© 2006 Prentice Hall, Inc.S6 – 1 Operations Management Chapter 8 - Statistical Process Control PowerPoint presentation to accompany Heizer/Render Principles.

© 2006 Prentice Hall, Inc. S6 – 1

Operations ManagementOperations ManagementChapter 8 - Statistical Process Control

PowerPoint presentation to accompany Heizer/Render Principles of Operations Management, 7eOperations Management, 9e


Outline

Statistical Process Control (SPC)Control Charts for VariablesThe Central Limit TheoremSetting Mean Chart Limits (x-Charts)Setting Range Chart Limits (R-Charts)Using Mean and Range ChartsControl Charts for AttributesManagerial Issues and Control Charts


Outline – Continued

Process CapabilityProcess Capability Ratio (Cp)

Process Capability Index (Cpk )

Acceptance SamplingOperating Characteristic CurveAverage Outgoing Quality


Learning Objectives

When you complete this supplement, you should be able to:Identify or Define:

Natural and assignable causes of variation

Central limit theorem Attribute and variable inspection Process control x-charts and R-charts


Learning Objectives


LCL and UCL P-charts and c-charts Cp and Cpk

Acceptance sampling OC curve


Learning Objectives


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


Variability is inherent in every processNatural or common causesSpecial or assignable causes

Provides a statistical signal when assignable causes are present

Detect and eliminate assignable causes of variation

Statistical Process Control (SPC)


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 Eliminate the bad causes Incorporate the good causes


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

Fre

qu

ency

Weight

#

## #

##

##

#

# # ## # ##

# # ## # ## # ##

Each of these represents one sample of five

boxes of cereal

Figure S6.1


Samples


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

The solid line represents the

distribution

Fre

qu

ency

WeightFigure S6.1


Samples


(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

Weight

Central tendency

Weight

Variation

Weight

Shape

Fre

qu

ency

Figure S6.1


Samples


(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

WeightTimeF

req

uen

cy Prediction

Figure S6.1


Samples


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

WeightTimeF

req

uen

cy Prediction

????

???

???

??????

???

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

Characteristics that can take any real value

May be in whole or in fractional numbers

Continuous random variables

Variables Attributes Defect-related

characteristics Classify products

as either good or bad or count defects

Categorical or discrete random variables


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

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

x = m

s n

sx =

2. 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


Process Control

Figure S6.2

Frequency

(weight, length, speed, etc.)Size

Lower control limit Upper control limit

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

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

(c) Out of control


Population and Sampling Distributions

Three population distributions

Beta

Normal

Uniform

Distribution of sample means

Standard deviation of the sample means

= sx =s

n

Mean of sample means = x

| | | | | | |

-3sx -2sx -1sx x +1sx +2sx +3sx

99.73% of all xfall within ± 3sx

95.45% fall within ± 2sx

Figure S6.3


Sampling Distribution

x = m(mean)

Sampling distribution of means

Process distribution of means

Figure S6.4


Steps In Creating Control Charts

1. Take samples from the population and compute the appropriate sample statistic

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

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

4. Investigate possible assignable causes and take any indicated actions

5. 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


Setting Chart Limits

For x-Charts when we know s

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 processz = number of normal standard deviations

sx = standard deviation of the sample means

= s/ ns = population standard deviationn = sample size


Setting Control LimitsHour 1

Sample Weight ofNumber Oat Flakes

1 172 133 164 185 176 167 158 179 16

Mean 16.1s = 1

Hour Mean Hour Mean1 16.1 7 15.22 16.8 8 16.43 15.5 9 16.34 16.5 10 14.85 16.5 11 14.26 16.4 12 17.3

n = 9

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

For 99.73% control limits, z = 3

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


17 = UCL

15 = LCL

16 = Mean

Setting Control Limits

Control Chart for sample of 9 boxes

Sample number

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

Variation due to assignable

causes

Variation due to assignable

causes

Variation due to natural causes

Out of control

Out of control



For x-Charts when we don’t know s

Lower control limit (LCL) = x - A2R

Upper control limit (UCL) = x + A2R

where R = average range of the samples

A2 = control chart factor found in Table S6.1 x = mean of the sample means


Control Chart Factors

Table S6.1

Sample Size Mean Factor Upper Range Lower Range

n A2 D4 D32 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



Process average x = 16.01 ouncesAverage range R = .25Sample size n = 5



UCLx = x + A2R= 16.01 + (.577)(.25)= 16.01 + .144= 16.154 ounces


From Table S6.1



UCLx = x + A2R= 16.01 + (.577)(.25)= 16.01 + .144= 16.154 ounces

LCLx = x - A2R= 16.01 - .144= 15.866 ounces


UCL = 16.154

Mean = 16.01

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



For R-Charts

Lower control limit (LCLR) = D3R

Upper control limit (UCLR) = D4R

whereR = average range of the samples

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



UCLR = D4R= (2.115)(5.3)= 11.2 pounds

LCLR = D3R= (0)(5.3)= 0 pounds

Average range R = 5.3 poundsSample size n = 5From Table S6.1 D4 = 2.115, D3 = 0

UCL = 11.2

Mean = 5.3

LCL = 0


Mean and Range Charts

(a)

These sampling distributions result in the charts below

(Sampling mean is shifting upward but range is consistent)

R-chart(R-chart does not detect change in mean)

UCL

LCL

Figure S6.5

x-chart(x-chart detects shift in central tendency)

UCL

LCL


Mean and Range Charts

R-chart(R-chart detects increase in dispersion)

