Pith. sign in

REVIEW 3 major objections 5 minor 44 references

School network reorganization under educational and spatial constraints using classical and quantum optimization

T0 review · 3 major / 5 minor · reviewed 2026-08-08 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗ Review JSON

Pith's one-line read The paper claims that school network consolidation can be exactly optimized by one integer program that encodes travel-time, capacity, and institutional constraints, and that real regional instances solve in under 0.2 seconds.

desk verdict Useful applied OR model for school dimensioning, but the objective function as printed rewards the opposite of what the text claims; needs a clean revision before its policy results are trusted. read the letter →

arxiv 2608.05427 v1 pith:ZXJADF2D submitted 2026-08-05 cs.CY quant-ph

Alessia Ciacco , Luigi Di Puglia Pugliese , Francesca Guerriero This is my paper · ORCID
classification cs.CYquant-ph
keywords SchoolnetworkreorganizationdimensioningIntegerLinearProgrammingHybridquantumoptimizationFacilityLocationProblemEducationalplanningReal-worldcasestudyTerritorialsustainability
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

The paper is trying to establish that school network reorganization—deciding which autonomous schools stay open and which are absorbed into a hub—can be captured in one integer linear programming model. The model combines the constraints that govern Italian school dimensioning: aggregations cannot cross provinces or mix educational levels, curricular mismatches are penalized, fragile territories get lower enrollment thresholds, travel time over the road network is capped by school type, and hub capacity is bounded. If the model is right, regional authorities gain a decision-support tool that is exact and fast: the complete Calabria public school network was solved to optimality in under 0.2 seconds for every one of 64 policy configurations, and a hybrid quantum-classical solver reproduced those optima in all configurations. The practical point is that consolidation policy can be tested scenario by scenario, with accessibility and territorial equity made explicit rather than left to administrative judgment.

What carries the argument

The load-bearing object is the capacitated facility location formulation (Variant II) of the school aggregation problem, with binary hub variables $z_j$ and assignment variables $y_{ij}$ defined only on the feasible arc set $\tilde{A}$. This set already requires same educational category and same province ($\delta_{ij}+\alpha_{ij}=2$), enrollment ordering ($s_i \le s_j$), and travel time within the type-specific thresholds $T^{\text{CI}}_{\max}$ or $T^{\text{USI}}_{\max}$, so the compatibility constraints are pushed into the variable domain. The assignment constraints then force each school to exactly one configuration, residual capacities bound each hub, and the reorganization target $\gamma$ caps the number of active hubs. The objective is a weighted sum of penalties for cross-municipal aggregation, curricular-track mismatch, and involvement of criticality-3 and criticality-4 schools, with rewards for intra-municipal and track-compatible aggregations.

What would settle it

Take the Calabria school network, recompute travel times from an independent road-network dataset, and re-solve one policy configuration; if the optimal aggregation plan changes or any chosen hub pair exceeds the stated thresholds, the claimed accessibility-aware optimality is not reproducible. For synthetic instances, set the travel-time thresholds externally rather than deriving them from the observed range and check whether the accessibility constraints bind; the paper's own near-zero correlations suggest they rarely do.

Watch

Extended reading notes

Core claim

The paper's central claim is that school dimensioning can be solved as a single compact integer program that simultaneously honors administrative geography, institutional compatibility, territorial fragility, road-network accessibility, capacity limits, and the required proportion of remaining autonomous schools. The compact formulation embeds the hard compatibility rules in the domain of the assignment variables, so the solver only considers admissible aggregations, and it reaches proven optimality on every tested instance: 512 parameter combinations at each of four synthetic sizes up to 1,000 schools, and 64 policy configurations for the complete Calabria public school network, with average runtimes around 0.1 seconds and always below 0.2 seconds in Calabria. Reformulated as a constrained quadratic model, the same problem was submitted to a hybrid quantum-classical solver, which in every tested configuration reproduced the classical optimum with zero standard deviation and a zero optimality gap. The authors interpret the quantum experiments as validation of the reformulation and as a proof-of-concept benchmark, not as evidence of a quantum speed advantage.

Load-bearing premise

The Calabria results assume that the road-network travel times, territorial-criticality indices, and enrollment figures used as inputs are accurate and complete, and that the thresholds $T^{\text{CI}}_{\max}=1200$ s and $T^{\text{USI}}_{\max}=2400$ s are the right policy limits; if those inputs or thresholds are wrong, the reported optimal aggregation plans are not optimal for the actual region.

