Solving Large Aircraft Landing Problems on Multiple Runways by Applying a Constraint Programming Approach

Transcription

1 Solving Large Aircraft Landing Problems on Multiple Runways by Applying a Constraint Programming Approach Amir Salehipour School of Mathematical and Physical Sciences, The University of Newcastle, Australia a.salehipour@newcastle.edu.au ABSTRACT 1. INTRODUCTION Aircraft Landing Problem is to assign an airport s runways to the arrival aircraft as well as to schedule the landing time of these aircraft. In this paper, due to the complexity of the problem, which is NP-hard, we develop an iterative-based heuristic by exploiting special characteristics of the problem. Computational results show the developed approach is quite competitive, and obtain optimal and near optimal solutions for instances with up to 500 aircraft in a reasonable amount of time. The algorithm outperforms the best known solutions available for the problem, while it s required computational time is at most 10% of that of the best available algorithm for the problem. KEYWORDS: Aircraft Scheduling, Mixed-Integer Programming, Iterative Heuristic The cost of air transportation is much higher than that of other transportations means. Nevertheless, a major volume of transportation is carried by aircraft. The cost associated with landing aircraft, if deviated from its target landing time, is considered to be a substantial amount. This turns minimization of this cost to be a very important saving area, and hence, has given the optimal aircraft landing schedule as one of the most important air related operational schedule. The objective of this paper is to minimize the landing cost of aircraft by appropriate scheduling. The aircraft landing problem consists of determining an optimal schedule of landing aircraft on airport s runways and also allocating the arriving aircraft to the available runaways with the objective of minimizing the total deviation of landing time from the target time (both earliness and tardiness). This problem is known to be NP-hard which makes impossible to solve large-scale instances in a reasonable amount of time. Therefore, the major contribution of this paper has been given to derive a practical solution approach to derive near-optimal, and preferably optimal solutions for problems up to 500 aircraft on multiple runways. Despite the efforts taken towards solution approaches for the problem, either exact or heuristics, there have not been a considerable progress regarding solving the problem to optimality. To the best of our knowledge, the optimal solutions have been existed for instances of up to 50 aircraft (Beasley et al., 2000; Pinol and Beasley, 2006). Recently, Salehipour et al., (2013) 1

2 extended that to instances with 100 aircraft, and report new best found solutions for instances with up to 200 aircraft. In the following we discuss the previous research on this problem by reviewing the most important research in this area. Ernst et al. (1999) developed a special purpose simplex algorithm for this problem, and obtained an optimal solution for instances with 44 aircraft and 4 runways. Beasley et al. (2000) presented a mixed-integer zero-one formulation of the problem for a single runway and extended it to multiple runways. They provided optimal solutions for instances up to 50 aircraft. Following their previous work, Beasley et al. (2004) studied the dynamic variant of the problem. Pinol and Beasly (2006) developed a scatter search meta-heuristic to solve the instances with 500 aircraft, although their research on finding optimal solution is limited to the instances up to 50 aircraft. Hansen (2004) considered a simplified variant of the problem by taking into account different assumptions, and reported results for instances with up to 20 aircraft. Bäuerle et al. (2007) developed a model for the landing aircraft based on the queuing theory concept. They solved instances with up to 75 aircraft on two runways. Soomer and Franx (2008) studied the single runway problem where airlines provide cost functions related to arrival delays for their flights. They implemented a local search heuristic to obtain reasonable solutions for instances of up to 100 aircraft. Artiouchine et al. (2008) described several special cases that can be solved in polynomial time. They provided a mixed integer programming formulation to solve special cases of the problem, and a general hybrid branch-and-cut framework to solve the problem with arbitrary time windows. Salehipour et al., (2009) considered the runway-dependent case and developed a Variable Neighborhood Descent (VND) algorithm. As these studies show, both exact and heuristic algorithms have been developed for the problem. Implementing the heuristic and meta-heuristic algorithms do not guarantee optimal solutions, nevertheless, the best found solutions to date have been reported by applying these algorithms. The paper is organized as follows. In Section 2 the problem is formulated and a mixed-integer programming formulation from the literature is discussed. Following the complexity of the problem, in Section 3 several properties of the problem are discussed. Later in the Section we show how applying these properties would result in deriving very high quality and near optimal solutions, specifically for larger instances. Section 4 is devoted to the computational results. Paper ends with the conclusion. 2. PROBLEM FORMULATION As previously stated, the aircraft landing problem is to determine an optimal allocation of aircraft to land on airport s runway, while an optimal schedule of landing is maintained. Minimizing the total cost of both early and delayed landings, with respect to the pre-specified target landing time is the objective of this problem. Both early and late landings (than target landing time) are penalized by a cost function associated with early and late landings. In most cases, these two costs are not equal and thus, to generalize these two costs we deal with different cost ratios. It is important to mention that while the target landing time can be determined before the landing is scheduled, the real landing time, i.e. landing schedule, is an operational-dependent time. This implies that an aircraft may not land on its target landing time; hence a cost penalty is incurred. Apparent from this target landing time, in reality a time window is considered for each aircraft s landing, defining earliest and latest landing times allowed for each aircraft. Besides, a safe landing requires a separation time between every two aircraft scheduled to land either on a same 2

