Skip to main content

Advertisement

SpringerLink
Log in

Time-dependent rural postman problem: time-space network formulation and genetic algorithm

  • Original Paper
  • Published:
Operational Research Aims and scope Submit manuscript

Abstract

In this paper, a new time-space network model is proposed for addressing the time-dependent rural postman problem (TDRPP) of a single vehicle. The proposed model follows the idea of arc-path alternation to form a feasible and complete route. Based on the proposed model, the time dependency of the TDRPP is better described to capture its dynamic process, compared to the existing methods using a piecewise constant function with limited intervals. Furthermore, the property of first-in-first-out (FIFO) can be satisfied with the time spent on each arc. We investigate the FIFO property for the considered time-dependent network and key optimality property for the TDRPP. Based on this property, a dedicated genetic algorithm (GA) is proposed to efficiently solve the considered TDRPP that suffers from computational intractability for large-scale cases. Comprehensive simulation experiments are conducted for various time-dependent networks to show the effectiveness of the proposed GA.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6

Similar content being viewed by others

References

  • Ahn BH, Shin JY (1991) Vehicle-routeing with time windows and time-varying congestion. J Opera Res Soc 42:393–400

    Article  Google Scholar 

  • Beasley J (1981) Adapting the savings algorithm for varying inter-customer travel times. Omega 9:658–659

    Article  Google Scholar 

  • Black D, Eglese R, Wøhlk S (2013) The time-dependent prize-collecting arc routing problem. Comput Oper Res 40:526–535

    Article  Google Scholar 

  • Boland NL, Savelsbergh MW (2019) Perspectives on integer programming for time-dependent models. Top 27:147–173

    Article  Google Scholar 

  • Calogiuri T, Ghiani G, Guerriero E, Mansini R (2019) A branch-and-bound algorithm for the time-dependent rural postman problem. Comput Oper Res 102:150–157

    Article  Google Scholar 

  • Colombi M, Corberán Á, Mansini R, Plana I, Sanchis JM (2017) The hierarchical mixed rural postman problem. Transp Sci 51:755–770

    Article  Google Scholar 

  • Corberán Á, Eglese R, Hasle G, Plana I, Sanchis JM (2021) Arc routing problems: a review of the past, present, and future. Networks 77:88–115

    Article  Google Scholar 

  • Corberán Á, Laporte G (2014) Arc routing: problems, methods, and applications. MOS-SIAM Series on (Optimization)

  • Corberán A, Prins C (2010) Recent results on arc routing problems: an annotated bibliography. Networks 56:50–69

    Google Scholar 

  • Cordeau JF, Ghiani G, Guerriero E (2014) Analysis and branch-and-cut algorithm for the time-dependent travelling salesman problem. Transp Sci 48:46–58

    Article  Google Scholar 

  • Fernández E, Laporte G, Rodríguez-Pereira J (2018) A branch-and-cut algorithm for the multidepot rural postman problem. Transp Sci 52:353–369

    Article  Google Scholar 

  • Fleischmann B, Gietz M, Gnutzmann S (2004) Time-varying travel times in vehicle routing. Transp Sci 38:160–173

    Article  Google Scholar 

  • Gendreau M, Ghiani G, Guerriero E (2015) Time-dependent routing problems: a review. Comput Oper Res 64:189–197

    Article  Google Scholar 

  • Guan M (1962) Graphic programming using odd and even points. Chinese Math. 1:237–277

    Google Scholar 

  • Gurobi (2018) Gurobi optimizer reference manual. http://www.gurobi.com

  • Halim AH, Ismail I (2019) Combinatorial optimization: comparison of heuristic algorithms in travelling salesman problem. Arch Comput Methods Eng 26:367–380

    Article  Google Scholar 

  • Hansen P, Mladenović N, Todosijević R, Hanafi S (2017) Variable neighborhood search: basics and variants. EURO J Comput Optim 5:423–454