Editorial extensions

If this is right

  • Authorities can rerun the optimization under many policy weights in seconds, which turns school dimensioning from a one-shot administrative act into a scenario-analysis exercise.
  • The aggregation plans are structurally stable: hub counts and cross-municipal aggregations change only modestly when penalty weights change, so different policy priorities do not force a completely different network.
  • The compact formulation scales to at least 1,000 schools with about 0.1 seconds average solution time, so it can serve as an exact baseline for future heuristics and for larger regions.
  • The constrained-quadratic reformulation is a faithful benchmark for quantum optimization: current hybrid hardware matches the classical optimum with zero gap, though not with a time advantage.
  • The synthetic instance generator supplies reproducible test cases for a problem that previously had no public benchmark dataset.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • The paper's travel-time constraints may not be seriously stressed in its own synthetic instances: thresholds are derived from the observed travel-time range (Equation 22), and the near-zero correlation between $\eta$ values and objective values suggests the accessibility constraints rarely bind; an external, tight threshold would test them.
  • In the Calabria data, no school reaches criticality level 4 and Vibo Valentia never produces a hub or an aggregation, so the territorial-protection and cross-municipal penalty terms are partly dormant in the validation; adding a genuinely fragile and fragmented province would exercise those components.
  • Because the hybrid quantum service is a black box, the perfect zero-variance match cannot separate the annealer's own contribution from classical preprocessing; instrumenting that split would clarify what the quantum hardware is actually doing.
  • A natural next step beyond the paper is to replace the static enrollment and travel-time inputs with projected demographic and infrastructure scenarios, making the aggregation plan a time-indexed sequence rather than a single snapshot.
Share X Bluesky LinkedIn Reddit HN

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

3 major / 5 minor

Summary. The paper proposes an Integer Linear Programming (ILP) framework for school network reorganization, with two equivalent formulations (a baseline model and a capacitated facility location reformulation) that jointly incorporate provincial and municipal boundaries, institutional and curricular compatibility, territorial criticality, travel-time accessibility, capacity limits, and a global reorganization target. The authors introduce a synthetic benchmark generator, evaluate the classical solver Gurobi on synthetic instances of sizes 250–1000 and on a real dataset covering all public schools in Calabria, and reformulate the model as a Constrained Quadratic Model solved with D-Wave's hybrid quantum solver. The quantum solver is reported to reproduce the classical optimal objective in all tested configurations, with zero optimality gap and zero variance across repeated runs.

Significance. If the model and the reported solutions were correct, the paper would represent a substantive contribution to educational planning: it integrates an unusually rich set of policy constraints into a single optimization model, provides a reproducible synthetic instance generator, reports a complete regional case study, and gives an honest proof-of-concept assessment of hybrid quantum optimization in which the quantum solver is explicitly not claimed to outperform classical exact methods. The authors also deserve credit for the transparent comparison protocol and for making the real-world dataset available on request. However, the central claim that the model 'rewards' intra-municipal and track-compatible aggregations is contradicted by the printed objective function, and the sensitivity analysis misattributes the corresponding coefficients C5 and C6. These internal inconsistencies mean that the reported 'optimal' aggregation plans do not demonstrably implement the stated policy objectives and that the main conclusions of the computational study are not supported as written.