3 runway or on different runways. In reality, this separation time is dependent on the aircraft s type (Heavy, Large, Small, Cargo, etc.), and not on the runway. Thus, the separation time for every two aircraft landing on a same runway should be different than landing on different runways. Without loss of generality, we assume the separation time for the latter case is 1 time unit. This assumption has been taken by others in the literature (Beasley et al., 2000; Pinol and Beasley, 2006; Salehipour et al., 2013). Pinol and Beasley (2006) presented an open formulation for the problem based on Beasley et al. (2000). By open, we mean different side constraints which may be applied according to the situation in hand. In fact, their mixed-integer formulation is quite simple and effective. Recently, Salehipour et al., (2013) have simplified the formulation of Beasley et al. (2000), and developed another mixed-integer programming formulation which is simpler in basis than that of Beasley et al. (2000). But neither an advantage in the computational time, nor in the reported gaps was obtained. To give a clear understanding of the problem, and the solution approach developed later, here we explain Beasely et al. s formulation (2000) together with the variables and parameters definitions. Decision Variables x i : The scheduled landing time of aircraft i (i = 1,, n). y ii : Takes 1 if aircraft i lands before aircraft j and otherwise 0. γ ii : Takes 1 if aircraft i is allocated to runway r (r = 1,, m) and otherwise 0. δ ii : Takes 1 if aircraft i and j land on a same runway and otherwise 0. a i : The delay of landing aircraft i (landing after the target time), thus a i = max (0, x i T i ). b i : The earliness of landing aircraft i (landing before the target time), thus b i = max(0, T i x i ). Parameters s ii : The separation time between two aircraft i and j when landing on a same runway. T i : The target landing time (target time) of aircraft i. E i : Earliest landing time of aircraft i. L i : Latest landing time of aircraft i. c i + : The cost of late landing of aircraft i. c i : The cost of early landing of aircraft i. The mathematical formulation is as follows. n Min z = i=1 (a i c + i + b i c i ) (1) S.t. E i x i L i i = 1,, n (2) x i T i = a i + b i i = 1,, n (3) T i x i a i T i E i i = 1,, n (4) 3

4 x i T i b i L i T i i = 1,, n (5) x j x i s ii δ ii + (1 δ ii ) My jj i, j = 1,, n, i j (6) y ii + y jj = 1 i, j = 1,, n, i j (7) δ ii γ ii + γ jj 1 i, j = 1,, n, i j, r = 1,, m (8) m r=1 γ ii = 1 i = 1,, n (9) δ ii = δ jj i, j = 1,, n, i j (10) y ii, γ ii, δ ii {0,1} i, j = 1,, n, i j, r = 1,, m (11) x i, a i, b i 0 i = 1,, n (12) Objective function (1) minimizes the total cost of landing not on target landing time. Constraints (2) ensure every aircraft lands in its time window. Coupling constraints (3), (4), and (5) link the decision variables x i and parameters T i to decision variables a i and b i. Constraints (6) impose if aircraft j lands right after aircraft i on a same runway, as least s ii time unit should be elapsed before aircraft j could be land, and otherwise 1 time unit should be elapsed. Constraints (7) ensure the feasibility of aircraft s landing. Coupling constraints (8) link the decision variables δ ii and γ ii. Assignment constraints (9) ensure allocation of each aircraft to only one runway. Constraints (10) ensure if δ ii = 1 then δ jj = 1. Constraints (11) and (12) ensure the decision variables δ ii, γ ii and y ii are binary, and the decision variables x i, a i and b i are nonnegative, respectively. Salehipour et al., (2013) applied this formulation, and implemented Cplex solver from ILOG over it. Changing parameters of Cplex enable them to report optimal solutions for instances with up to 100 aircraft and 4 runways in a short time. To date, their results are the first optimal solution for instances with 100 aircraft. However, due to the complexity behind the problem (according to Beasley et al this problem is NP-Hard), and in spite of this efficient formulation, and capability of Cplex in solving integer and mixed-integer programming, the optimal solutions for instances with more than 100 aircraft is very difficult to reach in a reasonable amount of time. Besides, no authors except Salehipour et al., (2013), and Pinol and Beasley (2006) have tried problems with more than 100 aircraft either by heuristic or by exact solution approaches. Thus, to solve larger instances we studied the problem in details and developed an iterative heuristic by introducing a set of constraints and iteratively modifying them which are capable of deriving very sharp upper bounds (UBs). The added constraints do not guarantee optimal solutions. However, these UBs are much better than the best found solutions reported by Pinol and Beasley (2006), and surprisingly by Salehipour et al., (2013). Furthermore, they could be reached much quicker, at least 9 times faster than that of Salehipour and Modarres s, and approximately 40 times faster than that of Pinol and Beasley s on a same machine. Another advantage of the proposed solution approach, is that it is purely lied on an exact solver (like Cplex from ILOG), which guarantees the robustness of the solution approach. 4

