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A novel MILP formulation and an efficient heuristic for the vehicle routing problem with lunch break

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Abstract

The vehicle routing problem with time windows and lunch break (VRPTW-LB) is an NP-hard combinatorial optimization problem that belongs to the vehicle routing problem family. In the VRPTW-LB, each vehicle must serve assigned customers within their availability periods and take a mandatory break within its time slot for a given duration. This paper proposes a new mixed integer linear programming (MILP) formulation for the VRPTW-LB. The new MILP formulation is based on the idea of the intersection of two intervals, which must be non-empty. It provides a flexible schedule for lunch breaks by defining the earliest and latest start times at which each break can be taken, instead of defining its exact start time. A comparative study of MILP formulations from the literature and the proposed one is performed, which are tested on benchmark instances from the literature. These MILP formulations are powerless to solve large-scale instances. To overcome this limitation, an efficient lunch break scheduling algorithm is proposed and embedded into a simulated annealing (SA) based heuristic. Computational results highlight the competitiveness of the new MILP formulation with respect to other MILP formulations and the efficiency of the proposed heuristic in obtaining high-quality solutions in short CPU running times.

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References

  • Bazirha, M., Kadrani, A., & Benmansour, R. (2020). Daily scheduling and routing of home health care with multiple availability periods of patients. In: Variable Neighborhood Search: 7th international conference, ICVNS 2019, Springer, pp 178–193.

  • Bazirha, M., Kadrani, A., & Benmansour, R. (2022). Optimization of the stochastic home health care routing and scheduling problem with multiple hard time windows. International Journal of Supply and Operations Management, 9(2), 235–250.

    Google Scholar 

  • Belhaiza, S., Hansen, P., & Laporte, G. (2014). A hybrid variable neighborhood tabu search heuristic for the vehicle routing problem with multiple time windows. Computers & Operations Research, 52, 269–281.

    Article  Google Scholar 

  • Beltrami, E. J., & Bodin, L. D. (1974). Networks and vehicle routing for municipal waste collection. Networks, 4(1), 65–94.

    Article  Google Scholar 

  • Buhrkal, K., Larsen, A., & Ropke, S. (2012). The waste collection vehicle routing problem with time windows in a city logistics context. Procedia-Social and Behavioral Sciences, 39, 241–254.

    Article  Google Scholar 

  • Clarke, G., & Wright, J. W. (1964). Scheduling of vehicles from a central depot to a number of delivery points. Operations Research, 12(4), 568–581.

    Article  Google Scholar 

  • Coelho, L. C., Gagliardi, J. P., Renaud, J., et al. (2016). Solving the vehicle routing problem with lunch break arising in the furniture delivery industry. Journal of the Operational Research Society, 67(5), 743–751.

    Article  Google Scholar 

  • Dantzig, G., Fulkerson, R., & Johnson, S. (1954). Solution of a large-scale traveling-salesman problem. Journal of the Operations Research Society of America, 2(4), 393–410.

    Article  Google Scholar 

  • Dotoli, M., & Epicoco, N. (2017). A vehicle routing technique for hazardous waste collection. IFAC-PapersOnLine, 50(1), 9694–9699.

    Article  Google Scholar 

  • Favaretto, D., Moretti, E., & Pellegrini, P. (2007). Ant colony system for a vrp with multiple time windows and multiple visits. Journal of Interdisciplinary Mathematics, 10(2), 263–284.

    Article  Google Scholar 

  • Glover, F. (1986). Future paths for integer programming and links to artificial intelligence. Computers & Operations Research, 13(5), 533–549.

    Article  Google Scholar 

  • Kirkpatrick, S., Gelatt, C. D., Jr., & Vecchi, M. P. (1983). Optimization by simulated annealing. Science, 220(4598), 671–680.

    Article  Google Scholar 

  • Kulkarni, R., & Bhave, P. R. (1985). Integer programming formulations of vehicle routing problems. European Journal of Operational Research, 20(1), 58–67.

    Article  Google Scholar 

  • Liu, R., Yuan, B., & Jiang, Z. (2017). Mathematical model and exact algorithm for the home care worker scheduling and routing problem with lunch break requirements. International Journal of Production Research, 55(2), 558–575.

    Article  Google Scholar 

  • Liu, W., Dridi, M., Fei, H., et al. (2021). Hybrid metaheuristics for solving a home health care routing and scheduling problem with time windows, synchronized visits and lunch breaks. Expert Systems with Applications, 183(115), 307.

    Google Scholar 

  • Mladenović, N., & Hansen, P. (1997). Variable neighborhood search. Computers & Operations Research, 24(11), 1097–1100.

    Article  Google Scholar 

  • Nickel, S., Schröder, M., & Steeg, J. (2012). Mid-term and short-term planning support for home health care services. European Journal of Operational Research, 219(3), 574–587.

    Article  Google Scholar 

  • Pourjavad, E., & Almehdawe, E. (2022). Optimization of the technician routing and scheduling problem for a telecommunication industry. Annals of Operations Research, 315(1), 371–395.

    Article  Google Scholar 

  • Renaud, J., Laporte, G., & Boctor, F. F. (1996). A tabu search heuristic for the multi-depot vehicle routing problem. Computers & Operations Research, 23(3), 229–235.

    Article  Google Scholar 

  • Sahoo, S., Kim, S., Kim, B. I., et al. (2005). Routing optimization for waste management. Interfaces, 35(1), 24–36.

    Article  Google Scholar 

  • Savelsbergh, M. W., & Sol, M. (1995). The general pickup and delivery problem. Transportation Science, 29(1), 17–29.

    Article  Google Scholar 

  • Solomon, M. M., & Desrosiers, J. (1988). Survey paper—time window constrained routing and scheduling problems. Transportation Science, 22(1), 1–13. https://doi.org/10.1287/trsc.22.1.1

    Article  Google Scholar 

  • Waters, C. (1988). Expanding the scope of linear programming solutions for vehicle scheduling problems. Omega, 16(6), 577–583.

    Article  Google Scholar 

  • Xiao, L., Dridi, M., & El Hassani, A. H. (2018). Mathematical model for the home health care scheduling and routing problem with flexible lunch break requirements. IFAC-PapersOnLine, 51(11), 334–339.

    Article  Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics and Operations Research, INSEA, Rabat, Morocco

    Mohammed Bazirha

  2. Ministry of Transport and Logistics, Road Transport Directorate, Rabat, Morocco

    Mohammed Bazirha

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  1. Mohammed Bazirha
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Correspondence to Mohammed Bazirha.

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Bazirha, M. A novel MILP formulation and an efficient heuristic for the vehicle routing problem with lunch break. Ann Oper Res (2023). https://doi.org/10.1007/s10479-023-05742-3

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