Iowa State University Capstones, Theses and Retrospective Theses and Dissertations Dissertations 1989 The ts atistical optimal design of Shewhart control charts with supplementary stopping rules Noel Artiles-Leon Iowa State University Follow this and additional works at: https://lib.dr.iastate.edu/rtd Part of the Industrial Engineering Commons, and the Statistics and Probability Commons Recommended Citation Artiles-Leon, Noel, "The ts atistical optimal design of Shewhart control charts with supplementary stopping rules " (1989). Retrospective Theses and Dissertations. 8911. https://lib.dr.iastate.edu/rtd/8911 This Dissertation is brought to you for free and open access by the Iowa State University Capstones, Theses and Dissertations at Iowa State University Digital Repository. It has been accepted for inclusion in Retrospective Theses and Dissertations by an authorized administrator of Iowa State University Digital Repository. For more information, please contact [email protected]. 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University Microfilms International A Bell & Howell Information Company 300 North Zeeb Road, Ann Arbor, Ml 48106-1346 USA 313/761-4700 800/521-0600 L I Order Number 8920108 The statistical optimal design of Shewhart control charts with supplementary stopping rules Artiles-Leon, Noel, Ph.D. Iowa State University, 1989 UMI SOON.ZeebRd. Ann Arbor, MI 48106 1 The statistical optimal design of Shewhart control charts with supplementary stopping rules by Noel Artiles-Leon A Dissertation Submitted to the Graduate Faculty in Partial Fulfillment of the Requirements for the Degree of DOCTOR OF PHILOSOPHY Major: Industrial Engineering Approved: Signature was redacted for privacy. In Charge of—itajor Work Signature was redacted for privacy. the Major Department Signature was redacted for privacy. the Graduate College Iowa State University Ames, Iowa 1989 ii TABLE OF CONTENTS PAGE I. INTRODUCTION 1 A. The Uses of Control Charts 1 B. Control Limits and Control Rules 3 C. Selection of Control Limits and Control Rules 5 II. LITERATURE REVIEW 12 A. Semieconomic Design of Control Charts 12 B. Economic Design of Control Charts 15 C. Control Charts with Warning Lines 33 D. Simplified Designs for Control Charts 41 III, PROBLEM STATEMENT, RESEARCH OBJECTIVES AND METHODOLOGY ... 43 A. Problem Statement 43 B. A General Methodology for Determining the Average Run Length 49 C. Robustness of the Most Powerful Control Schemes 53 IV. DETERMINATION AND EVALUATION OF THE MOST POWERFUL CONTROL SCHEMES 55 A. The { Rl[l,l,x,œ], RZ[2,3,y,œ] } Family of Control Schemes 55 B. The { Hi[1,1,x,œ], R3[3,4,y,œ] } Family of Control Schemes 72 C. The { R2[2,3,x,œ], R3[3,4,y,®] } Family of Control Schemes 8.2 D. The { Rl[l,l,w,=], R2[2,3.x,»], R3[3,4,y,»] } Family of Control Schemes 93 E. Comparison of Two Control Schemes 106 V. CONCLUSIONS AND RECOMMENDATIONS 114 VI. REFERENCES 117 VII. ACKNOWLEDGEMENTS 121 VIII. APPENDIX A: PROGRAMS P2311A.BAS AND P2311B.BAS 122 IX. APPENDIX B: PROGRAMS P3411A.BAS AND P3411B.BAS 136 X. APPENDIX C: PROGRAM GENSTATE.BAS AND SUBROUTINES CHECKST.INC 150 Ill XI, APPENDIX D: PROGRAMS P3423A.BAS AND P3423B.BAS 155 XII. APPENDIX E: PROGRAMS P342311A.BAS AND P342311B.BAS 170 XIII. APPENDIX F: SIMULATION PROGRAMS AND SUBROUTINES 181 iv LIST OF TABLES PAGE Table 1. Average run length for schemes suggested by Moore (with a single control limit at M+ka) 34 Table 2. Simple designs for control charts 42 Table 3. Optimal control limits for the { RlCl,l.x,®], R2[2,3,y,®] } family of control schemes (output from P2311A,BAS). Values of process mean and control limits are given in standard units 67 Table 4. Optimal control limit for the { R2[2,3,y,®] } family of control schemes. Values of the process mean and the control limit are given