Figure 1.
The scheduling process at cross-dock
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Lagrangian relaxation algorithm for the truck scheduling problem with products time window constraint in multi-door cross-dock
Institute of Industrial Engineering, School of Mechanical Engineering, Tongji University, Shanghai 201804, China |
Cross-docking is a kind of process that products are unloaded in front of the inbound doors, consolidated based on the downstream demand, and then directly transferred to the outbound doors without a long storage process during the transportation. In this paper, a multi-door cross-dock truck scheduling problem is investigated in which the scheduling and sequencing assignment of trucks need to be considered, with the objectives of minimizing the inner transportation cost in the cross-dock and the total truck waiting cost. The major contribution of this paper is that a novel product-related time window constraint and the temporary storage area are firstly introduced to adapt to different physical conditions of goods considering real-world requirements. Then, a Lagrangian relaxation algorithm is proposed which aims to decompose the relaxed problem into several easy-to-be-solved sub-problems. Besides, a subgradient algorithm is used at each iteration to further deal with these sub-problems. Finally, theory analysis and simulation experiments of different problem scales are carried out during the comparison with a Greedy algorithm to evaluate the performance of the proposed algorithm. Results indicate that the Lagrangian relaxation algorithm is able to achieve more satisfactory near-optimal solutions within an acceptable time.
Mathematics Subject Classification:
Primary: 90B35; Secondary: 90C11.
Citation:
Binghai Zhou, Yuanrui Lei, Shi Zong.
Lagrangian relaxation algorithm for the truck scheduling problem with products time window constraint in multi-door cross-dock. Journal of Industrial and Management Optimization, doi: 10.3934/jimo.2021151
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References:
| [1] |
A. Amini and R. Tavakkoli-Moghaddam,
A bi-objective truck scheduling problem in a cross-docking center with probability of breakdown for trucks, Computers & Industrial Engineering, 96 (2016), 180-191.
doi: 10.1016/j.cie.2016.03.023. |
| [2] |
H. Arbabi, M. M. Nasiri and A. Bozorgi-Amiri,
A hub-and-spoke architecture for a parcel delivery system using the cross-docking distribution strategy, Eng. Optim., 53 (2021), 1593-1612.
doi: 10.1080/0305215X.2020.1808973. |
| [3] |
M. T. Assadi and M. Bagheri,
Differential evolution and Population-based simulated annealing for truck scheduling problem in multiple door cross-docking systems, Computers & Industrial Engineering, 96 (2016), 149-161.
doi: 10.1016/j.cie.2016.03.021. |
| [4] |
P. Bodnar, R. de Koster and K. Azadeh,
Scheduling trucks in a cross-dock with mixed service mode dock doors, Transportation Science, 51 (2017), 112-131.
doi: 10.1287/trsc.2015.0612. |
| [5] |
N. Boysen and M. Fliedner,
Cross dock scheduling: Classification, literature review and research agenda, Omega, 38 (2010), 413-422.
doi: 10.1016/j.omega.2009.10.008. |
| [6] |
A. Chiarello, M. Gaudioso and M. Sammarra,
Truck synchronization at single door cross-docking terminals, OR Spectrum, 40 (2018), 395-447.
doi: 10.1007/s00291-018-0510-x. |
| [7] |
H. Chen, C. Chu and J. M. Proth,
An improvement of the Lagrangean relaxation approach for job shop scheduling: A dynamic programming method, IEEE Transactions on Robotics and Atomation, 14 (1998), 786-795.
doi: 10.1109/70.720354. |
| [8] |
J. Chen, K. Wang and Y. Huang, An integrated inbound logistics mode with intelligent scheduling of milk-run collection, drop and pull delivery and LNG vehicles, Journal of Intelligent Manufacturing, (2020), 1–9.
doi: 10.1007/s10845-020-01637-3. |
| [9] |
L. Chen, Y. Liu and A. Langevin,
A multi-compartment vehicle routing problem in cold-chain distribution, Comput. Oper. Res., 111 (2019), 58-66.
doi: 10.1016/j.cor.2019.06.001. |
| [10] |
U. Clausen, D. Diekmann, M. Pöting and C. Schumacher,
Operating parcel transshipment terminals: A combined simulation and optimization approach, Journal of Simulation, 11 (2017), 2-10.
doi: 10.1057/s41273-016-0032-y. |
| [11] |
H. Corsten, F. Becker and H. Salewski,
Integrating truck and workforce scheduling in a cross-dock: Analysis of different workforce coordination policies, Journal of Business Economics, 90 (2020), 207-237.
|
| [12] |
C. Dong, Q. Li, B. Shen and X. Tong,
Sustainability in supply chains with behavioral concerns, Sustainability, 11 (2019), 4051.
doi: 10.3390/su11154051. |
| [13] |
G. B. Fonseca, T. H. Nogueira and M. G. Ravetti,
A hybrid Lagrangian metaheuristic for the cross-docking flow shop scheduling problem, European J. Oper. Res., 275 (2019), 139-154.
doi: 10.1016/j.ejor.2018.11.033. |
| [14] |
T. Garai and H. Garg,
Multi-objective linear fractional inventory model with possibility and necessity constraints under generalised intuitionistic fuzzy set environment, CAAI Transactions on Intelligence Technology, 4 (2019), 175-181.
|
| [15] |
M. Gaudioso, M. F. Monaco and M. Sammarra,
A Lagrangian heuristics for the truck scheduling problem in multi-door, multi-product Cross-Docking with constant processing time, Omega, 101 (2021), 102255.
doi: 10.1016/j.omega.2020.102255. |
| [16] |
A. Golshahi-Roudbaneh, M. Hajiaghaei-Keshteli and M. M. Paydar,
Developing a lower bound and strong heuristics for a truck scheduling problem in a cross-docking center, Knowledge-Based Systems, 129 (2017), 17-38.
