Determine the central line and control limit Xbar
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. X = The average of the means of the samples i = Sample number g = Total number of samples g åXi X= j =1 g
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. R = Average of the measurement differences R for all samples Ri= Difference between the highest and lowest values in sample i g = Total number of samples g å Ri R= i =1 g
Determine the central line and control limit UCLx = x= + 3 x LCLx = x= - 3 x = UCLx = x + A 2 R LCL = x= - A R x 2 UCLR = = x + 3 R LCLR = = x - 3 R UCLR = D 4 R LCLR = D 3 R 3
Note: All factors are based on the normal distribution. Source: E. L. Grant, Statistical Quality Control, 6 th ed. pp. 57– 59 (Copyright © 1998 by The Mc. Graw-Hill Companies, Inc. 1988). Reprinted by permission of Mc. Graw-Hill, Inc. 4
Control Charts for Variables West Allis Industries -Continuous process -Produce 400 pieces per day -Quality characteristic is the diameter of bolts Krajewaksi, operation management 7 th, 2004 5
Control Charts for Variables Special Metal Screw Subgroup Number 1 1 0. 5014 2 0. 5021 3 0. 5018 4 0. 5008 5 0. 5041 Example 5. 1 Sample 2 3 0. 5022 0. 5009 0. 5041 0. 5024 0. 5026 0. 5035 0. 5034 0. 5024 0. 5056 0. 5034 4 0. 5027 0. 5020 0. 5023 0. 5015 0. 5039 Krajewaksi, operation management 7 th, 2004 R _ x 6
Control Charts for Variables Special Metal Screw Sample Number 1 2 3 4 5 Example 5. 1 1 0. 5014 0. 5021 0. 5018 0. 5008 0. 5041 Sample 2 3 4 0. 5022 0. 5009 0. 5027 0. 5041 0. 5024 0. 5020 0. 5026 0. 5035 0. 5023 0. 5027 – 0. 5009 0. 5034 0. 5024 0. 5015 0. 5056 0. 5034 0. 5039 Krajewaksi, operation management 7 th, 2004 R 0. 0018 _ x 0. 5018 = 0. 0018 7
Control Charts for Variables Special Metal Screw Sample Number 1 2 3 4 5 Example 5. 1 1 0. 5014 0. 5021 0. 5018 0. 5008 0. 5041 Sample 2 3 4 0. 5022 0. 5009 0. 5027 0. 5041 0. 5024 0. 5020 0. 5026 0. 5035 0. 5023 0. 5027 – 0. 5009 0. 5034 0. 5024 0. 5015 0. 5056 0. 5034 0. 5039 Krajewaksi, operation management 7 th, 2004 R 0. 0018 _ x 0. 5018 = 0. 0018 8
Control Charts for Variables Special Metal Screw Sample Number 1 2 3 4 5 Example 5. 1 1 0. 5014 0. 5021 0. 5018 0. 5008 0. 5041 Sample _ 2 3 4 R x 0. 5022 0. 5009 0. 5027 0. 0018 0. 5041 0. 5024 0. 5020 0. 5026 0. 5035 0. 5023 0. 5027 – 0. 5009 = 0. 0018 0. 5034 0. 5024 0. 5015 (0. 5014 0. 5056 0. 5034+ 0. 5022 0. 5039 + 0. 5009 + 0. 5027)/4 = 0. 5018 Krajewaksi, operation management 7 th, 2004 9
Control Charts for Variables Special Metal Screw Sample Number 1 2 3 4 5 Example 5. 1 1 0. 5014 0. 5021 0. 5018 0. 5008 0. 5041 Sample 2 3 0. 5022 0. 5009 0. 5041 0. 5024 0. 5026 0. 5035 0. 5034 0. 5024 0. 5056 0. 5034 4 0. 5027 0. 5020 0. 5023 0. 5015 0. 5047 R= Krajewaksi, operation management 7 th, 2004 R 0. 0018 0. 0021 0. 0017 0. 0026 0. 0022 0. 0021 = x= _ x 0. 5018 0. 5027 0. 5026 0. 5020 0. 5045 0. 5027 10
