DEVELOPING AN INTEGRATED QUANTITY AND QUALITY APPROACH FOR IMPROVING THE PERFORMANCE OF MULTISTAGE MANUFACTURING SYSTEMS CAO YONGXIN (M. Eng.) A THESIS SUBMITTED FOR THE DEGREE OF DOCTOR OF PHILOSOPHY DEPARTMENT OF MECHANICAL ENGINEERING NATIONAL UNIVERSITY OF SINGAPORE 2012 Acknowledgments It is beyond doubt that the work in this thesis cannot be completed without the support, advice, and encouragement of teachers, colleagues, friends, and family members. In this acknowledgement, I wish to express my sincere appreciation and thanks to their support. First and foremost, I would like to gratefully and sincerely thank my supervisor, Professor Velusamy Subramaniam, for his invaluable guidance, insightful comments, patience, strong encouragement and personal concerns both academically and otherwise throughout the course of my research. By meeting and discussing with Professor Subramaniam every week during the past five years, his comments, critiques and attitude towards research were deeply engraved into this thesis, and also significantly improved the quality of this research. I would like to express my special thanks to my colleagues and closest friends Chen Ruifeng, Chanaka D. Senanayake and Lin Yuheng, who have given me valuable suggestions for this research. My gratitude is also extended to other friends in Control and Mechatronics Lab: Yu Deping, Feng Xiaobing, Ganesh Kumar Meenashisundaram, Kok You Cheng, Huang Weiwei, Fu Yong, Yang Jianbo, Zhao Guoyong, Wan Jie, Weng Yulin, Zhao Meijun, Zhu Kunpeng, Wang Qing, Albertus Adiwahono, Zhou Longjiang, Dau Van Huan, Chao Shuzhe, Wu Ning and many others. They have provided me with helpful comments, great friendship and a warm community during the past few years in NUS. i I gratefully acknowledge that my PhD study and this research are financially supported by the research scholarship provided by the National University of Singapore. To my family, in particular, I would like to thank my parents for their unwavering faith in me, and my sisters for their strong support, of which I’m truly grateful. Finally, my deepest thanks to my wife Yun Li for her support, understanding and encouragement through these years. ii Table of Contents Acknowledgments i Summary vii List of Tables x List of Figures Introduction 1.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1.1 Quality failures . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1.2 Sampling plans . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1.3 Rework loops . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 1.3 xiii Performance improvement of manufacturing systems using the proposed integrated quantity and quality approach . . . . . . . . . . . . . . . . . 1.2.1 Determining suitable continuous sampling plans . . . . . . . . 1.2.2 Allocating appropriate buffer capacities in the system . . . . . . 1.2.3 Identifying bottleneck machines of multistage systems . . . . . 10 1.2.4 Positioning of inspection machines in multistage systems . . . . 11 Thesis outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 Performance analysis of manufacturing systems: state of the art 13 2.1 Fundamental modeling approaches . . . . . . . . . . . . . . . . . . . . 14 2.2 Analytical studies on manufacturing systems with sampled inspection . 18 2.3 Performance analysis of systems with both operational and quality failures 21 iii 2.4 Analysis of manufacturing systems with rework loops . . . . . . . . . . 23 2.5 Chapter summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 Analysis of manufacturing systems with continuous sampling plans 28 3.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 3.2 Modeling a single stage manufacturing system with continuous sampling plans . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3 Modeling two stage systems: building blocks for decomposition analysis of multistage systems . . . . . . . . . . . . . . . . . . . . . . . . . 36 3.3.1 Performance evaluation using single failure mode model . . . . 38 3.3.2 Performance evaluation using multiple failure mode model . . . 