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Statistical Quality Control (SQC)

(Chapter 6)

Production & Operations Management

INFO 335-71

Week 2

What is SQC?

Statistical Quality Control (SQC)

the term used to describe the set of statistical

tools used by quality professionals to evaluate

organizational quality.

SQC

1. Categories:

1. Descriptive Statistics (mean, variance, range, distribution)

2. Statistical Process Control

3. Acceptance Sampling – Not to worry about this for now

2. Causes of Variation

1. Common

2. Assignable

Descriptive Statistics

• The Mean- measure of central tendency

• The Range- difference between largest/smallest observations in a set of data

• Standard Deviation measures the amount of data dispersion around mean

• Distribution of Data shape • Normal or bell shaped or • Skewed

i ==

( )

−

=  =

Normal Distribution of Data

Control Charts

Center

14-17 17-1912-14

Bottling

Plant ~ 16

Fl. Oz.

UCL: μ + 3σ

LCL: μ - 3σ

Variables and Attributes

⚫ Variables

• Continuous measures (height, weight, volume etc.) • X-bar (centrality)

• Variance or standard deviation of operation provided • When variance or standard deviation of operation is

unknown

• R-bar (dispersion)

⚫ Attributes

• Discrete measures (complaints, defects etc.) • P-chart (proportions known, such as proportion of

defectives)

• C-chart (proportion unknown, just totals are known)

Control Charts for Variables

⚫ Use x-Bar charts to monitor the changes in the mean of a process (central tendencies)

⚫ Use R-bar charts to monitor the dispersion or variability of the process

⚫ System can show acceptable central tendencies but unacceptable variability

⚫ System can show acceptable variability but unacceptable central tendencies

Center line and control limit

formulas

n21

zσxLCL

zσxUCL

sample each w/in nsobservatio of# the is

(n) and means sample of # the is )( where

σ σ ,

...xxx x x

−=

= ++

k k

Time 1 Time 2 Time 3

Observation 1 15.8 16.1 16.0

Observation 2 16.0 16.0 15.9

Observation 3 15.8 15.8 15.9

Observation 4 15.9 15.9 15.8

Sample means (X-bar)

15.875 15.975 15.9

Sample ranges (R)

0.2 0.3 0.2

Constructing an x-Bar Chart: A quality control inspector at

the Cocoa Fizz soft drink company has taken three samples

with four observations each of the volume of bottles filled. If

the standard deviation of the bottling operation is .2 ounces,

use the below data to develop control charts with limits of 3

standard deviations for the 16 oz. bottling operation.

Sample Size = number of observations per sampling activity

Number of samples = number of times we went and collected samples

Solution and x-Bar Control Chart

15.92 3

15.915.97515.875 x =

++ =

15.62 4

.2 315.92zσxLCL

16.22 4

.2 315.92zσxUCL

= 

  

 −=−=

= 

  

 +=+=

Control limits for±3σ limits:

Center line (x-double bar):

Time 1 Time 2 Time 3

Observation 1 15.8 16.1 16.0

Observation 2 16.0 16.0 15.9

Observation 3 15.8 15.8 15.9

Observation 4 15.9 15.9 15.8

Sample means (X- bar)

15.875 15.975 15.9

Sample ranges (R)

0.2 0.3 0.2

15.92 + 3 * (.2/2) = 15.92 + 3*.1 = 15.92 + 0.3 = 16.22

Control Chart for Range (R)

Center Line and Control Limit

formulas:

Factors for three sigma control limits

0.00.0(.233)RDLCL

.532.28(.233)RDUCL

.233 3

0.20.30.2 R

===

= ++

Factor for x-Chart

A2 D3 D4

2 1.88 0.00 3.27

3 1.02 0.00 2.57

4 0.73 0.00 2.28

5 0.58 0.00 2.11

6 0.48 0.00 2.00

7 0.42 0.08 1.92

8 0.37 0.14 1.86

9 0.34 0.18 1.82

10 0.31 0.22 1.78

11 0.29 0.26 1.74

12 0.27 0.28 1.72

13 0.25 0.31 1.69

14 0.24 0.33 1.67

15 0.22 0.35 1.65

Factors for R-Chart Sample Size

(n)

n is not the total number of observations that you have sampled.

It is number of observations per sampling activity.