UCL

LCL

Figure S6.5

(b)

These sampling distributions result in the charts below

(Sampling mean is constant but dispersion is increasing)

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

UCL

LCL


Automated Control Charts


Control Charts for Attributes

For variables that are categoricalGood/bad, yes/no,

acceptable/unacceptable

Measurement is typically counting defectives

Charts may measurePercent defective (p-chart)Number of defects (c-chart)


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

UCLp = p + zsp^

LCLp = p - zsp^

where p = mean fraction defective in the samplez = number of standard deviationssp = standard deviation of the sampling distribution

n = sample size

^

p(1 - p)n

sp =^


p-Chart for Data EntrySample Number Fraction Sample Number FractionNumber of Errors Defective Number of Errors Defective

1 6 .06 11 6 .062 5 .05 12 1 .013 0 .00 13 8 .084 1 .01 14 7 .075 4 .04 15 5 .056 2 .02 16 4 .047 5 .05 17 11 .118 3 .03 18 3 .039 3 .03 19 0 .00

10 2 .02 20 4 .04Total = 80

(.04)(1 - .04)

100sp = = .02^

p = = .0480

(100)(20)


.11 –

.10 –

.09 –

.08 –

.07 –

.06 –

.05 –

.04 –

.03 –

.02 –

.01 –

.00 –

Sample number

Fra

ctio

n d

efec

tive

| | | | | | | | | |

2 4 6 8 10 12 14 16 18 20

p-Chart for Data Entry

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

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

UCLp = 0.10

LCLp = 0.00

p = 0.04


.11 –

.10 –

.09 –

.08 –

.07 –

.06 –

.05 –

.04 –

.03 –

.02 –

.01 –

.00 –

Sample number

Fra

ctio

n d

efec

tive

| | | | | | | | | |

2 4 6 8 10 12 14 16 18 20

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

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

UCLp = 0.10

LCLp = 0.00

p = 0.04

p-Chart for Data Entry

Possible assignable

causes present


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

where c = mean number defective in the sample

UCLc = c + 3 c LCLc = c - 3 c


c-Chart for Cab Company

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

|1

|2

|3

|4

|5

|6

|7

|8

|9

Day

Nu

mb

er d

efec

tive

14 –

12 –

10 –

8 –

6 –

4 –

2 –

0 –

UCLc = c + 3 c= 6 + 3 6= 13.35

LCLc = c - 3 c= 3 - 3 6= 0

UCLc = 13.35

LCLc = 0

c = 6


Patterns in Control Charts

Normal behavior. Process is “in control.”

Upper control limit

Target

Lower control limit

Figure S6.7


Upper control limit

Target

Lower control limit


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

Figure S6.7


Upper control limit

Target

Lower control limit


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

Figure S6.7


Upper control limit

Target

Lower control limit


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

Figure S6.7


Upper control limit

Target

Lower control limit


Run of 5 above (or below) central line. Investigate for cause. Figure S6.7


Upper control limit

Target

Lower control limit


Erratic behavior. Investigate.

Figure S6.7


Which Control Chart to Use

Using an x-chart and R-chart:Observations are variablesCollect 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



Using the p-chart:Observations are attributes that can

be categorized in two states We deal with fraction, proportion, or

percent defectivesHave several samples, each with

many observations

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

Attribute Data


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

Cp = Upper Specification - Lower Specification

6s

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




6s

Insurance claims process

Process mean x = 210.0 minutesProcess standard deviation s = .516 minutesDesign specification = 210 ± 3 minutes




6s



= = 1.938213 - 2076(.516)




6s



= = 1.938213 - 2076(.516)

Process is capable


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

Cpk = minimum of ,

UpperSpecification - xLimit

3s

Lowerx - Specification

Limit3s



New Cutting Machine

New process mean x = .250 inchesProcess standard deviation s = .0005 inchesUpper Specification Limit = .251 inchesLower Specification Limit = .249 inches



New Cutting Machine


Cpk = minimum of ,(.251) - .250

(3).0005



New Cutting Machine


Cpk = = 0.67.001

.0015

New machine is NOT capable

Cpk = minimum of ,(.251) - .250

(3).0005.250 - (.249)

(3).0005

Both calculations result in


Interpreting Cpk

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 goodsTake samples at random from a lot

(shipment) of items Inspect each of the items in the sampleDecide 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


Return whole shipment

The “Perfect” OC Curve

% Defective in Lot

P(A

cc

ept

Wh

ole

Sh

ipm

en

t)

100 –

75 –

50 –

25 –

0 –| | | | | | | | | | |

0 10 20 30 40 50 60 70 80 90 100

Cut-Off

Keep whole shipment


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 badConsumer (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


An OC Curve

Probability of

Acceptance

Percent defective

| | | | | | | | |0 1 2 3 4 5 6 7 8

100 –95 –

75 –

50 –

25 –

10 –

0 –

= 0.05 producer’s risk for AQL

= 0.10

Consumer’s risk for LTPD

LTPDAQL

Bad lotsIndifference zone

Good lots

Figure S6.9


OC Curves for Different Sampling Plans

n = 50, c = 1

n = 100, c = 2


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

AOQ = (Pd)(Pa)(N - n)

N


Average Outgoing Quality

1. If a sampling plan replaces all defectives

2. 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

(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)

Lower specification

limit

Upper specification

limit

Process mean, m Figure S6.10

© 2006 Prentice Hall, Inc.S6 – 1 Operations Management Chapter 8 - Statistical Process Control PowerPoint presentation to accompany Heizer/Render Principles.

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