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arxiv: 2508.19933 · v3 · submitted 2025-08-27 · 📡 eess.SY · cs.SY

Combined Stochastic and Robust Optimization for Electric Autonomous Mobility-on-Demand with Nested Benders Decomposition

Sten Elling Tingstad Jacobsen , Bal\'azs Kulcs\'ar , Anders Lindman This is my paper

Pith reviewed 2026-05-18 21:20 UTC · model grok-4.3

classification 📡 eess.SY cs.SY
keywords electric autonomous mobility-on-demandstochastic optimizationrobust optimizationmodel predictive controlnested benders decompositionfleet managementdemand forecastinguncertainty handling
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The pith

A combined stochastic and robust model predictive control framework for electric autonomous mobility-on-demand fleets reduces median passenger waiting times by up to 36 percent and electricity costs by more than 35 percent.

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

The paper develops a planning method that uses probabilistic forecasts of demand and conditions to anticipate typical operations while adding protective constraints that keep decisions feasible even in worst-case travel times and energy use. This mixed approach is cast as a large mixed-integer program and solved in real time by a Nested Benders Decomposition procedure that exploits the scenario-tree structure for parallel computation. High-fidelity simulations on San Francisco and Chicago networks show the resulting dispatch, rebalancing, and charging decisions produce shorter waits, fewer extreme delays, less empty driving, and lower energy expenditure than purely deterministic, reactive, or purely robust alternatives. A sensitivity study further links vehicle efficiency and battery size to the amount of charging infrastructure required for stable performance. The work therefore argues that jointly handling average-case variability and worst-case protection, together with efficient solution methods, is necessary for practical large-scale electric autonomous ride services.

Core claim

We formulate a multi-stage stochastic-robust model predictive control problem that incorporates spatio-temporal Bayesian neural network forecasts of demand, travel time, energy consumption, and charger availability, augments the model with robust constraints on energy and travel time to guard against adverse realizations, and solves the resulting large-scale mixed-integer linear program with a tailored Nested Benders Decomposition algorithm. In closed-loop high-fidelity simulations of San Francisco and Chicago, the combined approach yields up to 36 percent lower median passenger waiting times, nearly 20 percent lower 95th-percentile delays, 27 percent less rebalancing distance, and more than

What carries the argument

A multi-stage stochastic optimization model with added robust constraints on energy consumption and travel times, solved by Nested Benders Decomposition that decomposes the scenario tree for efficient parallel solution.

If this is right

  • Dispatch, rebalancing, and charging decisions can be coordinated in a single optimization that anticipates variability while protecting against worst-case energy and time outcomes.
  • Median passenger waiting times drop by up to 36 percent and 95th-percentile delays by nearly 20 percent relative to simpler controllers.
  • Rebalancing distance falls by 27 percent and electricity costs by more than 35 percent.
  • Energy-efficient vehicles maintain performance with smaller batteries, whereas less efficient vehicles need larger batteries and denser charging support.

Where Pith is reading between the lines

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

  • The same combined stochastic-robust structure could be tested on related fleet problems such as last-mile delivery or shared electric scooters where both typical and extreme conditions matter.
  • Online updating of the Bayesian forecasts from live vehicle data would likely increase the realized gains beyond the offline simulation results.
  • City planners could use the battery-size sensitivity results to set minimum vehicle-efficiency standards or charging-station density targets before large-scale EAMoD rollout.

Load-bearing premise

The Bayesian neural network forecasts of demand, travel time, energy consumption, and charger availability remain accurate enough outside the simulation that the computed plans retain their reported performance advantages in actual deployment.

What would settle it

Deploy the full closed-loop controller on a real electric autonomous fleet for several weeks and measure whether median waiting times, 95th-percentile delays, rebalancing distance, and electricity cost differ from the deterministic and robust baselines by the margins reported in simulation.

Figures

Figures reproduced from arXiv: 2508.19933 by Anders Lindman, Bal\'azs Kulcs\'ar, Sten Elling Tingstad Jacobsen.

