Chapter 6 Machine Capacity Allocation Experiments
6.5 Efficiency of Using Simplified Models for OT
The DES simulation is developed in a commercial software, Plant Simulation 8.1.
96
Each replication takes 2 seconds in average and each design needs 30 replications. For brute force method there are 415 designs in total, which include the unstable designs because brute force method could not identify unstable designs before running simulations. So, computation time of brute force method is 415*30*2 seconds, approximately equal to 7 hours.
For OT, the mathematical formulas of parametric decomposition method are implemented by Matlab 2010a. Evaluations of all 415 designs by QNA or JNA take less than one second of CPU time, approximately 0.8 second. In the comparison with computation time of DES simulation, computation time of QNA or JNA can be ignored. QNA considers heterogeneous SCVs and takes additional computation time of milliseconds compared with JNA, but acquires a great improvement on ranking of top designs. This deal is actually a real bargain and cost-effective.
97
Chapter 7 Conclusions
In this thesis, how selecting the simplified models affects ranking for OO is investigated and specify re-entrant line machine capacity allocation problem as the conveyor problem. Parametric decomposition method was exploited to the considered re-entrant line. Based on parametric decomposition method, we compared two simplified models: queueing network analyzer (QNA) and Jackson network approximation (JNA). The major difference is only the characterization of variability terms, QNA being heterogeneous SCVs and JNA being unity SCVs because of exponential assumptions.
To analyze the goodness of ranking by simplified models in theory, we developed a bound and ranking analysis, BRA, and took the first step to investigate the probability of correct ranking in case of single GI/G/m queue with two designs. In single GI/G/m queue, bound analysis showed that QNA is bounded by the upper and lower bound presented by Kingman, and Brumelle and Marchal respectively. In
addition, with the variation of QNA, the least variation of upper bound was derived.
This facilitates us to derive a better probability of correct ranking α. In our experiment, α is greater than 0.75 which is significantly difference with the probability of 0.5 like
tossing a coin. Based on single GI/G/m queue, BRA is extended to general re-entrant
98
queueing network with multiple workstations and amount of designs. Rank correlation, a statistic to measure the concordance of pair-wise comparisons in two quantitative indices, is introduced to quantify the goodness of ranking for the general cases.
Simulation studies demonstrated that rank correlation of QNA always outperforms JNA, especially significant for top-10 designs, which coincided with our BRA. Then, the original designs space is transformed by true ranking, and cluster each thirty designs into a group in this ordinal space. After grouping, it shows that heterogeneous SCVs benefit differentiation between groups and also make designs in a group better separated. That is why considering heterogeneous SCVs in simplified models improve the rank correlation. In this thesis, we investigated how selecting simplified models of different variability affects ranking for OO and support the validity of using ranking information by simplified models for optimization in aspects of theory and experiment.
99
Appendix
Ranking Analysis of QNA as Simplified Model in Other Cases
A.1 Ranking analysis of QNA as simplified model for Single GI/G/m queue under
the assumption of normal distribution of actual cycle time
We assume that CT1 and CT2 are normal distributions, which the means of CT1
100
Therefore, the probability of being a concordant pair is
P[CT1 < CT2|𝐿𝐵1 ≤ CT1 ≤ 𝑈𝐵1, 𝐿𝐵2 ≤ CT2 ≤ 𝑈𝐵2] positive. It implies that in normal distributions ranking two designs according to their approximated mean cycle times by QNA makes sure that the probability of being a concordant pair (Pc) is greater than the probability of being a discordant pair, Pc > 0.5.
101
References
[1] Xu, Jie, et al. "An ordinal transformation framework for multi-fidelity simulation optimization." Automation Science and Engineering (CASE), 2014 IEEE
International Conference on. IEEE, (2014): 385-390.
[2] Shanthikumar, J. George, and John A. Buzacott. "Open queueing network models of dynamic job shops." The International Journal Of Production Research 19.3 (1981): 255-266.
