Integrated Optimization of Aircraft Surface Operation and De-Icing Resources at Multi De-Icing Zones Airport

Since the surface operation of hub airports was facing severe congestion and aircraft delay under the de-icing mode, some practical problems of improving aircraft operation and support efficiency are put forward. A two-phase model is constructed to coordinate the scheduling of aircraft surface operation (ASO) and de-icing support resources operation (DSRO). An optimized method is used to schedule the aircraft surface operation. On the basis of the scheduling results of surface operation, an aircraft and de-icing resource collaborative scheduling (ADCS) mechanism is established to optimize the assignment of aircraft and de-icing resources. The algorithm combining receding horizon control strategy and CPLEX solver (RHC-CPLEX) is designed to solve the model. Computational experiments performed on case studies of Beijing DaXing airport show some potential improvements: Firstly, for the ASO model, the RHC-CPLEX algorithm can reduce the objective function value by more than 20% compared with the FCFS algorithm. And the results shows that not only the delay distributions under different snow conditions are reasonable, but the spatial distributions of the de-icing zones of aircraft are closer to the location of their apron and allocated runway. Secondly, for the DSRO model, the RHC-CPLEX algorithm can reduce the objective function value by more than 6% compared with the algorithm based on the principle of proximity and availability. The de-icing vehicles are used efficiently and the number of refilling de-icing fluid and the free time of the de-icing vehicles can be significantly reduced.

SECTION I. Introduction

Special weather near the airport is one of reasons for aircraft delay. Especially under snow conditions, aircraft on the surface has to de-icing before departure. Caused by this reason, various coordinated scheduling of aircraft support resources not only increases the complexity of surface traffic situation but the workload of Air Traffic Controllers (ATC), which always lead to severe congestion and aircraft delay in hub airports. In order to reduce aircraft delay under the de-icing mode, the traditional methods mainly include increasing the de-icing pads and de-icing vehicles [1]. But it is a waste when increasing all kinds of resource blindly. That is to say, it is necessary to provide proper operation rules, optimized models and reasonable algorithms to support the solution of this problem. In recent years, scholars have gradually discovered large-scale aircraft delay warning, the emergency-response mechanism based on ATC [2] and tactical aircraft plan adjustments based on airport [3], which are significantly effective and improved. However, from the perspective of aircraft support and passenger service, the aforementioned measures are just treated as passive and temporary mitigation measures. It is basically still needy for ATC combining with airport to continuously upgrade their abilities of supporting aircraft surface operation under the snow condition. Furtherly, these support units should complete the aircraft surface operation scheduling method oriented by aircraft operation needs, build a kind of aircraft support resource complement and optimization mechanism. By this way, passengers’ needs for convenient, efficient and comfortable travel will also be satisfied.

At present, airports in China are generally equipped with Airport Collaborative Decision Making (A-CDM) systems, which provide decision support for airport operations by sharing important data of airport stakeholders. Under the de-icing mode, aircraft need to send request to A-CDM system before departure, which includes target departure time information and de-icing request information. Then the ATC department obtains the relevant information the departure aircraft and assigns the off-block time, arranges the de-icing pads and runway usage time. The departure process comparison between the de-icing aircraft and the non-deicing aircraft is shown in Figure 1. The current A-CDM system has basically developed the Departure Management (DMAN) module without deicing operation. The DMAN module can share the Target Off-Block Time (TOBT) and Target Takeoff Time (TTOT) of the aircraft, which can optimize the aircraft departure process scheduling to help ATC make decisions [4]. Because the airport DMAN module has not covered the optimal scheduling function of the departure aircraft surface operation under the de-icing mode, ATC still needs to schedule off-block time and assign de-icing pad for departure aircraft according to controlling experience. If a large number of aircraft need to de-icing during the departure peak hours, this loss of function will cause aircraft queuing up and traffic congestion near the de-icing area, which will lead to subsequent departure delay.

FIGURE 1.

Comparison of aircraft departure process.

The aircraft ground support begins when aircraft landing and finishes when aircraft taking off again. There are many processes during aircraft ground support, mainly including passenger baggage loading and unloading, aircraft cleaning and refueling, toilet cleaning, passenger catering, etc. When snow conditions occur, the de-icing support needs to be additionally considered during the aircraft ground support process. Figure. 2 is a Gantt chart of the turnaround activity of an aircraft. The left half indicates whether it is related to de-icing, and the right half indicates whether it is related to ground support resources.

FIGURE 2.

Gantt chart of aircraft turnaround activities.

It can be seen from the Figure. 2 that de-icing activities are critical parts of ground support and need to be reasonably connected and coordinated with other activities of the aircraft, which reflects an important reason for studying de-icing resource scheduling: aircraft de-icing increases time and complexity of surface operation. Compared with other ground support activities, de-icing must be carried out within a certain period of time before taking off, which means it cannot be performed too early to ensure the deicing liquid is effective or too late to takeoff on time. In other words, scheduling of aircraft de-icing time is restricted by the off-block time and takeoff time at the same while. Although the A-CDM system provides some operational decision support for the scheduling of aircraft turn around by fully sharing the data of various stakeholders during the airport operation, which can alleviate the untimely scheduling of ground support resources and congestion. The A-CDM system has not yet had the function of automatic coordinated scheduling of relevant aircraft de-icing support resources when snow condition occurs. Most of the aircraft support resources are manually allocated by the airport operation controllers, which is very unstable. Therefore, it is of special significance to study the aircraft de-icing, which involves the coordination of scheduling between aircraft operation resource and support resource. Specific contributions and findings are as follows.

1. This paper focuses on coordinated scheduling of aircraft surface operation resources and de-icing support resources for airports with multi de-icing zones under de-icing mode. With given reference aircraft, we can give the aircraft operation resource usage results and de-icing resource usage results within an acceptable time frame after optimized scheduling.

2. This paper comprehensively analyzes the impact of snow conditions on the aircraft surface operation and support resource operation. Two optimization models were built to optimize the scheduling of aircraft surface operation resources and aircraft de-icing support resources respectively. A two-phase optimization framework combining aircraft surface operation and de-icing resource operation is proposed to integrated optimize the airport surface operation, by establishing a ADCS mechanism to take the output of ASO model as the input of DSRO model.

3. The efficient RHC-CPLEX algorithm was designed to solve the model. De-icing parameters for different snow conditions are used to conduct numerical experiments, and the results of optimal scheduling for multiple types of aircraft at multiple de-icing pads airports demonstrate the effectiveness of the optimization framework.

The remainder of this paper is organized as follows. Section II reviews the current scholars’ research results on aircraft surface operation and aircraft support resource scheduling operation under de-icing mode, and points out the shortcomings of existing research. Section III builds an optimal scheduling model for aircraft surface operation under de-icing mode. Section IV builds a coordinated scheduling mechanism for aircraft support resources based on the distinctive features of the aircraft support process. An optimal scheduling model for de-icing support resources is also built in this section. Section V explains the calculation process and critical steps of algorithm combining receding horizon control strategy and CPLEX solver. Section VI uses the actual operation data of Beijing DaXing Airport for numerical experiments, and deeply interprets the optimization scheduling results. Section VII discusses the conclusions.

SECTION II. Literature Review

In this section, we review the current research results of aircraft surface operation scheduling under the de-icing mode, and arrange the latest research results related to the scheduling of aircraft de-icing support resources.

The performance of aircraft surface operation scheduling in airport systems, especially in major airports with multiple runways and de-icing zones, is crucial to the efficiency of arrivals and departures. The surface operation scheduling of departing aircraft is of great significance in relieving airport surface congestion and reducing aircraft delays. Most recent studies of the departing aircraft scheduling problem have been conducted from an integrated optimization perspective, such as MIP models [5] to capture the important time such as off-block time as well as runway usage time [6]. Airport surfaces can also be considered as complex networks [7], and the off-block times and departure sequences of departing aircraft can be scheduled by reconstructing topological networks with given Origin-Destination (O-D) pairs in the ATFM pre-tactical phase [4]. Simulation techniques can also be used to perform departing aircraft scheduling with the cellular automata model [8], and the corresponding surface operations simulation software, which can simulate aircraft operations under different strategies could provide a reference for airport operations decisions [9]. Stochastic dynamic programming framework can be used for departure aircraft scheduling under uncertainty [10], while some departure metering procedures are developed to optimize aircraft operations [11]. A bilevel programming approach was also used to optimally schedule departing aircraft taxiing paths while determining the aircraft departure sequence [2]. The complexity nature of aircraft surface operation under the de-icing mode necessitates the extensive attention from scholars that have conducted some preliminary research on this problem. The related studies mainly focus on considering aircraft and de-icing pad assignments based on departure aircraft scheduling, and some scholars have applied multi-agent theory to model the aircraft ground operation scheduling process under de-icing mode. They proposed a multi-agent model consisting of airlines, airport ground service companies and airport controllers, established a non-cooperative game model for aircraft de-icing ground operations [12], analyzed aircraft delays in the de-icing process, and optimized the existing “First Come First Served” de-icing scheduling strategy [13]. Some scholars also conducted research on the adaptivity of multi-agent systems [14]. Multi-service-desk queuing theory can also be used to model multiple de-icing pads queues [15] and to analyze the de-icing scheduling and planning problem in combination with the game theory approach. A short-process-first scheduling method was proposed for the characteristics of the process of centralized aircraft surface de-icing [16], and the short-process-first de-icing scheduling method outperformed existing “First Come First Served” de-icing scheduling strategy in terms of the total aircraft de-icing delay time and the number of aircraft delays index. The stochastic dynamic programming algorithm was also used for aircraft surface operation scheduling under de-icing mode [17], which was used to complete the de-icing pads assignment of aircraft based on a completed data-driven airport de-icing operation optimization model built with the objective of minimizing the impact of airport de-icing operations on the environment and satisfying the de-icing pads capacity constraint [18]. Scholars also studied the formation mechanism of aircraft surface de-icing queuing delays, proposed Markov description of de-icing aircraft arrival rate and polynomial description of average de-icing service rate, established a de-icing queuing model based on the proposed extinction process, and analyzed de-icing aircraft delay parameters under different snow conditions at the airport with multiple de-icing pads [19]. Some scholars consider the de-icing pads as the key node of airport operation, and add the de-icing pads node to the mixed integer planning model of surface operation for departing aircrafts under regular weather [20], which considers the key time nodes of surface operation of departing aircraft, and considers the safety separation of de-icing pad usage as well as that of runway usage of aircraft to establish a mixed integer program model, which can control the delay time of de-icing of aircraft within a certain time range.