Figure 1
Figure 1. Figure 1: Overview of the scenario-based model predictive control (SMPC) framework using a Bayesian Neural Network (BNN), scenario tree generation and reduction, and Nested Benders Decomposition (NBD). Historical data is first used to train the BNN, which provides predictions of future demand and charging availability. These predictions are used to generate a multi-stage scenario tree, which is then reduced using th… view at source ↗
Figure 2
Figure 2. Figure 2: This diagram illustrates a multi-layer network representing an Electric Autonomous Mobility-on-Demand (EAMoD) system. Each layer corresponds to a different battery state of charge (SoC). When a vehicle travels in the network (illustrated by a dashed arrow with a vehicle icon), it consumes energy, leading to a decrease in its state of charge (SoC). In contrast, a dashed arrow with a charging station represe… view at source ↗
Figure 3
Figure 3. Figure 3: Scenario tree representation of the random variables, incorporating both the robust horizon and prediction horizon. Each scenario is defined as 𝜉 𝑠 𝑡 = (𝜆 𝑠 𝑖𝑗𝑡, 𝑘𝑐,𝑠 𝑖,𝑡 ). 𝑠 ∈ {1, 2,…, 𝑆} is defined as 𝜉 𝑠 𝑡 = [𝜆 𝑠 𝑖𝑗𝑡, 𝑘𝑐,𝑠 𝑖𝑡 ], where 𝜆 𝑠 𝑖𝑗𝑡 is the predicted travel demand and 𝑘 𝑐,𝑠 𝑖𝑡 the available chargers in scenario 𝑠. The cost function is then reformulated using the Sample Average Approximation (S… view at source ↗
Figure 4
Figure 4. Figure 4: An example of solving a multi-stage SMPC problem by decomposing it into a nested series of two-stage SMPC problems. The figure on the left illustrates the first and second stages of the SMPC problem, while the figure on the right shows a portion of the second and third stages. Both sections can be resolved using Benders Decomposition. Benders cuts derived from the duals of the subproblems (SP) ( [PITH_FUL… view at source ↗
Figure 5
Figure 5. Figure 5: Illustration of nested Benders decomposition. The master problem proposes trial solutions which are passed sequentially to subproblems at different stages. Each subproblem generates Benders cuts based on its solution and returns them to the previous stage, enabling iterative refinement across the scenario tree Within the NBD framework, multiple MPs are defined, and as the decision process progresses down t… view at source ↗
Figure 6
Figure 6. Figure 6: Electricity prices in €/kWh for three different pricing scenarios: Flat, Peak, and Solar Price, shown over a 24-hour period. 5.2. Travel Demand Prediction We conducted a comprehensive comparative analysis between BNF, our proposed prediction framework, and three established baselines: multi-output Gaussian Process regression with a spectral kernel (MOSK) [69], Long Short-Term Memory (LSTM) networks, and Ve… view at source ↗
Figure 7
Figure 7. Figure 7: Predicted travel demand with 95% prediction intervals for selected locations. Black dots show training data, white dots indicate test data, the red line is the median prediction, and blue shaded areas represent the 95% prediction intervals. In the nested Benders decomposition, one optimality gap is computed for each master problem and subproblem across the scenario tree. Since these gaps can vary considera… view at source ↗
Figure 8
Figure 8. Figure 8: Energy consumption of the different vehicles without auxiliary power and with a 40 kWh battery. 5.6. Sensitivity analysis of battery size and energy efficiency (San Francisco) Battery capacity significantly influences system performance, though the extent depends on vehicle efficiency ( [PITH_FULL_IMAGE:figures/full_fig_p021_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: Comparison of wait time metrics in San Francisco. Fig. a shows metrics across different fleet sizes and vehicle types with 100 chargers. Fig. b shows metrics across different number of charger and vehicle types with a fleet size of 450 vehicles. All plots assume a 40 kWh battery capacity. For clarity, 75th percentile wait times for Vehicle 3 are not shown in the plots, but their values are as follows: for … view at source ↗
Figure 10
Figure 10. Figure 10: Comparison of hourly energy charged and consumed across three different vehicle types with 40 kWh battery capacity for (a) San Francisco and 70 kWh for (b) Chicago. Each line represents the total energy (in kWh) aggregated across all vehicles in a fleet. Charging events are shown with dashed lines, and consumption with solid lines. The energy demand patterns in Chicago are more concentrated during peak pe… view at source ↗
Figure 11
Figure 11. Figure 11: Observed charge cycles (20–80% Depth of Discharge) for different vehicle types (1, 2, 3) and battery sizes (40, 70, 100 kWh) in (a) San Francisco and (b) Chicago. These distributions highlight the combined influence of vehicle characteristics and geographical location on battery charging patterns. However, a more pronounced improvement is seen in the upper percentiles. The 75th percentile drops from 329s … view at source ↗
Figure 12
Figure 12. Figure 12: Comparison of wait time metrics in Chicago. Fig. a shows metrics across different fleet sizes and vehicle types with 200 chargers. Fig. b shows metrics across different number of charger and vehicle types with a fleet size of 1400 vehicles. All plots assume a 70 kWh battery capacity. For clarity, 75th percentile wait times for Vehicle 3 are not shown in the plots, but their values are as follows: for vary… view at source ↗
read the original abstract

The electrification and automation of mobility are reshaping how cities operate on-demand transport systems. Managing Electric Autonomous Mobility-on-Demand (EAMoD) fleets effectively requires coordinating dispatch, rebalancing, and charging decisions under multiple uncertainties, including travel demand, travel time, energy consumption, and charger availability. We address this challenge with a combined stochastic and robust model predictive control (MPC) framework. The framework integrates spatio-temporal Bayesian neural network forecasts with a multi-stage stochastic optimization model, formulated as a large-scale mixed-integer linear program. To ensure real-time applicability, we develop a tailored Nested Benders Decomposition that exploits the scenario tree structure and enables efficient parallelized solution. Stochastic optimization is employed to anticipate demand and infrastructure variability, while robust constraints on energy consumption and travel times safeguard feasibility under worst-case realizations. We evaluate the framework using high-fidelity simulations of San Francisco and Chicago. Compared with deterministic, reactive, and robust baselines, the combined stochastic and robust approach reduces median passenger waiting times by up to 36% and 95th-percentile delays by nearly 20%, while also lowering rebalancing distance by 27% and electricity costs by more than 35%. We also conduct a sensitivity analysis of battery size and vehicle efficiency, finding that energy-efficient vehicles maintain stable performance even with small batteries, whereas less efficient vehicles require larger batteries and greater infrastructure support. Our results emphasize the importance of jointly optimizing predictive control, vehicle capabilities, and infrastructure planning to enable scalable, cost-efficient EAMoD operations.

