Determine the central line and control limit X-bar Chart A chart that tracks the changes in the means of the samples by plotting the means that were taken from a process. g X j i å = 1 Total number of samples Sample number The average of the means of the samples = g i X
Determine the central line and control limit R Chart A chart that tracks the change in the variability by plotting the range within each sample. The range is the difference between the lowest and highest values in that sample. g R i å = 1 Total number of samples Difference between the highest and lowest values in sample i Average of the measurement differences R for all samples = g Ri R
Determine the central line and control limit UCLx = x + 3x LCLx = x - 3x = UCLR = x + 3R LCLR = x - 3R = UCLx = x + A2R LCLx = x - A2R = UCLR = D4R LCLR = D3R
Note: All factors are based on the normal distribution. Source: E. L. Grant, Statistical Quality Control, 6th ed. pp. 57–59 (Copyright © 1998 by The McGraw-Hill Companies, Inc. 1988). Reprinted by permission of McGraw-Hill, Inc.
Krajewaksi, operation management 7th, 2004 Control Charts for Variables West Allis Industries The next series of slides presents Example 5.1. The series builds in steps to the conclusion of the Example showing the development of key equations along the way. The first example is of variables charting using x-bar and R-charts. Continuous process Produce 400 pieces per day Quality characteristic is the diameter of bolts Krajewaksi, operation management 7th, 2004 1
Krajewaksi, operation management 7th, 2004 Control Charts for Variables Subgroup Sample Number 1 2 3 4 R x 1 0.5014 0.5022 0.5009 0.5027 2 0.5021 0.5041 0.5024 0.5020 3 0.5018 0.5026 0.5035 0.5023 4 0.5008 0.5034 0.5024 0.5015 5 0.5041 0.5056 0.5034 0.5039 Special Metal Screw _ This slide and the next six show in a step-by-step fashion how the raw data is used to derive ranges and means for each observation. Example 5.1 Krajewaksi, operation management 7th, 2004 4
Krajewaksi, operation management 7th, 2004 Control Charts for Variables Sample Sample Number 1 2 3 4 R x 1 0.5014 0.5022 0.5009 0.5027 0.0018 0.5018 2 0.5021 0.5041 0.5024 0.5020 3 0.5018 0.5026 0.5035 0.5023 4 0.5008 0.5034 0.5024 0.5015 5 0.5041 0.5056 0.5034 0.5039 Special Metal Screw _ 0.5027 – 0.5009 = 0.0018 Example 5.1 Krajewaksi, operation management 7th, 2004 9
Krajewaksi, operation management 7th, 2004 Control Charts for Variables Sample Sample Number 1 2 3 4 R x 1 0.5014 0.5022 0.5009 0.5027 0.0018 0.5018 2 0.5021 0.5041 0.5024 0.5020 3 0.5018 0.5026 0.5035 0.5023 4 0.5008 0.5034 0.5024 0.5015 5 0.5041 0.5056 0.5034 0.5039 Special Metal Screw _ 0.5027 – 0.5009 = 0.0018 Example 5.1 Krajewaksi, operation management 7th, 2004 9
Krajewaksi, operation management 7th, 2004 Control Charts for Variables Sample Sample Number 1 2 3 4 R x 1 0.5014 0.5022 0.5009 0.5027 0.0018 0.5018 2 0.5021 0.5041 0.5024 0.5020 3 0.5018 0.5026 0.5035 0.5023 4 0.5008 0.5034 0.5024 0.5015 5 0.5041 0.5056 0.5034 0.5039 Special Metal Screw _ 0.5027 – 0.5009 = 0.0018 (0.5014 + 0.5022 + 0.5009 + 0.5027)/4 = 0.5018 Example 5.1 Krajewaksi, operation management 7th, 2004 9
Krajewaksi, operation management 7th, 2004 Control Charts for Variables Special Metal Screw Sample Sample Number 1 2 3 4 R x 1 0.5014 0.5022 0.5009 0.5027 0.0018 0.5018 2 0.5021 0.5041 0.5024 0.5020 0.0021 0.5027 3 0.5018 0.5026 0.5035 0.5023 0.0017 0.5026 4 0.5008 0.5034 0.5024 0.5015 0.0026 0.5020 5 0.5041 0.5056 0.5034 0.5047 0.0022 0.5045 R = 0.0021 x = 0.5027 _ To complete the exercise, the R-bar and grand mean values are calculated and added at the bottom of the appropriate columns. = Example 5.1 Krajewaksi, operation management 7th, 2004 11
Krajewaksi, operation management 7th, 2004 Control Charts for Variables Control Charts – Special Metal Screw R-Charts R = 0.0021 UCLR = D4R LCLR = D3R The data set is replaced by a worksheet showing the given values for R-bar and the appropriate equations for the R-chart. The following segment will show x-double bar. Example 5.1 Krajewaksi, operation management 7th, 2004 13