major comments (3)
  1. [Section 4.1, Eq. (1); Section 4.2, Eq. (9)] The objective function in Eq. (1) and its CFLP equivalent in Eq. (9) contain the terms +C5(1−β_ij)y_ij and +C6(1−θ_ij)y_ij, with C5, C6 > 0, under minimization. Since β_ij = 1 for cross-municipal pairs and θ_ij = 1 for track-incompatible pairs, these terms add positive cost exactly when the aggregations are intra-municipal (β = 0) and track-compatible (θ = 0). This is the opposite of the reward structure described in Section 4.1, where 'the fifth term rewards aggregations between schools belonging to the same municipality' and 'the sixth term rewards aggregations between schools with compatible curricular structures.' Consequently, the optimizer is penalized for the very consolidations the policy claims to favor, and the reported 'optimal' aggregation plans in Tables B.5, C.6, E.12, and F.14 do not demonstrably implement the stated policy objectives. The signs should be reversed (or the terms defined explicitly as negative rewards), and the entire computational campaign must be re-run under the corrected objective before the results can be interpreted.
  2. [Section 5.3.1.1 and Section 6.2.1] The sensitivity analysis repeatedly groups C1 and C6 as 'inter-municipality penalty coefficients' and C2 and C5 as 'curricular compatibility coefficients.' For example, the n=250 paragraph states that 'the strongest positive correlation is observed for the inter-municipality penalty coefficients C1 and C6,' and Section 6.2.1 reports that 'the geographical coefficients remain the principal drivers ... Pearson correlation coefficients equal to 0.57 for C6 and 0.35 for C1.' However, by the definitions in Section 4, C6 multiplies (1−θ_ij), a curricular-track term, and C5 multiplies (1−β_ij), a geographic term. The labels are therefore swapped, so the reported Pearson correlations do not measure what the text claims they measure. The conclusion that 'geographical coherence represents the primary determinant of the final aggregation strategy' is not supported by the presented statistics. The correlation analysis should be re-run with the correct mapping between coefficients and objective terms, and the associated discussion in Sections 5.3.1.2 and 6.2.1 should be revised accordingly.
  3. [Section 5.2.2, Eq. (22); Section 5.3.1.1; Section 6.1, Appendix E] The accessibility constraints are effectively never binding in the synthetic study: Section 5.3.1.1 reports Pearson correlations for η_CI and η_USI that are numerically equal to zero for all instance sizes, and the thresholds are derived from the observed travel-time range by Eq. (22), which guarantees a large feasible arc set regardless of policy. For the real case, the paper fixes T_ICmax = 1200 s and T_USImax = 2400 s (Appendix E caption) but does not provide the road-network travel-time matrix, the criticality classification, or the enrollment thresholds used for eligibility filtering. As a result, the claim that the model 'preserves accessibility' in the Calabria case cannot be checked, and the reported optimal plans depend on unverifiable inputs. The authors should publish the real-case data or at least summary statistics (e.g., distributions of travel times on candidate arcs and the fraction of arcs satisfying each threshold) and should discuss whether the thresholds are binding in the optimal solutions.
minor comments (5)
  1. [Section 3.6 and Section 4.1, constraint (5)] Section 3.6 states that the macro-level parameter defines the 'exact minimum proportion of schools that must remain active,' but constraint (5) is an upper bound (≤ γ n) on the number of active schools. The wording should be changed to 'maximum proportion' to match the mathematics and the subsequent interpretation of γ in Section 5.3.1.1.
  2. [Section 2] The sentence listing application areas repeats 'packing problems' twice (see 'packing problems [10, 11], personnel scheduling [12, 13, 14, 15], packing problems [10, 16]'); one occurrence should be removed.
  3. [Appendix E, caption of Table E.12] The caption says 'maximum execution times were fixed to TICmax = 1200 s and TISmax = 2400 s'; these are maximum admissible travel times, not execution times, and the notation in the main text uses T_CI_max and T_USI_max.
  4. [Section 6.2.1] The statement that the real dataset 'does not include school institutions classified as being at risk of dimensioning' is unclear and appears to conflate the criticality-level-4 coefficient C4 with the eligibility concept; please clarify the intended meaning.
  5. [Section 4 and Table 2] The symbol N is used both for the full set of schools (Section 3) and for the eligible subset (Section 4, Table 2), which forces the reader to rely on font distinctions that are not visible in all contexts. A distinct symbol for the eligible subset would improve clarity.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the ILP formulation and its classical/quantum solvers are evaluated on the same objective function, with no fitted parameter disguised as a prediction and no load-bearing self-citation.

full rationale

The paper's central claims are (i) an ILP/CQM formulation for school aggregation, (ii) synthetic benchmark generation, and (iii) a classical/quantum solver comparison against Gurobi on the same objective function. None of these steps reduces to its own inputs by construction. The objective coefficients C1...C6 are policy parameters, not fitted values; no constant is calibrated to reproduce a target solution. The quantum experiments are explicitly framed as a solver-consistency test: the hybrid solver is compared against the classical optimum on the identical objective (Section 6.2.2), and the paper states the experiments 'should be interpreted as a validation of the proposed quantum implementation rather than as evidence of computational superiority.' That is not a circular derivation of an empirical prediction. The few self-citations (e.g., [4], [5], [8], [15], [19], [41], [43]) appear in literature-review or methodological context and are not load-bearing for the model's validity. The travel-time thresholds in Eq. (22) are derived from the synthetic instance's own travel-time range, but this is an instance-generation choice, not a back-fitting of a target result; the paper even reports that accessibility constraints are rarely binding, which is a straightforward consequence of that choice rather than evidence of circularity. The separate concern raised in the skeptic headline — that Eq. (1) and Eq. (9) use positive signs for terms described as 'rewards' in Section 4.1 — is an internal correctness/interpretation issue, not a circularity issue, because the objective function is still evaluated as written and compared across solvers on that same function. Overall, the derivation chain is self-contained: the formulation is stated, solved, and compared on the stated objective, with no fitted input renamed as a prediction and no uniqueness theorem or prior-work conclusion imported to force the result.

