Mathematical Problems in Engineering

Mathematical Problems in Engineering / 2016 / Article

Research Article | Open Access

Volume 2016 |Article ID 6048326 | 21 pages | https://doi.org/10.1155/2016/6048326

Sectorization and Configuration Transition in Airspace Design

Academic Editor: Babak Shotorban
Received20 Feb 2016
Accepted24 May 2016
Published26 Jun 2016

Abstract

Current airspace is sectorized according to some predefined rules that are not flexible. To facilitate utilizing the airspace more efficiently, methods to design sectors need to be promoted. In this paper, we propose an undirected graph cut-based approach that employs a memetic local search-embedded constrained evolution algorithm, NSGA-II, to generate nondominated airspace configurations. We also propose a new concave hull-based method to automatically depict sector boundaries. In addition, we also study the configuration transition problem. We define the similarity of the two different configurations and calculate their similarity with a bisection diagram and a minimum cost flow algorithm. We build a forward network to represent configuration transitions across several consecutive time periods and use multiobjective dynamic programming to determine a series of nondominated configuration links from the first period to the end. We test our approaches by simulation in high-altitude airspace controlled by Beijing Area Control Center. The results show that our sectorization method outperforms the current configuration in practice, providing a lower sector number, lower intersector flow, more balanced workload distribution among the different sectors, and no constraint violations, so that the proposed approach shows its significant potential as practical applications for dynamic airspace configuration.

1. Introduction and Literature Review

Airspace sectors are basic controlling units in Air Transportation Systems (Figure 1). They were originally designed according to some predefined rules such as historical or geographic considerations or just according to experience. Sectors have essentially remained unchanged in terms of geometric shape and the total number of sectors inside a specific airspace. However, along with rapidly increasing air transportation, fixed sectors cannot accommodate varying traffic flows anymore; several problems have arisen, such as unbalanced workload distribution across different sectors, with overload in some sectors and very sparse flow density in others, and improper sector numbers, which means too many open sectors in off-peak time periods and too few sectors during busy times or too little flight time in a single sector for some flights.

Original ideas to deal with the problem of fixed airspace structure is the “Merge and Divide” operation, meaning combining two or more adjacent sectors together when the traffic flow is low and splitting one sector into several during peak hours or choosing one airspace structure from a predefined experienced structure set [14]. However, this approach is not flexible enough because the boundaries of these sectors remain unchanged across different time periods. A more advanced concept, called Dynamic Airspace Configuration (DAC), was therefore proposed [5]. In DAC, both the boundaries of the sectors and the number of sectors are allowed to change according to varying traffic situations.

One key issue in DAC is the sectorization problem, that is, how to divide an airspace into several sectors. The solution of a sectorization problem is always called the airspace configuration.

To the best of our knowledge, the work by Delahaye et al. [6] may be one of the earliest studies to systematically research the sectorization problem, in which the author utilized a genetic algorithm to generate an optimal airspace configuration. Since then, many approaches have been developed. To summarize, relevant methods can be sorted into three categories [7]:(i)Methods based on geometric computation.(ii)Methods by cells (grids) growth (gathering) or by directly clustering trajectory points.(iii)Methods based on undirected graph cuts.

In the geometric computation category [813], approaches combining Voronoi diagrams with genetic algorithms were proposed in [810]. Tang et al. [11] used several kinds of geometric cuts such as bisection cuts and kd trees to split the airspace and compare different cutting methods.

In the cell-growth category [11, 1420], Brinton directly clustered trajectory points to form sectors [14]. Yousefi and Donohue divided an airspace into three layers with different altitude ranges [16, 17]. Each layer was discretized into hexagonal cells, with information about the controller workload. The hexagonal cells were then gathered into sectors. Based on [16], Drew utilized a boundary-smoothing method to eliminating jagged boundary segments [18]. Klein also divided an airspace into hexagonal cells [19], but, in his approach, sectors grew up from a set of seeding cells.

The third category [2024] is in fact another kind of clustering approach, but it is based on a weighted undirected graph and uses one subgraph to represent a sector. Li et al. constructed a weighted graph model that accurately represents the air-route network [21]. The sectorization problem was then formulated as a graph cut problem and solved by iterative spectral bisection. Martinez et al. proposed a method based on a weighted graph and a grid [22] and also utilized spectral bisection to cut the graph. Chen and Zhang proposed a spectral clustering-based approach to clustering vertices [23]. The spectral clustering solution was further refined by the ODLB algorithm and another heuristic algorithm to get better performance in terms of workload balancing. Trandac et al. proposed a method based on Constraint Programming [24].

In [25], Zelinski gave a comprehensive comparison of different approaches. The results showed completely different sector shapes according to the different approaches. The performance of these approaches was evaluated, which revealed their strengths and weaknesses. To summarize, methods using geometric computation are simple and straightforward, but they are optimally inferior because the methods used to cut the plane or space are limited. Although the second category may be the best in terms of workload balancing, it can hardly handle other objectives or constraints in sectorization problems. Approaches based on undirected graph cuts show great potential in managing multiple objectives and constraints. However, these approaches have to face two critical issues. The first one is the validity of the graph cut method, which must ensure conformance to all constraints of the problem. The second one, which is much more challenging, is how to depict exact sector boundaries based on the generated subgraphs. In this paper, we focus on high-altitude airspace in en route areas. The top and bottom covers of this kind of airspace match completely, so that horizontal and vertical partitioning operations can be executed separately [7]. Because the operations for vertical partitioning are always much simpler than the horizontal ones [7, 10], in our work, we only carry out sectorization in the horizontal plane, and we call the sectorization problem solved here the 2D en route sectorization problem (2D ERSP).