in standard units 68 Table 5. Optimal average run lengths for the { R2[2,3,y,®] } family of control schemes 69 Table 6. Average-run-length comparisons between optimal and non- optimal control schemes .72 Table 7. Optimal control limits for the { R1[1,1,x,=], R3[3,4, y,=] } family of control schemes (output from P3411A.BAS). Values of control limits and process mean are given in standard units 75 Table 8. Optimal control limit for the { R3[3,4,y,=] } family of control schemes. Values of the process mean and the control limit are given in standard units 78 Table 9. Optimal average run lengths for the { R3[3,4,y,®] } family of control schemes 80 Table 10. Ratios of average run lengths: optimal control scheme { R2[2,3,y,<=] } to optimal control scheme { R3[3,4,y,™] }. 81 Table 11. Optimal control limits for the { R2[2,3,x,=], R3[3,4, y,®] } family of control schemes (output from P2334A,BAS). Values of control limits are given in standard units 88 Table 12. Optimal average run lengths for the { R2[2,3,x,®], R3[3,4,y,®] } family of control schemes 91 V Table 13. ratios of average run lengths: optimal control scheme { R3[3,4,y,»] } to optimal control scheme { R2[2,3,x,=], R3[3,4,y,»j } 92 Table 14. Control limits for the family of control schemes { R1[ 1,1 , R2[2,3,x,<»], R3[3,4,y,œ] } resulting in an in-control A.R.L. of 200. Control limits given in standard units 98 Table 15. Control limits for the { Rl[l,l,w,®], R2[2,3,x,®], R3[3,4,y,®] } family of control schemes resulting in an in-control A.R.L. of 400. Control limits are given in standard units 100 Table 16. Optimal control limits for the { Rl[l,l,w,®], R2[2,3,x,œ], R3[3,4,y,®] } family of control schemes. Values of control limits are given in standard units . 101 Table 17. Optimal average run lengths for the { Rl[l,l,w,®], R2[2,3,x,œ], R3[3,4,y,®] } family of control schemes. Values of control limits and process mean are given in standard units 104 Table 18. Ratios of average run lengths: optimal control scheme { R2[2,3,x,®], R3[3,4,y,®] } to optimal control scheme { Rl[l,l,w,®], R2[2,3,x,®], R3[3,4,y,m] } . 105 Table 19. Average run length as a function of the shift in the process mean for the control schemes SI = { Rl[l,l,3,®], R2[2,3, 2,»], R3[4,5, 1,®]. R4[8,8,0,"] } and S2 = { Rl[l,l,3.216,®], R2[2,3,1.962,=], R3[3.4,1.181,»] }. 107 Table 20. Control-scheme comparisons: Double exponential distribution Ill Table 21. Control-scheme comparisons: Cauchy distribution .... 112 Table 22. Control-scheme comparisons: A.R.M.A. process 113 vi LIST OF FIGURES PAGE Figure 1. Diagram of in-control and out-of-control states of a process 31 Figure 2. Power comparison of two control schemes 47 Figure 3. A control chart to detect a positive shift in the process mean 51 Figure 4. The { Rl[l,l,x,®], R2[2,3,y,®] } control scheme 56 Figure 5. Graph of the modified objective function vs. the inner control limit for ARLo = 200, 300, and 500, and k = 1 . 65 Figure 6. Control-limit combinations giving a fixed ARLo. Graph of X = f(ARLo, y), ARLo = 200, 300, 500 66 Figure 7. Optimal control limits for the { R2[2,3,y,®] } family of control schemes. Values of the control limit are given in standard units 70 Figure 8. The { Rl[l,l,x,®], R3[3,4,y,®] } control scheme 73 Figure 9. Graph of the modified objective function vs. the inner control limit for ARLo = 100, 200, ..., 500 and k = 1 , .77 Figure 10. Control-limit combinations giving a fixed in-control average run length, ARLo; ARLo = 100, 200, .... 500 ... 77 Figure 11. Optimal control limits for the { R3[3,4,y,œ] } family of control schemes. Values of the control limit are given in standard units 79 Figure 12. The { R2[2,3,x,"], R3[3,4,y,®] } control scheme 83 Figure 13. Optimal control limits for the { R2[2,3,x,a>], R3[3,4,y,m] } family of control schemes.