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Statistical Process Control

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Outline

STATISTICAL PROCESS CONTROL (SPC) Control Charts for Variables The Central Limit Theorem Setting Mean Chart Limits Setting Range Chart Limits (R-Charts) Using Mean and Range Charts Control Charts for Attributes Managerial Issues and Control Charts

Charts)-x(

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Outline

PROCESS CAPABILITY Process Capability Ratio (Cp)

Process Capability Index (Cpk

ACCEPTANCE SAMPLING Operating Characteristic (OC) Curves Average Outgoing Quality

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When you complete this chapter, you should be able to Identify or Define: Natural and assignable causes of variation Central limit theorem Attribute and variable inspection Process control LCL and UCL p-charts and C-charts

Learning Objectives

charts-Randcharts-x

pkp CandC ˆ

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When you complete this chapter, you should be able to Identify or Define: Acceptance sampling OC curve AQL and LTPD AOQ Producer’s and consumer’s risk

Learning Objectives - Continued

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Learning Objectives - Continued

When you complete this chapter, you should be able to Describe or explain: The role of statistical quality control

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Measures performance of a process Uses mathematics (i.e., statistics) Involves collecting, organizing, & interpreting data Objective: provide statistical signal when

assignable causes of variation are present Used to

Control the process as products are produced Inspect samples of finished products

Statistical Quality Control (SPC)

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StatisticalQuality Control

Process Control

Acceptance Sampling

Variables Charts

Attributes Charts

Types of Statistical Quality Control

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Natural and Assignable Variation

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Characteristics for which you focus on defects

Classify products as either ‘good’ or ‘bad’, or count # defects e.g., radio works or not

Categorical or discrete random variables

AttributesVariables

Quality Characteristics

Characteristics that you measure, e.g., weight, length

May be in whole or in fractional numbers

Continuous random variables

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Statistical technique used to ensure process is making product to standard

All process are subject to variability Natural causes: Random variations Assignable causes: Correctable problems

Machine wear, unskilled workers, poor material

Objective: Identify assignable causes Uses process control charts

Statistical Process Control (SPC)

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Process Control: Three Types of Process Outputs

Frequency

Lower control limit

Size(Weight, length, speed,

etc. )

Upper control limit

(b) In statistical control, but not capable of producing within control limits. A process in control (only natural causes of variation are present) but not capable of producing within the specified control limits; and

(c) Out of control. A process out of control having assignable causes of variation.

(a) In statistical control and capable of producing within control limits. A process with only natural causes of variation and capable of producing within the specified control limits.

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The Relationship Between Population and Sampling

Distributions

Uniform

Normal

BetaDistribution of sample means

x means sample of Mean

n

xx

Standard deviation of

the sample means

(mean)

x2 withinfall x all of 95.5%

x3 withinfall x all of 99.7%

x3 x2 x x x1 x2 x3

Three population distributions

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Sampling Distribution of Means, and Process Distribution

Sampling distribution of the means

Process distribution of the sample

)mean(

mx

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Process Control Charts

Plot of Sample Data Over Time

0

20

40

60

80

1 5 9 13 17 21

Time

Sam

ple

Val

ue

SampleValueUCL

Average

LCL

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Show changes in data pattern e.g., trends

Make corrections before process is out of control

Show causes of changes in data Assignable causes

Data outside control limits or trend in data

Natural causes Random variations around average

Control Chart Purposes

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X

As sample size gets large enough,

sampling distribution becomes almost normal regardless of population distribution.

Central Limit Theorem

XX

Theoretical Basis of Control Charts

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X

Mean

Central Limit Theorem

x

x

n

xx

nX X

Standard deviation

X X

Theoretical Basis of Control Charts

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ControlCharts

RChart

VariablesCharts

AttributesCharts

XChart

PChart

CChart

Continuous Numerical Data

Categorical or Discrete Numerical Data

Control Chart Types

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Statistical Process Control Steps

Produce GoodProvide Service

Stop Process

Yes

No

Take Sample

Inspect Sample

Find Out WhyCreateControl Chart

Start

Can we assign

causes?

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Type of variables control chart Interval or ratio scaled numerical data

Shows sample means over time Monitors process average Example: Weigh samples of coffee & compute

means of samples; Plot

X Chart

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Control Chart for Samples of 9 Boxes

Variation due to natural causes

17=UCL

16=Mean

15=LCL

Variation due to assignable causes

Variation due to assignable causes

Out of control

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

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X Chart Control Limits

Range for sample i

# Samples

Mean for sample i

From Table S6.1

RAxxLCL

RAxxUCL

n

R R

i

n

1i

n

xi

n

ix

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Factors for Computing Control Chart Limits

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 0.729 2.282 0

5 0.577 2.115 0

6 0.483 2.004 0

7 0.419 1.924 0.076

8 0.373 1.864 0.136

9 0.337 1.816 0.184

10 0.308 1.777 0.223

12 0.266 1.716 0.284 0.184

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Type of variables control chart Interval or ratio scaled numerical data

Shows sample ranges over time Difference between smallest & largest values

in inspection sample

Monitors variability in process Example: Weigh samples of coffee & compute

ranges of samples; Plot

R Chart

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R Chart Control Limits

Range for Sample i

# Samples

From Table S6.1

n

R R

R D LCL

R D UCL

i

n

1i

3R

4R

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Steps to Follow When Using Control Charts

1. Collect 20 to 25 samples of n=4 or n=5 from a stable process and compute the mean.

2. Compute the overall means, set approximate control limits,and calculate the preliminary upper and lower control limits.If the process is not currently stable, use the desired mean instead of the overall mean to calculate limits.