    Article  Google Scholar 

  • Horn ME (2000) Efficient modeling of travel in networks with time-varying link speeds. Networks Int J 36:80–90

    Google Scholar 

  • Huang T, Gong YJ, Kwong S, Wang H, Zhang J (2019) A niching memetic algorithm for multi-solution traveling salesman problem. IEEE Trans Evol Comput

  • Huang Y, Zhao L, Van Woensel T, Gross JP (2017) Time-dependent vehicle routing problem with path flexibility. Trans Res Part B Methodol 95:169–195

    Article  Google Scholar 

  • Ichoua S, Gendreau M, Potvin JY (2003) Vehicle dispatching with time-dependent travel times. Eur J Oper Res 144:379–396

    Article  Google Scholar 

  • Kaufman DE, Smith RL (1993) Fastest paths in time-dependent networks for intelligent vehicle-highway systems application. J Intell Transp Syst 1:1–11

    Google Scholar 

  • Kramer O (2017) Genetic algorithm essentials. Springer, Berlin

    Book  Google Scholar 

  • Meng L, Zhou X (2014) Simultaneous train rerouting and rescheduling on an n-track network: a model reformulation with network-based cumulative flow variables. Trans Res Part B Methodol 67:208–234

    Article  Google Scholar 

  • Mourao MC, Pinto LS (2017) An updated annotated bibliography on arc routing problems. Networks 70:144–194

  • Nossack J, Golden B, Pesch E, Zhang R (2017) The windy rural postman problem with a time-dependent zigzag option. Eur J Oper Res 258:1131–1142

    Article  Google Scholar 

  • Orloff C (1974) A fundamental problem in vehicle routing. Networks 4:35–64

    Article  Google Scholar 

  • Quirion-Blais O, Langevin A, Lehuédé F, Péton O, Trépanier M (2017) Solving the large-scale min-max k-rural postman problem for snow plowing. Networks 70:195–215

    Article  Google Scholar 

  • Rabbouch B, Saâdaoui F, Mraihi R (2019) Efficient implementation of the genetic algorithm to solve rich vehicle routing problems. Oper Res, pp 1–29. https://doi.org/10.1007/s12351-019-00521-0

  • Shakibayifar M, Sheikholeslami A, Corman F, Hassannayebi E (2020) An integrated rescheduling model for minimizing train delays in the case of line blockage. Oper Res 20:59–87

    Google Scholar 

  • Sharma D, Deb K, Kishore N (2011) Domain-specific initial population strategy for compliant mechanisms using customized genetic algorithm. Struct Multidiscip Optim 43:541–554

    Article  Google Scholar 

  • Sun J, Meng Y, Tan G (2015) An integer programming approach for the Chinese postman problem with time-dependent travel time. J Combin Optim 29:565–588

    Article  Google Scholar 

  • Sun J, Tan G, Hou G (2011a) A new integer programming formulation for the chinese postman problem with time dependent travel times. World Academy of Science, Engineering and Technology. Int J Comput Electrical Autom Control Inf Eng 5:410–414

    Google Scholar 

  • Sun J, Tan G, Qu H (2011b) Dynamic programming algorithm for the time dependent Chinese postman problem. J Inf Comput Sci 8:833–841

    Google Scholar 

  • Sung K, Bell MG, Seong M, Park S (2000) Shortest paths in a network with time-dependent flow speeds. Eur J Oper Res 121:32–39

    Article  Google Scholar 

  • Tagmouti M, Gendreau M, Potvin JY (2007) Arc routing problems with time-dependent service costs. Eur J Oper Res 181:30–39

    Article  Google Scholar 

  • Tagmouti M, Gendreau M, Potvin JY (2010) A variable neighborhood descent heuristic for arc routing problems with time-dependent service costs. Comput Ind Eng 59:954–963

    Article  Google Scholar 

  • Tagmouti M, Gendreau M, Potvin JY (2011) A dynamic capacitated arc routing problem with time-dependent service costs. Transp Res Part C Emerg Technol 19:20–28

