The Scientific World Journal

Volume 2014 (2014), Article ID 923859, 27 pages

http://dx.doi.org/10.1155/2014/923859

## The Airport Gate Assignment Problem: A Survey

Industrial Engineering Department, College of Engineering, King Saud University, P.O. Box 800, Riyadh 11421, Saudi Arabia

Received 6 August 2014; Accepted 2 September 2014; Published 20 November 2014

Academic Editor: Dehua Xu

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

#### Abstract

The airport gate assignment problem (AGAP) is one of the most important problems operations managers face daily. Many researches have been done to solve this problem and tackle its complexity. The objective of the task is assigning each flight (aircraft) to an available gate while maximizing both conveniences to passengers and the operational efficiency of airport. This objective requires a solution that provides the ability to change and update the gate assignment data on a real time basis. In this paper, we survey the state of the art of these problems and the various methods to obtain the solution. Our survey covers both theoretical and real AGAP with the description of mathematical formulations and resolution methods such as exact algorithms, heuristic algorithms, and metaheuristic algorithms. We also provide a research trend that can inspire researchers about new problems in this area.

#### 1. Introduction

The complexity of airport management has increased significantly. Flight delays or accidents might happen if operations were not handled well, and domino effect might happen to influence the whole operations of airport. In airports, the tasks related to gate assignment problem (AGAP) are one of the most important daily operations many researches have been published on with the aim of solving the problem in spite of its complexity. The objective of the task is assigning each flight (aircraft) to an available gate while maximizing both conveniences to passengers and the operational efficiency of airport. Large airlines typically need to manage different gates across an airport in the most efficient way in a dynamic operational environment. This requires a solution that provides the ability to change and update the gate assignment data on a real time basis. It should also provide robust and efficient disruption management, while maintaining safety, security, and cost efficiency.

Numerous methods have been developed to solve this problem since 1974. Steuart [1] proposed simple stochastic model to find the efficiency use of the gate positions. The research interest in this field was slow in development because there were less than 15 publications within 25 years. However, after 2000, the interest to develop solutions for this problem increased, until nowadays, though with small growth. The objective of this problem varied and depended on the point of view. The first is as an airport owner, which is the government. The objectives are to maximize the utilization of the available gates and terminal [1–4], minimize the number of gate conflicts [5], minimize the number of ungated flights [3, 6–9], and minimize the flights delay [10]. Another point of view is as an airlines owner. Their goals were to increase the customer satisfaction with minimizing the passenger walking distance between gates [3, 6, 7, 11–18] and minimizing the travelling distance from runway to the gate [19].

Dorndorf et al. [20] divided the objectives into five parts, which are reducing the number of the procedures for the costly aircraft towing, minimizing the passengers total walking distance, minimizing the deviations in the schedules, minimizing the number of ungated aircrafts, and maximizing the preferences (i.e., certain aircrafts should go for particular gates). They also defined three usually used constraints, which are the fact that only one aircraft can be gated in a defined amount of time, the fulfillment of the space restriction and service requirements, and the assurance of getting a minimum time between sequent aircrafts and a minimum ground time.

The solution approaches and the solving techniques are varied with no methods, until nowadays, that provide a robust technique for such problem. This study focuses on assessing the trend of solving gate assignment problem in light of the preceding four points. Specifically, this study will address the following research questions. Is this problem NP-hard? What formulation can be defined for such problem? How effective are the recent methods and techniques to solve the problem? What recommendation can be made based on the current findings with regard to research trends?

From a mathematical view, AGAP has been formulated as integer, binary, or mixed integer, general linear or nonlinear models. Specific formulation as binary or mixed binary quadratic models has also been suggested. Other well-known related problems in combinatorial optimization such as quadratic assignment problem (QAP), clique partitioning problem (CPP), and scheduling problem have been used to formulate AGAP. However, few publications on AGAP tackled stochastic or robust optimization.

