1 Introduction

With the rapid growth of airport traffic, runway capacity is no longer enough for flights to land at the estimated time of arrival (ETA) [1]. The aircraft landing problem (ALP) becomes a task that must be solved for an airport. Generally, the main task of the ALP is to decide the predetermined landing time for each flight and satisfy the separation criteria among the landing flights [2].

As an optimization model, the research of ALP mainly focuses on the following aspects:

(1) minimizing the total penalty; (2) minimizing the total delays or total fuel cost; and (3) maximizing the runway throughput.

Abhishek Awasthi et al.(2013) thought the objective of ALP was to optimize landing sequences and landing times for all the aircraft, and presented a polynomial algorithm to reduce the total penalty for given feasible landing sequence for the single runway [3]. Bennell J A. et al. (2017) adopted dynamic programing algorithm to solve the on-line ALP and the total penalty was took account of optimization objective [4]. Alain Faye (2015) [5], Sabar et al. [6] also solved the ALP by minimizing the total penalty for the single runway. Salehipour A. et al. (2013) dealt with this problem by allocating aircraft to land on the available runways to minimize the total delay [7]. Farhadi et al. [8] minimized the total fuel consumption by minimizing the deviation of aircraft’s start-times from their respective ready-times. Veresnikov et al. [9] developed a dynamic programming algorithm to minimize the total delay of aircraft landings. Wei-dong et al. [10] applied ant colony algorithm in flight landing scheduling problem to maximize runway throughput. Li Yang et al. (2018) developed a optimization algorithm based teaching–learning for runway scheduling under constrained position shifting to maximize runway throughput [11]. The above documents only selected one target from total penalty, total delay, total fuel consumption, or runway throughput to solve the ALP. In reality, air transportation involves some stakeholders such as air traffic controllers, airports, airlines and passengers. Different stakeholders care about different goals, for example, reducing delay is the concern of passengers and airlines, ensuring safety is the concern of all stakeholders, maximizing runway throughput is the concern of airports, and reducing fuel consumption is the concern of airlines. Therefore, the ALP model needs to optimize multiple conflicting objectives simultaneously and the ALP becomes a complex multi-objective optimization of aircraft landing problem (MOOALP). We explicitly consider the following objectives and constrains for solving ALP.

  • To ensure the safety, the scheduled landing time must meet the separation requirements.

  • To take full advantage of airspace and runway resources, all aircraft must land as soon as possible.

  • Minimize the total delay time of terminal arrivals.

  • To minimize the total fuel cost, the aircraft uses the most economical speed to land.

  • Scheduled landing time should be generated before the arranged flights enter the terminal.

  • Flight can be delayed by waiting at holding position, regulating speed, or making maneuvers.

If there are n flights, the solution space has n! combinations. In order to meet the requirements of the control task, an approximate or global optimal solution must be found as soon as possible. The simulated annealing can meet the request of control task because it accepts the poor solution with a certain probability so as to jump out of the local optimal solution and obtain the approximate or global optimal solution.

The remaining part of this paper is organized as follows. In Sect. 2, we describe the problem and model of MOOALP. In Sect. 3, based on FCFS service rules, the ALP is solved by SA. In Sect. 4, taking an airport as an example, we analyze the test results of FCFS and the SA. The discussion and conclusion are raised in Sect. 5 and 6, respectively.

2 MOOALP model

2.1 Problem description

Flights enter the terminal area from different routes and prepare for landing according to the instrument approach procedure. The initial points of instrument approach procedures are called the approach point, such as p001-p011 in Fig. 1. The approach points form the outer ring. The optimization algorithm of aircraft landing problem must be terminated when the flights arrival inner ring. The flights lying between the outer ring and inner ring need to be extracted from radar and be scheduled the landing time.

Fig. 1
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The spatiotemporal model of airport terminal area

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In order to guarantee the safety, ICAO (International Civil Aviation Organization) stipulates the time-based wake turbulence separation minima for arriving aircraft on the same runway. The time separation for three wake turbulence categories (LIGHT, MEDIUM, and HEAVY) is shown below [12].

  • MEDIUM aircraft behind HEAVY aircraft—2 min;

  • LIGHT aircraft behind a HEAVY or MEDIUM aircraft—3 min;

For the other situations, each airport has its own time separation stipulates, for instance, in our experiment airport is 1.5 min.

