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July  2017, 13(3): 1431-1448. doi: 10.3934/jimo.2016081

## Rescheduling optimization of steelmaking-continuous casting process based on the Lagrangian heuristic algorithm

 1 Department of Information and Control Engineering, Shenyang Jianzhu University, No. 9, Hunnan East Road, Hunnan New District, Shenyang City, Liaoning 110168, China 2 Department of Information Science and Engineering, Northeastern University, NO. 3-11, Wenhua Road, Heping District, Shenyang City, Liaoning 110004, China

* Corresponding author:Liangliang Sun

Received  January 2015 Published  October 2016

Fund Project: The research is financially sponsored by the National Natural Science Foundation Committee of China (Subject Numbers: 61503259), Hanyu Plan of Shenyang Jianzhu University and Research Funding from the Networked Control System Key Laboratory of the Chinese Academy of Sciences.

This study investigates a challenging problem of rescheduling a hybrid flow shop in the steelmaking-continuous casting (SCC) process, which is a major bottleneck in the production of iron and steel. In consideration of uncertain disturbance during SCC process, we develop a time-indexed formulation to model the SCC rescheduling problem. The performances of the rescheduling problem consider not only the efficiency measure, which includes the total weighted completion time and the total waiting time, but also the stability measure, which refers to the difference in the number of operations processed on different machines for the different stage in the original schedule and revised schedule. With these objectives, this study develops a Lagrangian heuristic algorithm to solve the SCC rescheduling problem. The algorithm could provide a realizable termination criterion without having information about the problem, such as the distance between the initial iterative point and the optimal point. This study relaxes machine capacity constraints to decompose the relaxed problem into charge-level subproblems that can be solved using a polynomial dynamic programming algorithm. A heuristic based on the solution of the relaxed problem is presented for obtaining a feasible reschedule. An improved efficient subgradient algorithm is introduced for solving Lagrangian dual problems. Numerical results for different events and problem scales show that the proposed approach can generate high-quality reschedules within acceptable computational times.

Citation: Liangliang Sun, Fangjun Luan, Yu Ying, Kun Mao. Rescheduling optimization of steelmaking-continuous casting process based on the Lagrangian heuristic algorithm. Journal of Industrial & Management Optimization, 2017, 13 (3) : 1431-1448. doi: 10.3934/jimo.2016081
##### References:
 [1] A. Atighehchian, M. Bijari and H. Tarkesh, A novel hybrid algorithm for scheduling steelmaking continuous casting production, Computers and Operations Research, 36 (2009), 2450-2461.   Google Scholar [2] H. Aytug, M. Lawley, K. McKay, S. Mohan and R. Uzsoy, Executing production schedules in the face of uncertainties: A review and some future directions, European Journal of Operational Research, 161 (2005), 86-110.  doi: 10.1016/j.ejor.2003.08.027.  Google Scholar [3] A. Bellabdaoui and J. Teghem, A mixed-integer linear programming model for the continuous casting planning, International Journal of Production Economics, 104 (2006), 260-270.  doi: 10.1016/j.ijpe.2004.10.016.  Google Scholar [4] D. Bertsekas, Nonlinear Programming, 2$^{nd}$ edition, Athena Scientific, Massachusetts, 1999. doi: 10.1007/978-1-4612-0873-0.  