Egekwu-2112

Statistical Quality Control/Statistical Process Control

Acceptance Sampling– Operating Characteristic Curve

Process Control Procedures– Variable data– Attribute or Characteristic data

Process Capability

Egekwu-2113

Basic Forms of Statistical Sampling for Quality Control

Sampling to accept or reject the immediate lot of product at hand (Acceptance Sampling).

Sampling to determine if the process is within acceptable limits (Statistical Process Control)

Egekwu-2114

Acceptance Sampling Purposes

– Determine quality level– Ensure quality is within predetermined level

Advantages– Economy– Less handling damage– Fewer inspectors– Upgrading of the inspection job– Applicability to destructive testing– Entire lot rejection (motivation for improvement)

Egekwu-2115

Acceptance Sampling

Disadvantages– Risks of accepting “bad” lots and rejecting “good”

lots– Added planning and documentation– Sample provides less information than 100-percent

inspection

Egekwu-2116

Statistical Sampling--Data

Attribute (Go no-go information)– Defectives--refers to the acceptability of product

across a range of characteristics.– Defects--refers to the number of unacceptable

conditions per unit--may be higher than the number of defectives.

Variable (Continuous)– Usually measured by the mean and the standard

deviation.

Egekwu-2117

Acceptance Sampling--Single Sampling Plan

A simple goal

Determine (1) how many units, n, to sample from a lot, and

(2) the maximum number of defective items, c, that can be found in the sample before the lot is rejected.

Egekwu-2118

Risk

Acceptable Quality Level (AQL)– Max. acceptable percentage of defectives defined by

producer. (Producer’s risk)

– The probability of rejecting a good lot. Lot Tolerance Percent Defective (LTPD)

– Percentage of defectives that defines consumer’s rejection point.

(Consumer’s risk)– The probability of accepting a bad lot.

Egekwu-21110

Example: Acceptance Sampling

Zypercom, a manufacturer of video interfaces, purchases printed wiring boards from an outside vender, Procard. Procard has set an acceptable quality level of 1% and accepts a 5% risk of rejecting lots at or below this level. Zypercom considers lots with 3% defectives to be unacceptable and will assume a 10% risk of accepting a defective lot.

Develop a sampling plan for Zypercom and determine a rule to be followed by the receiving inspection personnel.

Egekwu-21111

Sorting It Out

For this example, how do we determine

AQL?

?

LTPD?

?

Egekwu-21112

Example: Continued

LTPD

AQL =

.03

.01 = 3 n (AQL) = 3.286

c LTPD/AQL n AQL c LTPD/AQL n AQL0 44.890 0.052 5 3.549 2.6131 10.946 0.355 6 3.206 3.2862 6.509 0.818 7 2.957 3.9813 4.890 1.366 8 2.768 4.6954 4.057 1.970 9 2.618 5.426

How can we determine the value of n?What is our sampling procedure?

Exhibit TN 7.10

Egekwu-21113

Example: Continued

c = 6, from Table; c is also called acceptance numbern (AQL) = 3.286, from TableAQL = .01, given in problem

n(AQL/AQL) = 3.286/.01 = 328.6, or 329 (always round up)

Sampling Procedure:Take a random sample of 329 units from a lot. Reject the lot if more than 6 units are defective.

Egekwu-2119

Operating Characteristic Curve

n = 99c = 4

AQL LTPD

00.10.20.30.40.50.60.70.80.9

1

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

Percent defective

Pro

bab

ilit

y of

acc

epta

nce

=.10(consumer’s risk)

= .05 (producer’s risk)

Egekwu-211

Chance Versus Assignable Variation

Chance variation is variability built into the system.

Assignable variation occurs because some element of the system or some operating conditions are out of control.

Quality control seeks to identify when assignable variation is present so that corrective action can be taken.

Egekwu-211

Control Based on Attributes and Variables

Inspection for Variables: measuring a variable that can be scaled such as weight, length, temperature, and diameter.

