doi: 10.3934/jimo.2021094

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

* Corresponding author: Chao Meng

Received  August 2020 Revised  March 2021 Published  May 2021

Fund Project: The first author is supported by China Postdoctoral Science Foundation (No.2020M673279), National Natural Science Foundation of China (NSFC) (No.51675450), Sichuan Science and Technology Program (No.2020JDTD0012) and MOE (Ministry of Education in China) Project of Humanities and Social Sciences (No.18YJC630255)

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.

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 & Management Optimization, doi: 10.3934/jimo.2021094
References:
[1]

R. BaldacciA. MingozziR. 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.  Google Scholar

[2]

A. BevilaquaD. 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.  Google Scholar

[3]

U. BreunigR. BaldacciR. 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.  Google Scholar

[4]

U. BreunigV. SchmidR. 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.  Google Scholar

[5]

M.-C. ChenP.-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.  Google Scholar

[6]

Double 11 constantly refreshes the imagination of Chinese market, Global times, November 12, 2019 (015). Google Scholar

[7]

P. GrangierM. GendreauF. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[11]

W. JieJ. YangM. 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.  Google Scholar

[12]

H. LiL. ZhangT. 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.  Google Scholar

[13]

H. LiH. WangJ. 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.  Google Scholar

[14]

H. LiY. LiuX. 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.  Google Scholar

[15]

R. LiuL. TaoQ. 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.  Google Scholar

[16]

T. LiuZ. LuoH. 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.  Google Scholar

[17]

Z. Y. MaY. B. Ling and J. Li, 2E-VRP Optimization Algorithm with Optimal Cutting and Full Path Matching Cross, Computer Engineering, 41 (2015), 279-285.   Google Scholar

[18]

M. MarinelliA. 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.  Google Scholar

[19]

E. MorgantiL. 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.  Google Scholar

[20]

G. PerboliR. 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.  Google Scholar

[21]

F. A. SantosG. 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.  Google Scholar

[22]

M. SoysalJ. 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.  Google Scholar

[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.  Google Scholar

[24]

Y. Tan and Y. Zhu, Fireworks algorithm for optimization, International Conference in Swarm Intelligence, Berlin: Springer, 355–364. Google Scholar

[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.  Google Scholar

[26]

E. B. TirkolaeeA. GoliM. 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.  Google Scholar

[27]

E. B. TirkolaeeS. 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.   Google Scholar

[28]

E. B. TirkolaeeS. HadianG.-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.  Google Scholar

[29]

K. WangS. 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.  Google Scholar

[30]

X. M. YanZ. F. HaoH. HuangB 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.   Google Scholar

[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.  Google Scholar

show all references

References:
[1]

R. BaldacciA. MingozziR. 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.  Google Scholar

[2]

A. BevilaquaD. 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.  Google Scholar

[3]

U. BreunigR. BaldacciR. 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.  Google Scholar

[4]

U. BreunigV. SchmidR. 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.  Google Scholar

[5]

M.-C. ChenP.-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.  Google Scholar

[6]

Double 11 constantly refreshes the imagination of Chinese market, Global times, November 12, 2019 (015). Google Scholar

[7]

P. GrangierM. GendreauF. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[11]

W. JieJ. YangM. 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.  Google Scholar

[12]

H. LiL. ZhangT. 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.  Google Scholar

[13]

H. LiH. WangJ. 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.  Google Scholar

[14]

H. LiY. LiuX. 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.  Google Scholar

[15]

R. LiuL. TaoQ. 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.  Google Scholar

[16]

T. LiuZ. LuoH. 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.  Google Scholar

[17]

Z. Y. MaY. B. Ling and J. Li, 2E-VRP Optimization Algorithm with Optimal Cutting and Full Path Matching Cross, Computer Engineering, 41 (2015), 279-285.   Google Scholar

[18]

M. MarinelliA. 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.  Google Scholar

[19]

E. MorgantiL. 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.  Google Scholar

[20]

G. PerboliR. 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.  Google Scholar

[21]

F. A. SantosG. 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.  Google Scholar

[22]

M. SoysalJ. 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.  Google Scholar

[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.  Google Scholar

[24]

Y. Tan and Y. Zhu, Fireworks algorithm for optimization, International Conference in Swarm Intelligence, Berlin: Springer, 355–364. Google Scholar

[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.  Google Scholar

[26]

E. B. TirkolaeeA. GoliM. 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.  Google Scholar

[27]

E. B. TirkolaeeS. 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.   Google Scholar

[28]

E. B. TirkolaeeS. HadianG.-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.  Google Scholar

[29]

K. WangS. 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.  Google Scholar

[30]

