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Imitation-regularized Optimal Transport on Networks: Provable Robustness and Application to Logistics Planning

Koshi Oishi, Yota Hashizume, Tomohiko Jimbo, Hirotaka Kaji, Kenji Kashima

TL;DR

The paper addresses robust transport on networks under unforeseen disruptions by introducing imitation-regularized OT (I-OT) that minimizes $ \\sum_{x} C(x) P(x) + \alpha D_{KL}(P\|Q)$ subject to $P(x_0)=\nu_0$, $P(x_T)=\nu_T$. I-OT links to the Schrödinger bridge via $\mathfrak{M}_Q = \mathfrak{M}_{RB} Q$ and uses a Sinkhorn-like iteration, enabling efficient computation. A robust cost-set $\mathcal{C}$ is defined with $ \alpha \log (\sum_x Q(x) e^{(\tilde{C}(x)-C(x))/\alpha}) \le \epsilon$, yielding a worst-case value of $ \sum_x P(x) C(x) + \alpha D_{KL}(P\|Q) + \epsilon$, which formalizes robustness. Empirical validation on automotive parts logistics under Mt. Fuji demonstrates that I-OT achieves lower post-disaster costs than MaxEnt OT or standard OT, by leveraging prior knowledge to prioritize robust routes, with significant practical impact for resilient transportation planning.

Abstract

Transport systems on networks are crucial in various applications, but face a significant risk of being adversely affected by unforeseen circumstances such as disasters. The application of entropy-regularized optimal transport (OT) on graph structures has been investigated to enhance the robustness of transport on such networks. In this study, we propose an imitation-regularized OT (I-OT) that mathematically incorporates prior knowledge into the robustness of OT. This method is expected to enhance interpretability by integrating human insights into robustness and to accelerate practical applications. Furthermore, we mathematically verify the robustness of I-OT and discuss how these robustness properties relate to real-world applications. The effectiveness of this method is validated through a logistics simulation using automotive parts data.

Imitation-regularized Optimal Transport on Networks: Provable Robustness and Application to Logistics Planning

TL;DR

The paper addresses robust transport on networks under unforeseen disruptions by introducing imitation-regularized OT (I-OT) that minimizes subject to , . I-OT links to the Schrödinger bridge via and uses a Sinkhorn-like iteration, enabling efficient computation. A robust cost-set is defined with , yielding a worst-case value of , which formalizes robustness. Empirical validation on automotive parts logistics under Mt. Fuji demonstrates that I-OT achieves lower post-disaster costs than MaxEnt OT or standard OT, by leveraging prior knowledge to prioritize robust routes, with significant practical impact for resilient transportation planning.

Abstract

Transport systems on networks are crucial in various applications, but face a significant risk of being adversely affected by unforeseen circumstances such as disasters. The application of entropy-regularized optimal transport (OT) on graph structures has been investigated to enhance the robustness of transport on such networks. In this study, we propose an imitation-regularized OT (I-OT) that mathematically incorporates prior knowledge into the robustness of OT. This method is expected to enhance interpretability by integrating human insights into robustness and to accelerate practical applications. Furthermore, we mathematically verify the robustness of I-OT and discuss how these robustness properties relate to real-world applications. The effectiveness of this method is validated through a logistics simulation using automotive parts data.
Paper Structure (12 sections, 3 theorems, 35 equations, 3 figures, 1 table)

This paper contains 12 sections, 3 theorems, 35 equations, 3 figures, 1 table.

Table of Contents

  1. Introduction
  2. Preliminary: optimal transport and Schrödinger bridge over networks
  3. Optimal transport
  4. Schrödinger bridge
  5. Imitation-regularized optimal transport
  6. Formulation of I-OT
  7. Relation to Schrödinger bridge
  8. Provable robustness
  9. Application to logistics
  10. Formulation
  11. Results
  12. Conclusion

Key Result

Proposition 1

If all components of $\bm{M}^{T}$ are positive, there exists a unique pair of non-negative functions $\varphi, \hat{\varphi}$ that satisfies eq:phi_t and eq:phi_b. Moreover, the probability distribution $P^*[\nu_0,\nu_T]$ defined by eq:Pstar is the unique solution to eq:sch.

Figures (3)

  • Figure 1: Automotive parts transportation. (a) Node locations and magnitudes of supply and demand. The circular and square nodes denote the delivery destinations and supply sources, respectively. The blue nodes denote ports and the triangular node denotes Mt. Fuji. The size of each node represents the supply and demand quantities. (b) Digraph representation. The axes denote the position (units: km). The blue and red edges denote roads and maritime paths, respectively. The self-loop representing storage is omitted.
  • Figure 2: A priori risk information. The edge width is not an exact value as visibility is prioritized.
  • Figure 3: Logistics plan. The maritime edges are emphasized with thicker lines.

Theorems & Definitions (8)

  • Proposition 1
  • Remark 1
  • Theorem 1
  • proof
  • Remark 2
  • Theorem 2
  • proof
  • Remark 3