Uncertainty-aware integration of control with process operations and multi-parametric programming 9783030381363, 9783030381370

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Springer Theses Recognizing Outstanding Ph.D. Research

Vassilis M. Charitopoulos

Uncertainty-aware Integration of Control with Process Operations and Multi-parametric Programming Under Global Uncertainty

Springer Theses Recognizing Outstanding Ph.D. Research

Aims and Scope The series “Springer Theses” brings together a selection of the very best Ph.D. theses from around the world and across the physical sciences. Nominated and endorsed by two recognized specialists, each published volume has been selected for its scientific excellence and the high impact of its contents for the pertinent field of research. For greater accessibility to non-specialists, the published versions include an extended introduction, as well as a foreword by the student’s supervisor explaining the special relevance of the work for the field. As a whole, the series will provide a valuable resource both for newcomers to the research fields described, and for other scientists seeking detailed background information on special questions. Finally, it provides an accredited documentation of the valuable contributions made by today’s younger generation of scientists.

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Vassilis M. Charitopoulos

Uncertainty-aware Integration of Control with Process Operations and Multi-parametric Programming Under Global Uncertainty Doctoral Thesis accepted by University College London, London, UK

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Author Dr. Vassilis M. Charitopoulos Department of Chemical Engineering University College London London, UK

Supervisor Dr. Vivek Dua Department of Chemical Engineering University College London London, UK

ISSN 2190-5053 ISSN 2190-5061 (electronic) Springer Theses ISBN 978-3-030-38136-3 ISBN 978-3-030-38137-0 (eBook) https://doi.org/10.1007/978-3-030-38137-0 © Springer Nature Switzerland AG 2020 This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, expressed or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. This Springer imprint is published by the registered company Springer Nature Switzerland AG The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland

To my beloved mother Chrysanthi and my partner Vassilis

Supervisor’s Foreword

Vassilis’ research has focused on developing novel methods for mitigating the impact of uncertainty for optimal decision-making in process and energy systems engineering problems. The unique advances presented in this Thesis signal a major step towards the systematic treatment of uncertainty for highly interconnected industrial and energy systems that are emerging currently and in the foreseeable future. Uncertainty is ubiquitous in decision-making problems and in the face of the constantly increasing sustainability and economic considerations, the research outcomes of the present Thesis are of paramount importance. The work comprises key algorithmic developments related to the fields of process systems engineering, operations research, control engineering and decision-making under uncertainty and its timeliness and relevance is underlined by the Industry 4.0 inspired applications demonstrated within the thesis. In the first part of the thesis, novel computer algebra-based algorithms are presented for the solution of (mixed integer) optimisation problems which are susceptible to “global uncertainty” from a multi-parametric programming viewpoint. For the first time, theorems are established for the mathematical nature of the explicit solutions in multi-parametric linear programs under global uncertainty which provide transformative insights for mitigating uncertainty considerations in optimisation problems. The proposed algorithms are tested and showcased in a rich portfolio of flagship process systems engineering case studies such as energy planning, process scheduling and process design and model-based control. In the second part of thesis, the problem of integrating process planning, scheduling and control is examined and a rigorous mathematical framework for its “uncertainty-aware” solution is presented. Integrating control with operations is central to Industry 4.0 and smart manufacturing considerations. Complex, multi-scale, model-based optimisation problem is formulated and solved. It is showcased how by integrating control decisions with other operational levels, manufacturing processes can become more flexible, responsive and resilient to externalities. Using state-of-the-art chance-constrained programming and robust optimisation techniques a novel hybrid-framework is developed and applied to chemical industry case studies. The research work has been presented in six journal publications, several international conference vii

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Supervisor’s Foreword

proceedings and presentations. For his scientific excellence, Vassilis has received several awards and distinctions including the: UCL Dennis Rooke Prize (2016), UCL David’s Newton Prize (2019) and he has been selected as one of the best young researchers, globally, by the Institute of Chemical Engineers (2018;2019). London, UK October 2019

Dr. Vivek Dua

Abstract

Advanced decision-making in the process industries requires efficient use of information available at the different hierarchical levels. However, given the time and decision space sparsity, the consideration of such integrated problems poses a plethora of challenges. It is the goal of the present thesis to present some recent algorithmic and modelling developments with special focus on the uncertaintyaware integration of control with process operations. To this end, the thesis comprises two parts that are orchestrated towards the attainment of the aforestated goal. The first part discusses theoretical and algorithmic advances in the field of multi-parametric programming. Initially, the case of multi-parametric linear programs under simultaneous variations in the left-hand side, right-hand side of the constraints and the objective function’s coefficients is examined. For the first time theoretical characterisation of the explicit solution is proven while an algorithm for their exact computation is proposed. Later on, the aforementioned algorithm is extended to the mixed-integer case where problems of process synthesis and scheduling under global uncertainty are studied. Next, the concept of multi-setpoint explicit controllers and their potential in the context of enterprise-wide optimisation problems is introduced. While a prototype implementation of the aforementioned works is also discussed. The second part is dedicated to the development of a systematic framework for the uncertainty-aware integration of process planning, scheduling and control (iPSC) of continuous processes. Initially, a Traveling Salesman Problem based formulation is presented and a decomposition method for the deterministic case is proposed. Next, the multi-setpoint explicit controllers developed in the first part of the thesis, enable the development of a reactive closed-loop framework for the iPSC. Ultimately, proactive and reactive approaches are employed in order to instantiate the uncertainty aware iPSC while Monte Carlo simulations are conducted to evaluate the robustness of the proposed framework.

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Preface

Advances in computing power along with the widespread availability of data have led process industries to consider a new paradigm for automated and more efficient operations. This trend is commonly referred to as “Industry 4.0” and many reports indicate that it can result in cost reductions of up to 20% while improving asset utilisation by 30–40%. Despite the research effort dedicated to techniques for the interpretation and efficient employment of big-data within the industry, the development of mathematical modelling and optimisation techniques that serve the requirements for seamless communication among the different levels of decision making has received considerably less attention. McKinsey Global Institute estimates that operational optimisation within the Industry 4.0 can generate up to $3.7T of value in 2025. It is the goal of this project to present models and methodologies that can be employed towards the realisation of the Industry 4.0 vision within the process industries while at the same time investigate the impact of uncertainties in such highly integrated settings. Firstly, the algorithms presented in Part I of the thesis for the multi-parametric solution of (mixed integer) linear programs, can aid the study of problems of process synthesis, scheduling and planning of energy systems under uncertainty. The practical value of these algorithms lies in the computation and exact characterisation of the effect the uncertain variations have on the optimal solution. Novel model based control concepts are crucial for smart manufacturing and thus the multi-setpoint controllers that are presented in the thesis is of paramount importance. Case studies showcase that the solution times of the related optimisation problems can be reduced by 3–4 orders of magnitude compared to conventional approaches allowing thus for their online use. Secondly, the model and frameworks that are presented in Part II of the thesis on the uncertainty-aware integration of planning, scheduling and control (iPSC) constitute the first systematic study on the topic. The proposed framework can be applied to a wide variety of process industries ranging from the petrochemical to the biopharmaceutical sector. A recent report from the UK government indicates that manufacturing as of 2016 accounts for 10% of the UK’s total economic output (*£177 billion) compared to 17.4% in the 90s. Automation improvements were xi

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Preface

highlighted as the limiting factor in the manufacturing sector’s growth. The presented frameworks enable automation improvements that can lead to at least 4–10% overall profit increase, depending on the market volatility, as indicated by the case studies. A decomposition of the integrated problem that yields improved and operationally optimal solutions in fast computational times is proposed and is compared to the monolithic model formulation. The importance and practical applicability of the proposed framework is further underlined under the presence of uncertain operations where the “uncertainty-aware” solutions offer operational decisions which are more close to the reality when compared to the deterministic case. London, UK October 2019

Dr. Vassilis M. Charitopoulos

Parts of this thesis have been published in the following journal articles: 1. V. M. Charitopoulos, L. G. Papageorgiou, and V. Dua. Multi-parametric linear programming under global uncertainty. AIChE J., 63(9):3871–3895, 2017. 2. V. M. Charitopoulos, V. Dua, and L. G. Papageorgiou. Traveling salesman problem-based integration of planning, scheduling, and optimal control for continuous processes. Ind. Eng. Chem. Res., 56(39):11186–11205, 2017. 3. V. M. Charitopoulos, V. Dua, and L. G. Papageorgiou. Closed loop integration of planning, scheduling and control via exact multi-parametric nonlinear programming. Comput. Aid. Chem. Eng., 40:1273–1278, 2017. 4. V. M. Charitopoulos, L. G. Papageorgiou, and V. Dua. Multi-parametric mixed integer linear programming under global uncertainty. Comput. Chem. Eng., 116:279–295, 2018. 5. V. M. Charitopoulos, V. Dua, and L. G. Papageorgiou. Uncertainty aware integration of planning, scheduling and multi-parametric control. Comput. Aid. Chem. Eng., 44:1171–1176, 2018. 6. V. M. Charitopoulos, L. G. Papageorgiou and V. Dua. Closed-loop integration of planning, scheduling and multi-parametric nonlinear control. Comput. Chem. Eng., 122:172–192, 2019.

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Acknowledgements

I would like to thank both of my supervisors, Dr. Vivek Dua and Prof. Lazaros Papageorgiou for giving me the opportunity to embark on this exciting research journey. My introduction to the world of optimisation and process systems will always be due to Vivek, whose regular advice and thought-provoking comments have played a tremendous role in my evolution as a researcher. Thank you for all the opportunities, the patience and trust that you showed me all these years. My deepest gratitude also goes to Lazaros for his support in my research endeavors, the time he devoted to widen my horizons and knowledge on research topics as well as the opportunities to work on exciting projects throughout the course of this Ph.D. Your devotion in conducting impactful multi-disciplinary research together with your ethical conduct have been definitely one the greatest inspirations for me. I would like to express my gratitude to my mother for all the personal sacrifices she had to make to provide me with good education and a chance for a better living but most of all for the ethics she instilled in me. Stella and Marilena your existence in my life has been pivotal and if it was not for many of our discussions I probably would have not gone towards this direction in my life. Some people argue that a Ph.D. is not so about acumen but more about endurance and discipline to operate under pressure and in a sense a test of your limits. If it was not for my dearest friends I probably would not have reached the finishing line. Finally and possibly the most importantly, I would have to thank Vassilis for his endless support, love and tolerance that he had to show throughout these three years, you probably deserve this Ph.D. as much as I do. Thank you for being my significant other in life. Love you to Andromeda and back!

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Contents

1 Thesis Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1 Introduction and Research Questions . . . . . . . . . . . . . . . . . . . . . . 1.2 Thesis Scope and Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Part I

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Theoretical and Algorithmic Advances in Multi-parametric Programming Problems Under Global Uncertainty

2 Parametric Optimisation: 65 years of developments and status quo . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 The Early Days of Parametric Optimisation: 1952–1967 . . . . . 2.3 Dawn of Multi-parametric Optimisation, From Sensitivity Analysis to Parametric Nonlinear Programming and Beyond: 1968–1993 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.1 From Parametric to Multi-parametric Linear Programming . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.2 From Sensitivity Analysis to Parametric Nonlinear Programming . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.3 Sensitivity Analysis and Parametric Integer Programming . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4 Process Systems Engineering Led Multi-parametric Optimisation: 1993–Present . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4.1 Multi-parametric (Mixed Integer) Linear Programming 2.4.2 Multi-parametric (Mixed Integer) Quadratic Programming . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4.3 Multi-parametric Linear Complementarity Problems . . . 2.4.4 Multi-parametric (Mixed Integer) Nonlinear Programming . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4.5 Convex Problems . . . . . . . . . . . . . . . . . . . . . . . . . . .

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2.4.6 Non-convex Problems . . . . . . . . . . . . . . . . . . . . . . . . 2.4.7 Multi-parametric Dynamic Optimisation . . . . . . . . . . . 2.5 Major Application Areas . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5.1 Multi-parametric Multi-level Optimisation . . . . . . . . . . 2.5.2 Multi-parametric Model Predictive Control . . . . . . . . . 2.5.3 Enterprise Wide Optimisation Under Uncertainty . . . . . 2.5.4 Multi-objective Optimisation . . . . . . . . . . . . . . . . . . . 2.5.5 Other Application Areas . . . . . . . . . . . . . . . . . . . . . . . 2.6 Complexity Issues in Multi-parametric Optimisation Problems 2.6.1 Degeneracy Implications in mp-LPs/mp-QPs . . . . . . . . 2.6.2 Computational Complexity of mp-P Algorithms and Reduction Strategies . . . . . . . . . . . . . . . . . . . . . . 2.7 Software Implementations . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.8 Conclusions—Quo vandis? . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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4 Towards Exact Multi-setpoint Explicit Controllers for Enterprise Wide Optimisation . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Multi-parametric Algorithms and Explicit MPC . . . . . . . . . . . . . .

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3 Multi-parametric Linear and Mixed Integer Linear Programming Under Global Uncertainty . . . . . . . . . . . . . . . . . . 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Global Uncertainty in Multi-parametric Linear Programs . . . . . 3.2.1 A Computer Algebra Inspired Algorithm . . . . . . . . . . . 3.2.2 Cylindrical Algebraic Decomposition and Comparison Procedure for Overlapping CRs . . . . . . . . . . . . . . . . . 3.3 Global Uncertainty in General mp-MILPs . . . . . . . . . . . . . . . 3.4 Computational Studies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4.1 Parametric Linear Program with LHS Unbounded Parameter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4.2 Discontinuous Mp-LP with LHS Uncertainty . . . . . . . 3.4.3 Mp-LP Under Global Uncertainty . . . . . . . . . . . . . . . . 3.4.4 Mp-MILP Under Global Uncertainty . . . . . . . . . . . . . 3.4.5 Case Study 1: Process Synthesis Under Global Uncertainty . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4.6 Case Study 2: Process Scheduling Under Global Uncertainty . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.5 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.5.1 Scalability of the Proposed Algorithm . . . . . . . . . . . . . 3.5.2 Non-convexity of the Underlying Problem . . . . . . . . . 3.6 Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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4.2.1 Multi-setpoint Explicit Controllers via Multi-parametric Programming . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.2 An Algorithm for the Global Solution of mp-NLPs . . . . 4.2.3 Motivating Example . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3 Computational Studies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3.1 Single-Input Single-Output Isothermal CSTR . . . . . . . . 4.3.2 Methyl-Methacrylate Polymerisation Isothermal CSTR . . 4.3.3 Overall Scalability of the Algorithm . . . . . . . . . . . . . . . 4.4 Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Part II

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Uncertainty-Aware Integration of Planning, Scheduling and Control

5 Open-Loop Integration of Planning, Scheduling and Optimal Control: Overview, Challenges and Model Formulations . . . . . 5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1.1 Integration of Process Operations with Control . . . . . 5.1.2 Motivation and Problem Statement . . . . . . . . . . . . . . 5.2 Methodology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2.1 Objective of the iPSC . . . . . . . . . . . . . . . . . . . . . . . 5.2.2 Modelling the Planning and Scheduling Problem . . . . 5.2.3 Dynamic Optimisation (DO) . . . . . . . . . . . . . . . . . . . 5.2.4 Linking Variables Between DO, Scheduling and Planning Model . . . . . . . . . . . . . . . . . . . . . . . . . 5.2.5 Linking Equations Between DO and TSP Planning and Scheduling . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2.6 Monolithic and Decomposed Integration of Planning, Scheduling and optimal Control . . . . . . . . . . . . . . . . 5.3 Case Studies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3.1 SISO Multi-product CSTR . . . . . . . . . . . . . . . . . . . . 5.3.2 MIMO Multi-product CSTR . . . . . . . . . . . . . . . . . . . 5.3.3 MMA Polymerisation Process . . . . . . . . . . . . . . . . . 5.4 Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 Closed-Loop Integration of Planning, Scheduling and Multi-parametric Nonlinear Control . . . . . . . . . . . . . 6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2 Literature Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3 Mathematical Formulations . . . . . . . . . . . . . . . . . . . . . 6.3.1 Modelling the Closed-Loop Integrated Planning, Scheduling and Control . . . . . . . . . . . . . . . . . . 6.3.2 The Overall Closed-Loop Integrated Framework

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6.4 Case Studies . . . . . . . . . . . . . . . . . . . . . . 6.4.1 Single Input Single Output CSTR . 6.4.2 MMA Polymerisation Reactor . . . 6.5 Concluding Remarks . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . .

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7 A Hybrid Framework for the Uncertainty-Aware Integration of Planning, Scheduling and Explicit Control . . . . . . . . . . . . . . . . 7.1 Introduction and Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2 Modelling and Optimisation with Uncertainty Considerations . . 7.2.1 Robust Optimisation . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2.2 Stochastic Programming . . . . . . . . . . . . . . . . . . . . . . . . 7.2.3 Chance Constrained Programming . . . . . . . . . . . . . . . . 7.3 Mathematical Formulations . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.3.1 Rolling Horizon Strategy . . . . . . . . . . . . . . . . . . . . . . . 7.3.2 Robust Optimisation for Process Planning . . . . . . . . . . . 7.3.3 Chance-Constrained Process Scheduling . . . . . . . . . . . . 7.3.4 Overall Uncertainty-Aware iPSC Model . . . . . . . . . . . . 7.4 Computational Studies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.4.1 Problem Instance Classes and Tuning of the Proactive Module . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.4.2 Monte Carlo Simulations for Evaluation of the Proactive Module of the Uncertainty-Aware iPSC . . . . . . . . . . . . 7.5 Results and Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.5.1 Solution Quality of the Rolling Horizon Approach . . . . 7.5.2 Problem Class 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.5.3 Problem Class 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.5.4 Uncertainty Aware iPSC of MMA Polymerisation Process . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.6 Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 Conclusions and Future Work . . . . . . . . . . . . . . . . . . 8.1 Multi-parametric Programming Theory, Algorithms and Applications . . . . . . . . . . . . . . . . . . . . . . . . . . 8.1.1 Contributions . . . . . . . . . . . . . . . . . . . . . . . 8.1.2 Perspectives for Future Research . . . . . . . . 8.2 Integration of Control with Process Operations Under Uncertainty . . . . . . . . . . . . . . . . . . . . . . . . 8.2.1 Contributions . . . . . . . . . . . . . . . . . . . . . . . 8.2.2 Perspectives for Future Research . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Contents

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Appendix A: Background Material . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 225 Appendix B: PERSEUS: A Prototype Software Implementation . . . . . . . 237 Appendix C: Multi-parametric Linear and Mixed Integer Linear Programming Under Global Uncertainty: Further Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 247 Appendix D: Open-Loop Integration of Planning, Scheduling and Optimal Control: Further Results . . . . . . . . . . . . . . . . 255 Appendix E: Detailed iPSC Model for the Rolling Horizon Solution Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 265

List of Figures

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Fig. 2.3 Fig. 2.4 Fig. 3.1 Fig. 3.2 Fig. 3.3 Fig. 3.4 Fig. 3.5 Fig. 3.6

Fig. 3.7 Fig. 3.8 Fig. 3.9

Pictorial representation of the underlying interconnection in the integration of planning, scheduling and control . . . . . . Conceptual organisation and thesis outline . . . . . . . . . . . . . . . Graphical conceptualisation of the explicit solution of a (multi-)parametric programming problem . . . . . . . . . . . . Taxonomy of multi-parametric programming problem classes and the main related solution algorithms. Depending on whether the set of discrete variables (Y) of the mp-P is empty different strategies are followed . . . . . . Conceptual illustration of the “online via offline optimisation” framework . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Overview of multi-parametric programming problem classes and their indicative applications . . . . . . . . . . . . . . . . . . . . . . . Visual representation of the different cases where the CRINT can be identified . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . CR1 , CR2 and CRINT in the parametric space. . . . . . . . . . . . . Final non-overlapping CRs in the parametric space . . . . . . . . Outline of the proposed algorithm for mp-LP under global uncertainty . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Graphical definition of CR2 of parametric linear program with LHS unbounded parameter . . . . . . . . . . . . . . . . . . . . . . . . . . . Plot of the explicit objective function and the CRs in the parametric space of parametric linear program with LHS unbounded parameter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Visual representation of the final results for discontinuous mp-LP with LHS uncertainty . . . . . . . . . . . . . . . . . . . . . . . . . Separate Critical Regions of the mp-LP under global uncertainty example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Visualisation of the optimal partition of the parametric space for the mp-LP under global uncertainty example . . . . . . . . . .

.. ..

3 6

..

10

..

25

..

28

..

31

.. .. ..

52 53 57

..

58

..

67

..

67

..

72

..

74

..

75

xxiii

xxiv

Fig. 3.10 Fig. 3.11 Fig. 3.12 Fig. 4.1 Fig. 4.2 Fig. 4.3

Fig. 4.4 Fig. 4.5 Fig. 4.6

Fig. 4.7 Fig. 4.8

Fig. 4.9 Fig. 4.10 Fig. 4.11 Fig. 4.12

Fig. 4.13 Fig. 5.1 Fig. 5.2

Fig. 5.3 Fig. 5.4 Fig. 5.5

List of Figures

Visualisation of the different partitions of the parametric space for mp-MILP under global uncertainty example . . . . . . . . . . . Superstructure of process synthesis under global uncertainty case study . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 stage process for scheduling . . . . . . . . . . . . . . . . . . . . . . . . Interdependent levels of decision making with APC within the EWO scope . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Conceptual block diagram of functionalities involved between APC and RTO . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Conceptual representation of a multi-setpoint explicit controller (on the right) in comparison to the conventional paradigm of designing multiple explicit controllers for different setpoints (on the left) . . . . . . . . . . . . . . . . . . . . . . . . Overlapping CRs in motivating mp-NLP under global uncertainty example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Final optimal partition of the parametric space of the motivating mp-NLP under global uncertainty example . . . . . . SISO CSTR production scheme, the manipulated variable is the volumetric flowrate of the liquid ðQR Þ while the state variable is the concentration of the reactant (CR ) . . . . . . . . . . Explicit results for the SISO CSTR case study for PH ¼ 1 . . . Comparative plots of the solution computed by the multi-setpoint explicit controller versus online MPC for the SISO CSTR case study . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Final CRs of the SISO CSTR case study for PH ¼ 2 . . . . . . . Performance of the multi-setpoint explicit MPC of the SISO CSTR with PH ¼ 2 under disturbance rejection . . . . . . Plot of the optimal objective value against time for the MMA CSTR case study . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Comparative plots of the solution computed by the multi-setpoint explicit controller versus online MPC for the MMA CSTR case study . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Multi-setpoint tracking instance of the MMA polymerisation case study . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ISA 95 automation pyramid . . . . . . . . . . . . . . . . . . . . . . . . . . Conceptual visualisation of an arbitrary scheduling instance and the calculation of raw material cost during changeover periods (A1 ) and production periods (A2 ) . . . . . . . . . . . . . . . . Hybrid time representation of the TSP model for the levels of planning and scheduling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Different timescales involved in the iPSC problem and their interdependence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Flow of information across the different levels of decision making within the iPSC . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

..

77

.. ..

78 82

..

90

..

91

..

93

. . 100 . . 101

. . 103 . . 105

. . 106 . . 109 . . 110 . . 115

. . 116 . . 117 . . 124

. . 129 . . 130 . . 137 . . 140

List of Figures

Fig. 5.6 Fig. 5.7 Fig. 5.8 Fig. 5.9

Fig. 5.10 Fig. 6.1 Fig. 6.2 Fig. 6.3 Fig. 6.4 Fig. 6.5 Fig. 6.6

Fig. 6.7 Fig. 6.8

Fig. 6.9 Fig. 6.10 Fig. 6.11 Fig. 6.12 Fig. 6.13 Fig. 7.1

Graph of the linear metamodels built from sampling data for the transition from C to A . . . . . . . . . . . . . . . . . . . . . . . . . . . Dynamic interpretation of the production plan for the SISO CSTR case study . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Dynamic response of the system for the transition from B to F . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Comparative graph of the coolant flowrate (Fc ¼ fðtÞ) as computed by the decomposed and the monolithic solution approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Dynamics interpretation of the optimal production schedule for the first planning period of the MMA case study . . . . . . . Enterprise wide optimisation scope for process industries . . . Conceptual representation of the integrated planning, scheduling and control problem . . . . . . . . . . . . . . . . . . . . . . . Conceptual instance of product duplication . . . . . . . . . . . . . . Numerical integration of arbitrary transition curve via Simpson’s rule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Flowchart of the proposed closed loop implementation of the iPSC for continuous manufacturing processes . . . . . . . Comparison of the closed loop behavior of the state of the SISO CSTR (CR ) using explicit MPC (blue continuous line) and conventional MPC (red dashed line) for prediction horizon of unity (1st planning period) . . . . . . . . . . . . . . . . . . . . . . . . . Comparative plot of the rescheduled and nominal solution due to state deviation during the transition period . . . . . . . . . Gantt charts of the closed-loop iPSC for the SISO CSTR case study. The horizontal axis represent the time (h) and each week spans across 168 h with changeovers indicated using the “Y” label . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . CR ¼ fðtÞ plot indicative of the system’s dynamics over the entire planning horizon . . . . . . . . . . . . . . . . . . . . . . . Gantt charts of all the reschedules for the MMA case study where the gray blocks represent changeover periods . . . . . . . Y ¼ fðtÞ dynamic plots indicative of the system’s dynamics over the entire planning horizon for the MMA case study . . . Y ¼ fðtÞ dynamic plot indicating all the rescheduled actions across the planning horizon for the MMA case study. . . . . . . Visual representation of the effect of the rescheduling mechanism on the economic performance of the iPSC . . . . . . Rolling horizon strategy under demand uncertainty . . . . . . . .

xxv

. . 145 . . 148 . . 151

. . 152 . . 156 . . 160 . . 161 . . 163 . . 164 . . 166

. . 171 . . 173

. . 174 . . 175 . . 178 . . 179 . . 180 . . 181 . . 192

xxvi

Fig. 7.2

Fig. 7.3

Fig. 7.4

Fig. 7.5

Fig. 7.6

Fig. 7.7

Fig. 7.8

Fig. 7.9

Fig. 7.10

Fig. A.1 Fig. A.2

List of Figures

Cumulative level of protection against demand uncertainty for different assumed budgets. As illustrated the case of C ¼ p is the most conservative followed by C ¼ 0:5p and the least risk-averse C ¼ 0:1p þ 0:5 . . . . . . . . . . . . . . . . . . . . . . . . . . Overall methodology of quantifying the expected value of the resulting solution by the deterministic and the hybrid iPSC model through Monte Carlo simulation . . . . . . . . . . . . . . . . . Gantt charts of the different solutions computed with varying risk-averseness by employing the hybrid framework and the deterministic approach for the iPSC for problem class 1. The blocks filled with the color of label “Y” indicate changeover and idle times in the planning periods . . . . . . . . . . . . . . . . . . Problem Class 1: Monte Carlo simulation results for different assumed distributions and budgets of robust optimisation. The dotted lines in the plots indicate different realisations of the objective value (cost) across the MC simulations while the solid lines indicate the value computed by the a priori optimisation of the problem . . . . . . . . . . . . . . . . . . . . . . . . . . Histogram of the normalised random deviations for the demand of a product following the uniform, Gamma and Cauchy distributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Gantt charts of the different solutions computed with varying risk-averseness by employing the hybrid framework and the deterministic approach for the iPSC of problem class 2. The blocks filled with the color of label “Y” indicate changeover and idle times in the planning periods . . . . . . . . . . . . . . . . . . Problem Class 2: Monte Carlo simulation results for different assumed distributions and budgets of robust optimisation. The dotted lines in the plots indicate different realisations of the objective value (cost) across the MC simulations while the solid lines indicate the value computed by the a priori optimisation of the problem . . . . . . . . . . . . . . . . . . . . . . . . . . Representation of the effect of the degree of risk-averseness on the overall levels of backlog and inventory for the different problem classes of the case study . . . . . . . . . . . . . . . . . . . . . . Overview of the closed-loop response of the underlying dynamic system for the uncertainty-aware iPSC of the MMA polymerisation process . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Hierarchy of algebraic fields, rings and groups . . . . . . . . . . . . Stack over interval. The green lines represent the graph of the different functions that form three distinct non-intersecting sections. The cylinder over the interval (a; b) is partitioned into 4 two-dimensional cells, the sectors of the functions, and 3 one-dimensional cells, the sections of the functions . . . . . . . .

. . 201

. . 202

. . 205

. . 207

. . 208

. . 209

. . 211

. . 212

. . 213 . . 226

. . 233

List of Figures

Fig. Fig. Fig. Fig.

A.3 B.1 B.2 C.1

Fig. D.1

Main algorithmic steps of the CAD algorithm . . . . . . . . . . . . PERSEUS structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . PERSEUS’ graphical user interface . . . . . . . . . . . . . . . . . . . . Optimal partition of the parametric space for the mp-MILP under global uncertainty example . . . . . . . . . . . . . . . . . . . . . . Transition profiles from E to D . . . . . . . . . . . . . . . . . . . . . . .

xxvii

. . 234 . . 242 . . 244 . . 247 . . 255

List of Tables

Table 2.1 Table 2.2 Table 2.3 Table 3.1

Table 3.2 Table 3.4 Table 3.5 Table 3.6 Table 3.7 Table 3.8 Table 3.9 Table 3.10

Table 3.11

Indicative list of multi-parametric programming application in enterprise wide optimisation problems. . . . . . . . . . . . . . . Overview of complexity reduction techniques in multi-parametric programming problems . . . . . . . . . . . . . . . Overview of the available software implementations for the solution of multi-parametric programs . . . . . . . . . . . Summary of mp-MILP algorithms, the uncertainty classes that can be handled and the resulting number of explicit solutions per CR . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Possible outcomes in the definition of CRINT . . . . . . . . . . . Candidate solutions of parametric linear program with LHS unbounded parameter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Explicit solution of parametric linear program with LHS unbounded parameter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Explicit solutions and CRs of discontinuous mp-LP with LHS uncertainty . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Explicit solution of mp-LP under global uncertainty example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Explicit solutions of mp-MILP under global uncertainty example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Explicit results of process synthesis under global uncertainty case study . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Effect of the dimensionality of the uncertain parameters (nh ) on the number of CRs computed (nCR ) and the related solution time . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Multi-parametric expressions of the weighted sum objective function of the three stage scheduling problem along with the related sequencing decisions . . . . . . . . . . . . . . . . . . . . .

..

29

..

33

..

34

.. ..

48 51

..

65

..

66

..

71

..

73

..

76

..

80

..

81

..

83

xxix

xxx

Table 3.12

Table 4.1 Table 4.2 Table 4.3 Table 4.4 Table 4.5 Table 4.6 Table 4.7 Table 4.8 Table 4.9 Table 4.10 Table 4.11 Table 4.12

Table 4.13 Table 4.14 Table 4.15

Table 5.1 Table 5.2 Table 5.3

Table 5.4 Table 5.5

Table 5.6

List of Tables

Computational statistics of the proposed algorithm with respect to the dimensionality of the inequality constraints (ng ), continuous variables (nx ), binary variables (ny ) and uncertain parameters (nh ) . . . . . . . . . . . . . . . . . . . . . . . . . . . Overview of mp-NLP algorithms . . . . . . . . . . . . . . . . . . . . Candidate solutions of the motivating mp-NLP under global uncertainty example (I) . . . . . . . . . . . . . . . . . . . . . . . . . . . . Candidate solutions of the motivating mp-NLP under global uncertainty example (II) . . . . . . . . . . . . . . . . . . . . . . . . . . . Overview of final optimal explicit solutions of the motivating mp-NLP under global uncertainty example . . . . Data of SISO CSTR case study . . . . . . . . . . . . . . . . . . . . . . Candidate solutions of the SISO CSTR mp-MPC . . . . . . . . Final CRs and explicit solutions (control law and state evolution) for PH ¼ 1 of the SISO CSTR case study . . . . . Final explicit optimal solutions for PH ¼ 2 for the SISO CSTR case study . . . . . . . . . . . . . . . . . . . . . . . . . . . . CRs of SISO CSTR for PH ¼ 2 . . . . . . . . . . . . . . . . . . . . . MMA polymerisation reactor case study nomenclature . . . . Model parameters for the MMA polymerisation reactor case study . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Computational study of CPU times (s) required for different prediction horizons using BARON 14.4 as global optimisation solver . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Uncertain parameters of mp-MPC for MMA . . . . . . . . . . . . Candidate solutions for explicit controller of the MMA case study for PH ¼ 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Computational statistics of the proposed method for the design of multi-setpoint explicit controllers based on different systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Demand and cost data for the SISO CSTR case study . . . . Kinetic data for the SISO CSTR case study . . . . . . . . . . . . Problem statistics for the offline computation of the minimum transition time (smin i;j ) for the SISO CSTR case study . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Minimum transition times between products for the SISO CSTR case study . . . . . . . . . . . . . . . . . . . . . . . . . . . . Results for the SISO CSTR case study for planning horizon of 2 weeks. OCFE with 20 finite elements and 3 collocation points was employed for the discretisation of the DO part . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Results of the SISO CSTR case study with fixed sequencing decisions from the decomposed iPSC . . . . . . . . . . . . . . . . .

.. ..

84 92

..

98

..

99

. . 101 . . 102 . . 103 . . 104 . . 107 . . 108 . . 111 . . 111

. . 111 . . 112 . . 112

. . 117 . . 143 . . 144

. . 144 . . 144

. . 146 . . 147

List of Tables

Table 5.7 Table 5.8 Table 5.9 Table 5.10 Table 5.11

Table Table Table Table

5.12 5.13 5.14 5.15

Table 5.16 Table 6.1

Table 6.2 Table 6.3 Table 6.4 Table 7.1

Table 7.2

Table 7.3

Table C.1 Table C.2

xxxi

Comparative cost breakdown of iPSC solution for the SISO case study . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Minimum transition times for the SISO CSTR case study with OCFE of 45 finite elements and 3 collocation points . Design parameters of the MIMO multi-product CSTR case study . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Minimum transition times between products for the MIMO case study . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Results for the MIMO CSTR case study for planning horizon of 2 weeks. OCFE with 20 finite elements and 3 collocation points was employed for the discretisation of the DO part . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Data for MMA case study . . . . . . . . . . . . . . . . . . . . . . . . . . Steady state operating conditions for the MMA products . . Minimum transition times for the MMA case study . . . . . . Results of the computational performance between the monolithic and the decomposed integrated model using the proposed TSP and the time-slot (T-S) based formulation for the MMA case study . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Comparison between the decomposed TSP and T-S formulation for varying planning horizons. . . . . . . . . . . . . . Comparative results for the solutions computed by the closed loop iPSC and the open loop iPSC for the case of state deviation during transition period . . . . . . . . . . . . . . . . Steady state information about the different polymer grades of the MMA CSTR case study . . . . . . . . . . . . . . . . . . . . . . Economic data of the MMA case study . . . . . . . . . . . . . . . Overview of the impact of the rescheduling mechanism on different factors of the iPSC . . . . . . . . . . . . . . . . . . . . . . Rolling horizon iterations explained along with the related tuning parameters. The color coordination of the planning blocks in the graphical illustration has as follows: control horizon (CH): orange, relaxed/prediction horizon (PH): blue and fixed periods (FP): light blue . . . . . . . . . . . . . . . . . . . . Problem class 1: Computational statistics of the solutions computed by the full-space approach (FS) compared the ones by the different variants of RH strategy . . . . . . . . . . . Computational statistics with regards to the characteristics of the different solutions and problem classes examined through the MC simulations . . . . . . . . . . . . . . . . . . . . . . . . Critical regions of the mp-MILP under global uncertainty example (i) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Critical regions of the mp-MILP under global uncertainty example (ii) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . 147 . . 148 . . 150 . . 150

. . . .

. . . .

151 153 154 154

. . 155 . . 157

. . 173 . . 176 . . 176 . . 181

. . 193

. . 203

. . 206 . . 249 . . 250

xxxii

Table C.3 Table C.4 Table C.5 Table C.6 Table C.7 Table D.1

Table D.2

Table D.3

Table D.4 Table D.5 Table D.6

Table D.7 Table D.8

Table D.9 Table D.10

Table D.11 Table D.12 Table D.13 Table D.14

List of Tables

Critical regions of the mp-MILP under global uncertainty example (iii) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Processing times (h) of the scheduling case studies . . . . . . . Explicit results of the 2-stage scheduling case study with 3 products . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Explicit results of the 2-stage scheduling case study with 5 products . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Explicit results of the 3-stage scheduling case study with 4 products . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Slope and intercept coefficients of the linear metamodels correlating transition cost and transition time, i.e. CTi;j ¼ aTTi;j þ b, for the SISO CSTR case study . . . . . . . Results of SISO CSTR case study for the case that OCFE is employed with 45 finite elements and 3 collocation points . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Comparison between the decomposed TSP and T-S formulation for varying planning horizons for the SISO CSTR case study . . . . . . . . . . . . . . . . . . . . . . . . . . . . Economic data for the MIMO CSTR case study . . . . . . . . . Kinetic data for the MIMO CSTR case study . . . . . . . . . . . Slope and intercept coefficients of the linear metamodels correlating transition cost and transition time, i.e. CTi;j ¼ aTTi;j þ b, for the MIMO CSTR case study . . . . . . Demand data for the decomposed iPSC of the MIMO CSTR case study . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Comparison between the decomposed TSP and T-S formulation for varying planning horizons for the MIMO CSTR case study . . . . . . . . . . . . . . . . . . . . . . . . . . . Demand and cost data for the MMA case study . . . . . . . . . Slope and intercept coefficients of the linear metamodels correlating transition cost and transition time, i.e. CTi;j ¼ aTTi;j þ b, for the MMA case study . . . . . . . . . . . . Problem statistics for the offline computation of the minimum transition time (smin i;j ) for the MMA case study . . Results of the decomposed iPSC for the MMA case study . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Results of the monolithic iPSC for the MMA case study . . Comparison between the decomposed TSP and T-S formulation for the multiple customers case . . . . . . . . . . . .

. . 251 . . 252 . . 252 . . 252 . . 253

. . 256

. . 256

. . 257 . . 257 . . 258

. . 258 . . 259

. . 259 . . 259

. . 260 . . 260 . . 260 . . 261 . . 261

List of Algorithms

Algorithm 3.1 Algorithm 3.2 Algorithm 3.3 Algorithm 4.1 Algorithm Algorithm Algorithm Algorithm

6.1 6.2 6.3 7.1

Algorithm 8.1 Algorithm C.1

Comparison procedure for overlapping CRs which returns the final list of non-overlapping CRs (CRfin) . . . Algorithm for global mp-LPs . . . . . . . . . . . . . . . . . . . . . Algorithm for global mp-MILPs . . . . . . . . . . . . . . . . . . . Algorithm for explicit solution of mp-NLPs under global uncertainty . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Closed-loop iPSC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Rescheduling routine for production disruption . . . . . . . Rescheduling routine for transition disruption . . . . . . . . Rolling horizon algorithm for the approximate solution of uncertainty-aware iPSC . . . . . . . . . . . . . . . . . . . . . . . Sequential logic decomposition and CAD computations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Algorithm for mp-MILPs under global uncertainty . . . . .

.. .. .. . . . .

55 59 64

. 96 . 167 . 169 . 170

. . 194 . . 221 . . 248

xxxiii

Chapter 1

Thesis Background

Everything has been thought of before, but the problem is to think of it again. –Johann Wolfgang von Goethe

1.1 Introduction and Research Questions The process manufacturing industry is confronted nowadays, more than ever before, with several factors that threaten its economic stability, such as market globalisation, fluctuation in the products’ demand and increased cost of raw material, to name a few. Under the light of this ever-increasingly uncertain socioeconomic environment, the need to safeguard process profitability, sustainability and reliability becomes progressively urgent. The field of Process Systems Engineering has always been keen on addressing those concerns with the development of systematic and rigorous decision making tools that rely on the principles of mathematical programming and modelling. Traditionally, the decision making in the process industries is conducted in a hierarchical manner where decisions are made in a sequential and decoupled way. To deal with the aforementioned dynamic environment, chemical industries shift their design and operations towards a more integrated paradigm. Integration of the different layers of decision making is based on the principle that there exists a strong interdependence between the different problems and the sequential approach, despite leading to more tractable optimisation problems, might result in suboptimal or even infeasible solutions from a practical standpoint. Supply chain, process planning and scheduling, real time optimisation and advanced process control form the key levels of decision making typically encountered in a process industry. Each of these layers is concerned with different sets of decisions which span vastly different time scales. For example, supply chain decisions are strategic in principle and need to be examined with time horizons of several years ahead while process scheduling decisions are mostly operational and are considered on an hourly/daily basis. Due to the different time scales under consideration, the integration tends to © Springer Nature Switzerland AG 2020 V. M. Charitopoulos, Uncertainty-aware Integration of Control with Process Operations and Multi-parametric Programming Under Global Uncertainty, Springer Theses, https://doi.org/10.1007/978-3-030-38137-0_1

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lead to intractable optimisation problems. Recent developments and technological breakthroughs, such as Internet of Things and Big Data, point towards the dawn of a new era. In the future industry, or Industry 4.0, seamless communication among the devices across all the sites of a manufacturing company could be achieved thus allowing instant access and exploitation of data that in the current state-of-the-art are not taken into consideration within the decision making process. This thesis is concerned with the development of a framework that follows closely the Industry 4.0 paradigm. More specifically, the integration of process planning, scheduling and control (iPSC) is examined with the ultimate goal of creating an uncertainty-aware framework for their integration. Mathematically, the multiscale and interconnected nature of the underlying optimisation problem is given by Eq. (1.1). TP TS TC min iPSC =

xP , xS , xC

L(xP , xS , xC , θ)dt dτ dt t=0 τ =0 t=0

Subject to : dxP = dt dxS = dτ dxC = dt xP , xS ,

f P (xP , d P , xS , d S , xC , d C , θ P )

(1.1)

f S (xS , d S , xC , d C , θ S ) f C (xC , d C , θ C ) xC ∈ P × S × C

θ = θP × θC × θS where xP , xS , xC denote the decision variables of planning, scheduling and control respectively which encompass both discrete, i.e. 0–1, and continuous decisions. Moreover, their related admissible sets are indicated by P × S × C while its level involves its own set of parameter data, i.e. d P , d S , d C . As indicated by Eq. (1.1), the integrated problem involves multiple time scales (t, τ, t) which are related to one another and within the scope of the integrated problem the decisions of the upper levels are dependent on the previous level’s decisions through a number of functionals, i.e. f P , f S , f C . For instance, with regards to the functionalities of planning, scheduling and control the time scales range from months to hours and seconds while decisions such as production times, quality grades and inventory levels are highly interconnected as illustrated in Fig. 1.1. Considering the integrated problem as shown by Eq. (1.1), it is in the first instance an infinite dimensional problem which cannot be solved directly. To this end, a discretisation scheme is typically employed over the different horizons for the three problems and then the resulting finite-time equivalent problem is solved using state of the art optimisation solvers. It is easily understood that the more fine the discretisation the better the approximation of the infinite dimensional problem, however this

1.1 Introduction and Research Questions

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Fig. 1.1 Pictorial representation of the underlying interconnection in the integration of planning, scheduling and control

comes at the cost of rapid increase in the size of the optimisation problem making it more difficult to be solved efficiently in a timely manner. On top of that, due to the integration, moving from the lower level to the upper level, the related timescales are propagated exacerbating even more the computational effort. The disparity in time scales and decision spaces comes also with diverse kinds of uncertainty (θ P , θ C , θ S ) which naturally call for different types of treatment. As such, theoretical developments in the field of optimisation under uncertainty are of paramount importance. More specifically because of the disperse and diverse kinds of uncertainties that the decision maker is confronted with, in an integrated environment, the collaborative use of different modelling and solution techniques for optimisation under uncertainty will be pursued. For instance, at the level of process planning, uncertainty ranges from pricing policies to demand variations while in scheduling discrete types of uncertainty are more frequently encountered such as machine breakdowns. Systematic and additive disturbances lead to uncertain operations at a control level. It is understood thus, that one universal solution and modelling technique cannot be employed. Consideration of uncertainty in process systems engineering is of great importance as it can endanger the optimality or even the feasibility of a solution that was computed in a deterministic way. In an effort to avoid such occasions, a number of mathematical formulations and solution techniques have been proposed in the literature with the goal to create models which are robust towards uncertainty. Scenario based stochastic programming relies on the availability of historical data which can provide statistical information about the behavior of uncertain parameters. In stochastic programming, the unknown parameters are assumed to follow a discrete probability distribution and the decision variables are classified into two groups: “here and now” and “wait and see”. Depending on the instances that

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the uncertainty is expected to be revealed, the mathematical program is referred to as “two-stage” or “multi-stage” with the objective to minimise the cost of the initial and recourse actions. Robust optimisation (RO), assumes that all constraints of the optimisation should never be violated and aims to provide a solution that is feasible regardless of the extent of the actual uncertainty. Because of that, RO is often conceived as conservative or worst-case oriented. For the study of the effect of uncertain parameters on the optimal solution, the two main methodologies reported in the open literature are: sensitivity analysis and (multi-)parametric programming. The former provides information about the effect of uncertainty around the neighborhood of the nominal value while the latter characterises explicitly its effect on the optimal solution throughout the entire range of parametric variability. Multi-parametric programming, has proven to be an invaluable optimisation based methodology for the treatment of uncertainty on various topics in the literature of process systems engineering, from advanced model-based control strategies to process synthesis and design. Undoubtedly, the most prevalent application of multiparametric programming is the design of explicit controllers which allow for mitigation of the online computational burden associated with the control problem, in an offline step. The integrated problem under study in this thesis can be explored from a multi-parametric scope by considering the different uncertainties and the interdependent decisions between each layer as uncertain parameters. Following such a paradigm would allow for very fast calculations however given the dimensionality of such integrated problems their multi-parametric solution may prove to be prohibitive. Notwithstanding, the control layer of the integrated problem can be considered as the limiting one, because of the need to compute its related solutions in very rapid times. As such the use of explicit controllers in integrated problem can potentially alleviate the curse of dimensionality that is inherently present. Based on that, the main focus for the potential theoretical advancements will be the field of multi-parametric programming because of its potential use within the hybrid framework. Thereof the main research questions that this thesis attempts to address can be summarised as follows: 1. Can multi-parametric mixed integer algorithms be developed for the treatment of the different kinds of uncertainty in the integrated problem? 2. Can one develop novel explicit model based controllers in order to alleviate the computational complexity of the integration of control with process operations? 3. How can one model the problem of iPSC in a computationally efficient manner without compromising the quality of the solutions? 4. How can multi-parametric programming techniques be employed within the scope of the integrated problem so as to alleviate the related computational complexity? 5. Does one approach fit all towards the development of a framework for the uncertainty aware iPSC or is hybrid formulation necessary?

1.2 Thesis Scope and Outline

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1.2 Thesis Scope and Outline The research questions posed in the previous section, span two distinctly different topics and as such the thesis is organised in two parts. Part I discusses developments in the field of multi-parametric programming under global uncertainty while Part II investigates the development of an uncertainty-aware framework for the integration of planning, scheduling and model based control. Even though the two parts of the thesis are seemingly unrelated, their is a strong connection since as it turns out developments in multi-parametric control will be employed and effectively alleviate the computational complexity involved in the multi-scale integrated problem. In more details, the rest of the thesis is organised as follows: • Chapter 2, presents a thorough review of the field of multi-parametric programming which allows to identify limitations in the current state-of-the-art and motivate the research efforts subsequently pursued. • Chapter 3, introduces an algorithm for the solution of multi-parametric linear programming problems under global uncertainty while the theoretical properties of the explicit solution are established for the first time. The algorithm is then generalised to address multi-parametric mixed integer linear programming problems under global uncertainty and its application to problems of process synthesis and scheduling is examined. • Chapter 4, examines the application of a variant of the algorithm presented in Chap. 3, for multi-parametric nonlinear programming problems under global uncertainty. However, the main focus of this chapter is not the algorithm itself but the introduction of a novel concept for the design of “multi-setpoint explicit controllers”. The value of a such controllers for problems from enterprise wide optimisation is also discussed. • Chapter 5, provides a systematic overview of the efforts in integrating control with process operations. For the case of integrating planning, scheduling and optimal control a Traveling Salesman Problem (TSP) based formulation is discussed along with the development of both MINLP and MILP models. A comparative analysis, for the deterministic case, is then conducted with regards to the widely employed timeslot based formulations. • Chapter 6, investigates the case of closed-loop iPSC under dynamic disruptions. Employing the MILP TSP-based model introduced in Chap. 5 in conjunction with the multi-setpoint explicit controllers from Chap. 4 a control-relevant rescheduling mechanism is developed. • Chapter 7, generalises the closed-loop framework introduced in Chap. 6 so as to handle variations in the level of scheduling and planning apart from the dynamics in control. Proactive and reactive approaches are coupled and their efficiency is investigated via a number of Monte-Carlo simulations. • Finally, Chap. 8 summarises the main developments introduced in the thesis and suggests future research directions. The outline of the thesis is conceptually visualised in Fig. 1.2.

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Fig. 1.2 Conceptual organisation and thesis outline

As indicated by Fig. 1.2, the two parts of the thesis converge towards the development of an uncertainty-aware framework for the integrated planning, scheduling and control of process systems. In the first instance parallel advances within the two parts are presented and by coupling the multi-setpoint multi-parametric controllers (mp-MPC) from part I and the open-loop iPSC model from part II, a reactive strategy for the closed-loop iPSC is examined. The reactive strategy for the closed-loop iPSC can handle dynamic disruptions from the level of system’s dynamics while proactive approaches are employed in the final chapter of the thesis so as to handle different types of uncertainty related to the integrated problem. Finally as part of the research work from part I of the thesis, a prototype implementation of the proposed algorithms, i.e. Perseus, is briefly discussed.

Part I

Theoretical and Algorithmic Advances in Multi-parametric Programming Problems Under Global Uncertainty

The wronger answer to the righter question, is better than the righter answer to the wronger question. —George Stephanopoulos

Chapter 2

Parametric Optimisation: 65 years of developments and status quo

In this chapter, a histogram of the theoretical and algorithmic developments that led the current status of multi-parametric programming is drawn. For conceptual and organisational purposes, three distinct eras are identified and the related findings are discussed while key limitations in the state of the art are outlined. Based on these, the main developments in the field of multi-parametric programming presented in the thesis are motivated.

2.1 Introduction Consider the following generic mathematical programming problem as shown by Eq. (2.1), where the aim is to optimise its objective value (z) by manipulating its decision variables (x) subject to a number constraints (g). z(θ ) =min f(x, θ ) x

Subject to :

(2.1)

g(x, θ ) ≤ b(θ ) x ∈ Rnx , θ ∈ Rnθ Apart from its decision variables the mathematical program is typically dependent on a number of parameters, θ ∈ Rnθ , that are not known with certainty, such as catalyst’s activity, product demand or weather conditions. These uncertain parameters are most of the times, beyond the control of the decision maker and can have significant impact on the solution of the mathematical program. Multi-parametric programming (mp-P) is an optimisation based methodology which systematically characterises the effect of uncertain parameters on the optimal solution of mathematical programming Parts of this chapter have been reproduced with permission from: https://doi.org/10.1016/j. compchemeng.2018.04.015 © Springer Nature Switzerland AG 2020 V. M. Charitopoulos, Uncertainty-aware Integration of Control with Process Operations and Multi-parametric Programming Under Global Uncertainty, Springer Theses, https://doi.org/10.1007/978-3-030-38137-0_2

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Fig. 2.1 Graphical conceptualisation of the explicit solution of a (multi-)parametric programming problem

problems. Through multi-parametric programming, one aims to compute offline, the explicit optimal solution to a mathematical program which consists of two parts: • The optimisers and the optimal objective value as functions of the uncertain parameters, i.e. x(θ ) and z(θ ) respectively and, • The regions of the parametric space where each explicit solution remains optimal. These regions are also known and will be referred to for the rest of the thesis as critical regions (CRs). Conceptually the solution of uni-parametric program is shown in Fig. 2.1a where the CRs are parametric intervals while the multi-parametric case can be envisaged in Fig. 2.1b. The distinct feature of mp-P is the fact that, under the presence of uncertainty, the need for constant re-optimisation is replaced by efficient function evaluations that can be performed online whenever the uncertainty is realised. For this reason, mp-P has attracted the interest of many researchers and the main milestones in the history of mp-P are summarised in this chapter.

2.2 The Early Days of Parametric Optimisation: 1952–1967 Shortly after George B. Dantzig presented the simplex algorithm for the solution of linear programming problems (LPs) in 1947, the need to investigate the effect of single parametric (uni-parametric) variations on the optimal solutions emerged. There does not exist conclusive evidence as to who was the first one to systematically

2.2 The Early Days of Parametric Optimisation: 1952–1967

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examine uni-parametric LPs (p-LPs). Alan Manne in 1953, published a technical report on the solution of p-LPs with variations on the right hand side (RHS) of the constraints and because of that he is typically accredited with being the first one to study p-LPs [1]. On the other hand, Saul Gass in a series of communications supports that his group and more specifically William Orchard-Hays was the first one to study p-LPs with RHS variations as part of his Master thesis [2], in an attempt to modify the simplex procedure. According to Gal, Manne is known as the first one to deal with p-LPs because in his report he suggested the expression “Parametric Programming” [3, 4]. Formally, a general representation of p-LPs with RHS variations is given by Eq. (2.2) and both groups concluded that the optimal objective value is a piecewise affine function of the uncertain parameter over different parametric intervals. ⎧ ⎪ ⎪ ⎪ ⎪ ⎨ p-LPRHS

⎪ ⎪ ⎪ ⎪ ⎩

z(θ ) =

min cT x x

Subject to : Ax = b(θ ) x ∈ Rnx , θ ∈ R

(2.2)

Gass and Saaty in 1954 and 1955 presented their works on parametric programming for perturbations in the objective function’s coefficients (OFC) as shown by eq. (2.3) [5, 6]. ⎧ ⎪ z(θ ) = min cT (θ )x ⎪ ⎪ x ⎪ ⎨ Subject to : (2.3) p-LPOFC ⎪ Ax = b ⎪ ⎪ ⎪ ⎩ x ∈ Rnx , θ ∈ R Later, the same authors generalised their approach for n-parametric perturbations in the OFC and proved that the CRs of such case are convex [7]. Kelley identified the equivalence between the aforementioned parametric programming algorithm by Gass and Saaty and the “Primal-Dual algorithm” for the solution of LPs by Dantzig et al. [8], Kelley [9]. The 1960s for the parametric programming community are characterised by rather sparse activity with the exception of some works that primarily considered general parametric variations in p-LPs [10–12]. One might attribute this, to the emergence of alternative methodologies for solving optimisation problems under uncertainty, such as the paper of Charnes and Cooper that introduced “Chance constrained programming” in 1959 [13]. Another cornerstone of the same period was the work of Dantzig on the idea of two-stage programming where the decision maker optimises the expected value of the deterministic objective function and splits their decisions in: (i) here and now and (ii) wait and see [14]. Finally during this period, Wolfe presented the simplex algorithm for the solution of quadratic programs where he also examined the solution of parametric quadratic programs (p-QPs) with OFC uncertainty [15].

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2.3 Dawn of Multi-parametric Optimisation, From Sensitivity Analysis to Parametric Nonlinear Programming and Beyond: 1968–1993 The end of 1960s signaled a new distinct period for the field of parametric optimisation with a plethora of developments from linear to nonlinear and integer programming problems. The main cornerstones involve the first algorithm for the systematic solution of multi-parametric linear programs as well as numerous contributions from the sensitivity analysis of nonlinear programs that later on served as foundations for the solution of multi-parametric nonlinear programs.

