Fuzzy location and allocation Hub Network Design for Air Cargo Transportation Considering Sustainability and Time Window

Ali Mohammad Malekdar; Mohsen Akbarpour Shirazi

REVIEW 5 major objections 7 minor 2 references

Fuzzy location and allocation Hub Network Design for Air Cargo Transportation Considering Sustainability and Time Window

T0 review · 5 major / 7 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read This paper proposes a tri-objective, capacity-limited hub location and allocation model for air cargo that couples sustainability with service time windows and reports that greater demand uncertainty raises both costs and emissions.

desk verdict The hard time-window constraint indexes the wrong route and the uncertainty sensitivity result is forced by the defuzzification formula, so the paper's central claims don't hold as written. read the letter →

arxiv 2506.12064 v1 pith:CQZTUUTS submitted 2025-06-01 math.GM

Ali Mohammad Malekdar , Mohsen Akbarpour Shirazi This is my paper · ORCID

classification math.GM MSC 90B8090C2990C70

keywords hublocationproblemaircargotransportationsustainabilitytimewindowfuzzydemandmulti-objectiveoptimizationNSGA-IIMOPSO

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 proposes a capacity-limited hub location and allocation model for air cargo networks in which a fixed number of nodes are chosen as hubs and all other nodes are assigned to them. The model has three objectives: total transport and hub cost, aircraft pollutant emissions, and penalties for missing delivery time windows, and it treats demand as a trapezoidal fuzzy number converted into a deterministic expected value. The authors' central claim is that this tri-objective model can be solved by the epsilon-constraint method on small instances and by NSGA-II, MOPSO, and MOWOA on larger ones. Their main empirical finding is that as the uncertainty rate rises, both total system cost and environmental pollution increase, and under a TOPSIS comparison NSGA-II ranks highest among the metaheuristics.

What carries the argument

The carrying mechanism is the route-variable decomposition $Y_{ij}$, $W_{ikj}$, $V_{iklj}$ together with the fuzzy expected-value replacement $$\tilde{q}_{ij} \cong (1-\$\alpha$')\frac{$q^{1}$_{ij}+$q^{2}$_{ij}}{2} + \$\alpha$'\frac{$q^{3}$_{ij}+$q^{4}$_{ij}}{2},$$ which converts range-valued demand into a linear parameter controlled by the uncertainty rate $\alpha'$. This makes each objective, cost, emissions, and time-window penalty, a deterministic linear function of the same binary routing decisions. The time-window penalties use $\max\{0, E_{ij}-\tau\}$ and $\max\{0, \tau-L_{ij}\}$ for early and late arrivals, while constraint (14) enforces hub capacity and constraint (15) sets the hard maximum transfer time.

What would settle it

Re-solve the 10-node epsilon-constraint example after (a) replacing the expected-value defuzzification with another standard defuzzification and (b) rewriting constraint (15) so the travel-time sum follows the route variable's hub order; if the efficient solution set or the rising cost-and-emission trend changes, the reported results depend on those transcription choices rather than on the model's structure.

Watch

Extended reading notes

Core claim

On the paper's own terms, the central discovery is that a sustainable hub network for air cargo can be modeled as a tri-objective integer program in which the same route-choice variables, namely direct, one-hub, and two-hub transport, simultaneously determine cost, emissions, and on-time delivery. The emission objective follows the LTO and CCD aircraft emission accounting, the time-window objective charges early and late penalties against soft time bounds, and a separate constraint enforces a hard maximum travel time. The fuzzy expected-value transformation turns uncertain demand into a parameter the solver can handle. The paper reports seven efficient solutions for a 10-node instance solved by the epsilon-constraint method, and Pareto fronts from the metaheuristics; on larger instances NSGA-II has the best spread and lowest CPU time, MOPSO has the best spacing, and MOWOA has the largest front, with TOPSIS ranking NSGA-II first. Its empirical regularity is that increasing the uncertainty rate raises total costs and emissions because it raises expected demand under fixed aircraft capacity.

Load-bearing premise

The load-bearing premise is that the written model is faithful: range-valued demand can be replaced by its expected value, and the two-hub time-window constraint (15) adds travel times in the same hub order as the route variable it bounds.