doi: 10.1016/j.knosys.2017.05.006. |
| [17] |
O. Guemri, P. Nduwayo, R. Todosijević, S. Hanafi and F. Glover, Probabilistic tabu search for the cross-docking assignment problem, European J. Oper. Res., 277 (2019), 875–885.,
doi: 10.1016/j.ejor.2019.03.030. |
| [18] |
A. Gunawan, A. T. Widjaja, P. Vansteenwegen and V. F. Yu, A matheuristic algorithm for solving the vehicle routing problem with cross-docking, International Conference on Learning and Intelligent Optimization Springer, (2020), 9–15.
doi: 10.1007/978-3-030-53552-0_2. |
| [19] |
A. Gutierrez, L. Dieulle, N. Labadie and N. Velasco,
A hybrid metaheuristic algorithm for the vehicle routing problem with stochastic demands, Comput. Oper. Res., 99 (2018), 135-147.
doi: 10.1016/j.cor.2018.06.012. |
| [20] |
A. L. Ladier and G. Alpan,
Robust cross-dock scheduling with time windows, Computers & Industrial Engineering, 99 (2016), 16-28.
doi: 10.1016/j.cie.2016.07.003. |
| [21] |
A. L. Ladier and G. Alpan,
Crossdock truck scheduling with time windows: Earliness, tardiness and storage policies, J. Intell. Manufacturing, 29 (2018), 569-583.
|
| [22] |
S. M. Mousavi and B. Vahdani,
Cross-docking location selection in distribution systems: A new intuitionistic fuzzy hierarchical decision model, International J. Computational Intelligence Systems, 9 (2016), 91-109.
doi: 10.1080/18756891.2016.1144156. |
| [23] |
W. Nassief, I. Contreras and R. As'ad,
A mixed-integer programming formulation and Lagrangean relaxation for the cross-dock door assignment problem, International Journal of Production Research, 54 (2016), 494-508.
doi: 10.1080/00207543.2014.1003664. |
| [24] |
W. Nassief, I. Contreras and B. Jaumard, A comparison of formulations and relaxations for cross-dock door assignment problems, Comput. Oper. Res., 94 (2018), 76–88.,
doi: 10.1016/j.cor.2018.01.022. |
| [25] |
S. Niroomand, H. Garg and A. Mahmoodirad,
An intuitionistic fuzzy two stage supply chain network design problem with multi-mode demand and multi-mode transportation, ISA transactions, 107 (2020), 117-133.
doi: 10.1016/j.isatra.2020.07.033. |
| [26] |
C. Serrano, X. Delorme and A. Dolgui,
Scheduling of truck arrivals, truck departures and shop-floor operation in a cross-dock platform, based on trucks loading plans, International Journal of Production Economics, 194 (2017), 102-112.
doi: 10.1016/j.ijpe.2017.09.008. |
| [27] |
I. Seyedi, M. Hamedi and R. Tavakkoli-Moghaddam,
Truck Scheduling in a Cross-Docking Terminal by Using Novel Robust Heuristics, International Journal of Engineering, 32 (2019), 296-305.
|
| [28] |
M. Shakeri, M. Y. H. Low, S. J. Turner and E. W. Lee,
An efficient incremental evaluation function for optimizing truck scheduling in a resource-constrained crossdock using metaheuristics,, Expert Systems with Applications, 45 (2016), 172-184.
doi: 10.1016/j.eswa.2015.09.041. |
| [29] |
J. G. Urzúa-Morales, J. P. Sepulveda-Rojas, M. Alfaro, G. Fuertes, R. Ternero and M. Vargas,
Logistic modeling of the last mile: Case study santiago, chile,, Sustainability, 12 (2020), 648.
doi: 10.3390/su12020648. |
| [30] |
B. Vahdani and S. Shahramfard,
A truck scheduling problem at a cross-docking facility with mixed service mode dock doors, Engineering Computations, 12 (2019), 648.
|
| [31] |
R. H. Waliv, U. Mishra, H. Garg and H. P. Umap,
A nonlinear programming approach to solve the stochastic multi-objective inventory model using the uncertain information, Arabian Journal for Science and Engineering, 45 (2020), 6963-6973.
doi: 10.1007/s13369-020-04618-z. |
| [32] |
Z. Yong-wu,
A Quantity discount model for two-echelon supply chain coordination under stochastic demand, Systems Engineering-Theory & Practice, 7 (2006), 25-32.
|
| [33] |
B. H. Zhou and W. L. Liu,
A modified column generation heuristic for hybrid flow shop multiple orders per job scheduling problem, International Journal of Manufacturing Technology and Management, 33 (2019), 88-113.
doi: 10.1504/IJMTM.2019.100166. |
| [34] |
B. H. Zhou, M. Yin and Z. Q. Lu,
An improved Lagrangian relaxation heuristic for the scheduling problem of operating theatres, Computers & Industrial Engineering, 101 (2016), 490-503.
doi: 10.1016/j.cie.2016.09.003. |
| [35] |
W. Zhou and Y. H. JIN,
Fuzzy subgradient algorithm for solving Lagrangian relaxation dual problem, Control Decis., 11 (2004), 1213-1217.
|
| [36] |
L. Zhu, X. Ren, C. Lee and Y. Zhang,
Coordination contracts in a dual-channel supply chain with a risk-averse retailer, Sustainability, 9 (2017), 2148.
doi: 10.3390/su9112148. |
show all references
References:
| [1] |
A. Amini and R. Tavakkoli-Moghaddam,
A bi-objective truck scheduling problem in a cross-docking center with probability of breakdown for trucks, Computers & Industrial Engineering, 96 (2016), 180-191.
