1 quality and quality control

SPRING 2013 -2014

QUALITY MANAGEMENTMI 334

Instructor – Dr. Akshay Dvivedi

This material is for classroom discussion and teaching only

QUALITY MANAGEMENT

QUALITY AND QUALITY CONTROL

Mahatma Gandhi said ………

A Customer is the most important visitor on our premises.

He is not dependent on us.

We are dependent on him.

He is not an interruption on our work.

He is the purpose of it.

He is not an outsider on our business.

He is a part of it

We are not doing him a favor by serving him…

He is doing us a favor by giving us an opportunity to do.

Global Market ->(Competiveness) -> New Scenarios (Perform in terms of cost, quality, delivery, dependability, innovation and flexibility)

Organizational performance can be improved through the effective use of production capability, technology and operations strategy

Total Quality Management (TQM)

Quality Function Deployment (QFD)

Six Sigma

Business Process Re-engineering (BPR)

Just In Time (JIT)

Benchmarking

Performance Measurement



PRICE TO QUALITY

7.03 7.006.33

5.92 5.67 5.57

3.47

0

2

4

6

8

10

Japan Singapore Germany USA Thailand Taiwan India

Ratin

g


CUSTOMER ORIENTATION

7.94

6.46 6.45 6.245.78 5.67

3.37

0

2

4

6

8

10

Japan Singapore Germany USA Thailand Taiwan India

Ratin

g


FACTORS India's RANKING OUT OF 53

Technology Ranking 36

Global Technological Leadership 41

Science and Math Education Levels 11

Corporate R & D Spending 44

Indigenous Innovation 35

No. of Scientist and Engineers 3

Quality of Engineers 52

Primary Education ( Female) 48

Primary Education ( Male) 8

Secondary Education ( Female) 47

Secondary Education ( Male) 39


Success

• Product Testing

• Complaints

1960-1970

Success

• QA• Process• Documentation• Training• Qualification

1970-1980

Success

• Quality Manuals

• Process Manuals

• QA –Everybody's job

• QA standards ISO 9000-14000

1980-1990

Success

• Customer Satisfaction

• Strategic Planning

• People and Change Management

• Process Management

• Social Impact• Quality Awards

1990------->

Success

• Liberalization

• Globalization

• WTO

• Conformity

Standards

• Technical

Regulations

• Documentation

• Information

1990------->

Qu

alit

y Im

pro

vem

ent

*Saad GH, Siha S (2000),"Managing quality: critical links and a contingency model", International Journal of Operations & Production Management, 20 (10),pp. 1146-1164

Evaluation of Quality*

Quality has evolved from mere specifications, controls,

inspections, systems, and methods for regulatory

compliance to a harmonized relationship with business

strategies aimed at satisfying both the internal and

external customer


Products and their customer EXPECTATIONS

Automaker

Auto have the intended durability?

Parts within the manufacturing tolerances?

Auto’s appearance pleasing?

Lumber mill

Lumber within moisture content tolerances?

Lumber properly graded?

Knotholes, splits, and other defects excessive?


Services and Their Customer EXPECTATIONS

Hospital

Patient receive the correct treatments?

Patient treated courteously by all personnel?

Hospital environment support patient recovery?

Bank

Customer’s transactions completed with precision?

Bank comply with government regulations?

Customer’s statements accurate?


Dimensions of Product Quality (Garvin, 1990*)

Performance - will the product do the intended job?

Reliability - how often the product fails?

Durability - how long the product lasts?

Serviceability - how easy is to repair the product?

Aesthetics - what does the product look like?

Features - what does the product do?

Perceived quality - what is the reputation of a company or its products?

*Healthcare Forum Journal, September-October 1990, Vol. 33, #5


Dimensions of Service Quality

Reliability

Responsiveness

Competence

Courtesy

Communication

Credibility

Security


Quality is the fitness of use (Juran) - it is the value of

the goods and services as perceived by the supplier,

producer and customer

The efficient production of the quality that the market

expects (Deming)

Quality is conformance to requirements (P. Crosby)

Quality is what the customer says, it is (Feigenbaum)

Quality is the loss that a product costs to the society after

being shipped to the customer (Taguchi)


18

“The first job we have, is to turn out qualityproducts that consumers will buy and keep onbuying. If we produce it efficiently andeconomically, we will earn a profit, in which youwill share.”

- William Cooper ProcterGrandson of the founder of Procter and Gamble

(October 1887)

Modern ? Importance of Quality

The totality of features and characteristics of a product or

services that bear on its ability to satisfy stated or implied

needs of the customers (ASQC)

A quality system is the agreed on company wide and plant

wide operating work structure, documented in effective,

integrated , technical and managerial procedures for

guiding the co-coordinated actions of people, the machines,

or the information of company in the best and most practical

ways to assume customer quality satisfaction and

economical costs of quality (Feigenbaum)


THREE ISSUES

20

1. Productivity

2. Cost

3. Quality

21

Of these three determinants of the profitability the mostsignificant factor in determining the long range success or failureof an organization is Quality.

Good quality of product and service can provide:

1. Competitive edge

2. Reduce cost due to returns, rework, and scrap

3. Productivity and Profits

4. Generates Satisfied customers: Continued Patronage and word of mouth advertisement

Significant Issue

Specification: A set of conditions and requirements, of specific

and limited application, that provide a detailed description of the

procedure, process, material, product/service primarily in

procurement and manufacturing, e.g. ID - 3 ± 0.1 cm, OD - 5 ±

0.1 cm is specification limit

Standard: A prescribed set of conditions and requirements, of

general or broad application, established by authority or

agreement, to be satisfied by a material, product, process,

procedure, convention, test method; and/or the physical,

functional, performance/conformance characteristic thereof, e.g.

document that addresses the requirements of all QC.


Three Aspects of Quality


QUALITY

QUALITYOF

DESIGN

QUALITYOF

PERFORMANCE

QUALITYOF

CONFORMANCE


Quality of Design: Consumer's Perspective

Product must be designed to meet the requirement of the customer.

Product must be designed right first time and every time and while designing all aspects of customer expectations must be incorporated into the product.

Factors need to consider while designing the product are:

Cost

Profit policy of the company

Demand

Availability of the parts



Quality of Conformance: Manufacturer's Perspective

The product must be manufactured exactly as designed (defect finding, defect prevention, defect analysis, and rectification).

The difficulties encountered at the manufacturing stage must be conveyed to the designers for modification in design, if any.

Two-way communication between designer and manufacturing

Quality of Performance

The product must function as per the expectations of the customer. The two way communication between designers and customer is the key to have a quality product.


Cost of Quality

Running a company by profit alone is like driving a car by looking inthe rearview mirror. It tells you where you’ve been, not where you aregoing!

Dr. E. Deming

Cost of Quality: the cost of ensuring that the job is done right + the cost of not doing the job right.

Cost of Conformance + Cost of Non-Conformance

(Prevention & Appraisal) (Internal/External Defects)

27

What is the Cost of Quality?