Editorial extensions

If this is right

Because the three sustainability objectives stay linear, small air-cargo hub networks can be solved exactly with epsilon-constraint methods, giving planners a benchmark Pareto set.
For larger hub networks, the three metaheuristics keep run times within practical limits, so the approach scales beyond exact methods.
Raising the uncertainty rate in fuzzy demand raises expected shipment volumes; under fixed aircraft capacity this mechanically increases both transportation cost and emissions, so forecast quality directly shapes the sustainability assessment.
The discount factors $\alpha$ and $\beta$ shift the trade-off between cost and emissions, meaning the hub network design is sensitive to how inter-hub and spoke-hub discounts are negotiated.
NSGA-II is the recommended algorithm for this problem according to the paper's TOPSIS ranking, with the best spread and lowest average CPU time among the three metaheuristics compared.

Reading between the lines

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

A cleaner test of the uncertainty result would hold expected demand fixed while widening the fuzzy spread; the reported cost and pollution rise may be driven by the larger mean that the paper's uncertainty levels imply, not by fuzziness alone.
The TOPSIS ranking gives equal weight to the number of Pareto solutions, mean ideal distance, spacing, and CPU time; with unequal weights reflecting a planner's priorities, MOPSO or MOWOA could rank ahead of NSGA-II.
Modeling heterogeneous aircraft, as the paper suggests for future work, would require replacing the single ceiling-based emission term with trip-specific capacities and would likely weaken the monotone relationship between demand and pollution.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

You should know two things before spending time on this. First, the hard time-window constraint (15) is written for the reverse route: it attaches the travel time of i-k-l-j to V_il_kj, while V_iklj is the variable defined for the i-k-l-j route. The same index confusion shows up in (11) and (12), where V_iklj is bounded by Z_kl and Z_lj instead of Z_ik and Z_jl. These are not cosmetic typos; they change the feasible set. The reported allocations do not demonstrably satisfy the intended service-time limitation. Second, the main sensitivity result—that higher uncertainty increases cost and pollution—is built into the defuzzification formula (24). Since q_tilde is a convex combination of E1 and E2, any increase in alpha' shifts demand upward whenever E2 > E1. The "finding" is an artifact of the modeling choice, not something the data or the optimization reveals.

To give credit where it's due: the paper does assemble a legitimate new combination—tri-objective capacitated hub location with hard and soft time windows, fuzzy demand, and metaheuristic solution methods. The numerical work is standard but reasonably careful: ten instances, four algorithms, four comparison metrics, and a TOPSIS ranking. If the constraints were fixed, the model could be a usable extension to the literature.

But there are more soft spots. The paper claims three linear objective functions, yet Z3 is full of max{0,...} terms, and no linearization is provided. The fuzzy demand table in Table 2 is garbled (four identical entries). No real data are used; the small instance is random uniform. The TOPSIS weights are set to 0.25 each without justification, so the "NSGA-II is best" conclusion is not robust to that choice.

My take: this is a promising draft, not a publishable paper. The internal inconsistencies are load-bearing, and the central sensitivity claim is circular. It deserves a desk reject with an invitation to resubmit after rewriting the constraints, adding the linearization, and testing on real or realistically generated data. I would not spend referee time on the current version.

Referee Report

5 major / 7 minor

Summary. The paper proposes a tri-objective capacitated hub location and allocation model for air cargo transportation, incorporating fuzzy demand, hard and soft time windows, and sustainability objectives (economic cost, environmental emissions, and time-window penalties). It claims that the model is linear and solves it with the epsilon-constraint method and three metaheuristics (NSGA-II, MOPSO, MOWOA). The main reported findings are that higher demand uncertainty increases total cost and pollution, and that NSGA-II outperforms the other algorithms according to TOPSIS ranking.

