# Locating Movable Parcel Lockers under Stochastic Demands

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## Abstract

**:**

## 1. Introduction

## 2. Literature Review

## 3. Robust Optimization Model under Stochastic Demands

#### 3.1. Problem Description and Assumptions

#### 3.2. Optimization Model under Stochastic Demands

#### 3.2.1. Optimization Model under Deterministic Demands

**Lemma 1.**

**Proof.**

**Lemma 2.**

**Proof.**

**Scenario 1.**

**Scenario 2.**

**Scenario 3.**

#### 3.2.2. Robust Optimization

## 4. Experiments and Results

#### 4.1. Parameter Settings

#### 4.1.1. Purchase Cost

#### 4.1.2. Maintenance Cost

#### 4.1.3. Travel Cost

#### 4.1.4. Rent for Land

#### 4.2. The Robustness of Solutions

#### 4.3. The Impacts of Key Parameters on the Optimization Results

#### 4.3.1. Impacts on the Number of Self-Pickup Sites

#### 4.3.2. Impacts on the Number of Movable Parcel Locker Units

#### 4.3.3. Impacts on the Costs

#### 4.4. The Impacts of Mobility Restrictions on the Optimization Results

## 5. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

- ATKearney. Available online: https://www.atkearney.com (accessed on 18 March 2020).
- Wang, X.T.; Wang, H. A study on sustaining corporate innovation with E-commerce in China. Sustainability
**2019**, 11, 6604. [Google Scholar] [CrossRef] [Green Version] - E-commerce Europe. Available online: https://www.ecommerce-europe.eu/news-item/double-digit-growth-global-b2c-e-commerce-sales-2015/ (accessed on 18 March 2020).
- Francke, J.; Visser, J. Internet shopping and its impacts on mobility. Igarss
**2014**, 1, 1–5. [Google Scholar] - Syntun. Available online: http://www.syntun.com.cn/2019nian11yue1ri-11ri-qi-jian-kuang-huan-gou-wu-jie-quan-wang-da-bao-gao.html (accessed on 1 April 2020). (In Chinese).
- Xu, M.; Ferrand, B.; Roberts, M. The last mile of e-commerce-unattended delivery from the consumers and eTailers’ perspectives. Int. J. Electron. Market. Retail.
**2008**, 2, 20–38. [Google Scholar] [CrossRef] - GOV.UK. Available online: www.foresight.gov.uk (accessed on 9 February 2020).
- Song, L.; Cherrett, T.; McLeod, F.; Guan, W. Addressing the last mile problem: Transport impacts of collection and delivery points. Transp. Res. Rec.
**2009**, 2097, 9–18. [Google Scholar] [CrossRef] [Green Version] - Moroz, M.; Polkowski, Z. The last mile issue and urban logistics: Choosing parcel machines in the context of the ecological attitudes of the Y generation consumers purchasing online. Transp. Res. Procedia.
**2016**, 16, 378–393. [Google Scholar] [CrossRef] [Green Version] - He, Z.; Zhang, W.; Jia, N. Estimating carbon dioxide emissions of freeway traffic: A spatiotemporal cell-based model. IEEE Trans. Intell. Transp. Syst.
**2020**, 21, 1976–1986. [Google Scholar] [CrossRef] - Xiong, J.; He, Z.; Guan, W.; Ran, B. Optimal timetable development for community shuttle network with metro stations. Transp. Res. Part C Emerg. Technol.
**2015**, 60, 540–565. [Google Scholar] [CrossRef] - Zhou, L.; Wang, X.; Ni, L.; Lin, Y. Location-routing problem with simultaneous home delivery and customer’s pickup for city distribution of online shopping purchases. Sustainability
**2016**, 8, 828. [Google Scholar] [CrossRef] [Green Version] - Iwan, S.; Kijewska, K.; Lemke, J. Analysis of parcel lockers’ efficiency as the last mile delivery solution—The results of the research in Poland. Transp. Res. Procedia.
**2016**, 12, 644–655. [Google Scholar] [CrossRef] [Green Version] - Faugere, L.; Montreuil, B. Hyperconnected pickup & delivery locker networks. In Proceedings of the 4th International Physical Internet Conference, Graz, Austria, 4–6 July 2017. [Google Scholar]
- DHL. Available online: https://www.dpdhl.com/en/about-us.html (accessed on 31 March 2020).
