Mikhail Y. Kovalyov

Scheduling with batching: A review

There is an extensive literature on models that integrate scheduling with batching decisions. Jobs may be batched if they share the same setup on a machine. Another reason for batching occurs when a machine can process several jobs simultaneously. This paper reviews the literature on scheduling with batching, giving details of the basic algorithms, and referencing other significant results. Special attention is given to the design of efficient dynamic programming algorithms for solving these types of problems.

Scheduling a batching machine

textabstractWe study the problem of scheduling a chain-reentrant shop, in which each job goes for its processing first to a machine called the primary machine, then to a number of other machines in a fixed sequence, and finally back to the primary machine for its last operation. The problem is to schedule the jobs so as to minimize the makespan. This problem is unary NP-hard for a general number of machines. We focus in particular on the two-machine case that is also at least binary NP-hard. We prove some properties that identify a specific class of optimal schedules, and then use these properties in designing an approximation algorithm and a branch-and-bound type optimization algorithm. The approximation algorithm, of which we present three versions, has a worst-case performance guarantee of f32 along with an excellent empirical performance. The optimization algorithm solves large instances quickly. Finally, we identify a few well solvable special cases and present a pseudo-polynomial algorithm for the case in which the first and the last operations of any job (on the primary machine) are identical.

Scheduling jobs with position-dependent processing times

Bachman and Janiak provided a sketch of the proof that the problem 1∣ri,pi(v)=ai/v∣Cmax is NP-hard in the strong sense. However, they did not show how to avoid using harmonic numbers whose encoding is not pseudo-polynomial, which makes the proof incomplete. In this corrigendum, we provide a new complete proof.

Fixed interval scheduling: Models, applications, computational complexity and algorithms

The defining characteristic of fixed interval scheduling problems is that each job has a finite number of fixed processing intervals. A job can be processed only in one of its intervals on one of the available machines, or is not processed at all. A decision has to be made about a subset of the jobs to be processed and their assignment to the processing intervals such that the intervals on the same machine do not intersect. These problems arise naturally in different real-life operations planning situations, including the assignment of transports to loading/unloading terminals, work planning for personnel, computer wiring, bandwidth allocation of communication channels, printed circuit board manufacturing, gene identification and examining computer memory structures. We present a general formulation of the interval scheduling problem, show its relations to cognate problems in graph theory, and survey existing models, results on computational complexity and solution algorithms.

Single machine scheduling with batch deliveries

Abstract The single machine batch scheduling problem is studied. The jobs in a batch are delivered to the customer together upon the completion time of the last job in the batch. The earliness of a job is defined as the difference between the delivery time of the batch to which it belongs and its completion time. The objective is to minimize the sum of the batch delivery and job earliness penalties. A relation between this problem and the parallel machine scheduling problem is identified. This enables the establishment of complexity results and algorithms for the former problem based on known results for the latter problem.

Minimizing the total weighted completion time of deteriorating jobs

The paper deals with a single machine problem of scheduling jobs with start time dependent processing times to minimize the total weighted completion time. The computational complexity of this problem was unknown for ten years. We prove that the problem is NP-hard.

Single machine scheduling subject to deadlines and resource dependent processing times

The problem of scheduling n jobs on a single machine is studied. Each job has a deadline and a processing time which is a linear decreasing function of the amount of a common resource allocated to the job. The objective is to find simultaneously a sequence of the jobs and a resource allocation so as the deadlines are satisfied and the total weighted resource consumption is minimized. The problem is shown to be solvable in O(n log n) time if the resource is continuously divisible. If the resource is discrete, then the problem is proved to be binary NP-hard. Some special cases are solvable in O(n log n) time. A fully polynomial approximation scheme is presented for the general problem with discrete resource.

Bicriterion Single Machine Scheduling with Resource Dependent Processing Times

A bicriterion problem of scheduling jobs on a single machine is studied. The processing time of each job is a linear decreasing function of the amount of a common discrete resource allocated to the job. A solution is specified by a sequence of the jobs and a resource allocation. The quality of a solution is measured by two criteria, F1 and F2. The first criterion is the maximal or total (weighted) resource consumption, and the second criterion is a regular scheduling criterion depending on the job completion times. Both criteria have to be minimized. General schemes for the construction of the Pareto set and the Pareto set

A Fully Polynomial Approximation Scheme for Minimizing Makespan of Deteriorating Jobs

\epsilon

Single machine scheduling with a variable common due date and resource-dependent processing times

-approximation are presented. Computational complexities of problems to minimize F1 subject to F_2\le K