Significance. If the model were correctly formulated and the results verified, the paper would contribute to the sustainable hub location literature by combining time windows, fuzzy demand, and three sustainability pillars. The authors make a credible effort to incorporate EEA Tier 3 emission estimation and to benchmark multiple metaheuristics against an exact method. However, the load-bearing formulation errors and the circular sensitivity result prevent the paper from establishing its claims; the numerical comparisons are therefore not trustworthy.

major comments (5)

[Section 3, Eq. (3); Section 4, epsilon-constrained method] The abstract and Section 4 state that the model has three linear objective functions, but Z3 contains max{0, ...} terms and Z2 contains ceiling functions ⌈q̃_{ij}/φ⌉. Neither is linear, and no linearization with auxiliary variables is provided. Since the epsilon-constraint method as described requires solving a single-objective linear program at each stage, the claim of exact solution is unsupported, and the reported Pareto solutions from the epsilon-constraint method are not validated.
[Section 3, Eq. (15) and Eq. (11) with variable definitions] The variable V_{iklj} is defined as the route i→k→l→j and is used with transit time τ_{ik}+τ_{kl}+τ_{lj} in constraints (10)-(14) and in Z3. However, Eq. (15) writes the two-hub term as (τ_{ik}+τ_{kl}+τ_{lj}) V_{il_kj}, which is the transit time for i→k→l→j applied to the reverse route variable i→l→k→j. Hence the hard time-window constraint never bounds V_{iklj}, the actual two-hub route, and a solution using i→k→l→j faces no hard time bound. In addition, Eq. (11) enforces V_{iklj} ≤ Z_{kl}, where Z_{kl} is the assignment variable indicating that node k is assigned to hub l, not a hub-to-hub connection variable; this does not enforce the intended connectivity between hubs k and l. These inconsistencies change the feasible set and invalidate the reported allocations as solutions to the stated model.
[Section 3, Eq. (24) and Section 5, Fig. 5] The defuzzified demand is (1-α')E1 + α'E2 with E1=(q1+q2)/2 and E2=(q3+q4)/2. For a non-degenerate trapezoidal fuzzy number, E2 > E1, so increasing α' strictly increases the demand value used in every objective. The conclusion that higher 'uncertainty' raises cost and emissions is therefore a necessary consequence of the defuzzification formula, not an empirical finding about the system. Moreover, α' is a weighting parameter in the expected-value formula, not a measure of uncertainty magnitude; the paper does not justify equating it with the uncertainty rate.
[Section 5, Table 2] The fuzzy demand row in Table 2 reads 'q̃_{ij} q_{ij}^1 ~U(60,70) − q_{ij}^1 ~U(60,70) − q_{ij}^1 ~U(60,70) − q_{ij}^1 ~U(60,70)', which is garbled and would imply that all four trapezoidal parameters are identically distributed as U(60,70), making the fuzzy number degenerate. The actual membership function parameters are not clearly specified, so the sensitivity analysis in Fig. 5 lacks a precise data basis.
[Section 5, Table 8] The TOPSIS ranking of the metaheuristics uses equal weights of 0.25 for the four indices with no sensitivity analysis. Because the algorithms trade off different indices (e.g., MOWOA has the highest NPF, NSGA-II the highest MSI and lowest CPT), the ranking and the conclusion that NSGA-II is superior may depend on the arbitrary chosen weights; the paper provides no evidence that the ranking is robust.

minor comments (7)

[Section 3, Eq. (14)] Equation (14) contains a duplicated plus sign: '…V_{iklj} + + ∑∑∑ q̃_{ij} V_{ilkj} ≤ …'.
[Section 4, Table 1] The mutation probability values for NSGA-II (Pm = 0.7, 0.8, 0.9) are unusually high; the paper does not explain how the parameter levels were selected or whether the 'optima level' was used for the results in Table 7.
[References] Several references are duplicated: Ernst et al. (2009), Harley et al. (2020), and Krile et al. (2015) each appear twice.
[Section 3, Eq. (5)] The text describes Eq. (5) as stating 'a non-hub node can be connected to at most one hub node,' but the equation is an equality (∑_k Z_{ik} = 1), meaning exactly one hub assignment.
[Section 5, Table 3] Table 3 is difficult to read because of the layout; the row and column labels are not clearly separated, and the meaning of 'Direct' entries is not explained in the table itself.
[Section 5, Table 4] The sentence 'with increasing coefficients α and β, the network costs increase due to the rise in economic discount factors' is confusing; α and β are discount factors, and the phrase 'rise in economic discount factors' is ambiguous.
[Section 5, Tables 6-7] The paper does not report the number of replicates or any statistical significance tests for the metaheuristic results, so the comparisons of CPT and MSI are anecdotal.

