J0444J0444OPERATION MANAGEMENTOPERATION MANAGEMENT

SPCSPC

Pert11

Universitas Bina Universitas Bina NusantaraNusantara

Process Capability and Statistical Process Capability and Statistical Quality ControlQuality Control

Process VariationProcess Variation Process CapabilityProcess Capability Process Control ProceduresProcess Control Procedures

– Variable dataVariable data– Attribute dataAttribute data

Acceptance SamplingAcceptance Sampling– Operating Characteristic CurveOperating Characteristic Curve

Basic Forms of VariationBasic Forms of Variation Assignable variationAssignable variation is caused is caused

by factors that can be clearly by factors that can be clearly identified and possibly managed.identified and possibly managed.

Common variationCommon variation is inherent in is inherent in the production process. the production process.

Taguchi’s View of Taguchi’s View of VariationVariation

IncrementalCost of Variability

High

Zero

LowerSpec

TargetSpec

UpperSpec

Traditional View

IncrementalCost of Variability

High

Zero

LowerSpec

TargetSpec

UpperSpec

Taguchi’s View

Process CapabilityProcess Capability Process limitsProcess limits

Tolerance limitsTolerance limits

How do the limits relate to one How do the limits relate to one another? another?

Process Capability Index, Process Capability Index, CCpkpk

3

X-UTLor 3

LTLXmin=C pk

Shifts in Process Mean

Capability Index shows how well parts being produced fit into design limit specifications.

As a production process produces items small shifts in equipment or systems can cause differences in production performance from differing samples.

Types of Statistical SamplingTypes of Statistical Sampling

Attribute (Go or no-go information)Attribute (Go or no-go information)– DefectivesDefectives refers to the acceptability of refers to the acceptability of

product across a range of characteristics.product across a range of characteristics.– Defects Defects refers to the number of defects per refers to the number of defects per

unit which may be higher than the number of unit which may be higher than the number of defectives.defectives.

– pp-chart application-chart application

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

standard deviation.standard deviation.– X-bar and R chart applicationsX-bar and R chart applications

UCL

LCL

Samples over time

1 2 3 4 5 6

UCL

LCL

Samples over time

1 2 3 4 5 6

UCL

LCL

Samples over time

1 2 3 4 5 6

Normal Behavior

Possible problem, investigate

Possible problem, investigate

Statistical Process Control (SPC) Charts

Control Limits are based on the Control Limits are based on the Normal CurveNormal Curve

x

0 1 2 3-3 -2 -1z

Standard deviation units or “z” units.

Control LimitsControl Limits

We establish the Upper Control Limits We establish the Upper Control Limits (UCL) and the Lower Control Limits (LCL) (UCL) and the Lower Control Limits (LCL) with plus or minus 3 standard deviations. with plus or minus 3 standard deviations. Based on this we can expect 99.7% of Based on this we can expect 99.7% of our sample observations to fall within our sample observations to fall within these limits. these limits.

xLCL UCL

99.7%

Example of Constructing a Example of Constructing a pp-Chart: -Chart: Required DataRequired Data

1 100 42 100 23 100 54 100 35 100 66 100 47 100 38 100 79 100 1

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

Statistical Process Control Formulas:Statistical Process Control Formulas:Attribute Measurements (Attribute Measurements (pp-Chart)-Chart)

p =Total Number of Defectives

Total Number of Observations

ns )p-(1 p = p

p

p

z - p = LCL

z + p = UCL

s

s

Given:

Compute control limits:

1. Calculate the sample proportions, p (these are what can be plotted on the p-chart) for each sample.

Sample n Defectives p1 100 4 0.042 100 2 0.023 100 5 0.054 100 3 0.035 100 6 0.066 100 4 0.047 100 3 0.038 100 7 0.079 100 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

Example of Constructing a Example of Constructing a pp-chart: -chart: Step 1Step 1

2. Calculate the average of the sample proportions.

0.036=1500

55 = p

3. Calculate the standard deviation of the sample proportion

.0188= 100

.036)-.036(1=)p-(1 p = p ns

Example of Constructing a Example of Constructing a pp-chart: -chart: Steps 2&3Steps 2&3

4. Calculate the control limits.

3(.0188) .036

UCL = 0.0924LCL = -0.0204 (or 0)

p

p

z - p = LCL

z + p = UCL

s

s

Example of Constructing a Example of Constructing a pp-chart: Step 4-chart: Step 4

