Figure 1.
Schematic diagram of the 2E-VRPDB
-
Previous Article
Solving discrete linear fractional bilevel programs with multiple objectives at the upper level
- JIMO Home
- This Issue
-
Next Article
A dynamic analysis of a monopolist's product and process innovation with nonlinear demand and expected quality effects
doi: 10.3934/jimo.2021094
Online First
Online First articles are published articles within a journal that have not yet been assigned to a formal issue. This means they do not yet have a volume number, issue number, or page numbers assigned to them, however, they can still be found and cited using their DOI (Digital Object Identifier). Online First publication benefits the research community by making new scientific discoveries known as quickly as possible.
Readers can access Online First articles via the “Online First” tab for the selected journal.
A time-division distribution strategy for the two-echelon vehicle routing problem with demand blowout
| 1. | School of Mechanical Engineering, Southwest Jiaotong University, Chengdu 610031, China |
| 2. | Technology and Equipment of Rail Transit Operation and, Maintenance Key Laboratory of Sichuan Province, Chengdu 610031, China |
| 3. | Avic Chengdu Aircraft Industrial (Group)Co., Ltd, Chengdu 610031, China |
| 4. | School of Marketing, University of Southern Mississippi, Hattiesburg, MS 39406, USA |
Based on the rapid development of e-commerce, major promotional events and holidays can lead to explosive growth in market demand and place significant pressure on distribution systems. In this study, we considered a distribution system in which products are first transported to transfer satellites from a central depot and then delivered to customers from the transfer satellites. We modeled this distribution problem as a two-echelon vehicle routing problem with demand blowout (2E-VRPDB). We adopt a time-division distribution strategy to address massive delivery demand in two phases by offering incentives to customers who accept flexible delivery dates. We propose a hybrid fireworks algorithm (HFWA) to solve the 2E-VRPDB model. This model fuses an optimal cutting algorithm with an improved fireworks algorithm. To demonstrate the effectiveness and efficiency of the proposed HFWA, we conducted comparative analysis on a genetic algorithm and ant colony algorithm using a VRP example set. Finally, we applied the proposed model and HFWA to solve a distribution problem for the Jingdong Mall in Chengdu, China. The computational results demonstrate that the proposed approach can effectively reduce logistical costs and maintain a high service level.
Keywords:
Two-echelon vehicle routing problem,
demand blowout,
time-division distribution,
hybrid fireworks algorithm.
Mathematics Subject Classification:
Primary: 90B06, 62K05; Secondary: 68T37.
Citation:
Min Zhang, Guowen Xiong, Shanshan Bao, Chao Meng.
A time-division distribution strategy for the two-echelon vehicle routing problem with demand blowout. Journal of Industrial and Management Optimization, doi: 10.3934/jimo.2021094
![]()
References:
| [1] |
R. Baldacci, A. Mingozzi, R. Roberti and R. W. Clavo,
An exact algorithm for the two-echelon capacitated vehicle routing problem, Operations Research, 61 (2013), 298-314.
doi: 10.1287/opre.1120.1153. |
| [2] |
A. Bevilaqua, D. Bevilaqua and K. Yamanaka,
Parallel island based Memetic Algorithm with Lin-Kernighan local search for a real-life Two-Echelon Heterogeneous Vehicle Routing Problem based on Brazilian wholesale companies, Applied Soft Computing, 76 (2019), 697-711.
doi: 10.1016/j.asoc.2018.12.036. |
| [3] |
U. Breunig, R. Baldacci, R. F. Hartl and T. Vidal,
The electric two-echelon vehicle routing problem, Computers and Operations Research, 103 (2019), 198-210.
doi: 10.1016/j.cor.2018.11.005. |
| [4] |
U. Breunig, V. Schmid, R. F. Hartl and T. Vidal,
A large neighbourhood based heuristic for two-echelon routing problems, Computers and Operations Research, 76 (2016), 208-225.
doi: 10.1016/j.cor.2016.06.014. |
| [5] |
M.-C. Chen, P.-J. W and Y.-H. Hsu,
An effective pricing model for the congestion alleviation of e-commerce logistics, Computers and Industrial Engineering, 129 (2019), 368-376.
doi: 10.1016/j.cie.2019.01.060. |
| [6] |
Double 11 constantly refreshes the imagination of Chinese market, Global times, November 12, 2019 (015). |
| [7] |
P. Grangier, M. Gendreau, F. Lehuédé and L.-M. Rousseau,
An adaptive large neighborhood search for the two-echelon multiple-trip vehicle routing problem with satellite synchronization, European Journal of Operational Research, 254 (2016), 80-91.
doi: 10.1016/j.ejor.2016.03.040. |
| [8] |
M. Guan, M. Cha, Y. Li, Y. Wang and J. Yu, Predicting time-bounded purchases during a mega shopping festival, 2019 IEEE International Conference on Big Data and Smart Computing (BigComp), (2019), 1–8.
doi: 10.1109/BIGCOMP.2019.8679217. |
| [9] |
X. Guo, Y. J. L. Jaramillo, J. Bloemhof-Ruwaard and G. D. H. Claassen, On integrating crowdsourced delivery in last-mile logistics: A simulation study to quantify its feasibility, Journal of Cleaner Production, 241 (2019), 118365.
doi: 10.1016/j.jclepro.2019.118365. |
| [10] |
P. He and J. Li,
The two-echelon multi-trip vehicle routing problem with dynamic satellites for crop harvesting and transportation, Applied Soft Computing, 77 (2019), 387-398.
doi: 10.1016/j.asoc.2019.01.040. |
| [11] |
W. Jie, J. Yang, M. Zhang and Y. Huang,
The two-echelon capacitated electric vehicle routing problem with battery swapping stations: Formulation and efficient methodology, European Journal of Operational Research, 272 (2019), 879-904.
doi: 10.1016/j.ejor.2018.07.002. |
| [12] |
H. Li, L. Zhang, T. Lv and X. Chang,
The two-echelon time-constrained vehicle routing problem in linehaul-delivery systems, Transportation Research Part B: Methodological, 94 (2016), 169-188.
doi: 10.1016/j.trb.2016.09.012. |
| [13] |
H. Li, H. Wang, J. Chen and M. Bai,
Two-echelon vehicle routing problem with time windows and mobile satellites, Transportation Research Part B: Methodological, 138 (2020), 179-201.
doi: 10.1016/j.trb.2020.05.010. |
| [14] |
H. Li, Y. Liu, X. Jian and Y. Lu,
The two-echelon distribution system considering the real-time transshipment capacity varying, Transportation Research Part B: Methodological, 110 (2018), 239-260.
doi: 10.1016/j.trb.2018.02.015. |
| [15] |
R. Liu, L. Tao, Q. Hu and X. Xie,
Simulation-based optimisation approach for the stochastic two-echelon logistics problem, International Journal of Production Research, 55 (2017), 187-201.
doi: 10.1080/00207543.2016.1201221. |
| [16] |
T. Liu, Z. Luo, H. Qin and A. Lim,
A branch-and-cut algorithm for the two-echelon capacitated vehicle routing problem with grouping constraints, European Journal of Operational Research, 266 (2018), 487-497.
doi: 10.1016/j.ejor.2017.10.017. |
| [17] |
Z. Y. Ma, Y. B. Ling and J. Li,
2E-VRP Optimization Algorithm with Optimal Cutting and Full Path Matching Cross, Computer Engineering, 41 (2015), 279-285.
|
| [18] |
M. Marinelli, A. Colovic and M. Dell'Orco,
A novel Dynamic programming approach for Two-Echelon Capacitated Vehicle Routing Problem in City Logistics with Environmental considerations, Transportation Research Procedia, 30 (2018), 147-156.
doi: 10.1016/j.trpro.2018.09.017. |
| [19] |
E. Morganti, L. Dablancg and F. Fortin,
Final deliveries for online shopping: The deployment of pickup point networks in urban and suburban areas, Research in Transportation Business and Management, 11 (2014), 23-31.
