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Planning For Edge Failure in Fixed-Charge Flow Networks

Daniel Olson, Caleb Eardley, Sean Yaw

TL;DR

This paper extends Fixed-Charge Network Flow to account for the possibility that a designated edge may fail after initial edge procurement, by formulating a multi-objective problem that minimizes both the initial flow cost and the post-failure repaired flow cost. It introduces a MILP-based algorithm that iteratively computes the Pareto front of these objectives, using three coupled MILPs per iteration to generate successive trade-off solutions. The method is demonstrated on real CCS infrastructure data, showing how decision makers can select points along the Pareto front based on edge-failure risk and risk tolerance. The work advances practical recovery-aware planning for critical networks and suggests avenues for extending to multiple failure scenarios and scalable suboptimal approaches.

Abstract

The Fixed-Charge Network Flow problem is a well-studied NP-hard problem that has the goal of finding a flow in a network where fixed edge costs are incurred, regardless of the amount of flow hosted by the edge. In this paper, we consider scenarios where a designated edge in the network has the potential to fail after edges have already been purchased. If the edge does fail, procurement of additional edges may be required to repair the flow and compensate for the failed edge so as to maintain the original flow amount. We formulate a multi-objective optimization problem that aims to minimize the costs of both the initial flow as well as the repaired flow. We introduce an algorithm that finds the Pareto front between these two objectives, thereby providing decision makers with a sequence of solutions that trade off initial flow cost with repaired flow cost. We demonstrate the algorithm's efficacy with an evaluation using real-world CO2 capture and storage infrastructure data.

Planning For Edge Failure in Fixed-Charge Flow Networks

TL;DR

This paper extends Fixed-Charge Network Flow to account for the possibility that a designated edge may fail after initial edge procurement, by formulating a multi-objective problem that minimizes both the initial flow cost and the post-failure repaired flow cost. It introduces a MILP-based algorithm that iteratively computes the Pareto front of these objectives, using three coupled MILPs per iteration to generate successive trade-off solutions. The method is demonstrated on real CCS infrastructure data, showing how decision makers can select points along the Pareto front based on edge-failure risk and risk tolerance. The work advances practical recovery-aware planning for critical networks and suggests avenues for extending to multiple failure scenarios and scalable suboptimal approaches.

Abstract

The Fixed-Charge Network Flow problem is a well-studied NP-hard problem that has the goal of finding a flow in a network where fixed edge costs are incurred, regardless of the amount of flow hosted by the edge. In this paper, we consider scenarios where a designated edge in the network has the potential to fail after edges have already been purchased. If the edge does fail, procurement of additional edges may be required to repair the flow and compensate for the failed edge so as to maintain the original flow amount. We formulate a multi-objective optimization problem that aims to minimize the costs of both the initial flow as well as the repaired flow. We introduce an algorithm that finds the Pareto front between these two objectives, thereby providing decision makers with a sequence of solutions that trade off initial flow cost with repaired flow cost. We demonstrate the algorithm's efficacy with an evaluation using real-world CO2 capture and storage infrastructure data.
Paper Structure (5 sections, 3 equations, 6 figures)

This paper contains 5 sections, 3 equations, 6 figures.

Table of Contents

  1. Introduction
  2. Motivation
  3. Algorithm
  4. Evaluation
  5. Conclusions

Figures (6)

  • Figure 1: Sample flow network (top) with source $s$, sink $t$, displayed fixed costs, no variable costs, unit edge capacities, and a target flow amount of one. Panels (a)-(d) show a sequence of initial flows (top) and optimal repaired flows (bottom) when edge $(b,t)$ fails.
  • Figure 2: Overview of the objectives that the three MILPs optimize for when finding an initial and repaired flow during a single iteration of the algorithm. The initial and repaired flows found by MILP 3 form the next solution on the Pareto front.
  • Figure 3: A sample reduction from a CID problem instance (left) to a FCNF instance (right). A new source vertex is added to the FCNF instance that has directed edges from it to each of the CID instance source vertices. A new sink vertex is added that has directed edges to it from each of the CID instance sink vertices. The capacities and costs of the source and sink vertices in the CID instance are assigned to the new directed edges in the FCNF instance. The capacities and costs of the edges in the original CID instance have the same values in the FCNF instance. Non-source and non-sink vertices in CID instances do not have capacities or costs. This demonstrates that the CID problem is identical to the FCNF problem.
  • Figure 4: Nevada CCS data set and sample solution.
  • Figure 5: Progression of the initial and repaired flows, over the course of the execution of the algorithm, from the minimum cost flow (and its associated repaired flow) to minimum cost flow on the network with sink number two excluded.
  • ...and 1 more figures