Google Scholar [5] U. Brannlund, On Relaxation Methods for Nonsmooth Convex Optimization, Ph. D thesis, Royal Institute of Technology in Stockholm, 1993.  Google Scholar [6] P. Camerini, L. Fratta and F. Maffioli, On improving relaxation methods by modified gradient techniques, Mathematical Programming Study, 3 (1975), 26-34.   Google Scholar [7] H. Chen and P. Luh, An alternative framework to Lagrangian relaxation approach for job shop scheduling, European Journal of Operational Research, 149 (2003), 499-512.  doi: 10.1016/S0377-2217(02)00470-8.  Google Scholar [8] P. Cowling, D. Ouelhadj and S. Petrovic, Dynamic scheduling of steel casting and milling using multi-agents, Production Planning and Control, 15 (2004), 178-188.  doi: 10.1080/09537280410001662466.  Google Scholar [9] V. Demjanov and V. Somesova, Conditional subdifferentials of convex functions, Soviet Mathematics Doklady, 19 (1978), 1181-1185.   Google Scholar [10] J. Goffin and K. Kiwiel, Convergence of a simple subgradient level method, Mathematical Programming, 85 (1999), 207-211.  doi: 10.1007/s101070050053.  Google Scholar [11] B. Guta, Subgradient Optimization Methods in Integer Programming with an Application to a Radiation Therapy Problem, Ph. D thesis, Teknishe Universitat Kaiserlautern in Kaiserlauter, 2003. Google Scholar [12] I. Harjunkoski and I. Grossmann, A decomposition approach for the scheduling of a steel plant production, Computers and Chemical Engineering, 25 (2001), 1647-1660.  doi: 10.1016/S0098-1354(01)00729-3.  Google Scholar [13] T. Larsson, M. Patriksson and A. Stromberg, Conditional subgradient optimization --theory and applications, European Journal of Operational Research, 88 (1996), 382-403.  doi: 10.1016/0377-2217(94)00200-2.  Google Scholar [14] J. Li, X. Xiao, Q. Tang and C. Floudas, Production scheduling of a Large-scale steelmaking continuous casting process via unit-specific event-based continuous-time models: Short-term and medium-term scheduling, Industrial and Engineering Chemistry Research, 51 (2012), 7300-7319.  doi: 10.1021/ie2015944.  Google Scholar [15] P. Luh and D. Hoitomt, Scheduling of manufacturing systems using the Lagrangian relaxation technique, IEEE Transactions on Automatic Control, 38 (1993), 1066-1079.  doi: 10.1109/9.231461.  Google Scholar [16] P. Luh, D. Hoitomt, E. Max and K. Pattipati, Scheduling generation and reconfiguration for parallel machines, IEEE Transactions on Robotics and Automation, 6 (1990), 687-696.   Google Scholar [17] K. Mao, Q. Pan, X. Pang and T. Chai, A novel Lagrangian relaxation approach for the hybrid flowshop scheduling problem in a steelmaking-continuous casting process, European Journal of Operational Research, 236 (2014), 51-60.  doi: 10.1016/j.ejor.2013.11.010.  Google Scholar [18] K. Mao, Q. Pan, X. Pang and T. Chai, An effective Lagrangian relaxation approach for rescheduling a steelmaking-continuous casting process, Control Engineering Practice, 30 (2014), 67-77.  doi: 10.1016/j.conengprac.2014.06.003.  Google Scholar [19] K. Mao, Q. Pan, X. Pang, T. Chai and P. Luh, An Effective Subgradient Method for Scheduling a Steelmaking-Continuous Casting Process, IEEE Transactions on Automation Science and Engineering, 12 (2014), 1-13.  doi: 10.1109/TASE.2014.2332511.  Google Scholar [20] H. Missbauer, W. Hauber and W. Werner Stadler, A scheduling system for the steelmaking-continuous casting process: A case study from the steelmaking industry, International Journal of Production Research, 47 (2009), 4147-4172.  