doi: 10.1016/j.rtbm.2014.03.002. |
| [20] |
G. Perboli, R. Tadei and D. Vigo,
The two-echelon capacitated vehicle routing problem: Models and math-based heuristics, Transportation Science, 45 (2011), 364-380.
doi: 10.1287/trsc.1110.0368. |
| [21] |
F. A. Santos, G. R. Mateus and A. S. D. Cunha,
A branch-and-cut-and-price algorithm for the two-echelon capacitated vehicle routing problem, Transportation Science, 49 (2015), 355-368.
doi: 10.1287/trsc.2013.0500. |
| [22] |
M. Soysal, J. M. Bloemhof-Ruwaard and T. Bektas,
The time-dependent two-echelon capacitated vehicle routing problem with environmental considerations, International Journal of Production Economics, 164 (2015), 366-378.
doi: 10.1016/j.ijpe.2014.11.016. |
| [23] |
E. Swilley and R. E. Goldsmith,
Black Friday and Cyber Monday: Understanding consumer intentions on two major shopping days, Journal of Retailing and Consumer Services, 20 (2013), 43-50.
doi: 10.1016/j.jretconser.2012.10.003. |
| [24] |
Y. Tan and Y. Zhu, Fireworks algorithm for optimization, International Conference in Swarm Intelligence, Berlin: Springer, 355–364. |
| [25] |
E. B. Tirkolaee, A. Goli, A. Faridnia, M. Soltani and G.-W. Weber, Multi-objective optimization for the reliable pollution-routing problem with cross-dock selection using Pareto-based algorithms, Journal of Cleaner Production, 276 (2020), 122927.
doi: 10.1016/j.jclepro.2020.122927. |
| [26] |
E. B. Tirkolaee, A. Goli, M. Pahlevan and R. M. Kordestanizadeh,
A robust bi-objective multi-trip periodic capacitated arc routing problem for urban waste collection using a multi-objective invasive weed optimization, Waste Management and Research, 37 (2019), 1089-1101.
doi: 10.1177/0734242X19865340. |
| [27] |
E. B. Tirkolaee, S. Hadian and H. Golpra,
A novel multi-objective model for two-echelon green routing problem of perishable products with intermediate depots, Journal of Industrial Engineering and Management Studies, 6 (2019), 196-213.
|
| [28] |
E. B. Tirkolaee, S. Hadian, G.-W. Weber and I. Mahdavi,
A robust green traffic-based routing problem for perishable products distribution, Computational Intelligence, 36 (2020), 80-101.
doi: 10.1111/coin.12240. |
| [29] |
K. Wang, S. Lan and Y. Zhao,
A genetic-algorithm-based approach to the two-echelon capacitated vehicle routing problem with stochastic demands in logistics service, Journal of the Operational Research Society, 68 (2017), 1409-1421.
doi: 10.1057/s41274-016-0170-7. |
| [30] |
X. M. Yan, Z. F. Hao, H. Huang, B Li and S. Jiang,
Assignment-preference ant colony optimization for the two-echelon vehicle routing problem, Indian Pulp and Paper Technical Association, 30 (2018), 484-494.
|
| [31] |
T. T. Zhang and Z. F. Liu,
Fireworks algorithm for mean-VaR/CVaR models, Physica A: Statistical Mechanics and its Applications, 483 (2017), 1-8.
doi: 10.1016/j.physa.2017.04.036. |
show all references
References:
| [1] |
R. Baldacci, A. Mingozzi, R. Roberti and R. W. Clavo,
An exact algorithm for the two-echelon capacitated vehicle routing problem, Operations Research, 61 (2013), 298-314.
doi: 10.1287/opre.1120.1153. |
| [2] |
A. Bevilaqua, D. Bevilaqua and K. Yamanaka,
Parallel island based Memetic Algorithm with Lin-Kernighan local search for a real-life Two-Echelon Heterogeneous Vehicle Routing Problem based on Brazilian wholesale companies, Applied Soft Computing, 76 (2019), 697-711.
doi: 10.1016/j.asoc.2018.12.036. |
| [3] |
U. Breunig, R. Baldacci, R. F. Hartl and T. Vidal,
The electric two-echelon vehicle routing problem, Computers and Operations Research, 103 (2019), 198-210.
doi: 10.1016/j.cor.2018.11.005. |
| [4] |
U. Breunig, V. Schmid, R. F. Hartl and T. Vidal,
A large neighbourhood based heuristic for two-echelon routing problems, Computers and Operations Research, 76 (2016), 208-225.
doi: 10.1016/j.cor.2016.06.014. |
| [5] |
M.-C. Chen, P.-J. W and Y.-H. Hsu,
An effective pricing model for the congestion alleviation of e-commerce logistics, Computers and Industrial Engineering, 129 (2019), 368-376.
doi: 10.1016/j.cie.2019.01.060. |
| [6] |
Double 11 constantly refreshes the imagination of Chinese market, Global times, November 12, 2019 (015). |
| [7] |
P. Grangier, M. Gendreau, F. Lehuédé and L.-M. Rousseau,
An adaptive large neighborhood search for the two-echelon multiple-trip vehicle routing problem with satellite synchronization, European Journal of Operational Research, 254 (2016), 80-91.
doi: 10.1016/j.ejor.2016.03.040. |
| [8] |
M. Guan, M. Cha, Y. Li, Y. Wang and J. Yu, Predicting time-bounded purchases during a mega shopping festival, 2019 IEEE International Conference on Big Data and Smart Computing (BigComp), (2019), 1–8.
doi: 10.1109/BIGCOMP.2019.8679217. |
| [9] |
X. Guo, Y. J. L. Jaramillo, J. Bloemhof-Ruwaard and G. D. H. Claassen, On integrating crowdsourced delivery in last-mile logistics: A simulation study to quantify its feasibility, Journal of Cleaner Production, 241 (2019), 118365.
doi: 10.1016/j.jclepro.2019.118365. |
| [10] |
P. He and J. Li,
The two-echelon multi-trip vehicle routing problem with dynamic satellites for crop harvesting and transportation, Applied Soft Computing, 77 (2019), 387-398.
doi: 10.1016/j.asoc.2019.01.040. |
| [11] |
W. Jie, J. Yang, M. Zhang and Y. Huang,
The two-echelon capacitated electric vehicle routing problem with battery swapping stations: Formulation and efficient methodology, European Journal of Operational Research, 272 (2019), 879-904.
doi: 10.1016/j.ejor.2018.07.002. |
| [12] |
H. Li, L. Zhang, T. Lv and X. Chang,
The two-echelon time-constrained vehicle routing problem in linehaul-delivery systems, Transportation Research Part B: Methodological, 94 (2016), 169-188.
doi: 10.1016/j.trb.2016.09.012. |
| [13] |
H. Li, H. Wang, J. Chen and M. Bai,
Two-echelon vehicle routing problem with time windows and mobile satellites, Transportation Research Part B: Methodological, 138 (2020), 179-201.
doi: 10.1016/j.trb.2020.05.010. |
| [14] |
H. Li, Y. Liu, X. Jian and Y. Lu,
The two-echelon distribution system considering the real-time transshipment capacity varying, Transportation Research Part B: Methodological, 110 (2018), 239-260.
doi: 10.1016/j.trb.2018.02.015. |
| [15] |
R. Liu, L. Tao, Q. Hu and X. Xie,
Simulation-based optimisation approach for the stochastic two-echelon logistics problem, International Journal of Production Research, 55 (2017), 187-201.
doi: 10.1080/00207543.2016.1201221. |
| [16] |
T. Liu, Z. Luo, H. Qin and A. Lim,
A branch-and-cut algorithm for the two-echelon capacitated vehicle routing problem with grouping constraints, European Journal of Operational Research, 266 (2018), 487-497.
