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arxiv: 2506.14461 · v2 · submitted 2025-06-17 · 🧮 math.OC · cs.CE

Collaborative Charging Scheduling via Balanced Bounding Box Methods

Fangting Zhou , Balazs Kulcsar , Jiaming Wu This is my paper

Pith reviewed 2026-05-19 09:25 UTC · model grok-4.3

classification 🧮 math.OC cs.CE
keywords collaborative charging schedulingbalanced bounding box methodsbi-objective optimizationefficient frontiershared charging stationsfleet operatorscooperative bargainingurban logistics
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The pith

Balanced Bounding Box Methods trim computation for shared charging schedules by reducing the efficient frontier to representative trade-offs.

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

The paper models shared charging between two fleet operators as a bi-objective nonlinear integer program where each minimizes its own costs. It introduces Balanced Bounding Box Methods to approximate the efficient frontier by dropping some nearby competing solutions. This produces a smaller yet diverse set of options much faster than full enumeration. Cooperative bargaining then picks one balanced agreement from the reduced set. Case studies confirm the approach scales while keeping the key cost trade-offs intact.

Core claim

The Balanced Bounding Box Methods (B3Ms) efficiently derive the efficient frontier for the bi-objective collaborative scheduling model by selectively disregarding closely positioned and competing solutions, thereby reducing computational time while preserving the diversity and representativeness of the solutions over the frontier.

What carries the argument

Balanced Bounding Box Methods (B3Ms), which improve efficiency in bi-objective optimization by selectively disregarding closely positioned solutions on the efficient frontier to produce a reduced yet representative set.

If this is right

  • B3Ms significantly reduce computational time for the collaborative charging scheduling problem.
  • The reduced set maintains the integrity and representativeness of the original efficient frontier.
  • The methods supply a practical framework for other bi-objective nonlinear integer programs.
  • Cooperative bargaining applied to the reduced frontier yields a balanced collaboration outcome.

Where Pith is reading between the lines

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

  • The same trimming logic could be applied to real-time rescheduling when charger availability changes during operations.
  • Extension to three or more operators would test whether the time savings remain proportional as the number of objectives grows.
  • The reduced frontier might serve as input to simulation models that evaluate robustness under uncertain travel times.

Load-bearing premise

Selectively disregarding closely positioned and competing solutions on the efficient frontier will preserve enough diversity to avoid missing practically important cost trade-offs.

What would settle it

Run the complete Pareto front computation on one of the paper's numerical instances and check whether any high-impact trade-off points between the two operators' costs are absent from the reduced set returned by B3Ms.

Figures

Figures reproduced from arXiv: 2506.14461 by Balazs Kulcsar, Fangting Zhou, Jiaming Wu.

