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On feasibility cuts for chance-constrained multicommodity network design problems

Niels A. Wouda, Ward Romeijnders, Evrim Ursavas

TL;DR

FlowMIS generates strong feasibility cuts tailored to subproblems with a network flow structure, which results in reduced solution times for existing decomposition-based algorithms in the context of network design, and the ability to solve larger problems.

Abstract

Problem definition: We study efficient exact solution approaches to solve chance-constrained multicommodity network design problems under demand uncertainty, an important class of network design problems. The chance constraint requires us to construct a network that meets future commodity demand sufficiently often, which makes the problem challenging to solve. Methodology/results: We develop a solution approach based on Benders' decomposition, and accelerate the approach with valid inequalities and cut strengthening. We particularly investigate the effects of different subproblem formulations on the strength of the resulting feasibility cuts. We propose a new formulation that we term FlowMIS, and investigate its properties. Additionally, we numerically show that FlowMIS outperforms standard formulations: in our complete solution approach with all enhancements enabled, FlowMIS solves 67 out of 120 solved instances the fastest, with an average speed-up of 2.0x over a basic formulation. Implications: FlowMIS generates strong feasibility cuts tailored to subproblems with a network flow structure. This results in reduced solution times for existing decomposition-based algorithms in the context of network design, and the ability to solve larger problems.

On feasibility cuts for chance-constrained multicommodity network design problems

TL;DR

FlowMIS generates strong feasibility cuts tailored to subproblems with a network flow structure, which results in reduced solution times for existing decomposition-based algorithms in the context of network design, and the ability to solve larger problems.

Abstract

Problem definition: We study efficient exact solution approaches to solve chance-constrained multicommodity network design problems under demand uncertainty, an important class of network design problems. The chance constraint requires us to construct a network that meets future commodity demand sufficiently often, which makes the problem challenging to solve. Methodology/results: We develop a solution approach based on Benders' decomposition, and accelerate the approach with valid inequalities and cut strengthening. We particularly investigate the effects of different subproblem formulations on the strength of the resulting feasibility cuts. We propose a new formulation that we term FlowMIS, and investigate its properties. Additionally, we numerically show that FlowMIS outperforms standard formulations: in our complete solution approach with all enhancements enabled, FlowMIS solves 67 out of 120 solved instances the fastest, with an average speed-up of 2.0x over a basic formulation. Implications: FlowMIS generates strong feasibility cuts tailored to subproblems with a network flow structure. This results in reduced solution times for existing decomposition-based algorithms in the context of network design, and the ability to solve larger problems.
Paper Structure (21 sections, 7 theorems, 41 equations, 5 tables)

This paper contains 21 sections, 7 theorems, 41 equations, 5 tables.

Table of Contents

  1. Introduction
  2. Problem definition
  3. Solution approach
  4. Feasibility cuts
  5. FlowMIS.
  6. SNC.
  7. MIS.
  8. BB.
  9. Determining $M$, $\beta$, and $\gamma$.
  10. Metric inequalities.
  11. Cut properties in the single-commodity case
  12. Valid inequalities
  13. Numerical experiments
  14. Implementation details
  15. Experimental design
  16. ...and 6 more sections

Key Result

Theorem 1

Fix scenario $\bar{s} \in [S]$. The subset $Y_{\bar{s}} = \{ y \in \mathbb{B}^{|A|} \mid SP(y, \omega_{\bar{s}}) \text{ is feasible} \}$ can be described by finitely many linear inequalities (or feasibility cuts) of the form: where $C_{\bar{s}}$ is the (index) set of feasibility cuts, $\gamma_{r\bar{s}}$ a scalar, and $\beta_{r\bar{s}}$ a vector in $\mathbb{R}^{|A|}$.

Theorems & Definitions (7)

  • Theorem 1
  • Proposition 1
  • Theorem 2
  • Theorem 3
  • Theorem 4
  • Corollary 1
  • Proposition 2