Operations Management Statistical Process Control Supplement 6

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Operations Management Statistical Process Control Supplement 6. 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

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Operations ManagementStatistical Process ControlSupplement 6

© 2004 by Prentice Hall, Inc., Upper Saddle River, N.J. 07458

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

© 2004 by Prentice Hall, Inc., Upper Saddle River, N.J. 07458

Outline
• PROCESS CAPABILITY
• Process Capability Ratio (Cp)
• Process Capability Index (Cpk
• ACCEPTANCE SAMPLING
• Operating Characteristic (OC) Curves
• Average Outgoing Quality

© 2004 by Prentice Hall, Inc., Upper Saddle River, N.J. 07458

Learning Objectives

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

© 2004 by Prentice Hall, Inc., Upper Saddle River, N.J. 07458

Learning Objectives - Continued

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

© 2004 by Prentice Hall, Inc., Upper Saddle River, N.J. 07458

Learning Objectives - Continued

When you complete this chapter, you should be able to Describe or explain:

• The role of statistical quality control

© 2004 by Prentice Hall, Inc., Upper Saddle River, N.J. 07458

Statistical Quality Control (SPC)
• 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

© 2004 by Prentice Hall, Inc., Upper Saddle River, N.J. 07458

Statistical

Quality Control

Acceptance Sampling

Process Control

Variables Charts

Attributes Charts

Types of Statistical Quality Control

© 2004 by Prentice Hall, Inc., Upper Saddle River, N.J. 07458

Natural and Assignable Variation

© 2004 by Prentice Hall, Inc., Upper Saddle River, N.J. 07458

Variables

Attributes

Quality Characteristics
• 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
• Characteristics that you measure, e.g., weight, length
• May be in whole or in fractional numbers
• Continuous random variables

© 2004 by Prentice Hall, Inc., Upper Saddle River, N.J. 07458

Statistical Process Control (SPC)
• 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

© 2004 by Prentice Hall, Inc., Upper Saddle River, N.J. 07458

(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.

Frequency

Lower control limit

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.

Size

(Weight, length, speed, etc. )

Process Control: Three Types of Process Outputs

© 2004 by Prentice Hall, Inc., Upper Saddle River, N.J. 07458

Three population distributions

Distribution of sample means

Beta

Standard deviation of the sample means

Normal

Uniform

(mean)

The Relationship Between Population and Sampling Distributions

© 2004 by Prentice Hall, Inc., Upper Saddle River, N.J. 07458

Sampling distribution of the means

Process distribution of the sample

Sampling Distribution of Means, and Process Distribution

© 2004 by Prentice Hall, Inc., Upper Saddle River, N.J. 07458

Process Control Charts

© 2004 by Prentice Hall, Inc., Upper Saddle River, N.J. 07458

Control Chart Purposes
• 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

© 2004 by Prentice Hall, Inc., Upper Saddle River, N.J. 07458

Central Limit Theorem

As sample size gets large enough,

sampling distribution becomes almost normal regardless of population distribution.

Theoretical Basis of Control Charts

© 2004 by Prentice Hall, Inc., Upper Saddle River, N.J. 07458

Theoretical Basis of Control Charts

Central Limit Theorem

Standard deviation

Mean

© 2004 by Prentice Hall, Inc., Upper Saddle River, N.J. 07458

Control Chart Types

Continuous Numerical Data

Categorical or Discrete Numerical Data

Control

Charts

Variables

Attributes

Charts

Charts

R

P

C

X

Chart

Chart

Chart

Chart

© 2004 by Prentice Hall, Inc., Upper Saddle River, N.J. 07458

No

Produce Good

Start

Provide Service

Can we assign causes?

Take Sample

Yes

Inspect Sample

Stop Process

Create

Find Out Why

Control Chart

Statistical Process Control Steps

© 2004 by Prentice Hall, Inc., Upper Saddle River, N.J. 07458

X Chart
• 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

© 2004 by Prentice Hall, Inc., Upper Saddle River, N.J. 07458

Variation due to assignable causes

17=UCL

16=Mean

Variation due to natural causes

15=LCL

Variation due to assignable causes

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

Sample Number

Out of control

Control Chart for Samples of 9 Boxes

© 2004 by Prentice Hall, Inc., Upper Saddle River, N.J. 07458

From Table S6.1

Range for sample i

Mean for sample i

# Samples

X Chart Control Limits

© 2004 by Prentice Hall, Inc., Upper Saddle River, N.J. 07458

Factors for Computing Control Chart Limits

© 2004 by Prentice Hall, Inc., Upper Saddle River, N.J. 07458

R Chart
• 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

© 2004 by Prentice Hall, Inc., Upper Saddle River, N.J. 07458

From Table S6.1

Range for Sample i

# Samples

R Chart Control Limits

© 2004 by Prentice Hall, Inc., Upper Saddle River, N.J. 07458

Steps to Follow When Using Control Charts

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

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.

