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In this paper, we consider the problem of scheduling aircraft (plane) landings at an airport. This problem is one of deciding a landing time for each plane such that each plane lands within a predetermined time window and that separation criteria between the landing of a planeInthispaper,weconsidertheproblemofscheduling aircraft (plane) landings at an airport. This problem is one of deciding a landing time on a runway for each plane in a given set of planes such that each plane lands within a predetermined time window, and that separation criteria between the landing of a plane, and the landing of all successive planes, are respected.This paper is organized as follows. In Section 1, we set the problem in context. In Section 2, we present a mixed-integer zero-one formulation of the problem for the single runway case. We then discuss previous work on the problem in Section 3. In Section 4, we extend the formulation to the multiple runway case, and, in Section 5, we strengthen the linear programming (LP) relaxations of these formulations by introducing additional constraints. These formulations can be solved using LP-based tree search. An effective heuristic for the problem (for any number of runways) is presented in Section 6. Computational results for both the heuristic and the optimal algorithm for a number of test problems involving up to 50 planes and four runways are presented in Section 7.It is important to note here that, although throughout this paper we shall typically refer to planes landing, the models presented in this paper are applicable to problems involving just takeoffs only and to problems involving a mix of landings and takeoffs on the same runway. We should also stress here that we are dealing only with the static case. In other words, we are dealing with the off-line case where we have complete knowledge of the set of planes that are going to land. The dynamic, or on-line, case, where decisions about the landing times for planes must be made as time passes and the situation changes (planes land, new planes appear, etc.) is the subject of a separate paper (BEASLEY et al., 1995).

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We consider the two-dimensional cutting problem of cutting a number of rectangular pieces from a single large rectangle so as to maximize the value of the pieces cut. We develop a Lagrangean relaxation of a zero-one integer programming formulation of the problem and use it as a bound in a tree search procedure. Subgradient optimization is used to optimize the bound derived from the Lagrangean relaxation. Problem reduction tests derived from both the original problem and the Lagrangean relaxation are given. Incorporating the bound and the reduction tests into a tree search procedure enables moderately sized problems to be solved.