doi: 10.1016/j.cie.2016.03.023. |
| [2] |
H. Arbabi, M. M. Nasiri and A. Bozorgi-Amiri,
A hub-and-spoke architecture for a parcel delivery system using the cross-docking distribution strategy, Eng. Optim., 53 (2021), 1593-1612.
doi: 10.1080/0305215X.2020.1808973. |
| [3] |
M. T. Assadi and M. Bagheri,
Differential evolution and Population-based simulated annealing for truck scheduling problem in multiple door cross-docking systems, Computers & Industrial Engineering, 96 (2016), 149-161.
doi: 10.1016/j.cie.2016.03.021. |
| [4] |
P. Bodnar, R. de Koster and K. Azadeh,
Scheduling trucks in a cross-dock with mixed service mode dock doors, Transportation Science, 51 (2017), 112-131.
doi: 10.1287/trsc.2015.0612. |
| [5] |
N. Boysen and M. Fliedner,
Cross dock scheduling: Classification, literature review and research agenda, Omega, 38 (2010), 413-422.
doi: 10.1016/j.omega.2009.10.008. |
| [6] |
A. Chiarello, M. Gaudioso and M. Sammarra,
Truck synchronization at single door cross-docking terminals, OR Spectrum, 40 (2018), 395-447.
doi: 10.1007/s00291-018-0510-x. |
| [7] |
H. Chen, C. Chu and J. M. Proth,
An improvement of the Lagrangean relaxation approach for job shop scheduling: A dynamic programming method, IEEE Transactions on Robotics and Atomation, 14 (1998), 786-795.
doi: 10.1109/70.720354. |
| [8] |
J. Chen, K. Wang and Y. Huang, An integrated inbound logistics mode with intelligent scheduling of milk-run collection, drop and pull delivery and LNG vehicles, Journal of Intelligent Manufacturing, (2020), 1–9.
doi: 10.1007/s10845-020-01637-3. |
| [9] |
L. Chen, Y. Liu and A. Langevin,
A multi-compartment vehicle routing problem in cold-chain distribution, Comput. Oper. Res., 111 (2019), 58-66.
doi: 10.1016/j.cor.2019.06.001. |
| [10] |
U. Clausen, D. Diekmann, M. Pöting and C. Schumacher,
Operating parcel transshipment terminals: A combined simulation and optimization approach, Journal of Simulation, 11 (2017), 2-10.
doi: 10.1057/s41273-016-0032-y. |
| [11] |
H. Corsten, F. Becker and H. Salewski,
Integrating truck and workforce scheduling in a cross-dock: Analysis of different workforce coordination policies, Journal of Business Economics, 90 (2020), 207-237.
|
| [12] |
C. Dong, Q. Li, B. Shen and X. Tong,
Sustainability in supply chains with behavioral concerns, Sustainability, 11 (2019), 4051.
doi: 10.3390/su11154051. |
| [13] |
G. B. Fonseca, T. H. Nogueira and M. G. Ravetti,
A hybrid Lagrangian metaheuristic for the cross-docking flow shop scheduling problem, European J. Oper. Res., 275 (2019), 139-154.
doi: 10.1016/j.ejor.2018.11.033. |
| [14] |
T. Garai and H. Garg,
Multi-objective linear fractional inventory model with possibility and necessity constraints under generalised intuitionistic fuzzy set environment, CAAI Transactions on Intelligence Technology, 4 (2019), 175-181.
|
| [15] |
M. Gaudioso, M. F. Monaco and M. Sammarra,
A Lagrangian heuristics for the truck scheduling problem in multi-door, multi-product Cross-Docking with constant processing time, Omega, 101 (2021), 102255.
doi: 10.1016/j.omega.2020.102255. |
| [16] |
A. Golshahi-Roudbaneh, M. Hajiaghaei-Keshteli and M. M. Paydar,
Developing a lower bound and strong heuristics for a truck scheduling problem in a cross-docking center, Knowledge-Based Systems, 129 (2017), 17-38.
doi: 10.1016/j.knosys.2017.05.006. |
| [17] |
O. Guemri, P. Nduwayo, R. Todosijević, S. Hanafi and F. Glover, Probabilistic tabu search for the cross-docking assignment problem, European J. Oper. Res., 277 (2019), 875–885.,
doi: 10.1016/j.ejor.2019.03.030. |
| [18] |
A. Gunawan, A. T. Widjaja, P. Vansteenwegen and V. F. Yu, A matheuristic algorithm for solving the vehicle routing problem with cross-docking, International Conference on Learning and Intelligent Optimization Springer, (2020), 9–15.
doi: 10.1007/978-3-030-53552-0_2. |
| [19] |
A. Gutierrez, L. Dieulle, N. Labadie and N. Velasco,
A hybrid metaheuristic algorithm for the vehicle routing problem with stochastic demands, Comput. Oper. Res., 99 (2018), 135-147.
doi: 10.1016/j.cor.2018.06.012. |
| [20] |
A. L. Ladier and G. Alpan,
Robust cross-dock scheduling with time windows, Computers & Industrial Engineering, 99 (2016), 16-28.
doi: 10.1016/j.cie.2016.07.003. |
| [21] |
A. L. Ladier and G. Alpan,
Crossdock truck scheduling with time windows: Earliness, tardiness and storage policies, J. Intell. Manufacturing, 29 (2018), 569-583.
|
| [22] |
S. M. Mousavi and B. Vahdani,
Cross-docking location selection in distribution systems: A new intuitionistic fuzzy hierarchical decision model, International J. Computational Intelligence Systems, 9 (2016), 91-109.