doi: 10.1080/00207540801950136.  Google Scholar [21] A. Nedic and D. Bertsekas, Incremental Subgradient Methods for Nondifferentiable Optimization, SIAM Journal on Optimization, 12 (2001), 109-138.  doi: 10.1137/S1052623499362111.  Google Scholar [22] T. Nishi, Y. Hiranaka and M. Inuiguchi, Lagrangian relaxation with cut generation for hybrid flowshop scheduling problems to minimize the total weighted tardiness, Computers and Operations Research, 37 (2010), 189-198.  doi: 10.1016/j.cor.2009.04.008.  Google Scholar [23] T. Nishi, Y. Isoya Y and M. Inuiguchi, An integrated column generation and lagrangian relaxation for flowshop scheduling problems, Proceedings of the 2009 IEEE International Conference on Systems, Man and Cybernetics, (2009), 209-304.  doi: 10.1109/ICSMC.2009.5346159.  Google Scholar [24] D. Ouelhadj, P. Cowling and S. Petrovic, Utility and stability measures for agent-based dynamic scheduling of steel continuous casting, Journal of Scheduling, 12 (2009), 417-431.  doi: 10.1109/ROBOT.2003.1241592.  Google Scholar [25] D. Ouelhadj and S. Petrovic, A survey of dynamic scheduling in manufacturing systems, Journal of Scheduling, 12 (2009), 417-431.  doi: 10.1007/s10951-008-0090-8.  Google Scholar [26] D. Ouelhadj, S. Petrovic, P. Cowling and A. Meisels, Inter-agent cooperation and communication for agent-based robust dynamic scheduling in steel production, Advanced Engineering Informatics, 18 (2004), 161-172.  doi: 10.1016/j.aei.2004.10.003.  Google Scholar [27] D. Pacciarelli and M. Pranzo, Production scheduling in a steelmaking-continuous casting plant, Computers and Chemical Engineering, 28 (2004), 2823-2835.  doi: 10.1016/j.compchemeng.2004.08.031.  Google Scholar [28] Q. Pan, L. Wang, K. Mao, J. Zhao and M. Zhang, An Effective Artificial Bee Colony Algorithm for a Real-World Hybrid Flowshop Problem in Steelmaking Process, IEEE Transactions on Automation Science and Engineering, 10 (2013), 307-322.  doi: 10.1109/TASE.2012.2204874.  Google Scholar [29] H. Sherali, G. Choi and C. Tuncbilek, A Variable Target Value Method for Nondifferentiable Optimization, Operation Research Letters, 26 (2000), 1-8.  doi: 10.1016/S0167-6377(99)00063-2.  Google Scholar [30] L. Sun, Research on the Optimal Scheduling Method for the productive Process of Steelmaking-Refining-Continuous Casting, Ph. D thesis, Northeastern University in Shenyang, 2015. Google Scholar [31] L. Tang, J. Liu, A. Rong and Z. Yang, A review of planning and scheduling systems and methods for integrated steel production, European Journal of Operational Research, 133 (2001), 1-20.  doi: 10.1016/S0377-2217(00)00240-X.  Google Scholar [32] L. Tang, P. Luh, J. Liu and L. Fang, Steelmaking process scheduling using Lagrangian relaxation, International Journal of Production Research, 40 (2002), 55-70.   Google Scholar [33] L. Tang, G. Wang and Z. Chen, Integrated charge batching and casting width selection at Baosteel, Operations Research, 62 (2014), 772-787.  doi: 10.1287/opre.2014.1278.  Google Scholar [34] L. Tang, Y. Zhao and J. Liu, An Improved Differential Evolution Algorithm for Practical Dynamic Scheduling in Steelmaking-continuous Casting Production, IEEE Transactions on Evolutionary Computation, 18 (2014), 209-213.  doi: 10.1109/TEVC.2013.2250977.  Google Scholar [35] G. Vieira, J. Hermann and E. Lin, Rescheduling manufacturing systems: a framework of strategies, policies and methods, Journal of Scheduling, 6 (2003), 36-92.  doi: 10.1023/A:1022235519958.  