pith. sign in

arxiv: 2604.10307 · v1 · submitted 2026-04-11 · 🧮 math.OC

Hub location problems with asymmetric allocation

I. Espejo (1) , M. Landete (2) , M. Leal (2) , A. Mar\'in (3) ((1) Departamento de Estad\'istica e Investigaci\'on Operativa , Universidad de C\'adiz , Spain , (2) Centro de Investigaci\'on Operativa , Universidad Miguel Hern\'andez
show 3 more authors
(3) Departamento de Estad\'istica e Investigaci\'on Operativa Universidad de Murcia Spain.)
This is my paper

Pith reviewed 2026-05-10 15:36 UTC · model grok-4.3

classification 🧮 math.OC
keywords asymmetric hub locationinteger programmingvalid inequalitieshub networkslogistics optimizationsupply chaindecomposition methods
0
0 comments X

The pith

The (1,p)-AHLP models asymmetric hub allocation with a compact three-index formulation that solves efficiently on standard instances.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper introduces the Asymmetric Hub Location Problem where origins and destinations connect to hubs under distinct limits r and s. It focuses on the (1,p) variant with single assignment for origins and multi-assignment for destinations, motivated by applications such as humanitarian logistics. Two integer programming formulations are given, including a new compact three-index model that is smaller and stronger than the classical four-index version. Valid inequalities and decomposition techniques are added to improve performance. Computational results on common hub location benchmark datasets confirm that the approach is effective and efficient.

Core claim

This paper defines the (r,s)-AHLP in which origins and destinations obey separate allocation cardinalities and then specializes to the (1,p)-AHLP. It shows that this variant admits a compact three-index integer program that, when augmented with valid inequalities and solved via decomposition, yields high-quality solutions on the standard hub-location test sets used throughout the literature.

What carries the argument

The (1,p)-AHLP together with its compact three-index integer programming formulation and the valid inequalities that strengthen it.

If this is right

  • The three-index model has fewer variables and constraints than the four-index adaptation.
  • Valid inequalities tighten the linear-programming relaxation of the formulation.
  • Decomposition techniques accelerate exact solution of the resulting programs.
  • The framework directly supports asymmetric-flow applications in humanitarian relief and e-commerce fulfillment networks.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • Extending the same approach to other (r,s) pairs beyond (1,p) would enlarge the set of solvable asymmetric problems.
  • Optimal hub locations under asymmetric rules are likely to differ from those produced by symmetric models on the same data.
  • Applying the model to proprietary real-world networks could reveal whether the benchmark performance carries over to operational instances.

Load-bearing premise

The asymmetric allocation rules are captured exactly by the integer programs without introducing modeling gaps that would change solution quality on realistic instances.

What would settle it

Take a medium-sized instance from the standard hub datasets whose optimal value is already known; solve it with both the four-index and three-index models and check whether the three-index model recovers the same optimum in less time without any gap.

read the original abstract

Hub location problems are central to optimizing logistics, telecommunications, and transportation networks by consolidating flows through strategically placed hubs. While existing models assume symmetric allocation, where hubs handle incoming and outgoing flows uniformly, real-world applications often require asymmetric handling of origins and destinations. This paper introduces the Asymmetric Hub Location Problem ((r,s)-AHLP), a novel framework where origins and destinations may connect to hubs under distinct allocation limits (r and s, respectively). We then focus on the (1,p)-AHLP variant, where origins are single-assigned and destinations are multi-assigned, motivated by applications in humanitarian logistics and global supply chains (e.g., UN relief networks, e-commerce fulfillment). We propose two integer programming formulations: A four-index adaptation of classical models and a new compact three-index formulation. The latter reduces the size while improving effectiveness, supported by valid inequalities and decomposition techniques. The computational study, performed on standard datasets commonly used in hub location literature, demonstrates the high effectiveness and efficiency of the proposed solution methodology. Our work presents a robust methodological framework for the (1,p)-AHLP, significantly expanding the applicability of hub location theory to asymmetric flow contexts and providing a foundation for future studies on complex multi-level hub network design.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The paper introduces the (r,s)-Asymmetric Hub Location Problem ((r,s)-AHLP) and focuses on the (1,p)-AHLP variant, in which origins are assigned to a single hub while destinations may be assigned to multiple hubs. It presents a four-index formulation adapted from classical models and a new compact three-index integer programming formulation, along with valid inequalities and decomposition techniques. Computational experiments on standard hub location datasets (AP, CAB) are reported to demonstrate the effectiveness and efficiency of the three-index model plus inequalities.

Significance. If the three-index formulation correctly encodes the distinct r=1 and s>1 allocation limits without relaxation gaps and if the experiments properly instantiate asymmetric instances, the work extends hub location theory to a practically relevant asymmetric setting with applications in humanitarian logistics and supply chains. The compact formulation and inequalities represent a methodological contribution that could improve scalability over four-index models.

major comments (2)
  1. [Computational study] Computational study section: the manuscript must explicitly describe how the standard AP and CAB instances are adapted to enforce asymmetric allocation (single origin assignment with r=1 versus multi-destination assignment with s>1). Without this, the reported high effectiveness cannot be confirmed to validate the novel (1,p)-AHLP rather than the underlying symmetric p-hub model, directly affecting the central claim of the abstract.
  2. [Formulation] Formulation section: the three-index model is asserted to reduce size while improving effectiveness, but the manuscript should provide a formal proof or explicit argument that the valid inequalities close any modeling gaps introduced by the asymmetry (distinct r and s limits) and do not inadvertently relax the single-assignment constraint on origins.
minor comments (2)
  1. [Abstract] Abstract: the phrase 'standard datasets commonly used in hub location literature' should be accompanied by a brief statement on whether and how asymmetry is imposed, to avoid reader confusion.
  2. [Introduction] Notation: ensure consistent use of r and s throughout; the transition from general (r,s)-AHLP to the (1,p) focus should be stated with explicit values at first use.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the constructive comments. We address each major comment below and will revise the manuscript to incorporate the requested clarifications.

read point-by-point responses

  1. Referee: [Computational study] Computational study section: the manuscript must explicitly describe how the standard AP and CAB instances are adapted to enforce asymmetric allocation (single origin assignment with r=1 versus multi-destination assignment with s>1). Without this, the reported high effectiveness cannot be confirmed to validate the novel (1,p)-AHLP rather than the underlying symmetric p-hub model, directly affecting the central claim of the abstract.

