CHAPTER 10: QUALITY CONTROLSuman Niranjan

STATISTICAL PROCESS CONTROL

Quality conformance of a process Does the output of a process conform to the intent of

the design Determine if it is statistically acceptable

Take periodic samples and compare with a standard Use control charts

It is a time ordered plot of samples obtained from a process

THE STEPS IN CONTROL PROCESS

Effective control requires Define

What is to be controlled? Measure

Counted or measured? Compare

Any standard available? Evaluate

What is out of control? Correct

Corrective action Monitor results

Results analyzed

VARIATIONS AND CONTROL There are primarily two types of variations

Random variations Assignable variations

Random Variations Natural variation, usually occurred due to combination of many

minor factors Fixing all these little things might be very expensive Also known as common variability

Assignable Variation A variation whose cause can be identified and tied to specific

cause Also known as specific variability

Samples of a process are taken and checked for non-random variability Sample statistics The goal is to determine if a correctable source of variation is

present

VARIATIONS AND CONTROL

Process distribution Is a collection of data points One data point refers to one individual observation

Count Measurement

Normally distributed due to the central limit theorem Sampling distribution

Is a collection of data points, where each data point is a mean of several data points

Mean of several data points Mean of several counts Mean of several Measurements

Normally distributed due to the central limit theorem

VARIATIONS AND CONTROL

CONTROL CHARTS It is a time ordered plot of all the samples used to

distinguish between random and nonrandom variability Upper control limit Lower control limit

CONTROL CHARTS Type I error

Concluding a process is in not in control when it actually is in control

Producers risk? Type II error

Concluding a process is in control when it is out of control Buyers risk?

Sample number

UCL

LCL1 2 3 4

TYPE I AND TYPE II ERRORS

OKType I

(Producers Risk)

Type II(Buyers Risk)

OK

And the Conclusion Process is

In Control Out of Control

Out of Control

In Control

If a

Pro

cess is A

ctu

ally

TYPES OF CONTROL CHARTS: OVERVIEW

Control Charts for Variables: for continuous measures such as temperature, volume, etc. Mean Chart (or -Chart): detects shift of the

mean Range Chart (or R-Chart): detects change in

shape of distribution Control Charts for Attributes: for discrete

measures such as number of complaints, scratches, etc. p-Chart: measures percent defective c-Chart: measures # of defects per sample

X

FOR VARIABLES: MEAN CONTROL CHART

Approach 1: When the process mean ( ) and standard deviation ( ) are known:

The standard deviation of sample means (X-bar):

nx

:means sample ofdeviation Standard

x

x

zxLCL

zxUCL

:Limit ControlLower

:Limit ControlUpper

x = Standard deviation of distribution of sample meansn = Sample size

= Process Standard deviation

x = Average of sample means

x

Z = Standard normal deviate

FOR VARIABLES: MEAN CONTROL CHART

Approach 2: Uses the sample range as a measure of process variability, when mean and standard deviation are unknown The appropriate formulas for mean control limits are:

RAxLCL

RAxUCL

2

2

:Limit ControlLower

:Limit ControlUpper

= A factor from three sigma control limits table 10.3 2A

R = Average of sample ranges

RANGE CONTROL CHART

Range control chart is used to monitor process dispersion

RDLCL

RDUCL

R

3

4

:chart )( Range

The appropriate formulas for range control limits are:

= A factor from three sigma control limits table 10.3 34 ,DD

EXAMPLE 1 A quality inspector took five samples, each with four observations,

of the length of time for glue to dry. The analyst computed the mean of each sample and then computed the grand mean. All values are in minutes. Use this information to obtain three-sigma (i.e., z = 3) control limits for means of future times. It is known from previous experience that the standard deviation of the process is .02 minute.

What is UCL, LCL?

Answer: UCL =12.14, LCL =12.08

TABLE 10.3

EXAMPLE 2

Twenty samples of n = 8 have been taken from a cleaning operation. The average sample range for the 20 samples was .016 minute, and the average mean was 3 minutes. Determine three-sigma control limits for this process.

Answer: UCL = 3.006 min LCL = 2.994 min

EXAMPLE 3

Twenty-five samples of n = 10 observations have been taken from a milling process. The average sample range was .01 centimeter. Determine upper and lower control limits for sample ranges.

Answer: UCLr = 0.0178 LCLr = 0.0022

SOLVED EXAMPLE 2 Control charts for means and ranges. Processing new

accounts at a bank is intended to average 10 minutes each. Five samples of four observations each have been taken. Use the sample data in conjunction with Table 10.2 to construct upper and lower control limits for both a mean chart and a range chart. Do the results suggest that the process is in control?

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

MEAN AND RANGE CHARTS

UCL

LCL

LCL

R-chart Reveals increase

(process variability is increasing)SamplingDistribution

CONTROL CHART FOR ATTRIBUTES

p-Chart - Control chart used to monitor the proportion of defectives in a process

c-Chart - Control chart used to monitor the number of defects per unit

Attributes generate data that are counted.

USE OF P-CHARTS

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 (e.g. 15 samples of n = 20 observations each )

USE OF P-CHARTS

p

p

p

zpLCL

zpUCLn

ppp

p

)1( : ofdeviation Standard

nsobservatio # total

defectives of # total :defectivePercent

Construct p-Charts:

EXAMPLE 4 An inspector counted the number of defective monthly

billing statements of a company telephone in each of 20 samples. Using the following information, construct a control chart that will describe 99.74 percent of the chance variation in the process when the process is in control. Each sample contained 100 statements.

PLOTTING THE CONTROL LIMITS

USE OF C-CHARTS

Use only when the number of occurrences per unit of measure can be counted; non-occurrences cannot be counted. Scratches, chips, dents, or errors per item Cracks or faults per unit of distance Breaks or Tears per unit of area Bacteria or pollutants per unit of volume Calls, complaints, failures per unit of time

USE OF C-CHARTS

c

c

c

zcLCL

zcUCL

cc

c

: of deviation Standard

samples of #

defects of # total :sampleper defects of No.

Construct c-Charts:

EXAMPLE 5 Rolls of coiled wire are monitored using a c-chart.

Eighteen rolls have been examined, and the number of defects per roll has been recorded in the following table. Is the process in control? Plot the values on a control chart using three standard deviation control limits.

PLOT OF EXAMPLE 5

MANAGERIAL CONSIDERATIONS FOR CONTROL CHARTS

At what point in the process to use control charts

What size samples to take What type of control chart to use

Variables Attributes

NONRANDOM PATTERNS IN CONTROL CHARTS

PROCESS CAPABILITY

LowerSpecification

UpperSpecification

Process variability matches specifications

LowerSpecification

UpperSpecification

Process variability well within specifications

LowerSpecification

UpperSpecification

Process variability exceeds specifications

Processmean

Lowerspecification

Upperspecification

1350 ppm 1350 ppm

1.7 ppm 1.7 ppm

+/- 3 Sigma

+/- 6 Sigma

3 Sigma and 6 Sigma Quality

TOPICS NOT IN SYLLABUS

Problems from Process Variability Supplement chapter 10

Acceptance sampling