Error - “The sooner, the better”

Error elimination cost ratio for

development: production : delivery is 1:10:100

Stage Error Prevention Error Correction 1:10:100 Time CostProcess

DevelopmentAdaptive X1 Best Best

Early Review of Design/Process

X1 Slight Delay Low

Mass Production

Good QA X10 Good Expensive

QC After Production

X10 Conditional Delay

High Expenses

Delivered Service/Exchange X100 - Very High Expenses

Unhappy Customer Keeps

Bad Product

X100 - Loss of business

ExternalInternal

AppraisalPrevention

29

Four Cost Categories Related to Quality

• Prevention cost: Cost of planning and executing a project so it is error-free or within an acceptable error range.

• Prevention cost: Cost of planning and executing a project so it is error-free or within an acceptable error range.

• Appraisal cost: Cost of evaluating processes and their outputs to ensure quality.• Appraisal cost: Cost of evaluating processes and their outputs to ensure quality.

• Internal failure cost: Cost incurred to correct an identified defect before the customer receives the product.

• Internal failure cost: Cost incurred to correct an identified defect before the customer receives the product.

• External failure cost: Cost that relates to all errors not detected and corrected before delivery to the customer.

• External failure cost: Cost that relates to all errors not detected and corrected before delivery to the customer.

ExternalInternal

AppraisalPrevention

QUALITY COSTS

31

Normal Cost of Quality Distribution whenQuality System is NOT in Place

ExternalFailure Cost

InternalFailure Cost

Prevention Cost

Appraisal Cost

Cost ofQuality

32

Optimum Cost of Quality Distributionwhen Quality System is in Place

Cost ofQuality Internal Failure Cost

External Failure Cost

Appraisal Cost

Prevention Cost


Raw Materials,Parts, andSupplies

ProductionProcesses

Products andServices

Inputs Conversion Outputs

Control Chartsand

Acceptance Tests

Control Chartsand

Acceptance TestsControl Charts

Quality ofInputs

Quality ofOutputs

Quality ofPartially CompletedProducts

QC Throughout Production Systems

Pre-production Insp. In-line/In-process Insp. Pre-shipment Insp.Timely corrections of any non-conformities detected

Evaluation of the average product quality

Representative of whole batch

Inspection - generally refers to the activity of checking

products

“Activity such as measuring, examining, testing or gauging

one or more characteristics of a product or service, and

comparing the results with specified requirements in order

to establish whether conformity is achieved for each

characteristic” - ISO 2859

The inspected products can be the components used for

production, work-in-process inventory, or finished goods.

QC-INSPECTION

Cos

t

OptimalAmount of Inspection

QC-INSPECTION

Cost of inspectionCost of passing defectives

Total Cost

QC-INSPECTION

Inspection of lotsbefore/afterproduction

Inspection andcorrectiveaction duringproduction

Quality builtinto theprocess

The leastprogressive

The mostprogressive

How Much/How OftenWhere/When Centralized vs. On-site

Inputs Transformation Outputs

Production

Where to Inspect in the Process

Raw materials and purchased parts

Finished products

Before a costly operation

Before an irreversible process

Before a covering process

QC-INSPECTION

HOW?

DATA

Statistics

Statistics is the study of the collection, organization, analysis, interpretation, and presentation of data

COPY-ExactlyNO VARIATION

Statistical quality control (SQC) is the term used to describe the set of statistical tools used by quality professionals.

Descriptive statistics are used to describe quality characteristics and relationships (mean, standard deviation, the range etc.)

Statistical process control (SPC) involves inspecting a random sample of the output from a process and deciding whether the process is producing products with characteristics that fall within a predetermined range. SPC answers the question of whether the process is functioning properly or not.

Acceptance sampling is the process of randomly inspecting a sample of goods and deciding whether to accept the entire lot based on the results. Acceptance sampling determines whether a batch of goods should be accepted or rejected.

STATISTICAL CONTROL

Variation ? – Common Causes

Unavoidable common, or random, causes of variation

(random cause- can not be eliminated)

Patties in bakery – some are slightly larger

Natural random variation –

Average bottle of cold drink contains 300 ml liquid

(natural variation 298-302 ml)

Variation outside normal variation – problem with process

STATISTICAL CONTROL

Range

Variation ? – Common Causes

Inherent part of the process design and affects all items

Process operating under stable system of common

causes is in Statistical Control

Fluctuations in working conditions (temperature, pressure

etc.), vibrations in machines,

Management alone is responsible for common causes

85% of problems are due to common causes and can be

solved only by action on the part of management

STATISTICAL CONTROL

Variation ? Special Causes

Variation due to special/assignable causes

Not inherent in process

Not part of process as designed

Non-random pattern

Process out of control

Does not affect all items

Problem has a remedy

15% of all problems are due to special causes

Actions on part of both management and workers may reduce such cases

STATISTICAL CONTROL

The Statistical Control Process

Define

Measure

Compare

Evaluate

Correct

Monitor results

STATISTICAL CONTROL

HAVE YOU EVER…

Shot a rifle?

Played darts?

Played basketball?

44

What is System of point ofthese sports?

HAVE YOU EVER…

Shot a rifle?

Played darts?

Played basketball?

45

Ram

Arjun

Who is the better shot?

DISCUSSION

What do you measure in your process?

Why do those measures matter?

Are those measures consistently the same?

Why not?

46

VARIABILITY

Deviation = distance between observations and the mean (or average)

47

Ram

871089

Arjun

Observations

10

9

8

8

7

averages 8.4

Deviations

10 - 8.4 = 1.6

9 – 8.4 = 0.6

8 – 8.4 = -0.4

8 – 8.4 = -0.4

7 – 8.4 = -1.4

0.0

Deviation = distance between observations and the mean (or average)

48

76776 Arjun

Ram

Observations

7

7

7

6

6

averages 6.6

Deviations

7 – 6.6 = 0.4

7 – 6.6 = 0.4

7 – 6.6 = 0.4

6 – 6.6 = -0.6

6 – 6.6 = -0.6

0.0

Variability

VARIABILITY

Variance = average distance between observations and the mean (squared)

49

Arjun

Observations

10

9

8

8

7

averages 8.4

Deviations

10 - 8.4 = 1.6

9 – 8.4 = 0.6

8 – 8.4 = -0.4

8 – 8.4 = -0.4

7 – 8.4 = -1.4

0.0

Ram

871089

Squared Deviations

2.56

0.36

0.16

0.16

1.96

1.0 Variance

Variance

Average distance between observations and the mean (squared) Measures fluctuations of observations around the

mean

50

Observations

7

7

7

6

6

averages 6.6

Deviations

7 - 6.6 = 0.4

7 - 6.6 = 0.4

7 - 6.6 = 0.4

6 – 6.6 = -0.6

6 – 6.6 = -0.6

0.0

Squared Deviations

0.16

0.16

0.16

0.36

0.36

0.24 Variance

Variability

76776 Arjun

Ram

Standard deviation = square root of variance

51

Variance Standard Deviation

Ram 1.0 1.0

Arjun 0.24 0.4898979

But what good is a standard deviation

Ram

Arjun

Variability

VARIABILITY

52

The world tends to be bell-shaped

Most outcomes

occur in the middle

Fewer in the “tails”

