doi: 10.3934/jimo.2020022

A two-stage solution approach for plastic injection machines scheduling problem

1. 

Eskisehir Osmangazi University, Department of Industrial Engineering, Meselik Campus, 26040, Eskisehir, Turkey

2. 

Eskisehir Technical University, Department of Industrial Engineering, Iki Eylul Campus, 26555, Eskisehir, Turkey

* Corresponding author: Emine Akyol Ozer

Received  August 2018 Revised  March 2019 Published  February 2020

One of the most common plastic manufacturing methods is injection molding. In injection molding process, scheduling of plastic injection machines is very difficult because of the complex nature of the problem. For example, similar plastic parts should be produced sequentially to prevent long setup times. On the other hand, to produce a plastic part, its mold should be fixed on an injection machine. Machine eligibility restrictions should be considered because a mold can be usually fixed on a subset of the injection machines. Some plastic parts which have same shapes but different colors are used same mold so these parts can only be scheduled simultaneously if their mold has copies, otherwise resource constraints should be considered. In this study, a multi-objective mathematical model is proposed for parallel machine scheduling problem to minimize makespan, total tardiness, and total waiting time. Since NP-hard nature of problem, this paper presents a two-stage mathematical model and a two-stage solution approach. In the first stage of mathematical model, jobs are assigned to the machines and each machine is scheduled separately in the second stage. The integrated model and two-stage mathematical model are scalarized by using goal programming, compromise programming and Lexicographic Weighted Tchebycheff programming methods. To solve large-scale problems in a short time, a two-stage solution approach is also proposed. In the first stage of this approach, jobs are assigned to machines and scheduled by using proposed simulated annealing algorithm. In the second stage of the approach, starting time, completion time and waiting time of the jobs are calculated by using a mathematical model. The performance of the methods is demonstrated on randomly generated test problems.

Citation: Tugba Sarac, Aydin Sipahioglu, Emine Akyol Ozer. A two-stage solution approach for plastic injection machines scheduling problem. Journal of Industrial & Management Optimization, doi: 10.3934/jimo.2020022
References:
[1]

M. Afzalirad and M. Shafipour, Design of an efficient genetic algorithm for resource-constrained unrelated parallel machine scheduling problem with machine eligibility restrictions, Journal Of Intelligent Manufacturing, 29 (2018), 423-437.  doi: 10.1007/s10845-015-1117-6.  Google Scholar

[2]

E. Akyol Ozer and T. Sarac, MIP models and a matheuristic algorithm for an identical parallel machine scheduling problem under multiple copies of shared resources constraints, TOP, 27 (2019), 94-124.  doi: 10.1007/s11750-018-00494-x.  Google Scholar

[3]

O. Alagoz and M. Azizoglu, Rescheduling of identical parallel machines under machine eligibility constraints, European Journal of Operational Research, 149 (2003), 523-532.  doi: 10.1016/S0377-2217(02)00499-X.  Google Scholar

[4]

A. Baykasoglu and F. Ozsoydan, Dynamic scheduling of parallel heat treatment furnaces: A case study at a manufacturing system, Journal of Manufacturing Systems, 46 (2018), 152-162.  doi: 10.1016/j.jmsy.2017.12.005.  Google Scholar

[5]

G. Bektur and T. Sarac, A Mathematical Model and Heuristic Algorithms for an Unrelated Parallel Machine Scheduling Problem with Sequence-Dependent Setup Times, Machine Eligibility Restrictions and a Common Server, Computers & Operations Research, 103 (2019), 46-63.  doi: 10.1016/j.cor.2018.10.010.  Google Scholar

[6]

I. A. Chaudhry and P. R. Drake, Minimizing total tardiness for the machine scheduling and worker assignment problems in identical parallel machines using genetic algorithms, International Journal of Advanced Manufacturing Technology, 42 (2009), 581-594.  doi: 10.1007/s00170-008-1617-z.  Google Scholar

[7]

S. G. Dastidar and R. Nagi, Scheduling injection molding operations with multiple resource constraints and sequence dependent setup times and costs, Computers & Operations Research, 32 (2005), 2987-3005.  doi: 10.1016/j.cor.2004.04.012.  Google Scholar

[8]

R. Driessel and L. Moench, Scheduling jobs on parallel machines with sequence dependent setup times precedence constraints and ready times using variable neighborhood search, International Conference on Computers and Industrial Engineering, (2009), 273–278. doi: 10.1109/ICCIE.2009.5223515.  Google Scholar

[9]

E. B. Edis and C. Oguz, Parallel machine scheduling with flexible resources, Computers & Industrial Engineering, 63 (2012), 433-447.  doi: 10.1016/j.cie.2012.03.018.  Google Scholar

[10]

E.B. Edis and I. Ozkarahan, A combined integer/constraint programming approach to a resource constrained parallel machine scheduling problem with machine eligibility restrictions, Engineering Optimization, 43 (2011), 135-157.  doi: 10.1080/03052151003759117.  Google Scholar

[11]

E. B. Edis and I. Ozkarahan, Solution approaches for a real-life resource-constrained parallel machine scheduling problem, International Journal of Advanced Manufacturing Technology, 58 (2012), 1141-1153.  doi: 10.1007/s00170-011-3454-8.  Google Scholar

[12]

A. Ezugwu and F. Akutsah, An Improved Firefly Algorithm for the Unrelated Parallel Machines Scheduling Problem With Sequence-Dependent Setup Times, IEEE ACCESS, 6 (2018), 54459-54478.  doi: 10.1109/ACCESS.2018.2872110.  Google Scholar

[13]

B. GaciasC. Artigues and P. Lopez, Parallel machine scheduling with precedence constraints and setup times, Computers & Operations Research, 37 (2010), 2141-2151.  doi: 10.1016/j.cor.2010.03.003.  Google Scholar

[14]

R. Gokhale and M. Mathirajan, Scheduling identical parallel machines with machine eligibility restrictions to minimize total weighted flow time in automobile gear manufacturing, International Journal of Advanced Manufacturing Technology, 60 (2012), 1099-1110.   Google Scholar

[15]

T. KeskinturkM. B. Yildirim and M. Barut, An ant colony optimization algorithm for load balancing in parallel machines with sequence-dependent setup times, Computers & Operations Research, 39 (2012), 1225-1235.  doi: 10.1016/j.cor.2010.12.003.  Google Scholar

[16]