5 3. AN INTERATIVE HEURISTIC FOR AIRCRAFT LANDING PROBLEM In this section we develop a class of constraints for the aircraft landing problem. These constraints are developed according to the characteristic of the problem. We shall discuss an example to explicate more these constraints Example Assume the following aircraft landing data (Table 1) for 10 aircraft and two runways (n = 10 and m = 2). These data are of problem Airland1 from Beasley et al. (2000). Table 1. Aircraft landing data for problem of example Aircraft E i T i L i The separation times between every two aircraft landing on same runway are shown in Table 2. Table 2. The separation time for problem of example If someone would like to derive an intuitive solution for the problem, at first, aircraft should be sorted according to their target landing times (T i, i = 1,, n). This will result in Table 3. Sorted aircraft based on T i Aircraft E i T i L i

6 In this intuitive solution, aircraft 3 is the first aircraft to schedule for landing on runway 1 followed by aircraft 4. The separation time between these two aircraft (s ii ) is 8, which means aircraft 4 cannot land before time 106 on runway 1. Although this aircraft could be land not any sooner than time 99 on runway 2, of course at the cost of 7 units of deviation from its target landing time, i.e. T 4. Similarly, aircraft 5 and 6 can be scheduled to land on runway 1 on their target landing times, thus no deviation cost from their designated target landing times is considered. Aircraft 7 can be scheduled to land no sooner than time 143 on runway 1, assuming the previous aircraft land exactly on their target landing times. This would impose a cost of 5 unit delay for this aircraft. On the other hand, if we schedule this aircraft to land on runway 2 (different runway from aircraft 6), it can be scheduled to land on time 138 which leaves no delay for aircraft 7. Thus it would be beneficial to schedule aircraft 6 and 7 not to land on a same runway. We could import this into the mixed-integer programming formulation of Section 2 as γ 6r + γ 7r 1, r = 1,2. Adding a complete set of these constraints to the mixed-integer programming formulation would reduce the required computational time substantially, specifically for larger instances, as computation experiments show this. A closer look at the problem reveals that it would also be better not to land aircraft 7 and 8, and 9 and 1 on a same runway, thus γ 7r + γ 8r 1 and γ 9r + γ 1r 1 where r = 1,2. This results in a complete set of following constraints for the example: γ 6r + γ 7r 1, r = 1,2 γ 7r + γ 8r 1, r = 1,2 γ 9r + γ 1r 1, r = 1,2 Depicting the Gantt chart for this example reveals this fact (see Figure 1). Note that if we use more than two runways here, we yield a total objective function of 0, as aircraft 8 could be scheduled to land on its target time on runway 3. Runway Figure 1. The Gantt chart of aircraft landing schedule explained in the example Landing Time (not scaled) 6