39 3.3.3 Validation of the proposed methods for evaluating the performance of two stage systems . . . . . . . . . . . . . . . . . . . 3.3.4 3.4 3.5 29 40 Quantitative analysis on the effects of sampling plans on system performance . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 Determining the best sampling plan of multistage systems . . . . . . . . 47 3.4.1 Decomposition of multistage systems with sampling plans . . . 47 3.4.2 Determining the best sampling plan for maximizing profit . . . 51 3.4.3 Case studies . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53 Chapter summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56 Analysis of systems with machines having both operational and quality failures 58 4.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Modeling of two-machine, one-buffer lines with both operational and 58 quality failures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59 4.2.1 Inspection machine model . . . . . . . . . . . . . . . . . . . . 61 4.2.2 Processing machine model . . . . . . . . . . . . . . . . . . . . 61 4.2.3 Machine-buffer-inspection model . . . . . . . . . . . . . . . . 63 4.2.4 A method for solving the balance equations . . . . . . . . . . . 67 iv 4.3 Performance measures . . . . . . . . . . . . . . . . . . . . . . 70 4.2.6 Model validation . . . . . . . . . . . . . . . . . . . . . . . . . 71 Decomposition of multistage systems with both operational and quality failures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74 4.3.1 Decomposition model . . . . . . . . . . . . . . . . . . . . . . 76 4.3.2 Validation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81 4.4 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89 4.5 Chapter summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92 Analysis of manufacturing systems with rework loops 94 5.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94 5.2 A 3M1B Markov model for rework systems . . . . . . . . . . . . . . . 95 5.3 Decomposition of multistage systems with rework loops . . . . . . . . 103 5.4 4.2.5 5.3.1 Quality of material flow . . . . . . . . . . . . . . . . . . . . . 103 5.3.2 Decomposition analysis of multistage rework systems . . . . . 109 Results and Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . 117 5.4.1 Model validation . . . . . . . . . . . . . . . . . . . . . . . . . 117 5.4.2 Applications of the model . . . . . . . . . . . . . . . . . . . . 121 5.5 Extending of the model to systems with inspection errors . . . . . . . . 130 5.6 Chapter summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132 Conclusions and future research work 134 6.1 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134 6.2 Future research work . . . . . . . . . . . . . . . . . . . . . . . . . . . 137 6.2.1 Incorporating vendor selection into performance analysis of imperfect production systems . . . . . . . . . . . . . . . . . . . . 138 6.2.2 Manufacturing systems with machines having multiple quality failures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139 6.2.3 Workforce planning . . . . . . . . . . . . . . . . . . . . . . . 140 v 6.2.4 Integrated quantity and quality analysis of flexible systems . . . 142 6.2.5 Developing a single model for studying real systems with various issues such as sampling plans, rework and quality failures . 143 Author’s Publications 145 Bibliography 147 Appendices 160 Appendix A: Balance equations for the single stage model with sampling plans . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160 Appendix B: Balance equations for the machine-buffer-inspection model . . . 164 Appendix C: Balance equations for the 3M1B model . . . . . . . . . . . . . 