K. LiY. ShiaS. Yanga and B. Cheng, Parallel machine scheduling problem to minimize the makespan with resource dependent processing times, Applied Soft Computing, 11 (2011), 5551-5557.  doi: 10.1016/j.asoc.2011.05.005.  Google Scholar

[17]

X. LiH. ChehadeF. Yalaoui and L. Amodeo, Fuzzy logic controller based multi-objective meta-heuristics to solve parallel machines scheduling problem, Journal of Multiple-Valued Logic and Soft Computing, 18 (2012), 617-636.   Google Scholar

[18]

S. W. LinZ. J. LeeK. C. Ying and C. C. Lu, Minimization of maximum lateness on parallel machines with sequence-dependent setup times and job release dates, Computers & Operations Research, 38 (2011), 809-815.  doi: 10.1016/j.cor.2010.09.020.  Google Scholar

[19]

M. Liu and C. Wu, Scheduling algorithm based on evolutionary computing in identical parallel machine production line, Robotics and Computer Integrated Manufacturing, 19 (2003), 401-407.  doi: 10.1016/S0736-5845(03)00041-3.  Google Scholar

[20]

T. ParkT. Lee and C. O. Kim, Due-date scheduling on parallel machines with job splitting and sequence-dependent major/minor setup times, International Journal of Advanced Manufacturing Technology, 59 (2012), 325-333.  doi: 10.1007/s00170-011-3489-x.  Google Scholar

[21]

R. Ruiz and C. A. Romano, Scheduling unrelated parallel machines with resource-assignable sequence-dependent setup times, International Journal of Advanced Manufacturing Technology, 57 (2011), 777-794.  doi: 10.1007/s00170-011-3318-2.  Google Scholar

[22]

T. Sarac and A. Sipahioglu, Plastik enjeksiyon makinalarinin Çizelgelenmesi problemi, Journal of Industrial Engineering, 20 (2009), 2-14.   Google Scholar

[23]

L. Su, W. Y. Chang and F. D. Chou, Minimizing maximum lateness on identical parallel machines with flexible resources and machine eligibility constraints, International Journal of Advanced Manufacturing Technology, 56 (2011), 1195. doi: 10.1007/s00170-011-3236-3.  Google Scholar

[24]

I. T. TanevT. Uozumi and Y. Morotome, Hybrid evolutionary algorithm-based real-world flexible job shop scheduling problem: Application service provider approach, Applied Soft Computing, 5 (2004), 87-100.  doi: 10.1016/j.asoc.2004.03.013.  Google Scholar

[25]

A. K. Turker and C. Sel, A Hybrid approach on single server parallel machines scheduling problem with sequence dependent setup times, Journal of the Faculty of Engineering and Architecture of Gazi University, 26 (2011), 731-740.   Google Scholar

[26]

Y. Unlu and S. J. Mason, Evaluation of mixed integer programming formulations for non-preemptive parallel machine scheduling problems, Computers & Industrial Engineering, 58 (2010), 785-800.  doi: 10.1016/j.cie.2010.02.012.  Google Scholar

show all references

References:
[1]

M. Afzalirad and M. Shafipour, Design of an efficient genetic algorithm for resource-constrained unrelated parallel machine scheduling problem with machine eligibility restrictions, Journal Of Intelligent Manufacturing, 29 (2018), 423-437.  doi: 10.1007/s10845-015-1117-6.  Google Scholar

[2]

E. Akyol Ozer and T. Sarac, MIP models and a matheuristic algorithm for an identical parallel machine scheduling problem under multiple copies of shared resources constraints, TOP, 27 (2019), 94-124.  doi: 10.1007/s11750-018-00494-x.  Google Scholar

[3]

O. Alagoz and M. Azizoglu, Rescheduling of identical parallel machines under machine eligibility constraints, European Journal of Operational Research, 149 (2003), 523-532.  doi: 10.1016/S0377-2217(02)00499-X.  Google Scholar

[4]

A. Baykasoglu and F. Ozsoydan, Dynamic scheduling of parallel heat treatment furnaces: A case study at a manufacturing system, Journal of Manufacturing Systems, 46 (2018), 152-162.  doi: 10.1016/j.jmsy.2017.12.005.  Google Scholar

[5]

G. Bektur and T. Sarac, A Mathematical Model and Heuristic Algorithms for an Unrelated Parallel Machine Scheduling Problem with Sequence-Dependent Setup Times, Machine Eligibility Restrictions and a Common Server, Computers & Operations Research, 103 (2019), 46-63.  doi: 10.1016/j.cor.2018.10.010.  Google Scholar

[6]

I. A. Chaudhry and P. R. Drake, Minimizing total tardiness for the machine scheduling and worker assignment problems in identical parallel machines using genetic algorithms, International Journal of Advanced Manufacturing Technology, 42 (2009), 581-594.  doi: 10.1007/s00170-008-1617-z.  Google Scholar

[7]

S. G. Dastidar and R. Nagi, Scheduling injection molding operations with multiple resource constraints and sequence dependent setup times and costs, Computers & Operations Research, 32 (2005), 2987-3005.  doi: 10.1016/j.cor.2004.04.012.  Google Scholar

[8]

R. Driessel and L. Moench, Scheduling jobs on parallel machines with sequence dependent setup times precedence constraints and ready times using variable neighborhood search, International Conference on Computers and Industrial Engineering, (2009), 273–278. doi: 10.1109/ICCIE.2009.5223515.  Google Scholar

[9]

E. B. Edis and C. Oguz, Parallel machine scheduling with flexible resources, Computers & Industrial Engineering, 63 (2012), 433-447.  doi: 10.1016/j.cie.2012.03.018.  Google Scholar

[10]

E.B. Edis and I. Ozkarahan, A combined integer/constraint programming approach to a resource constrained parallel machine scheduling problem with machine eligibility restrictions, Engineering Optimization, 43 (2011), 135-157.  doi: 10.1080/03052151003759117.  Google Scholar

[11]

E. B. Edis and I. Ozkarahan, Solution approaches for a real-life resource-constrained parallel machine scheduling problem, International Journal of Advanced Manufacturing Technology, 58 (2012), 1141-1153.  doi: 10.1007/s00170-011-3454-8.  Google Scholar

[12]

A. Ezugwu and F. Akutsah, An Improved Firefly Algorithm for the Unrelated Parallel Machines Scheduling Problem With Sequence-Dependent Setup Times, IEEE ACCESS, 6 (2018), 54459-54478.  doi: 10.1109/ACCESS.2018.2872110.  Google Scholar