Circularity Check

1 steps flagged · score 6.0 of 10

The headline sensitivity result—higher uncertainty increases cost and pollution—is built into the defuzzification equation (24), so the 'finding' reduces to a monotonicity property of the chosen fuzzy replacement.

self definitional [Section 3, Eq. (24) and Section 5, text after Fig. 5]
"Considering the uncertainty in demand parameters in this paper and the use of fuzzy programming methods to control the mathematical model, changes in the objective function values due to uncertainty rates are shown in Fig. 5... Fig. 5 shows that with the increase in uncertainty rates, the demand values for nodes have increased. This increase in demand has led to higher product transfer costs through the air transport system and also increased the level of environmental pollutant emissions."

The paper's headline conclusion about uncertainty is not an emergent system property but a direct consequence of the defuzzification formula it adopts. In Eq. (24), the fuzzy demand is replaced by q̃_ij ≈ (1-α')·(q^1+q^2)/2 + α'·(q^3+q^4)/2. For a trapezoidal fuzzy number, the expected values satisfy E2 > E1, so q̃ is strictly increasing in α'. Both objective functions Z1 (cost) and Z2 (emissions) contain q̃ with positive coefficients only (no negative terms). Therefore any increase in α' necessarily increases Z1 and Z2. The paper reports exactly this as its key sensitivity result, without noting that it is an algebraic identity of its own modeling choice.

full rationale

The central modeling contribution—a capacitated multi-objective hub location network with fuzzy demand and time windows—is presented as a standard mathematical program, and the algorithmic comparison (NSGA-II, MOPSO, MOWOA) is an external computational experiment. No self-citations or imported uniqueness theorems appear. However, the paper's headline result about uncertainty is circular in the specific sense defined above: the defuzzification in Eq. (24) makes demand an increasing function of α', and the objectives are monotone in demand, so the reported sensitivity (higher α' → higher cost and pollution) is forced by the paper's own equation. The TOPSIS ranking with equal weights is an arbitrary but non-circular methodological choice. Separately, the reviewer's concern about constraint (15) mis-indexing V_il_kj versus V_iklj is a correctness/feasibility issue in the model, not a circularity issue, but it further undermines confidence in the reported feasible allocations. Overall, the formulation is self-contained, but the main advertised insight is definitionally predetermined; score 6 reflects partial circularity centered on the uncertainty sensitivity claim.

Assumptions & free parameters 4 free parameters · 5 assumptions · 0 invented entities

The central claims rest on several hand-chosen parameters (defuzzification level, big-M, TOPSIS weights, metaheuristic tuning) and on unstated modeling assumptions about fuzzy demand, emissions, and linearity. There are no invented physical entities.

free parameters (4)

alpha' (uncertainty level) = 0.1, 0.3, 0.5, 0.7, 0.9
User-chosen levels for the fuzzy defuzzification in Eq (24); the rise in cost and emissions with alpha' is a direct consequence of this parameter rather than an empirical finding.
Big-M constant in Eq (14) = unspecified
Introduced ad hoc to relax hub capacity constraints when a node is not selected as a hub; no value or tuning procedure is given, so the exact results depend on an unstated parameter.
TOPSIS evaluation weights = 0.25 for NPF, MSI, SM, CPT
Chosen by hand without sensitivity analysis; the ranking that places NSGA-II first depends on this arbitrary weighting.
Metaheuristic tuning parameters = Population 50-120, iterations 75-125, crossover 0.05-0.07, mutation 0.7-0.9, etc.
Selected from three candidate levels in Table 1; the reported algorithm comparison is conditional on these hand-picked settings.

assumptions (5)

domain assumption Hub nodes have limited capacity, demand is a trapezoidal fuzzy number, and the number of hubs is predetermined.
Stated assumptions in Section 3; these define the problem but are not validated against a real air cargo system.
ad hoc to paper The fuzzy expected value of demand is (1-alpha') E1 + alpha' E2 with E1=(q1+q2)/2 and E2=(q3+q4)/2.
Eq (24) is presented without citation to an established fuzzy programming method and without justification that this convex combination represents the decision-maker's uncertainty attitude.
domain assumption Aircraft emissions can be modeled additively as E_LTO + R_CCD(distance) for two pollutant groups, following EEA Tier 3.
Borrowed from EEA (2016) via the text; assumed to apply to the synthetic distances and demands used here.
ad hoc to paper The objective functions are linear and the epsilon-constraint method solves the model exactly.
The paper claims three linear functions, but Eq (3) contains max terms; no linearization is supplied, so the exact solution claim is unsupported.
domain assumption All nodes have sufficient flights and uniform fleet speed and capacity.
Stated in model assumptions; simplifies reality and affects how time windows are computed.

