Chapter 7

Statistical Quality Control

Quality Control Approaches

Statistical process control (SPC)Monitors the production process to prevent

poor quality

Statistical Process Control

Take periodic samples from a process

Plot the sample points on a control chart

Determine if the process is within limits

Correct the process before defects occur

Types Of Data

Attribute data Product characteristic evaluated with a

discrete choice– Good/bad, yes/no

Variable data Product characteristic that can be

measured– Length, size, weight, height, time, velocity

SPC Applied To Services

Nature of defect is different in services

Service defect is a failure to meet customer requirements

Monitor times, customer satisfaction

Service Quality Examples Hospitals

timeliness, responsiveness, accuracy Grocery Stores

Check-out time, stocking, cleanliness Airlines

luggage handling, waiting times, courtesy Fast food restaurants

waiting times, food quality, cleanliness

Process Control Chart

1 2 3 4 5 6 7 8 9 10

Sample number

Uppercontrollimit

Processaverage

Lowercontrollimit

Constructing a Control Chart Decide what to measure or count Collect the sample data Plot the samples on a control chart Calculate and plot the control limits on the control

chart Determine if the data is in-control If non-random variation is present, discard the data

(fix the problem) and recalculate the control limits

A Process Is In Control If

No sample points are outside control limits

Most points are near the process average

About an equal # points are above & below the centerline

Points appear randomly distributed

99.74 %

The Normal Distribution

95 %

= 0 1 2 3-1-2-3

Area under the curve = 1.0

Control Charts and the Normal Distribution

Mean

UCL

LCL

+ 3

- 3

Types Of Data

Attribute data (p-charts, c-charts)Product characteristics evaluated with a

discrete choice (Good/bad, yes/no, count)

Variable data (X-bar and R charts)Product characteristics that can be measured

(Length, size, weight, height, time, velocity)

Control Charts For Attributes

p ChartsCalculate percent defectives in a sample;an item is either good or bad

c ChartsCount number of defects in an item

p - ChartsBased on the binomial distribution

p = number defective / sample size, n

p = total no. of defectives total no. of sample observations

UCL = p + 3 p(1-p)/n

LCL = p - 3 p(1-p)/n

p-Chart Example

The Western Jean Company produced denim jean. The company wants to establish a p-chart to monitor the production process and main high quality. Western beliefs that approximately 99.74 percent of the variability in the production process (corresponding to 3-sigma limits, or z = 3.00) is random and thus should be within control limits, whereas 0.26 percent of the process variability is not random and suggest that the process is out of control.

p-Chart Example

The company has taken 20 sample (one per day for 20 days), each containing 100 pairs of jeans (n = 100), and inspected them for defects, the results of which are as follow.

Sample # Defects Sample # Defects1 6 11 122 0 12 103 4 13 144 10 14 85 6 15 66 4 16 167 12 17 128 10 18 149 8 19 20

10 10 20 18

p-Chart Calculations Proportion

Sample Defect Defective 1 6 .06 2 0 .00 3 4 .04

. . .

. . .20 18 .18 200

= 0.10

=

total defectives total sample observations 200 20 (100)

p =

100 jeans in each sample

LCL = p - 3 p(1-p) /n

= 0.10 + 3 0.10 (1-0.10) /100

= 0.010

UCL = p + 3 p(1-p) /n

= 0.10 + 3 0.10 (1-0.10) /100= 0.190

. .

00.020.040.060.08

0.10.120.140.160.18

0.2

0 2 4 6 8

10 12 14 16 18 20

Prop

ortio

n de

fect

ive

Sample number

c - Charts

Count the number of defects in an item

Based on the Poisson distribution

c = number of defects in an item

c = total number of defects number of samples

UCL = c + 3 c

LCL = c - 3 c

c-Chart ExampleThe Ritz Hotel has 240 rooms. The hotel’s housekeeping

department is responsible for maintaining the quality of the room’s appearance and cleanliness. Each individual housekeeper is responsible for an area encompassing 20 rooms. Every room in use is thoroughly clean and its supplies, toiletries, and so on are restocked each day. Any defects that the housekeeping staff notice that are not part the normal housekeeping service are supposed to be reported hotel maintenance.

c-Chart ExampleEvery room is briefly inspected each day by a

housekeeping supervisor. However, hotel management also conducts inspection for quality-control purposes. The management inspector not only check for normal housekeeping defects like clean sheets, dust, room supplies, room literature, or towels, but also for defects like an inoperative or missing TV remote, poor TV picture quality or reception, defective lamps, a malfunctioning clock, tears or stains in bedcovers or curtain, or a malfunctioning curtain pull.

c-Chart ExampleAn inspection sample include 12 rooms, i.e., one

room selected at random from each of the twelve 20-room blocks served by a housekeeper. Following are the results from 15 inspection samples conducted at random during a 1-month period.

