“Scheduling Aircraft Landings – The Static Case” J. E. Beasley, M. Krishnamoorthy, Y. M. Sharaiha, and D. Abramson Transportation Science, May 2000, p. 180-197 Article Review by Chris Hopkins Problem Statement This paper deals with the problem of scheduling aircraft landings at an airport. This problem involves determining a landing time on a runway for each plane in a given set of planes such that each plane lands within a pre-determined time window and that separation criteria between the landing of a plane and the landing of all successive planes are respected. The models presented in the paper are also applicable to problems involving takeoffs only and to problems involving a mix of landings and takeoffs on the same runway. The paper deals only with the static case of this problem, which is the case where there exists complete knowledge of the set of planes that are going to land. Problem Description When a plane enters within radar range of air traffic control (ATC) at an airport, a plane requires ATC to assign it a landing time and, if more than one runway is in use, assign it a runway on which to land. The landing time must lie within a specified time window, bounded by an earliest time and a latest time, with these times being different for different plane types. The earliest time represents the earliest a plane can land if it flies at its maximum airspeed. The latest time represents the latest a plane can land if it flies at its most fuel-efficient airspeed while also holding (circling) for the maximum allowable time. Each plane has a most economical, preferred speed, referred to as its cruise speed. A plane is said to be assigned by its preferred time, or target time, if it is required to fly for landing at its cruise speed. A cost is incurred if ATC requires the plane to slow down, hold, or speed up. The time between a particular plane landing and the landing of any successive plane must be greater than a specified minimum. Obviously, the practical problem of scheduling aircraft landings is more complex than this basic problem, so the authors address how their mixed-integer zero-one formulations deal with the following complexities: the associated control problem (can the planes be flown in such a manner as to enable the solution of the resulting decision model?), minimum separation times (based on FAA regulations), latest landing times (based on fuel considerations), runway allocation, and alternative objective functions. Solution Methodology The authors present an initial mixed-integer zero-one formulation of the static single runway aircraft landing problem. The authors discuss the use of alternative objective functions, but choose to minimize total plane costs, combining the cost of deviation before and after the target times. This is a nonlinear cost function, but it is composed of two linear portions, thus enabling the authors to linearize it and formulate the problem with a linear objective function. The formulation for the single runway problem can be found in Appendix A. Constraint 1 ensures that each plane lands within its time window. Constraint 2 enforces a landing order for the planes. Constraints 6, 7, and 12 ensure that separation times are enforced. Constraints 14-18 relate the αi, βi, and xi variables to each other and help ensure a linear objective function.

After discussing previous work for the single runway problem and the limited research that has been done for the multiple runway problem, the authors extend their single runway formulation to the multiple runway case. This case would appear to have greater value because most busy international airports have two or more runways. The extended formulation to handle the multiple runway problem can be found in Appendix B. Constraint 28 ensures that each plane lands on exactly one runway. Constraint 29 is a symmetry constraint that ensures that the indices are equal both ways if planes i and j land on the same runway. Constraint 30 ensures that the zero-one variables agree if two planes are landing on the same runway or if they are not landing on the same runway. In the United States, planes can land at the same time on parallel runways provided that the planes are landing far enough apart to conform to FAA regulations. With this in mind, constraints 31 and 33 replace constraints 7 and 8 from the single runway model. The authors then relax the zero-one variables to continuous variables so that they can make use of LP-based tree search in order to solve the multiple runway problem numerically. After relaxing the zero-one variables, the authors add several constraints that strengthen the value of the LP relaxation in continuous space. The extended formulation of the complete (strengthened) single/multiple runway formulation can be found in Appendix C. The authors developed a heuristic to find an upper bound to help tighten the time windows for each plane. This upper bound is also used to shorten the LP-based tree search. Although the heuristic gave good results, computational experience showed that sometimes the LP-based tree search found an improved feasible solution (upper bound) early in its search. When this occurred, they determined that it was better to restart the problem using the improved lower bound. The following is the final solution approach that the authors used: - Heuristic gives an upper bound (using complete single/multiple runway formulation) - Use the upper bound to tighten the time windows - Use tree search to re-solve the LP relaxation of the problem • if an improved feasible solution is found within P seconds of starting the tree search, terminate the tree search and restart with the new upper bound • otherwise, continue the tree search until normal termination (when an integer solution has been found and proven optimal). Applicability Separation times between planes landing and taking off exist due to aerodynamic considerations. A Boeing 747, for example, generates a great deal of air turbulence (wake vortices) and a plane flying too close behind could lose aerodynamic stability. For safety reasons, therefore, landing a Boeing 747 or having another plane take off after a Boeing 747 necessitates a large time delay. A light plane, on the other hand, generates little air turbulence and therefore only a relatively small delay is necessary before other planes can land or take off. The recent crash of an American Airlines plane in New York is an example of a plane accident that may have been the result of taking off too soon after a large plane. A lot of research has looked at the single runway model, but research for the multiple runway case has been limited to modeling the problem as a job shop scheduling problem or as a traveling salesman problem, with the resulting problem formulations being NP-Hard. The authors’ contribution to research in this area is that their formulation solves the multiple runway problem to optimality for moderate size problems.