    Authors: We agree that an explicit description of the instance adaptation is necessary to substantiate the claims for the (1,p)-AHLP. The current manuscript applies the standard AP and CAB datasets but does not detail the modifications. In the revised version we will add a dedicated paragraph (or subsection) explaining the precise changes: how origin-to-hub costs and assignment variables are restricted to enforce r=1, how destination assignments allow s>1, and how the flow and cost matrices are adjusted to reflect asymmetry while preserving the original distance data. This will confirm that the reported results pertain to the asymmetric model rather than the symmetric p-hub case. revision: yes

  2. Referee: [Formulation] Formulation section: the three-index model is asserted to reduce size while improving effectiveness, but the manuscript should provide a formal proof or explicit argument that the valid inequalities close any modeling gaps introduced by the asymmetry (distinct r and s limits) and do not inadvertently relax the single-assignment constraint on origins.

    Authors: We acknowledge that the manuscript would benefit from a clearer argument on this point. The valid inequalities were derived from the three-index formulation to tighten the relaxation while respecting the distinct r=1 and s>1 limits; they are obtained by projecting out certain flow variables and applying standard lifting techniques that preserve the single-origin assignment. To address the concern directly, the revised manuscript will include an explicit (non-formal) argument, supported by a small illustrative example, showing that any feasible solution to the strengthened LP satisfies the r=1 constraint and that no relaxation gap is introduced for the origin side. A full formal proof will be added if space permits; otherwise the argument will be placed in an appendix. revision: yes

Circularity Check

0 steps flagged

No circularity: new asymmetric model and IP formulations are defined directly from problem requirements

full rationale

The paper introduces the (r,s)-AHLP by explicitly stating the distinct allocation limits r for origins and s for destinations, then specializes to the (1,p) case. Both the four-index and compact three-index integer programs are assembled from standard hub-location variables and constraints plus valid inequalities derived from the new asymmetric structure. The computational study applies these formulations to standard AP/CAB datasets as external benchmarks. No equation or claim reduces to a self-definition, a fitted input relabeled as prediction, or a load-bearing self-citation; the derivation chain remains independent of its own outputs.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 1 invented entities

The central claim rests on the definition of the new asymmetric allocation model and the validity of the integer-programming formulations and inequalities. No numerical parameters are fitted to data. The only background assumptions are the standard modeling conventions of hub-location integer programs.

axioms (1)
  • domain assumption Hub location problems can be modeled as integer programs with flow-balance and assignment constraints.
    The paper adapts classical hub-location integer programs to the asymmetric setting.
invented entities (1)
  • (r,s)-Asymmetric Hub Location Problem no independent evidence
    purpose: To represent networks in which origins and destinations obey separate hub-allocation limits
    New combinatorial object introduced by the paper; no independent empirical validation supplied in the abstract.

pith-pipeline@v0.9.0 · 5594 in / 1362 out tokens · 34102 ms · 2026-05-10T15:36:09.173768+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Reference graph

Works this paper leans on

4 extracted references · 4 canonical work pages

  1. [1]

    Aflalo, Y., Bronstein, A., & Kimmel, R. (2015). On convex relaxation of graph isomorphism.Proceedings of the National Academy of Sciences,112(10), 2942–2947. Alumur, S., Campbell, J. F., Contreras, I., Kara, B. Y., Marianov, V., & O’Kelly, M. E. (2021). Perspectives on modeling hub location problems.European Journal of Operational Research,291(1), 1–17. A...

    work page 2015

  2. [2]

    Ghaffarinasab, N., & Kara, B. Y. (2019). Benders decomposition algorithms for two variants of the single allocation hub location problem.Networks and Spatial Economics,19, 83–108. Gilmore, P. C. (1962). Optimal and suboptimal algorithms for the quadratic assignment problem.Journal of the Society for Industrial and Applied Mathematics,10(2), 305–313. Gutek...

    work page 2019

  3. [3]

    M., & S´ anchez-Gil, M

    Puerto, J., Ramos, A., Rodr´ ıguez-Ch´ ıa, A. M., & S´ anchez-Gil, M. C. (2016). Ordered median hub location problems with capacity constraints.Transportation Research C,70, 142–156. Rostami, B., Errico, F., & Lodi, A. (2023). A convex reformulation and an outer approximation for a large class of binary quadratic programs.Operations Research,71(2), 471–48...

    work page 2016

  4. [4]

    Singh, R., Xu, J., & Berger, B. (2008). Global alignment of multiple protein interaction networks with application to functional orthology detection.Proceedings of the National Academy of Sciences, 105(35), 12763–12768. Skorin-Kapov, D., Skorin-Kapov, J., & O’Kelly, M. E. (1996). Tight linear programming relaxations of uncapacitatedp-hub median problems.E...

    work page 2008