Assumptions & free parameters 6 free parameters · 6 assumptions · 0 invented entities

The central claim rests on hand-chosen policy weights, synthetic distribution parameters, and domain assumptions about Italian regulation, criticality data, and road travel times. The most fragile inputs are the travel-time and fragility data, which are not shipped. The model itself is not circular in a fitting sense, but two assumptions are ad hoc: the reward interpretation of the C5/C6 terms and the instance-dependent travel-time thresholds in Eq. (22).

free parameters (6)
  • Policy penalty weights C1..C6 = C1,C2,C3,C5,C6 in {20,80}; C4=0
    Set by hand to represent policy scenarios; they directly determine the objective and all reported optima.
  • Autonomy target gamma = 0.925 and 0.95
    Global constraint on the fraction of schools remaining active; varied across experiments.
  • Travel-time threshold mixing parameters eta_CI, eta_USI = eta_CI in {0.3,0.4}; eta_USI in {0.6,0.7}
    Define T_max as convex combinations of observed min and max travel times per instance (Eq. 22).
  • Maximum hub capacity = 1500 students
    Capacity limit in constraints (6) and (11); chosen without sensitivity analysis.
  • Real-case travel-time thresholds = T_ICmax=1200 s, T_USImax=2400 s
    Fixed for the Calabria instance in Appendix E; no source or derivation is provided.
  • Synthetic generator distribution parameters = rho_CI=0.6; tracks 0.4/0.3/0.3; enrollments U[300,800]; v=1; municipality weights 5/4/2/1
    These hand-chosen distributions shape all synthetic instances on which scalability and sensitivity claims rest.
assumptions (6)
  • domain assumption Aggregations require same educational category and same province.
    Constraint (4) and arc set A-tilde in Section 4 impose this rule; it is presented as Italian regulation but not formally cited.
  • domain assumption Territorial criticality levels are correct and exogenous.
    Section 3.3 and 3.4 use a four-pillar fragility index to set eligibility thresholds; no data or validation is provided.
  • domain assumption Real road travel times in Calabria are accurate.
    Section 6.1 claims road-network travel times but does not state the computation method or data source.
  • ad hoc to paper The terms (C5(1-beta)+C6(1-theta)) implement rewards for compatible aggregations.
    The text in Section 4.1 says these terms reward intra-municipal and same-track unions, but with positive coefficients in a minimization they add cost.
  • ad hoc to paper Travel-time thresholds derived from the observed travel-time range are meaningful constraints.
    Eq. (22) sets thresholds from each instance's own min and max travel times; empirical correlations for eta are near zero, so accessibility rarely binds.
  • domain assumption Schools above the eligibility thresholds remain active and are not decision variables.
    Section 3.4 and 5.2.3 assume these schools stay active; if they could also be selected as hubs, the feasible region changes.

how reviews work

0 comments
Cite this review

Pith. "Pith review of School network reorganization under educational and spatial constraints using classical and quantum optimization." pith.science (2026). https://pith.science/paper/ZXJADF2D

@misc{pith2026260805427,
  author       = {Pith},
  title        = {Pith review of: School network reorganization under educational and spatial constraints using classical and quantum optimization},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/ZXJADF2D}},
  note         = {Machine review of arXiv:2608.05427}
}
read the original abstract

School network reorganization is a strategic planning problem that requires balancing demographic trends, territorial accessibility, educational requirements, and institutional constraints while ensuring an efficient allocation of public resources. This paper proposes an optimization framework for school dimensioning decisions based on a novel Integer Linear Programming formulation integrating geographical, administrative, and educational criteria. A synthetic benchmark generator is introduced to evaluate the scalability and computational performance of the model on artificial instances, while a real-world case study involving the complete public school network of the Calabria region (Italy) is conducted using actual institutional, territorial, and demographic data. The proposed approach effectively identifies optimal aggregation plans under different policy scenarios while preserving the structural characteristics of the educational system. Furthermore, the model is reformulated as a constrained quadratic model and implemented within a hybrid quantum optimization environment, demonstrating its compatibility with emerging quantum technologies. The results highlight the robustness of the proposed methodology and its potential as a decision-support tool for sustainable and equitable school network planning.