2.3.1 From Parametric to Multi-parametric Linear Programming Despite the research efforts during the previous era, a general algorithm for the solution of multi-parametric linear programs (mp-LPs) did not appear until the late 1960s when Tomas Gal as part of his Ph.D. dissertation studied the systematic treatment of mp-LPs [16]. The general mathematical form of an mp-LP is given by Eq. (2.4). ⎧ ⎪ ⎪ ⎪ ⎪ ⎨ mp-LP

⎪ ⎪ ⎪ ⎪ ⎩

z(θ ) =

min cT (θ )x x

Subject to : Ax ≤ b(θ ) x ∈ Rnx , θ ∈ Rnθ

(2.4)

In their seminal paper, Gal and Nedoma [17] presented for the first time an algorithm for mp-LPs with OFC and/or RHS uncertainty based on the simplex method and finding the associated parametric optimal bases. Further to the algorithm that involved pivoting over the feasible simplex bases and finding the corresponding regions of the parametric space, in their work one can find a connected graph solution approach. In Gal [18], a method for simultaneous variations in the OFC and RHS (also know as “RIM”) of the constraints was proposed and the resulting problem is an RIM-mpLP. It should be noted that independently of Gal and Nedoma around the same time Sokolova proposed another approach for the solution of mp-LPs [19]. Apart from Sokolova, another significant stream of research work on the relationship between sensitivity analysis and parametric programming was conducted by Dinkelbach as part of his Ph.D. thesis which was later on published as a monograph [20]. The author examines a wide variety of p-LPs with interesting insights for the case of parametric variations on the left hand side (LHS) of the constraints, i.e. the elements of the A-matrix, as well as simultaneous variations of a single parameter in the RHS, LHS and OFC. For the case of p-LPs with LHS uncertainty a simplex based algorithm was presented by Gal where the author provided analytical formulae of the explicit

2.3 Dawn of Multi-parametric Optimisation …

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characterisation of the optimal solution [21] while a special case of this topic was also studied by Barnett [22]. Notice that in the aforementioned work the parametric variations were restricted on either a single column or row of the A-matrix. Townsley and Candler [23] presented a parametric linear programming based algorithm for the approximate solution of QPs while parametric linear and quadratic programs were preliminarily studied also by Propoi and Yadykin [24]. Yu and Zeleny [25], based on advances from the multi-criteria simplex method, presented two algorithms for the solution of mp-LPs. The first one was an indirect algebraic method which locates the set of non-dominated extreme points while the second one was a direct geometric decomposition method. An alternative procedure for mp-LPs with RHS uncertainty can be found in Schechter [26] where the author dualises the mp-LP and then enumerates all the extreme points of the dual polyhedron so as to compute the “irredundant” piecewise representation of z(θ ) and the corresponding CRs. The last work presented during this era is due to Adler and Monteiro [27]. In their work, a geometric approach to the solution is presented and the concept of “optimal basis invariancy” is replaced with the one of “unique active sets”. An interesting aspect of the aforementioned work is that when the number of breakpoints is polynomial in terms of the size of the p-LP, it can be solved in polynomial time. This geometric approach as will be explained later was extended to the multi-parametric case during the forthcoming era.

2.3.2 From Sensitivity Analysis to Parametric Nonlinear Programming In contrast to the aim of parametric programming to explicitly characterise the effect of uncertain parameter(s) on the optimal solution throughout the entire range of parameter variability, sensitivity analysis aims to analyse the effect of parametric perturbations in an arbitrary neighborhood of the nominal parametric values. Similar to the efforts of Dinkelbach [20] who studied the p-LPs through the lenses of sensitivity analysis, during this era the works of Fiacco and coworkers [28, 29] focused on parametric nonlinear programming problems (p-NLPs). The general form of the p-NLPs is shown by Eq. (2.5). ⎧ ⎪ z(θ ) = min f(x, θ ) ⎪ ⎪ x ⎪ ⎨ Subject to : p-NLP ⎪ g(x, θ ) ≤ 0 ⎪ ⎪ ⎪ ⎩ x ∈ Rnx , θ ∈ R

(2.5)

and the functionals f, g where assumed to be at least C 2 . A result of these works, that would later on fuel plethora of developments by different research groups in the field of mp-P, is the “Basic Sensitivity Theorem”. This theorem played a key role on the formation of the mp-P literature as it is established today as it provided

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theoretical explicit characterisation of the optimal solution under parametric variations. In brief, the theorem states that given a nominal point in the parametric space, under certain conditions, one can approximate the optimal explicit solution with a first order Taylor expansion. The convexity of the underlying problem or its convex approximation is one of the prerequisites for the application of Fiacco’s methodology nonetheless. Apart from the group of Fiacco, the problem of solution stability of p-NLPs motivated the works from the group of Kojima [30, 31] who along with his coworkers studied the regularity of the KKT points of NLPs subject to parametric perturbations. They established sufficient conditions based on the parametrised version of Sard’s Lemma such that the optimal explicit objective value to consist of a union of piecewise arcs and closed loops. It is interesting to note that Kojima and Hirabayashi [32] who proved this, transformed the KKT-system of the NLP into a generalised equation following closely the work of Robinson [33, 34]. For the bi-parametric case, Hirabayashi et al. [35] continued the work of Kojima and proved that the set of KKT points of the NLP, if the Mangasarian-Fromovitz constraint qualification along with a regularity condition hold, is a two-dimensional topological manifold without a boundary. Theoretical and algorithmic foundations for p-NLPs were also provided by the group of Prof. Nozicka who Gal [3] in a historic perspective of parametric programming identifies as the first one to study p-NLPs around the mid 60s. Members of this group published a monograph dedicated to, mainly convex, p-NLPs where a number of different problem instances with RHS and/or OFC were studied [36]. The end of this era for the case of p-NLPs can be identified in the works of (i) Poore et al. [37], Tiahrt and Poore [38], Lundberg and Poore [39] who studied the continuity and singularity properties of p-NLPs by formulating the Fritz John first order conditions of the problem and classified the “critical points” of the resulting system of nonlinear equations and (ii) [40–42] who laid the foundations for the forthcoming developments in global optimisation techniques for multi-parametric programming problems.

2.3.3 Sensitivity Analysis and Parametric Integer Programming 2.3.3.1

Parametric Integer and Mixed Integer Linear Programming

With the development of solution techniques for integer and mixed integer programs, inevitably a number of researchers focused on deciphering the effect of parametric perturbations on their optimal solution, some from a sensitivity analysis and others from a parametric programming viewpoint. The earliest article that discusses sensitivity analysis in mixed integer programs dates back to 1968 when Jensen [43] discussed an intuitive approach using Gomory cuts for which little details were given. The general mathematical form of a p-(M)IP is given by Eq. (2.6).

2.3 Dawn of Multi-parametric Optimisation …

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⎧ z(θ ) = min dT (θ )y ⎪ ⎪ y ⎪ ⎪ ⎪ ⎪ ⎨Subject to : p-IP

⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩

⎧ z(θ ) = min cT (θ )x + dT (θ )y ⎪ ⎪ x,y ⎪ ⎪ ⎪ ⎪ ⎨Subject to : Ey ≤ b(θ ) p-MILP ⎪ Ax + Ey ≤ b(θ ) ⎪ ⎪ ny ⎪ y ∈ {0, 1} x ∈ Rnx , y ∈ {0, 1}ny ⎪ ⎪ ⎩ θ ∈R θ ∈R (2.6)

The first systematic way of performing sensitivity analysis so as to define what extent of parametric variation would result in violation of optimality of the nominal solution was proposed by Roodman [44]. The author employed Balas additive algorithm, a specialised version of the branch and bound (B&B) method, and while visiting different nodes sensitivity information was collected. Parametric variations in the RHS, apart from the OFC case, are assumed to lead relaxation of the nominally feasible region; this is due to the algorithm’s strategy to collect information about fathomed solutions that nominally are infeasible but can be feasible due to the parameter variations, i.e. “partial solutions”. In the same work, the author motivates the generalisation to the multi-parametric case however computational details are not provided. Following Roodman’s work, Piper and Zoltners [45] provided a refinement of his algorithm by providing more elaborate computer implementation of the routine while also allowing parameter variations to either tighten or relax the nominally feasible region. Roodman [46] then generalised his work to the mixed integer case as shown by Eq. (2.6) following an LP-based branch and bound routine. Another work that built on the method proposed by Roodman was presented by Loukakis and Muhlemann [47]. The aforementioned works can be categorised as enumeration-based approaches and as expected required extensive computational effort for the solution of p-(M)I(L)Ps. Apart from the enumeration-based methods, cutting planes based methods, e.g. Gomory cuts, were also proposed. The first paper along these lines was presented by Holm and Klein [48] who examined the case of p-MIPs with RHS uncertainty. The same authors derived sufficient conditions for testing the optimality of a nominal solution under RHS or OFC uncertainty [49]; however as the work of Jenkins [50] points out these conditions are only valid over a small range of parameter variability. Jenkins [50] also employed Gomory and knapsack facet cuts in solution routine of p-IPs with OFC uncertainty and employed the convexity of the parametric objective function to improve computational performance. Finally, the use of parametric lower and upper bounds in a B&B routine has also been proposed by Marsten and Morin [51] for p-IPs with RHS uncertainty while Geoffrion and Nauss [52] presented results for p-IPs with special structure and proposed a generalised B&B method. Bailey and Gillett [53] presented a contraction approach on the solution of p-IPs with RHS uncertainty in which Gomory cuts were added so as to exclude already visited integer solutions. Rountree and Gillett [54] proposed a hybrid scheme combining B&B with cutting planes for the solution of pMIPs with RHS uncertainty. Sergienko and Kozeratskaya [55] developed a strategy for the approximate solution of p-MIPs with RHS or OFC uncertainty. Schrage and

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Wolsey [56] outlined a method for the type of information that needs to be stored during the solution of the p-MIP using B&B or cutting plane method so as to compute the optimal value as a function of the uncertain RHS or OFC parameters; the authors did not provide the related computational times of the method’s test suite. Piper and Zoltners [57] also proposed a method along these lines where the K-best nominal solution are stored and upon the realisation of uncertainty their optimality is revised. The work of Ohtake and Nishida [58] is notably the most closely related to the algorithms proposed for mp-MIPs in the subsequent era. The authors studied pMILPs with RHS uncertainty and developed a B&B method where parametric cuts were employed to fathom inferior explicit solutions. Finally, overviews on the developments during this era for p-IPs and p-MIPs can be found in Geoffrion and Nauss [52], Jenkins [59] while an extended summary of works on post-optimality analysis for MIPs can be found in Greenberg [60]. 2.3.3.2

Parametric Mixed Integer Quadratic and Nonlinear Programming

The earliest work on the topic of p-MINLPs appears to be the conducted by Radke [61] as part of his Ph.D. thesis. However, the most commonly cited paper of this era is the work of McBride and Yormark [62] who treated the RHS parametric pure integer quadratic programs (p-IQPs) with linear constraints as shown by Eq. (2.7). ⎧ ⎧ z(θ ) = min cT y + f(x) z(θ ) = min yT Qy ⎪ ⎪ ⎪ ⎪ y x,y ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨Subject to : ⎨Subject to : p-MINLP p-IQP Ey ≤ b(θ ) Ax + Ey ≤ b(θ ) ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ny ⎪ ⎪ y ∈ {0, 1} x ∈ Rnx , y ∈ {0, 1}ny ⎪ ⎪ ⎪ ⎪ ⎩ ⎩ θ ∈R θ ∈R (2.7) The use of dynamic programming for the solution of p-MIQPs was also reported by Cooper [63]. Following the methodology of Schrage and Wolsey [56], Skorin-Kapov and Granot [64] examined the sensitivity analysis of p-MINLPs whose relaxation is a convex program that satisfies the Kuhn-Tucker constraint qualification and computed parametric ranges over which different expressions of the value function were computed, for alternative RHS perturbations. Pure integer nonlinear problems with RHS uncertainty were also examined by Chern et al. [65].

2.4 Process Systems Engineering Led Multi-parametric Optimisation: 1993–Present In the beginning of the 1990s a new era for the multi-parametric optimisation is distinguished with the key developments mainly driven by the process systems engineering community. The emergence of this era is identified initially in Carnegie Mellon Uni-

2.4 Process Systems Engineering Led Multi-parametric Optimisation: 1993–Present

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versity where Tasos Pertsinidis under the supervision of Prof. Gregory McRae conducted his Ph.D. research on p-MIPs [66]. Subsequently, the main center of theoretical and algorithmic developments was the group of Pistikopoulos at Imperial College London where the vast majority of mp-P algorithms were developed. From a theoretical perspective, the works of Fiacco and coworkers [28] played a pivotal role in the developments during this era as researchers employed Fiacco’s theoretical results so as to explicitly characterise the optimal solution of different classes of problem under parametric variations.

2.4.1 Multi-parametric (Mixed Integer) Linear Programming The beginning of this era dates back around the doctoral research of Pertsinidis [66, 67] who developed algorithms for p-MILPs and convex p-MINLPs with RHS uncertainty and he proposed two different strategies. For the solution of p-MILPs the author revisited the work of Jenkins [68] and presented two different types of sensitivity analysis. Acevedo and Pistikopoulos [69] then presented B&B based algorithms for the explicit solution of mp-MILPs as shown by Eq. (2.8). ⎧ ⎧ z(θ ) = min cT x + dT y z(θ ) = min dT (θ )y ⎪ ⎪ ⎪ ⎪ y x,y ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ Subject to : Subject to : ⎨ ⎨ mp-MILP mp-IP Ey ≤ b(θ) Ax + Ey ≤ b(θ ) ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ny ⎪ ⎪ y ∈ {0, 1} x ∈ Rnx , y ∈ {0, 1}ny ⎪ ⎪ ⎪ ⎪ ⎩ ⎩ nθ θ ∈R θ ∈ Rnθ (2.8) The basic idea of these procedures was to relax integrality conditions at the root node of the combinatorial tree and compute a parametric lower bound (for the minimisation case) from the solution of the resulting mp-LP. Progressing in the tree different nodes are visited, the integrality conditions are imposed and the mp-LP is solved again providing progressively tighter bounds to the optimal explicit solution. In the case that a node is infeasible, its “child nodes” are fathomed. The authors also provided a comparison procedure for the elimination of overlapping CRs. For the solution of mp-MILPs the decomposition approach of Dua and Pistikopoulos [70] has proven to be computationally advantageous compared to the rest. It involves an iterative scheme between the solution of a master MIP problem and slave mp-LPs until the master MIP is infeasible. Within this procedure, integer and parametric cuts are employed to prevent investigation of previously explored solutions. Multi-parametric pure integer linear programs (mp-IPs) were also studied in a series of papers for the RHS, OFC, LHS uncertainty and their simultaneous variations by Crema [71–76] where the main idea is to treat the parameters as variables and solve a series of MILPs along with a integer cuts so as to explore the full parametric space. Note that for the case of LHS, the author employed a linearisation similar to the technique of Glover [77].

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Almost 10 years after Adler and Monteiro [27] presented their geometric algorithm for the solution of p-LPs, Borrelli et al. [78] presented a refined version of the geometric algorithm for mp-LPs with RHS uncertainty, where rather than examining different optimal bases of the mp-LP, a direct exploration of the parametric space is employed and this way degenerate mp-LPs can be handled efficiently. Despite its efficiency it should be noted that the exploration in the parametric space is done by defining a small numerical tolerance which can lead to neglecting small CRs and as such practically the solution of an additional mp-QP is needed from a rigorous viewpoint. Another approach for the approximate solution of mp-LPs with RHS uncertainty can be found in Filippi [79]. Jones and Morari [80] proposed an algorithm for multi-parametric linear complementary problems, for the case that parameters appear linearly in the objective function and the RHS of the constraints. In their approach, lexicographic perturbation was employed to handle degeneracy in mp-LPs while the exploration of the parametric space is done along the geometric lines. In general, all the aforementioned algorithms are closely related to the simplex method for the solution of the original LPs and their common denominator is the search for “optimal basis invariancy” through one way or another. Interior point method (IPM) algorithms attracted a significant amount of interest from researchers for the solution of LPs and as a result new theoretical advances emerged. For mp-LPs, IPM based algorithms do not seek for “optimal basis invariancy” but “support set invariancy” and “optimal partition invariancy” [81, 82]. Li and Ierapetritou [83] proposed a methodology for the solution of the general mpMILPs with simultaneous LHS, RHS and OFC uncertainty. When LHS uncertainty was considered, the authors employed discretisation of the parametric space and projection algorithms. The underlying mp-LP algorithm was solved based on the optimality conditions of LP and identifying basic and non-basic variables at every instance of the parametric space. Finally, in order to examine if they had covered the original (generally) non-convex CR, evenly distributed points from the parametric space were sampled to check the validity of the results; it can be understood that results from this framework are heavily influenced from the level of discretisation of the parametric space as well as the choice of the projection algorithm and cannot guarantee the validity of the resulting explicit solutions. An algorithm for the global optimisation of mp-MILPs for RIM problems was proposed by Faísca et al. [84]. The authors followed the decomposition scheme of Pertsinidis et al. [67] and Dua and Pistikopoulos [70], where the integer vector is fixed after the solution of a master MINLP to global optimality resulting in a slave mp-LP. Despite the merits of the aforementioned algorithm, because of the non-convex nature of the parametric problem, the comparison procedure of overlapping CRs is not always computationally tractable and thus the authors for these cases store the corresponding solutions in a parametric envelope and the best one is chosen online through function evaluation. Mitsos and Barton [85] also presented a B&B based algorithm for p-MILPs. A framework for the global solutions of the general mp-MILPs, within pre-specified tolerance of -optimality, was presented by Wittmann-Hohlbein and Pistikopoulos [86] within which the authors considered also LHS uncertainty. The authors exploited the structure of the mp-LP with LHS uncertainty sub-problem,

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where they identified the bilinear terms from the multiplication of the uncertain parameters with the optimisation variables and employed McCormick relaxations in order to transform the LHS into approximate RHS uncertainty. Their approach included a spatial branch and bound routine on the optimisation variables and the uncertain parameters. As noted by the authors, the quality of the results by implementing this algorithm, is dependent on the partitioning scheme for the uncertain parameters but that results in higher computational times and number of CRs to be examined. This approach does not provide the exact solution for the case of LHS uncertainty. The same authors [87] proposed a two stage approach to handle the global uncertainty in mp-MILPs in which the conventional “worst case” robust counterpart was employed for the LHS uncertainty while the resulting partially robust mp-MILP was then solved in a decomposed fashion through iterations between master MINLPs (solved to global optimality) and slave mp-LPs. More recently, Oberdieck et al. [88] drawing on the work of Wittmann-Hohlbein and Pistikopoulos [89], in an attempt to reduce the number of parametric envelopes previously needed during the solution of RIM mp-MILPs, employed McCormick envelopes so as to develop convex envelopes of the bilinear terms that were present in the underlying parametric problem. In Khalilpour and Karimi [90], an algorithm for the systematic treatment of LHS single-parameter uncertainty in LPs was presented. The authors identified that the problem includes at its core the inversion of parametric matrices and the computational complexities arising by this fact. In order to handle this issue, a two stage algorithm was devised that uses the Flavell and Salkin [91] approximate method to find the location of breakpoints in the parametric intervals and the Woodbury formula [92] for the correctness of the result. Habibi et al. [93] examined the trade-off between –suboptimality in the explicit solution of mp-MILPs with RHS uncertainty and their solution’s computational complexity.

2.4.2 Multi-parametric (Mixed Integer) Quadratic Programming Multi-parametric (mixed integer) Quadratic Programming (mp-(MI)QP) problems form another important class of mp-P problems due to their application in model based control schemes for linear and hybrid systems respectively. A general formulation of mp-(MI)QPs is given by Eq. (2.9). ⎧ ⎧ ⎪ z(θ) = ⎪ Tx ⎪ z(θ ) = min (c + Hθ ) ⎪ ⎪ ⎪ ⎪ x ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ + 21 xT Qx ⎨ ⎨ mp-QP Subject to : mp-MIQP Subject to : ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ Ax ≤ b(θ ) ⎪ ⎪ ⎪ ⎪ ⎩ ⎪ n n ⎪ ⎪ x ∈ R x, θ ∈ R θ ⎩

min (c + Hθ)T ω x,y

+ 21 ωT Qω Ax + Ey ≤ b(θ) x ∈ Rnx , y ∈ {0, 1}ny ω = [x y]T , θ ∈ Rnθ

(2.9)

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The first algorithm for the exact solution of mp-QPs was devised by Dua [94] where the Karush-Kuhn-Tucker (KKT) conditions of optimality were solved explicitly and were later on applied in the seminal work of Bemporad et al. [95] leading to the concept of explicit model predictive control. Dua [94] in his doctoral work proved theoretically, for the mp-QPs, that their optimal objective value (z(θ )) is quadratic in its arguments, the optimisers x(θ ) are a set of affine/linear functions of the uncertain parameters and the resulting CRs are polytopic. Seron et al. [96] following a preliminary conference version of the famous paper of Bemporad et al. [95] also provided some insights about the underlying geometry of the explicit MPC problem. Tøndel et al. [97] proposed an mp-QP algorithm which differs from the one proposed by Bemporad et al. [95] on the way the parameter space is explored so as to define the CRs. The authors based on the concept of neighboring CRs and the common facet that is formed between them, proposed an alternative exploration strategy that led to less computational steps. The correctness of the algorithm proposed by Tøndel et al. [97] is dependent on a number of assumptions, related to Linear Independence Constraint Qualification (LICQ) and Strict Complementary Slackness (SCS), so that the “facet-to-facet” property holds. Later on, Spjøtvold et al. [98] proposed a modification of the way the parametric space is explored, so as to ensure that all the CRs have been computed. The authors employed the artificial cuts proposed by Bemporad et al. [95] in conjunction with the “facet-to-facet” property to compensate for the cases when the latter was violated. The aforementioned methods fall broadly under the category of “facet-exploration” solution approaches for mp-QPs. An algorithm for the approximate solution of mp-QPs was proposed by Bemporad and Filippi [99]. The authors employ a degree of −suboptimality to relax the first order KKT conditions with respect to dual feasibility with the aim to reduce the number of CRs involved in the final explicit solution. Another body of research forms the “graph-traversal” approaches where the main idea is that the nodes of the graph explored are the full dimensional CRs and the arcs connect adjacent CRs. Patrinos and Sarimveis [100, 101] approached the solution of the mp-QPs by enhancing the graph-traversal approaches together with the use of graphical derivatives. The authors proved that the solution mapping of the convex mp-QPs is proto-differentiable and proposed formulas for the computation of the graphical derivatives. It should be noted that this was the first method that did not rely on assumptions about non-degeneracy and secured the full exploration of the parametric space using the graph-traversal approach. Shortly after this work, Gupta et al. [102] presented their work that enumerates the combination of active constraints of an mp-QP but avoids their exhaustive enumeration by using a pruning criterion thus fathoming the “child-nodes” of the combinatorial tree. Feller et al. [103] employed the combinatorial algorithm of Gupta et al. [102] and the authors proposed an alternative pruning criterion for the exploration of the candidate active sets. They employed the idea of saturation matrix of a constraint polyhedron so as to prune infeasible candidate active sets, while symmetry in mp-QPs was also investigated so as to achieve better computational performance. Recently, Bemporad [104] proposed an algorithm based on non-negative least square computations for the solution of mp-QP problems with uncertain parameters on the RHS of the constraints.

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As mentioned earlier, the solution algorithms for the mp-QPs can be broadly classified into combinatorial and geometric based on the exploration strategy employed for the exploration of the parametric space. An algorithm that unifies the two strategies was proposed by Oberdieck et al. [105] where the authors showed that the explicit solution of the mp-QPs is given by a connected graph whose nodes correspond to different optimal active sets heavily relying on Gal’s result for the mp-LP case [21]. Independently, Ahmadi-Moshkenani et al. [106] proposed a method that shares a number of similarities to the connected graph approach where information about the underlying geometry of the adjacent CRs is exploited so as to enhance the pruning process of infeasible sets in the combinatorial’s approach solution loop [107, 108]. Remark 2.1 Even though the paper of Bemporad et al. [95] established theoretically and algorithmically a parametric framework for the design of explicit controllers, it is important to note that along these lines was the early work of Zafiriou [109]. Zafiriou’s approach was to enumerate all the possible active sets of the QP, a computationally strenuous task, and is a “semi-offline” approach compared to the completely offline optimisation introduce by Bemporad et al. [95]. A follow-up to the work of Zafiriou can be found in Mayne and Rakovic [110] where the authors used the concept of “reverse transformation” to perform the same task and compute the explicit control law. Reverse transformation coupled with dynamic programming was also proposed by Munoz de la Pena et al. [111]. A distinct implication when one considers the mixed integer case is that the exact CRs are not polytopic but rather quadratically constrained. This issue arises due to the potential non-convexity inferred by the combinatorial nature of the problem where different integer feasible mp-QPs can result in the overlapping partitions of the parametric space. When overlapping CRs are identified, a comparison procedure similar to the one presented by Acevedo and Pistikopoulos [112] needs to be followed however this time quadratic parametric functions are compared rendering the problem more challenging and non-convex. The first solution procedure of mp-MIQPs was presented by Dua et al. [113] and the authors proposed the use of “parametric envelopes” and no comparison procedure was employed but rather an online evaluation would return which of the overlapping solutions is the optimal one. Axehill and Morari [114, 115] proposed an alternative method for the solution of mpMIQPs coupling the idea of branch and bound and comparing overlapping parametric solutions over entire CR and directly requiring thus the solution of potentially nonconvex mp-QPs. Another work that presented preliminary results on the solution of mp-MIQPs along with comparison of quadratic parametric objective functions is that of Alessio and Bemporad [116] while a two-step online/offline procedure was proposed by Almér and Morari [117]. As mentioned earlier the comparison of quadratic functions can lead to non-convex problems and thus a natural way of dealing with this issue is the use of convex approximations, e.g. McCormick envelopes [118]. Based on this observation, Oberdieck and Pistikopoulos [119] presented the first algorithm for the “exact” solution of mp-MIQPs without the use of parametric envelopes.

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2.4.3 Multi-parametric Linear Complementarity Problems A special class of problems whose solution unifies mp-LPs and mp-QPs, with RHS and OFC uncertainty, is the multi-parametric linear complementarity problems (mpLCPs) which are defined by Eq. (2.10). ⎧ w − Mz = q + Qθ ⎪ ⎪ ⎪ ⎨w ≥ 0, q ≥ 0 mp-LCP ⎪ wT z = 0 ⎪ ⎪ ⎩ θ ∈ Rnθ

(2.10)

Recasting mp-LPs/QPs as mp-LCPs stems from their interesting property that the parametric KKT conditions of mp-LPs/QPs can be rewritten in the form of Eq. (2.10). Finding all the optimal bases of the mp-LCP is equivalent to finding all the optimal active sets and thus the full dimensional CRs that partition the parametric space. Similar to the algorithms of mp-LPs/QPs ones presented for mp-LCPs can be divided into: (i) geometric [80] and (ii) combinatorial [120]. The initial geometric based approach of Jones and Morari [80] was later revised using a two-step approach by Adelgren and Wiecek [121]. The solution of mp-LCPs was studied from a different perspective by Li and Ierapetritou [122] where the authors reformulated the problem as a multi-parametric mixed integer program.

2.4.4 Multi-parametric (Mixed Integer) Nonlinear Programming In contrast to the (mixed integer) linear and convex quadratic case, developing a strategy for the exact solution of general mp-(MI)NLPs is in general not always possible, apart from some very limiting and special cases. A general form of mp(MI)NLPs is given by Eq. (2.11). ⎧ ⎧ ⎪ z(θ) = ⎪ ⎪ ⎪ ⎪ z(θ) = min f(x, θ ) ⎪ ⎪ ⎪ ⎪ x ⎪ ⎪ ⎨ ⎨Subject to : Subject to : mp-NLP mp-MINLP ⎪ ⎪ g(x) ≤ b(θ) ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ ⎪ ⎪ x ∈ R nx , θ ∈ R nθ ⎪ ⎩

min f(x, y, θ ) x,y

g(x, y) ≤ b(θ) x ∈ Rnx , y ∈ {0, 1}ny θ ∈ Rnθ

(2.11) The convexity properties of the functions entailed in mp-(MI)NLPs are of great importance and based on them two distinct research currents can be identified as listed in the following.

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2.4.5 Convex Problems During this era the solution of p-MINLPs as shown by Eq. (2.7) for which binary and continuous variables are separable and the constraints are linear was examined first by Pertsinidis [66]. The function f(·) is assumed to convex and continuously differentiable to its arguments. The solution method for the p-MINLPs was based on outer-approximation concepts so as to iteratively approximate the original MINLP by a series of NLPs and MILPs for he could employ his p-MILP solution method and Fiacco’s sensitivity for the NLPs. Drawing on the Pertsinidis’ Ph.D., [123] presented an iterative algorithm for the p-MINLPs with RHS uncertainty. As mentioned above, the basic sensitivity theorem of Fiacco [28] is valid for convex NLPs or ones that are approximated by convex over/under-estimators. To this end, Dua and Pistikopoulos [124] were the first to study mp-(MI)NLPs with convex nonlinear objective functions and convex nonlinear constraints by exploiting the convex nature of the objective function. The authors generalised concepts from the works of Acevedo and Pistikopoulos [123], Pertsinidis et al. [67], Papalexandri and Dimkou [125], Dua and Pistikopoulos [126] where valid parametric lower and upper bounds were computed by the solution of a series of “primal” mp-LPs and “master” MINLPs. Based on the convexity property of the objective function, it was noted that the maximum error of the outer-approximation (mp-OA) by mp-LPs will appear on the vertices of the CRs where a prespecified tolerance was defined in order to refine the partition of the parametric space. For the formulation of the master MINLP the authors proposed three different strategies including (i) outer approximation, (ii) generalised Benders decomposition and the third one including parametric and integer cuts so as to avoid the exploration of already visited solutions. Shortly after, Johansen [127] proposed the use of quadratic approximations of the objective function and linear approximation of the convex constraints which resulted in the solution of a series of mp-QPs instead of mp-LPs as in Dua and Pistikopoulos [124]. Acevedo and Salgueiro [128] refined the mp-OA algorithm of Dua and Pistikopoulos [124] with a number of heuristics so as to alleviate the computational complexity. Johansen [129] following up his work with Grancharova for the linear case [130], proposed an approximate mp-NLP algorithm. The main idea is to directly partition the parametric space (Rnθ ) into a set of hypercubes (0 ⊂Rnθ ) and using linear interpolation the mp-NLP is solved in a series of fixed-point NLPs at each of the vertices of the hypercubes. Similar to the previous cases a prespecified tolerance is defined so as to partition the hypercubes in order to compute more accurate solutions. Partitions into simplices instead of hypercubes are proposed in the algorithm of Bemporad and Filippi [131]. Later on, Dominquez as part of his doctoral research examined a series of different scenarios met in the context of mp-NLPs, namely: (i) the case where the parametric space is defined by nonlinear inequalities; the author proceeded with successive linearisations and approximated the mp-NLP locally with mp-QPs/LPs [132] and (ii) a quadratic approximation based algorithm for the solution of convex mp-MINLPs [133]. Around the same period, Narciso [134] proposed as part of his doctoral research the geometric vertex algorithm a detailed description of which can be found in Dominguez et al. [135].

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2.4.6 Non-convex Problems For the non-convex case of mp-(MI)NLPs the amount of research work is relatively limited owing to the computational complexity of the underlying parametric problem. The global optimisation of non-convex mp-NLPs and mp-MINLPs with RHS uncertainty was initially discussed by Dua et al. [136] and the authors proposed four different parametric convex overestimators along with a B&B algorithm. Note that Fiacco [42] had proposed a solution technique for global optimisation for the case of non-convex multi-parametric separable NLPs restricted to a convex set. A distinctly different approach compared to the aforementioned is the one investigated by Fotiou as part of his doctoral research [137]. The author examined the case of where the nonlinear terms of the mp-NLPs are of polynomial nature and thus accept exact analytical solution. He studied RHS parametrisations for which he employed Cylindrical Algebraic Decomposition [138] and Homotopy Continuation methods [139] so as to analytically solve the mp-NLP. This work is closely related to the developments presented in this thesis, however in contrast to the approach followed by Fotiou where the generalised companion matrices were used to identify the partition of the parametric space where each explicit solution is optimal, in the present body of work the corresponding CRs will be explicitly and exactly computed. Hale [140] in her Ph.D. research proposed a predictor-corrector moving front based algorithm where the mpNLP was transformed into a system of simultaneous nonlinear equations based on the Fritz-John conditions. Following her work, the solutions of the resulting nonlinear systems were approximated based on parametrised triangulations [141]. Following up their work on convex mp-NLPs, Grancharova et al. [142] proposed the use of nonlinear interpolation for the explicit solution within their previous framework [127] along with heuristics but the accuracy of resulting explicit solutions was not encouraging. Recently, Leverenz et al. [143] presented subgradient-based approaches for mp-NLPs while Charitopoulos and Dua [144] presented the exact solution of multiparametric mixed integer polynomial problems with RHS uncertainty as well as for the case of exponential nonlinear terms [145].

2.4.7 Multi-parametric Dynamic Optimisation Undoubtedly, the application area that has fueled many of the developments in the field of mp-P is design of explicit MPC and therein lies the reason why the solution of mp-QPs compared to the other classes of problems has received far more attention. For the development of model based controller one has to employ some numerical technique, e.g. multiple shooting, control vector parametrisation or other variational approaches, so as to transform the infinite dimensional problem into a finite one. Another possibility however is to directly work on the infinite dimensional problem by combining parametric programming approaches with Calculus of variations. This class of problem is referred to as multi-parametric dynamic optimisation problems (mp-DO) and has received very limited attention compared to the other problem classes, mainly due to its intrinsic complexity.

Fig. 2.2 Taxonomy of multi-parametric programming problem classes and the main related solution algorithms. Depending on whether the set of discrete variables (Y) of the mp-P is empty different strategies are followed

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Early foundations for the solution of mp-DOs can be found in the work of Poore [146] where the author solved directly the continuous time optimal control problem and obtained the optimality conditions as piecewise continuous functions of the problem’s parameters. The different regions of optimality were characterised using bifurcation techniques while path constraints and high-order dynamics were not facilitated. Despite the valuable insights one could get from this work its complexity was prohibitive for the solution of problems with more than one parameter. A systematic methodology for the solution of mp-DOs for linear dynamic systems was presented for the first time by Sakizlis et al. [147] who employed the extended Euler-Lagrange necessary optimality conditions and identified the switching time points along with the nonlinear CRs that relate to the cases of constrained/unconstrained law. Later on Sakizlis et al. [148] presented two approaches for the design of explicit controllers for nonlinear dynamic systems. The first one followed the direct approach, where the continuous time systems was discretised, while the second was based on the variational approach and the use of the Hamiltonian function of the system for its parametric solution. Recently, Sun et al. [149] presented their work on combining NCO tracking and mp-DO for linear dynamic systems. Based on the aforementioned a taxonomy on the different basic ideas behind the solution of the various classes of mp-Ps is organised and visualised in Fig. 2.2.

2.5 Major Application Areas 2.5.1 Multi-parametric Multi-level Optimisation Optimisation techniques form a rigorous mathematical tool that aid decision making which naturally is organised in a multilevel fashion. Formulating and solving multilevel optimisation problems has been and continues to be considered as a formidable task for any but the simplest cases due to the inherent computational complexity. A general mathematical formulation of a multilevel optimisation problem involving both continuous (x) and discrete (y) variables is given by Eq. (2.12). ⎧ min f 1 (x1,...,L , y1,...,L , θ 1,...,L ) ⎪ ⎪ ⎪ x1 ,y1 ⎪ ⎪ ⎪ ⎪ Subject to : g1 (x1,...,L , y1,...,L ) ≤ b1 (θ 1,...,L ) ⎪ ⎪ ⎪ ⎪ ⎪ min f 2 (x1,...,L , y1,...,L , θ 1,...,L ) ⎪ ⎪ x2 ,y2 ⎪ ⎪ ⎪ ⎨ Subject to : g2 (x1,...,L , y1,...,L ) ≤ b2 (θ 1,...,L ) mp-MLVL .. ⎪ . ⎪ ⎪ ⎪ ⎪ ⎪ .. ⎪ ⎪ . ⎪ ⎪ ⎪ ⎪ ⎪ min f L (x1,...,L , y1,...,L , θ 1,...,L ) ⎪ ⎪ L ,yL x ⎪ ⎪ ⎩ Subject to : gL (x1,...,L , y1,...,L ) ≤ bL (θ 1,...,L ) (2.12)

2.5 Major Application Areas

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As indicated by Eq. (2.12), each level (l = 1, . . . , L) of the optimisation problem involves decision variables that are not controlled by it and effectively its decision space is a function of the other levels’ decisions. A similar intuitive explanation of the problem was expressed by Bard and Falk [150] who were the first to discuss the concept of parametric optimisation within a multilevel optimisation setting. The key idea in their work and in the works that followed, is to start from the bottom-most level, i.e. l = L and compute the related decisions as a function of the preceding level’s optimisation variables, i.e. [xL , yL ] = h(x1,...,L−1 , y1,...,L−1 ). It was not until the work of Ryu et al. [151] that mp-P was applied to solve bilevel problems with the following structures: (i) LP-LP, (ii) LP-QP, (iii) QP-QP. Similar concepts were studied by Faísca et al. [152, 153] where also multilevel and bilevel programs with mixed integer decisions on the upper level were solved using mp-P algorithms. In the aforementioned works the case of RHS parametrisations in all the levels were studied. Domínguez and Pistikopoulos [154] addressed the case of integer and mixed integer bilevel programs using reformulation linearisation techniques and convexifcation of the resulting nonlinear terms. Kassa and Kassa [155, 156] also worked on the solution of multilevel non-convex problems. Currently, the development of new solution procedures for bilevel MILPs (B-MILPs) and bilevel MIQPs (B-MIQPs) has been presented by Pistikopoulos and Avraamidou [157], Avraamidou and Pistikopoulos [158] along with their application to a wide range of problems closely related to enterprise wide optimisation [158, 159]. Finally, the solution of trilevel MILPs (T-MILPs) has appeared by the same authors [160] following the general ideas earlier introduced by the group.

2.5.2 Multi-parametric Model Predictive Control Inarguably the most prevalent application of mp-P is the design of explicit controllers (mp-MPC) [161–163]. The key idea behind mp-MPC can be explained as follows: instead of repeatedly performing the optimisation that the conventional MPC requires at every sample instance, let the state of the system at each sampling instance be treated as an (uncertain) parameter [161]. A generic mathematical formulation of the explicit MPC problem for discrete time systems with initial state at time point tk is given by Eq. (2.13), ⎧ P H −1 ⎪ ⎪ (x(t )) = min L(xt , ut ) + E(xN ) ⎪ k ⎪ u ⎪ t=0 ⎪ ⎪ ⎪ ⎪ Subject to : xt|t=0 = x(tk ) ⎪ ⎪ ⎪ ⎨ t = 0, 1, . . . , PH − 1 xt+1 = f(xt ,ut ) (mp − MPC) t = 0, 1, . . . , PH − 1 yt+1 = h(xt ,ut ) ⎪ ⎪ ⎪ ⎪ ⎪ ≤ α t = 0, 1, . . . , PH Ax ⎪ t ⎪ ⎪ ⎪ ⎪ Byt ≤ β t = 0, 1, . . . , PH ⎪ ⎪ ⎩ Cut ≤ γ t = 0, 1, . . . , PH

(2.13)

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where xt , ut , yt are the state, control input and system output vectors respectively at every sampling point, t, and are nx , nu , ny dimensional. A, B, C are matrices of appropriate dimensions and α, β, γ vectors of pertinent dimensions which represent inequality constraints for the state, output and control inputs while L: Rnx +nu → R is a stage cost and E : Rnx → R is a terminal cost function. The repetitive solution vector, of problem (2.13) provides the optimal cost (x(t   k )) and the optimisation i.e. the sequence of optimal control inputs u∗ = u1∗ , u2∗ , . . . , uP∗H −1 over the finite prediction horizon PH . In contrast with the conventional receding horizon policy, in which the optimisation problem has to be solved at every sampling instance, by using the mp-MPC strategy, the explicit control law is derived once and for all. The solution of the resulting mp-P problem effectively returns the optimal control inputs as explicit functions of the (uncertain) parameters, i.e. the state of the system whenever a measurement is available, along with the corresponding CRs as shown by Eq. (2.14). ⎧ ν1 (x(tk )) if x(tk ) ∈ CR1 ⎪ ⎪ ⎪ ⎪ ⎨ν2 (x(tk )) if x(tk ) ∈ CR2 (2.14) u∗ = . .. .. ⎪ ⎪ . ⎪ ⎪ ⎩ νω (x(tk )) if x(tk ) ∈ CRω For the case of MPC for linear systems, instead of solving at each sampling instance a quadratic program, the explicit MPC requires the offline solution of an mpQP while online, only simple function evaluations are required [164]. This concept is also known as “online via offline optimisation” and it is shown in Fig. 2.3.

Fig. 2.3 Conceptual illustration of the “online via offline optimisation” framework

2.5 Major Application Areas

29

Applications of explicit MPC range from energy systems, such as fuel cells [165] and periodic systems [166], to biomedical applications with regards to diabetes regulation [167], control of intravenous anesthesia [168] while also the explicit control of solvent gradient purification process has been reported in Papathanasiou et al. [169]. The use of multi-parametric dynamic programming (mp-DP) and robust optimisation techniques constitute another two significant areas within the scope of mp-MPC where extensive research work exists both for linear and hybrid systems [170–174]. A number of surveys have been published so as to organise the contributions on the design and application of mp-MPC [161, 164, 175, 176] while Dominguez and Pistikopoulos [177] focused on the developments regarding solely explicit nonlinear MPC. Oberdieck et al. [178] presented an extensive literature review where a more comprehensive list of specific methodologies for the design of mp-MPC can be found while a monograph with biomedical applications of mp-MPC has also been recently published [179].

2.5.3 Enterprise Wide Optimisation Under Uncertainty Multi-parametric programming in the context of enterprise wide optimisation has been employed from two scopes: (i) dealing with uncertainty and (ii) formulating computationally efficient integrated frameworks. In Table 2.1, an indicative list of

Table 2.1 Indicative list of multi-parametric programming application in enterprise wide optimisation problems Authors Contribution Sakizlis et al. [180] Ryu et al. [151] Li and Ierapetritou [181] Li and Ierapetritou [182] Wittmann-Hohlbein and Pistikopoulos [183] Kopanos and Pistikopoulos [184] Zhuge and Ierapetritou [185] Hadera et al. [186] Diangelakis [187] Diangelakis et al. [188] Charitopoulos et al. [189]

Incorporated the explicit solution of mp-MPC into process design Bilevel integration of supply distribution and planning under uncertainty Process scheduling under uncertainty Reactive batch scheduling under uncertainty Hybrid robust optimisation mp-P approach to batch scheduling under uncertainty Reactive state space scheduling using the mp-MPC approach Incorporated explicit solution of mp-MPC in scheduling formulation Solved energy-cost optimisation as an mp-MILP and embedded the solution in the scheduling problem Proposed a mp-P modelling and integration framework for design, scheduling and control Derived the process design dependent mp-MPC Closed-loop integration of planning, scheduling and mp-MPC

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the enterprise wide optimisation applications where multi-parametric programming has been reported, is given. By studying the list it becomes clear that mp-P has primarily been used for the efficient integration of process operations with model based control due to the dimensionality of the related control problem which allows for efficient mp-P solution.

2.5.4 Multi-objective Optimisation Multi-objective optimisation problems (MOO) have also been studied through the lenses of mp-P, since the early days of Gass and Saaty [7] and Yu and Zeleny [25] for linear problems. Trade-off analysis in multi-objective optimisation problems has been studied using parametric programming for linear problems [190]. Enkhbat et al. [125] proposed a parametric programming approach for the solution of multi-objective mixed integer nonlinear programs. Papalexandri and Dimkou [191] proposed parametric programming techniques for the solution of linear and convex nonlinear multi-objective problems using the weighted sum method and treated the weight as a parameter and its complement, i.e. θ and 1 − θ respectively. The problem of solving multi-objective MPC using mp-P has been reported in Bemporad and Muñoz de la Peña [192]. Optimal partition invariancy methods for biparametric quadratic multi-objective programs were proposed in Ghaffari-Hadigheh et al. [193] while the use of parametric - constraint method for the solution of MOOs with convex quadratic functions was briefly introduced by Oberdieck et al. [178]. Lastly, Charitopoulos and Dua [163] proposed a mp-P based unified framework for MOO under uncertainty (MO2 U2 ) and studied its application to energy systems.

2.5.5 Other Application Areas The problem of material design under uncertainty has been studied and solved as an mp-MILP by Dua and Pistikopoulos [194]. Flexibility analysis has been studied by Bansal et al. [195, 196] within an mp-P context. The use of mp-P as a means of accelarating the calculations involved in stochastic programming is another interesting application that has been proposed for both MILPs and convex MINLPs [197, 198]. The key idea is to compute the objective function explicitly in the uncertain parameters and thus substitute the use of integration schemes with function evaluations. Akbari and Barton [199] conducted flux balance analysis in metabolic networks through mp-LPs under uncertain concentrations of glucose and oxygen. Finally, potentially the most interesting and promising application of mp-P is with respect to machine learning problems such as support vector machines and some studies have already been reported by Karasuyama et al. [200], Zhou and Spanos [201]. Finally a summary of mp-P problem classes and their related applications is given in Fig. 2.4.

2.6 Complexity Issues in Multi-parametric Optimisation Problems

31

Adjustable Robust Optimisation Game theory

LPbased hybrid mpMPC

Process integration

Continuous time Hybrid mp-MPC

mpmultilevel NCO tracking

Reactive Scheduling Process Synthesis Process Design

mp-MILP

Material Design

mp-MOO

Multiparametric programming

mp-MIDO

mp-DO

mp-LP LPbased mpMPC

Continuous time mp-MPC

Flexibility analysis

mp-QP

mp-NLP

mp-MIQP

Real time optimisation mp-MINLP

Linear hybrid mp-MPC

Linear mp-MPC

Nonlinear mp-MPC Process Nonlinear Synthesis Hybrid mp-MPC

Fig. 2.4 Overview of multi-parametric programming problem classes and their indicative applications

2.6 Complexity Issues in Multi-parametric Optimisation Problems 2.6.1 Degeneracy Implications in mp-LPs/mp-QPs The case of degeneracy is widespread in many mp-LPs and results in non-unique mapping between the vector of parameters and the vector of optimisation variables along with non-unique partition of the parametric space. The aforementioned implications can lead to overlapping CRs and discontinuous optimisers which are highly undesirable especially for control applications that require smooth and stable control inputs. In terms of classification, degeneracy can be either (i) primal, (ii) dual or (iii) both.

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Primal degeneracy occurs when at an optimal solution the number of active constraints exceeds the number of optimisation variables or equivalently when at least one basic variable takes null value. From a geometric standpoint that means that both solutions are found at the same vertex of the primal polyhedron but the related set of shadow prices are not the same. This is typically an outcome of the existence of weakly redundant constraint, i.e. constraints that “lean” but do not intersect on the feasible region. In a similar way, dual degeneracy occurs when there exists different primal solutions satisfying the optimality conditions. A thorough introduction to multi-parametric linear programming and the issues arising from degeneracy is presented in the book of Gal [21]. The following approaches of dealing with degeneracy in mp-LPs/QPs have been proposed in the literature. Olaru and Dumur [202] proposed a parametrised vertex approach to handle degeneracy which guarantees the continuity and uniqueness of the optimiser, however the related computational cost can be considerably high even for problems of moderate size. Spjøtvold et al. [203] for dual degenerate mp-LPs modified the geometric algorithm of Borrelli et al. [78] and secured the continuity of the optimiser by always selecting the one with the least square norm, through the solution of an additional strictly convex mp-QP. The same authors later on extended this approach for strictly convex mp-QPs with RHS and OFC uncertainty [204]. Finally, the most efficient strategy for treating degeneracy appears to be the use of symbolic lexicographic perturbation as proposed by Jones et al. [205]. The method is called symbolic because the sufficiently small perturbation is not calculated in the objective function, at the end, but only its effect on the degenerate explicit solution is studied with the aim to render it non-degenerate and thus yield a full dimensional CR.

2.6.2 Computational Complexity of mp-P Algorithms and Reduction Strategies Solving (multi-)parametric optimisation problems is an inherently computationally challenging task as it eliminates completely the need for re-optimisation once the uncertainty is realised. Murty [206] provided theoretical proofs that the computational complexity of solving a p-LP is not bounded by a polynomial in the problem’s size but rather exhibits exponential complexity in the worst case. Carstensen [207] studied parametric 0-1 programs and proved that the number √ of parametric intervals when OFC uncertainty is considered, is of the order of 2 n , where n is the number of variables. For general mp-Ps universal theoretical proofs on the complexity of their solution are not available but in the worst case the solution burden grows exponentially in the number of the problem’s constraints. On top of that most of the solution algorithms for mp-Ps are not output sensitive, i.e. their complexity is not a function of only the size of the input and output of the problem [208]. For the class of mp-QPs, in the worst case their explicit solution involves 2ng combinations of active sets (where ng stands for the number of constraints) while computing the related CRs

2.6 Complexity Issues in Multi-parametric Optimisation Problems

33

can be seen as exploring a tree with maximum depth of 2ng and thus an upper bound ng −1 2 on the number of full dimensional CRs can be given by NCRs ≤ k!ngk . k=0

In order to overcome the inherent complexity issue of computing solutions to mp-Ps two different strategies have been widely proposed: (i) model reduction techniques: so as to a priori reduce the size of the problem under examination whilst computing its exact explicit solution and (ii) heuristics on how to approximate the explicit solution of an mp-P. An overview of the different complexity reduction techniques can be found in Table 2.2. Apart from the complexity of solving the mp-P problem itself, another significant body of work has identified the so called “point-location problem”. Practically, this problem is concerned with the speed of calculations that are required so as to identify in which CR a given parametric “point”, i.e. θ ∗ , belongs to. Especially for large problems that may involve thousands of CRs the point location problem Table 2.2 Overview of complexity reduction techniques in multi-parametric programming problems Authors Contribution Explicit solution approximation Johansen [209]

Singular value decomposition for reduction of the parametric set cardinality Rossiter and Grieder [210] Used interpolation of the input sequences to reduce the complexity of the mp-QP solution Scibilia et al. [211] Approximation of the explicit solution of linear mp-MPC using Delaunay tessellations Kvasnica et al. [212] Critical region free polynomial approximation of the explicit solution of mp-QPs Khan and Rossiter [213] Generalised function parametrisations for approximation of the explicit solution Kvasnica et al. [214] Unification of the CRs that result in saturated control input Holaza et al. [215] Approximate space partition and parametrised affine approximations of the explicit solution Cseko et al. [216] Used radial basis function based artificial neural network (RBF-ANN) to approximate the explicit solution of mp-MPC A priori model reduction techniques Johansen [209] Singular value decomposition for reduction of the parametric set cardinality Narciso and Pistikopoulos [217] Balanced truncation for reduction of the state dimensionality, while a maximum error in the predicted output is established Rivotti et al. [218] Nonlinear model reduction using balancing of empirical gramians. The order of the model was reduced via residualisation instead of truncation by setting the derivatives of the less important states equal to zero Lambert et al. [219] Developed N step ahead affine expressions that represent the original nonlinear system through a number of Monte Carlo simulations

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might prohibit the successful and timely implementation of the explicit solution to the system. To this end, a number of different approaches have been proposed with some of them being the formulation of a binary search tree, use of hash-tables or performing an orthogonal search tree search if the parametric space is partitioned in hyper-rectangles [130, 220, 221]. The concept of parallel computing in mp-P solution algorithms has recently been examined by Oberdieck and Pistikopoulos [222] and the authors proposed a userdefined parameter to manipulate the trade-off between information overhead and individual machine independence. The results indicated reasonably promising speedup in the solution times.

2.7 Software Implementations Currently for the solution of mp-Ps three toolboxes are available: (i) the MultiParametric Toolbox (MPT3) from the groups of Herceg et al. [223] , (ii) the Parametric OPtimisation (POP) toolbox from the group of Pistikopoulos [224] and (iii) the Hybrid Toolbox of [225]. MPT3 constitutes the new major re-release of the MPT that was initially released in 2004 [226] while the current POP is also a re-release of the version initially introduced by Pistikopoulos et al. [227]. The key features of the three toolboxes are given in Table 2.3. MPT3 builds on YALMIP and relies on a series of computational geometry libraries that allow for efficient polyhedral computations which are ubiquitous in the solution of mp-Ps. For the solution of mp-LPs and mp-QPs, MPT3 employs the mp-LCP algorithm of Jones and Morari [80] which features lexicographic perturbation and has proven to be very efficient in problems with degeneracy. It offers interfaces to state-of the-art solvers like CPLEX, NAG, GLPK and GUROBI to name a few. For the solution of mp-MILPs and mp-MIQPs it employs exhaustive enumeration procedure which can be computationally strenuous for any but small Table 2.3 Overview of the available software implementations for the solution of multi-parametric programs POP MPT3 Hybrid toolbox Language Free/Open source Dependencies mp-LPs/QPs mp-MILPs/MIQPs mp-(MI)NLPs Problem library Problem generator Complexity reduction Graphical user interface

MATLAB /– MATLAB   –   – 

MATLAB / MATLAB, YALMIP   – – –  –

MATLAB /– MATLAB   – – – – –

2.7 Software Implementations

35

problems. A very attractive feature MPT3 especially for high-dimensional problems is the ability to employ complexity reduction techniques so as to control the size of the output explicit solution [228]. Hybrid toolbox has been developed by the group of Bemporad and it is capable of performing design, verification and explicit control of hybrid systems [225, 229]. The toolbox includes a solver for the solution of mp-LPs and mp-QPs that follows the approach described in Tøndel et al. [97] while for the solution of their mixed integer counterparts the exhaustive enumeration is circumvented following a backward reachability analysis on feasible modes enumeration that has been proposed by Alessio and Bemporad [116]. In a parallel effort, the group of Pistikopoulos recently launched its own parametric optimisation toolbox (POP). The key difference between POP and MPT3, apart from the existence of an intuitive graphical user interface (GUI), is its ability to efficiently handle mp-MILPs and mp-MIQPs. POP for the solution of mp-LPs/QPs offers three different algorithms that the user can choose from: (i) the geometric of Borrelli et al. [78], (ii) the combinatorial of Gupta et al. [102] and (iii) the connectedgraph approach of Oberdieck et al. [105]. Solving mp-MILPs mp-MIQPs is also more efficient with POP since the decomposition approach is implemented on top of the exhaustive enumeration of MPT3. POP also offers interfaces to state-of theart solvers as well as the ability to call MPT3 and solve problems using the mpLCP algorithm. Finally, POP is part of bigger prototype software platform, namely PAROC (PARametric Optimisation and Control) which offers an integrated framework for high-fidelity modelling in gPROMS along with model reduction techniques for more efficient solution of the related mp-Ps [162]. Recently, the implementation of multi-parametric multilevel strategies in conjunction with POP was presented by Avraamidou et al. [230]. Remark 2.2 The development of FORTRAN 90-based code for the solution of mpNLPs had been reported by Hale and Qin [231]. Hale was working on the, Parametric Optimisation PAcKage (POPAK), where the algorithms presented during her doctoral research were implemented. The development of another MATLAB toolbox for the solution of mp-NLPs using the approximate methods of parametric space partitioning has been reported by Grancharova [232]. However, since these reports no further progress towards the release or the development status of the aforementioned toolboxes has been made.

2.8 Conclusions—Quo vandis? Undoubtedly, multi-parametric programming has experienced significant flourishing in its algorithmic and theoretical aspects over the last seven decades, the main reason for that being its applied value to many contemporary problems in the field of process systems engineering. Notwithstanding, multi-parametric programs with simultaneous variations in the OFC, RHS and LHS have received considerably less

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attention in the literature due to their intrinsic computational complexity while the theoretical characterisation of their objective value still remains an open question. Moreover, despite the fact that explicit MPC is one of the best studied areas of mp-P theory, the design of explicit controller for set-point tracking remains a formidable task, for nonlinear systems, as one would have to design one controller for each set-point target given the methodologies that have been presented in the literature until now [162]. To this end, in the forthcoming two chapters algorithmic and theoretical developments for mp-(MI)LPs under simultaneous variations in the OFC, RHS and LHS, i.e. “global uncertainty”, are presented along with a new concept for the design of multi-setpoint explicit controllers and their potential value for enterprise wide optimisation problems. Finally, in Appendix B motivated by the lack of solvers for the solution of mp-Ps under global uncertainty, along with the solution of a special class of analytical mp-NLPs as discussed in Sect. 2.7, Perseus is introduced, a prototype software implementation of the algorithms that will be presented later in the thesis.

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Chapter 3

Multi-parametric Linear and Mixed Integer Linear Programming Under Global Uncertainty

In this chapter, two algorithms for the exact explicit solution of mp-(MI)LPs under global uncertainty are presented. The algorithms comprise of two key steps: (i) analytical solution of the problem’s KKT system with the uncertain parameters and the integer variables treated as symbols using Gr obner ¨ Bases and (ii) the computation of the related possibly non-convex and discontinuous CRs using Cylindrical Algebraic Decompositions on the parametric space. Problems related to process synthesis and scheduling highlight the potential of the proposed work while for the first time the functional nature of the explicit solution is theoretically characterised and proven.