Scholars have mostly conducted research on the optimal scheduling of de-icing vehicles for the scheduling of aircraft support resources under the de-icing mode. Some works have been carried out to improve the efficiency of de-icing vehicle operations [21] and to reduce aircraft delays caused by untimely dispatch of de-icing vehicles [22]. Most commonly, the de-icing vehicle scheduling problem can be regarded as vehicle routing problem with time windows, and on the basis of airport data sharing, the minimum weighted sum of aircraft delays due to untimely de-icing vehicle scheduling and the total route length of de-icing vehicles is taken as the optimization objective to establish the de-icing vehicle scheduling model under the de-icing mode [23] to comprehensively improve the efficiency of aircraft support during the aircraft turnaround activities. Some scholars have also taken the perspective of path planning to optimize the operational routes related to the scheduling of aircraft support vehicles in terms of overall optimization of airport operational efficiency [24]. A simulation model that considers other constraints such as de-icing time window and aims at minimum aircraft delay time and de-icing waiting time was proposed to carry out simulation experiment [25]. Some scholars proposed an airport de-icing vehicle assignment model based on equipment group operating capacity coefficient and operating area priority, an improved Hungarian algorithm was proposed to solve the model [26]. An attribute decision optimization was proposed at the same year based on the complex multiple constraints of the de-icing operation process [27]. At the same time, some scholars also optimize the scheduling of de-icing vehicles and other support resources with the goal of reducing the impact of de-icing fluid on the environment [28], and develop a corresponding scheduling system [29] to ensure the smooth operation of aircraft de-icing process and improve the efficiency of aircraft surface de-icing.

The above-mentioned literature has carried out some preliminary studies on the scheduling of aircraft operations under ice and snow conditions and the scheduling of aircraft support resources under de-icing mode, and has achieved preliminary research results. However, it should be noted that the current research has the following shortcomings:

1. Most of the above studies of airport surface operations focus on the optimization process of aircraft operation resources and the optimization process of aircraft support resources, respectively, but in fact the two processes are clearly coupled and interdependent. The scheduling of aircraft support resources is strongly related to the scheduling of aircraft operation resources, and the scheduling of aircraft support resources needs to be based on the optimal scheduling of aircraft operation resources to achieve a more scientific, reasonable and feasible implementation effect.

2. They exclusively focus on scheduling aircraft and support resource at the airport with single de-icing zone, few studies focus on optimal scheduling of multiple types of aircraft and support resources under different snow conditions at busy airports with multiple de-icing zones. It is difficult to obtain the optimal aircraft operation scheduling results and aircraft support resource scheduling results in a timely and efficient manner when snow conditions occur near the airport.

3. The deicing time and usage of de-icing resource of different types of aircraft under different snow conditions are key parameters in the scheduling process. Few works have compared the scheduling results under different snow conditions and different deicing parameters.

From the literature review, we conclude that there is a lack of collaborative scheduling mechanism of airport surface operation resources and aircraft support resources, and the analyze of the optimal scheduling results of multiple types of flights under different snow conditions at the airport with multiple de-icing zones. Therefore, the research work of this paper has great theoretical research value and engineering practice significance.

SECTION III. Problem Description and Aircraft Surface Operation Model

A. Problem Description

Airport controllers need to assign de-icing zones and corresponding de-icing pads to each departing aircraft before that takeoff in winter snow conditions. Figure. 3 shows the airport structure of Beijing DaXing Airport. As shown in Figure 3, DaXing Airport has three de-icing zones, namely De-icing Zone No.1 on the left side of the entrance Runway 11L, De-icing Zone No.2 on the right side of the entrance of Runway 35R and De-icing Zone No.3 on the left side of the entrance of Runway 17L. De-icing pads are the spaces for de-icing operations. There are multiple opening de-icing pads in one de-icing zone, of which De-icing Zone No.1 has 6 opening de-icing pads, De-icing Zone No.2 has 4 de-icing opening pads and De-icing Zone No.3 has 2 opening de-icing pads.

FIGURE 3.

Airport structure of Beijing DaXing Airport.

In order to improve the efficiency of surface operations and reduce aircraft delays, we assign a de-icing pad to each departing aircraft and determine when the de-icing pad should be used. In this Section, we propose an optimal mixed integer programming (MIP) model for aircraft surface operations (ASO) under de-icing mode, which takes into account the use of runway resources by arrival aircraft compared to the traditional scheduling model for surface operations under de-icing mode [20], and we consider the aircraft surface operation as the process of assigning the aircraft to a specified route [30]. Figure. 4 gives an overview of the total operations of aircraft on airport surface.

FIGURE 4.

Arrival-Departure operations in airport. The aircraft goes through several phases: landing on the runway and taxiing to the gate, turnaround, off-block, taxiing between the gate and the de-icing pad, de-icing, taxiing between the de-icing pad and runway and take-off.

B. Aircraft Surface Operation Model

1) Input Data

${\mathcal{ N}}$ is the set of runways available for aircraft, where ${\mathcal{ N}}=\{n\left |{ {n_{1},} }\right.n_{2},n_{3},\cdots \}$ . ${\mathcal{ M}}$ is the set of de-icing pads available for aircraft, where ${\mathcal{ M}}=\{m\left |{ {m_{1},} }\right.m_{2},m_{3},\cdots \}$ . ${\mathcal{ F}}$ is the set of aircraft that requiring scheduled during the planning time period, where ${\mathcal{ F}}={\mathcal{ A}}\cup {\mathcal{ D}}\cup {\mathcal{ A}}{\mathcal{ D}}$ . ${\mathcal{ A}}$ is the set of arrival aircraft that land at the airport and stay until planning time period. ${\mathcal{ A}}{\mathcal{ D}}$ is the set of arrival-departing aircraft that arrive at the airport and depart from it after a turnaround duration, i.e., the aircraft have continuous consecutive voyage at this airport. ${\mathcal{ D}}$ is the set of departing aircraft that are parked at the airport at the beginning of the planning time period and depart later. In order to simplify the descriptions, we divide the set ${\mathcal{ A}}{\mathcal{ D}}$ into two sets, where ${\mathcal{ A}}^{\ast }$ denotes the set of arrival aircraft in the ${\mathcal{ A}}{\mathcal{ D}}$ set and ${\mathcal{ D}}^{\ast }$ denotes the set of aircraft in the ${\mathcal{ A}}{\mathcal{ D}}$ set, ${\mathcal{ A}}{\mathcal{ D}}={\mathcal{ A}}^{\ast }\cup {\mathcal{ D}}^{\ast }$ . ${\mathcal{ R}}$ is the set of route of the scheduling problem, where ${\mathcal{ R}}={\mathcal{ R}}_{A} \cup {\mathcal{ R}}_{D},{\mathcal{ R}}_{A}$ is the set of surface operation routes for arrival aircraft, ${\mathcal{ R}}_{D}$ is the set of surface operation routes for departing aircraft.

