# A Method to Optimize Routing Paths for City-Pair Airlines on Three-Layer Air Transport Networks
## 1 Introduction  
With the development of the economy, the air transportation system is becoming more important than ever before due to its intrinsic value, high speed, and high security. Without a doubt, a better understanding of the air transportation system is important for policy, administration, and efficiency reasons. Unfortunately, the air transportation system is a huge, complex system with emergence and self-organization, whose behaviours are hard to predict. Therefore, it is necessary to model the air transportation system.  
In the last twenty years, two human-made infrastructures, airline networks and air route networks, have been used to model air transportation system. Furthermore, complex network theory [1] has been used to study the air transportation system extensively, from topology analysis to functional analysis. However, these two infrastructures do not deeply consider the natural relation between traffic and transportation.  
According to a report from the International Civil Aviation Organization (ICAO), the number of city-pair airlines in the world has increased by approximately 30% from 2005 to 2014 [2]. Meanwhile, the increasing air traffic flows could finally exceed the capacity limit of the current air traffic management (ATM) system, resulting in the demand-capacity imbalances problem [3] and the air-traffic congestion problem. Additionally, the air-traffic congestion problem will dramatically increase the load of air-traffic control managers and impede the safety and the performance of flights.  
Generally, there are two ways to deal with the congestion problem of air transportation systems [4, 5]. The first is the construction of infrastructure, such as building new airportsand runways to increase the system's capacity--a long-term strategy. However, doing so would take a long time and would have a high cost due to the need to build new airports and runways. The second is to regulate and reshape the flying traffic flow using Traffic Management Initiatives (TMIs) [6]. TMIs can be classified into two categories: (1) strategic actions, which are the strategies taken before the aircraft has taken off; (2) tactical actions, which are the strategies taken after the aircraft is airborne. In this paper, we will focus on the strategic actions to alleviate the congestion problem.  
When one aircraft flies in the sky, it follows its flight plan route though airways, much like highways on the ground. Generally, there are at least two pre-planned routes for most airlines. However, in practice, air traffic managers usually take a default routing path for each city-pair airline (that is, a pair of airports) without considering the traffic flows on the air route network, resulting in the congestion of some air route segments. Therefore, it is necessary to select the optimization routing path on the air route network for each city-pair airline, in strategy stage, from the candidate pre-planned routing paths with minimization of the congestion of air route network.  
In this study, to understand the nature of traffic and transportation deeply, three-layer networks were built to model the air transportation system from physical facilities to traffic applications. In addition, to reduce the congestion of air route segments in the air route network, a simulated-annealing-based method is proposed to select optimized routing paths for each city-pair airline based on the three-layer air transport network.