[13]

B. GaciasC. Artigues and P. Lopez, Parallel machine scheduling with precedence constraints and setup times, Computers & Operations Research, 37 (2010), 2141-2151.  doi: 10.1016/j.cor.2010.03.003.  Google Scholar

[14]

R. Gokhale and M. Mathirajan, Scheduling identical parallel machines with machine eligibility restrictions to minimize total weighted flow time in automobile gear manufacturing, International Journal of Advanced Manufacturing Technology, 60 (2012), 1099-1110.   Google Scholar

[15]

T. KeskinturkM. B. Yildirim and M. Barut, An ant colony optimization algorithm for load balancing in parallel machines with sequence-dependent setup times, Computers & Operations Research, 39 (2012), 1225-1235.  doi: 10.1016/j.cor.2010.12.003.  Google Scholar

[16]

K. LiY. ShiaS. Yanga and B. Cheng, Parallel machine scheduling problem to minimize the makespan with resource dependent processing times, Applied Soft Computing, 11 (2011), 5551-5557.  doi: 10.1016/j.asoc.2011.05.005.  Google Scholar

[17]

X. LiH. ChehadeF. Yalaoui and L. Amodeo, Fuzzy logic controller based multi-objective meta-heuristics to solve parallel machines scheduling problem, Journal of Multiple-Valued Logic and Soft Computing, 18 (2012), 617-636.   Google Scholar

[18]

S. W. LinZ. J. LeeK. C. Ying and C. C. Lu, Minimization of maximum lateness on parallel machines with sequence-dependent setup times and job release dates, Computers & Operations Research, 38 (2011), 809-815.  doi: 10.1016/j.cor.2010.09.020.  Google Scholar

[19]

M. Liu and C. Wu, Scheduling algorithm based on evolutionary computing in identical parallel machine production line, Robotics and Computer Integrated Manufacturing, 19 (2003), 401-407.  doi: 10.1016/S0736-5845(03)00041-3.  Google Scholar

[20]

T. ParkT. Lee and C. O. Kim, Due-date scheduling on parallel machines with job splitting and sequence-dependent major/minor setup times, International Journal of Advanced Manufacturing Technology, 59 (2012), 325-333.  doi: 10.1007/s00170-011-3489-x.  Google Scholar

[21]

R. Ruiz and C. A. Romano, Scheduling unrelated parallel machines with resource-assignable sequence-dependent setup times, International Journal of Advanced Manufacturing Technology, 57 (2011), 777-794.  doi: 10.1007/s00170-011-3318-2.  Google Scholar

[22]

T. Sarac and A. Sipahioglu, Plastik enjeksiyon makinalarinin Çizelgelenmesi problemi, Journal of Industrial Engineering, 20 (2009), 2-14.   Google Scholar

[23]

L. Su, W. Y. Chang and F. D. Chou, Minimizing maximum lateness on identical parallel machines with flexible resources and machine eligibility constraints, International Journal of Advanced Manufacturing Technology, 56 (2011), 1195. doi: 10.1007/s00170-011-3236-3.  Google Scholar

[24]

I. T. TanevT. Uozumi and Y. Morotome, Hybrid evolutionary algorithm-based real-world flexible job shop scheduling problem: Application service provider approach, Applied Soft Computing, 5 (2004), 87-100.  doi: 10.1016/j.asoc.2004.03.013.  Google Scholar

[25]

A. K. Turker and C. Sel, A Hybrid approach on single server parallel machines scheduling problem with sequence dependent setup times, Journal of the Faculty of Engineering and Architecture of Gazi University, 26 (2011), 731-740.   Google Scholar

[26]

Y. Unlu and S. J. Mason, Evaluation of mixed integer programming formulations for non-preemptive parallel machine scheduling problems, Computers & Industrial Engineering, 58 (2010), 785-800.  doi: 10.1016/j.cie.2010.02.012.  Google Scholar

