# A note on solving the Fleet Quickest Routing Problem on a grid graph

• Mirko Di Giacomo
• Francesco Mason
• Marisa Cenci
Article

## Abstract

We address the Fleet Quickest Routing Problem (FQRP) on a grid graph, i.e. the problem of finding the paths $$n$$ vehicles have to perform to reach their destinations by moving from the lowest to the highest level (row) of a grid, respecting capacity constraints on the arcs and nodes in the shortest time possible. Traversing an arc of the grid graph takes one unit of time: in this way, a solution is optimal if a vehicle travels along one of the shortest paths it can find from its starting node to its destination without stopping, meeting the capacity constraints we mentioned above. Such a path is made of both horizontal and vertical steps, from the lowest to the highest level of the grid, without cycles. The existence of such a solution is guaranteed by a sufficient number of levels. If this not being, an optimal solution could require stopping vehicles somewhere or making them travel on a non-shortest path. In a previous work, it has been proved that $$m^{ *}$$ levels, where $$m^{ *}$$ is an increasing function of $$n$$, are sufficient to solve FQRP on grid graphs, when each vehicle performs all the horizontal steps of its path, if any, on one and only one level. The main contribution of this paper is the proof that, relaxing this property, the number of levels needed to guarantee a feasible and optimal solution is constant, i.e. it is 8. Whatever the number of vehicles, in a grid graph $$(8 \times n)$$ it is always possible to find $$n$$ shortest paths, one for each vehicle, which do not create conflicts and route every vehicle to its destination. Moreover, a routing rule that allows finding the solution to FQRP in every $$8 \times n$$ grid graph, whatever the destinations of the vehicles are, is provided. Practical relevance of FQRP mainly concerns the routing of Automated Guided Vehicles as well as the routing of aircrafts in airports.

## Keywords

Routing Logistics Graph theory Grid graph Transportation

## References

1. Aggarwal A, Kleinberg J, Williamson DP, Chuzhoy J, Kim DHK, Li S (2000) Node-disjoint paths on the mesh and a new trade-off in VLSI layout. SIAM J Comput 29(4):1321–1333Google Scholar
2. Andreatta G, De Giovanni L, Salmaso G (2010) Fleet quickest routing on grids: a polynomial algorithm. Int J Pure Appl Math 62(4):419–432Google Scholar
3. Balakrishnan H, Jung Y (2007) A framework for coordinated surface operations planning at Dallas-Fort Worth International Airport. In: AIAA guidance, navigation and control conference and exhibit, guidance, navigation, and control and co-located conferences. American Institute of Aeronautics and Astronautics, Reston, Virigina, pp 2382–2400. Available from: http://arc.aiaa.org/doi/pdf/10.2514/6.2007-6553. Accessed 15 Jan 2016
4. Cenci M, Di Giacomo M, Mason F (2017) A note on a mixed routing and scheduling problem on a grid graph. J Oper Res Soc 68:1363–1376Google Scholar
5. Chuzhoy J, Kim DHK (2015) On approximating node-disjoint paths in grids. In: LIPIcs-Leibniz International Proceedings in Informatics, pp 187–211Google Scholar
6. Chuzhoy J, Kim DHK, Nimavat R (2018) Improved approximation for node-disjoint paths in grids with sources on the boundary. http://arxiv.org/abs/1805.09956. Accessed 9 Apr 2019
7. Clare G, Richards A (2011) Optimization of taxiway routing and runway scheduling. IEEE Trans Intell Transp Syst 12(4):1000–1013. http://ieeexplore.ieee.org/xpls/abs_all.jsp?arnumber=5752242. Accessed 15 Jan 2016
8. De IPZ, Cutler M, Shiloach Y (1978) Permutation layout. Networks 8(3):253–278Google Scholar
9. Garcia J, Berlanga A, Molina J, Casar J (2005) Optimization of airport ground operations integrating genetic and dynamic flow management algorithms. AI Commun 18(2):143–164. http://www.researchgate.net/profile/Jose_Molina5/publication/220308927_Optimization_of_airport_ground_operations_integrating_genetic_and_dynamic_flow_management_algorithms/links/0912f509267636ce70000000.pdf. Accessed 15 Jan 2016
10. Gupta G, Malik W, Jung Y (2009) A mixed integer linear program for airport departure scheduling. In: 9th AIAA aviation technology, integration, and operations conference (ATIO), pp 21–23. http://arc.aiaa.org/doi/pdf/10.2514/6.2009-6933. Accessed 15 Jan 2016
11. Marín ÁG, Marin A (2006) Airport management: taxi planning. Ann Oper Res 143(1):191–202. http://link.springer.com/article/10.1007/s10479-006-7381-2. Accessed 15 Jan 2016
12. Pesic B, Durand N, Alliot J (2001) Aircraft ground traffic optimisation using a genetic algorithm. GECCO 2001. https://hal-enac.archives-ouvertes.fr/hal-00938003/. Accessed 15 Jan 2016
13. Roling PC, Visser HGH (2008) Optimal airport surface traffic planning using mixed-integer linear programming. Int J Aerosp Eng 2008(1):1–11. http://dl.acm.org/citation.cfm?id=1648386. Accessed 15 Jan 2016
14. Smeltink JW, Soomer MJ (2005) An optimisation model for airport taxi scheduling. In: Proc. 13th Conf. Math. Oper. Res., Citeseer, pp 1–25. http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.116.9774&rep=rep1&type=pdf

© Springer-Verlag GmbH Germany, part of Springer Nature 2019

## Authors and Affiliations

1. 1.Department of Business StudiesUniversity of Rome IIIRomeItaly
2. 2.Department of ManagementCa’ Foscari University of VeniceVeniceItaly