A Decision Diagram Approach for the Parallel Machine Scheduling Problem with Chance Constraints

TL;DR

The paper tackles CC-PMSP with uncertain processing and setup times by introducing a DD-based decomposition that partitions the problem into a master assignment problem and per-machine, per-scenario subproblems. It develops two DD formulations, DD-LJ (linear arc costs) and DD-JS (nonlinear costs to shrink diagrams), and derives No-Good and IIS cuts from the DDs to prune infeasible master solutions. The approach extends Lozano & Smith's two-stage cuts to CCP and demonstrates, across 405 instances, that DD-based decompositions outperform a strong IP baseline, with the best DD variant solving substantially more instances and achieving notably smaller average gaps. The work also investigates symmetry breaking and candidate-solution improvements, and provides a detailed evaluation of DD-LJ and DD-JS performance across VRP, ORS, and Equal data sets, confirming the superior effectiveness of DD-based methods in this scheduling context.

Abstract

The Chance-Constrained Parallel Machine Scheduling Problem (CC-PMSP) assigns jobs with uncertain processing times to machines, ensuring that each machine's availability constraints are met with a certain probability. We present a decomposition approach where the master problem assigns jobs to machines, and the subproblems schedule the jobs on each machine while verifying the solution's feasibility under the chance constraint. We propose two different Decision Diagram (DD) formulations to solve the subproblems and generate cuts. The first formulation employs DDs with a linear cost function, while the second uses a non-linear cost function to reduce the diagram's size. We show how to generate no-good and irreducible infeasible subsystem (IIS) cuts based on our DDs. Additionally, we extend the cuts proposed by Lozano & Smith (2018) to solve two-stage stochastic programming models. Our DD-based methodology outperforms traditional integer programming (IP) models designed to solve the CC-PMSP in several instances. Specifically, our best DD-based approach solves 55 more instances than the best IP alternative (from a total of 405) and typically achieves smaller gaps (50% vs. 120% gap on average).