(lower)

Fewer in the “tails” (upper)

Even very rare outcomes are

possible(probability > 0)

Even very rare outcomes are

possible(probability > 0)

VARIABILITY

53

Add up the dots on the dice

0

0.05

0.1

0.15

0.2

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18

Sum of dots

Pro

ba

bili

ty 1 die

2 dice

3 dice

Here is why:

Even outcomes that are equally likely (like dice), when you add them up, become bell shaped

“NORMAL” BELL SHAPED CURVE

54

Add up about 30 or most thingsand you start to be “normal”

Normal distributions are divided upinto 3 standard deviations on each side of the mean

Once you know that, you know a lot about what is going on

And that is what a standard deviation is good for

POTENTIAL REASONS OF VARIATIONS

The Operator Training, supervision, technique

55

The Method• Procedure, Set-up, temperature, etc.

The Material• Moisture content, blending, contamination

The Machine• Machine condition, inherent precision

Management• Poor Process management, Poor System

CAUSES OF VARIATIONS

Common Causes : Improvement in system

Variation inherent in process

Can be eliminated only through improvement in system

No pattern

56

Assignable Causes : Control of Process

• Variation due to identifiable factors

• Can be modified through operator or management action

• May exhibit a pattern

• Examples of special causes include: wrong tool, wrong production method, improper raw material, operator’s skill, wrong die etc.

SPC uses samples to identify that Assignable causes have occurred

STATISTICAL PROCESS CONTROL

The underlying concept of stat istical process control is based on a comparison of what is happening today with what happened previously.

57

• We take a snapshot of how the process typically performs or build a model of how we think the process will perform and calculate control limits for the expected measurements of the output of the process.

• Then we collect data from the process and compare the data to the controllimits.

The majority of measurements should fall within the control limits.

Measurements that fall outside the control limits are examined to see if they belong to the same population as our initial snapshot or model.

STATISTICAL QUALITY CONTROL

The purpose of stat istical quality control is to ensure, in a cost eff icientmanner, that the product shipped to customers meets their

specif ications.

58

• Inspecting every product is costly and inefficient, but the consequences of shipping non conforming product can be significant in terms of customer dissatisfaction.

• Statistical Quality Control is the process of inspecting enough product from given lots to ensure a specified quality level.

WHAT ARE SPC TECHNIQUES?

There are many ways to implement process control. Key monitoring and investigating tools include: Histograms Run Charts Pareto Charts Cause and Effect Diagrams Flow Diagram Scatter Diagrams Control Charts

59

60

Control Charts:Recognizing Sources of Variation

• Why Use a Control Chart?To monitor, control, and improve process performance over time by

studying variation and its source.

What Does a Control Chart Do?• Focuses attention on detecting and monitoring process variation over

time;• Distinguishes special from common causes of variation, as a guide to

local or management action;• Serves as a tool for ongoing control of a process;• Helps improve a process to perform consistently for higher quality, lower

cost, and higher effective capacity;• Provides a common language for discussing process performance.

61


How Do I Use Control Charts?

There are many types of control charts. The control charts that you or your team decides to use should be determined by the type of data that you have.

Data are of two typesVariablesAttributes

62


Variables “Things we measure”

Attributes “Things we count”

Length Blood Pressure DiameterWeight Volume Tensile strengthTime Temperature

Number or percent defective itemsNumber of defects per itemTypes of defects etc.

Mean

Range and Standard Deviation

Range – Difference between largest and smallest observation

Sample Std. Dev. – Measures variability (sq. root of variance)

(μ)

(μ)

BEFORE WE START….

Samplingdistribution

Processdistribution

Mean

Central Limit Theorem – if plotted statistic is a sample average, itwill tend to have a normal distribution.Thus, even if the parent population is not normally distributed,control charts for averages are based on normal distributions

BEFORE WE START….

294 296 290 300 302 304 306(ml)

294 296 290 300 302 304 306(ml)

Distribution of Data

Before We Start….

66

Control Charts: TypesControl Charts for Variables Data

_X and R charts : for sample averages and ranges_X and s charts : for sample averages and standard deviations

Md and R charts : for sample medians and ranges

X and Rm charts : for individual measures and moving ranges

Control Charts for Attributes Data

p charts : proportions of units nonconforming

np charts : number of units nonconforming

c charts : number of defects/nonconformities

u charts : number of defects/nonconformities per unit

67

Control Charts: Selection

• How Do I Select Control Charts?

Use the following tree diagram to determine which chart will best fit your situation.

Only the most common types of charts are addressed.

68

Control Chart Selection

Control Chart

Variables

n Large X, s

n small X, R

n = 1 X-chart, Moving Range

Attributes

Defective

n constant p or np

n variable p

Defects

n constant c or u

n variable u

69

Basis of Control Chart

CL = E θ θEstimates of θ

UCL = E θ + K σ θ S.D of ESTIMATOR θ

LCL= E θ - K σ θK: - No. of std. deviations of the sample statistics that the control limits are placed from the centerline.

If we assume normal distribution k=3, 99.74% fall in the range of UCL & LCL 0.26% or 0.0026 fraction out of range.K can be decided on % outside the limits. 0.2%--K=3.09

Type I error (false alarm)

Concluding a process is not in control when it actually is in control. Probability of Type I error is α

Type II error (lost opportunity)

Concluding a process is in control when it is not. Probability of Type II error is β

MeanLCL UCL

/2 /2

MeanLCL UCL

/2

β


In control Out of control

In control No Error Type I error

(producers risk)

Out of control Type II Error

(consumers risk)

No error

Type I and Type II Errors


Mean

95.44%

99.74%

Standard deviation

CL = μ (mean)UCL = μ + K σx ̄ (K=number of standard deviations of the sample statistic)LCL = μ - K σx ̄

σx ̄ = σ/√n where, σx̄ is std. dev of sample mean x̄

2

2

1 -(x-μ)f(x)= exp

2σ2 σx

X

X- X-Z = or

ZStandard normal distribution

BASIS OF CONTROL CHART

Sample Ave Dimension x̄ Sample Ave Dimension x̄ Sample Ave Dimension x̄

1 31.45 6 30.20 11 31.56

2 29.70 7 29.10 12 29.50

3 31.48 8 30.85 13 30.50

4 29.52 9 31.55 14 30.72

5 28.30 10 29.43 15 28.92

CL = 30 mmσx ̄ = 1.5/√5 = 0.671 mmUCL = 30 + 3(0.671) = 32.0123 mmLCL = 30 - 3(0.671) = 27.987 mm

A machining process for a particular dimension has process mean as 30 mm and standard deviation as 1.5mm. Construct a control chart for 3σ limits if samples of size 5 are randomly selected


CL = 120 kgσx ̄ = 8/√5 UCL = 120 + 3(8/√5) = 130.733 kgLCL = 120 - 3(8/√5) = 109.267 kg

a. Construct a control chart for average breaking strength of rope. Samples of size 5 are randomly chosen from the process.Given - Process mean and standard deviation are 120 kg and 8 kg respectively.b. If control limits are placed at 3 σ, what is probability of Type I error.

