An integrated bi-objective optimization model and improved genetic algorithm for vehicle routing problems with temporal and spatial constraints

Jiao-Yan Li; Xiao Hu; Zhong Wan

doi:10.3934/jimo.2018200

Article Contents

2020, Volume 16, Issue 3: 1203-1220. Doi: 10.3934/jimo.2018200

This issue Previous Article Optimal ordering policy for inventory mechanism with a stochastic short-term price discount Next Article Stability analysis for generalized semi-infinite optimization problems under functional perturbations

An integrated bi-objective optimization model and improved genetic algorithm for vehicle routing problems with temporal and spatial constraints

1.
School of Mathematics and Statistics, Central South University, Changsha, 410083, China
2.
Hunan University of Commerce, Shool of Computer and Information Engineering, Changsha, 410205, China

* Corresponding author: Zhong Wan
* Corresponding author: Zhong Wan

Received: May 31, 2017

Revised: August 31, 2017

Early access: December 2018

Published: March 2020

All the three authors are supported by the National Science Foundation of China (Grant No. 71671190)

Abstract Full Text(HTML) Figure(3) / Table(4) Related Papers Cited by

Abstract

Vehicle routing problem (VRP) is a typical and important combinatorial optimization problem, and is often involved with complicated temporal and spatial constraints in practice. In this paper, the VRP is formulated as an optimization model for minimizing the number of vehicles and the total transportation cost subject to constraints on loading plan, service time and weight capacity. The transportation cost consists of the rent charge of vehicles, fuel cost, and carbon tax. Owing to complexity of the built model, it is divided into two subproblems by a two-stage optimization approach: at the first stage, the number of vehicles is minimized, then the routing plan is optimized at the second stage. For solving the sequential subproblems, two correlated genetic algorithms are developed, which share the same initial population to reduce their computational costs. Numerical results indicate that the developed algorithms are efficient, and a number of important managerial insights are revealed from the model.

Keywords:

Mathematics Subject Classification: Primary: 90B06, 62K05; Secondary: 68T37.

Citation:

Full Text(HTML)

Figure 1. Example of crossover

Download: Full-size image PowerPoint slide

Figure 2. Minimal total costs by Algorithm 5

Download: Full-size image PowerPoint slide

Figure 3. Sensitivity analysis on speed of vehicles

Download: Full-size image PowerPoint slide

Table 1. Numerical results in case study

	NC	MVN	$v$	TD	TPV	WTV	VC	TFC	TCE	TC	CPU
1	19	6	50	1544.2	5.14	0.33	600	1160.5	7.09	1767.6	68.5
2	39	10	50	2208.7	3.40	0.28	1000	1654.2	10.1	2664.3	79.4
3	59	19	50	4368.9	4.60	0.32	1900	3145.7	19.2	5064.9	92.9
4	79	26	50	6070.6	4.66	0.32	2600	4379.8	26.8	7006.6	125.7
5	99	35	50	8636.1	4.93	0.27	3500	6133.3	37.5	9670.8	163.0
6	119	38	50	9743.4	5.12	0.25	3800	6970.8	42.6	10813.4	219.3
7	139	45	50	11470.4	5.09	0.43	4500	8213.9	50.2	12764.1	244.3
8	159	49	50	13913.2	5.68	0.44	4900	9938.1	60.8	14898.9	259.0
9	179	60	50	14883.3	4.96	0.50	6000	10673.7	65.3	16739.0	274.4
10	199	71	50	17599.0	4.96	0.62	7100	12544.3	76.7	19721.0	309.7

