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

Corentin Juvigny, Antonín Novák, Jan Mandík, Zdeněk Hanzálek

TL;DR

The paper introduces a novel RCPSP variant that combines time-of-use energy tariffs with a three-state energy machine for a single energy-intensive resource. It develops two monolithic models (CP and MILP) and a Logic-Based Benders Decomposition (LBBD) that splits the problem into a master energy-state MILP (SPACES) and a CP subproblem solving the RCPSP, achieving strong performance, especially on large and sparse instances. The authors normalize objectives with $lb_{TEC}$ and $lb_{RCPSP}$ and demonstrate that LBBD outperforms monolithic approaches, solving up to 1600 tasks in sparse cases and generalizing the methodology to RCPSP with blocking times/tardiness and flexible job shops. They show substantial improvements in sparsity-dominated scenarios and provide practical guidance on warmstarts, cuts, and lazy constraints, highlighting the method’s applicability to a broader class of energy-aware scheduling problems. The work offers a promising decomposition framework for TOU-aware scheduling and opens avenues for multi-machine energy constraints, additional machine states, energy storage, and price uncertainty in future research.

Abstract

In this paper, we investigate the Resource-Constrained Project Scheduling Problem (RCPSP) with time-of-use energy tariffs (TOU) and machine states, a variant of RCPSP for production scheduling where energy price is part of the criteria and one machine is highly energy-demanding and 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 pricing, 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 dealing with energy cost solved using Integer Linear Programming (ILP) with a subproblem handling the RCPSP resolved using CP. Both approaches surpass the monolithic compact ILP approach, but the LBBD significantly outperforms the CP when the ratio of energy-intensive tasks over the overall tasks is moderate, allowing for solving instances with up to 1600 tasks in sparse instances. Finally, we put forth a way of generalizing our LBBD approach to other problems sharing similar characteristics, and we applied it to a problem based on an RCPSP problem with blocking times & total weighted tardiness criterion and a flexible job shop.

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

TL;DR

The paper introduces a novel RCPSP variant that combines time-of-use energy tariffs with a three-state energy machine for a single energy-intensive resource. It develops two monolithic models (CP and MILP) and a Logic-Based Benders Decomposition (LBBD) that splits the problem into a master energy-state MILP (SPACES) and a CP subproblem solving the RCPSP, achieving strong performance, especially on large and sparse instances. The authors normalize objectives with and and demonstrate that LBBD outperforms monolithic approaches, solving up to 1600 tasks in sparse cases and generalizing the methodology to RCPSP with blocking times/tardiness and flexible job shops. They show substantial improvements in sparsity-dominated scenarios and provide practical guidance on warmstarts, cuts, and lazy constraints, highlighting the method’s applicability to a broader class of energy-aware scheduling problems. The work offers a promising decomposition framework for TOU-aware scheduling and opens avenues for multi-machine energy constraints, additional machine states, energy storage, and price uncertainty in future research.

Abstract

In this paper, we investigate the Resource-Constrained Project Scheduling Problem (RCPSP) with time-of-use energy tariffs (TOU) and machine states, a variant of RCPSP for production scheduling where energy price is part of the criteria and one machine is highly energy-demanding and 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 pricing, 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 dealing with energy cost solved using Integer Linear Programming (ILP) with a subproblem handling the RCPSP resolved using CP. Both approaches surpass the monolithic compact ILP approach, but the LBBD significantly outperforms the CP when the ratio of energy-intensive tasks over the overall tasks is moderate, allowing for solving instances with up to 1600 tasks in sparse instances. Finally, we put forth a way of generalizing our LBBD approach to other problems sharing similar characteristics, and we applied it to a problem based on an RCPSP problem with blocking times & total weighted tardiness criterion and a flexible job shop.
Paper Structure (28 sections, 2 theorems, 35 equations, 15 figures, 8 tables, 1 algorithm)

This paper contains 28 sections, 2 theorems, 35 equations, 15 figures, 8 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Paper contributions and outline
  3. Literature Review
  4. Problem statement
  5. Example
  6. Constraint programming model
  7. Variables
  8. Objective
  9. Constraints
  10. Logic-Based Benders decomposition
  11. RCPSP seen as a feasibility problem: case $\alpha = 1$ (i.e., TEC only)
  12. Master: modeled by ILP-SPACES
  13. Subproblem: modeled by CP
  14. Feasibility cuts
  15. No good cuts
  16. ...and 13 more sections

Key Result

Proposition 1

For any $(u, v) \in \mathcal{J}^2$, if $\mathcal{L}_{uv} \neq \emptyset$, then:

Figures (15)

  • Figure 1: Example of a 48-hour part of the energy profile with real electricity costs in the day-ahead market in the Czech Republic from June 2025 provided by Czech OTE.
  • Figure 2: Parameters of the transition power function $\IfNoValueTF{s, s^{\prime}} { P }{ P(s, s^{\prime}) }$ and transition time function $\IfNoValueTF{s, s^{\prime}} { T }{ T(s, s^{\prime}) }$, and the corresponding transition graph, where every edge from $s$ to $s^{\prime}$ is labeled by $\IfNoValueTF{s, s^{\prime}} { T }{ T(s, s^{\prime}) }$/$\IfNoValueTF{s, s^{\prime}} { P }{ P(s, s^{\prime}) }$.
  • Figure 3: Parameters for the example problem instance.
  • Figure 4: SPACES graph for the transition graph in Figure \ref{['fig:example-func-power-time']} and energy cost profile from Figure \ref{['fig:example-energy-profile']}.
  • Figure 5: Example project schedule with the optimal transition for $R_0$. Parameter $\alpha=0.75$ (more towards TEC): $C_{max}$$=12$, TEC $=172$.
  • ...and 10 more figures

Theorems & Definitions (4)

  • Proposition 1
  • proof
  • Proposition 2
  • proof