Xbar/R Chart for x1-x3

Transcription

1 Chapter 6 Selected roblem Solutios Sectio a) X-bar ad Rage - Iitial Study Chartig roblem 6- X-bar Rage UCL:. sigma 7.4 UCL:. sigma 5.79 Ceterlie 5.9 Ceterlie.5 LCL: -. sigma.79 LCL: -. sigma Test Results: X-bar Oe poit more tha. sigmas from ceter lie. Test Failed at poits: Test Results for R Chart:Oe poit more tha. sigmas from ceter lie. Test Failed at poits: 9 Xbar/R Chart for x-x Sample Mea 5 UCL7.4 Mea5.9 LCL.79 Subgroup Sample Rage UCL5.79 R.5 LCL. b. Removed poits 4, 6, 7,,, 5, 6, 9, ad ad revised the cotrol limits The cotrol limits are ot as wide after beig revised X-bar UCL7.96, CL5.78 LCL.6 ad R UCL 5.45, R-bar.8, LCL. The X-bar cotrol moved dow. Sample Mea Subgroup Xbar/R Chart for x-x Revised Cotrol Limits 5 UCL7.95 Mea5.78 LCL.6 Sample Rage UCL5.45 R.8 LCL

2 c) X-bar ad StDev - Iitial Study Chartig roblem 6- X-bar StDev UCL:. sigma 7.4 UCL:. sigma.5 Ceterlie 5.9 Ceterlie.88 LCL: -. sigma.77 LCL: -. sigma Test Results: X-bar Oe poit more tha. sigmas from ceter lie. Test Failed at poits: Test Results for S Chart:Oe poit more tha. sigmas from ceter lie. Test Failed at poits: 9 Xbar/S Chart for x-x Sample Mea 5 UCL7.4 Mea5.9 LCL.77 Subgroup Sample StDev UCL.5 S.88 LCL Removed poits 4, 6, 7,,, 5, 6, 9, ad ad revised the cotrol limits The cotrol limits are ot as wide after beig revised X-bar UCL7.95, CL5.78 LCL.6 ad S UCL.848, S-bar.9, LCL. The X-bar cotrol moved dow. Xbar/S Chart for x-x Revised Cotrol Limits Sample Mea Subgroup 5 UCL7.95 Mea5.78 LCL.6 Sample StDev UCL.848 S.9 LCL

3 785 x r 5 x chart a) UCL CL A r CL LCL CL A r ) ). R chart b) UCL D r 4 CL 4.86 LCL D r ˆ ˆ µ x r d.54.86) ) a) X-bar ad Rage - Iitial Study Chartig roblem 6-7 X-bar Rage UCL:. sigma.6476 UCL:. sigma.9567e- Ceterlie.6976 Ceterlie 9.4E-4 LCL: -. sigma.6446 LCL: -. sigma out of limits 5 out of limits Chart: Both Normalize: No 5 subgroups, size 5 subgroups excluded Estimated Xbar/R Chart for x Sample Mea Subgroup UCL.647 Mea.694 LCL.64 Sample Rage.... UCL.954 R.94 LCL

4 b) Xbar/S Chart for x Sample Mea Subgroup UCL.646 Mea.694 LCL.64 Sample StDev..5. UCL7.66E-4 S.67E-4 LCL c) There are several poits out of cotrol. The cotrol limits eed to be revised. The poits are, 5, 4,7,,, ad ; or outside the cotrol limits of the R chart: 6 ad 5 Sectio a) Idividuals ad MR) - Iitial Study Chartig roblem 5-8 Id.x MR) UCL:. sigma UCL:. sigma 9.68 Ceterlie 5.5 Ceterlie.9477 LCL: -. sigma 45. LCL: -. sigma out of limits out of limits Chart: Both Normalize: No subgroups, size subgroups excluded Estimated process mea 5.5 process sigma.69 mea MR.9477