doi: 10.1080/18756891.2016.1144156. |
| [23] |
W. Nassief, I. Contreras and R. As'ad,
A mixed-integer programming formulation and Lagrangean relaxation for the cross-dock door assignment problem, International Journal of Production Research, 54 (2016), 494-508.
doi: 10.1080/00207543.2014.1003664. |
| [24] |
W. Nassief, I. Contreras and B. Jaumard, A comparison of formulations and relaxations for cross-dock door assignment problems, Comput. Oper. Res., 94 (2018), 76–88.,
doi: 10.1016/j.cor.2018.01.022. |
| [25] |
S. Niroomand, H. Garg and A. Mahmoodirad,
An intuitionistic fuzzy two stage supply chain network design problem with multi-mode demand and multi-mode transportation, ISA transactions, 107 (2020), 117-133.
doi: 10.1016/j.isatra.2020.07.033. |
| [26] |
C. Serrano, X. Delorme and A. Dolgui,
Scheduling of truck arrivals, truck departures and shop-floor operation in a cross-dock platform, based on trucks loading plans, International Journal of Production Economics, 194 (2017), 102-112.
doi: 10.1016/j.ijpe.2017.09.008. |
| [27] |
I. Seyedi, M. Hamedi and R. Tavakkoli-Moghaddam,
Truck Scheduling in a Cross-Docking Terminal by Using Novel Robust Heuristics, International Journal of Engineering, 32 (2019), 296-305.
|
| [28] |
M. Shakeri, M. Y. H. Low, S. J. Turner and E. W. Lee,
An efficient incremental evaluation function for optimizing truck scheduling in a resource-constrained crossdock using metaheuristics,, Expert Systems with Applications, 45 (2016), 172-184.
doi: 10.1016/j.eswa.2015.09.041. |
| [29] |
J. G. Urzúa-Morales, J. P. Sepulveda-Rojas, M. Alfaro, G. Fuertes, R. Ternero and M. Vargas,
Logistic modeling of the last mile: Case study santiago, chile,, Sustainability, 12 (2020), 648.
doi: 10.3390/su12020648. |
| [30] |
B. Vahdani and S. Shahramfard,
A truck scheduling problem at a cross-docking facility with mixed service mode dock doors, Engineering Computations, 12 (2019), 648.
|
| [31] |
R. H. Waliv, U. Mishra, H. Garg and H. P. Umap,
A nonlinear programming approach to solve the stochastic multi-objective inventory model using the uncertain information, Arabian Journal for Science and Engineering, 45 (2020), 6963-6973.
doi: 10.1007/s13369-020-04618-z. |
| [32] |
Z. Yong-wu,
A Quantity discount model for two-echelon supply chain coordination under stochastic demand, Systems Engineering-Theory & Practice, 7 (2006), 25-32.
|
| [33] |
B. H. Zhou and W. L. Liu,
A modified column generation heuristic for hybrid flow shop multiple orders per job scheduling problem, International Journal of Manufacturing Technology and Management, 33 (2019), 88-113.
doi: 10.1504/IJMTM.2019.100166. |
| [34] |
B. H. Zhou, M. Yin and Z. Q. Lu,
An improved Lagrangian relaxation heuristic for the scheduling problem of operating theatres, Computers & Industrial Engineering, 101 (2016), 490-503.
doi: 10.1016/j.cie.2016.09.003. |
| [35] |
W. Zhou and Y. H. JIN,
Fuzzy subgradient algorithm for solving Lagrangian relaxation dual problem, Control Decis., 11 (2004), 1213-1217.
|
| [36] |
L. Zhu, X. Ren, C. Lee and Y. Zhang,
Coordination contracts in a dual-channel supply chain with a risk-averse retailer, Sustainability, 9 (2017), 2148.
doi: 10.3390/su9112148. |
Figure 2.
Several values of %gap (group2) based on the different value of slackness
Figure 3.
Values of %gap and %dev in group 2
Table 1.
Computer results for problems in group 1, b = 0.98, 0.95, and 0.90
| p=0.98 | p=0.95 | p=0.90 | ||||||||||||||||||||
