# Resource-constrained Project Scheduling with Time-of-Use Energy Tariffs and Machine States: A Logic-based Benders Decomposition Approach

arXiv: https://arxiv.org/abs/2601.06542 · HTML: https://mlsanity.com/paper/2601.06542/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 7 of its 75 references that are in the map.
- First posted on arXiv 10 Jan 2026.

## Builds on (most cited first, 7 of 7)

- [Constraint Programming and Constructive Heuristics for Parallel Machine Scheduling with Sequence-Dependent Setups and Common Servers](https://mlsanity.com/paper/2305.19888/) (2023)
- [Stochastic Planning and Scheduling with Logic-Based Benders Decomposition](https://mlsanity.com/paper/2012.14074/) (2020)
- [Bottleneck Identification in Resource-Constrained Project Scheduling via Constraint Relaxation](https://mlsanity.com/paper/2504.07495/) (2025)
- [Reinforcement Learning for Search Tree Size Minimization in Constraint Programming: New Results on Scheduling Benchmarks](https://mlsanity.com/paper/2508.20056/) (2025)
- [Power of Pre-Processing: Production Scheduling with Variable Energy Pricing and Power-Saving States](https://mlsanity.com/paper/1912.02430/) (2019)
- [Optimal Scheduling of Flexible Power-to-X Technologies in the Day-ahead Electricity Market](https://mlsanity.com/paper/2110.09800/) (2021)
- [Green Scheduling with Time-of-Use Tariffs and Machine States: Optimizing Energy Cost via Branch-and-Bound and Bin Packing Strategies](https://mlsanity.com/paper/2506.10405/) (2025)

## Abstract (arXiv)

In this paper, we investigate the Resource-Constrained Project Scheduling Problem (RCPSP) with Time-of-Use (TOU) energy tariffs and machine states, a variant of RCPSP for production scheduling, where energy price is part of the criteria and one highly energy-demanding machine can be in one of the following three states: proc, idle, or off. The problem involves scheduling all tasks, respecting precedence constraints and resource limitations, while minimizing the combination of the overall makespan and the Total Energy Cost (TEC), which varies according to the TOU tariffs, which can take negative values. We propose two novel approaches to solve it: a monolithic Constraint Programming (CP) approach and a Logic-Based Benders Decomposition (LBBD) approach. The latter combines a master problem handling the energy cost solved using Integer Linear Programming (ILP) with a subproblem handling the RCPSP, resolved using CP. Both approaches outperform the monolithic compact ILP counterpart, but the LBBD significantly outperforms the monolithic CP in most cases, especially when the makespan criterion is not included in the objective function, solving to optimality instances with up to 480 tasks. Finally, we propose a way to generalize our LBBD approach to other problems sharing similar characteristics, and applied it to various problems, such as an RCPSP with blocking times & total weighted tardiness criterion, or a flexible job shop.