Google Scholar [36] R. Xiong, Y. Fan and C. Wu, A dynamic job shop scheduling method based on Lagrangian relaxation, Tsinghua Science and Technology, 4 (1999), 1297-1302.   Google Scholar [37] H. Xuan and L. Tang, Scheduling a hybrid flowshop with batch production at the last stage, Computers and Operations Research, 34 (2007), 2718-2733.  doi: 10.1016/j.cor.2005.10.014.  Google Scholar [38] S. Yu and Q. Pan, A Rescheduling Method for Operation Time Delay Disturbance in Steelmaking and Continuous Casting Production Process, International Journal of Iron and Steel Research, 19 (2012), 33-41.  doi: 10.1016/S1006-706X(13)60029-1.  Google Scholar [39] H. Zhong, X Dong and H. Shi, Research on the load balancing scheduling problem of reentrant hybrid flowshops, Chinese High Technology Letters, 25 (2015), 70-81.   Google Scholar [40] H. Zhong, Y Zhu and S. Lin, A dynamic co-evolution compact genetic algorithm for E/T problem, The 17th IFAC Symposium on System Identification, (2015), 1433-1437.   Google Scholar

show all references

##### References:
 [1] A. Atighehchian, M. Bijari and H. Tarkesh, A novel hybrid algorithm for scheduling steelmaking continuous casting production, Computers and Operations Research, 36 (2009), 2450-2461.   Google Scholar [2] H. Aytug, M. Lawley, K. McKay, S. Mohan and R. Uzsoy, Executing production schedules in the face of uncertainties: A review and some future directions, European Journal of Operational Research, 161 (2005), 86-110.  doi: 10.1016/j.ejor.2003.08.027.  Google Scholar [3] A. Bellabdaoui and J. Teghem, A mixed-integer linear programming model for the continuous casting planning, International Journal of Production Economics, 104 (2006), 260-270.  doi: 10.1016/j.ijpe.2004.10.016.  Google Scholar [4] D. Bertsekas, Nonlinear Programming, 2$^{nd}$ edition, Athena Scientific, Massachusetts, 1999. doi: 10.1007/978-1-4612-0873-0.  Google Scholar [5] U. Brannlund, On Relaxation Methods for Nonsmooth Convex Optimization, Ph. D thesis, Royal Institute of Technology in Stockholm, 1993.  Google Scholar [6] P. Camerini, L. Fratta and F. Maffioli, On improving relaxation methods by modified gradient techniques, Mathematical Programming Study, 3 (1975), 26-34.   Google Scholar [7] H. Chen and P. Luh, An alternative framework to Lagrangian relaxation approach for job shop scheduling, European Journal of Operational Research, 149 (2003), 499-512.  doi: 10.1016/S0377-2217(02)00470-8.  Google Scholar [8] P. Cowling, D. Ouelhadj and S. Petrovic, Dynamic scheduling of steel casting and milling using multi-agents, Production Planning and Control, 15 (2004), 178-188.  doi: 10.1080/09537280410001662466.  Google Scholar [9] V. Demjanov and V. Somesova, Conditional subdifferentials of convex functions, Soviet Mathematics Doklady, 19 (1978), 1181-1185.   Google Scholar [10] J. Goffin and K. Kiwiel, Convergence of a simple subgradient level method, Mathematical Programming, 85 (1999), 207-211.  doi: 10.1007/s101070050053.  Google Scholar [11] B. Guta, Subgradient Optimization Methods in Integer Programming with an Application to a Radiation Therapy Problem, Ph. D thesis, Teknishe Universitat Kaiserlautern in Kaiserlauter, 2003. Google Scholar [12] I. Harjunkoski and I. Grossmann, A decomposition approach for the scheduling of a steel plant production, Computers and Chemical Engineering, 25 (2001), 1647-1660.  doi: 10.1016/S0098-1354(01)00729-3.  Google Scholar [13] T. Larsson, M. Patriksson and A. Stromberg, Conditional subgradient optimization --theory and applications, European Journal of Operational Research, 88 (1996), 382-403.  