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REVIEW 3 major objections 4 minor 52 references

A Coalition Game for On-demand Multi-modal 3D Automated Delivery System

T0 review · 3 major / 4 minor · reviewed 2026-08-11 · deepseek-v4-flash

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Pith's one-line read A single learned routing model solves mixed drone-robot pickup and delivery and reveals that most drone-robot coalitions are stable and can share costs fairly.

desk verdict Useful multi-modal VRP architecture with a serious game-theory proof gap and unvalidated coalition claims; the routing results deserve a careful look, but the cooperative-advantage conclusion should not be cited yet. read the letter →

arxiv 2412.17252 v2 pith:OVRD2C3I submitted 2024-12-23 cs.LG math.OC

Farzan Moosavi , Bilal Farooq This is my paper · ORCID
classification cs.LGmath.OC MSC 91A1290B0668T07
keywords coalitiongameCE-CPDPTWUAV-ADRdeliveryheterogeneousgraphattentionnetworktransformerdecoderdeepreinforcementlearninglast-milelogisticsShapleyvalue
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 argues that urban last-mile delivery is cheaper and more reliable when drones and sidewalk delivery robots operate as a cooperating fleet, and that this cooperation can be analysed as a coalition game with fair cost shares. To support that, it formulates the routing task as a collaborative electric capacitated pickup-and-delivery problem with time windows (CE-CPDPTW) on two overlapping urban networks, and solves it with an end-to-end reinforcement-learning policy built from a heterogeneous edge-enhanced graph attention encoder and a transformer decoder. On synthetic and Mississauga-based instances with wind, uneven demand, and tight time windows, the model is reported to beat classical solvers and attention-based baselines on cost while running in seconds. Using the same trained model's costs as the characteristic function, the coalition analysis finds that most tested drone-robot coalitions satisfy the paper's super-additivity condition (combined cost no greater than the sum of separate costs), the grand coalition usually has a non-empty core, and Shapley values can divide the savings. The intended payoff is a practical rule for when to combine drones and sidewalk robots and how to price cooperation.

What carries the argument

The central object is HetGAT, the paper's named model: a dual heterogeneous edge-enhanced graph attention encoder paired with a transformer decoder. The encoder builds separate node embeddings for the UAV and ADR networks, computes edge features from shortest-path travel times divided by each mode's speed and time-window slack, and masks message passing to temporal neighbours for UAVs and spatial neighbours for ADRs using thresholds ζ and μ. The decoder concatenates per-vehicle state (load, battery, elapsed time) with the global graph embedding, computes vehicle-customer compatibility scores, applies masks for capacity, battery, visited nodes, precedence, and restricted zones, and then emits node-vehicle assignments. The coalition layer then uses the trained model's cost on any subset of vehicles as the characteristic function and checks the conditions for a non-empty core and Shapley fairness.

What would settle it

Compute exact optimal costs for all sub-coalitions on the smallest CE-CPDPTW instances still solvable to optimality (for example, 10-20 requests with 2-4 vehicles), then re-run the super-additivity and core checks with those exact costs in place of the trained model's C(S). If a material share of the model's stable-coalition verdicts reverses, the paper's game-theoretic conclusions would not survive the swap.

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Extended reading notes

Core claim

The paper's central claim is that one learned model can solve CE-CPDPTW for a heterogeneous UAV-ADR fleet and, without retraining, can serve as the characteristic function C(S) of a cooperative game that reveals when collaboration pays. In the reported experiments, HetGAT produces lower total cost---monetized travel time, waiting, and delay penalties---than commercial exact solvers and the two transformer-based baselines on almost every tested scale, with solution times measured in seconds rather than hours. Treating the model's cost for a coalition as C(S), the paper reports that C(S1∪S2) ≤ C(S1)+C(S2) holds for most tested coalitions (which it calls super-additivity), that the core is non-empty in nearly every configuration except some large-network cases with a single UAV, and that Shapley allocation assigns larger marginal shares to UAVs in uniform demand but shifts toward ADRs when demand is clustered and heavy. The intended consequence is a managerial tool: identify which sub-coalitions are stable, then divide operational cost fairly so no mode prefers to defect.

Load-bearing premise

The coalition conclusions depend on the assumption that one model trained on a ten-vehicle mixed fleet gives cost estimates for smaller drone-only and robot-only coalitions that are accurate enough and not systematically biased, so that measured cost differences reflect real routing efficiency rather than model error.