| Show Table

DownLoad: CSV

Table 2. Routes of all the vehicles in the case $n = 19$

LV	LNC	DT	AT	WT	LW	LL	DD	TFC	TCEt
1	0	9:00	-	-	-	-	-	-	-
	6	12:08	11:48	20	0.32	1.12	130.2	86.6	0.53
	4	13:32	13:32	0	0.18	0.48	59.0	38.3	0.23
	0	-	15:20	-	0	0	1654.2	50.2	0.31
2	0	9:00	-	-	-	-	-	-	-
	5	10:19	10:19	0	1.26	0.52	56.4	43.7	0.27
	1	11:20	11:20	0	1.05	0.31	40.8	30.6	0.19
	10	13:29	12:28	61	0.59	0.05	46.0	32.0	0.20
	0	-	14:24	-	0	0	1654.2	22.6	0.14
3	0	9:00	-	-	-	-	-	-	-
	9	11:54	11:54	0	2.57	1.62	135.3	125.7	0.77
	17	13:11	13:10	0	1.91	1.35	53.7	45.7	0.28
	7	14:35	14:35	0	1.00	0.64	60.5	45.0	0.28
	0	-	15:13	0	0	0	1654.2	13.5	0.08
4	0	9:00	-	-	-	-	-	-	-
	2	10:57	10:37	20	1.14	1.47	71.1	54.1	0.33
	13	13:07	12:07	60	0.91	0.97	48.4	35.5	0.22
	19	14:22	14:22	0	0.38	0.42	51.8	34.8	0.21
	0	-	15:19	-	0	0	37.7	23.6	0.14
5	0	9:00	-	-	-	-	-	-	-
	18	11:55	11:15	39	2.31	2.03	120.9	92.4	0.57
	3	12:53	12:53	0	1.93	1.81	38.2	32.6	0.20
	12	13:44	13:44	0	1.28	1.54	33.2	25.9	0.16
	16	14:23	14:23	0	0.85	0.60	22.4	16.3	0.10
	0	-	15:11	-	0	0	29.7	18.6	0.11
6	0	9:00	-	-	-	-	-	-	-
	15	11:30	11:30	0	1.98	2.58	115.3	99.1	0.61
	8	13:03	13:03	0	1.74	1.82	67.2	55.9	0.35
	11	14:14	14:14	0	0.92	0.82	49.5	36.4	0.22
	14	15:55	15:55	0	0.33	0.55	73.6	49.0	0.30
	0	-	16:48	0	0	0	83.8	52.5	0.32

| Show Table

DownLoad: CSV

Table 3. Comparison of the optimal costs by different models

Case	NC	TD	TFC	TCE	CPU
Test1$^a$	20	1544.2	1160.5	7.1	68.5
Test1$^b$	20	1442.2	1214.1	7.5	102.8
Test2$^a$	40	2208.7	1654.2	10.1	79.4
Test2$^b$	40	2304.6	1815.2	11.1	111.2
Test3$^a$	60	4368.9	3145.7	19.2	92.9
Test3$^b$	60	4262.7	3351.5	20.5	148.6
Test4$^a$	80	6070.6	4379.8	26.8	125.7
Test4$^b$	80	6118.4	4811.0	29.4	188.6
Test5$^a$	100	8636.1	6133.3	37.5	163.0
Test5$^b$	100	8288.8	6517.7	39.8	252.7
Test6$^a$	120	9743.4	6970.8	42.6	219.3
Test6$^b$	120	9758.8	7693.3	46.9	346.5
Test7$^a$	140	11470.4	8213.9	50.2	244.3
Test7$^b$	140	11853.7	9320.9	57.0	359.1
Test8$^a$	160	13913.2	9938.1	60.8	259.0
Test8$^b$	160	13011.9	10230.9	62.5	383.3
Test9$^a$	180	14883.3	10673.7	65.3	274.4
Test9$^b$	180	15488.2	12178.9	74.4	466.5
Test10$^a$	200	17599.0	12544.3	76.7	309.7
Test10$^b$	200	17378.1	13664.8	83.5	526.5

| Show Table

DownLoad: CSV

Table 4. Vehicle parameters

V	VC	VL	VW	VH	WV	VP	$LC$
V1	Light	4.0	1.9	2.3	2.6	3	100
V2	Light	6.2	2	2	3.5	5	150
V3	Medium	9.6	2.3	2.7	25	25	300
V4	Medium	12.0	2.4	2.7	28	28	400
V5	Heavy	17.5	2.4	2.7	35	35	500

