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Open vehicle routing problem with split deliveries: mathematical formulations and a cutting-plane method

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Abstract

This study addresses the open vehicle routing problem with split deliveries, a variant of the classical vehicle routing problem that allows open routes and partitioned deliveries for customers (i.e., multiple vehicles may serve one customer). This approach may be beneficial for companies interested in reducing their logistics and distribution costs. This study is motivated by a company in the pharmaceutical industry, that seeks to explore the possibility of splitting their deliveries to improve quality indicators related to on-time deliveries and customer satisfaction. Two mixed-integer formulations of the problem are proposed. Additionally, a cutting-plane method is designed to improve the optimization performance. Computational experiments were conducted to validate the performance of the formulations. The second formulation effectiveness was confirmed by providing optimal solutions for instances of up to 30 nodes in a reasonable computational time. The incorporation of the cutting-plane method improves performance with a substantial reduction in the GAP. In the case study, this new approach shows its effectiveness in economic terms by providing savings of up to 20% of the current distribution costs.

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Funding

Funding was provided by FORDECYT-CONACYT (Grant No. 265667), PROFIDES-UAdeC, Fomento a la Investigación UP 2019 (Grant No. UP-CI-2019-ING-GDL-08).

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Correspondence to Samuel Nucamendi-Guillén.

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Appendix: Results of computational experiments

Appendix: Results of computational experiments

The summary results for the benchmark problems are presented in Tables 7, 8 and 9. In such tables, columns 1, 2, and 3 denote the name of the instance, the HC, and the size of the fleet reported. Column 4 shows the initial Lower Bound (LB), column 5 the Best Integer solution found, column 6 shows the final Lower Bound (LB), and column 7 the GAP in percentage. Column 8 reports the total elapsed CPU time (including the cutting plane phase) to obtain the integer solution. Finally, the last column indicates the number of split deliveries.

Table 7 Results obtained by the cutting plane and formulation 2 with HC=0
Table 8 Results obtained by the cutting plane and formulation 2 with HC=50
Table 9 Results obtained by the cutting plane and formulation 2 with HC=100

The complete results for the case study are presented in table 10. Columns 1, 2 and 3 denote the name of the instance, the HC and the size of the fleet reported. Column 4 shows the optimal solution not allowing split deliveries, whereas column 5 indicates the optimal solution allowing split deliveries. Column 6 reports the GAP in percentage. Finally, columns 7 and 8 report the CPU time required by the formulation to obtain the solution and the number of split deliveries, respectively.

Table 10 Detailed results for the case study

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Ruiz y Ruiz, E., García-Calvillo, I. & Nucamendi-Guillén, S. Open vehicle routing problem with split deliveries: mathematical formulations and a cutting-plane method. Oper Res Int J 22, 1017–1037 (2022). https://doi.org/10.1007/s12351-020-00580-8

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