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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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Operations management

Operations Management

Chapter 8 -

Statistical Process Control

PowerPoint presentation to accompany

Heizer/Render

Principles of Operations Management, 7e

Operations Management, 9e


Outline

  • 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


Outline continued

Outline – Continued

  • Process Capability

    • Process Capability Ratio (Cp)

    • Process Capability Index (Cpk )

  • Acceptance Sampling

    • Operating Characteristic Curve

    • Average Outgoing Quality


Learning objectives

  • 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 objectives1

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 objectives2

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 objectives3

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

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

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

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

    • Eliminate the bad causes

    • Incorporate the good causes


Samples

Each of these represents one sample of five boxes of cereal

#

#

#

#

#

Frequency

#

#

#

#

#

#

#

#

#

#

#

#

#

#

#

#

#

#

#

#

#

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


Samples1

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


Samples2

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


Samples3

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


Samples4

?

?

?

?

?

?

?

?

?

?

?

?

?

?

?

?

?

?

?

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

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

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


Central limit theorem

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


Process control

(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


Population and sampling distributions

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

Sampling distribution of means

Process distribution of means

x = m

(mean)

Sampling Distribution

Figure S6.4


Steps in creating control charts

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


Control 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

Control Charts for Variables


Setting chart limits

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


Setting control 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


Setting control limits1

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


Setting chart limits1

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


Control chart factors

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


Setting control limits2

Process average x = 16.01 ounces

Average range R = .25

Sample size n = 5

Setting Control Limits


Setting control limits3

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


Setting control limits4

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

Setting Control Limits


R chart

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


Setting chart limits2

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


Setting control limits5

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


Mean and range charts

(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


Mean and range charts1

(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

Mean and Range Charts

Figure S6.5


Automated control charts

Automated Control Charts


Control charts for attributes

Control 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 charts

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


P chart for data entry

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


P chart for data entry1

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


P chart for data entry2

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


Control limits for c charts

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 chart for cab company

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

|

1

|

2

|

3

|

4

|

5

|

6

|

7

|

8

|

9

Day

c-Chart for Cab Company


Patterns in control charts

Upper control limit

Target

Lower control limit

Patterns in Control Charts

Normal behavior. Process is “in control.”

Figure S6.7


Patterns in control charts1

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


Patterns in control charts2

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


Patterns in control charts3

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


Patterns in control charts4

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


Patterns in control charts5

Upper control limit

Target

Lower control limit

Patterns in Control Charts

Erratic behavior. Investigate.

Figure S6.7


Which control chart to use

  • 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

Which Control Chart to Use

Variables Data


Which control chart to use1

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 use2

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

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


Process capability ratio

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 capability ratio1

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 capability ratio2

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 capability ratio3

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


Process capability index

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


Process capability index1

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


Process capability index2

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


Process capability index3

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


Interpreting c pk

Cpk = negative number

Cpk = zero

Cpk = between 0 and 1

Cpk = 1

Cpk > 1

Interpreting Cpk

Figure S6.8


Acceptance sampling

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

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


The perfect oc curve

Keep whole shipment

100 –

75 –

50 –

25 –

0 –

Return whole shipment

P(Accept Whole Shipment)

Cut-Off

|||||||||||

0102030405060708090100

% Defective in Lot

The “Perfect” OC Curve


Aql and ltpd

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 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


An oc curve

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

Bad lots

An OC Curve

Figure S6.9


Oc curves for different sampling plans

n = 50, c = 1

n = 100, c = 2

OC Curves for Different Sampling Plans


Average outgoing quality

(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 quality1

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)


Spc and process variability

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

SPC and Process Variability

Figure S6.10


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