3.1 Introduction An1 abundance of problems from the field of process systems engineering (PSE), such as process planning, synthesis and scheduling are formulated as (MI)LPs and thus providing a solution technique for mp-(MI)LPs under global uncertainty can significantly enhance our understanding of the systems under study. Acevedo and Pistikopoulos [1] studied the problem of plant synthesis under demand uncertainty while uncertainty in process planning has also been formulated as a parametric problem [2]. Process scheduling forms another important class of problems that has been studied through parametric programming. Ryu et al. [3] studied the scheduling of zero-wait batch processes and they considered variable processing times after the employment of linearisation techniques. Jia and Ierapetritou [4] proposed a framework for RHS uncertainty in scheduling problems that leads to the solution of an mp-MILP problem. Li and Ierapetritou [5] provided a generalised framework for process scheduling under uncertainty where depending on the topology of the uncertainty (RHS, LHS, OFC) different mixed integer mp-P problems had to be solved. 1 Parts

of this chapter have been reproduced with permission from: https://doi.org/10.1016/j. compchemeng.2018.04.015; https://doi.org/10.1002/aic.15755. © Springer Nature Switzerland AG 2020 V. M. Charitopoulos, Uncertainty-aware Integration of Control with Process Operations and Multi-parametric Programming Under Global Uncertainty, Springer Theses, https://doi.org/10.1007/978-3-030-38137-0_3

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Despite the considerable attention that mp-P has drawn from the research community the solution of mp-MILPs under global uncertainty remains one of the least studied problems due to the computational complexity involved [6]. In Table 3.1 an updated summary of the proposed algorithms for mp-MILPs is presented along with the classes of uncertainty that can be handled. In the third column of Table 3.1, the average number of explicit solution per CR is given based on computational studies reported in the corresponding papers. To the best of the author’s knowledge, no previous research work has been proposed for the exact solution of mp-MILPs without the employment of projection or discretisation techniques or through a hybrid optimisation scheme. Hereto in this chapter, two algorithms for the exact solution of mp-(MI)LPs under global uncertainty based on the principles of symbolic manipulation and semi-algebraic geometry are proposed. A significant feature of the proposed algorithms lies in the exact computation of non-convex CRs where only one globally optimal explicit solution is stored and no need for online comparison is needed. The remainder of the chapter is organised as follows. Theoretical and algorithmic foundations are firstly established for the case of mp-LPs under global uncertainty in Sect. 3.2. Next in Sect. 3.3, the algorithm is modified in order to facilitate the solution of mp-MILPs under global uncertainty. The main computational steps of the proposed methodologies are illustrated through the solution of four motivating examples in Sect. 3.4. Case studies related to process synthesis and zero-wait scheduling are also examined next. In Sect. 3.5 a short discussion on the computational complexity of the proposed algorithms is conducted and lastly concluding remarks are drawn in Sect. 3.6.

Table 3.1 Summary of mp-MILP algorithms, the uncertainty classes that can be handled and the resulting number of explicit solutions per CR Algorithms Uncertainty class Explicit solutions RHS OFC LHS per CR Acevedo and Pistikopoulos [7] Dua and Pistikopoulos [8] Li and Ierapetritou [9] Faísca et al. [10] Wittmann-Hohlbein and Pistikopoulos [11] Oberdieck et al. [12] 1 The

   

 





1 

1 1 1 2 1 1.3

no. of CRs reported in this work is based on approximate projections in the parametric space and as will be indicated later on in the chapter its correctness is not always guaranteed

3.2 Global Uncertainty in Multi-parametric Linear Programs

49

3.2 Global Uncertainty in Multi-parametric Linear Programs The main idea of the proposed algorithm is to employ the first order Karush Kuhn Tucker (KKT) optimality conditions so as to create a square system of equations and solve this system analytically using symbolic manipulation so as to compute the optimisation variables and the Lagrange multipliers as explicit functions of the uncertain parameters. At the core of the proposed algorithm, Gröbner Bases theory [13] is found because of the ability it offers to solve systems of simultaneous equations analytically.

3.2.1 A Computer Algebra Inspired Algorithm Consider the general mp-LP problem with uncertain entries in the OFC, the LHS and the RHS of the constraints, i.e. global uncertainty, given by problem (3.1). ⎧ z(θ ) = min cT (θ )x ⎪ ⎪ x ⎪ ⎪ ⎪ ⎪ ⎨Subject to : mp-LP global A(θ )x ≤ b + F(θ ) ⎪ ⎪ ⎪ ⎪ x ∈ X  {x ∈ Rnx | xkmin ≤ xk ≤ xkmax , k = 1, . . . , nx } ⎪ ⎪ ⎩ θ ∈   {θ ∈ Rnθ | θlmin ≤ θl ≤ θlmax , l = 1, . . . , nθ } (3.1) where c ∈ Rnx is the cost coefficient vector, A ∈ Rm×nx is the technology matrix and b ∈ Rm and F ∈ Rm×nθ comprise the RHS of the problem’s m-dimensional constraint vector. The solution of problem (3.1), over the parametric range specified by the set , provides the optimisers, i.e. x(θ ) = arg min cT (θ )x and the optimal value, i.e. x

z(θ ), as explicit functions of the uncertain parameters along with the corresponding CRs where each solution remains optimal. Problem (3.1) are as follows: ⎧ ⎨∇x L(x, θ ) = 0 nθ nx   (P1 ) ⎩λj ( aj,k (θ) − bj − fj,l (θ )) = 0, ∀j = 1, . . . , m k=1

where L(x, θ ) = cT (θ )x +

l=1 m  j=1

λj (

nx  k=1

aj,k (θ )xk − bj −

nθ  l=1

fj,l (θ )) is the Lagrange

function of problem (3.1). Notice that in (P1 ) the inequality constraints is assumed to include also the constraints regarding the X- domain of the optimisation variables. The total number of equation is given by: nx + m, which is sufficient to compute analytically the Lagrange multipliers and the optimisation variables in terms of the

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uncertain parameters θ, using Gröbner Bases. Solving the system of equations analytically, the optimisation variables (x(θ )) and the Lagrange multipliers (λ(θ )) as functions of the problem’s uncertain parameters, i.e. θ , can be computed. Substituting the explicit functions back in the inequality constraints given by Eqs. (3.2), (3.3), the feasibility and optimality of the candidate solutions are evaluated. Note that Eq. (3.2), refers to the non-negativity of the Lagrange multipliers, which are now parametric in θ . Equation (3.3) refers to the satisfaction of the inequality constraints A(θ )x−b−F(θ ), denoted by g(·) for short, which after the substitution of the explicit expression of the optimisers (x(θ )) are now also parametric inequalities. λj (θ ) ≥ 0, j = 1, . . . , m =⇒ optimality conditions gj (θ ) ≤ 0, j = 1, . . . , m =⇒ feasibility conditions

(3.2) (3.3)

Definition 3.1 (Candidate solution) Within the context of the present work, a solution of problem (3.1) is said to be candidate if it satisfies the system of equations given by (P1 ). Note that the candidate solutions are composed of the optimisers (x(θ )) and the Lagrange multipliers (λ(θ )) which are explicit functions of the uncertain parameters. Some of the candidate solutions may be infeasible if the conditions (3.2) or (3.3) are violated ∀θ ∈  and global/local parametric optima for some θ ∈  otherwise. Substituting a candidate solution into Eqs. (3.2), (3.3), results in a set of parametric inequality constraints that form the Critical Region (CR) of the candidate feasible solution. Due to the global uncertainty that is considered, the parametric inequalities will comprise of fractional polynomial functions of the uncertain parameters. As explained in Appendix A, Cylindrical Algebraic Decomposition can be employed so as to solve the aforementioned system of polynomial inequalities and thus compute the related CRs. Definition 3.2 (Critical regions) Within the context of the present work and given the nature of the problem and the symbolic nature of the proposed algorithm, CRs are considered as the regions of the parametric space where conditions (3.2), (3.3) are satisfied for a specific candidate solution. A CR is defined uniquely by a specific set of active/inactive constraints and may not be continuous. After the final feasible explicit solutions have been computed along with their corresponding CRs, because of the non-convex nature of the parametric problem under study, there might be some CRs that share the same part of the parametric space (overlapping CRs). In order to provide at the end the globally optimal explicit set of solutions, i.e. explicit solutions that do not overlap, a comparison procedure has to be followed. The main challenge, is that the CRs involved are in general non-convex and can be discontinuous as well.

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Table 3.2 Possible outcomes in the definition of CRINT Case 1 CR1 ⊆ CR2 which means that all constraints of CR2 are redundant and CRINT = CR1 Case 2 CR1 ⊇ CR2 which means that all constraints of CR1 are redundant and CRINT = CR2 Case 3 The CRINT is defined by a set of active constraints from both CR1 and CR2 as both CRs have some non-redundant constraints

3.2.2 Cylindrical Algebraic Decomposition and Comparison Procedure for Overlapping CRs Identifying redundant constraints and computing the new CRs within the comparison procedure is a non-trivial task, especially for non-convex problems [14]. From a numerical perspective it involves the repetitive and sequential solution of a number of slack-minimisation problems so as to identify redundant and weakly-redundant constraints and proceed with the comparison of the overlapping CRs. A comparison procedure for explicit solutions valid in the same parametric space can be found in Acevedo and Pistikopoulos [1]. This procedure is applicable only for the case of convex CRs, i.e. when the CRs are defined as a set of linear inequality constraints. In general, while solving an mp-(MI)LP problem under global uncertainty it can happen that two different parametric solutions, i.e. z1 (θ ) and z2 (θ ) to be feasible in the same parametric space. The comparison procedure aims to identify the regions where: z1 (θ) − z2 (θ) ≤ 0

(3.4)

z2 (θ) − z1 (θ) ≤ 0

(3.5)

and

given that z1 (θ ) is valid in CR1 and z2 (θ ) is valid in CR2 . The first step is to compute CRINT = CR1 ∩ CR2 , i.e. their overlap in the parametric space.

3.2.2.1

Computation of CRINT and Redundant Constraints

Excluding the case that CRINT = ∅ there are three possible outcomes in the definition of the CRINT which are described in Table 3.2. In Fig. 3.1 the different cases for the definition of the CRINT can be envisaged. For illustration purposes assume that the following two randomly generated CRs, given by Eqs. (3.6), (3.7), are under examination. It was chosen to illustrate a case that one of the CRs is convex the other one non-convex and their overlap (CRINT ) is nonconvex as well, in order to underline the salient feature of the proposed algorithm, i.e. computing exact non-convex CRs. Graphically, in the parametric space CR1 and CR2 are presented in Fig. 3.2.

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3 Multi-parametric Linear and Mixed Integer Linear Programming …

Fig. 3.1 Visual representation of the different cases where the CRINT can be identified

 CR1 =

0 ≤ θ1 , θ2 , θ3 ≤ 1 θ1 − θθ21 + 25θ3 ≥ 25

 0 ≤ θ1 , θ2 , θ3 ≤ 1 CR2 = θ3 − θ1 ≥ 0.5θ2

(3.6)

(3.7)

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Fig. 3.2 CR1 , CR2 and CRINT in the parametric space

CR1 is non-convex while CR2 is convex as polyhedral and thus previously proposed methods for computing their potential overlap are not applicable without some kind of convex approximation. Moreover, identifying redundant constraints and computing the “dominant” CRs infers a problem of solving inequalities which are quantified by logical operators (∃, ∀, ∧, ¬ etc.). It can be understood that posing the problem of computing the overlap between two CRs is equivalent to posing the question “is there any range of uncertain parameters for which any inequalities that form the CRs are simultaneously satisfied?”. This question can be in turn postulated as the following quantified mathematical formula: {∃θ|CRi ∧ CRj , for i = j} where ∧ stands for the “logical and” operator. One of the most widely known and used algorithms for the solution of quantified systems of inequalities is the Cylindrical Algebraic Decomposition (CAD) algorithm [15, 16]. In brief, one by computing the CAD of a system of inequalities after a number of projections in the decision space (the parametric space in the case of interest for the present work), partitions the space into a set of, typically non-convex, regions where each inequality retains a constant sign. For example, in a region where an inequality retains positive sign, i.e. its numerical value results in a positive quantity, but for feasibility purposes it should result in negative values one can identify the regions where the related primal and dual feasibility constraints are violated/satisfied. By doing so, one can evaluate whether a set of inequalities is satisfied within certain regions and at the end compute the final solution to the system of inequalities (in this case, a CR itself, an overlap

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3 Multi-parametric Linear and Mixed Integer Linear Programming …

among different CRs or the region of the parametric space where an explicit solution dominates another). For a detailed exposition on the subject of cylindrical algebraic decomposition the interested reader is referred to the tutorial article of Jirstrand [15]. In the present work Mathematica [17] was employed for the analytic solution of the mp-MILP under global uncertainty. Specifically, for the comparison procedure the command “Reduce” was employed which involves an implementation of the CAD algorithm. From a technical points of view, “Reduce” is a command in Mathematica that qualifies sets of conditional arguments within a given set of parameters and computes a new set within which these conditional statements are satisfied. Essentially, “Reduce” is given the primal/dual feasibility conditions, which in the case under study is a set of symbolic inequalities, and computes the regions of the parametric space where they are satisfied. An example of conditional statement is given below in Sect. 3.2.2.2 where the notion of the dominance criterion is explained. A detailed exposition on the specifics of the function can be found in Strzebonski [16] where the author details the different strategies employed internally in Mathematica. For example in the definition of the intersection of two CRs (CRINT ), “Reduce” identifies the redundant constraints of both CRs and computes the region of parametric space where both CRs exists; for the case that the CRs do not overlap the output of “Reduce” is a “False” statement equivalent to the argument CRINT = ∅. Defining the CRINT thus infers computing the CAD of the parametric space where both CR1 and CR2 are always valid and a part of its mathematical expression is given by Eq. (3.8). In Fig. 3.1 the meshed area of the parametric space represents the overlap of the two CRs.  ⎧ ⎪ 0 ≤ θ2 ≤ θ1 (20 + θ1 ) ⎪ ⎪ 0 < θ1 ≤ 0.049 ⎪ 2 ⎪ ≤ θ3 ≤ 1 0.2 − 0.04θ1 + 0.04θ ⎪ ⎪ θ1  ⎪ ⎪ ⎪ ⎪ 0 ≤ θ2 ≤ 1 ⎪ ⎪ 0.049 ≤ θ1 ≤ 0.099 ⎪ 2 ⎪ ≤ θ3 ≤ 1 0.2 − 0.04θ1 + 0.04θ ⎨ θ1 CRINT = .. (3.8) ⎪ . ⎪ ⎪ ⎪ ⎪ . ⎪ ⎪ ⎪.. ⎪  ⎪ ⎪ ⎪ ⎪ 0 ≤ θ2 < 2 − 2θ1 ⎪ ⎪ ⎩0.5 ≤ θ1 ≤ 1 θ1 + 0.5θ2 ≤ θ3 ≤ 1 The redundant constraints from each CR can be computed as RCCRi = {θ| θ ∈ (CRi ∧ (¬CRINT ))}, ∀i = 1, 2 using CAD computations.

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Algorithm 3.1 Comparison procedure for overlapping CRs which returns the final list of non-overlapping CRs (CRfin ) Input: List of CR (step 19, Algorithm 3.2) Output: CRfin 1: for i, j = 1, length(CR) : 2: if CRi = ∅ : 3: if j > i : 4: if CRj = ∅ : 5: if CRint := {θ|CRi ∧ CRj } = ∅ : 6: CRicut := {θ |CRint ∧ (zi (θ) ≥ zj (θ))} CRjcut := {θ |CRint ∧ (zi (θ) ≤ zj (θ))} 7: if CRicut = ∅ : 8: CRi := {θ |CRi ∧ ¬CRicut } 9: if CRi = ∅ : 10: Append CRi to CRfin 11: end if 12: elif CRjcut = ∅ : 13: CRj := {θ|CRj ∧ ¬CRjcut } 14: if CRj = ∅ : 15: Append CRj to CRfin 16: end if 17: end if 18: end if 19: end if 20: end if 21: end if 22: end for 23: return CRfin

3.2.2.2

Computation of CRREST and the Final Non-overlapping CRs

After the definition of the CRINT the dominance criterion can be expressed by the conditional inequality (3.9). z1 (θ ) − z2 (θ ) ≤ 0, θ ∈ CRINT

(3.9)

As a next step, excluding the case that CRINT = ∅, the comparison procedure is continued and a new set of conditional statements is qualified, given by Eq. (3.9). The output of this step is used so as to define the CRRESTi , given by Eqs. (3.10), (3.11), while the two modified CRs after the comparison procedure no longer overlap. CRREST1 = {θ | θ ∈ (CRINT ∧ (z1 (θ ) ≤ z2 (θ))} CRREST2 = {θ | θ ∈ (CRINT ∧ (z1 (θ ) ≥ z2 (θ))}

(3.10) (3.11)

Following the comparison procedure for the previous illustrative case, assume that z1 (θ ) − z2 (θ ) = −2θ1 − 19θ2 + 2θ3 − 68. In order to identify the dominant solution for the illustrative case the related CAD is computed in order to evaluate Eq. (3.9).

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The output of “Reduce” in the present case is a new set of inequalities, namely CRRest ; this is the fraction of CRINT in which z1 (θ ) ≤ z2 (θ). More specifically in the case, the explicit solution of CR1 is always dominant in the overlap of the two CRs and thus CRREST1 ≡ CRINT while CRREST2 = ∅. After the CRREST regions are computed the final CRs can be computed as follows: CR1fin = {θ | θ ∈ (CR1 ∧ (¬CRREST2 )}

(3.12)

={θ| θ ∈ (CR2 ∧ (¬CRREST1 )}

(3.13)

CR2fin

Finally, the two CRs that no longer overlap are presented graphically in Fig. 3.3, the mathematical expression of CR2 is given by Eq. (3.14) while the mathematical expression of CR1 remains the same as the one given by Eq. (3.6). Notice that z1 (θ) is globally optimal in CR1fin and z2 (θ ) is globally optimal in CR2fin .

CR2fin

 ⎧ ⎧ & θ3 ≥ 0 θ2 = 0 ⎪ ⎪ ⎪ ⎪ ⎪ = 0 θ ⎪ 1 ⎪ ⎪ ⎪ ⎪ 0 ≤ θ ≤ 1 & 0.5θ2 ≤ θ3 ⎪ ⎪ 2 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪  ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨θ > 0 θ1 (θ1 + 20) < θ2 ≤ 1 ⎪ 1 ⎪ ⎪ θ3 ≤ 1 ⎪ θ1 + 0.5θ2 ≤ θ3 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨0 ≤ θ2 ≤ 1 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ θ1 = 0.0498 0.5θ2 + 0.0498 ≤ θ3 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ ⎩ ⎪ ⎪ θ3 < 0.198 + 0.8019θ2 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎧⎧ ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨θ1 > 0.0992 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ θ1 (θ1 (θ1 + 0.48θ2 − 0.272) ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ −0.0769θ2 + 0.0153) + 0.003θ2 < 0) ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ = ⎪ ⎪ ⎨ ⎪  ⎪ ⎪ ⎪ ⎪ 0.04θ2 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪0.04θ1 + θ3 < 0.2 + θ1 & θ1 + 0.5θ2 ≤ θ3 ⎪ 0 < θ1 < 0.0498 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ θ2 < θ1 (θ1 + 20) ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ θ2 ≥ 0 ⎪ ⎪ ⎪  ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ 0.0498 < θ1 < 0.0992 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ ⎪ ⎪ ⎪ ⎪ ⎪ θ 2 ≤1 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨θ3 < 0.40318θ2 + 0.196 ⎪ ⎪⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ 0.5θ2 + 0.0992 ≤ θ3 ⎪ ⎪ ⎪ ⎪ ⎪ ⎩⎪ ⎩ ⎪ ⎪ θ1 = 0.0992 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎧ ⎪ ⎪ ⎪⎨ ⎪0 < θ1 < 0.0498 ⎪ ⎪ ⎪ ⎪ ⎪ θ2 = θ1 (20 + θ1 ) ⎪ ⎪ ⎩⎪ ⎩ θ1 + 0.5θ2 ≤ θ3 < 1

(3.14) In Algorithm 3.1 a programming routine that allows for the implementation of the comparison procedure is outlined. Notice that the command “length” refers to a high-level routine which finds the biggest dimension of a list of elements which for the cases investigated in the thesis refers to the number of alternative explicit solutions.

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Fig. 3.3 Final non-overlapping CRs in the parametric space

Remark 3.3 Another major difficulty in the global solution of general mp-LPs arises from the ability to invert parametric matrices when LHS is considered [18]. This difficulty, is because typically for the solution of general mp-LPs, one would have to solve, at least for initialisation, an NLP (with the uncertain parameters treated as free variables, thus resulting in bilinear terms), locate a feasible point, perform the required sensitivity calculations and then perturb until the entire parametric space is covered. In the present work, the optimal parametric bases are not explicitly visited. On the contrary, in this work the problem is solved analytically and the entire parametric space is explored implicitly and thus it is not needed to perform inversion of parametric matrices which is a formidable task from a computational standpoint. In Algorithm 3.2 the main steps of the proposed methodology are described while in Fig. 3.4, the outline of the algorithm is given. In Algorithm 3.2, steps (1)–(6) account for the formulation of the problem, where an empty list (LIST) is initiated for the storage of the candidate solutions that result from the step (4) of the algorithm. In the case that the problem is not infeasible, i.e. step (7), the candidate solutions are added to the LIST. As mentioned earlier in the chapter, each candidate solution corresponds to a triplet {x(θ ), z(θ ), λ(θ )} and so, the LIST will have a length equal to the number of candidate solutions computed from step (4) based on the Gröbner Bases calculations. At steps (9)–(18), the first major loop of the algorithm initiates for each of the candidate solutions where the corresponding CAD is computed based on the primal and dual feasibility conditions; if the CAD is empty, i.e. there is no θ ∈  such that the primal and dual feasibility conditions are met, the corresponding CR of the candidate solution is empty (steps (12)–(13)) and candidate solution is removed from further consideration; otherwise, the CAD provides the

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Fig. 3.4 Outline of the proposed algorithm for mp-LP under global uncertainty

3.2 Global Uncertainty in Multi-parametric Linear Programs

59

mathematical expression of the CR where the candidate solution is feasible (steps (14)–(15)). Once the CRs for the candidate feasible solutions have been computed, the second loop of the algorithm begins where overlapping CRs are identified (step (19)) and the comparison procedure as explained in Algorithm 3.1 is followed. Note that again this step involves CAD calculations. The algorithm finally terminates and returns the final list of explicit solutions along with the corresponding CRs, i.e. F.

Algorithm 3.2 Algorithm for global mp-LPs Input: Output: 1: 2: 3: 4: 5: 6: 7: 8: 9: 10: 11: 12: 13: 14: 15: 16: 17: 18: 19: 20: 21: 22: 23: 24: 25:

mp-LP problem F

(x(θ ), λ(θ), z(θ)) ← (void, void, ∞) LIST ← ∅ Formulate the Lagrangian of problem (3.1) Solve problem (P1 ) using Groebner Bases for (x(θ ), λ(θ), z(θ)) if problem (P1 ) is infeasible then : F =∅ else : Add (x(θ ), λ(θ), z(θ)) to LIST while κ ≤ length(LIST) do : for j = 1, m : Substitute xκ (θ), λκ (θ) in inequalities (3.2) − (3.3) if inequalities (3.2) − (3.3) are violated ∀ θ ∈  : CRκ  ∅ and (xκ (θ), λκ (θ)) is infeasible solution else CRκ  {θ ∈ |λκ,j (θ) ≥ 0 ∧ gj (xκ (θ)) ≤ 0} Add element (xκ (θ), zκ (θ), CRκ ) to F end if end for end while for each CRκ check CRκ ∩ CRκ  :  if CRκω ∩ CRκω (κ  =κ) = ∅ then : Perform dominance criterion according to Algorithm 3.1 end if end for end if return F

Theorem 3.4 Optimal Explicit Solution of global mp-LP Let θ be a vector of uncertain parameters and A(θ ) = Anom + A∗ θ be affine mappings with respect to θ , where Anom is the nominal part of the constraint matrix. Let also strict complementary slackness hold for each value of theta. Assume also ˘ ) matrix is invertible so that the problem’s solution is not undefined or that the A(θ ˘ )) should not be null, which practically unbounded. For the inverse to exist the det(A(θ means excluding roots of the resulting polynomial computed by the condition that ˘ det(A(θ)) = 0. When global uncertainty is considered in multi-parametric linear programming problems, the optimisers (x(θ )) and the Lagrange multipliers (λ(θ )), are piecewise fractional polynomial functions of the uncertain parameters, i.e. θ.

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Proof The Lagrangian and its gradient for problem (3.1) are given by Eqs. (3.15), (3.16): L = cT (θ ) x +

m 

λi (Ai (θ )x − bi − Fi (θ ))

(3.15)

λi Ai (θ) = cT (θ ) + λT A(θ )

(3.16)

i=1

∇x L = c (θ) + T

m   i=1

where λi, Ai (θ), bi , Fi (θ ) correspond to the elements associated with the i-th row of problem (3.1) and ∇x is the Nabla operator in the vector of optimisation variables. Because problem (3.1) is linear, the gradient of the Lagrange function with respect to the optimisation variables is an expression explicit in λ and θ , i.e. ∇x L = f (λ, θ ). The first order KKT conditions for problem (3.1) are as follows: ∇x L = cT (θ) + λT A(θ ) = 0 λi (Ai (θ)x − bi − Fi (θ )) = 0, ∀i = 1, . . . , m

(3.17) (3.18)

˘ F˘ denote the Lagrange multipliers, technology matrix, RHS ele˘ A(θ ˘ ), b, Let λ, ments respectively corresponding to active constraints, i.e. the constraints for which the corresponding Lagrange multipliers are non-zero, based on the strict complimentary slackness assumption. For active constraints Eq. (3.17) becomes: T ˘ cT (θ ) + λ˘ A(θ) =0

(3.19)

˘ ) is a square matrix of nx dimension Note that for the case of active constraints, A(θ and thus can be inverted under the assumption that the determinant of the matrix is non-zero. To this end, assume that there exist θ such that the determinant is non˘ zero and therefore A(θ) is invertible. Using the Cramer’s rule [19] the inverse of the parametric matrix can be expressed as follows: ˘ ˘ −1 (θ ) = adj(A(θ)) A ˘ )) det(A(θ

(3.20)

where adj(·) and det(·) denote the adjugate and the determinant of a matrix respectively. Solving Eq. (3.19) and using Cramer’s rule, i.e. Eq. (3.20), for λ˘ it results in: det(A˘ i (θ )) λ˘ i = −ci (θ) (3.21) ˘ )) det(A(θ which proves that the λ(θ ) is a fractional polynomial function of θ . The continuity of λ(θ) will be commented on in Theorem 3.6. Notice that in the above formula

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61

˘ i (θ)) refers to the determinant of the A matrix if the i-th column is the term det(A substituted with the RHS of the corresponding equation (Eq. (3.19)) which in this case is: −cT (θ ). Corollary 3.5 The expressions of both the numerator and the denominator in Eq. (3.21) are polynomial in aij (θ), i, j = 1, . . . , m, where aij (θ ) are the perturbed elements of the corresponding matrices. It follows that the fraction of two polynomials results in a fractional polynomial function. From Eq. (3.18) and because of the assumption for strict complementary slackness for the active constraints: ˘ )) ˘ )x − b˘ − F(θ ˘ )=0→x=A ˘ −1 (θ )(b˘ + F(θ A(θ ˘ i (θ )) det(A → xi (θ ) = (b˘ i + F˘ i (θ )) ˘ det(A(θ)) which proves that the optimiser x(θ ) is a fractional polynomial function of θ .

(3.22) 

Theorem 3.6 Critical regions of the global mp-LP The critical regions of a multi-parametric linear programming problem, when global uncertainty is considered, are semi-algebraic sets defined by fractional polynomial functions and are not necessarily convex nor continuous. Within a CR the corresponding optimiser is continuous but not necessarily continuous in the original parametric space, . A CR is the region of parametric space where each parametric solution remains optimal and feasible. Having said that the generic definition of a CR would be as follows: ˘ CR = {θ ∈ | λ(θ) ≥0 ∧ g(x) ≤ 0}

(3.23)

˘ where λ(θ) is the vector of Lagrange multipliers of the active constraints and g(x) is the vector of inactive constraints. Substituting Eqs. (3.21) and (3.22) in Eq. (3.23) leads to: CR = {θ ∈ | −ci (θ)

˘ i (θ )) ˘ i (θ)) det(A det(A Fi (θ) ≤ 0} ≥0 ∧ Ai (θ )(b˘ i + F˘ i (θ)) − bi − ˘ ˘ det(A(θ )) det(A(θ))

(3.24)

Based on Theorem 3.4 the Lagrange multipliers are fractional polynomial functions of θ and the vector of inactive constraints form a set of fractional polynomial inequalities of θ and thus the CRs are semi-algebraic sets defined by fractional polynomial functions and are not necessarily convex. As mentioned in Theorem 3.4, for the inverse of the parametric matrix to exist, the determinant of the parametric matrix corresponding to the active constraints should be non-zero. Based on

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the definition of semi-algebraic sets, such condition can be expressed in the definition of a CR and thus: the optimiser and the Lagrange multipliers are continuous within the corresponding CR but not necessarily continuous on the entire parametric space . 

3.3 Global Uncertainty in General mp-MILPs Consider now the mp-MILPs under global uncertainty. Without loss of generality assume the case where the equality constraints are replaced by opposing inequality constraints thus leading to the form of problem (Pmaster ). ⎧ ⎪ ⎪ ⎪ ⎪ ⎨

z(θ )

= min cT (θ)x + dT (θ )y x,y

(Pmaster ) = Subject to : A(θ )x + W(θ)y ≤ b(θ ) ⎪ ⎪ x ∈ X ⊆ Rnx , y ∈ {0, 1}ny ⎪ ⎪ ⎩ θ ∈  ⊆ Rnθ

(3.25)

Problem (Pmaster ) is an mp-MILP that involves uncertain parameters on the RHS, LHS and OFC. The key idea is to treat both the uncertain parameters and the binary variables as symbols and thus reduce (Pmaster ) to an mp-LP under global uncertainty at the first stage. Another idea would be to follow a decomposition scheme similar to Dua and Pistikopoulos [8] where the decision maker would iterate between the a Master MILP and slave symbolic mp-LPs; however this option is not explored in the present work as results from the case studies indicate the dimensionality of the binary variables does not affect significantly the computational complexity of the proposed scheme. The idea for the relaxation of the binary variables as uncertain parameters has been used in previous works [14, 20–22]. Treating the binary variables as uncertain parameters bounded between their respective lower and upper bound results in a relaxed mp-MILP (R-mp-MILP) which can be solved analytically. ⎧ z(θ ) = min cT (θ )x + dT (θ )y ⎪ ⎪ x,y ⎪ ⎪ ⎪ ⎪ ⎨Subject to : A(θ )x + W(θ)y ≤ b(θ ) (R − mpMILP) x ∈ X  {x ∈ Rnx | xmin ≤ xk ≤ xmax , k = 1, . . . , nx } k k ⎪ ⎪ ⎪ ny ⎪ y ∈ [0, 1] ⎪ ⎪ ⎩ θ ∈   {θ ∈ Rnθ | θlmin ≤ θl ≤ θlmax , l = 1, . . . , nθ }

(3.26)

The R-mp-MILP is an augmented mp-LP under global uncertainty where apart from the uncertain parameters the relaxed binary variables are also considered. Formulating the first order KKT conditions for the R-mp-MILP leads to the system of Eq. (3.27).

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63

⎧ ⎨∇x L(x, y, θ ) = 0 ny nx   (P) ⎩λj (y, θ )( aj,k (θ)xk + wj,k (θ )yk − bj (θ)) = 0, ∀j = 1, . . . , m k=1

k=1

where L(x, y, θ, λ) = cT (θ)x + dT (θ)y+ λT (

m   j=1

k = 1nx aj,k (θ)xk

+



(3.27)

j = 1m

ny 

w j,k

k=1

(θ)yk −bj (θ )) is the Lagrangian function of the R-mp-MILP problem. Solving (P)

analytically results in the explicit parametric expressions of the optimisation variables, i.e. x(y, θ ) and the Lagrange multipliers, i.e. λ(y, θ ) which will be used in the next step to evaluate the optimality and feasibility conditions, i.e. the non-negativity of the Lagrange multipliers and the satisfaction of the inactive constraints. The set of solutions computed at this step are called “candidate solutions”. Candidate solutions, include solutions that can be locally or globally optimal or infeasible due to constraint violation or integrality conditions. In the evaluation of the candidate solutions the first step is to consider the non-negativity of the Lagrange multipliers which would lead to the rejection of infeasible solutions. Notice that by doing so, it is avoided to visit every possible integer node and thus reduce the computational burden. As next step, the integrality conditions are imposed on the binary variables, i.e. y ∈ [0, 1]ny → y ∈ {0, 1}ny ; as a result now the Lagrange multipliers and the vector of optimisation variables are functions of the uncertain parameters. i.e. x(θ ), λ(θ) and the feasibility and optimality qualification is performed so as to compute the final “integer feasible solutions”. At the end of this step, for the “integer feasible solutions” the corresponding CRs are given by Eqs. (3.2), (3.3). Remark 3.7 When global uncertainty is considered in mp-MILPs the explicit optimisers and the optimal objective value, i.e. x(θ ) and z(θ ), are fractional polynomial functions of the uncertain parameters continuous within their respective CR but not necessarily continuous in the entire parametric space. The corresponding CRs are in general non-convex and possibly discontinuous [23]. Similar to the case of mp-LPs some CRs may co-exist in the same space and thus requiring the comparison procedure introduced in Sect. 3.2.2 so as to decide on the dominant CR in the common parametric space. A flowchart of the main steps for the exact solution of general mp-MILPs under global uncertainty is given in Algorithm 3.3 while a more elaborate description is given by Algorithm C.1 Remark 3.8 Despite the fact that in the proposed algorithm only binary variables are mentioned, the algorithm is applicable to integer variables too, as illustrated in a the work of Dua [21].

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3 Multi-parametric Linear and Mixed Integer Linear Programming …

Algorithm 3.3 Algorithm for global mp-MILPs Input: Output: 1: 2: 3: 4: 5: 6: 7: 8: 9: 10: 11: 12: 13: 14: 15: 16: 17: 18: 19: 20: 21: 22: 23: 24: 25: 26: 27: 28: 29: 30: 31: 32: 33: 34: 35: 36: 37:

mp-MILP problem : List of explicit solutions and their corresponding CRs (x(y, θ ), λ(y, θ), z(y θ)) ← (void, void, ∞) LIST ← ∅ Formulate the 1st order KKT conditions of problem (3.26) Solve problem (P) using GrR obner Bases for (x(y, θ), λ(y, θ)) if problem (P) is infeasible then : =∅ else : Add (x(y, θ), λ(y, θ)) to LIST while κ ≤ length(LIST) : for j = 1, m : Substitute (λκj (y, θ)) in inequalities (3.2) if inequalities (3.2) hold for some θ ∈  then : Keep element (xκ (y, θ), λκ (y, θ)) to LIST else : Remove element (xκ (y, θ ), λκ (y, θ)) from LIST end if end for end while while κ ≤ length(LIST) : y ∈ [0, 1]ny → y ∈ 0, 1ny (Integrality conditions on y) for j = 1, m : Substitute (xκ (y, θ), λκj (y, θ)) in inequalities (3.2) − (3.3) if inequalities (3.2) − (3.3) hold for some θ ∈  then : for ω = 1, 2ny : CRκω  {θ ∈ |λκj (θ) ≥ 0 ∧ gj (xκ (θ)) ≤ 0}  for each CRκω check CRκω ∩ CRκω :  ω ω if CRκ ∩ CRκ  = ∅ then : Proceed with comparison procedure as Algorithm (3.1) end if end for Add element (xκω (θ), zκω (θ), CRκω ) to  else CRκω  ∅ and (xκω (θ), λωκ (θ)) is infeasible solution end if end for end while end if return 

3.4 Computational Studies In this section, numerical examples are examined to illustrate the main computational steps as well as the advantages of the proposed methodology compared to the ones that have been proposed so far in the literature. Two case studies dealing with process synthesis and process scheduling under global uncertainty are also studied. All the

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65

computations were conducted on a Dell workstation with 3.70 GHz processor, 16 GB RAM and Windows 7 64-bit operating system while Mathematica 10 was chosen as computer algebra system (CAS).

3.4.1 Parametric Linear Program with LHS Unbounded Parameter In order to validate the proposed methodology the present example from Gal [24] was chosen which involves an unbounded single parameter in the first row of technology matrix. The corresponding parametric problem is given by Eqs. (3.28), (3.32): z(θ ) = max 10x1 + 30x2 + 14x3 + 5x4

(3.28)

Subject to : (1 − θ )x1 + 5x2 + 2x3 ≤ 21 (3 + 2θ )x1 + 11x2 + 4x3 + 2x4 ≤ 47

(3.29) (3.30)

x

xi ≥ 0, ∀i = 1, . . . , 4 θ ∈R

(3.31) (3.32)

Following the proposed algorithm, 14 candidate solutions are computed and given in Table 3.4.

Table 3.4 Candidate solutions of parametric linear program with LHS unbounded parameter x1

x2

x3

x4

λ1

λ2

λ3

λ4

λ5

1

0

5

–2

0

17

–5

–8−27θ

0

0

λ6 –15

2

0

0

21 2

2

0

0

0

21 5

0

0

–3

0

4

0

0

1−6θ 1−θ

0

0

89θ+16 21θ+4

0

5(9θ−4) 2(θ−1) 27θ+8 4θ+1

0

6

21 1−θ 5 4θ+1 4 21θ+4

89θ+16 21θ+4

0

0

1 2 5(2θ−1) 2(θ−1) 14θ+1 4θ+1 20(3θ−1) 21θ+4

(6θ−1) 2 (9θ−4) 2

15 2

3

5 2 2 5 89θ+16 21θ+4

1−6θ 1−θ 5(9θ−4) 2(θ−1)

7

0

0

0

47 2

0

8

0

47 11

0

0

0

9

0

0

47 4

0

0

10

47 2θ+3

0

0

0

0

5 2 5 2 5 2 7θ+3 4θ+1 10(3θ+2) 21θ+4 5 2 30 11 7 2 10 2θ+3

11

0

21 5

0

0

6

0

−2(3θ + 0 2)

–2

–5

12

0

0

21 2

0

7

0

−7θ − 3 5

0

–5

13

0

0

0

0

0

0

–10

–30

–14

–5

14

21 1−θ

0

0

0

10 1−θ

0

0

10(3θ+2) 1−θ

2(7θ+3) 1−θ

–5

5

0 0 0

0

2(27θ+8) −21θ−4

5(2θ−1) 2 20(3θ−1) 11 (14θ+1) 2

− 25

–4

0

0

− 34 11

5 11

17 2 20(3θ−1) −2θ−3

0

2

2(14θ+1) −2θ−3

1−10θ 2θ+3

0

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3 Multi-parametric Linear and Mixed Integer Linear Programming …

Table 3.5 Explicit solution of parametric linear program with LHS unbounded parameter Solution no x1 x2 x3 x4 z(θ) CR 1

− θ21 −1

2 3

47 2θ +3 5 4θ +1

4

0

0

0

0 0

0

21 5

0

89θ +16 2(θ −1)

89θ +16 2(4θ +1)

0 0 2 5

5(89θ −68) 2(θ −1) 470 2θ +3 623θ +162 4θ +1 319 2

θ ∈ (−∞, − 16 89 ] 1 θ ∈ [− 16 89 , − 14 ] 1 1 θ ∈ [− 14 , 6 ]

θ ∈ [ 16 , +∞)

Remark 3.9 In the present example, the uncertain set is unbounded. In principle, the proposed algorithm can facilitate parametric unboundedness but for larger problems that leads to excessive computations because of the increase in the possible CRs. As shown in Table 3.4, from the 14 candidate solutions the 1st, 3rd, 7th, 8th, 11th − 14th violate the non-negativity of the Lagrange multipliers and are subsequently discarded. After the 13th step of the proposed algorithm where the evaluation of the non-negativity of the Lagrange multipliers and the negativity of the inequality constraints is performed, 4 solutions are feasible and since none of them overlap, they are the final explicit solutions. In Table 3.5, the parametric solutions and the corresponding CRs are provided and are in agreement with the ones reported in Gal [24]. In order to demonstrate the computation of the CRs the steps of computing CR2 are demonstrated. CR2 corresponds to the 10th candidate solution and is formulated by the intersection of the parametric ranges defined by the inequalities (3.2), (3.3) which are as given by Eq. (3.33): ⎧ 89θ+16 − 2θ+3 ≤ 0 ⎪ ⎪ ⎪ ⎪ 47 ⎪ ≤0 − 2θ+3 ⎪ ⎪ ⎪ ⎨ 10 ≥ 0 2θ+3 20−60θ ⎪ ≥0 ⎪ 2θ+3 ⎪ ⎪ 28θ+2 ⎪ ⎪ ≥0 − ⎪ 2θ+3 ⎪ ⎩ 5−10θ ≥0 2θ+3

→ θ ∈ [−

1 16 ,− ] 89 14

(3.33)

For the sake of clarity, in Fig. 3.5 the different parametric ranges where each constraint is satisfied are given. Notice that the set of constraints that leads to the definition of the CRs are the Lagrange multipliers of the active constraints and the inactive constraints. Within Fig. 3.5, above the parametric ranges the corresponding inequality is defined as well as the resulting parametric range where it is valid. As can be observed from the Fig. 3.5, CR2 is defined as the space where all the constraints are satisfied and for illustration purposes is marked with a red rectangle. In the inequalities given by Eq. (3.33) the point θ = − 23 is point where the corresponding explicit solution cannot be defined and is not included in CR2 . Finally, in Fig. 3.6 the graph of the objective function across the corresponding CRs for this example is given. Despite the fact that each explicit solution may not be continuous in original

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Fig. 3.5 Graphical definition of CR2 of parametric linear program with LHS unbounded parameter

Fig. 3.6 Plot of the explicit objective function and the CRs in the parametric space of parametric linear program with LHS unbounded parameter

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3 Multi-parametric Linear and Mixed Integer Linear Programming …

parametric space, each continuity is preserved within its corresponding CR. Next, the case where more than one parameter are considered in the mp-LP and global uncertainty is treated.

3.4.2 Discontinuous Mp-LP with LHS Uncertainty The present example is a modified version of the one presented by Dinkelbach [25]. This example was chosen because as shown later on, the explicit solution involves fragmented CRs and a discontinuous objective function in the original parametric space. z(θ1 , θ2 ) = min 2x1 − x2

(3.34)

Subject to : (3 − θ2 )x1 + (2 − θ1 )x2 ≤ 2 −θ1 x1 − θ2 x2 ≤ −1

(3.35) (3.36) (3.37)

x1,2 ≤ 0 − 25 ≤ θ1,2 ≤ 25

(3.38) (3.39)

x

This example involves two uncertain parameters located in both rows and columns of the A matrix. First, the first order KKT conditions are calculated and the corresponding system of polynomial equations in x, λ, θ is solved using Mathematica. In principle, the procedure followed is to compute the Gröbner Bases (GB) of the polynomial system of equations and then proceed with back-substitutions so as to solve the triangular system that arises. The Gröbner Basis of this example using lexicographic ordering for the monomials with x lex λ, is given by Eq. (3.40). ⎧ ⎪ λ2 λ3 λ4 = 0 ⎪ ⎪ ⎪ ⎪ ⎪λ1 λ3 λ4 = 0 ⎪ ⎪ ⎪ ⎪ ⎪2θ1 λ1 λ2 λ4 + θ2 λ1 λ2 λ4 − 3λ1 λ2 λ4 = 0 ⎪ ⎪ ⎪ ⎪ θ1 λ1 λ2 λ3 + 2θ2 λ1 λ2 λ3 − 2λ1 λ2 λ3 = 0 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨λ4 x2 = 0 GB := θ2 λ2 λ3 x2 − λ2 λ3 = 0 ⎪ ⎪ ⎪ ⎪2λ1 λ3 + θ1 λ1 λ3 x2 − 2λ1 λ3 x2 = 0 ⎪ ⎪ ⎪ ⎪ ⎪−θ1 λ1 λ2 − 2θ2 λ1 λ2 + 2λ1 λ2 + θ1 2 λ1 λ2 x1 − 2θ1 λ1 λ2 x1 − θ2 2 λ1 λ2 x1 + 3θ2 λ1 λ2 x1 = 0 ⎪ ⎪ ⎪ ⎪ λ3 x1 = 0 ⎪ ⎪ ⎪ ⎪ ⎪ −λ + θ1 λ2 x1 + θ2 λ2 x2 = 0 ⎪ ⎪ ⎪ 2 ⎩ 2λ1 + θ2 λ1 x1 − 3λ1 x1 + θ1 λ1 x2 − 2λ1 x2 = 0

(3.40) To illustrate how the solution of the system is computed, a candidate solution will be computed herein and following similar procedure the rest of the candidate

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69

solutions are computed. Note that this procedure is done for illustration purposes and simply to highlight the role of Gröbner Bases theory in the proposed algorithm while in the implementation done in Mathematica the command “Solve” is employed. From the last row of the bracket in Eq. (3.40), using λ1 as a common factor and λ2 for the row above the last one the system is factorised as shown in Eq. (3.41).

GBfactored

⎧ λ2 λ3 λ4 = 0 ⎪ ⎪ ⎪ ⎪ ⎪ λ1 λ3 λ4 = 0 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪2θ1 λ1 λ2 λ4 + θ2 λ1 λ2 λ4 − 3λ1 λ2 λ4 = 0 ⎪ ⎪ ⎪ ⎪ θ1 λ1 λ2 λ3 + 2θ2 λ1 λ2 λ3 − 2λ1 λ2 λ3 = 0 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨λ4 x2 = 0 := θ2 λ2 λ3 x2 − λ2 λ3 = 0 ⎪ ⎪ ⎪ ⎪2λ1 λ3 + θ1 λ1 λ3 x2 − 2λ1 λ3 x2 = 0 ⎪ ⎪ ⎪ ⎪ ⎪−θ1 λ1 λ2 − 2θ2 λ1 λ2 + 2λ1 λ2 + θ1 2 λ1 λ2 x1 − 2θ1 λ1 λ2 x1 − θ2 2 λ1 λ2 x1 + 3θ2 λ1 λ2 x1 = 0 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪λ3 x1 = 0 ⎪ ⎪ ⎪ ⎪ λ2 (−1 + θ1 x1 + θ2 x2 ) = 0 ⎪ ⎪ ⎩ λ1 (2 + θ2 x1 − 3x1 + θ1 x2 − 2x2 ) = 0

(3.41) Now, let us consider the scenario for which x1 = 0 and λ3 = 0 such that the row 3 from the bottom is always satisfied. For x1 = 0, the system (3.41) allows us to compute x2 from the last row as shown in Eq. (3.42). ⎧ λ2 λ3 λ4 = 0 ⎪ ⎪ ⎪ ⎪ ⎪ λ1 λ3 λ4 = 0 ⎪ ⎪ ⎪ ⎪ ⎪ 2θ1 λ1 λ2 λ4 + θ2 λ1 λ2 λ4 − 3λ1 λ2 λ4 = 0 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪θ1 λ1 λ2 λ3 + 2θ2 λ1 λ2 λ3 − 2λ1 λ2 λ3 = 0 ⎪ ⎪ ⎪ ⎪ ⎪ ⎨λ4 x2 = 0 θ2 λ2 λ3 x2 − λ2 λ3 = 0 ⎪ ⎪ ⎪ ⎪2λ1 λ3 + θ1 λ1 λ3 x2 − 2λ1 λ3 x2 = 0 ⎪ ⎪ ⎪ ⎪ −θ1 λ1 λ2 − 2θ2 λ1 λ2 + 2λ1 λ2 + θ1 2 λ1 λ2 x1 − 2θ1 λ1 λ2 x1 − θ2 2 λ1 λ2 x1 + 3θ2 λ1 λ2 x1 = 0 ⎪ ⎪ ⎪ ⎪ ⎪ λ3 x1 = 0 ⎪ ⎪ ⎪ ⎪ ⎪ x2 = θ1 ⎪ ⎪ 2 ⎪ ⎩ x1 = 0

(3.42) Having computed the explicit expression of x2 = f(θ ), λ3 can now be computed from the 7th row of the system (3.42), under the scenario that λ1 = 0 and the procedure continues until the entire system is solved, i.e. the candidate solution has been retrieved.

70

3 Multi-parametric Linear and Mixed Integer Linear Programming … ⎧ λ2 λ3 λ4 = 0 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪λ1 λ3 λ4 = 0 ⎪ ⎪ ⎪ ⎪ ⎪2θ1 λ1 λ2 λ4 + θ2 λ1 λ2 λ4 − 3λ1 λ2 λ4 = 0 ⎪ ⎪ ⎪ θ1 λ1 λ2 λ3 + 2θ2 λ1 λ2 λ3 − 2λ1 λ2 λ3 = 0 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨λ4 x2 = 0

θ2 λ2 λ3 x2 − λ2 λ3 = 0 ⎪ ⎪ 2 2 ⎪ −θ ⎪ 1 λ1 λ2 − 2θ2 λ1 λ2 + 2λ1 λ2 + θ1 λ1 λ2 x 1 − 2θ1 λ1 λ2 x 1 − θ2 λ1 λ2 x 1 + 3θ2 λ1 λ2 x 1 = 0 ⎪ ⎪ ⎪x = 1 ⎪ ⎪ θ2 ⎪ 2 ⎪ ⎪ ⎪ x = 0 ⎪ 1 ⎪ ⎪ ⎪ ⎪ λ1 = 0 ⎪ ⎪ ⎪ ⎩ θ (2θ +1) λ3 = − 1 θ 2 2

(3.43)

The procedure described above for the computation of the candidate solutions given the Gröbner Bases is rather for illustration purposes and for the sake of clarity rather than implementation purposes. In fact, in the literature of Gröbner Bases theory, one can find rules in order to optimise the performance of the reduction procedure; such details are beyond the scope of the thesis but the interested reader is referred to the book of Buchberger and Winkler [26]. Following the steps of the proposed algorithm, two explicit solutions are computed while their corresponding CRs as given in Table 3.6. As shown in Fig. 3.7, CR1 (colored gray) is discontinuous and fragmented in two parts both involving the same explicit solution. The reason why this example was modified and solved is twofold: first, as illustrated in Fig. 3.7a, the discontinuity poses a significant challenge on the implementation of the already existing algorithms for mp-LPs as the “neighboring” property does not hold in the discontinuous instance of the parametric space. Moreover, one when solving the present example might consider that there exist 3 CRs and not 2, leading thus in unnecessary increase of the dimensions of the explicit solution. This is due to the fact that, the algorithms proposed so far in the literature of mp-P are based on solution of problems with a valued parameter vector, i.e. one has to first find a feasible point in the parametric space, collect the post-optimal information needed and then repeat this procedure until the entire feasible parametric space is covered. Following this algorithmic routine however, one would have to perform two steps for the identification of a single CR for the case of discontinuous regions.

4θ1 +5θ2 −7 θ1 2 −2θ1 −(θ2 −3)θ2

− θ12

2θ1 +θ2 −3 θ1 2 −2θ1 −(θ2 −3)θ2

1 θ2

θ1 +2θ2 −2 θ1 2 −2θ1 −(θ2 −3)θ2

0

1

2

zi (θ1 , θ2 )

x2i (θ1 , θ2 )

x1i (θ1 , θ2 )

i

Table 3.6 Explicit solutions and CRs of discontinuous mp-LP with LHS uncertainty ⎧ ⎧⎧ ⎨−11 < θ1 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ θ ⎧ 3 ⎪ 0 < θ1 ≤ 2 ⎪ ⎪ ⎪ ⎨ θ2 < 0 θ1 ≥ −2θ2  ⎪ ⎪ 2 < θ1 ≤ 25 ⎪ ⎪ ⎩ θ1 + 2θ2 ≤ 2

CRi

3.4 Computational Studies 71

72

3 Multi-parametric Linear and Mixed Integer Linear Programming …

Fig. 3.7 Visual representation of the final results for discontinuous mp-LP with LHS uncertainty

3.4.3 Mp-LP Under Global Uncertainty To illustrate the case of global uncertainty in general mp-LPs the following example is taken from Li and Ierapetritou [9] (L&I):

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73

Table 3.7 Explicit solution of mp-LP under global uncertainty example i zi CRi zi (L&I) ⎧ ⎪ ⎨ 0 ≤ θ1 ≤ 5 1 0 0 0 ≤ θ2 ≤ 5 ⎪ ⎩−5 ≤ θ ≤ 5 3 ⎧⎧ ⎪−5 ≤ θ1 ≤ − 15 ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ −1 ≤ θ2 ≤ 5 ⎪ ⎪ ⎪ ⎪ ⎩ 1 ⎪ ⎪ ⎨ −5 ≤ θ3 ≤ − θ1 2 θ1 θ1 ⎧ ⎪ ⎪ ⎪ ⎪− 15 ≤ θ1 ≤ 0 ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ −1 ≤ θ2 ≤ 5 ⎪ ⎪ ⎩⎪ ⎩−5 ≤ θ ≤ 5 3 ⎧ ⎪ 0 ≤ θ1 ≤ 5 ⎨ 3 −θ1 θ2 −θ1 θ2 −1 ≤ θ2 ≤ 0 ⎪ ⎩−5 ≤ θ ≤ 5 3 ⎧⎧ ⎪⎨ ⎪−5 ≤ θ1 ≤ −1 ⎪ ⎪ ⎪ ⎪ −1 ≤ θ2 ≤ 5 ⎪ ⎪ ⎪ ⎪ ⎪ ⎩− 1 ≤ θ < 1 ⎪ ⎪ 3 θ1 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎧ ⎪ 1 ⎪ ⎪⎪ ⎨ ⎨−1 ≤ θ1 ≤ − 5 1+θ1 +θ2 +θ1 θ2 θ3 1+θ1 +θ2 +θ1 θ2 θ3 4 −5 ≤ θ2 ≤ −1 1−θ3 1−θ3 ⎪ ⎪ ⎪ ⎩1 < θ ≤ − 1 ⎪ ⎪ 3 θ1 ⎪ ⎪ ⎪ ⎪ ⎪ ⎧ ⎪ ⎪ ⎪ ⎪− 15 ≤ θ1 ≤ 5 ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ −5 ≤ θ2 ≤ −1 ⎪ ⎪ ⎪ ⎩1 < θ ≤ 5 ⎩⎪ 3

z(θ1 , θ2 , θ3 ) = min θ1 x1 + x2 x

Subject to : −x1 + x2 + x3 =θ2 x1 −θ3 x2 + x4 = 1 xi ≥ 0, ∀i = 1, . . . , 4 − 5 ≤ θq ≤ 5, ∀q = 1, 2, 3

CRi (L&I) ⎧ ⎪ ⎨−5 ≤ θ1 ≤ 0 −1 ≤ θ2 ≤ 5 ⎪ ⎩θ θ ≥ −1 1 3 ⎧ ⎪ ⎨0 ≤ θ1 ≤ 5 −1 ≤ θ2 ≤ 0 ⎪ ⎩−5 ≤ θ ≤ 5 3 ⎧ ⎪ ⎨0 ≤ θ1 ≤ 5 0 ≤ θ2 ≤ 5 ⎪ ⎩−5 ≤ θ ≤ 5 3

⎧ ⎪ ⎨0 ≤ θ1 ≤ 5 −5 ≤ θ2 ≤ −1 ⎪ ⎩1 ≤ θ ≤ 5 3

(3.44) (3.45) (3.46) (3.47) (3.48) (3.49)

In this example the most generic case of an mp-LP problem is considered, i.e. uncertainty in the RHS, LHS and OFC simultaneously. Following Algorithm 3.2, the results are given in Table 3.7 and a comparison with results from Li and Ierapetritou [9] is presented. As observed, the CRs computed are different from the ones computed by Li and Ierapetritou [9]; this is because of the core difference between the two approaches: following the approach proposed by Li and Ierapetritou [9], the accuracy of the

74

3 Multi-parametric Linear and Mixed Integer Linear Programming …

Fig. 3.8 Separate Critical Regions of the mp-LP under global uncertainty example

approximation for the non-convex CRs arising from the LHS uncertainty is highly dependent on the projection methodology that is employed as well as the degree of discretisation of the parametric space, while in the proposed methodology exact non-convex CRs are computed from the analytic solution of the global mp-LP. For better understanding, in Fig. 3.8, the different CRs are presented and their non-convex nature is clearly shown; in Fig. 3.9a, the entire final parametric space is presented. This example was validated using GAMS [27] for evenly distributed parametric points of the 3D space in order to evaluate the correctness of the proposed methodology compared to the one by Li and Ierapetritou [9] and it was clearly illustrated that

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75

Fig. 3.9 Visualisation of the optimal partition of the parametric space for the mp-LP under global uncertainty example

the proposed methodology indeed computes exact non-convex CRs; especially for CR1 and CR4 . Note that the results computed complement the argument of Theorem 3.4 about the continuity of the optimiser and the CRs and this can be envisaged in Fig. 3.9b, where the discontinuous CR4 at θ3 = 1 is “joined” by CR2 and thus the optimiser of CR4 is continuous ∀θ ∈ CR4 but not continuous for θ3 = 1.

