ENGM 620: Quality Management

Session 8 – 23 October 2012

• Control Charts, Part I– Variables

Statistical Thinking

• All work occurs in a system of interconnected processes

• All process have variation

• Understanding variation and reducing variation are important keys to success

Variability

• A certain amount of variability is inescapable

• Therefore, no two products are identical

• The larger the variability, the greater the probability that the customer will perceive its existence

Sources of Variability

Include:

• Differences in materials

• Differences in the performance and operation of the manufacturing equipment

• Differences in the way the operators perform their tasks

Variability and Statistics

• Variability is difference from the target• Characteristics of quality must be measurable

Therefore,• Variability is described in statistical terms• We will use statistical methods in our quality

improvement activities

Recall: Types of Errors

• Type I error – Producers risk– Probability that a good product will be rejected

• Type II error– Consumers risk– Probability that a nonconforming product will be

available for sale

• Type III error– Asking the wrong question

Types of Errors

HO HA

HO

HA

Truth

Accept

NoError

Type I

Type II

NoError

A Parable

Where should we put the additional armor?

Data on Quality Characteristics

• Attribute data– Discrete– Often a count of some type

• Variable data– Continuous– Often a measurement, such as length,

voltage, or viscosity

Terms

• Specifications

• Target (or Nominal) Value

• Upper Specification Limit

• Lower Specification Limit

• Random Variation

• Non-random Variation

• Process stability

Terms

• Nonconforming: failure to meet one or more of the specifications

• Nonconformity: a specific type of failure

• Defect: a nonconformity serious enough to significantly affect the safe or effective use of the produce or completion of the service

Nonconforming vs. Defective

• A nonconforming product is not necessarily unfit for use

• A nonconforming product is considered defective if if it has one or more defects

Classroom Exercise

• For a product or service in your job:– Name a quality characteristic– Give an example of a nonconformity that is

not a defect– Give an example of a defect

Types of Inspection

• Receiving

• In Process

• Final

• None

• One Hundred Percent

• Acceptance Sampling

Quality Design & Process Variation

60 80 100 120 140

60 140

14060

Lower Spec Limit

Upper Spec Limit

AcceptanceSampling

Statistical ProcessControl

ExperimentalDesign

Variation and Control

• A process that is operating with only common causes of variation is said to be in statistical control.

• A process operating in the presence of special or assignable cause is said to be out of control.

Finding Trends and Special Causes

• Inspection does not tell you about a problem until it becomes a problem

• We need a mechanism to help us spot special causes when they occur

• We need mechanism to help us determine when we have a trend in the data

Statistical Process Control

• Originally developed by Walter Shewhart in 1924 at the Bell Telephone Laboratories

• Late 1920s, Harold Dodge and Harry Romig developed statistically based acceptance sampling

• Not recognized by industry until after World War II

Definition

• Statistical Process Control (SPC):– “a methodology for monitoring a process to

identify special causes of variation and signal the need to take corrective action when it is appropriate”

(Evans and Lindsay)

Statistical Process Control Tools

• The magnificent seven

• The tool most often associated with Statistical Process Control is Control Charts

Common Causes

Special Causes

Histograms do not take into account changes over time.

Control charts can tell us when a process changes

Control Chart Applications• Establish state of statistical control• Monitor a process and signal when it goes out of

control• Determine process capability

• Note: Control charts will only detect the presence of assignable causes. Management, operator, and engineering action is necessary to eliminate the assignable cause.

Capability Versus Control

Control

Capability

Capable

Not Capable

In Control Out of Control

IDEAL

Commonly Used Control Charts

• Variables data– x-bar and R-charts

– x-bar and s-charts

– Charts for individuals (x-charts)

• Attribute data– For “defectives” (p-chart, np-chart)

– For “defects” (c-chart, u-chart)

Control Charts

We assume that the underlying distribution is normal with some mean and some constant but unknown standard deviation .

Letx

x

ni

i

n

1

Distribution of x

Recall that x is a function of random variables, so it also is a random variable with its own distribution. By the central limit theorem, we know that

where,

x N x ( , )

xnx

Control Charts

xx

x

x

Control Charts

x

x

UCL

LCL

UCL & LCL Set atProblem: How do we estimate & ?

3 x

Control Charts

xx

m

m

ii

1

)(1 fm

RR

m

i

Control Charts

xx RALCL 2

xx RAUCL 2

RDLCLR 3

RDUCLR 4

xx

m

m

ii

1

)(1 fm

RR

m

i

Example

• Suppose specialized o-rings are to be manufactured at .5 inches. Too big and they won’t provide the necessary seal. Too little and they won’t fit on the shaft. Twenty samples of 2 rings each are taken. Results follow.