X

X- 130.733-120Z = 3.00

8 5


Probability of Type I error will be .0026

This table gives a probability that a statistic is less than Z. This equates to the area of the distribution below Z.


CL = 120 kgσx ̄ = 8/√5 UCL = 120 + 3(8/√5) = 130.733 kgLCL = 120 - 3(8/√5) = 109.267 kg

c. If process mean shifts to 125 kg, what is the probability that process is in control and hence making a type II error

1

X

2

X

X- 130.733-125Z = 1.60

8 5

X- 109.267-125Z = 4.40

8 5

β= 0.9452



0.0548

Effect of CL on errors in inference making

Type I error (α) reduces when control limits are placed farther apart.

Control Limits > = 4 sigma , Type I error is negligible.

μ ±k σControl limits =


79

Operating Characteristic Curve

An operating characteristic (OC) curve is a measure of goodness of a

control chart's ability to detect changes in process parameters.

Specifically, it is a plot of the probability of the Type II error versus

the shifting of a process parameter value from its in-control value.

OC curves enable us to determine the chances of not detecting a shift

of a certain magnitude in a process parameter on a control chart.

80


81

Operating Characteristic CurveSamples of size 5 are randomly chosen from a process whose

mean and standard deviation are estimated to be 120 kg and 8 kg, respectively. Construct the operating characteristic curve for increases in the process mean from 120 kg.

CL = 120 kg, σx ̄ = 8/√5 UCL = 120 + 3(8/√5) = 130.733 kgLCL = 120 - 3(8/√5) = 109.267 kg X

X- 130.733-120Z = 3.00

8 5

1

X

2

X

X- 130.733-125Z = 1.60

8 5

X- 109.267-125Z = 4.40

8 5

82


Processmean

Z-value at UCL

Z1

Area above UCL

Z-value at LCLZ2

Area below LCL

Probability of nondetection,

b

123.578127.156130.733134.311137.888141.466

2.001.00.00

-1.00-2.00-3.00

.0228

.1587

.5000

.8413

.9772

.9987

-4.00-5.00-6.00-7.00-8.00-9.00

.0000

.0000

.0000

.0000

.0000

.0000

.9772

.8413

.5000

.1587

.0228

.0013

83


84

Control Chart Performance

Average Run Length• The average run length (ARL) is a measure of the

performance of the control chart. This denotes the number of samples on average, required to detect an out-of-control signal.

• Let p = probability that any point exceeds the control limits. Then,

Process in control (p=α), i.e., ARL = 1/α = 1/0.0026 = 385 (False Alarm)

Process out-of-control (p=1-β), i.e., ARL = 1/(1-β)

3σ Limits

85

Control Chart Performance

Magnitude of Process Shift α Error β Error ARL Remarks

0 0.0026 - 385 Process in Control

0.5 σ - 0.9938 161.2 Process Out of Control




3.0 σ - 0.5000 2 Process Out of Control

For a process in control we For an out of control process it is

Prefer the ARL to be large desirable for the ARL to be smallBecause an observation we want to detect the out of controlPlotting outside the control condition.Limit represent false alarm.

86

Effect of CL on errors in inference making

Type I error (α) reduces when control limits are placed farther apart.

Control Limits > = 4 sigma , Type I error is negligible.

Type II error (β) α [1/ α]

Type II error increases when Type I error decreases.

Type II error α [1/ n]

μ ±k σ

kμ ± σx / n

Control limits = increase in sample size draw limits to be closer

87

Effect of sample size on errors in inference making

n1<n2<n3

Control Charts (what they reveal):

Purpose: to monitor process output to see if it is random

When to take corrective action

Pattern of the plot diagnose causes ---- indicates possible remedial actions

When to leave process alone?

Process capability (ability of process to produce within desirable specifications)

Possible means for quality improvement


Sample number

UCL

LCL

1 2 3 4

Process should remain random at all times !

Observations from Sample Distribution


MEAN AND RANGE CHARTS

UCL

LCL

UCL

LCL

R-chart

x-Chart Detects shift

Does notdetect shift

(process mean is shifting upward)

SamplingDistribution


x-Chart

UCL

Does notreveal increase

UCL

LCL

LCL

R-chart Reveals increase

(process variability is increasing)

SamplingDistribution

MEAN AND RANGE CHARTS


92

Warning Limits on Control Charts

Warning limits (if used) are typically set at 2 standard deviations from the mean.

If one or more points fall between the warning limits and the control limits, or close to the warning limits the process may not be operating properly.

Good thing: Warning limits often increase the sensitivity of the control chart.Bad thing: Warning limits could result in an increased risk of false alarms.

Statistical Control -Control Charts for Mean and Range

94

General Rules for Out of Control Situations

Natural Pattern:They are indicative of a process that is in control; i.e. they demonstrate the presence of a stable system of common cause.

Sudden Shift:Sudden change or jump occurs because of changes – in process setting as temp, pr., depth of cut, change in customer behavior, no. of tellers etc.

95


Gradual Shift :

- Change in the Q. of coming material overtimeX

- Change in the maintenance program

new operator

R decrease in worker skill

gradual improvement in vendors supply

96


Trending Pattern:

X Tool wear, die wear, gradual deterioration of Machines

R Operator skill

97


Cyclic Pattern:

repetitive periodic behavior

X : Periodic change in temp, rotation of operator, seasonal variation

R : Operators fatigue periodic maintenance

98

Control Chart Construction

• Select the process to be charted;

• Decide what to measure and count; (Pareto sizing)• Determine sampling method and plan;

• How large a sample needs to be selected? (Time, Cost ---- Information)

• Samples under the same technical conditions: (machine, operator, lot) ----(Rational Sub-grouping)

Frequency of sampling will depend on whether you are able to discern patterns in the data. Consider hourly, daily, shifts, monthly, annually, lots, and so on. Once the process is “in control”, you might consider reducing the frequency with which you sample.

99


• Initiate data collection:Run the process untouched, and gather sampled data.Record data on an appropriate Control Chart sheet or other

graph paper. Include any unusual events that occur.

• Calculate the appropriate statistics and control limits:Use the appropriate formulas.

• Construct the control chart(s) and plot the data.

100


101

Control Chart Interpretation

What is Process Control?

Process Control is the active changing of the process based on the results of process monitoring.

Once the process monitoring tools have detected an out-of-control situation,

the person responsible for the process makes a change to bring the process back into control.

102

Control Chart Interpretation

What to do if the process is "Out of Control"?

If the process is out-of-control, the process engineer looks for an assignable cause by following the out-of-control action plan (OCAP) associated with the control chart.

