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Chapter 6 - Statistical Quality Control (SQC)

Operations Management 6th Edition

R. Dan Reid & Nada R. Sanders

Learning Objectives

 Describe categories of statistical quality control (SQC).

 Identify and describe causes of variation.

 Explain the use of descriptive statistics in measuring quality characteristics.

 Describe the use of control charts.

 Identify the differences between x-bar and R- charts.

Learning Objectives - cont'd

 Identify the differences between p- and c-charts.

 Explain the meaning of process capability and the process capability index.

 Explain the concept Six Sigma.

 Explain the process of acceptance sampling and describe the use of operating characteristic (OC) curves.

 Identify decisions that managers must make when implementing SPC.

 Describe the challenges inherent in measuring quality in service organizations.

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.

3 Categories of SQC

1. Statistical process control (SPC) inspecting a random sample of an output from process, within range and functioning properly

2. Descriptive statistics the mean, standard deviation, and range  Involve inspecting the output from a process

 Quality characteristics are measured and charted

 Helps identify in-process variations

3. Acceptance sampling used to randomly inspect a batch of goods to determine acceptance/rejection

 Does not help to catch in-process problems

Sources of Variation

 Variation exists in all processes.

 Variation can be categorized as either:  Common or Random causes of variation, or

 Random causes that we cannot identify

 Unavoidable, i.e.; slight differences in process variables like diameter, weight, service time, temperature

 Assignable causes of variation  Causes can be identified

 Eliminate cause i.e.; poor employee training, worn tool, machine needing repair

Descriptive Statistics

x x

1i i∑

 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

( ) 1n

Xx σ

−

− = ∑ =

Distribution of Data

Normal distributions Skewed distribution 8

SPC Methods-Developing Control Charts

Control Charts (aka process or QC charts) show sample data plotted on a graph with CL, UCL, and LCL

Control chart for variables are used to monitor characteristics that can be measured, e.g. length, weight, diameter, time

Control charts for attributes are used to monitor characteristics that have discrete values and can be counted, e.g. % defective, # of flaws in a shirt, etc.

Setting Control Limits 10

Control Charts for Variables

 Use x-Bar and R-bar charts together

 Used to monitor different variables

 x-Bar and R-bar charts reveal different problems

 What is the statistical control difference from one chart to the next?

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

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.

n21

zσxLCL

zσxUCL

sample each w/in nsobservatio of# the is (n) and means sample of # the is )( where

n 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

Center line and control limit formulas

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):

x-Bar Control Chart 15

Control Chart for Range (R)

0.00.0(.233)RDLCL

.532.28(.233)RDUCL

.233 3

0.20.30.2 R

===

= ++

Center Line and Control Limit formulas:

Factors for three sigma control limits 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)

R-Bar Control Chart 17

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

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 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 ( ) ( ) 0.1023(.064).09σzpLCL

.2823(.064).09σzpUCL

0.64 20

(.09)(.91) n

)p(1p σ

.09 100

9 Inspected Total

Defectives# pCL

=−=−=−=

=+=+=

== −

====

Solution:

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

P- Control Chart 21

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

02.252.232.2ccLCL

6.652.232.2ccUCL

2.2 10 22

samples of # complaints#

=−=−=−=

=+=+=

===

Solution:

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

that is defective

C- Control Chart 23

Process Capability

Product Specifications

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

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

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

Relationship Between Process Variability & Specification Width

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)

Machine σ USL-LSL 6σ

A .05 .4 .3

B .1 .4 .6

C .2 .4 1.2

The table below shows the information gathered from production runs on each machine. Are they all acceptable?

Solution:  Machine A

 Machine B

Cp=

 Machine C

Cp=

1.33 6(.05)

.4 6σ

LSLUSL Cp ==

−

Computing the Cpk Value at Cocoa Fizz

.33 .3 .1

Cpk

3(.1) 15.815.9

, 3(.1)

15.916.2 minCpk

 

  

 −− =

 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

±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

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

AQL, LTPD, Consumer’s Risk (α) & Producer’s Risk (β)

 AQL is the small % of defects that consumers are willing to accept; order of 1-2%

 LTPD is the upper limit of the percentage of defective items consumers are willing to tolerate

 Consumer’s Risk (β) is the chance of accepting a lot that contains a greater number of defects than the LTPD limit; Type II error

 Producer’s Risk (α) is the chance a lot containing an acceptable quality level will be rejected; Type I error

Developing OC Curves

 OC curves graphically depict discriminating power of a sampling plan

 Cumulative binomial tables like partial table below are used to obtain probabilities of accepting a lot given varying levels of lot defectives

 Top of the table shows value of p (proportion of defective items in lot), Left hand column shows values of n (sample size) and x represents the cumulative number of defects found

Table 6-2 Partial Cumulative Binomial Probability Table (see Appendix C for complete table) Proportion of Items Defective (p)

