# Flight Level Assignment Using Graph Coloring
## 4 Discussion  
We study the problem of behavior optimization in the context of flight level assignment. Apart from being a relevant problem due to the expected growth (both in size and complexity) of the air traffic networks, it is very interesting due to the existence of different individual and social goals: the incentive for fuel consumption minimization by the flight operators, and the safety constraints demanded by regulators and by society as a whole. In this paper, we use a spectrum graph coloring model for behavior optimization, which relies on a factorized optimization model and realistic utility models for the individual and social goals. After modeling the problem, we propose an adaptation of hill-climber and simulated annealing techniques, including some versions which are more greedy than the ones we had used before, and we evaluate their relative performance in a setting representing the Spanish air space. The experiments show that there is an inherent trade-off between fuel consumption and safety, and that the proposed techniques are suitable for finding flight level assignments that take into account both individual and social goals. The results also reveal that the greedy versions of the algorithms are able to reach acceptable solutions in much less time, which enables an additional trade-off between fuel consumption and response time, which can be very significant for real-time flight level assignment. This motivates us to pursue as our immediate future work a real-time operation model, with flight level assignments dynamically depending on the actual position of aircrafts within the routes in real time. In the same line, we will study the performance of message-passing optimization algorithms like belief propagation [33], which have demonstrated a better real-time response. Moreover, the study of message-passing optimization algorithms would be especially suitable if we assume distributed settings where there are a number of agents (for example, airlines) that compete to achieve optimum FLs. We are also interested in introducing risk aversion as a parameter in the utility model, since recent history has shown that different airlines follow very different policies in this regard. Finally, we plan to apply the proposed techniques in other research problems. One of such problems is the management of routes for unmanned aircrafts or drone operations in urban areas [34]. Another important problem that we plan to explore with these techniques is the reduction of pollution in cities by means of ride-sharing [35]. All authors have made relevant contributions to this work. The proposal of the problem was proposed by [PERSON] and developed in numerous meetings involving [PERSON] and [PERSON], who later led different parts of the work. Although all the authors actively participated in all tasks including writing and editing the manuscript, the theoretical part of the paper was leaded by [PERSON] and [PERSON], while [PERSON], [PERSON] and [PERSON] leaded the programming and experimental tasks. All authors have reviewed, read and approved the manuscript. All authors have read and agreed to the published version of the manuscript. [PERSON] and [PERSON] are partially funded by MINECO Project TIN2016-80622-P (AEI/FEDER, UE), by Project PID2019-104855 RB-100/AEI/10.13039/501100011033 of the Spanish Ministry of Science and Innovation, by Project SBPLY/19/180501/000171 of the Junta de Comunidades de Castilla-La Mancha and FEDER and by Project UCeNet (CM/JIN/2019-031) of the Comunidad de Madrid and University of Alcala. [PERSON] and D.O. are partially supported by Project PID2019-104129 GB-100/ AEI/10.13039/501100011033 of the Spanish Ministry of Science and Innovation. [PERSON] is also funded by the predoctoral contract PRE2018-085668 of the Spanish Ministry of Science, Innovation, and Universities. D.O. is also partially supported by H2020-MSCA-RISE project 734922--CONNECT. [PERSON] research visit to Universidad de Alcala was funded by the MIT-Spain Internship Program. The authors declare no conflict of interest.