Scenario-Sampling Rolling-Horizon Framework

• The framework is a dynamic optimization policy that uses simulated future scenarios to hedge against myopic decisions and forecast errors.
• It integrates static or two-stage subproblem solvers within a receding-horizon control loop to update decisions continuously while balancing computation and accuracy.
• Applied in domains like dynamic routing, inventory planning, and energy management, it delivers robust performance under uncertainty.

A scenario-sampling rolling-horizon framework is a class of anticipatory online optimization policies designed to handle dynamic decision-making under uncertainty. These frameworks combine (i) scenario-based modeling of possible future evolutions of exogenous processes, (ii) periodic resolution of finite-horizon subproblems on a rolling basis, and (iii) commitment of only the first (or first few) actions at each decision epoch. They are distinguished by their explicit construction and exploitation of simulated or statistically-generated scenarios to hedge against myopic bias or forecast error, and by the integration of static or two-stage subproblem solvers within a dynamic, receding-horizon control loop. These techniques have become central in applications such as dynamic vehicle routing, production and inventory planning, energy management, and simulation-based verification.

1. Key Elements and General Structure

A typical scenario-sampling rolling-horizon (SS-RH) policy operates as follows. At each decision epoch:

• The current system state is observed (e.g., outstanding tasks, current demands, worker/vehicle locations).
• A set of scenarios is generated, often by sampling from empirical or estimated distributions. These scenarios represent plausible future realizations of uncertain parameters or events over a fixed look-ahead window.
• For each scenario, the subproblem (e.g., assignment, routing, scheduling, recourse actions) is formulated, incorporating both observed and sampled data.
• The scenario-augmented optimization is solved, either as a deterministic approximation, explicit multi-scenario/multistage stochastic program, or via scenario aggregation.
• Only the next action(s) are implemented, and the process is repeated as system state evolves.

The framework is event-driven or periodically triggered, supporting real-time reactivity and continual adaptation to new information. The design of the sampling procedure, scenario tree or scenario set, and finite-horizon subproblem structure are critical elements distinguishing variants across domains (see (Wu et al., 16 Jan 2026, Schlenkrich et al., 2024, Spinelli et al., 2024, Cortés et al., 19 Mar 2025, Gioia et al., 2022)).

2. Scenario Generation and Sampling Procedures

Scenario sampling is central to these frameworks. Major approaches include:

• Monte Carlo sampling from empirical or probabilistic models, e.g., Gaussian forecast error models for renewable generation and loads in microgrid operation (Cortés et al., 19 Mar 2025, Schlenkrich et al., 2024).
• Virtual task augmentation in dynamic routing: At each epoch, synthetic tasks are generated, with attributes sampled uniformly from the empirical range of real tasks—to serve as proxies for anticipated future arrivals and to enrich the optimization's look-ahead (Wu et al., 16 Jan 2026).
• Nonparametric conditional density estimation for building scenario trees in multi-stage settings, as in waste collection where fill rates are estimated by kernel methods and assembled into a tree structure using dynamic stochastic approximation (Spinelli et al., 2024).
• Historical resampling and structure-aware scenario trees, such as constructing trees by drawing from month-specific, family-correlated, or bimodal marginal demand distributions in assemble-to-order planning (Gioia et al., 2022).
• Enumerator-based sampling for simulation-based verification, where automata-theoretic data structures allow uniform sampling or enumeration of all legal scenarios of arbitrary (rolling) horizon up to prescribed constraints (Mancini et al., 2021).

The number of scenarios, their branching structure, and the statistical fidelity of the sampled process are all key determinants of computational tractability and policy quality.

3. Rolling-Horizon Control Logic

SS-RH frameworks use a receding (rolling) time window to re-optimize actions as system state and forecasts update. Important features include:

• Receding finite windows: Only a limited forecast horizon (e.g., 12 periods, 6 days, or variable-length event intervals) is considered at each optimization step (Cortés et al., 19 Mar 2025, Schlenkrich et al., 2024, Spinelli et al., 2024).
• Event-driven reoptimization: Epochs correspond not to uniform time intervals but to discrete "events"—such as new task arrivals, workers becoming idle, or other state changes—triggering subproblem resolution (see (Wu et al., 16 Jan 2026)).
• Non-anticipativity and recursion: Only first-stage/initial-period actions are implemented; future plans are revised as each new epoch is reached.
• State updates and warm starts: Where possible, subproblems are warm-started using prior epoch solutions, with inventories, positions, or resources rolled over (Spinelli et al., 2024).

Aggregation heuristics (e.g., frequency-based assignment confidence) and hyperparameter tuning (e.g., window length, scenario count) help balance look-ahead depth with solution time and myopia.

