Multiobjective Gate Assignment Based on Passenger Walking Distance and Fairness

Passenger walking distance is an important index of the airport service quality. How to shorten the walking distance and balance the airlines’ service quality is the focus of much research on airport gate assignment problems. According to the problems of airport passenger service quality, an optimization gate assignmentmodel is established.The gate assignmentmodel is based onminimizing the total walking distance of all passengers and balancing the average walking distance of passengers among different airlines. Lingo is used in the simulation of a large airport gate assignment. Test results show that the optimization model can reduce the average walking distance of passenger effectively, improve the number of flights assigned to gate, balance airline service quality, and enhance the overall service level of airports and airlines. The model provides reference for the airport gate preassignment.


Introduction
Airport gate is a main component of airport resource.Rational and efficient gate assignment is an important way to improve airport operation efficiency and passenger service level.Airport gate is divided into contact gate (a gate with an aerobridge) and remote stand (on the apron).The type and layout lead to different distance from gate to security check, baggage hall, and transit counters.The distance between different areas has a direct impact on passenger activities in the terminal.How to optimize the gate assignment from the perspective of passengers becomes a hot research area at home and abroad.
At present, the main research findings of gate assignment from the perspective of passengers took the shortest passenger walking distance and the minimum embarking and transit time as objective function to optimize gate assignment; for example, Braaksma, [1], Babic et al. [2], Mangoubi and Mathaisel [3], Yan and Huo [4], Bolat [5], Yan et al. [6], and Cheng's [7] research findings showed that reasonable gate assignment could reduce passenger walking distance appropriately.In 1998, Haghani and Chen [8] took the number of transfer passengers of different flights and the distance between different gates into account comprehensively while minimizing passenger walking distance in a terminal.Assuming that all passengers were converted into transit passengers and had taken the shortest transit time as the objective function, Xu and Bailey [9] established quadratic mixed 0-1 integer programming model of gate assignment through virtual assumption.In further research, some scholars began to consider minimizing the number of flights, which are assigned to remote stands, and the passenger walking distance/time, such as Pintea et al. [10], Ding et al. [11,12], and so on; some scholars considered minimizing walking distance together with delay costs, like Zhu et al. [13], balancing usage of airport gates, like Wei and Liu [14,15], passenger waiting time, like Hu and Paolo [16], and fuel consumption of aircraft taxiing, like Maharjan and Matis [17], respectively.
Optimization gate assignment from the perspective of passengers can reduce passenger walking distance and improve passenger service levels to a certain extent.However, there are some deficiencies in research findings.Firstly, in some large hub airporsts, the proportion of transit passengers is large and the actual walking distance of transit passengers is not equal to the actual distance between two gates.The walking distance is related with the layout of transit counters and transit halls.In the research, ignoring transit passengers can cause the model to be inaccurate.Secondly, civil airport service quality issued by the Civil Aviation Administration in 2006 requires that the number of passengers embarking/disembarking through aerobridges should be above 80%.But in most current research, the proportion of passengers is not taken into account.Thirdly, most current researches do not consider the balance of passenger walking distance between different airlines, which can lead to reducing the passenger service level and can be unfair for some airlines, especially for small airlines.
Optimizing gate assignment can improve passengers' satisfaction and balance the service quality of each airline.In this research, we propose a new model which is different from previous researches; the gates are categorized into contact gate and remote stand in the mode, the proportion of passenger embarking/disembarking through aerobridges is taken into account, and the model considers the fairness between airlines besides reducing the overall passengers' walking distance.
The paper is organized as follows.The gate assignment model is detailed in Section 2. Section 3 briefs the simulation software and analyzes the results under different conditions in detail.Some conclusions are drawn in the last section.

Gate Assignment Model
2.1.Description of Gate Assignment.Gate assignment is to arrange a reasonable gate for each arrival-departure flight timely according to the flight plan, which is submitted by every airline.Safety operation of aircraft and gate is the premise of gate assignment.
Passenger walking distance in a large airport is composed of three parts: arrival passenger walking distance, departure passenger walking distance, and transit passenger walking distance.The arrival passenger walking distance refers to the distance from gate to baggage hall; the departure passenger walking distance refers to the distance from security check to gate; the transit passenger walking distance refers to the distance from gate to transit counter and then to the next flight gate.The arrival-departure transit passengers are known collectively as transit passengers in the paper.The walking distance of transit counter passengers includes the arrival passengers' distance from gate to transit counter and the departure passengers' distance from transit counter to gate.
Minimizing and balancing the walking distance of all passengers from different airlines are goals to model gate assignment in the paper.Then Lingo software is adopted to verify the effectiveness of a model in order to improve the service level of airport and airline.

Model Assumptions
(1) Gate assignment is a continuous operation course.In order to reduce the scale of the problem, the paper selects some time intervals for gate assignment.
(2) The capacity of gates can meet the demand of all flights in the research time; it means that every flight can be assigned to a gate.
(3) The arrival-departure flight performed by the same aircraft is assigned to the same gate and it used the same flight number.
(4) All information, such as flight plan, aircraft basic information, the usage status of gates, and so on, is known in research time.
(5) Only the gate assignment of domestic flights is considered in the paper.

