### Article

## Single machine serial-batching scheduling with independent setup time and deteriorating job processing times

This paper investigates the scheduling problems of a single serial-batching machine with independent setup time and deteriorating job processing times. With the assumption of deteriorating jobs, the job processing times are described by an increasing function of their starting times. All the jobs are first partitioned into serial batches and then processed on a single serial-batching machine. Before each batch is processed, an independent constant setup time is required. Two optimization algorithms are proposed to solve the problems of minimizing the makespan and the total number of tardy jobs, respectively. Specifically, for the problem of minimizing the total completion time, two special cases with the smallest and the largest number of batches are studied, and an optimization algorithm is also presented for the special case without setup time.

The scheduling problem of minimizing total tardiness on a single machine is known to be *NP*-hard in the ordinary sense. In this paper, we consider the special case of the problem when the processing times p_j and the due dates d_j of the jobs are oppositely ordered: p_1 >= p_2>=...>=p_n and d_1.

The following special case of the classical NP-hard scheduling problem (Formula presented.) is considered. There is a set of jobs (Formula presented.) with identical processing times (Formula presented.) for all jobs (Formula presented.). All jobs have to be processed on a single machine. The optimization criterion is the minimization of maximum lateness (Formula presented.). We analyze algorithms for the makespan problem (Formula presented.), presented by Garey et al. (SIAM J Comput 10(2):256–269, 1981), Simons (A fast algorithm for single processor scheduling. In: 19th Annual symposium on foundations of computer science (Ann Arbor, Mich., 1978, 1978) and Benson’s algorithm (J Glob Optim 13(1):1–24, 1998) and give two polynomial algorithms to solve the problem under consideration and to construct the Pareto set with respect to the criteria (Formula presented.) and (Formula presented.). The complexity of the presented algorithms is (Formula presented.) and (Formula presented.), respectively, where (Formula presented.) is the accuracy of the input-output parameters. © 2016 Springer-Verlag Berlin Heidelberg

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, we present a modification of dynamic programming algorithms (DPA), which we denote as graphical algorithms (GrA). For some single machine scheduling problems, it is shown that the time complexity of the GrA is less than the time complexity of the standard DPA. Moreover, the average running time of the GrA is often essentially smaller. A GrA can also solve large-scale instances and instances, where the parameters are not integer. For some problems, GrA has a polynomial time complexity in contrast to a pseudo-polynomial complexity of a DPA.

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.