167 vi Summary The fierce competition in the global market pressurizes manufacturers to improve productivity, quality and other performance for financial survival. Improving manufacturing performance is a challenging task due to the complex configurations of multistage systems and the existence of various uncertainties. Uncertainties such as machine breakdowns and random processing times substantially undermine the performance. Quality failures, the scrap of defects, inspection strategies, and rework loops further complicate system modeling and performance prediction. This thesis incorporates the study of these uncertainties into the analysis of multistage manufacturing systems, and proposes an integrated quantity and quality approach for evaluating the performance. Using the proposed approach, several managerial problems, which are often encountered in manufacturing plants, are solved for improving both quantitative and qualitative performance simultaneously. This thesis first investigates manufacturing systems with continuous sampling plans. This is a critical inspection strategy often adopted in industrial factories. An analytical method is proposed for modeling single-stage, two-stage and multistage systems and predicting both quantitative and qualitative performance. Using the proposed method, the effects of sampling parameters pertaining to various performance measures are studied quantitatively in numerical experiments. This method is further used as a mathematical tool in determining the best sampling plan for maximizing the performance of manufacturing systems. Experimental results also demonstrate the computational efficiency of the proposed method compared to simulation. vii Analysis of manufacturing systems with uncertainties in both operational and quality failures is another major study of this thesis. Using Markov chains to represent both types of failures, an integrated quantity and quality method is proposed for evaluating the performance of such systems. This model also characterizes the roles of an inspection station in real manufacturing systems, i.e., to detect defective parts as well as to monitor the quality status of a processing machine. Manufacturers may use this model to optimize system configurations such as buffer capacities. In systems with imperfect product quality, when defective parts are detected by inspection, these defects may then be delivered back to various stages for rework. Analytical modeling of such multiple rework loop systems is lacking in the published literature. This thesis proposes an analytical method for the performance analysis of rework systems. This model is capable of identifying various bottlenecks and studying bottleneck migration characteristics in rework systems. Such bottleneck analysis benefits industrial practitioners in continuously improving the system performance. In addition, developing analytical models using Markov chains requires a large number of states to characterize various uncertainties such as operational and quality failures in manufacturing systems. Much computational effort is involved in solving the balance equations of these analytical models. The thesis also develops a mathematical method for reducing the computational effort in obtaining the solution. Experimental results demonstrate that this method leads to greater computational efficiency compared to simulation. Keywords: Multistage Manufacturing Systems; Performance Evaluation; Markov Chains; Quantitative and Qualitative Performance; Decomposition; Continuous Sampling Plans; Rework; Bottleneck Identification viii List of Tables 3.1 State transition for the continuous sampling plan . . . . . . . . . . . . . 34 3.2 Experiment parameters of two stage systems . . . . . . . . . . . . . . . 42 3.3 Simulation results of the two stage systems . . . . . . . . . . . . . . . 43 3.4 Parameters of multistage systems . . . . . . . . . . . . . . . . . . . . . 50 3.5 Comparison of results from the analytical model and simulation . . . . 51 3.6 The best sampling plans of cases and . . . . . . . . . . . . . . . . . 54 3.7 The best sampling plans of Cases 19 and 20 . . . . . . . . . . . . . . . 56 4.1 Balance equation groups . . . . . . . . . . . . . . . . . . . . . . . . . 64 4.2 CPU time comparison of analytical methods and simulation . . . . . . . 