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REVIEW 2 major objections 5 minor 41 references

Collaborative Last-Mile Delivery: A Multi-Platform Vehicle Routing Problem With En-route Charging

T0 review · 2 major / 5 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read Routing trucks, drones, and robots together as one plan is feasible and scales to hundreds of customers.

desk verdict New three-platform VRP variant with a serious flaw: the charging constraints don't enforce physical causality, so the headline charging savings are unsupported. read the letter →

arxiv 2505.23584 v1 pith:O5PMRE3O submitted 2025-05-29 cs.MA cs.AIcs.RO

Sumbal Malik , Majid Khonji , Khaled Elbassioni , Jorge Dias This is my paper
classification cs.MAcs.AIcs.RO MSC 90B0690C1190C59
keywords vehicleroutingproblemlast-miledeliverytruck-drone-robotcollaborationen-routechargingmixed-integerlinearprogrammingmulti-visitsortieflexibledockingmakespanminimization
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

This paper sets out to show that last-mile delivery can be planned jointly for a fleet of trucks, drones, and ground robots, with trucks acting as mobile launch pads and charging stations. The central object is a vehicle routing problem called VRP-DR, formulated as a mixed-integer linear program (MILP) that minimizes a weighted combination of operational cost and makespan. Because the MILP only solves small instances tractably, the paper adds a three-phase heuristic, FINDER, and reports that it reaches near-optimal objective values on 5 to 35 customer instances while cutting computation from thousands of seconds to a fraction of a second. On larger instances the experiments report that the full fleet shortens makespan by about 4.8% versus truck-only delivery, that multi-visit sorties reduce cost by about 6.6% versus single-visit sorties, and that en-route charging cuts both cost and makespan by a few percent. The intended significance is a planning model that captures realistic features—multi-visit, multi-trip, flexible docking, cyclic and acyclic operations, and charging during transit—that earlier single-platform models omit.

What carries the argument

The load-bearing object is the sortie, defined as a triplet (i, l, k): a drone or robot leaves a truck at node i, serves an ordered sequence l of customers up to sortie capacity m, and rejoins a truck at node k, with the launch truck and recovery truck named explicitly in the decision variables. This single object carries multi-visit service, multi-trip operation, flexible docking, and cyclic versus acyclic movement, because multiple sorties can share a vehicle and the launch and recovery trucks may differ. Energy is handled by load-dependent consumption formulas—a linear drone model with decreasing payload and a linearized quadratic robot gait model—capped by battery capacity, with en-route charging amounts limited by the carrying truck's available travel time. FINDER, the heuristic, decomposes the problem into three phases: nearest-neighbor truck route construction, synchronized auxiliary-vehicle assignment against the truck timeline, and cheapest-insertion of leftover customers, which is what makes instances up to 300 customers computationally accessible.

What would settle it

Run a small instance with two trucks, one drone, and one sortie in which the drone's only charging opportunity occurs at a node the drone reaches before the truck; solve the MILP and replay the chosen plan in chronological order. If the battery level ever goes below zero at takeoff, the en-route charging model is not physically enforcing recharge.

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Extended reading notes

Core claim

The paper's central claim is that the VRP-DR, a synchronized multi-platform routing problem, can be captured by a MILP whose objective is Z = α·(truck, drone, and robot travel costs plus fixed deployment costs) + (1−α)·Γ, with Γ the makespan, and solved in practice by the FINDER heuristic. The model treats each drone or robot sortie as a triplet (launch node, ordered customer sequence, recovery node), explicitly indexes launch and recovery trucks so drones need not return to the truck that launched them, and allows each auxiliary vehicle to serve several customers per trip, to fly or drive multiple trips, and to recharge while carried by a truck. Experiments with one truck, one drone, and one robot show the heuristic within about 22% of the exact objective on small to medium instances while being orders of magnitude faster, and the system-level comparisons report the headline gains: a 4.82% makespan reduction for the entire fleet over truck-only operation, a 6.64% cost reduction from multi-visit over single-visit sorties, and roughly 1.8% cost and 2.9% makespan improvements from en-route charging.

Load-bearing premise

The energy accounting lets a drone or robot recharge at any node labeled with a truck, without tracking which truck physically carries it and in what order, so the model can allow later or off-route charging to pay for earlier energy use.