Figure 1.  Gantt Chart of Toy Example
Figure 2.  Number of Non-Dominated Solutions
Table 1.  Literature Review
Ref. $ \alpha $ $ \beta $ $ \gamma $ Solution Methods
[24] $ FJ_c $ Resource Number of Tardy Jobs Hybrid evolutionary algorithm
[8] $ P_m $ $ r_j $, prec, $ s_{ij} $ $ \sum w_j T_j $ Variable neighborhood search
[7] $ P_m $ Resource, $ s_{ij} $ Inventory holding, backlogging and setup time Work center based decomposition approach
[15] $ P_m $ $ s_{ij} $ Average relative percentage of imbalance GA and ant colony optimization
[16] $ P_m $ $ r_j $, $ s_{ij} $ $ C_{max} $, $ \sum T_j $ 0-1 MIP model and GA with a fuzzy logic controller
[18] $ P_m $ $ r_j $, $ s_{ij} $ Maximum tardiness($ T_{max} $) Greedy Algorithm
[25] $ P_m $ $ s_{ij} $ $ C_{max} $ A hybrid GA and tabu search approach
[3] $ P_m $ $ M_j $ $ \sum C_j $ Optimizing Algorithm
[23] $ P_m $ $ M_j $ $ T_{max} $ A network flow mathematical programming model and a heuristic approach
[17] $ P_m $ $ M_j $ $ C_{max} $ Simulated annealing algorithm
[11] $ P_m $ Resource, $ M_j $ $ C_{max} $ Integer programming and constraint programming model
[21] $ R_m $ Resource $ s_{ij} $ $ \sum C_j $, total amount of resources assigned Mixed integer programming model
[11] $ P_m $ Resource, $ M_j $ $ C_{max} $ Integer programming, constraint programming model
[10] $ P_m $ Resource, $ M_j $ $ C_{max} $ A Combined Integer/Constraint Programming Approach
[9] $ P_m $ Resource $ C_{max} $ Integer programming, constraint programming model
[1] $ R_m $ $ M_j $ $ C_{max} $ An integer programming model, GA
[12] $ R_m $ $ s_{ij} $ $ C_{max} $ Firefly algorithm
[4] $ P_m $ $ r_j $, $ s_{ij} $, $ M_j $ Energy consumption and annual income A multi-start and constructive search algorithm
[2] $ P_m $ Resource, $ s_{ij} $, $ M_j $ $ \sum w_j C_j $ MIP models and a matheuristic algorithm
Ref. $ \alpha $ $ \beta $ $ \gamma $ Solution Methods
[24] $ FJ_c $ Resource Number of Tardy Jobs Hybrid evolutionary algorithm
[8] $ P_m $ $ r_j $, prec, $ s_{ij} $ $ \sum w_j T_j $ Variable neighborhood search
[7] $ P_m $ Resource, $ s_{ij} $ Inventory holding, backlogging and setup time Work center based decomposition approach
[15] $ P_m $ $ s_{ij} $ Average relative percentage of imbalance GA and ant colony optimization
[16] $ P_m $ $ r_j $, $ s_{ij} $ $ C_{max} $, $ \sum T_j $ 0-1 MIP model and GA with a fuzzy logic controller
[18] $ P_m $ $ r_j $, $ s_{ij} $ Maximum tardiness($ T_{max} $) Greedy Algorithm
[25] $ P_m $ $ s_{ij} $ $ C_{max} $ A hybrid GA and tabu search approach
[3] $ P_m $ $ M_j $ $ \sum C_j $ Optimizing Algorithm
[23] $ P_m $ $ M_j $ $ T_{max} $ A network flow mathematical programming model and a heuristic approach
[17] $ P_m $ $ M_j $ $ C_{max} $ Simulated annealing algorithm
[11] $ P_m $ Resource, $ M_j $ $ C_{max} $ Integer programming and constraint programming model
[21] $ R_m $ Resource $ s_{ij} $ $ \sum C_j $, total amount of resources assigned Mixed integer programming model
[11] $ P_m $ Resource, $ M_j $ $ C_{max} $ Integer programming, constraint programming model
[10] $ P_m $ Resource, $ M_j $ $ C_{max} $ A Combined Integer/Constraint Programming Approach
[9] $ P_m $ Resource $ C_{max} $ Integer programming, constraint programming model
[1] $ R_m $ $ M_j $ $ C_{max} $ An integer programming model, GA
[12] $ R_m $ $ s_{ij} $ $ C_{max} $ Firefly algorithm
[4] $ P_m $ $ r_j $, $ s_{ij} $, $ M_j $ Energy consumption and annual income A multi-start and constructive search algorithm
[2] $ P_m $ Resource, $ s_{ij} $, $ M_j $ $ \sum w_j C_j $ MIP models and a matheuristic algorithm
Table 2.  Results of Small-scale ($ n $ = 8, $ m $ = 2) Problems
Test No 1-1 2-1 3-1 4-1 5-1 6-1 7-1 8-1
M0-WGP $ f_1 $ 384 273 326 394 346 335 318 280
$ f_2 $ 299 99 296 564 116 295 232 102
$ f_3 $ 4 7 3 0 0 0 0 0
$ z $ 0.1 0.07 0.09 0.24 0.07 0.08 0.11 0.05
$ t(sec.) $ 196 181 299 880 153 175 82 95
M0-CP $ f_1 $ 384 273 343 388 326 383 328 280
$ f_2 $ 299 99 281 506 157 258 222 102
$ f_3 $ 4 7 3 87 14 0 16 0