While the goal of combinatorial optimization research is to find an algorithm that guarantees an optimal solution in polynomial time with respect to the problem size, the main interest in practice is to find a nearly optimal or at least good-quality solution in a reasonable amount of time. Many approaches to solve the GAP have been proposed, varying from Brand and Bound (B&B) to highly esoteric optimization methods. The majority of these methods can be broadly classified as either “exact” algorithms or “heuristic” algorithms. Exact algorithms are those that yield an optimal solution. As discussed in Section 3.1 different exact solution techniques have been used to solve the GAP and in some research, the authors used some optimization programming languages like CPLEX and AMPL.

Basically the GAP is a QAP and it is an NP-hard problem as shown in Obata [21]. Since the AGAP is NP-hard, researchers have suggested various heuristic and metaheuristics approaches for solving the GAP. With heuristic algorithms, theoretically there is a chance to find an optimal solution. That chance can be remote because heuristics often reach a local optimal solution and get stuck at that point. But metaheuristics or “modern heuristics” introduce systematic rules to deal with this problem. The systematic rules avoid local optima or give the ability of moving out of local optima. The common characteristic of these metaheuristics is the use of some mechanisms to avoid local optima. Metaheuristics succeed in leaving the local optimum by temporarily accepting moves that cause worsening of the objective function value. Sections 3.2 and 3.3 addressed the heuristic and metaheuristics approaches for solving the GAP. Some papers presenting good overviews as well as annotated bibliographies on the topic of GAP and a good literature on the AGAP and the use of metaheuristics for AGAP are Dorndorf et al. [20, 22] and Cheng et al. [23].

This paper surveys a large number of models and techniques developed to deal with GAP. In Section 2, we detail the models formulations of the problem. In Section 3, we addressed the resolution methods used to solve the problem. We conclude in Section 4, and we represent the research trends.

#### 2. Formulations of AGAP and Related Problems

Many researchers formulated the AGAP as an integer, binary, or mixed integer linear or nonlinear model and some of them formulated it as binary or mixed binary quadratic models, whereas some of the researchers have formulated the AGAP as well-known related problems in combinatorial optimization such as quadratic assignment problem (QAP), clique partitioning problem (CPP), and scheduling problem or even as a network representation. However, some of the researchers formulated the AGAP as a robust optimization model. In this section, according to the way of how the researchers deal with the gate assignment problem, a classification for the AGAP has been made as follows.

##### 2.1. Selected AGAP Formulations

###### 2.1.1. Integer Linear Programming Formulations (IP)

Lim et al. [24] formulated the AGAP as an integer programming model and developed two models with time windows. The first model was devoted to minimization of the passenger walking distance (travel time) while the second model optimized the gate assignments with cargo handling costs: Both of these objectives put a penalty function due to a delay. These two objectives used the constraints, as follows: where , , and are binary and is integer.

Constraint (3) ensures that each flight must be assigned to exactly one gate. Constraints (4)-(5) state that a binary variable can be equal to one if flight is assigned to gate () and flight is assigned to gate (). Constraint (6) further specifies the necessary condition that must be equal to one if and . Constraints (7) and (8) ensure that the flight must land and depart within the specified time window. Constraint (9) indicates that = 1 if , which means when flight departs before or right at the time when some gate opens for flight . Constraint (10) states that if , which means when flight departs after some gate opens for flight . Constraint (11) specifies that one gate cannot be occupied by two different flights simultaneously.

In the first model and according to the linearity of the objective function and constraints, they used a standard IP solver (CPLEX) to find the optimal solution, whereas in the second model authors used several heuristic algorithms, namely, the “Insert Move Algorithm,” the “Interval Exchange Move Algorithm,” and a “Greedy Algorithm” to generate solutions. The generated solutions then have been improved using a tabu search (TS) and memetic algorithm. The results showed that the used heuristics performed better than the IP solver (CPLEX) in both CPU time and solutions quality.

Diepen et al. [25, 26] formulated the AGAP as integer linear programming model with a relaxation for the integrality. After relaxing the integrality, the resulting relaxed LP was exploited to obtain solutions of ILP by using column generation (CG). The problem was divided into two phases, planning and attaching. The first phase was the planning section and it was easier to model and calculate. Their objective is to minimize the cost of a gate plan. They proposed the following model: subject to where This constraint defined the high price penalty () when the flights were not assigned to the gates. This penalty appeared since the planner should do the assignment manually. In addition, they added another constraint regarding the assignment since there was possibility that a long stay flight could be split into two parts. The extra flights and that refer to the arrival and departure of flight were added to the previous constraints: where where and denotes the total number of preferences.