- Zenezini, G.; Lagorio, A.; Pinto, R.; Marco, A.D.; Golini, R. The Collection-and-Delivery Points implementation process from the courier, express and parcel operator’s perspective. IFAC-PapersOnLine
**2018**, 51, 594–599. [Google Scholar] [CrossRef] - Industry News. “Contactless distribution” brings about a turnaround of logistics industry. Green Packag.
**2020**, 2, 21–22. (In Chinese) [Google Scholar] - Ducret, R. Parcel deliveries and urban logistics: Changes and challenges in the courier express and parcel sector in Europe—The French case. Res. Transp. Bus. Manag.
**2014**, 11, 15–22. [Google Scholar] [CrossRef] - BEE EXPRESS. Available online: https://a.eqxiu.com/s/B7D37aS7?share_level=1&from_user=20200326f3c10538&from_id=d44a19c6-d&share_time=1585190638775&from=singlemessage (accessed on 2 April 2020). (In Chinese).
- Deutsch, Y.; Golany, B. A parcel locker network as a solution to the logistics last mile problem. Int. J. Prod. Res.
**2018**, 56, 251–261. [Google Scholar] [CrossRef] - Lemke, J.; Iwan, S.; Korczak, J. Usability of the parcel lockers from the customer perspective: The research in Polish cities. Transp. Res. Procedia.
**2016**, 16, 272–287. [Google Scholar] [CrossRef] [Green Version] - Kedia, A.S.K.; Kusumastuti, D.; Nicholson, A. Acceptability of Collection and Delivery Points from consumers’ perspective: A qualitative case study of Christchurch city. Case Stud. Transp. Policy.
**2017**, 5, 587–595. [Google Scholar] [CrossRef] - Lachapelle, U.; Burke, M.; Brotherton, A.; Leung, A. Parcel locker systems in a car dominant city: Location, characterisation and potential impacts on city planning and consumer travel access. J. Transp. Geogr.
**2018**, 71, 1–14. [Google Scholar] [CrossRef] - Liu, S.; Lin, B.; Wang, J.; Wu, J. Modeling the multi-period and multi-classification-yard location problem in a railway network. Symmetry
**2018**, 10, 135. [Google Scholar] [CrossRef] [Green Version] - Guo, X.; Song, R.; He, S.; Bi, M.; Jin, G. Integrated optimization of stop location and route design for community shuttle service. Symmetry
**2018**, 10, 678. [Google Scholar] [CrossRef] [Green Version] - Guo, X.; Song, R.; He, S.; Hao, S.; Zheng, L.; Jin, G. A multi-objective programming approach to design feeder bus route for high-speed rail stations. Symmetry
**2019**, 11, 514. [Google Scholar] [CrossRef] [Green Version] - Ji, S.F.; Luo, R.J.; Peng, X.S. A probability guided evolutionary algorithm for multi-objective green express cabinet assignment in urban last-mile logistics. Int. J. Prod. Res.
**2019**, 57, 3382–3404. [Google Scholar] [CrossRef] - Lee, H.; Chen, M.W.; Pham, H.T.; Choo, S. Development of a decision making system for installing unmanned parcel lockers: Focusing on residential complexes in Korea. KSCE J. Civ. Eng.
**2019**, 23, 2713–2722. [Google Scholar] [CrossRef] - Tan, K.C.; Cheong, C.Y.; Goh, C.K. Solving multiobjective vehicle routing problem with stochastic demand via evolutionary computation. Eur. J. Oper. Res.
**2006**, 177, 813–839. [Google Scholar] [CrossRef] - Martí, J.M.C.; Tancrez, J.S.; Seifert, R.W. Carbon footprint and responsiveness trade-offs in supply chain network design. Int. J. Prod. Econ.
**2015**, 166, 129–142. [Google Scholar] [CrossRef] - Wen, M.L.; Qin, Z.F.; Kang, R.; Yang, Y. The capacitated facility location-allocation problem under uncertain environment. J. Intell. Fuzzy Syst.