| Show Table

DownLoad: CSV

Related Papers

Cited by

References

	R. Akcelik and M. Besley , Operating cost, fuel consumption, and emission models in aaSIDRA and aaMOTION, 25th Conference of Australian Institutes of Transport Research (CAITR 2003), 2003 (2003) , 1-15.
	J. P. Arnaout , G. Arnaout and J. EIKhoury , Simulation and optimization of ant colony optimization algorithm for the stochastic uncapacitated location-allocation problem, Journal of Industrial and Management Optimization, 12 (2016) , 1215-1225. doi: 10.3934/jimo.2016.12.1215.
	R. Baldacci , N. Christofides and A. Mingozzi , An exact algorithm for the vehicle routing problem based on the set partitioning formulation with additional cuts, Mathematical Programming, 115 (2008) , 351-385. doi: 10.1007/s10107-007-0178-5.
	M. Barth and K. Boriboonsomsin , Real-world carbon dioxide impacts of traffic congestion, Transportation Research Record: Journal of the Transportation Research Board, 2058 (2008) , 163-171.
	T. Bektas and G. Laporte , The pollution-routing problem, Transportation Research Part B: Methodological, 45 (2011) , 1232-1250.
	J. F. Cordeau, G. Laporte, M. W. P. Savelsbergh, et al. Vehicle routing, Transportation, Handbooks in Operations Research and Management Science, 14 (2006), 367-428.
	G. B. Dantzig and J. H. Ramser , The truck dispatching problem, Management Science, 6 (1959) , 80-91. doi: 10.1287/mnsc.6.1.80.
	J. H. Holland, Adaptation in Natural and Artificial Systems: An Introductory Analysis with Applications to Biology, Control, and Artificial Intelligence, University of Michigan Press, Ann Arbor, Mich., 1975
	J. Homberger and H. Gehring , Two evolutionary metaheuristics for the vehicle routing problem with time windows, INFOR: Information Systems and Operational Research, 37 (1999) , 297-318.
	J. Homberger and H. Gehring , A two-phase hybrid metaheuristic for the vehicle routing problem with time windows, European Journal of Operational Research, 162 (2005) , 220-238.
	S. Huang and Z. Wan , A new nonmonotone spectral residual method for nonsmooth nonlinear equations, Journal of Computational and Applied Mathematics, 313 (2017) , 82-101. doi: 10.1016/j.cam.2016.09.014.
	S. Huang , Z. Wan and X. H. Chen , A new nonmonotone line search technique for unconstrained optimization, Numerical Algorithms, 68 (2015) , 671-689. doi: 10.1007/s11075-014-9866-4.
	S. Huang , Z. Wan and J. Zhang , An extended nonmonotone line search technique for large-scale unconstrained optimization, Journal of Computational and Applied Mathematics, 330 (2018) , 586-604. doi: 10.1016/j.cam.2017.09.026.
	O. Jabali , T. Woensel and A. G. de Kok , Analysis of travel times and CO$_2$ emissions in time-dependent vehicle routing, Production and Operations Management, 21 (2012) , 1060-1074.
	I. Kara, B. Y. Kara and M. K. Yetis, Energy minimizing vehicle routing problem, Combinatorial Optimization and Applications, Springer Berlin Heidelberg, 2007, 62-71. doi: 10.1007/978-3-540-73556-4_9.
	M. Lai, H. Yang, S. Yang, et al., Cyber-physical logistics system-based vehicle routing optimization, Journal of Industrial and Management Optimization, 10 (2014), 701-715. doi: 10.3934/jimo.2014.10.701.
	Y. X. Li and Z. Wan, Bi-level programming approach to optimal strategy for VMI problems under random demand, ANZIAM Journal, 59 (2017), 247-270, doi: 10.1017/S1446181117000384.