This constraint defined the flight preferences; for example, a flight should be assigned to the same gate due to the ownership or security. The coefficient 0.5 refers to the extra flight defined in constraint (15).

They checked the solution’s optimality using pricing problem (minimum reduced cost) since they had dual multipliers , , and for constraints (15), (16), and (19), respectively: The second phase was a matter of assignment in physical gate. They made the rules to solve this phase as follows.(i)Sort the gates based upon the quality.(ii)Sort the gate plans from the highest on the total number of departing passengers that are on the flights in that gate plan.(iii)Assign the gate plan to the best gate considering the highest number of departing passengers, assign the next gate plan to the next-best gate, and so on.In [26], Diepen et al. used the solution obtained from their assignment of gates as an input to solve the bus-planning problem in the same airport.

###### 2.1.2. Binary Integer Programming

In 2009, Tang et al. [27] formulated the AGAP as a binary integer programing model as below. The output model was used to generate a lower bound to their original problem: subject to

*Parameter Variables*:time inconsistency value indicating that the flight is assigned at the time point (starting time); if is equal to the original time point, then ;:space inconsistency value indicating that the flight is assigned to the gate; if equals the original gate, then .

The following sets have been defined::considered flights;:available gates;:gates that the th flight can be assigned to;:time points in which the th flight can be assigned to the th gate;:flights that can be assigned to the th gate so that their time windows will cover the th time point;:all time points (i.e., the time points from the planning time at each stage to the end of daily operations);:time points (starting times) assigned to the th flight, where the resulting time windows cover the th time point;:conflicting flight pairs for the th adjacent gate pair;:flights included in the th conflicting flight pair for the th adjacent gate pair;:adjacent gate pairs.Equation (23) is the flight constraint, indicating that every flight is exactly assigned to a gate. Equation (24) is the gate constraint, ensuring that every gate is assigned to at most one flight at any time. Constraint (25) is related gate adjacency, denoting that two conflicting flights cannot be concurrently assigned to an adjacent gate pair. Constraint (26) indicates that the assignment variables are either zero or one.

Kumar et al. [18] presented a binary integer programing model that produced a feasible gate plan in the light of all the business constraints: subject to Constraint (28) ensures that turn is assigned to at most one gate. Constraint (29) states that turn is assigned to a gate only if its equipment type is among the types which the assigned gate can accommodate. Constraint (30) restricts the number of ungated turns to less than or equal to the allowed number . Constraint (31) shows that, at any given time, at most one turn is assigned to one gate. Constraint (32) ensures that adjacency constraints are observed. Constraints (33)-(34) enforce LIFO restrictions. Constraint (35) guarantees that pushback restrictions are observed. Constraint (36) confirms that no turn is towed if towing is not allowed. Finally, constraints (37)–(39) certify that if a long turn is towed, the variable is set to be 1.

Mangoubi and Mathaisel [11] also developed a binary integer model to minimize the passenger total walking distance and proposed a heuristic method to find the solution. The heuristic method result has been compared with the results from a standard IP solver and the comparison results showed that the heuristic method was superior to the LP solver; the average walking distance using the LP is 527 feet while heuristic is 558 feet. The developed model is introduced as follows: where Transfer passenger walking distances are determined from a uniform probability distribution of all intergate walking distances. The expected walking distance if is the distance between gate and gate is subject to Constraint (43) shows that each flight is assigned to at most one gate. Constraint (44) ensures that no two planes are assigned to the same gate concurrently. Constraint (45) determines the conflict constraint for each gate . Constraint (46) is written to consider only the constraint generated by the last plane of two or more flights arriving with no departure in between. Constraint (47) ensures that flights are assigned to nearby gates. Constraint (48) assigns flight to gate , where is the gate with the minimum total passenger walking distance for flight .

