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Fortifying Distribution Network Nodes Subject to Network-Based Disruptions

Pelin Keşrit, Bahar Çavdar, Joseph Geunes

TL;DR

This work addresses fortification of distribution networks against uncertain disturbances by formulating a nonlinear, stochastic knapsack with precedence constraints over a tree topology. It proves NP-Hardness for general trees, derives polynomial-time methods for serial systems, and develops a heuristic Network Search Algorithm (NSA) for general trees that leverages KKT insights and multiple initialization strategies. The NSA demonstrates near-optimal performance with low computation times compared to a global solver, and the analysis includes convexity considerations, KKT structure, and envelope-based bounds. The findings advance resilient planning under budgeted fortification and offer practical tools for quick, high-quality solutions in infrastructure networks.

Abstract

We consider a distribution network for delivering a natural resource or physical good to a set of nodes, each of which serves a set of customers, in which disruptions may occur at one or more nodes. Each node receives flow through a path from a source node, implying that the service at a node is interrupted if one or more nodes on the path from a source node experience a disruption. All network nodes are vulnerable to a future disturbance due to a potential natural or man-made disaster, the severity of which follows some measurable probability distribution. For each node in the network, we wish to determine a fortification level that enables the node to withstand a disturbance up to a given severity level, while minimizing the expected number of customers who experience a service interruption under a limited fortification budget. We formulate this problem as a continuous, nonlinear knapsack problem with precedence constraints, demonstrate that this optimization problem is $\mathcal{NP}$-Hard for general tree networks and general disturbance severity distributions, and provide a polynomial-time solution algorithm for serial systems, which forms the basis for an effective heuristic approach to problems on tree networks. Our computational test results demonstrate the ability of the proposed heuristic methods to quickly find near-optimal solutions.

Fortifying Distribution Network Nodes Subject to Network-Based Disruptions

TL;DR

This work addresses fortification of distribution networks against uncertain disturbances by formulating a nonlinear, stochastic knapsack with precedence constraints over a tree topology. It proves NP-Hardness for general trees, derives polynomial-time methods for serial systems, and develops a heuristic Network Search Algorithm (NSA) for general trees that leverages KKT insights and multiple initialization strategies. The NSA demonstrates near-optimal performance with low computation times compared to a global solver, and the analysis includes convexity considerations, KKT structure, and envelope-based bounds. The findings advance resilient planning under budgeted fortification and offer practical tools for quick, high-quality solutions in infrastructure networks.

Abstract

We consider a distribution network for delivering a natural resource or physical good to a set of nodes, each of which serves a set of customers, in which disruptions may occur at one or more nodes. Each node receives flow through a path from a source node, implying that the service at a node is interrupted if one or more nodes on the path from a source node experience a disruption. All network nodes are vulnerable to a future disturbance due to a potential natural or man-made disaster, the severity of which follows some measurable probability distribution. For each node in the network, we wish to determine a fortification level that enables the node to withstand a disturbance up to a given severity level, while minimizing the expected number of customers who experience a service interruption under a limited fortification budget. We formulate this problem as a continuous, nonlinear knapsack problem with precedence constraints, demonstrate that this optimization problem is -Hard for general tree networks and general disturbance severity distributions, and provide a polynomial-time solution algorithm for serial systems, which forms the basis for an effective heuristic approach to problems on tree networks. Our computational test results demonstrate the ability of the proposed heuristic methods to quickly find near-optimal solutions.

Paper Structure

This paper contains 24 sections, 7 theorems, 24 equations, 4 figures, 7 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Literature Review
  3. Network Resilience
  4. Knapsack with Precedence Constraints
  5. Problem Definition and Model Properties
  6. Problem Formulation
  7. Model Properties and Reformulation
  8. KKT Conditions
  9. Polynomially Solvable Special Cases
  10. Series Systems
  11. Tree Networks with a Uniform Distribution
  12. Heuristic Solution Method for General Trees
  13. Computational Study
  14. Conclusion
  15. Appendix
  16. ...and 9 more sections

Key Result

Proposition 1

In an optimal solution for the network fortification problem, $x_i \geq x_j$ for all $i \in \mathcal{P}(j)$ where node $i$ is a predecessor of node $j$, for any $c_j$, $j\in \{1,\dots,n\}$.

Figures (4)

  • Figure 1: A general tree topology for a distribution network.
  • Figure 2: Upper concave envelope of the triangular distribution with $\beta=\frac{1}{2}$.
  • Figure 3: $\Delta \Bar{z}\%$ with $n=50$ and triangular (0,$\beta$,1) distribution.
  • Figure 4: S-curve distribution with $\beta=\frac{2}{3}$.

Theorems & Definitions (12)

  • Proposition 1
  • Corollary 1
  • Proposition 2
  • Lemma 1
  • Lemma 2
  • Lemma 3
  • Corollary 2
  • proof
  • proof
  • proof
  • ...and 2 more