A real-time schedule method for Aircraft Landing Scheduling problem based on Cellular Automation
Introduction
The Aircraft Landing Scheduling (ALS) problem, which requires determining the landing time for a given set of aircrafts [1], is an important problem for air traffic control. When entering the radar range of an air traffic controller, the aircraft's flight number, altitude and speed are transmitted to controllers within the air traffic control tower. Accordingly, controllers will give the aircrafts instructions regarding the approaching corridor to use and the required speed and altitude of them. The ultimate goal of these instructions is to arrange the safe and effective landings of a continuous flow of aircrafts onto the assigned runway(s). As the first step, it is necessary to schedule the landing time for each aircraft in the flow to land, subject to the constraints raised by the control and aerodynamic considerations such as the earliest and latest arrival time for each aircraft, and the minimum separation times between successive aircrafts. The schedule must be made in a very short time since the aircrafts cannot “wait in the sky” for long time for the controller to work out the best schedule. In practice, it is typically required that a schedule should be decided in one radar period of the air traffic controller, which merely takes a few seconds. Within such a short and strict time period, the ALS problem with only a few aircrafts is already challenging. Unfortunately, the ALS problem is a typical NP-hard problem [2], and it will become more and more difficult as the number of aircrafts increases. On one hand, a larger number of aircrafts makes it more difficult to schedule the landing plans. On the other hand, the more the aircrafts, the more urgent it is to determine the schedule so that the aircrafts can landing as soon as possible. Hence, there is an urgent need to develop an effective real-time approach to provide useful support to controllers on the ALS problems.
In practice, the most widely used strategy for the ALS problem is the First-Come-First-Served (FCFS) rule that schedules the aircrafts in the same order as they enter the radar range. Obviously, this simple rule, though very efficient, can seldom lead to good landing schedule. Hence, different scientific communities have conducted intensive studies on the ALS problem in the past decades. For example, advanced algorithms have been proposed following both the linear programming [1] and meta-heuristics (such as genetic algorithm (GA) [3] and ant colony optimization (ACO) [4]) routines. However, these studies mainly focus on finding the OPTIMAL solution (i.e., the landing schedule), rather than to attain a solution that is as good as possible within a limited time budget. Since the ALS problem is in nature NP-hard, it is unsurprising that these advanced algorithms do not promise to meet the real-time requirement even for moderate number of aircraft. Although seeking the global optimal solution is definitely an important issue, it might not be appropriate if our aim is to develop an approach that can be really used for real-world problems. It was this consideration that motivated the work presented in this paper.
The ALS problem can be viewed as involving two types of sub-problems. Firstly, one needs to determine the order of landing of the aircrafts (namely the landing sequence). Then, the exact landing time needs be decided for each aircraft based on the landing sequence. Ideally, these two sub-problems should be simultaneously solved by direct optimization of the landing time for all the aircrafts, because the optimal landing times must correspond to the optimal landing sequence. However, such a direct optimization routine is very costly and may be computationally prohibitive in practice. Fortunately, given a landing sequence, the optimal landing times corresponding to it can be obtained efficiently by some existing methods [7]. Hence, the main challenge stems from seeking a good or even the optimal landing sequence, which is a combinatorial optimization problem.
In this paper, we propose an algorithm called Cellular-Automata-based Optimization (CAO) algorithm for the ALS problem. Although the major novelty of the CAO algorithm lies in a novel method for the search of the optimal landing sequence, the CAO algorithm addresses the ALS problem as a whole and its major steps is summarized as below:
- (1)
A one-dimensional Cellular Automation (CA) model is adopted to efficiently obtain a good landing sequence. The exact landing time is not considered in this phase.
- (2)
Apply a simple stochastic local search to the landing sequence obtained in (1) to further improve it. During this phase, each landing sequence is evaluated based on the optimal landing times corresponding to it. The optimal landing sequence as well as its optimal landing times are thereby obtained simultaneously.
The rest of this paper is organized as follows. Section 2 presents the formal definition of the ALS problem, and briefly reviews related work. Section 3 describes our CAO algorithm in detail. Experimental study is presented in Section 4 to evaluate the efficacy of the CAO algorithm. Finally, conclusion and discussion are presented in Section 5.
Section snippets
Problem definition
Upon entering the radar range (radar horizon) of air traffic controller (ATC) at an airport, an aircraft requires ATC to assign it a landing time. The landing time should be determined based on the following considerations:
First, an aircraft must land in a given time window, bounded by the earliest time (E) and the latest time (L). The earliest time can be obtained if the aircraft flies at its maximum speed, and the latest time is determined by the limited fuel capacity of the aircraft (i.e.,
CAO algorithm
Previous works [1], [7], [12], [14] have shown that the aircraft landing optimization process can be divided into 2 steps: searching for a good permutation and optimizing the landing time based on the permutation. Following this framework, here we propose a Cellular-Automaton-based Optimization (CAO) to address the ALS problem. The flow chart of CAO is shown in Fig. 1. As it can be seen, our method consists of two parts: (i) a CA-based model is devised to simulate the landing process to find a
Benchmark and computing platform
To validate the efficacy of our approach, experiments on benchmark (OR-Library) problems are carried out. The benchmark we choose is the OR-Library which can be obtained at http://people.brunel.ac.uk/∼mastjjb/jeb/jeb.html. It was first proposed by Beasley [1], [12]. In the literature, this benchmark has been adopted by most related work [1], [4], [7], [12], [13], [14]. The experiments were programmed with C++ and run on a PC with 1.6 GHz CPU.
Parameter selection
As discussed in Section 3, three key parameters, α, β
Discussions
CAO is fast due to the CA part can get a good starting point for the further optimization parts. Compared to the methods used before, CAO is superior in terms of both the quality of the solutions and the speed of computation. However, one major drawback of the CAO is that the CA model is essentially a greedy search method, and the CAO relies on the heuristic search process to search for the global optimal solution. This may work well on small-size problem. However, even with a good landing
Conclusion
In this paper, we investigated the single runway Aircraft Landing Scheduling Problem. A novel algorithm called CAO was proposed. The CAO consists of three main components, a CA model, a local search, and a deterministic operator called RO. It starts from using the CA model to generate a good initial landing sequence of aircrafts, then the local search operator is applied to the initial sequence to further improve it, the RO operator is adopted to determine the optimal landing times based on a
Acknowledgments
This paper is supported by the National Basic Research Program of China (Grant No.2011CB707000) and the Foundation for Innovative Research Groups of the National Natural Science Foundation of China (Grant No. 60921001).
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