3.4.4 Mp-MILP Under Global Uncertainty This example is taken from Wittmann-Hohlbein and Pistikopoulos [28] and involves uncertain entries in the RHS, OFC and LHS as shown by Eq. (3.50). z(θ ) = min θ1 x1 + x2 + y1 x,y

Subject to : x1 + θ3 x2 + x4 = 1 + θ1 y2 −x1 + x2 + x3 = θ2 + 2y1 y2 − y1 ≤ 0 xi ≥ 0, ∀i = 1, . . . , 4 y1,2 ∈ {0, 1} − 5 ≤ θ1,2,3 ≤ 5

(3.50)

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3 Multi-parametric Linear and Mixed Integer Linear Programming …

Table 3.8 Explicit solutions of mp-MILP under global uncertainty example x1

x2

x3

x4

y1

y2

z(θ )

θ ∈ CR1

1

0

θ2 + 3

0

1

0

θ1 + 1

θ ∈ CR2

1

0

θ1

θ ∈ CR3 θ ∈ CR5

θ1 −θ2 θ3 −2θ3 +1 θ3 +1 −θ2 θ3 −2θ3 +1 θ3 +1 −θ2 θ3 −2θ3 +1 θ3 +1

θ2 +θ1 +3 θ3 +1 θ2 +3 θ3 +1 θ2 +1 θ3 +1

θ ∈ CR6

0

θ ∈ CR7

0

θ ∈ CR8

−θ2 − 2

0

θ ∈ CR9

−θ2

0

θ ∈ CR4

θ2 + 1

0

0

0

0

0

1

1

0

0

1

0

0

0

0

0

0

θ2 + 2

0

1

0

1

0

θ2

0

0

0

0

0

θ2 + 3

1

0

−θ1 θ2 − 2θ1 + 1

0

θ2 + 1

0

0

−θ1 θ2

θ12 −θ1 ((θ2 +2)θ3 −2)+θ2 +θ3 +4 θ3 +1 θ1 (−θ2 )θ3 −2θ1 θ3 +θ1 +θ2 +θ3 +4 θ3 +1 −θ1 θ2 θ3 +θ1 +θ2 +1 θ3 +1

The solution of the problem returns 6 candidate set of solutions and after the integrality conditions are imposed 13 integer candidate solutions are considered; note that the 13 integer candidate solutions are now parametric only in θ. Qualifying with the primal and dual optimality conditions 13 explicit solutions and CRs are computed and the comparison procedure follows next. At this step, a number of different integer solutions were found to be cost-wise identical and thus dominance in these case cannot be proven. For these cases, two different scenarios were considered. In the first one the solutions of the integer vector y = [1 1] were preferred over those with integer vector y = [1 0] and vice versa but for the sake of space only the first scenario is reported in Table 3.8. Note that the final explicit solutions are in general fractional polynomial functions of θ and the CRs are non-convex with a number of them discontinuous as shown in Fig. 3.10, e.g. CR5 . For the sake of space the mathematical expression of CR5 is omitted as it was found to be three pages long. The explicit mathematical expressions of the CRs given in Tables C.1, C.2 and C.3 show that CRs are not necessarily convex while in the present example the order of polynomials involved are up to 3. Finally, it is worth noticing that even though CR4 and CR5 are individually fragmented, at the final representation of the parametric space in Fig. C.1 the feasible solution set is compact and the objective function continuous across the different regions.

3.4.5 Case Study 1: Process Synthesis Under Global Uncertainty The present case study is a variant of a process synthesis problem presented in Biegler et al. [29]. Within the synthesis problem, uncertainty in product demand, operation cost and conversion rate, namely θ1 , θ2 and θ3 respectively is considered. As shown in Fig. 3.11, the process refers to the production of a chemical C (x5 ) which can be achieved either through process unit II or III; for the production of C, a chemical B

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77

Fig. 3.10 Visualisation of the different partitions of the parametric space for mp-MILP under global uncertainty example

(x2,3 ) needs to be converted. B, can be either purchased directly from the market (x4 ) or manufactured through process I with raw material A (x1 ) as feed. The corresponding MILP under global uncertainty is formulated as an mp-MILP as follows: z(θ ) = min 2.5x1 + (4+θ1 )x2 + 5.5x3 + 10y1 + 15y2 + 20y3 − 18x5 x,y

Subject to : 0.9x1 − x2 − x3 + x4 = 0 x5 = 0.82x2 +θ3 x3

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3 Multi-parametric Linear and Mixed Integer Linear Programming …

Fig. 3.11 Superstructure of process synthesis under global uncertainty case study

2 ≤ x5 ≤ 5+θ2 x1 ≤ 16y1 xi ≤ 30yi , i = 1, 2 y2 + y3 ≥ 1 x4 ≤ 14 0.4x1 ≤ 5+θ2 xi ≥ 0, ∀i = 1, . . . , 5 yi ∈ {0, 1}, ∀i = 1, 2, 3 0 ≤ θ1,2 ≤ 5 0.75 ≤ θ3 ≤ 0.95 The LHS uncertainty involved is located in the second equality constraint and represents uncertainty in the conversion coefficient. Solving problem (P7 ) results in 97 candidate solutions. Evaluating with the optimality and integrality conditions results in 3 integer feasible solutions. Two of these solutions are found to overlap and the comparison procedure is employed, resulting in two final optimal solutions which are given in Table 3.9 along with their corresponding CRs. The first explicit solution dictates the direct purchase of chemical B from the market and its conversion to the final product through the reactor III

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3.4.6 Case Study 2: Process Scheduling Under Global Uncertainty In order to illustrate the generality and applicability of the proposed algorithm the case of process scheduling under global uncertainty is examined. Scheduling problems have been studied in the past using multi-parametric programming techniques [3, 5, 28], however the case of simultaneous variations on the LHS, RHS and OFC has yet to be treated. The point of departure is the multi-stage zero-wait batch scheduling problem formulation as proposed by Ryu et al. [3]. The model employs a time slot based formulation for the sequencing decisions among different products. At each time slot (s) only one product (i) can be manufactured and the corresponding assignment is modeled using the binary variable ysi . The model assumes unlimited intermediate storage and thus the objective is to minimise the makespan of the process (CN,J ). z(θ) = min CS,J

(3.51)

Subject to : N 

ysi = 1

∀i

(3.52)

ysi = 1

∀s

(3.53)

s N  i

Csj ≥ Cs,j−1 +



Csj ≥ Cs−|J|,j +

iN ysi Pij

N 

ysi Pij

∀j > 1, s ∀i ≥ |J|, j

(3.54) (3.55)

i

Csj ≥ 0, ∀s, j ysi ∈ {0, 1}, ∀s, j

(3.56) (3.57)

Equations (3.52), (3.53) are used to ensure that only one product can be processed at a time in each stage while Eqs. (3.54), (3.55) are employed to compute the completion time of the time slot s in stage j (Csj ). The processing time of product i in stage j (Pij ) is considered as uncertain while equipment availability can be included by adding a new vector of uncertain parameters on the RHS of Eqs. (3.54), (3.55) . Another type of uncertainty on the LHS of Eqs. (3.54), (3.55) can be included if a time proportional to the completion time is considered as a buffer for maintenance or other reason, i.e. θsj Csj .

3.4.6.1

Two-Stage Scheduling Problem Under Global Uncertainty

Initially only 3 products and 2 stages are considered with the corresponding data given in the Appendix in Table C.4. It is assumed that the processing time for product B at stage j2 is uncertain and uniformly distributed as 4 ≤ θ1 ≤ 8. There exist two “buffer”

0

0

1

2

x1

(θ2 +5) θ3

0

x2

x4

x5

0

(θ2 +5) θ3

5 + θ2

1.05θ2 + 5.26 1.05θ2 + 5.26 5 + θ2

x3

[0 1 0]

[0 0 1]

y

Table 3.9 Explicit results of process synthesis under global uncertainty case study

(θ1 +4)(θ2 +5) θ3

− 18θ2 − 75

−12.21θ2 − 41.052

z(θ)

CRs ⎧ ⎪ ⎪0 ≤ θ2 ⎧⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪0.342 ≤ θ1 ⎪ ⎪ ⎪ ⎪⎪ ⎨θ ≤ 1.5 ⎪ ⎪ ⎪ ⎪ 1 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ θ ⎪ ⎪ 3 ≤ 0.691 ⎪ ⎪ ⎪⎪ ⎪ ⎪ ⎨ ⎨⎩ +0.173θ 1 θ ≤5 ⎪ 2 ⎪ ⎪ ⎪ ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪1.5 ≤ θ1 ⎪ ⎪ ⎪ ⎪⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪⎪θ1 ≤ 5 ⎪ ⎩ ⎩θ ≤ 0.95 ⎪ ⎪ 3 ⎪ ⎪ ⎩ 0.75 ≤ θ3 θ2⎧≤⎧ ⎪⎪ ⎪ ⎪0.342 ≤ θ1 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪θ1 ≤ 1.5 ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ 0.691+ ⎪ ⎪ ⎪ ⎪ ⎪0.173θ ≤ θ ⎪ ⎪⎪ ⎪ 1 3 ⎪ ⎪ ⎪ ⎨⎩θ ≤ 0.95 3 5 ⎪ ⎪ ⎪ ⎧ ⎪ ⎪ ⎪⎪ ⎪0 ≤ θ1 ⎪ ⎪ ⎪⎪ ⎨θ ≤ 0.342 ⎪ ⎪ 1 ⎪ ⎪ ⎪⎪0.75 ≤ θ ⎪ 3 ⎪ ⎪ ⎪ ⎩⎪ ⎩ θ3 ≤ 0.95

80 3 Multi-parametric Linear and Mixed Integer Linear Programming …

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81

Table 3.10 Effect of the dimensionality of the uncertain parameters (nθ ) on the number of CRs computed (nCR ) and the related solution time nθ nCR CPU(s) 2 3 4 5 6

1 4 4 5 5

3.83 8.46 9.24 11.56 240.38

times proportional to the completion time of the third slot of the first stage (Cs3j1 ) and the first time slot of second stage (Cs1j2 ) both uniformly distributed as 0.8 ≤ θ2,3 ≤ 1.2. Following the proposed algorithm in 2.46s, four globally optimal explicit solutions are found and their expressions are given in Table C.5. As shown in Table C.5, two optimal integer configurations of the schedule are computed throughout the range of parameter variability: C → A → B and B → A → C. Remark 3.10 Insights like this are of great importance for responsive and effective process operations as it becomes explicitly known that even if there is a significant degree of variability in the processing time of product B there is no need to change the task sequencing. For instance, in a scenario that θ1 = 6, i.e. the processing time of B at stage 2 is six hours and θ2,3 = 1 the optimal solution computed numerically by CPLEX 24.9.2 is in total agreement with the explicit solution and the optimal sequencing is C → A → B with a resulting makespan of 15 h. However, when fixing that A should be the first product to be processed the optimal sequencing is A → C → B with a makespan of 31 h, which is more than two times of the previously discussed sequence of products. A number of instances like this were investigated with the uncertain parameters been drawn randomly within their respective ranges and in no case when product A was fixed to be the first in scheduling resulted in less makespan than the two sequences computed by the explicit solution. The use of multi-parametric programming in scheduling problems is appealing due to the ability to compute offline schedules that can be readily employed once the uncertainty is realised, thus leading to more responsive operations. To this end, the effect of the dimensionality of uncertain parameters on the solution time of the proposed algorithm was examined and the corresponding results are shown in Table 3.10. Breaking down the computational burden associated with the dimensionality of the uncertain parameters it should be highlighted that the CPU time (s) needed by the first computational step of the algorithm is not affected (computation of the candidate solutions). However, the second major computational step (computation of the CRs and the comparison procedure) scales quite rapidly. This is due to the nature of the CAD computations which scale in the worst case doubly exponentially with respect to the indeterminates, i.e. the vector of uncertain parameters.

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3 Multi-parametric Linear and Mixed Integer Linear Programming …

Fig. 3.12 3 stage process for scheduling

Next, the case of 5 products scheduling of the two stage manufacturing process was studied in order to test the proposed algorithm for the case of increased dimensionality of the integer vector. This instance, involves 25 binary variables, 21 constraints and 31 continuous variables and in 4,048.3s a total of 234,600 candidate solutions were computed out of which 25,920 candidate solutions were linearly independent and thus considered for the next steps of the algorithm. The computation of the integer feasible candidate solutions returns 1,136 explicit solutions together with the related CRs in 1,900s. The final partition of the parametric space involves 3 overlapping CRs with explicit solutions that result in the same explicit objective value, CS,J (θ ) = 21 + θ2 and thus no CR can be proven to be dominant. The sequencing decisions involved in the overlapping CRs are two alternatives, more specifically, the two integer optimal sequences are: D → E → C → A → B and D → A → C → E → B. The explicit solutions are given in Table C.6.

3.4.6.2

Multi-objective Three Stage Scheduling Problem Under Global Uncertainty

Finally, the scheduling of a 3 stage process as indicated in Fig. 3.12 was examined. Related data and a more detailed description of case study can be found in the work of Ryu et al. [3] and Table C.4. The manufacture of four products was considered and the uncertainty has as follows: θ1 ∈ [10, 15] as the processing time of product B in stage 2, θ2 ∈ [0.9, 1.1] to model the possibility of a buffer time that is proportional to the completion time of time slot 4 in stage 1, θ3 ∈ [0, 4] to model equipment availability of the mixer. Lastly, a modification of the objective in a weighted sum multi-objective sense is considered where θ4 ∈ [2] indicates the different preferences of the decision maker with respect to minimising the completion time of the fourth time slot of stage 3 and 1. The algebraic model with the incorporated uncertainty is given by Eqs. (C.1), (C.13) and

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83

Table 3.11 Multi-parametric expressions of the weighted sum objective function of the three stage scheduling problem along with the related sequencing decisions CRs Objective value Schedule CRs Objective value Schedule CR1 CR2 CR3 CR4 CR5 CR6 CR7 CR8 CR9 CR10 CR11 CR12

41θ4 θ2

+ 14(1 − θ4 )

ABCDE ABDCE ABCDE ABCDE ACBDE ACDEB ACDBE ABCDE ABCDE ACBDE ACDEB ACDBE

CR13 CR14 CR15 CR16 CR17 CR18 CR19 CR20 CR21 CR22 CR23 CR24

12(1 − θ4 ) −

θ4 (−θ1 −θ3 −16) θ2

6(1 − θ4 ) −

(−θ3 −42)θ4 θ2

6(1 − θ4 ) −

(−θ3 −38)θ4 θ2

6(1 − θ4 ) −

θ4 (−θ1 −θ3 −20) θ2

12(1 − θ4 ) −

θ4 (−θ1 −θ3 −16) θ2

ABCED ABEDC BADEC BAECD CADBE CAEBD ACEDB ACEDB ABCED ABEDC ABCED ABEDC

can be found in the supplementary material. Following the proposed algorithm, the KKT system is solved and 688,320 candidate solutions are returned in 13,781.9s. Some of the solutions involve linearly dependent solutions sets, by neglecting these solutions the final full-dimensional candidate solutions are 36,863 which are explicit only in (θ, y). Screening the candidate solutions for dual and primal feasibility and computing their CRs takes 808.78s and the output involves 144 CRs. After 177.4s the comparison procedure has removed overlapping CRs that can be proven to be inferior and the final optimal explicit solution involves 24 CRs and the corresponding multi-parametric expressions of the optimisers. In Table 3.11 the explicit weighted sum function is given along with the related scheduling sequence. An example of the mathematical expressions that define the related CRs is given in Eq. (3.58), for the case of CR22 . While the detailed explicit solutions to the scheduling problem are given in Table C.7.

CR22

⎧ 0.9 ≤ θ2 ≤ 1.1 ⎪ ⎪ ⎪ θ2 ⎨ ≤ θ4 ≤ 1 := θ2 +0.67 ⎪ 0 ≤ θ3 ≤ 12 − θ1 ⎪ ⎪ ⎩ 10 ≤ θ1 ≤ 12

(3.58)

3.5 Discussion Having demonstrated the applicability of the proposed algorithm for general mp(MI)LPs under global uncertainty, in this section computational aspects and limitations will briefly be discussed.

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3 Multi-parametric Linear and Mixed Integer Linear Programming …

Table 3.12 Computational statistics of the proposed algorithm with respect to the dimensionality of the inequality constraints (ng ), continuous variables (nx ), binary variables (ny ) and uncertain parameters (nθ ) ng nx ny nθ Candidate solutions Total CPU (s) P1 P2 P3 P4 P5 P6 P7 P8 P9 P10 P11 P12 P13 P14

6 5 6 13 7 5 7 10 8 5 15 11 21 23

4 2 4 7 2 2 4 2 2 2 5 17 31 35

0 0 0 0 2 2 2 2 2 2 3 9 25 16

1 7 3 3 2 2 3 3 4 6 3 5 3 4

14 8 9 91 17 8 6 30 26 8 97 202 25,920 36,863

0.835 2.645 5.635 69.546 0.35 0.18 1.24 2.54 1.95 1.46 15.41 8.46 5,948.3 14,768.08

3.5.1 Scalability of the Proposed Algorithm Computing the exact explicit solution for mp-(MI)LPs under global uncertainty is one of the most general and challenging problems and as a result it is computationally intensive. The proposed algorithm was tested on a number of numerical examples and case studies. In Table 3.12, a summary of the problems’ statistics is provided along with the number of candidate solutions that were found. The number of candidate solutions that are parametric in y and θ grows rapidly with the number of constraints and continuous variables with more dependence on the number of constraints. This is expected since Gröbner Bases based algorithms are known to be doubly exponential on the worst case with regards to the system’s indeterminates. An interesting observation during the tests of the proposed algorithm is the symmetry involved in the initial candidate solutions and exploiting this symmetry, through an early elimination of symmetric solutions, would probably benefit the scalability of the proposed algorithm. On the other hand, as illustrated in the case studies, the number of uncertain parameters and binary variables does not affect greatly the scalability of the proposed algorithm and the reason is twofold: (i) within the proposed algorithm both of them are treated as symbols until a certain step, leaving the initial computation of the candidate solutions unaffected; (ii) for the candidate solutions computed, not all the integer nodes are explored as some of them are rejected based on the primal or dual feasibility conditions of the problem.

3.5 Discussion

85

Especially for one case study, the proposed algorithm required less than 20 comparisons between overlapping solutions while the same example for half range of uncertainty required in the best case the solution of 3331 MINLPs and one mp-LP following the algorithm proposed in Wittmann-Hohlbein and Pistikopoulos [11].2 This instance is indicative of the significant reduction in computational effort in comparison to approximation based techniques presented in the literature. Finally, as observed from the results throughout the chapter, computer algebra systems tend to spend a considerable amount of memory and computational time for the precision of the results; this problem is also know in the literature of symbolic computations as the “floating point format problem” [31]. Since the algorithm presented herein was implemented in Mathematica 10 [17], such decisions were beyond the scope of the thesis but constitute an observation that may optimise the performance of a custom-made implementation of the algorithm.

3.5.2 Non-convexity of the Underlying Problem As introduced in the Sect. 3.2 and illustrated through the computational studies, the underlying optimisation problem can be highly non-convex. The main reason is the presence of bilinear terms that appear as product between the uncertain parameters and the continuous/integer variables. As illustrated in Wittmann-Hohlbein and Pistikopoulos [28], in order to overcome this issue, global optimisation techniques should be employed that could lead to computationally intractable problems for a modest size example. The case of bilinear terms is undoubtedly one of the most well studied problem in the global optimisation literature and remains still a rather active field of research because of its frequent occurrence as part of important applications. In the present work, the treatment of bilinear terms is done through symbolic manipulation of the uncertain parameters. Furthermore, as shown, the problem can be discontinuous at some instances which further exacerbates the computational effort required. Although bilinear terms pose a tough difficulty in the solution of mp-MILPs under global uncertainty, a possibly even more tough problem nested within the solution is the definition of overlapping CRs and the comparison procedure that needs to be employed for its treatment. As discussed previously, in the most general case the optimisers and thus the optimal explicit value is a fractional polynomial function of the uncertain parameters. Previous works have proposed to store overlapping solutions in “parametric envelopes” where two solutions are stored and the best one is chosen via online function evaluation. Although this could be a possible solution, it is not the optimal one as it still requires an additional evaluation procedure for the decision maker. In order to overcome this issue, the conventional polyhedral based definition of CRs was generalised as “semi-algebraic sets”. Defining the CRs 2 Details

al. [30].

about the related case study and its explicit solution can be found in Charitopoulos et

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3 Multi-parametric Linear and Mixed Integer Linear Programming …

as semi-algebraic sets where a certain number of conditions hold, in conjunction with the symbolic manipulation it was possible to efficiently compare overlapping solutions, characterising the overlap and most importantly computing the exact nonconvex CRs. This is due to the fact that a semi-algebraic set can be manipulated in a disjunctive way and thus divide a large complex CR into more simple one to ease the complexity of the calculations and at the end reconnect them as a union.

3.6 Concluding Remarks In this chapter, two novel algorithms for the solution of general mp-(MI)LPs that are subject to global uncertainty have been presented. The main motivation was the extended existence of uncertainty in optimisation problems either in the extrinsic data of the system, e.g. demands, resources availability, prices etc., that lead mostly to RHS and OFC uncertainty but also intrinsic data of the system such as stoichiometric coefficients or transition times that result in LHS uncertainty. Multi-parametric programming can handle, nowadays, uncertainty in the RHS and OFC but the main bottleneck is when uncertainty is present on the LHS and to address this was the main aim of this chapter. Using symbolic manipulation software to analytically solve the square system of equations derived by the KKT conditions, the exact solution of the general mp(MI)LPs was computed together with the corresponding non-convex CRs. Through a number of computational studies the applicability and generality of the proposed framework as well as some instances that the proposed framework outperforms in accuracy and/or computational complexity than other proposed algorithms in the literature was demonstrated. Of particular interest for the thesis is the solution of scheduling problems under global uncertainty for which however only moderate instances were computationally possible to be solved. Nonetheless, it should be reported that despite the merits of this work, its applicability at the moment is highly dependent upon the mathematical software used to conduct the calculations and as a result the size of problems that can be solved are of small to medium scale; note that for the case of LHS in mp-P problems only small size problems have been solved because of the computational complexity involved. In the following chapter, the proposed algorithm is refined so as to facilitate the solution of mp-NLPs under global uncertainty while its application to explicit MPC problems is discussed.

References 1. Acevedo J, Pistikopoulos EN (1997) A multiparametric programming approach for linear process engineering problems under uncertainty. Ind Eng Chem Res 36(3):717–728

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2. Pistikopoulos EN, Dua V (1998) Planning under uncertainty: a parametric optimization approach. In: 3rd international conference on foundations of computer-aided process operations. AIChE 3. Ryu J-H, Dua V, Pistikopoulos EN (2007) Proactive scheduling under uncertainty: a parametric optimization approach. Ind Eng Chem Res 46(24):8044–8049 4. Jia Z, Ierapetritou MG (2006) Uncertainty analysis on the righthand side for MILP problems. AIChE J 52(7):2486–2495 5. Li Z, Ierapetritou MG (2007) Process scheduling under uncertainty using multiparametric programming. AIChE J 53(12):3183–3203 6. Oberdieck R, Diangelakis NA, Papathanasiou MM, Nascu I, Pistikopoulos EN (2016) Pop– parametric optimization toolbox. Ind Eng Chem Res 55(33):8979–8991 7. Acevedo J, Pistikopoulos EN (1999) An algorithm for multiparametric mixed-integer linear programming problems. Oper Res Lett 24(3):139–148 8. Dua V, Pistikopoulos EN (2000) An algorithm for the solution of multiparametric mixed integer linear programming problems. Ann Oper Res 99(1–4):123–139 9. Li Z, Ierapetritou MG (2007) A new methodology for the general multiparametric mixedinteger linear programming (MILP) problems. Ind Eng Chem Res 46(15):5141–5151. https:// doi.org/10.1021/ie070148s 10. Faísca NP, Kosmidis VD, Rustem B, Pistikopoulos EN (2009) Global optimization of multiparametric MILP problems. J Global Optim 45(1):131–151 11. Wittmann-Hohlbein M, Pistikopoulos EN (2012) On the global solution of multi-parametric mixed integer linear programming problems. J Global Optim 57(1):51–73 12. Oberdieck R, Wittmann-Hohlbein M, Pistikopoulos EN (2014) A branch and bound method for the solution of multiparametric mixed integer linear programming problems. J Global Optim 59(2–3):527–543 13. Buchberger B (2001) Gröbner bases and systems theory. Multidimens Syst Signal Process 12(3–4):223–251 14. Charitopoulos VM, Dua V (2016) Explicit model predictive control of hybrid systems and multiparametric mixed integer polynomial programming. AIChE J 62(9):3441–3460 15. Jirstrand M (1995) Cylindrical algebraic decomposition-an introduction. Linköping University, Linköping 16. Strzebonski A (2000) Solving systems of strict polynomial inequalities. J Symb Comput 29(3):471–480 17. Wolfram (2014) Mathematica 10.0, 1998-2014 Wolfram research 18. Khalilpour R, Karimi I (2014) Parametric optimization with uncertainty on the left hand side of linear programs. Comput Chem Eng 60:31–40 19. Bellman R (1997) Introduction to matrix analysis, vol 19. SIAM, Philadelphia 20. Gueddar T, Dua V (2012) Approximate multi-parametric programming based B&B algorithm for MINLPs. Comput Chem Eng 42:288–297 21. Dua V (2015) Mixed integer polynomial programming. Comput Chem Eng 72:387–394 22. Charitopoulos VM, Papageorgiou LG, Dua V (2017) Nonlinear model-based process operation under uncertainty using exact parametric programming. Engineering 3(2):202–213 23. Charitopoulos VM, Papageorgiou LG, Dua V (2017) Multi-parametric linear programming under global uncertainty. AIChE J 63(9):3871–3895 24. Gal T (1995) Postoptimal analyses, parametric programming and related topics. Walter de Gruyter, Berlin 25. Dinkelbach W (1970) Sensitivitätsanalysen und parametrische programmierung. J Appl Math Mech 50(6):434–435 26. Buchberger B, Winkler F (1998) Gröbner bases and applications, vol 251. Cambridge University Press, Cambridge 27. McCarl B, Meeraus A, Van der Eijk P (2012) Mccarl expanded GAMS user guide version 23:6 28. Wittmann-Hohlbein M, Pistikopoulos EN (2012) A two-stage method for the approximate solution of general multiparametric mixed-integer linear programming problems. Ind Eng Chem Res 51(23):8095–8107

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29. Biegler LT, Grossmann IE, Westerberg AW (1997) Systematic methods for chemical process design. Prentice Hall, Old Tappan 30. Charitopoulos VM, Papageorgiou LG, Dua V (2018) Multi-parametric mixed integer linear programming under global uncertainty. Comput Chem Eng 31. Fousse L, Hanrot G, Lefèvre V, Pélissier P, Zimmermann P (2007) MPFR: A multiple-precision binary floating-point library with correct rounding. ACM Trans Math Softw 33(2):13

Chapter 4

Towards Exact Multi-setpoint Explicit Controllers for Enterprise Wide Optimisation

In this chapter, the applicability of the algorithm presented in the previous chapter, is showcased for the solution of a special class of multi-parametric programming nonlinear problems. The main focus of the chapter, however is not the algorithmic advances themselves, but the study of the concept of “multi-setpoint explicit controller” and their potential use in problems of enterprise wide optimisation in which the grade of control is involved.

4.1 Introduction The need for rigorous and reliable optimisation models for efficient and profitable decision making in the process industries has been underlined by a considerable amount of researchers throughout the years [1]. In the recent years, the need for better and tighter integration among the different layers of decision making has been highlighted again through the concept of Enterprise Wide Optimisation (EWO) [2]. Given the current increasingly dynamic and competitive market conditions it is crucial for modern process industries to operate efficiently so as to maximise their profitability [3]. EWO is promising more efficient operations due to better use of information that is readily available among the different levels of decision making; however it comes at a cost. Because of its multi-scale nature, EWO results in computationally challenging problems that are difficult to solve. Specifically, the consideration of control objectives within the EWO framework usually leads to (mixed integer) nonlinear programming problems which pose an additional degree of complication. On the other hand, advanced process control (APC) problems within the context of EWO are ubiquitous. As shown in Fig. 4.1, two of the most important functionalities of the process industries exchange directly information with the APC. Real time optimisation (RTO) and process scheduling are interdependent with the decisions made by the APC level as there is reciprocal information flow. © Springer Nature Switzerland AG 2020 V. M. Charitopoulos, Uncertainty-aware Integration of Control with Process Operations and Multi-parametric Programming Under Global Uncertainty, Springer Theses, https://doi.org/10.1007/978-3-030-38137-0_4

89

90

4 Towards Exact Multi-setpoint Explicit Controllers …

Fig. 4.1 Interdependent levels of decision making with APC within the EWO scope

RTO deals with methods that manipulate the underlying dynamical systems of a process so as to optimise the profitability by manipulating quantities such as feed composition or efficiencies. Process scheduling deals with the optimal allocation of limited resources for the completion of competing tasks. Both of them however need and provide information to the level of APC so as to guarantee optimal decision making; as a result different methods on their integration have been proposed by the research community. As indicated in Fig. 4.1, the three problems involve different time scales ranging from days to seconds and due to the consideration of continuous dynamics that need to be discretised, their integration leads to large scale typically non-convex problems which are difficult to tackle. Another common pitfall, especially for the integration of RTO and APC, is the use of different models for the same system. This inconsistency stems from the need to solve control problems at fast rates, with APC employing a linearised version of the original nonlinear model that RTO considers. As a result issues with non-reachable states or suboptimal trajectories arise [4]. Darby et al. [5] conducted a review on the integration of RTO and model predictive control (MPC) and suggested that for their integration to be successful common vulnerabilities such as model mismatch and conflicting objectives between the two problems should be eliminated. Of course when real industrial processes are considered, where model degradation and other sources of mismatch manifest, functionalities such as data reconciliation and parameter estimation are involved in the integration of RTO and APC as shown in Fig. 4.2. The formulations of the RTO problem with respect to MPC can be in general classified in three groups: (i) static RTO (s-RTO), (ii) dynamic RTO (d-RTO) and (iii) economic MPC (e-MPC). The first refer to two-layered schemes were the RTO problem provides the reference trajectories to MPC [6]. While s-RTO is solved periodically once new information becomes available or steady state is achieved, D-RTO considers explicitly the transient behavior of the plant and requires the solution of dynamic rather than steady state optimisation problems. On the other hand, e-MPC [7] constitutes a family of single-layer strategies that allow for economic considerations within the feedback control structure. Inclusion of the gradient of the economic objective function with the cost function MPC was proposed by De Souza et al. [8] as another single layer strategy of integrating RTO and MPC. Even though their approach was shown to efficiently handle disturbance rejection, the way the weighted sum of the objective function is formulated might affect the

4.1 Introduction

91

Fig. 4.2 Conceptual block diagram of functionalities involved between APC and RTO

stability of the dynamic scheme. Adetola and Guay [9] proposed an alternative way for the integration in which steady state identification along with set-based parameter estimation and robust MPC are integrated in one/two-layer schemes for the case of uncertain constrained nonlinear systems. Furthermore, for the case of uncertain systems [10] studied different model adaptation strategies and argued in favor of the modifier adaptation method. The present work is motivated by the aforementioned and attempts to address common vulnerabilities at the control level from a multiparametric programming perspective. A special class of multi-parametric nonlinear problems (mp-NLPs) that involve uncertain parameters on the right hand side (RHS), left hand side (LHS), the objectives function’s coefficients (OFC) is examined and analytical solutions are computed. The algorithm presented in Chap. 3 is refined and a novel framework for the design of multi-setpoint explicit controllers is proposed with EWO considerations. The scope of this chapter is limited to the design of the EWO-oriented controllers rather than their integration with the RTO or scheduling layers. Nonetheless, their integration with the scheduling layer will be showcased and discussed in detail in Chap. 6. The remainder of the chapter is organised in the following manner: in Sect. 4.2 some preliminary background concepts are introduced and then in Sect. 4.2.2 the proposed algorithm is presented along with the concept of multi-setpoint explicit controllers. In Sect. 4.3 two case studies are examined so as to elucidate the main computational steps of the proposed methodology and lastly, in Sect. 4.4 concluding remarks are drawn.

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4 Towards Exact Multi-setpoint Explicit Controllers …

4.2 Multi-parametric Algorithms and Explicit MPC Despite the fact that explicit MPC is one of the best studied areas of mp-P theory, the design of explicit nonlinear controllers for set-point tracking remains a rather difficult task as one would have to design one controller for each set-point target given the methodologies that have been presented in the literature until now [11]. The main reason for that is the lack of algorithms that can facilitate the existence of parametric variations in the OFC even for convex mp-NLPs as indicated by Table 4.1.

4.2.1 Multi-setpoint Explicit Controllers via Multi-parametric Programming In this section the idea behind the design of a multi-setpoint explicit controller for nonlinear systems is presented. Within the context of explicit MPC, one considers as uncertain parameters the initial states of the system at each sampling instance and thus an mp-P problem with RHS uncertainty is formulated [21–23]. The solution of this mp-P leads to the computation of the control law, i.e. the optimal control input as explicit function of the state of the system together with the regions where each expression holds as discussed earlier in Sect. 2.5.2. Enterprise wide optimisation problems, especially when continuous processes are considered, require instantaneous information sharing across the different levels of decision making. The setpoints that model based controllers need to track, are usually provided by the grade of RTO or as recent research works indicate the functionality of scheduling [5, 24]. When multiple set-points need to be considered there are two ways of designing the explicit controller(s). The first one, is to solve nsp mpP problems, where nsp is the number of set-points considered and thus design nsp

Table 4.1 Overview of mp-NLP algorithms mp-NLP algorithm Uncertainty RHS OFC Dua and Pistikopoulos [12] Johansen [13] Acevedo and Salgueiro [14] Johansen [15] Dua et al. [16] Fotiou et al [17] Narciso [18] Dominguez and Pistikopoulos [19] Charitopoulos and Dua [20]

        

– – – – – – – – –

LHS

Remarks

– – – – – – – – –

Convex Convex Convex Convex Non-convex Polynomial Convex Convex Polynomial

4.2 Multi-parametric Algorithms and Explicit MPC

93

Fig. 4.3 Conceptual representation of a multi-setpoint explicit controller (on the right) in comparison to the conventional paradigm of designing multiple explicit controllers for different setpoints (on the left)

explicit controllers. The second alternative, which is the one presented in this chapter, is to design a multi-setpoint explicit controller; the idea is to solve only one mp-P problem for nsp set-points and create a “multi-layer” controller as shown conceptually in Fig. 4.3. Traditionally, for the case of nonlinear explicit MPC that were designed for setpoint tracking, one would solve the corresponding mp-P with the setpoint as a given parameter and then if a new setpoint needed to be tracked another mp-P would arise and so on. However, given the increasingly volatile market conditions the process industry is facing, the need to operate at the optimal conditions leads to new setpoints thus hindering the use of mp-MPC. As shown in Fig. 4.3, in this work by treating the setpoints as uncertain parameters allowed to vary within prespecified bounds, the aforementioned drawback of mp-MPC can be addressed and the consideration of multiple setpoints through the solution of a single mp-NLP is enabled. Designing a multi-setpoint explicit controller mathematically can be expressed as in Eq. (4.1). ⎧ P H −1 ⎪ ⎪ ⎪ Υ (x(tk ), xsp ) = min L(xt , ut , xsp ) + E(xN , xsp ) ⎪ ⎪ u ⎪ t=0 ⎪ ⎪ ⎪ Subject to : xt|t=0 = x(tk ) ⎪ ⎪ ⎪ ⎪ ⎪ t = 0, 1, . . . , PH − 1 xt+1 = f (xt ,ut ) ⎪ ⎨ t = 0, 1, . . . , PH − 1 yt+1 = h(xt ,ut ) ⎪ ⎪ ⎪ ≤ α t = 0, 1, . . . , PH Ax t ⎪ ⎪ ⎪ ⎪ ⎪ Byt ≤ β t = 0, 1, . . . , PH ⎪ ⎪ ⎪ ⎪ Cu ≤ γ t = 0, 1, . . . , PH ⎪ t ⎪ ⎪ ⎩ nsp xsp ∈ R

(4.1)

94

4 Towards Exact Multi-setpoint Explicit Controllers …

The notation adopted is identical to the mathematical formulation introduced in Sect. 2.5.2, apart from the following: treating the set-points as uncertain parameters results in an mp-P problem with simultaneous RHS and OFC uncertainty. As shown by Eq. (4.1), apart from the vector of the initial states (x(tk )) the various set-points (xsp ) are considered as uncertain parameters as well. Furthermore, the symbolic nature of the proposed algorithm that is employed in this chapter, allows for the following degree of freedom. One can consider the various set-points as an uncertain parameter bounded as shown by Eq. (4.2). lo up ≤ xsp ≤ xsp xsp

(4.2)

up

lo , xsp represent the lower and upper bounds on the set-points set for the where xsp controller. However, within a computer algebra environment, one can actually perform computations either in continuous or a discrete sets fashion as indicated by Eq. (4.3). Thus the uncertain parameters involved in the mp-P problem for the design of the multi-setpoint explicit controller can be treated in either way. n

1 2 , xsp , . . . , xspsp } xsp ∈ {xsp

(4.3)

It is interesting to notice that, within the context of EWO, being able to design this kind of controllers is of great importance because the set-points targets are calculated dynamically by the layer of real-time optimisation (RTO). Thus, having at hand a universal controller that can readily substantiate the decisions made by the RTO or the layer of process scheduling can be of great importance. More specifically, for the integration of scheduling and control it is assumed that the production rate and the changeover times are explicitly dependent on the dynamic state of the system. One by manipulating the dynamic state of the system can enforce the manufacture of a certain grade of products while ensuring the minimisation of off-spec material and in a multi-product setting this translates into multiple setpoints that the system should be controlled for. On the other hand, given the volatile market conditions that contemporary industries have to endure, their economic performance depends heavily on a responsive RTO-control integration. Within this class of problems, one would solve a steady state problem that involves the dynamics of the production system and the maximisation of profit as the key objective. The solution of the latter would be a number of setpoints to be passed to the layer of advanced process control so as to regulate the system accordingly. In the following section, the algorithm from Chap. 3 is revisited to facilitate the design of such multi-setpoint explicit controllers.

4.2 Multi-parametric Algorithms and Explicit MPC

95

4.2.2 An Algorithm for the Global Solution of mp-NLPs The problem under study in this chapter is the multi-parametric nonlinear program as shown by Eq. (4.4). z(θ ) =

min f (x, θ )

x∈Rnx

Subject to : g(x, θ ) ≤ 0 θ ∈ Rnθ

(4.4)

where θ stands for the vector of uncertain parameters which is nθ −dimensional, x is the nx −dimensional vector of continuous decision variables, g(x, θ ) is the vector of inequality constraints which can be either linear or nonlinear and f is the objective function which can be either linear or non-transcendental nonlinear mapping Rnx ×nθ → R, both of which are assumed to be at least C 2 (twice continuously differentiable). Based on the aforementioned and to the best of the author’s knowledge no previous research work has been done on the solution of mp-NLPs that involve simultaneous variations on the RHS and OFC. The algorithm proposed herein, is able to solve mp-NLPs of moderate size under the assumption that the nonlinear terms involved are not transcendental, i.e. they have a closed-form analytical solution in an algebraic sense. It should be mentioned that this algorithm can be considered as a generalisation of a previous work presented by the author for multi-parametric Mixed Integer Polynomial programming (mp-MIPOPT) problems [20] and the one proposed by Fotiou et al. [25] for the polynomial case. Both of the aforementioned algorithms were devised only for the case that uncertain parameters are considered on the RHS of the constraints and the latter does not provide the final optimal partition of the parametric space. The key idea of the mp-NLP algorithm proposed is along the lines of Algorithm 3.2 and can be summarised as follows. Given an mp-NLP, formulate the first order KKT conditions and solve the resulting square system of nonlinear equations using Gr¨obner bases while treating the uncertain parameters as symbols. This step results in a set of candidate solutions which are parametric in θ and include infeasible solutions, local and global optima. For the candidate solutions computed, qualify with the primal and dual feasibility together with a constraint qualification and remove the infeasible explicit solutions. Finally, perform a comparison procedure and keep only the globally optimal solutions along with their corresponding CRs. Remark 4.1 The proposed algorithm can facilitate problems that also involve LHS uncertainty as will be illustrated later in the chapter. However, the solution of such problems tends to be computationally intensive and thus restrictive for the applicability of the proposed algorithm to large-scale systems given the current state of the art in computer algebra systems (CAS). In Algorithm 4.1, an outline of the proposed methodology is presented. The same comparison procedure as the one presented in Chap. 3 is followed. It is interesting

96

4 Towards Exact Multi-setpoint Explicit Controllers …

to note that even though the aforementioned algorithms were initially developed for polynomial systems, recent developments allow for their employment on systems of non-polynomial equations [26–28]. The key idea is to substitute the original nonpolynomial terms with a new set of variables and equations that ensure the agreement of the computed solution.

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

21

Input: f, g, x, θ, Θ Output: x(θ), CRs Formulate 1st order KKT conditions of mp-NLP Solve the 1st order KKT conditions of mp-NLP using Gro ¨bner Bases TEMP ← solutions, i.e. x(θ), λ(θ), z(θ) if TEMP = ∅ then mp-NLP is infeasible. else for ( i ∈ range(1, . . . , Length[TEMP]) ) { Evaluate with a first order constraint qualification, e.g. Linear Independence Constraint Qualification (LICQ) Evaluate with primal and dual feasibility conditions (Cylindrical Algebraic Decomposition computation): CR = {∃ θ such that [λi (θ) ≥ 0] ∧ [gi (θ) ≤ 0]} if CR = ∅ then Candidate solution i is infeasible and discard from TEMP. else Candidate solution i is feasible and append CRi to TEMP. i+=1 for ( (k, j) ∈ range(1, . . . , Length[TEMP]) ∧ k = j ) { Identify if any overlapping CRs exist (Cylindrical Algebraic Decomposition computation): CRint = {θ|CRk ∧ CRj } if CRint = ∅ then The two CRs are not overlapping else Follow the comparison procedure from Algorithm 3.2 so as to remove the overlaps Collect the final non-overlapping CRs and the corresponding explicit solutions, i.e. x(θ)

4.2.3 Motivating Example To illustrate the main computational steps of the proposed methodology the following modified numerical example from Dominguez et al. [29] that involves 3 uncertain parameters on the RHS, LHS of the constraints and the OFC is examined:

4.2 Multi-parametric Algorithms and Explicit MPC

97

z(θ ) = min x + 2x12 − 5x1 + x22 − 3θ1 x2 − 6 x1, x2

Subject to : 2x1 + x2 ≤ 2.4 − θ2

(4.5)

0.5θ3 x1 + x2 ≤ 1.5 x1,2 ≥ 0, x2 ≥ 0 0 ≤ θ1 ≤ 6, 0 ≤ θ2 ≤ 4, 0 ≤ θ3 ≤ 2 Initially, the first order KKT system of problem (4.5) is formulated and results in the square system of Eqs. (4.6), (4.10) ∇x L(x, λ, θ ) = 0

(4.6)

λ1 (2x1 + x2 − 2.5 + θ2 ) = 0 λ2 (0.5θ3 x1 + x2 − 1.5) = 0

(4.7) (4.8)

λ3 (−x1 ) = 0 λ4 (−x2 ) = 0

(4.9) (4.10)

where L(x, λ, θ ) is the Lagrangian function of problem (4.5). Equations (4.6), (4.10) postulate a square system of equations which are solved analytically using symbolic manipulation techniques so as to compute the Lagrange multipliers and the optimisation variables as functions of the uncertain parameters, i.e. λ(θ ) and x(θ ) respectively. The solution of the KKT systems using Gr¨obner bases takes 0.078s and returns 15 candidate solutions which are shown in Tables 4.2, 4.3. These candidate solutions satisfy the square KKT system however they have to be evaluated with primal and dual feasibility conditions so as to keep only feasible ones. In order to evaluate the primal and dual feasibility of the candidate solutions, following the proposed algorithm, the cylindrical algebraic decomposition of the corresponding disjunctions is computed, which practically poses the following question: “is there a region in the parametric space within which the candidate solution has positive Lagrange multipliers and satisfy the inequality constraints?”. If the outcome of this step is an empty set, then the candidate solution violates the principles of optimality or feasibility, otherwise it results in a set of inequality constraints that form the CR of the candidate solution. For the motivating example it takes 5.33s for all the CADs to be computed and 9 of them are non-empty, thus from the 15 candidate solutions 9 are feasible. Even though these 9 solutions satisfy the primal and dual feasibility conditions, that does not guarantee their global optimality because of the non-convex nature of the underlying parametric problem and for that reason the algorithm continues with identification of possible overlapping CRs. It was found that CR10 and CR11 were overlapping as illustrated in Fig. 4.4 where the overlap (CRint ) is indicated as the black region between the two meshed CRs in the parametric space. In order to eliminate the overlap the comparison procedure is followed and the following logic preposition is given for the CAD computation as shown by Eqs. (4.11), (4.12).

0

0

0.7863

0.7863

5

6

7

8

0

2(θ2 −1) θ3 −4

3 θ3

11

12

13

15

14



0.041 θ3 θ34 − 72θ1 θ3 + 16θ32 + 72θ3 + 304 + θ3 + 8θ3 + 36 

0.041 −θ3 θ34 − 72θ1 θ3 + 16θ32 + 72θ3 + 304 + θ3 + 8θ3 + 36

0.5(−2θ2 θ3 +5θ3 −12) θ3 −4

0.25(5 − 2θ2 )

10



0.083 − θ34 − 72θ1 θ3 + 16θ32 + 72θ3 + 304 − θ32 − 8 

0.083 θ34 − 72θ1 θ3 + 16θ32 + 72θ3 + 304 − θ32 − 8

0

√ −0.57735 −6θ1 − 4θ2 + 27 − 2   √ 0.333 1.732 −6θ1 − 4θ2 + 27 − 6

9

1.5θ1   √ 0.5 2.309 −6θ1 − 4θ2 + 27 − 2θ2 + 13   √ 0.166 −6.928 −6θ1 − 4θ2 + 27 − 6θ2 + 39

0

1.5θ1

0

0

0

0

3θ1 θ33 −24θ1 θ32 +48θ1 θ3 +24θ22 +2θ2 θ33 (θ3 −4)3 −8θ2 θ32 +16θ2 θ3 −112θ2 −5θ33 +22θ32 +16θ3 −72 (θ3 −4)3

  √ 0.33 −6.92 −6θ1 − 4θ2 + 27 + 9θ1 + 6θ2 − 39 √ 2.309 −6θ1 − 4θ2 + 27 + 3θ1 + 2θ2 − 13

0.0312 −12θ22 + 92θ2 − 75

0

0

0

0

0

0

4

3θ1 + 2θ2 − 5

0

3

0 0

0.5(5 − 2θ2 )

1.5θ1

−2.119

λ1

1.5

0

−2.119

2

x2

1

Cand. x1 sol.

Table 4.2 Candidate solutions of the motivating mp-NLP under global uncertainty example (I)

98 4 Towards Exact Multi-setpoint Explicit Controllers …

−

12

15

14

13

0 0 3(θ1 − 1) 0 0 0 0 0 0 0 0

1 2 3 4 5 6 7 8 9 10 11

0.083(−θ3 θ34 − 72θ1 θ3 + 16θ32 + 72θ3 + 304 + 36θ1 − θ33 − 8θ3 − 36) 0.083(θ3 θ34 − 72θ1 θ3 + 16θ32 + 72θ3 + 304 + 36θ1 − θ33 − 8θ3 − 36)

  2 6θ1 θ32 −48θ1 θ3 +96θ1 +12θ22 3 (θ3 −4)   2 4θ2 θ32 −8θ2 θ3 −56θ2 −15θ32 +96θ3 −132 − 3   (θ3 −4) 2 5θ32 −12θ3 −27 3 θ3

λ2

Cand. Sol.

0

0

0

0

0 0 0.5(3θ1 θ3 − 3θ3 − 10) 6θ1 + 4θ2 − 15 –5 –5 0 0 0 0 0

λ3

Table 4.3 Candidate solutions of the motivating mp-NLP under global uncertainty example (II) λ4

0

0

10θ32 −3θ1 θ33 +−24θ3 −54 θ33

0

−3θ1 0 0 0 −3θ1 0 −3θ1 0 0 0 0.0312(92θ2 − 96θ1 − 12θ22 − 75)

4.2 Multi-parametric Algorithms and Explicit MPC 99

100

4 Towards Exact Multi-setpoint Explicit Controllers …

Fig. 4.4 Overlapping CRs in motivating mp-NLP under global uncertainty example

∃θ | {θ1 , θ2 , θ3 } ∈ CRint ∧ zCR10 (θ ) ≤ zCR11 (θ)

(4.11)

∃θ | {θ1 , θ2 , θ3 } ∈ CRint ∧ zCR10 (θ ) ≥ zCR11 (θ)

(4.12)

or

where zCRi (θ ) stands for the explicit optimal value within the CRi . The outcome of this computational step is the region of the parametric space where each CR is dominant; in the particular case it was found that CR10 was dominant throughout the entire overlapping region and thus the latter was removed from CR11 . In order to remove the overlap from CR11 the following expression was qualified via the CAD computation as shown by Eq. (4.13). ∃θ| {θ1 , θ2 , θ3 } ∈ CR11 ∧ CRcutoff

(4.13)

where and CRcutoff is the part of the overlap that needs to be removed. Equation (4.13) practically means “does θ exist so that CR11 is true (in the sense of satisfying the constraints that form it) while the CRcutoff is not true?”. Once no overlapping CRs

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101

Table 4.4 Overview of final optimal explicit solutions of the motivating mp-NLP under global uncertainty example Mathematical expression of critical regions Explicit objective value ⎧ ⎪ ≤ 4. 2.6667 ≤ θ 1 ⎨ CR1 = 0 ≤ θ2 ≤ 1 z1 (θ) = −3.75 − 4.5θ1 ⎪ ⎩ 10 ≤ θ ≤ 2 3 3θ1 −3 ⎧⎧ √ 0.666667 19 − 18θ1 + θ2 ≥ 3.833 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪⎨0 ≤ θ1 ≤ 2.0053025 · 10−16 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪1.5θ1 + θ2 ≤ 0.927401 ⎪⎪ ⎪ ⎪ ⎩ ⎪ ⎪ 0 ≤ θ3 ≤ 2 ⎪ ⎪ ⎪ z2 (θ) = 0.0156(5 − 2θ2 )3 ⎨ √ CR2 = ⎧ + 0.125(5 − 2θ2 )2 ⎪ 0.3212 81.86 − 77.5θ1 ⎪ ⎪ ⎪ ⎪ −1.25(5 − 2θ2 ) − 6 ⎪ ⎪⎪ ⎪ + θ2 ≥ 3.833 ⎪ ⎨ ⎪⎪ ⎪ ⎪ ⎪ 0 ≤ θ1 ≤ 0.8333 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪θ2 ≤ 2.5 ⎪⎪ ⎪ ⎪ ⎪ ⎩ ⎩⎪ 0 ≤ θ3 ≤ 2

Fig. 4.5 Final optimal partition of the parametric space of the motivating mp-NLP under global uncertainty example

can be identified the algorithm terminates and final optimal solutions are collected. In Table 4.4, a collection of the mathematical expressions of the CRs along with the explicit expression of their optimal objective value is given while in Fig. 4.5 the final optimal partition of the parametric space can be envisaged.

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4 Towards Exact Multi-setpoint Explicit Controllers …

4.3 Computational Studies In this section two case studies are examined and the corresponding multi-setpoint explicit controllers are designed and tested. All the computations were conducted on a Dell workstation with 3.70 GHz processor, 16GB RAM and Windows 7 64-bit operating system while Mathematica 10 was chosen as CAS.

4.3.1 Single-Input Single-Output Isothermal CSTR First, a case study involving a SISO multi-product CSTR is examined. The chemical reaction exhibits multiple steady states and can lead to the production of five different products, namely, A, B, C, D, E, based on the concentration (CR ) and the volumetric flow of the reactant (QR ). The related steady states given in Table 4.5, where x and u denote the CR and QR respectively. The MPC is designed with the state variable of the system being the concentration of the reactant, while the control input is k

the volumetric flow of the reactant. The reaction is 3rd order irreversible, i.e. R → 3P, −R R = kC R3 . The dynamic model of the system is given by Eq. (4.14) QR dCR = (C0 − CR ) + R R dt V

(4.14)

where C0 denotes the concentration of the reactant in the feed stream, V is the reactor’s volume and k is the reaction’s kinetic constant (Fig. 4.6). In order to design the multi-setpoint explicit controller the system’s model is transformed into its algebraic equivalent. In this work, for the sake of simplicity Euler integration is followed and the problem is formulated as shown in Eqs. (4.15), (4.21).

Table 4.5 Data of SISO CSTR case study Product xiss ( mol L ) A B C D E

0.0967 0.2 0.3032 0.393 0.5

uiss ( Lh ) 10 100 400 1000 2500

4.3 Computational Studies

103

Fig. 4.6 SISO CSTR production scheme, the manipulated variable is the volumetric flowrate of the liquid (QR ) while the state variable is the concentration of the reactant (CR ) Table 4.6 Candidate solutions of the SISO CSTR mp-MPC QR

λ1

λ2

λ3 0

1

0

0

θ14 − θ13 − 50θ12 + 50θ1 θ2 + 50θ1 − 50θ2 12500000

2

3000

0

0

10θ13 − 10θ14 + 497θ12 − 500θ1 θ2 − 494θ1 + 500θ2 − 3 125000000

−2θ2

0

0

0

0

0

3

−

4

−



10000 θ13 − 50θ1 θ1 − 1 10000 θ13 − 50θ1 θ1 − 1

min J(θ) = u

 + 50θ2

PH 



x(t) − xref 2

(4.15)

t=0

Subject to :

 x(t + 1) = x(t) + h

(4.16)

u(t) (C0 − x(t)) − kx3 (t) , 0 ≤ t ≤ PH − 1 V (4.17)

0 ≤ x(t) ≤ 1, 0 ≤ t ≤ PH 0 ≤ u(t) ≤ 3000, 0 ≤ t ≤ PH

(4.18) (4.19)

x(t|t=0 ) = θ1

(4.20)

= θ2

(4.21)

x

ref

Following the proposed algorithm for the solution of mp-NLPs the corresponding KKT system is formulated and solved using Mathematica 10. It takes 0.092s to compute 4 candidate solutions which are parametric in θ1 , θ2 as shown in Table 4.6. Evaluating with the primal and dual optimality conditions, 3 explicit solutions are left and next the final non-overlapping explicit solutions are computed following the comparison procedure and dominance criterion. The final globally optimal explicit solutions along with their corresponding CRs are given in Table 4.7 while the final

104

4 Towards Exact Multi-setpoint Explicit Controllers …

Table 4.7 Final CRs and explicit solutions (control law and state evolution) for PH = 1 of the SISO CSTR case study CRs Mathematical expression Explicit solution ⎧ ⎪ ≤ 0.503 0.0967 ≤ θ 1 ⎪ ⎪ ⎪ 3 ⎪  ⎪ ⎨ 0.02θ1 + θ2 ≤ θ1 θ3 C R (t = 1) = θ1 − 501 CR1 θ2 ≥ 0.0967  ⎪ Q R (t = 0) = 0 ⎪ ⎪ ⎪ 0.503 ≤ θ1 ≤ 1 ⎪ ⎪ ⎩ θ ≤ 0.5 2 ⎧  ⎪ ⎪ ⎪θ2 ≥ 0.0967 θ1 = 1 ⎪ ⎪ ⎪ 0 ≤ θ1 ≤ 0.0912 ⎪ ⎪     ⎪ ⎨ 0.0912 ≤ θ ≤ 0.4994 3−10θ13 +497θ1 C R (t = 1) = 1 500 CR2 θ2 ≤ 0.5 3 ⎪ Q R (t = 0) = 3000 ⎪0.994θ1 + 0.006 ≤ 0.02θ1 + θ2 ⎪ ⎪ ⎪ ⎪ ⎪ 0.0967 ≤ θ2 ⎪ ⎪ ⎩ 0 ≤ θ ≤ 0.0912 1 ⎧ ≤ 0.0967 0.0913 < θ ⎪ 1 ⎪ ⎪ ⎪ ⎪ 0.0967 ≤ θ ≤ 0.002(3 + 497θ1 − 10θ13 ) ⎪ 2 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪  ⎪ ⎪  ⎪ ⎨ 0.499 ≤ θ1 ≤ 0.503 C R (t = 1) = θ2    CR3 0.02 50θ1 − θ13 ≤ θ2 ≤ 0.5 10000 θ13 −50θ1 +50θ2 ⎪ Q (t = 0) = ⎪ R 1−θ1 ⎪ ⎪ ⎪ ⎧ ⎪ ⎪ ⎪⎪0.0967 ≤ θ1 ≤ 0.4994 ⎪ ⎨ ⎪   ⎪ ⎪ ⎪ 0.02 50θ1 − θ13 ≤ θ2 ⎪ ⎪ ⎩⎪ ⎩≤ 0.002 −10θ 3 + 497θ + 3 1 1

partition of the parametric space is given in Fig. 4.7a. The total time of computation for this case was 1.32s. In order to validate the design of the multi-setpoint explicit controller a transition instance between two steady states was examined. The multi-setpoint explicit controller’s predictions were in perfect agreement with the results of the corresponding NLP using BARON 14.4 solver in GAMS. In Fig. 4.8 the state and control input plots are given while a plot of the optimal objective function in the parametric space can be envisaged in Fig. 4.7b. The same problem was revisited with PH = 2. Considering a prediction horizon of 2 results in 32 candidate solutions parametric in θ1 and θ2 , which for the sake of space are not presented. Evaluating with the primal and dual feasibility, results in 5 feasible explicit solutions and next the comparison procedure is followed so as to keep only the globally optimal solutions. During the comparison procedure 3 overlaps were found while at the end the final optimal explicit representation of the parametric space consists of 5 CRs which are given in Table 4.9 while visually are presented in Fig. 4.9. The total time of computation for this case was 65.81s.