**2015**, 29, 2217–2226. [Google Scholar] [CrossRef] [Green Version] - Sun, H.L.; Zhou, Z.J.; Xue, Y.F. Emergency location-routing problem with uncertain demand under path risk. Shanghai Jiaotong Daxue Xuebao
**2013**, 47, 962–966. (In Chinese) [Google Scholar] - Bieniek, M. A note on the facility location problem with stochastic demands. Omega
**2015**, 55, 53–60. [Google Scholar] [CrossRef] - Albareda-Sambola, M.; Fernández, E.; Saldanha-Da-Gama, F. Heuristic solutions to the facility location problem with general Bernoulli demands. INFORMS J. Comput.
**2017**, 29, 737–753. [Google Scholar] [CrossRef] - Bertsimas, D.; Sim, M. The price of robustness. Oper. Res.
**2004**, 52, 35–53. [Google Scholar] [CrossRef] - Gabrel, V.; Lacroix, M.; Murat, C.; Remli, N. Robust location transportation problems under uncertain demands. Discrete Appl. Math.
**2014**, 164, 100–111. [Google Scholar] [CrossRef] - Huang, M.; Ren, L.; Lee, L.H.; Wang, X.W.; Kuang, H.B.; Shi, H.B. Model and algorithm for 4PLRP with uncertain delivery time. Inf. Sci.
**2016**, 330, 211–225. [Google Scholar] [CrossRef] [Green Version] - Lin, D.S.; Zhang, Z.Y.; Wang, J.X.; Liang, X.; Shi, Y.Q. Low-carbon logistics distribution center location with uncertain demand. Kongzhi Yu Juece Control Decis.
**2020**, 35, 492–500. [Google Scholar] - Tanonkou, G.A.; Benyoucef, L.; Xie, X.L. A scenario analysis of a location problem with uncertain demand. Int. J. Comput. Appl. Technol.
**2008**, 32, 290–297. [Google Scholar] [CrossRef] - Zhang, B.; Ma, Z.J.; Jiang, S. Location-routing-inventory problem with stochastic demand in logistics distribution systems. In Proceedings of the 2008 International Conference on Wireless Communications, Networking and Movable Computing, WiCOM 2008, Dalian, China, 12–14 October 2008. [Google Scholar]
- Li, H.B.; Yan, J.; Ren, M.M. Bender’s algorithm for facility location problem with uncertain demand. In Proceedings of the Innovative Computing and Information, ICCIC 2011, Wuhan, China, 17–18 September 2011; Dai, M., Ed.; Springer: Berlin/Heidelberg, Germany, 2011. [Google Scholar]
- Brenna, M.; Lazaroiu, G.C.; Roscia, M.; Saadatmandi, S. Dynamic model for the EV’s charging infrastructure planning through finite element method. IEEE Access.
**2020**, 8, 102399–102408. [Google Scholar] [CrossRef] - Balinski, M. Integer Programming: Methods, Uses, Computation. In 50 Years of Integer Programming 1958–2008; Jünger, M., Naddef, D., Pulleyblank, W.R., Rinaldi, G., Liebling, T.M., Nemhauser, G.L., Reinelt, G., Wolsey, L.A., Eds.; Springer: Berlin/Heidelberg, Germany, 2010; pp. 133–197. [Google Scholar]
- Gong, Y.M. Integer programming. In Operations Research Course, 2nd ed.; Hu, Y.Q., Guo, Y.H., Eds.; Tsinghua University Press: Beijing, China, 2003. (In Chinese) [Google Scholar]
- MathWorks. Available online: https://www.mathworks.com/help/optim/ug/intlinprog.html?s_tid=srchtitle (accessed on 6 April 2020).
- Longo, M.; Foiadelli, F.; Yaïci, W. Simulation and optimisation study of the integration of distributed generation and electric vehicles in smart residential district. Int. J. Energy Environ. Eng.
**2019**, 10, 271–285. [Google Scholar] [CrossRef] [Green Version] - Beijing Municipal Commission of Development and Reform. Available online: http://www.bjmy.gov.cn/art/2019/5/31/art_3334_267683.html (accessed on 2 April 2020). (In Chinese)
- The People’s Government of Beijing Municipality. Available online: http://www.bjdch.gov.cn/n2001806/n2917336/n2917339/c7722611/content.html (accessed on 2 April 2020). (In Chinese)
- Phommixay, S.; Doumbia, M.L.; Lupien St-Pierre, D. Review on the cost optimization of microgrids via particle swarm optimization. Int. J. Energy Environ. Eng.
**2020**, 11, 73–89. [Google Scholar] [CrossRef] [Green Version]