Figures

Figures reproduced from arXiv: 2608.05427 by the authors.

Figure 1
Figure 1. Schematic representation of the main institutional configurations of the Italian school system, [PITH_FULL_IMAGE:figures/full_fig_p009_1.png] view at source ↗

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

44 extracted references · 38 canonical work pages

  1. [1]

    URL:https://ec.europa.eu/e urostat/web/interactive-publications/demography-2026, accessed: 9 July 2026

    Eurostat, Demography of europe – 2026 edition, 2025. URL:https://ec.europa.eu/e urostat/web/interactive-publications/demography-2026, accessed: 9 July 2026

    work page 2026

  2. [2]

    URL:https://economy-finance.ec.europa.eu/economic-surveilla nce-eu-member-states/country-pages-including-country-reports/country-r eport-italy_en, accessed 9 July 2026

    European Commission, 2025 Country Report – Italy, Technical Report, European Com- mission, 2025. URL:https://economy-finance.ec.europa.eu/economic-surveilla nce-eu-member-states/country-pages-including-country-reports/country-r eport-italy_en, accessed 9 July 2026

    work page 2025

  3. [3]

    URL:https://www.istat.it/, accessed: 9 July 2026

    ISTAT, Indicatori demografici – Anno 2024, Technical Report, Istituto Nazionale di Statistica, 2025. URL:https://www.istat.it/, accessed: 9 July 2026

    work page 2024

  4. [4]

    Ciacco, F

    A. Ciacco, F. Guerriero, E. Osaba, Steiner traveling salesman problem with quantum annealing, in: Proceedings of the Genetic and Evolutionary Computation Conference Companion, 2025, pp. 2412–2418

    2025

  5. [5]

    Ciacco, L

    A. Ciacco, L. D. P. Pugliese, F. Guerriero, Traveling salesman problem with a prepro- cessing method for classical and quantum optimization, arXiv preprint arXiv:2603.23290 (2026)

    arXiv 2026

  6. [6]

    J. B. Holliday, D. Blount, E. Osaba, K. Luu, Advanced quantum annealing approach to vehicle routing problems with time windows, arXiv preprint arXiv:2503.24285 (2025)

    arXiv 2025

  7. [7]

    Osaba, E

    E. Osaba, E. Garrote, P. Miranda-Rodriguez, A. Ciacco, I. Cabanes, A. Mancisidor, Quantum-assisted automatic path-planning for robotic quality inspection in industry 4.0, in: 2025 IEEE International Conference on Quantum Computing and Engineering (QCE), volume 2, IEEE, 2025, pp. 388–389

    work page 2025

  8. [8]

    Ciacco, F

    A. Ciacco, F. Guerriero, F. P. Saccomanno, Quantum annealing for the two-level facility location problem, Future Generation Computer Systems 174 (2026) 107961

    work page 2026

Show all 44 references
  1. [9]

    Malviya, B

    G. Malviya, B. AkashNarayanan, J. Seshadri, Logistics network optimization using quantum annealing, in: International Conference on Emerging Trends and Technologies on Intelligent Systems, Springer, 2023, pp. 401–413

    2023

  2. [10]

    M. G. de Andoin, E. Osaba, I. Oregi, E. Villar-Rodriguez, M. Sanz, Hybrid quantum- classical heuristic for the bin packing problem, in: Proceedings of the Genetic and Evolutionary Computation Conference Companion, 2022, pp. 2214–2222

    2022

  3. [11]

    Cellini, A

    L. Cellini, A. Macaluso, M. Lombardi, Qal-bp: an augmented lagrangian quantum approach for bin packing, Scientific Reports 14 (2024) 5142

    2024

  4. [12]