Editorial extensions

If this is right

  • If the MILP and FINDER outputs are correct, logistics planners can treat the entire fleet as one optimization problem and get plans that finish about 4.8% sooner than truck-only routes in the tested 20 to 300 customer instances.
  • Multi-visit sorties, letting one drone or robot serve several customers per launch, lower operational cost by roughly 6.6% compared with one-customer-per-trip, with no meaningful makespan penalty.
  • En-route charging is worth roughly 1.8% of cost and 2.9% of makespan over depot-only charging in the tested settings, and the model quantifies when adding drones has diminishing returns.
  • The three-phase heuristic turns a problem the exact solver cannot finish in reasonable time into a few seconds for 100 customers and about half an hour for 300 customers.
  • Flexible docking lets a drone or robot end a sortie at any truck whose route passes the recovery node, so fleet utilization improves without dedicated vehicle pairings.

Reading between the lines

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

  • The charging gains should be read as optimistic until battery bookkeeping is tied to actual truck-vehicle pairings in chronological order; the displayed MILP constraints sum charging over nodes and can let energy added after a flight or at another truck pay for it.
  • The 22% heuristic gap is measured only where an exact solution exists (5 to 35 customers); for larger instances the heuristic's quality is an extrapolation rather than a measured bound.
  • The same sortie-triplet formulation could be extended to pickup-and-delivery, multiple depots, or customer time windows without changing the core decomposition, since those features only alter feasibility checks inside each phase.
  • A fairer charging test would compare plans under identical customer locations but with charging only during the drone's or robot's actual on-truck intervals; the paper's fixed-seed experiments allow such a direct check.
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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

2 major / 5 minor

Summary. The paper proposes a collaborative last-mile delivery problem (VRP-DR) involving trucks, drones, and robots, with multi-visit sorties, multi-trip operations, flexible docking, and en-route charging. It formulates a MILP that minimizes a weighted sum of operational cost and makespan, and develops a three-phase heuristic called FINDER for large instances. Experiments on instances from 5 to 300 customers compare the MILP and heuristic, quantify the effect of collaborative modes, single- vs multi-visit, en-route charging, flexible docking, and several sensitivity parameters. The headline reported effects are roughly 4.8% makespan savings for the full fleet over truck-only delivery, about 6.6% cost savings for multi-visit over single-visit operation, and about 1.8% cost and 2.91% makespan savings from en-route charging.

Significance. If the formulation were physically sound, the paper would address a relevant and timely extension of truck-drone-robot routing. The manuscript has clear strengths: it tackles a genuinely complex integrated problem, provides a readable MILP skeleton, develops a decomposition heuristic, and benchmarks the heuristic against an exact solver on many instances. However, the central modeling of en-route charging and flexible docking is not physically coherent: vehicle identity and chronological battery state are not enforced, so the charging and docking experiments do not measure the features they claim to measure. The reported savings can arise from acausal bookkeeping rather than from feasible operations. Because these issues affect the core contribution, the paper as it stands is not publishable; a corrected version would require substantial reformulation and new experiments.

major comments (2)
  1. [§3.3, Eqs. (34)–(37) and (41)–(42)] The en-route charging constraints do not enforce that a drone or robot is physically onboard the truck that charges it. In Eq. (34), the charging amount C^d_vt is bounded only by truck t visiting node v; there is no variable or constraint linking drone d to truck t at node v. Consequently, Eq. (36) lets the total energy consumed by drone d across all sorties be offset by charging at any node and any truck in the network, including nodes the drone never visits and charges that occur after the energy has been consumed. The battery constraints in Eqs. (41)–(42) are not state-of-charge constraints: each reduces to cumulative charging up to node label v being no greater than cumulative consumption up to v, and they never require the battery level to remain nonnegative between sorties. As a result, the 1.80% cost and 2.91% makespan savings attributed to en-route charging in Sec. 5.6 are not evidence for feasible recharging; they can be produced by acausal bookkeeping.
  2. [§3.3, Eqs. (9)–(14) and (44)–(47)] The model never tracks the physical location of an individual drone or robot over time. Although y^{ti,tk}_{ilkd} and z^{ti,tk}_{ilkr} carry launch-truck and recovery-truck indices, no constraint ensures that drone d is on truck ti before launch, remains with truck tk after recovery, or is not engaged in two sorties simultaneously. Any truck that visits the launch node can serve as ti, and any truck that visits the recovery node can serve as tk, regardless of where drone d actually is at that moment. The flexible docking scenarios in Sec. 5.7 and the multi-trip analysis in Sec. 5.5 are therefore not validated: the model permits physically impossible launch/recovery sequences and overlapping sorties for the same vehicle.
minor comments (5)
  1. [Table 2] The entry for R reads "Set of drones"; it should read "Set of robots".
  2. [§3.3 and Table 3] The sortie capacity m appears in the definition of L but no value is reported in Table 3 or in the experimental setup; since the size of L drives the MILP complexity, the paper should state m and, ideally, report its influence.
  3. [Eq. (29)] Equation (29) overloads the big-M constant M as a distance threshold for inaccessible nodes; using the same symbol for two different purposes is confusing and should be replaced by a separate parameter.
  4. [§3.3, end of synchronization paragraph] The text says that constraints (44) and (45) govern the synchronization of return times, but Eqs. (44)–(45) actually govern launch times; this appears to be a typo.
  5. [§5.2.1, Table 4] The paper describes an average gap of about 22% as "near-optimal"; even putting aside the model-validity issue, the authors should discuss whether this gap is acceptable for the intended practical use.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the paper's MILP, heuristic, and experiments form an explicit derivation chain with no fitted inputs, self-cited uniqueness theorems, or definitional equivalences.

