POP – Parametric Optimization Toolbox

Multiobjective Optimization of Mixed-Integer Linear Programming Problems: A Multiparametric Optimization Approach

Industrial process systems need to be optimized, simultaneously satisfying financial, quality and safety criteria. To meet all those potentially conflicting optimization objectives, multiobjective optimization formulations can be used to derive optimal trade-off solutions. In this work, we present a framework that provides the exact Pareto front of multiobjective mixed-integer linear optimization problems through multiparametric programming. The original multiobjective optimization program is reformulated through the well-established ϵ-constraint scalarization method, in which the vector of scalarization parameters is treated as a right-hand side uncertainty for the multiparametric program. The algorithmic procedure then derives the optimal solution of the resulting multiparametric mixed-integer linear programming problem as an affine function of the ϵ parameters, which explicitly generates the Pareto front of the multiobjective problem. The solution of a numerical example is analytically presented to exhibit the steps of the approach, while its practicality is shown through a simultaneous process and product design problem case study. Finally, the computational performance is benchmarked with case studies of varying dimensionality with respect to the number of objective functions and decision variables.

Multiparametric Programming in Process Systems Engineering: Recent Developments and Path Forward

The inevitable presence of uncertain parameters in critical applications of process optimization can lead to undesirable or infeasible solutions. For this reason, optimization under parametric uncertainty was, and continues to be a core area of research within Process Systems Engineering. Multiparametric programming is a strategy that offers a holistic perspective for the solution of this class of mathematical programming problems. Specifically, multiparametric programming theory enables the derivation of the optimal solution as a function of the uncertain parameters, explicitly revealing the impact of uncertainty in optimal decision-making. By taking advantage of such a relationship, new breakthroughs in the solution of challenging formulations with uncertainty have been created. Apart from that, researchers have utilized multiparametric programming techniques to solve deterministic classes of problems, by treating specific elements of the optimization program as uncertain parameters. In the past years, there has been a significant number of publications in the literature involving multiparametric programming. The present review article covers recent theoretical, algorithmic, and application developments in multiparametric programming. Additionally, several areas for potential contributions in this field are discussed, highlighting the benefits of multiparametric programming in future research efforts.

Integrated Data-Driven Process Monitoring and Explicit Fault-Tolerant Multiparametric Control

We propose a novel active fault-tolerant control strategy that combines machine learning based process monitoring and explicit/multiparametric model predictive control (mp-MPC). The strategy features (i) data-driven fault detection and diagnosis models by using the support vector machine (SVM) algorithm, (ii) ranking via a nonlinear, kernel-dependent, SVM-based feature selection algorithm, (iii) data-driven regression models for fault magnitude estimation via the random forest algorithm, and (iv) a parametric optimization and control (PAROC) framework for the design of the explicit/multiparametric model predictive controller. The resulting explicit control strategies correspond to affine functions of the system states and the magnitude of the detected fault. A semibatch process, an example for penicillin production, is presented to demonstrate how the proposed framework ensures smart operation for which rapid switches between a priori computed explicit control action strategies are enabled by continuous process monitoring information.