    Venturelli, D

    D. Venturelli, D. Marchand, G. Rojo, Job shop scheduling solver based on quantum annealing, in: Proc. of ICAPS-16 Workshop on Constraint Satisfaction Techniques for Planning and Scheduling (COPLAS), 2016, pp. 25–34. 43

    2016

  5. [13]

    URL: https://doi.org/10.1038/s41598-024-67168-6

    L.F.PérezArmas, S.Creemers, S.Deleplanque, Solvingtheresourceconstrainedproject scheduling problem with quantum annealing, Scientific Reports 14 (2024) 16784. URL: https://doi.org/10.1038/s41598-024-67168-6. doi:10.1038/s41598-024-67168 -6

    2024 doi

  6. [14]

    C. Li, Z. Liu, Y. Song, H. Liu, H. Liu, X. Liu, Quantum computing approaches to optimizeemployeeschedulinginmulti-taskcallcenters, in: TheInternationalConference on Smart Manufacturing, Industrial & Logistics Engineering (SMILE), Springer, 2023, pp. 3–9

    2023

  7. [15]

    Ciacco, F

    A. Ciacco, F. Guerriero, E. Osaba, Quantum annealing for staff scheduling in educa- tional environments, in: 2026 International Conference on Quantum Communications, Networking, and Computing (QCNC), IEEE, 2026, pp. 630–637

    2026

  8. [16]

    Garcia-de Andoin, I

    M. Garcia-de Andoin, I. Oregi, E. Villar-Rodriguez, E. Osaba, M. Sanz, Compara- tive benchmark of a quantum algorithm for the bin packing problem, arXiv preprint arXiv:2207.07460 (2022)

    2022 arXiv

  9. [17]

    Venturelli, A

    D. Venturelli, A. Kondratyev, Reverse quantum annealing approach to portfolio opti- mization problems, Quantum Machine Intelligence 1 (2019) 17–30

    2019

  10. [18]

    Rosenberg, P

    G. Rosenberg, P. Haghnegahdar, P. Goddard, P. Carr, K. Wu, M. L. De Prado, Solving the optimal trading trajectory problem using a quantum annealer, in: Proceedings of the 8th workshop on high performance computational finance, 2015, pp. 1–7

    2015

  11. [19]

    Ciacco, F

    A. Ciacco, F. Guerriero, G. Macrina, Review of quantum algorithms for medicine, finance and logistics, Soft Computing 29 (2025) 2129–2170

    2025

  12. [20]

    F. A. Quinton, P. A. S. Myhr, M. Barani, P. Crespo del Granado, H. Zhang, Quantum annealing applications, challenges and limitations for optimisation problems compared to classical solvers, Scientific Reports 15 (2025) 12733

    2025

  13. [21]

    Gilbert, J

    V. Gilbert, J. Rodriguez, S. Louise, Benchmarking quantum annealers with near-optimal minor-embedded instances, in: 2024 IEEE International Conference on Quantum Com- puting and Engineering (QCE), volume 1, IEEE, 2024, pp. 531–537

    2024

  14. [22]

    R. J. O’brien, Model for planning the location and size of urban schools, Socio-economic Planning Sciences 2 (1969) 141–153

    1969

  15. [23]

    M. I. Henig, M. Haimovich, Y. Oron, Dynamic capacity planning of public schools in changing urban communities, Socio-Economic Planning Sciences 20 (1986) 73–82. doi:10.1016/0038-0121(86)90017-4

    1986 doi

  16. [24]

    J. J. Greenleaf, T. P. Harrison, A mathematical programming approach to elementary school facility decisions, Socio-Economic Planning Sciences 21 (1987) 67–80. doi:10.1 016/0038-0121(87)90002-9

    1987

  17. [25]

    J. E. Bruno, P. W. Andersen, Analytical methods for planning educational facilities in an era of declining enrollments, Socio-Economic Planning Sciences 16 (1982) 121–131. 44

    1982

  18. [26]

    R. L. Church, A. T. Murray, Modeling school utilization and consolidation, Journal of Urban Planning and Development 119 (1993) 23–38. doi:10.1061/(ASCE)0733-948 8(1993)119:1(23)

    1993 doi

  19. [27]

    D. S. Lemberg, R. L. Church, The school boundary stability problem over time, Socio- Economic Planning Sciences 34 (2000) 159–176. doi:10.1016/S0038-0121(99)00019-1

    2000 doi

  20. [28]