Symbol Definition
: flight set,  = { According to the objective function (2), gate assignment may result in longer walking distance of some airlines' passengers while some others are shorter.The objective function (2) can lead to unbalanced passenger walking distance among airlines and reduce the airlines' service quality.Therefore, in order to improve the service quality of the entire airport, it is necessary to balance passenger walking distance of each airline from the viewpoint of airline fairness. Consider The objective function ( 3) is to minimize the ratio between the difference and the average walking distance of all passengers, where    represents the average passenger walking distance of airline   in research period;  represents the average walking distance of all passengers in research period;     represents the ratio of the difference and walking distance of all passengers, where the difference is the average passenger walking distance of airline   and all passengers.
Equation ( 7) is to restrain the proportion of passengers who are required to embark/disembark aircraft through aerobridges.Civil airport service quality, which was issued by the Civil Aviation Administration in 2006, requires that the number of passengers that embark/disembark through aerobridges should be above 80%.Equations ( 8) and ( 9) indicate that each flight has one and only one gate to be assigned.That is, for flight   , in the gate set , there is only one gate   to make    ,  = 1.
Equation ( 10) is used to judge whether the two flights are assigned to the same gate.When    ,  ×    ,  = 1,    ,  = 1, it indicates that the later arrival flight   and the front flight   are arranged in the same gate; otherwise,    ,  = 0.
Equation (11) requires that the two flights which were assigned to the same gate should meet certain safety interval.According to (10) when    ,  = 1, it needs to meet    −    ≥ ; the front and later flights should meet the minimum safety interval.When    ,  = 0, the two flights are not assigned in the same gate; it need not meet safety interval.Therefore, value , which is big enough, is introduced to ensure the inequality holds.
Equation (12) means that the gate type should match the aircraft type.When    ,  = 1, flight   is assigned to gate   ; it should meet    ≤    .When    ,  = 0, there is no relationship between flight   and gate   .
Equation ( 13) is a positive integer constraint.

Simulations
The decision variables in the gate assignment model are 0 and 1, belonging to 0-1 planning of integer programming problem.Due to nonlinear constraints involved in the model, the model is called integer nonlinear programming (INLP).The paper uses Lingo software to simulate and verify the model.The Global (global optimization algorithm) and Multistart (more initial point algorithm) built-in Lingo are specifically used to solve nonlinear programming (Scharge [18]).In addition, Lingo can be connected with EXCEL, database, and other software conveniently; it also can easily input and output the simulation results.Another important superiority of Lingo is convenient to describe large-scale optimization problems concisely and intuitionisticly.Therefore, the paper uses Lingo software to simulate and verify the effectiveness of the model.The simulation data of domestic flights to be assigned in a typical time interval (8:00-11:00) in a large airport is shown in Tables 1 and 2. The minimum time interval  = 15 minutes; this is the time when the two flights are to be assigned to the same gate continuously.The constant value  = 300.The paper uses Lingo 11.0 and selects Global Solver (Global optimization solve) and Global set strategy (Branching: Rel Violation; Box Selection: Worst Bound; Reformulation: High) to verify the effectiveness of models.The paper uses Lingo to simulate the results of the random assignment, the objective function (2) ( 1 optimal) and the objective function (3) ( 2 optimal), respectively.The simulation result is shown in Table 3, where  1 and  2 represent the value of the objective function (2) and the objective function (3), respectively.The smaller  1 is, the shorter passenger walking distance is.The smaller  2 is, the fairer between airlines is. represents the proportion of passengers embarking/disembarking through aerobridge (referring to passing rate); the larger  is, the more passengers embarking/disembarking through aerobridge are. represents the overall average passenger walking distance. 1 - 4 represents the average passenger walking distance of four airlines, respectively. max represents the maximum difference of average passenger walking distance between airlines.(the unit of  1 , ,  1 - 4 , and  max is meters.) With Table 3 and Figures 1 and 2, we can draw the conclusions.
(1) When  1 is optimal, the value of  1 is a minimum.It means that the total passenger walking distance is the shortest.The maximum value of  is 1, which means the passing rate is 100%.But the difference of average passenger walking distance between four airlines is comparatively large and the value of  2 (0.299) is also the largest; it means that the ration between the average and total passenger walking distance of airlines is large; the largest ration is approaching 30%.(2) When  2 is optimal, the value of  2 is approximately zero and the average passenger walking distance of four airlines (Figure 2) is basically flat.That is, the gate assignment is the fairest.But the average passenger distance is 180.8 meters higher than the value of optimal  1 .and the passing rate is only 80% ( = 0.800), which is the lowest in the three simulation groups.
(3) The three values of  1 ,  2 , and  in random gate assignment are relatively concentrated.But compared with the value of optimal  1 , the value of  1 in random gate assignment is high and the value of  is low.Compared with the value of optimal  2 , the value of  2 in random gate assignment is high.And the difference of average passenger walking distance between airlines is the largest, which is 156.5 meters.
The gate assignment schedule is much unfair to airline  4 because the average walking distance is larger than other airlines distinctly.
From the above simulation results, the three groups all have shortcomings.To find a set of ideal solution, the paper takes the objective function (2) as primary objective and transfers the objective function (3) into constraint.Assuming  2 ≤ 0.10 and  2 ≤ 0.20, the simulation results can be acquired (Table 4).
According to Table 4 and Figures 3 and 4, when the objective function (2) is the objective and  2 ≤ 0.10, all the indexes of simulation are in the state of ideal according to the five group simulation results.Compared with the random gate assignment, the simulation result of  2 ≤ 0.10 is as follows.(1) The total walking distance of passengers is 1,368,320 meters and decreases by 34.5%; (2) the passing rate is 96.4% and improves by 13.3%; (3) the total average passenger walking distance is 268.5 meters and decreases by 141.6 meters; (4) the simulation results show that it is relatively fair among airlines.The largest difference of average walking distance among airlines is only 20.3 meters; it is more inferior to the random gate assignment.
Flight Gantt chart of the random gate assignment and the situation of  2 ≤ 0.10 are shown in Figures 5 and 6.Distribution of passengers and the average distance of gate are shown in Figure 7.
It is convenient for passengers to embark/disembark the aircraft through aerobridge because the distance is close and passengers will not be influenced by weather.The average distance from gate to baggage hall, security check, and transit

