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

  • * Corresponding authors: Xiliang Sun and Wanjie Hu

    * Corresponding authors: Xiliang Sun and Wanjie Hu 
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  • 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.

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

    Citation:

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  • Figure 1.  Overview of the 3EM-ULSND problem

    Figure 2.  Overview of the 3EM-ULSND problem

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

    Figure 4.  Model decomposition and algorithmic flowchart

    Figure 5.  Pareto-optimal front obtained with MOPSO

    Figure 6.  Pareto-optimal front obtained with MOPSO

    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
     | Show Table
    DownLoad: CSV

    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
     | Show Table
    DownLoad: CSV

    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%
     | Show Table
    DownLoad: CSV

    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 1NFS-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 12.51.89 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).
     | Show Table
    DownLoad: CSV
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