4.3 Computational Studies

105

Fig. 4.7 Explicit results for the SISO CSTR case study for PH = 1

The corresponding explicit solutions for prediction horizon 1 and 2 are given in Table 4.8 and 4.9 respectively. It is interesting to note that compared to the prediction horizon of unity, in the three new CRs which can be seen as a fragmentation of CR3 for the case of PH = 1 the nature of the control law is different. While in the first explicit controller the control law has three modes (idle, deadbeat and settling) the second explicit controller due to the increase of prediction horizon offer two more control modes. The case of PH = 3 was also investigated but this time the computational effort was exponentially increased to 4652.12s and the resulting partition of the parametric space involved 6 CRs. Further to that, because the proposed algorithm designs explicit controllers that retain the globally optimal control law their stability was found to be improved when compared to conventional online MPC and the use of fast local NLP solvers. Finally, the case of disturbance rejection while tracking two different setpoints sequentially was examined. For continuous manufacturing processes, disturbances such as fluctuations on the feed composition, temperature as well as the flowrate of the reactants can lead to significant deviations from the desired open-loop state trajectory. The explicit multi-setpoint controller was coded in GAMS 24.7.4 and a feed composition disturbance was imposed as d = 10−3 sin(i) where i stands for the MPC-iteration counter. It takes 10−3 s for the explicit controller to compute the optimal dynamic trajectories and in Fig. 4.10 a visual representation is given. A number of different starting points were tested along with different combination of setpoints and all instances were found to be feasible and provide good performance of the control system.

106

4 Towards Exact Multi-setpoint Explicit Controllers …

Fig. 4.8 Comparative plots of the solution computed by the multi-setpoint explicit controller versus online MPC for the SISO CSTR case study

4.3.2 Methyl-Methacrylate Polymerisation Isothermal CSTR As a next case study the design of explicit controller for the optimal grade transition of a polymerisation reactor was selected. The main nonlinear terms involved in the model include square roots and bilinear terms. The free radical polymerisation reaction takes place in an isothermal CSTR that operates at its steady state temperature of 335 K, where methyl-methacrylate (MMA) is produced using as initiator azobis (isobutyronitrile) and as solvent toluene [30, 31]. The mathematical model is given by Eqs. (4.22)–(4.26). The system has 4 states, i.e. the concentrations of the monomer

4.3 Computational Studies

107

Table 4.8 Final explicit optimal solutions for PH = 2 for the SISO CSTR case study ⎧ θ3 ⎪ C R (t = 1) = θ1 − 501 ⎪ ⎪ 

3 ⎪ ⎪ ⎨ θ13 θ3 1 C (t = 2) = − − − 501 + θ1 θ R 1 50 50 if (θ1 , θ2 ) ∈ CR1 then ⎪ ⎪ ⎪ Q R (t = 0) = 0 ⎪ ⎪ ⎩ Q R (t = 1) = 0 ⎧ 3 ⎪C R (t = 1) = θ1 − θ1 ⎪ 50 ⎪ ⎪ ⎪ ⎨C R (t = 2) = θ2 if (θ1 , θ2 ) ∈ CR2 then Q R (t = 0) = 0 ⎪



⎪  2  2 2  2 ⎪ ⎪ ⎪ ⎩ Q R (t = 1) = − 4 θ1 θ1 −50 θ1 θ1 −50 −125000 −6250000θ2 θ13 −50θ1 +50   ⎧ 3−10θ13 +497θ1 ⎪ ⎪ 500 ⎪C R (t = 1) =   ⎪ ⎪ ⎪ (10θ13 −497θ1 −3)3 + 3(10θ13 −497θ1 +497) +100θ − 3(θ1 −1) ⎨ −2θ13 + 1 62500000 2500 5 if (θ1 , θ2 ) ∈ CR3 then C R (t = 2) = 100 ⎪ ⎪ ⎪ ⎪ Q R (t = 0) = 3000 ⎪ ⎪ ⎩  ⎧ Q R (t = 1) = 3000 3 3−10θ1 +497θ1 ⎪ ⎪C R (t = 1) = ⎪ 500 ⎪ ⎪ ⎪ C R (t = 2) = θ2 ⎪ ⎪ ⎨ Q R (t = 0) = 3000 if (θ1 , θ2 ) ∈ CR4 then 9 7 +900θ 6 −7410270θ 5 −89460θ 4 ⎪ 1 1 ⎪ Q R (t = 1) = −1000θ1 +149100θ  1  1 ⎪ ⎪ 25 10θ13 −497θ1 +497 ⎪ ⎪ ⎪ +247763203θ13 +2223081θ12 −6212486581θ1 +6250000000θ2 −37499973 ⎪ ⎩   25 10θ13 −497θ1 +497 ⎧ C R (t = 1) = θ2 ⎪ ⎪ ⎪ ⎪ ⎨C R (t = 2) = θ2   if (θ1 , θ2 ) ∈ CR5 then 10000 θ13 −50θ1 +50θ2 ⎪ Q R (t = 0) = ⎪ 1−θ 1 ⎪ ⎪ 10000θ 3 ⎩ Q R (t = 1) = 1−θ2 2

(Cm ) and the initiator (Cl ), the molar concentration of the dead chains (D0 ) and the mass concentration of the dead chains (Dl ). The manipulated variable is the flowrate of the initiator (Fl ) and one output, i.e. the molecular weight of the polymer produced (y). Based on different steady states that the system exhibits it is possible to produce different polymeric grades which correspond to different molecular weights. The notation of the case study is given in Table 4.10, while the values of the model’s parameters are given in Table 4.11.  2f  kl Cl F(Cmin − Cm ) dCm = −(kp + kfm ) Cm + dt kTd + kTc V dCl Fl Clin − FCl = − kl Cl dt V

(4.22) (4.23)

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4 Towards Exact Multi-setpoint Explicit Controllers …

Table 4.9 CRs of SISO CSTR for PH = 2 CRs Mathematical expression ⎧⎧ ⎪ ⎪ ⎨0.09673 < θ1 ≤ 0.50516 ⎪ ⎪ ⎪ ⎪ 0.0967 ≤ θ2 ≤ ⎪ ⎪ ⎪ ⎩1.6 · 10−7 θ 9 − 150θ 7 + 7500θ 5 − 250000θ 3 + 6.25 · 106 θ  ⎨⎪ 1 1 1 1 1 CR1 := ⎪ ⎪  ⎪ ⎪ ⎪ ⎪ 0.505116 ≤ θ1 ≤ 1 ⎪ ⎪ ⎩ 0.0967 ≤ θ2 ≤ 0.5 ⎧ ⎧ ⎪0.502538 ≤ θ1 ≤ 0.505116 ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ≤ 0.5 θ 1.6 · 10−7 θ19 + 0.0012θ15 + θ1 ≤ 0.000024θ17 2 ⎪ ⎪ ⎪ ⎪ ⎩ ⎪ ⎪ +0.04θ13 + θ2 ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎧ ⎪ 0.0967181 ≤ θ1 ≤ 0.0967362 CR2 := ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ 0.0967 ≤ θ2 ⎪ ⎨ ⎪ ⎧ ⎪ ⎪ 3 ⎪0.02θ1 + θ2 ≤ θ1 ⎪ ⎪ ⎨θ1 ≤ 0.502538 ⎪ ⎪ ⎪ ⎪ ⎪ 5 −7 9 ⎪ ⎪ ⎪ ⎪ ⎪⎪1.6 · 10 θ1 + 0.0012θ1 + θ1 ≤ ⎪ ⎪ ⎩ ⎩⎩0.000024θ 7 + 0.04θ 3 + θ 2 1 1 ⎧ ⎪ ≤ 0.08579 0 ≤ θ 1 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ 0.0967 ≤ θ2 ≤ 0.5 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨⎧ 0.08579 ≤ θ1 ≤ 0.498968 ⎪ ⎪ CR3 := ⎪ ⎪ 7 −7 9 −7 6 ⎪ ⎪ ⎪ ⎨1.6 · 10 θ1 − 0.000023856θ1 − 1.44 · 10 θ1 ⎪ ⎪ ⎪ 4 − 0.039522θ 3 − 0.00035569θ 2 ⎪ + 0.000014313θ ⎪ 1 1 1 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ + 0.988034θ1 + 0.011964 + 0.00118564θ15 ≤ θ2 ⎪ ⎪ ⎪ ⎪ ⎪ ⎩⎩ ⎧ θ2 ≤ 0.5 ⎧⎧ ⎪ ⎪ ⎪ ⎨0.08579 ≤ θ1 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ θ1 ≤ 0.09126 ⎪ ⎪ ⎪ ⎪⎪ ⎪ ⎪ ⎩0.0967 ≤ θ ⎪ ⎪ ⎧ ⎪ ⎪ 2 ⎪ ⎪ ⎪ ⎪ ⎪−θ19 + 149.1θ17 + 0.9θ16 − 7410.27θ15 − ⎪ ⎨ ⎨ ⎪ ⎪ ⎪ 4 3 2 ⎧ ⎪ 89.46θ1 + 247013θ1 + 2223.08θ1 ⎪ ⎪ ⎪ ⎪ 0.08579 ≤ θ1 ⎪ ⎪⎪ ⎪ ⎪ ⎪ ⎪ ⎪⎩−6.1752 · 106 θ1 + 6.25 × 106 θ2 ≤ 74775 ⎪ ⎨ ⎨ ⎪ θ ⎪ 1 ≤ 0.498968 ⎪ ⎪ CR4 := ⎪ ⎪ ⎪ ⎪ 0.994θ ⎪ 1 + 0.006 < ⎪ ⎪ ⎪ ⎪ ⎩⎪ ⎩ ⎪ 3+θ ⎪ 0.02θ ⎪ 2 ⎪ 1 ⎪ ⎪ ⎪ ⎪ ⎪ ⎧ ⎪ ⎪ 3 ⎪ ⎪ ⎪ ⎨0.006 + 0.994θ1 ≤ 0.02θ1 + θ2 ⎪ ⎪ ⎪ 0.498968 ≤ θ1 ≤ 0.4994889 ⎪ ⎪ ⎪ ⎩⎪ ⎩θ ≤ 0.5  ⎧ 2 ⎪ 0.09126 ≤ θ1 ≤ 0.49949 ⎪ 3 ⎪0.02θ1 + θ2 ≤ 0.994θ1 + 0.006 ⎪ ⎪ θ1 ≤ 0.02θ13 + θ2 ⎪ ⎪ ⎪ ⎨ ⎧ CR5 := ⎪ ⎪ ⎪ ⎨0.49949 ≤ θ1 ≤ 0.502538 ⎪ ⎪ ⎪ ⎪ θ1 ≤ 0.02θ13 + θ2 ⎪ ⎪ ⎩⎪ ⎩θ ≤ 0.5 2

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109

Fig. 4.9 Final CRs of the SISO CSTR case study for PH = 2

dD0 2f  kl Cl = (0.5kTc + kTd ) Cm + kfm dt kTd + kTc



2f  kl Cl FD0 Cm − kTd + kTc V  2f  kl Cl dDl FDl = Mm (kp + kfm ) Cm − dt kTd + kTc V Dl y= D0

(4.24)

(4.25) (4.26)

Since none of the nonlinear terms of the system is transcendental, the proposed methodology for the design of the explicit controller can be employed. Similar to the previous case study, the infinite dimensional optimal control problem is transformed into a finite one with the employment of a discretisation scheme. The present case study has been studied by a considerable number of researchers and it has been reported that its optimal control through numerical schemes is rather challenging due to numerical instabilities that may arise during the discretisation [31]. To this end, forward Euler discretisation is employed with a step size of h = 36s as trade-off

110

4 Towards Exact Multi-setpoint Explicit Controllers …

Fig. 4.10 Performance of the multi-setpoint explicit MPC of the SISO CSTR with PH = 2 under disturbance rejection

between computational complexity and stability of the integration scheme. A number of simulations were conducted so as to decide on a step size that can be small enough so as to avoid oscillatory behavior but not too small so as to avoid an extenuating repetitive solution that would result in further computational effort. Moreover, as will be shown in the results the nature of proposed methodology, i.e. the analytical and not numerical solution of the optimisation problems, enhances the robustness of the solutions computed as numerical instabilities are circumvented. At the beginning, the set of differential equations is discretised and then the optimisation problem is formulated and solved in Mathematica 10. Following the proposed methodology the global optimality of the explicit solutions is guaranteed whereas using readily available numerical solvers for online implementation of globally opti-

4.3 Computational Studies

111

Table 4.10 MMA polymerisation reactor case study nomenclature Cm (kmol/m3 ) State: concentration of monomer Cl (kmol/m3 ) State: concentration of initiator 3 D0 (kmol/m ) State: dead chains molar concentration Dl (kg/m3 ) State: dead chains mass concentration Fl (m3 /h) Input: flow rate of initiator y = Dl /D0 Output: molecular weight

Table 4.11 Model parameters for the MMA polymerisation reactor case study F = 10.0 m3 /h Monomer flowrate V = 10.0 m3 Reactor volume f  = 0.58 Initiator efficiency m3 kmol·h m3 kTd = 1.09 × 1011 kmol·h m3 kTc = 1.33 × 1010 kmol·h Clin = 8.00 kmol/m3 Cmin = 6.00 kmol/m3 m3 kfm = 2.45 × 103 kmol·h kl = 1.02 × 10−1 h−1

kp = 2.50 × 106

Mm = 100.12 kg/kmol

Propagation rate constant Termination by disproportionation rate constant Termination by coupling rate constant Inlet initiator concentration Inlet monomer concentration Chain transfer to monomer rate constant Initiation rate constant Molecular weight of monomer

Table 4.12 Computational study of CPU times (s) required for different prediction horizons using BARON 14.4 as global optimisation solver Prediction horizon (Hp ) CPU times (s) 1 2 5 10 20 a The

602.676 1905.328 3600a 3600a 3600a solver failed to converge to a globally optimal solution within 3600 CPU times

mal solutions results in rather exhaustive computational times. For illustrative purposes the results of a comparative study using BARON 14.4 solver in GAMS with varying prediction horizons is provided in Table 4.12. In order to design the explicit controller we consider 8 uncertain parameters, 4 for each state and 4 for each family of set-points. In Table 4.13, the notation for the uncertain parameters is given. It can be argued that having these parameters as continuous

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4 Towards Exact Multi-setpoint Explicit Controllers …

Table 4.13 Uncertain parameters of mp-MPC for MMA Uncertain parameter Bounds θ1 θ2 θ3 θ4 θ5 θ6 θ7 θ8

Correlation

∈ [0, 5] ∈ [0, 0.5] ∈ [0, 0.05] ∈ [0, 300] ∈ [0, 5] ∈ [0, 0.5] ∈ [0, 0.05] ∈ [0, 300]

Cm|t=0 Cl|t=0 D0|t=0 Dl|t=0 Set-points for Cm Set-points for Cl Set-points for D0 Set-points for Dl

Table 4.14 Candidate solutions for explicit controller of the MMA case study for PH = 1 Fl

λ1

λ2

λ3

λ4

λ5

λ6

λ7

λ8

λ9

λ10

1

125θ6 − 124

0

0

0

0

0

0

0

0

0

0

2

0.4

−0.01582θ2 + 0.016θ6 −0.0000512

0

0

0

0

0

0

0

0

0

3

0

0

0.01582θ2 −0.016θ6

0

0

0

0

0

0

0

0

4

−124θ2

0

0

0

−2θ6

0

0

0

0

0

0

5

62.5 − 124θ2

0

0

0

0

0

0

0

2θ6 − 1 0

0

may lead to unnecessary computations but on the contrary one could argue that in the context of enterprise wide optimisation where the supervisory controller receives data from the RTO layer the constant use of the same set-points is not guaranteed. In that case a conventional explicit controller would have to be re-designed from the beginning (solution of the corresponding mp-P problem, storage of the explicit solutions, possibly, in a micro-chip) whereas following the methodology proposed herein, there would be no need for that, under the assumption that the bounds used initially are the feasible range of the system. The corresponding mp-NLP problem consists of 1 optimisation variable, the control input, 10 constraints and 8 uncertain parameters for the case of prediction horizon of 1. Note that even though the optimisation variable is one the variables for which analytical solution is sought, are eleven i.e. the optimisation variable and the Lagrange multipliers of the constraints. Solving the mp-NLP results in 5 candidate solutions which are shown in Table 4.14. An interesting observation is that the underlying control law is linear function of the uncertain parameters and most specifically of the concentration of the initiator at each sampling instance and the set point for that state even though the original optimisation problem is highly nonlinear.

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113

The explicit solutions are substituted back to the original constraints and a set of parametric inequalities arise. If a candidate solution satisfies the primal and dual feasibility conditions it is considered at the next step; otherwise, it is flagged as infeasible and discarded from further consideration. For example, in this case study, the fifth candidate solution is dual infeasible as there is no value of θ6 that would result in a positive λ8 . Note that the same would apply for the 4th candidate solution but θ6 is allowed to take the value of zero. Next, the parametric inequalities for the feasible candidate solutions are considered. Note that each parametric inequality defines a region in the parametric space where it is always satisfied. The intersection of these regions is the CR of the candidate solution. As mentioned, because the problem is non-convex it is possible that some of the solutions might overlap; in order to compute at the end only the globally optimal parametric solutions the comparison procedure is employed. Overall, 3 overlaps where identified among the CRs. An example of a set of inequalities defining the overlap of the CR1 and CR2 is given by Eq. (4.27). The mathematical definition of CR2 is given by Eq. (4.28) and for the sake of space the rest of CRs are omitted due to the length of their mathematical expressions. It is not surprising that the mathematical expressions defining the CRs include square roots and polynomial expressions which are found in the original dynamic model as well. Once the final optimal explicit solutions have been computed they are coded in GAMS and the comparison of the performance of the multi-setpoint explicit MPC and the conventional online MPC is conducted. Similar to the previous case study, the online MPC problem is formulated in GAMS and solved using BARON 14.4 as the global NLP solver. In Fig. 4.12, the graphs of the output as well as the control input of the system indicate a perfect agreement with the solution from BARON thus validating the correctness of the explicit solution. Indicative of the stability that the explicit controller holds is the plot of the objective function, in Fig. 4.11, that strongly resembles the typical Lyapunov-like graphs. Finally, the case of two different setpoints for tracking in a sequential manner was examined. Initially the kg and was steered to the reactor was operated at the steady state of y = 15, 000 kmol kg where it was regulated for 9 h for the production steady state of y = 45, 000 kmol of the corresponding polymer grade. In a similar manner, the controller regulates kg ) where another polymer grade is the system to the next setpoint (y = 19, 250 kmol produced in steady state. The aforementioned instance is shown in Fig. 4.13.

CRint := CR1 ∩ CR2 =

⎧ 0 ≤ θ3 ⎪ ⎪ ⎪ ⎪ ⎪ 0 ≤ θ4 ⎪ ⎪ ⎪ ⎪ ⎪0 ≤ θ5 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨0 ≤ θ 7

≤ 0.05 ≤ 300 ≤5 ≤ 0.05

⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ 0 ≤ θ4 ≤ 300 ⎪ ⎪ ⎪ ⎪ ⎪ θ2 + 0.0024268 = 1.0114θ6 ⎪ ⎪ ⎪ ⎪ ⎪ 0 ≤ θ2 ≤ 0.199802 ⎪ ⎪ ⎩ 0 ≤ θ1 ≤ 2.72345

⎧ ⎪ 4.989 ≤ θ1 ≤ 5 ⎪ ⎨ ≤ 1617.7 + 40280.2 θ12 ⎪ ⎪ ⎩θ2 ≤ 1.48197 θ2

16144.7 θ1

+ θ2

1

(4.27)

114

⎧ 0 ≤ θ3 ⎪ ⎪ ⎪ 0 ≤ θ4 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪0 ≤ θ5 ≤ 5 ⎪ ⎪ ⎨0.0032+ CR2 = 0.988θ2 ≤ θ6 ⎪ ⎪ θ6 ≤ 0.5 ⎪ ⎪ ⎪ ⎪ 0 ≤ θ7 ≤ 0.05 ⎪ ⎪ ⎪ ⎪ 0 ⎪ ⎩ ≤ θ8 ≤ 300

4 Towards Exact Multi-setpoint Explicit Controllers …

⎧ ⎧⎧ ⎨0.42 ≤ θ2 ≤ 0.5 ⎪ ⎪  ⎪ ⎪ ⎪ ⎪ ⎪ 0.00113θ ⎪ ⎪ ⎪  2 + 0.0000457 θ2 + θ3 ≤ 0.05 ⎪ ⎪ ⎩ ⎪ ⎪ ⎪ ⎨ 4.6722 θ2 + θ4 ≤ 303.03 ⎪ ⎪ ⎪ ⎪ θ1 = 1.8769 ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨0 ≤ θ2 ≤ 0.42 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩⎩θ3 ≤ 0.05 ⎪ ⎪ ⎪ θ4 ≤ 300 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ −5 θ 2 θ + ⎪ ⎪ 1 2 ⎪2.43 · 10 ⎪ ⎧ ⎧⎧ 1.48 ⎪ ⎪ ⎨ ⎪ < θ2 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ θ2 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ 1 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ 1.72 < θ1 < 1.877 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ 1.87695 ≤ θ1 ≤ 5 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨2.319 · 10−5 θ 2 + 0.446 < ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪  1  ⎪ ⎪ ⎪ ⎪ θ + 2.49 θ θ ≤ 303.03 ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ 4 2 1 8.85 · 10−16 6.86 · 1022 θ14 + 2.64 · 1026 θ12 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ ⎪ θ ≤ 0.5 ⎪ ⎪ ⎪ 2 ⎪ +θ ⎪ ⎪ ⎪ ⎪ ⎪ 2 ⎪ ⎪ ⎪ ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ 0 ≤ θ1 ≤ 1.72 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ 0.00113θ2 + θ3 ≤ 0.05 ⎪ ⎪ ⎪ 2.319 · 10−5 θ12 + 0.446 < ⎪ ⎪ ⎪ ⎪ ⎪⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ 8.85 · 10−16 6.86 · 1022 θ14 + 2.64 · 1026 θ12 ⎪ ⎪⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪⎪ ⎪ ⎩ ⎪ ⎩ +θ2 ⎪ ⎪ ⎪ ⎪ θ4 ≤ 300 ⎪ ⎪⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ 1.72 < θ ⎪ ⎪⎪ ⎪ 1 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ −5 θ 2 + 0.446 < ⎪ ⎪ ⎪ ⎪ 2.319 · 10 ⎪ ⎪ ⎪⎨ ⎪ 1 ⎪ ⎪ ⎪ ⎪ ⎪ −16 6.86 · 1022 θ 4 + 2.64 · 1026 θ 2 + θ ⎪ ⎪ ⎪ ⎪ ⎪ 2 ⎪⎪ ⎪8.85 · 10 1 1 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪⎪ 1.48 ⎪ ⎪ ⎪ ⎪ ⎩⎩θ2 ≤ 2 ⎪ ⎪ θ1 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎧ ⎪ ⎧⎧ ⎪ ⎪ ⎪1.877 < θ1 ≤ 5 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ θ2 > 1.48 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ θ12 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪⎪ ⎪  ⎪ ⎪ ⎨ ⎩ ⎪ ⎪ θ4 + 2.49 θ2 θ1 ≤ 303.03 ⎪ ⎪ 4 2 −16 ⎪ ⎪ 22 26 6.86 · 10 θ1 + 2.64 · 10 θ1 ⎪8.85 · 10 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪  ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ 0 ≤ θ < 1.879 ⎪ ⎪ ⎪ 1 ⎨ ⎪ ⎩ ⎪ ⎪ ⎪ θ4 ≤ 300 θ ⎪ 3≤0.05 ⎪ ⎪ ⎪ ⎪ ⎪+θ2 ≤ 2.319 · 10−5 θ 2 + 0.446 ⎪ ⎪ ⎪ ⎪ 1 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ 1.87 < θ1 ≤ 4.989 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨⎧θ4 ≤ 300 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ 1.48 ⎪ ⎪ ⎨θ1 > 4.989 ⎪ ⎪θ2 ≤ 2 ⎪ ⎪ ⎪ ⎪ ⎪ θ1 ⎪ ⎪ ⎪ ⎪ 40280.2−16144.7θ1 + 1617.74 ≤ θ ⎪ ⎪ ⎪ ⎩ ⎩ 2 ⎩ ⎩ θ12

(4.28)

4.3.3 Overall Scalability of the Algorithm Apart from the explicit controllers for the aforementioned cases, a number of nonlinear systems were examined (one of them being the MIMO CSTR which will be formally introduced in Sect. 5.3.2) so as to conduct a study on the scalability of the presented algorithm and the related results are given in Table 4.15. The results of Table 4.15 indicate that as expected the consideration of LHS uncertainty in design of mp-MPC, despite being a quite desirable feature from a practical point of view, limits the application of the algorithm to small-scale systems and/or short prediction

4.3 Computational Studies

115

Fig. 4.11 Plot of the optimal objective value against time for the MMA CSTR case study

horizons. Nonetheless, the value of explicitly designing controllers in such a manner (with LHS uncertainty) can result in more robust operations and new insights about the impact that model intrinsic uncertainties, e.g. kinetic parameters etc., can have on the stability of the system. On the other hand, the type of nonlinear terms encountered plays also a significant role in the related computational effort since the MMA case study for PH = 2 resulted in termination of the algorithm due to exceeding memory requirements. This is because for the MMA system, the square root terms is polynomialised thus resulting in additional variables for which the symbolic computations need to be made. Overall, compared to problem classes investigated in Chap. 3 the consideration of analytical nonlinear systems (or systems whose nonlinear terms can be polynomialised) limits, for the current state of the art, the proposed algorithm to small scale systems.

4.4 Concluding Remarks Motivated by the crucial role that model based control plays within the context of EWO, an algorithm for the solution of a class of mp-NLPs that accept analytical solution and involve OFC, RHS and LHS uncertainty has been presented. First, Gr¨obner bases are employed to compute the optimisation variables as a function of the uncertain parameters based on the KKT system of the mp-NLP and then CADs are computed that allow us to develop the final optimal partition of the parametric space. Based on the algorithm, a novel methodology for the design of multi-setpoint explicit controllers with EWO considerations was proposed. The methodology enhances the

116

4 Towards Exact Multi-setpoint Explicit Controllers …

Fig. 4.12 Comparative plots of the solution computed by the multi-setpoint explicit controller versus online MPC for the MMA CSTR case study

scope of explicit MPC as it is shown that it is possible to design once and for all an explicit controller that can track instantaneously new setpoints and possibly new parameter estimations of the model employed. In the forthcoming chapters of Part II of the thesis, the aforementioned controllers play a pivotal role in the development of an efficient uncertainty-aware framework for the integration of planning, scheduling and control.

4.4 Concluding Remarks

117

Fig. 4.13 Multi-setpoint tracking instance of the MMA polymerisation case study Table 4.15 Computational statistics of the proposed method for the design of multi-setpoint explicit controllers based on different systems Case study Problem statistics Uncertainty CPU (s) SISO CSTR (PH = 1) SISO CSTR (PH = 2) SISO CSTR (PH = 2) MMA CSTR (PH = 1) MMA CSTR (PH = 1) MMA CSTR (PH = 2) MIMO CSTR (PH = 1) MIMO CSTR (PH = 2) MIMO CSTR (PH = 3)

nx 1

ng 2

nθ 2

OFC, RHS

1.32

2

4

2

OFC, RHS

65.8

3

6

2

OFC, RHS

4,652

1

10

8

OFC, RHS

1.86

1

10

9

4.86

2

20

8

OFC, RHS, LHS OFC, RHS

Memory limit

2

8

4

OFC, RHS

2.76

4

16

4

OFC, RHS

192.75

6

24

4

OFC, RHS

5,929

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4 Towards Exact Multi-setpoint Explicit Controllers …

References 1. Cutler CR, Perry R (1983) Real time optimization with multivariable control is required to maximize profits. Comput Chem Eng 7(5):663–667 2. Grossmann IE (2005) Enterprise-wide optimization: A new frontier in process systems engineering. AIChE J 51(7):1846–1857 3. Bauer M, Craig IK (2008) Economic assessment of advanced process control - A survey and framework. J Process Control 18(1):2–18 4. Young RE (2006) Petroleum refining process control and real-time optimization. IEEE Control Syst 26(6):73–83 5. Darby ML, Nikolaou M, Jones J, Nicholson D (2011) RTO: an overview and assessment of current practice. J Process Control 21(6):874–884 6. Pontes KV, Wolf IJ, Embirucu M, Marquardt W (2015) Dynamic real-time optimization of industrial polymerization processes with fast dynamics. Ind Eng Chem Res 54(47):11881– 11893 7. Ellis M, Durand H, Christofides PD (2014) A tutorial review of economic model predictive control methods. J Process Control 24(8):1156–1178 8. De Souza G, Odloak D, Zanin AC (2010) Real time optimization (RTO) with model predictive control (MPC). Comput Chem Eng 34(12):1999–2006 9. Adetola V, Guay M (2010) Integration of real-time optimization and model predictive control. J Process Control 20(2):125–133 10. Chachuat B, Srinivasan B, Bonvin D (2009) Adaptation strategies for real-time optimization. Comput Chem Eng 33(10):1557–1567 11. Pistikopoulos EN, Diangelakis NA, Oberdieck R, Papathanasiou MM, Nascu I, Sun M (2015) PAROC-an integrated framework and software platform for the optimisation and advanced model-based control of process systems. Chem Eng Sci 136:115–138 12. Dua V, Pistikopoulos EN (1999) Algorithms for the solution of multiparametric mixed-integer nonlinear optimization problems. Ind Eng Chem Res 38(10):3976–3987 13. Johansen TA (2002) On multi-parametric nonlinear programming and explicit nonlinear model predictive control. In: Proceedings of the 41st IEEE conference on decision and control, vol 3. IEEE, pp 2768–2773 14. Acevedo J, Salgueiro M (2003) An efficient algorithm for convex multiparametric nonlinear programming problems. Ind Eng Chem Res 42(23):5883–5890 15. Johansen TA (2004) Approximate explicit receding horizon control of constrained nonlinear systems. Automatica 40(2):293–300 16. Dua V, Papalexandri KP, Pistikopoulos EN (2004) Global optimization issues in multiparametric continuous and mixed-integer optimization problems. J Global Optim 30(1):59–89 17. Fotiou IA, Parrilo PA, Morari M (2005) Nonlinear parametric optimization using cylindrical algebraic decomposition. In: 44th IEEE conference on decision and control and 2005 European control conference. IEEE, pp 3735–3740 18. Narciso DA (2009) Developments in nonlinear multiparametric programming and control. PhD thesis, Imperial College London 19. Dominguez LF, Pistikopoulos EN (2013) A quadratic approximation-based algorithm for the solution of multiparametric mixed-integer nonlinear programming problems. AIChE J 59(2):483–495 20. Charitopoulos VM, Dua V (2016) Explicit model predictive control of hybrid systems and multiparametric mixed integer polynomial programming. AIChE J 62(9):3441–3460 21. Patrinos P, Sarimveis H (2010) A new algorithm for solving convex parametric quadratic programs based on graphical derivatives of solution mappings. Automatica 46(9):1405–1418 22. Pistikopoulos EN (2012) From multi-parametric programming theory to MPC-on-a-chip multiscale systems applications. Comput Chem Eng 47:57–66 23. Sun M, Chachuat B, Pistikopoulos EN (2016) Design of multi-parametric NCO tracking controllers for linear dynamic systems. Comput Chem Eng 92:64–77

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Part II

Uncertainty-Aware Integration of Planning, Scheduling and Control

Chapter 5

Open-Loop Integration of Planning, Scheduling and Optimal Control: Overview, Challenges and Model Formulations

Traditionally, planning, scheduling and optimal control problems are solved in a decoupled way, neglecting their strong interdependence. Integrated Planning, Scheduling and optimal Control (iPSC) aims to address this issue. In this chapter, current developments on the topic of integrating control with process operations are reviewed and a new approach for the iPSC of continuous processes aiming to reduce model and computational complexity is proposed. The resulting problem is a mixed integer program for which different solution strategies are employed and analysed.1

5.1 Introduction Decision making in the process industries is organised in a hierarchical manner and actions are taken in a decentralised fashion with common objectives being the maximisation of profitability, sustainability and safety. Three of the most important functionalities in the chain of command of the industry, involve medium-term planning, production scheduling and advanced process control. Process planning aims to organise the production and meet demands over long time horizons; a planning horizon typically spans from weeks to months and key decisions include the determination of sales, backlog and inventory levels. After the planning decisions have been made, a detailed schedule of production is carried out for each planning week. Production scheduling deals with the optimal resource allocation of the manufacture in greater detail than planning; major decisions include, optimal production and changeover times, sequencing of tasks and so forth. On a lower level in the decision pyramid, advanced process control (APC) is located. APC, involves optimal and

1 Parts of this chapter have been reproduced with permission from: https://doi.org/10.1021/acs.iecr.

7b01122. © Springer Nature Switzerland AG 2020 V. M. Charitopoulos, Uncertainty-aware Integration of Control with Process Operations and Multi-parametric Programming Under Global Uncertainty, Springer Theses, https://doi.org/10.1007/978-3-030-38137-0_5

123

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5 Open-Loop Integration of Planning, Scheduling and Optimal Control …

model based control strategies that are responsible for the regulation of the process dynamics. For the case of continuous manufacturing systems, such as polymerisation processes, APC aims to secure the optimal operation between transition of different production regimes as well as to maintain the desirable steady state operating conditions. The ever increasing need to maintain profitable and efficient operations in the industry along with recent developments such as parallel computing and more efficient IT infrastructure indicate the dawn of a new era in the way decision making is conducted. Concepts such as enterprise wide optimisation (EWO), smart manufacturing [1] and Industry 4.0 [2] underline the need for efficient use of information available among closely related functionalities of the supply chain like medium term planning, production scheduling and optimal control.

5.1.1 Integration of Process Operations with Control In Fig. 5.1, the automation pyramid based on the ISA 95 standards is envisaged. Based on the ISA 95 standards the levels of hierarchy that are of interest in this chapter, are the three top ones [3]. As shown in Fig. 5.1, these three problems involve different time scales and there is reciprocal information flow. Because of the interdependence existing among these three problems, developing an integrated approach to model and solve the problem is likely to yield better solutions and thus enhance process operations. Indeed, research work from the literature indicate that integration of different decision layers of the automation pyramid can lead to more efficient operations [3–6]. Until recently, the integration was studied from two different perspectives: (i) integration of medium-term planning and scheduling and (ii) integration of scheduling and control. For the problem of integration between planning and scheduling a number of modelling approaches have been proposed in the literature with some

Fig. 5.1 ISA 95 automation pyramid

5.1 Introduction

125

researchers formulating the problem in a monolithic way while others employing aggregated modelling techniques or surrogate models for the problem of scheduling [7]. Regardless of the modelling approach employed, due to the different time scales the computational effort required for the solution of the problem grows rapidly with the length of the planning horizon under consideration. To this effect, solution techniques such as Lagrangian relaxation, rolling horizon and bi-level decomposition have been proposed [8, 9]. Integration of process scheduling and optimal control has been studied recently for both continuous and batch processes [10]. The interactions between scheduling and control can dramatically affect the economically optimal operation of the manufacture, since decisions such as production time, sequencing of tasks and transition time are dependent on the dynamic status of the system. To be more specific, the production rate for a manufacturing process can be seen as a function of the state variables of the underlying control problem; the role of the controller is to keep the system around the desirable steady state and reject disturbances that might occur because otherwise the production time would be disrupted, jeopardizing the feasibility of the schedule under execution. The optimal sequencing of tasks relies mostly on the transition times and so the control policy can have considerable impact on the schedule as the transition time is practically the time that the controller needs to stir the system from one steady state to the next. The simultaneous cyclic scheduling and dynamic optimisation of a single multiproduct CSTR was studied by Flores-Tlacuahuac and Grossmann [11]. The authors proposed a time-slot based formulation for the scheduling part of the problem, where equal number of slots and products were considered. Zhuge and Ierapetritou [12] proposed a framework for the closed loop implementation of the integrated solution of scheduling and control (iSC) for continuous processes that shared many similarities with model predictive control (MPC). An alternative methodology for the closed-loop iSC has been reported by Chu and You [13]. The authors, in an offline step, generated a number of PI-controllers for each possible transition and studied the integrated problem as the optimal simultaneous scheduling and controller selection. In order to achieve fast computational times, the resulting MINLP with fractional objective function was solved using Dinkelbach’s algorithm. In general, the iSC is formulated as a mixed integer dynamic optimisation problem (MIDO) and then following a numerical integration scheme is discretised into large scale mixed integer nonlinear programming problem (MINLP) which can be computationally demanding. To ease the computational burden for the case of iSC of a single multi-product CSTR the use of Benders decomposition has been proposed as an alternative solution technique [14]. Aiming to reduce the time needed for the solution Zhuge and Ierapetritou [15] suggested the use of multi-parametric model predictive control within the context of iSC. First, the original nonlinear dynamics of the underlying production system were linearised and then the explicit controller was designed through the solution of the corresponding multi-parametric program. The explicit solutions were then incorporated in the scheduling model as a set of big-M constraints and the overall iSC was modelled as an MILP. Later on, the same authors proposed the use of

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5 Open-Loop Integration of Planning, Scheduling and Optimal Control …

fast MPC [16] and in that work piecewise affine approximations of the nonlinear dynamics were employed. One of the main bottlenecks in the integrated problem is the time-scale separation among the different layers of decision making. To this end, Baldea et al. [17] proposed the use of lower-dimensional dynamic models and embedded them in the scheduling formulation as a set of soft constraints. In their time scale-bridging approach, a scheduling oriented MPC was also employed so as to synchronise the calculations between MPC and scheduling. For further discussion the interested reader is referred to some recent reviews on the topic of iSC [3, 9, 10]. Recently, the development of frameworks for the integrated planning, scheduling and control (iPSC) has started to receive attention. Prompted by the potential economic benefits that the iSC has indicated, the investigation of an even more integrated decision making framework that takes full advantage of all the possible interactions involved in the three problems constitutes a promising direction. Within the iPSC framework, the decisions involved in planning, scheduling and control are made simultaneously with a common objective. It can be understood that iPSC poses many challenges despite the potential benefits; the time-scale problem is further exacerbated when planning decisions are considered, the way that the three different problems are effectively linked constitutes another major factor that affects model complexity. The iPSC of single unit continuous manufacturing processes, for short term planning periods, was studied and formulated as an MINLP by Gutierrez-Limon et al. [18]. The authors employed the scheduling and planning model of Dogan and Grossmann [8] while for the control part of the iPSC a nonlinear MPC (NMPC) scheme was used. Following this work, the transition times are fixed offline based on heuristics and thus are not considered as decision variables. Note that despite the use of a NMPC scheme in the aforementioned work, disturbance rejection was not considered. The same authors proposed a reactive heuristic strategy for the iPSC under the presence of unforeseen events [5]. Shi et al. [19] considered the integration of planning, scheduling and dynamic optimisation of continuous manufacturing processes. The authors, proposed a technique to create a flexible-recipe framework in which pairs of transition times and cost are determined offline. In order to alleviate the computational complexity, they decomposed the problem, thus rendering it a mixed integer linear programming (MILP) problem, for which a bi-level decomposition solution procedure was also proposed. Nonetheless in their work, no disturbances were considered and the dynamic optimisation provides only open-loop related information about the real dynamics of the process. Again the model of Dogan and Grossmann [8] was used as a basis for the integrated planning and scheduling and the authors proposed an alternative way to calculate the inventory balance in every production period, based on the trapezoid rule.

5.1 Introduction

127

5.1.2 Motivation and Problem Statement The objective of this chapter is to provide a computationally efficient framework for the integration of planning, scheduling and optimal control. Motivated by the ever lasting need for more efficient and reliable operations in the process industries it is attempted to explore the interdependence of the decisions between the three different hierarchical levels which up until recently were considered in a decoupled manner. In general, a single unit continuous manufacturing process is considered which based on multiple steady states, that the underlying dynamic system exhibits, can produce a number of different products. Concisely, the iPSC problem can be summarised as follows: Given: • • • • •

A single stage, multi-product continuous manufacturing process Production planning horizon Demands for each product at the end of each planning period Process dynamics and steady state operating conditions of the process Raw material, operating and inventory costs and product prices.

Determine: • • • • •

Dedicated production time and cost for each product in every planning period Optimal production sequences Optimal change-over times and cost Inventory and backlog level for each product at the end of each period Optimal dynamic trajectories of the process.

Objective: Maximise the production profit of the process; i.e. the revenue, minus inventory, backlog, operation, production and transition costs under deterministic assumptions. Since in this chapter no disturbances are considered at the level of dynamics, the optimal control part of the integrated problem is concerned with open-loop policies and thus the analysis is restricted to open-loop stable systems and the term optimal control is used in a dynamic optimisation sense. Since the iPSC involves the optimal control of the process the problem is formulated as a MIDO which is discretised into an MINLP. The degrees of freedom for the optimisation within the iPSC involve among others: (i) production amounts within each planning period, (ii) assignment of products, (iii) transition times for the changeovers between products, (iv) optimal dynamic trajectories and (v) backlog of unmet demand. The remainder of the chapter is organised as follows: in Sect. 5.2 the main parts of the methodology developed for the integrated problem are described. Next, in Sect. 5.3 the proposed methodology is illustrated and compared through three different case studies with the time-slot based formulation for the integrated problem highlighting the computational savings achieved. Finally, in Sect. 5.4 concluding remarks are given.

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5 Open-Loop Integration of Planning, Scheduling and Optimal Control …

5.2 Methodology The majority of research work conducted for the integration of control with operations has focused on time-slot based formulations. In the present work, a Travelling Salesman Problem (TSP) based framework for the integrated planning and scheduling problem that has two distinct features is proposed. First, the time representation used herein is hybrid, i.e. the planning periods are modelled as discrete time points while within each planning period a continuous time representation is followed. Secondly, following the proposed TSP formulation the notion of time-slots is replaced by implicitly unique pairs of products which can be produced in a sequential manner within each planning period.

5.2.1 Objective of the iPSC As discussed in the previous section, the main reason that the iPSC is formulated in a simultaneous and not sequential manner is to take advantage of the information available across the different decision levels. Another reason and key objective of the present study is to provide a way of reflecting the process dynamics into process economics. The objective function of the integrated problem is the maximisation of profit and is calculated as the revenue minus the costs associated with the inventory, raw material consumption, backlogged demand and operation. Equation (5.1) represents the revenue (1 ) of the production as the summation of the sales of every product i, for every customer c, for every planning period (Scip ) multiplied by its respective selling price (Pi ). 1 =

 c

Pi Scip

(5.1)

p

i

Equation (5.2) accounts for the total operational cost (2 ) over the planning oper horizon which is calculated as the product of the unitary operational cost (Ci ) and the amount produced (Qip ). 2 =

 oper Ci Qip

(5.2)

p

i

The total inventory cost is given by Eq. (5.3) and is calculated as the product of the unitary inventory cost (Cinv i ) and the inventory level (Vip ) of product i at the end of every planning period p.  Cinv (5.3) 3 = i Vip i

p

5.2 Methodology

129

When demand cannot be met, in the present work backlog is allowed under a penalty. The cost associated with backlogged demand is calculated by Eq. (5.4). 4 =

 CBic Bcip c

i

(5.4)

p

where CBic stands for the unitary backlog cost and Bcip denotes the demand of customer c at the end of planning period p for product i that is backlogged. Another way planning, scheduling and optimal control interact is the calculation of the changeover cost. In principle, the changeover cost is calculated as the product multiplication of a unitary changeover cost by the binary variable that indicates the changeover occurrence. However, following the holistic approach that the iPSC dictates, the changeover cost is practically the utilisation of resources for the achievement of the next steady state, which from a control perspective is translated into the values of the control input during the transition period. Apart from that, the changeover costs account for any off-spec products that are manufactured during the transient period. In Fig. 5.2, a conceptual graph for an arbitrary schedule is given. As shown in Fig. 5.2, the calculation of the raw material consumption cost during transition periods is computed as the integral of the arbitrary curve. The corresponding cost is numerically calculated following orthogonal collocation on finite elements (OCFE) with Radau IV quadrature as shown by Eq. (5.5). A1 =

  ocfe Trans Craw umijpfecp · Tijp · hocfe · cpNcp m m

i

j

p

fe

(5.5)

cp

Fig. 5.2 Conceptual visualisation of an arbitrary scheduling instance and the calculation of raw material cost during changeover periods (A1 ) and production periods (A2 )

130

5 Open-Loop Integration of Planning, Scheduling and Optimal Control …

where hocfe is the discrete time step and cp,cp is the collocation matrix of the OCFE method. The calculation of raw material consumption cost during the production of a product is simply calculated as the area of the rectangle defined by the production ss ). For that time (Tip ) and the pre-specified steady state value of the control input (umi reason the aforementioned cost is calculated as shown by Eq. (5.6). A2 =

 ss Craw m umi Tip m

i

(5.6)

p

Overall the objective function to be maximised is the profit of the process which is given by Eq. (5.7). PROF =1 − 2 − 3 − 4 − A1 − A2

(5.7)

5.2.2 Modelling the Planning and Scheduling Problem For the integration of planning and scheduling in the present work a TSP model is employed. The model was originally proposed by Liu et al. [20] and proved to have computational benefits in comparison with the time-slot based formulation. Similar to the classic TSP problem where binary variables indicate the path from one city to another, the changeovers between different products are modelled in a similar fashion. Following this model, the planning horizon is divided into typically equal-length planning periods which are modelled as discrete time points and within each planning horizon continuous time representation is employed for the detailed schedule. Next, the equations that formulate the model are presented. A visual representation of the hybrid time formulation involved in the TSP model is given in Fig. 5.3. As shown by Fig. 5.3, demands for each product are satisfied at the end of each planning period and the inventory and backlog levels are calculated at those time points too. Within each time point (planning period), continuous time representation

Fig. 5.3 Hybrid time representation of the TSP model for the levels of planning and scheduling

5.2 Methodology

131 up

of total duration θp is employed for the modelling of decisions on a scheduling level. Notice that, a changeover between two adjacent periods, e.g. week p-1 and week p, occurs at the beginning of planning period p. In Fig. 5.3, the blocks with the stripes are used to denote changeovers while the rest denote production of a product.

5.2.2.1

Allocation Constraints

Only one product can be processed first in every planning period. To model this, the binary variable Fip is introduced in Eq. (5.8), which if equal to 1 indicates that product i is the first one to be produced in planning period p and zero otherwise. N 

Fip = 1

∀p

(5.8)

i

where i is the index of products, N is the total number of products and p is the planning period. Equation (5.9) ensures that only one product can be processed last in every planning period. N  Lip = 1 ∀p (5.9) i

Similar to Eq. (5.8) the binary variable Lip is used to indicate whether or not product i is the last one to be processed in planning period p. Equation (5.10) dictates that a product can be processed first only if it has been assigned to the planning period. Fip ≤ Eip

∀i, p

(5.10)

where Eip is a binary variable that indicates whether product i is assigned for production in planning period p. To ensure that a product can be processed last, only if it has been assigned to the planning period, Eq. (5.11) is used. Lip ≤ Eip

∀i, p

(5.11)

Notice, that the binary variables Fip and Lip are indicative of the sequence of tasks performed within each planning period as the order of the rest of tasks is computed implicitly by the changeovers.

5.2.2.2

Sequencing Constraints

When moving from the production of a product i to another product j, a changeover occurs. To model changeovers within the same planning period the binary variable Zijp is used if product i precedes product j in planning period p.

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5 Open-Loop Integration of Planning, Scheduling and Optimal Control … N 

Zijp = Ejp − Fjp

∀j, p

(5.12)

i=j

Equation (5.12) denotes that a product if assigned to a planning period, unless the first to be processed, results in a changeover with another assigned product. Next through Eq. (5.13) it is ensured that a product unless the last to be processed, if assigned to the planning period, results in a changeover with another assigned product. N  Zijp = Eip − Lip ∀i, p (5.13) j=i

Finally, to model changeovers across adjacent weeks Eqs. (5.14)–(5.15) are used. N 

Zfijp = Fjp

∀j, p > 1

(5.14)

i N 

Zfijp = Li,p−1

∀i, p > 1

(5.15)

j

where the binary variable Zfijp is used to model changeovers between adjacent planning periods.

5.2.2.3

Symmetry Breaking Constraints

In order to avoid the enumeration of symmetric solutions and exclude infeasible production sub-cycles the integer variable Oip (order index) is introduced and denotes the order at which the product i is processed during planning period p. Equations (5.16)–(5.18) ensure the exclusion of such subcycles. Ojp − (Oip + 1) ≥ −M(1 − Zijp ) Oip ≤ M · Eip Fip ≤ Oip ≤

N  i

Eip

∀i, j = i, p

∀i, p ∀i, p

(5.16) (5.17)

(5.18)

5.2 Methodology

5.2.2.4

133

Timing Constraints

At the core of the proposed model is the hybrid time formulation. Production (Tip ) trans and transition times (Tijp ) are modelled in a continuous way. Lower and upper bounds on processing time are imposed through Eq. (5.19). θplo Eip ≤ Tip ≤ θpup Eip

∀i, p

(5.19)

up

where θ plo , θp constitute lower and upper bounds respectively. The changeover time across two adjacent periods can be split into two parts in different time periods based on Eq. (5.20) which is adopted from the work of Kopanos et al. [21]. CT1p + CT2p−1 =

 i

trans Tijp · Zfijp

∀p > 1

(5.20)

j

The time balance within each planning period is then given by Eq. (5.21). N  i

Tip +

N  N  i

trans (Zijp · Tijp ) + CT1p|p>1 + CT2p|p 1

(5.30)

cp=1

The numerical solution of the state profiles across the discretised domain is computed by Eq. (5.31). ocfeinit ocfe trans xnijpfecp = xnijpfe + Tijp · hocfe

ncp 

cpp cp x˙ nijpfecpp

∀n, i, j, fe, cp

(5.31)

cpp=1

In general, variable steps (hocfe ) and continuity constraints on the profiles of the control variables may be employed, if required for the problem under study; however in the present work such cases were not considered.

5.2.4 Linking Variables Between DO, Scheduling and Planning Model Tracking the dynamic trajectory of the system can be translated into a regime of production. In the context of iPSC, changeovers can be defined as the time needed by the system to move from one steady state to the next one. Similarly, the production time can be defined as the time that the system is stabilised around the desirable steady state in order to manufacture a certain amount of products that would satisfy the demand. Ideally, one would have to perform a discretisation of the dynamics across the entire dynamic trajectory but in the present work since no process disturbances are

136

5 Open-Loop Integration of Planning, Scheduling and Optimal Control …

considered it is assumed that once the steady state is achieved the system remains in steady state unless a changeover occurs. Therefore, the discretisation of the dynamics is employed only for the transient periods of changeovers. In general, the linking between the DO and the scheduling and planning model is achieved with Eqs. (5.32)– (5.35). ocfeinit in = xnijpfe ∀n, i, j, i = j, fe = 1 xnijp in ocfe = umijpfecp umijp fin ocfe xnijp = xnijpN fe Ncp fin umijp

=

ocfe umijpN fe Ncp

(5.32)

∀m, i, j, i = j, fe = 1, cp = 1

(5.33)

∀n, i, j, i = j

(5.34)

∀m, i, j, i = j

(5.35)

in in fin fin , umijp , xnijp and umijp . The first two variThe new variables introduced here are xnijp ables define the value of the state and control inputs at the beginning of the transition and the last two are used in a similar fashion for the end of the transition. Equation ocfeinit ) at the beginning (5.32) ensures that the value of the discretised state variable (xnijpfe of the first finite element is equal to the initial condition of the system at the beginning of the transition, while Eq. (5.33) is employed for the control input. At the end of fin ) and Eq. the transition the state of the system must have reached a certain value (xnijp (5.34) imposes that value of the discretised state variable at the last collocation point (Ncp ) of the last finite element (Nfe ) is such that the transition terminates. Finally, Eq. (5.35) has similar functionality with Eq. (5.34) but for the discretised control input. A conceptual graph about the linking between DO and scheduling is given in Fig. 5.4 where the multi-scale nature of the problem is illustrated. On the top graph, the planning scale is modelled as discrete time points each of which is then expanded in a continuous time domain for the scheduling scale. The scheduling scale is then further analysed into the control scale and it is shown how the process moves from the production of product i to product j. At the beginning, the system is regulated around the steady state of product i until a changeover occurs in which case a transition from ss ss ) to the steady state of the next product, e.g. j, (xnj ) is the steady state of product i (xni initiated. On the bottom graph, the dynamics of the system throughout this transition are envisaged. Notice, that the system at the beginning of the transition starts from ocfeinit ss ) and thus the discretised state variable of the first finite element, i.e. xnijpfe| , (xni fe=1 must be equal to that value. During the dynamic optimisation, the time is discretised in finite elements (intervals between consecutive blue dashed lines in Fig. 5.4) and within each finite element a certain number of collocation points are defined (orange dots).

5.2 Methodology

137

Fig. 5.4 Different timescales involved in the iPSC problem and their interdependence

5.2.5 Linking Equations Between DO and TSP Planning and Scheduling Now that the linking variables between DO and scheduling have been established, it remains to provide the physical meaning of these variables in terms of production in fin and xnijp , correspond to the initial and final values of the scheduling. Since, xnijp transitions between different products it follows that they are also associated with the steady state operation of the system when production takes place. Indeed, the initial value of the state variables at the beginning of a transition should be equal to the steady state of the product that was being processed before the changeover occurred. Similarly, since the transition between two products is terminated once the system has reached the next steady state, the final value of the state variables at the end of the transition must be equal to the steady state of the following product. The same holds for the control variables. Since in the present work a TSP based planning and scheduling model is adapted, in contrast with the time-slot based formulations, the binary variable that can be used to express mathematically the aforementioned conditions is the Zijp for transitions that occur within the same planning period and Zfijp for transitions between adjacent planning periods. The final set of constraints that constitute the iPSC model are given by Eqs. (5.36)–(5.43).

138

5 Open-Loop Integration of Planning, Scheduling and Optimal Control … in ss xnijp = xni · Zijp

∀n, i, j, i = j, p = 1

in ss umijp = umi · Zijp ∀m, i, j, i = j, p = 1 in ss xnijp = xni · (Zijp + Zfijp ) ∀n, i, j, i = j, p > 1 in ss umijp = umi · (Zijp + Zfijp ) ∀m, i, j, i = j, p > 1 fin ss xnijp = xnj · Zijp ∀n, i, j, i = j, p = 1 fin ss umijp = umj · Zijp ∀m, i, j, i = j, p = 1 fin ss xnijp = xnj · (Zijp + Zfijp ) fin umijp

=

ss umj

· (Zijp + Zfijp )

∀n, i, j, i = j, p > 1 ∀m, i, j, i = j, p > 1

(5.36) (5.37) (5.38) (5.39) (5.40) (5.41) (5.42) (5.43)

Equation (5.36)–(5.37) impose that the initial values of the state and control variables respectively are equal to the steady state values of product i, which is the precedent product in the changeover for the first planning period. After the first planning period a changeover might occur at the beginning of this period, in which case the binary variable Zfijp is activated, to indicate this transition from the previous week to the current one. Because of that, in Eqs. (5.38)–(5.39) the summation of the two changeover binary variables, i.e. Zijp , Zfijp is included. Similarly, Eqs. (5.40)– (5.43) impose that at the end of each transition the state and control input of the system must reach the next steady state.

