Optimization with Multi-sourced Information and Unknown Reliability: A Distributionally Robust Approach

In problems that involve input parameter information gathered from multiple data sources with varying reliability, incorporating decision makers’ trust on different sources in optimization models can potentially improve solution performance. In this work, we propose a novel multi-reference distributionally robust optimization (MR-DRO) framework, where the model inputs are uncertain and their probability distributions can be statistically inferred from multiple information sources. Via nonparametric data fusion, we construct a Wasserstein ambiguity set to minimize the worst-case expected cost of a stochastic objective function, accounting for both uncertainty and unknown reliability of several given information sources. We reformulate the MR-DRO model as a linear program given linear objective and constraints in the original problem. We also incorporate a dynamic trust update mechanism that adjusts the trust for each source based on its performance over time. In addition, we introduce the concept of probability dominance to identify sources with dominant trust. Via computational studies using resource allocation and portfolio optimization instances, we demonstrate the effectiveness of the MR-DRO approach compared to traditional optimization frameworks relying on a single data source. Our results highlight the significance of integrating (dynamic) decision maker’s trust in optimization under uncertainty, particularly when given diverse and potentially conflicting input data.

💡 Research Summary

The paper introduces a Multi‑Reference Distributionally Robust Optimization (MR‑DRO) framework that explicitly incorporates heterogeneous data sources and their unknown reliabilities into stochastic optimization. Traditional stochastic and robust optimization assume either a known probability distribution or a single uncertainty set, ignoring the fact that real‑world decisions often rely on multiple forecasts or measurements that differ in accuracy and bias. To address this, the authors propose a non‑parametric data‑fusion scheme: for each source h, past prediction errors are used to correct the current forecast, producing revised predictions ˆξ⁽ʰ⁾. A trust weight t_h, initialized uniformly, is updated after each observation based on the alignment between a source’s past errors and realized outcomes, following a Bayesian‑inspired rule that ensures Σ_h t_h = 1 and 0 ≤ t_h ≤ 1.

These trust‑weighted predictions are aggregated into a discrete empirical distribution ˆPᴴᴵ consisting of H·I atoms (H sources, I past events). A Wasserstein ambiguity set B_ε(ˆPᴴᴵ) is then defined as all probability measures within radius ε of this empirical distribution. The MR‑DRO problem seeks a decision x ∈ X that minimizes the worst‑case expected loss over this set:
 inf_{x∈X} sup_{Q∈B_ε(ˆPᴴᴵ)} E_Q