Very Large-Scale Neighborhood Search in Airline Fleet Scheduling

Many discrete optimization problems of practical interest cannot be solved to optimality in the available time. A practical approach to these problems is to use heuristic algorithms, which do not guarantee the optimality of the solution but which, for many problems, can find nearly optimal solutions within a reasonable amount of computational time. The literature on heuristic algorithms often distinguishes between two broad classes: constructive algorithms and improvement algorithms. A constructive algorithm builds a solution from scratch by assigning values to one or more decision variables at a time. An improvement algorithm starts with a feasible solution and iteratively tries to improve it. Neighborhood search algorithms, also called local search algorithms, are a wide class of improvement algorithms that at each iteration search the “neighborhood” of the current solution to find an improved solution. A neighborhood search algorithm for a discrete optimization problem P (where we wish to minimize an objective function over a finite discrete set of objects) starts with a feasible solution x of the problem. For each feasible solution x, an associated neighborhood of x, denoted N(x), is a set of feasible solutions that can be obtained by perturbing the solution x using some prespecified scheme. The elements of N(x) are called neighbors of x. The neighborhood search iteratively obtains a sequence x, x, x, . . . of feasible solutions. At the kth iteration, the algorithm determines a solution x with a lower objective function value than x, if one exists. The algorithm terminates when it finds a solution that is at least as good as any of its neighbors; such a solution is called a locally optimal solution. Typically, multiple runs of the neighborhood search algorithm are performed with different starting solutions, and the best locally optimal solution is selected. Figure 1 illustrates one run of the neighborhood search algorithm. A comprehensive discussion of neighborhood search algorithms can be found in [1]. A critical issue in the design of a neighborhood search algorithm is the choice of the neighborhood structure—that is, the manner in which the neighborhood N(x) is defined for the solution x. This choice largely determines whether the solutions obtained will be highly accurate or will have poor local optima. As a rule of thumb, the larger the neighborhood, the better will be the quality of the locally optimal solutions, and the greater the accuracy of the final solution. At the same time, searching larger neighborhoods requires more time at each iteration. Because of the many runs of a neighborhood search algorithm usually performed, longer execution times per run lead to fewer runs within a specified time. For this reason, a larger neighborhood can produce a more effective heuristic algorithm only if the larger neighborhood can be searched in a very efficient manner. Most of the neighborhood search algorithms in the literature use small neighborhoods, explicitly enumerating all neighbors in the search. The algorithms we describe here search very large neighborhoods, and we refer to them as very largescale neighborhood (VLSN) search algorithms. VLSN algorithms open up myriad possibilities for neighborhood search. For some problems, we can search a large neighborhood in a time roughly comparable to that needed for a small neighborhood, but with far better solution quality. A survey of VLSN search techniques can be found in [2]. We illustrate the VLSN search technique here for airline scheduling, a problem for which a small neighborhood search is not likely to be effective [3].