| L | G | L | G | L | G | |||||||||||||||||
| I | #t×#d | s | IP | LR | %gap | CPU | UB | %dev | IP | LR | %gap | CPU | UB | %dev | IP | LR | %gap | CPU | UB | %dev | ||
| 1 | 15 | 1.683 | 1.683 | 5.32 | 0.55 | 1.815 | 10.43 | 1.98 | 1.98 | 4.97 | 0.56 | 2.187 | 10.43 | 1.643 | 1.643 | 5.27 | 0.46 | 1.99 | 11.99 | |||
| 2 | 20 | 1.701 | 1.701 | 4.82 | 0.5 | 1.89 | 8.37 | 2.1 | 2.1 | 4.38 | 0.33 | 2.278 | 8.45 | 1.785 | 1.785 | 5.39 | 0.42 | 1.754 | 8.03 | |||
| 3 | 4×2 | 30 | 1.729 | 1.729 | 3.44 | 0.53 | 2.004 | 15.27 | 2.059 | 2.059 | 3.55 | 0.42 | 2.358 | 14.54 | 1.997 | 1.997 | 3.59 | 0.51 | 1.769 | 14.69 | ||
| 4 | 15 | 2.56 | 2.56 | 9.55 | 0.66 | 2.45 | 14.81 | 2.56 | 2.56 | 9.27 | 0.68 | 2.816 | 14.52 | 2.765 | 2.765 | 9.83 | 0.84 | 3.013 | 14.96 | |||
| 5 | 20 | 2.26 | 2.26 | 8.15 | 0.8 | 3.064 | 13.02 | 2.659 | 2.659 | 7.03 | 0.8 | 3.225 | 12.76 | 2.872 | 2.872 | 6.89 | 0.8 | 2.838 | 13.4 | |||
| 6 | 5×3 | 30 | 2.498 | 2.498 | 10.55 | 0.65 | 2.648 | 12.76 | 2.602 | 2.602 | 9.02 | 0.8 | 3.152 | 12.76 | 2.862 | 2.862 | 11 | 0.96 | 3.089 | 12.89 | ||
| 7 | 15 | 2.709 | 2.709 | 6.58 | 1.59 | 3.954 | 26.35 | 3.078 | 3.078 | 6.33 | 1.61 | 4.163 | 26.62 | 2.924 | 2.924 | 6.46 | 1.07 | 3.83 | 27.95 | |||
| 8 | 20 | 3.108 | 3.108 | 4.32 | 1.64 | 3.533 | 25 | 3.205 | 3.205 | 4.32 | 1.06 | 4.257 | 24.27 | 2.66 | 2.66 | 4.62 | 1.33 | 4.087 | 23.06 | |||
| 9 | 6×3 | 30 | 2.495 | 2.495 | 7.55 | 1.6 | 3.735 | 26.32 | 3.08 | 3.08 | 6.29 | 1.02 | 4.016 | 26.06 | 2.957 | 2.957 | 6.54 | 1.27 | 4.257 | 27.1 | ||
| 10 | 15 | 3.31 | 3.31 | 4.76 | 1.92 | 4.153 | 13.37 | 3.755 | 3.988 | 4.21 | 2.35 | 4.72 | 13.37 | 4.068 | 4.068 | 4.04 | 1.88 | 4.153 | 12.84 | |||
| 11 | 20 | 3.425 | 3.624 | 6.35 | 1.93 | 4.337 | 14.81 | 3.89 | 3.897 | 4.96 | 1.91 | 4.518 | 15.11 | 4.155 | 4.17 | 6.05 | 2.38 | 3.433 | 15.87 | |||
| 12 | 7×3 | 30 | 3.257 | 3.471 | 3.69 | 3.13 | 3.747 | 21.32 | 3.572 | 3.732 | 3.55 | 2.66 | 4.625 | 21.98 | 3.375 | 3.471 | 3.41 | 3.19 | 4.163 | 22.2 | ||
| 13 | 15 | 3.18 | 3.381 | 7.79 | 5.86 | 4.848 | 29.61 | 3.58 | 3.636 | 6.04 | 7.1 | 5.051 | 30.53 | 3.09 | 3.09 | 7.61 | 7.1 | 3.788 | 30.84 | |||
| 14 | 20 | 3.155 | 3.301 | 6.23 | 4.89 | 4.847 | 29.92 | 3.42 | 3.628 | 6.23 | 7.41 | 5.049 | 30.53 | 3.171 | 3.301 | 6.17 | 4.94 | 3.786 | 29 | |||
| 15 | 7×4 | 30 | 3.178 | 3.204 | 6.83 | 5.38 | 4.262 | 32.8 | 3.595 | 3.641 | 5.38 | 8.41 | 5.014 | 31.24 | 3.285 | 3.386 | 6.62 | 8.41 | 4.061 | 32.49 | ||
| 16 | 15 | 3.65 | 3.826 | 5.76 | 4.4 | 5.089 | 24.08 | 4.058 | 4.159 | 5.7 | 4.27 | 5.472 | 24.08 | 3.588 | 3.701 | 5.81 | 5.34 | 5.472 | 25.28 | |||
| 17 | 20 | 3.425 | 3.425 | 5.58 | 5.79 | 5.329 | 21.76 | 4.117 | 4.228 | 5.03 | 4.45 | 5.438 | 22.91 | 4.489 | 4.608 | 5.53 | 5.57 | 4.894 | 23.14 | |||
| 18 | 8×3 | 30 | 3.785 | 3.85 | 6.41 | 8.54 | 4.745 | 22.79 | 4.095 | 4.184 | 5.57 | 10.05 | 5.214 | 23.49 | 4.258 | 4.435 | 7.19 | 8.37 | 4.432 | 24.19 | ||
| 19 | 15 | 3.455 | 3.485 | 4.57 | 8.09 | 5.375 | 35.21 | 3.986 | 4.1 | 4.76 | 7.03 | 5.973 | 36.3 | 3.698 | 3.772 | 4.57 | 7.03 | 5.973 | 35.94 | |||
| 20 | 20 | 3.576 | 3.678 | 5.81 | 6.36 | 4.802 | 34.09 | 4.168 | 4.18 | 4.76 | 7.64 | 5.856 | 33.42 | 3.259 | 3.344 | 5.09 | 9.17 | 6.09 | 33.09 | |||
| 21 | 8×4 | 30 | 3.385 | 3.428 | 6.14 | 9.16 | 5.446 | 32.42 | 4.15 | 4.18 | 4.76 | 11.33 | 5.856 | 33.42 | 3.857 | 3.929 | 4.52 | 11.33 | 5.739 | 33.09 | ||
| Instance is short for |
||||||||||||||||||||||
| p=0.98 | p=0.95 | p=0.90 | ||||||||||||||||||||