    F. Caro, T. Shirabe, M. Guignard, A. Weintraub, School redistricting: Embedding gis tools with integer programming, Journal of the Operational Research Society 55 (2004) 836–849

    2004

  21. [29]

    Antunes, D

    A. Antunes, D. Peeters, A dynamic optimization model for school network planning, Socio-Economic Planning Sciences 34 (2000) 101–120. doi:10.1016/S0038-0121(99)0 0014-2

    2000 doi

  22. [30]

    Teixeira, A

    J. Teixeira, A. Antunes, D. Peeters, An optimization-based study on the redeployment of a secondary school network, Environment and Planning B: Planning and Design 34 (2007) 296–315. doi:10.1068/b31173

    2007 doi

  23. [31]

    Haase, S

    K. Haase, S. Müller, Management of school locations allowing for free school choice, Omega 41 (2013) 847–855. doi:10.1016/j.omega.2012.10.008

    2013 doi

  24. [32]

    M. R. Alifi, H. Hayati, S. H. Supangkat, Optimization of school network using location- allocation analysis: Case study bandung, indonesia, in: 2017 IEEE Region 10 Sympo- sium (TENSYMP), IEEE, 2017, pp. 1–6

    2017

  25. [33]

    Barbara, D

    M. Barbara, D. Rey, A. Akbarnezhad, Optimizing location of new public schools in town planning considering supply and demand, Journal of Urban Planning and Development 147 (2021) 04021057. doi:10.1061/(ASCE)UP.1943-5444.0000767

    2021

  26. [34]

    de Armas, H

    J. de Armas, H. Ramalhinho, M. Reynal-Querol, Improving the accessibility to public schools in urban areas of developing countries through a location model and an analytical framework, PLOS ONE 17 (2022) e0262520. doi:10.1371/journal.pone.0262520

    2022 doi

  27. [35]

    Bruno, E

    G. Bruno, E. Esposito, A. Genovese, C. Piccolo, Institutions and facility mergers in the italian education system: Models and case studies, Socio-Economic Planning Sciences 53 (2016) 23–32

    2016

  28. [36]

    Hannum, X

    E. Hannum, X. Liu, F. Wang, Estimating the effects of educational system consolidation: The case of china’s rural school closure initiative, Economic Development and Cultural Change 70 (2021) 485–528

    2021

  29. [37]

    Bhatnagar, N

    A. Bhatnagar, N. B. Bolia, A sustainable decision-making framework for school consol- idation policy, Regional Science Policy & Practice 15 (2023) 1037–1063

    2023

  30. [38]

    T.Klein, R.Aue, J.Ortega, Schoolchoicewithindependentversusconsolidateddistricts, Games and Economic Behavior 147 (2024) 170–205. 45

    2024

  31. [39]

    S. Kim, T. Kim, J. Jeon, Optimal prioritization model for school closure decisions considering educational accessibility in shrinking regions, Sustainability 17 (2025) 4057. doi:10.3390/su17094057

    2025 doi

  32. [40]

    URL:https: //www.gurobi.com/documentation/

    Gurobi Optimization, LLC, Gurobi Optimizer Reference Manual, 2024. URL:https: //www.gurobi.com/documentation/

    2024

  33. [41]

    Ciacco, L

    A. Ciacco, L. D. P. Pugliese, F. Guerriero, Cutting-plane methodology via quantum op- timization for solving the traveling salesman problem, arXiv preprint arXiv:2604.20321 (2026)

    2026 arXiv

  34. [42]

    Bertuzzi, D

    A. Bertuzzi, D. Ferrari, A. Manzalini, M. Amoretti, Evaluation of quantum and hybrid solvers for combinatorial optimization, in: Proceedings of the 21st ACM International Conference on Computing Frontiers, 2024, pp. 232–239

    2024

  35. [43]

    Ciacco, F

    A. Ciacco, F. Guerriero, E. Osaba, Steiner traveling salesman problem with time win- dows and pickup–delivery: Integrating classical and quantum optimization, Computers & Operations Research (2026) 107589

    2026

  36. [44]

    Pearson, Note on regression and inheritance in the case of two parents, Proceedings of the Royal Society of London 58 (1895) 240–242

    K. Pearson, Note on regression and inheritance in the case of two parents, Proceedings of the Royal Society of London 58 (1895) 240–242. doi:10.1098/rspl.1895.0041. 46

Pith tools

Reviewed August 8, 2026 · model on record in the stance chip above.