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# Operations Management - PowerPoint PPT Presentation

Operations Management. Chapter 8 - Statistical Process Control. PowerPoint presentation to accompany Heizer/Render Principles of Operations Management, 7e Operations Management, 9e . Statistical Process Control (SPC) Control Charts for Variables The Central Limit Theorem

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#### Presentation Transcript

Operations Management

Chapter 8 -

Statistical Process Control

PowerPoint presentation to accompany

Heizer/Render

Principles of Operations Management, 7e

Operations Management, 9e

• 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 – Continued

• Process Capability

• Process Capability Ratio (Cp)

• Process Capability Index (Cpk )

• Acceptance Sampling

• Operating Characteristic Curve

• Average Outgoing Quality

• Natural and assignable causes of variation

• Central limit theorem

• Attribute and variable inspection

• Process control

• x-charts and R-charts

### Learning Objectives

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

Identify or Define:

### Learning Objectives

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

Identify or Define:

• LCL and UCL

• P-charts and c-charts

• Cp and Cpk

• Acceptance sampling

• OC curve

### Learning Objectives

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

Identify or Define:

• AQL and LTPD

• AOQ

• Producer’s and consumer’s risk

### Learning Objectives

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

Describe or Explain:

• The role of statistical quality control

### Statistical Process Control (SPC)

• 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

### Natural 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 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

• Incorporate the good causes

Each of these represents one sample of five boxes of cereal

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Frequency

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Weight

### Samples

To 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

Figure S6.1

The solid line represents the distribution

Frequency

Weight

### Samples

To measure the process, we take samples and analyze the sample statistics following these steps

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

Figure S6.1

Central tendency

Variation

Shape

Frequency

Weight

Weight

Weight

### Samples

To measure the process, we take samples and analyze the sample statistics following these steps

(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

Figure S6.1

Prediction

Frequency

Time

Weight

### Samples

To measure the process, we take samples and analyze the sample statistics following these steps

(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

Figure S6.1

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Prediction

Frequency

Time

Weight

### Samples

To measure the process, we take samples and analyze the sample statistics following these steps

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

Figure S6.1

### Control Charts

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

### Types of Data

Variables

Attributes

• Characteristics that can take any real value

• May be in whole or in fractional numbers

• Continuous random variables

• Defect-related characteristics

• Classify products as either good or bad or count defects

• Categorical or discrete random variables

The mean of the sampling distribution (x) will be the same as the population mean m

x = m

s

n

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

sx =

### Central 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

(a) In statistical control and capable of producing within control limits

Frequency

Upper control limit

Lower control limit

(b) In statistical control but not capable of producing within control limits

(c) Out of control

Size

(weight, length, speed, etc.)

### Process Control

Figure S6.2

Three population distributions

Mean of sample means = x

Beta

Standard deviation of the sample means

Normal

= sx =

Uniform

s

n

|||||||

-3sx-2sx-1sxx+1sx+2sx+3sx

95.45% fall within ± 2sx

99.73% of all x

fall within ± 3sx

### Population and Sampling Distributions

Distribution of sample means

Figure S6.3

Sampling distribution of means

Process distribution of means

x = m

(mean)