$ z $ 0.1 0.07 0.09 0.29 0.09 0.08 0.15 0.05
$ t(sec.) $ 322 73 212 522 111 183 85 74
M0-LWT $ f_1 $ 384 273 343 388 326 383 328 280
$ f_2 $ 299 99 281 506 157 258 222 102
$ f_3 $ 4 7 3 87 14 0 16 0
$ z $ 0.1 0.07 0.09 0.29 0.09 0.08 0.15 0.05
$ t(sec.) $ 343 97 323 996 186 194 284 119
M1-M2-WGP $ f_1 $ 423 313 345 433 331 335 310 266
$ f_2 $ 465 130 333 796 291 295 485 141
$ f_3 $ 0 0 0 114 0 0 0 0
$ z $ 0.14 0.09 0.10 0.42 0.11 0.08 0.19 0.05
$ t(sec.) $ 1 1 1 4 1 1 1 1
M1-M2-CP $ f_1 $ 450 320 345 393 380 335 350 351
$ f_2 $ 636 144 333 701 237 295 479 208
$ f_3 $ 0 7 0 167 0 0 0 13
$ z $ 0.19 0.1 0.1 0.42 0.11 0.08 0.19 0.1
$ t(sec.) $ 2 1 0 2 1 1 1 1
M1-M2-LWT $ f_1 $ 450 320 345 393 390 335 350 351
$ f_2 $ 636 144 333 701 237 295 479 208
$ f_3 $ 0 7 0 167 10 0 0 13
$ z $ 0.19 0.1 0.1 0.42 0.12 0.08 0.19 0.1
$ t(sec.) $ 8 3 6 14 4 3 5 3
SA-M3 $ f_1 $ 473 320 411 366 402 383 328 295
$ f_2 $ 456 82 349 717 335 258 222 99
$ f_3 $ 90 16 37 112 127 0 16 0
$ z $ 0.19 0.08 0.13 0.37 0.25 0.08 0.15 0.05
$ t(sec.) $ 13 12 13 13 12 12 12 13
Test No 1-1 2-1 3-1 4-1 5-1 6-1 7-1 8-1
M0-WGP $ f_1 $ 384 273 326 394 346 335 318 280
$ f_2 $ 299 99 296 564 116 295 232 102
$ f_3 $ 4 7 3 0 0 0 0 0
$ z $ 0.1 0.07 0.09 0.24 0.07 0.08 0.11 0.05
$ t(sec.) $ 196 181 299 880 153 175 82 95
M0-CP $ f_1 $ 384 273 343 388 326 383 328 280
$ f_2 $ 299 99 281 506 157 258 222 102
$ f_3 $ 4 7 3 87 14 0 16 0
$ z $ 0.1 0.07 0.09 0.29 0.09 0.08 0.15 0.05
$ t(sec.) $ 322 73 212 522 111 183 85 74
M0-LWT $ f_1 $ 384 273 343 388 326 383 328 280
$ f_2 $ 299 99 281 506 157 258 222 102
$ f_3 $ 4 7 3 87 14 0 16 0
$ z $ 0.1 0.07 0.09 0.29 0.09 0.08 0.15 0.05
$ t(sec.) $ 343 97 323 996 186 194 284 119
M1-M2-WGP $ f_1 $ 423 313 345 433 331 335 310 266
$ f_2 $ 465 130 333 796 291 295 485 141
$ f_3 $ 0 0 0 114 0 0 0 0
$ z $ 0.14 0.09 0.10 0.42 0.11 0.08 0.19 0.05
$ t(sec.) $ 1 1 1 4 1 1 1 1
M1-M2-CP $ f_1 $ 450 320 345 393 380 335 350 351
$ f_2 $ 636 144 333 701 237 295 479 208
$ f_3 $ 0 7 0 167 0 0 0 13
$ z $ 0.19 0.1 0.1 0.42 0.11 0.08 0.19 0.1
$ t(sec.) $ 2 1 0 2 1 1 1 1
M1-M2-LWT $ f_1 $ 450 320 345 393 390 335 350 351
$ f_2 $ 636 144 333 701 237 295 479 208
$ f_3 $ 0 7 0 167 10 0 0 13
$ z $ 0.19 0.1 0.1 0.42 0.12 0.08 0.19 0.1
$ t(sec.) $ 8 3 6 14 4 3 5 3
SA-M3 $ f_1 $ 473 320 411 366 402 383 328 295
$ f_2 $ 456 82 349 717 335 258 222 99
$ f_3 $ 90 16 37 112 127 0 16 0
$ z $ 0.19 0.08 0.13 0.37 0.25 0.08 0.15 0.05
$ t(sec.) $ 13 12 13 13 12 12 12 13
Table 3.  Results of Small-scale ($ n $ = 8, $ m $ = 3) Problems
Test No 9-1 10-1 11-1 12-1 13-1 14-1 15-1 16-1
M0-WGP $ f_1 $ 296 331 391 239 154 283 183 262
$ f_2 $ 311 325 393 135 191 331 124 117
$ f_3 $ 2 0 0 2 0 0 0 0
$ z $ 0.08 0.13 0.14 0.06 0.06 0.08 0.07 0.04
$ t(sec.) $ 618 2211 1774 337 415 497 439 246
M0-CP $ f_1 $ 296 331 370 198 154 283 143 262
$ f_2 $ 311 325 326 122 191 331 94 117
$ f_3 $ 2 0 70 24 0 0 17 0
$ z $ 0.08 0.13 0.15 0.06 0.06 0.08 0.08 0.04
$ t(sec.) $ 691 1227 1414 401 403 539 252 169
M0-LWT $ f_1 $ 296 331 370 198 154 283 143 262
$ f_2 $ 311 325 326 122 191 331 94 117
$ f_3 $ 2 0 70 24 0 0 17 0
$ z $ 0.08 0.13 0.15 0.06 0.06 0.08 0.08 0.04
$ t(sec.) $ 1045 1231 1980 354 402 876 298 108
M1-M2-WGP $ f_1 $ 341 492 409 286 223 282 243 295
$ f_2 $ 495 870 428 344 431 500 295 342
$ f_3 $ 14 327 155 8 0 0 0 0
$ z $ 0.12 0.58 0.23 0.13 0.14 0.12 0.15 0.09
$ t(sec.) $ 1 2 0 1 1 1 1 1
M1-M2-CP $ f_1 $ 334 411 470 297 230 345 189 530
$ f_2 $ 488 791 567 263 430 600 273 1073
$ f_3 $ 7 274 305 89 0 7 21 0
$ z $ 0.11 0.5 0.34 0.16 0.14 0.15 0.16 0.26
$ t(sec.) $ 1 1 1 1 0 0 0 24
M1-M2-LWT $ f_1 $ 341 411 470 297 230 338 189 295
$ f_2 $ 495 791 567 263 430 600 273 344
$ f_3 $ 14 274 305 89 0 0 21 0
$ z $ 0.12 0.5 0.34 0.16 0.14 0.15 0.16 0.09
$ t(sec.) $ 2 6 3 3 2 2 2 1
SA-M3 $ f_1 $ 369 416 469 253 185 336 143 309
$ f_2 $ 317 571 639 128 186 407 94 116
$ f_3 $ 50 282 304 80 44 85 17 0
$ z $ 0.11 0.46 0.36 0.11 0.16 0.18 0.08 0.05
$ t(sec.) $ 12 14 12 13 12 14 14 13
