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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. AtighehchianM. 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. AytugM. LawleyK. McKayS. 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. CameriniL. 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. CowlingD. 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. LarssonM. 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. LiX. XiaoQ. 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. LuhD. HoitomtE. 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. MaoQ. PanX. 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. MaoQ. PanX. 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. MaoQ. PanX. PangT. 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. MissbauerW. 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. NishiY. 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. NishiY. 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. OuelhadjP. 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. OuelhadjS. PetrovicP. 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. PanL. WangK. MaoJ. 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. SheraliG. 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. TangJ. LiuA. 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. TangP. LuhJ. Liu and L. Fang, Steelmaking process scheduling using Lagrangian relaxation, International Journal of Production Research, 40 (2002), 55-70.   Google Scholar

[33]

L. TangG. 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. TangY. 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. VieiraJ. 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. XiongY. 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. ZhongX 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. ZhongY 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. AtighehchianM. 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. AytugM. LawleyK. McKayS. 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. CameriniL. 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. CowlingD. 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. LarssonM. 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. LiX. XiaoQ. 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. LuhD. HoitomtE. 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. MaoQ. PanX. 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. MaoQ. PanX. 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. MaoQ. PanX. PangT. 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. MissbauerW. 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. NishiY. 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. NishiY. 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. OuelhadjP. 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. OuelhadjS. PetrovicP. 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. PanL. WangK. MaoJ. 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. SheraliG. 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. TangJ. LiuA. 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. TangP. LuhJ. Liu and L. Fang, Steelmaking process scheduling using Lagrangian relaxation, International Journal of Production Research, 40 (2002), 55-70.   Google Scholar

[33]

L. TangG. 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. TangY. 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. VieiraJ. 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. XiongY. 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. ZhongX 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. ZhongY 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

Figure 1.  Steelmaking-continuous casting process
Figure 2.  Connection between started operations and statuses of an operatio
Figure 3.  Illustration of the four performance indexes for the revised scheduling of SCC
Table 1.  The results obtained by SSLRA
Cast vs. charge(SLLRA)LBUBGap (%) Time (s)
2 vs.59374569836724.70299.5
2 vs.61182728139889215.45313.3
2 vs.71497377173291313.59327.8
2 vs.81837244225238618.43323.7
3 vs.51923113218372511.93309.4
3 vs.6248775327112458.24355.2
3 vs.73294573369024510.72377.2
3 vs.83999272529474224.47466.1
4 vs.5310002332748485.34378.5
4 vs.6413474945912349.94449.2
4 vs.75368271754542228.85598.4
4 vs.86650012908276526.78739.6
5 vs.5464882351193749.19504.7
5 vs.66168391999811638.30663.2
5 vs.781029271192475332.052199.5
5 vs.8103737521358372123.637824
Average4106654753257017.601008.08
Cast vs. charge(SLLRA)LBUBGap (%) Time (s)
2 vs.59374569836724.70299.5
2 vs.61182728139889215.45313.3
2 vs.71497377173291313.59327.8
2 vs.81837244225238618.43323.7
3 vs.51923113218372511.93309.4
3 vs.6248775327112458.24355.2
3 vs.73294573369024510.72377.2
3 vs.83999272529474224.47466.1
4 vs.5310002332748485.34378.5
4 vs.6413474945912349.94449.2
4 vs.75368271754542228.85598.4
4 vs.86650012908276526.78739.6
5 vs.5464882351193749.19504.7
5 vs.66168391999811638.30663.2
5 vs.781029271192475332.052199.5
5 vs.8103737521358372123.637824
Average4106654753257017.601008.08
Table 2.  The results obtained by DCSLA
Cast vs. charge(DCSLA)LBUBGap (%) Time (s)
2 vs.59374569541511.751.6
2 vs.6118272812946218.642.2
2 vs.7149737715198471.483.2
2 vs.8183724419976368.031.9
3 vs.5192311320037434.022
3 vs.6248775325056320.713.2
3 vs.7329457333494251.641.5
3 vs.8399927241864434.471.5
4 vs.5310002331538461.712.1
4 vs.6413474942004741.562.7
4 vs.7536827154383621.293.8
4 vs.8665001267394361.333.1
5 vs.5464882347536282.202.4
5 vs.6616839163745223.233.9
5 vs.78102927919373611.865.9
5 vs.810373752113764638.818.8
Average410665443151223.923.11
Cast vs. charge(DCSLA)LBUBGap (%) Time (s)
2 vs.59374569541511.751.6
2 vs.6118272812946218.642.2
2 vs.7149737715198471.483.2
2 vs.8183724419976368.031.9
3 vs.5192311320037434.022
3 vs.6248775325056320.713.2
3 vs.7329457333494251.641.5
3 vs.8399927241864434.471.5
4 vs.5310002331538461.712.1
4 vs.6413474942004741.562.7
4 vs.7536827154383621.293.8
4 vs.8665001267394361.333.1
5 vs.5464882347536282.202.4
5 vs.6616839163745223.233.9
5 vs.78102927919373611.865.9
5 vs.810373752113764638.818.8
Average410665443151223.923.11
Table 3.  Computational results of DCSLA for SCC rescheduling
ETEventsEV-1 (s)EV-2 (s)EV-3 (min)DG (%) Time (s)IN
1R1-T2-M1002212.9476.63162
2R1-T3-M1001511.8566.31115
3R1-T1-M2001613.6475.92141
4R1-T2-M2001913.0958.7128
5R1-T3-M2002111.5544.76103
6R1-T1-M300188.68121.91196
7R1-T2-M3001112.97134.73187
8R1-T3-M3001011.3283.66165
9R2-T2-M100197.749.3263
10R2-T3-M100128.318.6960
11R2-T1-M200169.2212.8152
12R2-T2-M200197.939.3361
13R2-T3-M200228.888.8968
14R2-T1-M300149.1215.959
15R2-T2-M300168.457.6342
16R2-T3-M300118.678.6974
Average0016.3110.2746.49104
(ET: Event Type, IN: Number of Iterations, DG: Duality Gap, EV: Evaluation Values)
ETEventsEV-1 (s)EV-2 (s)EV-3 (min)DG (%) Time (s)IN
1R1-T2-M1002212.9476.63162
2R1-T3-M1001511.8566.31115
3R1-T1-M2001613.6475.92141
4R1-T2-M2001913.0958.7128
5R1-T3-M2002111.5544.76103
6R1-T1-M300188.68121.91196
7R1-T2-M3001112.97134.73187
8R1-T3-M3001011.3283.66165
9R2-T2-M100197.749.3263
10R2-T3-M100128.318.6960
11R2-T1-M200169.2212.8152
12R2-T2-M200197.939.3361
13R2-T3-M200228.888.8968
14R2-T1-M300149.1215.959
15R2-T2-M300168.457.6342
16R2-T3-M300118.678.6974
Average0016.3110.2746.49104
(ET: Event Type, IN: Number of Iterations, DG: Duality Gap, EV: Evaluation Values)
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