doi: 10.1016/j.ejor.2017.10.017. |
| [17] |
Z. Y. Ma, Y. B. Ling and J. Li,
2E-VRP Optimization Algorithm with Optimal Cutting and Full Path Matching Cross, Computer Engineering, 41 (2015), 279-285.
|
| [18] |
M. Marinelli, A. Colovic and M. Dell'Orco,
A novel Dynamic programming approach for Two-Echelon Capacitated Vehicle Routing Problem in City Logistics with Environmental considerations, Transportation Research Procedia, 30 (2018), 147-156.
doi: 10.1016/j.trpro.2018.09.017. |
| [19] |
E. Morganti, L. Dablancg and F. Fortin,
Final deliveries for online shopping: The deployment of pickup point networks in urban and suburban areas, Research in Transportation Business and Management, 11 (2014), 23-31.
doi: 10.1016/j.rtbm.2014.03.002. |
| [20] |
G. Perboli, R. Tadei and D. Vigo,
The two-echelon capacitated vehicle routing problem: Models and math-based heuristics, Transportation Science, 45 (2011), 364-380.
doi: 10.1287/trsc.1110.0368. |
| [21] |
F. A. Santos, G. R. Mateus and A. S. D. Cunha,
A branch-and-cut-and-price algorithm for the two-echelon capacitated vehicle routing problem, Transportation Science, 49 (2015), 355-368.
doi: 10.1287/trsc.2013.0500. |
| [22] |
M. Soysal, J. M. Bloemhof-Ruwaard and T. Bektas,
The time-dependent two-echelon capacitated vehicle routing problem with environmental considerations, International Journal of Production Economics, 164 (2015), 366-378.
doi: 10.1016/j.ijpe.2014.11.016. |
| [23] |
E. Swilley and R. E. Goldsmith,
Black Friday and Cyber Monday: Understanding consumer intentions on two major shopping days, Journal of Retailing and Consumer Services, 20 (2013), 43-50.
doi: 10.1016/j.jretconser.2012.10.003. |
| [24] |
Y. Tan and Y. Zhu, Fireworks algorithm for optimization, International Conference in Swarm Intelligence, Berlin: Springer, 355–364. |
| [25] |
E. B. Tirkolaee, A. Goli, A. Faridnia, M. Soltani and G.-W. Weber, Multi-objective optimization for the reliable pollution-routing problem with cross-dock selection using Pareto-based algorithms, Journal of Cleaner Production, 276 (2020), 122927.
doi: 10.1016/j.jclepro.2020.122927. |
| [26] |
E. B. Tirkolaee, A. Goli, M. Pahlevan and R. M. Kordestanizadeh,
A robust bi-objective multi-trip periodic capacitated arc routing problem for urban waste collection using a multi-objective invasive weed optimization, Waste Management and Research, 37 (2019), 1089-1101.
doi: 10.1177/0734242X19865340. |
| [27] |
E. B. Tirkolaee, S. Hadian and H. Golpra,
A novel multi-objective model for two-echelon green routing problem of perishable products with intermediate depots, Journal of Industrial Engineering and Management Studies, 6 (2019), 196-213.
|
| [28] |
E. B. Tirkolaee, S. Hadian, G.-W. Weber and I. Mahdavi,
A robust green traffic-based routing problem for perishable products distribution, Computational Intelligence, 36 (2020), 80-101.
doi: 10.1111/coin.12240. |
| [29] |
K. Wang, S. Lan and Y. Zhao,
A genetic-algorithm-based approach to the two-echelon capacitated vehicle routing problem with stochastic demands in logistics service, Journal of the Operational Research Society, 68 (2017), 1409-1421.
doi: 10.1057/s41274-016-0170-7. |
| [30] |
X. M. Yan, Z. F. Hao, H. Huang, B Li and S. Jiang,
Assignment-preference ant colony optimization for the two-echelon vehicle routing problem, Indian Pulp and Paper Technical Association, 30 (2018), 484-494.
|
| [31] |
T. T. Zhang and Z. F. Liu,
Fireworks algorithm for mean-VaR/CVaR models, Physica A: Statistical Mechanics and its Applications, 483 (2017), 1-8.
doi: 10.1016/j.physa.2017.04.036. |
Figure 2.
Initial second level solution schematic diagram
Figure 3.
Explosive operation Ⅰ (2-opt)
Figure 4.
Explosive operation II
Figure 5.
3-opt operation
Figure 6.
Flow chart of the HFWA
Figure 7.
Optimal results of three algorithms
Figure 8.
Jindong distribution of self-pickup points in Jinniu District
Table 1.
List of parameters and descriptions
| Sets and Parameters | Description |
| D | Set of depots, |
| S | Set of satellites, |
| C | Set of customers, |
| G | Set of first-level delivery vehicles, |
| H | Set of second-level delivery vehicles, |
| M | A large enough number |
| T |
Working hours per day |
| d |
The distance of the |
| q |
Demand of customer c |
| cap |
The capacity of the first-level vehicle |
| cap |
The capacity of the second-level vehicle |
| t |
The deadline of customer c |
| b |
Compensation per unit cargo for accepting flexible delivery |
| b |
Delay cost per delivery |
| a |
Fixed cost of the first-level delivery vehicle per delivery |
| a |
Fixed cost of the second-level delivery vehicle per delivery |
| c |
Unit distance cost of the first-level delivery vehicle per delivery |
| c |
Unit distance cost of the second-level delivery vehicle per delivery |
| c |
The labor cost of the first-level delivery vehicle per delivery |
| c |
The labor cost of the second -level delivery per delivery |
| f |
If customer c chooses flexible delivery, fc =1; otherwise, fc =0 |
| T |
Time required to complete standard delivery |
| Sets and Parameters | Description |
| D | Set of depots, |
| S | Set of satellites, |
| C | Set of customers, |
| G | Set of first-level delivery vehicles, |
| H | Set of second-level delivery vehicles, |
| M | A large enough number |
| T |
Working hours per day |
| d |
The distance of the |
| q |
Demand of customer c |
| cap |
The capacity of the first-level vehicle |
| cap |
The capacity of the second-level vehicle |
| t |
The deadline of customer c |
| b |
Compensation per unit cargo for accepting flexible delivery |
| b |
Delay cost per delivery |
| a |
Fixed cost of the first-level delivery vehicle per delivery |
| a |
Fixed cost of the second-level delivery vehicle per delivery |
| c |
Unit distance cost of the first-level delivery vehicle per delivery |
| c |
Unit distance cost of the second-level delivery vehicle per delivery |
| c |
The labor cost of the first-level delivery vehicle per delivery |
| c |
The labor cost of the second -level delivery per delivery |
| f |
If customer c chooses flexible delivery, fc =1; otherwise, fc =0 |
| T |
Time required to complete standard delivery |
Table 2.
List of variables and descriptions
| Variables | Description |
| x |
First-level distribution vehicle g travels the |
| y |
Second -level distribution vehicle h travels the |
| z |
Customer c cargo comes from satellite s, z |
| w |
The actual load of first level vehicle g to satellite s; decision variable |
| l |
The total demand of satellite s |
| t |
Time of first-level vehicle g arriving at satellite s |
| t |
Time of arrival of second-level vehicle h to satellite s |
| t |
The longest time for the first-level vehicle to complete the distribution task |
| time |
The actual delivery time of the customer c |
| dtime |
The delay time for customer c |
| U1 |
Restrict the occurrence of sub-tour in the first-level vehicles |
| U2 |
Restrict the occurrence of sub-tour in the second-level vehicles |
| u |
Intermediate variable, no actual meaning |
| u |
Intermediate variable, no actual meaning |
| Variables | Description |
| x |
First-level distribution vehicle g travels the |
| y |
Second -level distribution vehicle h travels the |
| z |
Customer c cargo comes from satellite s, z |
| w |
The actual load of first level vehicle g to satellite s; decision variable |
| l |
The total demand of satellite s |
| t |
Time of first-level vehicle g arriving at satellite s |
| t |
Time of arrival of second-level vehicle h to satellite s |
| t |
The longest time for the first-level vehicle to complete the distribution task |
| time |
The actual delivery time of the customer c |
| dtime |
The delay time for customer c |
| U1 |
Restrict the occurrence of sub-tour in the first-level vehicles |
| U2 |
Restrict the occurrence of sub-tour in the second-level vehicles |
| u |
Intermediate variable, no actual meaning |
| u |
Intermediate variable, no actual meaning |
Table 3.
