OutlineManagerial Issues Sampling (technique for data collection) Use appropriate Control ChartsControl Charts for Variables Setting Mean Chart Limits ( x-Charts) Setting Range Chart Limits (R-Charts)Control Charts for Attributes P-Charts: For single attributes C-Charts: For multiple attributes

Decide UCL and LCL for each control chartProcess Capability and Acceptance Sampling

Show changes in data pattern e.g., trends

Make corrections before process is out of control

Find causes of changes in data Assignable causes

Data outside control limits or trend in data Natural causes

Random variations around average

Purposes of Control Chart

Statistical Process Control (SPC) for Quality Management

A process used to monitor standards, making measurements and taking corrective action as a product or service is being produced or deliveredUses mathematics (i.e., statistics) methods to evaluate processSPC is suitable for managing process performanceBecome the backbone of modern quality control in both theory and practice

Natural (Normal) Variations

Comprised of a myriad of small sources that are always present in a process and affect all elements of the process. Vibration Humidity Temperature Lighting Other uncontrollable factors

Usually is difficult or costly to control

Assignable (Abnormal) Variations

Caused by the controllable quality problems in a product or process. Poor product design Machines out of order Tools wear out Poor incoming materials Low skills and qualification of workers Workers’ fatigue Unpleasant working conditions Poor training

Produce GoodProvide Service

Stop Process

Yes

No

Assign.Variation?Take Sample

Inspect Sample

Find Out WhyCreate

Control Chart

Start

Statistical Process Control Steps

Characteristics for which you focus on defectsClassify products as either ‘good’ or ‘bad’, or count # defectsCategorical or discrete random variables

AttributesAttributesVariablesVariables

Two Types of Quality Characteristics

•Characteristics that can be measured continuously, e.g., weight, length•May be in whole or in fractional numbers•Continuous random variables

ControlCharts

RChart

VariablesCharts

AttributesCharts

XChart

PChart

CChart

Continuous Numerical Data

Categorical or Discrete Numerical Data

Control Chart Types

Sampling Techniques in Quality Control

Why sampling Too costly to inspect all outcomes from a

process

Sample size: SPC usually uses average of a small number of items as a sample Individual pieces tend to be too erratic to make

trends quickly visible

Serve as the input of all control chartsBoth sampling rule and sample sizes affect the cost and accuracy of quality control

Monitoring the Weights of Oat Flakes (Example S1, p222)

Purpose of sampling and the sampling rule The weights of boxes of Oat Flakes within a large

production lot are sampled each hour

Sample Frequency Sampling every hour

Sample size In each sampling, 9 boxes are randomly selected

and weighted

Confidence and number of standard deviation = 2 for 95.5% confidence; = 3 for 99.73%

confidence

X

Mean

x

x

n

xx

n

Standard deviation (STD)

X X

Central Limit Theorem

n x

i

n

ix

Normalization of Sample Distributions

Uniform

Normal

Beta

(mean)

x2 withinfall x all of 95.5%

x3 withinfall x all of 99.7%

x3 x2 x x x1 x2 x3

Three population distributions

Relationship of Confidence and Number of STD ()

Properties of normal distribution

x2 withinfall x lal of 95.5%

x3 withinfall x lal of 99.7%

x

x

Type of variables control chart Interval or ratio scaled numerical

data

Shows sample means over timeMonitors process averageExample: Weigh samples of coffee & compute means of samples; Plot

X Chart

X Chart and Control Limits (Formula 1)   If the process mean and standard deviation are known:

 

where: _

X = average mean of samples

Z = number of standard deviations

x = standard deviation of sample means

x = process standard deviation,

n = number of observations in a sample

XX

XX

ZXLCL

ZXUCL

nX

X

Sample Range at Time i

# Samples

Sample Mean at Time i

From Table S6.1

RAxxLCL

RAxxUCL

n

R R

i

n

1i

n

xi

n

ix

X Chart and Control Limits (Formula 2)

Factors for Computing Control Chart Limits (3 sigma, p.223)

SampleSize, n

MeanFactor, A2

UpperRange, D4

LowerRange, D3

2 1.880 3.268 0

3 1.023 2.574 0

4 0.729 2.282 0

5 0.577 2.115 0

6 0.483 2.004 0

7 0.419 1.924 0.076

8 0.373 1.864 0.136

9 0.337 1.816 0.184

10 0.308 1.777 0.2230.184

Super Cola (Example S2, p224)

Super Cola bottles soft drinks labeled ”net weight 16 ounces.” An overall 16.01 ounces has been found by taking several batches of samples, in which each sample contained 5 bottles. The average range of the process is 0.25 ounce. Determine the upper and lower control limits for averages in this process.

Sample Range at Time i

# Samples

From Table S6.1

R Chart Control Limits

n

R R

R D LCL

R D UCL

i

n

1i

3R

4R

Loading Trucks (Example S3, p225)

The average range of a process for loading trucks is 5.3 pounds. If the sample size is 5, determine the upper and lower control limits for the R-Chart.

X-bar and R Charts Complement Each Other

Three Types of Output for Variable

Frequency

Lower control limit

SizeWeight, length, speed, etc.

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.

(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.

Control chart for attributes with scaled categorical data (e.g., good-bad)Normally measure the percent of defective in a sampleAssume the outcome of each sample follows binomial distributionExample: Count number defective chairs & divide by

total chairs inspected in each sample plot the result along the time line Chair is either defective or not defective

p Chart

Control limit of p Charts

# Defective Items in Sample i

Size of sample i

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

i

k

1i

i

k

1i

n

xp

)1(

)1(

n

ppzpLCL

n

ppzpUCL

p

p

Mosier (Example S4, p227)

Data-entry clerks at Mosier key in thousands of insurance records each day. Samples of the work of 20 clerks are shown in the table. One hundred records by each clerk were carefully examined and the number of errors counted. The fraction in each sample was then computed as p-bar.Set the control limits to include 99.73% of the random variation in the entry process when it is in control.

Attributes control chart for discrete data

Shows the number of nonconformities (defects) in a unit (unit may be chair, steel sheet, car etc). UCL and LCL are not sensitive to the sample size Assume the defect number is Poison distribution

Example:

Derive the average number of defects (scratches, chips etc.) in each chair of a sample of 100 chairs

Plot the average number along the timeline

c - Chart

Control Limits of c-Charts

# Defects in Unit i

# Units Sampled

Use 3 for 99.7% limits

k

c c

i

k

1i

cZcLCL

cZcUCL

c

c

Red Top Cap (Example S5, p229)

Red Top Cab Company receives several complaints per day about the behavior of its drivers. Over a 9-day period (where days are the units of measure), the owner received the following number of calls from rate passengers: {3, 0, 8, 9, 6, 7, 4, 9, 8} for a total of 54 complaints. Compute the UCL and LCL limits at 99.7% confidence.

Managerial Issues and Control Charts

Three major decisions regarding control chartSelect the points in the process that need SPC Which process point is critical Which point tends to be out of control

Select appropriate chart and UCL/LCLSet clear and specific SPC policies for workers to follow

Process Capability Cpk

population process theof deviation standard

mean process x where

3

Limition SpecificatLower x

or , 3

x Limit ion SpecificatUpper of minimum

pkC

Measure difference between actual and desire output quality

Application of Process Capacity:

Technology selection

Performance evaluation

Meanings of Cpk Measures

Cpk = negative number

Cpk = zero

Cpk = between 0 and 1

Cpk = 1

Cpk > 1

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

What Is Acceptance Sampling?

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

Operating Characteristics Curve

% Defective in Lot

P(Accept Whole Shipment)

100%

0%

Cut-Off1 2 3 4 5 6 7 8 9 100

Return whole shipment

Keep whole shipment

OC Curve100% Inspection

OC Curve with Less than 100% Sampling

P(Accept Whole Shipment)

100%

0%

% Defective in LotCut-Off1 2 3 4 5 6 7 8 9 100

Return whole shipment

Keep whole shipment

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

Supplier/Producer's risk () Probability of rejecting a good lot (type I

error) Probability that a lot get rejected when

fraction defective is AQL

Buyer/Consumer's risk (ß) Probability of accepting a bad lot (type II

error) Probability of accepting a lot when fraction

defective is LTPD

Producer’s & Consumer’s Risk

Acceptable quality level (AQL) Quality level of a good lot from producer’s

standard Producer (supplier) does not want lots with

fewer defects than AQL rejected

Lot tolerance percent defective (LTPD) Quality level of a bad lot from buyer’s

standard Consumer (buyer) does not want lots with

more defects than LTPD accepted

AQL & LTPD

An Operating Characteristic (OC) Curve Showing Risks

= 0.10

Consumer’s risk for LTPD

Probability of Acceptance

Percent Defective0 1 2 3 4 5 6 7 8

10095

75

50

25

10

0

= 0.05 producer’s risk for AQL

Bad lotsIndifference zoneGood lots

LTPDAQL

Set of procedures for inspecting incoming materials or finished goodsIdentifies Type of sample Sample size (n) Criteria (c) used to reject or accept a lot

Producer (supplier) & consumer (buyer) must negotiate

What Is an Acceptance Plan?