5.2.6 Monolithic and Decomposed Integration of Planning, Scheduling and optimal Control 5.2.6.1

Monolithic iPSC

Integrating the decision levels of planning, scheduling and optimal control for continuous manufacturing processes results in a MIDO problem, where the decisions are made simultaneously across all three levels under a common objective, the maximisation of profit. As presented in the previous sections, in the present work the model of Liu et al. [20] is employed for the problem of simultaneous planning and scheduling and then OCFE is used for the discretisation of the dynamics of the system for the optimal control. The optimal control is linked using the constraints and auxiliary variables presented in Sect. 5.2.5. Overall, the monolithic model for the iPSC consists of the following: Monolithic iPSC :

max PROF = 1 − 2 − 3 − 4 − A1 − A2 Subject to : Eqs. (5.8) − (5.25) (TSP Planning − Scheduling) Eqs. (5.28) − (5.31) (Dynamic Optimisation) Eqs. (5.32) − (5.43) (Linking constraints)

5.2 Methodology

5.2.6.2

139

Decomposed iPSC

To alleviate the computational burden associated with the monolithic formulation of the iPSC, in the present work a decomposition of the problem is proposed through the solution of offline DO problems and the development of linear metamodels to associate the transition time with transition cost. The decomposition is based on the grounds of the following two observations: 1. Formulating and solving the monolithic iPSC results typically in large scale non-convex MINLPs problems which are difficult to solve and the use of global optimisation solvers is rather difficult due to the scale of the MINLPs [9, 19]. Within the process industry, being in a position to provide on time good solutions is crucial and with the monolithic MINLPs such requirement may not be always satisfied. Upon inspection, apart from the time-scale problem that is inherent to the iPSC, the solution of the underlying dynamic optimisation problem constitutes the main bottleneck because of the consideration of the transition times as decision variables. In the literature, ways for dealing with this problem include, fixing the transition time matrix in an offline step based on heuristics [18] or creating pre-computed recipes that include couples of transition time and cost [19]. The disadvantage of following the aforementioned alternatives are for the first one that it does not reflect a true integration among the levels of iPSC and the latter assumes that the decision maker will always face one of the pre-specified realisations of transition time and cost. 2. To the best of the author’s knowledge, in none of the research works conducted for the iPSC of continuous processes the effect of disturbance at control level has been accounted for. In order to compensate for the effect of disturbances, one would have to employ a feedback control mechanism. To this effect, once a disturbance is realised, the decision maker would possibly need to re-schedule. It can be understood then, that the monolithic iPSC would have to be solved repeatedly, in decreased decision space (as some of the decisions would have already been fixed), but at a fast computational time. This cannot be done given the scale and nonconvexity of the problem as will be illustrated later on by the case studies, given the current computational power at hand. It follows that the feedback mechanism / closed-loop control should be considered in an outer loop rather than within the iPSC formulation. In Fig. 5.5, the flow of information across the different levels of decision making can be envisaged. As shown, the main information shared between the level of optimal control and scheduling is the transition times and the transition cost. Of course, the dynamics of the system given a certain transition time can be determined for the open loop “control” case, which is the one of interest of this work. In order to exploit this interaction, in the literature, a method of creating pre-computed pairs of transition time and cost has been proposed [19]. The drawbacks of following such an approach is the increase in the number of binary variables introduced in the iPSC model formulation for the selection of the specific pair of transition time-cost as well as it does not necessarily treat the case where a feedback controller is used and a

140

5 Open-Loop Integration of Planning, Scheduling and Optimal Control …

Fig. 5.5 Flow of information across the different levels of decision making within the iPSC

realisation of the system’s dynamics other than the pre-computed ones is observed. To this effect, linear metamodels are developed so as to carry the aforementioned information between the levels of optimal control and scheduling. The reason for that is two-fold: (1) through the use of the metamodels the transition times are still allowed to be continuous decision variables of the integrated problem, allowing for instances where the closed loop behaviour differs from the open loop thus resulting in different transition times than the minimum ones and (2) the procedure of building the linear metamodels is carried out offline and the problems associated are NLPs of moderate size for which global optimisation solvers can possibly be employed. As will be shown in the next section, through the case studies examined, the law that correlates transition time and transition cost is of logarithmic nature for large time scales but for shorter ones it can be approximated precisely by the linear metamodels. As far as the dynamic open-loop trajectories are concerned, with the transition time fixed the control and state variables can be calculated through the solution of the related optimal control problem in a subsequent step. However, for the deterministic case which is under study in the present work the decomposed model will always choose the minimum transition times because it infers minimum cost and so that particular dynamic trajectory is stored and employed. For the decomposed framework of the iPSC (decomposed iPSC for short), first the minimum transition times for any possible transition are computed based on problem (Offline DO part I). Offline DO part I : τijmin =

min

xocfe , uocfe

Subject to :

Tijtrans Eqs. (5.28) − (5.31) (Dynamic Optimisation) Eqs. (5.32) − (5.43) (Linking constraints)

Offline DO part I, is a non-convex NLP even for the case that linear dynamics govern the production system, since within the numerical integration, the time is

5.2 Methodology

141

considered as the objective variable. This can be better understood if the bilinear xnijpfecp ) in terms between the variable Tijtrans and the numerical value of the ODE (P Eqs. (5.30)–(5.31). After the transition time is computed for each possible product combination, the procedure of building the linear metamodels follows. In the present work, the transition time is allowed to vary up to 3 times the minimum transition time. Problem (Offline DO part II), is again an NLP and creates the data samples that will be used for the linear metamodels. Offline DO part II :

min

xocfe , uocfe

CTijtran

Subject to : Eqs. (5.28) − (5.31) (Dynamic Optimisation) Tijtrans ∈ [τijmin , 3τijmin ] (Variable transition time) In order to create a sufficient number of data points, a sampling method needs to be followed. After choosing the number of points (nsample ) the following computational routine (Meta_sample) is performed. Meta_sample: Tijtrans = τimin j While iter ≤ nsample :

EndWhile

max

Tijtrans = Tijtrans + τijmin · ntsample Solve (Offline DO part II) iter = iter + 1

where tmax denotes the maximum variance from the minimum transition time allowed, e.g. in the present work tmax = 2. The output of this computational step, which is performed offline, is a set of points correlating transition time and cost for each transition. With the sample data computed, the next step is to create the metamodels. In the present work, the Statsmodels 0.6.1 module from Python was employed for the creation of the linear metamodels [25]. A general form of the linear metamodels is given by Eq. (5.44). + βij CTijtran = αij Ttrans ij

∀i, j ∈ I, i = j

(5.44)

where αij , βij are N × N matrices, denoting the slope and intercept of the linear metamodel respectively. Once the metamodels and the minimum transition times are computed the next step is to form the final decomposed iPSC model. Ideally, the transition times should be kept as decision variables within the decomposed iPSC while allowing again the changeovers between two adjacent planning periods to be split. When the transition times are decision variables, bilinear terms of the form trans trans · Zijp or Tijp · Zfijp appear. Exact linearisation of such bilinear terms can be Tijp achieved based on the following [26]:

142

5 Open-Loop Integration of Planning, Scheduling and Optimal Control … translin1

Tijp

trans ≥ Tijp + (Zijp − 1) · τmax

∀i, j ∈ I, i = j, p

(5.45)

translin Tijp 2 translin Tijp 1

trans ≥ Tijp + (Zfijp − 1) · τmax

∀i, j ∈ I, p > 1

(5.46)

trans ≤ Tijp

∀i, j ∈ I, i = j, p

(5.47)

CT1p + CT2p−1 =

translin Tijp 2

(5.48)

translin Tijp 2

∀i, j ∈ I, p > 1

trans ≤ Tijp

∀i, j ∈ I, p > 1

(5.49) translin

translin

In Eqs. (5.45)–(5.49), new variables are introduced, Tijp 1 , Tijp 2 , which are the linear counterparts of the aforementioned bilinear terms. Notice, that this is achieved either when Zijp or Zfijp is equal to 1, thus indicating the corresponding changeover between products. In addition to that, the transition time between products should be bounded from below based on Eq. (5.50). trans ≥ τijmin Tijp

∀i, j ∈ I, i = j, p

(5.50)

trans trans · Zijp or Tijp · Zfijp ) is also The linear counterpart of the bilinear terms (Tijp used in the metamodels. The objective of the decomposed iPSC is the same as the one used in the monolithic formulation with the exception of the term A1 which is substituted by Eq. (5.51).

B1 =

 Translin Translin αij (Tijp 1 + Tijp|p>1 2 ) + βij (Zijp + Zfijp ) p

i

(5.51)

j=i

Overall, the decomposed iPSC model is an MILP and is formulated as follows: Decomposed iPSC :

max PROF = 1 − 2 − 3 − 4 − B1 − A2 Subject to : Eqs. (5.8) − (5.25) (TSP Planning − Scheduling) Eqs. (5.45) − (5.50) (Time linearisation)

5.3 Case Studies A number of case studies are presented in the next section to demonstrate the advantages of the proposed modelling framework for the iPSC. The first one examines the iPSC of a single-input single-output (SISO) multi-product CSTR; the second one, a multiple-input multiple-output (MIMO) non-isothermal multi-product CSTR while the third one studies methyl methacrylate (MMA) polymerisation process. The pattern followed in this section is as follows: first, a comparison between the monolithic and decomposed iPSC models is given, for both the TSP and the time-slot based formulation, and then the decomposed approach is tested under different planning horizons, namely 4, 6 and 12 weeks.

5.3 Case Studies

143

All the optimisation problems, for the monolithic approach, are formulated as MINLPs and solved using GAMS 24.4.1 [27] on a Dell workstation with 3.70 GHz processor, 16GB RAM and Windows 7 64-bit operating system. The optimisation problems corresponding to the decomposed approach are formulated as MILPs and modelled in GAMS 24.4.1 and solved using CPLEX 12.6.1. Finally, for the comparison between the TSP and the time-slot based model for the iPSC some modifications need to be made with respect to the existence of idle time within the planning period, backlog of unmet demand and inventory calculation. The time-slot based model was proposed by Dogan and Grossmann [8] (E-D &G) and the corresponding model is given in Appendix D. As far as the inventory calculation is concerned the TSP model proposed by Liu et al. [20], considers inventory calculation on a weekly basis whereas E-D&G employ a linear overestimation of the inventory curve; to this effect, Eqs. (D.14)–(D.17) are replaced by Eq. (5.24). Next, in E-D&G, the demand serves as lower bound to the level of sales at the end of each planning period and backlog is not considered; in order to account for backlog, Eq. (D.18) is replaced by Eq. (5.23). Finally, idle time is allowed between planning period by modifying Eq. (D.11) as shown in Eq. (5.52). TNe s p +

 i



s τij Zijp ≤ T1,p+1

∀p = 1, . . . , Np − 1

(5.52)

j=i

5.3.1 SISO Multi-product CSTR First, the iPSC of a multi-product SISO CSTR that produces five different products, namely, A, B, C, D, E, based on the concentration and the volumetric flow of the reactant, is considered. Details about the related model have been presented in Sect. 4.3.1. Assuming that, V = 5000L, C0 = 1 mol/L and k = 2L2 /mol2 h the aim of the iPSC is to maximise the profit of the process while satisfying the demand as shown in Table 5.1. The system based on the given design parameters can exhibit multiple steady state at which different products can be produced. The rate of production for each product C −Css can be calculated as ri = Qiss C0 ( 0C0 i ). The related kinetic information is provided in Table 5.2.

Table 5.1 Demand and cost data for the SISO CSTR case study Planning period Operation cost ($/m3 ) 3 Demand (m ) P1 P2 A B C D E

400 3000 7000 15000 31000

0 8000 1200 0 20000

0.13 0.22 0.35 0.29 0.25

Price ($/m3 ) 200 150 130 125 120

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5 Open-Loop Integration of Planning, Scheduling and Optimal Control …

Table 5.2 Kinetic data for the SISO CSTR case study i (mol/L) i (L/h) Products xss uss A B C D E

0.0967 0.2000 0.3032 0.3930 0.5000

ri (mol/h)

10 100 400 1000 2500

9.033 80.000 278.720 607.000 1250.000

min ) for Table 5.3 Problem statistics for the offline computation of the minimum transition time (τi,j the SISO CSTR case study Solver CONOPT3 BARON ANTIGONE

Type Solution status Constraints Cont. Var. CPU (s)a a Cumulative

NLP Locally optimal 204 266 1.059

NLP Optimal 204 266 23,452

NLP Optimal 204 266 149.632

computational time for all the transitions

Table 5.4 Minimum transition times between products for the SISO CSTR case study min (h) τi,j A B C D E A B C D E

– 20.99 24.6 25.72 26.34

0.21 – 3.62 4.74 5.38

0.47 0.26 – 1.13 1.76

0.85 0.61 0.31 – 0.64

1.64 1.43 1.21 0.87 –

Because of the order of the reaction the corresponding optimisation problem is formulated as an MINLP. Initially, we consider two planning periods, each one spanning a week (168h) and employ 20 finite elements with 3 collocation points for the discretisation of the corresponding dynamic system. Following the proposed framework, the minimum transition times and the linear metamodels are calculated offline. In Table 5.3, a comparison between the performance of BARON 14.4, CONOPT3 and ANTIGONE is provided. The computation of the minimum transition times results in the solution of 20 NLPs and Table 5.4 provides the related times using ANTIGONE. The linear metamodels are built next following the sampling technique described in Sect. 5.2.6.2. While building the linear metamodels, the majority has a value of R2 ≥ 0.99 but some fail to reach this desired threshold like the one shown in Fig. 5.6 which corresponds to the transition from product C to product A. The total number of metamodels that have R2 ≤ 0.99 are four out the twenty but for the sake of computational complexity we allow this approximation as it still reflects the impact that

5.3 Case Studies

145

Fig. 5.6 Graph of the linear metamodels built from sampling data for the transition from C to A

the dynamics of the system have in the production scheduling. Note that piecewise linear approximation could have been employed but that would lead in increase of the number of binary variables needed. In general it was noticed based on the simulations that were conducted, that the law that governs the relation between transition time and cost is of logarithmic nature for large time scales; however, from a production scheduling perspective for the instance for the transition from C to A a delay of the transition of the order of hours would probably result in rescheduling thus the effect of offset from the metamodel can be circumvented. In any case, as mentioned in Sect. 5.2.6.2, the aim of building the corresponding metamodels is to provide a computationally favorable correlation between the process dynamics and process economics within the context of iPSC. In Table D.1, the corresponding coefficients of the slope and the intercept of the linear metamodels are given. Once the metamodels are built, the monolithic and the decomposed integrated problems are formulated and solved in GAMS. Results for the SISO CSTR case study for planning horizon of 2 weeks are shown in Table 5.5. The results from Table 5.5 indicate as expected that the monolithic model for the iPSC has inferior performance when compared to the decomposed approach. As far as the comparison between TSP and the time slot formulation (T-S) is concerned, even though the integration through the TSP results in larger number of variables and constraints (because of the different linking constraints employed for the case of TSP), the TSP model needs less than half the number of binary variables to model the scheduling decisions. This advocates towards the better computational performance of the TSP over the T-S formulation for the iPSC. Moreover, despite the integration, suboptimal solutions may be computed because of the nonconvex nature of the problem. From a scheduling point of view the solution is suboptimal, since the monolithic iPSC chooses to perform a changeover between the adjacent

146

5 Open-Loop Integration of Planning, Scheduling and Optimal Control …

Table 5.5 Results for the SISO CSTR case study for planning horizon of 2 weeks. OCFE with 20 finite elements and 3 collocation points was employed for the discretisation of the DO part Model Monolithic TSP Monolithic T-S Decomposed iPSC Type No. Eq. Con. Var. Bin. Var. Solver Profit ($) CPU (s) Optimal Schedule trans (h) Ti,j,p Ti,p (h)

MINLP 8,557 3,170 10,841 3,551 105 300 SBB/CONOPT3 8,862,971 274.780 995.539 P1 : B → C → D → A → E P2 : B → C → E P1 : 0.57, 0.68, 57.88, 3.68 P2 : 12.1 , 0.57, 2.63 P1 : 19.27, 25.11, 24.71, 0, 24.8 P2 : 118.23, 4.3, 16

MILP (TSP) MILP (T-S) 287 581 256 488 105 300 CPLEX 12.6.1 9,190,688 0.078 3.5 P1: A→ B→ C→ D→ E P2: E→ C→ B P1 : 0.21, 0.25, 0.30, 0.86 P2 : 1.76, 3.62 P1 : 44.28, 37.5, 25.11, 24.71, 24.8 P2 : 16, 4.305, 100

planning periods even though product E is manufactured in both weeks and also the transition among products are not the optimal ones. In the solution computed by the monolithic approach it is noted that the transition times are not the minimum ones and this is because of the difficulty to identify the best possible transition due to the combinatorics involved. For example, in the first planning period the transition from D to A instead of 25.72 h which is minimum transition time is computed as 57.88 h and that results in not sufficient production time for A and backlog of the related demand. One could argue that the solution of the decomposed iPSC is affected by the approximation involved in the metamodels. For that reason, the monolithic TSP model was solved again with all the binary decision fixed based on the solution of the decomposed iPSC and as expected the solutions computed among the two are identical as shown in Table 5.6. Table 5.6 shows that indeed the solution computed by the decomposed iPSC is correct and feasible for the original monolithic model. Notice, that even with all the binary variables fixed, it takes approximately 150s for SBB to compute the solution of the MINLP. It appears that the main bottleneck in the monolithic iPSC is the nonconvex DO part where the transition times are treated as decision variables. From a mathematical perspective this results in nonconvex bilinear terms which require a global optimisation scheme, e.g. the use of a spatial branch and bound. The comparative profit breakdown of the two solutions is given in Table 5.7. Figure 5.7, depicts the production plan computed in terms of the evolution of the state of the system and control input. From an operational point of view, the sequence computed by the decomposed iPSC model is validated as feasible from the full-space monolithic iPSC. The monolithic iPSC computes less profit than the decomposed iPSC model partly because as indicated by Table 5.7 the transition cost is increased, due to the

5.3 Case Studies

147

Table 5.6 Results of the SISO CSTR case study with fixed sequencing decisions from the decomposed iPSC Model Monolithic TSP Decomposed iPSC Type No. Eq. Con. Var. Bin. Var. Solver Profit ($) CPU (s) Optimal Schedule trans (h) Ti,j,p Ti,p (h)

MINLP 8,557 10,841 105 SBB/CONOPT3 9,161,172 148.32 P1: A→ B→ C→ D→ E P2: E→ C→ B P1 : 0.47, 0.57, 0.68, 1.94 P2 : 3.96, 8.14 P1 : 35.3, 37.5, 25.1,24.7, 24.8 P2 : 16, 4.305, 100

MILP (TSP) 277 241 105 CPLEX 12.6.1 9,190,688 0.078 P1: A→ B→ C→ D→ E P2: E→ C→ B P1 : 0.21, 0.25 , 0.30, 0.86 P2 : 1.76, 3.62 P1 : 44.2, 37.5, 25.1, 24.7, 24.8 P2 : 16, 4.305, 100

Table 5.7 Comparative cost breakdown of iPSC solution for the SISO case study Monolithic TSP Decomposed iPSC Sales ($) Operational cost ($) Inventory cost ($) Transition cost ($) Production cost ($) Backlog cost ($)

10,784,287 22,437 0 67,609 1,526,354 6,713

10,791,000 22,442 0 51,143 1,526,726 0

prolonged transitions computed by the model. The biggest difference however is identified by the backlog cost since the increased transition times as computed by the monolithic do not allow the plant to meet its nominal demand and a significant part of it is being backlogged and penalised. Next, the same example is examined but this time, the discretisation of the time for the DO part is performed with 45 finite elements and 3 collocation point as design parameters for the OCFE. By doing so, the trade-off between the degree of the discretisation scheme and the optimal solution is under investigation. The minimum transition times and linear metamodels are computed again, using CONOPT3 and it takes 10.65s to converge. Table 5.8, provides the minimum transition times and as can be noticed the transition times are improved when compared to the ones computed with 20 finite elements. The monolithic and decomposed iPSC are solved again in GAMS and the results are provided in Table D.2. Increasing the number of finite elements employed for the discretisation of the continuous dynamics on the one hand can potentially lead to more

148

5 Open-Loop Integration of Planning, Scheduling and Optimal Control …

(a) Dynamic profile of the reactant’s concentration over the planning horizon for the SISO CSTR case study.

(b) Dynamic profile of the reactant’s flowrate over the planning horizon for the SISO CSTR case study.

Fig. 5.7 Dynamic interpretation of the production plan for the SISO CSTR case study Table 5.8 Minimum transition times for the SISO CSTR case study with OCFE of 45 finite elements and 3 collocation points min (h) τi,j A B C D E A B C D E

– 20.709 24.463 25.388 26.01

0.208 – 3.569 4.682 5.307

0.46 0.252 – 1.212 1.738

0.761 0.552 0.303 – 0.67

1.613 1.414 1.162 0.852 –

5.3 Case Studies

149

profitable operation but it also exacerbates the computational burden associated with the monolithic solution of the iPSC. Even in that case though, as observed in Table D.2 the solution of the monolithic formulation does not guarantee that the benefits of the integration can be exploited. Similar to the previous results, where the OCFE was employed with 20 finite elements, because of the non-convexity that characterises the problem again longer transition times and unnecessary changeovers between products occur that hinder the potential benefits of the integration. On the contrary, solving the corresponding DOs offline does not result in exhaustive computational times and the decomposed iPSC framework does not get affected performance-wise. Finally, variable planning horizons are considered and the performance between the TSP and T-S formulation for the decomposed iPSC is compared. As shown in Table D.3, as the planning horizon increases the TSP model for the decomposed iPSC outperforms the T-S model. For the case of planning horizon of 12 weeks, for the same solution, TSP decomposed iPSC takes 4.4 s while the T-S decomposed iPSC needs 3440.21 s.

5.3.2 MIMO Multi-product CSTR The next case study examines the iPSC of a non-isothermal MIMO CSTR where the decomposition reaction of the form A→R takes place under the kinetic law: −Rb = kr Cb . The reaction is exothermic and as such a cooling jacket is used; further details about the kinetics and design can be found in the book of Camacho and Alba [28]. The system is controlled through the flow rate of the liquid (Fl ) and the coolant (Fc ) while the corresponding states are the concentration of the decomposition product (Cb ) and the temperature of the liquid (Tl ). The calculation of the steady state conditions is carried out in an offline step and the related results are given in Table D.5. The utility price for the coolant Fc is $10/m3 while cost of product liquid Fl is $10/m3 . The dynamic model of the system is derived based on the mass and energy balances as shown in Eq. (5.53), respectively where the concentration of reactant A is assumed to be constant. Further details about the system can be found in the book of Camacho and Alba [28] and in Zhuge and Ierapetritou [16]. The rest of the data about the case study are given in Tables D.4–D.5; the cost of backlog is calculated as half of the selling price of the related product (Table 5.9). d(Vl Cb ) =Vl kr (Ca0 − Cb ) − Fl Cb dt d(Vl ρl Cpl Tl ) = Fl ρl Cpl Tl0 − Fl ρl Cpl Tl + Fc ρc Cpc (Tc0 − Tc ) dt + Vl kr (Ca0 − Cb )H (5.53) The minimum transition times as calculated by the offline DO, for discretisation with OCFE with 20 finite elements and 3 collocation points, are shown in Table 5.10.

150

5 Open-Loop Integration of Planning, Scheduling and Optimal Control …

Table 5.9 Design parameters of the MIMO multi-product CSTR case study kr reaction constant Vl volume of the tank ρl liquid density ρc coolant density Cpl specific heat of liquid

26 h1 24L3 800 mkg3 1000 mkg3 3 kgkJ· K

Cpc specific heat of coolant

4.19

Tl0 entering liquid temperature Tc0 inlet coolant temperature Tc outlet coolant temperature Ca0 initial concentration of the reactant

283K 273K 303K 4 mol L

kJ kg · K

Table 5.10 Minimum transition times between products for the MIMO case study A B C D E F G min (h) τi,j A B C D E F G H

– 0.0184 0.0243 0.0288 0.0324 0.0330 0.0341 0.0358

0.0138 – 0.0053 0.0103 0.0141 0.0145 0.0216 0.0176

0.0285 0.0184 – 0.0049 0.0081 0.0087 0.0388 0.0119

0.0512 0.0422 0.0304 – 0.0320 0.0239 0.0622 0.0375

0.0441 0.0334 0.0185 0.0113 – 0.0011 0.0541 0.0112

0.0523 0.0424 0.0260 0.0140 0.0227 – 0.0627 0.0302

0.0362 0.0324 0.0274 0.0255 0.0242 0.0252 – 0.0245

H 0.0677 0.0556 0.0486 0.0410 0.0298 0.027 0.0767 –

The coefficients for the linear metamodels that correlate transition time and cost are also given in Table D.7. The transition times computed indicate system’s fast dynamics where rapid changeovers between manufacturing conditions are achieved. This particular instance is very insightful as ideally, one would like to be in a position to have the solution of the integrated problem at hand in times faster than the transition times. This is because, assuming that the system is subject to disturbances at the level of process dynamics, then deviation from the pre-computed transition time/profile may lead to economic loss of the process. The dynamic response of the system as shown in Fig. 5.8, for the case of transition from product B to product F, the fast transitions are achieved with rapid rate of change of the manipulated variables while the profiles of the state variables remain rather smooth. Formulating and solving the monolithic and decomposed iPSC the results shown in Table 5.11, are computed. The multi-product MIMO CSTR case study, differs from the SISO CSTR case study that was previously studied in the following two ways: (1) it has increased dimensionality in terms of control and state variables and (2) the number of products is also increased from 5 to 8. The first difference, affects mostly the DO part of the

5.3 Case Studies

151

(a) Dynamic profile of the concentration of the decomposition product.

(b) Dynamic profile of the temperature of the liquid.

(c) Dynamic profile of the flow rate of the coolant.

(d) Dynamic profile of the flow rate of the liquid.

Fig. 5.8 Dynamic response of the system for the transition from B to F Table 5.11 Results for the MIMO CSTR case study for planning horizon of 2 weeks. OCFE with 20 finite elements and 3 collocation points was employed for the discretisation of the DO part Model Type

Monolithic TSP

Monolithic T-S

MINLP

Decomposed iPSC MILP (TSP)

MILP (T-S)

No. Eq.

45,703

8,965

647

4,157

Con. Var.

60,465

10,865

601

3,361

Bin. Var.

240

1,152

240

1,152

Solver Profit ($) CPU (s)

SBB/CONOPT3

CPLEX 12.6.1

3,075,778 660.15

7,710,974 832.25

0.765

50.857

Optimal

P1 : B → H → A → G

P1: D→ F→ E→ C→ B→ A

Schedule

P2 : B → C → G → A → F

P2: A→ B→ C→ D→ F→ E→ H→ G

P1 : 0.057, 0.212, 0.037

P1 : 0.014, 0.001, 0.008, 0.018

trans (h) Ti,j,p

P2 : 0.586, 0.019, 0.028, 0.796, 0.053 P2 : 0.013, 0.018, 0.030, 0.014, 0.001, 0.030, 0.025 Ti,p (h)

P1 : 22.8, 6.4, 28.5, 57.6

P1 : 24, 85, 4.8, 24.7, 2.6, 22.8, 28.5

P2 : 1.4, 2.8, 12.5, 1.9, 95.6

P2 : 1.9, 1.4, 0.17, 4, 10.6, 66.3, 6.4, 70.1

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Fig. 5.9 Comparative graph of the coolant flowrate (Fc = f(t)) as computed by the decomposed and the monolithic solution approach

iPSC while the second affects the combinatorial nature on the level of planningscheduling of the iPSC. This effect, becomes apparent after the observation of the results in Table 5.11. The solution computed through the monolithic approach, is clearly suboptimal because not only the transition times computed are not the minimum possible ones but also because there is a number of inconsistencies at a planning and scheduling level. More specifically, again a changeover occurs across the adjacent planning weeks which should not have occurred since product B is produced in both weeks; next, the entire available time is not consumed especially in week 1 where a great amount of demand, e.g. for product E, is backlogged despite the availability of processing time. Because of the great difference computed between the two solution approaches as shown in Table 5.11, again the integer decisions as computed by the decomposed approach are fixed in the monolithic model and it is solved again in “reduced space”. Not surprisingly, with the exception of some transition times which were computed with larger values by the monolithic model, the solutions are identical in terms of planning and scheduling decision while the dynamic profiles are the same as well. In Fig. 5.9, a comparative graph with the flow rate of the coolant as computed by the monolithic and the decomposed model can be envisaged. Figure 5.9, shows that the monolithic approach computes a less optimal solution for the transition from product H to product G as in the beginning the first control move is 0 and then reaches the value of 1000 m3 /h. Next, the computational performance of the T-S and TSP models for the decomposed iPSC is compared based on three different planning periods, i.e. 4, 6 and 12 weeks. The related demand over the planning period for each product is given in Table D.7. As shown in Table D.8, increasing the number of products has a significant impact on the computational performance of both models since the number of binary variables is increased. However, even in that case the computational performance of the TSP formulation is better than the one achieved by the T-S formulation.

5.3 Case Studies

153

Table 5.12 Data for MMA case study 3

Qm V Ciin Mm Cmin f∗

1 0.1 8 100.12 6 0.58

Monomer feed stream [ mh ] Reactor volume [m3 ] Feed stream initiator concentration [ kmol ] m3 kg Monomer molecular weight [ kmol ] Feed stream monomer concentration [ kmol ] m3 Initiator efficiency

ktc

1.328 × 1010

ktd ki

1.093 × 1011 1.025 × 10−1

m Termination by coupling rate constant [ kmol ·h]

kp

2.495 × 106

kfm

2.452 × 103

3

3

m Termination by disproportion rate constant [ kmol ·h] 1 Initiation rate constant [ h ] 3

m Propagation rate constant [ kmol ·h]

3

m Chain transfer to monomer rate constant [ kmol ·h]

5.3.3 MMA Polymerisation Process Next, the iPSC of the isothermal methyl-methacrylate polymerisation process [29] is considered, where different polymer grades are produced. The model and related information have been outlined earlier in Sect. 4.3.2. The dynamic model of the system is derived based on the following assumptions [29]: (i) constant density and heat capacity of the reacting mixture and the coolant, (ii) insulated reactor and cooling system, (iii) no polymer in the inlet streams, (iv) no gel effect caused by the low polymer conversion, (v) constant reactor volume, (vi) negligible flow rate of the initiator solution compared to the flow rate of the monomer stream, (vii) negligible inhibition and chain transfer to solvent reactions, (viii) quasi-steady state and long chain hypothesis. Data about the kinetic and design information of the MMA case study are given in Table 5.12. The system consists of four different states, namely: Cm is the monomer concentration, Cl represents the concentration of the initiator, D0 denotes the molar concentration of dead polymer chain while Dl the corresponding mass concentration. The system is controlled through the manipulation of the initiator flow rate (Fl ) while the measurable output (y) is the average molecular weight of each polymer grade. Based on the multiple steady states that the system exhibits [30], products of different quality can be produced. In the present work 5 different polymer grades are produced and the corresponding information about the steady state operating conditions are given in Table 5.13 while the related demand and cost data are given in Table D.9. Note that the backlog cost is calculated as CBi,c = 0.5 · Pi and the 3 inventory cost is Cinv i = 0.1($/m · h). The dynamic optimisation of the corresponding system is rather demanding as it results in numerical issues, which stem from the different numerical scales associated with the state and control variables. Following the proposed framework, first the pairwise minimum transition times are computed offline resulting in 20 DO problems.

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Table 5.13 Steady stateoperating for  conditions   theMMA products     kg kmol kmol kmol ss ss ss kg ss Cm m 3 Cl D D y 3 3 3 0 l kmol m m m A B C D E

3.078 3.725 3.978 4.201 4.583

0.148 0.0615 0.0426 0.0302 0.0157

0.0195 0.0091 0.0067 0.0051 0.00315

292.546 227.699 202.380 180.064 141.866

15000 25000 30000 35000 45000

Table 5.14 Minimum transition times for the MMA case study min (h) τi,j A B C D A B C D E

– 2.65 2.84 2.93 3.00

3.64 – 1.28 1.48 1.61

4.45 2.61 – 1.17 1.40

 Fl

5.18 3.58 2.65 – 1.24

m3 h



0.2048 0.0847 0.0586 0.0416 0.0217

E 6.51 5.13 4.45 3.72 –

The DO problems are further discretised using OCFE with 3 collocation points and 20 finite elements and the corresponding NLP problem are formulated and solved in GAMS using BARON 14.4 and CONOPT3 as solvers. The results of the minimum transition times are given in Table 5.14 while an example of the profiles of the control input and system output during the transition from product E to D is shown in Fig. D.1. The coefficients for the linear metamodels for the MMA case study can be found in Table D.12. It is should be mentioned that both BARON and CONOPT3 converge to the same solution but despite the fact that CONOPT3 is much faster it requires the provision of a good initialisation point as noted while performing the numerical experiments. The computational statistics are given in Table D.11. The same problems were solved employing OCFE with 45 finite elements with slightly improved results computed which for the sake of space are not reported herein. After the minimum transition times have been computed, with transition time trans min min ∈ [τi,j , 3τi,j ] a series of optimisation up to three times the minimum one, i.e. Ti,p problems are solved with the objective to minimise the corresponding transition cost, i.e. the utilisation of control input. Once the linear meta-models have been computed offline, both the monolithic and the decomposed one are formulated and solved in GAMS and the corresponding statistics are given in Table 5.15. In Table D.13, the decision made by the decomposed iPSC are shown, while in Fig. 5.10 the systems-dynamic interpretation of the corresponding schedule for the first planning period can be envisaged. In Table D.14, the decision computed by the monolithic approach are given in more details. Studying Table D.13, justifies why the solution computed by the monolithic models (both the TSP and the T-S) is clearly suboptimal when compared to the

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155

Table 5.15 Results of the computational performance between the monolithic and the decomposed integrated model using the proposed TSP and the time-slot (T-S) based formulation for the MMA case study Model Monolithic TSP Monolithic T-S Decomposed TSP Decomposed T-S Horizon Type No. Eq. Con. Var. Bin. Var. Solver Profit ($) CPU (s) Gap (%)

2 Weeks MINLP 35,067 37,767 105 SBB/CONOPT3 11,423 8,253 0.26

MINLP 7,890 10,811 300 11,423 10,323 0.26

MILP 287 236 105 CPLEX 12.6.1 15,439 0.39 0

MILP 1,083 951 300 15,439 1.5 0

decomposed one. First similarly to the previous case studies, from a scheduling perspective unnecessary change-over occurs between planning period 1 and 2, as A is produced in both periods and the change-over sequence within each planning period differs from the one computed by the decomposed one. The latter, results in less available production time and an amount of demand to be backlogged, thus incurring additional costs. Another interesting observation is the change-over performed at the second planning period from A to E, from the decomposed iPSC. One by reading Table 5.14, would probably expect to have a changeover between A and B as the transition time is less; however, the optimiser chooses the transition between A and E and this is due to the less transition cost involved. This particular instance, stands for a very clear example of where optimal control, planning and scheduling interconnect and how the proposed decomposition captures these interdependences without strenuous computational times. In order to demonstrate the computational potential of the proposed decomposition the same case study was examined with planning horizons of 4, 6 and 12 weeks and a comparison between the TSP and T-S formulation was conducted. As shown in Table 5.16, for short planning horizons the TSP and T-S perform similarly but as larger planning horizons are considered the TSP outperforms the T-S formulation. This can be justified by both the less number of equations and variables generated and the symmetry breaking constraints used. Finally, the case of multiple customers for planning horizon of 8 weeks is examined with a total of 10 customers. The related computational results are given in Table D.14.

5.4 Concluding Remarks In this chapter, the integrated Planning, Scheduling and optimal Control (iPSC) problem for continuous manufacturing processes was introduced. A TSP based model was employed for the decision on the levels of planning and scheduling and was

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Fig. 5.10 Dynamics interpretation of the optimal production schedule for the first planning period of the MMA case study

compared to the time-slot based models adapted by the majority of the research works for the integrated problem. As shown from the three different case studies, the TSP and T-S based integration perform similarly for small planning horizons while for larger planning horizons the TSP based model outperforms the T-S considerably. New features for the integrated problem are studied such as multiple customers, inventory capacity as well as backlog and idle production time within the planning period are allowed. Aiming to reduce the computational complexity of the iPSC, a decomposition based on linear metamodels was proposed and tested under the assumption that we account for the open loop performance of the underlying dynamic system. The linear metamodels, associate the transition cost with the transition time between the different products manufactured.

5.4 Concluding Remarks

157

Table 5.16 Comparison between the decomposed TSP and T-S formulation for varying planning horizons Planning 4 Weeks 6 Weeks 12 Weeks horizon Model Type Constraints Cont. Var. Binary Var. Solver Profit ($) CPU (s) Gap (%)

TSP T-S MILP MILP 583 2,185 516 1,901 235 600 CPLEX 12.6.1 29,563 29,563 0.328 235.031 0 0

TSP MILP 879 776 365

T-S MILP 3,287 2,851 900

TSP MILP 1,767 1,556 755

T-S MILP 6,593 5,701 1,800

45,831 1.654 0

45,831 1700 0

71,570 9.251 0

71,569 3600 0.03

Computational studies indicate that when disturbances are not considered at the level of control then the solutions computed from the decomposed iPSC and the monolithic formulations when solved to global optimality are equivalent. The case studies examined highlighted the improved solutions computed by the decomposed iPSC as compared to the monolithic iPSC with computational savings that can reach up to 3500 times faster solutions. Employing the decomposed iPSC model allows to determine operationally optimal solutions that are not realised by the monolithic MINLP model due to numerical difficulties of the non-convexity of the model. In the following chapter, the assumption with regards the occurrence of dynamic disturbances is dropped and the closed-loop case is treated.

References 1. Davis J, Edgar T, Porter J, Bernaden J, Sarli M (2012) Smart manufacturing, manufacturing intelligence and demand-dynamic performance. Comput Chem Eng 47:145–156 2. Lee J, Bagheri B, Kao H-A (2015) A cyber-physical systems architecture for industry 4.0based manufacturing systems. Manuf Lett 3:18–23 3. Harjunkoski I, Nystrom R, Horch A (2009) Integration of scheduling and control-theory or practice? Comput Chem Eng 33(12):1909–1918 4. Grossmann IE (2005) Enterprise-wide optimization: a new frontier in process systems engineering. AIChE J 51(7):1846–1857 5. Gutiérrez-Limón MA, Flores-Tlacuahuac A, Grossmann IE (2016) A reactive optimization strategy for the simultaneous planning, scheduling and control of short-period continuous reactors. Comput Chem Eng 84:507–515 6. Sarimveis H, Patrinos P, Tarantilis CD, Kiranoudis CT (2008) Dynamic modeling and control of supply chain systems: a review. Comput Oper Res 35(11):3530–3561. ISSN 0305-0548 7. Maravelias CT, Sung C (2009) Integration of production planning and scheduling: overview, challenges and opportunities. Comput Chem Eng 33(12):1919–1930 8. Dogan ME, Grossmann IE (2006) A decomposition method for the simultaneous planning and scheduling of single-stage continuous multiproduct plants. Ind Eng Chem Res 45(1):299–315

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9. Chu Y, You F (2015) Model-based integration of control and operations: overview, challenges, advances, and opportunities. Comput Chem Eng 83:2–20 10. Baldea M, Harjunkoski I (2014) Integrated production scheduling and process control: a systematic review. Comput Chem Eng 71:377–390 11. Flores-Tlacuahuac A, Grossmann IE (2006) Simultaneous cyclic scheduling and control of a multiproduct CSTR. Ind Eng Chem Res 45(20):6698–6712 12. Zhuge J, Ierapetritou MG (2012) Integration of scheduling and control with closed loop implementation. Ind Eng Chem Res 51(25):8550–8565 13. Chu Y, You F (2012) Integration of scheduling and control with online closed-loop implementation: fast computational strategy and large-scale global optimization algorithm. Comput Chem Eng 47:248–268 14. Chu Y, You F (2013) Integration of production scheduling and dynamic optimization for multi-product CSTRs: generalized benders decomposition coupled with global mixed-integer fractional programming. Comput Chem Eng 58:315–333 15. Zhuge J, Ierapetritou MG (2014) Integration of scheduling and control for batch processes using multi-parametric model predictive control. AIChE J 60(9):3169–3183 16. Zhuge J, Ierapetritou MG (2015) An integrated framework for scheduling and control using fast model predictive control. AIChE J 61(10):3304–3319 17. Baldea M, Du J, Park J, Harjunkoski I (2015) Integrated production scheduling and model predictive control of continuous processes. AIChE J 61(12):4179–4190 18. Gutierrez-Limon MA, Flores-Tlacuahuac A, Grossmann IE (2014) MINLP formulation for simultaneous planning, scheduling, and control of short-period single-unit processing systems. Ind Eng Chem Res 53(38):14679–14694 19. Shi H, Chu Y, You F (2015) Novel optimization model and efficient solution method for integrating dynamic optimization with process operations of continuous manufacturing processes. Ind Eng Chem Res 54(7):2167–2187 20. Liu S, Pinto JM, Papageorgiou LG (2008) A TSP-based MILP model for medium-term planning of single-stage continuous multiproduct plants. Ind Eng Chem Res 47(20):7733–7743 21. Kopanos GM, Puigjaner L, Maravelias CT (2010) Production planning and scheduling of parallel continuous processes with product families. Ind Eng Chem Res 50(3):1369–1378 22. Biegler LT (2007) An overview of simultaneous strategies for dynamic optimization. Chem Eng Process 46(11):1043–1053 23. Carey GF, Finlayson BA (1975) Orthogonal collocation on finite elements. Chem Eng Sci 30(5):587–596 24. Hairer E, Wanner G (1999) Stiff differential equations solved by Radau methods. J Comput Appl Math 111(1):93–111 25. Seabold J, Perktold J (2010) Statsmodels: econometric and statistical modeling with Python 26. Glover F (1975) Improved linear integer programming formulations of nonlinear integer problems. Manag Sci 22(4):455–460 27. Rosenthal E (2008) GAMS-A user’s guide. GAMS Development Corporation, Washington, DC, USA 28. Camacho EF, Alba CB (2013) Model predictive control. Springer, London 29. Daoutidis P, Soroush M, Kravaris C (1990) Feedforward/feedback control of multivariable nonlinear processes. AIChE J 36(10):1471–1484 30. Verazaluce-Garcia JC, Flores-Tlacuahuac A, Saldívar-Guerra E (2000) Steady-state nonlinear bifurcation analysis of a high-impact polystyrene continuous stirred tank reactor. Ind Eng Chem Res 39(6):1972–1979

Chapter 6

Closed-Loop Integration of Planning, Scheduling and Multi-parametric Nonlinear Control

In this chapter, motivated by the need for efficient closed-loop implementation of the control objectives set within the integrated planning, scheduling and control (iPSC) problem a novel framework that enables its online solution under dynamic disturbances is presented. Utilising the concept of multi-setpoint explicit controllers, from Chap. 4, a rigorous rescheduling mechanism that mitigates the impact of the dynamic disruptions on the operational decisions of planning and scheduling is developed. The overall closed-loop problem is formulated as mixed integer linear program with the control problem integrated via an outer loop. The benefits of the proposed framework are highlighted through two case studies and the results indicate the importance of considering dynamic disruptions within the scope of the integrated problem.1

6.1 Introduction Volatile global market environment, increasing competition and the need for reduction in cost and environmental impact are only a few of the reasons that have led the process industries to seek more responsive and integrated operations. Enterprise Wide Optimisation (EWO) aims to address the aforementioned challenges and provide the industries with tools that can serve as means for enhanced profitability and more sustainable operations [1]. Within the EWO scope one seeks for more integrated decision making via the coordinated optimisation of the supply chain functionalities so as to holistically guarantee the efficient information sharing and optimal operations among the different levels of decision making. A conceptual representation of the EWO scope is given in Fig. 6.1, where the different levels of decision making and the key decisions involved are summarised. 1 Parts

of this chapter have been reproduced with permission from: https://doi.org/10.1016/j. compchemeng.2018.06.021. © Springer Nature Switzerland AG 2020 V. M. Charitopoulos, Uncertainty-aware Integration of Control with Process Operations and Multi-parametric Programming Under Global Uncertainty, Springer Theses, https://doi.org/10.1007/978-3-030-38137-0_6

159

160

6 Closed-Loop Integration of Planning, Scheduling and Multi-parametric …

Fig. 6.1 Enterprise wide optimisation scope for process industries

The efficient online computation of the decisions involved in the iPSC under uncertain conditions remains an open challenge. The main aim of this chapter is to propose a novel framework for the closed-loop iPSC under dynamic disturbances and illustrate how the consideration of uncertain operating conditions accentuates the need for integration among the different levels of decision making. Building largely on the developments previously presented in Chap. 5 for the open-loop case and together with the multi-parametric controllers which were introduced in Chap. 4, a novel framework for the closed-loop implementation of the iPSC is proposed. The key elements of the proposed framework involve: (i) linear metamodels that correlate transition times and costs based on closed-loop simulations of the underlying dynamic systems, (ii) the implementation of novel multi-parametric nonlinear model predictive controllers and (iii) an optimisation based algorithm for the efficient rescheduling that mitigates the impact of disturbances on the online implementation of the integrated problem. The remainder of the chapter is structured as follows: first a literature review is presented on the topic of integrating control with operations under uncertainty along with an overview of reactive approaches and the need for a closed-loop framework for iPSC is underlined. Next, the new elements of the proposed framework are introduced. Subsequently, the overall closed-loop framework is presented in detail with its necessity and efficiency been shown through two case studies. Finally, the chapter concludes with discussion on some of the interesting findings of the present work along with a number of concluding remarks.

6.2 Literature Review

161

6.2 Literature Review Integrating control and operations has attracted significant amount of attention from the research community because of the potential benefits that result from the exploitation of their underlying synergies [2, 3]. Control relevant decisions provide an important set of data, such as transition times and production rates, which are crucial for modelling and solving in an optimal manner the scheduling problem. On the other hand, sequencing decisions are needed by the control decision layer so as to proceed with the manipulation of the dynamics of the production system. As discussed in the previous chapter these interactions have been examined by a number of researchers who studied the integration of cyclic scheduling and control under the assumption of demand periodicity. However, when the demand is not assumed to follow a periodic pattern, the integration of planning along with scheduling and control becomes necessary. As shown in Fig. 6.2 the different problems communicate via a number of interconnected decisions and there is information flow throughout. Even though the problem of integrating control with operations has received considerable attention from the research community, no previous work has considered the iPSC under dynamic disturbances, i.e. closed-loop iPSC. For the online implementation of the iPSC to be effective and realistic one would have to account for dynamic disruptions at the level of control and develop a uncertainty-aware framework so as to secure optimal operations and real

Fig. 6.2 Conceptual representation of the integrated planning, scheduling and control problem

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6 Closed-Loop Integration of Planning, Scheduling and Multi-parametric …

time execution. Upon the realisation of dynamic disruptions, the neighboring level to control, i.e. production scheduling, is likely to get affected and thus its revision should be considered. The need for efficient reactive policies in the context of process scheduling has been long underlined in the open literature [4]. A least impact heuristic proposed by Kanakamedala et al. [5] was among the first works published that considered the problem of reactive scheduling in multi-product batch plants. The reactive scheduling of batch processes was also studied by Vin and Ierapetritou [6] and the authors considered two different kinds of disturbances, rush orders and machine breakdowns. In their work, the rescheduling mechanism is developed by the means of a repetitive solution of a reduced MILP problem every time new information about disruptions becomes available to the plant. The degree of deviation from the original schedule is also controlled through the use of a penalty in the objective function. Scheduling disruptions and reactive policies were studied by Mendez and Jaime [7], where the authors used continuous time representation for multistage batch facilities and formulated the rescheduling problem as an MILP. The impact of rescheduling penalties in the objective function on the quality of the reschedule solution was also studied by Kopanos et al. [8]. Novas and Henning [9] proposed a reactive scheduling framework based on a combination of constraint programming and explicit object oriented domain model that resulted in nearly optimal solutions at relatively low computational times. The use of multi-parametric programming has also been reported in the literature as a way of developing a reactive policy in scheduling problems by several researchers [10, 11]. Recently, Maravelias and co workers [12, 13] in a series of papers based on the state-space interpretation of the scheduling of batch processes studied the effect of different factors on the derivation of periodic and reactive schedules. Due to the integrated nature of the problem it is reasonable to expect an immediate effect on the scheduling and planning decisions whenever the dynamics of the system are disturbed. In the following section, the new mathematical developments for the real-time closed-loop implementation of the iPSC decisions are reviewed along with an MIP-based rescheduling mechanism.

6.3 Mathematical Formulations In this section the main methodological components of the proposed framework are presented. Firstly the open-loop problem, from Chap. 5, is revised in order to facilitate rescheduling considerations. The overall closed-loop framework along with the corresponding algorithmic steps is presented next.

6.3 Mathematical Formulations

163

Fig. 6.3 Conceptual instance of product duplication

6.3.1 Modelling the Closed-Loop Integrated Planning, Scheduling and Control The decomposition of the iPSC through the use of linear metamodels and the offline derivation of the minimum transition times is valid under a number of deterministic assumptions throughout the three levels of integration. However, when dynamic disruptions are considered, such as variations in the feed’s composition or the temperature of the reactor’s jacket, the need to account for possible rescheduling instances must be addressed. To this end the model presented in the previous chapter is modified accordingly. When dynamic disruptions are identified during the production of a product, it should be allowed to resume the production so as to fulfill the remaining demand. Product duplication is employed so as to facilitate this issue as shown in Fig. 6.3. Through product duplication, a duplicate product is created and inherits all the relevant information from the original one. Next a dynamic set is created, II (i, j, p) which denotes the set of products that are considered for duplication on a specific planning period, during which the disruption occurred. Moreover, the following sets are considered: IR (i) which is the set of only the original products and ID (i, p) which is the set of the dummy products that represent the disturbance occurrence during period p. Eq. (6.1) is the modified version of Eq. (5.23) for the calculation of backlog, while for inventory calculations Eq. (5.24) is replaced by Eq. (6.2). Bcip = Bci,p−1 + Dcip − Scip −



∀c, i ∈ IR , p

Scjp

(6.1)

j∈II

Vip = Vi,p−1 + Qip −

 c

Scip +

  Qjp − Scjp j∈II

j∈II

∀i ∈ IR , p

(6.2)

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6 Closed-Loop Integration of Planning, Scheduling and Multi-parametric …

Fig. 6.4 Numerical integration of arbitrary transition curve via Simpson’s rule

Notice that Eq. (6.1) is considered only for the original products and not the duplicate ones while the effect of the duplicate ones is taken into account through the second summation term which is controlled by the dynamic set II (i, j, p). The inventory calculations are modified in a similar manner. The minimum and maximum production times are relaxed for the case of disturbance modelling and thus Eq. (6.3) arises instead of Eq. (5.19). Disturbances do not result manufacture of products thus Eq. (6.4) is only employed for any product except for the ones that belong in ID . θplo Eip ≤ Tip ≤ θpup Eip Qip = ri Tip

∀i ∈ / ID , p

∀i ∈ / ID , p

(6.3)

(6.4)

As will be discussed in the next section, the closed-loop implementation of the iPSC is enabled via the use of novel multi-parametric controllers which communicate with the open-loop iPSC via a number of ways. One of them is via the information the controllers offer back to the integrated problem about the approximate transition cost between products in the case of rescheduling. In general the transition cost is the integral of the control actions (u(t)) over the transition period ([0, Tijtrans ]), i.e.  Tijtrans u(t)dt; in order to allow for fast calculations the controller is programmed 0 to compute the numerical integral of the transition based on Simpson’s rule which provides a good trade-off between numerical accuracy and computational expense  Ttrans as shown in Fig. 6.4. Thus, the quantity CTij  0 ij udt, represents the numerical approximate value of the transition cost as computed by the multi-parametric controller.

6.3 Mathematical Formulations

165

This results in the modification of Eq. (5.51) about the calculation of the cumulative transition cost as shown in Eq. (6.5). TC =

 p i∈I / D j=i

Translin1

αij (Tijp

Translin

+ Tijp|p>1 2 ) + βij (Zijp + Zfijp|p>1 ) +



CTij (Zijp + Zfijp|p>1 )

i∈ID j=i

(6.5)

6.3.2 The Overall Closed-Loop Integrated Framework Real processes are subject to a number of disturbances that endanger the feasibility and optimality of operations. For continuous manufacturing processes, fluctuations on the feed composition, feed temperature as well as the flow rate of the reactants can lead to significant deviations from the desired open-loop state trajectory and result in the production of off-spec material. In the context of iPSC the different timescales involved, lead to the formulation of large scale MINLPs that pose an additional degree of complication that exacerbates the computational requirements and thus prohibit its online solution. In order to alleviate the computational complexity the proposed strategy involves an offline step where linear metamodels that correlate the transition time and cost are built, along with the related nominal minimum transition times. By doing so, this aspect of the interdependence between scheduling and control is exploited and at the same time the control timescale is de-dimensionalised thus reducing the computational complexity. Furthermore, it can happen that the data provided about the transition time and cost from the optimal control simulations can lead to inconsistencies when compared to the closed-loop behavior of the system due to potential model mismatch or because of the different objectives considered. To this end, the data used for the derivation of the linear metamodels as well as the minimum transition times are computed via a number of closed-loop simulations of the underlying dynamic system through the use of the multi-setpoint explicit controller that was introduced earlier. This was also done in order to simulate what would happen in a real process where the mechanisms that dictate the transition costs and times may not be sufficiently captured by open-loop DO runs but from historical data. For the online part, the solution of an MILP (open-loop iPSC problem) is considered and then the corresponding control problem that tracks online the open-loop solution. Conceptually, the proposed strategy for the closed loop solution is given in Fig. 6.5. Firstly, the mp-MPC for the underlying dynamic system is designed. As shown in Fig. 6.5 in an offline step the linear metamodels that correlate transition time and cost are computed based on the closed-loop simulations of the system. The linear coefficient of the metamodels are then used in the (open-loop iPSC) problem which is an MILP. The two parts (open-loop iPSC and (mp)-MPC) are coupled through the decision variables Eip , Oip and Zijp , ZFijp . More specifically, the set of assigned products to the planning period is created and renewed in every planning period,

166

6 Closed-Loop Integration of Planning, Scheduling and Multi-parametric …

Fig. 6.5 Flowchart of the proposed closed loop implementation of the iPSC for continuous manufacturing processes

i.e. Iactive (p)= {i|Eip = 1}. Notice that every time a rescheduling takes place the set of assigned products is revised accordingly. Next, the production sequence is made known to the controller through the integer variable Oip and the binary variables that indicate the changeovers (Zijp and ZFijp ) are employed for the derivation of the desired set-points. In Algorithm 6.1, the main steps for the integration of closed loop control are given. Initially a nominal production plan and the relevant schedules are computed by solving the open-loop iPSC (Algorithm 6.1, Step 1). The main computational loop iterates over all the planning periods under consideration (Algorithm 6.1, Step 2). The sequencing decisions and the set of assigned products are developed and then the schedule is supervised in a logic manner. A scalar k is used to track the order of the product that is currently being processed (Algorithm 6.1, Steps 3–6). Then following the proposed framework the information goes to the outer control loop where the mpMPC is employed and the system is regulated around the desired set-point subject to the quality bound (xq ). If the system stays within this limit (Algorithm 6.1, Step 10) then the next instance is normally sampled; otherwise if the threshold/quality bound is violated that means that the product’s current production time has to be fixed, set the current measurement as the state of the disturbance and go to Algorithm 6.2 to initiate the rescheduling (Algorithm 6.1, Steps 10–16). If the nominal production time has been accomplished without disruptions then the set-point of the mp-MPC is set as the steady state of the next product (Algorithm 6.1, Step 20) and start the changeover. Similar to the previous steps there has been set a threshold () for which a disturbance during a transition is supposed to lead to negligible disruption and thus no rescheduling is triggered (Algorithm 6.1, Steps 21–33); otherwise a rescheduling needs to be initiated and the steps outlined in Algorithm 6.3 should be followed.

6.3 Mathematical Formulations

167

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6 Closed-Loop Integration of Planning, Scheduling and Multi-parametric …

As shown in Algorithm 6.1 whether the disturbance is detected during a production or a transition period calls for different strategies. During the production period the role of control is to regulate the system around the desired steady state given a quality bound for allowable deviation (xq ), which in the context of iPSC reflects a specific product grade, while during the transition period there is a need for setpoint tracking control. In the case that disturbance is detected during the production time of a product, it is assumed that its production is instantly interrupted as the disturbance exceeds the prespecified threshold (); note that in the present work the convention of Zhuge and Ierapetritou [14] is followed and these parameters are assumed to be determined via heuristics based on process knowledge. Practically this happens because in real processes there will always exist some noise and plantmodel mismatch and without those tolerances there would be excessive need for rescheduling. Once the disturbance is detected during the production period, it is reasonable to consider the need of re-assignment of that product within the same planning period and this is achieved via a product duplication. The steps outlined in Algorithm 6.2 are then followed in order to decide upon the rescheduled optimal sequence that allows for resume of the production. That is, the current measurement ss ) and the of the system is assumed to be the steady state of a dummy product (xd_name mp-MPC is employed to simulate all the potential transitions to the other products and compute transition times and costs (Algorithm 6.2, Steps 1–14). Next, the set of products that have already been processed in the current period is constructed (Ipast ) and the relevant timing and sequencing decisions are fixed (Algorithm 6.2, Steps 15–17). Notice that by fixing the timing decisions, production decisions are also fixed given the assumption about constant production rate. Since the disruption was detected during a production period, it should be allowed for the model to choose the resume of production and to do so the product duplication concept is used and define an alias element in the products set that is set to be active only for the current planning period and then the solution of the reduced iPSC follows (Algorithm 6.2, Steps 19–23). The term “reduced iPSC” is employed since the open-loop iPSC model is solved in reduced decision space, i.e. with the past decisions fixed.

