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HOW TO DEVELOP AND USE X BAR AND R CONTROL CHARTS?

AN EXAMPLE

Example: Control Charts for Variable DataSample 1 2 3 4 5 6 7 8 9 10 Slip Ring Diameter (cm) 1 2 3 4 5 5.02 5.01 4.94 4.99 4.96 5.01 5.03 5.07 4.95 4.96 4.99 5.00 4.93 4.92 4.99 5.03 4.91 5.01 4.98 4.89 4.95 4.92 5.03 5.05 5.01 4.97 5.06 5.06 4.96 5.03 5.05 5.01 5.10 4.96 4.99 5.09 5.10 5.00 4.99 5.08 5.14 5.10 4.99 5.08 5.09 5.01 4.98 5.08 5.07 4.99 X 4.98 5.00 4.97 4.96 4.99 5.01 5.02 5.05 5.08 5.03 50.09 R 0.08 0.12 0.08 0.14 0.13 0.10 0.14 0.11 0.15 0.10 1.15

DETERMINE CENTERLINE The centerline should be the population mean, Since it is unknown, we use X Double bar, or the grand average of the subgroup averages.

X =

m

X m

i

i=1

DETERMINE CONTROL LIMITS

Xbar chart The normal curve displays the distribution of the sample averages. A control chart is a time-dependent pictorial representation of a normal curve. Processes that are considered under control will have 99.73% of their graphed averages fall within 6.

UCL & LCL calculation

UCL = X + 3 LCL = X 3 = standard deviation

DETERMINING THE VALUES OF THE CONTROL LIMITS USING RANGE

R

=

m

R m

i=1

i

UCL

= X + A

2

R

LCL

= X A

2

R

3-Sigma Control Chart FactorsSample size n 2 3 4 5 6 7 8 X-chart A2 1.88 1.02 0.73 0.58 0.48 0.42 0.37 R-chart D3 0 0 0 0 0 0.08 0.14 D4 3.27 2.57 2.28 2.11 2.00 1.92 1.86

DETERMINE CONTROL LIMITS

R chart The range chart shows the spread or dispersion of the individual samples within the subgroup. If the product shows a wide spread, then the individuals within the subgroup are not similar to each other. Equal averages can be deceiving.

Calculated similar to x-bar charts; Use D3 and D4

CalculationFrom Table above: Sigma X-bar = 50.09 Sigma R = 1.15 m = 10 Thus; X-Double bar = 50.09/10 = 5.009 cm R-bar = 1.15/10 = 0.115 cmNote: The control limits are only preliminary with 10 samples. It is desirable to have at least 25 samples.

CONTROL LIMITS UCLx-bar = X-D bar + A2 R-bar = 5.009 + (0.577)(0.115) = 5.075 cm LCLx-bar = X-D bar - A2 R-bar = 5.009 (0.577)(0.115) = 4.943 cm UCLR = D4R-bar = (2.114)(0.115) = 0.243 cm LCLR = D3R-bar = (0)(0.115) = 0 cm n=5 For A2, D3, D4: see Table

X-bar Chart5.10 5.08 5.06UCL

X bar

5.04 5.02 5.00 4.98 4.96 4.94 0 1 2 3 4 5 6 7 8 9 10 11LCL CL

Subgroup

R Chart0.25 0.20UCL

Range

0.15 0.10 0.05

CL

LCL

0.00 0 1 2 3 4 5 6 7 8 9 10 11

Subgroup

WHAT DO YOU DO NOW ? Revise Control Limits ? The concept of Trial Control Limits

HOW TO REVISE CONTROL LIMITS ? One or two points outside ? Quite a few points outside ?

THREE CATEGORIES OF VARIATION Within-piece variation One portion of surface is rougher than another portion.

Piece-to-piece variation Variation among pieces produced at the same time.

Time-to-time variation Service given early would be different from that given later in the day.

SOURCES OF VARIATION Equipment Tool wear, machine vibration,

Material Raw material quality

Operator Operator performs- physical & emotional

CONTROL CHART VIEWPOINT Variation due to Common or chance causes Assignable causes

Control chart may be used to discover assignable causes

TYPICAL OUT-OF-CONTROL PATTERNS Point outside control limits Sudden shift in process average Cycles Trends Hugging the center line Hugging the control limits Instability20

Shift in Process Average

Identifying Potential Shifts

Cycles

8-3 Introduction to Control Charts8-3.4 Analysis of Patterns on Control Charts

8-3 Introduction to Control Charts8-3.4 Analysis of Patterns on Control Charts

Trend

Process in Control When a process is in control, there occurs a natural pattern of variation. Natural pattern has: About 34% of the plotted point in an imaginary band between 1 on both side CL. About 13.5% in an imaginary band between 1 and 2 on each side of CL. About 2.5% of the plotted point in an imaginary band between 2 and 3 on both side CL.

The Normal Distribution = Standard deviation Mean -3 -2 -1 +1 +2 +3 68.26% 95.44% USL 99.74%

LSL

-3

CL

+3

Control Chart Design Issues Basis for sampling Sample size Frequency of sampling Location of control limits

29

Setting Control Limits

Pre-ControlLTL Red Zone UTL Red Zone

Green Zone

nominal value

Yellow Zones

31