Parameter settings
| Algorithm | Parameter | Value |
| HFWA | Fireworks population size | 5 |
| The number of explosion sparks | 2 | |
| Upper limit of the number of explosion sparks | 50 | |
| Variation spark number | 2 | |
| Number of iterations | 1000 | |
| ACO | Number of ants | 50 |
| Pheromone heuristic factor | 1 | |
| Fitness heuristic factor | 9 | |
| Pheromone volatile factor | 0.1 | |
| Constant coefficient | 1 | |
| Number of iterations | 1000 | |
| GA | Population size | 50 |
| Cross factor | 0.8 | |
| Mutation factor | 0.2 | |
| Number of iterations | 1000 |
| Algorithm | Parameter | Value |
| HFWA | Fireworks population size | 5 |
| The number of explosion sparks | 2 | |
| Upper limit of the number of explosion sparks | 50 | |
| Variation spark number | 2 | |
| Number of iterations | 1000 | |
| ACO | Number of ants | 50 |
| Pheromone heuristic factor | 1 | |
| Fitness heuristic factor | 9 | |
| Pheromone volatile factor | 0.1 | |
| Constant coefficient | 1 | |
| Number of iterations | 1000 | |
| GA | Population size | 50 |
| Cross factor | 0.8 | |
| Mutation factor | 0.2 | |
| Number of iterations | 1000 |
Table 4.
The optimization results of ACO algorithm
| No | Standard test |
n | ACO | Optimal solution (km) |
|||
| Optimal (km) | Average (km) | Time (s) | GAP (%) | ||||
| 1 | Set2a_E-n22-k4-s6-17 | 22 | 422.93 | 422.93 | 50.7 | 1.40% | 417.07 |
| 2 | Set2a_E-n22-k4-s8-14 | 22 | 387.84 | 387.84 | 50.5 | 0.75% | 384.96 |
| 3 | Set2a_E-n22-k4-s9-19 | 22 | 479.05 | 484.18 | 49.1 | 1.80% | 470.6 |
| 4 | Set2a_E-n22-k4-s10-14 | 22 | 377.56 | 377.56 | 49.1 | 1.63% | 371.5 |
| 5 | Set2a_E-n33-k4-s1-9 | 33 | 753.75 | 768.28 | 74.9 | 3.23% | 730.16 |
| 6 | Set2a_E-n33-k4-s2-13 | 33 | 761.76 | 776.57 | 75 | 6.60% | 714.63 |
| 7 | Set2a_E-n33-k4-s3-17 | 33 | 745.38 | 759.23 | 74.9 | 5.36% | 707.48 |
| 8 | Set2a_E-n33-k4-s7-25 | 33 | 790.55 | 804.92 | 75 | 4.45% | 756.85 |
| 9 | Set2a_E-n33-k4-s14-22 | 33 | 797.87 | 802.72 | 75.7 | 2.42% | 779.05 |
| 10 | Set2b_E-n51-k5-s2-4-17-46 | 51 | 618.69 | 637.24 | 124.9 | 16.57% | 530.76 |
| 11 | Set2b_E-n51-k5-s2-17 | 51 | 665.23 | 684.06 | 120.8 | 11.34% | 597.49 |
| 12 | Set2b_E-n51-k5-s4-46 | 51 | 613.78 | 627.29 | 120.2 | 15.64% | 530.76 |
| No | Standard test |
n | ACO | Optimal solution (km) |
|||
| Optimal (km) | Average (km) | Time (s) | GAP (%) | ||||
| 1 | Set2a_E-n22-k4-s6-17 | 22 | 422.93 | 422.93 | 50.7 | 1.40% | 417.07 |
| 2 | Set2a_E-n22-k4-s8-14 | 22 | 387.84 | 387.84 | 50.5 | 0.75% | 384.96 |
| 3 | Set2a_E-n22-k4-s9-19 | 22 | 479.05 | 484.18 | 49.1 | 1.80% | 470.6 |
| 4 | Set2a_E-n22-k4-s10-14 | 22 | 377.56 | 377.56 | 49.1 | 1.63% | 371.5 |
| 5 | Set2a_E-n33-k4-s1-9 | 33 | 753.75 | 768.28 | 74.9 | 3.23% | 730.16 |
| 6 | Set2a_E-n33-k4-s2-13 | 33 | 761.76 | 776.57 | 75 | 6.60% | 714.63 |
| 7 | Set2a_E-n33-k4-s3-17 | 33 | 745.38 | 759.23 | 74.9 | 5.36% | 707.48 |
| 8 | Set2a_E-n33-k4-s7-25 | 33 | 790.55 | 804.92 | 75 | 4.45% | 756.85 |
| 9 | Set2a_E-n33-k4-s14-22 | 33 | 797.87 | 802.72 | 75.7 | 2.42% | 779.05 |
| 10 | Set2b_E-n51-k5-s2-4-17-46 | 51 | 618.69 | 637.24 | 124.9 | 16.57% | 530.76 |
| 11 | Set2b_E-n51-k5-s2-17 | 51 | 665.23 | 684.06 | 120.8 | 11.34% | 597.49 |
| 12 | Set2b_E-n51-k5-s4-46 | 51 | 613.78 | 627.29 | 120.2 | 15.64% | 530.76 |
Table 5.
The optimization results of GA algorithm
| No. | Standard test |
n | GA | Optimal solution (km) |
|||
| Optimal (km) | Average (km) | Time (s) | GAP (%) | ||||
| 1 | Set2a_E-n22-k4-s6-17 | 22 | 417.07 | 438.04 | 472.4 | 0.00% | 417.07 |
| 2 | Set2a_E-n22-k4-s8-14 | 22 | 387.84 | 399.37 | 468.9 | 0.75 % | 384.96 |
| 3 | Set2a_E-n22-k4-s9-19 | 22 | 475.62 | 492.41 | 474.9 | 1.07 % | 470.6 |
| 4 | Set2a_E-n22-k4-s10-14 | 22 | 377.56 | 383.11 | 472.1 | 1.63 % | 371.5 |
| 5 | Set2a_E-n33-k4-s1-9 | 33 | 730.16 | 764.25 | 502.2 | 0.00 % | 730.16 |
| 6 | Set2a_E-n33-k4-s2-13 | 33 | 725.04 | 747.5 | 484.6 | 1.46 % | 714.63 |
| 7 | Set2a_E-n33-k4-s3-17 | 33 | 732.37 | 760.82 | 488.5 | 3.52 % | 707.48 |
| 8 | Set2a_E-n33-k4-s7-25 | 33 | 763.58 | 790.26 | 491.8 | 0.89 % | 756.85 |
| 9 | Set2a_E-n33-k4-s14-22 | 33 | 782.04 | 792.21 | 510.7 | 0.38 % | 779.05 |
| 10 | Set2b_E-n51-k5-s2-4-17-46 | 51 | 599.66 | 631.44 | 860.2 | 12.98 % | 530.76 |
| 11 | Set2b_E-n51-k5-s2-17 | 51 | 641.66 | 671.12 | 695.5 | 7.39 % | 597.49 |
| 12 | Set2b_E-n51-k5-s4-46 | 51 | 604.92 | 620.14 | 656.7 | 13.97 % | 530.76 |
| No. | Standard test |
n | GA | Optimal solution (km) |
|||
| Optimal (km) | Average (km) | Time (s) | GAP (%) | ||||
| 1 | Set2a_E-n22-k4-s6-17 | 22 | 417.07 | 438.04 | 472.4 | 0.00% | 417.07 |
| 2 | Set2a_E-n22-k4-s8-14 | 22 | 387.84 | 399.37 | 468.9 | 0.75 % | 384.96 |
| 3 | Set2a_E-n22-k4-s9-19 | 22 | 475.62 | 492.41 | 474.9 | 1.07 % | 470.6 |
| 4 | Set2a_E-n22-k4-s10-14 | 22 | 377.56 | 383.11 | 472.1 | 1.63 % | 371.5 |
| 5 | Set2a_E-n33-k4-s1-9 | 33 | 730.16 | 764.25 | 502.2 | 0.00 % | 730.16 |
| 6 | Set2a_E-n33-k4-s2-13 | 33 | 725.04 | 747.5 | 484.6 | 1.46 % | 714.63 |
| 7 | Set2a_E-n33-k4-s3-17 | 33 | 732.37 | 760.82 | 488.5 | 3.52 % | 707.48 |
| 8 | Set2a_E-n33-k4-s7-25 | 33 | 763.58 | 790.26 | 491.8 | 0.89 % | 756.85 |
| 9 | Set2a_E-n33-k4-s14-22 | 33 | 782.04 | 792.21 | 510.7 | 0.38 % | 779.05 |
| 10 | Set2b_E-n51-k5-s2-4-17-46 | 51 | 599.66 | 631.44 | 860.2 | 12.98 % | 530.76 |
| 11 | Set2b_E-n51-k5-s2-17 | 51 | 641.66 | 671.12 | 695.5 | 7.39 % | 597.49 |
| 12 | Set2b_E-n51-k5-s4-46 | 51 | 604.92 | 620.14 | 656.7 | 13.97 % | 530.76 |
Table 6.