2.2 Multi-objective optimization model

In this work, four objectives are considered to build the MOOALP.

Minimize the aircraft total fuel cost resulting from the start-time deviation the ready-time.

$$ f_{1} {\text{ = min}}\sum\limits_{{{\text{i}} = 1}}^{P} {[{\text{g}} \times {\text{max(}}0,\,T{\text{i}} - xi) + h \times {\text{max}}(0,xi - Ti)]} $$
(1)

where P is the number of aircraft; Ti is the preferred landing time for aircraft i; xi is the scheduled landing time for aircraft i; g is the penalty cost (≥ 0) per unit of time before the preferred landing time Ti for aircraft i; h is the penalty cost (≥ 0) per unit of time after preferred landing time Ti for aircraft i.

Minimize the total delays.

$$ f_{{2}}\, =\,{\text{min}}\sum\nolimits_{{\text{i}}}\,=\,1^{P} {[} {\text{max(0,}}x_{i} - T_{i} {)]} $$
(2)

Maximize throughput by minimizing the makespan.

$$ f_{3} \; = \;\min [\max (x_{i} ,\;i\; = \;1,\;...,\;P)]. $$
(3)

Maximize fairness by minimizing the variance of delay.

$$ f_{4} {\text{ = min[stadev}}^{2} {]} $$
(4)
$$ {\text{stadev}}^{2} = \frac{1}{P - 1}\sum\limits_{i = 1}^{P} {\left( {(x_{i} - T_{i} ) - \sum\limits_{i = 1}^{P} {(x_{i} - T_{i} )} {/}P} \right)}^{2} $$
(5)

Because these four objectives sometimes may conflict with each other, the importance of these goals must be weighed according to the actual situation. In the rush hour, aircraft are expected to land as soon as possible, so the third objective will have greater weight. During periods of low traffic, the aircraft are expected to land in the most economical way, so the first objective will have greater weight.

The MOOALP mathematical model can be represented as follows:

$$ \mathop F\limits_{s.t.} \; = \;\min (w_{1} f_{1} + w_{2} f_{2} + w_{3} f_{3} + w_{4} f_{4} ) $$
(6)
$$ \sum\limits_{k = 1}^{4} {\mathop w\nolimits_{k} } \; = \;1 $$
(7)
$$ \delta_{ij} + \delta_{ji} = 1,\; = {1},{2},...P,\; = {1},{2},...P,\;i \ne j $$
(8)
$$ \delta_{ij} \in \;{{\{ 0,}}\;{{1\} ,}}\; = {1},{2},...P,\; = \;{1},{2},...P,\;i \ne j $$
(9)
$$ E_{i} \le \;x_{i} \le L_{i} ,\; = \;{1},\;{2},\;...\;P $$
(10)
$$ x_{j} \ge x_{i} + S_{ij} \delta (L_{i} \; - \;E_{j} )\delta_{ji} \; = \;{1},{2},...P,\; = \;{1},{2},...P,\;i \ne j $$
(11)

where, wk represents the weight of fk, \(\mathrm{k}\) k = 1,2,3,4; \({\updelta }_{\mathrm{ij}}\) \(\delta_{ij}\) is the decision variable, if aircraft i lands before j, \({\updelta }_{\mathrm{ij}}=1\) \(\delta_{ij} { = 1}\), otherwise \(\delta_{ij} { = 0}\); Ei is the earliest landing time for aircraft i; Li is the latest landing time for aircraft i; Sij is the minimum time separation between aircraft i and aircraft j.

The objective function (6) minimizes the sum of four objectives for optimizing MOOALP. Constrain (7) ensures the sum of these weights is one. Constrains (8–9) define the landing sequence. It means that flight i must land before flight j or flight j must land before flight i. Constraint (10) ensures that each flight is scheduled in its time window. The time window is bounded by the earliest landing time and the latest landing time. Constrain (11) means that the landing time must satisfy the minimum time separation.

3 Algorithm of MOOALP model

3.1 FCFS algorithm

3.1.1 FCFS service rules

FCFS is a kind of service rule. It means that the flight is arranged to land according to their estimated arriving time to the runway threshold. The first coming flight will be arranged to the first landing [13,14,15].