The 2D ERSP is a typical multiobjective optimization problem (MOOP) with several objectives, such as balancing the workload across different sectors and minimizing intersector flow, and several constraints, such as no overload in any sector and no reentry of any single sector for any flight.

The first main contribution of this paper is that we propose an approach that comprehensively handles multiple objectives and ensures no violations against any constraint. This approach belongs to the undirected graph cut category, and it is based on the reasonable airspace model and the Constrained NSGA-II [26] accompanied by memetic local search [27]. In addition, thanks to the evolution procedure, we do not need to predefine desired sector numbers as many previous studies have had to do.

Different from geometric computation-based and cell-growth-based approaches, undirected graph cut-based methods cannot naturally generate sector boundaries after vertex clustering. However, many studies of this type directly neglect the boundary depiction problem, only giving vertex clustering results. Among the very limited results in undirected graph cut methods that seriously consider the boundary depiction problem, Chen and Zhang used Voronoi polygons of vertices to form sector boundaries [23]. Li et al. proposed a method based on a shortest-path algorithm [21], and Trandac et al. used constrained triangulation [24] to form sector boundaries. These two studies both used manual selection. In fact, to the best of our knowledge, except for the methods employing Voronoi polygons, other boundary depiction methods serving undirected graph cut-based approaches generally require manual selection or manual modification.

Hence, the second main contribution of this paper is its proposal of a new concave hull-based approach to automatically depict exact sector boundaries.

In the DAC, although airspace configuration should remain fixed for a particular time period for practical reasons, such as the demand of air traffic controllers [24], it can and should be changed to match varying traffic situations. Generally, the change of airspace configuration should be carried out in conjunction with the change in air traffic controllers [2]. Hence, another important issue in DAC is how to avoid sharp changes in airspace configuration in order to maintain steady air traffic. This problem is called the configuration transition problem (CTP). Unfortunately, like the boundary depiction problem, the CTP is also seldom reported in the existing literatures. To our best knowledge, only the results in [4, 28] presented several rules to minimize the workload caused by the configuration transition.

The third main contribution of this paper is that it represents a serious study of the CTP. We first define the similarity of the two configurations and calculate their similarity with a bisection diagram and a minimum cost flow algorithm [29]. Next, we build a forward network to represent configuration transitions across several consecutive time periods. Then, we take advantage of multiobjective dynamic programming to find a series of nondominated configuration links from the first period to the end.

This paper is organized as follows. In Section 2, we give a comprehensive description of the 2D ERSP, including its objectives and constraints. In Section 3, we discuss how to model the airspace network into an undirected graph and how to integrate traffic information into the model. In Section 4, approaches used to solve the 2D ERSP and to depict sector boundaries are presented. In Section 5, procedures to solve the CTP are stated in detail. The performance of the proposed approaches is tested and analyzed in Section 6. Conclusions and future works are discussed in the last section.

2. Description of the 2D ERSP

We describe the 2D ERSP by three objectives and five constraints that are commonly mentioned in the majority of the literatures working on the sectorization problem.

The three objectives are as follows.

Obj1. Balancing intrasector workload across different sectors.

Obj2. Minimizing aggregate intersector workload (coordination workload).

Obj3. Minimizing aggregate occurrences of the violations of the minimum staying time constraint.

Obj1 and Obj2 are general objectives that are nearly mentioned in all relevant literatures. Nevertheless, Obj1 always acts as the core objective, while Obj2 is sometimes not really considered when solving the sectorization problem in previous studies. Besides, Obj3 is sometimes considered as a constraint in literatures like [20]. However, when it acts as a constraint, it is always treated as “soft constraint”; that is, it may not be strictly satisfied. Therefore, we model it as an objective aiming at reducing its violations.

Along with these three objectives, the 2D ERSR has five constraints as follows.

Con1. Con1 is the connectivity constraint; that is, any sector must be connected, and no sector can consist of two or more separate parts. This is the basic constraint which can never be violated.

Con2. Con2 is the convexity constraint; that is, no flight can enter one single sector more than once. Practically, this is also a strict constraint. However, it cannot be ensured in some cases.

Con3. Con3 is the instantaneous flow constraints; that is, no sector can be overload, and the number of flights in one single sector at any moment cannot exceed a threshold. This constraint is often neglected in some literatures.

Con4. Con4 is the minimum distance between fixes and boundaries (MDFB) constraint; that is, the distance between any fix and any segment of the sector boundary should not be below a minimum threshold. Few literatures have had proposed powerful methods for Con4.

Con5. The sector boundary should be as compact as possible. Jagged boundary is unwanted.

3. Airspace Model

3.1. Initial Undirected Graph Construction

Basic elements in airspace are waypoints, navaids, ordinary intersections, and air routes connecting them. This natural network structure makes it very convenient to represent airspace by a graph.

We model airspace into an undirected graph , where is the set of vertices that a vertex stands for a waypoint, or a navaid, or an ordinary intersection, and is the set of edges. We set , if is connected with by a segment of air route or if the Voronoi polygons of and have a common side.