103

Control Chart Representing Limits, Special Causes, Common Causes

Upper control limits

Lower control limits

Centre Line

Stable process

Special cause

Unstable process

Common causeUpper control limits

Lower control limits

Centre Line

Stable process

Special cause

Unstable process

Common cause

104

Control Charts for Variable Type of Data

and R charts

In the x bar chart the sample means are plotted in order tocontrol the mean value of a variable.

In R chart, the sample ranges are plotted in order to controlthe variability of a variable

THE CHART

Control Charts applicable to quality measurements that

possesses continuous probability distribution . This sampling

is commonly referred to as “sampling by variables.”

The chart helps the quality control person decide whether

the center (or average, or the location of central tendency) of

the measurement has shifted.

x

x


Deciding whether the center of the distribution of quality

measurements has shifted up or down may not be enough.

It is frequently of interest to decide if the variability of the

process measurements has significantly increased or

decreased.

A process that suddenly starts turning out highly variable

products could cause severe problems in the operations.

In R chart, the sample ranges are plotted in order to control

the variability of a variable

THE R -CHART


= mean of ‘g’ samples (g=number of samples)

Centre line, UCL and LCL are calculated using following formulae:

iXn

n

iiX

1

= mean of the ith samplen = sample size, Xi = ith data

iX

g

g

iiX

1X X


Sample Mean

Centerline (X bar)

Centerline (R chart)

XX iiiR)min()(max

Ri = range of ith sampleXmax(i) = maximum value of the data in ith sampleXmin (i) = minimum value of the data in ith sample

Centre line, UCL and LCL are calculated using following formulae:


(Upper control limit for X bar chart)

(Lower control limit for X bar chart)

RAXUCLx 2

RAXLCLx 2

A2 is constant and its value can be obtained from standard table. Its value depend upon the size of sample.

RDUCLR 4

RDLCLR 3

(Upper control limit for R chart)

(Lower control limit for R chart)

D3 and D4 are constants and their values can be obtained from standard tables. Their values depend upon the size of sample.


n A2 D3 D4

2 1.880 0 3.267

3 1.023 0 2.575

4 0.729 0 2.282

5 0.577 0 2.115

6 0.483 0 2.004

7 0.419 0.076 1.924

8 0.373 0.136 1.864

9 0.337 0.184 1.816

10 0.308 0.223 1.777

Factors for Calculating Limits for and R Charts

n = observations in an sample


The mean values and ranges of data from 16 samples (sample size = 5) aregiven as under:

S.NoMean of Sample

Range S.NoMean of Sample



Range

1 10 4 5 9 5 9 10 4 13 12 4

2 15 4 6 11 6 10 11 6 14 12 3

3 12 5 7 11 4 11 12 5 15 11 3

4 11 4 8 9 4 12 13 4 16 15 4

Upper Control Limit of x bar chart = 11.5 + A2*4.3125 = 13.98

Lower Control Limit of x bar chart = 11.5 - A2*4.3125 = 9.01

Average of mean values of 15 samples = = 11.5 (Center Line of x bar Chart)16

16

1i

X

A2 = 0.577 for sample size 5


X-Bar Chart

Sample data at S.N 2, and 16 are slightly above the UCL. Sample dataat S. No. 5 and 8 are slightly below LCL.Efforts must be made to find the special causes and revised limits areadvised to calculate after deleting these data.

CL= 11.5

LCL = 9.01

UCL = 13.98


RDUCLR 4

RDLCLR 3

= 4.3125

= 2.115*4.3125 = 9.12

= 0*4.3125 = 0

2

3

4

5

6

7

8

9

10

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16

All the data are within theLCL and UCLVariability of the processdata is not an issue

R Chart

CL

UCL


g

ii 1

R87

R 3.48g 25

R 4

R 3

UCL D R (2.114)(3.48) 7.357

LCL D R (0)(3.48) 0

Consider a process by which coils are manufactured. Samples of size 5 arerandomly selected from the process, and the resistance values (in ohms) of thecoils are measured. The data values are given in Table

For a sample of size 5, D4 = 2.114 and D3 = 0. The trial control limits for theR-chart are calculated as follows:

Statistical Control -Control Charts for Mean and RangeSample Observation (ohms) X R Comments

123456789

10111213141516171819202122232425

20,22,21,23,2219,18,22,20,2025,18,20,17,2220,21,22,21,2119,24,23,22,2022,20,18,18,1918,20,19,18,2020,18,23,20,2121,20,24,23,2221,19,20,20,2020,20,23,22,2322,21,20,22,2319,22,19,18,1920,21,22,21,2220,24,24,23,2321,20,24,20,2120,18,18,20,2020,24,22,23,2320,19,23,20,1922,21,21,24,2223,22,22,20,2221,18,18,17,1921,24,24,23,2320,22,21,21,2019,20,21,21,22

21.6019.8020.4021.0021.6019.4019.0020.4022.0020.0021.0021.6019.4021.2022.8021.2019.2022.4020.2022.0021.8018.6023.0020.8020.60

3482542542334244244334323

New vendor

High TemperatureWrong Die

Sum=521.00 Sum=87


g

ii 1

X521.00

X 20.840g 25

2X

2X

UCL X A R 20.84 (0.577)(3.48) 22.848

LCL X A R 20.84 (0.577)(3.48) 18.832


R 4

R 3

UCL D R (2.114)(3.273) 6.919

LCL D R (0)(3.273) 0

459X 20.864

22

2X

2X

UCL X A R 20.864 (0.577)(3.273) 22.753

LCL X A R 20.864 (0.577)(3.273) 18.975

The revised control limits on the R-chart are

The revised center line on the X BAR chart

The revised control limits on the

Sample 15 falls close to UCL on the chart. On further investigation, nospecial causes could be identified for this sample. So, the revised limits will beused for future observations until a subsequent revision takes place.

117

Control Chart InterpretationSet of rules to determine "Out of Control"

General rules for detecting out of control or non-random situations

Any Point Above +3 Sigma------------------------------------------------------------------------ +3 LIMIT

2 Out of the Last 3 Points Above +2 Sigma---------------------------------------------------------------------- +2 LIMIT

4 Out of the Last 5 Points Above +1 Sigma---------------------------------------------------------------------- +1 LIMIT

8 Consecutive Points on This Side of Control Line=========================================== CENTER LINE

8 Consecutive Points on This Side of Control Line------------------------------------------------------------------------ -1 LIMIT

4 Out of the Last 5 Points Below - 1 Sigma-------------------------------------------------------------------------- -2 LIMIT

2 Out of the Last 3 Points Below -2 Sigma------------------------------------------------------------------------- -3 LIMIT

Any Point Below -3 Sigma

TREND RULE :6 in a row trending up or down. 14 in a row alternating up and down

118

Control Charts for Variable Type of DataVariable Sample Size

A change in the sample size has an impact on the control limits for theX bar and R charts

An increase in the sample size n reduces the width of the control limits.

