Revisiting Continuous p-Hub Location Problems with the L1 Metric

TL;DR

We revisit continuous $p$-hub location problems under the $\ell^1$ (Manhattan) metric, deriving a closed-form, convex solution for the one-dimensional case and an analytic treatment for up to two hubs in two dimensions. For larger two-dimensional instances, we propose a two-step approximation: discretize to a large-scale $p$-median problem (solved efficiently via Elloumi’s formulation and Benders decomposition) and then refine with SPSA-based simulation optimization, scalable to multiple service providers through Monte Carlo integration. A Virginia Beach case study on deploying public-access AEDs demonstrates meaningful reductions in mean delivery distance, with results highlighting the strong impact of volunteer availability and robustness to reasonable shifts in volunteer distributions. The work advances both analytical understanding of low-hub configurations and practical, scalable methods for urban hub placement in real-world, continuous settings. Overall, the combination of exact results for simple cases and a robust two-stage approximation yields actionable insights for urban logistics and life-saving applications.

Abstract

Motivated by emerging urban applications in commercial, public sector, and humanitarian logistics, we revisit continuous $p$-hub location problems in which several facilities must be located in a continuous space such that the expected minimum Manhattan travel distance from a random service provider to a random customer through exactly one hub facility is minimized. In this paper, we begin by deriving closed-form results for a one-dimensional case and two-dimensional cases with up to two hubs. Subsequently, a simulation-based approximation method is proposed for more complex two-dimensional scenarios with more than two hubs. Moreover, an extended problem with multiple service providers is analyzed to reflect real-life service settings. Finally, we apply our model and approximation method using publicly available data as a case study to optimize the deployment of public-access automated external defibrillators in Virginia Beach.

Figures (19)

• Figure 1: Illustration of a Two-Hub Situation
• Figure 2: Illustration of the Integration Area Division of $\mathbf{x}$
• Figure 3: Illustration of Constant Distance Difference Lines for Different $b$ Values
• Figure 4: Illustration of the Integration Area of $\mathbf{y}$ Given $\mathbf{x}$ in Area 2 to Area 7
• Figure 5: Benders Decomposition Computation Times for Large $n$ Values

Theorems & Definitions (18)

• Proposition 1
• Corollary 1
• Proposition 2
• Corollary 2
• Proposition 3
• Proposition 4
• Theorem 1
• proof
• Proposition 5
• Proposition 6