The following data is given: wake turbulence category for $f\in {\mathcal{ F}}$ , the assigned parking position for $f\in {\mathcal{ F}}$ , maximum acceptable delay time $Q_{f}^{\textrm {DL}}$ for $f\in {\mathcal{ F}}$ , the historical departure taxi time for $f\in {\mathcal{ D}}\cup {\mathcal{ D}}^{\ast }$ including the taxi time $Q_{fm}^{\textrm {PD}}$ from the parking position to the de-icing pad $m$ and the taxiing time $Q_{mn}^{\textrm {DR}}$ from the de-icing pad $m$ to the runway $n$ , the maximum surface operation time $Q^{\textrm {MA}}$ for $f\in {\mathcal{ D}}\cup {\mathcal{ D}}^{\ast }$ after the aircraft is de-iced and before takeoff, the de-icing pad occupancy time $Q_{f}^{ED}$ which is the sum of the de-icing preparation time $Q^{\textrm {MO}}$ and the de-icing time $Q_{f}^{\textrm {DI}}$ for $f\in {\mathcal{ D}}\cup {\mathcal{ D}}^{\ast }$ . To simplify the model, we assumed the taxi time from a defined parking position to a defined de-icing position and the taxi time from a defined de-icing position to a defined runway as a constant value.

For each route $r\in {\mathcal{ R}}_{A}$ , the runway information of the arrival aircraft is included in the route $r$ . If the path $r$ contains runway $n$ , then $\sigma _{r}^{n} =1$ , otherwise $\sigma _{r}^{n} =0$ , and only one runway is included in the route $r$ , where $\sum \limits _{n\in {\mathcal{ N}}} {\sigma _{r}^{n} =1}$ . For each route $r\in {\mathcal{ R}}_{D}$ , the runway and de-icing pad information is included in the route $r$ . If the route $r$ contains runway $n$ , then $\sigma _{r}^{n} =1$ , otherwise $\sigma _{r}^{n} =0$ , and only one runway included in the route $r$ , where $\sum \limits _{n\in {\mathcal{ N}}} {\sigma _{r}^{n} =1}$ . If the route $r$ contains de-icing pad $m$ , then $\sigma _{r}^{m} =1$ , otherwise $\sigma _{r}^{m} =0$ , and only one de-icing pad is included in the route $r$ , where $\sum \limits _{m\in {\mathcal{ M}}} {\sigma _{r}^{m} =1}$ .

Further, we obtain the following parameters. The earliest runway usage time $T_{f}^{\text {E}}$ for $f\in {\mathcal{ A}}\cup {\mathcal{ A}}^{\ast }$ , which can be generally considered as the target landing time $T_{f}^{\text {TL}}$ . The latest runway usage time $T_{f}^{\text {L}}$ is equal to the sum of the earliest landing time $T_{f}^{\text {E}}$ and the maximum acceptable delay time $Q_{f}^{\textrm {DL}}$ for $f\in {\mathcal{ A}}\cup {\mathcal{ A}}^{\ast }$ . We can get earliest runway usage time $T_{f}^{\text {E}}$ for $f\in {\mathcal{ D}}\cup {\mathcal{ D}}^{\ast }$ , where $T_{f}^{\text {E}} =T_{f}^{\text {TB}} +Q_{f}^{\text {PD}} +Q_{f}^{\text {ED}} +Q_{f}^{\text {DR}}$ . The latest runway usage time $T_{f}^{\text {L}}$ is equal to the sum of the earliest landing time $T_{f}^{\text {E}}$ and the maximum acceptable delay time $Q_{f}^{\textrm {DL}}$ for $f\in {\mathcal{ D}}\cup {\mathcal{ D}}^{\ast }$ .

2) Decision Variables

We propose six types of decision variables for the characteristics of the aircraft surface operation problem under de-icing mode. The process of using the resources of surface operation is considered as an assignment between aircraft and scheduling routes as work of Guépet’s [30]. The sets, parameters and variables of the ASO model are shown in Table 1.

TABLE 1 Sets, Parameters and Variables of ASO Model

3) Constraints

The constraints of the ASO model including de-icing resource and airport surface operation resource usage constraints, aircraft operation time window constraints, and aircraft safety separation constraints. We define the de-icing resource and airport surface operation resource usage constraints as follows:

$$\begin{align*} \sum \limits _{r\in {\mathcal{ R}}_{D}} x_{f}^{r}& =1,\quad \forall f\in {\mathcal{ D}}\cup {\mathcal{ D}}^{\ast } \tag{1}\\ \sum \limits _{r\in {\mathcal{ R}}_{A}} x_{f}^{r} &=1,\quad \forall f\in {\mathcal{ A}}\cup {\mathcal{ A}}^{\ast } \tag{2}\\ a_{fg} +a_{gf} &=1,\quad \forall f,g\in {\mathcal{ F}},f\ne g \tag{3}\\ b_{fg} +b_{gf} &=1,\quad \forall f,g\in {\mathcal{ D}}\cup {\mathcal{ D}}^{\ast },f\ne g \tag{4}\end{align*}$$

Equation (1) constraints each departing aircraft has one surface operation route. Equation (2) constraints that each arrival aircraft has one surface operation route. Equation (3) constraints that there is only one sequence of runway usage between two aircraft. Equation (4) constraints that there is only one sequence of de-icing pad occupancy between two aircraft.

Let each aircraft be subject to time windows that are defined as follows.

$$\begin{align*}&T_{f}^{\text {L}}\ge t_{f}^{\text {R}} \ge T_{f}^{\text {E}},\quad \forall f\in {\mathcal{ F}} \tag{5}\\ & t_{f}^{\text {OB}} \ge T_{f}^{\text {TB}},\quad \forall f\in {\mathcal{ D}}\cup {\mathcal{ D}}^{\ast } \tag{6}\\ & t_{f}^{\text {R}} -t_{f}^{\text {MS}} -Q_{f}^{\text {ED}} -B(3-x_{f}^{r} -\sigma _{r}^{n} -\sigma _{r}^{m})-Q_{mn}^{\text {DR}} \le Q_{f}^{\text {MA}}, \\ &\quad \forall f\in {\mathcal{ D}}\cup {\mathcal{ D}}^{\ast },m\in {\mathcal{ M}},n\in {\mathcal{ N}},r\in {\mathcal{ R}}_{D} \tag{7}\\ & t_{f}^{\text {MS}} \ge t_{f}^{\text {OB}} +Q_{fm}^{\text {PD}} -B(2-x_{f}^{r} -\sigma _{r}^{m}) \\ &\quad \forall f\in {\mathcal{ D}}\cup {\mathcal{ D}}^{\ast },m\in {\mathcal{ M}},r\in {\mathcal{ R}}_{D} \tag{8}\\ & t_{f}^{R} \ge t_{f}^{\text {MS}} +Q_{mn}^{\text {DR}} +Q_{f}^{\text {ED}} -B(3-x_{f}^{r} -\sigma _{r}^{n} -\sigma _{r}^{m}) \\ &\quad \forall f\in {\mathcal{ D}}\cup {\mathcal{ D}}^{\ast },m\in {\mathcal{ M}},n\in {\mathcal{ N}},r\in {\mathcal{ R}}_{D} \tag{9}\end{align*}$$

Equation (5) constraints that the runway usage time of the aircraft is restricted to the runway usage time window. Equation (6) constraints that the actual off-block time of departing aircraft is restricted to be no earlier than the target off-block time. Equation (7) constraints that the operation time duration of departing aircraft after de-icing to takeoff should be no larger than $Q_{f}^{\text {MA}}$ to prevent deicing fluid failure. Equation (8) constraints that the time when the departing aircraft start occupying the de-icing pad is restricted to be no earlier than the sum of its $t_{f}^{\text {OB}}$ and its $Q_{fm}^{\text {PD}}$ . Equation (9) constraints that the takeoff time of departing aircraft is restricted to be no earlier than the sum of its $t_{f}^{\text {MS}}$ , the de-icing pad occupancy time $Q_{f}^{\text {ED}}$ and taxi time $Q_{mn}^{\text {DR}}$ .

The aircraft operating separation constraints are defined as follows.

$$\begin{align*}t_{g}^{\textrm {MS}} &\ge t_{f}^{\textrm {MS}} +Q_{f}^{\textrm {ED}} -\textrm {B}(5-x_{f}^{r} -x_{f}^{r'} -\sigma _{r}^{m} -\sigma _{r'}^{m} -b_{fg}), \\ &\qquad \forall f,g\in {\mathcal{ D}}\cup {\mathcal{ D}}^{\ast },f\ne g,\forall r,r'\in {\mathcal{ R}}_{D},m\in {\mathcal{ M}} \tag{10}\\ t_{g}^{\text {R}} &\ge t_{f}^{\text {R}} +L_{fg} -\text {B}(5-x_{f}^{r} -x_{g}^{r'} -\sigma _{r}^{n} -\sigma _{r'}^{n} -a_{fg}), \\ &\qquad \forall f,g\in {\mathcal{ F}},f\ne g,\forall r,r'\in {\mathcal{ R}},n\in {\mathcal{ N}} \tag{11}\end{align*}$$

Equation (10) constraints that the time separation between entering and leaving the de-icing pad should meet the relevant requirements if two of departing aircraft are assigned to the same de-icing pad. Equation (11) constraints the runway usage time separation between two aircraft should meet the relevant requirements if two of them are assigned to use the same runway.