Krajewaksi, operation management 7th, 2004 Control Charts for Variables Table 5.1 Control Chart Factors Factor for UCL Factor for Factor Size of and LCL for LCL for UCL for Sample x-Charts R-Charts R-Charts (n) (A2) (D3) (D4) 2 1.880 0 3.267 3 1.023 0 2.575 4 0.729 0 2.282 5 0.577 0 2.115 6 0.483 0 2.004 7 0.419 0.076 1.924 8 0.373 0.136 1.864 9 0.337 0.184 1.816 10 0.308 0.223 1.777 The data set is replaced by a worksheet showing the given values for R-bar and the appropriate equations for the R-chart. The following segment will show x-double bar. Example 5.1 Krajewaksi, operation management 7th, 2004 13
Krajewaksi, operation management 7th, 2004 Control Charts for Variables Control Charts—Special Metal Screw R-Charts R = 0.0021 D4 = 2.282 D3 = 0 UCLR = D4R LCLR = D3R Removing the Table, the values are now shown on the worksheet. Example 5.1 Krajewaksi, operation management 7th, 2004 16
Krajewaksi, operation management 7th, 2004 Control Charts for Variables Control Charts—Special Metal Screw R-Charts R = 0.0021 D4 = 2.282 D3 = 0 UCLR = D4R LCLR = D3R In this slide the values are substituted into the UCL equation and the resulting value is shown. UCLR = 2.282 (0.0021) = 0.00479 in. Example 5.1 Krajewaksi, operation management 7th, 2004 17
Krajewaksi, operation management 7th, 2004 Control Charts for Variables Control Charts—Special Metal Screw R-Charts R = 0.0021 D4 = 2.282 D3 = 0 UCLR = D4R LCLR = D3R In a similar fashion, this slide shows the substitution and calculation for the LCL. The arrows will disappear after a few seconds leaving only the calculations. UCLR = 2.282 (0.0021) = 0.00479 in. LCLR = 0 (0.0021) = 0 in. Example 5.1 Krajewaksi, operation management 7th, 2004 18
Krajewaksi, operation management 7th, 2004 Control Charts for Variables Control Charts—Special Metal Screw R-Charts R = 0.0021 D4 = 2.282 D3 = 0 UCLR = D4R LCLR = D3R This slide can be used to review and discuss the calculations. UCLR = 2.282 (0.0021) = 0.00479 in. LCLR = 0 (0.0021) = 0 in. Example 5.1 Krajewaksi, operation management 7th, 2004 19
Range Chart - Special Metal Screw This slide shows the blank control chart for the R-chart with the mean and control limit values already plotted. This is a screen shot from the OM Explorer software. The process Range is comfortably in control. (The picky person might suggest there is not enough variability in the process, but with so few observations it is difficult to draw such conclusions.) Krajewaksi, operation management 7th, 2004 20
Krajewaksi, operation management 7th, 2004 Control Charts for Variables Control Charts—Special Metal Screw X-Charts R = 0.0021 x = 0.5027 = UCLx = x + A2R LCLx = x - A2R = We return to the calculations, this time for the x-bar control limits. Example 5.1 Krajewaksi, operation management 7th, 2004 22
Krajewaksi, operation management 7th, 2004 Control Charts for Variables Table 5.1 Control Chart Factors Factor for UCL Factor for Factor Size of and LCL for LCL for UCL for Sample x-Charts R-Charts R-Charts (n) (A2) (D3) (D4) 2 1.880 0 3.267 3 1.023 0 2.575 4 0.729 0 2.282 5 0.577 0 2.115 6 0.483 0 2.004 7 0.419 0.076 1.924 8 0.373 0.136 1.864 9 0.337 0.184 1.816 10 0.308 0.223 1.777 Control Charts—Special Metal Screw X-Charts R = 0.0021 x = 0.5027 = UCLx = x + A2R LCLx = x - A2R = We return to the calculations, this time for the x-bar control limits. Example 5.1 Krajewaksi, operation management 7th, 2004 22
Krajewaksi, operation management 7th, 2004 Control Charts for Variables Control Charts—Special Metal Screw x-Charts R = 0.0021 A2 = 0.729 x = 0.5027 = UCLx = x + A2R LCLx = x - A2R = The A2 value is now shown with the other required values for the calculations. Example 5.1 Krajewaksi, operation management 7th, 2004 24
Krajewaksi, operation management 7th, 2004 Control Charts for Variables Control Charts—Special Metal Screw x-Charts R = 0.0021 A2 = 0.729 x = 0.5027 = UCLx = x + A2R LCLx = x - A2R = Here the required values are substituted in the UCL equation and the value is calculated. The arrows on this slide will disappear after a few seconds and the second calculation will appear. UCLx = 0.5027 + 0.729 (0.0021) = 0.5042 in. Example 5.1 Krajewaksi, operation management 7th, 2004 25
Krajewaksi, operation management 7th, 2004 Control Charts for Variables Control Charts—Special Metal Screw x-Charts R = 0.0021 A2 = 0.729 x = 0.5027 = UCLx = x + A2R LCLx = x - A2R = This slide completes the substitution and calculation for the LCL. UCLx = 0.5027 + 0.729 (0.0021) = 0.5042 in. LCLx = 0.5027 – 0.729 (0.0021) = 0.5012 in. Example 5.1 Krajewaksi, operation management 7th, 2004 26
Control chart (Ex 5-2)