    Article  Google Scholar 

  • Tan G, Sun J (2011) An integer programming approach for the rural postman problem with time dependent travel times. In: International Computing and Combinatorics Conference. Springer, pp 414–431

  • Tan G, Sun J, Hou G (2013) The time-dependent rural postman problem: polyhedral results. Optim Methods Softw 28:855–870

    Article  Google Scholar 

  • Vincent FY, Lin SW (2015) Iterated greedy heuristic for the time-dependent prize-collecting arc routing problem. Comput Ind Eng 90:54–66

    Article  Google Scholar 

  • Wang HF, Wen YP (2002) Time-constrained Chinese postman problems. Comput Math Appl 44:375–387

    Article  Google Scholar 

  • Yang L, Zhou X (2014) Constraint reformulation and a lagrangian relaxation-based solution algorithm for a least expected time path problem. Transp Res Part B Methodol 59:22–44

    Article  Google Scholar 

  • Yu B (2020) Data set of time-dependent rpp. https://github.com/momoyby/TDRPP

  • Yu G, Yang Y (2019) Dynamic routing with real-time traffic information. Oper Res 19:1033–1058

    Google Scholar 

  • Zanotti R, Mansini R, Ghiani G, Guerriero E (2019) A kernel search approach for the time-dependent rural postman problem. In: WARP3 Proceedings, Pizzo, Italy

Download references

Acknowledgements

This research is supported by the National Natural Science Foundation of China under Grant 61703372 and the Outstanding Foreign Scientist Project in Henan Province under Grant GZS2019008.

Author information

Authors and Affiliations

  1. School of Electrical Engineering, Zhengzhou University, Science Road 100, Zhengzhou, 450001, China

    Jianbin Xin, Benyang Yu & Heshan Wang

  2. Dipartimento di Ingegneria, Università Degli Studi Roma Tre, via della Vasca Navale, 79 - 00146, Roma, Italy

    Andrea D’Ariano

  3. Department of Transport and Planning, Delft University of Technology, Mekelweg 2, 2628 CN , Delft, The Netherlands

    Meng Wang

Authors
  1. Jianbin Xin
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Benyang Yu
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Andrea D’Ariano
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Heshan Wang
    View author publications

    You can also search for this author in PubMed Google Scholar

  5. Meng Wang
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Heshan Wang.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Appendix

Appendix

  • The total time spent \(f_{\pi }(t_{s})\) can be calculated by Algorithm 3. In Algorithm 3, the term \(\{v^{k}_{1},v^{k}_{2},...,v^{k}_{n_k}\}\) represents the node sequence corresponding to transition path \(p_{k}\), while \(n_k\) represents the number of nodes included in \(p_{k}\).

  • The number of constraints and decision variables of the proposed TSN model for each scenario is provided in Table 11.

  • A description of the VNS and ACO algorithms used in this paper can be found in Hansen et al. (2017); Halim and Ismail (2019). In order to compare with GA fairly, the maximum fitness evaluation times of the two algorithms are 144000(360*400). The neighborhood structure used by the VNS algorithm in the shaking and improvement procedure is “2-opt move”, “Insertion-1 move” and “Insertion-2 move”. The ant colony size and iteration number of the ACO algorithm are 360 and 400, which are consistent with those of the GA. The other parameters of ACO are optimized by the cross validation, and the specific parameters are shown in Table 12.

figure c
Table 11 The number of constraints and decision variables corresponding to the time-space network model
Full size table
Table 12 VNS and ACO algorithm parameters
Full size table

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Xin, J., Yu, B., D’Ariano, A. et al. Time-dependent rural postman problem: time-space network formulation and genetic algorithm. Oper Res Int J 22, 2943–2972 (2022). https://doi.org/10.1007/s12351-021-00639-0

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s12351-021-00639-0

Keywords

Search

Navigation