Test No 9-1 10-1 11-1 12-1 13-1 14-1 15-1 16-1
M0-WGP $ f_1 $ 296 331 391 239 154 283 183 262
$ f_2 $ 311 325 393 135 191 331 124 117
$ f_3 $ 2 0 0 2 0 0 0 0
$ z $ 0.08 0.13 0.14 0.06 0.06 0.08 0.07 0.04
$ t(sec.) $ 618 2211 1774 337 415 497 439 246
M0-CP $ f_1 $ 296 331 370 198 154 283 143 262
$ f_2 $ 311 325 326 122 191 331 94 117
$ f_3 $ 2 0 70 24 0 0 17 0
$ z $ 0.08 0.13 0.15 0.06 0.06 0.08 0.08 0.04
$ t(sec.) $ 691 1227 1414 401 403 539 252 169
M0-LWT $ f_1 $ 296 331 370 198 154 283 143 262
$ f_2 $ 311 325 326 122 191 331 94 117
$ f_3 $ 2 0 70 24 0 0 17 0
$ z $ 0.08 0.13 0.15 0.06 0.06 0.08 0.08 0.04
$ t(sec.) $ 1045 1231 1980 354 402 876 298 108
M1-M2-WGP $ f_1 $ 341 492 409 286 223 282 243 295
$ f_2 $ 495 870 428 344 431 500 295 342
$ f_3 $ 14 327 155 8 0 0 0 0
$ z $ 0.12 0.58 0.23 0.13 0.14 0.12 0.15 0.09
$ t(sec.) $ 1 2 0 1 1 1 1 1
M1-M2-CP $ f_1 $ 334 411 470 297 230 345 189 530
$ f_2 $ 488 791 567 263 430 600 273 1073
$ f_3 $ 7 274 305 89 0 7 21 0
$ z $ 0.11 0.5 0.34 0.16 0.14 0.15 0.16 0.26
$ t(sec.) $ 1 1 1 1 0 0 0 24
M1-M2-LWT $ f_1 $ 341 411 470 297 230 338 189 295
$ f_2 $ 495 791 567 263 430 600 273 344
$ f_3 $ 14 274 305 89 0 0 21 0
$ z $ 0.12 0.5 0.34 0.16 0.14 0.15 0.16 0.09
$ t(sec.) $ 2 6 3 3 2 2 2 1
SA-M3 $ f_1 $ 369 416 469 253 185 336 143 309
$ f_2 $ 317 571 639 128 186 407 94 116
$ f_3 $ 50 282 304 80 44 85 17 0
$ z $ 0.11 0.46 0.36 0.11 0.16 0.18 0.08 0.05
$ t(sec.) $ 12 14 12 13 12 14 14 13
Table 4.  Results of Medium-scale ($ n $ = 40, $ m $ = 2) Problems
Test No 1-1 2-1 3-1 4-1 5-1 6-1 7-1 8-1
M0-GP $ f_1 $ 2365 2135 1928 2115 1574 1805 - 1817
$ f_2 $ 22190 20719 15318 19944 4590 8989 - 10340
$ f_3 $ 396 94 130 124 0 0 - 0
$ z $ 0.27 0.23 0.19 0.28 0.07 0.11 - 0.12
M0-CP $ f_1 $ 2358 2274 1965 1625 1491 1772 1656 1821
$ f_2 $ 18290 17933 15682 14654 4761 5670 8345 9159
$ f_3 $ 280 416 364 301 18 56 0 78
$ z $ 0.23 0.25 0.22 0.22 0.07 0.08 0.10 0.12
M0-LWT $ f_1 $ 5776 8558 7284 2547 2764 5648 2013 4304
$ f_2 $ 21808 33944 26740 56966 7363 16720 19845 14963
$ f_3 $ 577 1458 657 1070 91 240 0 420
$ z $ 0.43 0.74 0.56 0.80 0.16 0.37 0.22 0.33
M1-M2-WGP $ f_1 $ 2077 2022 1779 1771 1447 1789 1474 1463
$ f_2 $ 21191 19565 17681 16175 4779 15204 5719 13613
$ f_3 $ 76 178 4 198 0 0 0 12
$ z $ 0.22 0.23 0.19 0.23 0.06 0.16 0.07 0.14
M1-M2-CP $ f_1 $ 1979 1992 1865 1546 1467 1635 1440 1663
$ f_2 $ 15322 20602 14408 13934 3348 6396 4255 7751
$ f_3 $ 467 128 185 306 0 2 0 186
$ z $ 0.21 0.23 0.18 0.21 0.05 0.08 0.05 0.12
M1-M2-LWT $ f_1 $ 5232 5076 6301 5395 2624 3838 3360 2862
$ f_2 $ 19227 18025 22470 18168 6768 11926 10643 8291
$ f_3 $ 593 489 349 382 83 13 0 41
$ z $ 0.38 0.37 0.45 0.43 0.14 0.22 0.19 0.15
SA-M3 $ f_1 $ 1819 1561 1623 1346 1473 1621 1399 1776
$ f_2 $ 8153 5314 3941 4380 2609 1347 2687 3086
$ f_3 $ 479 82 137 290 0 0 0 357
$ z $ 0.14 0.07 0.07 0.1 0.04 0.03 0.04 0.08
$ t(sec.) $ 36 32 38 40 30 31 36 36
Test No 1-1 2-1 3-1 4-1 5-1 6-1 7-1 8-1
M0-GP $ f_1 $ 2365 2135 1928 2115 1574 1805 - 1817
$ f_2 $ 22190 20719 15318 19944 4590 8989 - 10340
$ f_3 $ 396 94 130 124 0 0 - 0
$ z $ 0.27 0.23 0.19 0.28 0.07 0.11 - 0.12
M0-CP $ f_1 $ 2358 2274 1965 1625 1491 1772 1656 1821
$ f_2 $ 18290 17933 15682 14654 4761 5670 8345 9159
$ f_3 $ 280 416 364 301 18 56 0 78
$ z $ 0.23 0.25 0.22 0.22 0.07 0.08 0.10 0.12
M0-LWT $ f_1 $ 5776 8558 7284 2547 2764 5648 2013 4304
$ f_2 $ 21808 33944 26740 56966 7363 16720 19845 14963
$ f_3 $ 577 1458 657 1070 91 240 0 420
$ z $ 0.43 0.74 0.56 0.80 0.16 0.37 0.22 0.33
M1-M2-WGP $ f_1 $ 2077 2022 1779 1771 1447 1789 1474 1463
$ f_2 $ 21191 19565 17681 16175 4779 15204 5719 13613
$ f_3 $ 76 178 4 198 0 0 0 12
$ z $ 0.22 0.23 0.19 0.23 0.06 0.16 0.07 0.14
M1-M2-CP $ f_1 $ 1979 1992 1865 1546 1467 1635 1440 1663
$ f_2 $ 15322 20602 14408 13934 3348 6396 4255 7751
$ f_3 $ 467 128 185 306 0 2 0 186
$ z $ 0.21 0.23 0.18 0.21 0.05 0.08 0.05 0.12
M1-M2-LWT $ f_1 $ 5232 5076 6301 5395 2624 3838 3360 2862
$ f_2 $ 19227 18025 22470 18168 6768 11926 10643 8291
$ f_3 $ 593 489 349 382 83 13 0 41