| L | G | L | G | L | G | |||||||||||||||||
| I | #t×#d | s | IP | LR | %gap | CPU | UB | %dev | IP | LR | %gap | CPU | UB | %dev | IP | LR | %gap | CPU | UB | %dev | ||
| 1 | 15 | 1.683 | 1.683 | 5.32 | 0.55 | 1.815 | 10.43 | 1.98 | 1.98 | 4.97 | 0.56 | 2.187 | 10.43 | 1.643 | 1.643 | 5.27 | 0.46 | 1.99 | 11.99 | |||
| 2 | 20 | 1.701 | 1.701 | 4.82 | 0.5 | 1.89 | 8.37 | 2.1 | 2.1 | 4.38 | 0.33 | 2.278 | 8.45 | 1.785 | 1.785 | 5.39 | 0.42 | 1.754 | 8.03 | |||
| 3 | 4×2 | 30 | 1.729 | 1.729 | 3.44 | 0.53 | 2.004 | 15.27 | 2.059 | 2.059 | 3.55 | 0.42 | 2.358 | 14.54 | 1.997 | 1.997 | 3.59 | 0.51 | 1.769 | 14.69 | ||
| 4 | 15 | 2.56 | 2.56 | 9.55 | 0.66 | 2.45 | 14.81 | 2.56 | 2.56 | 9.27 | 0.68 | 2.816 | 14.52 | 2.765 | 2.765 | 9.83 | 0.84 | 3.013 | 14.96 | |||
| 5 | 20 | 2.26 | 2.26 | 8.15 | 0.8 | 3.064 | 13.02 | 2.659 | 2.659 | 7.03 | 0.8 | 3.225 | 12.76 | 2.872 | 2.872 | 6.89 | 0.8 | 2.838 | 13.4 | |||
| 6 | 5×3 | 30 | 2.498 | 2.498 | 10.55 | 0.65 | 2.648 | 12.76 | 2.602 | 2.602 | 9.02 | 0.8 | 3.152 | 12.76 | 2.862 | 2.862 | 11 | 0.96 | 3.089 | 12.89 | ||
| 7 | 15 | 2.709 | 2.709 | 6.58 | 1.59 | 3.954 | 26.35 | 3.078 | 3.078 | 6.33 | 1.61 | 4.163 | 26.62 | 2.924 | 2.924 | 6.46 | 1.07 | 3.83 | 27.95 | |||
| 8 | 20 | 3.108 | 3.108 | 4.32 | 1.64 | 3.533 | 25 | 3.205 | 3.205 | 4.32 | 1.06 | 4.257 | 24.27 | 2.66 | 2.66 | 4.62 | 1.33 | 4.087 | 23.06 | |||
| 9 | 6×3 | 30 | 2.495 | 2.495 | 7.55 | 1.6 | 3.735 | 26.32 | 3.08 | 3.08 | 6.29 | 1.02 | 4.016 | 26.06 | 2.957 | 2.957 | 6.54 | 1.27 | 4.257 | 27.1 | ||
| 10 | 15 | 3.31 | 3.31 | 4.76 | 1.92 | 4.153 | 13.37 | 3.755 | 3.988 | 4.21 | 2.35 | 4.72 | 13.37 | 4.068 | 4.068 | 4.04 | 1.88 | 4.153 | 12.84 | |||
| 11 | 20 | 3.425 | 3.624 | 6.35 | 1.93 | 4.337 | 14.81 | 3.89 | 3.897 | 4.96 | 1.91 | 4.518 | 15.11 | 4.155 | 4.17 | 6.05 | 2.38 | 3.433 | 15.87 | |||
| 12 | 7×3 | 30 | 3.257 | 3.471 | 3.69 | 3.13 | 3.747 | 21.32 | 3.572 | 3.732 | 3.55 | 2.66 | 4.625 | 21.98 | 3.375 | 3.471 | 3.41 | 3.19 | 4.163 | 22.2 | ||
| 13 | 15 | 3.18 | 3.381 | 7.79 | 5.86 | 4.848 | 29.61 | 3.58 | 3.636 | 6.04 | 7.1 | 5.051 | 30.53 | 3.09 | 3.09 | 7.61 | 7.1 | 3.788 | 30.84 | |||
| 14 | 20 | 3.155 | 3.301 | 6.23 | 4.89 | 4.847 | 29.92 | 3.42 | 3.628 | 6.23 | 7.41 | 5.049 | 30.53 | 3.171 | 3.301 | 6.17 | 4.94 | 3.786 | 29 | |||
| 15 | 7×4 | 30 | 3.178 | 3.204 | 6.83 | 5.38 | 4.262 | 32.8 | 3.595 | 3.641 | 5.38 | 8.41 | 5.014 | 31.24 | 3.285 | 3.386 | 6.62 | 8.41 | 4.061 | 32.49 | ||
| 16 | 15 | 3.65 | 3.826 | 5.76 | 4.4 | 5.089 | 24.08 | 4.058 | 4.159 | 5.7 | 4.27 | 5.472 | 24.08 | 3.588 | 3.701 | 5.81 | 5.34 | 5.472 | 25.28 | |||
| 17 | 20 | 3.425 | 3.425 | 5.58 | 5.79 | 5.329 | 21.76 | 4.117 | 4.228 | 5.03 | 4.45 | 5.438 | 22.91 | 4.489 | 4.608 | 5.53 | 5.57 | 4.894 | 23.14 | |||
| 18 | 8×3 | 30 | 3.785 | 3.85 | 6.41 | 8.54 | 4.745 | 22.79 | 4.095 | 4.184 | 5.57 | 10.05 | 5.214 | 23.49 | 4.258 | 4.435 | 7.19 | 8.37 | 4.432 | 24.19 | ||
| 19 | 15 | 3.455 | 3.485 | 4.57 | 8.09 | 5.375 | 35.21 | 3.986 | 4.1 | 4.76 | 7.03 | 5.973 | 36.3 | 3.698 | 3.772 | 4.57 | 7.03 | 5.973 | 35.94 | |||
| 20 | 20 | 3.576 | 3.678 | 5.81 | 6.36 | 4.802 | 34.09 | 4.168 | 4.18 | 4.76 | 7.64 | 5.856 | 33.42 | 3.259 | 3.344 | 5.09 | 9.17 | 6.09 | 33.09 | |||
| 21 | 8×4 | 30 | 3.385 | 3.428 | 6.14 | 9.16 | 5.446 | 32.42 | 4.15 | 4.18 | 4.76 | 11.33 | 5.856 | 33.42 | 3.857 | 3.929 | 4.52 | 11.33 | 5.739 | 33.09 | ||
| Instance is short for |
||||||||||||||||||||||
Table 2.
