OPSM 301 Operations Management

Class 17:

Quality: Statistical process control

Koç University

Zeynep Aksinzaksin@ku.edu.tr

Announcement

Group Case Assignment 2 See Web page to download a copy Due Thursday in class or Friday in my or

Buse’s office

Statistical Quality Control Objectives

1.Reduce normal variation (process capability)– If normal variation is as small as desired, Process

is capable– We use capability index to check for this

2.Detect and eliminate assignable variation (statistical process control)– If there is no assignable variation, Process is in

control– We use Process Control charts to maintain this

Natural Variations

Also called common causes

Affect virtually all production processes

Expected amount of variation, inherent due to:- the nature of the system - the way the system is managed - the way the process is organised and operated

can only be removed by- making modifications to the process - changing the process

Output measures follow a probability distribution

For any distribution there is a measure of central tendency and dispersion

Assignable Variations

Also called special causes of variation

Exceptions to the system

Generally this is some change in the process

Variations that can be traced to a specific reason

considered abnormalities

often specific to a

certain operator

certain machine

certain batch of material, etc.

The objective is to discover when assignable causes are present

Eliminate the bad causes

Incorporate the good causes

1. Process Capability

Design requirements:Diameter: 1.25 inch ±0.005 inch

Specification Limits

Lower specification Limit:LSL=1.25-0.005=1.245Upper Specification Limit:USL=1.25+0.005=1.255

Example:Producing bearings for a rotating shaft

Relating Specs to Process LimitsProcess performance (Diameter of the products produced=D):Average 1.25 inchStd. Dev: 0.002 inch

Fre

qu

ency

Fre

qu

ency

DiameterDiameter1.25

Question:What is the probability That a bearing does not meet specifications?(i.e. diameter is outside (1.245,1.255) )

006.0)5.2(1)5.2()002.0

25.1255.1()255.1(

006.0)5.2()5.2()002.0

25.1245.1()245.1(

NORMSDISTzPzPDP

NORMSDISTzPzPDP

P(defect)=0.006+0.006=0.012 or 1.2% This is not good enough!!

Process capability

What can we do to improve capability of our process? What should be to have Six-Sigma quality?

We want to have: (1.245-1.25)/ = 6 =0.00083 inch We need to reduce variability of the process. We cannot change specifications

easily, since they are given by customers or design requirements.

•If P(defect)>0.0027 then the process is not capable of producing according to specifications.

•To have this quality level (3 sigma quality), we need to have:•Lower Spec: mean-3 •Upper Spec:mean+3

If we want to have P(defect)0, we aim for 6 sigma quality, then, we need: Lower Spec: mean-6 Upper Spec:mean+6

Six Sigma Quality

Process Capability Index Cpk

Shows how well the parts being produced fit into the range specified by the design specifications

Want Cpk larger than one

3

X-USLor

3

LSLXmin=Cpk

183.0)002.03

25.1255.1,

002.03

245.125.1min(

xxC pk

For our example:

Cpk tells how many standard deviations can fit between the mean and the specification limits. Ideally we want to fit more, so that probability of defect is smaller

Process Capability Index Cp

Process Interval = 6

Specification interval = US –LS

Cp= (US-LS) / 6

Process Interval = 60

Specification Interval = US – LS = 60

Cp= (US-LS) / 6 = 60 / 60 = 1

Process IntervalSpecification Interval

99.73%

USLS

100 160

= 10

Process Capability Index Cp

Process Interval = 6 = 30

Specification Interval = US – LS =60

Cp= (US-LS) / 6 =2

Specification Interval6 Process Interval

3 Process Interval

USLS

100 160 = 5

99.73%

99.99998%

Process Mean Shifted

USLS

100 160

= 10

130

Cpk = min{ (US - )/3, ( - LS)/3 }

Cpk = min(2,0)=0

Specification

3 Process

70

2. Statistical Process Control: Control Charts

Can be used to monitor ongoing production process quality

Can be used to monitor ongoing production process quality

970

980

990

1000

1010

1020

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

LCL

UCL

Setting Chart Limits

For x-Charts when we know

Upper control limit Upper control limit (UCL)(UCL) = x + z = x + zxx

Lower control limit Lower control limit (LCL)(LCL) = x - z = x - zxx

where x =mean of the sample means or a target value set for the processz =number of normal standard deviations (=3)