[3] G. R. Bitran and D. Tirupati , “Multi-product Queueing Networks with Deterministic Routing: Decomposition Approach and the Notion of Interference”,
Management Science,Vol.34, No.1(1988): 75-100.
[4] Connors, Daniel P., Gerald E. Feigin, and David D. Yao. "A queueing network model for semiconductor manufacturing." Semiconductor Manufacturing, IEEE
Transactions on 9.3 (1996): 412-427.
[5] Jackson, James R. "Job shop-like queueing systems." Management science 10.1 (1963): 131-142.
[6] Kelly, Frank P. "Networks of queues with customers of different types." Journal
of Applied Probability (1975): 542-554.
[7] Gordon, William J., and Gordon F. Newell. "Closed queuing systems with exponential servers." Operations research 15.2 (1967): 254-265.
102
[8] Baskett, Forest, et al. "Open, closed, and mixed networks of queues with different classes of customers." Journal of the ACM (JACM) 22.2 (1975):
248-260.
[9] H. Kobayashi. "Application of Diffusion Approximations to Queueing Networks.
Part I. Equilibrium Queue Distributions," J. Assoc. Comput. Mach., 21 (1974), 316-328.
[10] Reiser, M., and H. Kobayashi. "Accuracy of the diffusion approximation for some queuing systems." IBM Journal of Research and development 18.2 (1974):
110-124.
[11] Reiser, Martin, and Stephen S. Lavenberg. "Mean-value analysis of closed multichain queuing networks." Journal of the ACM (JACM) 27.2 (1980):
313-322.
[12] P. Schweitzer and A. Seidmann, “Optimizing processing rates for flexible manufacturingsystems.” Managment Science, 37 (4), (1991): 454–466.
[13] W. Whitt, “The Queueing Network Analyzer”, Bell System Tech. J. 62 (1983):
2779-2815.
[14] Segal, Moshe, and Ward Whitt. "A queueing network analyzer for manufacturing." Teletraffic Science for New Cost-Effective Systems, Networks
and Services, ITC 12 (1989): 1146-1152.
103
[15] Boxma, Onno Johan, AHG Rinnooy Kan, and Mario van Vliet. "Machine allocation problems in manufacturing networks." European Journal of
Operational Research 45.1 (1990): 47-54.
[16] W. Whitt, “Towards Better Multi-class Parametric-decomposition Approximation for Open Queueing Networks”, Annals of Operations Research 48 (1994)
221-248.
[17] Kouvelis, Panos, Chester Chambers, and Dennis Z. Yu. "Manufacturing operations manuscripts published in the first 52 issues of POM: Review, trends, and opportunities." Production and Operations Management 14.4 (2005):
450-467.
[18] Spinellis, Diomidis, Chrissoleon Papadopoulos, and J. MacGregor Smith. "Large production line optimization using simulated annealing." International Journal
of Production Research 38.3 (2000): 509-541.
[19] Tempelmeier, Horst. "Practical considerations in the optimization of flow production systems." International Journal of Production Research 41.1 (2003):
149-170.
[20] Banks, J. and Carson,J. S., 1984, Discrete-Event System Simulation (Englewood Cliffs, NJ: Prentice Hall)
[21] Johnson, Rachel T., John W. Fowler, and Gerald T. Mackulak. "A discrete event
104
simulation model simplification technique." Simulation Conference, 2005
Proceedings of the Winter. IEEE (2005): 5-8
[22] Huang, Edward, et al. "Multi-fidelity Model Integration for Engineering Design." Proceedings of Computer Science 44 (2015): 336-344.
[23] Brooks, Roger J., and Andrew M. Tobias. "Simplification in the simulation of manufacturing systems." International Journal of Production Research 38.5 (2000): 1009-1027.