$ z $ 0.38 0.37 0.45 0.43 0.14 0.22 0.19 0.15
SA-M3 $ f_1 $ 1819 1561 1623 1346 1473 1621 1399 1776
$ f_2 $ 8153 5314 3941 4380 2609 1347 2687 3086
$ f_3 $ 479 82 137 290 0 0 0 357
$ z $ 0.14 0.07 0.07 0.1 0.04 0.03 0.04 0.08
$ t(sec.) $ 36 32 38 40 30 31 36 36
Table 5.  Results of Medium-scale ($ n $ = 40, $ m $ = 6) Problems
Test No 9-1 10-1 11-1 12-1 13-1 14-1 15-1 16-1
M0-WGP $ f_1 $ 1539 1896 1515 - 645 709 - 1336
$ f_2 $ 17499 25024 19640 - 3198 4627 - 14430
$ f_3 $ 1092 1396 3426 - 0 3 - 1094
$ z $ 0.27 0.43 0.51 - 0.03 0.05 - 0.29
M0-CP $ f_1 $ 1487 2179 1515 1798 756 784 815 1219
$ f_2 $ 18635 24170 19422 20895 3262 3674 6330 10974
$ f_3 $ 1530 2531 3777 1862 66 7 94 869
$ z $ 0.31 0.55 0.54 0.43 0.04 0.05 0.08 0.23
M0-LWT $ f_1 $ 4093 6059 10466 16743 1002 1947 1726 3677
$ f_2 $ 17052 23743 44256 70611 3124 7355 6522 14022
$ f_3 $ 1434 1912 3808 5674 156 333 126 830
$ z $ 0.38 0.62 1.0 1.0 0.06 0.16 0.12 0.33
M1-M2-WGP $ f_1 $ 1261 1608 1205 1136 598 578 528 633
$ f_2 $ 11303 15041 12840 10871 2202 4029 2336 3878
$ f_3 $ 1350 2117 1138 1645 0 3 2 6
$ z $ 0.23 0.41 0.23 0.29 0.02 0.04 0.02 0.04
M1-M2-CP $ f_1 $ 1278 1674 1217 1069 627 619 531 712
$ f_2 $ 11048 16595 12358 9831 2197 2454 1988 3554
$ f_3 $ 1486 2512 1140 1795 92 50 5 190
$ z $ 0.25 0.47 0.23 0.3 0.04 0.04 0.02 0.06
M1-M2-LWT $ f_1 $ 4806 8617 4168 5018 841 906 727 1253
$ f_2 $ 20346 34350 16584 20148 2316 2578 1798 3606
$ f_3 $ 1711 2793 1427 1619 11 10 56 79
$ z $ 0.46 0.91 0.39 0.51 0.03 0.04 0.04 0.07
SA-M3 $ f_1 $ 1249 1807 1047 977 876 651 639 738
$ f_2 $ 9711 14617 6886 5216 2845 1635 1121 1539
$ f_3 $ 1941 2752 1318 1327 315 28 39 457
$ z $ 0.27 0.48 0.20 0.21 0.08 0.03 0.03 0.08
$ t(sec.) $ 231 232 198 235 34 33 37 38
Test No 9-1 10-1 11-1 12-1 13-1 14-1 15-1 16-1
M0-WGP $ f_1 $ 1539 1896 1515 - 645 709 - 1336
$ f_2 $ 17499 25024 19640 - 3198 4627 - 14430
$ f_3 $ 1092 1396 3426 - 0 3 - 1094
$ z $ 0.27 0.43 0.51 - 0.03 0.05 - 0.29
M0-CP $ f_1 $ 1487 2179 1515 1798 756 784 815 1219
$ f_2 $ 18635 24170 19422 20895 3262 3674 6330 10974
$ f_3 $ 1530 2531 3777 1862 66 7 94 869
$ z $ 0.31 0.55 0.54 0.43 0.04 0.05 0.08 0.23
M0-LWT $ f_1 $ 4093 6059 10466 16743 1002 1947 1726 3677
$ f_2 $ 17052 23743 44256 70611 3124 7355 6522 14022
$ f_3 $ 1434 1912 3808 5674 156 333 126 830
$ z $ 0.38 0.62 1.0 1.0 0.06 0.16 0.12 0.33
M1-M2-WGP $ f_1 $ 1261 1608 1205 1136 598 578 528 633
$ f_2 $ 11303 15041 12840 10871 2202 4029 2336 3878
$ f_3 $ 1350 2117 1138 1645 0 3 2 6
$ z $ 0.23 0.41 0.23 0.29 0.02 0.04 0.02 0.04
M1-M2-CP $ f_1 $ 1278 1674 1217 1069 627 619 531 712
$ f_2 $ 11048 16595 12358 9831 2197 2454 1988 3554
$ f_3 $ 1486 2512 1140 1795 92 50 5 190
$ z $ 0.25 0.47 0.23 0.3 0.04 0.04 0.02 0.06
M1-M2-LWT $ f_1 $ 4806 8617 4168 5018 841 906 727 1253
$ f_2 $ 20346 34350 16584 20148 2316 2578 1798 3606
$ f_3 $ 1711 2793 1427 1619 11 10 56 79
$ z $ 0.46 0.91 0.39 0.51 0.03 0.04 0.04 0.07
SA-M3 $ f_1 $ 1249 1807 1047 977 876 651 639 738
$ f_2 $ 9711 14617 6886 5216 2845 1635 1121 1539
$ f_3 $ 1941 2752 1318 1327 315 28 39 457
$ z $ 0.27 0.48 0.20 0.21 0.08 0.03 0.03 0.08
$ t(sec.) $ 231 232 198 235 34 33 37 38
Table 6.  Results of Large-scale ($ n $ = 100, $ m $ = 2) Problems
Test No 1-1 2-1 3-1 4-1 5-1 6-1 7-1 8-1
M0-WGP $ f_1 $ - - - - - - - -
$ f_2 $ - - - - - - - -
$ f_3 $ - - - - - - - -
$ z $ - - - - - - - -
M0-CP $ f_1 $ - - - - - - - -
$ f_2 $ - - - - - - - -
$ f_3 $ - - - - - - - -
$ z $ - - - - - - - -
M0-LWT $ f_1 $ 5499 7943 - - 5224 5683 - -
$ f_2 $ 371810 394667 - - 394518 388008 - -
$ f_3 $ 1399 5206 - - 942 1450 - -
$ z $ 0.71 0.89 - - 0.72 0.76 - -
M1-M2-WGP $ f_1 $ 5737 8535 6578 6779 5073 5495 5340 5873
$ f_2 $ 133487 275567 218122 226738 114247 154587 1222567 151816
$ f_3 $ 1928 6108 3476 3618 310 1371 64 1643
$ z $ 0.35 0.75 0.55 0.53 0.24 0.39 0.23 0.4
M1-M2-CP $ f_1 $ - - - - 5196 - - 6133
$ f_2 $ - - - - 122797 - - 148961
$ f_3 $ - - - - 847 - - 1784
$ z $ - - - - 0.29 - - 0.41
M1-M2-LWT $ f_1 $ 4975 7435 5715 - - 5068 5098 6142
