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A Genetic Algorithm for Multi-Capacity Fixed-Charge Flow Network Design

Caleb Eardley, Dalton Gomez, Ryan Dupuis, Michael Papadopoulos, Sean Yaw

TL;DR

A genetic algorithm designed to quickly find high-quality flow solutions to the Multi-Capacity Fixed-Charge Network Flow problem is presented, which uses a novel solution representation scheme that eliminates the need to repair invalid flow solutions.

Abstract

The Multi-Capacity Fixed-Charge Network Flow (MC-FCNF) problem, a generalization of the Fixed-Charge Network Flow problem, aims to assign capacities to edges in a flow network such that a target amount of flow can be hosted at minimum cost. The cost model for both problems dictates that the fixed cost of an edge is incurred for any non-zero amount of flow hosted by that edge. This problem naturally arises in many areas including infrastructure design, transportation, telecommunications, and supply chain management. The MC-FCNF problem is NP-Hard, so solving large instances using exact techniques is impractical. This paper presents a genetic algorithm designed to quickly find high-quality flow solutions to the MC-FCNF problem. The genetic algorithm uses a novel solution representation scheme that eliminates the need to repair invalid flow solutions, which is an issue common to many other genetic algorithms for the MC-FCNF problem. The genetic algorithm's efficiency is displayed with an evaluation using real-world CO2 capture and storage infrastructure design data. The evaluation results highlight the genetic algorithm's potential for solving large-scale network design problems.

A Genetic Algorithm for Multi-Capacity Fixed-Charge Flow Network Design

TL;DR

A genetic algorithm designed to quickly find high-quality flow solutions to the Multi-Capacity Fixed-Charge Network Flow problem is presented, which uses a novel solution representation scheme that eliminates the need to repair invalid flow solutions.

Abstract

The Multi-Capacity Fixed-Charge Network Flow (MC-FCNF) problem, a generalization of the Fixed-Charge Network Flow problem, aims to assign capacities to edges in a flow network such that a target amount of flow can be hosted at minimum cost. The cost model for both problems dictates that the fixed cost of an edge is incurred for any non-zero amount of flow hosted by that edge. This problem naturally arises in many areas including infrastructure design, transportation, telecommunications, and supply chain management. The MC-FCNF problem is NP-Hard, so solving large instances using exact techniques is impractical. This paper presents a genetic algorithm designed to quickly find high-quality flow solutions to the MC-FCNF problem. The genetic algorithm uses a novel solution representation scheme that eliminates the need to repair invalid flow solutions, which is an issue common to many other genetic algorithms for the MC-FCNF problem. The genetic algorithm's efficiency is displayed with an evaluation using real-world CO2 capture and storage infrastructure design data. The evaluation results highlight the genetic algorithm's potential for solving large-scale network design problems.

Paper Structure

This paper contains 7 sections, 1 theorem, 14 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Problem Formulation
  3. Non-Integer Linear Program
  4. Genetic Algorithm
  5. Optimal Search Viability
  6. Evaluation
  7. Conclusion

Key Result

Theorem 1

For every problem instance, there exists a set of $d_{ek}$ values such that the optimal flow found by the resulting LP is also an optimal flow for the ILP.

Figures (4)

  • Figure 1: Counterexample to Claim \ref{['cl:rationale']} with the displayed fixed costs ($a_e$), variable costs ($b_e$), capacities ($c_e$), and a capture target of three. In this instance, the optimal ILP solution is $20$ with a flow of two units on $e_1$ and $e_3$ and one unit on $e_2$ and $e_4$. The corresponding optimal LP solution is $21$ with a flow of one unit on $e_1$ and $e_3$ and two units on $e_2$ and $e_4$.
  • Figure 2: CCS datasets consisting of sources (red), sinks (blue), and possible pipeline routes.
  • Figure 3: Solution cost versus running time for the genetic algorithm and optimal ILP.
  • Figure 4: Solution cost versus target flow amount for the genetic algorithm and optimal ILP.

Theorems & Definitions (3)

  • Claim 1
  • Theorem 1
  • proof