Editorial extensions

If this is right

  • Operators can route a mixed UAV-ADR fleet with a single learned policy that respects battery, capacity, precedence, time-window, recharging, obstacle, and wind constraints, replacing per-instance mixed-integer solves with one forward pass.
  • Because most tested coalitions satisfy the cost inequality C(S1∪S2) ≤ C(S1)+C(S2), delivery companies can expect a stable grand coalition in which no subset of vehicles can do the same job more cheaply on its own.
  • Shapley cost shares give each mode a defensible marginal contribution to the coalition, which can be used to write rental or revenue-sharing agreements between drone and robot fleet owners.
  • Fleet-composition planning can target balanced drone-robot mixes, since the reported coalition gains are largest in the middle range and near-zero or unstable for all-ADR large fleets.

Reading between the lines

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

  • The paper's own coalition analysis (Section 5.5.2) uses the grand-coalition-trained model as the characteristic function for all sub-coalitions, so the core and Shapley results are conditional on that model's cross-fleet accuracy; verifying the same inequalities with per-coalition retraining or exact small-instance costs would strengthen or qualify the managerial claims.
  • A controlled ablation isolating the temporal adjacency mask from distance-only edge weights would show how much of the reported edge over plain attention comes from this design choice, which the paper compares only as a full package.
  • The same dual-encoder design could be lifted to other two-mode pairings, such as truck-drone or courier-bike fleets, by swapping the two edge-feature networks, making the method a template for mixed-fleet cooperation analysis rather than a UAV-ADR-specific solver.
  • Before any real payment is set from the Shapley numbers, the characteristic function would need validation on exact small instances, because a biased cost estimator can turn a stable-looking game into an artifact even when all the stated inequalities hold.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 4 minor

Summary. The paper addresses the electric capacitated pickup and delivery problem with time windows (CE-CPDPTW) for a mixed fleet of UAVs and autonomous delivery robots operating over two urban networks. It proposes an end-to-end multi-agent reinforcement learning model with a heterogeneous edge-enhanced graph attention encoder and transformer decoder, and evaluates it on synthetic instances and a Mississauga case study against Gurobi, OR-Tools, AM, and HetAM. The trained model is then used as a characteristic function for a coalition game, and the paper claims that most tested fleets form a super-additive game with a non-empty core and meaningful Shapley cost allocations. The central claims are therefore two-fold: the routing architecture achieves high-quality solutions, and the coalition analysis demonstrates an advantage of cooperation.

Significance. If the routing results are taken at face value, the paper contributes a realistic multi-modal formulation, a substantial architectural contribution, and a broad experimental study that includes wind and time-window uncertainty; the comparisons in Tables 4 and 5 provide useful evidence for the proposed encoder-decoder. However, the cooperative-game component is not established: the proof in Section 4.3 is circular, and the characteristic function is evaluated with an unvalidated frozen model. The claimed non-empty cores and Shapley shares therefore do not follow from the presented evidence, so the paper's central 'advantage of cooperation' conclusion is currently unsupported. The paper's strengths are the problem scope, model design, and experimental breadth rather than formal guarantees; no machine-checked proofs or released code are provided.

major comments (3)
  1. [Section 4.3, Eqs. (47)-(49), Algorithm 1] The proof of super-additivity is circular and verifies a different inequality than the one derived. Equation (48) assumes C_opt(S1∪S2) ≤ C_opt(S1)+C_opt(S2), which is exactly the property to be proved for the cooperative game. Substituting the upper bounds C_opt(S1) ≤ C_RL(S1) and C_opt(S2) ≤ C_RL(S2) yields Eq. (49) with C_opt(S1∪S2) on the left, not C_RL(S1∪S2). Algorithm 1, however, checks C_RL(S1∪S2) ≤ C_RL(S1)+C_RL(S2) at line 8. No result relates C_RL(S1∪S2) to C_opt(S1∪S2) in the required direction, and comparing upper bounds can neither prove nor disprove subadditivity of the RL cost game. Hence the claimed proof of super-additivity and non-empty core is invalid.
  2. [Section 5.5.2 and Algorithm 1, lines 1-2] The characteristic function is computed by applying a single model trained on the 10-vehicle grand coalition to every sub-coalition, with no retraining or validation of these extrapolated costs. Algorithm 1 states C(S)=C_opt(D,R) for any coalition, but Section 5.5.1 and Figure 14 show that generalization over fleet sizes is uneven, and the text around Figure 15 explicitly attributes irregularities to 'a misfit in the generalization of larger to smaller fleets.' Since super-additivity, core emptiness, and Shapley values are all comparisons of such C(S) values, a non-uniform bias across fleet compositions can create or destroy the reported effects. Without per-coalition cost estimates whose fidelity is checked, the core and Shapley results in Section 5.5.2 are artifacts.
  3. [Tables 4 and 5, Eq. (50)] The empirical routing claim rests on single point estimates: no confidence intervals, standard deviations, or random seeds are reported in Tables 4 and 5. In addition, Eq. (50) defines the gap relative to the best observed solution ('Obj_best'), so the values labeled 'Gap' and the 'optimal baseline solution' in Section 5.3 are not optimality gaps for instances where no exact solution is available. The reported rankings may be correct, but the evidence is weaker than presented; the comparison should be rerun over multiple seeds and the gap definition clarified.
minor comments (4)
  1. [Eq. (1) and Algorithm 1] The inequality in Eq. (1), C(S1∪S2)≤C(S1)+C(S2), is subadditivity for cost games, yet the paper repeatedly calls it 'super-additive'; Algorithm 1 even uses 'sub-additive' in Step 2. The terminology should be made consistent.
  2. [Equations (23)-(26)] The set C in Equations (23)-(26) denotes charging stations but is not defined in Table 2 and conflicts notationally with the characteristic function C(S) used in Section 2.2.1; please introduce a distinct symbol.
  3. [Section 5.2 and Table 3] The training-time figures in Section 5.2 (21, 55, and 112 minutes per epoch) and the 'Running time' column in Table 3 are not clearly labeled as per-epoch running times, and no software or hardware versions are given; please state these details to make the comparison reproducible.
  4. [Figures 15 and 16] Figure 16's caption spells 'Shapely' instead of 'Shapley', and there are numerous typographical errors (e.g., 'di fferent', 'UA Vs') throughout; a careful proofread is needed.