The optimization results of HFWA algorithm
| No. | Standard test |
n | HFWA | Optimal solution (km) |
|||
| Optimal (km) | Average (km) | Time (s) | GAP (%) | ||||
| 1 | Set2a_E-n22-k4-s6-17 | 22 | 417.07 | 417.07 | 103.7 | 0.00 % | 417.07 |
| 2 | Set2a_E-n22-k4-s8-14 | 22 | 384.96 | 386.69 | 106.2 | 0.00 % | 384.96 |
| 3 | Set2a_E-n22-k4-s9-19 | 22 | 470.6 | 472.84 | 107.1 | 0.00 % | 470.6 |
| 4 | Set2a_E-n22-k4-s10-14 | 22 | 371.5 | 376.35 | 104.2 | 0.00 % | 371.5 |
| 5 | Set2a_E-n33-k4-s1-9 | 33 | 730.16 | 734.76 | 188.9 | 0.00 % | 730.16 |
| 6 | Set2a_E-n33-k4-s2-13 | 33 | 714.63 | 724.6 | 191.1 | 0.00 % | 714.63 |
| 7 | Set2a_E-n33-k4-s3-17 | 33 | 707.48 | 712.08 | 192.1 | 0.00 % | 707.48 |
| 8 | Set2a_E-n33-k4-s7-25 | 33 | 756.85 | 765.18 | 192.1 | 0.00 % | 756.85 |
| 9 | Set2a_E-n33-k4-s14-22 | 33 | 779.05 | 781.95 | 194.7 | 0.00 % | 779.05 |
| 10 | Set2b_E-n51-k5-s2-4-17-46 | 51 | 530.76 | 557.82 | 593.6 | 0.00 % | 530.76 |
| 11 | Set2b_E-n51-k5-s2-17 | 51 | 597.49 | 622.8 | 490 | 0.00 % | 597.49 |
| 12 | Set2b_E-n51-k5-s4-46 | 51 | 530.76 | 549.47 | 499.2 | 0.00 % | 530.76 |
| No. | Standard test |
n | HFWA | Optimal solution (km) |
|||
| Optimal (km) | Average (km) | Time (s) | GAP (%) | ||||
| 1 | Set2a_E-n22-k4-s6-17 | 22 | 417.07 | 417.07 | 103.7 | 0.00 % | 417.07 |
| 2 | Set2a_E-n22-k4-s8-14 | 22 | 384.96 | 386.69 | 106.2 | 0.00 % | 384.96 |
| 3 | Set2a_E-n22-k4-s9-19 | 22 | 470.6 | 472.84 | 107.1 | 0.00 % | 470.6 |
| 4 | Set2a_E-n22-k4-s10-14 | 22 | 371.5 | 376.35 | 104.2 | 0.00 % | 371.5 |
| 5 | Set2a_E-n33-k4-s1-9 | 33 | 730.16 | 734.76 | 188.9 | 0.00 % | 730.16 |
| 6 | Set2a_E-n33-k4-s2-13 | 33 | 714.63 | 724.6 | 191.1 | 0.00 % | 714.63 |
| 7 | Set2a_E-n33-k4-s3-17 | 33 | 707.48 | 712.08 | 192.1 | 0.00 % | 707.48 |
| 8 | Set2a_E-n33-k4-s7-25 | 33 | 756.85 | 765.18 | 192.1 | 0.00 % | 756.85 |
| 9 | Set2a_E-n33-k4-s14-22 | 33 | 779.05 | 781.95 | 194.7 | 0.00 % | 779.05 |
| 10 | Set2b_E-n51-k5-s2-4-17-46 | 51 | 530.76 | 557.82 | 593.6 | 0.00 % | 530.76 |
| 11 | Set2b_E-n51-k5-s2-17 | 51 | 597.49 | 622.8 | 490 | 0.00 % | 597.49 |
| 12 | Set2b_E-n51-k5-s4-46 | 51 | 530.76 | 549.47 | 499.2 | 0.00 % | 530.76 |
Table 7.
The results of existing literature
| No. | Standard test |
n | [18] and [11] | Optimal solution (km) |
|||
| Optimal (km) | GAP (%) | Optimal (km) | GAP (%) | ||||
| 1 | Set2a_E-n22-k4-s6-17 | 22 | 417.07 | 0.00 % | 417.07 | 0.00 % | 417.07 |
| 2 | Set2a_E-n22-k4-s8-14 | 22 | 384.96 | 0.00 % | 384.96 | 0.00 % | 384.96 |
| 3 | Set2a_E-n22-k4-s9-19 | 22 | 470.6 | 0.00 % | 470.6 | 0.00 % | 470.6 |
| 4 | Set2a_E-n22-k4-s10-14 | 22 | 371.5 | 0.00 % | 371.5 | 0.00 % | 371.5 |
| 5 | Set2a_E-n33-k4-s1-9 | 33 | 743.22 | 1.79 % | 730.16 | 0.00 % | 730.16 |
| 6 | Set2a_E-n33-k4-s2-13 | 33 | 710.48 | -0.58 % | 714.63 | 0.00 % | 714.63 |
| 7 | Set2a_E-n33-k4-s3-17 | 33 | - | - | 707.48 | 0.00 % | 707.48 |
| 8 | Set2a_E-n33-k4-s7-25 | 33 | 756.85 | 0.00 % | 756.85 | 0.00 % | 756.85 |
| 9 | Set2a_E-n33-k4-s14-22 | 33 | - | - | 779.05 | 0.00 % | 779.05 |
| 10 | Set2b_E-n51-k5-s2-4-17-46 | 51 | 577.16 | 8.74 % | 530.76 | 0.00 % | 530.76 |
| 11 | Set2b_E-n51-k5-s2-17 | 51 | - | - | 597.49 | 0.00 % | 597.49 |
| 12 | Set2b_E-n51-k5-s4-46 | 51 | - | - | 530.76 | 0.00 % | 530.76 |
| No. | Standard test |
n | [18] and [11] | Optimal solution (km) |
|||
| Optimal (km) | GAP (%) | Optimal (km) | GAP (%) | ||||
| 1 | Set2a_E-n22-k4-s6-17 | 22 | 417.07 | 0.00 % | 417.07 | 0.00 % | 417.07 |
| 2 | Set2a_E-n22-k4-s8-14 | 22 | 384.96 | 0.00 % | 384.96 | 0.00 % | 384.96 |
| 3 | Set2a_E-n22-k4-s9-19 | 22 | 470.6 | 0.00 % | 470.6 | 0.00 % | 470.6 |
| 4 | Set2a_E-n22-k4-s10-14 | 22 | 371.5 | 0.00 % | 371.5 | 0.00 % | 371.5 |
| 5 | Set2a_E-n33-k4-s1-9 | 33 | 743.22 | 1.79 % | 730.16 | 0.00 % | 730.16 |
| 6 | Set2a_E-n33-k4-s2-13 | 33 | 710.48 | -0.58 % | 714.63 | 0.00 % | 714.63 |
| 7 | Set2a_E-n33-k4-s3-17 | 33 | - | - | 707.48 | 0.00 % | 707.48 |
| 8 | Set2a_E-n33-k4-s7-25 | 33 | 756.85 | 0.00 % | 756.85 | 0.00 % | 756.85 |
| 9 | Set2a_E-n33-k4-s14-22 | 33 | - | - | 779.05 | 0.00 % | 779.05 |
| 10 | Set2b_E-n51-k5-s2-4-17-46 | 51 | 577.16 | 8.74 % | 530.76 | 0.00 % | 530.76 |
| 11 | Set2b_E-n51-k5-s2-17 | 51 | - | - | 597.49 | 0.00 % | 597.49 |
| 12 | Set2b_E-n51-k5-s4-46 | 51 | - | - | 530.76 | 0.00 % | 530.76 |
Table 8.