Second Method for the x-Bar

Chart Using R-bar & A2 Factor

Use this method, Control limits solution, when

sigma for the process distribution is not known:

( )

( ) 15.75.2330.7315.92RAxLCL

16.09.2330.7315.92RAxUCL

.233 3

0.20.30.2 R

=−=−=

=+=+=

= ++

Control Charts for Attributes –

P-Charts & C-Charts

Attributes are discrete events: yes/no or pass/fail

⚫ Use P-Charts for quality characteristics that are discrete and involve yes/no or good/bad decisions

• Number of leaking caulking tubes in a box of 48 • Number of broken eggs in a carton

⚫ Use C-Charts for discrete defects when there can be more than one defect per unit

• Number of flaws or stains in a carpet sample cut from a production run

• Number of complaints per customer at a hotel 13

P-Chart Example: A production manager for a tire

company has inspected the number of defective tires

in five random samples with 20 tires in each sample.

The table below shows the number of defective tires

in each sample of 20 tires. Calculate the 3-sigma

control limits.

Sample Number of

Defective Tires

Number of Tires in each

Sample

Proportion Defective

1 3 20 .15

2 2 20 .10

3 1 20 .05

4 2 20 .10

5 1 20 .05

Total 9 100 .09

Solution:

( )

( ) 0.1023(.064).09σzpLCL

.2823(.064).09σzpUCL

0.064 20

(.09)(.91)

)p(1p σ

.09 100

Inspected Total

Defectives# pCL

=−=−=−=

=+=+=

== −

====

P-Charts are used when both the total sample size and the number of defects can be computed

C-Chart Example: The number of weekly

customer complaints are monitored in a large

hotel using a c-chart. Develop three sigma

control limits using the data table below. Week Number of

Complaints

1 3

2 2

3 3

4 1

5 3

6 3

7 2

8 1

9 3

10 1

Total 22

Solution:

02.252.232.2ccLCL

6.652.232.2ccUCL

2.2 10

samples of #

complaints# CL

=−=−=−=

=+=+=

===

C-Charts are used when you can compute only the number of defects but not the proportion

that is defective

Memory Card

Continuous or Discrete Variable?

Continuous Variable

Discrete Variable

R-Bar Chart

Standard Deviation of Operation

Given?

X-Bar Chart

Yes

X-Bar Chart

Can we calculate

proportion?

Yes

P-Chart

C-Chart

Process Capability

Product Specifications

• Preset product or service dimensions, tolerances: bottle fill might be 16 oz. ±.2 oz. (15.8oz.-16.2oz.).

This should be the 3-sigma limits.

• Upper Specification Limit (USL) = 16 + .2 = 16.2oz. • Lower Specification Limit (LSL) = 16 - .2 = 15.8oz.

• Based on how the product is to be used or what the customer expects

Process Capability

⚫ Design (for±3σ limits) ⚫ Actual

⚫ Superimposed

16.0

16.2

15.6

15.9

16.1

15.7

USL

LSL

UCL

LCL

6σ

LSLUSL Cp =

− =

x y

⚫ Risk

>= 1

Cp > 1

X √

Process Capability

⚫ Superimposed

USL

LSL

μ μ

UCL = μ + 3σ

LCL = μ - 3σ

p q

r s

 

  

 =



  

 −− =

s ,

q min

3σ

LSLμ ,

3σ

μUSL minCpk

Process Capability - cont’d

Process Capability – Cp and Cpk

• Assessing capability involves evaluating process variability relative to preset product or service specifications

• Cp assumes that the process is centered in the specification range

• Cpk helps to address a possible lack of centering of the process 6σ

LSLUSL

width process

width ionspecificat Cp

− ==

 

  

 −− =

3σ

LSLμ ,

3σ

μUSL minCpk

Relationship Between Process

Variability & Specification Width

⚫ Three possible ranges for Cp

• Cp = 1, as in Fig. (a), process variability just meets specifications

• Cp ≤ 1, as in Fig. (b), process not capable of producing within specifications

• Cp ≥ 1, as in Fig. (c), process exceeds minimal specifications

⚫ One shortcoming, Cp assumes that the process is centered on the specification range

⚫ Cp=Cpk when process is centered

Computing the Cp Value at Cocoa Fizz: 3 bottling

machines are being evaluated for possible use at the

Fizz plant. The machines must be capable of meeting

the design specification of 15.8-16.2 oz. with at least

a process capability index of 1.0 (Cp≥1)

The table below shows the information

gathered from production runs on each

machine. Are they all acceptable?

Solution:

• Machine A

• Machine B Cp= .4/.6 = 2/3 = .67

• Machine C Cp= .4/1.2 = 1/3 = .33

Machine σ USL-LSL 6σ

A .05 .4 .3

B .1 .4 .6

C .2 .4 1.2

1.33 6(.05)

6σ

LSLUSL Cp ==

−

Computing the Cpk Value at

Cocoa Fizz

⚫ Design specifications call for a

target value of 16.0 ±0.2 OZ.

(USL = 16.2 & LSL = 15.8)

⚫ Observed process output has now

shifted and has a µ of 15.9 and a

σ of 0.1 oz.