Sample # Defects Sample # Defects1 12 11 122 8 12 103 16 13 144 14 14 175 10 15 156 117 98 149 13

10 15

c - Chart Calculations Count # of defects per roll in 15 rolls of denim fabric

Sample Defects1 122 83 16. .

. .15 15 190

c = 190/15 = 12.67

UCL = c + 3 c = 12.67 + 3 12.67 = 23.35LCL = c - 3 c = 12.67 - 3 12.67 = 1.99

Example c - Chart

.

0

3

6

9

12

15

18

21

24

0 2 4 6 8 10 12 14

Sample number

Num

ber o

f def

ects

Control Charts For Variables

Mean chart (X-Bar Chart)Measures central tendency of a sample

Range chart (R-Chart)Measures amount of dispersion in a sample

Each chart measures the process differently. Both the process average and process variability must be in control for the process to be in control.

Example: Control harts for Variable Data

The Goliath Tool Company produces slip-ring bearings, which look like flat doughnut or washer, they fit around shafts or rods, such as drive shaft in machinery or motor. In the production process for a particular slip-ring bearing the employees has taken 10 samples (during a 10 day period) of 5 slip-ring bearing (i.e., n = 5). The individual observation from each sample are shown as followed:

Example: Control Charts for Variable Data Slip Ring Diameter (cm)Sample 1 2 3 4 5 X R

1 5.02 5.01 4.94 4.99 4.96 4.98 0.082 5.01 5.03 5.07 4.95 4.96 5.00 0.123 4.99 5.00 4.93 4.92 4.99 4.97 0.084 5.03 4.91 5.01 4.98 4.89 4.96 0.145 4.95 4.92 5.03 5.05 5.01 4.99 0.136 4.97 5.06 5.06 4.96 5.03 5.01 0.107 5.05 5.01 5.10 4.96 4.99 5.02 0.148 5.09 5.10 5.00 4.99 5.08 5.05 0.119 5.14 5.10 4.99 5.08 5.09 5.08 0.15

10 5.01 4.98 5.08 5.07 4.99 5.03 0.10 50.09 1.15

Constructing an Range Chart

UCLR = D4 R = (2.11) (.115) = 0.24

LCLR = D3 R = (0) (.115) = 0

where R = R / k = 1.15 / 10 = .115 k = number of samples = 10 R = range = (largest - smallest)

0

0.05

0.1

0.15

0.2

0.25

1 2 3 4 5 6 7 8 9 10

Sample number

Ran

geExample R-Chart

UCL

R

LCL

Constructing A Mean Chart

UCLX = X + A2 R = 5.01 + (0.58) (.115) = 5.08

LCLX = X - A2 R = 5.01 - (0.58) (.115) = 4.94

where X = average of sample means = X / n = 50.09 / 10 = 5.01

R = average range = R / k = 1.15 / 10 = .115

4.92

4.94

4.96

4.98

5.00

5.02

5.04

5.06

5.08

5.101 2 3 4 5 6 7 8 9 10

Sample number

Sam

ple

aver

age

Example X-bar Chart

UCL

X

LCL

Variation Common Causes

Variation inherent in a processCan be eliminated only through improvements in the system

Special CausesVariation due to identifiable factorsCan be modified through operator or management action

UCL

LCL LCL

UCL

Sample observationsconsistently below thecenter line

Sample observationsconsistently above thecenter line

Control Chart Patterns

Control Chart Patterns

LCL LCL

UCL UCL

Sample observationsconsistently increasing

Sample observationsconsistently decreasing

Sample Size Determination

Attribute control charts50 to 100 parts in a sample

Variable control charts2 to 10 parts in a sample