Distribution network node coordinates
| Node | X | Y | Node | X | Y | Node | X | Y |
| D | 20639 | 18019 | 16 | 14533 | 5098 | 34 | 13728 | 8485 |
| S1 | 807 | 16768 | 17 | 13623 | 8242 | 35 | 12730 | 4622 |
| S2 | 33084 | 19137 | 18 | 13573 | 3064 | 36 | 12968 | 4261 |
| 1 | 11066 | 5223 | 19 | 15874 | 7803 | 37 | 12611 | 3804 |
| 2 | 10481 | 7350 | 20 | 11212 | 6558 | 38 | 15604 | 5195 |
| 3 | 16356 | 5406 | 21 | 13396 | 3457 | 39 | 16340 | 4248 |
| 4 | 14197 | 6453 | 22 | 11813 | 9258 | 40 | 15342 | 6701 |
| 5 | 9591 | 5209 | 23 | 15606 | 5339 | 41 | 12902 | 3362 |
| 6 | 12664 | 5236 | 24 | 17577 | 5196 | 42 | 12149 | 5210 |
| 7 | 10772 | 5910 | 25 | 10496 | 5801 | 43 | 13221 | 5054 |
| 8 | 15321 | 5178 | 26 | 17472 | 3701 | 44 | 13451 | 7415 |
| 9 | 15239 | 5209 | 27 | 13839 | 7873 | 45 | 15543 | 7984 |
| 10 | 13556 | 7147 | 28 | 16555 | 8635 | 46 | 13025 | 4248 |
| 11 | 16660 | 4104 | 29 | 9347 | 6362 | 47 | 17460 | 4241 |
| 12 | 12438 | 3987 | 30 | 9547 | 8857 | 48 | 10895 | 4818 |
| 13 | 13850 | 6882 | 31 | 17942 | 3752 | 49 | 13704 | 10362 |
| 14 | 15196 | 8050 | 32 | 11042 | 5921 | 50 | 12929 | 4844 |
| 15 | 12864 | 4804 | 33 | 11387 | 5026 |
| Node | X | Y | Node | X | Y | Node | X | Y |
| D | 20639 | 18019 | 16 | 14533 | 5098 | 34 | 13728 | 8485 |
| S1 | 807 | 16768 | 17 | 13623 | 8242 | 35 | 12730 | 4622 |
| S2 | 33084 | 19137 | 18 | 13573 | 3064 | 36 | 12968 | 4261 |
| 1 | 11066 | 5223 | 19 | 15874 | 7803 | 37 | 12611 | 3804 |
| 2 | 10481 | 7350 | 20 | 11212 | 6558 | 38 | 15604 | 5195 |
| 3 | 16356 | 5406 | 21 | 13396 | 3457 | 39 | 16340 | 4248 |
| 4 | 14197 | 6453 | 22 | 11813 | 9258 | 40 | 15342 | 6701 |
| 5 | 9591 | 5209 | 23 | 15606 | 5339 | 41 | 12902 | 3362 |
| 6 | 12664 | 5236 | 24 | 17577 | 5196 | 42 | 12149 | 5210 |
| 7 | 10772 | 5910 | 25 | 10496 | 5801 | 43 | 13221 | 5054 |
| 8 | 15321 | 5178 | 26 | 17472 | 3701 | 44 | 13451 | 7415 |
| 9 | 15239 | 5209 | 27 | 13839 | 7873 | 45 | 15543 | 7984 |
| 10 | 13556 | 7147 | 28 | 16555 | 8635 | 46 | 13025 | 4248 |
| 11 | 16660 | 4104 | 29 | 9347 | 6362 | 47 | 17460 | 4241 |
| 12 | 12438 | 3987 | 30 | 9547 | 8857 | 48 | 10895 | 4818 |
| 13 | 13850 | 6882 | 31 | 17942 | 3752 | 49 | 13704 | 10362 |
| 14 | 15196 | 8050 | 32 | 11042 | 5921 | 50 | 12929 | 4844 |
| 15 | 12864 | 4804 | 33 | 11387 | 5026 |
Table 9.
Demand and delivery time of customer points
| Node | Demand (packages) | Time (days) | Node | Demand (packages) | Time (days) | Node | Demand (packages) | Time (days) |
| 1 | 91 | 3 | 18 | 46 | 3 | 35 | 302 | 6 |
| 2 | 224 | 3 | 19 | 49 | 3 | 36 | 94 | 6 |
| 3 | 215 | 3 | 20 | 85 | 3 | 37 | 249 | 6 |
| 4 | 53 | 6 | 21 | 277 | 6 | 38 | 248 | 3 |
| 5 | 39 | 6 | 22 | 84 | 6 | 39 | 125 | 3 |
| 6 | 164 | 6 | 23 | 268 | 6 | 40 | 130 | 3 |
| 7 | 316 | 6 | 24 | 80 | 3 | 41 | 25 | 3 |
| 8 | 112 | 3 | 25 | 306 | 3 | 42 | 75 | 6 |
| 9 | 192 | 3 | 26 | 115 | 3 | 43 | 175 | 6 |
| 10 | 74 | 3 | 27 | 65 | 3 | 44 | 256 | 6 |
| 11 | 247 | 6 | 28 | 83 | 6 | 45 | 307 | 6 |
| 12 | 84 | 6 | 29 | 203 | 6 | 46 | 42 | 3 |
| 13 | 166 | 6 | 30 | 156 | 6 | 47 | 186 | 3 |
| 14 | 230 | 3 | 31 | 116 | 6 | 48 | 100 | 6 |
| 15 | 293 | 3 | 32 | 273 | 3 | 49 | 57 | 6 |
| 16 | 316 | 6 | 33 | 192 | 3 | 50 | 110 | 6 |
| 17 | 180 | 6 | 34 | 181 | 3 |
| Node | Demand (packages) | Time (days) | Node | Demand (packages) | Time (days) | Node | Demand (packages) | Time (days) |
| 1 | 91 | 3 | 18 | 46 | 3 | 35 | 302 | 6 |
| 2 | 224 | 3 | 19 | 49 | 3 | 36 | 94 | 6 |
| 3 | 215 | 3 | 20 | 85 | 3 | 37 | 249 | 6 |
| 4 | 53 | 6 | 21 | 277 | 6 | 38 | 248 | 3 |
| 5 | 39 | 6 | 22 | 84 | 6 | 39 | 125 | 3 |
| 6 | 164 | 6 | 23 | 268 | 6 | 40 | 130 | 3 |
| 7 | 316 | 6 | 24 | 80 | 3 | 41 | 25 | 3 |
| 8 | 112 | 3 | 25 | 306 | 3 | 42 | 75 | 6 |
| 9 | 192 | 3 | 26 | 115 | 3 | 43 | 175 | 6 |
| 10 | 74 | 3 | 27 | 65 | 3 | 44 | 256 | 6 |
| 11 | 247 | 6 | 28 | 83 | 6 | 45 | 307 | 6 |
| 12 | 84 | 6 | 29 | 203 | 6 | 46 | 42 | 3 |
| 13 | 166 | 6 | 30 | 156 | 6 | 47 | 186 | 3 |
| 14 | 230 | 3 | 31 | 116 | 6 | 48 | 100 | 6 |
| 15 | 293 | 3 | 32 | 273 | 3 | 49 | 57 | 6 |
| 16 | 316 | 6 | 33 | 192 | 3 | 50 | 110 | 6 |
| 17 | 180 | 6 | 34 | 181 | 3 |
Table 10.