Random assignment
The ratio of distance difference and passing rate The total walking distance of passengers (m) counter is shorter; the total walking distance of passengers assigned to the gate is shorter.Thus the passenger will feel comfortable.We can draw the conclusions from Figure 5 to Figure 7: (1) the number of flights assigned to the remote stands (G009, G010) is only one, and this gate assignment schedule can improve the passing rate; (2) two flights are reduced to be assigned to gate which is near the remote stand; one flight is added to be assigned to G001, G003, G004, and G008 gate, respectively; (3) the gate, where the average walking distance is short, is assigned efficiently.Making  effective use of a gate can reduce the walking distance and improve the service level of passengers.In summary, the simulation optimization results can not only reduce the average passenger walking distance effectively and improve passing rate, but also reduce the difference of average walking distance of passengers among airlines and enhance the overall passenger service quality of airports and airlines.

Conclusions
The paper presents a new idea for the airport gate assignment problem.Unlike the previous researches, it takes the restraint of passenger passing rate and airlines' fairness into account under the premise of airport safety operation.Combining with the objective of minimizing the whole passengers' walking distances, the paper builds a multiobjective optimization model of gate assignment.Lingo software is used to verify the effectiveness of model by simulating a large airport gate assignment.According to the test results, we can draw some conclusions.
(1) The assignment can ensure the passengers passing rate by setting ( 7).
(2) The two objectives are interactional between each other.And decision makers can get a set of suitable results by adjusting the value range of the second objective.
(3) Compared to the random assignment, this model can reduce the whole passengers' walking distances and improve the fairness between airlines at the same time.
(4) The research scope of the paper is only part of the domestic flights.How to combine with international flights and effective resource schedule should be further researched.

Figure 7 :
Figure 7: Distribution of passengers and the average distance of gate.
1 ,  2 , . . .,   },  is the total number of flights in research period.  (1 ≤  ≤ ) is flight number which is ordered by the arrival time of flights; the bigger  means the later flight   arrives at the airport.   : size of the aircraft which executes flight   .The bigger    is, the larger aircraft is.The smaller    is, the smaller aircraft is.: airline set,  = { 1 ,  2 , . . .,   },  is the total number of airlines in research period.  (1 ≤  ≤ ) is airline code.   : the flights set of airline   .   : size of gate   ; the bigger    means the larger aircraft can be parked on the gate; the smaller    means the smaller aircraft can be parked.   : arrival time of flight   ; the unit is minute.   : departure time of flight   ; the unit is minute.: minimum time interval of two flights which are assigned to the same gate; the unit is minute.    : distance of arrival passenger walking from gate   to baggage hall.    : distance of departure passenger walking from security checking to gate   .    : distance between gate   and transit counter.    : number of arrival passengers of flight   .    : number of departure passengers of flight   .    : number of transit passengers of flight   .Minimizing the total walking distance of all passengers in research period is one of the goals in the paper.Consider min  1 = ∑    ,  (        +         +         ) , (2) where         represents the total walking distance of arrival passengers of flight   walking from gate   to baggage hall;         represents the total walking distance of departure passengers of flight   walking from security checking point to gate   ;         represents the total walking distance of transit passengers of flight   .
Note: L, M, and S represent large, middle, and small aircrafts, respectively.A1∼A4 represent different airlines.

Table 2 :
Data of available gates.
Note: L, M, and S represent large, middle, and small gates, respectively.

Table 3 :
Comparison of simulation results between random assignment, optimization  1 , and optimization  2 .

Table 4 :
Comparison of simulation results under five different conditions.