It is an expansion to existing literatures considering the situation of common side of Voronoi polygons when constructing the graph, in which only air routes are treated as edges of the undirected graph. One of the major reasons of introducing these extra edges is to cover the shortcomings in the connectivity check in the existing graph cut-based methods. In Figure 2(a), we assume a regional airspace consisting of three vertices and two route segments. If we want to cluster and into one sector and split alone into another sector, the connectivity constraint in existing methods would be violated while it is satisfied in the expanded graph (see Figure 2(b)) proposed here, because and are treated as connected to the common side of their Voronoi polygons. This situation is possible in practice, such as a regional hub-spoke structure in which all surrounding fixes are connected to the hub fix with air routes while no routes exist between these surrounding fixes. To relax the burden of the controller in charge of the airspace near the hub fix, one general choice is to divide the airspace into more parts so that some route-unconnected fixes may be included in one sector. This dividing action could lead connectivity violation based on existing type of graph, while it still functions well if our expanded graph is employed.

3.2. Graph Simplification and Modification

We set four rules to simplify and modify the initial undirected graph.

Rule 1. If any fix is only crossed by one air route and there is no difference between the inbound course and the outbound course, directly delete this fix. In this situation, the flow density on both sides of the fix is nearly the same so that it is not necessary to consider these two route segments separately.

Rule 2. If several fixes are less than twice of the MDFB to each other, merge them as a fix union. This rule is to avoid any generated sector boundaries crossing the gap between two fixes that are too near to each other. In other words, it is to satisfy the MDFB constraint. For any fix union, we represent it with a “union-point” at the geometric center of the union and define the radius of the “union-point” as , where is the largest distance from the “union-point” to any member fixes in the fix union. Edges completely inside one fix union are deleted; edges connecting fixes belonging to different unions are represented by edges to connect the corresponding “union-points.” After Rule 2, we set each “union-point” as a vertex of the modified undirected graph.

Rule 3. If any edge is less than the distance of MDFB from vertex that is not one of its endpoints, we split into two segments, and .

Rule 4. If any edge representing Voronoi neighborhood intersects with any other edge representing air route, delete . This action is to avoid generating unnecessary vertices.

3.3. Traffic Information Representation

Many previous studies only employ flow density information, such as the number of flights or the number of trajectory points, to describe the traffic situation. Although the flow density is indeed the most important factor when evaluating the workload of air traffic controllers, it is oversimplified in some extent. In fact, several comprehensive metrics [3032] evaluating airspace complexity or air traffic controllers’ workload have been proposed. These metrics can be identified into four types: flow density, altitude change, speed change, and course change [2, 31, 32]. Accurately calculating each metric is very time-consuming, especially when solving the sectorization problem that needs iterative optimization. Hence, simplification or approximation must be used if we want to consider all these metrics simultaneously. There are examples such as representing several metrics with one integrated metric [2, 8], ignoring several metrics in the optimizing objectives [14], limiting the size of the target airspace and using much simpler airspace model [8], and precalculating the metrics in some basic airspace cells and directly using these values in the optimization process [15]. In the proposed approach, we introduce ten matrices to store these four types of information.

(1) Flow Density Matrix . This matrix stores the number of flights flying through each edge of , where is the number of vertices in . If flights fly through , . Besides, if and , we still set , where is a small positive value, to remain the connectivity of and .

(2) Symmetric Altitude Change Matrix . stores the information about altitude changes. Suppose a set of trajectory points of flight that are projected on , denoted as . We calculate the standard variance of the altitudes of , denoted as . Define the set of flights flying through as , and we havewhere is the count of trajectory points in . Other elements in with or , and are all set to 0. As is an undirected graph, is symmetric.

(3) Asymmetric Altitude Change Matrix . also stores the information about altitude changes. We divide trajectory points that are projected on edge into two parts, and . The trajectory points nearer to than are put into , while the others are put into . Then, we calculate the standard variances of these two sets, denoted as and . Hence,

(4) Symmetric Speed Change Matrix . Similar to those notations in , we define and thus

Other elements in with or and are all set to 0.

(5) Asymmetric Speed Change Matrix . Similar to those notations in ,

(6) Symmetric Trajectory Point Count Matrix . stores the count of trajectory points that are projected on each edge:

(7) Asymmetric Trajectory Point Count Matrix . Consider

(8) Flight Course Change Matrix . is the number of flights considered in the sectorization problem and is the number of vertices in . Recall the rules in Section 3.2. In fact, any vertex in represents a fix union. Hence, for any flight-vertex pair, such as and , we set as the aggregate course change times when flight flies through .

(9) Vertex Arrival Time Matrix . The element is defined as the earliest arriving time of flight at any fix that belongs to .

(10) Vertex Leaving Time Matrix . The element is defined as the latest leaving time of flight from any fix that belongs to .

4. Solve the 2D ERSP

4.1. Mathematical Expression

As stated before, the airspace is modeled as an undirected graph, . Solving the 2D ERSP is to cut into subgraphs, , where ,

Each subgraph stands for one sector.

4.1.1. Objectives

(1) Balancing the Intrasector Workload. Consider is the intraworkload of sector and is the mean value of , . is a weighted and scaled summation of the values of four metrics of sector , including information about the flow density, the altitude change, the speed change, and the course change: where , , , and are weights of these four parts.