3.1.2 Scheduled landing time with FCFS rules

Let array \(X = [x(A_{1} ),x(A_{2} ), \cdots ,x(A_{p} )]\) represents the scheduled landing time of each flight, T(Ai) be the estimated landing time of flight Ai, SAi-1,Ai be the safety separation between Ai-1 and Ai. The scheduled landing time x(Ai) is calculated as follows:

$$ \left. {\begin{array}{*{20}c} {x(A_{1} ) = T(A_{1} )} \\ {x(A_{2} ) = \max [T(A_{2} ),(x(A_{1} ) + S_{{A_{1},A_{2} }} )]} \\ \vdots \\ {x(A_{i} ) = \max [T(A_{i} ),(x(A_{{i - 1}} ) + S_{{A_{{i - 1}} ,A_{i} }} ]} \\ \vdots \\ {x(A_{p} ) = \max [T(A_{p} ),(x(A_{{p - 1}} ) + S_{{A_{{p - 1}} ,A_{p} }} ]} \\ \end{array} } \right\} $$
(12)

3.2 Simulated annealing for MOOALP

The FCFS rules cannot get the global optimal solution because the MOOALP belongs to combinatorial optimization. In 1983, S. Kirkpatrick et al. proposed simulated annealing from the annealing process in metallurgy to solve combinatorial optimization problems. When the temperature rises the atoms become disordered and their internal energy increases. During the temperature dropping, the atoms can reach an equilibrium state at every temperature. At the low-temperature state, the atoms reach the ground state and the internal energy decreases to a minimum. According to the Metropolis criterion, the probability that atoms approach equilibrium at temperature T is exp(-ΔE/(k*T)), where, E is the internal energy at temperature T, ΔE is the variation of internal energy, k is Boltzmann's constant [16]. When SA is used to solve combinatorial optimization problems, the internal energy E and temperature T correspond to the objective function value and control parameter, respectively. The pseudo-code of SA is as follows:

figure a

4 Experiments and results

The two methods of FCFS and SA are provided to solve MOOALP. In Fig. 1, there are 11 flights between the outer ring and inner ring. The information of 11 flights is shown in Table 1. The objective function includes four sub-objectives and their weights combinations represent different stakeholder’s interest. To incorporate the different stakeholder’s interest, we set 13 weight combinations.

Table 1 The information of 11 flights between the outer ring and inner ring
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4.1 Results of FCFS

The simulation results of FCFS are unrelated to objective function, and the simulation results are shown in Fig. 2 and Table 2.

Fig. 2
figure 2

The arrival sequence and the scheduled landing instant of FCFS

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Table 2 Results of test1 of FCFS
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The symbols “ο” and “*” represent ELT (Estimated Landing Time) and SLT (Scheduled Landing Time), respectively. The line segment represents the landing interval from the earliest time to the latest time. The rightmost column shows the information of data number, flight number, delay time, and aircraft category. The scheduled landing instant, for example (12:48:12), equals the program start time adding the scheduled landing time in Table 2. The scheduled landing instant of CSN1428 can be calculated as flowing:

$$ 12:28:30\; + \;19{\text{m}}:42{\text{s}}\,1182{\text{s}}\; = \;12:48:12 $$

When the weights take the following values, the total delay time, total fuel cost, makespan, maximum of delay, and variance of delay of FCFS are presented in Table 3.

Table 3 Using FCFS to calculate performance indicators when the weights take the following value
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4.2 Results of SA

The simulation results of SA are related to weight of objective function, and the simulation results of SA of test1 are shown in Fig. 3 and Table 4. The manifestations of simulation results of test 2–13 are similar with test 1.

Fig. 3
figure 3

The arrival sequence and the scheduled landing instant of SA

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Table 4 Results of test1 by SA
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The meaning of symbol and the computing method of scheduled landing instant are same with the Fig. 2.

When the weights take the following values, the total delay time, total fuel cost, makespan, maximum of delay, and variance of delay of SA are presented in Table 5.

Table 5 Using SA to calculate the performance indicators when the weights take the following value
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4.3 Test analysis

In Tables 3 and 5, different weight combination represents different stakeholder’s interest. For example, the optimization goal of Test 1 is minimizing total fuel cost and it is the concern of airlines. Those weight combinations belong to the Pareto front problem. The expert evaluation method is used to select the satisfactory solutions for all stakeholders.