doi: 10.1016/0377-2217(94)00200-2.  Google Scholar [14] J. Li, X. Xiao, Q. Tang and C. Floudas, Production scheduling of a Large-scale steelmaking continuous casting process via unit-specific event-based continuous-time models: Short-term and medium-term scheduling, Industrial and Engineering Chemistry Research, 51 (2012), 7300-7319.  doi: 10.1021/ie2015944.  Google Scholar [15] P. Luh and D. Hoitomt, Scheduling of manufacturing systems using the Lagrangian relaxation technique, IEEE Transactions on Automatic Control, 38 (1993), 1066-1079.  doi: 10.1109/9.231461.  Google Scholar [16] P. Luh, D. Hoitomt, E. Max and K. Pattipati, Scheduling generation and reconfiguration for parallel machines, IEEE Transactions on Robotics and Automation, 6 (1990), 687-696.   Google Scholar [17] K. Mao, Q. Pan, X. Pang and T. Chai, A novel Lagrangian relaxation approach for the hybrid flowshop scheduling problem in a steelmaking-continuous casting process, European Journal of Operational Research, 236 (2014), 51-60.  doi: 10.1016/j.ejor.2013.11.010.  Google Scholar [18] K. Mao, Q. Pan, X. Pang and T. Chai, An effective Lagrangian relaxation approach for rescheduling a steelmaking-continuous casting process, Control Engineering Practice, 30 (2014), 67-77.  doi: 10.1016/j.conengprac.2014.06.003.  Google Scholar [19] K. Mao, Q. Pan, X. Pang, T. Chai and P. Luh, An Effective Subgradient Method for Scheduling a Steelmaking-Continuous Casting Process, IEEE Transactions on Automation Science and Engineering, 12 (2014), 1-13.  doi: 10.1109/TASE.2014.2332511.  Google Scholar [20] H. Missbauer, W. Hauber and W. Werner Stadler, A scheduling system for the steelmaking-continuous casting process: A case study from the steelmaking industry, International Journal of Production Research, 47 (2009), 4147-4172.  doi: 10.1080/00207540801950136.  Google Scholar [21] A. Nedic and D. Bertsekas, Incremental Subgradient Methods for Nondifferentiable Optimization, SIAM Journal on Optimization, 12 (2001), 109-138.  doi: 10.1137/S1052623499362111.  Google Scholar [22] T. Nishi, Y. Hiranaka and M. Inuiguchi, Lagrangian relaxation with cut generation for hybrid flowshop scheduling problems to minimize the total weighted tardiness, Computers and Operations Research, 37 (2010), 189-198.  doi: 10.1016/j.cor.2009.04.008.  Google Scholar [23] T. Nishi, Y. Isoya Y and M. Inuiguchi, An integrated column generation and lagrangian relaxation for flowshop scheduling problems, Proceedings of the 2009 IEEE International Conference on Systems, Man and Cybernetics, (2009), 209-304.  doi: 10.1109/ICSMC.2009.5346159.  Google Scholar [24] D. Ouelhadj, P. Cowling and S. Petrovic, Utility and stability measures for agent-based dynamic scheduling of steel continuous casting, Journal of Scheduling, 12 (2009), 417-431.  doi: 10.1109/ROBOT.2003.1241592.  Google Scholar [25] D. Ouelhadj and S. Petrovic, A survey of dynamic scheduling in manufacturing systems, Journal of Scheduling, 12 (2009), 417-431.  doi: 10.1007/s10951-008-0090-8.  Google Scholar [26] D. Ouelhadj, S. Petrovic, P. Cowling and A. Meisels, Inter-agent cooperation and communication for agent-based robust dynamic scheduling in steel production, Advanced Engineering Informatics, 18 (2004), 161-172.  doi: 10.1016/j.aei.2004.10.003.  Google Scholar [27] D. Pacciarelli and M. Pranzo, Production scheduling in a steelmaking-continuous casting plant, Computers and Chemical Engineering, 28 (2004), 2823-2835.  