4) Objectives

The objective function of the ASO model is shown in Equation (12). The objective function is divided into two parts, the first part indicates the total weighted delay time of the aircraft, and the second part indicates the total weighted operation time of the departing aircraft. We can assign different values to the weights according to the importance of the two parts respectively, and minimize the weighted sum of the two parts in the optimization process, which can effectively improve the efficiency of the surface operation. The airport surface operation is more complicated under de-icing mode, but it is also within the range of airport surface operation scheduling, and the weighted sum of the two as the objective function can also achieve the optimization effect under de-icing mode. It has been proven that optimizing both objectives simultaneously can improve passenger satisfaction, save operational costs, and reduce congestion in airport surface operations [31].

$$\begin{equation*} \min \Phi \tag{12}\end{equation*}$$ where $\Phi =\alpha \sum \limits _{f\in {\mathcal{ F}}} {S_{f}} +(1-\alpha)\sum \limits _{f\in {\mathcal{ D}}\cup {\mathcal{ D}}^{\ast }} {E_{f}},S_{f}$ and $E_{f}$ are the delay time cost and the surface operation time cost of the aircraft, respectively. For each aircraft $f\in {\mathcal{ A}}\cup {\mathcal{ A}}^{\ast },S_{f}$ is the weighted value of the difference between the actual runway usage time and the target runway usage time, $S_{f} =\mu _{f} \sum \limits _{f\in {\mathcal{ A}}\cup {\mathcal{ A}}^{\ast }} {(t_{f}^{\text {R}} -T_{f}^{\text {TL}})}$ . For each aircraft $f\in {\mathcal{ D}}\cup {\mathcal{ D}}^{\ast },S_{f}$ is the weighted value of the difference between the actual runway usage time and the earliest runway usage time, $S_{f} =\mu _{f} \sum \limits _{f\in {\mathcal{ D}}\cup {\mathcal{ D}}^{\ast }} {(t_{f}^{\text {R}} -T_{f}^{\text {E}})}$ . $\mu _{f}$ is the delay cost factor of aircraft $f$ , where $\mu _{f} =G/P_{f}$ . $P_{f}$ is the priority factor of aircraft $f$ . $G$ is the maximum value in the priority table, the calculation methods of $P_{f}$ and $G$ can be found in the literature [32]. For each aircraft $f\in {\mathcal{ D}}\cup {\mathcal{ D}}^{\ast },E_{f}$ is the difference between the actual runway usage time and the actual off-block time, which is the total surface operation time, where $E_{f} =\sum \limits _{f\in {\mathcal{ D}}\cup {\mathcal{ D}}^{\ast }} {(t_{f}^{R} -t_{f}^{\text {OB}})}$ .

SECTION IV. Aircraft and De-Icing Resource Collaborative Scheduling Mechanism and De-Icing Support Resource Operation Model

Deicing resource assignment is the classical optimal scheduling problem with many tasks and few resources [23]. In this paper, we only consider the optimal scheduling of key de-icing resources, i.e., the de-icing vehicles. The de-icing demand of the aircraft is the tasks and the de-icing vehicles are the available resources. We optimally schedule the de-icing vehicles assignment process on the basis of the correlation between the de-icing support resources and the aircraft.

A. Aircraft and De-Icing Resource Collaborative Scheduling Mechanism

In this section, we propose an aircraft and de-icing resource collaborative scheduling (ADCS) mechanism and analyze its intension. For a de-icing vehicle, the operation process at an airport surface can be described as follows: a de-icing vehicle with full de-icing fluid leaves the de-icing vehicles’ parking position and proceeds to the de-icing pad where the de-icing task is assigned. After completing the de-icing task, it waits in place or proceeds to the de-icing pad where the next de-icing task is assigned or returns to refill the de-icing fluid. The de-icing process of a de-icing vehicles cannot be interrupted. When a de-icing task is completed, if the remaining de-icing fluid of the vehicle is not enough to complete the next task, it cannot go to execute the next task, the vehicle should return to the parking position of de-icing vehicles to refill the de-icing fluid.

As shown in Figure 3, the parking position of the de-icing vehicles at Beijing DaXing Airport is adjacent to the De-icing Zones No. 1 and No. 2. Airport operation controllers often use the Principle of Proximity and Availability (PPA) to assign de-icing vehicles [25]. Although this method is simple and efficient, there are often situations where aircraft wait for vehicles and the scheduling of vehicles is unreasonable, resulting in inefficient use of vehicles and poor utilization of de-icing fluid.

For each departing aircraft, the result from ASO model can give the start occupancy time of the de-icing pad as well as the departure time of the aircraft. The collaborative scheduling of de-icing resources can be understood as maximizing the efficiency of de-icing resources usage on the basis of the optimal scheduling results of ASO model. We divide the integrated optimal scheduling process of the airport surface operation under the de-icing mode into two phases. The aircraft de-icing task set is inputted to optimize the assignment process of de-icing vehicles. The de-icing vehicles are assigned to complete the de-icing task of departing aircraft to achieve the goal of improving the overall airport surface operation efficiency. The scheduling mechanism of aircraft de-icing resources is shown in Figure. 5.

FIGURE 5.

Aircraft and De-icing Resource Collaborative Scheduling Mechanism.

B. De-Icing Support Resource Operation Model

In the following, the input parameters of the DSRO model and the associated sets are described. The decision variables, constraints and objective functions of the optimal scheduling model related to the DSRO model are defined. Our proposed DSRO model is based on the existing de-icing vehicle scheduling model [23] and the AGV scheduling model [33]. The de-icing vehicle scheduling model proposed by Norin is based on the aircraft de-icing task set and minimizes the delay caused by aircraft waiting for scheduled de-icing vehicles. This is similar to the ADCS mechanism we have proposed in section IV-A, where the ASO model generates an aircraft de-icing task set as an input to the DSRO model. We also note that the de-icing fluid in de-icing vehicles is similar to the power required for AGV operation, and the replenishment process of the de-icing fluid of de-icing vehicles is similar to the charging process of the AGV. Compared with the previous de-icing vehicles scheduling model [23], our proposed DSRO model refers the idea of the mature AGV scheduling model and adds the de-icing fluid refilling decision variables.

1) Input Data

a: Data of De-Icing Vehicles

${\mathcal{ V}}$ is the set of de-icing vehicles that can be scheduled, for each vehicle $v\in {\mathcal{ V}}$ , the following data is given: the de-icing liquid capacity $W$ and the average travel speed $\kappa$ .

b: Input From Phase One

${\mathcal{ E}}$ is the set of de-icing tasks need to be completed of departing aircraft during the planning time period, where ${\mathcal{ E}}=\{e\left |{ {e_{1},} }\right.e_{2},e_{3},\cdots,e_{\left |{ {\mathcal{ E}} }\right |} \}$ . $e_{0}$ indicate the traveling task from the de-icing vehicles’ parking position to the location of its first de-icing task. $e_{\left |{ {\mathcal{ E}} }\right |+1}$ indicate the suppositionally finishing task. ${\mathcal{ E}}^{+}={\mathcal{ E}}\cup \{e_{0},e_{\left |{ {\mathcal{ E}} }\right |+1} \}$ . For each de-icing task $e\in {\mathcal{ E}}^{+},\chi _{e}$ is the number of de-icing vehicles required. $\gamma _{ee'}$ is the distance between the location of the task $e$ and the location of the task $e'$ . $\phi _{e}$ is the task completion time required for task $e$ . For each de-icing task $e\in {\mathcal{ E}},\psi _{e}$ is the maximum amount of de-icing fluid per de-icing vehicle required. $\phi _{e}$ is the completion time of task $e$ . $Y_{e}$ is the start occupancy time of the de-icing pad of task $e$ .

2) Decision Variables

We consider the assignment variables of de-icing tasks and de-icing vehicles, de-icing vehicles execution tasks sequence decision variables, de-icing position reaching time decision variables, de-icing liquid decision variables and de-icing fluid refilling decision variables. The sets, parameters and variables of the DSRO model are shown in Table 2.

