On {\lambda}-Cent-Dians and Generalized-Center for Network Design: Formulations and Algorithms

TL;DR

This paper investigates the $\lambda$-centdian problem in network design, proposing formulations, algorithms, and decomposition methods to optimize sub-networks under constraints, extending previous work with new solution concepts and scalability analysis.

Contribution

It introduces a mathematical formulation and solution procedures for the generalized $\lambda$-centdian problem, including bilevel modeling, Pareto-optimality parametrization, and scalable Benders decomposition.

Findings

Effective mixed integer linear formulations for $\lambda$-centdian.

A Benders decomposition approach for $\lambda ext{ in }[0,1]$.

Analysis of solution quality using inequality measures.

Abstract

In this paper, we study the $\lambda$-centdian problem in the domain of Network Design. The focus is on designing a sub-network within a given underlying network while adhering to a budget constraint. This sub-network is intended to efficiently serve a collection of origin/destination demand pairs. We extend the work presented in \cite{bucarey2024on}, providing an algorithmic perspective on the generalized $\lambda$-centdian problem. In particular, we provide a mathematical formulation for $\lambda\geq 0$ and discuss the bilevel structure of this problem for $\lambda>1$. Furthermore, we describe a procedure to obtain a complete parametrization of the Pareto-optimality set based on solving two mixed integer linear formulations by introducing the concept of maximum $\lambda$-cent-dian. We evaluate the quality of the different solution concepts using some inequality measures. Finally, for $\lambda\in[0,1]$, we study the implementation of a Benders decomposition method to solve it at scale.