We define as the set of intrasector edges of sector , where and are the set of intersector edges between and other sectors, where ; thenwhere is the total course change times of flights when flying inside sector and is the total number of flights flying through . Hence, stands for average course change times of single flight in sector . Suppose the flight set considered in the sectorization problem as , and we have

Because of different ranges of the four parts in (9), and a more meaningful understanding of the summation, we normalize them to the range of . We set four standard values, , , , and , that are large enough and do not change during the optimization. Then, we define

(2) Minimizing the Aggregate Coordination Workload. We use the density of intersector flow to stand for the coordination workload. This objective can be expressed bywhere is the intersector flow density between and . It is the number of flights flying across and :

(3) Minimizing the Occurrences of the Violations of the Minimum Staying Time Constraint. If any flight stays in the sector less than a minimum time threshold after it enters a sector, the minimum staying time constraint is violated. Obj3 is to minimize the occurrences of these situations.

As we do not depict exact sector boundaries during the optimization process, it is impossible to accurately calculate the staying time of any flight. We approximate the accurate staying time as follows.

Still suppose the flight path of flight as , , where is the th vertex along the path. Then, we calculate the staying time of flight in sector as follows: where and are the first and the last vertices inside sector along . The first term of (15) stands for the time spent to reach after entering the sector. The second term stands for the time spent to leave the sector after leaving . The third term stands for the time spent in . The fourth term stands for other times spent inside sector .

We define the set of all considered flights as and set , if any flight violates the minimum staying time constraint in sector . Hence, Obj3 can be expressed as

4.1.2. Constraints

(1) Connectivity Constraint. Each subgraph representing one sector must be self-connected.

(2) Flight-Route-Based Convexity Constraint. For any flight , , . If , , for , , .

(3) Instantaneous Flow Constraint. Similar to the expressions above, we suppose and as the first and last vertex inside sector along . Hence, the entry time of sector can be denoted as , which is the midtime of the time leaving and the time reaching . Similarly, we denote the departure time of sector as . With the entry time and the departure time for every flight, we can calculate the number of flights in any sector at any moment. Suppose the maximum instantaneous flight number in sector as and the instantaneous flow threshold as , and Con3 can be expressed as

Con4 and Con5 are not considered in the optimization phase. But they are ensured by the airspace modeling in Section 3 and the boundary depiction method in Section 4.3.

4.2. Constrained NSGA-II Based Evolution

Some objectives and constraints of the 2D ERSP cannot be written in explicit mathematical formulas [2, 24]. For example, consider the constraint that no flights can enter any single sector more than once. It is affected by the trajectory of flights and the boundary of a sector, which is too complex to express this constraint by an explicit inequality. Hence, it is impossible to construct the 2D ERSP into a typical form of mathematical programming below:

As evolution algorithms can still work even if the constraints can only be expressed by text descriptions, they are good choices to solve the 2D ERSP. The Constrained Nondominated Sorting Genetic Algorithm II (CNSGA-II) [26] is a fast and elitist multiobjective genetic algorithm that is very appropriate for the 2D ERSP, considering its characteristics of multiobjectives and the strong capability in handling nonclassic constraints. We utilize the CNSGA-II accompanied with the memetic local search [27] to solve the 2D ERSP. The main flow diagram of the evolution algorithm is shown in Figure 3.

4.2.1. Chromosome

Every chromosome has genes representing vertices in graph . The value of each gene is the index of the sector to which the related vertex belongs. For example, in Figure 4, the third vertex is assigned to sector 2 and the 50th vertex is assigned to sector 5.

We set a common largest possible index of sectors for every individual; that is, the possible maximum number of sectors is fixed. On one hand, this number is large enough so that it has no impact on the results. On the other hand, because we only set the possible maximum number, indexes of sectors in different individuals are totally independent, which means that two sectors with the same index in different individuals may refer to different sectors.

4.2.2. Reproduction

Tournament selection is used to select individuals in parent generation.

4.2.3. Crossover

Two individuals can crossover only when they have large enough hamming distance. In single crossover operation, two randomly chosen genes are exchanged. To avoid generating individuals that violates connectivity constraint as much as possible, the crossover can be granted only when at least one of the vertices (genes) is moved to an adjacent sector or stays in original sector after the crossover.

4.2.4. Mutation

In single mutation operation, one gene is randomly selected. The value of the gene is randomly changed and it is bounded by the value of the maximum sector number. However, similar to the crossover operation, only when the related vertex is changed to an adjacent sector, the mutation operation can be granted.

4.2.5. Memetic Local Search

The memetic algorithm [27] is a local search method embedded in one global search iteration of evolution algorithms that can help improving the individuals in one generation. In this paper, two heuristic polices of memetic local search are applied.

Policy A. For any sector, if , where is an upper bound, we split the sector into two or more parts. For the subgraph corresponding to sector , we build its similarity matrix , where is the number of vertices that belong to . For any element , , , suppose the indices of its two related vertices in graph as and , and then . We add up all elements of each row of to build a diagonal matrix , where . We define the Laplacian matrix of , . Besides, we introduce three lemmas in the Spectral Graph Theory [33].

Lemma 1. For a weighed undirected graph with nonnegative edge weights, its Laplacian matrix has at least one eigenvalue with value of 0.

Lemma 2. The number of isolated subgraphs of a weighed undirected graph with nonnegative edge weights equals the number of 0 eigenvalues of its Laplacian matrix.