⚫ Cpk is less than 1, revealing that

the process is not capable

.33 .3

.1 Cpk

3(.1)

15.815.9 ,

3(.1)

15.916.2 minCpk

 

  

 −− =

 

  

 −− =

3σ

LSLμ ,

3σ

μUSL minCpk

.3/.3 = 1 .1/.3 = 1/3 = .33

±6 Sigma versus ± 3 Sigma

⚫ In 1980’s, Motorola coined “six- sigma” to describe their higher quality efforts

⚫ Six-sigma quality standard is now a benchmark in many industries

• Before design, marketing ensures customer product characteristics

• Operations ensures that product design characteristics can be met by controlling materials and processes to 6σ levels

• Other functions like finance and accounting use 6σ concepts to control all of their processes

PPM Defective for ±3σ versus ±6σ quality

SQC in Services

⚫ Service Organizations have lagged behind manufacturers in the

use of statistical quality control

⚫ Statistical measurements are required and it is more difficult to

measure the quality of a service

• Services produce more intangible products • Perceptions of quality are highly subjective

⚫ A way to deal with service quality is to devise quantifiable

measurements of the service element

• Check-in time at a hotel • Number of complaints received per month at a restaurant • Number of telephone rings before a call is answered • Acceptable control limits can be developed and charted

SQC Across the Organization

SQC requires input from other organizational functions, influences their success, and used in

designing and evaluating their tasks

• Marketing – provides information on current and future quality standards

• Finance – responsible for placing financial values on SQC efforts

• Human resources – the role of workers change with SQC implementation. Requires workers with right skills

• Information systems – makes SQC information accessible for all.

Practice Problem

Backup Slides

Process Capability

⚫ Superimposed

USL

LSL

CT μ

μ + 3σ

μ - 3σ

RAx 2

−

x y

p q

r s

6σ

LSLUSL Cp =

− =

 

  

 =



  

 −− =

s ,

q min

3σ

LSLμ ,

3σ

μUSL minCpk

Acceptance Sampling

Defined: the third branch of SQC refers to the process of

randomly inspecting a certain number of items from a lot

or batch in order to decide whether to accept or reject the

entire batch

⚫ Different from SPC because acceptance sampling is

performed either before or after the process rather than

during

• Sampling before typically is done to supplier material • Sampling after involves sampling finished items before

shipment or finished components prior to assembly

⚫ Used where inspection is expensive, volume is high, or

inspection is destructive

Acceptance Sampling Plans

⚫ Goal of Acceptance Sampling plans is to determine the criteria for

acceptance or rejection based on:

• Size of the lot (N) • Size of the sample (n) • Number of defects above which a lot will be rejected (c) • Level of confidence we wish to attain

⚫ There are single, double, and multiple sampling plans

• Which one to use is based on cost involved, time consumed, and cost of passing on a defective item

⚫ Can be used on either variable or attribute measures, but more

commonly used for attributes

Operating Characteristics (OC)

Curves

⚫ OC curves are graphs which show the probability of accepting a lot given various proportions of defects in the lot

⚫ X-axis shows % of items that are defective in a lot- “lot quality”

⚫ Y-axis shows the probability or chance of accepting a lot

⚫ As proportion of defects increases, the chance of accepting lot decreases

⚫ Example: 90% chance of accepting a lot with 5% defectives; 10% chance of accepting a lot with 24% defectives

Implications for Managers

⚫ How much and how often to inspect? • Consider product cost and product volume • Consider process stability • Consider lot size

⚫ Where to inspect? • Inbound materials • Finished products • Prior to costly processing

⚫ Which tools to use? • Control charts are best used for in-process production • Acceptance sampling is best used for inbound/outbound;

attribute measures

• Control charts are easier to use for variable measures

Practice Problem - Service at a bank: The Dollars Bank

competes on customer service and is concerned about

service time at their drive-by windows. They recently

installed new system software which they hope will meet

service specification limits of 5±2 minutes and have a

Capability Index (Cpk) of at least 1.2. They want to also

design a control chart for bank teller use.

They have done some sampling recently (sample size: 4 customers)

and determined that the process mean has shifted to 5.2 with a

Sigma of 1.0 minutes.

Control Chart limits for ±3 sigma limits

1.2 1.5

1.8 Cpk

3(1/2)

5.27.0 ,

3(1/2)

3.05.2 minCpk

 

  

 −− =

1.33

1.0 6

3-7

6σ

LSLUSL Cp =

 

  

 =

−

minutes 6.51.55.0 4

1 35.0zσXUCL xx =+=



  

 +=+=

minutes 3.51.55.0 4

1 35.0zσXLCL xx =−=



  

 −=−=