Computational results for the regular delivery model
| Level | Vehicle NO. | Standard delivery vehicle route |
| First-level vehicle delivery | 1S | D-S1-D |
| 2S | D-S1-D | |
| 3S | D-S1-D | |
| 4S | D-S1-D | |
| Second-level vehicle delivery | 1 | S1-21-18-41-S1 |
| 2 | S1-37-12-S1 | |
| 3 | S1-35-6-S1 | |
| 4 | S1-42-33-48-1-S1 | |
| 5 | S1-32-20-S1 | |
| 6 | S1-25-5-S1 | |
| 7 | S1-29-2-S1 | |
| 8 | S1-30-22-49-34-S1 | |
| 9 | S1-17-27-S1 | |
| 10 | S1-44-10-13-S1 | |
| 11 | S1-4-40-14-S1 | |
| 12 | S1-45-19-28-S1 | |
| 13 | S1-24-47-31-26-S1 | |
| 14 | S1-11-39-S1 | |
| 15 | S1-3-23-S1 | |
| 16 | S1-38-8-S1 | |
| 17 | S1-7-S1 | |
| 18 | S1-9-S1 | |
| 19 | S1-16-43-S1 | |
| 20 | S1-50-15-36-S1 | |
| 21 | S1-46-S1 | |
| Delivery time (days) | 7 | |
| Delay cost (yuan) | 32240 | |
| Compensation cost (yuan) | 0 | |
| Total cost (yuan) | 2159128 | |
| Level | Vehicle NO. | Standard delivery vehicle route |
| First-level vehicle delivery | 1S | D-S1-D |
| 2S | D-S1-D | |
| 3S | D-S1-D | |
| 4S | D-S1-D | |
| Second-level vehicle delivery | 1 | S1-21-18-41-S1 |
| 2 | S1-37-12-S1 | |
| 3 | S1-35-6-S1 | |
| 4 | S1-42-33-48-1-S1 | |
| 5 | S1-32-20-S1 | |
| 6 | S1-25-5-S1 | |
| 7 | S1-29-2-S1 | |
| 8 | S1-30-22-49-34-S1 | |
| 9 | S1-17-27-S1 | |
| 10 | S1-44-10-13-S1 | |
| 11 | S1-4-40-14-S1 | |
| 12 | S1-45-19-28-S1 | |
| 13 | S1-24-47-31-26-S1 | |
| 14 | S1-11-39-S1 | |
| 15 | S1-3-23-S1 | |
| 16 | S1-38-8-S1 | |
| 17 | S1-7-S1 | |
| 18 | S1-9-S1 | |
| 19 | S1-16-43-S1 | |
| 20 | S1-50-15-36-S1 | |
| 21 | S1-46-S1 | |
| Delivery time (days) | 7 | |
| Delay cost (yuan) | 32240 | |
| Compensation cost (yuan) | 0 | |
| Total cost (yuan) | 2159128 | |
Table 11.
Computational results for the TDD model
| Delivery method | Level | Vehicle NO. | TDD vehicle route |
| Standard delivery | First-level vehicle delivery | 1S | D-S1-D |
| 2S | D-S1-D | ||
| Second-level vehicle delivery | 1 | S1-26-47-24-8-9-S1 | |
| 2 | S1-38-19-20-S1 | ||
| 3 | S1-32-1-S1 | ||
| 4 | S1-2-25-S1 | ||
| 5 | S1-33-41-18-S1 | ||
| 6 | S1-46-15-S1 | ||
| 7 | S1-10-27-S1 | ||
| 8 | S1-14-34-S1 | ||
| 9 | S1-40-3-S1 | ||
| 10 | S1-39-S1 | ||
| Flexible delivery | First-level vehicle delivery | 1S* | D-S1-D |
| 2S* | D-S1-D | ||
| 3S* | D-S1-D | ||
| Second-level vehicle delivery | 11 | S1-23-13-S1 | |
| 12 | S1-17-44-S1 | ||
| 13 | S1-45-28-49-S1 | ||
| 14 | S1-22-31-11-S1 | ||
| 15 | S1-16-43-S1 | ||
| 16 | S1-6-37-12-S1 | ||
| 17 | S1-21-36-50-S1 | ||
| 18 | S1-42-35-48-S1 | ||
| 19 | S1-7-5-S1 | ||
| 20 | S1-29-30-S1 | ||
| Delivery time (days) | 6 | ||
| Delay cost (yuan) | 0 | ||
| Compensation cost (yuan) | 2239.5 | ||
| Total cost (yuan) | 1937381 | ||
| Delivery method | Level | Vehicle NO. | TDD vehicle route |
| Standard delivery | First-level vehicle delivery | 1S | D-S1-D |
| 2S | D-S1-D | ||
| Second-level vehicle delivery | 1 | S1-26-47-24-8-9-S1 | |
| 2 | S1-38-19-20-S1 | ||
| 3 | S1-32-1-S1 | ||
| 4 | S1-2-25-S1 | ||
| 5 | S1-33-41-18-S1 | ||
| 6 | S1-46-15-S1 | ||
| 7 | S1-10-27-S1 | ||
| 8 | S1-14-34-S1 | ||
| 9 | S1-40-3-S1 | ||
| 10 | S1-39-S1 | ||
| Flexible delivery | First-level vehicle delivery | 1S* | D-S1-D |
| 2S* | D-S1-D | ||
| 3S* | D-S1-D | ||
| Second-level vehicle delivery | 11 | S1-23-13-S1 | |
| 12 | S1-17-44-S1 | ||
| 13 | S1-45-28-49-S1 | ||
| 14 | S1-22-31-11-S1 | ||
| 15 | S1-16-43-S1 | ||
| 16 | S1-6-37-12-S1 | ||
| 17 | S1-21-36-50-S1 | ||
| 18 | S1-42-35-48-S1 | ||
| 19 | S1-7-5-S1 | ||
| 20 | S1-29-30-S1 | ||
| Delivery time (days) | 6 | ||
| Delay cost (yuan) | 0 | ||
| Compensation cost (yuan) | 2239.5 | ||
| Total cost (yuan) | 1937381 | ||
Table 12.
