Solutions to Problems 1,2 and 7 followed by 3,4,5,6 and 8.

Transcription

1 DSES-423 Quality Control Spring 22 Solution to Homework Assignment #2 Solutions to Problems 1,2 and 7 followed by 3,4,,6 and The cause-and-effect diagram below was created by a department of the 3M Company to analyze the variability in the adhesion of a particular product, Test Methods Plate cleaning Machine Coating weight variability Dwell time Humidity Roll down Oven temperature Oven air flow Shelf life Caliper Adhesion variability Storage conditions Adhesive formulation Concentration Linear characteristics Miscellaneous Miing method Raw Materials We have added concentration as a branch off adhesion, and miing method as a branch off concentration. Any number of causes could be added to eisting branches and sub-branches.

2 2. Following are MINITAB-produced Pareto diagrams for each week and a Pareto diagram for the three-week total. Discussion follows the graphs. Pareto Diagram for Week Defect Cum % Sand Co reb rea k Brok en Shift Drop Mis-r un O thers Pareto Diagram for Week Defect Cum % Sand C o rebreak Br oken Mis-ru n Drop Shift Crush Oth er s Pareto Diagram for Week Defect Cum % Sand Co reb rea k Brok en Drop Crush Run-out O thers

3 Pareto Diagram for the 3 Week Total Defect Cum % Sand C orebreak Broken Drop Crush Run-out Others The primary causes in order of frequency of occurrence, and hence presumably in importance are sand, corebreak, and broken. The reason for plotting both the individual weekly totals and the entire three-week total is to see whether there is consistency over time with respect to order of frequency. Here, the order of the three most frequent causes is the same for each week as well as the entire period, and the percentages are remarkably consistent from week to week as well as for the period. If the causes and/or their frequencies showed high variability, this would reveal something different about the process. For eample, suppose different types of castings were produced from week to week and the primary causes were also changing. This would suggest that different causes are associated with different casting types.

4 7. The X-bar and S charts appear below. The process appears to be in control (as before for the X-bar and R charts). Note, however, that on the X-bar the limits are slightly different, due of course to the different method of estimating the process standard deviation. Here, the estimate of the standard deviation is ˆ σ = s c = 2.73/.9213 = 2.93, 4 slightly less than the r-based estimate. Xbar/S Chart for Problem 7 (Eercise -7) 1 3.SL=14.78 Sample Mean 1 Subgroup 1 2 X= SL=.974 Sample StDev SL=6.12 S= SL=. Calculation of the limits for a.998 probability limit S-chart follows. From eqn. (31) and Table 3.1, Notes, 2 2 s χ.999 s χ [ LCLs, UCLs],, 2.93,2.93, c4 n 1 c4 n , [ ] Compare this with the Shewhart limits above. Note that although small, there is a positive lower limit, and the upper limit is greater than the Shewhart upper limit. Here, these differences are small, and there is no difference with respect to out-of-control indications. In some cases, however, the difference in the limits may be just enough to cause a difference with respect to points being within the limits on one type chart while being outside of the limits on the other.

5 4-24. The pattern in Figure (a) matches the control chart in Figure (2). The pattern in Figure (b) matches the control chart in Figure (4). The pattern in Figure (c) matches the control chart in Figure (). The pattern in Figure (d) matches the control chart in Figure (1). The pattern in Figure (e) matches the control chart in Figure (3).

6 -1. XLS : CHAP.XLS : worksheet E -1 (a) n ; 34.; R bar Chart for Bearing ID (all samples in calculations) UCL = bar CL = LCL = Sample No. R chart for Bearing ID (all samples in calculations) 12 1 UCL = R 6 CL = Sample No. LCL = The process is not in statistical control; is beyond the upper control limit for both Sample No. 12 and Sample No. 1. With these two samples ecluded from the control limit calculations: 33.6; R 4.; ˆ R / d 4. /

7 -bar Chart for Bearing ID (samples 12, 1 ecluded) UCL =36.2 -bar CL = LCL = Sample No. R chart for Bearing ID (samples 12, 1 ecluded) 12 1 UCL = R 6 CL = LCL = Sample No. (b) pˆ Pr{ LSL} Pr{ USL} Pr{ 2} Pr{ 4} ( 7.7) 1 (3.29) (a) MTB : Stat : Control Charts : Xbar/R Chart: E-2V Xbar/R Chart for Output Voltage 1 UCL=14.93 Sample Mean 1 Subgroup 1 2 Mean=1.38 LCL=.822 Sample Range 1 1 UCL=14.26 R=6.2 LCL= The process is in statistical control with no out-of-control signals, runs, trends, or cycles.

8 -2 continued (b) n 4, 1.38, R 6.2, ˆ R / d 6.2 / X 2 Actual specs are 3 V V ˆ USL LSL 3 34 PCR.48, so the process is capable. 6 ˆ 6(3.4 / 1) Also: MTB : Stat : Quality Tools : CAPA for E-2V Using transformation i = (observed voltage on unit i 3) 1, USL T = +, LSL T = Process Capability Analysis for Output Voltage USL Target LSL Mean Process Data Sample N StDev (Within) StDev (Overall). * LSL USL Within Overall Potential (Within) Capability Cp CPU CPL Cpk (c) MTB : Graph : Prob Plot for E-2V N o r m a l P r o b a b i l i t y P l o t f o r O u t u t V o l t a g e 9 9 n t e c e r P O u t p u t V o l t a g e, i = ( o b s V - 3 ) 1 2 A normal probability plot of the transformed output voltage shows the distribution is close to normal.

9 -4. (a) MTB : Stat : Control Charts : Xbar/R Chart: E-4Th Xbar/R Chart for Board Thickness.64 UCL=.6389 Sample Mean Mean=.629 LCL=.621 Subgroup Sample Range UCL=.2368 R=.92 LCL= Test Results for Xbar Chart TEST 1. One point more than 3. sigmas from center line. Test Failed at points: 22 Test Results for R Chart TEST 1. One point more than 3. sigmas from center line. Test Failed at points: 1 Assuming an assignable cause is found, remove samples 1 and 22: Revised X-bar Chart for Board Thickness.64 1 Sample 1 removed UCL=.638 Sample Mean.63 Mean= Sample 22 removed 1 LCL= Sample Number 2 2

10 -4 (a) continued Revised R Chart for Board Thickness.3 1 Sample Range.2.1 UCL=.2183 R=8.48E-4. LCL= 1 1 Sample Number 2 2 (b) ˆ R / d.92 / (c) Natural tolerance limits are: 3 ˆ.629 3(.4) [.6133,.647] (d) ˆ USL LSL.1 (.1) C P.93 6 ˆ 6(.4) Also, MTB : Stat : Quality Tools : CAPA for E-4Th Process Capability Analysis for Board Thickness USL Target LSL Mean Process Data Sample N StDev (Within).64 * StDev (Overall).622 LSL USL Within Overall Potential (Within) Capability Cp CPU CPL Cpk

11 -11. i i 1 i 1 n 6 items/subgroup; 1; S 7; m subgroups (a) S i i 1 7 i 1 i 1 2; S 1. m m UCL AS (1.) LCL AS (1.) 18.7 UCL BS 1.97(1.) 2.9 S LCL BS.3(1.).4 S i (b) natural process tolerance limits: S 1. 3 ˆ [1.27, 24.73] c.91 4 (c) ˆ USL - LSL 4. ( 4.) C P.84, so the process is not capable. 6 ˆ 6(1.8) 23 2 (d) pˆ Pr{ USL} 1 Pr{ USL} 1 1 (1.9) rework or %. 1 2 pˆ Pr{ LSL} ( 3.16).78 scrap, or.78% 1.8 Total = 2.88% +.78% = 2.98%

12 -11 continued (e) pˆ 1 1 (2.3) rework, or.68% pˆ ( 2.3).68 scrap, or.68% 1.8 Total =.68% +.68% = 1.136% Centering the process would reduce rework, but increase scrap. A cost analysis is needed to make the final decision. An alternative would be to work to improve the process by reducing variability.