x =standard deviation of the sample means

=/ n =population standard deviationn =sample size

Setting Control Limits

Hour 1sample item Weight of

Number Oat Flakes1 172 133 164 185 176 167 158 179 16

Mean 16.1 = 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 17.3n = 9n = 9

For For 99.73%99.73% control limits, z control limits, z = 3= 3

Sample size

Setting Control Limits

Hour 1Hour 1

SampleSample Weight ofWeight ofNumberNumber Oat FlakesOat Flakes

11 1717

22 1313

33 1616

44 1818

55 1717

66 1616

77 1515

88 1717

99 1616

MeanMean 16.116.1

== 11

HourHour MeanMean HourHour MeanMean

11 16.116.1 77 15.215.2

22 16.816.8 88 16.416.4

33 15.515.5 99 16.316.3

44 16.516.5 1010 14.814.8

55 16.516.5 1111 14.214.2

66 16.416.4 1212 17.317.3n = 9n = 9

For For 99.73%99.73% control limits, z control limits, z = 3= 3

UCLUCLxx = x + z = x + zxx = 16 + 3(1/3) = 17= 16 + 3(1/3) = 17

LCLLCLxx = x - z = x - zxx = = 16 - 3(1/3) = 1516 - 3(1/3) = 15

17 = UCL17 = UCL

15 = LCL15 = LCL

16 = Mean16 = Mean

Setting Control Limits

Control Chart Control Chart for sample of for sample of 9 boxes9 boxes

Sample numberSample number

|| || || || || || || || || || || ||11 22 33 44 55 66 77 88 99 1010 1111 1212

Variation due Variation due to assignable to assignable

causescauses

Variation due Variation due to assignable to assignable

causescauses

Variation due to Variation due to natural causesnatural causes

Out of Out of controlcontrol

Out of Out of controlcontrol

R – ChartR – 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

Setting Chart LimitsSetting Chart Limits

For R-ChartsFor R-Charts

Lower control limit (LCLLower control limit (LCLRR) = D) = D33RR

Upper control limit (UCLUpper control limit (UCLRR) = D) = D44RR

wherewhere

RR ==average range of the samplesaverage range of the samples

DD33 and D and D44 ==control chart factorscontrol chart factors

Setting Control LimitsSetting Control Limits

UCLUCLRR = D= D44RR

= (2.115)(5.3)= (2.115)(5.3)= 11.2 pounds= 11.2 pounds

LCLLCLRR = D= D33RR

= (0)(5.3)= (0)(5.3)= 0 pounds= 0 pounds

Average range R = 5.3 poundsAverage range R = 5.3 poundsSample size n = 5Sample size n = 5DD44 = 2.115, D = 2.115, D33 = 0 = 0

UCL = 11.2UCL = 11.2

Mean = 5.3Mean = 5.3

LCL = 0LCL = 0

Mean and Range ChartsMean and Range Charts

(a)(a)

These These sampling sampling distributions distributions result in the result in the charts belowcharts below

(Sampling mean is (Sampling mean is shifting upward but shifting upward but range is consistent)range is consistent)

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

UCLUCL

LCLLCL

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

UCLUCL

LCLLCL

Mean and Range ChartsMean and Range Charts

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

UCLUCL

LCLLCL

(b)(b)

These These sampling sampling distributions distributions result in the result in the charts belowcharts below

(Sampling mean (Sampling mean is constant but is constant but dispersion is dispersion is increasing)increasing)

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

UCLUCL

LCLLCL

Process Control and Improvement

LCL

UCL

Out of Control In Control Improved

Six Sigma Quality

Six Sigma

a vision; a philosophy; a symbol; a metric; a goal; a methodology All of the

Above

Six Sigma : Organizational Structure

Champion– Executive Sponsor

Master Black Belts– Process Improvement Specialist– Promotes Org / Culture Change

Black Belts– Full Time– Detect and Eliminate Defects– Project Leader

Green Belts– Part-time involvement

Six Sigma Quality: DMAIC Cycle (Continued)

5. Control (C)

Customers and their prioritiesProcess and its performanceCauses of defects

Remove causes of defectsMaintain quality

1. Define (D)

2. Measure (M)3. Analyze (A)

4. Improve (I)

Process Control and Capability: Review

Every process displays variability: normal or abnormal Do not tamper with process “in control” with normal variability Correct “out

of control” process with abnormal variability Control charts monitor process to identify abnormal variability Control charts may cause false alarms (or missed signals) by mistaking

normal (abnormal) for abnormal (normal) variability Local control yields early detection and correction of abnormal Process “in control” indicates only its internal stability Process capability is its ability to meet external customer needs Improving process capability involves changing the mean and reducing

normal variability, requiring a long term investment Robust, simple, standard, and mistake - proof design improves process

capability Joint, early involvement in design improves quality, speed, cost