7 3.2. A Constraint Programming Approach for Aircraft Landing Problem The previous example and Figure 1 carry a fundamental observation regarding the solution approach, i.e. the iterative heuristic, provided here in this paper. Observation 1. Assume r > 1 and all T i (i = 1,, n) are sorted in ascending order. Thus T i1 T i2 T in 1 T in. For each two sorted aircraft i k and i k+1, if T ik+1 < T ik + s ik i k+1 then a better schedule, not necessarily optimal, can be found, if we can schedule these two aircraft to land on different runways. Proof. Trivial. A direct implementation of the observation 1 is that it can cut the solution space which will simplify the solution process. The drawback of the new constraints is that they do not guarantee an optimal solution. This is because these constraints imposed two arbitrary aircraft to land on two different runways, while they may land on a same runway (when the cost of deviation from target landing time if landing on a same runway is less than the total deviation of these two aircraft from their target landing times if landing on two different runways). Nevertheless, the quality of solutions applied by this approach and the reported computational times are very good. Observation 2. Following Observation 1, if two aircraft j and k may not land on the same runway, a constraint of the form γ jj + γ kk 1, r can be derived. Proof. Trivial. Observation 3. Assume > 0, generating constraints via T ik+1 + λ < T ik + s ik i k+1 may result in an improved objective function value, if and only if λ = 0 does not result to optimal solution. Proof. Adding a positive value (λ) to the left of T ik+1 < T ik + s ik i k+1 limits the constraints generation process. Thus a reduced number of constraints are generated. Note that the larger the value λ the less the number of generated constraints. According to the mixed-integer programming formulation and the explanation given above, the reduced number of constraints guarantees improved objective function value, of course, at the cost of higher computational time. According to the experiments performed, adding these constraints to the mixed-integer programming formulation of Section 2 can develop a landing schedule of aircraft which has a very competitive quality. Nevertheless, if not optimal, this initial schedule could easily be improved using different values for parameter λ. Note that increasing the value of λ will increase the required computational time. Thus someone can make a trade-off between the quality of solution and the computational time. Based on the Observation 3, the outline of the developed solution approach is illustrated in Algorithm 1. Algorithm 1. The iterative heuristic for aircraft landing problem Until stopping condition is met do Step 1. Generate constraints by applying Observation 2. Step 2. Incorporate generated constraints into mixed-integer programming formulation (Section 2); Step 3. Solve the mixed-integer programming formulation. 7

8 4. COMPUTATIONAL RESULTS In this section we report the computational results of the iterative heuristic solution approach proposed in Section 3. For comparison purposes we shall report the results of Pinol and Beasley (2006), and that of Salehipour et al., (2013). The results of Pinol and Beasley (2006) were obtained by applying a Scatter Search (SS) meta-heuristic while the results of Salehipour et al., (2013) were obtained by applying a hybrid Simulated Annealing + Variable Neighborhood Search (SA+VND) with an effective construction heuristic. We have solved a set of 13 instances from the literature (Pinol and Beasley, 2006) ranging from 10 to 500 aircraft and 2 to 5 runways. We benefitted from Cplex 12.1 from ILOG to solve to optimality instances with up to 100 aircraft. All computational results have been carried out on a personal computer with 2 GHz CPU and 512 MB memory (The same PC specification as appeared in Pinol and Beasley, 2006 to carry out direct comparison). Table 4 reports the complete computational results of the first 9 instances where optimal solutions were reported by Salehipour et al., (2013). In this table, we have reported the computational results of the developed iterative heuristic procedure, and the exact optimal solution. Throughout this table, the first column is the instance name, the second and the third columns are the number of aircraft and the number of runways for each instance, respectively. The remaining columns are associated with the objective function value and the total computational time in seconds for each reported solution approaches. Those gaps in the last column are gaps from objective function of iterative heuristic approach calculated over the optimal solution. Note that except four cases where the iterative heuristic approach did not report optimal solution, it yielded optimal solutions for the all remaining 16 cases, thus on 80% of cases (out of 20 cases with up to 100 aircraft). These optimal solutions were shown in bold. This is also notable that the iterative heuristic approach is approximately 10 times faster than the optimal approach. Table 4. Results for instances with up to 100 aircraft using exact and iterative heuristic procedures (Salehipour et al., 2013) This paper Instance n m Optimal Iterative heuristic Obj T (s) Obj T (s) GAP(%) Airland Airland Airland Airland Airland Airland Airland Airland Airland Average