Control Charts for Variables Control Charts – Special Metal Screw R-Charts R = 0. 0021 UCLR = D 4 R LCLR = D 3 R Example 5. 1 Krajewaksi, operation management 7 th, 2004 11
Control Charts Table 5. 1 Control Chart Factors for Variables Size of Sample (n ) Factor for UCL and LCL for x-Charts (A 2 ) Factor for LCL for R-Charts (D 3 ) Factor UCL for R-Charts (D 4 ) 2 3 4 5 6 7 8 9 10 1. 880 1. 023 0. 729 0. 577 0. 483 0. 419 0. 373 0. 337 0. 308 0 0 0. 076 0. 136 0. 184 0. 223 3. 267 2. 575 2. 282 2. 115 2. 004 1. 924 1. 864 1. 816 1. 777 Example 5. 1 Krajewaksi, operation management 7 th, 2004 12
Control Charts for Variables Control Charts—Special Metal Screw R-Charts R = 0. 0021 UCLR = D 4 R LCLR = D 3 R Example 5. 1 Krajewaksi, operation management 7 th, 2004 D 4 = 2. 282 D 3 = 0 13
Control Charts for Variables Control Charts—Special Metal Screw R-Charts R = 0. 0021 UCLR = D 4 R LCLR = D 3 R D 4 = 2. 282 D 3 = 0 UCLR = 2. 282 (0. 0021) = 0. 00479 in. Example 5. 1 Krajewaksi, operation management 7 th, 2004 14
Control Charts for Variables Control Charts—Special Metal Screw R-Charts R = 0. 0021 UCLR = D 4 R LCLR = D 3 R D 4 = 2. 282 D 3 = 0 UCLR = 2. 282 (0. 0021) = 0. 00479 in. LCLR = 0 (0. 0021) = 0 in. Example 5. 1 Krajewaksi, operation management 7 th, 2004 15
Control Charts for Variables Control Charts—Special Metal Screw R-Charts R = 0. 0021 UCLR = D 4 R LCLR = D 3 R D 4 = 2. 282 D 3 = 0 UCLR = 2. 282 (0. 0021) = 0. 00479 in. LCLR = 0 (0. 0021) = 0 in. Example 5. 1 Krajewaksi, operation management 7 th, 2004 16
Range Chart Special Metal Screw Krajewaksi, operation management 7 th, 2004 17
Control Charts for Variables Control Charts—Special Metal Screw X-Charts R = 0. 0021 x= = 0. 5027 = UCLx = x + A 2 R LCL = x= - A R x Example 5. 1 2 Krajewaksi, operation management 7 th, 2004 18
Control Charts Table 5. 1 Control Chart Factors for Variables Factor for UCL Factor for Factor Size of Charts—Special and LCL for Control Metal Screw UCL for Sample x-Charts R-Charts (n ) (A 2) R = 0. 0021 (D 3) (D 4 ) X-Charts = x 2 1. 880 = 0. 5027 0 3. 267 3 1. 023 0 2. 575 4 x 0. 729 0 2. 282 2 5 0. 577 0 2. 115 2 6 x 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 = UCL = x + A R LCL = x= - A R Example 5. 1 Krajewaksi, operation management 7 th, 2004 19
Control Charts for Variables Control Charts—Special Metal Screw x-Charts R = 0. 0021 x= = 0. 5027 A 2 = 0. 729 = UCLx = x + A 2 R LCL = x= - A R x Example 5. 1 2 Krajewaksi, operation management 7 th, 2004 20
Control Charts for Variables Control Charts—Special Metal Screw x-Charts R = 0. 0021 x= = 0. 5027 A 2 = 0. 729 = UCLx = x + A 2 R LCL = x= - A R x 2 UCLx = 0. 5027 + 0. 729 (0. 0021) = 0. 5042 in. Example 5. 1 Krajewaksi, operation management 7 th, 2004 21
Control Charts for Variables Control Charts—Special Metal Screw x-Charts R = 0. 0021 x= = 0. 5027 A 2 = 0. 729 = UCLx = x + A 2 R LCL = x= - A R x 2 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 7 th, 2004 22
Control chart (Ex 5 -2) 23
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