TABLE 2 Sets, Parameters and Variables of DSRO Model

3) Constraints

a: De-Icing Task Assignment Constraints

$$\begin{align*} \sum \limits _{v\in {\mathcal{ V}}} \sum \limits _{e\in {\mathcal{ E}}^{+}} {u_{ee'}^{v}} &=\chi _{e'},\quad \forall e'\in {\mathcal{ E}}^{+},e\ne e^{\prime } \tag{13}\\ u_{ee'}^{v} +u_{e'e}^{v}& \le 1,\quad \forall v\in {\mathcal{ V}}, \forall e,e'\in {\mathcal{ E}},e\ne e' \tag{14}\\ \sum \limits _{e\in {\mathcal{ E}}} {u_{e_{0} e}^{v}}& =1,\quad \forall v\in {\mathcal{ V}},e\ne e' \tag{15}\\ \sum \limits _{e\in {\mathcal{ E}}} u_{ee_{\left |{ {\mathcal{ E}} }\right |+1}}^{v} &=1,\quad \forall v\in {\mathcal{ V}},e\ne e' \tag{16}\end{align*}$$

Equation (13) constraints that the number of de-icing vehicles assigned to each de-icing task needs to meet the requirements. Equation (14) constraints that any two de-icing tasks have only one sequence of execution. Equation (15)–(16) constraints that each de-icing vehicle should start from the de-icing vehicles’ parking position and execute the suppositionally finishing task finally.

b: Time Separation Constraint for De-Icing Tasks

$$\begin{align*}&t_{ve'} -t_{ve} +\text {B}(1-u_{ee}^{\prime v} +z_{v}^{e})\ge \phi _{e} +\gamma _{ee'} /\kappa \\ &\qquad \forall v\in {\mathcal{ V}},\forall e,e'\in {\mathcal{ E}}^{+},e\ne e' \tag{17}\\ & t_{ve'} -t_{ve} +\text {B}(2-u_{ee}^{v_{'}} -z_{v}^{e})\ge \phi _{e} \\ &\quad +(\gamma _{ee'} +\eta _{e} +\eta _{e'})/\kappa +Q^{\text {BC}}\quad \forall v\in {\mathcal{ V}}, \\ &\quad \forall e,e'\in {\mathcal{ E}}^{+},e\ne e' \tag{18}\\ & t_{ve'} -\text {B}\left({1-\sum \limits _{e\in {\mathcal{ E}}^{+}} {u_{ee}^{\prime v}} }\right)\le Y_{e'},\quad \forall v\in {\mathcal{ V}},\forall e'\in {\mathcal{ E}} \tag{19}\end{align*}$$

Equation (17) constrains that if two tasks $e$ and $e'$ are assigned by the same de-icing vehicle, the task $e$ is executed immediately preceding task $e'$ , and the de-icing vehicle does not return to the de-icing vehicles’ parking position to refill the de-icing fluid, the tasks execution interval needs to meet the time separation restrictions. Equation (18) constrains that if two tasks $e$ and $e'$ are assigned by the same de-icing vehicle, the task $e$ is executed immediately preceding task $e'$ , and the de-icing vehicle returns to the de-icing vehicles’ parking position to refill the de-icing fluid, the tasks execution interval needs to meet the time separation restrictions. Equation (19) constrains the time when the de-icing vehicle reach the task position to prevent unnecessary aircraft delays.

c: De-Icing Liquid Capacity Limit Constraint

$$\begin{align*}W&\ge o_{v}^{e} \ge 0,\quad \forall v\in {\mathcal{ V}} \tag{20}\\ o_{v}^{e'}& \ge o_{v}^{e} -B(1-u_{ee'}^{v} +z_{v}^{e})-\psi _{e'},\quad \forall v\in {\mathcal{ V}}, \\ &\quad \forall e,e'\in {\mathcal{ E}}^{+},e\ne e' \tag{21}\\ o_{v}^{e'} &\ge W-B(2-u_{ee'}^{v} -z_{v}^{e})-\psi _{e'},\quad \forall v\in {\mathcal{ V}}, \\ &\quad \forall e,e'\in {\mathcal{ E}},e\ne e' \tag{22}\end{align*}$$

Equation (20) constrains the de-icing fluid is restricted to be no more than $W$ . Equation (21) constrains that if the de-icing vehicle does not return to the parking position of de-icing vehicles to refill the de-icing fluid after executing the task $e$ , the de-icing fluid remaining in that vehicle needs to meet the restrictions. Equation (22) constrains that if the de-icing vehicle returns to the parking position of de-icing vehicles to refill the de-icing fluid after executing the task $e$ , the de-icing fluid remaining in that vehicle needs to meet the restrictions.

4) Objectives

Our proposed DSRO objective function is shown in Equation (23), which covers the three objectives of minimizing the total length of the de-icing vehicle traveling path, reducing the free time of the de-icing vehicle, and reducing the number of the de-icing fluid refilling times. The proof is given and the rationality of the objective function is analyzed in the following.

$$\begin{equation*} \min \Gamma \tag{23}\end{equation*}$$ where $\begin{aligned} \Gamma =\sum \limits _{v\in {\mathcal{ V}}} \left({t_{ve_{\left |{ {\mathcal{ E}} }\right |+1}} -{\begin{array}{cccccccccccccccccccc} {t_{ve_{0}}} \\ \end{array}}}\right) \end{aligned}$ , $\begin{aligned} t_{ve_{\left |{ {\mathcal{ E}} }\right |+1}} -{\begin{array}{cccccccccccccccccccc} {t_{ve_{0}}} \\ \end{array} } \end{aligned}$ is the operating cycle of the de-icing vehicle $v$ during the planning time period.

Proof 1:

It may be assumed that for any de-icing vehicle $v$ which is assigned $w$ tasks during the planning time period, and $w\ge 2$ , the last task is task $e_{w}$ . The de-icing fluid will be refilled $\lambda$ times during the planning period. The distance between the parking position of the de-icing vehicles and the nearest de-icing pad is $\delta$ . The time of $v$ reaching the location of task $w$ is $t_{ve_{w} }$ . The time of $v$ reaching the location of the task $w$ should be no earlier than the sum of the time to complete the task $w-1$ and the travel time. The time $v$ to reaching the location of the task $w-1$ should be no earlier than the sum of the time to complete the task $w-2$ and the travel time. Then $t_{ve_{w} } \ge \sum \limits _{i=1}^{w-1} {\phi _{e_{i}}} +\frac {\sum \limits _{i=2}^{w-\lambda } {\gamma _{e_{i-1} e_{i}}} +2\lambda \cdot \delta }{\kappa }+t_{ve_{0}}$ , where $\phi _{e_{i}},\kappa$ are parameters, $t_{ve_{1}} \ge 0$ . $t_{ve_{w}}$ is only related to $\sum \limits _{i=2}^{w-\lambda } {\gamma _{e_{i-1} e_{i}}}$ and $t_{ve_{0}}$ . Because $t_{ve_{\left |{ {\mathcal{ E}} }\right |+1}} =t_{ve_{w}} +\phi _{e_{w}}$ , $t_{ve_{\left |{ {\mathcal{ E}} }\right |+1}} -t_{ve_{0}} \ge \sum \limits _{i=1}^{w} {\phi _{e_{i}}} +\frac {\sum \limits _{i=2}^{w-\lambda } {\gamma _{e_{i-1} e_{i}}} +2\lambda \cdot \delta }{\kappa }$ . $\sum \limits _{i=2}^{w-\lambda } {\gamma _{e_{i-1} e_{i}}}$ is the total length of the path traveled by $v$ to execute the task, and $\lambda \cdot \delta$ is the total length traveled by $v$ to refill the de-icing fluid. Therefore, minimizing the operating cycle of the de-icing vehicle $v$ during the planning time period can reduce the total traveling length.

Proof 2:

It may be assumed that for any de-icing vehicle $v$ which is assigned $w$ tasks during the planning time period, and $w\ge 2$ , the last task is task $e_{w}$ . The de-icing fluid will be refilled $\lambda$ times during the planning period. The distance between the parking position of the de-icing vehicles and the nearest de-icing pad is $\delta$ . The time of $v$ reaching the location of task $w$ is $t_{ve_{w} }$ . The time of $v$ reaching the location of the task $w$ should be no earlier than the sum of the completion time of task $w-1$ , the travel time, and the free time duration after completing the task $w-1$ . The time of $v$ reaching the location of the task $w-1$ should be no earlier than the sum of the completion time of task $w-2$ and the travel time, and so on, then $t_{ve_{w} } \ge \sum \limits _{i=1}^{w-1} {\phi _{e_{i}}} +\frac {\sum \limits _{i=2}^{w-\lambda } {\gamma _{e_{i-1} e_{i}}} +2\lambda \cdot \delta }{\kappa }+t_{ve_{0}} +\sum \limits _{i=1}^{w-1} {t_{ve_{i} }^{\text {EMPTY}}}$ , where $\phi _{e_{i}},\kappa$ are parameters, according to Proof 1, the above equation can be simplified to $t_{ve_{\left |{ {\mathcal{ E}} }\right |+1}} -t_{ve_{0}} \ge \sum \limits _{i=1}^{w} {\phi _{e_{i}}} +\frac {\sum \limits _{i=2}^{w-\lambda } {\gamma _{e_{i-1} e_{i}}} +2\lambda \cdot \delta }{\kappa }+\sum \limits _{i=1}^{w-1} {t_{ve_{i}}^{\text {EMPTY}}}$ . $t_{ve_{w}}$ is only related to $\sum \limits _{i=2}^{w-\lambda } {\gamma _{e_{i-1} e_{i}}}$ and $\sum \limits _{i=1}^{w-1} {t_{ve_{i} }^{\text {EMPTY}}}$ . Therefore, minimizing the sum of the operating cycles of all de-icing vehicles during the planning time period can reduces the total free time of de-icing vehicles and improves the efficiency of de-icing vehicles usage.

SECTION V. Solution Approaches

During actual airport operations, air transport decision makers are more concerned about the efficiency of solving the ASO model as well as the DSRO model. Therefore, the computation time of the algorithm is critical. A hybrid algorithm combining heuristic methods and accurate algorithms may have more potential than traditional heuristics or accurate algorithms for the complexity of the problem [34]. Therefore, we propose an algorithm combining RHC strategy and CPLEX solver to solve the above model. We give the specific procedure of the algorithm in the following.

