the sample statistic to calculate control limits and draw the control chart

3. Plot sample results on the control chart and determine the state of the process (in or out of control)

4. Investigate possible assignable causes and take any indicated actions

5. Continue sampling from the process and reset the control limits when necessary


Manual and AutomatedManual and AutomatedControl ChartsControl Charts


Control Charts for AttributesControl 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)


Control Limits for p-ChartsControl 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 + zp^

LCLp = p - zp^

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

n =sample size

^

p(1 - p)n

p =^


p-Chart for Data Entryp-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)

100p = = .02^

p = = .0480

(100)(20)


.11 –

.10 –

.09 –

.08 –

.07 –

.06 –

.05 –

.04 –

.03 –

.02 –

.01 –

.00 –

Sample number

Fra

ctio

n d

efec

tive

| | | | | | | | | |

2 4 6 8 10 12 14 16 18 20

p-Chart for Data Entryp-Chart for Data Entry

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

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

UCLp = 0.10

LCLp = 0.00

p = 0.04


.11 –

.10 –

.09 –

.08 –

.07 –

.06 –

.05 –

.04 –

.03 –

.02 –

.01 –

.00 –

Sample number

Fra

ctio

n d

efec

tive

| | | | | | | | | |

2 4 6 8 10 12 14 16 18 20

p-Chart for Data Entryp-Chart for Data Entry

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

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

UCLp = 0.10

LCLp = 0.00

p = 0.04

Possible assignable

causes present


Control Limits for c-ChartsControl 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

where c =mean number defective in the sample

UCLc = c + 3 c LCLc = c - 3 c


c-Chart for Cab Companyc-Chart for Cab Company

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

|1

|2

|3

|4

|5

|6

|7

|8

|9

Day

Nu

mb

er d

efec

tive

14 –

12 –

10 –

8 –

6 –

4 –

2 –

0 –

UCLc = c + 3 c= 6 + 3 6= 13.35

LCLc = c - 3 c= 6 - 3 6= 0

UCLc = 13.35

LCLc = 0

c = 6


Managerial Issues andManagerial Issues andControl ChartsControl Charts

Select points in the processes that Select points in the processes that need SPCneed SPC

Determine the appropriate charting Determine the appropriate charting techniquetechnique

Set clear policies and proceduresSet clear policies and procedures

Three major management decisions:Three major management decisions:


Which Control Chart to UseWhich Control Chart to Use

Table S6.3

Variables DataUsing an x-Chart and R-Chart

1. Observations are variables

2. 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

3. Track samples of n observations each.



Table S6.3

Attribute DataUsing the p-Chart

1. Observations are attributes that can be categorized as good or bad (or pass–fail, or functional–broken), that is, in two states.

2. We deal with fraction, proportion, or percent defectives.

3. There are several samples, with many observations in each. For example, 20 samples of n = 100 observations in each.



Table S6.3

Attribute DataUsing a c-Chart

1. Observations are attributes whose defects per unit of output can be counted.

2. We deal with the number counted, which is a small part of the possible occurrences.

3. Defects may be: number of blemishes on a desk; complaints in a day; crimes in a year; broken seats in a stadium; typos in a chapter of this text; or flaws in a bolt of cloth.


Patterns in Control ChartsPatterns in Control Charts

Normal behavior. Process is “in control.”

Upper control limit

Target

Lower control limit

Figure S6.7


Upper control limit

Target

Lower control limit


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

Figure S6.7


Upper control limit

Target

Lower control limit


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

Figure S6.7


Upper control limit

Target

Lower control limit


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

Figure S6.7


Upper control limit

Target

Lower control limit


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

Figure S6.7


Upper control limit

Target

Lower control limit


Erratic behavior. Investigate.

Figure S6.7


Process CapabilityProcess 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


Process Capability RatioProcess Capability Ratio

Cp = Upper Specification - Lower Specification

6

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




6

Insurance claims process

Process mean x = 210.0 minutesProcess standard deviation = .516 minutesDesign specification = 210 ± 3 minutes




6



= = 1.938213 - 2076(.516)




6



= = 1.938213 - 2076(.516)

Process is capable


Process Capability IndexProcess 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 ,

UpperSpecification - xLimit

Lowerx - Specification

Limit



New Cutting Machine

New process mean x = .250 inchesProcess standard deviation = .0005 inchesUpper Specification Limit = .251 inchesLower Specification Limit = .249 inches



New Cutting Machine


Cpk = minimum of ,(.251) - .250

(3).0005



New Cutting Machine


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


Interpreting Interpreting CCpkpk

Cpk = negative number

Cpk = zero

Cpk = between 0 and 1

Cpk = 1

Cpk > 1

Figure S6.8


Acceptance SamplingAcceptance 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


Acceptance SamplingAcceptance 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)


Operating Characteristic Operating Characteristic CurveCurve

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


Return whole shipment

The “Perfect” OC CurveThe “Perfect” OC Curve

% Defective in Lot

P(A

cc

ept

Wh

ole

Sh

ipm

en

t)

100 –

75 –

50 –

25 –

0 –| | | | | | | | | | |

0 10 20 30 40 50 60 70 80 90 100

Cut-Off

Keep whole shipment


An OC CurveAn OC Curve

Probability of

Acceptance

Percent defective

| | | | | | | | |0 1 2 3 4 5 6 7 8

100 –95 –

75 –

50 –

25 –

10 –

0 –

= 0.05 producer’s risk for AQL

= 0.10

Consumer’s risk for LTPD

LTPDAQL

Bad lotsIndifference zone

Good lots

Figure S6.9


AQL and LTPDAQL 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


Producer’s and Consumer’s Producer’s and Consumer’s RisksRisks

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 () Probability of accepting a bad lot

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


OC Curves for Different OC Curves for Different Sampling PlansSampling Plans

n = 50, c = 1

n = 100, c = 2


Average Outgoing QualityAverage 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)

N


Average Outgoing QualityAverage Outgoing Quality

1. If a sampling plan replaces all defectives

2. If we know the incoming percent defective for the lot

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

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


Automated InspectionAutomated Inspection

Modern technologies allow virtually 100% inspection at minimal costs

Not suitable for all situations


SPC and Process VariabilitySPC and Process Variability

(a) Acceptance sampling (Some bad units accepted)

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

(c) Cpk >1 (Design a process that is in control)

Lower specification

limit

Upper specification

limit

Process mean, Figure S6.10


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