doi: 10.1016/j.compchemeng.2004.08.031.  Google Scholar [28] Q. Pan, L. Wang, K. Mao, J. Zhao and M. Zhang, An Effective Artificial Bee Colony Algorithm for a Real-World Hybrid Flowshop Problem in Steelmaking Process, IEEE Transactions on Automation Science and Engineering, 10 (2013), 307-322.  doi: 10.1109/TASE.2012.2204874.  Google Scholar [29] H. Sherali, G. Choi and C. Tuncbilek, A Variable Target Value Method for Nondifferentiable Optimization, Operation Research Letters, 26 (2000), 1-8.  doi: 10.1016/S0167-6377(99)00063-2.  Google Scholar [30] L. Sun, Research on the Optimal Scheduling Method for the productive Process of Steelmaking-Refining-Continuous Casting, Ph. D thesis, Northeastern University in Shenyang, 2015. Google Scholar [31] L. Tang, J. Liu, A. Rong and Z. Yang, A review of planning and scheduling systems and methods for integrated steel production, European Journal of Operational Research, 133 (2001), 1-20.  doi: 10.1016/S0377-2217(00)00240-X.  Google Scholar [32] L. Tang, P. Luh, J. Liu and L. Fang, Steelmaking process scheduling using Lagrangian relaxation, International Journal of Production Research, 40 (2002), 55-70.   Google Scholar [33] L. Tang, G. Wang and Z. Chen, Integrated charge batching and casting width selection at Baosteel, Operations Research, 62 (2014), 772-787.  doi: 10.1287/opre.2014.1278.  Google Scholar [34] L. Tang, Y. Zhao and J. Liu, An Improved Differential Evolution Algorithm for Practical Dynamic Scheduling in Steelmaking-continuous Casting Production, IEEE Transactions on Evolutionary Computation, 18 (2014), 209-213.  doi: 10.1109/TEVC.2013.2250977.  Google Scholar [35] G. Vieira, J. Hermann and E. Lin, Rescheduling manufacturing systems: a framework of strategies, policies and methods, Journal of Scheduling, 6 (2003), 36-92.  doi: 10.1023/A:1022235519958.  Google Scholar [36] R. Xiong, Y. Fan and C. Wu, A dynamic job shop scheduling method based on Lagrangian relaxation, Tsinghua Science and Technology, 4 (1999), 1297-1302.   Google Scholar [37] H. Xuan and L. Tang, Scheduling a hybrid flowshop with batch production at the last stage, Computers and Operations Research, 34 (2007), 2718-2733.  doi: 10.1016/j.cor.2005.10.014.  Google Scholar [38] S. Yu and Q. Pan, A Rescheduling Method for Operation Time Delay Disturbance in Steelmaking and Continuous Casting Production Process, International Journal of Iron and Steel Research, 19 (2012), 33-41.  doi: 10.1016/S1006-706X(13)60029-1.  Google Scholar [39] H. Zhong, X Dong and H. Shi, Research on the load balancing scheduling problem of reentrant hybrid flowshops, Chinese High Technology Letters, 25 (2015), 70-81.   Google Scholar [40] H. Zhong, Y Zhu and S. Lin, A dynamic co-evolution compact genetic algorithm for E/T problem, The 17th IFAC Symposium on System Identification, (2015), 1433-1437.   Google Scholar
Steelmaking-continuous casting process
Connection between started operations and statuses of an operatio
Illustration of the four performance indexes for the revised scheduling of SCC
The results obtained by SSLRA
 Cast vs. charge(SLLRA) LB UB Gap (%) Time (s) 2 vs.5 937456 983672 4.70 299.5 2 vs.6 1182728 1398892 15.45 313.3 2 vs.7 1497377 1732913 13.59 327.8 2 vs.8 1837244 2252386 18.43 323.7 3 vs.5 1923113 2183725 11.93 309.4 3 vs.6 2487753 2711245 8.24 355.2 3 vs.7 3294573 3690245 10.72 377.2 3 vs.8 3999272 5294742 24.47 466.1 4 vs.5 3100023 3274848 5.34 378.5 4 vs.6 4134749 4591234 9.94 449.2 4 vs.7 5368271 7545422 28.85 598.4 4 vs.8 6650012 9082765 26.78 739.6 5 vs.5 4648823 5119374 9.19 504.7 5 vs.6 6168391 9998116 38.30 663.2 5 vs.7 8102927 11924753 32.05 2199.5 5 vs.8 10373752 13583721 23.63 7824 Average 4106654 7532570 17.60 1008.08
