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Solving Unsplittable Network Flow Problems with Decision Diagrams

Hosseinali Salemi, Danial Davarnia

TL;DR

A novel decision diagram (DD)-based framework is developed that decomposes the underlying two-stage formulation into a master problem that contains the combinatorial requirements and a subproblem that models a continuous network flow problem.

Abstract

In unsplittable network flow problems, certain nodes must satisfy a combinatorial requirement that the incoming arc flows cannot be split or merged when routed through outgoing arcs. This so-called "no-split no-merge" requirement arises in unit train scheduling where train consists should remain intact at stations that lack necessary equipment and manpower to attach/detach them. Solving the unsplittable network flow problems with standard mixed-integer programming formulations is computationally difficult due to the large number of binary variables needed to determine matching pairs between incoming and outgoing arcs of nodes with no-split no-merge constraint. In this paper, we study a stochastic variant of the unit train scheduling problem where the demand is uncertain. We develop a novel decision diagram (DD)-based framework that decomposes the underlying two-stage formulation into a master problem that contains the combinatorial requirements, and a subproblem that models a continuous network flow problem. The master problem is modeled by a DD in a transformed space of variables with a smaller dimension, leading to a substantial improvement in solution time. Similarly to the Benders decomposition technique, the subproblems output cutting planes that are used to refine the master DD. Computational experiments show a significant improvement in solution time of the DD framework compared with that of standard methods.

Solving Unsplittable Network Flow Problems with Decision Diagrams

TL;DR

A novel decision diagram (DD)-based framework is developed that decomposes the underlying two-stage formulation into a master problem that contains the combinatorial requirements and a subproblem that models a continuous network flow problem.

Abstract

In unsplittable network flow problems, certain nodes must satisfy a combinatorial requirement that the incoming arc flows cannot be split or merged when routed through outgoing arcs. This so-called "no-split no-merge" requirement arises in unit train scheduling where train consists should remain intact at stations that lack necessary equipment and manpower to attach/detach them. Solving the unsplittable network flow problems with standard mixed-integer programming formulations is computationally difficult due to the large number of binary variables needed to determine matching pairs between incoming and outgoing arcs of nodes with no-split no-merge constraint. In this paper, we study a stochastic variant of the unit train scheduling problem where the demand is uncertain. We develop a novel decision diagram (DD)-based framework that decomposes the underlying two-stage formulation into a master problem that contains the combinatorial requirements, and a subproblem that models a continuous network flow problem. The master problem is modeled by a DD in a transformed space of variables with a smaller dimension, leading to a substantial improvement in solution time. Similarly to the Benders decomposition technique, the subproblems output cutting planes that are used to refine the master DD. Computational experiments show a significant improvement in solution time of the DD framework compared with that of standard methods.
Paper Structure (17 sections, 6 theorems, 12 equations, 9 figures, 8 tables, 3 algorithms)

This paper contains 17 sections, 6 theorems, 12 equations, 9 figures, 8 tables, 3 algorithms.

Table of Contents

  1. Introduction
  2. Literature Review on Train Scheduling
  3. Literature Review on Decision Diagrams
  4. Contributions
  5. Background on DDs
  6. Overview
  7. Continuous DD Models
  8. DD-BD Formulation for the SGUFP
  9. MIP Formulation
  10. DD-BD: Master Problem Formulation
  11. DD-BD: Subproblem Formulation
  12. Computational Experiments
  13. Test Instances
  14. Numerical Results
  15. Conclusion
  16. ...and 2 more sections

Key Result

Proposition 1

Any bounded mixed integer set of the form $\mathcal{P} \subseteq \mathbb{Z}^n \times {\mathbb R}$ is DD-representable w.r.t. $I=\{n+1\}$. $\square$

Figures (9)

  • Figure 1: Pie chart for ton-miles of freight shipments by mode within the U.S. in 2018. Multiple modes includes mail. Air and truck-air with the share of $0.1\%$ are omitted.
  • Figure 2: The exact, relaxed, and restricted DDs representing $\mathcal{P}$ in Example \ref{['ex: IP']}. Solid and dotted arcs indicate one and zero arc labels, respectively. Numbers next to arcs represent weights.
  • Figure 3: The last two iterations of solving the master problem in Example \ref{['ex: MIP']}
  • Figure 4: Illustration of network $G=(V' \cup \{s,t\},A)$
  • Figure 5: Comparison of DD-BD, BD, and MIP models when $|V'|=40$ under five scenarios
  • ...and 4 more figures

Theorems & Definitions (9)

  • Example 1
  • Proposition 1
  • Example 2
  • Proposition 2
  • Theorem 1
  • Theorem 2
  • Proposition 3
  • Proposition 4
  • Example 3