Figure 1
Figure 1. Figure 1: Illustration of collaborative charging scheduling problem In this setup, the two companies (i.e., the fleet operators, represented in green and orange) have access to four charging stations available for rent [PITH_FULL_IMAGE:figures/full_fig_p005_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Balanced box method: (a) unbounded search space, (b) incorporating non-collaborative point Given these constraints, the initial and final points of the efficient frontier (𝑧 𝑇 and 𝑧 𝐵) are determined by solving the following two equations: 𝑧 𝑇 ∶= lexmin 𝓍∈𝒳 { 𝑧1 (𝓍), 𝑧2 (𝓍) ∶ 𝑧(𝓍) ∈ 𝑅((0, 𝑧Non 1 ), (0, 𝑧Non 2 ))} (21) and 𝑧 𝐵 ∶= lexmin 𝓍∈𝒳 { 𝑧2 (𝓍), 𝑧1 (𝓍) ∶ 𝑧(𝓍) ∈ 𝑅((0, 𝑧Non 1 ), (0, 𝑧Non 2 ))} , (22) res… view at source ↗
Figure 3
Figure 3. Figure 3: Balanced box methods (a) original, (b) B3M with 𝑧 n ≈ e , (c) B3M with 𝑧 n ∼ e closeness according to the strict conditions of Definition 2.1 and terminates the search in the current region if the conditions are met. As shown in [PITH_FULL_IMAGE:figures/full_fig_p013_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: B3M1, 𝑧 n ≈ e , (a) investigating 𝑅𝐵 , (b) investigating 𝑅𝑇 , (c) new rectangles [PITH_FULL_IMAGE:figures/full_fig_p013_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: B3M1, 𝑧 n ≈ e , special scenario, (a) investigating 𝑅𝐵 , (b) investigating 𝑅𝑇 , (c) new rectangles are identified, as shown in [PITH_FULL_IMAGE:figures/full_fig_p014_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: The flowchart of B3M1 Zhou et al.: Preprint submitted to Elsevier Page 14 of 32 [PITH_FULL_IMAGE:figures/full_fig_p014_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: B3M2, 𝑧 n ∼ e , (a) investigating 𝑅𝐵 , (b) investigating 𝑅𝑇 , (c) new rectangles As depicted in [PITH_FULL_IMAGE:figures/full_fig_p015_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: The flowchart of B3M2 4.2. Cooperative bargaining The solution approaches proposed in Section 4.1 can derive an efficient frontier. This section is dedicated to the identification of a final solution from the obtained frontier. Given that the two objectives in the bi-objective optimization are neither fully aligned nor entirely opposed, it is possible for players to collaborate in order to reach a mutually… view at source ↗
Figure 9
Figure 9. Figure 9: Illustration of cooperative bargaining where 𝑧1 and 𝑧2 denote the costs incurred by both companies in the non-cooperative scenario, indicating the maximum cost that they would be willing to accept for such collaboration. In the symmetric case, the exponents are both equal to 1/2 and sum to 1, thereby ensuring normalization. These exponents represent the bargaining power of each player. As 𝜋 increases, play… view at source ↗
Figure 10
Figure 10. Figure 10: Geographical distribution of charging stations and EVs [PITH_FULL_IMAGE:figures/full_fig_p019_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: Hourly fluctuations in electricity prices vs. public charging fees Two fleet operators collaborate within the system, denoted as 𝐾. Each fleet operator 𝑘 manages a group of EVs, 𝐼𝑘 , while Göteborg Energi provides a set of chargers, 𝐽. The service period under consideration spans one day, which is divided into 24 one-hour intervals (𝑇 ) to allow for precise time-specific scheduling adjustments. Each charg… view at source ↗
Figure 12
Figure 12. Figure 12: Distribution of EVs and chargers in test cases. Subfigures (a)–(f) show electric vehicle (EV) spatial distributions with different fleet sizes (10, 20, 50) under uniform and clustered configurations. Subfigures (g)–(j) show charger layouts with two configurations (uniform and centralized) and varying charger counts (5 or 10). To distinguish between different test cases, we adopt the naming convention Dist… view at source ↗
Figure 13
Figure 13. Figure 13: Comparative analysis of results across various tolerance ranges 𝜖 This discrepancy arises because B3M1 disregards the solution by comparing it with the upper-left or lower-right points, which should be recorded as solutions. As a result, the search area is expanded in this particular instance. 5.4. Distribution of non-dominated points The non-dominated solutions for the three cases are presented in Figs. … view at source ↗
Figure 14
Figure 14. Figure 14: Non-dominated solutions of Case #1 with different methods In [PITH_FULL_IMAGE:figures/full_fig_p024_14.png] view at source ↗
Figure 15
Figure 15. Figure 15: Non-dominated solutions of Case #2 with different methods (a) All the solutions (b) B3M1 with 𝜖 = 1% (c) B3M1 with 𝜖 = 2% (d) B3M1 with 𝜖 = 5% (e) B3M2 with 𝜖 = 1% (f) B3M2 with 𝜖 = 2% (g) B3M2 with 𝜖 = 5% [PITH_FULL_IMAGE:figures/full_fig_p025_15.png] view at source ↗
Figure 16
Figure 16. Figure 16: Non-dominated solutions of Case #3 with different methods 5.5. Sensitivity analysis This section examines the effects of varying levels of value of time (VOT, 𝜆) on the optimization process and the balance between competing objectives [PITH_FULL_IMAGE:figures/full_fig_p025_16.png] view at source ↗
Figure 17
Figure 17. Figure 17: Impact of VOT levels (𝜆) on optimized results using B3Ms As shown in [PITH_FULL_IMAGE:figures/full_fig_p026_17.png] view at source ↗
Figure 18
Figure 18. Figure 18: Impact of changes in unit energy cost (𝜃) on optimized results using B3Ms the square markers represent non-dominated points in the solution space. The red squares indicate final solutions for specific parameter intervals, with the parameter values labeled next to each point. These red squares demonstrate the impact of parameter ranges on the trade-offs between objectives. A comparison of the final solutio… view at source ↗
Figure 19
Figure 19. Figure 19: Final solution of Case #1 with different methods Zhou et al.: Preprint submitted to Elsevier Page 27 of 32 [PITH_FULL_IMAGE:figures/full_fig_p027_19.png] view at source ↗
Figure 20
Figure 20. Figure 20: Final solution of Case #2 with different methods (a) Generalized Nash bargaining on all solutions (b) Distance function on all solu￾tions (c) Generalized Nash bargaining on B3M1 with 𝜖 = 2% (d) Distance function on B3M1 with 𝜖 = 2% [PITH_FULL_IMAGE:figures/full_fig_p028_20.png] view at source ↗
Figure 21
Figure 21. Figure 21: Final solution of Case #3 with different methods Both the Nash bargaining method and the distance function (𝛼-norm) approach provide frameworks for optimizing bi-objective problems, each offering unique flexibility. The Nash bargaining method considers the relative bargaining power of each party, optimizing outcomes based on their strategic influence. In contrast, the distance function (𝛼-norm) allows for… view at source ↗
Figure 22
Figure 22. Figure 22: Spatiotemporal distribution of EV charging scheduling results 6. Conclusion This paper addresses the collaborative scheduling problem in which two companies share charging stations, utilizing a bi-objective optimization approach to minimize costs for both parties through a collaborative scheduling model. We present the balanced box method to generate all non-dominated solutions and develop B3Ms to efficie… view at source ↗
read the original abstract

Electric mobility faces several challenges, most notably the high cost of infrastructure development and the underutilization of charging stations. The concept of shared charging offers a promising solution. The paper explores sustainable urban logistics through horizontal collaboration between two fleet operators and addresses a scheduling problem for the shared use of charging stations. To tackle this, the study formulates a collaborative scheduling problem as a bi-objective nonlinear integer programming model, in which each company aims to minimize its own costs, creating inherent conflicts that require trade-offs. The Balanced Bounding Box Methods (B3Ms) are introduced in order to efficiently derive the efficient frontier, identifying a reduced set of representative solutions. These methods enhance computational efficiency by selectively disregarding closely positioned and competing solutions, preserving the diversity and representativeness of the solutions over the efficient frontier. To determine the final solution and ensure balanced collaboration, cooperative bargaining methods are applied. Numerical case studies demonstrate the viability and scalability of the developed methods, showing that the B3Ms can significantly reduce computational time while maintaining the integrity of the frontier. These methods, along with cooperative bargaining, provide an effective framework for solving various bi-objective optimization problems, extending beyond the collaborative scheduling problem presented here.

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

2 major / 1 minor

Summary. The manuscript formulates a collaborative charging scheduling problem between two fleet operators as a bi-objective nonlinear integer program, where each operator minimizes its own costs. It introduces Balanced Bounding Box Methods (B3Ms) to approximate the efficient frontier by selectively pruning closely positioned competing solutions while aiming to retain diversity. Cooperative bargaining methods are then applied to select a final balanced solution. Numerical case studies are presented to show that B3Ms reduce computational time while maintaining frontier integrity, with claims of applicability to other bi-objective problems.

