An efficient polynomial-time approximation scheme for parallel multi-stage open shops

TL;DR

This paper introduces an efficient polynomial-time approximation scheme for parallel multi-stage open shop scheduling problems, optimizing job processing times with no preemption, especially when the number of shops and stages are fixed.

Contribution

It develops an EPTAS for parallel multi-stage open shops with constant number of shops and stages, combining categorization, scaling, and linear programming rounding techniques.

Findings

EPTAS achieves near-optimal makespan within a factor of (1+ε).

Method effectively handles complex scheduling with multiple stages and shops.

Algorithm runs in polynomial time for fixed parameters.

Abstract

Various new scheduling problems have been arising from practical production processes and spawning new research areas in the scheduling field. We study the parallel multi-stage open shops problem, which generalizes the classic open shop scheduling and parallel machine scheduling problems. Given m identical k-stage open shops and a set of n jobs, we aim to process all jobs on these open shops with the minimum makespan, i.e., the completion time of the last job, under the constraint that job preemption is not allowed. We present an efficient polynomial-time approximation scheme (EPTAS) for the case when both m and k are constant. The main idea for our EPTAS is the combination of several categorization, scaling, and linear programming rounding techniques. Jobs and/or operations are first scaled and then categorized carefully into multiple types so that different types of jobs and/or operations are scheduled appropriately without increasing the makespan too much.