**Figure 5.**The impacts of key parameters on the optimization results. (

**a**) impacts on the number of self-pickup sites; (

**b**) impacts on the number of movable parcel locker units; (

**c**) impacts on the generalized cost; and (

**d**) impacts on the real cost.

**Figure 6.**Distribution network and shortest paths from each demand point to the depot. (

**a**) distribution network under the normal situation; (

**b**) shortest paths under the normal situation; (

**c**) distribution network under mobility restrictions; and (

**d**) shortest paths at night under mobility restrictions.

Sets | Descriptions |
---|---|

I | Demand point set |

${I}^{\prime}$ | Set of demand points that cannot be selected as self-pickup sites |

${I}^{\u2033}$ | Set of demand points that must be selected as self-pickup sites |

E | Set of edges weighted by the shortest distance between demand points |

Parameters | Descriptions |

${c}^{\mathrm{f}}$ | Purchase and maintenance costs of a movable parcel locker unit |

${c}_{i}^{\mathrm{s}}$ | Land rental rate for a movable parcel locker unit at site i and the transportation cost of a unit from the depot to site i |

${c}_{i}$ | The generalized cost for one locker |

A | Number of lockers equipped on a movable parcel locker unit |

W | A very large positive number |

${l}_{ij}$ | Distance between demand point j and self-pickup site i |

r | Maximum walking distance acceptable to customers |

${d}_{j}$ | Customer’s delivery demand at the demand point j except for the demand met by fixed parcel lockers |

M | A very large positive number |

$\overline{{d}_{\mathit{\u0237}}}$ | Average delivery demand at the demand point j |

$\Delta {d}_{j}$ | Maximum variation in demand at the demand point j |

${\Gamma}_{i}$ | A parameter adjusting the trade-off between robustness and risk. It is the number of demand points with demand variation at the self-pickup site i |

Decision variables | Descriptions |

${x}_{i}$ | Total number of lockers set up at the self-pickup site i |

${n}_{i}$ | Number of movable parcel locker units placed at the self-pickup site i |

${y}_{ij}$ | Binary variable, which is 1 if the demand point j is allocated to the self-pickup site i and, otherwise, 0. |

Items | Numerical Values |
---|---|

Power consumption during driving | 400 W |

Power consumption during parking | 40 W |

Battery capacity | 1.2 kWh |

Charge for commercial electricity | ¥0.78/kWh |

Communication cost | ¥0.04/parcel |

Price of battery | ¥400 |

Rent for land | About ¥3/${\mathrm{m}}^{2}$/day |

Charging efficiency | 80% |

${\mathit{\Gamma}}_{\mathit{i}}$ | Probability Bound | Optimal Cost (¥) | Relative Cost Ratio (%) | Lockers | Relative Locker Ratio (%) | Optgap | Time (s) |
---|---|---|---|---|---|---|---|

4 | $3.38\times {10}^{-1}$ | 4520.80 | 20.66 | 6096 | 20.71 | 0.00 | 5.53 |

8 | $1.62\times {10}^{-1}$ | 4535.08 | 21.05 | 6115 | 21.09 | 0.00 | 4.62 |

13 | $4.62\times {10}^{-2}$ | 4553.85 | 21.55 | 6140 | 21.58 | 0.00 | 7.59 |

18 | $7.70\times {10}^{-3}$ | 4571.93 | 22.03 | 6164 | 22.06 | 0.00 | 1.94 |

23 | $8.91\times {10}^{-4}$ | 4591.43 | 22.55 | 6190 | 22.57 | 0.00 | 4.43 |

27 | $9.98\times {10}^{-5}$ | 4605.78 | 22.93 | 6209 | 22.95 | 0.00 | 17.8 |

33 | $1.71\times {10}^{-6}$ | 4631.14 | 23.61 | 6243 | 23.62 | 0.00 | 8.17 |

37 | $6.14\times {10}^{-8}$ | 4646.19 | 24.01 | 6263 | 24.02 | 0.00 | 3.12 |

40 | $2.14\times {10}^{-9}$ | 4653.76 | 24.21 | 6273 | 24.22 | 0.00 | 1.33 |

43 | $1.24\times {10}^{-10}$ | 4665.74 | 24.53 | 6289 | 24.53 | 0.00 | 51.19 |

46 | $1.18\times {10}^{-12}$ | 4676.99 | 24.83 | 6304 | 24.83 | 0.00 | 16.89 |

49 | $2.50\times {10}^{-14}$ | 4684.58 | 25.04 | 6314 | 25.03 | 0.00 | 3.75 |

Scenarios | Self-Pickup Sites | Sub-Demand Points | Lockers | Cost (¥) |
---|---|---|---|---|

Normal situation | 2 | 2 6 | 244 | 264.89 |

4 | 14 | 301 | ||

7 | 7 | 113 | ||

8 | 8 | 60 | ||

9 | 59 | 333 | ||

10 | 310 | 250 | ||

11 | 11 | 132 | ||

Movement restrictions | 1 | 14 | 301 | 283.04 |

2 | 2 6 | 244 | ||

3 | 39 | 296 | ||

7 | 57 | 289 | ||

8 | 8 | 60 | ||

10 | 10 | 111 | ||

11 | 11 | 132 |

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**MDPI and ACS Style**

Wang, Y.; Bi, M.; Lai, J.; Chen, Y.
Locating Movable Parcel Lockers under Stochastic Demands. *Symmetry* **2020**, *12*, 2033.
https://doi.org/10.3390/sym12122033

**AMA Style**

Wang Y, Bi M, Lai J, Chen Y.
Locating Movable Parcel Lockers under Stochastic Demands. *Symmetry*. 2020; 12(12):2033.
https://doi.org/10.3390/sym12122033

**Chicago/Turabian Style**

Wang, Yang, Mengyu Bi, Jianhui Lai, and Yanyan Chen.
2020. "Locating Movable Parcel Lockers under Stochastic Demands" *Symmetry* 12, no. 12: 2033.
https://doi.org/10.3390/sym12122033