Example of Constructing a Example of Constructing a pp-Chart: Step 5-Chart: Step 55. 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

Example of x-Bar and R Charts: Example of x-Bar and R Charts: Required DataRequired Data

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

Example of x-bar and R charts: Step 1. Example of x-bar and R charts: Step 1. Calculate sample means, sample ranges, Calculate sample means, sample ranges,

mean of means, and mean of ranges.mean of means, and mean of ranges.Sample Obs 1 Obs 2 Obs 3 Obs 4 Obs 5 Avg Range

1 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

Example of x-bar and R charts: Step 2. Example of x-bar and R charts: Step 2. Determine Control Limit Formulas and Determine Control Limit Formulas and

Necessary Tabled ValuesNecessary Tabled Valuesx Chart Control Limits

UCL = x + A R

LCL = x - A R2

2

R Chart Control Limits

UCL = D R

LCL = D R4

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 1.82

10 0.31 0.22 1.7811 0.29 0.26 1.74

Example of x-bar and R charts: Steps 3&4. Calculate x-Example of x-bar and R charts: Steps 3&4. Calculate x-bar Chart and Plot Valuesbar Chart and Plot Values

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 15Sample

Mea

ns

UCL

LCL

Example of x-bar and R charts: Steps Example of x-bar and R charts: Steps 5&6. Calculate R-chart and Plot 5&6. Calculate R-chart and Plot ValuesValues

0

0.46504

)2204.0)(0(R D= LCL

)2204.0)(11.2(R D= UCL

3

4

0.000

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

RUCL

LCL

Basic Forms of Statistical Sampling Basic Forms of Statistical Sampling for Quality Controlfor Quality Control

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

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

Acceptance SamplingAcceptance Sampling PurposesPurposes

– Determine quality levelDetermine quality level– Ensure quality is within predetermined levelEnsure quality is within predetermined level

AdvantagesAdvantages– EconomyEconomy– Less handling damageLess handling damage– Fewer inspectorsFewer inspectors– Upgrading of the inspection jobUpgrading of the inspection job– Applicability to destructive testingApplicability to destructive testing– Entire lot rejection (motivation for Entire lot rejection (motivation for

improvement) improvement)

Acceptance SamplingAcceptance Sampling

DisadvantagesDisadvantages– Risks of accepting “bad” lots and Risks of accepting “bad” lots and

rejecting “good” lotsrejecting “good” lots– Added planning and documentationAdded planning and documentation– Sample provides less information Sample provides less information

than 100-percent inspection than 100-percent inspection

Acceptance Sampling: Acceptance Sampling: Single Sampling PlanSingle Sampling Plan

A simple goalA simple goal

Determine (1) how many units, Determine (1) how many units, nn, , to sample from a lot, and (2) the to sample from a lot, and (2) the maximum number of defective maximum number of defective items, items, cc, that can be found in the , that can be found in the sample before the lot is rejected.sample before the lot is rejected.

RiskRisk

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

defined by producer.defined by producer. (Producer’s risk)(Producer’s risk)

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

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

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

Operating Characteristic Operating Characteristic CurveCurve

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

Prob

abili

ty o

f acc

epta

nce

=.10(consumer’s risk)

= .05 (producer’s risk)

Example: Acceptance Example: Acceptance Sampling ProblemSampling Problem

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.

Example: Step 1. What is Example: Step 1. What is given and what is not? given and what is not?

In this problem, AQL is given to be 0.01 and LTDP is given to be 0.03. We are also given an alpha of 0.05 and a beta of 0.10.

What you need to determine your sampling plan is “c” and “n.”

Example: Step 2. Determine Example: Step 2. Determine “c”“c”

First divide LTPD by AQL.LTPDAQL

= .03.01

= 3

Then find the value for “c” by selecting the value in the TN7.10 “n(AQL)”column that is equal to or just greater than the ratio above.

Exhibit TN 7.10

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

So, c = 6.

Example: Step 3. Example: Step 3. Determine Sample SizeDetermine Sample Size

c = 6, from Tablen (AQL) = 3.286, from TableAQL = .01, given in problem

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

Now given the information below, compute the sample size in units to generate your sampling plan.

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