Circularity Check

2 steps flagged · score 6.0 of 10

Coalition-game proof assumes super-additivity and the cost oracle is a single grand-coalition-trained model, so the core/Shapley conclusions partially reduce to the paper's own inputs.

  1. self definitional [Section 4.3, Equations 47-49 and Algorithm 1]
    "We prove this by contradiction, as if we assume the super-additive game is satisfied, and we then find the condition that meets Equation 1. ... Equation 48: Copt(S1∪ S2)≤ Copt(S1) + Copt(S2). If by the assumption, this statement is true, afterwards, using Equation 47 for each coalition, Copt(S1)≤ CR(S1) and Copt(S2)≤ CR(S2), and substituting in Equation 48, the sub-additive property of the cooperative coalition will hold as Equation 49 Copt(S1∪ S2)≤ CR(S1) + CR(S2) ..."

    Equation 48 is precisely Equation 1, the definition of super-additivity, applied to the true optimal costs. The proof assumes Equation 48 as its antecedent, substitutes upper bounds Copt(Si)≤CR(Si) into the right-hand side, obtains Equation 49, and then declares that super-additivity is proved. The property to be established is therefore the assumption, not a derived consequence. Moreover, Equation 49 has Copt(S1∪S2) on the left, not CR(S1∪S2), so it cannot justify the empirical Algorithm 1 check CR(D+R)≤CR(D)+CR(R). The claimed core existence and cooperative advantage rest entirely on this assumed premise.

  2. fitted input called prediction [Algorithm 1, lines 1-3 and Section 5.5.2]
    "Initialization: Trained RL actor model for ten agents, C(K)≃ Copt(D, R), ∀ D⊆ Nd and R⊆ Nr for the coalition ... Input: Initialize D and R for generalized trained model C(S) = Copt(D, R) for any coalition S⊆ Nd∪ Nr. ... To evaluate the cost function for each coalition, we use generalization on the pre-trained model to obtain cost values to avoid the high computational expense of training for each coalition."

    The characteristic function that feeds all super-additivity, core, and Shapley computations is defined, by initialization, to equal Copt for every coalition, but in practice every C(S) is produced by a single model trained only on the ten-agent grand coalition and then evaluated on smaller fleets. No sub-coalition is solved or calibrated independently. Thus the 'prediction' of cooperative gain is arithmetic on the extrapolated outputs of one fitted oracle, not on true optimal costs. The paper itself concedes in Section 5.5.2 that irregularities arise from 'a misfit in the generalization of larger to smaller fleets', so the non-uniform bias is acknowledged; under that concession the reported cores and Shapley shares are artifacts of the oracle rather than validated properties of CE-CPDPTW.

full rationale

The routing-performance part of the paper is largely self-contained: HetGAT is compared against external baselines (Gurobi, OR-Tools, AM, HetAM) on the same instances, and those comparisons do not reduce to a fitted parameter renamed as a prediction. The circularity is concentrated in the coalition-game contribution, which the abstract and conclusions use to 'signify the advantage of cooperation'. Section 4.3's proof assumes Equation 48, which is exactly the super-additivity property it claims to prove, and then derives only a statement about Copt(union) versus RL costs of the parts, which cannot validate the RL-vs-RL comparison in Algorithm 1. Separately, Algorithm 1 and Section 5.5.2 set the characteristic function equal to Copt for every coalition while actually using the output of one grand-coalition-trained model, and the paper explicitly concedes a generalization misfit for smaller fleets. The core and Shapley conclusions are therefore not supported by an independent derivation or calibrated oracle. Because the central cooperative-advantage claim partially reduces to an assumed premise and to the outputs of a single fitted model, but the routing results retain independent empirical content, a score of 6 is appropriate.