Figure S6.4

### Steps In Creating Control Charts

Take samples from the population and compute the appropriate sample statistic

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

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

Investigate possible assignable causes and take any indicated actions

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

• 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

### Control Charts for Variables

For x-Charts when we know s

Upper control limit (UCL) = x + zsx

Lower control limit (LCL) = x - zsx

wherex=mean of the sample means or a target value set for the process

z=number of normal standard deviations

sx=standard deviation of the sample means

=s/ n

s=population standard deviation

n=sample size

### Setting Chart Limits

Hour 1

SampleWeight ofNumberOat Flakes

117

213

316

418

517

616

715

817

916

Mean16.1

s =1

HourMeanHourMean

116.1715.2

216.8816.4

315.5916.3

416.51014.8

516.51114.2

616.41217.3

n = 9

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

LCLx = x - zsx = 16 - 3(1/3) = 15 ozs

### Setting Control Limits

For 99.73% control limits, z = 3

Variation due to assignable causes

Out of control

17 = UCL

Variation due to natural causes

16 = Mean

15 = LCL

Variation due to assignable causes

||||||||||||

123456789101112

Out of control

Sample number

### Setting Control Limits

Control Chart for sample of 9 boxes

For x-Charts when we don’t know s

Upper control limit (UCL) = x + A2R

Lower control limit (LCL) = x - A2R

whereR=average range of the samples

A2=control chart factor found in Table S6.1

x=mean of the sample means

### Setting Chart Limits

Sample Size Mean Factor Upper Range Lower Range

n A2D4D3

21.8803.2680

31.0232.5740

4.7292.2820

5.5772.1150

6.4832.0040

7.4191.9240.076

8.3731.8640.136

9.3371.8160.184

10.3081.7770.223

12.2661.7160.284

### Control Chart Factors

Table S6.1

Process average x = 16.01 ounces

Average range R = .25

Sample size n = 5

### Setting Control Limits

Process average x = 16.01 ounces

Average range R = .25

Sample size n = 5

UCLx = x + A2R

= 16.01 + (.577)(.25)

= 16.01 + .144

= 16.154 ounces

From Table S6.1

### Setting Control Limits

Process average x = 16.01 ounces

Average range R = .25

Sample size n = 5

UCLx = x + A2R

= 16.01 + (.577)(.25)

= 16.01 + .144

= 16.154 ounces

UCL = 16.154

Mean = 16.01

LCLx = x - A2R

= 16.01 - .144

= 15.866 ounces

LCL = 15.866

### R – 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

Upper control limit (UCLR) = D4R

Lower control limit (LCLR) = D3R

where

R=average range of the samples

D3 and D4=control chart factors from Table S6.1

### Setting Chart Limits

For R-Charts

Average range R = 5.3 pounds

Sample size n = 5

From Table S6.1 D4= 2.115, D3 = 0

UCLR = D4R

= (2.115)(5.3)

= 11.2 pounds

UCL = 11.2

Mean = 5.3

LCLR = D3R

= (0)(5.3)

= 0 pounds

LCL = 0

### Setting Control Limits

(a)

These sampling distributions result in the charts below

(Sampling mean is shifting upward but range is consistent)

UCL

(x-chart detects shift in central tendency)

x-chart

LCL

UCL

(R-chart does not detect change in mean)

R-chart

LCL

### Mean and Range Charts

Figure S6.5

(b)

These sampling distributions result in the charts below

(Sampling mean is constant but dispersion is increasing)

UCL

(x-chart does not detect the increase in dispersion)

x-chart

LCL

UCL

(R-chart detects increase in dispersion)

R-chart

LCL

Figure S6.5

### Control Charts for Attributes

• For variables that are categorical

• Measurement is typically counting defectives

• Charts may measure

• Percent defective (p-chart)

• Number of defects (c-chart)

p(1 - p)

n

sp =

UCLp = p + zsp

^

^

LCLp = p - zsp

^

wherep=mean fraction defective in the sample

z=number of standard deviations

sp=standard deviation of the sampling distribution

n=sample size

^

### Control 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

SampleNumberFractionSampleNumberFraction

Numberof ErrorsDefectiveNumberof ErrorsDefective

16.06116.06

25.05121.01

30.00138.08

41.01147.07

54.04155.05

62.02164.04

75.051711.11

83.03183.03

93.03190.00

102.02204.04

Total = 80

(.04)(1 - .04)

100

80

(100)(20)

sp = = .02

p = = .04

^

### p-Chart for Data Entry

UCLp= 0.10

.11 –

.10 –

.09 –

.08 –

.07 –

.06 –

.05 –

.04 –

.03 –

.02 –

.01 –

.00 –

LCLp= 0.00

UCLp = p + zsp= .04 + 3(.02) = .10

^

Fraction defective

p = 0.04

LCLp = p - zsp = .04 - 3(.02) = 0

^

||||||||||

2468101214161820

Sample number

### p-Chart for Data Entry

UCLp= 0.10

.11 –

.10 –

.09 –

.08 –

.07 –

.06 –

.05 –

.04 –

.03 –

.02 –

.01 –

.00 –

LCLp= 0.00

UCLp = p + zsp= .04 + 3(.02) = .10

^

Fraction defective

p = 0.04

LCLp = p - zsp = .04 - 3(.02) = 0

^

||||||||||

2468101214161820

Sample number

### p-Chart for Data Entry

Possible assignable causes present

UCLc = c + 3 c

LCLc = c - 3 c

wherec=mean number defective in the sample

### Control 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

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

UCLc= c + 3 c

= 6 + 3 6

= 13.35

UCLc= 13.35

14 –

12 –

10 –

8 –

6 –

4 –

2 –

0 –

Number defective

LCLc= c - 3 c

= 3 - 3 6

= 0

c = 6

LCLc= 0

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Day

### c-Chart for Cab Company

Upper control limit

Target

Lower control limit

### Patterns in Control Charts

Normal behavior. Process is “in control.”