Lemma 3. For a weighed undirected graph with nonnegative edge weights, denoted as , suppose the number of 0 eigenvalues of its Laplacian matrix as . Then, the dimension of the eigenspace of 0 eigenvalues is . This linear subspace is spanned by such a group of base vectors: , where is the indicator vector of th isolated subgraph, denoted as . If a vertex is in , the corresponding element in is 1; otherwise, it is 0.

With these three lemmas, we calculate the eigenvalues of and the corresponding eigenvectors. If the multiplicity of 0 eigenvalues, , is larger than 1, we just divide into parts according to the base vectors. If , we get the eigenvector corresponding to the second smallest eigenvalue, called Fiedler Vector [34]. Then, the vertices of are clustered into two groups. One is with the positive elements of Fiedler Vector and the other is with the negative ones. Vertices corresponding to the elements with value of 0 are randomly assigned to one of these two groups. Hence, when , we split into two parts.

Policy B. Policy B is operated after the division by Policy A. For any individual, if for sector , where is a lower bound, we treat the flow density in this sector as too low. Thus, we merge into one of its adjacent sectors, assumed as . Values of genes (vertices) belonging to are changed to the index of .

To avoid loss of diversity and being trapped in regional optima, the memetic local search is operated with gradually changing intensity and frequency [35]. The frequency refers to the gap of generations between two callings of the local search, and the intensity refers to the proportion of individuals in one single generation that are applied with the local search. At the beginning of the evolution, the qualities of the individuals are still low, and, in general, we set low frequency and intensity. Along with the evolution, the searching space approaches the optimal solutions, and thus we need more accurate search. Therefore, the frequency and intensity of the local search both increase.

4.2.6. Fitness Functions

Fitness functions correspond to the expressions in formulas (8), (13), and (16).

4.2.7. Constraint Check

Connectivity. Similar to those stated in “Policy A” of the memetic local search, for each subgraph (sector), we calculate its Laplacian matrix and check the multiplicity of 0 eigenvalues. If the number is larger than 1, we record one more time of violation of the connectivity constraint for current individual.

Flight-Route-Based-Convexity. Still suppose the flight path of flight as . If and , we check if , . Once the condition , is met, we record one more time of the violation of the convexity constraint for current individual.

Instantaneous Flow Constraint. We check the validation of inequality (17). If inequality (17) is unsatisfied, we record one more time of the violation of the instantaneous flow constraint for current individual.

4.2.8. Sort and Elitism

In the CNSGA-II, the elitism operation is executed for every generation. The dominating relationship between two individuals (solutions), and , is defined below:(1), if is feasible (no constraint violation) while is not.(2)When solutions and are both feasible, if the values of all fitness functions of are not more than that of and at least one of such values is smaller than that of .(3)When solutions and are both infeasible, we compare their times of constraint violations in the following order of decreasing priority: connectivity > convexity > instantaneous flow. That is, , if the connectivity constraint violating times of solution is less than that of solution . Or, if the connectivity constraint violating times are equal, we continue to compare the violating times of the convexity constraint and so on.(4)When solution and are both infeasible and they are equivalent in terms of the violating times of all three constraints, if the values of all fitness functions of are not more than that of and at least one such value is smaller than that of .

4.2.9. Initial Generation

Most previous studies using evolutionary algorithms always concern about how to refine the “genetic operations” (such as the crossover or mutation rules) or try to propose heuristics to improve the evolution procedure. However, for those problems that the gross optima or a boundary of the gross optima cannot be preliminarily determined (such as the 2D ERSP), no matter how excellent a heuristic or an improved operation seems to be, we are unable to determine whether the gross optima can be found with the help of these heuristics. In these problems, we cannot guarantee reaching the gross optima from any type of initial generations. Hence, we have to realize that different initial generations have different impact on final solutions. However, discussions about the initial generation of evolution algorithms and its impact to the final solutions are absent in relevant literatures.

In this paper, we propose four types of initial generations and compare them with the corresponding nondominated solution sets produced by the CNSGA-II.

The first type is the most general choice that just randomly produces the initial generation. In following text, we call this case Randomly Initialization, or RI.

The second type takes advantage of the current configuration in practice. We represent the current configuration with a chromosome and randomly mutate half of the genes several times to produce the initial generation. In following text, we call this case Current Initialization, or CI.

Thirdly, we employ a method based on the normalized spectral clustering algorithm (NSCA) [36]. The NSCA is a data clustering algorithm that shows its talent in the graph -cut problem if each data point is represented by one vertex in a weighted graph and the weights of edges stand for similarities between two data points [37]. We choose the NSCA because of two objectives of the 2D ERSP, balancing intrasector workload and minimizing gross intersector workload, which are very similar to the goals of the undirected graph -cut problem. We first change the values of elements in matrix with value of to 0 and call the modified density matrix . Then, is defined as the similarity matrix of ; that is, and we construct a diagonal matrix , where . Following steps of the method based on the NSCA are listed in Procedure 1. Note that may equal to 0 if no aircraft flies through any routes connecting to vertex . This may lead singularity of and cause an error in the 4th step in Procedure 1. Hence, we must prune and firstly, deleting rows and columns where . The method in Procedure 1 outputs one configuration of the airspace. We also use a chromosome to represent this solution and produce the initial generation by randomly mutating half genes of this individual several times. In this method, we need to set desired cluster number for the NSCA. However, this number is not the final counts of sectors because of possible change in the following evolution procedure. Hence, we directly set it as the count of sectors in current configuration. In following text, we call this case NSCA Initialization, or NI.