5 There are o poits beyod the cotrol limits. The process appears to be i cotrol. I ad MR Chart for hardess Idividual Value UCL6.89 Mea5.5 LCL45. Subgroup Movig Rage 5 UCL9.6 R.947 LCL b) ˆ µ x 5.5 ˆ mr d Sectio a) Assumig a ormal distributio with µ.4.5 ad r d 6. X LSL) LSL µ ˆ ˆ ).45)

6 X USL ˆ µ > USL) > ˆ >.48 >.).) Therefore, the proportio ocoformig is give by XLSL) X>USL) b) USL LSL CR. 6 ˆ) 6.48) CR K USL x x LSL mi, ˆ ˆ mi,.48).48) mi.4,.5.4 [ ] Sice CR exceeds uity, the atural tolerace limits lie iside the specificatio limits ad very few defective uits will be produced. CR K CR the process appears to be cetered. s ˆ c a) Assumig a ormal distributio with µ ad X LSL) LSL ˆ µ ˆ ).6 4

7 b X USL ˆ µ > USL) > ˆ 7 > 4.75 >.8).8) robability of producig a part outside the specificatio limits is USL LSL 7 CR. 6 ˆ) 64.75) CR K USL x x LSL mi, ˆ ˆ 7 7 mi, 4.75) 4.75) mi.6,.9.6 [ ] Sice CR exceeds uity, the atural tolerace limits lie iside the specificatio limits ad very few defective uits will be produced. The estimated proportio ocoformig is give by XLSL) X>USL) Assumig a ormal distributio with µ 5.6 ad 7.7 USL LSL CR 6 ˆ) CR K USL x x LSL mi, ˆ ˆ mi, 7.7) 7.7) mi ) [.474,.5] Sice the process capability ratios are less tha uity, the process capability appears to be poor.

8 Sectio U Chart for defects 5 Sample Cout 4.SL.8 U.94 -.SL Sample Number 5 Samples 5 ad 4 have out-of-cotrol poits. The limits eed to be revised. b) 4 U Chart for defects_ Sample Cout UCL.46 U.79 LCL Sample Number The cotrol limits are calculated without the out-of-cotrol poits. There are o poits out of cotrol for the revised limits C Chart for defects Sample Cout.SL9.6 C SL Sample Number 5

9 Sectio 6-9 There are two poits beyod the cotrol limits. They are samples 5 ad 4. The U chart ad the C chart both detected out-of-cotrol poits at samples 5 ad 4. ˆ a) x X 7.44) , µ X µ ˆ. x 4.7.7).7) 4.7) The probability that this shift will be detected o the ext sample is p ARL p. b) 9. 8 R 6.75 ˆ.8 ˆ d.59 x X 5.6) 6-. a) X µ x 4.9.6).6) 4.9) , µ The probability that this shift will be detected o the ext sample is p ARL p.548 b) 8. 5 Sectio a) ˆ.695 b) The process appears to be out of cotrol at the specified target level.

10 CUSUM Chart for diameter Upper CUSUM Cumulative Sum Lower CUSUM Subgroup Number E- Supplemetal 6-4. a) X-bar ad Rage - Iitial Study X-bar Rage UCL:. sigma 64.8 UCL:. sigma.4597 Ceterlie 64 Ceterlie.764 LCL: -. sigma 6.98 LCL: -. sigma out of limits out of limits Chart: Both Normalize: No Estimated process mea 64 process sigma.494 mea Rage.764 Xbar/R Chart for diameter Sample Mea Subgroup UCL64. Mea64. LCL6.98 Sample Rage UCL.448 R.74 LCL