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

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Steps to Follow When Using Control Charts - continued

4. Investigate points or patterns that indicate the process is out of control. Assign causes for the variations.

5. Collect additional samples and revalidate the control limits.

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Mean and Range Charts Complement Each Other

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Type of attributes control chart Nominally scaled categorical data

e.g., good-bad

Shows % of nonconforming items Example: Count # defective chairs & divide by

total chairs inspected; Plot Chair is either defective or not defective

p Chart

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p Chart Control Limits

# Defective Items in Sample i

Size of sample i

z = 2 for 95.5% limits; z = 3 for 99.7% limits

sample each of size where

n

xp and

k

nn

n)p(1p

zpLCL

n)p(1p

zpUCL

i

k

1i

i

k

1ii

k

1i

p

p

nn

ppp

)1(

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P-Chart for Data Entry Example

0.000.010.020.030.040.050.060.070.080.090.100.110.12

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

Sample Number

Perc

ent D

efec

tive

UCLp=0.10

LCLp=0.00

0.04p

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Type of attributes control chart Discrete quantitative data

Shows number of nonconformities (defects) in a unit Unit may be chair, steel sheet, car etc. Size of unit must be constant

Example: Count # defects (scratches, chips etc.) in each chair of a sample of 100 chairs; Plot

c Chart

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c Chart Control Limits

# Defects in Unit i

# Units Sampled

Use 3 for 99.7% limits

k

c c

i

k

1i

ccLCL

ccUCL

c

c

3

3

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Patterns to Look for in Control Charts

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Deciding Which Control Chart to Use

Using an X 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 each. Using the P-Chart:

We deal with fraction, proportion, or percent defectives Observations are attributes that can be categorized in two

states Have several samples, each with many observations Assume a binomial distribution unless the number of

samples is very large – then assume a normal distribution.

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Deciding Which Control Chart to Use

Using a C-Chart: Observations are attributes whose defects

per unit of output can be counted The number counted is often a small part of

the possible occurrences Assume a Poisson distribution Defects such as: number of blemishes on a

desk, number of typos in a page of text, flaws in a bolt of cloth

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Process Capability Ratio, Cp

process the of deviation standard

6σionSpecificat LowerionSpecificat Upper

pC

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Process Capability Cpk

population process the of deviation standard mean process x where

LimitionSpecificatLower x

or , x Limit ionSpecificatUpper

of minimum

3

3pkC

Assumes that the process is:• under control• normally distributed

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Meanings of Cpk Measures

Cpk = negative number

Cpk = zero

Cpk = between 0 and 1

Cpk = 1

Cpk > 1

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Form of quality testing used for incoming materials or finished goods e.g., purchased material & components

Procedure Take one or more 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

What Is Acceptance Sampling?

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Set of procedures for inspecting incoming materials or finished goods

Identifies Type of sample Sample size (n) Criteria (c) used to reject or accept a lot

Producer (supplier) & consumer (buyer) must negotiate

What Is an Acceptance Plan?

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

Shows the relationship between the probability of accepting a lot & its quality

Operating Characteristics Curve

45 % Defective in Lot

P(Accept Whole Shipment)

100%

0%

Cut-Off1 2 3 4 5 6 7 8 9 100

Return whole shipment

Keep whole shipment

OC Curve100% Inspection

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OC Curve with Less than 100% Sampling

P(Accept Whole Shipment)

100%

0%

% Defective in LotCut-Off

1 2 3 4 5 6 7 8 9 100

Return whole shipment

Keep whole shipment

Probability is not 100%: Risk of keeping bad shipment or returning good one.

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Acceptable quality level (AQL) Quality level of a good lot Producer (supplier) does not want lots with

fewer defects than AQL rejected

Lot tolerance percent defective (LTPD) Quality level of a bad lot Consumer (buyer) does not want lots with

more defects than LTPD accepted

AQL & LTPD

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Producer's risk () Probability of rejecting a good lot Probability of rejecting a lot when fraction

defective is AQL

Consumer's risk (ß) Probability of accepting a bad lot Probability of accepting a lot when fraction

defective is LTPD

Producer’s & Consumer’s Risk

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An Operating Characteristic (OC) Curve Showing Risks

= 0.05 producer’s risk for AQL

= 0.10

Consumer’s risk for LTPD

Probability of Acceptance

Percent Defective

Bad lotsIndifference zoneGood lots

LTPDAQL

0 1 2 3 4 5 6 7 8

10095

75

50

25

10

0

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OC Curves for Different Sampling Plans

1 2 3 4 5 6 7 8 9 100

% Defective in Lot

P(Accept Whole Shipment)

100%

0%

LTPDAQL

n = 50, c = 1

n = 100, c = 2

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Average Outgoing Quality

N

)nN)(P)(P(AOQ ad

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

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Negotiate between producer (supplier) and consumer (buyer)

Both parties attempt to minimize risk Affects sample size & cut-off criterion

Methods MIL-STD-105D Tables Dodge-Romig Tables Statistical Formulas

Developing a Sample Plan

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Statistical Process Control - Identify and Reduce Process

VariabilityLower

specification limit

Upper specification

limit

(a) Acceptance sampling –[ Some bad units accepted; the “lot” is good or bad]

(b) Statistical process control – [Keep the process in “control”]

(c) cpk >1 – [Design a process that is in control]