Figure S6.7

Upper control limit

Target

Lower control limit

### Patterns in Control Charts

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

Figure S6.7

Upper control limit

Target

Lower control limit

### Patterns in Control Charts

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

Figure S6.7

Upper control limit

Target

Lower control limit

### Patterns in Control Charts

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

Figure S6.7

Upper control limit

Target

Lower control limit

### Patterns in Control Charts

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

Figure S6.7

Upper control limit

Target

Lower control limit

### Patterns in Control Charts

Erratic behavior. Investigate.

Figure S6.7

• Using an x-chart 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

Variables Data

### Which Control Chart to Use

Attribute Data

• Using the p-chart:

• Observations are attributes that can be categorized in two states

• We deal with fraction, proportion, or percent defectives

• Have several samples, each with many observations

### Which Control Chart to Use

Attribute Data

• 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

• Defects such as number of blemishes on a desk, number of typos in a page of text, flaws in a bolt of cloth

### Process 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

Upper Specification - Lower Specification

6s

Cp =

### Process Capability Ratio

• 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

Process mean x = 210.0 minutes

Process standard deviation s = .516 minutes

Design specification = 210 ± 3 minutes

Upper Specification - Lower Specification

6s

Cp =

### Process Capability Ratio

Insurance claims process

Process mean x = 210.0 minutes

Process standard deviation s = .516 minutes

Design specification = 210 ± 3 minutes

Upper Specification - Lower Specification

6s

Cp =

213 - 207

6(.516)

= = 1.938

### Process Capability Ratio

Insurance claims process

Process mean x = 210.0 minutes

Process standard deviation s = .516 minutes

Design specification = 210 ± 3 minutes

Upper Specification - Lower Specification

6s

Cp =

213 - 207

6(.516)

= = 1.938

### Process Capability Ratio

Insurance claims process

Process is capable

UpperSpecification - xLimit

3s

Lowerx -SpecificationLimit

3s

Cpk = minimum of ,

### Process 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

New process mean x = .250 inches

Process standard deviation s = .0005 inches

Upper Specification Limit = .251 inches

Lower Specification Limit = .249 inches

### Process Capability Index

New Cutting Machine

New process mean x = .250 inches

Process standard deviation s = .0005 inches

Upper Specification Limit = .251 inches

Lower Specification Limit = .249 inches

(.251) - .250

(3).0005

Cpk = minimum of ,

### Process Capability Index

New Cutting Machine

New process mean x = .250 inches

Process standard deviation s = .0005 inches

Upper Specification Limit = .251 inches

Lower Specification Limit = .249 inches

(.251) - .250

(3).0005

.250 - (.249)

(3).0005

Cpk = minimum of ,

.001

.0015

Cpk = = 0.67

### Process Capability Index

New Cutting Machine

Both calculations result in

New machine is NOT capable

Cpk = negative number

Cpk = zero

Cpk = between 0 and 1

Cpk = 1

Cpk > 1

Figure S6.8

### Acceptance 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

### Operating Characteristic Curve

• 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

Keep whole shipment

100 –

75 –

50 –

25 –

0 –

Return whole shipment

P(Accept Whole Shipment)

Cut-Off

|||||||||||

0102030405060708090100

% Defective in Lot

### AQL 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 Risks

• 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 (b)

• Probability of accepting a bad lot

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

100 –

95 –

75 –

50 –

25 –

10 –

0 –

 = 0.05 producer’s risk for AQL

Probability of Acceptance

|||||||||

012345678

Percent defective

 = 0.10

AQL

LTPD

Consumer’s risk for LTPD

Good lots

Indifference zone

Figure S6.9

n = 50, c = 1

n = 100, c = 2

(Pd)(Pa)(N - n)

N

AOQ =

### 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

### Average Outgoing Quality

If a sampling plan replaces all defectives

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 defective or the lowest average quality and is called the average outgoing quality level (AOQL)

Lower specification limit

Upper specification limit

(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)

Process mean, m

Figure S6.10