Inspection of Attributes: determining the existence of a characteristic such as acceptable-defective, timely-late, and right-wrong.

Egekwu-211

Control Charts: Assumptions

Developed in 1920s to distinguish between chance variation in a system and variation caused by the system’s being out of control - assignable variation.

Egekwu-211

Control Charts - Assumptions continued

Repetitive operation will not produce exactly the same outputs.

Pattern of variability often described by normal distribution.

Random samples that fully represent the population being checked are taken.

Sample data plotted on control charts to determine if the process is still under control.

Egekwu-211

Control Chart with Limits Set at Three Standard Deviations

Egekwu-21115

Control Limits

If we establish control limits at +/- 3 standard deviations, then

we would expect 99.7% of our observations to fall within these limits

xLCL UCL

UCL

LCL

UCL

LCL

UCL

LCL

What other evidence(s) might prompt investigation?

StatisticalProcessControl

©The McGraw-Hill Companies, Inc., 1998

Egekwu-21117

Attribute Data: Constructing a p-Chart

Sample n Defectives1 100 42 50 23 100 54 100 35 75 66 100 47 100 38 50 89 100 1

10 100 211 100 312 100 213 100 214 100 815 100 3

Egekwu-21118

Statistical Process Control--Attribute Measurements (P-Charts)

p =Total Number of Defectives

Total Number of Observations

= p (1- p)

npS

UCL = p + Z

LCL = p - Z p

p

s

s

(Std Deviation)

= [ p ]

1. Calculate the sample proportion, p, for each

sample.Sample n Defectives p

1 100 4 0.042 50 2 0.043 100 5 0.054 100 3 0.035 75 6 0.086 100 4 0.047 100 3 0.038 50 7 0.149 100 1 0.01

10 100 2 0.0211 100 3 0.0312 100 2 0.0213 100 2 0.0214 100 8 0.0815 100 3 0.03

19©The McGraw-Hill Companies, Inc., 1998Irwin/McGraw-Hill

2. Calculate the average of the sample proportions.

0.04=1375

55 = p

3. Calculate the standard deviation of the sample proportion

.020467= 91.6667

.04)-.04(1=

n

)p-(1 p = ps

20©The McGraw-Hill Companies, Inc., 1998

Irwin/McGraw-Hill

4. Calculate the control limits.

3(.020467) .04

UCL = 0.1014

LCL = -0.0214 (or 0)

UCL = p + Z

LCL = p - Z p

p

s

s

21©The McGraw-Hill Companies, Inc., 1998

Irwin/McGraw-Hill

Egekwu-21122

p-Chart (Continued)

5. Plot the individual sample proportions, the average

of the proportions, and the control limits

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

Observation

p

UCL

LCL

Egekwu-21123

Variable Data Example: x-Bar and R Charts

Sample Obs 1 Obs 2 Obs 3 Obs 4 Obs 51 10.68 10.689 10.776 10.798 10.7142 10.79 10.86 10.601 10.746 10.7793 10.78 10.667 10.838 10.785 10.7234 10.59 10.727 10.812 10.775 10.735 10.69 10.708 10.79 10.758 10.6716 10.75 10.714 10.738 10.719 10.6067 10.79 10.713 10.689 10.877 10.6038 10.74 10.779 10.11 10.737 10.759 10.77 10.773 10.641 10.644 10.72510 10.72 10.671 10.708 10.85 10.71211 10.79 10.821 10.764 10.658 10.70812 10.62 10.802 10.818 10.872 10.72713 10.66 10.822 10.893 10.544 10.7514 10.81 10.749 10.859 10.801 10.70115 10.66 10.681 10.644 10.747 10.728

Egekwu-21124

Calculate sample means, sample ranges, mean of means, and mean of ranges.