()   Input , and desired cluster number, .
()   Prune and by deleting rows and columns where . Define the pruned matrices
     as and . Define the set of deleted vertices as where and the set of
     remained vertices as . Define the dimension of and as .
()   Calculate the Laplacian matrix, .
()   Compute the normalized Laplacian matrix, .
()   Compute the eigenvector corresponding to the smallest eigenvalues of .
()   Construct a matrix , each of its column corresponds to the eigenvector of one of the
      smallest eigenvalue.
()   Normalize every row of , meaning normalizing into ,
     , where and stand for the th row of and respectively,
     is the element at the th row and th column of .
()   For each row of , set a corresponding vector   .
()   Cluster vectors into clusters using k-means cluster. The index of the
      cluster for each row is the index of the sector for the th vertex of .
() For any vertex in , assign it to the sector to which its nearest vertex in belongs.

In the fourth type, half individuals in the initial generation are produced by CI and the other half is produced by NI. In following text, we call this case Mix Initialization, or MI.

4.3. Sector Boundary Depiction

As stated in Section 1, the boundary depiction is much more challenging in graph cut-based approaches than that in geometric based and “cell gathering” based approaches. The methods directly using Voronoi polygons of vertices to form the boundaries are simple and straightforward. But they may not be very suitable in the sectorization problem for two reasons. Firstly, the shape of sectors is not strictly convex so that Voronoi polygons may not fit the outline of concave vertex cluster very well for its characteristic of convexity. Secondly, the MDFB constraint cannot be ensured by Voronoi polygons.

We propose a concave hull-based method to depict the sector boundaries in this section. We firstly construct the concave hull for each vertex cluster (representing one sector) in Procedure 2 with necessary explanations below.

() Initialize, input , , , and
() For
()   Set
()   If is adjacent to the outer-boundary of the target airspace
()    Obtain (See explain in Section 4.3.1), and set
()    For each point
()    Find point , that , and is nearest to
()    
()    End
()  Obtain (See explain in Section 4.3.1)
()  
() End
() Else
()  
() End
() Construct using -shape algorithm (See explain in Section 4.3.1)
() End
4.3.1. Concave Hull Generation

Input Data. Input information of the concave hull generation method is in three categories.

The first is the cut of :

The second is the set of midpoints of edges connecting two neighboring sectors, . For any sector , define the set of midpoints of edges connecting it and other sectors as . Hence, we get that and if and are neighboring sectors.

The third is a set of points located on the outerboundary of the target airspace. We suppose points on the outerboundary of the target airspace are distributed clockwise for the purpose of unification and define the point set as , , , and if is not next to the outerboundary.

Obtain . For any sector (subgraph ) adjacent to the outerboundary, we find out one or more couples of points in , and denote the subset of as . For any point with corresponding edge , the following conditions must be satisfied:(i)Both endpoints of , denoted as and , are next to the outerboundary.(ii)The Voronoi polygons of and have one common side and this side intersects with the outerboundary.

Note that, as the shape of any sector is not strictly convex, there may be more than one couple of points in . In Figure 5(a), we assume that and belong to sector and and belong to the other two different sectors, respectively. In this situation, is made up of and . In Figure 5(b), we assume that sector consists of , , and , which surround . The shape of is concave, but still conforms to the convexity constraint because no air routes connect or . In this situation, , , , and are all elements of . with more than two couples of points is also possible which is not illustrated here. Note that , if and are neighboring sectors.

Obtain . For any sector adjacent to the outerboundary, sort clockwise (as points in all belong to , sorting clockwise means sorting with increasing index in ) as , where is the number of couples . Then, we define We are aware that two neighboring points in may not be consecutive in . We definewhere is the index of in and is the index of in .

Construct Concave Hull of . We use the concave hull of , denoted as , to fit the outline of the vertex cluster of sector . -shape algorithm [38] is applied here to generate the concave hull. The basic idea of -shape algorithm is to iteratively search -extreme points and -neighbors and to connect the found -neighbors sequentially to generate -hull. As large amount of extra definitions and notations must be introduced to formulate -shape algorithm clearly, we do not present the details here.

4.3.2. Concave Hull Amendment

is a promising candidate to be the boundary of sector because all vertices of are inside it and it fits the shape of very well. However, two more considerations make it necessary to recheck . The first is the compactness of the boundary; that is, fewer segments are preferred on the boundary. The second is about the MDFB constraint; that is, the distance between some segments of and some vertices of may be less than the MDFB. Hence, we propose the Concave Hull Amendment Method (CHAM) to deal with these two considerations (Procedure 3).

() For
()   Get path(s) on , set as (See Definition 4 in Section 4.3.2)
()   For each path ,
()    Define vertices along as and corresponding
      (note that when corresponding cross-sector edge has not been designed)
()    Set , .
()    While
()     If cross-sector edges referred by and are both designed
()      If line segment is infeasible (See Definition 5 in Section 4.3.2)
()        Execute the OABD to (See Explain in Section 4.3.2)
()    End
()    , go to line ()
()   End
()   While line segment is feasible (See Definition 5 in Section 4.3.2)
()    
()    If cross-sector edges referred by and are both designed
()     break
()    End
()   End
()   If
()    Execute the OABD to (See Explain in Section 4.3.2).
()    Set the cross-sector edge referred by as designed, remain
()    , go to line ()
()   End
()   Else
()    
()    Draw line segment ,
()    Set cross-sector edges referred by as designed
()    Set as the intersections of related cross-sector edges
      and line segment
()    , reset , go to line ()
()   End
()  End
() End
() End

There are two ideas in the CHAM. The first is to replace the polygonal line connecting several vertices on the concave hull with one line segment from the first vertex to the last one. The second idea is to avoid obstacle circles centering at the vertices of with radius of that vertex (refer to Section 3.1 for the definition of the radius of a vertex). We call modified concave hulls after the CHAM boundary polygons and present two definitions about the concave hull and the boundary polygon first.