Vanderstraeten and Bergeron [28] formulated the GAP as a binary integer model but with the objective of minimizing the off-gate events and they developed a new heuristic, which is the “Affectation Directe des Avions aux Portes (ADAP),” to solve the developed model. A real case has been studied in an Air Canada terminal. A new heuristic was applied to real data at Toronto International Airport. The developed model was as follows: subject to Constraint (50) ensures that each flight is assigned to at most one gate. Constraint (51) defines the occupation time at any gate. Constraint (52) determines the neighboring constraint. Constraint (53) expresses the binary constraint for all decision variables. The results showed that using the developed method resulted in no more than 30 events ever being handled off gate while the manual procedure obtained events up to 50 of the 300 events being handled off gate.

Bihr [12] developed a binary integer model to minimize the passenger walking distance and applied this model to solve a sample problem using primal-dual simplex algorithm. As a result, he obtained a total walking distance of 22,640. The developed model is introduced as follows: where the are the elements of the matrix product of and number of passengers arriving on flight and departing from gate ; number of passengers − distance units from gate to gate ; or subject to In 2002, Yan et al. [29] formulated the static GAP as a binary integer programing model to serve as a basis of real time gate assignments in a simulation framework developed to analyze the effects of stochastic flight delays on static gate assignments. The presented model is as follows: subject to Constraint (57) ensures that each flight is assigned to at most one gate. Constraint (58) certifies that no two planes are assigned to the same gate concurrently. Constraint (59) is related to the binary constraint for all decision variables. Two greedy heuristics were used to solve the model and their results were compared with the insights of the optimization method. The simulation framework was tested to solve certain real case instances from CKS airport. The results of the used methods were 24,562,588 for the optimization model and 27,833,552 and 30,166,809 (meters) for the two greedy heuristics.

###### 2.1.3. Mixed Integer Linear Programming (MILP)

Bolat [30] formulated a mixed integer program for the AGAP with the objective of minimizing the range of slack times (slack time is an idle time between two successive utilizations of the gate). Certain instances, with more than 20 gates, have been considered according to airplane types, gate types, terminal types, and utilization levels: subject to The results related to expected average utilizations were, respectively, 88.54%, 67.13%, and 45.57% over heavily utilized, normally utilized, and underutilized problems. Concerning the average number of flights, results were 10%, 7.59%, and 5.15% per gate.

In 2001, Bolat [31] presented a framework for the GAP that transformed the nonlinear binary models (it will be discussed in Section 2.1.4 according to our classification) into an equivalent linear binary model with the objective of minimizing the range or the variance of the idle times. The framework consists of five mathematical models, where two of the five models were formulated as a mixed integer linear programming and the others as a mixed integer nonlinear programming. Models P1 to P4 were defined for homogenous gate while model P5 was defined for heterogeneous gate: Using the presented framework, nonlinear model P1 (model P1 will be discussed in Section 2.1.4 according to our classification) was transformed to the following mixed integer linear model, which is model P2.

*Model P2*. Consider
subject to
Similarly, for model P3 (Section 2.1.4), the resultant model was model P4 that is a mixed binary model as in model P2, but with two additional real variables as follows.

*Model P4*. Consider
subject to
Different instances have been studied according to the number of the gates: small (five gates), medium (10 gates), and large (20 gates). Instances with more than 20 gates were not considered. The results were as follows: average numbers of flight were 26.125, 52.25, and 105.417 and the average utilizations were 45.725, 66.548, and 88.871% according to the gate size, respectively.

Şeker and Noyan [9] formulate the GAP as a mixed integer program with the objective of minimizing the number of conflicts and at the same time minimizing the total semideviation between idle time and buffer time: Another model was developed as a mixed integer program for the same objective function. The model was the same as the previous model but with some differences: where These two models have the same constraints properties, while objective (89) has the following additional constraints:

###### 2.1.4. Mixed Integer Nonlinear Programming

Li [5] formulated the GAP as a nonlinear binary mixed integer model hybrid with a constraint programing in order to minimize the number of gate conflicts of any two adjacent aircrafts assigned to the same gate. The developed model has been solved using CPLEX software: where where : scheduled arriving time, : scheduled departure time, and : buffer time (constant). Consider In another work, Li [32] defined the objective as These two models have the same constraints; all constraints are as follows. Constraint (107) indicates that each aircraft is assigned to at most only one gate. Constraint (108) represents a method to compute the auxiliary variable from . Constraint (109) ensures that one gate can only be assigned at most one aircraft at the same time. Some additional constraints in the real operations are ignored. Constraint (110) represents binary value of the decision variables.