Delivery time sensitivity analysis
| Standard | Penalty | Standard | TDD second- | Penalty | Compensation | Total cost |
| delivery | (yuan) | delivery | level arrival | (yuan) | cost | of delivery |
| time (day) | cost (yuan) | time (day) | (yuan) | (yuan) | ||
| 2 | 40300 | 1881455 | 4 | 6250 | 2239.5 | 1783100 |
| 5 | 3750 | 2239.5 | 1781518 | |||
| 6 | 1250 | 2239.5 | 1779718 | |||
| 3 | 32240 | 1868263 | 4 | 5000 | 2239.5 | 1782724 |
| 5 | 2500 | 2239.5 | 1779725 | |||
| 6 | 0 | 2239.5 | 1775722 | |||
| 4 | 24180 | 1862604 | 5 | 2500 | 2239.5 | 1779704 |
| 6 | 0 | 2239.5 | 1777541 | |||
| 7 | 0 | 2239.5 | 1777541 |
| Standard | Penalty | Standard | TDD second- | Penalty | Compensation | Total cost |
| delivery | (yuan) | delivery | level arrival | (yuan) | cost | of delivery |
| time (day) | cost (yuan) | time (day) | (yuan) | (yuan) | ||
| 2 | 40300 | 1881455 | 4 | 6250 | 2239.5 | 1783100 |
| 5 | 3750 | 2239.5 | 1781518 | |||
| 6 | 1250 | 2239.5 | 1779718 | |||
| 3 | 32240 | 1868263 | 4 | 5000 | 2239.5 | 1782724 |
| 5 | 2500 | 2239.5 | 1779725 | |||
| 6 | 0 | 2239.5 | 1775722 | |||
| 4 | 24180 | 1862604 | 5 | 2500 | 2239.5 | 1779704 |
| 6 | 0 | 2239.5 | 1777541 | |||
| 7 | 0 | 2239.5 | 1777541 |
| [1] |
Shengyang Jia, Lei Deng, Quanwu Zhao, Yunkai Chen. An adaptive large neighborhood search heuristic for multi-commodity two-echelon vehicle routing problem with satellite synchronization. Journal of Industrial and Management Optimization, 2022 doi: 10.3934/jimo.2021225 |
| [2] |
Ming-Yong Lai, Chang-Shi Liu, Xiao-Jiao Tong. A two-stage hybrid meta-heuristic for pickup and delivery vehicle routing problem with time windows. Journal of Industrial and Management Optimization, 2010, 6 (2) : 435-451. doi: 10.3934/jimo.2010.6.435 |
| [3] |
Yaw Chang, Lin Chen. Solve the vehicle routing problem with time windows via a genetic algorithm. Conference Publications, 2007, 2007 (Special) : 240-249. doi: 10.3934/proc.2007.2007.240 |
| [4] |
Bin Feng, Lixin Wei, Ziyu Hu. An adaptive large neighborhood search algorithm for Vehicle Routing Problem with Multiple Time Windows constraints. Journal of Industrial and Management Optimization, 2021 doi: 10.3934/jimo.2021197 |
| [5] |
Magfura Pervin, Sankar Kumar Roy, Gerhard Wilhelm Weber. A two-echelon inventory model with stock-dependent demand and variable holding cost for deteriorating items. Numerical Algebra, Control and Optimization, 2017, 7 (1) : 21-50. doi: 10.3934/naco.2017002 |
| [6] |
Puspita Mahata, Gour Chandra Mahata. Two-echelon trade credit with default risk in an EOQ model for deteriorating items under dynamic demand. Journal of Industrial and Management Optimization, 2021, 17 (6) : 3659-3684. doi: 10.3934/jimo.2020138 |
| [7] |
Magfura Pervin, Sankar Kumar Roy, Gerhard Wilhelm Weber. Multi-item deteriorating two-echelon inventory model with price- and stock-dependent demand: A trade-credit policy. Journal of Industrial and Management Optimization, 2019, 15 (3) : 1345-1373. doi: 10.3934/jimo.2018098 |
| [8] |
Mehmet Önal, H. Edwin Romeijn. Two-echelon requirements planning with pricing decisions. Journal of Industrial and Management Optimization, 2009, 5 (4) : 767-781. doi: 10.3934/jimo.2009.5.767 |
| [9] |
Erfan Babaee Tirkolaee, Alireza Goli, Mani Bakhsi, Iraj Mahdavi. A robust multi-trip vehicle routing problem of perishable products with intermediate depots and time windows. Numerical Algebra, Control and Optimization, 2017, 7 (4) : 417-433. doi: 10.3934/naco.2017026 |
| [10] |
Yanju Zhou, Zhen Shen, Renren Ying, Xuanhua Xu. A loss-averse two-product ordering model with information updating in two-echelon inventory system. Journal of Industrial and Management Optimization, 2018, 14 (2) : 687-705. doi: 10.3934/jimo.2017069 |
| [11] |
Honglin Yang, Qiang Yan, Hong Wan, Wenyan Zhuo. Bargaining equilibrium in a two-echelon supply chain with a capital-constrained retailer. Journal of Industrial and Management Optimization, 2020, 16 (6) : 2723-2741. doi: 10.3934/jimo.2019077 |
| [12] |
Jianbin Li, Niu Yu, Zhixue Liu, Lianjie Shu. Optimal rebate strategies in a two-echelon supply chain with nonlinear and linear multiplicative demands. Journal of Industrial and Management Optimization, 2016, 12 (4) : 1587-1611. doi: 10.3934/jimo.2016.12.1587 |
| [13] |
Qingguo Bai, Fanwen Meng. Impact of risk aversion on two-echelon supply chain systems with carbon emission reduction constraints. Journal of Industrial and Management Optimization, 2020, 16 (4) : 1943-1965. doi: 10.3934/jimo.2019037 |
| [14] |
Biswajit Sarkar, Arunava Majumder, Mitali Sarkar, Bikash Koli Dey, Gargi Roy. Two-echelon supply chain model with manufacturing quality improvement and setup cost reduction. Journal of Industrial and Management Optimization, 2017, 13 (2) : 1085-1104. doi: 10.3934/jimo.2016063 |
| [15] |
Zhenkai Lou, Fujun Hou, Xuming Lou. Optimal ordering and pricing models of a two-echelon supply chain under multipletimes ordering. Journal of Industrial and Management Optimization, 2021, 17 (6) : 3099-3111. doi: 10.3934/jimo.2020109 |
| [16] |
Jiao-Yan Li, Xiao Hu, Zhong Wan. An integrated bi-objective optimization model and improved genetic algorithm for vehicle routing problems with temporal and spatial constraints. Journal of Industrial and Management Optimization, 2020, 16 (3) : 1203-1220. doi: 10.3934/jimo.2018200 |
| [17] |
Huai-Che Hong, Bertrand M. T. Lin. A note on network repair crew scheduling and routing for emergency relief distribution problem. Journal of Industrial and Management Optimization, 2019, 15 (4) : 1729-1731. doi: 10.3934/jimo.2018119 |
| [18] |
Mostafa Abouei Ardakan, A. Kourank Beheshti, S. Hamid Mirmohammadi, Hamed Davari Ardakani. A hybrid meta-heuristic algorithm to minimize the number of tardy jobs in a dynamic two-machine flow shop problem. Numerical Algebra, Control and Optimization, 2017, 7 (4) : 465-480. doi: 10.3934/naco.2017029 |
| [19] |
Tao Zhang, W. Art Chaovalitwongse, Yue-Jie Zhang, P. M. Pardalos. The hot-rolling batch scheduling method based on the prize collecting vehicle routing problem. Journal of Industrial and Management Optimization, 2009, 5 (4) : 749-765. doi: 10.3934/jimo.2009.5.749 |
| [20] |
Bariş Keçeci, Fulya Altıparmak, İmdat Kara. A mathematical formulation and heuristic approach for the heterogeneous fixed fleet vehicle routing problem with simultaneous pickup and delivery. Journal of Industrial and Management Optimization, 2021, 17 (3) : 1069-1100. doi: 10.3934/jimo.2020012 |
2020 Impact Factor: 1.801
Tools
Article outline
Figures and Tables
[Back to Top]