A. Receding Horizon Control Strategy

Receding horizon control strategy, has been shown to be an effective optimization strategy for large-scale optimization problems with complex constraints [35]. In the RHC strategy framework, the original optimization problem is partitioned into several subproblems. The RHC strategy is required to look forward $C$ time horizons, i.e., when optimizing the current kth receding horizon, the optimal scheduling receding forward $C$ time horizons, but only implements the results in the kth horizon and repeats the same process on the next optimization stage.

B. Algorithm Design

1) Algorithm for ASO Model

Before we describe the proposed RHC-CPLEX algorithm, some notations are introduced. $H_{\textrm {SY}}$ is the length of an operating interval. $C$ is the length of receding horizon. $T_{0} (k)$ denotes the beginning time of the receding horizon at the kth operating interval, the operating time interval is $[T_{0} (k),T_{0} (k)+C\cdot H_{\textrm {SY}})$ when the optimization scheduling is at the kth stage. $\Omega (k)$ is the set of aircraft participating in the optimization scheduling of the kth stage. $\Theta (k)$ is the set of aircraft that have completed scheduling after the completion of the kth optimization scheduling stage. $\Pi (k)$ is the set of aircraft that have not completed scheduling after the completion of the kth optimization scheduling stage. $\Upsilon (k)$ is the set of aircraft that $T_{f}^{\textrm {TB}}$ or $T_{f}^{TL}$ in interval $[T_{0} (k),T_{0} (k)+C\cdot H_{\textrm {SY}})$ . $\Phi (k)$ is the weighted sum of the surface operation time and the delay time of aircraft that completed scheduling in the kth optimization scheduling stage. For each aircraft $f\in {\mathcal{ D}}\cup {\mathcal{ D}}^{\ast },T_{f}^{\textrm {TB}} (k)$ denotes target off-block time is at the kth operating stage, $t_{f}^{\textrm {OB}} (k)$ denotes the scheduled off-block time is in the $[T_{0} (k),T_{0} (k)+C\cdot H_{\textrm {SY}})$ interval after the kth optimization scheduling stage, $t_{f}^{\textrm {R}} (k)$ denotes the scheduled runway usage time is in the $[T_{0} (k),T_{0} (k)+C\cdot H_{\textrm {SY}})$ interval after the kth optimization scheduling stage, $t_{f}^{\textrm {MS}} (k)$ denotes the scheduled de-icing pad start occupancy time is in the $[T_{0} (k),T_{0} (k)+C\cdot H_{\textrm {SY}})$ interval after the kth optimization scheduling stage. For each aircraft $f\in {\mathcal{ A}}\cup {\mathcal{ A}}^{\ast },T_{f}^{\textrm {TL}} (k)$ denotes target landing time is at the kth operating stage, $t_{f}^{R} (k)$ denotes the scheduled landing time of aircraft $f$ is in the $[T_{0} (k),T_{0} (k)+C\cdot H_{\textrm {SY}})$ interval after the kth optimization scheduling stage.

• Step 1:

  Generate the initial aircraft queue in the order of the values of $T_{f}^{\textrm {E}}$ , set the $T_{f}^{\textrm {TB}}$ of the first aircraft to $T_{0} (0)$ if the first aircraft $f\in {\mathcal{ D}}\cup {\mathcal{ D}}^{\ast }$ , set the $T_{f}^{\textrm {TL}}$ of the first aircraft to $T_{0} (0)$ if the first aircraft $f\in {\mathcal{ A}}\cup {\mathcal{ A}}^{\ast }$ . Let $k=0$ , set the values of $\alpha$ and $C$ , initialize $\Omega (k),\Theta (k)$ and $\Upsilon (k)$ .

• Step 2:

  After completion the kth stage of aircraft scheduling, those aircraft with $t_{f}^{\textrm {R}} (k)\le T_{0} (k)+H_{\textrm {SY}}$ are put into the set $\Theta (k)$ and $\Phi (k)$ is calculated with reference to Equation (24).

  $$\begin{equation*} \Phi (k)=\alpha \sum \limits _{f\in \Theta (k)} {S_{f}} +(1-\alpha)\sum \limits _{f\in \Theta (k)} {E_{f}} \tag{24}\end{equation*}$$

  Freezing the existing scheduling results of those aircraft with $t_{f}^{\textrm {R}} (k) > T_{0} (k)+H_{\textrm {SY}}$ , and put them into the set $\Pi (k)$ and update the constraints with reference to Equation (25)–(27).

  $$\begin{align*}t_{f}^{\text {OB}} (k+1)&\ge t_{f}^{\text {OB}} (k),\quad \forall f\in \Pi (k)\cap ({\mathcal{ D}}\cup {\mathcal{ D}}^{\ast }) \tag{25}\\ t_{f}^{\text {MS}} (k+1)&\ge t_{f}^{\text {MS}} (k),\quad \forall f\in \Pi (k)\cap ({\mathcal{ D}}\cup {\mathcal{ D}}^{\ast }) \tag{26}\\ t_{f}^{\text {R}} (k+1)&\ge t_{f}^{\text {R}} (k),\quad \forall f\in \Pi (k)\cap {\mathcal{ F}} \tag{27}\end{align*}$$

  Let $T_{0} (k+1)=T_{0} (k)+H_{\textrm {SY}}$ . Put the aircraft $f\in {\mathcal{ D}}\cup {\mathcal{ D}}^{\ast }$ which $T_{f}^{\textrm {TB}}$ is in the $[T_{0} (k)+C\cdot H_{\textrm {SY}},T_{0} (k+1)+C\cdot H_{\textrm {SY}})$ interval into set $\Upsilon (k+1)$ . Put the aircraft $f\in {\mathcal{ A}}\cup {\mathcal{ A}}^{\ast }$ which $T_{f}^{\textrm {TL}}$ is in the $[T_{0} (k)+C\cdot H_{\textrm {SY}},T_{0} (k+1)+C\cdot H_{\textrm {SY}})$ interval into set $\Upsilon (k+1)$ .

• Step 3:

  The aircraft that in $\Omega (k+1)$ is optimally scheduled in k +1st stage by CPLEX solver with reference to Equation (28), where $\Omega (k+1)=\Pi (k)\cup \Upsilon (k+1)$

  $$\begin{align*} \min {\begin{array}{cccccccccccccccccccc} \\ \end{array}}{\begin{array}{cccccccccccccccccccc} {\alpha \sum \limits _{f=1}^{\left |{ {\Omega (k+1)} }\right |} {S_{f}}} \\ \end{array}}+(1-\alpha)\sum \limits _{f=1}^{\left |{ {\Omega (k+1)} }\right |} {E_{f}} \tag{28}\end{align*}$$

• Step 4:

  Let $k=k+1$ , if $\sum \limits _{0}^{k} {\left |{ {\Theta (k)} }\right |} =\left |{ {\mathcal{ F}} }\right |$ , go to Step 5. Otherwise, go to Step 2.

• Step 5:

  Calculate $\Phi$ with reference to Equation (29).

  $$\begin{equation*} \Phi =\sum \limits _{0}^{k} {\Phi (k)} \tag{29}\end{equation*}$$

The pseudo-code for the RHC-CPLEX algorithm to solve the ASO model is shown in Table 3.

TABLE 3 Pseudo Code of RHC-CPLEX Algorithm to Solve the ASO Model

2) Algorithm for DSRO Model

Before the proposed RHC-CPLEX algorithm for solving the DSRO model is described, some notations are also introduced. $H_{\textrm {SY}},T_{0} (k)$ and $C$ are the same as described in Section IV-B1. $\zeta (k)$ is the set of de-icing tasks participating in the kth optimization stage. $\Psi (k)$ is the set of de-icing tasks that have completed assignment after the kth optimization stage. $\Lambda (k)$ is the set of de-icing tasks that have not completed assignment after the kth optimization stage. $\vartheta (k)$ is the set of de-icing tasks that $Y_{e}$ in interval $[T_{0} (k-1)+C\cdot H_{\textrm {SY}},T_{0} (k)+C\cdot H_{\textrm {SY}})$ . $Y_{e} (k)$ denotes the start time of de-icing task $e$ is in the $[T_{0} (k),T_{0} (k)+C\cdot H_{\textrm {SY}})$ interval. $\omega _{e} (k)$ denotes the finish time of de-icing task $e$ is in the $[T_{0} (k),T_{0} (k)+C\cdot H_{\textrm {SY}})$ interval. $\Gamma (k)$ is the sum of the operating cycles of the de-icing vehicles that completed their assigned tasks in the kth optimization stage. $\Delta (k)$ is the set of de-icing vehicles that have completed all de-icing tasks assigned to them in the kth optimization assignment stage.

• Step 1:

  Generate the initial de-icing task queue in the order of the values of $Y_{e}$ . Set the $Y_{e}$ of the first de-icing task to $T_{0} (0)$ . Let $k=0$ , set the value of $C$ , initialize $\Psi (k),\zeta (k)$ and $\vartheta (k)$ .