Significance. If the B3Ms can be rigorously shown to approximate the Pareto set without omitting practically relevant trade-offs in discrete bi-objective integer programs, the framework could provide useful computational tools for shared EV infrastructure planning and horizontal collaboration in urban logistics. The numerical demonstrations of runtime reduction are potentially valuable for practitioners, but the absence of theoretical approximation guarantees limits the work's broader methodological contribution to multi-objective optimization.

major comments (2)
  1. [Abstract] Abstract (paragraph on B3Ms): The central claim that selectively disregarding closely positioned solutions 'preserves the diversity and representativeness of the solutions over the efficient frontier' lacks any formal bound (e.g., Hausdorff distance or worst-case deviation) on how much the reduced frontier can deviate from the true Pareto set. This is load-bearing for the integrity claim, especially for a discrete feasible set that may contain clusters or sharp bends.
  2. [Numerical case studies] Numerical case studies: The studies are asserted to demonstrate viability and scalability with reduced computational time, but no baseline comparisons, error bars, instance data, or quantitative metrics on frontier quality (e.g., coverage of trade-offs) are referenced, undermining verification of the performance claims.
minor comments (1)
  1. [Abstract] The abstract supplies no equations, model formulation details, or proof sketch for the B3Ms pruning rule, which would aid initial assessment of correctness.

Simulated Author's Rebuttal

2 responses · 1 unresolved

We thank the referee for the constructive comments. We address each major point below and have revised the manuscript to clarify limitations and strengthen the numerical evaluation where possible.

read point-by-point responses

  1. Referee: [Abstract] Abstract (paragraph on B3Ms): The central claim that selectively disregarding closely positioned solutions 'preserves the diversity and representativeness of the solutions over the efficient frontier' lacks any formal bound (e.g., Hausdorff distance or worst-case deviation) on how much the reduced frontier can deviate from the true Pareto set. This is load-bearing for the integrity claim, especially for a discrete feasible set that may contain clusters or sharp bends.

    Authors: We agree that the manuscript provides no formal approximation guarantees such as Hausdorff distance bounds or worst-case deviation results. B3Ms are introduced as a heuristic that prunes nearby solutions via bounding boxes to reduce the frontier size while retaining representative trade-offs, with this property supported only by the numerical case studies. In the revision we have added an explicit limitations paragraph in the methodology section stating that the approach lacks theoretical guarantees and is validated empirically for the EV scheduling instances. We also outline future research directions for deriving such bounds on restricted problem classes. revision: partial

  2. Referee: [Numerical case studies] Numerical case studies: The studies are asserted to demonstrate viability and scalability with reduced computational time, but no baseline comparisons, error bars, instance data, or quantitative metrics on frontier quality (e.g., coverage of trade-offs) are referenced, undermining verification of the performance claims.

    Authors: We acknowledge that the original numerical section lacked explicit baselines, statistical variability measures, and quantitative frontier-quality metrics. The revised version now includes direct runtime comparisons against the epsilon-constraint method and NSGA-II, reports mean runtimes with standard deviations over ten replications, supplies the full instance data in an appendix, and adds hypervolume and trade-off coverage ratios to quantify frontier quality. These changes allow independent verification of the claimed reductions in computational time while preserving frontier integrity. revision: yes

standing simulated objections not resolved
  • Providing rigorous theoretical approximation guarantees (e.g., Hausdorff distance or worst-case deviation bounds) for Balanced Bounding Box Methods on arbitrary discrete bi-objective nonlinear integer programs.

Circularity Check

0 steps flagged

No circularity: B3Ms introduced as independent algorithmic pruning rule

full rationale

The paper formulates a bi-objective nonlinear integer program and presents Balanced Bounding Box Methods (B3Ms) as a new algorithmic procedure that selectively prunes closely positioned solutions on the efficient frontier. No equations, fitted parameters, or self-citations are shown to reduce the claimed runtime reduction or frontier integrity to a tautological redefinition of the input data or prior results by the same authors. The derivation chain consists of standard multi-objective optimization steps followed by an independent heuristic rule whose correctness is asserted via numerical case studies rather than by construction. This is the most common honest non-finding for an algorithmic contribution paper.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract-only review yields no explicit free parameters, axioms, or invented entities; the central claim rests on the unstated assumption that the bi-objective model accurately captures real operator costs and that the bounding-box heuristic preserves frontier quality.

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