$ f_2 $ 371810 394667 381471 - - 388008 405443 393012
$ f_3 $ 797 3158 1725 - - 999 336 1841
$ z $ 0.67 0.80 0.72 - - 0.72 0.66 0.79
SA-M3 $ f_1 $ 3536 4105 3452 3849 3124 3530 4285 4252
$ f_2 $ 18409 24746 26997 23597 15347 20652 9298 26957
$ f_3 $ 192 158 241 86 0 173 0 108
$ z $ 0.05 0.07 0.07 0.06 0.04 0.06 0.04 0.07
$ t(sec.) $ 66 70 72 82 64 65 74 68
Test No 1-1 2-1 3-1 4-1 5-1 6-1 7-1 8-1
M0-WGP $ f_1 $ - - - - - - - -
$ f_2 $ - - - - - - - -
$ f_3 $ - - - - - - - -
$ z $ - - - - - - - -
M0-CP $ f_1 $ - - - - - - - -
$ f_2 $ - - - - - - - -
$ f_3 $ - - - - - - - -
$ z $ - - - - - - - -
M0-LWT $ f_1 $ 5499 7943 - - 5224 5683 - -
$ f_2 $ 371810 394667 - - 394518 388008 - -
$ f_3 $ 1399 5206 - - 942 1450 - -
$ z $ 0.71 0.89 - - 0.72 0.76 - -
M1-M2-WGP $ f_1 $ 5737 8535 6578 6779 5073 5495 5340 5873
$ f_2 $ 133487 275567 218122 226738 114247 154587 1222567 151816
$ f_3 $ 1928 6108 3476 3618 310 1371 64 1643
$ z $ 0.35 0.75 0.55 0.53 0.24 0.39 0.23 0.4
M1-M2-CP $ f_1 $ - - - - 5196 - - 6133
$ f_2 $ - - - - 122797 - - 148961
$ f_3 $ - - - - 847 - - 1784
$ z $ - - - - 0.29 - - 0.41
M1-M2-LWT $ f_1 $ 4975 7435 5715 - - 5068 5098 6142
$ f_2 $ 371810 394667 381471 - - 388008 405443 393012
$ f_3 $ 797 3158 1725 - - 999 336 1841
$ z $ 0.67 0.80 0.72 - - 0.72 0.66 0.79
SA-M3 $ f_1 $ 3536 4105 3452 3849 3124 3530 4285 4252
$ f_2 $ 18409 24746 26997 23597 15347 20652 9298 26957
$ f_3 $ 192 158 241 86 0 173 0 108
$ z $ 0.05 0.07 0.07 0.06 0.04 0.06 0.04 0.07
$ t(sec.) $ 66 70 72 82 64 65 74 68
Table 7.  Results of Large-scale ($ n $ = 100, $ m $ = 8) Problems
Test No 9-1 10-1 11-1 12-1 13-1 14-1 15-1 16-1
M0-WGP $ f_1 $ - - - - - - - -
$ f_2 $ - - - - - - - -
$ f_3 $ - - - - - - - -
$ z $ - - - - - - - -
M0-CP $ f_1 $ - - - - - - - -
$ f_2 $ - - - - - - - -
$ f_3 $ - - - - - - - -
$ z $ - - - - - - - -
M0-LWT $ f_1 $ - - - - - - - -
$ f_2 $ - - - - - - - -
$ f_3 $ - - - - - - - -
$ z $ - - - - - - - -
M1-M2-WGP $ f_1 $ 1865 4259 1637 4430 1048 578 1093 1744
$ f_2 $ 49222 144672 38766 156241 15984 4029 17236 399726
$ f_3 $ 3562 17320 2274 22013 1 3 0 2472
$ z $ 0.24 0.95 0.15 1.0 0.03 0.04 0.03 0.25
M1-M2-CP $ f_1 $ 1862 4431 2063 4710 1415 2107 1291 2050
$ f_2 $ 56691 158564 59000 186431 25063 41366 21170 42777
$ f_3 $ 3674 19700 4637 22460 1388 1930 125 1954
$ z $ 0.25 1.0 0.27 1.0 0.12 0.20 0.04 0.22
M1-M2-LWT $ f_1 $ 11668 - 13751 - 2141 16990 2203 14119
$ f_2 $ 123224 - 142194 - 15990 178534 17515 140421
$ f_3 $ 3611 - 4665 - 0 3370 71 2534
$ z $ 0.48 - 0.54 - 0.04 0.66 0.05 0.57
SA-M3 $ f_1 $ 1617 4096 1443 3695 1113 1746 1095 1350
$ f_2 $ 25013 109062 18602 93653 3156 11361 5052 9681
$ f_3 $ 2138 15728 2054 13899 0 1535 88 878
$ z $ 0.14 0.84 0.11 0.70 0.01 0.13 0.02 0.09
$ t(sec.) $ 90 276 275 267 72 173 69 277
Test No 9-1 10-1 11-1 12-1 13-1 14-1 15-1 16-1
M0-WGP $ f_1 $ - - - - - - - -
$ f_2 $ - - - - - - - -
$ f_3 $ - - - - - - - -
$ z $ - - - - - - - -
M0-CP $ f_1 $ - - - - - - - -
$ f_2 $ - - - - - - - -
$ f_3 $ - - - - - - - -
$ z $ - - - - - - - -
M0-LWT $ f_1 $ - - - - - - - -
$ f_2 $ - - - - - - - -
$ f_3 $ - - - - - - - -
$ z $ - - - - - - - -
M1-M2-WGP $ f_1 $ 1865 4259 1637 4430 1048 578 1093 1744
$ f_2 $ 49222 144672 38766 156241 15984 4029 17236 399726
$ f_3 $ 3562 17320 2274 22013 1 3 0 2472
$ z $ 0.24 0.95 0.15 1.0 0.03 0.04 0.03 0.25
M1-M2-CP $ f_1 $ 1862 4431 2063 4710 1415 2107 1291 2050
$ f_2 $ 56691 158564 59000 186431 25063 41366 21170 42777
$ f_3 $ 3674 19700 4637 22460 1388 1930 125 1954
$ z $ 0.25 1.0 0.27 1.0 0.12 0.20 0.04 0.22
M1-M2-LWT $ f_1 $ 11668 - 13751 - 2141 16990 2203 14119
$ f_2 $ 123224 - 142194 - 15990 178534 17515 140421
$ f_3 $ 3611 - 4665 - 0 3370 71 2534
$ z $ 0.48 - 0.54 - 0.04 0.66 0.05 0.57
SA-M3 $ f_1 $ 1617 4096 1443 3695 1113 1746 1095 1350
$ f_2 $ 25013 109062 18602 93653 3156 11361 5052 9681
$ f_3 $ 2138 15728 2054 13899 0 1535 88 878
$ z $ 0.14 0.84 0.11 0.70 0.01 0.13 0.02 0.09
$ t(sec.) $ 90 276 275 267 72 173 69 277
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