Definition 4 (path(s) on the concave hull). One scans the vertices of clockwise. If is adjacent to the outerboundary of the target airspace, whenever we encounter a sequence of vertices , where , or others belonging to , we define such a sequence as a path on the concave hull. If is not adjacent to the outerboundary, that is, it is surrounded by other sectors, we simply define a path on as a clockwise closed walk of all vertices of .

Definition 5. A line segment connecting any two vertices on the boundary polygon of one sector is defined as feasible if it does not intersect with any inner-edges of any sector and it does not go through any obstacle circles centering at the vertices of .

Next, we present another definition to avoid redepiction of common boundary between two neighboring sectors.

Definition 6. In the CHAM, once a segment on the boundary polygon is depicted, set all cross-sector edges that intersect with this segment as designed. Define the intersection of this segment and a cross-sector edge, , as the boundary crossing point of , denoted as . When one cross-sector edge has not been designed, its default boundary crossing point is set as its midpoint, that is, the intersection of this edge and the concave hull. To ensure unification, when one endpoint of a depicted segment lies on the outerboundary of the target airspace, we also define a virtual cross-sector edge corresponding to this endpoint as designed and set the boundary crossing point as the endpoint itself.

We also give two properties about the concave hull.

Property 1. For any sector , line segments on do not intersect with any inner-edges of any sector. This property is obvious from the definition and the generating process of -shape [38].

Property 2. The distance between any vertex on the concave hull, representing the midpoint of a cross-sector edge, and any vertex of is at least MDFB.

Proof. For any vertex on the concave hull, if the corresponding cross-sector edge represents a real air-route segment, the edge is at least MDFB away from any vertex of according to Rule 3 in Section 3.2 except its two endpoints. In addition, as all edges of are at least twice as long as MDFB, their midpoints are at least MDFB away from the two endpoints. For one vertex on the concave hull that is the midpoint of a sector-cross edge only standing for Voronoi neighboring relationship, it locates on the Voronoi polygons of both endpoints of this edge, which means that the nearest vertices of to this midpoint are just the two endpoints of the edge. Considering the possible distance between this midpoint and the two endpoints is at least MDFB, if any other vertex of is less than MDFB from this midpoint, a contradictory occurs.

These two properties show two pieces of important information when we construct the boundary polygons:(1)For any two consecutive vertices on the concave hull of a sector, if we can find a line segment or a polygonal line that does not go through any obstacle circles, this line segment or polygonal line can be set as boundary segment(s) of that sector.(2)The boundary polygons after the CHAM can cross any vertices on the concave hulls.

We show the procedures of the CHAM in Table 1 with necessary explanations below.


Time periodSectorization methodObj1Obj2Obj3Con1Con2Con3 NosDensity gapRT (s)

12:00–14:00Current in practice0.8145674238010164218
Only NSCA1.0518471133061166027
RI0.31584048410092521934
CI0.31243605900092232940
NI0.34043755800082356942
MI0.43293866800082512964

16:00–18:00Current in practice1.1466707234042166705
Only NSCA1.623042886003166732
RI0.3148525106100102409926
CI0.4508608130000103007937
NI0.4259499104000102521924
MI0.450251191000102619940

20:00–22:00Current in practice1.3666710247032167198
Only NSCA2.071240372015169150
RI0.4514684206200122375936
CI0.3487719218000121902934
NI0.4142592134000122258930
MI0.3993598136000122017915

2:00–24:00Current in practice1.3850427166012167604
Only NSCA1.623042886123169058
RI0.4318468205200122635944
CI0.410033579000122239937
NI0.3678350105000122159924
MI0.2982351118000122073930

Obstacle Avoidance Boundary Design (OABD). We illustrate the OABD with Figure 6. Firstly, we assume two consecutive vertices on the concave hull as and . is assumed to intrude one or more obstacle circles (Figure 6(a)). The OABD examines the first obstacle (such as in Figure 6) intruded by . Then, we draw two tangents from and to the obstacle circle, respectively. The intersection of these two tangents is denoted as . Thus, the boundary between and is now (Figure 6(b)). Next, we examine if still intrudes any other obstacle (such as in Figure 6). If it does, we truncate at the boundary of the second obstacle ( in Figure 6(c)). Then, we draw two tangents of originating from and , respectively, and get their intersection ( in Figure 6(c)). The boundary between and is now changed to . The OABD continues until the curve (1 iteration) or ( iterations, ) is free of all obstacles (Figure 6(c)). If too many iterations are needed in the OABD, the generated sector boundaries may be jagged, which is prohibited. Throughout our simulation, such situations do not appear. Relevant discussions will be in Section 6.5.

In the OABD, there are always two intersections when we draw tangents, such as and , in Figure 7. As and always locate symmetrically on opposite sides of the obstacle circle, one of the generated polygonal lines definitely intersects with inner-edges of sectors while the other does not. The one without intersections with inner-edges is our choice (such as in Figure 7).

