[Paper Review] A novel multi-stage multi-scenario multi-objective optimisation framework for adaptive robust decision-making under deep uncertainty

Babooshka Shavazipour, Theodor J. Stewart|arXiv (Cornell University)|Dec 18, 2023

Risk and Portfolio Optimization4 citations

TL;DR

This paper proposes a novel multi-stage multi-scenario multi-objective optimisation framework for adaptive robust decision-making under deep uncertainty, using a T-stage model and a two-stage moving horizon approach to generate dynamic-robust Pareto optimal solutions. The T-stage model outperforms the moving horizon approach by considering the full planning horizon simultaneously, yielding more robust solutions in a sequential portfolio selection case study with nine scenario paths.

ABSTRACT

Many real-world decision-making problems involve multiple decision-making stages and various objectives. Besides, most of the decisions need to be made before having complete knowledge about all aspects of the problem leaves some sort of uncertainty. Deep uncertainty happens when the degree of uncertainty is so high that the probability distributions are not confidently knowable. In this situation, using wrong probability distributions leads to failure. Scenarios, instead, should be used to evaluate the consequences of any decisions in different plausible futures and find a robust solution. In this study, we proposed a novel multi-stage multi-scenario multi-objective optimisation framework for adaptive robust decision-making under deep uncertainty. Two approaches, named multi-stage multi-scenario multi-objective and two-stage moving horizon, have been proposed and compared. Finally, the proposed approaches are applied in a case study of sequential portfolio selection under deep uncertainty, and the robustness of their solutions is discussed.

Motivation & Objective

• To address the challenge of multi-stage, multi-objective decision-making under deep uncertainty, where probability distributions are unknowable.
• To develop a framework that supports adaptive robust decision-making by identifying solutions resilient across multiple plausible future scenarios.
• To compare the performance of a full T-stage optimisation model against a two-stage moving horizon approach in generating dynamic-robust solutions.
• To evaluate the robustness of solutions in a sequential portfolio selection problem under deep uncertainty.
• To explore the trade-offs between computational complexity and solution robustness in multi-stage decision frameworks.

Proposed method

• Proposes a multi-stage multi-scenario multi-objective optimisation framework that models decisions across multiple time stages and uncertain scenarios without relying on probability distributions.
• Employs goal programming to handle multiple conflicting objectives and generate Pareto optimal solutions under deep uncertainty.
• Uses scenario planning to structure uncertainty, defining plausible future states without assigning probabilities.
• Applies a T-stage model that optimises all stages simultaneously, ensuring constraints are satisfied across the entire horizon.
• Implements a two-stage moving horizon approach that re-optimises at each stage, looking one step ahead and updating scenario sets as new information becomes available.
• Compares solution robustness using total profit as a performance metric across different scenario paths in a three-stage portfolio selection case study.

Figure 3: Scenarios of the two-stage structure in comparison with meta-scenarios of the three-stage structure

Experimental results

Research questions

• RQ1 How does a full T-stage multi-scenario multi-objective optimisation model compare to a two-stage moving horizon approach in generating robust solutions under deep uncertainty?
• RQ2 What is the trade-off between solution robustness and computational complexity when increasing the number of stages in multi-stage decision models?
• RQ3 Can the T-stage model generate more robust solutions than the moving horizon approach by anticipating future uncertainties across the entire planning horizon?
• RQ4 How does updating scenario sets during the planning process affect the robustness and feasibility of solutions in dynamic decision-making under deep uncertainty?
• RQ5 What role do decision-maker preferences play in selecting a single Pareto optimal solution from the robust solution set?

Key findings

• The T-stage model consistently generates more robust solutions than the two-stage moving horizon approach because it considers the entire planning horizon simultaneously and satisfies all constraints at once.
• In the three-stage portfolio selection case study with nine scenario paths, the T-stage model achieved superior robustness in terms of total profit across all scenarios.
• The two-stage moving horizon approach offers computational and cognitive advantages by allowing scenario set updates and reducing problem complexity at each stage.
• The number of scenario paths grows exponentially with the number of stages, making the T-stage model computationally expensive for large horizons.
• The T-stage model's optimal solution is no worse than those from the moving horizon approach, confirming its theoretical superiority in robustness.
• The study identifies future research opportunities in enhancing the robustness of moving horizon approaches through weighted objectives and determining optimal lookahead horizons.

Figure 4: Scenarios of the 2 $\times$ two-stage structure