full rationale

The paper's central chain is: problem definition -> MILP constraints -> exact solutions via Gurobi -> FINDER heuristic -> experimental comparisons. No step reduces to its own inputs. The objective function (Eq. 1) and all constraints are stated explicitly, and the numerical claims in Sections 5.3-5.6 are outputs of solving that MILP plus a separately implemented heuristic, not results fitted to reach the stated conclusions. The heuristic is benchmarked against the exact MILP solution using the explicit gap formula in Eq. (48), which is an independent standard of solution quality. En-route charging savings in Section 5.6 are obtained by comparing model runs under en-route charging versus no-charge modes, not by assuming the savings. The strongest concern is technical correctness of Eqs. (34)-(42), which may allow acausal charging that does not track which truck physically carries which drone or robot; however, this is a feasibility/modeling flaw, not circularity, because the derivation does not assume the conclusions it draws. There are no load-bearing self-citations and no imported uniqueness theorems. Accordingly, no circular step is identified.

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

The central model rests on many assumed operational parameters. The most fragile are the energy and charging bookkeeping assumptions, which are not temporally or physically coupled to vehicle locations and use inconsistent units.

free parameters (5)
  • Sortie capacity m = not specified in experiments
    Defines the maximum number of customers per drone or robot sortie and the size of the ordered-sequence set L. No value is given, making instance definitions incomplete and affecting tractability.
  • Objective weighting alpha = 0.5
    Chosen by hand; directly sets the trade-off between cost and makespan and therefore shapes all reported objective values.
  • Drone energy coefficient alpha_d = 128
    Taken from prior work [12]; converts weight-distance product into claimed energy units, but units are not reconciled with battery capacity in mAh.
  • Robot energy constants k1, k2 = 0.1, 0.2
    Assumed constants with no source; control the mechanical and electrical power model in Eqs. (22)-(24).
  • Charging rates Cd_rate, Cr_rate = 5000, 4000 mAh
    Chosen operating parameters; charging amounts in the model are bounded by these rates and by truck travel time.
assumptions (6)
  • domain assumption Each customer has demand for exactly one parcel and deliveries are not split.
    Stated in Section 3.2; simplifies demand but limits realism.
  • domain assumption Trucks have sufficient capacity for all parcels and carried vehicles, and launch and retrieval setup time is negligible.
    Section 3.2; removes truck capacity and handling-time constraints from the model.
  • domain assumption Speeds of trucks, drones, and robots are constant and known, with Manhattan distances for trucks and robots and Euclidean distances for drones.
    Sections 3.2 and 5.1; these choices affect all travel times, costs, and experimental results.
  • domain assumption The energy consumption models for drone (Eq. 19) and robot (Eqs. 20-24) are accepted from prior literature and assumed compatible with battery capacity in mAh.
    The paper relies on these models without validating units, calibration, or conversion factors.
  • standard math MTZ subtour elimination and big-M linearization are valid for the truck routes and robot energy product.
    Used in Eqs. (8) and (25)-(28); standard techniques, but big-M values are not specified.
  • ad hoc to paper The set L of all ordered customer sequences is enumerable and constrained only by sortie capacity m.
    Modeling choice that makes the MILP size grow rapidly; the paper does not state m or any pruning rule.