X. M. YanZ. F. HaoH. HuangB 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.   Google Scholar

[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.  Google Scholar

Figure 1.  Schematic diagram of the 2E-VRPDB
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, $ D=\{d_0\} $
S Set of satellites, $ S=\{s_1 $, $s _2 $$,…, s_{ns} $}, and the total number is $ ns $
C Set of customers, $ C=\{c_1 $, $ c_2 $, …, $ c_{nc} \}$, and the total number is $ nc $
G Set of first-level delivery vehicles, $G=\{g _1 $, $ g_2 $, …, $g _{ng} \}$, and the total number is $ ng $
H Set of second-level delivery vehicles, $H=\{h _1 $, $h _2 $, …, $ h_{nh} \}$, and the total number is $ nh $
M A large enough number
T$ _0 $ Working hours per day
d$ _{ij} $ The distance of the $ (i, j) $ edge
q$ _i $ Demand of customer c$ _i $
cap$ _1 $ The capacity of the first-level vehicle
cap$ _2 $ The capacity of the second-level vehicle
t$ _c $ The deadline of customer c
b$ _1 $ Compensation per unit cargo for accepting flexible delivery
b$ _2 $ Delay cost per delivery
a$ _1 $ Fixed cost of the first-level delivery vehicle per delivery
a$ _2 $ Fixed cost of the second-level delivery vehicle per delivery
c$ _g $ Unit distance cost of the first-level delivery vehicle per delivery
c$ _h $ Unit distance cost of the second-level delivery vehicle per delivery
c$ _1 $ The labor cost of the first-level delivery vehicle per delivery
c$ _2 $ The labor cost of the second -level delivery per delivery
f$ _c $ If customer c chooses flexible delivery, fc =1; otherwise, fc =0
T$ _s $ Time required to complete standard delivery
Sets and Parameters Description
D Set of depots, $ D=\{d_0\} $
S Set of satellites, $ S=\{s_1 $, $s _2 $$,…, s_{ns} $}, and the total number is $ ns $
C Set of customers, $ C=\{c_1 $, $ c_2 $, …, $ c_{nc} \}$, and the total number is $ nc $
G Set of first-level delivery vehicles, $G=\{g _1 $, $ g_2 $, …, $g _{ng} \}$, and the total number is $ ng $
H Set of second-level delivery vehicles, $H=\{h _1 $, $h _2 $, …, $ h_{nh} \}$, and the total number is $ nh $
M A large enough number
T$ _0 $ Working hours per day
d$ _{ij} $ The distance of the $ (i, j) $ edge
q$ _i $ Demand of customer c$ _i $
cap$ _1 $ The capacity of the first-level vehicle
cap$ _2 $ The capacity of the second-level vehicle
t$ _c $ The deadline of customer c
b$ _1 $ Compensation per unit cargo for accepting flexible delivery
b$ _2 $ Delay cost per delivery
a$ _1 $ Fixed cost of the first-level delivery vehicle per delivery
a$ _2 $ Fixed cost of the second-level delivery vehicle per delivery
c$ _g $ Unit distance cost of the first-level delivery vehicle per delivery
c$ _h $ Unit distance cost of the second-level delivery vehicle per delivery
c$ _1 $ The labor cost of the first-level delivery vehicle per delivery
c$ _2 $ The labor cost of the second -level delivery per delivery
f$ _c $ If customer c chooses flexible delivery, fc =1; otherwise, fc =0
T$ _s $ Time required to complete standard delivery
Table 2.  List of variables and descriptions
Variables Description
x$ _{ijg} $ First-level distribution vehicle g travels the $ (i, j) $ edge, $x _{ijg} $$ \in \{0,1\}$; decision variable
y$ _{ijg} $ Second -level distribution vehicle h travels the $ (i, j) $ edge, y$ _{ijg} $ $ \in \{0,1\}$; decision variable
z$ _{cs} $ Customer c cargo comes from satellite s, z$ _{cs} $ $ \in \{0,1\}$; decision variable
w$ _{sg} $ The actual load of first level vehicle g to satellite s; decision variable
l$ _s $ The total demand of satellite s
t$ _{sg} $ Time of first-level vehicle g arriving at satellite s
t$ _{ch} $ Time of arrival of second-level vehicle h to satellite s
t$ _1 $ The longest time for the first-level vehicle to complete the distribution task
time$ _c $ The actual delivery time of the customer c
dtime$ _c $ The delay time for customer c
U1$ _{ig} $ Restrict the occurrence of sub-tour in the first-level vehicles
U2$ _{ih} $ Restrict the occurrence of sub-tour in the second-level vehicles
u$ _{sg} $ Intermediate variable, no actual meaning
u$ _{cp} $ Intermediate variable, no actual meaning
Variables Description
x$ _{ijg} $ First-level distribution vehicle g travels the $ (i, j) $ edge, $x _{ijg} $$ \in \{0,1\}$; decision variable
y$ _{ijg} $ Second -level distribution vehicle h travels the $ (i, j) $ edge, y$ _{ijg} $ $ \in \{0,1\}$; decision variable
z$ _{cs} $ Customer c cargo comes from satellite s, z$ _{cs} $ $ \in \{0,1\}$; decision variable
w$ _{sg} $ The actual load of first level vehicle g to satellite s; decision variable
l$ _s $ The total demand of satellite s
t$ _{sg} $ Time of first-level vehicle g arriving at satellite s
t$ _{ch} $ Time of arrival of second-level vehicle h to satellite s
t$ _1 $ The longest time for the first-level vehicle to complete the distribution task
time$ _c $ The actual delivery time of the customer c
dtime$ _c $ The delay time for customer c
U1$ _{ig} $ Restrict the occurrence of sub-tour in the first-level vehicles
U2$ _{ih} $ Restrict the occurrence of sub-tour in the second-level vehicles
u$ _{sg} $ Intermediate variable, no actual meaning
u$ _{cp} $ 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
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