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Cite this review

Pith. "Pith review of Fuzzy location and allocation Hub Network Design for Air Cargo Transportation Considering Sustainability and Time Window." pith.science (2026). https://pith.science/paper/CQZTUUTS

@misc{pith2026250612064,
  author       = {Pith},
  title        = {Pith review of: Fuzzy location and allocation Hub Network Design for Air Cargo Transportation Considering Sustainability and Time Window},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/CQZTUUTS}},
  note         = {Machine review of arXiv:2506.12064}
}

read the original abstract

Hub location Problems seek to find hub facilities and assign non-hub nodes to them in such a way that the flow between origin and destination should be effectively established according to the desired goal. In general, in the literature of location, it is assumed that the time horizon of hub network design is a single time horizon. In the last two decades these problems have attracted special attention in the field of facility location problems and have wide applications in different fields including air cargo transportation. Cargo transportation is one of the most important economic sectors of any country. There are different ways to transport cargo, but air transport is preferred because it has high speed and security, so it is suitable for transporting goods related to technology, food, medicines, etc. In this article designing a hub network for air cargo transportation, taking into account hard and soft time windows along with considering the limited capacity for each hub under uncertainty is discussed. The proposed model is a developed model of an existing model in literature. Our Study has three linear functions: economic, environmental and social. In this article method of Fuzzy programming has been used to control the non-deterministic demand parameter. Results of model that has been solved by epsilon limitation method, NSGA-II, MOPSO and MOWOA algorithm show that as the uncertainty rate increases, the total costs of the system as well as the amount of environmental pollution increases. The reviews indicate the high performance of the NSGA-II algorithm in solving the proposed model.

Figures

Figures reproduced from arXiv: 2506.12064 by the authors.

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Reference graph

Works this paper leans on

2 extracted references · 2 canonical work pages

[1]

Ahmadianfar, I., Adib, A., & Taghian, M. (2017). Optimization of multi -reservoir operation with a new hedging rule: application of fuzzy set theory and NSGA -II. Applied Water Science, 7, 3075-3086. Alumur, S. A., Campbell, J. F., Contreras, I., Kara, B. Y., Marianov, V., & O’Kelly, M. E. (2021). Perspectives on modeling hub location problems. European J...

work page 2017
[2628]

Krile, S., Krile, M., & Průša, P. (2015). Non - linear mini -max problem of multi -stop flight routes. Transport, 30(3), 361-371. Krile, S., Krile, M., & Průša, P. (2015). Non - linear mini -max problem of multi -stop flight routes. Transport, 30(3), 361-371. Lin, C. C., Lin, J. Y., & Chen, Y. C. (2012). The capacitated p-hub median problem with integral ...

work page 2015

Pith tools

Reviewed August 7, 2026 · model on record in the stance chip above.

[1] [1]

Ahmadianfar, I., Adib, A., & Taghian, M. (2017). Optimization of multi -reservoir operation with a new hedging rule: application of fuzzy set theory and NSGA -II. Applied Water Science, 7, 3075-3086. Alumur, S. A., Campbell, J. F., Contreras, I., Kara, B. Y., Marianov, V., & O’Kelly, M. E. (2021). Perspectives on modeling hub location problems. European J...

work page 2017

[2] [2628]

Krile, S., Krile, M., & Průša, P. (2015). Non - linear mini -max problem of multi -stop flight routes. Transport, 30(3), 361-371. Krile, S., Krile, M., & Průša, P. (2015). Non - linear mini -max problem of multi -stop flight routes. Transport, 30(3), 361-371. Lin, C. C., Lin, J. Y., & Chen, Y. C. (2012). The capacitated p-hub median problem with integral ...

work page 2015