APPENDIX A (Static Single Runway Model) The following is the complete formulation of the Static Single Runway Model:1 P

Minimize∑ ( g i i =1

s.t.

i

+ hi

i

)

Ei ≤ xi ≤ Li i = 1, K , P ij + ji = 1 i = 1, K , P; j = 1, K , P; j > i ij

= 1 ∀ (i, j ) ∈ W ∪ V

(6)

x j ≥ xi + S ij ∀ (i, j ) ∈ V x j ≥ xi + S ij

ij

− ( Li − E j )

(7) ji

≥ Ti − xi i = 1, K , P 0 ≤ i ≤ (Ti − Ei ) i = 1, K , P i = 1, K , P i ≥ ( x i − Ti ) 0 ≤ i ≤ ( Li − Ti ) i = 1, K , P xi = Ti − i + i i = 1, K , P i

(1) (2)

∀ (i, j ) ∈ U

(12) (14) (15) (16) (17) (18)

where P = the number of planes Ei = the earliest landing time for plane i (i = 1, …, P) Li = the latest landing time for plane i (i = 1, …, P) Ti = the target (preferred) landing time for plane i (i =1, …, P) Sij = the required separation time (≥ 0) between plane i landing and plane j landing (where plane i lands before plane j) i=1, …, P; j=1, …, P; i≠j gi = the penalty cost (≥ 0) per unit of time landing before Ti for plane i (i=1, …, P) hi = the penalty cost (≥ 0) per unit of time landing after Ti for plane i (i=1, …, P) xi = the landing time for plane i (i=1, …, P) αi = how soon plane i (i=1, …, P) lands before Ti βi = how soon plane i (i=1, …, P) lands after Ti U = the set of pairs (i, j) of planes for which uncertain whether plane i lands before plane j V = the set of pairs (i, j) of planes for which i definitely lands before j (but for which the separation constraint is not automatically satisfied) W = the set of pairs (i, j) of planes for which i definitely lands before j (and for which the separation constraint is automatically satisfied) 1 if plane i lands before plane j (i = 1,..., P; j = 1,..., P; i ≠ j ) δij =  0 otherwise

1

Beasley, J. E., M. Krishnamoorthy, Y. M. Sharaiha, and D. Abramson, “Scheduling Aircraft Landings – The Static Case,” Transportation Science. Vol. 34: 183-185.

APPENDIX B (Extended Formulation for the Multiple Runway Model)

In conjunction with the objective function and constraints 1, 2, 6, and 14-18 from the Single Runway Model, the following additional constraints formulate of the Multiple Runway Model:2 R

∑y r =1

ir

= 1 i = 1, K , P

z ij = z ji

(28)

i = 1, K , P; j = 1, K , P; j > i

(29)

z ij ≥ y ir + y jr − 1 i = 1, K , P; j = 1, K , P; j > i; r = 1, K , R (30) x j ≥ xi + S ij z ij + sij (1 − z ij ) ∀ (i, j ) ∈ V x j ≥ xi + S ij z ij + sij (1 − z ij ) − ( Li + max(S ij , sij ) − E j ) where

2

(31) ji

∀ (i, j ) ∈ U

Replaces (7) (33) Replaces (8)

sij = the required separation time (0 ≤ sij ≤ Sij) between plane i landing and plane j landing (where plane i lands before plane j and they land on different runways) i=1, …, P; j=1, …, P; i≠j R = the number of runways 1 if planes i and j land on the same runway (i = 1, K , P; j = 1, K , P; i ≠ j ) zij =  0 otherwise 1 if plane i (i = 1, K , P) lands on runway r (r = 1, K , R) yir =  0 otherwise

Ibid, 189.

APPENDIX C (Extended Formulation for the Complete Single/Multiple Runway Model)

In conjunction with the extended formulation of the Multiple Runway Model, the following additional constraints formulate the Complete Single/Multiple Runway Model:3 ≥ ( x j − xi ) /( L j − Ei ) ∀ (i, j ) ∈ U

ij P

P

i =1

j =1, j ≠ i

∑ ∑

i

= P( P − 1) / 2

+ i ∀ (i, j ) ∈ U with Ti < T j T j − Ti + i ) + ( j + j ) ≥ [ S ij − (T j − Ti )] ij + [(T j − Ti ) + S ji ]

≥ 1−

ij

(

ij

(40) (41)

i

− max{[S ij − (T j − Ti )], [(T j − Ti ) + S ji ]}(1 − z ij ) ∀ (i, j ) ∈ U

(42) ji

(43)

with Ti < T j and (T j − Ti ) < S ij

ji

+ z ij ≤ 1 ∀ (i, j ) ∈ U *

(45)

ji

+ (1 − z ij ) ≤ 1 ∀ (i, j ) ∈ U * *

(47)

P

P

∑ ∑z i =1 j =1, j ≠ i

ij

≥K

(48)

where Ei = max[Ei, Ti – ZUB / gi] i=1, …, P Li = min[Li, Ti + ZUB / hi] i=1, …, P U* = [(i,j)|(i,j) ∈ U and (Ej + Sji) > Li] U** = [(i,j)|(i,j) ∈ U and (Ej + sji) > Li] K is derived by examining the different ways of landing P planes on R runways. If we say that m1 planes land on runway 1, m2 on runway 2, …, mR on runway R, then an appropriate value for K is the optimal solution of: R

Minimize ∑ (mr ) 2 − P r =1

R

s.t.

∑m r =1

3

Ibid, 190-192.

r

=P