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

© 2004 by Prentice Hall, Inc., Upper Saddle River, N.J. 07458

Steps to Follow When Using Control Charts - continued
• Investigate points or patterns that indicate the process is out of control. Assign causes for the variations.
• Collect additional samples and revalidate the control limits.

© 2004 by Prentice Hall, Inc., Upper Saddle River, N.J. 07458

Mean and Range Charts Complement Each Other

© 2004 by Prentice Hall, Inc., Upper Saddle River, N.J. 07458

p Chart
• Type of attributes control chart
• Nominally scaled categorical data
• Shows % of nonconforming items
• Example: Count # defective chairs & divide by total chairs inspected; Plot
• Chair is either defective or not defective

© 2004 by Prentice Hall, Inc., Upper Saddle River, N.J. 07458

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

# Defective Items in Sample i

Size of sample i

p Chart Control Limits

© 2004 by Prentice Hall, Inc., Upper Saddle River, N.J. 07458

P-Chart for Data Entry Example

UCLp=0.10

LCLp=0.00

© 2004 by Prentice Hall, Inc., Upper Saddle River, N.J. 07458

c Chart
• 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

© 2004 by Prentice Hall, Inc., Upper Saddle River, N.J. 07458

c Chart Control Limits

Use3 for 99.7% limits

# Defects in Unit i

# Units Sampled

© 2004 by Prentice Hall, Inc., Upper Saddle River, N.J. 07458

Patterns to Look for in Control Charts

© 2004 by Prentice Hall, Inc., Upper Saddle River, N.J. 07458

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.

© 2004 by Prentice Hall, Inc., Upper Saddle River, N.J. 07458

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

© 2004 by Prentice Hall, Inc., Upper Saddle River, N.J. 07458

Process Capability Ratio, Cp

© 2004 by Prentice Hall, Inc., Upper Saddle River, N.J. 07458

Process Capability Cpk
• Assumes that the process is:
• under control
• normally distributed

© 2004 by Prentice Hall, Inc., Upper Saddle River, N.J. 07458

Cpk = negative number

Cpk = zero

Cpk = between 0 and 1

Cpk = 1

Cpk > 1

Meanings of Cpk Measures

© 2004 by Prentice Hall, Inc., Upper Saddle River, N.J. 07458

What Is Acceptance Sampling?
• 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

© 2004 by Prentice Hall, Inc., Upper Saddle River, N.J. 07458

What Is an Acceptance Plan?
• 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

© 2004 by Prentice Hall, Inc., Upper Saddle River, N.J. 07458

Operating Characteristics Curve
• Shows how well a sampling plan discriminates between good & bad lots (shipments)
• Shows the relationship between the probability of accepting a lot & its quality

© 2004 by Prentice Hall, Inc., Upper Saddle River, N.J. 07458

0

1

2

3

4

5

6

7

8

9

10

OC Curve100% Inspection

P(Accept Whole Shipment)

100%

Keep whole shipment

Return whole shipment

0%

Cut-Off

% Defective in Lot

© 2004 by Prentice Hall, Inc., Upper Saddle River, N.J. 07458

P(Accept Whole Shipment)

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

100%

Keep whole shipment

Return whole shipment

0%

Cut-Off

0

1

2

3

4

5

6

7

8

9

10

% Defective in Lot

OC Curve with Less than 100% Sampling

© 2004 by Prentice Hall, Inc., Upper Saddle River, N.J. 07458

AQL & LTPD
• 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

© 2004 by Prentice Hall, Inc., Upper Saddle River, N.J. 07458

Producer’s & Consumer’s Risk
• 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

© 2004 by Prentice Hall, Inc., Upper Saddle River, N.J. 07458

100

95

75

50

25

10

0

 = 0.05 producer’s risk for AQL

Probability of Acceptance

= 0.10

Percent Defective

Consumer’s risk for LTPD

0 1 2 3 4 5 6 7 8

AQL

LTPD

Good lots

Indifference zone

An Operating Characteristic (OC) Curve Showing Risks

© 2004 by Prentice Hall, Inc., Upper Saddle River, N.J. 07458

P(Accept Whole Shipment)

n = 50, c = 1

100%

n = 100, c = 2

0%

AQL

LTPD

0

1

2

3

4

5

6

7

8

9

10

% Defective in Lot

OC Curves for Different Sampling Plans

© 2004 by Prentice Hall, Inc., Upper Saddle River, N.J. 07458

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

© 2004 by Prentice Hall, Inc., Upper Saddle River, N.J. 07458

Developing a Sample Plan
• 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

© 2004 by Prentice Hall, Inc., Upper Saddle River, N.J. 07458

Lower 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]

© 2004 by Prentice Hall, Inc., Upper Saddle River, N.J. 07458