Facet inducing inequalities for single-machine scheduling problems

We report new results for a time-indexed formulation of nonpreemptive single-machine scheduling problems. We give complete characterizations of all facet inducing inequalities with integral coefficients and right-hand side 1 or 2. Our results may lead to improved cutting plane algorithms for single-machine scheduling problems

A polyhedral approach to single-machine scheduling problems

We report new results for a time-indexed formulation of nonpreemptive single- machine scheduling problems. We give complete characterizations of all facet induc- ing inequalities with integral coecients and right-hand side 1 or 2 for the convex hull of the set of feasible partial schedules, i.e., schedules in which not all jobs have to be started. Furthermore, we identify conditions under which these facet inducing inequalities are also facet inducing for the original polytope, which is the convex hull of the set of feasible complete schedules, i.e., schedules in which all jobs have to be started. To obtain insight in thee ectiveness of these classes of facet-inducing inequalities, we develop a branch-and-cut algorithm based on them. We evaluate its performance on the strongly NP-hard single machine scheduling problem of mini- mizing the weighted sum of the completion times subject to release dates.mathematical economics and econometrics ;

Combining column generation and Lagrangean relaxation : an application to a single-machine common due date scheduling problem

Column generation has proved to be an effective technique for solving the linear programming relaxation of huge set covering or set partitioning problems, and column generation approaches have led to state-of-the-art so-called branch-and-price algorithms for various archetypical combinatorial optimization problems. Usually, if Lagrangean relaxation is embedded at all in a column generation approach, then the Lagrangean bound serves only as a tool to fathom nodes of the branch-and-price tree. We show that the Lagrangean bound can be exploited in more sophisticated and effective ways for two purposes: to speed up convergence of the column generation algorithm and to speed up the pricing algorithm. Our vehicle to demonstrate the effectiveness of teaming up column generation with Lagrangean relaxation is an archetypical single-machine common due date scheduling problem. Our comprehensive computational study shows that the combined algorithm is by far superior to two existing purely column generation algorithms: it solves instances with up to 125 jobs to optimality, while purely column generation algorithm can solve instances with up to only 60 jobs

Semiparametrically point-optimal hybrid rank tests for unit roots

We propose a new class of unit root tests that exploits invariance properties in the Locally Asymptotically Brownian Functional limit experiment associated to the unit root model. The invariance structures naturally suggest tests that are based on the ranks of the increments of the observations, their average and an assumed reference density for the innovations. The tests are semiparametric in the sense that they are valid, that is, have the correct (asymptotic) size, irrespective of the true innovation density. For a correctly specified reference density, our test is point-optimal and nearly efficient. For arbitrary reference densities, we establish a Chernoff–Savage-type result, that is, our test performs as well as commonly used tests under Gaussian innovations but has improved power under other, for example, fat-tailed or skewed, innovation distributions. To avoid nonparametric estimation, we propose a simplified version of our test that exhibits the same asymptotic properties, except for the Chernoff–Savage result that we are only able to demonstrate by means of simulations

Evolutionary air traffic flow management for large 3D-problems

We present an evolutionary tool to solve free-route Air Traffic Flow Management problems within a three-dimensional air space. This is the first evolutionary tool which solves free-route planning problems involving a few hundred aircraft. We observe that the importance of the recombination operator increases as we scale to larger problem instances. The evolutionary algorithm is based on a variant of the elitist recombinationalgorithm. We show a theoretical analysis of the problem, and present the results of experiments