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Pith. "Pith review of Collaborative Last-Mile Delivery: A Multi-Platform Vehicle Routing Problem With En-route Charging." pith.science (2026). https://pith.science/paper/O5PMRE3O

@misc{pith2026250523584,
  author       = {Pith},
  title        = {Pith review of: Collaborative Last-Mile Delivery: A Multi-Platform Vehicle Routing Problem With En-route Charging},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/O5PMRE3O}},
  note         = {Machine review of arXiv:2505.23584}
}
abstract

The rapid growth of e-commerce and the increasing demand for timely, cost-effective last-mile delivery have increased interest in collaborative logistics. This research introduces a novel collaborative synchronized multi-platform vehicle routing problem with drones and robots (VRP-DR), where a fleet of $\mathcal{M}$ trucks, $\mathcal{N}$ drones and $\mathcal{K}$ robots, cooperatively delivers parcels. Trucks serve as mobile platforms, enabling the launching, retrieving, and en-route charging of drones and robots, thereby addressing critical limitations such as restricted payload capacities, limited range, and battery constraints. The VRP-DR incorporates five realistic features: (1) multi-visit service per trip, (2) multi-trip operations, (3) flexible docking, allowing returns to the same or different trucks (4) cyclic and acyclic operations, enabling return to the same or different nodes; and (5) en-route charging, enabling drones and robots to recharge while being transported on the truck, maximizing operational efficiency by utilizing idle transit time. The VRP-DR is formulated as a mixed-integer linear program (MILP) to minimize both operational costs and makespan. To overcome the computational challenges of solving large-scale instances, a scalable heuristic algorithm, FINDER (Flexible INtegrated Delivery with Energy Recharge), is developed, to provide efficient, near-optimal solutions. Numerical experiments across various instance sizes evaluate the performance of the MILP and heuristic approaches in terms of solution quality and computation time. The results demonstrate significant time savings of the combined delivery mode over the truck-only mode and substantial cost reductions from enabling multi-visits. The study also provides insights into the effects of en-route charging, docking flexibility, drone count, speed, and payload capacity on system performance.

Figures

Figures reproduced from arXiv: 2505.23584 by the authors.

Figure 1
Figure 1. Schematic illustration of the proposed VRP-DR [PITH_FULL_IMAGE:figures/full_fig_p013_1.png] view at source ↗
Figure 2
Figure 2. Illustration of multiple scenarios for drone and robot launch and recovery oper [PITH_FULL_IMAGE:figures/full_fig_p014_2.png] view at source ↗
Figure 3
Figure 3. Objective values: exact vs heuris￾tic (small/medium instances) 5 10 15 20 25 30 35 # Customer Nodes 0 1000 2000 3000 4000 5000 6000 7000 Computation Time (s) Exact Solution Heuristic Solution [PITH_FULL_IMAGE:figures/full_fig_p033_3.png] view at source ↗
Figures from the paper (9 more)
Figure 5
Figure 5. Figure 5: Graphical representation of the objective values and computation time of the [PITH_FULL_IMAGE:figures/full_fig_p035_5.png]
Figure 6
Figure 6. Figure 6: Impact of varying collaborative modes on operational cost and makespan com [PITH_FULL_IMAGE:figures/full_fig_p036_6.png]
Figure 7
Figure 7. Figure 7: Impact of single-visit and multi-visit of drones and robots on operational cost [PITH_FULL_IMAGE:figures/full_fig_p038_7.png]
Figure 8
Figure 8. Figure 8: Impact of single-trip and multi-trips of drones and robots on makespan [PITH_FULL_IMAGE:figures/full_fig_p039_8.png]
Figure 9
Figure 9. Figure 9: Comparison of en-route charging and no-charging modes in terms of cost and [PITH_FULL_IMAGE:figures/full_fig_p040_9.png]
Figure 10
Figure 10. Figure 10: Visualization of flexible docking scenarios [PITH_FULL_IMAGE:figures/full_fig_p041_10.png]
Figure 11
Figure 11. Figure 11: The makespan results with varying numbers of drones [PITH_FULL_IMAGE:figures/full_fig_p041_11.png]
Figure 12
Figure 12. Figure 12: Impact of fleet speed on makespan ment is due to the truck’s central role in synchronizing auxiliary vehicle operations. A faster truck minimizes waiting times for drones and robots, enabling them to dispatch and rendezvous more efficiently. For N = 50, the makespan d…
Figure 13
Figure 13. Figure 13: (a) Impact of varying drone payload capacity on makespan (b) Impact of [PITH_FULL_IMAGE:figures/full_fig_p044_13.png]

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