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A Production Routing Problem with Mobile Inventories

Raian Lefgoum, Sezin Afsar, Pierre Carpentier, Jean-Philippe Chancelier, Michel de Lara

Abstract

Hydrogen is an energy vector, and one possible way to reduce CO 2 emissions. This paper focuses on a hydrogen transport problem where mobile storage units are moved by trucks between sources to be refilled and destinations to meet demands, involving swap operations upon arrival. This contrasts with existing literature where inventories remain stationary. The objective is to optimize daily routing and refilling schedules of the mobile storages. We model the problem as a flow problem on a time-expanded graph, where each node of the graph is indexed by a time-interval and a location and then, we give an equivalent Mixed Integer Linear Programming (MILP) formulation of the problem. For small to medium-sized instances, this formulation can be efficiently solved using standard MILP solvers. However, for larger instances, the computational complexity increases significantly due to the highly combinatorial nature of the refilling process at the sources. To address this challenge, we propose a two-step heuristic that enhances.

A Production Routing Problem with Mobile Inventories

Abstract

Hydrogen is an energy vector, and one possible way to reduce CO 2 emissions. This paper focuses on a hydrogen transport problem where mobile storage units are moved by trucks between sources to be refilled and destinations to meet demands, involving swap operations upon arrival. This contrasts with existing literature where inventories remain stationary. The objective is to optimize daily routing and refilling schedules of the mobile storages. We model the problem as a flow problem on a time-expanded graph, where each node of the graph is indexed by a time-interval and a location and then, we give an equivalent Mixed Integer Linear Programming (MILP) formulation of the problem. For small to medium-sized instances, this formulation can be efficiently solved using standard MILP solvers. However, for larger instances, the computational complexity increases significantly due to the highly combinatorial nature of the refilling process at the sources. To address this challenge, we propose a two-step heuristic that enhances.

Paper Structure

This paper contains 35 sections, 6 theorems, 57 equations, 9 figures, 3 tables.

Table of Contents

  1. Introduction
  2. Literature review
  3. Problem notations and assumptions
  4. Problem description on a time-expanded flow graph
  5. Definition of the time-expanded graph
  6. Definition of nodes
  7. Definition of arcs
  8. Flows in the time-expanded graph
  9. Problem modeling on graph ${\cal G}^{\textsf{te}}$
  10. Decision variables
  11. Arc and node constraints
  12. Arc constraints
  13. Constraints at nodes in $\mathbb{D}{\times}\underline{\mathbb{J}}$
  14. Constraints at nodes in $\mathbb{D}{\times}\overline{\mathbb{J}}$
  15. Constraints at nodes in $\mathbb{S} {\times} \underline{\mathbb{J}}$
  16. ...and 20 more sections

Key Result

Lemma 2

The optimization problem flow_model_problem_formulation, which includes nonlinear constraints—specifically eq:flow_arc, eq:emptying_bounded1, eq:swap_destination, eq:demand_satisfaction2, eq:swap_time, eq:emtying_bounded2, eq:permutation_constraints, eq:forced_arc_source— is equivalent to a MILP, in

Figures (9)

  • Figure 1: Storage unit delivery (left) and back to source after swap (right)
  • Figure 2: The day division at a destination $d$: a truck picks up a storage unit from the source $s$ at $h_0$, transports it to the destination $d$, performs the swap, and then returns the collected storage unit back to a source $s'$. During the first part of the day, the (blue) mobile storage originally located at destination $d$ satisfies the demand, and during the second part of the day, the newly arrived mobile storage (red) fulfills the demand
  • Figure 3: Graphic display of a time-expanded graph with one source -- where the maximum number of unit storages at source is 2 --- and two destinations. The arcs belonging to $\underline{\mathcal{E}}$ are in green, to $\overline{\mathcal{E}}$ are in orange and to $\widetilde{\mathcal{E}}$ are in blue
  • Figure 4: Display on a time-expanded graph (with one source, two destinations and an horizon of two days) of a storage flow solution $\varphi^{\mathsf{s}}$. We only display the subset of arcs where the storage flow is equal to one. As developed in Sect. \ref{['sec:flow_to_planning']}, transport planning, represented here arc paths (sequence of arcs with the same color), can be derived from $\varphi^{\mathsf{s}}$.
  • Figure 5: Example of a demand profile for a weekday at a destination with an average daily demand of [kg]85
  • ...and 4 more figures

Theorems & Definitions (9)

  • Definition 1
  • Lemma 2
  • Remark 3
  • Lemma 4
  • Lemma 5
  • Lemma 6
  • Lemma 7
  • Definition 8
  • Proposition 9