Mathematical Problems in Engineering

Volume 2014, Article ID 236756, 8 pages

http://dx.doi.org/10.1155/2014/236756

## An Efficient Approximation Algorithm for Aircraft Arrival Sequencing and Scheduling Problem

^{1}School of Economics and Management, Tongji University, Shanghai 200092, China^{2}Foshan Shuyuan Science and Technology Company Limited, Foshan, Guangdong 528200, China

Received 23 June 2014; Accepted 20 August 2014; Published 31 December 2014

Academic Editor: Chunlin Chen

Copyright © 2014 Weimin Ma et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

The aircraft arrival sequencing and scheduling (ASS) problem is a salient problem in airports' runway scheduling system, which proves to be nondeterministic polynomial (NP) hard. This paper formulates the ASS in the form of a constrained permutation problem and designs a new approximation algorithm to solve it. Then the numerical study is conducted, which validates that this new algorithm has much better performance than ant colony (AC) algorithm and CPLEX, especially when the aircraft types are not too many. In the end, some conclusions are summarized.

#### 1. Introduction

With the rapid development of airline industry, serious congestions and frequent delays have been hitting most major airports in the world, especially in the United States and Europe [1]. How to enhance the air traffic capacity and reduce the delay becomes a severe problem [2, 3].

In 1998, runway has been identified as the primary bottleneck in air traffic [4]; that is, even small enhancements to runway throughput will significantly reduce the delay. However, building more runways is often considered not a realistic option because of practical constraints and huge investment costs. Therefore, many researches and technologists resort to a promising approach, which is to more optimally schedule the aircraft arrival sequence so that the runway can land as many aircraft as possible within a period of time. The optimization process is formulated in this paper as the aircraft arrival sequencing and scheduling (ASS) problem (see in Section 2).

However, ASS is inherently hard to solve [5]; it is nondeterministic polynomial (NP) hard [6, 7]. To cope with it, two methods are often adopted, which are mixed integer programming (MIP) and ant colony (AC) algorithm [5, 8–10].

ASS can be expressed by MIP formulations. In 1992, Brinton [11] has introduced, as far as we know, the first MIP formulations and designed an implicit enumeration (IE) algorithm to optimize it. In 1993, another MIP is presented by Abela et al. [12] for single-runway ASS. A branch and bound (B&B) algorithm is developed to solve it. Back in 1999, the MIP is presented not only in single but also in multiple runway [8]. Beasley et al. [5] give an improved B&B algorithm by employing linear programming (LP) based tree search. Then Bennell et al. [13] provide an extensive literature overview for ASS.

Ant colony (AC) algorithm [9, 10, 14] is another effective method for ASS. It is originally proposed by Dorigo in 1992 [15–17]. In 2002, Randall [10] presents its first application in ASS, which shows great advantages. Then AC is used to generate initial solutions and to incorporate local search heuristic for single- and multiple-runway ASS [9, 18]. Back in 2010, AC is developed to tackle the real-time ASS based on the receding horizon control by Zhan et al. [14]. Experimental results validate that AC is robust, effective, and efficient for ASS.

In this paper, rather than using the above two methods, we develop a new approximation algorithm for ASS. The core idea is to find the lower bound solution of the ASS problem. Then the algorithm presents solutions infinitely approaching this bound. We compare the performance of this new algorithm with AC and MIP (by CPLEX). Computational results verify that this new algorithm returns much better solution and costs less time than AC and MIP, especially when there are several aircraft types.

This paper is organized as follows. In Section 2, some constraints for ASS are introduced. The approximation algorithm is proposed in Section 3. In Section 4, ant colony (AC) and MIP are designed for ASS. In Section 5, numerical study is conducted to compare the performance of this new algorithm with AC and MIP (by CPLEX), while some conclusion is summarized in Section 6.

#### 2. Basic Concepts

##### 2.1. Aircraft Sequencing and Scheduling (ASS) Problem

ASS aims to make the most use of runway, that is, to minimize the makespan of the landing sequence so that to land as many aircraft as possible within a period of time. The objective function is where is the landing time of the last aircraft in sequence . For the th aircraft in sequence , its landing time is achieved by where and insure the time-window constraint and MST insures the minimum separation time constraint. These two constraints, as well as some other constraints, are described below. If is not satisfied, equals .

###### 2.1.1. Minimum Separation Time (MST) Constraints

MST is a hard constraint to insure safety. When an aircraft flies in the air, it generates wake-vortex (WV). However, WV may result in the instability of the following aircraft (to shake or to lift) [19]. To avoid this, a MST is strictly kept between them.

Table 1 illustrates a typical MST table concerning three main types of aircraft. Generally, a smaller aircraft followed by a larger aircraft requires much shorter MST than the other way around. For example, a small aircraft has to wait for 196 s after the landing of a heavy aircraft. However, when a heavy aircraft lands after a small aircraft, the MST is only 60 s. One reason is that larger aircraft commonly generates and tolerates more turbulent air, while smaller aircraft generates and tolerates less.