Standardized Control ChartsWhen the sample size varies, the control limits on an X bar and an R-chart will change . With fluctuating control limits, the rules for identifying out-of-control conditions become difficult to apply—that is, except for Rule 1 (which assumes a process to be out of control when an observation plots outside the control limits). One way to overcome this drawback is to use a standardized control chart. To standardize a statistic

Subtract its mean from its valueDivide with its SD

119


g

i ii 1

ig

ii 1

n XX X sam ple m ean

n

g2

i i^i 1

ig

ii 1

(n 1)ss sa m p le S D

(n 1)

ii ^

i

X XStandardized value for the mean Z

/ n

+3

-3

0Zi values are plotted on a CC with CL=0 UCL & LCL +3 & -3

Estimate of process SD is the square root of the weighted average of the sample variance

ni – sample size for sample isi – standard deviationg – no of samples

120


+3

-3

0Ki values are plotted on a CC with CL=0 UCL & LCL +3 & -3

ii ^

Rr R a n g e 'R i', E s t im a to r o f P ro c e s s S D

i 2

i 2 33

r dRange 'K ' Mean 'd ', SD 'd '

d

121


0X

00 0 0X

00 0 0X

CL X

3UCL X X A

n3

LCL X X An

Control limits for given target or standard

^

2

R

d

^

2 0R d given

For R chart

CLR = d2s0UCLR = D2s0LCLR = D1s0

Care must be taken while interpreting CC based on standard values-Target value is specified as too high or too low

0 given process mean (target)s0 st.devX

122


0X

0 0X

0 0X

CL X 21.0

UCL X A 21.0 (1.342)(1.0) 22.342

LCL X A 21.0 (1.342)(1.0) 19.658

Refer to the coil resistance data, Let's suppose the targetvalues for the average resistance and standard deviation are21.0 and 1.0 ohms, respectively. The sample size is 5. Thecenter line and the control limits for the

The center line and control limits for the R-chart areCLR = d2s0 = (2.326)(1.0) = 2.326UCLR = D2s0 = (4.918)(1.0) = 4.918LCLR = D1s0 = (0)(1.0) = 0

- chart are as follows:X

123


The process seems to be out of control with respect to the given standard. Samples 5 and 8 are above the upper control limit, and a majority of the points lie above the center line. Only six of the points plot below the center line. thus it reveals that the process is not capable of meeting the company guidelines Sigma as 1.0. The estimated process standard deviation (calculated after the process was brought to control) is

R 3.50σ = = = 1.505

d 2 .3262

124


Several points are out-side the control limits—four points below and twopoints above. the revised center line for the X bar chart was found tobe 20.864. Our target center line is now 21.0. Adjustingcontrollable process parameters could possibly shift the averagelevel up to 21.0. However, the fact that there are, points outsideboth the upper and lower control limits signifies that processvariability is the issue here.

22.34219.658

125


Suppose we have g preliminary samples at our disposition, each of size n, and let si be the standard deviation of the ith sample. Then the average of the g standard deviations is

SBUCLR 4 (Upper control limit for ‘s’ chart)

SBLCLR 3 (Lower control limit for ‘s’ chart)

g

i

isg

S1

1(Center Line for ‘s’ chart)

and s charts

Centre line, upper, & lower control limit for s charts are calculated.The formulae used are as following:

Where B3 and B4

are constants and their values can be obtained from standard tables. These values depend upon the size of sample.

SD provides a better measure of variabilityGenerally n greater than 10

126


N A3 B3 B4

2 2.659 0 3.267

3 1.954 0 2.568

4 1.628 0 2.266

5 1.427 0 2.089

6 1.287 0.030 1.970

7 1.182 0.118 1.882

8 0.185 1.815

9 0.239 1.761

10 0.284 1.716

Factors for Calculating Limits for and S Charts

127


= mean of ‘g’ samples

Centre line, upper, & lower control limit for x bar charts are calculated.The formulae used are as following:

(Upper control limit for X bar chart)

(Lower control limit for X bar chart)

Where A3 is constants and its value can be obtained from standard table. Its value depend upon the size of sample.

iXn

n

iiX

1

= mean of the ith samplen = sample size,

Xi = ith data

iX

CLX

x

g

ii

g

1X X

SAXUCLx 3

SAXLCLx 3

128

Control Charts for Variable Type of DataExample

SN Mean SD SN Mean SD SN Mean SD SN Mean SD SN Mean SD

1 55.6 9.63 7 46.8 6.72 13 44 14.35 19 50.2 7.6 25 44.6 8.96

2 61 8.63 8 44.2 8.53 14 51.6 5.18 20 44 8.46 26 46.8 6.5

3 45.2 7.4 9 50.8 11.95 15 53.2 5.36 21 50 5.15 27 49.2 3.19

4 46.2 4.09 10 48.4 6.19 16 52.4 9.48 22 47 5.15 28 45.6 7.96

5 46.8 7.22 11 51.2 6.83 17 50.6 3.44 23 50.6 5.55 29 57.6 14.38

6 49.8 8.76 12 49.4 5.46 18 56 7 24 48.8 6.5 30 51.4 6.8

Mean values and SD of data from 30 samples (sample size = 5) are shown inthe table below:

Average of Ranges of 30 samples =

Upper Control Limit of s chart = B4*7.41 (B4 = 2.089 for sample size 5)= 15.479

Lower Control Limit of s chart = B3*7.41 (B3 = 0 for sample size 5) = 0

= 7.41 (Center Line of s Chart)

g

i

isg

S1

1

129

Control Charts for Variable Type of DataExample

SN Mean SD SN Mean SD SN Mean SD SN Mean SD SN Mean SD

1 55.6 9.63 7 46.8 6.72 13 44 14.35 19 50.2 7.6 25 44.6 8.96

2 61 8.63 8 44.2 8.53 14 51.6 5.18 20 44 8.46 26 46.8 6.5

3 45.2 7.4 9 50.8 11.95 15 53.2 5.36 21 50 5.15 27 49.2 3.19

4 46.2 4.09 10 48.4 6.19 16 52.4 9.48 22 47 5.15 28 45.6 7.96

5 46.8 7.22 11 51.2 6.83 17 50.6 3.44 23 50.6 5.55 29 57.6 14.38

6 49.8 8.76 12 49.4 5.46 18 56 7 24 48.8 6.5 30 51.4 6.8

Mean values and SD of data from 30 samples (sample size = 5) are shown inthe table below:

Upper Control Limit of x bar chart = 49.63 + A3*7.41 = 60.21 (A3 = 1.427 for sample size 5)

Lower Control Limit of x bar chart = 49.63 - A3*7.41 = 39.05 (A3 = 1.427 for sample size 5)

= 49.63 (Center Line of x bar Chart)Average of mean values of 30 samples = 30

30

1i

X

130

40

45

50

55

60

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30

X Bar chart

Sam

ple

Mean

-2

2

6

10

14

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30

S chart

Sam

ple

SD

Sample data at S.N 2, isslightly above the UCL.Efforts must be made to findthe special causes andrevised limits are advised tocalculate after deleting thesedata.