6.3 Mathematical Formulations

169

Another case involves the disturbance detection and rejection during the transition time from one product to the other. In that case, there is no need to account for product duplication and only a new dummy product is introduced that represents the current state of the system after the disturbance is realised. Similar to Algorithm 6.2, the ability of mp-MPC to run simulations in very fast times is exploited and the potential transition times and costs are computed (Algorithm 6.3, Steps 1–14). The output of these computations are passed to the model and the past decisions are fixed (Algorithm 6.3, Steps 15–18) and the reduced iPSC is solved again to define the optimal rescheduling action (Algorithm 6.3, Steps 19–21). A summary of these steps is given in Algorithm 6.3.

170

6 Closed-Loop Integration of Planning, Scheduling and Multi-parametric …

Overall, the integrated problem is formulated and solved as an MILP and the closed-loop is achieved through the rescheduling mechanism outlined in this section via the use of the proposed mp-MPC.

6.4 Case Studies In this section the closed loop implementation of iPSC is illustrated through two case studies, where each planning period is assumed to be equal to one week. In all the case studies presented, it is assumed that every system exhibits multiple steady states at which a specific product is produced in a single CSTR. Moreover, a short discussion on the results is conducted at the end of each case study. All the optimisation problems, are formulated and solved using GAMS 24.7.4, on a Dell workstation with 3.70 GHz processor, 16GB RAM and Windows 7 64-bit operating system using CPLEX 12.6.1 for the solution of the MILPs and BARON 16.8.24 [15] for the solution of NLPs. BARON was chosen for the comparison between the NLMPC and mp-MPC scheme because it is a global optimisation solver and the explicit solutions computed for the design of the mp-MPC controllers are globally optimal as well.

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171

Fig. 6.6 Comparison of the closed loop behavior of the state of the SISO CSTR (CR ) using explicit MPC (blue continuous line) and conventional MPC (red dashed line) for prediction horizon of unity (1st planning period)

6.4.1 Single Input Single Output CSTR For the design of multi-setpoint explicit controller the algorithm outlined in Chap. 4 was employed. The mp-MPC was designed for prediction horizons of unity and two; the final globally optimal explicit solutions along with their corresponding CRs are given in Table 4.7, while the final partitions of the parametric space are given in Figs. 4.7a–4.9. Once the multi-setpoint explicit controller is designed its performance in comparison to the use of conventional MPC within the context of iPSC is investigated. First, the case of no imposed disturbances is considered and then at specific time an additive disturbance is imposed and the dynamic behavior of the closed loop system is evaluated.

6.4.1.1

Case 1: No Additive Disturbance Imposed

First the closed loop behavior for prediction horizon of unity and then two is evaluated for the mp-MPC, the threshold values are set to ω = 10−6 ,  = ±5% and no imposed additive disturbance is considered, while a planning horizon of two weeks is employed. When the mp-MPC is employed, it takes 0.1705 CPU s for the nominal iPSC solution to be computed. For the case of the conventional NLMPC the same results are computed in 291.92 CPU s, a rather considerable difference in the computational effort. It is important to note that based on Fig. 6.6, the dynamic responses of the system as computed by the mp-MPC and the conventional NLMPC are identical. The same instance of the case study was investigated, by employing prediction horizon of 2 for both the explicit and the conventional MPC schemes. Regarding the dynamic response and the stability of the underlying control system, the results

172

6 Closed-Loop Integration of Planning, Scheduling and Multi-parametric …

indicate no significant difference when compared to the ones computed for prediction horizon of unity. As far as the online computational complexity is concerned, for the case of the multi-setpoint explicit MPC it takes 0.1707 s for the whole iPSC to be solved and validated in a closed loop manner while the same problem takes 831.24 s using the conventional MPC using BARON 16.3.4 and optimality tolerance of 10−5 . As mentioned earlier in the chapter, process systems are subject to a number of disturbances that may affect significantly the performance of the process. For the case that no disturbances are accounted for, the solution of the open loop and the closed loop iPSC are identical as demonstrated in case 1. However, under the effect of disturbances the need of feedback control mechanism becomes crucial. In the next two cases the effect of the implementation of the closed-loop strategy under the occurrence of disturbances that lead to state deviation is investigated.

6.4.1.2

Case 2: State Deviation During the Transition Period

In this case it is assumed that the nominal iPSC has been solved and the optimal decisions begin to be applied to the plant. During the first planning period, a disturbance is detected 12 min after the beginning of the transition from product A to product B and its magnitude exceeds the prespecified threshold. The rescheduling mechanism is triggered and first the current state of the system is passed to the mp-MPC according to the steps outlined in Algorithm 6.3. It takes 8ms for the mp-MPC to compute the candidate transition times and costs which are subsequently passed to the iPSC rescheduling model in order to compute the next optimal step. In this instance, the rescheduling mechanism dictates the change in the nominal sequence and instead of B the system is driven to the production of E while the production of B is set to be the last of the planning period. A graphical representation of this instance is given in Fig. 6.7. On the other hand, if the iPSC solution was applied without any feedback mechanism that would effectively close the loop, the pre-computed nominal control would have been applied to the system regardless of the disturbance occurrence. The impact of the disturbance on the open-loop framework was simulated by fixing all the relevant decisions and as shown in Fig. 6.7, it results in significantly extended transition time from product A to B which results in turn in considerable reduction of the production time and thus increase in the backlog of the corresponding unmet demand. A summary of the results is given in Table 6.1.

6.4.1.3

Case 3: Multiple Disturbances Over the Planning Horizon

In this case 8 products and 4 planning periods were considered. Solving the nominal problem, no disturbance is assumed and it takes 9.984 s for CPLEX 12.6.3 to compute the optimal solution. Within the first planning period, during the transition from H to C, 4.2 min after the transition has started (nominal transition time is 6.6 min), a state deviation is realised which exceeds the threshold and is equal to 0.04 mol L

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173

Fig. 6.7 Comparative plot of the rescheduled and nominal solution due to state deviation during the transition period Table 6.1 Comparative results for the solutions computed by the closed loop iPSC and the open loop iPSC for the case of state deviation during transition period Open loop no disturbance

Closed loop with disturbance

Open loop with disturbance

iPSC solution

A→B→C→D→E E→C→B→A

A→E→D→C→B B→A→C→E

A → B → C → D → E E→C→B→A

Profit (rmu)

9.18 · 106

9.174 · 106

8.65 · 106

Backlog cost (rmu) 62,331

68,729

338,677.65

Total CPU (s)

0.27 (mp-MPC)

0.152 (mp-MPC)

0.152 (mp-MPC)/831.24 (NLMPC)

resulting in a concentration of 0.329 mol , after its realisation the proposed framework L is employed and the need for rescheduling is examined. First, the explicit controller is used for a simulation between all the remaining products of the set Iactive ( p) and then the reduced iPSC is solved (7.083 s) for the remainder of the planning horizon. In this case, the optimal solution dictates that it is preferable to keep on the prolonged transition rather than switching production to another product. The result of the extended transition is the decrease of production time of product G and its subsequent increase during planning period 2. Moving on to period 2, another dynamic disruption is realised during the transition . from G to C (after 1.4 h of its start) and the system is led to xd2 = 0.2306 mol L Following Algorithm 6.1, this measurement is passed to mp-MPC for the calculation of the potential transition times and costs. This instance of the closed-loop integration is quite interesting as the rescheduling mechanism for the current planning period generates a completely different solution for the remaining iPSC and this can be visualised in Gantt chart that is given in Fig. 6.8.

Planning periods

174

6 Closed-Loop Integration of Planning, Scheduling and Multi-parametric … A

p1 p2

E

p3

F

p4

F

B

G

H

C

E

D A

C

D

G

E

H

B 160

140

120

100

80

60

40

20

A Y F B H C D G E

G

H

B

0

C

180

Planning periods

time (h)

A

p1 p2

E

p3

F

p4

F

B

G

H

C

E

D A

C

D

G

E

H

B 160

140

120

100

80

60

40

20

A Y F B H C D G E

G

H

B

0

C

180

Planning periods

time (h)

A

p1 p2

E

p3

F

F G

H

C

D

D

G

E

H

B 160

140

120

100

80

60

40

A Y F B H C D G E

G A

B

20

0

C

H

C

E

p4

B

180

Planning periods

time (h)

p1

A

p2

E

p3

F

p4

F G

H

C

G

H 40

D

E

B 60

80

100

A Y F B H C D G E D'

G A

B

D 20

C

H

C

E 0

B

F 120

140

160

D' 180

time (h)

Fig. 6.8 Gantt charts of the closed-loop iPSC for the SISO CSTR case study. The horizontal axis represent the time (h) and each week spans across 168 h with changeovers indicated using the “Y” label

As shown in Fig. 6.8 the disruption results in decrease of production of product C that in order to be rectified the sequencing decisions in planning period p3 were revised. Finally, during period 4, the case of dynamic disruption during production period is examined. While product D is being produced there is a discrepancy in the system’s dynamics. Due to this discrepancy during the production period the steps outlined in Algorithm 6.2 are followed and a duplicate product of D is created and the reduced iPSC is solved again with fixed the past decisions. As shown also in Fig. 6.8, it was computed that the optimal corrective move would be to return in the

6.4 Case Studies

175

Fig. 6.9 CR = f(t) plot indicative of the system’s dynamics over the entire planning horizon

production of D. A graphical interpretation of the system’s dynamics throughout the 4 planning periods is given in Fig. 6.9, where also the last rescheduling instance is illustrated in more details.

6.4.2 MMA Polymerisation Reactor Next the closed-loop iPSC of an isothermal methyl-methacrylate (MMA) polymerisation CSTR is studied for which the model (Eqs. (4.22)–(4.26)) and parameters presented in Sect. 4.3.2 were used. In order to demonstrate the merits of the proposed framework it is assumed that there are 16 polymer grades that can be produced and their corresponding steady state information is given in Table 6.2; notice that for the sake of space the values are given with truncated decimal points while for the numerical calculation the precision was up to 10 decimal places. With regards to the economic data a summary of the selling prices and costs is given in Table 6.3 while inventory and backlog costs are calculated as 10 and 30% of the product’s selling price, Pi . The multi-setpoint explicit controller from Sect. 4.3.2 is employed. After the explicit controller has been designed the explicit solutions and their CRs are coded in GAMS in the form of conditional statements where they are integrated with the open loop iPSC model so as to validate the computed solution and reject systematic disturbances; thus closing the loop in the iPSC.

176

6 Closed-Loop Integration of Planning, Scheduling and Multi-parametric …

Table 6.2 Steady state information about the different polymer grades of the MMA CSTR case study Product Css Css Dss Dss Yss Flss m l 0 l A B C D E F G H I J K L M N O P

3.2285 3.0780 3.3667 3.3331 3.4635 3.5552 3.7257 3.8815 3.9786 4.1154 4.2015 4.3238 4.4013 4.4760 4.5302 4.5830

0.1216 0.1487 0.1009 0.1056 0.0885 0.0780 0.0615 0.0491 0.0426 0.0346 0.0302 0.0248 0.0217 0.0191 0.0173 0.0157

0.0163 0.0195 0.0138 0.0144 0.0123 0.0111 0.0091 0.0075 0.0067 0.0057 0.0051 0.0044 0.0040 0.0036 0.0033 0.0031

277.47 292.54 263.64 261.01 253.95 244.76 227.69 212.09 202.38 188.68 180.06 167.81 160.05 152.57 147.15 141.86

Table 6.3 Economic data of the MMA case study Product OCi [ rmu Pi [ rmu kg ] kg ] A B C D E F G H I J K L M N O P

263 188 163 226 220 210 190 240 230 290 205 210 183 155 149 134

388.5 252.8 247.8 293 330.5 260 290 350 395 325 310 316 220 260 300 324

17,000 15,000 19,000 18,500 20,500 22,000 25,000 28,000 30,000 33,000 35,000 38,000 40,000 42,000 43,500 45,000

ri [ kg h ] 100.529 122.937 83.457 87.325 73.165 64.504 50.833 40.635 35.214 28.607 24.997 20.500 17.996 15.814 14.359 13.040

0.1675 0.2049 0.1391 0.1455 0.1219 0.1075 0.0847 0.0677 0.0587 0.0476 0.0416 0.0341 0.0299 0.0263 0.0239 0.0217

6.4 Case Studies

6.4.2.1

177

Discussion of Results

The overall closed-loop iPSC problem is formulated as an MILP with 15,334 equations, 10,376 continuous variables and 3,485 binary variables while the solution time for the nominal case takes 26.12 s using CPLEX 12.6.3 in GAMS. It is worth to note that the optimisation dictates to run the plant in a continuous way through the planning period so as to avoid the related long start-up/shut-down times. In this case study minimum production time of 4 h is imposed, i.e. for a product to be assigned to a planning period at least 4 h should be allocated for its production. As mentioned earlier in the chapter, the main requirement for the efficient online implementation of the closed-loop iPSC solution is that the computational time needed for the solution of the optimisation problem is less than sampling time of the underlying control system. For this case study, the sampling time is 36 s and thus it is requires to solve the iPSC in times less than 36 s. The parameters for the rescheduling framework in this case study were set as ω = 10−6 ,  = ±5%. The first rescheduling action takes place during the production of product O, when a disturbance that exceeds the threshold is realised and the rescheduling mechanism is triggered. Following the steps of Algorithm 6.1 a new duplicate product is generated and the reduced iPSC solved with fixed the past decisions, in this case the production time of O that was achieved and the relevant binary decisions with respect the order of O in the schedule. Solving the reduced iPSC takes 28.83 s, slightly increased when compared to the nominal case but this increase is due to the introduction of the two dummy products in the reduced iPSC, i.e. the duplicate of O and the dummy disturbance product. As shown in the Gantt chart in Fig. 6.10, the reschedule results in interrupting the production of O and move to the production of N while compared to the nominal plan in period p1 the additional production of product G was chosen too. The next disturbance is detected during the transition from J to I and once again the rescheduling mechanism is activated but this time the system is controlled towards the completion of the nominal transition while the related computational time is 7.96 s. Similarly to planning period 1, in period 2 another reschedule takes place due to dynamic disruption during the transition from product E to product G and the system is resumed to its nominal schedule with the completion of the original transition. During period 3 the no dynamic disturbance that exceeded the relevant threshold was detected. Period 4 involved a reschedule during the production of product K, 5.36 h after its initialisation and the rescheduling framework resulted in resuming its production. Finally, during period 6 another instance of dynamic disruption during the production of O was detected. The graphs of the system’s dynamics are given in Fig. 6.11 while the closed loop results from a scheduling perspective are given in Fig. 6.12. Next based on the solution of the MMA case study, a discussion on the impact that the rescheduling mechanism has on the overall iPSC is conducted and some key correlations are underlined.

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6 Closed-Loop Integration of Planning, Scheduling and Multi-parametric …

Fig. 6.10 Gantt charts of all the reschedules for the MMA case study where the gray blocks represent changeover periods

6.4 Case Studies

179

Fig. 6.11 Y = f(t) dynamic plots indicative of the system’s dynamics over the entire planning horizon for the MMA case study

180

6 Closed-Loop Integration of Planning, Scheduling and Multi-parametric …

Fig. 6.12 Y = f(t) dynamic plot indicating all the rescheduled actions across the planning horizon for the MMA case study

6.4.2.2

Impact of the Rescheduling Mechanism on the Overall iPSC

The integrated nature of the iPSC gives rise to several synergies as well as hidden interdependences that can prove to be rather crucial for the optimal operations. In Table 6.4, a number of indicating factors from the iPSC problem are given against the different rescheduling runs that took place in the MMA case study. The general trends, as expected, is that the profit and overall production time to overall transition time ratio are decreasing functions of the rescheduling runs while on the other hand opposite behavior is exhibited by the transition cost and backlog cost of the overall plan. By having a closer look at the results presented in Table 6.4 it can be understood that different rescheduling runs have different impact on the factors under examination. Firstly, the reduction in profit during the second rescheduling run of the period P1 is significantly greater than the decrease from the first rescheduling. The key difference between the two reschedules is the timing that the disturbance was detected. In the first instance, the disturbance was detected early during the first planning period but during a production time while the second reschedule took place during the transition from product J to product I. It seems that disturbances during transition periods have greater impact on the profit due to increased production of off-spec material. The timing of the disturbance seems to play, in this case, a secondary role as the first planning period had since the nominal plan an end time of 168 h which is the upper bound on time per planning period. The next interesting observation is that the transition cost during the first reschedule decreases slightly when compared to the nominal case. One would expect the transition cost to be monotonically increasing function of the rescheduling runs but from studying the economic data of the case study the following can explain the unexpected drop in transition cost. From the occurrence of the disruption in the dynamics, the transition form product O to N is benefited and thus happens in less time and thus cost. Moreover, product G that is inserted for production is the one that has the fastest transition time/less transition cost from product I and also its price is similar to the price of product O whose production time was reduced. Finally, the greatest change in the indicating factors is observed in the last rescheduling during the sixth period and there are a number of reasons that result in this. Demand that is backlogged during the end of the planning horizon does not

6.4 Case Studies

181

Table 6.4 Overview of the impact of the rescheduling mechanism on different factors ofthe iPSC Tip i trans trans Tijp +TFijp

Rescheduling run

Profit (rmu)

Transition cost (rmu)

Backlog cost (rmu)



Nominal

1.876 · 106

714.63

553,094.75

4.86

1st Reschedule P1

1.851 · 106

691.83

562,172.88

4.62

2nd Reschedule P1

1.811 · 106

714.63

599,572.34

4.51

Reschedule P2

1.767 · 106

753.58

639,595.41

4.32

P3

1.767 · 106

753.58

639,595.41

4.32

Reschedule P4

1.748 · 106

775.78

651,992.14

4.23

P5

1.748 · 106

775.78

651,992.14

4.23

Reschedule P6

1.703 · 106

850.34

692,827.74

4.11

i,j

Fig. 6.13 Visual representation of the effect of the rescheduling mechanism on the economic performance of the iPSC

allow for recovery of the lost sales since the model cannot allocate the backlogged demand in subsequent periods as in the reschedulings that happened in the earlier periods. Overall the following can be characterised as important factors that affect the optimal operation of the production facility within the iPSC scope: (i) the timing of the disturbance, i.e. if it happens at the beginning/middle/end of the planning period, (ii) the occurrence of the disturbance, i.e. if the disruption happens during the production or the transition time can result in loss of production but also in extended waste due to off-spec materials, (iii) the existence of idle time during each planning period and (iv) the complex trade-offs that stem from the economics of the iPSC, e.g. it might be more preferable to lose production of a product that has lower selling price when compared to a more premium grade. Finally, a graphical illustration of the economic indicators is given in Fig. 6.13.

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6.5 Concluding Remarks A novel framework for the closed-loop implementation of iPSC with rescheduling considerations has been presented. The key features of this framework involve: (i) a TSP model for the decisions at the planning and scheduling level where immediate sequence was used instead of the time-slots that have been proposed in the literature, (ii) a meta-modelling approach that allows the decision maker to take into consideration the transition cost for each transition and (iii) a novel model based controller for the closed-loop implementation of the control strategy. The case studies illustrate that the use of multi-setpoint explicit controllers within the closed-loop framework enable real-time implementation of the decisions made by the integrated problem in the face of dynamic disruptions, something that had not appeared in the literature so far. The case studies examined indicate that under the consideration of uncertain process dynamics, the inherent interdependence of the integrated problem manifests itself, since a disruption in the process dynamics can result in major changes on the decisions from the levels of scheduling and planning in order to grant optimal operations. In the next chapter, the proposed closed-loop reactive framework for the rejection of dynamic disturbances is coupled with proactive approaches so as to hedge against diverse types of uncertainty within the iPSC and a hybrid framework for the uncertainty-aware iPSC is proposed.

References 1. Grossmann IE (2012) Advances in mathematical programming models for enterprise-wide optimization. Comput Chem Eng 47:2–18 2. Dias LS, Ierapetritou MG (2016) Integration of scheduling and control under uncertainties: review and challenges. Chem Eng Res Des 116:98–113 3. Chu Y, You F (2015) Model-based integration of control and operations: overview, challenges, advances, and opportunities. Comput Chem Eng 83:2–20 4. Aytug H, Lawley MA, McKay K, Mohan S, Uzsoy R (2005) Executing production schedules in the face of uncertainties: a review and some future directions. Eur J Oper Res 161(1):86–110 5. Kanakamedala KB, Reklaitis GV, Venkatasubramanian V (1994) Reactive schedule modification in multipurpose batch chemical plants. Ind Eng Chem Res 33(1):77–90 6. Vin JP, Ierapetritou MG (2000) A new approach for efficient rescheduling of multiproduct batch plants. Ind Eng Chem Res 39(11):4228–4238 7. Mendez CA, Jaime C (2004) An MILP framework for batch reactive scheduling with limited discrete resources. Comput Chem Eng 28(6):1059–1068 8. Kopanos GM, Capon-Garcia E, Espuna A, Puigjaner L (2008) Costs for rescheduling actions: a critical issue for reducing the gap between scheduling theory and practice. Ind Eng Chem Res 47(22):8785–8795 9. Novas JM, Henning GP (2010) Reactive scheduling framework based on domain knowledge and constraint programming. Comput Chem Eng 34(12):2129–2148 10. Li Z, Ierapetritou MG (2008) Reactive scheduling using parametric programming. AIChE J 54(10):2610–2623

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11. Kopanos GM, Pistikopoulos EN (2014) Reactive scheduling by a multiparametric programming rolling horizon framework: a case of a network of combined heat and power units. Ind Eng Chem Res 53(11):4366–4386 12. Subramanian K, Maravelias CT, Rawlings JB (2012) A state-space model for chemical production scheduling. Comput Chem Eng 47:97–110 13. Gupta D, Maravelias CT, Wassick JM (2016) From rescheduling to online scheduling. Chem Eng Res Des 116:83–97 14. Zhuge J, Ierapetritou MG (2012) Integration of scheduling and control with closed loop implementation. Ind Eng Chem Res 51(25):8550–8565 15. Sahinidis NV (1996) BARON: a general purpose global optimization software package. J Glob Optim 8(2):201–205

Chapter 7

A Hybrid Framework for the Uncertainty-Aware Integration of Planning, Scheduling and Explicit Control

In this chapter the integrated planning, scheduling and control (iPSC) of process systems under uncertain conditions throughout the three levels of decision making is examined. The planning problem is explored in a rolling horizon fashion coupled with demand forecasts. Proactive and reactive approaches are employed to handle the effect of stochastic variations. Depending on the nature of the uncertain parameters robust optimisation and chance constrained programming are employed. For the closed-loop implementation of the control, multi-setpoint explicit controllers are designed. The proposed framework is tested on the iPSC of a polymerisation process. Finally, Monte Carlo simulations are conducted to highlight the benefits of the “uncertainty-aware” solutions when compared to the deterministic ones.

7.1 Introduction and Motivation The developments introduced in the previous chapters of the thesis have established a framework for the closed-loop iPSC which can handle efficiently dynamic disruptions at the level of control. Even though the integrated problem can lead to improved operations because of the synergies between the different levels of decision making, uncertainty is ubiquitous and its impact on operations can be tremendous. Shobrys and White [1] identified and discussed the key business tactics that prevent from the efficient integration of planning, scheduling and control. A control theoretic multilevel framework for the integration of planning, scheduling and rescheduling can be found in Mitra et al. [2]. The problems of planning and scheduling under uncertainty have been analysed in the surveys conducted by Mula et al. [3] and Aytug et al. [4] respectively among others. Li and Ierapetritou [5] reviewed developments on the field of process scheduling under uncertainty and identified challenges with respect to the conservatism, computational efficiency and stability of the resulting schedules. A comprehensive review on the problem of planning and scheduling across different sectors can be found in Verderame et al. [6]. More recently, Dias and © Springer Nature Switzerland AG 2020 V. M. Charitopoulos, Uncertainty-aware Integration of Control with Process Operations and Multi-parametric Programming Under Global Uncertainty, Springer Theses, https://doi.org/10.1007/978-3-030-38137-0_7

185

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7 A Hybrid Framework for the Uncertainty-Aware Integration …

Ierapetritou [7] reviewed the developments on the topic of integrating control with operations under uncertainty while Grossmann et al. [8] surveyed recent applications of optimisation under uncertainty methods to enterprise wide optimisation problems. The need for computing near-optimal solutions in rapid computational time within the context of process planning and scheduling has long been recognised. Early research works employed hierarchical solution approaches [9]. Wilkinson [10] studied and formulated the production planning problem as an MILP and for its solution a rolling horizon procedure was employed with the first period modelled in detail and the subsequent ones in an aggregate manner. Shortly after the appearance of the state-task network and resource-task network representations of the scheduling problems for multi-product, multi-purpose batch plants, Dimitriadis et al. [11] studied the use of rolling horizon for the solution of large problem instances. The key idea is to either approach with detail the first or the last planning period (forward and backward rolling horizon, respectively) and then apply an aggregation to the remainder periods and solve the resulting problem to optimality in a receding policy. The authors identified that the type of aggregation but also the type of variables to be fixed after the solution of the detailed problem can have significant impact on the feasibility of the overall solution. The use of Lagrangian relaxation in conjunction with rolling horizon for the integration of planning and scheduling has also been reported by Sand et al. [12] while a shrinking two-stage approach for batch process scheduling can be found in Balasubramanian and Grossmann [13]. Wu and Ierapetritou [14] reported a hierarchical approach to the integration of planning with scheduling of batch plants and in their work the planning periods are grouped into a three different “planning buckets” with increasing level of demand uncertainty each. Rolling horizon in the context of integrating planning and scheduling of refinery operation was also employed by Chunpeng and Gang [15]. In their work robust optimisation was employed to account for demand uncertainty. Li and Ierapetritou [16] examined how capacity considerations of the subsequent aggregate periods can lead to improved solutions for the integrated planning and scheduling problems while alternative ways of employing enhanced rolling horizon schemes without compromising their computational efficiency can be found in Sand and Engell [17], Verderame and Floudas [18]. Tiacci and Saetta [19] investigated the use of rolling horizon for the integrated planning and scheduling problem under demand variations. The authors provided a number of model reformulations with valid inequalities and simplifying assumptions, e.g. with regards the incorporation of changeover times in future periods. In their work which examined a wood manufacture, it was deemed that changeover times are not as significant as their related costs and thus their calculation over future planning periods was incorporated in the objective function of the model. An auto-adaptive method with exponential smoothing was employed to simulate the uncertain demand in the problem. Alem et al. [20] investigated the integrated lot-sizing and scheduling problems under demand uncertainty and proposed a number of heuristics on when it is preferable to employ robust optimisation or scenario based stochastic approaches. Among the first ones to incorporate uncertainty consideration in the integrated scheduling and control problem were Mitra et al. [2] who studied the use of different optimisation under uncertainty techniques for the proactive optimisation of

7.1 Introduction and Motivation

187

the problem. The application of two-stage stochastic programming for integration of cyclic scheduling and control of polymerisation processes was presented by TerrazasMoreno et al. [21] where the product demand was considered as uncertain. Chu and You [22] also applied two-stage stochastic programming for integration of cyclic scheduling and control of batch processes and implemented two variants of the generalised Benders decomposition to alleviate the computational complexity. Despite the decomposition, for a total of 100 scenarios the best computational time achieved was 3.3 h making its online implementation to the production system not practical. Developments on the reactive integration of planning, scheduling and control were surveyed in Chap. 6. As indicated by the literature review that was conducted the examination of a hybrid framework for the integration of planning, scheduling and control under uncertainty remains an open challenge. So far, the use of either proactive or reactive methods has been proposed but no work has been done on their potential synergistic combination mostly hindered by the need to compute the integrated solution in computational times sufficiently small for its online implementation. It is the purpose of this chapter to present a study for the development of such a hybrid framework for the uncertainty-aware integration of planning, scheduling and control. The remainder of the chapter is structured as follows: first, the key methodologies for optimisation under uncertainty are reviewed. Next, the main elements of the proposed framework are introduced. Subsequently, the overall closed-loop framework is presented in detail with its necessity and efficiency been shown through a polymerisation case study. Lastly, a short discussion and concluding remarks are drawn.

7.2 Modelling and Optimisation with Uncertainty Considerations Incorporating uncertainty considerations in mathematical programs has been an active field of research due to its significance on real life applications, where the feasibility and optimality of the computed solutions should be reasonably guaranteed [23, 24]. In order to explicitly account for the stochasticities in optimisation problems a number of methodologies have been proposed ranging from the use of interval mathematics [25], flexibility methods [26–28], stochastic programming approaches [29] to robust optimisation [30] and chance constrained programming methods [31]. Depending on the risk-outlook of the decision maker and the availability of historic data, three are the main optimisation under uncertainty techniques that have been widely employed: (i) robust optimisation, (ii) stochastic programming and (iii) chance constrained programming.

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7 A Hybrid Framework for the Uncertainty-Aware Integration …

7.2.1 Robust Optimisation Robust optimisation (RO) is a set-induced methodology where the decision maker assumes that all the uncertainty realisations belong to a set and then optimises so as to secure the feasibility of the computed solution [32]. As expected this uncertaintyimmune solution can be overly conservative and this is the reason why robust optimisation has been typically referred to as a worst-case/pessimistic methodology. min max c(ζ )T x x

ζ

Subject to :

(7.1) A(ζ )x ≤ d(ζ ) ζ ∈ U, x ∈ Rnx

Equation (7.1) provides a formulation of an uncertain LP in a RO context where the aim is to minimise the objective function by manipulating the degrees of freedom (x) while maximizing, i.e. optimising towards the worst case, the effect of uncertainty (ζ ). Notice that the uncertain parameters are assumed to belong to a prespecified “set”, i.e. U. The field of RO was pioneered by Soyster [33] even though around the same period similar attempts to address robustness in uncertain LPs can be identified in the works of Friedman and Reklaitis [34]; the uncertainty set introduced by these works is commonly referred to as “box-uncertainty set”. “Ellipsoidal-uncertainty sets” were introduced by Ben-Tal and Nemirovski [30, 35] in an effort to alleviate the conservativeness of the robust solution while the resulting robust-counterpart was formulated as a quadratic program. A distinctly different uncertainty set using the concept of “uncertainty-budgets” was introduced by Bertsimas and Sim [36] and the authors managed to preserve the linearity of the original problem whilst providing a flexible level of conservativeness by adjusting the user-specified budgets. The aforementioned works concentrated mostly on the solution of uncertain LPs while the extension to the MILP cases is due to the works of Lin et al. [37] and Janak et al. [38]. Other classes of problems that have been considered within a robust optimisation spectrum include semidefinite programs [39], quadratically constrained programs [40], conic programs [41] while finally the use of robust optimisation in conjunction with probability distribution information has also been reported [42]. Ben-Tal et al. [43] in an effort to alleviate the conservativeness resulting from the worst-case oriented nature of classic RO, introduced the concept of Adjustable Robust Optimisation. Following this paradigm, it is assumed that some decisions related to the problem can be adjusted so to hedge against the uncertainty after its realisation. However this adjustability in robust programs comes at the cost of solution complexity, since in the general case the resulting robust counterpart is NP-hard depriving robust optimisation of its attractive feature, i.e. its computational tractability. The authors proposed the use of linear affine decision rules so as to reduce the computational complexity and proved that in the case of fixed recourse, i.e. when the adjustments do not depend on the uncertainty, the resulting robust counterpart is a tractable problem, e.g.

7.2 Modelling and Optimisation with Uncertainty Considerations

189

LP or semi-definite program. Decision-dependent uncertainties motivated by mainly process scheduling problems, have been studied by Lappas and Gounaris [44] and Vujanic et al. [45] where the uncertainty set is conditional on the implementation of a specific scheduling action. The optimality of the affine policies in multistage robust optimisation for a special class of control-related problems was proven by Bertsimas et al. [46]. For a more exhaustive exposition the interested reader is referred to the review articles of Ben-Tal and Nemirovski [35], Li et al. [47], Gorissen et al. [48], Yaniko˘glu et al. [49].

7.2.2 Stochastic Programming Stochastic programming (SP) is a scenario-based approach which requires the existence or assumption of an underlying probability distribution that the uncertainty follows as well as the confection of recourse so as to hedge against the uncertainty [50]. In the stochastic programming setting, the uncertainty is supposed to be revealed in a number of instances or as they are widely referred to stages. For instance, if the uncertainty is revealed once and for all, then the decision maker is optimising using two-stage stochastic programming with their decisions being partitioned into here and now/first-stage, i.e. decisions that need to made before the realisation of the uncertainty, and wait and see/second-stage, i.e. recursive actions to mitigate the effect of the uncertainty after its realisation. A generic mathematical formulation of a two-stage stochastic linear program is given by Eq. (7.2). min c T x1 +

x1 ∈X1

 ps (dsT x2s ) s∈S

Subject to :

(7.2)

Wxs2 ≤ hs − Ts x1 ∀s ∈ S x1 ∈ X1 ; x2 ∈ X2 where x1 denote the first stage decisions, x2s are the second stage decisions which are discretised over the different number of scenarios (s ∈ S) each of which has a specific probability of realisation (ps ). The extension of (7.2) to the multistage case is trivial and for the sake of brevity omitted from the discussion. Despite its attractive feature of allowing for recourse, solving a mathematical program with a continuous probability distribution is in general computationally intractable. To overcome this issue, a number of different scenarios are sampled so as to discretise the underlying probability distribution, using sample-based methods, moment matching methods or simulation based ones. The number of scenarios highly affects the computational complexity of the resulting optimisation problem restricting its use to strategic problems where their fast and repetitive solution is not typically required such as supply chain planning [51], conceptual design and optimisation of chemical processes [52] or oil-field planning optimisation to name a few [53]. Ways of reducing the complex-

190

7 A Hybrid Framework for the Uncertainty-Aware Integration …

ity of solving stochastic programs involve scenario reduction techniques and problem decomposition techniques, e.g. L-shaped method [54]. Recently, Li and Floudas [55] proposed an MILP based scenario reduction technique that minimises the probabilistic distance between the original and the scenario-based distribution while Calfa et al. [56] proposed the use of statistical methods for distribution matching so as to generate a reduced multistage scenario tree.

7.2.3 Chance Constrained Programming Chance constrained programming (CCP) forms an alternative methodology for optimisation under uncertainty. Within the CCP framework the decision maker aims to compute a solution that will be feasible with high probability, i.e. the percentage of constraint violation given the values of the decision variables under any uncertainty realisation would be very low. In general a chance constrained program arises through the incorporation of either of the following probabilistic constraints: Pr (gi (x, θ ) ≤ 0, ∀i, . . . I) ≥ ρ Pr (gi (x, θ ) ≤ 0) ≥ ρi ∀i, . . . , I

(7.3) (7.4)

In Eqs. (7.3), (7.4) the operator Pr(·) stands for the probability measure while the parameter ρ takes a user-specified value that reflects the level of reliability required. What Eq. (7.3) implies, differs significantly from Eq. (7.4) as the former specifies a level of reliability to the related constraints as a whole while the latter assigns individual probabilities of constraint satisfaction on the different constraints. The concept of CCP was pioneered by Charnes and Cooper [57] and Miller and Wagner [58] for the cases of single and joint CCPs respectively. In comparison to RO and SP, CCP has received considerably less attention from the process systems engineering community. Maranas [59] studied the optimal molecular design under property prediction uncertainty through the use of joint CCP and derived the deterministic equivalent of the probabilistic problem as a convex MINLP. The nonlinearities in his model arose from the need of multi-variate integration when joint CCPs were considered. Petkov and Maranas [60, 61] also applied CCP to the problems of planning and scheduling of batch plants under demand uncertainty as well as the problem of optimisation of metabolic pathways under uncertainty. Lakhdar et al. [62] studied the medium-term planning of biopharmaceutical processes under uncertain yields through individual CCPs and solved the resulting MILP problem. Individual chance constraints that accounted for isosceles triangular distribution have also been applied to the integrated upstream and downstream optimisation of biopharmaceuticals [63]. A common assumption of the aforementioned works is the stability of the underlying probability distribution function that allows for analytical calculation of the inverse cumulative distribution function [64, 65]. When the stability assumption is missing the derivation of the deterministic counterpart becomes in general not

7.2 Modelling and Optimisation with Uncertainty Considerations

191

possible and thus approximation techniques are typically employed such as sample average approximation [66], convex approximation [67] and analytical approximations through the use of inequalities such as the Chebyshev’s or the Bonferonni [68]. Recently, Li and Li [69] proposed an iterative framework for the optimal approximation of individual chance constraint programs through robust optimisation. In their work, robust optimisation is employed as a means of providing contracting uncertainty sets that eventually lead to the same probability of constraint satisfaction as CCP. An overview of applications of CCP to process systems engineering problems can be found in Li et al. [64]. Due to the need for fast computational times of the integrated solutions so as to allow its online implementation on the production system, the use of scenario based stochastic programming was deemed not appropriate. To this end, the present study will focus on the employement of RO and CCP since these methodologies results in much more computationally tractable problems. In the next section, the main mathematical formulations are outlined.

7.3 Mathematical Formulations In this section the key mathematical formulations for the hybrid framework are presented. For the demand variations on the level of planning so as to secure computational tractability, robust optimisation is employed and the demand is assumed to belong a time-varying budget-polyhedral uncertainty set similar to the works of Bertsimas and Thiele [70]. Chance constraints are considered on the level of scheduling by assuming that the production rate is a normal random variable with known mean and variance. Scheduling variations about the transition times and unforeseen production disruptions are treated by the rescheduling mechanism outlined in Chap. 6. Finally, on the level of control the explicit controllers introduced in Chap. 4 are employed with considerations of LHS uncertainty which can be considered another source of disruption, e.g. catalytic activity or reactant concentration variation.

7.3.1 Rolling Horizon Strategy The integrated problem as shown in the previous chapters is a multi-scale problem whose monolithic solution can lead to extensive computational times. Within the thesis the incorporation of linear metamodels was proposed. Nonetheless, computing detailed schedules and control actions for future planning periods based on demand forecasts that are inevitably subject to variability might result to unnecessary complexity. On the other hand, considering the iPSC problem within a rolling horizon framework follows more closely the current industrial practice where the schedules and plans are updated once more accurate demand forecasts become available [3, 13]. As outlined in the previous section various researchers have resorted to

192

7 A Hybrid Framework for the Uncertainty-Aware Integration …

Fig. 7.1 Rolling horizon strategy under demand uncertainty

rolling horizon-based solution of the integrated planning and scheduling problems either to achieve better computational times for the deterministic case [71] or within a stochastic environment [14]. Within a rolling horizon setting, only the decisions related to the control horizon (CH) are fixed while a less detailed version of the scheduling model is employed for the remainder prediction horizon (PH). Typically, the CH is chosen as the number of planning periods for which the stochastic parameters, e.g. demand, are known with high certainty while for the remaining periods, i.e. p > CH, only an expectation of them is known which is subject to variations. These concepts can be envisaged in Fig. 7.1 (Table 7.1). Despite their merit of enabling for reduced computational times rolling horizon schemes provide an approximation of the true optimal solution. This can be better understood if the receding horizon control example is considered. Control problems are inherently infinite dimensional because of the time variable and as such cannot be solved using finite precision computers. To this end, the control community has

7.3 Mathematical Formulations

193

Table 7.1 Rolling horizon iterations explained along with the related tuning parameters. The color coordination of the planning blocks in the graphical illustration has as follows: control horizon (CH): orange, relaxed/prediction horizon (PH): blue and fixed periods (FP): light blue iteration Graphical illustration PH CH LH FP 1

4

2

2

0

2

6

4

2

2

3

8

6

0

4

4

8

8

0

6

studied extensively the concept of receding horizon control (another name for the rolling horizon scheme) where the decision maker considers a finite number of steps into the future and solves the problem iteratively by fixing their decisions for one step and then resolves the problem while the horizon is moved forward too. It is common knowledge in the control community that the stability and recursive feasibility of the resulting solution very much depends on the length of the look-ahead horizon and broadly speaking, the longer the prediction horizon the better the quality of the solution as the original infinite dimensional problem is better approximated [72]. By equivalence, in the case of rolling horizon for the iPSC the length of prediction horizon as well as the integrality considerations have an important role on the quality of the resulting solution. To this effect, three different variations of the rolling horizon iPSC (iPSC_RH) will be examined. In the first variant of the strategy (iPSC_RH I), the integrality condition of the binary variables will be sustained and thus the rolling horizon decomposition will only be left to alleviate the computational complexity of the integrated problem; notice that this is the case that the condition CH = PH is imposed. In the second variant (iPSC_RH II), for planning periods p > CH the integrality conditions on the binary variables Eip , Fip , Lip , Zijp , ZFijp will be dropped. However by doing so, it is highly possible that the system will overestimate its production capacity and as a result increase the backlog for future periods. A way to alleviate this issue was proposed by Clark and Clark [73] where the authors incorporated a “slack” component in their resource availability calculations. Similar to buff ) their work, in the third variant (iPSC_RH III) the calculation of a buffer-time (tip which will be incorporated in the time balance of the iPSC model will be employed as products assigned to planning periods will incur at least one changeover for the buff can be computed as shown planning period that are outside the control horizon tip by Eq. (7.5).

194

7 A Hybrid Framework for the Uncertainty-Aware Integration …

  (0.5 τijmin REip + 0.5 τjimin REjp ) buff tip =

j=i

j=i

|J|

∀i, CH < p ≤ PH

(7.5)

where τijmin is the product pairwise minimum transition time and is computed according to offline simulations detailed in Chap. 5 and REip ∈ [0, 1] is a relaxed copy of the original binary variable related to the assignment of product i to period p (Eip ). In Algorithm 7.1 the main computational steps of the rolling-horizon scheme are ˜ and PH ˜ are user-defined parameters for the of the look-ahead outlined where LP periods and the initial prediction horizon respectively.

1 2 3 4 5 6 7 8 9

˜ PH = PH, ˜ FP = 0 LP = LP, while FP ≤ PH do if PH ≤ |P| then Demand realisation for p ≤ CH Choose integrality or relaxation strategies as outlined in Section 7.3.1 Solve model iPSC RH (I) or (II) or (III) Update control horizon as: CH = CH + LP Fix scheduling and planning decisions for p ≤ FP Update prediction horizon as: PH = PH + LP

15

else Demand realisation for p ≤ CH Choose integrality or relaxation strategies as outlined in Section 7.3.1 Solve model iPSC RH (I) or (II) or (III) Update control horizon as: CH = CH + LP Fix scheduling and planning decisions for p ≤ FP

16

Update fixed periods as: FP = FP + LP

10 11 12 13 14

7.3.2 Robust Optimisation for Process Planning Before applying robust optimisation to the iPSC model that was presented in the Chaps. 5, 6, it is important to address the following issue: the product demand, which is assumed to be uncertain within the scope of the present chapter, appears only in the equality constraint that calculates the backlog as shown by Eqs. (7.6), (7.7). Bip = Bi,p−1 + Dip − Sip ∀i, p

(7.6)

Vip = Vi,p−1 + Qip − Sip ∀i, p

(7.7)

7.3 Mathematical Formulations

195

This results in two implications: (i) robust optimisation methodologies require inequality constraints [32] and (ii) variations in the demand affect both the costs related to backlog but also inventory. In order to deal with this issue, it is proposed to eliminate the sales variable (Sip ) from Eqs. (7.6) and (7.7) and derive Eq. (7.8) instead. Bip − Vip + Qip = Bi,p−1 − Vi,p−1 + Dip ∀i, p

(7.8)

Through Eq. (7.8) it is now easy to employ a convex piece-wise overestimation (ip ) of the related inventory and backlog costs as shown by Eqs. (7.9), (7.11) similar to the procedure employed by Bertsimas and Thiele [70]. ip = max(cii Vi , cbi Bip )

ip ≥ cbi [Bi,0 +

p 

(7.9)

(Diτ − Pr iτ )] ∀i, p

(7.10)

(Diτ − Pr iτ )] ∀i, p

(7.11)

τ =1

ip ≥ cii [Vi,0 −

p  τ =1

The advantages of such a reformulation are not restricted to just addressing the requirements of robust optimisation but also in the fact that the demand now appears in both the inventory and backlog calculations in a recursive manner. The recursive characteristics of Eqs. (7.10), (7.11) are of big importance when considering the robust optimisation for demand uncertainty, as it allows each planning period to explicitly account for past demand realisations, i.e. τ ∈ [1, p] ∀p and thus avoid overconservative solutions. Key to the derivation of the robust counterpart model is the selection of the uncertainty set. From the reviewed sets in Sect. 7.2, the “budgeted” uncertainty set from the work of Bertsimas and Sim [36] for two reasons: (i) the deterministic reformulation preserves the linearity of the model and (ii) it provides a good trade-off between solution robustness and conservativeness. First, the product demand is assumed to be an uncertain parameter that belongs to a symmetric and ¯ ip − D ˆ ip , D ¯ ip + D ˆ ip ] with known mean value (D ¯ ip ) bounded interval, i.e. Dip ∈ [D ˆ ip ). Based on these features, it is possible to define the random and variance (D variable that causes the perturbations in the values of demand as the following scaled ¯ D −D deviation:zip = ipDˆ ip ∈ [−1, 1]. It is easy to show that now the uncertain demand ip is effectively calculated by Eq. (7.12). ¯ ip + D ˆ ip zip Dip = D

(7.12)

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7 A Hybrid Framework for the Uncertainty-Aware Integration …

Following the methodology of Bertsimas and Sim [36] the uncertainty set to whom the random variable of the scaled deviation belongs to, i.e. z ∈ U, can be defined as shown by Eq. (7.13).  U = |zip | ≤ 1,

p 

 |ziτ | ≤ ip ∀i, p, τ ≤ p

(7.13)

τ =1

What Eq. (7.13) dictates is that at each time period the maximum number of uncertain parameters that are expected to be realised as their worst expectation is restricted by a user specified parameter, i.e. ip , which is widely referred to in the literature as “uncertainty budget”. In order to clarify the use of the uncertainty budget consider that its value can range from 0, which refers to the optimistic case that none of the uncertain parameters will take their worst-case value, while the extreme conservative case that all parameters are expected to vary is imposed by ip = p ∀i, p which coincides to the worst-case robust optimisation as consider by Soyster [33]. Once the uncertainty set is defined Eqs. (7.10), (7.11) are reformulated to incorporate the uncertainty set specification, that is:  ip ≥ cbi Bi,0 −  ip ≥ cvi Vi,0 +

p 

Pr iτ +

p 

τ =1

τ =1

p 

p 

Pr iτ −

τ =1

 ¯ iτ + D ˆ iτ ziτ ) (D

∀i, p ∧ z ∈ U

(7.14)

∀i, p ∧ z ∈ U

(7.15)

 ¯ iτ + D ˆ iτ ziτ ) (D

τ =1

Incorporating Eqs. (7.14), (7.15) directly to the iPSC model would result in an semiinfinite program that cannot be easily addressed since they require the inequalities to hold for infinite many realisations of the uncertain parameters with the uncertain set. To this end, Eqs. (7.14), (7.15) are re-written using their worst case expectations. For the sake of space only the inventory constraints will be reformulated and similar steps can be applied for the backlog ones. For the inventory constraints, the worst-case expectation will happen when the minimum value of demand is realised as shown by Eq. (7.16).  ip ≥ cvi Vi,0 +

p  τ =1

Pr iτ −

p  τ =1

 ¯ iτ − min D z∈U

p 

 ˆ iτ ziτ D

∀i, p

(7.16)

τ =1

Notice that the worst case value of uncertainty is clearly incurred for zip ∈ [0, 1] thus there is no need to consider the symmetric range. The inner minimisation is feasible and bounded for any value of ip and so is its dual by strong duality arguments [74]. The inner minimisation problem and its dual is written as follows:

7.3 Mathematical Formulations p 

min z

197

ˆ iτ z iτ D

max − ip ϕip − ϕ,y

τ =1

Subject to :

p 

yipτ

τ =1

Subject to :

p 

ˆ iτ ∀τ ≤ p ϕip + yipτ ≥ D ϕip , yipτ ≥ 0

Duality

zip ≤ 1 zip ≥ 0

⇐⇒

ziτ ≤ ip ∀τ ≤ p

τ =1

(7.17) Since it suffices for the dual problem to have a solution, to satisfy its constraints for a pair of (ϕ, y), the maximisation term can be dropped and thus result in the linear robust counterpart of Eq. (7.16) as shown by the system of Eqs. (7.18)–(7.21), for both the inventory and the backlog constraints. ip ≥ cbi [Bi,0 −

p 

Qiτ +

τ =1

ip ≥ cii [Vi,0 +

p  τ =1

Qiτ +

p 

¯ iτ + yipτ ) + ip ϕip ] ∀i, p (D

(7.18)

τ =1 p 

¯ iτ + yi pτ ) + i p ϕi p ] ∀i, p (−D

(7.19)

τ =1

ˆ iτ ∀i, p, τ ≤ p ϕip + yip τ ≥ D ϕip + yipτ ≥ 0 i, p, τ ≤ p

(7.20) (7.21)

7.3.3 Chance-Constrained Process Scheduling On the level of process scheduling, the different production rates (rip ) are assumed to be uncertain. Even though the production rates can be seen as functions of actions taken at the control level and thus deal with such disruptions in a purely reactive manner it is preferred to make the model “aware” of the probability that these variations are likely to occur and thus also provide a physical meaning to the rescheduling parameter that was employed in Chap. 6. To this end, chance-constrained programming has been selected and more specifically under the assumption that the rates are not correlated, individual chance constraints will be considered. This was chosen based on the following observation: production disruptions on different products tend to incur different reduction in the overall profit based on their product margin. It is thus reasonable to impose higher probability levels of constraint satisfaction with regards to the more premium products (high product margin) and lower levels for the products that offer less margin. For the purposes of this study the “quantile-

198

7 A Hybrid Framework for the Uncertainty-Aware Integration …

method” for deriving the deterministic equivalent of the probabilistic constraints is employed under the assumption that the random production rates follow a “stable distribution” [31]. Definition 7.1 (Stable Distribution [31]) A statistical distribution is called stable when it can be completely characterised by two parameters U and V and the convolution of any K distributions [(x − U1 )/V1 ], . . . , [(x − U K )/VK ] is again of the form [(x − U )/V ]. The two parameters U and V refer in the majority of the cases to the mean and standard deviation of the distributions. Characteristic examples of stable distributions include the normal, Poisson, Chisquared and binomial among others. By assuming stably distributed uncertain production rates and following the quantile-method the deterministic equivalent of the chance constraint that involves production rates is derived as follows: 1. The original constraint under consideration is shown by Eq. (7.22), where the production amount (Qip ) is given as the product of the production rate (rip ) and the production time (Tip ). Qip = rip Tip

∀i, p

(7.22)

2. In order to apply the CCP method the equality needs to be relaxed to its equivalent inequality. Logically since the production scheduling is timed-constrained the related relaxation would be in the form of Eq. (7.23). Qip ≤ rip Tip

∀i, p

(7.23)

3. The chance constraint related to Eq. (7.23) by requiring a minimum level of constraint satisfaction, i.e. ρi , is shown by Eq. (7.24). Pr(Qip ≤ rip Tip ) ≥ ρi

∀i, p

(7.24)

4. Assuming that the production rates are normally distributed with known mean μ(rip ) and variance σ (rip ) the inequality inside the probability can be normalised by subtracting the the mean and dividing by the variance term on both sides as indicated by Eq. (7.25). Pr(

ri p Ti p − μ(rip Tip ) Qip − μ(rip Tip ) ≤ ) ≥ ρi σ (rip Tip ) σ (rip Tip )

∀i, p

(7.25)

5. The right-hand side term of the inequality inside the probability can now be seen as a normal random variable (δ) with mean 0 and variance 1, i.e. a standarised normal variable and by re-arranging the terms of Eq. (7.25), the resulting chance constraint can be expressed by Eqs. (7.26), (7.27).

7.3 Mathematical Formulations

199

Qip − μ(rip Tip ) ) ≥ ρi ∀i, p ⇐⇒ σ (rip Tip ) Qip − μ(rip Tip ) Pr(δ ≤ ∀i, p ) ≤ 1 − ρi σ (rip Tip )

1 − Pr(δ ≤

(7.26) (7.27)

6. Since δ follows the standarised normal distribution which is stable, following the quantile method the probability operator can be replaced by the related operator of the cumulative distribution function (Φ) and subsequently apply its inverse transformation as illustrated in Eqs. (7.28), (7.29). Φ(

Qip − μ(rip Tip ) ) ≤ 1 − ρi ∀i, p ⇐⇒ σ (rip Tip ) Qip − μ(rip Tip ) ≤ Φ −1 (1 − ρi ) ∀i, p σ (rip Tip )

(7.28) (7.29)

7. Finally, since Φ −1 (1 − ρi ) = −Φ −1 (ρi ), the deterministic equivalent of the initial chance constraint is shown by Eq. (7.30). Qip ≤ μ(rip )Tip − Φ −1 (ρi )σ (rip )Tip

∀i, p

(7.30)

The term Φ −1 (ρi ) is essentially the quantile of the per-specified reliability level ρi and thus Eq. (7.30) remains a linear inequality constraint preserving the computational tractability of the overall model.

7.3.4 Overall Uncertainty-Aware iPSC Model Since the sales variables have now been eliminated, the problem of maximising profit is turned into minimising the total cost which comprises of the transition costs, production cost and the convex piece-wise overestimation of the inventory and backlog costs as shown below by the problem “Full-space iPSC” (iPSC FS). min Cost = TC + PC +



ip

(iPSC FS)

i, p

Subject to : Eqs. (5.8)−(5.21) Eqs. (5.45)−(5.50) Eqs. (7.17)−(7.20), (7.29) For the sake of space the detailed iPSC_RH model are given in detail in Appendix E. Below the summary of the optimisation following the first two variants of the RH approach is given, depending on the length of the control horizon (CH).

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7 A Hybrid Framework for the Uncertainty-Aware Integration …

min Cost = TC + PC +



ip

(iPSC RH I/II)

i,p

Subject to : Eqs. (E.1)−(E.31) Eqs. (7.17)−(7.20), (7.29) While the third variant of the model for the RH approach is in overall formulated as follows:  ip (iPSC RH III) min Cost = TC + PC + i,p

Subject to : Eqs. (E.1)−(E.24) Eqs. (E.26)−(E.31) Eqs. (7.17)−(7.20), (7.29)

7.4 Computational Studies All the optimisation problems were formulated and solved using GAMS 24.7.4, on a Dell workstation with 3.70 GHz processor, 16 GB RAM and Windows 7 64-bit operating system using CPLEX 12.6.1 for the solution of the MILPs.

7.4.1 Problem Instance Classes and Tuning of the Proactive Module The proactive module involves a number of tuning parameters that reflect the risk attitude of the decision maker when hedging against the uncertainty. First, the budget parameter of the robust optimisation was assigned three different sets of values, i.e.: ip = 0.1 p + 0.5, 0.5p, p. Secondly, depending on the product margin the parameter reliability parameter (ρi ) was set to be a monotonically increasing function of the margin ranging from 0.65 to 0.95. An illustrative graph of the cumulative level of protection employed (across the planning horizon) via the various budgets that were examined is given in Fig. 7.2. Finally, two different problem classes were examined. In Class 1, the plant is assumed to be nominally operating at an average 65% of its total capacity while Class 2 assumes an average of 90% of the plant’s capacity.

7.4 Computational Studies

201

Fig. 7.2 Cumulative level of protection against demand uncertainty for different assumed budgets. As illustrated the case of  = p is the most conservative followed by  = 0.5p and the least risk-averse  = 0.1p + 0.5.

