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The Time-Dependent Vehicle Routing Problem with Time Windows and Road-Network Information

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Abstract

Most time-dependent vehicle routing problems are based on a similar modeling paradigm: travel time information is represented by travel time functions between pairs of points of interest (e.g., depot or customers). Only a few papers investigate how these functions can be computed using the available travel time information. Furthermore, most of them neglect the possibility that different paths could be selected in the road network depending on the compromises they offer between cost (distance) and travel time. In this paper, we propose a new setting where travel time functions are defined on road-network arcs. We consider the Time-Dependent Vehicle Routing Problem with Time Windows and solve it with a branch-and-price algorithm. As far as we know, this is the first exact approach for a time-dependent vehicle routing problem when travel time functions are initially defined on the segments of a road-network.

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Acknowledgements

The first author was supported by the Labex IMobS3, by the European Fund for Regional Development (FEDER Auvergne region), and by the Auvergne Region. The authors thank the reviewers for their comments that permitted to greatly improve the quality of this paper.

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Authors and Affiliations

  1. Mines Saint-Etienne, Univ Clermont Auvergne, CNRS, UMR 6158 LIMOS, Centre CMP, F - 13541, Gardanne, France

    Hamza Ben Ticha, Nabil Absi & Dominique Feillet

  2. LIMOS, Institut Supérieur d’Informatique de Modélisation et leurs Applications, ISIMA, Campus des Cèzeaux, Aubière Cedex, France

    Alain Quilliot

  3. School of Industrial Engineering, Eindhoven University of Technology, Eindhoven, The Netherlands

    Tom Van Woensel

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  1. Hamza Ben Ticha
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  2. Nabil Absi
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  3. Dominique Feillet
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  4. Alain Quilliot
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  5. Tom Van Woensel
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Correspondence to Dominique Feillet.

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This article is part of the Topical Collection on Decomposition at 70

Appendix. Experiments on larger instances

Appendix. Experiments on larger instances

In this Appendix, we report preliminary results on larger instances.

1.1 A.1 Instance Generation

The new instances are generated based on real data from the road network of the central urban area of the city of Aix-en-Provence (a city-commune in the south of France). Spatial data is extracted from OpenStreetMapⒸ (www.openstreetmap.org/). We obtain a road-network graph with 5437 nodes and 19500 arcs (see Fig. 5). Each arc is defined with a length and a maximum allowed speed. Costs are set as road segment lengths.

Fig. 5
figure 5

Road Network of the central urban area of Aix-en-Provence (France)

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Time periods and speed profiles are defined as described in Section 6.1. Road segment types are defined according to maximum allowed speeds. For highways, motorways, and arterial roads (characterized with a high maximum allowed speed), the road segment type is set to “normal”. For streets, boulevards, and roads in the center of the city, the road segment type is set to “congestion-bound”. For small roads and living streets (characterized with a low maximum allowed speed), the road segment type is set to “congestion-free”.

Based on this road network, we generate instances with |C|∈{5, 10, 25}, with three instances for each value of |C|. Depot and customer locations, time windows, customer demands, service times, and vehicle capacity are defined in the same way as for NEWLET instances. We call these instances AIX instances.

1.2 A.2 Experiments

Table 5 reports the results obtained for AIX instances. Headings are the same as in previous tables, except Column “Ins”, which indicates the instance index. Note that results are not reported for min-time graphs. Indeed, due to the complexity of the proposed algorithm (see Section 4.3), we could not generate complete min-time graphs in a reasonable amount of time for this road-network.

Table 5 Computing times and solution values for AIX instances
Full size table

In these instances, the size of the road-network and the small density of customer nodes in the network are representative of what can be expected in real distribution systems. Especially, the road-network is much larger than customer-based graphs. We observe that solving the TDVRPTWRN becomes more complicated. Only instances with a limited number of customers can be solved in a reasonable amount of time. However, we also observe that the benefits are there. Solving the TDVRPTWRN enables improving solution costs for all the instances, in amounts largely greater than those obtained on NEWLET instances. The saving is 5.8% on average and reaches 11.8%. We also see that the improvement in solution costs decreases when the number of customer nodes increases. The average improvement is 9.9% for instances with 5 customers and goes down to 2.1% for instances with 25 customers.

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Ben Ticha, H., Absi, N., Feillet, D. et al. The Time-Dependent Vehicle Routing Problem with Time Windows and Road-Network Information. SN Oper. Res. Forum 2, 4 (2021). https://doi.org/10.1007/s43069-020-00049-6

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