MIE364H1S Methods of Quality Control and Improvement

Course Instructor: Prof. V. Makis Handout #1

A procedure of setting up control charts using historical data (should be recent history). Consider ),( XR charts. Data: mXX ,...,1 and mRR ,...,1 .

1. R chart with 3 sigma limits: Calculate R , RCLnDRUCLnDRLCL === ),(),( 43 . If a point plots outside the control limits:

a. A cause was found and was removed or disappeared delete the point b. A cause was found and cannot be removed keep the point c. No cause was found but not much information is available delete the

point d. No cause was found and we are quite sure that there was none keep the

point After deleting the points, recalculate newnewnew UCLandLCLR ,, and plot the remaining points again. If you still have points outside the control limits after several iterations (say, 2), there is a major problem with the process, causes should be found using other SPC tools and removed. If all the points are between the limits, or you decide to keep the

points outside (b. or d.), find)(

ˆ2 nd

RFINAL=σ and construct X chart using all historical data

(unless n = 1). If you delete points on X chart, do not changeσ̂ . The rules a. to d. apply also to X chart. 0ˆ FINALXµ = . Use the charts for future production and revise regularly. Data in the table are the measurements of four consecutive units on an assembly line taken every 30 minutes until 20 subgroups are obtained.

Subgroup x1 x2 x3 x4 xbar R s

1 72 84 79 49 71 35 15.47042 56 87 33 42 54.5 54 23.643183 55 73 22 60 52.5 51 21.702534 44 80 54 74 63 36 16.85235 97 26 48 58 57.25 71 29.680246 83 89 91 62 81.25 29 13.275927 47 66 53 58 56 19 8.0415598 88 50 84 69 72.75 38 17.231279 57 47 41 46 47.75 16 6.70199

10 13 10 30 32 21.25 22 11.3541511 26 39 52 48 41.25 26 11.5289512 46 27 63 34 42.54 36 15.758613 49 62 78 87 69 38 16.8720714 71 63 82 55 67.75 27 11.5289515 71 58 69 70 67 13 6.05530116 67 69 70 94 75 27 12.7279217 55 63 72 49 59.75 23 9.97914518 49 51 55 76 57.75 27 12.4197419 72 80 61 59 68 21 9.83192120 61 74 62 57 63.5 17 7.325754

Sample

Sam

ple

Ran

ge

191715131197531

80

70

60

50

40

30

20

10

0

_R=31.3

UCL=71.40

LCL=0

R Chart of x1, ..., x4

Sample

Sam

ple

Mea

n

191715131197531

90

80

70

60

50

40

30

20

__X=59.44

UCL=82.24

LCL=36.64

1

Xbar Chart of x1, ..., x4

Sample

Sam

ple

Mea

n

191715131197531

90

80

70

60

50

40

__X=61.45

UCL=84.25

LCL=38.64

Xbar Chart of x1, ..., x4

Sample

Sam

ple

StD

ev

191715131197531

35

30

25

20

15

10

5

0

_S=13.90

UCL=31.50

LCL=0

S Chart of x1, ..., x4

Sample

Sam

ple

Mea

n

191715131197531

90

80

70

60

50

40

30

20

__X=59.44

UCL=82.07

LCL=36.81

1

Xbar Chart of x1, ..., x4

Sample

Sam

ple

Mea

n

191715131197531

90

80

70

60

50

40

__X=61.45

UCL=84.08

LCL=38.82

Xbar Chart of x1, ..., x4

(XBAR, S) Charts with Variable Sample (Group) Size Data Display

Group Data Group Data Group Data Group Data Group Data 1 72 5 97 9 57 13 49 17 94 1 84 5 26 9 47 13 62 17 55 1 79 5 48 9 41 13 78 17 63 2 49 5 58 10 46 13 87 17 72 2 56 5 83 10 13 14 71 17 49 2 87 5 89 10 10 14 63 18 49 2 33 6 91 10 30 14 82 18 51 2 42 6 62 10 32 15 55 18 55 3 55 6 47 11 26 15 71 18 76 3 73 6 66 11 39 15 58 19 72 3 22 7 53 11 52 15 69 19 80 3 60 7 58 11 48 15 70 20 61 4 44 8 88 11 46 16 67 20 59 4 80 8 50 12 27 16 69 20 61 4 54 8 84 12 63 16 70 20 74 4 74 8 69 12 34 20 62 20 57

Sample

Sam

ple

StD

ev

191715131197531

40

30

20

10

0

_S=15.13

UCL=29.79

LCL=0.46

S Chart of C2

Tests performed with unequal sample sizes

Sample

Sam

ple

Mea

n

191715131197531

100

90

80

70

60

50

40

30

20

__X=59.44

UCL=78.91

LCL=39.97

1

Xbar Chart of C2

Tests performed with unequal sample sizes

Use the estimate of σ from the S chart, delete 10X , calculate new control limits for the X chart and plot again. No change in S chart.