Assumptions & free parameters 5 free parameters · 6 assumptions · 0 invented entities

The central result rests on a set of hand-set reward coefficients, adjacency thresholds, and synthetic vehicle parameters that shape the trained policy. The game-theoretic conclusion additionally depends on two unvalidated assumptions: that the RL surrogate cost is a valid upper bound for every coalition, and that a 10-agent trained model faithfully represents arbitrary sub-coalition fleets. No new physical entities are introduced.

free parameters (5)
  • alpha1, alpha2 (UAV/ADR per-minute operating cost weights) = 0.6 $/min, 0.1 $/min
    Hand-set in Eq. 5; no sensitivity analysis, though they enter the reward used to train the policy.
  • alpha3, alpha4 and early penalty (time-window penalty weights) = 0.05 $/min (late), 0.05 $/min (delivery delay), 0.01 early; emergency variants 0.1/0.2
    Chosen by the authors to encode service priority; the trained policy and all cost comparisons depend on these values.
  • Temporal and spatial adjacency thresholds zeta, mu, and density rho = zeta in [60,80] min, mu in [1,3] km, rho in [0.4,0.7]
    Sampled uniformly during training; these directly define which neighbors the GAT encoder attends to and are central to the claimed encoder improvement.
  • lambda (battery penalty coefficient) = 1
    Scales the added penalty when battery drops below threshold; the description and Eq. 22 disagree on whether the penalty is conditional.
  • Vehicle parameters in experiments (speeds, capacities, batteries, thresholds, charging times) = UAV 20 m/s, ADR 8.3 m/s; capacities 5 and 10; battery 6.5 kJ and 4.5 kJ; thresholds 30% and 20%
    These define the synthetic instances; results are not shown to be robust to their variation.
assumptions (6)
  • standard math Shapley value, core, and super-additivity definitions (Eqs. 1-4) are standard and correct.
    Used without proof; these are textbook results.
  • domain assumption The UAV and ADR energy consumption models (Stolaroff et al. 2018; Kimon and George 2015; Xiao and Whittaker 2014) accurately capture real battery behavior.
    These feed into MDP battery transitions and therefore into costs and route feasibility.
  • domain assumption The synthetic demand model (Poisson arrivals, uniform locations, random time windows) and the OSM-based Mississauga network are representative of real urban delivery.
    Generalization claims depend on this representativeness.
  • ad hoc to paper The RL solution cost for any coalition is an upper bound on the true optimal cost, and the inequality chain in Eqs. 47-49 transfers super-additivity to the RL game.
    This is the invalid step: Eq. 48 assumes the conclusion and Eq. 49 never bounds C_R(S1 union S2). This axiom is what the paper implicitly uses to prove cooperation stability.
  • ad hoc to paper A model trained on ten agents can be frozen and used to compute C(S) for arbitrary sub-coalitions without retraining.
    Algorithm 1 and Section 5.5.2 rely on this to construct the characteristic function; no validation that the surrogate is unbiased across coalition sizes.
  • domain assumption The stochastic wind and obstacle-density parameters follow the cited models (Johnson 1985; Liu et al. 2023) and are representative of real conditions.
    Used in robustness tests; no separate validation.

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Cite this review

Pith. "Pith review of A Coalition Game for On-demand Multi-modal 3D Automated Delivery System." pith.science (2026). https://pith.science/paper/OVRD2C3I

@misc{pith2026241217252,
  author       = {Pith},
  title        = {Pith review of: A Coalition Game for On-demand Multi-modal 3D Automated Delivery System},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/OVRD2C3I}},
  note         = {Machine review of arXiv:2412.17252}
}
read the original abstract

We introduce a multi-modal autonomous delivery optimization framework as a coalition game for a fleet of UAVs and ADRs operating in two overlaying networks to address last-mile delivery in urban environments, including high-density areas and time-critical applications. The problem is defined as multiple depot pickup and delivery with time windows constrained over operational restrictions, such as vehicle battery limitation, precedence time window, and building obstruction. Utilizing the coalition game theory, we investigate cooperation structures among the modes to capture how strategic collaboration can improve overall routing efficiency. To do so, a generalized reinforcement learning model is designed to evaluate the cost-sharing and allocation to different modes to learn the cooperative behaviour with respect to various realistic scenarios. Our methodology leverages an end-to-end deep multi-agent policy gradient method augmented by a novel spatio-temporal adjacency neighbourhood graph attention network using a heterogeneous edge-enhanced attention model and transformer architecture. Several numerical experiments on last-mile delivery applications have been conducted, showing the results from the case study in the city of Mississauga, which shows that despite the incorporation of an extensive network in the graph for two modes and a complex training structure, the model addresses realistic operational constraints and achieves high-quality solutions compared with the existing transformer-based and classical methods. It can perform well on non-homogeneous data distribution, generalizes well on different scales and configurations, and demonstrates a robust cooperative performance under stochastic scenarios across various tasks, which is effectively reflected by coalition analysis and cost allocation to signify the advantage of cooperation.