4. Static and Stochastic Subproblem Formulations

At each epoch, scenario-augmented subproblems are solved. The formulations include:

• Deterministic augmented static problems: For example, the event-driven DTOP-SC setting solves, in parallel per scenario, a Heterogeneous-Trajectory Team Orienteering Problem with Time Windows (HT-TOPTW) on the union of real and virtual tasks (Wu et al., 16 Jan 2026).
• Two-stage or multi-stage stochastic programs, where scenario trees drive recourse modeling over the reoptimization window (Schlenkrich et al., 2024, Spinelli et al., 2024, Gioia et al., 2022).
• Mixed-Integer Programming (MIP) or convex/linear relaxations: Depending on tractability, the subproblem may be modeled as a MIP (with time limits for offline benchmarking), a convex quadratic program, or a MILP (as in microgrid management, which uses models of varying fidelity (Cortés et al., 19 Mar 2025)).
• Heuristic and metaheuristic solvers: Highly combinatorial subproblems (e.g., DTOP-SC) are tackled using adaptive large neighborhood search (ALNS) for rapid, high-quality approximate solutions (Wu et al., 16 Jan 2026).

The interaction between the scenario structure and subproblem model determines the policy's anticipatory strength.

5. Computational Properties and Theoretical Performance

Theoretical and empirical properties of SS-RH frameworks include:

• Scalability: Parallelization across independent scenarios is standard (e.g., S=15 in (Wu et al., 16 Jan 2026)), and window truncation or tree pruning ensures manageable per-epoch complexity even in large-scale applications (Schlenkrich et al., 2024, Spinelli et al., 2024, Cortés et al., 19 Mar 2025).
• Mitigation of myopic bias: Myopic or deterministic policies, which ignore future uncertainty, are systematically outperformed by scenario-sampling rolling-horizon control, especially as dynamism or uncertainty increases. Statistically significant reductions in optimality gap, cost, or lost sales are reported (Wu et al., 16 Jan 2026, Schlenkrich et al., 2024, Gioia et al., 2022).
• Computational efficiency: Across domains, per-epoch solution times range from sub-second (vehicle routing, microgrid convex relaxations) to a few minutes (production planning with 30 scenarios), and trade-offs between horizon/window size, scenario count, and optimality gap are quantified (Spinelli et al., 2024, Cortés et al., 19 Mar 2025).
• Bounded suboptimality: Worst-case analysis proves optimality in degenerate cases (e.g., zero travel cost in waste collection), but in general no absolute performance ratio exists (Spinelli et al., 2024).

Empirical results consistently demonstrate that rolling-horizon scenario-sampling frameworks deliver robust, implementable policies at a fraction of the computational burden of full-horizon, multistage stochastic programs.

6. Application Domains and Benchmarks

Prominent areas of application include:

• Dynamic vehicle/task assignment: Crowdsourcing and micro-mobility platforms optimize profit over spatial-temporal networks using virtual task scenarios and ALNS-based slotting (Wu et al., 16 Jan 2026).
• Production and inventory planning: Capacitated lot-sizing under forecast evolution is managed by sampling scenario demand trajectories and continually re-optimizing, outperforming classical MRP systems especially under high utilization or frequent demand updates (Schlenkrich et al., 2024).
• Stochastic inventory-routing: Waste collection with uncertain generation utilizes scenario trees (kernel-based, DSA) for anticipation, with rolling-horizon windowing yielding practical policies near the multistage optimum (Spinelli et al., 2024).
• Microgrid energy management: Real-time receding-horizon optimization addresses renewable intermittency and phase imbalance, using sampled forecast error scenarios for evaluation (Cortés et al., 19 Mar 2025).
• Simulation-based verification: Uniform random sampling of admissible input scenarios for constrained CPS verification is integrated into rolling-horizon simulation-based model checking (Mancini et al., 2021).
• Assemble-to-order (ATO) systems: Scarce, imbalanced, and seasonally correlated demand is handled by small scenario trees or mixture models embedded in rolling-horizon stochastic programming (Gioia et al., 2022).

Each domain adapts the general SS-RH paradigm to the properties of its uncertainty and decision structure.

7. Limitations, Extensions, and Practical Guidelines

Limitations include:

• Scenario explosion: In multi-stage settings, the number of scenarios grows exponentially with horizon and branching, necessitating truncation or approximate tail modeling (Gioia et al., 2022, Spinelli et al., 2024).
• No global optimality guarantee in general due to the rolling-horizon myopia and approximate scenario construction; however, empirical stabilization and near-offline performance are standard.
• Domain-specific modeling: Effective scenario sampling relies on good data and/or domain-tailored stochastic modeling (e.g., MMFE for demand update, kernel methods for conditional waste generation).
• Parameter calibration: Window length, scenario count, and horizon truncation require empirical or computational tuning to balance quality and tractability.

Guidelines (distilled from reported managerial insights):

• Under low uncertainty and load, simple deterministic or MRP policies suffice, but as forecast error, decision dynamism, or resource tightness increases, stochastic scenario-sampling rolling-horizon methods become superior (Schlenkrich et al., 2024, Gioia et al., 2022).
• Small scenario trees with tailored scenario generation (e.g., correlated, seasonal, or mixture distributions) capture most stochastic benefits with low complexity (Gioia et al., 2022).
• Parallelization and warm-starting exploit independent scenario/subproblem decomposability for scalable deployment (Wu et al., 16 Jan 2026, Spinelli et al., 2024).

The scenario-sampling rolling-horizon framework thus serves as a general paradigm for dynamic, anticipatory decision-making under uncertainty, synthesizing statistical scenario modeling, finite-horizon optimization, and online receding control to jointly address predictive uncertainty and operational tractability across a range of complex domains.