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 / 2 minor

Summary. The paper proposes a combined stochastic and robust optimization framework for Electric Autonomous Mobility-on-Demand (EAMoD) using spatio-temporal Bayesian neural network forecasts integrated into a multi-stage stochastic-robust MILP. Solved via Nested Benders Decomposition for efficiency, it is tested in high-fidelity simulations of San Francisco and Chicago, reporting up to 36% reduction in median waiting times, 20% in 95th-percentile delays, 27% less rebalancing, and 35% lower electricity costs versus baselines, along with sensitivity analysis on battery and efficiency.

Significance. If validated beyond internal simulations, the results would be significant for EAMoD system design, highlighting advantages of hybrid stochastic-robust MPC over individual approaches and the practicality of the decomposition method for large-scale problems. The sensitivity findings on vehicle efficiency and battery size offer actionable insights for infrastructure. The work builds on standard techniques but applies them innovatively to this domain with computational advances.

major comments (2)
  1. [Results] The performance gains (e.g., 36% median wait time reduction) are demonstrated in closed-loop simulations that reuse the BNN forecasts for both optimization input and evaluation. This does not isolate the impact of forecast accuracy, which is the load-bearing assumption for the claimed benefits when deployed outside the simulator.
  2. [Methods] Details on the construction of the scenario tree and the robust uncertainty sets derived from the BNN predictions are insufficient to evaluate potential biases in the stochastic-robust combination.
minor comments (2)
  1. [Abstract] Clarify whether the 'up to 36%' and other metrics are for San Francisco, Chicago, or both, and under what demand conditions.
  2. Consider adding a table summarizing the key parameters of the BNN and the optimization model for reproducibility.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the constructive feedback and positive evaluation of the work's significance. We address the two major comments point by point below, agreeing that both warrant revisions to improve clarity and strengthen the claims.

read point-by-point responses

  1. Referee: [Results] The performance gains (e.g., 36% median wait time reduction) are demonstrated in closed-loop simulations that reuse the BNN forecasts for both optimization input and evaluation. This does not isolate the impact of forecast accuracy, which is the load-bearing assumption for the claimed benefits when deployed outside the simulator.

    Authors: We agree that the closed-loop simulations employ the same BNN forecasts for both the MPC optimization and the evaluation environment. This design evaluates the framework's performance when forecasts are representative of the simulated dynamics, which is a common approach for assessing predictive control methods. However, the referee correctly identifies that this does not fully isolate the effect of forecast accuracy or test robustness to forecast errors in deployment. In the revised manuscript, we will add a new subsection with sensitivity experiments that introduce controlled noise to the BNN predictions or use held-out historical data for evaluation. These results will quantify how performance degrades with forecast mismatch and better support the deployment claims. revision: yes

  2. Referee: [Methods] Details on the construction of the scenario tree and the robust uncertainty sets derived from the BNN predictions are insufficient to evaluate potential biases in the stochastic-robust combination.

    Authors: We concur that the current description of scenario tree construction and robust set derivation lacks sufficient detail for independent assessment of biases. The revised manuscript will include an expanded Methods subsection that explicitly describes: (i) how posterior samples from the spatio-temporal BNN are used to generate the multi-stage scenario tree, including branching factors, scenario reduction technique, and probability assignment; and (ii) the precise formulation of the robust uncertainty sets for energy consumption and travel times (e.g., using BNN-derived prediction intervals or worst-case bounds) and how they are combined with the stochastic scenarios via the hybrid objective and constraints. This will allow readers to evaluate potential biases in the stochastic-robust integration. revision: yes

Circularity Check

0 steps flagged

No circularity in derivation; standard stochastic-robust MPC applied to EAMoD with external simulation validation

full rationale

The paper's core derivation formulates a multi-stage stochastic-robust MILP that takes spatio-temporal BNN forecasts of demand, travel time, energy, and charger availability as exogenous inputs, then applies robust constraints on worst-case realizations and solves via Nested Benders Decomposition. Performance metrics (waiting times, delays, rebalancing distance, electricity costs) are generated by forward simulation against deterministic/reactive/robust baselines rather than by fitting or re-deriving the same quantities inside the model. No equations reduce by construction to their own inputs, no load-bearing self-citations are invoked to justify uniqueness, and the BNN component is treated as a separate forecasting module whose accuracy is assumed for the optimization step. The framework is therefore self-contained against external simulation benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract-only review provides no explicit free parameters, axioms, or invented entities; the framework relies on standard assumptions of stochastic programming and robust optimization that are not detailed here.

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Forward citations

Cited by 1 Pith paper

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