• Step 2:

  After the completion the kth optimization stage, those de-icing tasks with $\omega _{e} (k)\le T_{0} (k)+H_{\textrm {SY}}$ are put into the set $\Psi (k)$ , those de-icing vehicles that have completed all de-icing tasks in the kth stage are put into set $\Delta (k)$ , and $\Gamma (k)$ is calculated with reference to Equation (30).

  $$\begin{equation*} \Gamma (k)=\sum \limits _{v\in \Delta (k)} (t_{ve_{\left |{ {\Psi (k)} }\right |+1}} - {t_{ve_{0}}}) \tag{30}\end{equation*}$$

  Freezing the existing assignment results of those de-icing tasks with $\omega _{e} (k) > T_{0} (k)+H_{\textrm {SY}}$ , put them into the set $\Lambda (k)$ and update the constraints with reference to Equation (31).

  $$\begin{equation*} t_{ve'} (k+1)\ge t_{ve} (k)+\phi _{e},\quad \forall e\in \Lambda (k),e'\in \zeta (k+1) \tag{31}\end{equation*}$$

  Let $T_{0} (k+1)=T_{0} (k)+H_{\textrm {SY}}$ . Put the de-icing task which $Y_{e}$ is in the $[T_{0} (k)+C\cdot H_{\textrm {SY}},T_{0} (k+1)+C\cdot H_{\textrm {SY}})$ interval into set $\vartheta (k+1)$ .

• Step 3:

  The de-icing task that in $\zeta (k+1)$ is optimally assignment in k+1st stage by CPLEX solver with reference to Equation (32), where $\zeta (k+1)=\Lambda (k)\cup \vartheta (k+1)$

  $$\begin{align*} \min \sum \limits _{v\in {\mathcal{ V}}} \left({t_{ve_{\left |{ {\zeta (k+1)} }\right |+1}} -{\begin{array}{cccccccccccccccccccc} {t_{ve_{0}}} \\ \end{array}}}\right) \tag{32}\end{align*}$$

• Step 4:

  Let $k=k+1$ ,if $\sum \limits _{0}^{k} {\left |{ {\Psi (k)} }\right |} =\left |{ {\mathcal{ E}} }\right |$ , go to Step 5. Otherwise, go to Step 2.

• Step 5:

  Calculate $\Gamma$ with reference to Equation (33).

  $$\begin{equation*} \Gamma =\Gamma (k) \tag{33}\end{equation*}$$

The pseudo-code for the RHC-CPLEX algorithm to solve the DSRO model is shown in Table 4.

TABLE 4 Pseudo Code of RHC-CPLEX Algorithm to Solve the DSRO Model

SECTION VI. Numerical Experiments

In this section, aircraft using DaXing Airport De-icing Zone No. 1, No. 2 and No. 3 and four runways are used as experimental subjects. The optimized scheduling results under different snow scenarios are presented. The calculation results of the RHC-CPLEX algorithm are compared with those of the CPLEX solver only, FCFS and PPA.

A. Experimental Environment

Beijing DaXing Airport has four runways, Rwy17R/35L, Rwy 17L/35R, Rwy 36L/18R and Rwy 11L/29R. Currently, Rwy 17L/35R and Rwy 11L/29R are mainly used for takeoff and the other two are mainly used for landing. Their relative positions are shown in the Figure. 6. The runway separation is based on the safety regulations implemented by FAA (Federal Aviation Administration) and ICAO (International Civil Aviation Organization) [36]

FIGURE 6.

Relative position between runways of DaXing Airport.

Based on the real aircraft operation data of DaXing Airport from December 2020 to February 2021 under de-icing mode, the hourly traffic flow is counted to analyze the aircraft flow characteristics. We selected the typical flight days based on the principle of 90% to 95% and excluding the effect of adverse contingencies [5]. Real operational data for 10 typical days were selected, and the range of data in each typical day was 6 consecutive hours of the airport’s busy operating period. Among the 10 typical days we have chosen, there are 4 typical days of light snow, 3 typical days of moderate snow, one typical day of sticky snow and two typical days of heavy snow condition. 2268 aircraft are involved in the 10 typical days, and the aircraft data format is shown in Table 5. The relevant characteristics of the aircraft in these 10 cases are shown in Figure. 8. As shown in the Figure. 3, DaXing Airport has 15 apron areas. In 10 typical days, the heat map of aircraft using each apron space is plotted as shown in the Figure. 7, and we find that the frequency of using apron is at a low level except for Apron No.1- No.4, so the experimental subjects are aircraft using aprons Apron No.1- No.4, totaling 2187 aircraft, and the proportional distribution of aircraft at each ramp is shown in Figure. 8. When selecting the typical day of aircraft operation, only Runway 35L, 35R, 11L, 36L operation was considered. The airport can easily make the framework described in this paper when using other runways for operation, and it is not repeated here.

TABLE 5 Aircraft Data Format

FIGURE 7.

Parking position occupancy heat map.

FIGURE 8.

Aircraft characteristics.

B. Computational Performance

The RHC-CPLEX is coded in MATLAB R2016a and run on a PC with Intel i7-9700KF8C8T, 3.60 GHz and 16.0 GB RAM.

C. Multidimensional Analysis of Experimental Results

1) Algorithm Solution Results and Performance Analysis

a: Comparison of Algorithms for Solving ASO Models

The statistics of the de-icing operation time parameters for aircraft under different snow conditions at DaXing Airport in winter 2021 are shown in Table 6. In the experimental process, we configured a CPLEX solver time limit, and when the solving time exceeded 1500s, the solver selected the current optimal solution as the solution to the problem.

TABLE 6 De-Icing Time of Departure Aircraft Under Different Snow Conditions (Unit: s)

The average taxiing time from each parking position to De-icing Zone No.1-No.3 and that from three de-icing zones to each runway at DaXing Airport is used as model inputs. We take the key parameters $H_{\textrm {SY}}$ of RHC-CPLEX algorithm as 1800s and 2400s, and $C$ as 2 and 3, respectively. We set $\alpha$ to 0.3 as an example to compare the solution results of RHC-CPLEX algorithm with CPLEX only and FCFS algorithm under different snow conditions, as shown in Appendix A.

As we can see from Appendix A, although the computation time is significantly longer, the RHC-CPLEX algorithm can reduce the objective function by 26%, 28%, 27%, and 29% on average compared with the FCFS algorithm under the four snow scenarios. The RHC-CPLEX algorithm has better solution results than CPLEX only. Although the reduction of the objective function is less obvious, it can lead to an optimal solution in limited and shorter time. It significantly reduces the computational cost compared to the method which expanding the sample using a linear incremental function [1]. Numerous experiments have shown that the runtime of scheduled formulation can generally be controlled within 6s/aircraft [37]. Therefore, the RHC-CPLEX algorithm can meet the demand for aircraft operation scheduling decisions for airports with multiple de-icing zones and multiple runway systems at the tactical level.

We can also see from Appendix A that the larger the parameter $H_{\textrm {SY}}$ or $C$ in the RHC-CPLEX algorithm, the longer the algorithm takes to solve, but the results obtained are not necessarily better. The parameters of the algorithm corresponding to the best solution vary for different typical days. Although the parameter selection affects the objective value, the difference of the results is small. Therefore, considering the actual application, the decision maker can choose the parameter combination suitable for the actual scheduling scenario by considering the algorithm solution time and the solution result.

b: Comparison of Algorithms for Solving DSRO Models

There are 26 de-icing vehicles with a maximum capacity of 5160 liters of de-icing fluid, which are parked at the centralized vehicles parking position before de-icing operation, as shown in the Figure. 3. With reference to the existing operation mode of DaXing Airport (Two-vehicle de-icing, $\forall e\in E,\chi _{e} =2$ ), the amount of de-icing fluid required for each vehicle under different snow conditions are counted as shown in Table 7.

TABLE 7 Deicing Fluid for Multi Type Aircraft Under Different Snow Conditions

The travel distances between the de-icing pads and that from the de-icing vehicles’ parking positions to de-icing pads are used as model inputs. We take the key parameters $H_{\textrm {SY}}$ of RHC-CPLEX algorithm as 1800s and 2400s, and the key parameters $C$ as 2 and 3, respectively. We set $\alpha$ to 0.3 as an example to compare the solution results of RHC-CPLEX algorithm with CPLEX only and PPA algorithm under different snow conditions, as shown in Appendix B. From Appendix B, RHC-CPLEX algorithm can reduce the objective function $\Gamma$ by 6.2%, 7.2%, 7.0% and 6.2% on average under four snow conditions, although the operation time is significantly longer. Compared with CPLEX only, the RHC-CPLEX algorithm has better solution results. Although the reduction of the objective function is less obvious, it can lead to an optimal solution in limited and shorter time. The decision makers can also choose the optimal combination of algorithm parameters according to the actual scheduling scenarios.

We counted the total number of de-icing vehicles refills, the total traveled distance and the total free time for each typical day. We calculated the average value for 10 typical days. The solution results of RHC-CPLEX algorithm with CPLEX only and PPA algorithm under different snow conditions are shown in Table 8–​Table 10.

TABLE 8 Average Total Number of De-Icing Vehicles Refills Under Different Snow Conditions

TABLE 9 Average Total Travel Distance of De-Icing Vehicles Under Different Snow Conditions

TABLE 10 Average Total Free Time of De-Icing Vehicles Under Different Snow Conditions

From Table 8, it can be seen that the results of the RHC-CPLEX algorithm with various parameter settings can reduce the number of de-icing fluid refills. The results of the PPA algorithm are not significantly different from those of the RHC-CPLEX algorithm under light and moderate snow conditions, and the number of de-icing vehicles refills is kept at a low level. This is because the de-icing process does not require a lot of de-icing fluid under light and moderate snow conditions. The initial full load of de-icing fluid of the de-icing vehicles is sufficient to complete the de-icing task during the planning time period. The RHC-CPLEX algorithm can significantly reduce the number of de-icing fluid refills under sticky and heavy snow conditions. We can find that the de-icing process requires a large amount of de-icing fluid volume under sticky and heavy snow conditions. The volume of de-icing tasks and the capacity of de-icing fluid of vehicles are limited. Therefore, almost all de-icing vehicles are refilled during the planning period under sticky and heavy snow conditions.

From Table 9-​Table 10, the results of the RHC-CPLEX algorithm with various parameter settings can reduce the traveled distance and free time between tasks under different snow conditions. The reduction of vehicle travel distance is caused by the reduction of the number of de-icing fluid refills. Our proposed algorithm can reduce the average free time of vehicles and improve the utilization efficiency of de-icing vehicles. Meanwhile, the results can verify that the model can reduce the total traveled distance while reducing the free time and the number of de-icing fluid refills.

2) Analysis of RHC-CPLEX Algorithm Scheduling Results for ASO Model

a: Comparison of Aircraft Scheduling Results Under Different Conditions

We use the RHC-CPLEX algorithm, which has superior computational performance and set the key parameter in the algorithm to1800s and 2, respectively. We set the weight in the objective function to 0.3. Box plots of the resulting for each of 10 typical days are shown in Figure. 9–​Figure. 11. Box plots show the maximum queuing length and mean queuing times at runway, de-icing zone and on the apron. We can see that the number of delayed aircraft of runways and the number of delayed aircraft of the de-icing zones are much smaller than that on the apron under different snow conditions. This is because the scheduling model includes the aircraft surface operation time as part of the objective function. It is possible to achieve a shorter surface operation time including the de-icing delay time and the runway delay time. The queuing process of aircraft at de-icing zones and runways can be understood as occurring on the apron.

FIGURE 9.

Box plots of maximum runway queuing and mean queuing times from 10 independent simulation runs.

FIGURE 10.

Box plots of maximum de-icing queuing and mean queuing times from 10 independent simulation runs.

FIGURE 11.

Box plots of maximum apron delay and mean delay times from 10 independent simulation runs.

We can see in Figure. 9–​Figure. 11. that regardless of the snow conditions, the average aircraft apron delay is much larger than the de-icing zone delay and runway delay. As the snow conditions increase, the apron delay gradually increases, while the aircraft de-icing zone delay and runway delay do not change significantly and basically remain within one minute. The above results prove that under this optimized scheduling framework, aircraft delays are mainly absorbed by the way of waiting at the aircraft’s parking position. The aircraft surface operation delays are smaller, which not only meets the real operation demand, but also avoids the large-scale congestion at the surface.

b: Variation of Aircraft Scheduling Result With Changing Values of $\alpha$

We use the RHC-CPLEX algorithm, set the algorithm parameters $H_{\textrm {SY}}$ to 1800s and $C$ to 2, adjust the value of $\alpha$ . We take the aircraft flow of a typical day as an example to compare the solution results of the algorithm under different snow conditions. The relationship between the delay time of the apron, the delay time of the de-icing zone and the delay time of the runway under different snow conditions is studied, and the curve is plotted as shown in Figure. 12.

FIGURE 12.

The variation of aircraft delay with the change of $\alpha$ .

From Figure. 12, it can be seen that the delay time does not change significantly under different snow conditions with $\alpha$ changes, which proves that the two objectives are set reasonably, and the total aircraft operation time and delay time on the surface can be optimized while reducing the total aircraft delay time. We can also see that the aircraft delay time on the apron is much greater than the de-icing zone delay time and the runway entrance delay time, further indicating that our proposed model can reasonably allocate the aircraft delay time to prevent massive congestion of aircraft at the airport surface.

c: Joint Apron-Runway-De-Icing Zone Assignment Results

We use the RHC-CPLEX algorithm, which has superior computational performance and set the key parameter $H_{\textrm {SY}}$ and $C$ in the algorithm to1800s and 2, respectively. We set the weight $\alpha$ in the objective function to 0.3. A typical day under moderate snow condition is used as an example for analysis, the joint apron-runway-deicing zone assignment results are shown in Figure. 13.

FIGURE 13.

De-icing zone distribution results of the apron-runway-deicing zone assignment. Each shows the number of aircraft in each deicing zone from Apron No.1- Apron No.4 that uses Rwy35R (top four) and Rwy11L (bottom four), respectively.

By analyzing the results of the joint apron-runway-de-icing apron assignment, we can see that aircraft’s de-icing zone usage is spatially distributed closer to their apron location and the assigned runway. Further, some aircraft departing from Apron No.1 and No.2 are assigned to De-icing Zone No. 3 for de-icing when using Rwy 35R compared to Apron No.3 and No.4, because the model optimizes the total time of aircraft operations at the airport surface, while apron 3 and 4 are relatively far away from De-icing Zone No. 3. No aircraft were assigned to De-icing Zone No. 1 and De-icing Zone No. 2 when departing using Runway 11L from Apron 1 to 4. This is because the model constraining the time period between de-icing and takeoff to ensure the de-icing fluid is active during takeoff.

SECTION VII. Conclusion

This paper proposed an integrated optimization framework to schedule aircraft surface operation and de-icing support resources for airports with multi de-icing zones. For the aircraft surface operation problem, we have optimized the de-icing process for aircraft departures by adding consideration for the occupancy of de-icing pads. We optimize the scheduling of the aircraft off-block time, de-icing time, and runway usage sequence. For the de-icing resource scheduling problem, we optimize the task assignment, travel route of vehicles, and de-icing fluid consumption. DSRO can solve the problem of untimely allocation of de-icing resources and unreasonable allocation of de-icing resources under the existing technology can be solved based on the optimization results of the ASO model. The ADCS mechanism to integrate aircraft operation scheduling and de-icing support resource scheduling. The proposed framework can support for decision-making at the phase of the tactical stage operation at airport system with the use of the air traffic data.

Comprehensive case study of Beijing DaXing airport shows that RHC-CPLEX algorithm has significantly better optimization effect and computing performance compared with the traditional FCFS algorithm and using CPLEX only under the selected typical daily aircraft flow and different snow conditions. The RHC-CPLEX algorithm has the potential to provide real-time support for air transport decision makers. The analysis of the solution results of the ASO model shows that the optimization scheme of aircraft surface operation scheduling is reasonable and effective. ASO model can optimize the aircraft surface delay distribution and make the parking position absorb most of the departure aircraft surface delays, which can avoid large-scale congestion on the surface and meet the actual operation requirements. DSRO model can not only ensure that the de-icing vehicles can complete the de-icing task on time, but also assign de-icing resources more reasonably and improve the usage efficiency of the de-icing vehicles.

The findings can provide some significant reference for air transport decision makers under the aircraft de-icing operation mode, such as determining the off-block time of aircraft, the runway usage time of aircraft, the start of deicing time of aircraft, the task routes of de-icing vehicles, etc. It has the potential to support decision-making and improve the efficiency, safety and cost-effectiveness of airport surface operations. The airlines operation controller can make full use of the operational scheduling data to compare or integrate the differences between the aircraft target time and scheduled time nodes to manage the aircraft operation at the stage of airport.

Our proposed framework is based on a fixed aircraft taxiing route and does not take into account the uncertainties in the aircraft de-icing process, and the potential conflict between vehicles is also very important to be taken into account in the model. In subsequent work, we will make further efforts to focus on the integrated management of airport surface operation and scheduling under de-icing mode. Additionally, the taxiways scheduling of aircraft (i.e., rerouting and retiming) problem will also be paid close attention in the future work. Moreover, we will consider the uncertainty parameters for de-icing of aircraft under different snow conditions and the conflicts between them while scheduling the de-icing vehicles, and take into account the duty scheduling of related personnel to complete the integrated and collaborative scheduling of the airport surface operation.

Appendix A

We use the RHC-CPLEX algorithm, CPLEX solver and FCFS algorithm to solve the ASO model, and the results are compared as shown below. See Table 11.

TABLE 11

Appendix B

We use the RHC-CPLEX algorithm, CPLEX solver and PPA algorithm to solve the DSRO model, and the results are compared as shown below. See Table 12.

TABLE 12