Sample Obs 1 Obs 2 Obs 3 Obs 4 Obs 5 Avg Range1 10.68 10.689 10.776 10.798 10.714 10.732 0.1162 10.79 10.86 10.601 10.746 10.779 10.755 0.2593 10.78 10.667 10.838 10.785 10.723 10.759 0.1714 10.59 10.727 10.812 10.775 10.73 10.727 0.2215 10.69 10.708 10.79 10.758 10.671 10.724 0.1196 10.75 10.714 10.738 10.719 10.606 10.705 0.1437 10.79 10.713 10.689 10.877 10.603 10.735 0.2748 10.74 10.779 10.11 10.737 10.75 10.624 0.6699 10.77 10.773 10.641 10.644 10.725 10.710 0.13210 10.72 10.671 10.708 10.85 10.712 10.732 0.17911 10.79 10.821 10.764 10.658 10.708 10.748 0.16312 10.62 10.802 10.818 10.872 10.727 10.768 0.25013 10.66 10.822 10.893 10.544 10.75 10.733 0.34914 10.81 10.749 10.859 10.801 10.701 10.783 0.15815 10.66 10.681 10.644 10.747 10.728 10.692 0.103

Averages 10.728 0.220400

Egekwu-21125

Control Limit Formulas & Factor Table

x Chart Control Limits

UCL = x + A R

LCL = x - A R

2

2

R Chart Control Limits

UCL = D R

LCL = D R

4

3

n A2 D3 D42 1.88 0 3.273 1.02 0 2.574 0.73 0 2.285 0.58 0 2.116 0.48 0 2.007 0.42 0.08 1.928 0.37 0.14 1.869 0.34 0.18 1.8210 0.31 0.22 1.7811 0.29 0.26 1.74

Exhibit TN 7.7

Egekwu-21126

x-Bar Chart

10.601

10.856

=).58(0.2204-10.728RA - x = LCL

=).58(0.2204-10.728RA + x = UCL

2

2

10.550

10.600

10.650

10.700

10.750

10.800

10.850

10.900

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

Sample

Mea

ns

UCL

LCL

Egekwu-21127

R-Chart

0

0.46504

)2204.0)(0(R D= LCL

)2204.0)(11.2(R D= UCL

3

4

UCL

LCL0.000

0.100

0.200

0.300

0.400

0.500

0.600

0.700

0.800

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

Sample

R

Egekwu-21128

Process Capability

Process limits - determined from manufacturing process data.

Tolerance limits - specified in engineering design drawing

How do the limits relate to one another?

Egekwu-211

Process Capability

TQM’s emphasis on “making it right the first time” has resulted in organizations emphasizing the ability of a production system to meet design specifications rather than evaluating the quality of outputs after the fact with acceptance sampling.

Process Capability measures the extent to which an organization’s production system can meet design specifications.

Egekwu-211

Engineering Tolerance Versus Process Capability

Egekwu-211

Process Capability Depends On:

Location of the process mean. Natural variability inherent in the process. Stability of the process. Product’s design requirements.

Egekwu-211

Natural Variation Versus Product Design Specifications

(Mean out ofsync.)

(Look for moreeconomical means of production)

Egekwu-211

Process Capability Ratio (Text calls it Index)

6

LSL - USL

system production theof deviations standard 6

rangeion specificatdesign sproduct'pC

Cp < 1: process not capable of meeting design specs

Cp > 1: process capable of meeting design specs

As rule of thumb, many organizations desire a Cp ratio of at least 1.5.

Six sigma quality (fewer than 3.4 defective parts per million) corresponds to a Cp (index/ratio) of 2.

Egekwu-211

Effect of Production System Variability on Cp

Egekwu-21129

Process Capability Index, Cpk

3

X-UTLor

3

LTLXmin=C pk

Shifts in Process Mean

Egekwu-21130

Taguchi’s View of Variation

IncrementalCost of Variability

High

Zero

LowerSpec

TargetSpec

UpperSpec

Traditional View

IncrementalCost of Variability

High

Zero

LowerSpec

TargetSpec

UpperSpec

Taguchi’s View