	W. Liang W , B. Çatay and R. Eglese , Finding a minimum cost path between a pair of nodes in a time-varying road network with a congestion charge, European Journal of Operational Research, 236 (2014) , 915-923. doi: 10.1016/j.ejor.2013.10.044.
	W. Liang and R. Eglese , Minimum cost VRP with time-dependent speed data and congestion charge, Computers and Operations Research, 56 (2015) , 41-50. doi: 10.1016/j.cor.2014.10.007.
	W. Maden , R. Eglese and D. Black , Vehicle routing and scheduling with time-varying data: A case study, Journal of the Operational Research Society, 61 (2010) , 515-522.
	M. Maiti M, S. Maity and A. Roy, An improved genetic algorithm and its application in constrained solid TSP in uncertain environments. Facets of Uncertainties and Applications, Springer, New Delhi, 2015,177-200.
	D. Männel and A. Bortfeldt , A hybrid algorithm for the vehicle routing problem with pickup and delivery and three-dimensional loading constraints, European Journal of Operational Research, 254 (2016) , 840-858. doi: 10.1016/j.ejor.2016.04.016.
	E. Mardaneh , R. Loxton , Q. Lin and P. Schmidli , A mixed-integer linear programming model for optimal vessel scheduling in offshore oil and gas operations, Journal of Industrial and Management Optimization, 13 (2017) , 1601-1623. doi: 10.3934/jimo.2017009.
	A. McKinnon, CO$-2$ emissions from freight transport: An analysis of UK data, Logistics Research Network-2007 Conference Global Supply Chains: Developing Skills, Capabilities and Networks, 2007.
	A. McKinnon, M. Browne, A. Whiteing and M. Piecyk (Eds.), Green Logistics: Improving the environmental sustainability of logistics, Kogan Page Publishers, 2015 (2015), 215-228.
	S. Mima and P. Criqui, The future of fuel cells in a long term inter-technology competition framework, The Economic Dynamics of Fuel Cell Technologies, Springer Berlin Heidelberg, 2003, 43–78.
	D. M. Miranda and S. V. Conceic$\tilde{a}$o , The vehicle routing problem with hard time windows and stochastic travel and service time, Expert Systems with Applications, 64 (2016) , 104-116.
	A. Palmer, The Development of an Integrated Routing and Carbon Dioxide Emissions Model for Goods Vehicles, Ph.D. thesis, Cranfield University, School of Management, 2007.
	C. Prins , A simple and effective evolutionary algorithm for the vehicle routing problem, Computers and Operations Research, 31 (2004) , 1985-2002. doi: 10.1016/S0305-0548(03)00158-8.
	C. Schreyer, C. Schneider, M. Maibach, et al. External Costs of Transport: Update Study, final report, Infras, 2004.
	P. Shaw, Using constraint programming and local search methods to solve vehicle routing problems, Principles and Practice of Constraint Programming-CP98, Springer, Berlin Heidelberg, 1998,417–431.
	S. R. Thangiah , K. E. Nygard and J. P. L. Gideon , A genetic algorithm system for vehicle routing with time windows, Artificial Intelligence Applications, 1991. Proceedings., Seventh IEEE Conference on IEEE, 1 (1991) , 322-328.
	P. Toth and D. Vigo, Vehicle Routing: Problems, Methods, and Applications, Society for Industrial and Applied Mathematics, 2014. doi: 10.1137/1.9781611973594.
	X. Zhang , S. Huang and Z. Wan , Optimal pricing and ordering in global supply chain management with constraints under random demand, Appl. Math. Modelling, 40 (2016) , 10105-10130. doi: 10.1016/j.apm.2016.06.054.
	X. Zhang, S. Huang and Z. Wan, Stochastic Programming Approach to Global Supply Chain Management Under Random Additive Demand, Operational Research, 2017. doi: 10.1007/s12351-016-0269-2.