9 For those instances, that the iterative heuristic approach could not report optimal solution, optimal solutions can be reached by changing parameter λ (Section 3.2). These were reported in Table 5 together with the value of λ at which optimal solution is found, and the time at when this best objective function is observed for the first time. Table 5. Best found solution found by iterative heuristic approach and different values for λ Instance n m Obj T(s) λ Airland Airland Airland Airland Here, the best value of λ for each case was obtained by trial and error. Table 6. Results for instances with up to 500 aircraft using iterative heuristic approach, SA+VND and SS meta-heuristics SA+VND (Salehipour et Iterative heuristic SS (Pinol and Beasley, 2006) Instance n m al., 2013) Obj T(s) Obj T(s) Obj T(s) Airland Airland Airland Airland Average Table 6 reports the computational experiments of iterative heuristic approach, the SA+VND, and the SS meta-heuristics. Again here the first column is the instance name, the second and the third columns are the number of aircraft and the number of runways for each instance, respectively. Those remaining columns are associated with the objective function value and the total computational time in seconds for each reported solution approaches. Except from three cases where the iterative heuristic approach could not report the best found solution as either SA+VND or SS did (see Table 7 for new solutions by changing value of λ), for remaining 13 cases it reports either the best available solutions or new best found solutions (on 81.25% of cases). Apart from this capability it is almost 9 times faster than SA+VND of Salehipour et al., (2013), and 40 times faster than that of SS as Figure 2 depicts this. 9

10 Table 7. Best found solution found by iterative heuristic approach and different values for λ Instance n m Obj T(s) λ Airland Airland Airland Here, the best value of λ for each case was obtained by trial and error. CPU time in seconds Computational time of three approaches for large problems ,2 150,3 150,4 150,5 200,2 200,3 200,4 200,5 250,2 250,3 250,4 250,5 500,2 500,3 500,4 500,5 No. of Aircraft, No. of Runways Cutting Approach SA+VND SS Figure 2. A comparison of computational times of three different solution approaches According to tables 4 and 6, the performance of the iterative heuristic approach is very competitive on all instances, specifically when the size of instances increases rapidly (see tables 6 and 7 and also Figure 2 for a better understanding). This is the case where both SA+VND and SS could not reach the objective function values found by the iterative heuristic approach approximately after 20 and 75 minutes, respectively. Considering all these together, it is easy to show the superiority of the developed after approach in deriving optimal and near optimal solutions. To the best of our knowledge, and according to the published research on the problem, the reported results of this paper are the new best found solutions for most instances of the problem. For the remaining instances, reported solutions are the same as literature. 10

11 5. CONCLUSION In this paper we discussed an exact-based solution approach for the aircraft landing problem with the objective of minimizing the total deviation of landing time from the target time (both earliness and tardiness) where multiple runways are available. The proposed solution approach included a class of constraints to derive very high quality solutions for instances with up to 500 aircraft. The computational results show that this solution approach is very competitive, and can obtain very high quality solutions for instances of up to 500 aircraft in a short time, which is at least 10 times faster than the best solution approaches available for the problem. As the future directions, authors are working on improving the iterative heuristic procedure to generate constraints that without requiring changing parameter λ could find optimal solutions. REFERENCES Artiouchine, K., Baptiste, P., and Durr, C. (2008), Runway sequencing with holding patterns, European Journal of Operational Research, 189(3): Bäuerle N., Engelhardt-Funke O. and Kolonko M. (2007), On the waiting time of arriving aircraft and the capacity of airports with one or two runways, European Journal of Operational Research 177(2): Beasley J. E., Krishnamoorthy M., Sharaiha Y. M. and Abramson D. (2000), Scheduling Aircraft Landings-The Static Case, Transportation Science, 34(2): Beasley, J. E., Krishnamoorthy, M., Sharaiha, Y. M., and Abramson, D. (2004), Displacement problem and dynamically scheduling aircraft landings, Journal of the Operational Research Society, 55(1): Ernst, A. T., Krishnamoorthy, M, and Storer, R. H. (1999), Heuristic and exact algorithms for scheduling aircraft landings, Networks, 34(3): Hansen J. V. (2004), Genetic search methods in air traffic control, Computers & Operations Research 31(3): Pinol, H. and Beasley, J. E. (2006), Scatter Search and Bionomic Algorithms for the aircraft landing problem, European Journal of Operational Research 171: Salehipour, A., Modarres, M., and Naeni, Leila M. (2013), An Efficient Hybrid Meta-Heuristic for Aircraft Landing Problem, Computers & Operations Research 40(1): Salehipour, A., Naeni, Leila M., and Kazemipoor H. (2009), Scheduling Aircraft Landings by Applying a Variable Neighborhood Descent Algorithm: Runway-dependent landing time case, Journal of Applied Operational Research 1(1): Soomer, M. J., and Franx, G. J. (2008), Scheduling aircraft landings using airlines preferences, European Journal of Operational Research 190: