Multi-objective optimization model for planning metro-based underground logistics system network: Nanjing case study

Xiliang Sun; Wanjie Hu; Xiaolong Xue; Jianjun Dong

doi:10.3934/jimo.2021179

Article Contents

2023, Volume 19, Issue 1: 170-196. Doi: 10.3934/jimo.2021179

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Multi-objective optimization model for planning metro-based underground logistics system network: Nanjing case study

1.
School of Management, Harbin Institute of Technology, Harbin 150001, China
2.
College of Architecture and Civil Engineering, Beijing University of Technology, Beijing 100124, China
3.
School of Management, Harbin Institute of Technology, Harbin 150001, China
4.
College of Civil Engineering, Nanjing Tech University, Nanjing 211800, China

^* Corresponding authors: Xiliang Sun and Wanjie Hu
^* Corresponding authors: Xiliang Sun and Wanjie Hu

Received: March 31, 2021

Revised: July 31, 2021

Early access: October 2021

Published: January 2023

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Abstract

Utilizing rail transit system for collaborative passenger-and-freight transport is a sustainable option to conquer urban congestion. This study proposes effective modeling and optimization techniques for planning a city-wide metro-based underground logistics system (M-ULS) network. Firstly, a novel metro prototype integrating retrofitted underground stations and newly-built capsule pipelines is designed to support automated inbound delivery from urban logistics gateways to in-city destinations. Based on four indicators (i.e. unity of freight flows, regional accessibility, environmental cost-saving, and order priority), an entropy-based fuzzy TOPSIS evaluation model is proposed to select appropriate origin-destination flows for underground freight transport. Then, a mixed integer programming model, with a well-matched solution framework combining multi-objective PSO algorithm and A* algorithm, are developed to optimize the location-allocation-routing (LAR) decisions of M-ULS network. Finally, real-world simulation based on Nanjing metro case is conducted for validation. The best facility configurations and flow assignments of the three-tier M-ULS network are reported in details. Results confirm that the proposed algorithm has good ability in providing high-quality Pareto-optimal LAR decisions. Moreover, the Nanjing M-ULS project shows strong economic feasibility while bringing millions of Yuan of annual external benefit to the society and environment.

Keywords:

Urban freight transport, underground logistics system (ULS),
rail transit,
network design,
location-routing problem,
multi-objective optimization,
case study.

Mathematics Subject Classification: Primary: 90B06, 90B50; Secondary: 90C27.

Citation:

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Figure 1. Overview of the 3EM-ULSND problem

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Figure 2. Overview of the 3EM-ULSND problem

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Figure 3. Illustration of Pareto front in tri-objective optimization problem

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Figure 4. Model decomposition and algorithmic flowchart

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Figure 5. Pareto-optimal front obtained with MOPSO

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Figure 6. Pareto-optimal front obtained with MOPSO

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Table 1. Model parameters and values

Note	Definition	Attribute
Notation of indices
$ S $	set of LPWs, i.e., set of metro lines	indexed by $ s $
$ M $	set of SCs, i.e., set of candidate location of UDs	indexed by $ m $
$ N $	set of metro stations, i.e., set of candidate location of NFSs	indexed by $ i $
$ L $	set of metro interchanges, i.e., set of activated IFSs	indexed by $ j $
$ K $	set of metro line arcs between two adjacent metro stations	indexed by $ k $
$ H $	set of arcs between metro stations and SCs	indexed by $ h $
$ R $	set of arcs between two SCs	indexed by $ r $
Exogenous parameters
$ g_m^s $	size of delivery orders from LPW $ s $ to SC $ m $ (original value)	[0, 15] parcel per-day
$ d_m^s $	size of delivery orders from LPW $ s $ to SC $ m $ (after evaluated)	–
$ c $	freight travel cost by LGV	$ \yen $ 0.2 per-parcel per-km
$ \alpha $	ratio of the freight travel cost by metro to the freight travel cost by LGV	10%
$ \beta $	ratio of the freight travel cost by CPs to the freight travel cost by LGV	25%
$ w $	underground transfer cost at IFS	$ \yen $0.1 per-parcel
$ {\lambda _1} $	fixed cost for CP construction	$ \yen $4$ \times $$ 10^7 $ per-km
$ {\lambda _2} $	fixed cost for NFS retrofit	$ \yen $1$ \times $$ 10^8 $
$ {\lambda _3} $	fixed cost for UD construction	$ \yen $2$ \times $$ 10^7 $
$ {\eta _{NFS}} $	allowable level for low load operations at UD	60%
$ {\eta _{CP}} $	allowable level for low load operations at CP	50%
$ {\sigma _{NFS}} $	penalty cost due to low load operations of UD	$ \yen $2 per-parcel
$ {\sigma _{CP}} $	penalty cost due to low load operations of CP	$ \yen $1.5 per-parcel
$ [{R_{max}}] $	maximal road travel distance from UD to SC	2km
$ \left[ {{Q_{max}}} \right] $	order handling capacity of NFS	1$ \times $$ 10^5 $ parcel per-day
$ [{Z_{max}}] $	transport capacity of CP	4$ \times $$ 10^4 $ parcel per-day
$ [{G_{max}}] $	order handling capacity of UD	3.5$ \times $$ 10^4 $ parcel per-day
$ \left[ {{T_{max}}} \right] $	order transfer capacity of IFS	1.5$ \times $$ 10^6 $ parcel per-day
$ Eu $	Euclidean distance of arc $ k $, arc $ r $ and arc $ h $, respectively	–
$ \theta $	depreciation coefficient of M-ULS network facilities	1/25550
Binary variables
$ {X_i} $	1, if metro station $ i $ is selected as NFS
$ {\delta _{smj}} $	1, if $ d_m^s $ is transferred at IFS $ j $
$ {Y_m} $	1, if SC $ m $ is selected as UD
$ {W_h} $	1, if arc $ h $ is selected as CP
$ {U_{smi}} $	1, if the trip of $ d_m^s $ on the second-tier M-ULS network is assigned by NFS $ i $	0-1 variable
$ {V_{smm'}} $	1, if the trip of $ d_m^s $ on the third-tier M-ULS network is assigned by UD $ {m'} $
$ {\xi _{smr}} $	1, if $ d_m^s $ traverses on arc $ r $ via road segment
$ {{\bf{O}}_{smk}} $	1, if $ d_m^s $ traverses on arc $ k $ via metro segment
$ {{\bf{T}}_{smh}} $	1, if $ d_m^s $ traverses on arc $ h $ via CP segment

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Table 2. Number of model variables and constraints

	Number at most	Nanjing metro case
Variables $ {X_i} $, $ {Y_m} $, Constraint 22	$ \left\\| N \right\\| + {\rm{3}} \times \left\\| M \right\\| $	1,402
Variable $ {\delta _{smj}} $, Constraints 18, 23	$ \left\\| S \right\\| \times \left\\| M \right\\| \times \left( {{\rm{6}} + \left\\| L \right\\|} \right) $	35,200
Variables $ {W_h} $, $ {W_r} $	$ \left\\| M \right\\| \times \left\\| N \right\\| + C_{\left\\| M \right\\|}^{\rm{2}} $	132,660
Variables $ {U_{smi}} $, $ {V_{smm'}} $, Constraint 19	$ \left\\| S \right\\| \times \left\\| M \right\\| \times \left( {{\rm{2}}\left\\| N \right\\| + {\rm{2}}\left\\| M \right\\| - {\rm{2}}} \right) $	1,833,920
Variables $ {\xi _{smr}} $, Constraints 16, 21	$ \left\\| M \right\\| + C_{\left\\| M \right\\|}^{\rm{2}} + \left\\| S \right\\| \times \left\\| M \right\\| \times C_{\left\\| M \right\\|}^{\rm{2}} + \left\\| S \right\\| \times P_{\left\\| M \right\\|}^{\rm{2}} $	170,850,460
Variable $ {{\bf{O}}_{smk}} $, Constraint 15	$ \left\\| S \right\\| \times \left\\| M \right\\| \times \left( {\left\\| N \right\\| - {\rm{1}}} \right) + \left\\| L \right\\| + \left\\| N \right\\| $	142,649
Variable $ {{\bf{T}}_{smh}} $, Constraints 17, 20	$ {\rm{2}} \times \left\\| S \right\\| \times {\left\\| M \right\\|^{\rm{2}}} \times \left\\| N \right\\| + \left\\| M \right\\| \times \left\\| N \right\\| $	127,037,680
Sum of constraints and variables		300,033,971

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Table 3. Evaluation outputs of Nanjing M-ULS network flows

	LPW 1	LPW 2	LPW 3	LPW 4
Accessed metro line	Line 1	Line 2	Line 3	Line 4
Total demand orders ($ \times $10$ ^3 $ parcel per-day)	1,237	1,028	1,141	654
Average value of $ R{C_{sm}} $	0.3025	0.269	0.3207	0.3436
Average value of $ R{C_{sm}} $	0.3025	0.269	0.3207	0.3436
Maximum value of $ R{C_{sm}} $	0.9471	0.9226	0.8901	0.9284
Average value of $ a_{m1}^s $ ($ \times $10$ ^3 $ parcel·km)	58.47	60.2	58.01	21.83
Average value of $ a_{m2}^s $ (kmh)	36.59	37.98	41.31	22.84
Average value of $ a_{m3}^s $ ($ \yen $ per-day)	1,332	1,997	2,189	462
Average value of $ a_{m4}^s $ ($ \yen $ per-day)	13,921	10,380	8,990	3,897
Size of orders inputted into metro	704	647	621	446
Served SC number	255	265	264	271
Utilization rate of metro line	93.9%	86.3%	82.8%	59.5%
Fulfillment rate of underground logistics	57.9%	60.2%	60%	61.6%

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Table 4. Best configurations of Nanjing M-ULS network

	ID	Station full name	$ {N_{SC}} $¹	$ {N_{UD}} $²	$ \sum {d_m^s} $³	$ \overline {d_m^s} $⁴	$ {L_{CP}} $⁵	$ {R_{UD}} $⁶
Line 1	NFS-1	Er-Qiao-Gong-Yuan	11	6	40.5	6.8	5.5	4.63
	NFS-2	Ba-Dou-Shan	17	5	44.9	9	8.87	11.23
	NFS-3	Yan-Zi-Ji	17	9	80.5	8.9	13.14	6.6
	NFS-4	Xin-Mo-Fan-Ma-Lu	11	4	98.5	24.6	7.44	8.18
	NFS-5	Xuan-Wu-Men	9	3	77.8	25.9	3.36	3.87
	NFS-6	Zhang-Fu-Yuan	8	6	67.6	11.3	4.29	1.68
	NFS-7	San-Shan-Jie	4	3	37	12.3	3.66	0.84
	NFS-8	Zhong-Hua-Men	6	2	35	17.5	3.95	4.87
	NFS-9	Ruan-Jian-Da-Dao	9	4	44.9	11.2	6.01	4.51
	NFS-10	Hua-Shen-Miao	8	5	39.4	7.9	4.66	2.26
	NFS-11	Sheng-Tai-Lu	18	4	93.1	23.3	9.58	12.85
	NFS-12	Zhu-Shan-Lu	32	12	95.4	8	20.2	17.29
	NFS-13	Nan-Jing-Jiao-Yuan	9	4	35.1	8.8	4.34	5.23
Line 2	NFS-14	Qing-Lian-Jie	8	1	32.1	32.1	0.29	10.04
	NFS-15	You-Fang-Qiao	15	4	71.8	18	3.09	11.65
	NFS-16	Yuan-Tong	6	3	44.8	14.9	3.39	2.07
	NFS-17	Xiong-Long-Da-Jie	13	7	84.6	12.1	8.6	6.29
	NFS-18	Yun-Jing-Lu	14	7	90.5	12.9	10.99	6.45
	NFS-19	Virtual station	8	4	49.6	12.4	3.48	3.19
	NFS-20	Ming-Gu-Gong	19	9	79.3	8.8	13.83	9.47
	NFS-21	Xia-Ma-Fang	5	1	34	34	2.17	3.27
	NFS-22	Ma-Qun	13	3	55.6	18.5	2.45	10.65
	NFS-23	Xian-Lin-Zhong-Xin	14	8	44.8	5.6	12.54	4.49
Line 3	NFS-24	Virtual station	7	2	38.4	19.2	2.53	5.55
	NFS-25	Fu-Qiao	9	6	59.4	9.9	5.65	2.46
	NFS-26	Virtual station	2	1	34.8	34.8	0.29	0.36
	NFS-27	Ka-Zi-Men	11	7	74.1	10.6	9.41	2.82
	NFS-28	Hong-Yun-Da-Dao	5	2	41.6	20.8	1.65	1.72
	NFS-29	Tian-Yuan-Xi-Lu	26	9	98.4	10.9	13.02	12.26
	NFS-30	Cheng-Xin-Da-Dao	24	9	97.8	10.9	11.06	16.12
Line 4	NFS-31	Hui-Tong-Lu	16	6	85.1	14.2	11.63	9.38
	NFS-32	Wang-Jia-Wan	4	1	33.4	33.4	1.48	2.7
	NFS-33	Gang-Zi-Cun	7	3	31.5	10.5	3.23	3.9
	NFS-34	Yun-Nan-Lu	11	6	96	16	9.76	2.61
	IFS-1 & NFS-35	Nan-Jing-Zhan	13	9	83.8	9.3	11	3.09
	IFS-2 & NFS-36	Gu-Lou	7	5	91	18.2	6.93	0.99
	IFS-3 & NFS-37	Xin-Jie-Kou	8	4	89.3	22.3	4.91	2.55
	IFS-4 & NFS-38	Nan-Jing-Nan-Zhan	8	3	67.4	22.5	3.01	4.54
	IFS-5 & NFS-39	Da-Xing-Gong	8	4	46	11.5	2.38	1.92
	IFS-6	Jin-Ma-Lu	–	–	–	–	–	–
	IFS-7	Ji-Ming-Si	–	–	–	–	–	–

	ID	Station full name	$ {T_{1{\rm{st}}}} $	$ {T_{2{\rm{nd}}}} $	$ {T_{3{\rm{rd}}}} $	$ {P_{NFS}} $	$ {P_{CP}} $	$ {V_{IFS}} $
Line 1	NFS-1	Er-Qiao-Gong-Yuan	21.6	1.39	2.08	39	13.2	–
	NFS-2	Ba-Dou-Shan	20	3.78	7.64	30.2	11	–
	NFS-3	Yan-Zi-Ji	37.8	4.41	6.28	0	11.1	–
	NFS-4	Xin-Mo-Fan-Ma-Lu	56.3	6.49	19.69	0	0	–
	NFS-5	Xuan-Wu-Men	27.7	3.4	7.1	0	0	–
	NFS-6	Zhang-Fu-Yuan8	42.9	2.29	1.57	0	8.7	–
	NFS-7	San-Shan-Jie	13.2	2.33	0.64	46	7.7	–
	NFS-8	Zhong-Hua-Men	16.4	3.8	9.11	50	2.5	–
	NFS-9	Ruan-Jian-Da-Dao	19.4	4.44	2.09	30.2	8.8	–
	NFS-10	Hua-Shen-Miao	22.5	2.48	1.1	41.2	12.1	–
	NFS-11	Sheng-Tai-Lu	53.2	12.18	27.86	0	0	–
	NFS-12	Zhu-Shan-Lu	59.4	9.99	14.86	0	12	–
	NFS-13	Nan-Jing-Jiao-Yuan	12.5	1.89	3.04	49.8	11.2	–
Line 2	NFS-14	Qing-Lian-Jie	17.9	0.52	13.54	55.8	0	–
	NFS-15	You-Fang-Qiao	40.1	3.29	5.16	0	2	–
	NFS-16	Yuan-Tong	17.1	2.34	1.30	30.4	5.1	–
	NFS-17	Xiong-Long-Da-Jie	33.3	4.87	5.53	0	7.9	–
	NFS-18	Yun-Jing-Lu	56.3	6.45	8.99	0	7.1	–
	NFS-19	Virtual station	20.2	2.92	2.97	20.8	7.6	–
	NFS-20	Ming-Gu-Gong	38.3	7.19	8.4	0	11.2	–
	NFS-21	Xia-Ma-Fang	3.69	5.67	52	0	–
	NFS-22	Ma-Qun	35.3	2.68	8.16	8.8	1.5	–
	NFS-23	Xian-Lin-Zhong-Xin	27.9	4.41	2.41	30.4	14.4	–
Line 3	NFS-24 Virtual station	13.7	2.41	5.69	43.2	0.8	–
	NFS-25	Fu-Qiao	25.6	3.48	2.24	1.2	10.1	–
	NFS-26	Virtual station	11	0.61	0.86	50.4	0	–
	NFS-27	Ka-Zi-Men	40.5	3.67	1.15	0	9.4	–
	NFS-28	Hong-Yun-Da-Dao	18	2.06	2.71	36.8	0	–
	NFS-29	Tian-Yuan-Xi-Lu	51.2	6	4.89	0	9.1	–
	NFS-30	Cheng-Xin-Da-Dao	53.4	6.58	7.59	0	9.1	–
Line 4		NFS-31 Hui-Tong-Lu	33.5	11.11	5.42	0	5.8	–
	NFS-32	Wang-Jia-Wan	15.3	2.65	4.76	53.2	0	–
	NFS-33	Gang-Zi-Cun	15.2	1.17	4.33	57	9.5	–
	NFS-34	Yun-Nan-Lu	53.6	9.16	4.06	0	4	–
	IFS-1 & NFS-35	Nan-Jing-Zhan	36.2	5.02	1.13	0	10.7	871
	IFS-2 & NFS-36	Gu-Lou 40.4	7.34	1.58	0	1.8	759
	IFS-3 & NFS-37	Xin-Jie-Kou	47.6	4.93	4.01	0	0	1,259
	IFS-4 & NFS-38	Nan-Jing-Nan-Zhan	24	2.96	9.24	0	0	804
	IFS-5 & NFS-39	Da-Xing-Gong	16.4	1.87	1.95	28	8.5	1,090
	IFS-6	Jin-Ma-Lu	–	–	–	–	–	565
	IFS-7	Ji-Ming-Si	–	–	–	–	–	622
¹ number of SCs allocated to NFS; ² number of UDs covered by NFS; ³ total size of orders handled by NFS ($\times$10$^3$ parcel per-day); ⁴ average size of orders handled by UD ($\times$10$^3$ parcel per-day); ⁵ length of CP segments connected to NFS (km); ⁶ average service radius of UD (km). ⁷ transport cost of NFS orders on first-tier M-ULS network ($\times$10$^3$ $\yen$ per-day); ⁸ transport cost of NFS orders on second-tier M-ULS network ($\times$10$^3$ $\yen$ per-day); ⁹ transport cost of NFS orders on third-tier M-ULS network ($\times$10$^3$ $\yen$ per-day); ¹⁰ penalty cost of NFS ($\times$10$^3$ $\yen$ per-day); ¹¹ penalty cost of CP segments connected to NFS ($\times$10$^3$ $\yen$ per-day); ¹² size of orders transferred at IFS ($\times$10$^3$ parcel per-day).

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