Figures

Figures reproduced from arXiv: 2412.17252 by the authors.

Figure 1
Figure 1. Aerial network for UAVs, including a restricted zone marked as a red arced region [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. The reinforcement learning framework. Agents take a sequence of actions based on the reward feedback from the [PITH_FULL_IMAGE:figures/full_fig_p007_2.png] view at source ↗
Figure 3
Figure 3. Case study network. Shortest Path (Avoid obstacle) Direct Path (No obstacle) [PITH_FULL_IMAGE:figures/full_fig_p014_3.png] view at source ↗
Figures from the paper (14 more)
Figure 4
Figure 4. Figure 4: UAV Network Routing. This figure shows a top view of part of Mississauga with the pattern of the urban infrastructures [PITH_FULL_IMAGE:figures/full_fig_p014_4.png]
Figure 5
Figure 5. Figure 5: Overview of the methodology mechanism is provided to guide the multi-head attention layer to account for proximity nodes rather than irrelevant ones. The proximity nodes are defined based on the edge weight values in the graph. It can be defined for each node as a set …
Figure 6
Figure 6. Figure 6: The encoder architecture and graph attention adjacency mechanism [PITH_FULL_IMAGE:figures/full_fig_p020_6.png]
Figure 7
Figure 7. Figure 7: The decoder architecture strategies and masking scheme representation [PITH_FULL_IMAGE:figures/full_fig_p021_7.png]
Figure 8
Figure 8. Figure 8: UAV and ADR network coverage during the delivery service [PITH_FULL_IMAGE:figures/full_fig_p023_8.png]
Figure 9
Figure 9. Figure 9: Training convergence curve in 100 epochs for comparing encoder design [PITH_FULL_IMAGE:figures/full_fig_p027_9.png]
Figure 10
Figure 10. Figure 10: The learning curve for the various delivery cases from low to high critical [PITH_FULL_IMAGE:figures/full_fig_p028_10.png]
Figure 11
Figure 11. Figure 11: Training convergence curve in 100 epochs for prior-based decoding [PITH_FULL_IMAGE:figures/full_fig_p030_11.png]
Figure 12
Figure 12. Figure 12: Multi-modal assignment distribution in the existence of uncertain conditions [PITH_FULL_IMAGE:figures/full_fig_p032_12.png]
Figure 13
Figure 13. Figure 13: Generation from sparse to fully connected and dense graph network. [PITH_FULL_IMAGE:figures/full_fig_p033_13.png]
Figure 14
Figure 14. Figure 14: Generalizing on different problem scales. The overall result shows an acceptable solution generated from each of the scales to others; most notably, the generalization on smaller networks from the trained model of larger networks produces better results. This is proba…
Figure 15
Figure 15. Figure 15: Core coalition counter-plot for five agents in each mode. The sidebar shows the cost di [PITH_FULL_IMAGE:figures/full_fig_p036_15.png]
Figure 16
Figure 16. Figure 16: Cost allocation for all coalitions. The first number on the left show UAV and second ADR. [PITH_FULL_IMAGE:figures/full_fig_p037_16.png]
Figure 17
Figure 17. Figure 17: Cost allocation for non-homogeneous demand distribution. [PITH_FULL_IMAGE:figures/full_fig_p038_17.png]

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Reference graph

Works this paper leans on

52 extracted references · 49 canonical work pages

  1. [1]

    , year 2022

    author Zhang et al., K. , year 2022 . title Transformer-based reinforcement learning for pickup and delivery problems with late penalties . journal IEEE Trans. on ITS volume 23 , pages 24649--24661

    work page 2022

  2. [2]

    , et al., year 2014

    author Alexander, S. , et al., year 2014 . title Multi-period cooperative vehicle routing games . journal Contributions to Game Theory and Management volume 7 , pages 349--359

    work page 2014

  3. [3]

    , year 2017

    author Boeing, G. , year 2017 . title Osmnx: New methods for acquiring, constructing, analyzing, and visualizing complex street networks . journal Computers, Environment and Urban Systems volume 65 , pages 126--139

    work page 2017

  4. [4]

