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A hybrid optimization framework for the General Continuous Energy-Constrained Scheduling Problem

Roel Brouwer, Marjan van den Akker, Han Hoogeveen

TL;DR

This work develops a hybrid approach for the case where the objective is a step-wise increasing function of completion time, using local search, linear programming and O(n) lower bounds, and argues the broad applicability of this framework.

Abstract

We present a hybrid optimization framework for a class of problems, formalized as a generalization of the Continuous Energy-Con\-strained Scheduling Problem (CECSP), introduced by Nattaf et al. (2014). This class is obtained from challenges concerning demand response in energy networks. Our framework extends a previously developed approach. A set of jobs has to be processed on a continuous, shared resource. Consequently, a schedule for a job does not only contain a start and completion time, but also a resource consumption profile, where we have to respect lower and upper bounds on resource consumption during processing. In this work, we develop a hybrid approach for the case where the objective is a step-wise increasing function of completion time, using local search, linear programming and O(n) lower bounds. We exploit that the costs are known in the local search and use bounds to assess feasibility more efficiently than by LP. We compare its performance to a mixed-integer linear program. After that, we extend this to a hybrid optimization framework for the General CECSP. This uses an event-based model, and applies a decomposition in two parts: 1) determining the order of events and 2) finding the event times, and hence the start and completion times of jobs, together with the resource consumption profiles. We argue the broad applicability of this framework.

A hybrid optimization framework for the General Continuous Energy-Constrained Scheduling Problem

TL;DR

This work develops a hybrid approach for the case where the objective is a step-wise increasing function of completion time, using local search, linear programming and O(n) lower bounds, and argues the broad applicability of this framework.

Abstract

We present a hybrid optimization framework for a class of problems, formalized as a generalization of the Continuous Energy-Con\-strained Scheduling Problem (CECSP), introduced by Nattaf et al. (2014). This class is obtained from challenges concerning demand response in energy networks. Our framework extends a previously developed approach. A set of jobs has to be processed on a continuous, shared resource. Consequently, a schedule for a job does not only contain a start and completion time, but also a resource consumption profile, where we have to respect lower and upper bounds on resource consumption during processing. In this work, we develop a hybrid approach for the case where the objective is a step-wise increasing function of completion time, using local search, linear programming and O(n) lower bounds. We exploit that the costs are known in the local search and use bounds to assess feasibility more efficiently than by LP. We compare its performance to a mixed-integer linear program. After that, we extend this to a hybrid optimization framework for the General CECSP. This uses an event-based model, and applies a decomposition in two parts: 1) determining the order of events and 2) finding the event times, and hence the start and completion times of jobs, together with the resource consumption profiles. We argue the broad applicability of this framework.
Paper Structure (24 sections, 1 theorem, 4 equations, 7 figures, 2 tables, 1 algorithm)

This paper contains 24 sections, 1 theorem, 4 equations, 7 figures, 2 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Problem description
  3. Hybrid optimization framework
  4. Event-based model
  5. Decomposition and local search
  6. Assessing the quality of event orders
  7. Solving the LP
  8. Max flow on a bipartite graph
  9. Simple lower bounds
  10. Applicability of the framework for the general problem
  11. Objectives
  12. Type I: Linear functions
  13. Type II: Minmax criteria (and extensions)
  14. Type III: Piece-wise linear functions
  15. Non-linear functions
  16. ...and 9 more sections

Key Result

Theorem 1

For any feasible schedule $\mathcal{S}$ with resource consumption profiles $p_j(t)$ for all jobs $J_j, j \in \{1, ..., n\}$ and objective value $W$ that follows a given event order $\mathcal{E}$, a feasible schedule $\mathcal{S}'$ exists with the same order of events $\mathcal{E}$ and objective valu

Figures (7)

  • Figure 4: Example instance with three jobs
  • Figure 5: Graphical representation of part of an event order and associated intervals
  • Figure 6: Example with three jobs and four intervals
  • Figure 7: Average runtimes of evaluation methods for event orders
  • Figure 8: Average runtimes of discussed approaches
  • ...and 2 more figures

Theorems & Definitions (1)

  • Theorem 1