Variable Neighborhood Search Heuristics Research Articles

PERMUTATION-BASED GENETIC, TABU, AND VARIABLE NEIGHBORHOOD SEARCH HEURISTICS FOR MULTIPROCESSOR SCHEDULING WITH COMMUNICATION DELAYS

The multiprocessor scheduling problem with communication delays that we consider in this paper consists of finding a static schedule of an arbitrary task graph onto a homogeneous multiprocessor system, such that the total execution time (i.e. the time when all tasks are completed) is minimum. The task graph contains precedence relations as well as communication delays (or data transferring time) between tasks if they are executed on different processors. The multiprocessor architecture is assumed to contain identical processors connected in an arbitrary way, which is defined by a symmetric matrix containing minimum distances between every two processors. The solution is represented by a feasible permutation of tasks. In order to obtain the objective function value (i.e. schedule length, makespan), the feasible permutation has to be transformed into the actual schedule by the use of some heuristic method. For solving this NP-hard problem, we develop basic tabu search and variable neighborhood search heuristics, where various types of reduced Or-opt-like neighborhood structures are used for local search. A genetic search approach based on the same solution space is also developed. Comparative computational results on random graphs with up to 500 tasks and 8 processors are reported. On average, it appears that variable neighborhood search outperforms the other metaheuristics. In addition, a detailed performance analysis of both the proposed solution representation and heuristic methods is presented.

An Interior Point Algorithm for Minimum Sum-of-Squares Clustering

An exact algorithm is proposed for minimum sum-of-squares nonhierarchical clustering, i.e., for partitioning a given set of points from a Euclidean m-space into a given number of clusters in order to minimize the sum of squared distances from all points to the centroid of the cluster to which they belong. This problem is expressed as a constrained hyperbolic program in 0-1 variables. The resolution method combines an interior point algorithm, i.e., a weighted analytic center column generation method, with branch-and-bound. The auxiliary problem of determining the entering column (i.e., the oracle) is an unconstrained hyperbolic program in 0-1 variables with a quadratic numerator and linear denominator. It is solved through a sequence of unconstrained quadratic programs in 0-1 variables. To accelerate resolution, variable neighborhood search heuristics are used both to get a good initial solution and to solve quickly the auxiliary problem as long as global optimality is not reached. Estimated bounds for the dual variables are deduced from the heuristic solution and used in the resolution process as a trust region. Proved minimum sum-of-squares partitions are determined for the first time for several fairly large data sets from the literature, including Fisher's 150 iris.

Variable neighborhood search for the p-median

Consider a set L of potential locations for p facilities and a set U of locations of given users. The p-median problem is to locate simultaneously the p facilities at locations of L in order to minimize the total transportation cost for satisfying the demand of the users, each supplied from its closest facility. This model is a basic one in location theory and can also be interpreted in terms of cluster analysis where locations of users are then replaced by points in a given space. We propose several new Variable Neighborhood Search heuristics for the p-median problem and compare them with Greedy plus Interchange, and two Tabu Search heuristics.