11 The process is i cotrol. R.764 b) µ x 64 ˆ. 4 d.69 USL LSL c) CR. 64 6ˆ 6.4) The process does ot meet the miimum capability level of CR.. d) USL x x LSL CR k mi, mi, ˆ ˆ.4).4) mi [.64,.64]. 64 e) I order to make this process a six-sigma process, the variace would have to be decreased such that CR k.. The value of the variace is foud by solvig CR k x LSL. for : Therefore, the process variace would have to be decreased to.5).5. f) x X 64.) X µ x.88.96).96).88) The probability that this shift will be detected o the ext sample is p ARL 5.87 p a) Chart - Iitial Study Chart UCL:. sigma.867 Ceterlie. LCL: -. sigma.6 out of limits Estimated mea. sigma.89

12 Chart for def. UCL.9 roportio... Sample Number LCL.6 There are o poits beyod the cotrol limits. The process is i cotrol. b) Chart - Iitial Study Sample Size, Chart UCL:. sigma.7674 Ceterlie. LCL: -. sigma.466 out of limits Estimated mea. sigma.46 Chart for def.9 UCL.764 roportio LCL.46 Sample Number There is oe poit beyod the upper cotrol limit. The process is out of cotrol. The revised limits are: Chart - Revised Limits Sample Size, Chart UCL:. sigma.774 Ceterlie.66 LCL: -. sigma.4979 out of limits Estimated mea.66

13 sigma.796 There are o poits beyod the cotrol limits. The process is ow i cotrol..9 Chart for def UCL.77 roportio LCL.49 Sample Number c) A larger sample size with the same umber of defective items will result i more arrow cotrol limits. The cotrol limits correspodig to the larger sample size are more sesitive ARL /p where p is the probability a poit falls outside the cotrol limits. a) µ µ ad p X > UCL) X LCL) µ µ µ > / > ) > ) 4) ) [ 4)] Therefore, ARL /p / b) µ µ ) µ / whe.9775 [ ].75

14 .5866 ] [.844 5)] [ ) 5) ) ) ) / / ) ) > > > > whe LCL X UCL X µ µ µ µ Therefore, ARL /p / c) µ µ.5 ] [.5 6)] [ ) 6) ) ) ) / / ) ) > > > > whe LCL X UCL X µ µ µ µ Therefore, ARL /p /.5.. d) The ARL is decreasig as the magitude of the shift icreases from to to. The ARL will decrease as the magitude of the shift icreases sice a larger shift is more likely to be detected earlier tha a smaller shift a) X-bar ad Rage - Iitial Study Chartig xbar X-bar Rage UCL:. sigma 4.68 UCL:. sigma.4847 Ceterlie 9.49 Ceterlie.75 LCL: -. sigma 8.8 LCL: -. sigma out of limits 9 out of limits Estimated process mea 9.49 process sigma.5559 mea Rage.75

15 roblem 6-5 X-bar , ad 9. Rage subgroup.4847 There are poits beyod the cotrol limits. The process is out of cotrol. The poits are 4, 8,,, 5,.75 b) Revised cotrol limits are give i the table below: X-bar ad Rage - Iitial Study Chartig Xbar X-bar Rage UCL:. sigma 4.58 UCL:. sigma.69 Ceterlie 9.88 Ceterlie.77 LCL: -. sigma 9.98 LCL: -. sigma out of limits out of limits Estimated process mea 9.88 process sigma.596 mea Rage.77 z There are o poits beyod the cotrol limits the process is ow i cotrol. The process stadard deviatio estimate is give by R d 6. USL LSL 4 8 c) CR ) USL x x LSL CRk mi, mi, 59. ) 59. ) mi [ 84.,. ] 4. Sice CR exceeds uity, the atural tolerace limits lie iside the specificatio limits ad very few defective uits will be produced. CR is slightly larger tha CR k idicatig that the process is somewhat off ceter. d) I order to make this process a six-sigma process, the variace would have to be decreased such that CR k.. The value of the variace is foud by solvig CR k x LSL. for :

16 Therefore, the process variace would have to be decreased to.).9. e) x.59 p 9.98 X 4.58 µ 9.7) X µ x.4.55).55).4).55) [.4)].994 [.8785].8 The probability that this shift will be detected o the ext sample is p ARL 5. p 877.