Computer results for problems in group 2 (slackness = 10)
| Instances | #trucks×#doors | Lagrangian relaxation algorithm | Greedy algorithm | ||||
| LR | %gap | CPU time | UB | %dev | |||
| 1 | 9×3 | 4.759 | 5.21 | 18 | 5.965 | 18.8 | |
| 2 | 9×4 | 4.856 | 5.28 | 19.76 | 6.382 | 24.5 | |
| 3 | 9×5 | 4.661 | 4.38 | 30.13 | 6.8 | 39.5 | |
| 4 | 10×3 | 5.527 | 5.25 | 42.36 | 6.876 | 14.43 | |
| 5 | 10×4 | 5.595 | 4.76 | 15.57 | 7.014 | 21.56 | |
| 6 | 10×5 | 5.055 | 5.75 | 44.57 | 7.609 | 43.35 | |
| 7 | 11×5 | 6.017 | 7.86 | 77.65 | 8.618 | 31.41 | |
| 8 | 11×6 | 5.997 | 9.09 | 80.49 | 9.149 | 38.69 | |
| 9 | 15×6 | 8.223 | 9.09 | 177.84 | 12.43 | 37.39 | |
| 10 | 15×7 | 8.21 | 7.42 | 124.02 | 13.09 | 47.64 | |
| 11 | 20×10 | 10.89 | 9.91 | 265.06 | 20.73 | 71.41 | |
| Instances | #trucks×#doors | Lagrangian relaxation algorithm | Greedy algorithm | ||||
| LR | %gap | CPU time | UB | %dev | |||
| 1 | 9×3 | 4.759 | 5.21 | 18 | 5.965 | 18.8 | |
| 2 | 9×4 | 4.856 | 5.28 | 19.76 | 6.382 | 24.5 | |
| 3 | 9×5 | 4.661 | 4.38 | 30.13 | 6.8 | 39.5 | |
| 4 | 10×3 | 5.527 | 5.25 | 42.36 | 6.876 | 14.43 | |
| 5 | 10×4 | 5.595 | 4.76 | 15.57 | 7.014 | 21.56 | |
| 6 | 10×5 | 5.055 | 5.75 | 44.57 | 7.609 | 43.35 | |
| 7 | 11×5 | 6.017 | 7.86 | 77.65 | 8.618 | 31.41 | |
| 8 | 11×6 | 5.997 | 9.09 | 80.49 | 9.149 | 38.69 | |
| 9 | 15×6 | 8.223 | 9.09 | 177.84 | 12.43 | 37.39 | |
| 10 | 15×7 | 8.21 | 7.42 | 124.02 | 13.09 | 47.64 | |
| 11 | 20×10 | 10.89 | 9.91 | 265.06 | 20.73 | 71.41 | |
Table 3.
Computer results for problems in group 2 (slackness = 15)
| Instances | #trucks×#doors | Lagrangian relaxation algorithm | Greedy algorithm | ||||
| LR | %gap | CPU time | UB | %dev | |||
| 1 | 9×3 | 4.742 | 5.17 | 24.5 | 5.892 | 19.3 | |
| 2 | 9×4 | 4.813 | 4.97 | 22.69 | 6.382 | 24.5 | |
| 3 | 9×5 | 4.559 | 5.65 | 36.25 | 6.66 | 40.72 | |
| 4 | 10×3 | 5.456 | 5.09 | 40.71 | 6.674 | 16.1 | |
| 5 | 10×4 | 5.195 | 4.76 | 31.97 | 7.142 | 30.92 | |
| 6 | 10×5 | 5.355 | 5.3 | 42.97 | 7.609 | 35.32 | |
| 7 | 11×5 | 6.017 | 6.54 | 55.32 | 8.618 | 33.86 | |
| 8 | 11×6 | 5.6 | 7.45 | 91.46 | 9.149 | 41.4 | |
| 9 | 15×6 | 8.403 | 8.26 | 125.77 | 12.43 | 35.68 | |
| 10 | 15×7 | 8.237 | 6.65 | 81.64 | 13.18 | 49.35 | |
| 11 | 20×10 | 10.64 | 8.26 | 164.22 | 20.69 | 68.47 | |
| Instances | #trucks×#doors | Lagrangian relaxation algorithm | Greedy algorithm | ||||
| LR | %gap | CPU time | UB | %dev | |||
| 1 | 9×3 | 4.742 | 5.17 | 24.5 | 5.892 | 19.3 | |
| 2 | 9×4 | 4.813 | 4.97 | 22.69 | 6.382 | 24.5 | |
| 3 | 9×5 | 4.559 | 5.65 | 36.25 | 6.66 | 40.72 | |
| 4 | 10×3 | 5.456 | 5.09 | 40.71 | 6.674 | 16.1 | |
| 5 | 10×4 | 5.195 | 4.76 | 31.97 | 7.142 | 30.92 | |
| 6 | 10×5 | 5.355 | 5.3 | 42.97 | 7.609 | 35.32 | |
| 7 | 11×5 | 6.017 | 6.54 | 55.32 | 8.618 | 33.86 | |
| 8 | 11×6 | 5.6 | 7.45 | 91.46 | 9.149 | 41.4 | |
| 9 | 15×6 | 8.403 | 8.26 | 125.77 | 12.43 | 35.68 | |
| 10 | 15×7 | 8.237 | 6.65 | 81.64 | 13.18 | 49.35 | |
| 11 | 20×10 | 10.64 | 8.26 | 164.22 | 20.69 | 68.47 | |
Table 4.