 Cast vs. charge(SLLRA) LB UB Gap (%) Time (s) 2 vs.5 937456 983672 4.70 299.5 2 vs.6 1182728 1398892 15.45 313.3 2 vs.7 1497377 1732913 13.59 327.8 2 vs.8 1837244 2252386 18.43 323.7 3 vs.5 1923113 2183725 11.93 309.4 3 vs.6 2487753 2711245 8.24 355.2 3 vs.7 3294573 3690245 10.72 377.2 3 vs.8 3999272 5294742 24.47 466.1 4 vs.5 3100023 3274848 5.34 378.5 4 vs.6 4134749 4591234 9.94 449.2 4 vs.7 5368271 7545422 28.85 598.4 4 vs.8 6650012 9082765 26.78 739.6 5 vs.5 4648823 5119374 9.19 504.7 5 vs.6 6168391 9998116 38.30 663.2 5 vs.7 8102927 11924753 32.05 2199.5 5 vs.8 10373752 13583721 23.63 7824 Average 4106654 7532570 17.60 1008.08
The results obtained by DCSLA
 Cast vs. charge(DCSLA) LB UB Gap (%) Time (s) 2 vs.5 937456 954151 1.75 1.6 2 vs.6 1182728 1294621 8.64 2.2 2 vs.7 1497377 1519847 1.48 3.2 2 vs.8 1837244 1997636 8.03 1.9 3 vs.5 1923113 2003743 4.02 2 3 vs.6 2487753 2505632 0.71 3.2 3 vs.7 3294573 3349425 1.64 1.5 3 vs.8 3999272 4186443 4.47 1.5 4 vs.5 3100023 3153846 1.71 2.1 4 vs.6 4134749 4200474 1.56 2.7 4 vs.7 5368271 5438362 1.29 3.8 4 vs.8 6650012 6739436 1.33 3.1 5 vs.5 4648823 4753628 2.20 2.4 5 vs.6 6168391 6374522 3.23 3.9 5 vs.7 8102927 9193736 11.86 5.9 5 vs.8 10373752 11376463 8.81 8.8 Average 4106654 4315122 3.92 3.11
 Cast vs. charge(DCSLA) LB UB Gap (%) Time (s) 2 vs.5 937456 954151 1.75 1.6 2 vs.6 1182728 1294621 8.64 2.2 2 vs.7 1497377 1519847 1.48 3.2 2 vs.8 1837244 1997636 8.03 1.9 3 vs.5 1923113 2003743 4.02 2 3 vs.6 2487753 2505632 0.71 3.2 3 vs.7 3294573 3349425 1.64 1.5 3 vs.8 3999272 4186443 4.47 1.5 4 vs.5 3100023 3153846 1.71 2.1 4 vs.6 4134749 4200474 1.56 2.7 4 vs.7 5368271 5438362 1.29 3.8 4 vs.8 6650012 6739436 1.33 3.1 5 vs.5 4648823 4753628 2.20 2.4 5 vs.6 6168391 6374522 3.23 3.9 5 vs.7 8102927 9193736 11.86 5.9 5 vs.8 10373752 11376463 8.81 8.8 Average 4106654 4315122 3.92 3.11
Computational results of DCSLA for SCC rescheduling
 ET Events EV-1 (s) EV-2 (s) EV-3 (min) DG (%) Time (s) IN 1 R1-T2-M1 0 0 22 12.94 76.63 162 2 R1-T3-M1 0 0 15 11.85 66.31 115 3 R1-T1-M2 0 0 16 13.64 75.92 141 4 R1-T2-M2 0 0 19 13.09 58.7 128 5 R1-T3-M2 0 0 21 11.55 44.76 103 6 R1-T1-M3 0 0 18 8.68 121.91 196 7 R1-T2-M3 0 0 11 12.97 134.73 187 8 R1-T3-M3 0 0 10 11.32 83.66 165 9 R2-T2-M1 0 0 19 7.74 9.32 63 10 R2-T3-M1 0 0 12 8.31 8.69 60 11 R2-T1-M2 0 0 16 9.22 12.81 52 12 R2-T2-M2 0 0 19 7.93 9.33 61 13 R2-T3-M2 0 0 22 8.88 8.89 68 14 R2-T1-M3 0 0 14 9.12 15.9 59 15 R2-T2-M3 0 0 16 8.45 7.63 42 16 R2-T3-M3 0 0 11 8.67 8.69 74 Average 0 0 16.31 10.27 46.49 104 (ET: Event Type, IN: Number of Iterations, DG: Duality Gap, EV: Evaluation Values)
 ET Events EV-1 (s) EV-2 (s) EV-3 (min) DG (%) Time (s) IN 1 R1-T2-M1 0 0 22 12.94 76.63 162 2 R1-T3-M1 0 0 15 11.85 66.31 115 3 R1-T1-M2 0 0 16 13.64 75.92 141 4 R1-T2-M2 0 0 19 13.09 58.7 128 5 R1-T3-M2 0 0 21 11.55 44.76 103 6 R1-T1-M3 0 0 18 8.68 121.91 196 7 R1-T2-M3 0 0 11 12.97 134.73 187 8 R1-T3-M3 0 0 10 11.32 83.66 165 9 R2-T2-M1 0 0 19 7.74 9.32 63 10 R2-T3-M1 0 0 12 8.31 8.69 60 11 R2-T1-M2 0 0 16 9.22 12.81 52 12 R2-T2-M2 0 0 19 7.93 9.33 61 13 R2-T3-M2 0 0 22 8.88 8.89 68 14 R2-T1-M3 0 0 14 9.12 15.9 59 15 R2-T2-M3 0 0 16 8.45 7.63 42 16 R2-T3-M3 0 0 11 8.67 8.69 74 Average 0 0 16.31 10.27 46.49 104 (ET: Event Type, IN: Number of Iterations, DG: Duality Gap, EV: Evaluation Values)
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