    Learning to Solve Vehicle Routing Problems: A Survey

    author Bogyrbayeva, A. , author Meraliyev, M. , author Mustakhov, T. , author Dauletbayev, B. , . title Learning to solve vehicle routing problems: a survey (2022) . journal arXiv preprint arXiv:2205.02453

    work page Pith review arXiv 2022

  5. [5]

    , author Elkind, E

    author Chalkiadakis, G. , author Elkind, E. , author Wooldridge, M. , year 2022 . title Computational aspects of cooperative game theory . publisher Springer Nature

    work page 2022

  6. [6]

    , author Zhang, W

    author Chu, H. , author Zhang, W. , author Bai, P. , author Chen, Y. , year 2021 . title Data-driven optimization for last-mile delivery . journal Complex & Intelligent Systems , pages 1--14

    work page 2021

  7. [7]

    , author Daniele, P

    author Colajanni, G. , author Daniele, P. , author Nagurney, A. , year 2023 . title Centralized supply chain network optimization with uav-based last mile deliveries . journal Transportation Research Part C: Emerging Technologies volume 155 , pages 104316

    work page 2023

  8. [8]

    , author Kharkar, A

    author Drori, I. , author Kharkar, A. , author Sickinger, W.R. , author Kates, B. , author Ma, Q. , author Ge, S. , author Dolev, E. , author Dietrich, B. , author Williamson, D.P. , author Udell, M. , year 2020 . title Learning to solve combinatorial optimization problems on real-world graphs in linear time , in: booktitle 2020 19th IEEE International Co...

    work page 2020

Show all 52 references
  1. [9]

    , author Mohamed, M

    author Elsayed, M. , author Mohamed, M. , year 2020 . title The impact of airspace regulations on unmanned aerial vehicles in last-mile operation . journal Transportation Research Part D: Transport and Environment volume 87 , pages 102480

    2020

  2. [10]

    , author Farid, A

    author Fellek, G. , author Farid, A. , author Gebreyesus, G. , author Fujimura, S. , author Yoshie, O. , year 2023 . title Graph transformer with reinforcement learning for vehicle routing problem . journal IEEJ Transactions on Electrical and Electronic Engineering volume 18 ,...

    2023

  3. [11]

    , year 2023

    author Fortune, B. , year 2023 . title Delivery robots market size, share, and growth . howpublished https://www.marketsandmarkets.com/Market-Reports/delivery-robot-market-263997316.html/ . note Accessed: 2024-10-20

    2023

  4. [12]

    , author del Blanco, C.R

    author Fuertes, D. , author del Blanco, C.R. , author Jaureguizar, F. , author Navarro, J.J. , author Garc \' a, N. , year 2023 . title Solving routing problems for multiple cooperative unmanned aerial vehicles using transformer networks . journal Engineering Applications of A...

    2023

  5. [13]

    , author Chen, M

    author Gao, L. , author Chen, M. , author Chen, Q. , author Luo, G. , author Zhu, N. , author Liu, Z. , year 2020 . title Learn to design the heuristics for vehicle routing problem . journal arXiv preprint arXiv:2002.08539

    2020 arXiv

  6. [14]

    , author Bozanta, A

    author Jahanshahi, H. , author Bozanta, A. , author Cevik, M. , author Kavuk, E.M. , author Tosun, A. , author Sonuc, S.B. , author Kosucu, B. , author Ba s ar, A. , year 2022 . title A deep reinforcement learning approach for the meal delivery problem . journal Knowledge-Base...

    2022

  7. [15]

    , author Yu, W

    author James, J. , author Yu, W. , author Gu, J. , year 2019 . title Online vehicle routing with neural combinatorial optimization and deep reinforcement learning . journal IEEE Transactions on Intelligent Transportation Systems volume 20 , pages 3806--3817

    2019

  8. [16]

    , year 1985

    author Johnson, G.L. , year 1985 . title Wind energy systems . publisher Citeseer

    1985

  9. [17]

    , author George, J

    author Kimon, P. , author George, J. , year 2015 . title Handbook of unmanned aerial vehicles

    2015

  10. [18]

    , author Van Hoof, H

    author Kool, W. , author Van Hoof, H. , author Welling, M. , year 2018 . title Attention, learn to solve routing problems! journal arXiv preprint arXiv:1803.08475

    2018 arXiv

  11. [19]

    , author Guo, P

    author Lei, K. , author Guo, P. , author Wang, Y. , author Wu, X. , author Zhao, W. , year 2022 . title Solve routing problems with a residual edge-graph attention neural network . journal Neurocomputing volume 508 , pages 79--98

    2022

  12. [20]