Part Measurements R Chart X ChartNo. 1 2 x R UCL R UCL LCL Xbar

1 0.502 0.504 0.503 0.002 0.0077 0.002 0.5052 0.4964 0.5032 0.495 0.497 0.496 0.002 0.0077 0.002 0.5052 0.4964 0.4963 0.492 0.496 0.494 0.004 0.0077 0.004 0.5052 0.4964 0.4944 0.501 0.498 0.500 0.003 0.0077 0.003 0.5052 0.4964 0.5005 0.507 0.508 0.508 0.001 0.0077 0.001 0.5052 0.4964 0.5086 0.504 0.504 0.504 0.000 0.0077 0.000 0.5052 0.4964 0.5047 0.497 0.496 0.497 0.001 0.0077 0.001 0.5052 0.4964 0.4978 0.493 0.496 0.495 0.003 0.0077 0.003 0.5052 0.4964 0.4959 0.502 0.501 0.502 0.001 0.0077 0.001 0.5052 0.4964 0.502

10 0.498 0.500 0.499 0.002 0.0077 0.002 0.5052 0.4964 0.49911 0.505 0.507 0.506 0.002 0.0077 0.002 0.5052 0.4964 0.50612 0.502 0.499 0.501 0.003 0.0077 0.003 0.5052 0.4964 0.50113 0.495 0.497 0.496 0.002 0.0077 0.002 0.5052 0.4964 0.49614 0.499 0.496 0.498 0.003 0.0077 0.003 0.5052 0.4964 0.49815 0.503 0.507 0.505 0.004 0.0077 0.004 0.5052 0.4964 0.50516 0.507 0.509 0.508 0.002 0.0077 0.002 0.5052 0.4964 0.50817 0.503 0.501 0.502 0.002 0.0077 0.002 0.5052 0.4964 0.50218 0.497 0.493 0.495 0.004 0.0077 0.004 0.5052 0.4964 0.49519 0.504 0.508 0.506 0.004 0.0077 0.004 0.5052 0.4964 0.50620 0.505 0.503 0.504 0.002 0.0077 0.002 0.5052 0.4964 0.504

Avg = 0.501 0.002x R

Std. = 0.0047

X-Bar Control Charts

X-Bar Chart

0.490

0.495

0.500

0.505

0.510

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

x

X-bar charts can identify special causes of variation, but they are only useful if the processis stable (common cause variation).

Control Limits for Range

UCL = D4R = 3.268*.002 = .0065

LCL = D3 R = 0

R Chart

0.000

0.002

0.004

0.006

0.008

0.010

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

Observation

Ran

ge

37

Why Monitor Both Process Mean and Process Variability?

LowerSpecification

Limit

UpperSpecification

Limit

Process Doing OK

Mean shift in process

Increase in process variance

X-bar R

X-bar R

X-bar R

Process Over Time Control Charts

38

Teminology

• Causes of Variation:– Assignable Causes

• Keep the process from operating predictably

• Things that we can do something about

– Common / Chance Causes• Random, inherent

variation in the process

• Meaning of Control:– In Specification

• Meets customer constraints on product

– In Statistical Control• No Assignable

Causes of variation present in the process

Shift in Process Average

Identifying Potential Shifts

Cycles

Trend

Western Electric Sensitizing Rules:

• One point plots outside the 3-sigma control limits

• Two of three consecutive points plot outside the 2-sigma warning limits

• Four of five consecutive points plot beyond the 1-sigma limits

• A run of eight consecutive points plot on one side of the center line

Additional sensitizing rules:

• Six points in a row are steadily increasing or decreasing

• Fifteen points in a row with 1-sigma limits (both above and below the center line)

• Fourteen points in a row alternating up and down• Eight points in a row in both sides of the center

line with none within the 1-sigma limits• An unusual or nonrandom pattern in the data• One of more points near a warning or control limit

Special Variables Control Charts

• x-bar and s charts

• x-chart for individuals

X-bar and S charts

• Allows us to estimate the process standard deviation directly instead of indirectly through the use of the range R

• S chart limits:– UCL = B6σ = B4*S-bar– Center Line = c4σ = S-bar– LCL = B5σ = B3*S-bar

• X-bar chart limits– UCL = X-doublebar +A3S-bar– Center line = X-doublebar– LCL = X-doublebar -A3S-bar

X-chart for individuals

• UCL = x-bar + 3*(MR-bar/d2)

• Center line = x-bar

• LCL = x-bar - 3*(MR-bar/d2)

Next Class

• Homework– Ch. 11 Disc. Questions 5, 7– Ch. 11 Problems 6, 11

• Preparation– Chapter 11, Process Capability