We did an investigation and asked air traffic controllers, airport authorities, airline representatives, and the passengers to rank the five parameters. If the stakeholders think that the parameter is most important, the weight of the parameter is set to 1. In the subdominant place, the weight of the parameter is set to 2, and so on. The results of the investigation are shown in Table 6.

Table 6 Investigation results of different stakeholders
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To make the data have the same measurement scale, MOOALP results are normalized according to the following formula:

$$ z_{ij} = (x_{ij} - \overline{x}_{i} )/\sqrt {\sum\nolimits_{j = 1}^{n} {} (x_{ij} - \overline{x}_{i} )^{2} /(n - 1)} . $$
(13)

where, \(x_{ij}\) is the original value of parameter i in test j; \(\overline{x}_{i}\) is the average value of parameter i; n is the test time; \(z_{ij}\) is the normalized value of \({\mathrm{x}}_{\mathrm{ij}}\) \(x_{ij}\). The normalized results of Table 5 are shown in Table 7.

Table 7 the normalized results of Table 5
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The formula for comprehensive evaluation of test j by the ith stakeholder is as follows.

$$ y_{ij} { = }z_{1j} \cdot \mu_{1i} { + }z_{2j} \cdot \mu_{2i} { + }z_{3j} \cdot \mu_{3i} { + }z_{4j} \cdot \mu_{4i} { + }z_{5j} \cdot \mu_{5i} \, $$
(14)

where yij is the ith stakeholder’s comprehensive score about test j (j = 1,2,…,13), i (i = 1, 2, 3, or 4) respectively represents air traffic controllers, airports, airlines, and passengers.

Comprehensive scores of different stakeholders are shown in Table 8.

Table 8 Comprehensive scores of different stakeholders
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In Table 8, the sums of the comprehensive scores about different stakeholders are presented in the last column. To all interested stakeholders, the smaller the sum is, the more acceptable the test result will be. Test 13 of the SA has the smallest values. In theory, the weights of test 13 are the most reasonable.

4.4 Comparison about FCFS and SA

From Tables 3 and 5, we can see that performance indicators of SA are better than FCFS except test 3. It means that FCFS algorithm can be selected to solve ALP when airport capacity maximization is only considered.

Using the weight of test 13 as the objective function is the first choice for multi-objective optimization. When the weights take values of test13, the performance of FCFS and SA is compared from the computing time, the total delay time, the total fuel cost, the makespan, the maximum delay and the objective function value. The comparison results are shown in the Figs. 4, 5, 6, 7, 8, and 9.

Fig. 4
figure 4

The running time of FCFS and SA

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Fig. 5
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The total delay time of FCFS and SA

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Fig. 6
figure 6

The total fuel cost of FCFS and SA

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Fig. 7
figure 7

The makespan of FCFS and SA

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Fig. 8
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The maximum delay of FCFS and SA

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Fig. 9
figure 9

The objective function value of FCFS and SA

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From the Fig. 4, it is clear that the operation speed of FCFS is faster than that of SA. The running time of SA is much less than that of the aircraft flying to the inner ring, so the running time of SA can meet the control requirements. The total delay time, total fuel cost, maximum of delay and objective function value of FCFS algorithm exceed those of SA algorithm (Figs. 5, 6, 7, 8, and 9), but the total delay time of SA and FCFS is are little difference when the number of flights is less than 10 sorties (Fig. 5). SA has an obvious advantage only when there are more flights.

5 Conclusion

In this paper we discussed the aircraft landing problem and constructed the multi-objective optimization model from minimizing total fuel cost, minimizing total delays, maximizing throughput, and maximizing fairness to solve ALP. The optimization model fully considers the interests of air traffic controllers, airports, airlines and passengers. According to the different weights of objectives, the first-come-first-serve and simulated annealing are used to solve the multi-objective optimization problem. The different weight combination represents different stakeholder’s interest. Those weight combinations belong to the Pareto front problem. The expert evaluation method is used to select the satisfactory solutions for all stakeholders. Although we can always get the optimal weights for specific examples according to the expert's comprehensive score, these parameters are still manual settings. How to get different weights according to different arrival traffic flow by computer programs automatically is the further research.