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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

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

• Process Capability

• Process Capability Ratio (Cp)

• Process Capability Index (Cpk )

• Acceptance Sampling

• Operating Characteristic Curve

• Average Outgoing Quality

Learning Objectives

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

Identify or Define:

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

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

Identify or Define:

• AQL and LTPD

• AOQ

• Producer’s and consumer’s risk

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

Describe or Explain:

• The role of statistical quality control

• 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

• 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”

• 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

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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

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

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

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

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

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

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 control limits

Mean of sample means = x

Beta

Standard deviation of the sample means

Normal

= sx =

Uniform

s

n

| | | | | | |

-3sx -2sx -1sx x +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 control limits

Process distribution of means

x = m

(mean)

Sampling Distribution

Figure S6.4

Steps In Creating Control Charts control limits

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 control limits

• 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 control limitss

Upper control limit (UCL) = x + zsx

Lower control limit (LCL) = x - zsx

where x = 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 control limits

Sample Weight of Number Oat Flakes

1 17

2 13

3 16

4 18

5 17

6 16

7 15

8 17

9 16

Mean 16.1

s = 1

Hour Mean Hour Mean

1 16.1 7 15.2

2 16.8 8 16.4

3 15.5 9 16.3

4 16.5 10 14.8

5 16.5 11 14.2

6 16.4 12 17.3

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 control limits

Out of control

17 = UCL

Variation due to natural causes

16 = Mean

15 = LCL

Variation due to assignable causes

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

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

Out of control

Sample number

Setting Control Limits

Control Chart for sample of 9 boxes

For x-Charts when we don’t know control limitss

Upper control limit (UCL) = x + A2R

Lower control limit (LCL) = x - A2R

where R = 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 control limitsUpper Range Lower Range

n A2D4D3

2 1.880 3.268 0

3 1.023 2.574 0

4 .729 2.282 0

5 .577 2.115 0

6 .483 2.004 0

7 .419 1.924 0.076

8 .373 1.864 0.136

9 .337 1.816 0.184

10 .308 1.777 0.223

12 .266 1.716 0.284

Control Chart Factors

Table S6.1

Process average x control limits= 16.01 ounces

Average range R = .25

Sample size n = 5

Setting Control Limits

Process average x control limits= 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 control limits= 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 control limits

• 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 control limits(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 control limits= 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) control limits

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) control limits

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 control limits

Control Charts for Attributes control limits

• For variables that are categorical

• Measurement is typically counting defectives

• Charts may measure

• Percent defective (p-chart)

• Number of defects (c-chart)

p control limits(1 - p)

n

sp =

UCLp = p + zsp

^

^

LCLp = p - zsp

^

where p = 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

Sample Number Fraction Sample Number Fraction control limits

Number of Errors Defective Number of Errors Defective

1 6 .06 11 6 .06

2 5 .05 12 1 .01

3 0 .00 13 8 .08

4 1 .01 14 7 .07

5 4 .04 15 5 .05

6 2 .02 16 4 .04

7 5 .05 17 11 .11

8 3 .03 18 3 .03

9 3 .03 19 0 .00

10 2 .02 20 4 .04

Total = 80

(.04)(1 - .04)

100

80

(100)(20)

sp = = .02

p = = .04

^

p-Chart for Data Entry

UCL control limitsp= 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

^

| | | | | | | | | |

2 4 6 8 10 12 14 16 18 20

Sample number

p-Chart for Data Entry

UCL control limitsp= 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

^

| | | | | | | | | |

2 4 6 8 10 12 14 16 18 20

Sample number

p-Chart for Data Entry

Possible assignable causes present

UCL control limitsc = c + 3 c

LCLc = c - 3 c

where c = 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 control limits= 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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5

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9

Day

c-Chart for Cab Company

Upper control limit control limits

Target

Lower control limit

Patterns in Control Charts

Normal behavior. Process is “in control.”

Figure S6.7

Upper control limit control limits

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 control limits

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 control limits

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 control limits

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 control limits

Target

Lower control limit

Patterns in Control Charts

Erratic behavior. Investigate.

Figure S6.7

• Using an x-chart and R-chart: control limits

• 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 Use control limits

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 control limits

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 control limits

• 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 control limits

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 control limits= 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 control limits= 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 control limits= 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

Upper control limitsSpecification - xLimit

3s

Lowerx - Specification Limit

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 control limits= .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 control limits= .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 control limits= .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

C control limitspk = negative number

Cpk = zero

Cpk = between 0 and 1

Cpk = 1

Cpk > 1

Interpreting Cpk

Figure S6.8

Acceptance Sampling control limits

• 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 control limits

• 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 control limits

100 –

75 –

50 –

25 –

0 –

Return whole shipment

P(Accept Whole Shipment)

Cut-Off

| | | | | | | | | | |

0 10 20 30 40 50 60 70 80 90 100

% Defective in Lot

The “Perfect” OC Curve

AQL and LTPD control limits

• 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 control limits

• 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 control limits–

95 –

75 –

50 –

25 –

10 –

0 –

 = 0.05 producer’s risk for AQL

Probability of Acceptance

| | | | | | | | |

0 1 2 3 4 5 6 7 8

Percent defective

 = 0.10

AQL

LTPD

Consumer’s risk for LTPD

Good lots

Indifference zone

An OC Curve

Figure S6.9

n control limits = 50, c = 1

n = 100, c = 2

OC Curves for Different Sampling Plans

( control limitsPd)(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 control limits

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 control limits

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