As mentioned in Section 2.1.3, Bolat [31] proposed two models formulated as a mixed integer linear program which have been transformed from a mixed integer nonlinear program. The proposed mixed integer nonlinear program was as follows:

*Model P1.* Consider
subject to
Bolat [31] also proposed two alternative formulations for homogenous and heterogeneous gates. The proposed extended formulation for the homogenous gates was as follows:

*Model P3*. Consider
subject to
In addition, the proposed extended formulation for the heterogeneous gates was as follows:

*Model P5.* Consider
subject to
As mentioned in Section 2.1.3, different instances have been studied according to the number of the gates: small (five gates), medium (10 gates), and large (20 gates). Instances with more than 20 gates were not considered. The results were as follows: average numbers of flight were 26.125, 52.25, and 105.417 and the average utilizations were 45.725, 66.548, and 88.871% according to the gate size, respectively.

###### 2.1.5. Quadratic Programming

*(**1) Quadratic Mixed Binary Programming.* Zheng et al. [33] formulated the GAP as a mixed binary quadratic program with minimizing the slack time overall variance as the objective function; an assumption has been stated such that the flights are sequenced with the smallest arrival time. The proposed mixed binary quadratic model was as follows:
subject to
where the indices in (143)–(151) denote , . Equation (143) represents the objective function with the aim of minimizing overall variance of slack time. Constraint (144) imposes the assignment of every flight to one gate. Constraint (145) obliges every flight to have at most one immediate precedent flight. Constraint (146) enforces every flight to have at most one immediate succeeding flight. Constraints (147) and (148) define the first and last slack time of each gate, and constraint (149) defines the other slack times. Constraint (150) stipulates that the flight can be assigned to the gate when the preceding flight has departed for dwell time. Constraint (151) indicates that the different type of gate allows parking different type of flight.

Solutions were obtained using tabu search based on some initial (starting) solutions; the results were compared with those of a random algorithm developed in the literature. Using data from Beijing International Airport (10 gates and 100 of flights between 6:00 and 16:00), the initial solutions using metaheuristic and random algorithm were 9821 and 15775, respectively.

Bolat [34] formulated the AGAP as a mixed binary quadratic programming model to minimize the variance of idle times and used branch and bound algorithm and proposed two heuristics which were “single pass heuristic” (SPH) and “heuristic branch and bound” (HBB) for solving the proposed model. The proposed mixed binary quadratic model was stated as follows: subject to Real instances, from King Khalid International Airport (72 generated sets), were used. During the initial phase, the proposed heuristic methods gave an average improvement of 87.39% on the number of remote assigned flights, whereas the average improvement on the number of towed aircrafts during the real time phase was 76.19%.

Xu and Bailey [14] formulated the GAP as a mixed binary quadratic programming model (Model 1) and the objective was to minimize the passenger connection time. The proposed model (Model 1) was reformulated (linearized) into another model (Model 2) in which the objective function and the constraints have been linearized (the resultant model was a mixed binary integer model). Model 1 and Model 2 are listed below.

*Model 1.* Consider
subject to
where objective function (163) seeks to minimize the total connection times by passengers. Constraint (164) specifies that every flight must be assigned to one gate. Constraint (165) indicates that every flight can have at most one flight immediately followed at the same gate. Constraint (166) indicates that every flight can have at most one preceding flight at the same gate. Constraints (167) and (168) stipulate that a gate must open for boarding on a flight during the time between its arrival and departure and also must allow sufficient time for handling the passenger boarding, which is assumed to be proportional to the number of passengers going on board. Constraint (169) establishes the precedence relationship for the binary variable and the time variables and and is only effective when . It stipulates that if flight is assigned immediately before flight to the same gate , the gate must open for flight earlier than for flight . Therefore, it ensures each gate only serves one flight at any particular time. Constraint (170) further states that the aircraft can only arrive at the gate when the previous flight has departed for certain time.

