### Article

## Online heuristic for the preemptive single machine scheduling problem of minimizing the total weighted completion time

The preemptive single machine scheduling problem of minimizing the total weighted completion time with arbitrary processing times and release dates is an important NP-hard problem in scheduling theory. In this paper we present an efficient high-quality heuristic for this problem based on the WSRPT (Weighted Shortest Remaining Processing Time) rule. The running time of the suggested algorithm increases only as a square of the number of jobs. Our computational study shows that very large size instances might be treated within extremely small CPU times and the average error is always less than 0.1%.

In this paper, the single-track railway scheduling problem with two stations and several segments of the track is considered. Two subsets of trains are given, where trains from the first subset go from the first station to the second station, and trains from the second subset go in the opposite direction. The speed of trains over each segment is the same. A polynomial time reduction from the problem under consideration to a special case of the single-machine equal-processing-time scheduling problem with setup times is presented. Different polynomial time algorithms are developed for special cases with divers objective functions under various constraints. Moreover, several theoretical results which can be ranked in a series of similar investigations of NP-hardness of equal-processing-time single-machine scheduling problems without precedence relations are obtained.

This proceedings publication is a compilation of selected contributions from the “Third International Conference on the Dynamics of Information Systems” which took place at the University of Florida, Gainesville, February 16–18, 2011. The purpose of this conference was to bring together scientists and engineers from industry, government, and academia in order to exchange new discoveries and results in a broad range of topics relevant to the theory and practice of dynamics of information systems. Dynamics of Information Systems: Mathematical Foundation presents state-of-the art research and is intended for graduate students and researchers interested in some of the most recent discoveries in information theory and dynamical systems. Scientists in other disciplines may also benefit from the applications of new developments to their own area of study.

In this paper, the single-track railway scheduling problem with two stations and several segments of the track is considered. Two subsets of trains are given, where trains from the first subset go from the first station to the second station, and trains from the second subset go in the opposite direction. The speed of trains over each segment is the same. A polynomial time reduction from the problem under consideration to a special case of the single-machine equal-processing-time scheduling problem with setup times is presented. Different polynomial time algorithms are developed for special cases with divers objective functions under various constraints. Moreover, several theoretical results which can be ranked in a series of similar investigations of NP-hardness of equal-processing-time single-machine scheduling problems without precedence relations are obtained.

The preemptive single machine scheduling problem of minimizing the total weighted completion time with equal processing times and arbitrary release dates is one of the four single machine scheduling problems with an open computational complexity status. In this paper we present lower and upper bounds for the exact solution of this problem based on the assignment problem. We also investigate properties of these bounds and worst-case behavior.

A form for an unbiased estimate of the coefficient of determination of a linear regression model is obtained. It is calculated by using a sample from a multivariate normal distribution. This estimate is proposed as an alternative criterion for a choice of regression factors.