73 4.3 Parameters for a six-machine production line (group 1) . . . . . . . . . 82 4.4 Parameters of a twenty-machine production line (group 2) . . . . . . . 83 4.5 Comparison of decomposition and simulation results . . . . . . . . . . 84 4.6 Computational time against the number of machines in the production line 86 4.7 Comparison of simulation and decomposition . . . . . . . . . . . . . . 87 4.8 Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90 5.1 Blocked states of machine M3 . . . . . . . . . . . . . . . . . . . . . . 99 5.2 Blocked states of machine M1 . . . . . . . . . . . . . . . . . . . . . . 99 5.3 Balance equation groups . . . . . . . . . . . . . . . . . . . . . . . . . 100 5.4 Outgoing quality of each machine in the rework subsystem of MJ . . . . 107 5.5 Experiment parameters . . . . . . . . . . . . . . . . . . . . . . . . . . 119 5.6 Comparison of results from the analytical model and simulation . . . . 120 5.7 Parameters of Cases A and B . . . . . . . . . . . . . . . . . . . . . . . 125 ix Mandroli S.S., Shrivastava A.K. and Ding Y., 2006. A survey of inspection strategy and sensor distribution studies in discrete-part manufacturing processes. IIE Transactions, 38 (4), 309-328. Meerkov S.M. and Zhang L., 2010. Product quality inspection in Bernoulli lines: Analysis, bottlenecks, and design, International Journal of Production Research, 48 (16), 4745-4766. Montgomery D.C., 2009, Introduction to statistical quality control 6th ed., Hoboken, NJ: John Wiley. Mok P.Y., 2009. A decision support system for the production control of a semiconductor packaging assembly line, Expert Systems with Applications, 36(3), 4423-4430. Nahas N., Daoud A.K. and Nourelfath M., 2006. A new approach for buffer allocation in unreliable production lines. International Journal of Production Economics, 103 (2), 873-881. Penn M. and Raviv T., 2007. Optimizing the quality control station configuration. Naval Research Logistics, 54(3), 301-314. Panagiotidou S. and Tagaras G., 2008. Evaluation of maintenance policies for equipment subject to quality shifts and failures. International Journal of Production Research, 46 (20), 5761-5779. Price W., Gravel M. and Nsakanda A.L., 1994. Review of optimization models of Kanban-based production systems. European Journal of Operational Research, 75 (1), 1-12. 156 Rau H., Chu Y.-H. and Cho K.-H., 2005. Layer modelling for the inspection allocation problem in re-entrant production systems. International Journal of Production Research, 43 (17), 3633-3655. Sarker B.R., Jamal A.M.M. and Mondal S., 2008. Optimal batch sizing in a multi-stage production system with rework consideration, European Journal of Operational Research, 184 (3), 915-929. Senanayake C.D. and Subramaniam V., 2012. Analysis of a two-stage, flexible production system with unreliable machines, finite buffers and non-negligible setups. Flexible Services and Manufacturing Journal, doi:10.1007/s10696-011-9115-2. Shi L. and Men S., 2003. Optimal buffer allocation in production lines. IIE Transactions, 35 (1), 1-10. Shi C. and Gershwin S.B., 2009. An efficient buffer design algorithm for production line profit maximization. International Journal of Production Economics, 122 (2), 725-740. Shiau Y., 2002. Inspection resource assignment in a multistage manufacturing system with an inspection error model, International Journal of Production Research, 40 (8), 1787-1806. Shina S.G., 2002. Six Sigma for electronics design and manufacturing, Boston: McGraw-Hill. Tan B., 2001. A three station merge system with unreliable stations and a shared buffer. Mathematical and Computer Modeling, 33, 1011-1026. 