    , author Kunze, O

    author Li, F. , author Kunze, O. , year 2023 . title A comparative review of air drones (uavs) and delivery bots (sugvs) for automated last mile home delivery . journal Logistics volume 7 , pages 21

    2023

  13. [21]

    , author Liu, H

    author Li, J. , author Liu, H. , author Lai, K.K. , author Ram, B. , year 2022 . title Vehicle and uav collaborative delivery path optimization model . journal Mathematics volume 10 , pages 3744

    2022

  14. [22]

    , author Xin, L

    author Li, J. , author Xin, L. , author Cao, Z. , author Lim, A. , author Song, W. , author Zhang, J. , year 2021 . title Heterogeneous attentions for solving pickup and delivery problem via deep reinforcement learning . journal IEEE Transactions on Intelligent Transportation ...

    2021

  15. [23]

    , author Shin, H.S

    author Liu, R. , author Shin, H.S. , author Tsourdos, A. , year 2023 . title Edge-enhanced attentions for drone delivery in presence of winds and recharging stations . journal Journal of Aerospace Information Systems volume 20 , pages 216--228

    2023

  16. [24]

    , year 2019

    author Liu, Y. , year 2019 . title An optimization-driven dynamic vehicle routing algorithm for on-demand meal delivery using drones . journal Computers & Operations Research volume 111 , pages 1--20

    2019

  17. [25]

    , author Li, J

    author Ma, Y. , author Li, J. , author Cao, Z. , author Song, W. , author Guo, H. , author Gong, Y. , author Chee, Y.M. , year 2022 . title Efficient neural neighborhood search for pickup and delivery problems . journal arXiv preprint arXiv:2204.11399

    2022 arXiv

  18. [26]

    , author Xu, L

    author Mak, S. , author Xu, L. , author Pearce, T. , author Ostroumov, M. , author Brintrup, A. , year 2023 . title Fair collaborative vehicle routing: A deep multi-agent reinforcement learning approach . journal Transportation Research Part C: Emerging Technologies volume 157...

    2023

  19. [27]

    , author Wen, H

    author Mao, X. , author Wen, H. , author Zhang, H. , author Wan, H. , author Wu, L. , author Zheng, J. , author Hu, H. , author Lin, Y. , year 2023 . title Drl4route: A deep reinforcement learning framework for pick-up and delivery route prediction , in: booktitle Proceedings ...

    2023

  20. [28]

    , author Saha, S

    author Mehra, A. , author Saha, S. , author Raychoudhury, V. , author Mathur, A. , year 2023 . title Deliverai: Reinforcement learning based distributed path-sharing network for food deliveries . journal arXiv preprint arXiv:2311.02017

    2023 arXiv

  21. [29]

    , author Oroojlooy, A

    author Nazari, M. , author Oroojlooy, A. , author Snyder, L. , author Tak \'a c, M. , year 2018 . title Reinforcement learning for solving the vehicle routing problem . journal Advances in neural information processing systems volume 31

    2018

  22. [30]

    , author Guajardo, M

    author Osicka, O. , author Guajardo, M. , author van Oost, T. , year 2020 . title Cooperative game-theoretic features of cost sharing in location-routing . journal International Transactions in Operational Research volume 27 , pages 2157--2183

    2020

  23. [31]

    , year 2021

    author Osler, Hoskin, H. , year 2021 . title Drone law in canada . https://www.osler.com/osler/media/Osler/infographics/CG5049_Drone-Law-Canada.pdf. note accessed: 2023-11-30

    2021

  24. [32]

    , author Heimfarth, A

    author Ostermeier, M. , author Heimfarth, A. , author H \"u bner, A. , year 2023 . title The multi-vehicle truck-and-robot routing problem for last-mile delivery . journal European Journal of Operational Research volume 310 , pages 680--697

    2023

  25. [33]

    , author Chevrier, R

    author Rabecq, B. , author Chevrier, R. , year 2022 . title A deep learning attention model to solve the vehicle routing problem and the pick-up and delivery problem with time windows . journal arXiv preprint arXiv:2212.10399

    2022 arXiv

  26. [34]

    , et al., year 1991

    author Roger, B.M. , et al., year 1991 . title Game theory: analysis of conflict . journal The President and Fellows of Harvard College, USA volume 66

    1991

  27. [35]

    , author Rodin, E

    author Ruland, K. , author Rodin, E. , year 1997 . title The pickup and delivery problem: Faces and branch-and-cut algorithm . journal Computers & mathematics with applications volume 33 , pages 1--13

    1997

  28. [36]

    , author Gluza, V

    author Samouh, F. , author Gluza, V. , author Djavadian, S. , author Meshkani, S. , author Farooq, B. , year 2020 . title Multimodal autonomous last-mile delivery system design and application , in: booktitle 2020 IEEE International Smart Cities Conference (ISC2) , organizatio...