.05 .10 .15 .20 .25 .30 .35 .40 .45 .50 n x 5 0 .7738 .5905 .4437 .3277 .2373 .1681 .1160 .0778 .0503 .0313

Pac 1 .9974 .9185 .8352 .7373 .6328 .5282 .4284 .3370 .2562 .1875 AOQ .0499 .0919 .1253 .1475 .1582 .1585 .1499 .1348 .1153 .0938

Example: Constructing an OC Curve

 Lets develop an OC curve for a sampling plan in which a sample of 5 items is drawn from lots of N=1000 items

 The accept /reject criteria are set up in such a way that we accept a lot if no more that one defect (c=1) is found

 Using Table 6-2 and the row corresponding to n=5 and x=1

 Note that we have a 99.74% chance of accepting a lot with 5% defects and a 73.73% chance with 20% defects

Average Outgoing Quality (AOQ)

 With OC curves, the higher the quality of the lot, the higher is the chance that it will be accepted

 Conversely, the lower the quality of the lot, the greater is the chance that it will be rejected

 The average outgoing quality level of the product (AOQ) can be computed as follows: AOQ=(Pac)p

 Returning to the bottom line in Table 6-2, AOQ can be calculated for each proportion of defects in a lot by using the above equation

 This graph is for n=5 and x=1 (same as c=1)  AOQ is highest for lots close to 30% defects

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

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

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

4 1.0

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 =−=



  

 −=−=

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.

Chapter 6 Highlights

 SQC refers to statistical tools that can be sued by quality professionals. SQC an be divided into three categories: traditional statistical tools, acceptance sampling, and statistical process control (SPC).

 Descriptive statistics are used to describe quality characteristics, such as the mean, range, and variance. Acceptance sampling is the process of randomly inspecting a sample of goods and deciding whether to accept or reject the entire lot. Statistical process control involves inspecting a random sample of output from a process and deciding whether the process in producing products with characteristics that fall within preset specifications.

Chapter 6 Highlights – cont'd

 Two causes of variation in the quality of a product or process: common causes and assignable causes. Common causes of variation are random causes that we cannot identify. Assignable causes of variation are those that can be identified and eliminated.

 A control chart is a graph used in SPC that shows whether a sample of data falls within the normal range of variation. A control chart has upper and lower control limits that separate common from assignable causes of variation. Control charts for variables monitor characteristics that can be measured and have a continuum of values, such as height, weight, or volume. Control charts for attributes are used to monitor characteristics that have discrete values and can be counted.

Chapter 6 Highlights – cont'd

 Control charts for variables include x-Bar and R-charts. X-bar charts monitor the mean or average value of a product characteristic. R-charts monitor the range or dispersion of the values of a product characteristic. Control charts for attributes include p-charts and c-charts. P-charts are used to monitor the proportion of defects in a sample, C-charts are used to monitor the actual number of defects in a sample.

 Process capability is the ability of the production process to meet or exceed preset specifications. It is measured by the process capability index Cp which is computed as the ratio of the specification width to the width of the process variable.

Chapter 6 Highlights – cont'd

 The term Six Sigma indicates a level of quality in which the number of defects is no more than 2.3 parts per million.

 The goal of acceptance sampling is to determine criteria for the desired level of confidence. Operating characteristic curves are graphs that show the discriminating power of a sampling plan.

 It is more difficult to measure quality in services than in manufacturing. The key is to devise quantifiable measurements for important service dimensions.

Chapter 6 - Statistical Quality Control (SQC)
Learning Objectives
Learning Objectives - cont'd
What is SQC?
3 Categories of SQC
Sources of Variation
Descriptive Statistics
Distribution of Data
SPC Methods-Developing Control Charts
Setting Control Limits
Control Charts for Variables
Control Charts for Variables
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.
Solution and x-Bar Control Chart
x-Bar Control Chart
Control Chart for Range (R)
R-Bar Control Chart
Second Method for the x-Bar Chart Using R-bar & A2 Factor
Control Charts for Attributes–P-Charts & C-Charts
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 control limits.
P- Control Chart
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.
C- Control Chart
Process Capability
Process Capability - cont’d
Relationship Between Process Variability & Specification Width
Relationship Between Process Variability & Specification Width
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)
Computing the Cpk Value at Cocoa Fizz
±6 Sigma versus ± 3 Sigma
Acceptance Sampling
Acceptance Sampling Plans
Operating Characteristics (OC) Curves
AQL, LTPD, Consumer’s Risk (α) & Producer’s Risk (β)
Developing OC Curves
Example: Constructing an OC Curve
Average Outgoing Quality (AOQ)
Implications for Managers
SQC in Services
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.
SQC Across the Organization
Chapter 6 Highlights
Chapter 6 Highlights – cont'd
Chapter 6 Highlights – cont'd
Chapter 6 Highlights – cont'd