157 Tan B. and Gershwin S.B., 2009. Analysis of a general Markovian two-stage continuousflow production system with a finite buffer. International Journal of Prodution Economics, 120, 329-339. Tempelmeier H. and Burger M. 2001. Performance evaluation of unbalanced flow lines with general distributed processing times, failures and imperfect production. IIE Transactions, 33 (4), 293-302. Theta, 2012. Stent laser cutting machine, http://www.thetalaser.com/thetalaser/html/ English/products1.html Tolio T. and Matta A., 1998. Method for performance evaluation of automated flow lines, CIRP Annals - Manufacturing Technology, 47 (1), 373-376. Tolio T., Matta A. and Gershwin S.B., 2002. Analysis of two-machine lines with multiple failure modes. IIE Transactions, 34 (1), 51-62. Van Volsem S., Dullaert W. and Van Landeghem H., 2007. An evolutionary algorithm and discrete event simulation for optimizing inspection strategies for multi-stage processes. European Journal of Operational Research, 179 (3), 621-633. Vasudevan K.K., Lote R., Williams E.J. and Ugen O.M., 2008. Iterative use of simulation and scheduling methodologies to improve productivity, Winter Simulation Conference (WSC), 1896-1903. Veatch M. H., 2000. Inspection strategies for multistage production systems with timevarying quality, International Journal of Production Research, 38 (4), 837-853. Wang J., Li J., Arinez J., Biller S. and Huang N., 2010. Quality analysis in flexible manufacturing systems with batch productions: performance evaluation and non- 158 monotonic properties. IEEE Transactions on Automation Science and Engineering, 7(3), 671-676. Webber C.A. and Current J.C., 1993. A multiobjective approach to vendor selection, European Journal of Operational Research, 68(2) 173-184. Wright C.M. and Mehrez A., 1998. An overview of representative research of the relationships between quality and inventory. Omega, 26 (1), 29-47. Yamashita H. and Altiok T., 1998. Buffer capacity allocation for a desired throughput in production lines. IIE Transactions, 30 (10), 883-891. Yu K.Y. and Bricker D.L., 1993. Analysis of a markov chain model of a multistage manufacturing system with inspection, rejection, and rework. IIE Transactions, 25 (1), 109-112. Zhao X., Gong Q. and Wang J., 2002. On control strategies in a multi-stage production system. International Journal of Production Research, 40 (5), 1155-1171. 159 Appendices Appendix A: Balance equations for the single stage model with sampling plans In Section 3.2, a Markov model was proposed for a single stage manufacturing system with continuous sampling plans. The balance equations of this model are listed as follows: State (1,1,ig ,0) ≤ ig < cg P(1, 1, ig , 0)µ I = P(1, 3, ig , 0)µ P f H , ≤ ig < cg (A-1) State (1,2,0,0) P(1, 2, 0, 0)µ I = P(1, 4, 0, 0)µ P f L (A-2) State (1,3,0,0) P(1, 3, 0, 0)(µ P f H + pq ) = P(0, 3, 0, 0)rq (A-3) State (1,3,1,0) P(1, 3, 1, 0)(µ P f H + pq ) = P(1, 1, 0, 0)µ I (1 − ΦI ) I I + P(1, 1, 0, 1)µ (1 − Φ ) 160 (A-4) State (1,3,ig ,0) ≤ ig < cg P(1, 3, ig , 0)(µ P f H + pq ) = P(1, 1, ig − 1, 0)µ I (1 − ΦI ), ≤ ig < cg (A-5) State (1,4,0,0) P(1, 4, 0, 0)(µ P f L + pq ) = P(1, 1, cg − 1, 0)µ I (1 − ΦI ) I (A-6) I + P(1, 2, 0, 0)µ (1 − Φ ) State (1,1,0,1) P(1, 1, 0, 1)µ I = P(1, 3, 0, 1)µ P f H (A-7) State (1,2,0,1) (A-8) P(1, 2, 0, 1) = State (1,3,0,1) P H q P(1, 3, 0, 1)(µ f + p ) = cg −1 P(1, 1, ig , 0)µ I ΦI + P(1, 2, 0, 0)µ I ΦI ∑ g (A-9) i =0 State (1,4,0,1) (A-10) P(1, 4, 0, 1) = State (-1,1,ig ,0) ≤ ig < cg P(−1, 1, ig , 0)µ I = P(−1, 3, ig , 0)µ P f H , ≤ ig < cg 161 (A-11) State (-1,2,0,0) P(−1, 2, 0, 0)µ I = P(−1, 4, 0, 0)µ P f L (A-12) State (-1,3,0,0) P(−1, 3, 0, 0)µ P f H = P(1, 3, 0, 0)pq (A-13) State (-1,3,1,0) P(−1, 3, 1, 0)µ P f H = P(1, 3, 1, 0)pq + P(−1, 1, 0, 0)µ I ΦII (A-14) I + P(−1, 1, 0, 1)µ Φ II State (-1,3,ig ,0) ≤ ig < cg P(−1, 3, ig , 0)µ P f H = P(1, 3, ig , 0)pq + P(−1, 1, ig − 1, 0)µ I ΦII , ≤ ig < cg (A-15) State (-1,4,0,0) P(−1, 4, 0, 0)µ P f L = P(1, 4, 0, 0)pq + P(−1, 1, cg − 1, 0)µ I ΦII (A-16) + P(−1, 2, 0, 0)µ I ΦII State (-1,1,0,1) P(−1, 1, 0, 1)µ I = P(−1, 3, 0, 1)µ P f H (A-17) State (-1,2,0,1) (A-18) P(−1, 2, 0, 1) = 162 State (-1,3,0,1) P(−1, 3, 0, 1)µ P f H = P(1, 3, 0, 1)pq + P(−1, 2, 0, 0)µ I (1 − ΦII ) + cg −1 g I ∑ P(−1, 1, i , 0)µ (1 − Φ II (A-19) ) ig =0 State (-1,4,0,1) (A-20) P(1, 4, 0, 1) = State (0,3,0,0) P(0, 3, 0, 0)rq = P(1, 1, 0, 1)µ I ΦI + P(−1, 1, 0, 1)µ I (1 − ΦII ) 163 (A-21) Appendix B: Balance equations for the