[24] Kao, Yu-Ting, Chun-Ming Chang, and Shi-Chung Chang. "Do we still need daily production target setting in fully automated fabs?." e-Manufacturing and Design
Collaboration Symposium (eMDC), 2014. IEEE, (2014): 1-4
[25] Kao, Yu-Ting, Shi-Chung Chang, and Chun-Ming Chang. "Target setting with consideration of target-induced operation variability for performance improvement of semiconductor fabrication." Automation
Science and Engineering (CASE), 2014 IEEE International Conference on. IEEE (2014):
774-779
[26] Hu, Ming-Der, and Shi-Chung Chang. "Translating overall production goals into distributed flow control parameters for semiconductor manufacturing." Journal
of manufacturing systems 22.1 (2003): 46-63.
[27] Danping, Lin, and Carman KM Lee. "A review of the research methodology for
105
the re-entrant scheduling problem." International Journal of Production
Research 49.8 (2011): 2221-2242.
[28] Park, Youngshin, Sooyoung Kim, and Chi-Hyuck Jun. "Performance analysis of re-entrant flow shop with single-job and batch machines using mean value analysis." Production Planning & Control 11.6 (2000): 537-546.
[29] Whitt, Ward. "The Marshall and Stoyan bounds for IMRL/G/1 queues are tight."
Operations Research Letters 1.6 (1982): 209-213.
[30] Whitt, Ward. "Approximations for the GI/G/m queue." Production and
Operations Management 2.2 (1993): 114-161.
[31] Shanthikumar, J. George, and David D. Yao. "On server allocation in multiple center manufacturing systems." Operations Research 36.2 (1988): 333-342.
[32] Dallery, Yves, and Kathryn E. Stecke. "On the optimal allocation of servers and workloads in closed queueing networks." Operations Research 38.4 (1990):
694-703.
[33] Bitran, Gabriel R., and Devanath Tirupati. "Tradeoff curves, targeting and balancing in manufacturing queueing networks." Operations Research 37.4 (1989): 547-564.
[34] Frenk, Hans, et al. "Improved algorithms for machine allocation in manufacturing systems." Operations Research 42.3 (1994): 523-530.
106
[35] Bispo, Carlos F., and Sridhar Tayur. "Managing simple re-entrant flow lines:
theoretical foundation and experimental results." IIE transactions 33.8 (2001):
609-623.
[36] Sadre, Ramin, and Boudewijn R. Haverkort. "Decomposition-based queueing network analysis with FiFiQueues." Queueing Networks. Springer US, (2011):
643-699.
[37] Bolch, Gunter, et al. Queueing networks and Markov chains: modeling and
performance evaluation with computer science applications. John Wiley & Sons,
2006.
[38] Huang, Edward, et al. "Multi-fidelity Model Integration for Engineering Design." Procedia Computer Science 44 (2015): 336-344.
[39] J. F. C. Kingman, "Inequalities in the theory of queues." Journal of the Royal
Statistical Society. Series B (Methodological) (1970): 102-110.
[40] Marshall, Kneale T. "Some inequalities in queuing." Operations Research 16.3 (1968): 651-668.
[41] Brumelle, Shelby L. "Some inequalities for parallel-server queues." Operations
Research 19.2 (1971): 402-413.
[42] Ho, Yu-Chi, R_S Sreenivas, and P. Vakili. "Ordinal optimization of DEDS."
Discrete event dynamic systems 2.1 (1992): 61-88.
107
[43] Ho, Yu-Chi. "An explanation of ordinal optimization: soft computing for hard problems." Information Sciences 113.3 (1999): 169-192.
[44] Xu, Jie, et al. "Efficient multi-fidelity simulation optimization." Proceedings of
the 2014 Winter Simulation Conference. IEEE Press (2014): 3940-3951.
[45] Chang, Shi-Chug. "Demand-driven, iterative capacity allocation and cycle time
estimation for re-entrant lines." Decision and Control, 1999. Proceedings of the
38th IEEE Conference on. Vol. 3. IEEE, (1999): 2270-2275
[46] Kendall, Maurice George. "Rank correlation methods." (1948).