Data-Driven Chance Constrained Mixed Integer Nonlinear Bilevel Optimization via Copulas

Process supply chains have at their core the functionalities of planning and scheduling as means to ensure their profitability and responsive operations. At the same time, challenges in real-world demand data, such as multivariate data dependency and correlations, noisy data distributions, and parameter uncertainties, significantly complicate the decision-making process, making it difficult to identify optimal solutions. This necessitates the adoption of data-driven optimization approaches to effectively account for the dependency structures inherent in the observed data, thereby efficiently exploring solution spaces and identifying high-quality outcomes. This work proposes a data-driven probabilistic framework that integrates chance constrained programming (CCP) and copulas. CCP is an optimization-based method that enforces probabilistic constraints under specified risk thresholds. Copulas are a statistical technique designed to estimate data while capturing the dependency structure among multiple uncertain parameters with inherent correlations. The proposed framework is capable of accurately modeling variable dependencies across different scenarios, especially when the data exhibit complex distributions or nontrivial dependencies. We focus on three integrated case studies originating from the planning and scheduling of process supply chains and crude oil scheduling. We combine our proposed estimation framework with the Data-driven Optimization of bilevel Mixed-Integer NOnlinear problems (DOMINO) framework, which is a data-driven gray-box algorithm for addressing bilevel optimization problems, to derive decisions with guaranteed demand satisfaction rates. Computational experiments demonstrate that our proposed copula-based chance constrained optimization framework can incorporate demand correlation and achieve a higher joint demand satisfaction rate, lower total costs, and higher efficiency.