machine-buffer-inspection model As discussed in Section 4.2, the balance equations for the machine-buffer-inspection model are classified into 10 groups as in Table 4.1. These balance equations are as follows: Group 1. α1 = 0, α2 = P(x, 0, 0, 0)(r1 + r2 ) = P(x, 0, 1, 0)p1 + P(x, 0, 0, 1)p2 , ≤ x ≤ N−1 (B-1) P(0, 0, 0, 0) = (B-2) P(N, 0, 0, 0) = (B-3) Group 2. α1 = 0, α2 = P(x, 0, 0, 1)(r1 + µ2 + p2 ) = P(x, 0, 0, 0)r2 + P(x, 0, 1, 1)p1 (B-4) + P(x+1, 0, 0, 1)µ2 , ≤ x ≤ N−1 P(0, 0, 0, 1)r1 = P(0, 0, 1, 1)p1 + P(1, 0, 0, 1)µ2 (B-5) P(N, 0, 0, 1) = (B-6) Group 3. α1 = 1, α2 = P(x, 0, 1, 0)(p1 + µ1 + r2 + pq ) = P(x−1, 0, 1, 0)µ1 (B-7) + P(x, 0, 0, 0)r1 + P(x, 0, 1, 1)p2 , ≤ x ≤ N−1 P(0, 0, 1, 0) = (B-8) P(N, 0, 1, 0)r2 = P(N−1, 0, 1, 0)µ1 + P(N, 0, 1, 1)p2 (B-9) Group 4. α1 = 1, α2 = P(x, 0, 1, 1)(p1 + p2 + µ1 + µ2 + pq ) = P(x−1, 0, 1, 1)µ1 + P(x, 0, 1, 0)r2 (B-10) + P(x+1, 0, 1, 1)µ2 + P(x, 0, 0, 1)r1 , 164 ≤ x ≤ N−1 P(0, 0, 1, 1)(p1 + µ1 + pq ) = P(1, 0, 1, 1)µ2 + P(0, 0, 0, 1)r1 (B-11) + P(0, 0, , 1)r q P(N, 0, 1, 1)(p2 + µ2 ) = P(N−1, 0, 1, 1)µ1 + P(N, 0, 1, 0)r2 (B-12) + P(N, 0, 0, 1)r1 Group 5. α1 = 02 , α2 = P(x, xb , 02 , 0)(r1 + r2 ) = P(x, xb , −1, 0)p1 + P(x, xb , 02 , 1)p2 , (B-13) ≤ x ≤ N−1, ≤ xb ≤ x P(0, 0, 02 , 0) = P(N, xb , 02 , 0) = 0, (B-14) ≤ xb ≤ N (B-15) Group 6. α1 = 02 , α2 = P(x, xb , 02 , 1)(r1 + µ2 + p2 ) = P(x, xb , 02 , 0)r2 + P(x, xb , −1, 1)p1 + P(x+1, xb , 02 , 1)µ2 , (B-16) ≤ x ≤ N−1, ≤ xb ≤ x P(0, 0, 02 , 1)r1 = P(0, 0, −1, 1)p1 + P(1, 0, 02 , 1)µ2 (B-17) P(N, xb , 02 , 1) = 0, (B-18) ≤ xb ≤ N Group 7. α1 = −1, α2 = P(x, 0, −1, 0)(p1 + µ1 + r2 ) = P(x, 0, 1, 0)pq + P(x, 0, 02 , 0)r1 + P(x, 0, −1, 1)p2 , (B-19) ≤ x ≤ N−1 P(x, xb , −1, 0)(p1 + µ1 + r2 ) = P(x, xb , −1, 1)p2 + P(x, xb , 02 , 0)r1 + P(x−1, xb −1, −1, 0)µ1 , (B-20) ≤ x ≤ N−1, ≤ xb ≤ x P(0, 0, −1, 0) = (B-21) P(N, 0, −1, 0) = (B-22) 165 P(N, xb , −1, 0)r2 = P(N, xb , −1, 1)p2 (B-23) + P(N−1, xb −1, −1, 0)µ1 , ≤ xb ≤ N Group 8. α1 = −1, α2 = P(x, 0, −1, 1)(p1 + p2 + µ1 + µ2 ) = P(x, 0, 1, 1)pq + P(x, 0, −1, 0)r2 + P(x+1, 0, −1, 1)µ2 + P(x, 0, , 1)r1 , (B-24) ≤ x ≤ N−1 P(x, xb , −1, 1)(p1 + p2 + µ1 + µ2 ) = P(x−1, xb −1, −1, 1)µ1 + P(x+1, xb , −1, 1)µ2 + P(x, xb , −1, 0)r2 + P(x, xb , 02 , 1)r1 , (B-25) ≤ x ≤ N−1, ≤ xb ≤ x P(0, 0, −1, 1)(p1 + µ1 ) = P(0, 0, 1, 1)pq + P(1, 0, −1, 1)µ2 (B-26) + P(0, 0, , 1)r1 P(N, 0, −1, 1) = (B-27) P(N, xb , −1, 1)(p2 + µ2 ) = P(N−1, xb −1, −1, 1)µ1 + P(N, xb , −1, 0)r2 , (B-28) ≤ xb ≤ N Group 9. α1 = 01 , α2 = N q P(0, 0, , 1)r = P(0, 0, , 1)r1 + ∑ P(x, x, −1, 1)µ2 (B-29) x=1 Group 10. α1 = 03 , α2 = N P(0, 0, 03 , 1)r1 = ∑ P(x, x, 02, 1)µ2 x=1 166 (B-30) Appendix C: Balance equations for the 3M1B model In Section 5.2, a 3M1B Model is proposed for manufacturing systems with rework loops. The balance equations of this model are classified into groups as in Table 5.3. These equations are listed as follows: Group 1. α1 = 0, α2 = 0, α3 = P(0, 0, 0, 0)(r1 + r2 + r3 ) = P(0, 1, 0, 0)p1 + P(0, 0, 0, 1)p3 (C-1) P(x, 0, 0, 0)(r1 + r2 + r3 ) = P(x, 1, 0, 0)p1 + P(x, 0, 1, 0)p2 (C-2) + P(x, 0, 0, 1)p3 , ≤ x ≤ N−3 P(N−2, 0, 0, 0)(r1 + r2 + r3 ) = P(N−2, 0, 1, 0)p2 (C-3) + P(N−2, 0, 0, 1)p3 P(N−1, 0, 0, 0)(r1 + r2 + r3 ) = P(N−1, 0, 1, 0)p2 (C-4) P(N, 0, 0, 0)(r1 + r2 + r3 ) = P(N, 0, 1, 0)p2 (C-5) Group 2. α1 = 0, α2 = 0, α3 = P(0, 0, 0, 1)(r1 + r2 + p3 + µ3 h3 ) = P(0, 1, 0, 1)p1 + P(0, 0, 0, 0)r3 (C-6) P(x, 0, 0, 1)(r1 + r2 + p3 + µ3 h3 ) = P(x, 1, 0, 1)p1 + P(x, 0, 1, 1)p2 (C-7) + P(x, 0, 0, 0)r3 + P(x−1, 0, 0, 1)µ3 h3 , ≤ x ≤ N−3 P(N−2, 0, 0, 1)(r1 + r2 + p3 + µ3 h3 ) = P(N−2, 0, 1, 1)p2 (C-8) + P(N−2, 0, 0, 0)r3 + P(N−3, 0, 0, 1)µ3 h3 P(N−1, 0, 0, 1)(r1 + r2 ) = P(N−1, 0, 1, 1)p2 + P(N−1, 0, 0, 0)r3 (C-9) + P(N−2, 0, 0, 1)µ3 h3 P(N,0, 0, 1)(r1 + r2 ) = P(N, 0, 1, 1)p2 + P(N, 0, 0, 0)r3 167 (C-10) Group 3. α1 = 0, α2 = 1, α3 = P(0, 0, 1, 0)(r1 + r3 ) = P(0, 1, 1, 0)p1 + P(0, 0, 1, 1)p3 (C-11) + P(1, 0, 1, 0)µ2 P(x, 0, 1, 0)(r1 + p2 + r3 + µ2 ) = P(x, 1, 1, 0)p1 + P(x, 0, 0, 0)r2 (C-12) + P(x, 0, 1, 1)p3 + P(x+1, 0, 1, 0)µ2 , ≤ x ≤ N−3 P(N−2, 0, 1, 0)(r1 + p2 + r3 + µ2 ) = P(N−2, 0, 0, 0)r2 (C-13) + P(N−2, 0, 1, 1)p3 + P(N−1, 0, 1, 0)µ2 P(N−1, 0, 1, 0)(r1 + p2 + r3 + µ2 ) = P(N−1, 0, 0, 0)r2 (C-14) + P(N, 0, 1, 0)µ2 P(N,0, 1, 0)(r1 + p2 + r3 + µ2 ) = P(N, 0, 0, 0)r2 (C-15) Group 4. α1 = 0, α2 = 1, α3 = P(0, 0, 1, 1)(r1 + p3 + µ3 h3 ) = P(0, 1, 1, 1)p1 + P(0, 0, 0, 1)r2 (C-16) + P(0, 0, 1, 0)r3 + P(1, 0, 1, 0)µ2 P(x, 0, 1, 1)(r1 + p2 + p3 + µ2 + µ3 h3 ) = P(x, 1, 1, 1)p1 + P(x, 0, 0, 1)r2 + P(x, 0, 1, 0)r3 + P(x+1, 0, 1, 1)µ2 + P(x−1, 0, 1, 1)µ3 h3 , (C-17) ≤ x ≤ N−3 P(N−2, 0, 1, 1)(r1 + p2 + p3 + µ2 + µ3 h3 ) = P(N−2, 0, 0, 1)r2 + P(N−2, 0, 1, 0)r3 + P(N−1, 0, 1, 1)µ2 + P(N−3, 0, 1, 0)µ3 h3 168 (C-18) P(N−1, 0, 1, 1)(r1 + p2 + µ2 ) = P(N−1, 0, 0, 1)r2 (C-19) + P(N−1, 0, 1, 0)r3 + P(N, 0, 1, 1)µ2 + P(N−2, 0, 1, 1)µ3 h3 P(N,0, 1, 1)(r1 + p2 + µ2 ) = P(N, 0, 0, 1)r2 + P(N, 0, 1, 0)r2 (C-20) Group 5. α1 = 1, α2 = 0, α3 = P(0, 1, 0, 0)(p1 + r2 + r3 + µ1 l1 ) = P(0, 0, 0, 0)r1 + P(0, 1, 0, 1)p3 (C-21) P(x, 1, 0, 0)(p1 + r2 + r3 + µ1 l1 ) = P(x, 0, 0, 0)r1 + P(x, 1, 1, 0)p2 (C-22) + P(x, 1, 0, 1)p3 + P(x−1, 1, 0, 0)µ1 l1 , ≤ x ≤ N−3 P(N−2, 1, 0, 0)(r2 + r3 ) = P(N−2, 0, 0, 0)r1 + P(N−2, 1, 1, 0)p2 (C-23) + P(N−2, 1, 0, 1)p3 + P(N−3, 1, 0, 0)µ1 l1 P(N−1, 1, 0, 0)(r2 + r3 ) = P(N−1, 0, 0, 0)r1 + P(N−1, 1, 1, 0)p2 (C-24) + P(N−1, 1, 0, 1)p3 P(N, 1, 0, 0)(r2 + r3 ) = P(N, 1, 1, 0)p2 + P(N, 0, 0, 0)r1 (C-25) Group 6. α1 = 1, α2 = 0, α3 = P(0, 1, 0, 1)(p1 + r2 + p3 + µ1 l1 + µ3 h3 ) = P(0, 0, 0, 1)r1 (C-26) + P(0, 1, 0, 0)r3 P(x, 1, 0, 1)(p1 + r2 + p3 + µ1 l1 + µ3 h3 ) = P(x, 0, 0, 1)r1 (C-27) + P(x, 1, 1, 1)p2 + P(x, 1, 0, 0)r3 + P(x−1, 1, 0, 1)(µ1 l1 + µ3 h3 ), 169 ≤ x ≤ N−3 P(N−2, 1, 0, 1)(r2 + p3 + µ3 h3 ) = P(N−2, 0, 0, 1)r1 + P(N−2, 1, 1, 1)p2 + P(N−2, 1, 0, 0)r3 (C-28) + P(N−3, 1, 0, 1)(µ1 l1 + µ3 h3 ) P(N−1, 1, 0, 1)(r2 + p3 + µ3 h3 ) = P(N−1, 0, 0, 1)r1 + P(N−1, 1, 1, 1)p2 + P(N−1, 1, 0, 0)r3 (C-29) + P(N−2, 1, 0, 1)µ3 h3 P(N, 1, 0, 1)r2 = P(N, 0, 0, 1)r1 + P(N, 1, 1, 1)p2 (C-30) + P(N, 1, 0, 0)r3 + P(N−1, 1, 0, 1)µ3 h3 Group 7. α1 = 1, α2 = 1, α3 = P(0, 1, 1, 0)(p1 + r3 + µ1 l1 ) = P(0, 1, 0, 0)r2 + P(0, 0, 1, 0)r1 (C-31) + P(0, 1, 1, 1)p3 + P(1, 1, 1, 0)µ2 P(x, 1, 1, 0)(p1 + p2 + r3 + µ1 l1 + µ2 ) = P(x, 1, 0, 0)r2 + P(x, 0, 1, 0)r1 + P(x, 1, 1, 1)p3 + P(x+1, 1, 1, 0)µ2 + P(x−1, 1, 1, 0)µ1 l1 , (C-32) ≤ x ≤ N−3 P(N−2, 1, 1, 0)(p2 + r3 + µ2 ) = P(N−2, 0, 1, 0)r1 + P(N−2, 1, 0, 0)r2 + P(N−2, 1, 1, 1)p3 (C-33) + P(N−1, 1, 1, 0)µ2 + P(N−3, 1, 1, 0)µ1 l1 P(N−1, 1, 1, 0)(p2 + r3 + µ2 ) = P(N−1, 0, 1, 0)r1 + P(N−1, 1, 0, 0)r2 + P(N−1, 1, 1, 1)p3 (C-34) + P(N, 1, 1, 0)µ2 P(N, 1, 1, 0)(p2 + r3 + µ2 ) = P(N, 0, 1, 0)r1 + P(N, 1, 0, 0)r2 170 (C-35) Group 8. α1 = 1, α2 = 1, α3 = P(0, 1, 1, 1)(p1 + p3 + µ1 l1 + µ3 h3 ) = P(0, 1, 0, 1)r2 (C-36) + P(0, 0, 1, 1)r1 + P(0, 1, 1, 0)r3 + P(1, 1, 1, 1)µ2 P(x, 1, 1, 1)(p1 + p2 + p3 + µ1 l1 + µ2 + µ3 h3 ) = P(x, 1, 0, 1)r2 + P(x, 0, 1, 1)r1 + P(x, 1, 1, 0)r3 + P(x+1, 1, 1, 1)µ2 + P(x−1, 1, 1, 1)(µ1 l1 + µ3 h3 ), (C-37) ≤ x ≤ N−3 P(N−2, 1, 1, 1)(p2 + p3 + µ2 + µ3 h3 ) = P(N−2, 0, 1, 0)r2 + P(N−2, 0, 1, 1)r1 + P(N−2, 1, 1, 0)r3 (C-38) + P(N−1, 1, 1, 1)µ2 + P(N−3, 1, 1, 1)(µ1 l1 + µ3 h3 ) P(N−1, 1, 1, 1)(p2 + p3 + µ2 + µ3 h3 ) = P(N−1, 0, 1, 0)r2 + P(N−1, 0, 1, 1)r1 + P(N−1, 1, 1, 0)r3 (C-39) + P(N, 1, 1, 1)µ2 + P(N−2, 1, 1, 1)µ3 h3 P(N, 1, 1, 1)(p2 + µ2 ) = P(N, 1, 0, 1)r2 + P(N, 0, 1, 1)r1 (C-40) + P(N, 1, 1, 0)r3 + P(N−1, 1, 1, 1)µ3 h3 171 [...]... increase production rate but affect quality control (Khouja, 2003) It is necessary to incorporate quality control into quantity control of manufacturing systems for simultaneously improving performance measures Therefore, this thesis proposes an integrated quantity and quality approach for performance analysis of multistage manufacturing systems, and the outline of this approach is described in Fig 1.1... evaluating various performance measures i.e., throughput, WIP and quality is also one of major focuses of this thesis 7 1.2 Performance improvement of manufacturing systems using the proposed integrated quantity and quality approach The analytical methods