**6. Discussion**

The methods presented in the previous sections allow us to find provably optimal or heuristic solutions to FQRP-G. The problem is relevant for the design and the operation of automated transportation systems where the routing network consists of intersecting horizontal and vertical lanes, vehicles move between opposite sides (e.g., from bottom to top) and a network of sensors supports safe and efficient operations: port container terminals, automated warehouses, train terminals, etc., are some significant examples that can be approximated by such routing networks. By solving FQRP-G, the number of horizontal lanes and a set of routes is determined that can be seamlessly operated without intermediate stops nor deviations from static shortest paths (efficient routes) and without any collision (safe routes). In real-time, the sensing network and related logic monitor the operations and manage further conflicts just in case they arise due to unpredicted events (vehicle breakdowns, network interruptions, etc.), making the overall routing system robust against disruptions, and further reducing the risk of collisions.

While the number of vertical lanes is often determined by the facility layout, the number of required horizontal lanes should be carefully dimensioned at the design phase, in order to minimize the cost of the underlying transportation and sensing infrastructure. In this work, we have devised and tested four possible approaches to solve FQRP-G and determine the minimum number of levels, given vehicles' initial positions and final destinations: an exact method (ILP) based on solving an integer linear programming formulation of the problem by standard solvers, and three heuristics (A, B1 and B2) that prioritize vehicles based on the properties of a graph summarizing C-conflicts between vehicles.

Experiments on benchmarks of random and ad hoc instances show that, from a computational point of view, ILP is able to find the proven minimum number of horizontal lanes (with related vehicle routes) for instances of up to 300 vehicles, even if running times seem to be suitable for real-time operations of up to about 100 vehicles. For larger random instances, Heuristic A always provides, in negligible running time, feasible routes with at most two additional horizontal lanes, if compared to the optimal solutions, while heuristics B1 and B2 often fail in finding a set of non-conflicting vehicle paths.

From a network design perspective, it is interesting to notice that the optimal solution for the tested instances (up to 300 vehicles) always requires no more than 3 levels (4 in 2 out of 80 cases), thus suggesting that the size of the transportation network can be set to a relatively small number of horizontal lanes. Even more interestingly, our experimental results show that, at the cost of a few additional horizontal lanes, Heuristic A can be run to produce feasible seamless routes for the case where, due to limited computational resources, solving ILP is unpractical. Moreover, Heuristic A has the advantage of providing simple Manhattan paths that can be run on a network with one-way lanes and leads to a simpler network to design, monitor and maintain, as well as to smoother, safer and simpler routes to operate. The drawback is that the number of levels required by Heuristic A may be very large with respect to the optimal one, as our experiments on ad hoc instances show: however, such instances (with long chains of vehicles in C-conflict paths) seem to be extreme cases and, in fact, they never occurred in random experiments. Moreover, they ge<sup>t</sup> solved by ILP in less than 40 s, even for the larger 233 vehicles instance, with optimal solutions requiring, as for random instances, no more than four horizontal lanes. It follows that an automated grid transportation network can be conveniently designed with a relatively small number of horizontal lanes and operated through ILP or Heuristic A (depending on instance size and available computational resources), leaving to the sensor network and to the run-time collision detection and avoidance system (based, e.g., on more general methods for collision-free routing presented in literature) the rare cases where the proposed methods do not find feasible solutions to FQRP-G.

Our experiments with ILP show that a grid routing network with four horizontal lanes has always been able to accommodate routing paths according to the requirements of FQRP-G. In case an exact solution method (like ILP) is not conveniently available, the proposed heuristic would require at most five horizontal lanes in almost all of the FQRP-G instances. For the residual cases, a grid network with five horizontal lanes may not guarantee nonstop routing on Manhattan paths for all vehicles: in such events, the envisioned routing system can be integrated with state-of-the-art algorithms for multi-agent pathfinding, like the ones presented in the literature review, in order optimize any required space-time deviations from nominal shortest routes.