    2020

  29. [37]

    , author Wardoyo, R

    author Santiyuda, G. , author Wardoyo, R. , author Pulungan, R. , author Vincent, F.Y. , year 2024 . title Multi-objective reinforcement learning for bi-objective time-dependent pickup and delivery problem with late penalties . journal Engineering Applications of Artificial In...

    2024

  30. [38]

    , author Sol, M

    author Savelsbergh, M.W. , author Sol, M. , year 1995 . title The general pickup and delivery problem . journal Transportation science volume 29 , pages 17--29

    1995

  31. [39]

    , year 1967

    author Shapley, L.S. , year 1967 . title On balanced sets and cores . journal Naval research logistics quarterly volume 14 , pages 453--460

    1967

  32. [40]

    , year 1971

    author Shapley, L.S. , year 1971 . title Cores of convex games . journal International journal of game theory volume 1 , pages 11--26

    1971

  33. [41]

    , author Kim, M

    author Son, J. , author Kim, M. , author Choi, S. , author Park, J. , year 2023 . title Solving np-hard min-max routing problems as sequential generation with equity context . journal arXiv preprint arXiv:2306.02689

    2023 arXiv

  34. [42]

    , author Meshcheryakov, A

    author Soroka, A. , author Meshcheryakov, A. , author Gerasimov, S. , year 2023 . title Deep reinforcement learning for the capacitated pickup and delivery problem with time windows . journal Pattern Recognition and Image Analysis volume 33 , pages 169--178

    2023

  35. [43]

    , author Samaras, C

    author Stolaroff, J.K. , author Samaras, C. , author O’Neill, E.R. , author Lubers, A. , author Mitchell, A.S. , author Ceperley, D. , year 2018 . title Energy use and life cycle greenhouse gas emissions of drones for commercial package delivery . journal Nature communications...

    2018

  36. [44]

    , author Hutchinson, E.B

    author Sudbury, A.W. , author Hutchinson, E.B. , year 2016 . title A cost analysis of amazon prime air (drone delivery) . journal Journal for Economic Educators volume 16 , pages 1--12

    2016

  37. [45]

    , author McAllester, D

    author Sutton, R.S. , author McAllester, D. , author Singh, S. , author Mansour, Y. , year 1999 . title Policy gradient methods for reinforcement learning with function approximation . journal Advances in neural information processing systems volume 12

    1999

  38. [46]

    , author Cucurull, G

    author Veli c kovi \'c , P. , author Cucurull, G. , author Casanova, A. , author Romero, A. , author Lio, P. , author Bengio, Y. , year 2017 . title Graph attention networks . journal arXiv preprint arXiv:1710.10903

    2017 arXiv

  39. [47]

    , year 1992

    author Williams, R.J. , year 1992 . title Simple statistical gradient-following algorithms for connectionist reinforcement learning . journal Machine learning volume 8 , pages 229--256

    1992

  40. [48]

    , author Whittaker, W

    author Xiao, X. , author Whittaker, W. , year 2014 . title Energy considerations for wheeled mobile robots operating on a single battery discharge . journal Robot. Inst., Carnegie Mellon Univ., Pittsburgh, PA, USA, Tech. Rep., CMU-RI-TR-14-16

    2014

  41. [49]

    , author Li, M

    author Zhang, K. , author Li, M. , author Wang, J. , author Li, Y. , author Lin, X. , year 2023 a. title A two-stage learning-based method for large-scale on-demand pickup and delivery services with soft time windows . journal Transportation Research Part C: Emerging Technolog...

    2023

  42. [50]

    , author Lin, X

    author Zhang, K. , author Lin, X. , author Li, M. , year 2023 b. title Graph attention reinforcement learning with flexible matching policies for multi-depot vehicle routing problems . journal Physica A: Statistical Mechanics and its Applications volume 611 , pages 128451

    2023

  43. [51]

    , author Hafezalkotob, A

    author Zibaei, S. , author Hafezalkotob, A. , author Ghashami, S.S. , year 2016 . title Cooperative vehicle routing problem: an opportunity for cost saving . journal Journal of Industrial Engineering International volume 12 , pages 271--286

    2016

  44. [52]

    , author Zheng, M

    author Zong, Z. , author Zheng, M. , author Li, Y. , author Jin, D. , year 2022 . title Mapdp: Cooperative multi-agent reinforcement learning to solve pickup and delivery problems , in: booktitle Proceedings of the AAAI Conference on Artificial Intelligence , pp. pages 9980--9988

    2022

Pith tools

Reviewed August 11, 2026 · model on record in the stance chip above.