*Model 2.* Consider
subject to
where constraints (186) and (187) state that a binary variable can be equal to one if flight is assigned to gate () and flight is assigned to gate 1 (). Constraint (188) further gives the necessary condition which is that must be equal to one if and .

The B&B and tabu search algorithm were used to solve the generated instances (seven instances, up to 400 flights and 50 gates for 5 consecutive working days). The results of the analyzed instances showed an average saving of the connection time of 24.7%.

*(**2) Binary Quadratic Programming.* Ding et al. [6, 35] developed a binary quadratic programming model for the overconstrained AGAP to minimize the number of ungated flights. A greedy algorithm was designed to obtain an initial solution, which has been improved using tabu search (TS). The developed model was stated as follows:
subject to
where constraint (192) ensures that every flight must be assigned to one and only one gate or assigned to the apron. Constraint (193) specifies that the departure time of each flight is later than its arrival time. Constraint (194) says that an assigned gate cannot admit overlapping the schedule of two flights.

In 2005, Ding et al. [7] developed a binary quadratic programming model for the overconstrained AGAP to minimize the number of ungated flights. The developed model was as follows: subject to where constraint (199) ensures that every flight must be assigned to one and only one gate or assigned to the apron and constraint (200) requires that flights cannot overlap if they are assigned to the same gate.

Using the same case study by Ding et al. [6, 35], a greedy algorithm was designed to obtain an initial solution, which has been improved using simulated annealing (SA) and a hybrid of simulated annealing and tabu search (SA-TS).

###### 2.1.6. Multiple Objective AGAP Formulations

Hu and Di Paolo [36] mathematically formulated the multiobjective GAP (MOGAP) as a minimization problem and solved this problem using a new genetic algorithm with uniform crossover. The developed MOGAP model was presented as follows: subject to Wei and Liu [16] considered the AGAP as a fuzzy model and adopted a hybrid genetic algorithm to solve the developed model. The main objectives were minimizing passengers’ total walking distance and gates idle times variance. They developed the following model: subject to where objective function (212) reflects the total walking distance of passengers. is 0-1 variable; if flight is assigned to gate ; otherwise it is 0; describes the number of passengers transferring from flight to , and is walking distance for passenger from gate to . Objective function (213) is used as a surrogate for the variance of idle times. The actual number of assignments is and the number of nondummy idle times is . Constraint (214) indicates that every flight must be assigned to one gate. Constraint (215) shows that flights that have overlap schedule cannot be assigned to the same gate, where is the least safe time between continuous aircrafts assigned to the same gate. Constraint (216) denotes that is a binary variable.

In 2001, Yan and Huo [2] formulated the AGAP as a model with two objectives: minimizing (3) the walking distance, and (4) the waiting time for the passengers. The proposed mathematical model is binary integer linear programming: subject to where objective (217a) represents the minimum total passenger walking distance. Objective (217b) represents the minimum total passenger waiting time. Constraint (217c) denotes that every flight must be assigned to one and only one gate. Constraint (217d) ensures that at most one aircraft is assigned to every gate in every time window.

Column generation approach, simplex method, and B&B algorithm were used to solve the proposed problem, which was a case study in Chiang Kai-Shek Airport, Taiwan. The problem consisted of 24 gates (of which two were temporary; eight out of 24 gates were only available for the wide type of aircrafts, whereas the rest were available for the other types) and 145 flights. The results showed that the obtained solution (7,300,660 s the best feasible solution found so far) was away from the optimal one by 0.077% (5595s).

Wipro Technologies [17] proposed a binary multiple objective integer quadratic programming model for the AGAP with a quadratic objective function. The proposed model was reformulated into a mixed binary integer linear programming model (linear objective functions and constraints). The proposed model has been solved using greedy heuristic, SA, and TS (MIP solvers based B&B cannot solve the proposed model within a reasonable time). The developed model was represented as follows.

*Generic Model.* Consider
subject to