Computer results for problems in group 2 (slackness = 20)
| Instances | #trucks×#doors | Lagrangian relaxation algorithm | Greedy algorithm | ||||
| LR | %gap | CPU time | UB | %dev | |||
| 1 | 9×3 | 5.035 | 4.53 | 18.16 | 6.003 | 13.12 | |
| 2 | 9×4 | 4.764 | 5.32 | 27.23 | 6.173 | 29.83 | |
| 3 | 9×5 | 4.758 | 6.02 | 26.69 | 6.899 | 34.3 | |
| 4 | 10×3 | 5.466 | 4.92 | 29.03 | 6.674 | 16.1 | |
| 5 | 10×4 | 5.087 | 5.56 | 27.07 | 7.142 | 32.45 | |
| 6 | 10×5 | 5.075 | 4.9 | 40.41 | 7.609 | 42.79 | |
| 7 | 11×5 | 6.037 | 7.41 | 62.67 | 8.618 | 32.18 | |
| 8 | 11×6 | 5.777 | 7.41 | 62.09 | 9.139 | 46.64 | |
| 9 | 15×6 | 8.012 | 7.41 | 133.29 | 12.4 | 36.94 | |
| 10 | 15×7 | 8.045 | 5.82 | 159.65 | 13.22 | 45.71 | |
| 11 | 20×10 | 10.85 | 7.41 | 226.27 | 20.73 | 66.9 | |
| Instances | #trucks×#doors | Lagrangian relaxation algorithm | Greedy algorithm | ||||
| LR | %gap | CPU time | UB | %dev | |||
| 1 | 9×3 | 5.035 | 4.53 | 18.16 | 6.003 | 13.12 | |
| 2 | 9×4 | 4.764 | 5.32 | 27.23 | 6.173 | 29.83 | |
| 3 | 9×5 | 4.758 | 6.02 | 26.69 | 6.899 | 34.3 | |
| 4 | 10×3 | 5.466 | 4.92 | 29.03 | 6.674 | 16.1 | |
| 5 | 10×4 | 5.087 | 5.56 | 27.07 | 7.142 | 32.45 | |
| 6 | 10×5 | 5.075 | 4.9 | 40.41 | 7.609 | 42.79 | |
| 7 | 11×5 | 6.037 | 7.41 | 62.67 | 8.618 | 32.18 | |
| 8 | 11×6 | 5.777 | 7.41 | 62.09 | 9.139 | 46.64 | |
| 9 | 15×6 | 8.012 | 7.41 | 133.29 | 12.4 | 36.94 | |
| 10 | 15×7 | 8.045 | 5.82 | 159.65 | 13.22 | 45.71 | |
| 11 | 20×10 | 10.85 | 7.41 | 226.27 | 20.73 | 66.9 | |
Table 5.
Computer results for problems in group 2 (slackness = 30)
| Instances | #trucks×#doors | Lagrangian relaxation algorithm | Greedy algorithm | ||||
| LR | %gap | CPU time | UB | %dev | |||
| 1 | 9×3 | 4.76 | 4.8 | 17.01 | 5.965 | 19.3 | |
| 2 | 9×4 | 4.647 | 5.08 | 22.76 | 6.108 | 30.38 | |
| 3 | 9×5 | 4.684 | 3.9 | 22.59 | 6.8 | 39.5 | |
| 4 | 10×3 | 5.943 | 4.9 | 35.42 | 6.344 | 6.74 | |
| 5 | 10×4 | 5.375 | 4.76 | 42.1 | 7.082 | 26.54 | |
| 6 | 10×5 | 5.315 | 4.76 | 47.71 | 7.609 | 36.34 | |
| 7 | 11×5 | 6.057 | 4.76 | 50.29 | 8.618 | 35.51 | |
| 8 | 11×6 | 5.817 | 6.54 | 77.44 | 9.149 | 46.99 | |
| 9 | 15×6 | 8.221 | 6.54 | 114.18 | 12.37 | 40.67 | |
| 10 | 15×7 | 8.527 | 4.29 | 148.43 | 13.3 | 47.33 | |
| 11 | 20×10 | 10.79 | 5.66 | 302.54 | 23.68 | 70.76 | |
| Instances | #trucks×#doors | Lagrangian relaxation algorithm | Greedy algorithm | ||||
| LR | %gap | CPU time | UB | %dev | |||
| 1 | 9×3 | 4.76 | 4.8 | 17.01 | 5.965 | 19.3 | |
| 2 | 9×4 | 4.647 | 5.08 | 22.76 | 6.108 | 30.38 | |
| 3 | 9×5 | 4.684 | 3.9 | 22.59 | 6.8 | 39.5 | |
| 4 | 10×3 | 5.943 | 4.9 | 35.42 | 6.344 | 6.74 | |
| 5 | 10×4 | 5.375 | 4.76 | 42.1 | 7.082 | 26.54 | |
| 6 | 10×5 | 5.315 | 4.76 | 47.71 | 7.609 | 36.34 | |
| 7 | 11×5 | 6.057 | 4.76 | 50.29 | 8.618 | 35.51 | |
| 8 | 11×6 | 5.817 | 6.54 | 77.44 | 9.149 | 46.99 | |
| 9 | 15×6 | 8.221 | 6.54 | 114.18 | 12.37 | 40.67 | |
| 10 | 15×7 | 8.527 | 4.29 | 148.43 | 13.3 | 47.33 | |
| 11 | 20×10 | 10.79 | 5.66 | 302.54 | 23.68 | 70.76 | |
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