All the data are within theLCL and UCL in S Chart.Hence variability of theprocess data is not an issue to

worry.


131

Control Charts for Variable Type of Data Example

SampleSample Mean

XSample Standard

Deviation, sSample

Sample MeanX

Sample Standard

Deviation, s

1 36.4 4.6 11 36.7 5.3

2 35.8 3.7 12 35.2 3.5

3 37.3 5.2 13 38.8 4.7

4 33.9 4.3 14 39.0 5.6

5 37.8 4.4 15 35.5 5.0

6 36.1 3.9 16 37.1 4.1

7 38.6 5.0 17 38.3 5.6

8 39.4 6.1 18 39.2 4.8

9 34.4 4.1 19 36.8 4.7

10 39.5 5.8 20 37.7 5.4

Data for Magnetic Coating Thickness (in microns)

132


The control limits for the s-chart are (sample size – 4)

4

3

UCL =B = (2.266)(4.790) =10.854

LCL =B = (0)(4.790) = 0

s

s

s

s

133


The center line of the X Bar Chart is

20

=1 743.5C L = = = = 37.175

20 20

i

iX

XX

3

3

UCL = + A s = 37.175 +(1.628)(4.790) = 44.973

LCL = - A s = 37.175 - (1.628)(4.790) = 29.377

X

X

X

X

134


Assuming the thickness of the coating to be normallydistributed, what proportion of the product will not meetspecifications?

s 4.790σ = = = 5.199

c 0.92134

1

33.5-37.175= =-0.71

5.199z

2

42.5-37.175= =1.02

5.199z

Hence, the proportion of product not meeting specifications is 0.2389+0.1539=0.3928.

135

CLs = C4s0UCLs= B6s0LCLs = B5s0

0 0X

0 0X

UCL = X + A

LCL = X - A

Limits for s chart

Limits for X bar chart


Control limits for given target or standard

136

Control chart for attributesAttribute: - Quality characteristic for which a numerical value is not specified.

Taste, paint quality, SF, Quality of output,Nonconformity (defect)

Nonconforming (defective)Attribute - at all levels, cost effectiveVariable - at lowest level

Control Charts for Attribute Type of Data

When observations can be placed into two categories

Good or bad

Pass or fail

Operate or don’t operate

When the data consists of multiple samples of several observations each

p-charts calculates the percent defective in sample

P-CHARTS PROPORTIONS OF UNITS NONCONFORMING


138


Centre line, upper, & lower control limit for p charts are calculated.The formulae used are as following:

(Upper control limit for p chart)

(Lower control limit for p chart)

samplesallinitemsofnumbertotal

samplesallindefectivesofsump

n

pppUCL

)1(3

n

pppLCL

)1(3

= centre line of p chart

Where n is the sample size. Sample size in p chart must be 50

Sometimes LCL in p chart becomes negative, in such cases LCL should be taken as 0

139

Control Charts for Attribute Type of DataExample

Data for defective CDs from 20 samples (sample size = 100) are shown in thetable below:

Sample No. No. of Defective CDs = x

Proportion Defective =

x/sample size

Sample No. No. of Defective CDs = x

Proportion Defective =

x/sample size

1 4 .04 11 6 .06

2 3 .03 12 5 .05

3 3 .03 13 4 .04

4 5 .05 14 5 .05

5 6 .06 15 4 .04

6 5 .05 16 7 .07

7 2 .02 17 6 .06

8 3 .03 18 8 .08

9 5 .05 19 6 .06

10 6 .06 20 8 .08

12.100

)051.1(051.3051.0

)1(3

n

pppUCL

002.100

)051.1(051.3051.0

)1(3

n

pppLCL

051.02000

101

allsamplesofsum

defectivesofsumCL =

140


There is important observation that is clearly visible from the data points thatthere is an increasing trend in the average proportion defectives beyondsample number 15. Also, data show cyclic pattern. Process appears to be outof control and also there is strong evidence that data are not fromindependent source.

P CHART

0.00

0.02

0.04

0.06

0.08

0.10

0.12

0.14

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

MEANP

UCLP

LCLP

P

Some cases - particular items being subjected to inspection

may have more than one defect, e.g., no. of scratches,

cracks, etc.

c-charts counts the number of defects in an item.

c-charts are used only when the number of occurrence per

unit of measure can be counted.

THE C -CHART

Centre line, upper, & lower control limit for c charts are calculated usingfollowing formulae:

= centre line of c chartsamplesallinitemsofnumbertotal

samplesallindefectsofsumc


(Upper control limit for c chart)

(Lower control limit for c chart)

ccUCL 3

ccLCL 3

The c-Chart


THE C -CHART

Data for defects on TV set from 20 samples (sample size = 10) are shown inthe table below:

Sample No.

No. of Defects

Sample No.

No. of Defects

Sample No.

No. of Defects

Sample No.

No. of Defects

1 4 6 4 11 6 16 5

2 4 7 4 12 5 17 4

3 5 8 6 13 4 18 6

4 6 9 8 14 7 19 6

5 4 10 7 15 6 20 6

289.1235.5335.53 ccUCL

0589.135.5335.53 ccLCL

5.3520

107

samplesofnumber

defectsofsumCL =


Cyclic trend !

0

2

4

6

8

10

12

14

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

Def

ects

The c-Chart


145

Variable Sample Size:100% inspection, a change in the rate of production

may cause the sample size to changeChange in the unit cost of inspectionChange in the available inspection personnel


Control Limits for individual samples

i

i

n

pppLCL

n

pppUCL

)1(3

)1(3

Sample proportion nonconforming is Sample Size = ni

p

146

Variable Sample Size:


147



148


Control Limits based on Average sample Size

n

pppLCL

n

pppUCL

g

nn

g

ii

)1(3

)1(3

SizeSampleAve. 1


149


Standardized CC

i

p

)/np(1p

ppZ

n

p)p(1σ-SD

p)pE(-Mean

ingnonconformproportionSample


ni – Size of ith sampleSame UCL and LCLat 3 times SDCL is at 0

150


Standardized CC


151


Standardized CC


152


Standardized CC


153


np chart: chart for the number of nonconforming

np-charts calculates the number of defective in sample. np-charts are used when observations can be placed in twocategories such as yes or no, good or bad, pass or fail etc.

It is easier to relate the number of defectives than to theproportion defective. However, the np charts should not beused when the sample size varies as the Centre line and thecontrol limits will vary.

154


Centre line, upper, & lower control limit for np charts are calculated.The formulae used are as following:

(Upper control limit for np chart)

(Lower control limit for np chart)

samplesofnumbertotal

defectivesofsumpn

ppnpnUCL 13

= centre line of np chart

Where n is the sample size. Sample size in np chart must be 50

Sometimes LCL in np chart becomes negative, in such cases LCL should be taken as 0

ppnpnLCL 13

155

Control Charts for Attribute Type of DataExample

Data for the number of defective products from 20 samples (sample size =300) are shown in the table below:

Sample No. No. of Defective

Sample No. No. of Defective

1 10 11 6

2 12 12 19

3 8 13 10

4 9 14 8

5 6 15 7

6 11 16 4

7 13 17 11

8 10 18 10

9 8 19 6

10 9 20 7

159.18)3002.91(2.932.9)1(3

ppnpnUCL

2.920

184

samplesofNumber

defectivesofsumCL =

241.0)3002.91(2.932.9)1(3

ppnpnLCL

300

2.9

n

pnp

156


Sample data at S.N 12, is above the UCL. Efforts must be made to find thespecial causes and revised limits are advised to calculate after deleting thesedata.