4.3.3. Vacuum Space Handle

After the CHAM, there still remains several vacuum spaces at the intersecting areas of three or more sectors, such as quadrilateral ABCD in Figure 8(a). These vacuum spaces contain no vertices or edges and are not assigned to any sector. In this section, we propose a heuristic method to assign vacuum spaces. Firstly, we represent a vacuum space with a polygon, calling the vacuum polygon. The vacuum polygons can be sorted into four categories according to types of their sides:(I)Triangles with no side from the OABD.(II)Polygons with more than three sides and no side that is from the OABD.(III)Quadrangles with two sides from the OABD.(IV)Other types.

For vacuum polygons in Category (I), we propose two choices, one is directly erasing the opposite side of the biggest inner angle (Figure 9(a)), and the other is to find the “most vertical” midline and erase those two sides on which the midpoint does not locate (Figure 9(b)).

For vacuum polygons in Category (II), we first give the following property.

Property 3. The vacuum polygon in Category (II) is convex.

Proof. Suppose there is a concave vacuum polygon in Category (II) (such as the one in Figure 8(b)). We consider the vertex causing the concavity (point in Figure 8(b)). As this vertex is on a cross-sector edge, one of the endpoints of this cross-sector edge locates inside the vacuum polygon because of the concavity, obviously causing a contradictory against the definition of the vacuum space. In another situation, it locates outside the polygon while the cross-sector edge must intersect with one extra side of the polygon. However, neither of the two endpoints of this extra side is the vertex that causes the concavity; that is, a cross-sector edge between two sectors intersects with the boundary of an irrelevant sector. It is also impossible. Hence, we show that the vacuum polygons in Category (II) must be convex.

With Property 3, we can calculate the geometric center of a polygon in Category (II), which must locate inside the polygon because of its convexity (point in Figure 8(b)). Then, we connect the center to every vertices of the polygon and erase original sides of the polygon (Figure 8(c)).

For polygons in Category (III), which stand for an intersecting area of three sectors, we find the vertex produced by the OABD (point in Figure 9(c)), draw the diagonal from it, and erase those two opposite sides of the vertex produced by the OABD.

For polygons in Category (IV), situation of their sides may be very complex. We directly assign the entire vacuum space to one sector, whose boundary segments are sides of the polygon after OABD.

5. Configuration Transition

The methods presented from Sections 24 are for the sectorization problem in single time period. This section focuses on solving the CTP. First of all, we present the definition of the similarity of two different configurations followed by its calculation method.

5.1. Similarity of Two Configurations

We still use graph-based notations to denote configurations. We still model the whole target airspace as , and define and as two different configurations:

We construct a bisection diagram (Figure 10(a)), where nodes on each side stand for the sectors in each configuration. The weight of an edge connecting two nodes on opposite sides is the similarity of two sectors, which is defined as the number of same vertices of these two sectors:

To calculate the similarity of and , we need to get the maximum match of the bisection diagram, which ensures a monoprojection for every node on the side with less nodes while the aggregate weights of selected edges are maximized. Without loss of generality, we suppose (it is similar when ), the similarity of and is defined as:where .

This is a typical 0-1 programming problem, which can be solved with network flow algorithms. We modify the bisection diagram (Figure 10(b)), adding one virtual source node to the side with less nodes and connect the source node with each node on the same side. Besides, we also add a virtual sink node to the side with more nodes, connecting the sink node with each node on the same side. The weights on all newly added edges are 0 so that they have no impact on the result. The weights on original edges are also changed. For an edge connecting and , its weight is changed to from , where is a constant. So the objective function is changed to the form of . Because is known, original maximization problem in (24) is now transformed to a minimization problem. The supply of the source node and the demand of the sink node are both equal to and the supply/demand at other nodes are all 0. Finally, the capacities on all edges are set as 1. Based on these modifications, we build a typical network that applies to any generic minimum cost flow algorithms [29]. We do not list the procedures of the minimum cost flow algorithms in detail. Readers can easily find them in [29] or any other textbooks on network flow or Discrete Optimization.

5.2. Nondominated Configuration Links

If we want to make a configuration plan for next several time periods, we should consider the quality of selected configurations in terms of the values in Obj1 to Obj3 and the similarities between configurations of consecutive time periods. Hence, finding an optimal configuration plan is another multiobjective problem.

We build a forward network to represent the CTP (Figure 10(c)). Each column of nodes represents one of several nondominated configurations in one period. A node in the network is denoted as , , where is the number of time periods and , where is the number of nondominated configurations in period . Each edge in the network stands for a possible transition from the previous period to the next period. Edges standing for transitions with too low similarity are truncated.

We define two kinds of parameters in the network. The first one is the weight of edge , denoted as , set as the inverse of the similarity of configuration and . The second one is the values of Obj1 to Obj3 of node , denoted as .

Definition 7. One defines a route in the forward network from one node to any node in the last period as a configuration link. A link without its starting node is called a path.

For node , one denotes the cost along a path from it to any node in the last period as , where is the sum of similarities along the path and the other three items correspond to the sum of values of three objective functions of nodes along the path. Hence, one can define the dominating relationship between two paths.

Definition 8. Assuming two paths form the same time period to the last period as and , , if and , .

From Definition 8, we can define a nondominated set of paths from node , denoted as . Then, we assume another path from one node in previous period, , which goes across as , and the part after along is . Then, we can get the following relationships: