A flow-based formulation for parallel machine scheduling using decision diagrams

TL;DR

This paper introduces a novel flow-based formulation for parallel machine scheduling that leverages decision diagrams and decomposition techniques to improve solution bounds and computational efficiency.

Contribution

It presents a new decision diagram-based formulation and a branch-and-price algorithm that outperform classical models in solving scheduling problems.

Findings

Stronger LP relaxation bounds than classical formulations.

Effective branch-and-price framework for complex instances.

Successful computational experiments on literature instances.

Abstract

We present a new flow-based formulation for identical parallel machine scheduling with a regular objective function and without idle time. The formulation is constructed with the help of a decision diagram that represents all job sequences that respect specific ordering rules. These rules rely on a partition of the planning horizon into, generally non-uniform, periods and do not exclude all optimal solutions, but they constrain solutions to adhere to a canonical form. The new formulation has numerous variables and constraints, and hence we apply a Dantzig-Wolfe decomposition in order to compute the linear programming relaxation in reasonable time; the resulting lower bound is stronger than the bound from the classical time-indexed formulation. We develop a branch-and-price framework that solves several instances from the literature for the first time. We compare the new formulation with the time-indexed and arc-time-indexed formulation by means of a series of computational experiments.