proposed in this thesis are capable of predicting both quantitative and qualitative performance measures of multistage manufacturing systems These methods... on the system performance Based on the proposed integrated quantity and quality approach, analytical methods are also explored for improving the quantitative and qualitative performance of multistage systems In the following subsections, the author shall elaborate on the major quality characteristics (which are also shown in Fig 1.1) of manufacturing systems: 3 Manufacturing systems Modeling Random... pertaining to performance analysis of multistage manufacturing systems is presented in Chapter 2 In Chapter 3, the manufacturing systems with continuous sampling plans are investigated An analytical model is formulated for performance evaluation of such systems and subsequently used to determine the best sampling plans for improving system performance In Chapter 4, an integrated quantity and quality model... determine the best sampling plans to meet system requirements on both qualitative and quantitative performance (Mandroli et al., 2006) To the best of the author’s knowledge, this is the first research to incorporate continuous sampling plans into integrated quality and quantity analysis of manufacturing systems 2.3 Performance analysis of systems with both operational and quality failures Quality failures of. .. evaluate and compare the performance of systems with a large number of sampling combinations, and thus a quick mathematical tool is required for estimating the performance 8 The proposed analytical method provides reliable estimates of both quantitative and qualitative performance measures in a short time It is applicable for performing quantitative analysis of the effects of sampling plans on throughput and. .. summary of the key findings and provides several future research opportunities 12 Chapter 2 Performance analysis of manufacturing systems: state of the art The major objective of modeling manufacturing systems is to predict the system performance (e.g., production rate, WIP, quality, etc.) and conduct what-if analysis for performance improvement Analytical modeling methods, i.e., Markov theories and decomposition... collar manufacturing (Vasudevan et al., 2008; Li, 2004) The existence of rework loops in manufacturing systems complicates performance analysis For example, improving the performance of a bottleneck in rework systems may lead to sophisticated phenomena such as bottleneck migration (Li and Meerkov, 2009) The performance of rework systems vary significantly depending on the number and the location of rework... determining the best sampling plan to simultaneously improve both quantitative and qualitative performance of a manufacturing system Therefore, analyzing manufacturing systems with sampling plans is another major focus of this thesis 1.1.3 Rework loops Real manufacturing systems may experience substantial defects These defects generate waste in the form of yield loss, additional material handling costs,... 140 6.2 Workforce planning for a production line 141 xiii Chapter 1 Introduction 1.1 Motivation The effect of globalization and the easy accessibility of world markets has created turbulence in the business environment, thereby pressurizing manufacturers to enhance manufacturing performance for financial survival (Anand and Kodali, 2009) In the past decades, manufacturers of many industries . DEVELOPING AN INTEGRATED QUANTITY AND QUALITY APPROACH FOR IMPROVING THE PERFORMANCE OF MULTISTAGE MANUFACTURING SYSTEMS CAO YONGXIN (M. Eng.) A THESIS SUBMITTED FOR THE DEGREE OF DOCTOR OF. on the system performance. Based on the pro- posed integrated quantity and quality approach, analytical methods are also explored for improving the quantitative and qualitative performance of multistage. incorporate quality control into quantity control of manufacturing systems for simultaneously im- proving performance measures. Therefore, this thesis proposes an integrated quantity and quality approach