Statistical Process Control

An Old Storyabridged from ‘Right First Time’

• An entrepreneur knows of a business opportunity of supplying plastic discs between diameter 39.5 mm to 40.5 mm. Each disc costs about Rs. 28 and the margin is Rs. 2

• As a matter of caution, he moulds a sample lot of 32 discs, measures their diameter, finds all are within the range, except one which is on the fringe.

• Decides that now production can start• Asking the production manager to be careful, he

produces first lot of 10,000 discs, sends to the customer with an invoice of Rs. 300,000, hoping for a profit of Rs. 20,000

An Old StoryContinued ….

• Three days later, the discs are rejected, with a note stating that they have more than 5% defectives. (i.e. too small or too large discs)

• Determined not to give up, the entrepreneur develops two gauges, puts a few spinsters to sort out discs. They find 9300 good discs.

• Finally sends 9300 discs. • This whole business of inspection costs him 3

days & Rs. 21,000 considering the cost of rejected discs & wages.

• The customer however accepts this lot and places a trial order of 100,000 discs.

An Old StoryContinued ….

• Determined not to loose money this time, he cautions the production manager to be extra careful

• A quality inspector is recruited to check the diameter of sample discs every hour and report if any defective is found.

• The moment a defective is found, the production manager swings into action, adjusting a few parameters to bring the deviation in control.

• After spending about 6 days & nearly Rs 28,00,00, the lot of 100,000 discs is sent with an invoice for Rs. 30,00,000

An Old StoryContinued ….

After 4 days,

…

…

…

…

A large parcel of 100,000 discs arrives in place of a cheque for Rs. 30,00,000

An Old StoryContinued ….

• What Went Wrong?– Error 1 : Threw away data on process capability– Error 2 : Inspection is not quality control– Error 3 : Introduction of variability in the process

• 4 Questions for SPC– Can we make it right? Process Capability– Are we making it right? Process Control– Have we made it right? Quality Assurance– Can we make it better? Process Improvement

Some Facts

• Any process, however well controlled, produces an output with some variation within.

• Most process variation follows a pattern called normal distribution

• An important property of normal distribution is that nearly 99.5% of output is within Mean ± 3 * SD

• For a process to be suitable, this variation should be smaller than specifications

An Old StoryContinued ….

• What Should have been done• Use the data of 32 discs to determine the Mean

and SD of the process– For a sufficiently large sample (at least 20), Sample

Mean = Population Mean. Sample SD = Population SD / √(sample size)

– Generally Population Mean ± 3* Population SD covers nearly 99.5% of population

• Determine if process is capable to meet specifications

• Use Process Control Charts

Types of Control Charts• Control Charts for Variables

– Mean X and R Chart

• Control Charts for Attributes– Control Chart for fraction defectives

• p Chart & np chart

– Control Chart for non-conformities• c Chart• u chart• demerit system

• Cumsum & Moving Average Charts

General Procedure for Use of Control Charts

• Determine Process Capability

• Determine Sample Size

• Determine Center-Line and Upper and Lower Control Limits

• Take Periodic Sample and Plot the results on the chart

Control Charts for Variables

• Mean X & R chart– How to plot Mean X Chart– How to plot R chart– Interpretation

Mean X Chart R Chart

RAxULC

xCL

RAxLCL

*

*

2

2

RD

R

RD

*UCL

CL

*LCL

4

3

Control Charts for Variables

• Mean X & R chart– How to plot Mean X Chart– How to plot R chart– Interpretation

Mean X Chart R Chart

SDdxULC

xCL

SDdxLCL

*

*

2

2

SDD

SDd

SDD

*2UCL

*CL

*1LCL

2

Table for Mean X & R Charts

n A2 D3 D4 d2 D1 D2

2 1.880 0.000 3.267 1.128 0 3.69

3 1.023 0.000 2.574 1.693 0 4.36

4 0.729 0.000 2.282 2.059 0 4.7

5 0.577 0.000 2.115 2.326 0 4.92

6 0.483 0.000 2.004 2.534 0 5.08

7 0.419 0.076 1.924 2.704 0.2 5.2

8 0.373 0.136 1.864 2.847 0.39 5.31

9 0.337 0.184 1.816 2.970 0.55 5.39

10 0.308 0.223 1.777 3.078 0.69 5.47

An Example• From Process Capability Study, Sample Size 4

– Mean X = 40.1– Mean R = 0.25

An Example• For Range Chart

– LCL = 0– CL = 0.25– UCL = 0.57

• For Mean X Chart– LCL = 40.1-0.73*0.25 =39.92– CL = 40.1– UCL = 40+0.73*0.25 = 40.28

• Population is likely to be between– LCL = 40.1-0.73*0.25*2 =39.735– UCL = 40.1+0.73*0.25*2 = 40.465

Interpretation of charts

• Cyclic Pattern

• Mixture- Two or more overlapping distributions

• A Shift (5 or more consecutive readings on one side of mean)

• Trend

• Stratification

Type I & Type II errors

• Type I error – False alarm when process is in control– Risk of committing this error is denoted by α

• Type II error– No alarm when process has gone out of

control – Risk of committing this error is denoted by β

• Estimate SD of process from average range• Calculate PCR as

• If PCR > 1, process is capable to meet requirements.

Process Capability

SD

LSLUSL

*6

Acceptance Sampling

• A method to evaluate quality post-production. Used extensively to check in-coming quality

• From a lot of N pieces, a sample of n pieces is drawn & inspected. The lot is rejected if more than a defectives are found in the lot.

• When using this method, there are 2 risks, viz. Producers risk – α and Consumer’s risk – β

• For a given sampling Plan, the OC curve shows the risks