-2

0

2

4

6

8

10

12

14

16

18

20

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

Defe

ctiv

es

157


u charts: Chart for number of nonconformity per unit

u-chart counts the number of defect per sample. The u chart is used whenit is not possible to have a sample size of a fixed size.UCL and LCL with change in sample size but CL remains constant

158

Control Charts for Attribute Type of DataVariable Sample SizeCentre line, upper, & lower control limit for u charts are calculated.The formulae used are as following:

(Upper control limit for u chart)

(Lower control limit for uc chart)

= centre line of u chart

in

uuUCL 3

in

uuLCL 3

samplesallinitemsofnumbertotal

samplesallindefectsofsumu

k

ii

k

ii

n

c

1

1=

ci =number of defects in ith samplek = number of samplesni = size of ith samples

159

ExampleData for defects in a product from 20 samples are shown in the table below.The number of products varies from sample to sample. Construct a controlfor number of defects per 100 products.

Sample No.

No. of productsInspected per unit

No. of Defects

Sample No.

No. of productsinspected

No. of Defects

Sample No.

No. of productsinspected

No. of Defects

1 200 5 8 150 10 15 100 6

2 300 14 9 150 6 16 200 8

3 250 8 10 250 10 17 200 5

4 150 8 11 300 9 18 100 5

5 250 12 12 250 16 19 300 14

6 100 6 13 200 12 20 200 8

7 200 20 14 250 10 TOTAL 4100 or 41

1 unit = 100

192

27.92

68.4368.4

1UCL

092.02

68.4368.4

1LCL in

uuUCL 368.4

41

192

1

1

k

ii

k

ii

n

cu

160

S No. Sample size ni

No. of Defects per 100 product

UCL* LCL* S No. Sample size ni

No. of Defects per 100 product

UCL* LCL*

1 2.0 2.50 9.274 0.092 11 3.0 3.00 8.4 0.9

2 3.0 4.67 8.431 0.935 12 2.5 6.40 8.8 0.6

3 2.5 3.20 8.789 0.577 13 2.0 6.00 9.3 0.1

4 1.5 5.33 10.0 0.0 14 2.5 4.00 8.8 0.6

5 2.5 4.80 8.8 0.6 15 1.0 6.00 11.2 0.0

6 1.0 6.00 11.2 0.0 16 2.0 4.00 9.3 0.1

7 2.0 10.00 9.3 0.1 17 2.0 2.50 9.3 0.1

8 1.5 6.67 10.0 0.0 18 1.0 5.00 11.2 0.0

9 1.5 4.00 10.0 0.0 19 3.0 4.67 8.4 0.9

10 2.5 4.00 8.8 0.6 20 2.0 4.00 9.3 0.1

-2

0

2

4

6

8

10

12

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

Defec

ts

*UCL & LCL – Values rounded off

161

• c and u charts do not differentiate with the types of

nonconformity

• There is a need to have a system that assigns weights to

nonconformities according to their relative degree of

severity.

• Charts for demerits per unit (U chart) takes care of the

deficiency of the c and u charts.

CHART FOR DEMERITS PER UNIT (U-CHART)

162

Class 1 defects- very serious.

Defects that lead directly to severe injury or to catastrophic economic loss.

Class 2 defects- serious.

Defects that lead to significant injury or significant economic loss.

Class 3 defects- major.

Defect that can cause major problems with normal use of a product or service

rendered.

Class 4 defects- minor

Defect that can cause minor problems with normal use of a product or service

rendered.


163

For the ANSI/ASQC standard, a weight system of 100, 50, 10 and 1

could, for example, be chosen for the categories of very serious,

serious, major and minor, respectively.


164

Suppose we have four categories of nonconformities. Let the

sample size be ‘n’ and let c1, c2, c3 and c4 denote the total number

of nonconformities in a sample for four categories. Let w1, w2,

w3, and w4 denote the weights assigned to each category.

It is assumed that nonconformities in each category are

independent of defects in the other categories. Furthermore, it is

also assumed that the occurrence on nonconformities in any

category is represented by a Poisson distribution.


165

For a sample of size ‘n’, the total number of demerits is given

by

D = w1c1 + w2c2 + w3c3+ w4c4 (1)

The demerits per unit for the sample are given by

(2)


166

The centre line of the U-chart is given by

(3)

Where, represent the average number of

nonconformities per unit in their respective classes.


167

The estimated standard deviation of U is given by

(4)

The control limits for the U-chart are given by

(5)

If the lower control limit is calculated to be less than zero, it is converted to zero.


168

Example:

Customer Survey

Twenty random samples, each involving 10 customers (Sample

size), are taken in which customers are asked about the number of

serious, major, and minor nonconformities that they have

experienced. Clear definitions of each category are provided. The

weights assigned to a serious, major, and minor nonconformity

are 50, 10, and 1, respectively. Construct a control chart for the

number of demerits per unit.


169

SampleSerious

Nonconformities c1

Major Nonconformities

c2

Minor Nonconformities

c3

Total Demerits

D

Demerits per unit

U

1 1 4 2 92 9.2

2 0 3 8 38 3.8

3 0 5 10 60 6.0

4 1 2 5 75 7.5

5 0 6 2 62 6.2

6 0 0 8 8 0.8

7 0 7 5 75 7.5

8 1 1 1 61 6.1

9 1 3 2 82 8.2

10 0 4 12 52 5.2

11 1 5 3 103 10.3

12 2 0 2 102 10.2

13 0 0 9 9 0.9

14 0 6 8 68 6.8

15 1 12 10 180 18.0

16 0 5 7 57 5.7

17 0 1 1 11 1.1

18 1 2 5 75 7.5

19 0 5 6 56 5.6

20 0 3 8 38 3.8

Total 9 74 114

Table: Data for nonconformities in a department store customer survey

170

Solution:For each sample, the total number of demerits given by equation (1) isshown in Table. The table also shows the number of demerits per unit U,given by equation (2). To find the center line Ū, the average number ofnonconformities per unit for each category is calculated.For (Serious” nonconformities

Similarly,

57.0200

114

37.0200

74

3

2

u

u


171

Using eq. (3), the center line of the U-chart is

The estimated standard deviation of U, using eq. (4) is

Hence, the control limits (from eq. (5)) are

L


172Note: Figure shows (U-chart) all the point within the control limits

Figure U-chart for department store customer survey.

UCL= 18.142

CL= 6.52

LCL= 0


THANKS ……

1 quality and quality control

Documents