7.4.2 Monte Carlo Simulations for Evaluation of the Proactive Module of the Uncertainty-Aware iPSC In order to evaluate the advantages of considering proactive approaches (RO and CCP) within the uncertainty-aware iPSC, the robustness of the resulting solution needs to be implemented in a stochastic environment so as to compute and compare its expected value (EV) with the deterministic version of the iPSC. To this end, once the related solutions have been computed, the binary variables related to sequencing and assignment of products in planning periods are fixed along with the relevant production times and a number of Monte Carlo (MC) simulations (nMC ) are conducted with the expected value of the solution being calculated as shown by Eq. (7.31).  EV =

n∈Nmc

Costn

nMC

(7.31)

A graphical illustration of the proposed scheme is shown by Fig. 7.3. Typically the user-defined parameter nMC is computed using a trial and error approach with the goal to result in a stabilised normalised EV [63, 75]; in the case of the present study the it was found that 250 MC iterations were sufficient for the EV to be stabilised around a value. At each MC iteration a different value of the uncertain parameters related to the MC and DMC proactive module is realised, i.e. rip ip . More specifically, for the uncertain ¯ ¯ demand the assumed distributions were (i) uniform DMC ip = U(−0.2Dip , +0.2 Di p ) MC MC ¯ ¯ and (ii) Gamma Dip = Γ (0.2, 1)Di p and (iii) Cauchy Dip = C(0.1, 0.05)Dip . For the production rate a uniform distribution of 10% variance was considered, i.e. MC = U(−0.1¯ri p , +0.1¯rip ). In order to abide by the assumption about not correlated rip deviations, product and planning period specific normalised variations were drawn at each iteration by manipulating the random number seed. However the same seeds were imposed for the comparison between deterministic and hybrid approaches so as

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7 A Hybrid Framework for the Uncertainty-Aware Integration …

Fig. 7.3 Overall methodology of quantifying the expected value of the resulting solution by the deterministic and the hybrid iPSC model through Monte Carlo simulation

to secure meaningful results. Finally, the uniform, Gamma and Cauchy distributions were chosen for the simulation of the uncertain demand so as to investigate the impact of both simultaneously expect order cancellations and rush orders (uniform/Cauchy) and purely increase in the demand (Gamma).

7.5 Results and Discussion In this section, the key findings from the computational studies will be overviewed with the aim of identifying the benefits and drawbacks of the uncertainty-aware solution as well as a number of insights about the underlying trade-offs of the integrated problem.

7.5.1 Solution Quality of the Rolling Horizon Approach In order to evaluate the solution quality of the RH approach and its variants that were discussed in Sect. 7.3.1 a series of problems were solved with planning horizon ranging from 6 periods to 12 and 16 products. The related computational statistics can be found in Table 7.2. As indicated in Table 7.2, when considering planning horizons greater than 6 periods the computational effort required by the FS approach is prohibitive for the online implementation of the iPSC solution; this is due to the logic requirement to provide solutions in times smaller than the sampling instance of the model based controller so as to allow for reactive actions. On the other hand, the RH (I) strategy in the first

Strategy Bin. Var. Con. Var. Eq. CPU (s) Obj. Val. Gap (%)

FS 3,485 10,222 15,334 20.3 30,125 0

RH (II)

4.3 30,125 0

RH (I)

14.2 30,125 0

6 planning periods

4.5 30,125 0

RH (III)

FS 4,743 13,828 20,842 109.9 48,427 0 73.2 48,427 0

RH (I)

8 planning periods

12.3 48,696 0

RH (II)

11.5 48,427 0

RH (III)

FS 7,259 21,040 31,858 1000* 84,270 0.02

480 89,303 0

RH (I)

34.5 92,930 0

RH (II)

12 planning periods

35.1 89,890 0

RH (III)

Table 7.2 Problem class 1: Computational statistics of the solutions computed by the full-space approach (FS) compared the ones by the different variants of RH strategy

7.5 Results and Discussion 203

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7 A Hybrid Framework for the Uncertainty-Aware Integration …

instance reduces the computational effort but due to the integrality constraints its solution times grow rapidly with the length of the planning horizon. The variants RH (II) and (III) scale favorably with the increase of plannning periods under consideration while the RH (III) for the case of 8 planning periods computes the same solution as the FS approach in almost ten times faster. For the 12 period case the FS approach results in the best objective value (with 0.02% gap due to time limit of 1000 s) while among the three variant of the RH approach the best trade-off between solution time and objective value is given by RH (III).

7.5.2 Problem Class 1 In this problem class, the nominal product demand requires approximately 65% of the plant’s capacity while the instance considered herein comprises of 6 planning periods. Before analysing the results of the MC simulations for the different distributions assumed it would be helpful to examine the Gantt charts of the different solutions based on the varying levels of risk-averseness. Firstly in Fig. 7.4 one can notice that due to the characteristics of the problem class, the deterministic solution which ignores the uncertainty, results in excessive idle time in planning periods p3 and p4 so not to build up inventory and avoid its related cost. Next, Fig. 7.4b illustrates the planning and scheduling decisions when mild risk-averseness is employed within the hybrid framework. Because the model is “aware” of the potential demand and production rate variations it starts to build up inventory and reduces the idle time. By increasing the conservativeness of the budget parameters in Fig. 7.4c, d it is shown that the idle time vanishes and instead the products that hold high-margin/highbacklog cost are being produced in excess while low-margin products begin to be backlogged. As outlined in Sect. 7.4.2, once the solutions from the different approaches (deterministic/hybrid) have been computed the binary variables and the related production times are fixed and solution is evaluated in the stochastic environment through the MC simulations. A detailed overview of the statistical characteristics of the solution of each approach for this problem class can be found in Table 7.3. In the first instance uniform distribution was employed to simulate the demand uncertainty and production rate uncertainty. As illustrated by Fig. 7.5a, even though the deterministic solution compared to the solutions computed by the hybrid framework (for different budgets) is on average 90% lower, this observation can be misleading. When the decisions are fixed and evaluated through the MC simulations, the deterministic solution becomes the most costly one followed by the least risk-averse solution of the hybrid framework (ip = 0.1p + 0.5). On the other hand, the solutions computed with the “medium” and “high” risk-averseness provide the lowest expected value as shown by Fig. 7.5a. The highly-risk averse solution with  = p overestimates quite significantly the resulting EV, while the solution with  = 0.5p provides the closest “robust” bound to the resulting EV which from a managerial point of view can be desirable. Similar trends in terms of EV were identified when the Gamma and Cauchy

7.5 Results and Discussion

205

Fig. 7.4 Gantt charts of the different solutions computed with varying risk-averseness by employing the hybrid framework and the deterministic approach for the iPSC for problem class 1. The blocks filled with the color of label “Y” indicate changeover and idle times in the planning periods

Uniform Gamma Cauchy Uniform Gamma Cauchy

Uniform Gamma Cauchy Uniform Gamma Cauchy

1

1

2

2

Distribution

Class

421,737 44,105 366,549 1,010,102 1,054,449 406,452 209,449 1,468,922 971,275 261,940 236,800 1,220,341 1,599,783 1,226,532 90,549 1,111,858 2,409,299 2,175,193 537,261 976,832 2,716,980 2,097,409 346,651 1,131,632 2,423,002 Hybrid approach with ip = p (High risk-aversion) 533,200 420,141 37,392 373,635 919,009 1,047,277 400,816 229,273 1,456,137 952,318 256,369 245,290 1,196,738 1,770,204 1,290,312 82,792 1,193,517 2,378,471 2,238,866 533,343 1,079,223 2,778,232 2,126,522 340,269 1,196,961 2,449,532

454,312

Hybrid approach withip = 0.5p (Medium risk-aversion) Worst case Obj. Value Mean Std. Dev. Best case Worst case

567,250 107,668 369,749 1,901,546 1,241,940 459,117 32,054 1,692,123 1,420,930 310,032 459,894 1,691,530 578,789 1,129,878 137,883 862,282 2,847,582 2,075,854 572,707 581,144 2,637,894 2,243,257 384,106 1,086,481 2,587,998 Hybrid approach with ip = 0.1p + 0.5 (Low risk-aversion) 244,050 454,343 75,062 352,090 1,425,431 1,112,941 431,317 127,511 1,548,444 1,128,139 287,726 272,016 1,392,147 1,084,572 1,121,415 116,378 944,207 2,609,753 2,099,620 553,037 785,655 2,653,738 2,108,874. 368,090 1,038,780 2,444,944

30,125

Deterministic approach (No risk-aversion) Obj. Value Mean Std. Dev. Best case

Table 7.3 Computational statistics with regards to the characteristics of the different solutions and problem classes examined through the MC simulations

206 7 A Hybrid Framework for the Uncertainty-Aware Integration …

7.5 Results and Discussion Fig. 7.5 Problem Class 1: Monte Carlo simulation results for different assumed distributions and budgets of robust optimisation. The dotted lines in the plots indicate different realisations of the objective value (cost) across the MC simulations while the solid lines indicate the value computed by the a priori optimisation of the problem

207

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7 A Hybrid Framework for the Uncertainty-Aware Integration …

Fig. 7.6 Histogram of the normalised random deviations for the demand of a product following the uniform, Gamma and Cauchy distributions

distributions were assumed for product demand variations as can be envisaged by Fig. 7.5b, c however the gap between the EV and the “nominal” solution computed by the hybrid framework is quite large. For the case of the Gamma distribution this is expected as in this setting the product demand was constantly rising since no negative variations can be imposed by the distribution; this resulted in accumulation of backlogged demand. The Cauchy distribution instance served as a means to investigate the impact of underestimating the expected variation of the uncertainty as the fabricated distribution resulted in variance of ~35% instead of 20% as well as the ratio of demand increase to decrease was significantly greater than 1. A histogram of the normalised random deviations from the three different distributions is given in Fig. 7.6.

7.5.3 Problem Class 2 This problem class served as a basis to investigate the case where the nominal demand requires approximately 90% of the plant’s capacity. In such cases where the system under consideration is operating at its limits, the value of optimisation under uncertainty would intuitively be quite high. The results of this case however indicate that due to the plant’s limited capacity the flexibility of the system to hedge a priori against the uncertainty is quite limited and can result in worse expected values when the system is set to operate in the stochastic environment. Similar to the previous case Fig. 7.7a is the Gantt chart of the deterministic solution, which in contrast to

7.5 Results and Discussion

209

Fig. 7.7 Gantt charts of the different solutions computed with varying risk-averseness by employing the hybrid framework and the deterministic approach for the iPSC of problem class 2. The blocks filled with the color of label “Y” indicate changeover and idle times in the planning periods

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7 A Hybrid Framework for the Uncertainty-Aware Integration …

the previous problem class does not involve any idle time but also results in high levels of inventory and backlog as illustrated by Fig. 7.9b. In Fig. 7.7b, c and 7.7d the Gantt charts of the hybrid framework solutions with different uncertainty budgets are shown. The main observation here is that hybrid framework results in interchanges between inventory of high-value products, e.g. product O, with the backlog of lower value products, e.g. product M, by comparing Fig. 7.7a with Fig. 7.7d. The results from the MC simulations for this problem class have the opposite trends compared to the previous class. By observing Fig. 7.8 and its subplots one can easily identify that the less risk-averse solutions achieve better (lower) expected value. Having a closer look at the statistics in Table 7.3, there exists a paradox where the deterministic solution even though as a solution is characterised by the biggest variance, it offers better EV than the medium and high risk averse solutions of the hybrid framework. The low-risk averse solution for the cases of the uniform and Cauchy distribution provides marginally the best EV of the iPSC (followed by the deterministic solution) with lower variance too. A potential explanation for the results of this class can be found in the very limited degrees of freedom the proactive approaches have at hand so as to hedge against the uncertainty. In contrast to the problem class 1, in this case, the cost of hedging against the uncertainty beforehand is quite higher since there is no idle time and whatever additional production time is decided on will result in reduction of the production time of another product and consequently the relevant backlog. Towards this direction advocate also the results with regards the backlog and inventory levels for the different solutions of this class as shown by Fig. 7.9b.

7.5.4 Uncertainty Aware iPSC of MMA Polymerisation Process Finally the implementation of the uncertainty aware iPSC of the MMA process which has been introduced in Chaps. 5 and 6 is considered. Considering a case similar to problem class 2, a 6 week planning horizon in which the demand forecast is updated every time the horizon is moved CH step forward. The MMA CSTR is assumed to be subject to dynamic disruptions along with variations in the inlet concentration of the initiator (±10% of its nominal value) which is assumed to be measured at the same rate as the control sampling time [76]. The methodology employed in Chap. 4 is followed for prediction horizon of unity, 3 CRs and their related control laws are computed involving idle, deadbeat and settling mode. For the sake of space the complete expressions are omitted and one expression is given by Eq. (7.32), so as to illustrate the fractional polynomial nature of the control law when LHS is considered. u0 =

988.88(1.0114θ6 − θ2 ) θ9

(7.32)

7.5 Results and Discussion Fig. 7.8 Problem Class 2: Monte Carlo simulation results for different assumed distributions and budgets of robust optimisation. The dotted lines in the plots indicate different realisations of the objective value (cost) across the MC simulations while the solid lines indicate the value computed by the a priori optimisation of the problem

211

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7 A Hybrid Framework for the Uncertainty-Aware Integration …

Fig. 7.9 Representation of the effect of the degree of risk-averseness on the overall levels of backlog and inventory for the different problem classes of the case study

7.5 Results and Discussion

213

Fig. 7.10 Overview of the closed-loop response of the underlying dynamic system for the uncertainty-aware iPSC of the MMA polymerisation process

where θ9 is the inlet concentration of the initiator, while θ2 and θ6 are the state measurement and the set-point for the of the initiator concentration inside the CSTR. The overall closed-loop iPSC problem is formulated as an MILP with 9,656 equations, 8,075 continuous variables and 3,104 binary variables while the solution time for the nominal case takes 5.23s using CPLEX 12.6.3 and the iPSC_RH (III) approach. In this case study minimum production time of 2 h is imposed, i.e. for a product to be assigned to a planning period at least 2 h should be allocated for its production. As mentioned earlier in the chapter, the main requirement for the efficient online implementation of the closed-loop iPSC solution is that the computational time needed for the solution of the optimisation problem is less than sampling time of the underlying control system. For this case study, the sampling time is 36s and thus it is requires to solve the iPSC in times less than 36s. The parameters for the rescheduling framework, from Chap. 6, were set as ω = 10−6 ;  = ±5%. The first rescheduling action takes place during the production of product O, when a disturbance that exceeds the threshold is realised and the rescheduling mechanism is triggered. Following the steps of Algorithm 1 a new duplicate product is generated and the reduced iPSC solved with fixed the past decisions, in this case the production time of O that was achieved and the relevant binary decisions with respect the order of O in the schedule. Solving the reduced iPSC takes 7.86s, slightly increased when compared to the nominal case but this increase is due to the introduction of the two dummy products in the reduced iPSC, i.e. the duplicate of O and the dummy disturbance product. The next disturbance is detected during the transition from J to I and once again the rescheduling mechanism is activated but this time the system is controlled towards the completion of the nominal transition while the related computational time is 4.96s. Similar to planning period 1, in period 2 another reschedule takes place due to dynamic disruption during the transition from product N to product O and the system is resumed to its nominal schedule with the completion of the original transition. During period 3 the no dynamic disturbance that exceeded the relevant threshold was detected. Period 4 involved a reschedule again during the transition from product A to product N but within 1.79s the rescheduling mechanism had computed the corrective actions and the transition was resumed. Finally, during period 6 another instance of dynamic disruption during the production of M was detected, due to variation in the inlet concentration of the initiator, and in this case the production was immediately resumed. The graphs of the system’s dynamics are given in Fig. 7.10.

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7 A Hybrid Framework for the Uncertainty-Aware Integration …

7.6 Concluding Remarks In this chapter reactive and proactive approaches were employed so as to investigate a hybrid framework for the uncertainty aware integration of planning, scheduling and explicit control. Since scenario-based stochastic programming approaches tend to result in large solution times which render the online implementation of the iPSC problem not possible, depending on the type of the uncertainty RO and CCP approaches were employed. The proactive module of RO and CCP is coupled with the reactive module that was introduced in Chap. 6 comprising thus the proposed hybrid framework for the uncertainty-aware iPSC. The results from the computational studies indicate that by considering a RH based solution approach the online implementation of the iPSC solution is possible for planning horizons of up to 12 periods and 16 products while for larger instances alternative decompositions need to be investigated. Furthermore, the nominal plant capacity utilisation level was concluded to be a crucial factor on the appropriate selection of tuning parameters for the proactive module mainly with regards the budget parameters of RO for demand uncertainty. Variations on the level of scheduling with respect to production rates do not appear to have as big impact as the product demand variations mainly due to their implicit relationship with the systems dynamics which in the present work were also considered by the reactive module. Overall, further investigation on different solution approaches and proactive methods is needed so as to rigorously postulate the hybrid framework for the uncertainty-aware iPSC and these research directions are proposed in the next and final chapter of the thesis.

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Chapter 8

Conclusions and Future Work

In this final chapter, the main contributions of the thesis are summarised and future research directions are drawn for each of the two topics that were discussed, i.e. multiparametric programming and uncertainty-aware integration of process operations with control.

8.1 Multi-parametric Programming Theory, Algorithms and Applications 8.1.1 Contributions In Chap. 3, two algorithms were presented for the first time in the literature for the exact solution of mp-LPs and mp-MILPs under global uncertainty. In contrast to the algorithms that have already been proposed, computations of the exact nonconvex CRs were performed reducing in that way drastically the number of convex approximate ones. Further to the development of a rigorous procedure, two theorems were provided for the characterisation of the explicit optimal solution and CRs. Case studies from process synthesis and zero-wait multi-stage batch scheduling were examined and highlighted the potential practical value of the proposed algorithm. In Chap. 4, the solution of a special class of mp-NLPs under global uncertainty was investigated. The key focus of this chapter was the development of novel multisetpoint explicit controllers that can be of great use in EWO problems that involve the control level in them. Apart from the state of the system at each sampling instance, the set-points and the model parameters were considered as uncertain paving the way for controllers that can be seamlessly integrated with the grade of RTO, scheduling as well as parameter estimation.

© Springer Nature Switzerland AG 2020 V. M. Charitopoulos, Uncertainty-aware Integration of Control with Process Operations and Multi-parametric Programming Under Global Uncertainty, Springer Theses, https://doi.org/10.1007/978-3-030-38137-0_8

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8 Conclusions and Future Work

Finally, the algorithms presented in this part of the thesis have been implemented in Perseus, a Python based platform for which more details are available in Appendix B.

8.1.2 Perspectives for Future Research • Algorithmic refinements: The algorithm introduced in Chaps. 3, 4 can be of great value for many problems in the field of PSE but it comes at the cost of an almost doubly-exponential complexity. With regards to the underlying architecture of the algorithm it is extremely memory intensive with instances where the CPU usage was ~10% the related RAM usage peaked at almost 99.9% causing the computation to stop due to memory limitations on a 16GB RAM machine. However, there exist alternative methods for computing the Groebner Bases that involve Linear Algebra calculation which can re-allocate the computational burden towards CPU usage [1, 2]. Such algorithms are available either as standalone [3] or they can be used through Maple [4]. Further to this, the parallelisation of the while loop in Algorithm 3.2 (steps 9–18) can yield better computational results. Finally, the use of heuristics for the CAD computations (definition of CRs and comparison procedure) has been reported to result in more efficient computational times [5]. Preliminary experimentation has shown that decomposition of a long logic statement into its elements and then computing sequentially their CADs, as shown by Algorithm 8.1, can provide reasonable speed-ups in the calculations. • Multi-parametric Dynamic Programming: As outlined in Chap. 2, dynamic programming principles have been employed for the alleviation of the complexity related to multi-stage multi-parametric programs, such as explicit MPC. The multi-setpoint explicit controller presented in Chap. 4 can benefit for such type of sequential decomposition. • Multi-parametric Global Optimisation: The algorithms presented in the thesis deal with highly non-convex multi-parametric programming problems for which numerical solution can be formidable. However, as its was theoretically proven in Chap. 3, at least for the linear and mixed integer linear cases, the exact computation of the explicit solution involves optimising over fractional polynomials. An interesting research direction would be the development of tailor-made convex envelopes for the global solution of the related fractional polynomial problem, related to the definition of CRs and their comparison [6]. The explicit solution can be then retrieved by either inversion of the parametric optimal bases or by generalising the method of [7] to the multi-parametric case.

8.2 Integration of Control with Process Operations Under Uncertainty

221

Algorithm 8.1 Sequential logic decomposition and CAD computations Input: Standard atomic formula of the parametric primal and dual feasibility conditions, i.e.:  = Q1 (λ1 (θ) ≥ 0) ∧ · · · ∧Qm (λm (θ) ≥ 0)∧Qm+1 (g1 (θ) ≤ 0) ∧ · · · ∧Qm+m (gm (θ) ≤ 0)ψ(θ1 , . . . , θn ) Output:CR 1: for i = 1, m : 2: Compute CRi = CAD[Qi (λi (θ) ≥ 0) ∧ . . . Qm+i (gm+i (θ) ≤ 0)ψ(θ)] 3: if CRi = ∅ : 4: if i > 1 : 5: Compute CRi = CAD[CRi−1 ∧ CRi ] 6: if CRi = ∅: 7: GoTo step 1 8: else : 9: GoTo step 18 10: endif 11: else : 12: GoTo step 1 13: endif 14: else : 15: GoTo step 18 16: endif 17: endfor 18: return CR

8.2 Integration of Control with Process Operations Under Uncertainty 8.2.1 Contributions In Chap. 5, the main developments in the field of integrating control with operations were reviewed and a comparative analysis between the time slot based formulation and a TSP based formulation was conducted. Following this comparison, for the deterministic case the use of linear metamodels that correlate transition time and cost and the offline storage of the related control moves was proposed together with a decomposition for the alleviation of the computational complexity of the deterministic integrated problem. In Chap. 6, the deterministic assumption on the level of system dynamics was removed. In order to efficiently deal with dynamic disruptions, the multi-setpoint mp-MPC from Chap. 4 was employed and an optimisation based rescheduling framework was developed. The necessity and benefits of multi-setpoint mp-MPC within the context of closed loop iPSC were also underlined through the comparison with conventional MPC. Based on the case studies it was shown that neglecting uncertainty at the level of system dynamics can lead to significant degradation of system’s performance.

222

8 Conclusions and Future Work

The rescheduling framework presented in Chap. 6, involved the use of some tuning parameters and considered uncertainty only at the control level. Building largely on it, in Chap. 7, a hybrid framework for the uncertainty aware integration of planning, scheduling and multi-parametric control was presented. Depending on the nature of uncertainty, proactive approaches of robust optimisation and chance constrained programming were coupled with reactive ones. The results indicate an interesting trade-off between the nominal plant capacity utilisation level and the tuning of the proactive module of the hybrid framework while the reactive module aids towards the alleviation of the conservativeness due to the proactive methods.

8.2.2 Perspectives for Future Research • Multi-level optimisation: Within the thesis a monolithic integrated decision making paradigm was followed where a single player formulates the optimisation problem under study and postulates a universal objective function. However, as in many multi-level optimisation problems, the objective of the different levels can be conflicting or from a practical standpoint a central decision maker does not have sufficient expertise to make those decisions all together [8]. A characteristic example of the implications resulting from the multi-level optimisation of iPSC is the trade-off between minimising transition times and obtaining smooth and robust control profiles. The first objective, i.e. minimum transition times, is typically sought by the scheduling level while the second objective, i.e. smooth and stable control inputs, is required by the control level. Formulating instances like the aforementioned in a game theoretic way can provide invaluable insights on the derivation of optimal solutions for the integrated problem. • Optimisation under uncertainty: In the thesis the use of proactive and reactive approaches based on the availability of past realisations of the uncertain parameters was examined. Even though the results from the case studies investigated showcased the benefits of considering the uncertainty-aware integrated problem the systematic development of a framework for the classification and characterisation of uncertainty still remains an open problem. Apart from the classification and optimal treatment of the uncertainty, another major issue is the conservatism and dimensionality of the resulting deterministic equivalent problem. The adjustable robust optimisation paradigm for the proactive module of the iPSC would reduce the level of conservativeness, however as reported in the literature the monolithic/full-space solution of the resulting problem can pose computational challenges. Another promising research avenue would be the field of data-driven optimisation under uncertainty and how can such developments construct less conservative uncertainty sets [9]. • Decomposition techniques: The computational burden associated with the solution of the iPSC can be prohibitive from its successful online application to real processes. In Chap. 7, the use of rolling horizon approaches was examined and the results showed reasonable reduction in solution times while preserving good

8.2 Integration of Control with Process Operations Under Uncertainty

223

quality of solutions. Nonetheless, RH approaches are hierarchical and rely purely on the time decomposition in order to reduce the computational complexity. Preliminary studies on the application of Benders Decomposition (BD) on the integrated problem do not indicate promising results due to the mathematical structure of the underlying problem. Column and constraint generation technique on the other hand has been reported to provide fast solution times in problems that BD does not behave well, thus making its application on the iPSC a promising direction [10]. • Application areas: The aim of the thesis was to develop and present a framework for the iPSC but its application to industrially relevant problem remains still an open issue. Given that contemporary industries shift their production paradigm from batch to continuous the investigation of industrial case studies from the field of (bio-)pharmaceutical [11] or refinery operations [12] can fuel further developments on the topic.

References 1. Faugere JC (1999) A new efficient algorithm for computing Gröbner Bases (F4). J Pure Appl Algebr 139(1):61–88 2. Faugere JC (2002) A new efficient algorithm for computing Gröbner Bases without reduction to zero (f5). In: International symposium on symbolic and algebraic computation ’02. ACM Press 3. Faugere JC (2010) FGb: a library for computing Gröbner bases. In: International conference in mathematical software. Lecture notes in computer science, vol 6327. Springer, Berlin, pp 84–87 4. Bregman A, Achim A, Ahad P (1992) The MAPLE software system 5. Davenport JH, England M (2016) Need polynomial systems be doubly-exponential? In: International congress on mathematical software. Springer, pp 157–164 6. Tuy H, Thach PT, Konno H (2004) Optimization of polynomial fractional functions. J Global Optim 29(1):19–44 7. Khalilpour R, Karimi I (2014) Parametric optimization with uncertainty on the left hand side of linear programs. Comput Chem Eng 60:31–40 8. Shobrys DE, White DC (2000) Planning, scheduling and control systems: why can they not work together. Comput Chem Eng 24(2–7):163–173 9. Bertsimas D, Gupta V, Kallus N (2018) Data-driven robust optimization. Math Program 167(2):235–292 10. Zeng B, Zhao L (2013) Solving two-stage robust optimization problems using a column-andconstraint generation method. Oper Res Lett 41(5):457–461 11. Lee SL, Connor TF, Yang X, Cruz CN, Chatterjee S, Madurawe RD, Moore CM, Lawrence XY, Woodcock J (2015b) Modernizing pharmaceutical manufacturing: from batch to continuous production. J Pharm Innov 10(3):191–199 12. Pinto J, Joly M, Moro L (2000) Planning and scheduling models for refinery operations. Comput Chem Eng 24(9–10):2259–2276

Appendix A

Background Material

Symbolic solution of polynomial systems of equations and inequalities is the main focus of Part I of the thesis. In this chapter, some key concepts from computer algebra are briefly summarised along with the formal definitions of Gröbner Bases for the analytical solution of polynomial equations and Cylindrical Algebraic Decomposition for the solution of polynomial inequalities. Most of the material presented in this chapter is adopted from the books of Cox et al. [1] and Bochnak et al. [2] and the interested reader is directed to these books for a more detailed exposition on the aforementioned topics.

A.1 A.1.1

Computer Algebra Preliminary Concepts Algebraic Rings, Fields and Closures

Definition A.1 (Ring) A ring K is defined as a set equipped with two binary operators “+” and “∗”, representing addition and multiplication, whose elements satisfy the following conditions: (i) distributivity, (ii) associativity, (iii) additive identity, (iv) additive inverse and (v) additive commutativity. Definition A.2 (Commutative ring) If the elements of K also obey the commutative multiplication condition, i.e. x1 , x2 ∈ K : x1 · x2 = x2 · x1 , then the ring is called commutative ring. An example of commutative ring is the set of integers Z. Definition A.3 (Field) A field is a ring K which apart from the aforementioned properties, the condition of inverse and identity holds both for the addition and the multiplication operation. In that sense one can understand that the set of rational numbers Q is a field. In Fig. A.1, the hierarchy of the rings and fields is shown along with their required laws; as it can be seen, apart from the groups and Abelian groups © Springer Nature Switzerland AG 2020 V. M. Charitopoulos, Uncertainty-aware Integration of Control with Process Operations and Multi-parametric Programming Under Global Uncertainty, Springer Theses, https://doi.org/10.1007/978-3-030-38137-0

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Fig. A.1 Hierarchy of algebraic fields, rings and groups

which are included for the sake of completeness, rings are the most general sets with the fields being the most specific ones. For the solution of a system of polynomial equations it is important to perform the relevant calculations over a field were the solutions are defined, or as such a field is known within the computer algebra community an “algebraically closed field”. Definition A.4 (Algebraically closed field) A field K is algebraically closed, if any univariate polynomial with degree at least one and coefficients in K has a root in K, i.e. ∃x ∈ K : g(x) = 0. The most prominent example of an algebraic field that is not closed is the field of real numbers (R) since the equation z2 + 1 = 0 is a polynomial with coefficient that belongs to R but there does not exist a real solution to satisfy that equation. However, the field of complex numbers C is an algebraically closed field, as derived by the Fundamental Theorem of Algebra, and is also known as the “algebraic closure” of R.

A.1.2

Polynomials, Ideals and Varieties β

β

Definition A.5 (Monomial/Power product) Let K be any field, the term x1 1 · · · xn n is called monomial or power product. All the exponents of the monomial are positive i = 1, . . . , n , and its total degree is defined as the summation integers, i.e. βi ∈ Z+ , of its exponents, i.e. βi . i

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227

Definition A.6 (Polynomial) Let K be any field and let K[x1 , . . . , xn ] be the ring of polynomials in n-indeterminates. A polynomial can be described as a sum of terms β β of the form: αx1 1 · · · xn n with α ∈ K. Definition A.7 (Ideals) Let K be a field and K[x1 , x2 , . . . , xn ] is a ring over the field of n-variable polynomials. Let G be a finite subset of the field, i.e. G = {g1 , g2 , . . . , gt }, then an ideal I can be generated by G as follows: I ={

n 

ui gi | ui ∈ R[x1 , x2 , . . . , xn ], gi ∈ K, ∀i = 1, . . . , t

i=1

Alternatively a subset I ⊂ K[x1 , x2 , . . . , xn ] is an ideal if: (i) 0 ∈ I (ii) If g1 , g2 ∈ I , then g1 + g2 ∈ I (iii) If g1 ∈I and u1 ∈ R[x1 , x2 , . . . , xn ], then g1 u1 ∈I For notational purposes, the ideal generated by a set of polynomials G = {g1 , g2 , . . . , gt } is denoted either as I (G) or as g1 , . . . , gt . Finally to get a better understanding of what an ideal is, as an analogy to linear algebra concepts, an Ideal is similar to a subspace. Definition A.8 (Variety) Let K be an algebraically closed field and g1 , . . . , gt a set of polynomials in K[x1 , x2 , . . . , xn ]. As variety or affine variety the set of all solutions to the system of polynomials is defined, i.e.: V(g1 , . . . , gt ) = {(α1 , . . . , αn ) ∈ K| gi (α1 , . . . , αn ) = 0 ∀i = 1, . . . , t} Example A.9 The variety V(x2 + y2 − 1, x − 3y2 ) ⊆ R2 is the intersection of a unit circle in the xy-plane and a parabola. Proposition A.10 If g1 , . . . , gt and h1 , . . . , hs are bases of the same ideal in K[x1 , x2 , . . . , xn ], i.e. I(G)=I(H), then their respective varieties are the same, i.e. V(G) = V(H). Example A.11 (Ideal, varieties and alternative bases) Let us examine the following system of polynomial equations: g1 = x12 − x2 + x1 g2 = x1 + x2 − 2

228

Appendix A: Background Material

The ideal generated by this set of polynomials is: I (G) = {u1 (x12 − x2 + x1 ) + u2 (x1 + x2 − 2), u1 , u2 ∈ C[x1 , x2 ]} It is the aim of the example to illustrate how two alternative bases can form the same ideal and as such let us claim that the following representation leads to the same ideal I :  = {h1 (x2 + 2x1 − 2) + h2 (2x2 x2 − 2x2 − x2 + x1 x2 + 2x1 ), h1 , h2 ∈ C[x1 , x2 ]} I (G) 1 1 2 1

 = {˜g1 , g˜ 2 } = {x2 + 2x1 − 2, 2x2 x2 − 2x2 − x2 + x1 x2 + 2x1 }. where G 1 1 2 1 For the two representations to be equivalent and generate the same ideal it suffices to prove that there exists a mathematical relationship between {g1 , g2 } and {g˜ 1 , g˜ 2 }. To this end one can resort in polynomial division but for the purposes of this example it can be easily verified that the following relationships hold: g˜ 1 = g1 + g2 g˜ 2 = 2x2 g1 − x1 g2

(A.1)

Based on Eq. (A.1) it can be seen that any polynomial can be expressed by either representation of the ideal and also that both bases result in the same variety as shown by Eq. (A.2).  = {(−1 − V(I ) = V(G) = V(G)

√ √ √ √ 3, 3 + 3), (−1 + 3, 3 − 3)}

(A.2)

The existence of ideals, for problems that accept analytical solution, is guaranteed by the Hilbert Basis theorem [2], which also guarantees the termination of algorithms that are used for the computation of Gröbner bases. The concept of ideals is central in Gröbner bases computations, one of the main component of the algorithms that have been presented in the thesis. This is due to the fact that a given ideal may accept different bases and a Gröbner basis is a useful type of basis when it comes to solving polynomial equations as it has a triangular form that enables sequential elimination.

A.2

Gröbner Bases

As introduced earlier, the basis that generates an ideal is not unique. Because the representation of the ideal I holds a significant role on the level of difficulty one faces when solving systems of polynomials, it is important to compute the bases that have favorable properties and a special type of such bases are the Gröbner bases. Before providing the reader with the formal definition of what a Gröbner basis is, it is important to establish some key concepts first.

Appendix A: Background Material

A.2.1

229

Basic Notions of Gröbner Bases

Definition A.12 (Monomial ordering) A monomial ordering or term order, is defined with regards to a set of power-products (Tn ) and imposes a total order ≺ on the set in compliance with the conditions below: 1. 1 ≺ xβ , ∀ xβ ∈ Tn 2. If xα ≺ xβ → xα xγ ≺ xβ xγ , ∀ xγ ∈ Tn There exists a number of monomial orderings with the most popular and efficient for equation solving being the lexicographic one [3]. In the lexicographic monomial ordering, a monomial xα is greater than xβ if α1 >β1 or if αi = βi , ∀i = 1, . . . , m and αm+1 > βm+1 . The fact that xα is greater than xβ lexicographically is denoted as follows: xα lex xβ . Another popular monomial ordering is the Graded reverse lexicographic, where xα grevlex xβ if the degree of xα is greater or equal than xβ and the last non-zero entry of the vector of integers α − β is negative. However, it should be noted that depending on the reason that Gröbner bases are employed different monomial orderings might be more efficient. Example A.13 (Ordering monomials) Let two monomials be γ = x3 y2 z8 and δ = xy9 z2 . If the variables are ordered as (x, y, z) then: γ lex δ

γ ≺grevlex δ

Definition A.14 Assume that is a defined monomial ordering and let g = αα1 xα + αα2 xα + · · · + ααm xα 1

2

m

with xα xα · · · xα 1

2

m

α with α ∈ Zm = 0. Then the following are defined: + and α 1 α (i) LT(g) : αα1 x is the leading term of g (ii) LC(g) : αα1 is the leading coefficient of g 1 (iii) LM(g) : xα is the leading monomial of g. 1

A.2.2

Gröbner Bases Computation

Definition A.15 (Gröbner basis [4]) A set of non-zero polynomials G = {g1 , . . . , gt } contained in an ideal I , is called a Gröbner basis for I if and only if for all g ∈ I such that g = 0, there exists i ∈ {1, . . . , t} such that lp(gi ) divides lp(g), where lp(· ) stands for the leading power-product of a polynomial function. Roughly speaking, within Gröbner bases theory a set of polynomial V is transformed into an other set of polynomials G which is equivalent to the former but has certain favorable computational properties. At the core of Gröbner bases theory the Buchberger algorithm is found [4] which is employed for the computation of

230

Appendix A: Background Material

the Gröbner basis of a specific set of polynomials. Buchberger introduced within the algorithm the concept of S-polynomials as well as provided a theorem for the proposed algorithm which for the sake of space are not discussed in the present thesis; however, the interested reader can refer to the book of Buchberger and Winkler [5] for further exposition on the subject. Except for Buchberger’s algorithm for the calculation of Gröbner bases, Faugère has proposed two variants of an alternative algorithm, namely F4 [6] and F5 [7]. The algorithm employs concepts from Linear Algebra where polynomials are represented in a matrix form allowing for successive truncated Gröbner bases to be created and parallelisation on the polynomial reductions. Finally, computer algorithms that allow Gröbner bases computations and thus solution of multivariable polynomial equations can be found in the commercial packages of Mathematica and Maple while freely available computer algebra systems include SageMath, SymPy and Singular. With respect to the computational complexity of computing Gröbner Bases, given a system of s equations and n variables with the highest degree polynomial being d O(n) the arithmetic complexity bound is 22 for the general case and s O(1) d O(n) for the case of a problem with finitely many solutions [8].

A.3

Cylindrical Algebraic Decomposition (CAD)

The idea of cylindrical algebraic decomposition was introduced in 1975 by Collins [9] in order to address the problem of quantifier elimination over real closed fields. In the present thesis, cylindrical algebraic decomposition is applied for the computation of non-convex partitions in the parametric space so it would be useful to proceed to the following definitions.

A.3.1

Semi-algebraic Sets and Quantified Formulas

Definition A.16 (Semi-Algebraic Sets [2]) Let K[x1 , x2 , . . . , xn ] denote the ring of polynomials in n-indeterminates with real coefficients. A subset S of Rn is called semi-algebraic if it can be constructed by finitely many applications of the union, intersection and complementation operations, starting from sets of the form: {x ∈ Rn | g(x) ≤ 0}, where g ∈ R [X] Alternatively it can be said that a semi-algebraic set of Rn is defined by Boolean operations (conjunctions, disjunctions and negations) of conditional expressions involving a finite number of polynomials. Definition A.17 (Standard atomic formula) A standard atomic formula is a formula that involves a functional relationship over a ring of polynomial in any of the following ways:

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231

g(x) = 0, g(x) = 0, g(x) < 0, g(x) > 0, g(x) ≤ 0, g(x) ≥ 0 A standard formula can be created as the combination of standard atomic formulae with variables’ quantifiers (∃, ∀) and logic propositional connectives, e.g. ∨, ∧, ¬. Definition A.18 (Quantifier free formula) A standard formula that does not involve any quantifiers on its arguments (variables) is defined as a quantifier free formula. Definition A.19 (Quantifier elimination problem) Let  be a standard formula over the field K as shown by Eq. (A.3)  = (Qi xi )(Qi+1 xi+1 ) . . . (Qn xn )ψ(x1 , . . . , xn )

(A.3)

where Qi can be either a universal or an existential quantifier and ψ is a quantifier free formula. Let also, x1 , . . . , xi be the free variables, i.e. free of quantifiers, and xi , . . . , xn be the bound variables. The problem of eliminating quantifiers involves the computation of a quantifier free formula in the free variables which is equivalent to . Example A.20 (Quantified formulas and variables) From the following formulas in Eqs. (A.4), (A.5): (∀x)[(x ≥ 0) → (x2 + γ x − β ≥ 0)]

(A.4)

(∀x)(∃y)[x < y ]

(A.5)

3

the first one involves one bound variable, x, and two free variables γ , β while for the second one its two variables are quantified and thus bound.

A.3.2

Cylindrical Algebraic Decomposition

The first one to prove that there exists an algorithmic solution to the problem of quantifier elimination was Alfred Tarski in 1951; however despite of the methods theoretical value, its complexity with respect to the number of variables was nonelementary [10]. [9] proposed the use of cylindrical algebraic decomposition to eliminate the quantifiers by evaluating sequentially the polynomial inequalities. It can be understood from the above definitions and propositions, that one can perform CAD calculations to solve system of polynomial inequalities.

232

A.3.3

Appendix A: Background Material

Basic Notions of Cylindrical Algebraic Decomposition

In order to introduce the notion of CAD some formal definitions are required first. Definition A.21 (Decomposition) Let X ⊆ Kn be a set. A decomposition of X is a finite compilation of a number of disjoint regions whose union form X. Definition A.22 (Algebraic Decomposition) A decomposition is algebraic if any of the resulting partitions is a semi-algebraic set, meaning a region that is defined by a collection of polynomial equations and inequalities. Definition A.23 (Cell) A set of polynomials in n-indeterminates (x1 , x2 , . . . , xn ) induces an algebraic decomposition of Kn into a number of regions. The regions over which a polynomial has constant sign are called cells, their respective boundaries are the zeros of the polynomials and constitutes a connected subset of Kn . Definition A.24 (Cylinder) Let R be a connected subset of Kn . The set Z (R) = R × Kn = {(γ , x)|γ ∈ R, x ∈ Kn } is called a cylinder over R. Definition A.25 (Section, Sector) Let g = {g1 , . . . , gk } be a vector of continuous real valued functions over a region R ⊂ Kn . A g– section over Z (R) is called the set that is defined by Eq. (A.6). {(α, f(α))| α ∈ R}

(A.6)

A (gi − gi+1 )– sector of Z (R), under the condition that gi < gi+1 , is the set defined by Eq. (A.7). {(α, β)| α ∈ R, gi (α) < β < gj (α)}

(A.7)

Definition A.26 (Stack) Let R be a connected subset of Kn . A stack over R is a partition which comprises of gi – sections and (gi , gi+1 )– sectors with, gi < gi+1 and g0 = −∞ and gk = +∞ . Notice that by definition (A.26) based on the strict inequalities imposed on the graphs of the real valued functions, it is not permitted to have intersections among each other over R. In Fig. A.2, a graphical illustration of a stack over an interval (α, β) is presented so as to elucidate on the latest definition. Definition A.27 (Cylindrical Decomposition [11]) A decomposition D over Rn is cylindrical if : • For n = 1, i.e. the region R is one dimensional, D constitutes a stack over R0 (the singleton set), or • For n > 1, there is a cylindrical decomposition D over Rn−1 such that for each region R of D , some subset of D is a stack over R

Appendix A: Background Material

233

Fig. A.2 Stack over interval. The green lines represent the graph of the different functions that form three distinct non-intersecting sections. The cylinder over the interval (α, β) is partitioned into 4 two-dimensional cells, the sectors of the functions, and 3 one-dimensional cells, the sections of the functions

g1

(g1- section)

(g1-g2 sector) g2

(g2- section)

(g2-g3 sector)

α

β

g3 (g3- section)

Definition (A.27) implies that the cells of the decomposition D are arranged in such a manner that their projections, under a given variable ordering, are either equal or disjoint. Definition A.28 (Cylindrical Algebraic Decomposition [12]) A cylindrical algebraic decomposition D is a decomposition that is both algebraic and cylindrical according to the definition of algebraicity and cylindricity introduced above.

A.3.4

Computation of the CAD

One by computing the CAD over a set of polynomial inequalities decomposes the related space into a number of cells. The cells are regions where each polynomial has an invariant sign. In that way one can determine whether the condition imposed by the inequalities holds true by sampling a point in each cell. The original algorithm of Collins [9] involves three main phases, the projection, base and the lifting phase. Projection Phase Initially, during the projection phase the polynomials involved in the system under study are recursively processed until univariate ones exist and a decomposition of Rn is achieved based on these. In more details, given a variable ordering, the “critical points” of the involved multivariate polynomials are identified. The critical points can either be: (i) intersections among polynomials or (ii) self-crossings and turning points (change of curvature) of the polynomials. Most algorithms, employ resultant and discriminant calculations where the projection variable (xn ) is eliminated and the roots of resulting polynomial in the remaining variables, form the coordinates

234

Appendix A: Background Material

in the projected space Rn−1 . There exist different strategies for the projection phase [9, 12, 13] while also the use of equational constraints during the projection phase remains to the date an active and promising direction [14, 15]. Base Phase During this phase, the computations that take place are in R1 , i.e. for the univariate polynomials their real roots are computed and the boundaries of the aforementioned cells of the decomposition are defined. Lifting Phase Finally, in the lifting phase starting from the decomposition on R1 using the projection polynomials a decomposition on R2 is performed and so on until the CAD of Rn is completed. In Fig. A.3, a graphical illustration of the main CAD steps is given. During the lifting phase, within each cell the algorithm employs a “sample point” to evaluate the signs of the n-variate polynomials so as to assure the sign-invariance property of CAD [12]. Even though the algorithmic complexity of Collin’s algorithm in the worst case, for an input of t polynomials in n indeterminates with maximum 2n+8 nx +6 degree d and coefficient length l, is O((2d)2 t 2 l3 ) there has been research effort dedicated towards the alleviation of that complexity. In general, the complexity of most CAD algorithms is doubly-exponential on the number of variables. Strzebonski [16] proposed the use of validated numerics during the lifting phase instead of algebraic numbers and considerable computational savings were reported. Finally, based on the theory of triangular decompositions and regular chains a new algorithm for CAD computations and quantifier elimination was presented by Maza [17]. Of course, there exist many more details in the computational steps of the CAD algorithms but since in the present thesis the main focus was to employ CAD as a tool and not to refine or develop new algorithms for its computation, these details lie outside of the scope of this thesis. For a more detailed exposition on the topic of solving systems of polynomial inequalities via CAD the interested reader is referred to Jirstrand [18].

Fig. A.3 Main algorithmic steps of the CAD algorithm

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It is interesting to note that even though the aforementioned algorithms were initially developed for polynomial systems, recent developments allow for their employment on systems of non-polynomial equations [19, 20, 21]. The key idea is to substitute the original non-polynomial terms with a new set of variables and equations that ensure the agreement of the computed solution.

References 1. Cox D, Little J, O’shea D (2007) Ideals, varieties, and algorithms, vol 3. Springer, Berlin 2. Bochnak J, Coste M, Roy M-F (2013) Real algebraic geometry, vol 36. Springer Science & Business Media, Berlin 3. Lazard D (1983) Gröbner bases, Gaussian elimination and resolution of systems of algebraic equations. Springer, Berlin, pp 146–156. https://doi.org/10.1007/3-540-12868-9_99 4. Buchberger B (2006) Bruno Buchberger’s PhD thesis 1965: an algorithm for finding the basis elements of the residue class ring of a zero dimensional polynomial ideal. J Symb Comput 41(3–4):475–511 5. Buchberger B, Winkler F (1998) Gröbner bases and applications, vol 251. Cambridge University Press, Cambridge 6. Faugere JC (1999) A new efficient algorithm for computing Gröbner Bases (F4). J Pure Appl Algebr 139(1):61–88 7. Faugere JC (2002) A new efficient algorithm for computing Gröbner Bases without reduction to zero (f5). In: International symposium on symbolic and algebraic computation ’02. ACM Press, Philippines 8. Bardet M, Faugere J-C, Salvy B (2004) On the complexity of Gröbner basis computation of semi-regular overdetermined algebraic equations. In: Proceedings of the international conference on polynomial system solving, pp 71–74 9. Collins GE (1975) Quantifier elimination for real closed fields by cylindrical algebraic decomposition. In: Automata theory and formal languages 2nd GI conference. Springer, pp 134–183 10. Tarski A (1998) A decision method for elementary algebra and geometry. In: Quantifier elimination and cylindrical algebraic decomposition. Springer, Berlin, pp 24–84 11. Arnon DS, Collins GE, McCallum S (1984) Cylindrical algebraic decomposition i: The basic algorithm. SIAM J Comput 13(4):865–877 12. Collins GE, Hong H (1991) Partial cylindrical algebraic decomposition for quantifier elimination. J Symb Comput 12(3):299–328 13. Brown CW (2001) Improved projection for cylindrical algebraic decomposition. J Symb Comput 32(5):447–465 14. England M, Davenport JH (2016) The complexity of cylindrical algebraic decomposition with respect to polynomial degree. In: International workshop on computer algebra in scientific computing. Springer, pp 172–192 15. Davenport JH, England M (2016) Need polynomial systems be doublyexponential? In: International congress on mathematical software. Springer, pp 157–164 16. A. Strzebonski. Cylindrical algebraic decomposition using validated numerics. J Symb Comput, 41(9):1021–1038, 2006. 17. Chen C, Maza MM (2016) Quantifier elimination by cylindrical algebraic decomposition based on regular chains. J Symb Comput 75:74–93 18. Jirstrand M (1995) Cylindrical algebraic decomposition-an introduction. Linköping University, Linköping

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19. Jeffrey DJ, Rich AD (1994) The evaluation of trigonometric integrals avoiding spurious discontinuities. ACM Trans Math Softw 20(1):124–135 20. Franze G, Famularo D, Casavola A (2010) A receding horizon control strategy for constrained non-polynomial systems via semi-algebraic methods. In: IFAC proceedings, vol 43. Elsevier, pp 741–746 21. Papachristodoulou A, Prajna S (2005) Analysis of non-polynomial systems using the sum of squares decomposition. Springer, Berlin, pp 580–580

Appendix B

PERSEUS: A Prototype Software Implementation

This chapters provides a discussion on the software developments conducted within the focus of this thesis and the prototype implementation of Perseus, a parametric platform written in Python. Firstly, a short overview of the existing computer algebra codes and there abilities in terms Gröbner Bases and CAD computations is presented. Then, the software structure of Perseus is presented and finally ongoing and future developments are briefly discussed.

B.1

P ERSEUS: Parametric Algorithms for Explicit Optimisation of Resilient Systems Engineering Under Stochasticities

As indicated by the literature review in Part I of the thesis, the software implementation of multi-parametric programming algorithms has received increasing attention the last few years. All of the research work done employs numerical techniques and addresses a wide variety of class of problem that are however restricted to parametric variations on the RHS of the constraints and in some cases also the OFC. A considerable amount of the present thesis has been dedicated in the development of new multi-parametric programming algorithms for the most general case: multiparametric programs under global uncertainty. A considerable departure from the conventional numeric solution of mp-Ps was made and that led to the need of implementation of the new algorithms. In order to decide upon the programming language that Perseus would be coded a review of the candidate languages was made and the key factors that were taken into consideration were: (i) software portability, (ii) ease of maintenance, (iii) availability of algebraic geometry libraries and (iv) to a big extend free distribution. Next an overview of the available libraries for Gröbner Bases and CAD computations is given.

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B.2 B.2.1

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Computer Algebra Implementations GiNaC/GiNaCRA

GiNaC is entirely written in C++ and initially was intended for quantum field calculations. Bauer et al. [1] argued that the limited use of computer algebra systems is partially due to obstacles in the combination of symbolic/ numerical and graphical programs; to this end they claimed that the object oriented and modularised programming paradigm that C++ offers can aid towards the abatement of the aforementioned. GiNaCRA is an open source freely distributed C++ library, which depends on the non-standard library GiNaC [2]. GiNaCRA offers data types and subroutines that are vital for a CAD implementation such as real algebraic number data representation, computation of the roots of univariate polynomials. Currently it includes an implementation for the computation of univariate CADs but has not yet expanded to the multivariate case [3].

B.2.2

M AXIMA

Maxima is probably one of the most historic computer algebra systems, given that its ancestor MACSYMA was among the first ones to be developed. MACSYMA was a product of the MIT’s artificial intelligent group in the 1960s [4]. Maxima is written in Lisp and is freely distributed and open source under GNU Public License. With respect to Grobner Bases computations Maxima offers an implementation of the F4 algorithm while currently it does not have any capabilities of performing CAD computations.

B.2.3

Macaulay2

Macaulay2 is a CAS mostly used for algebraic geometry and commutative algebra that is free and open source under GNU General Public License. It uses its own high level language that allows for definition of classes, functions and control structures. One can employ Macaulay2 for the computation of Grobner Bases and other polynomial manipulations, however in a computational study that was performed for computation of Grobner Bases Singular indicated better computational behavior [5].

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B.2.4

239

M ATHEMAGIX

Mathemagix is free computer algebra system which started to be developed in the late nineties by Joris van der Hoeven and his laboratory [6]. Mathemagix has its own compiled programming language, which is strongly typed and allows for signatures, operator overloading and classes. Mathemagix has the ability to be interfaced easily from a C++ program and conversely allows for use of several C++ libraries which can make its run-time performance quite efficient. An important feature of Mathemagix is its interface with FGb, which is a fast C-library for the computation of Grobner Bases using the F-series of algorithms proposed by Faugere [7]. Even though Mathemagix is currently under active development its ability to solve symbolically polynomial systems is limited and there is no library for CAD computations.

B.2.5

Mathematica

Mathematica is a commercial computer algebra software that offers a wide range of computing capabilities ranging from Astronomy calculations to Machine Learning. Mathematica has been actively developed over the last three decades and has its own implementation for the calculation of Gröbner Bases as well as for CAD calculations. Strzebo´nski [8, 9] has developed an efficient version of the Partial CAD algorithm by Collins and Hong [10]. Currently the use adjacency criteria on top of validated numerics is investigated for the CAD calculations in order to reduce the computational effort required [11].

B.2.6

Maple

Maple, similar to Mathematica, is a commercial computer algebra software that enjoys many features. For the solution of systems of polynomial equations Maple has an implementation of Faugere’s FGb program [7]. For the case of CAD computation two are the main packages that are available in Maple, SyNRAC and Regular Chains. SyNRAC [12] is a symbolic/numeric toolbox in Maple, primarily written in Maple and C language. This toolbox was developed by Fujitsu Laboratories Ltd. and a “Maple workbook” including its library can be downloaded from their website.1 SyNRAC employs interval arithmetics in some of the phases of the CAD computations alleviating in that way some of the burden that stems from symbolic computations. More recently, Chen and Maza [13] presented an incremental approach to CAD computations based on regular chains. Briefly, a regular chain is a triangular set of polynomials in a closed field that is either non-empty or its iterated resultant 1 http://www.fujitsu.com/jp/group/labs/en/resources/tech/announced-tools/synrac/.

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is non-empty. The Regular Chains library is available at the group’s website and is written in Maple language.2

B.2.7

Q EPCAD B

Qepcad B is the ancestor of Qepcad, a program that was written by Hoon Hong for the partial cylindrical algebraic decomposition algorithm presented in Collins and Hong [10]. Qepcad B is mostly written in C (with some elements coded in C++) and it uses the Saclib library for computer algebra functions. The McCallum projection variants are employed for the projection phase of the CAD while the lifting phase involves the use of validated numerics [14, 15]. The current status of Qepcad B remains open but the latest release (Qepcad B v1.69) was issued in 2012 [16]. Another potential drawback that Qepcad B bares is that it is only available for Sun Solaris and Linux OS which limits its portability as a component of a bigger code.

B.2.8

Reduce/Redlog

Redlog3 is part of the open-source CAS Reduce which is freely available under FreeBSD License. For the solution of systems of polynomial equations Reduce offers a refined implementation of the Buchburger algorithm as described in Faugere et al. [17]. Redlog offers an implementation for quantifier elimination using virtual substitution methods as well as a CAD computation routine. However, the solution of polynomial inequalities with parametric coefficients is not supported and only numerical coefficients are allowed [18].

B.2.9

S INGULAR

Singular is a CAS for polynomial computations that is free and open source under GNU General Public License [19] has its own C-like language while the vast majority of its functions are written is C/C++. Singular offers one of the fastest implementations for the computation of Grobner Bases with the options to employ either the classic algorithm of Buchberger or by calling the FGLM algorithm [17]. It is interesting to note that the FGLM algorithm can be faster compared to the one proposed by Buchberger since it computes a lexicographical Grobner basis from a reduced Grobner basis with a different ordering.

2 http://www.regularchains.org/index.html. 3 http://www.redlog.eu/.

Appendix B: PERSEUS: A Prototype Software Implementation

B.2.10

241

SageMath

SageMath is another open-source mathematical software project that involves computer algebra capabilities and is freely available under GPLv3.4 Its purpose is to “create a viable free open source alternative to Magma, Maple, Mathematica and Matlab”. As mentioned, SageMath can be more precisely characterised as a mathematical software project rather than a standalone system as it serves as a coherent mechanism among Python and its libraries with, GAP, FLINT, R, QEPCADB and many others [20]. SageMath boasts of a rather active developing community and new versions are released every 6 months. Considering the above Python was chosen as the programming language that Perseus was written. The key reasons behind this decisions were: (i) there has been an increasingly growing interest from the Python community to enhance the capabilities of the SymPy module of Python that forms its platform for symbolic computations, (ii) SageMath is written mostly in Python and provides a free competitive alternative to the commercial platforms like Mathematica and Maple. Finally, by developing Perseus in Python a better integration with SageMath can be achieved so as to directly take advantage of the latest additions to SageMath one of which can its of CAD implementation.

B.3

Software Structure

In this section the structure of Perseus is described along with its main functions, their inputs and outputs. Accounting for ease of code maintenance and versatility, Perseus was coded in a modular fashion while at its current state is mostly independent of commercial software. In Fig. B.1 the main data flows and computational routines can be envisaged.

B.3.1

Modelling of the mp-P Problem

Software implementations of multi-parametric programming algorithms require the user to extract the problem’s matrices and vectors from the abstract model. However this procedure can be cumbersome for problems with a high number of constraints and optimisation variables. To overcome this issue, in Perseus the user is asked for three inputs: (i) a GAMS file including the model of optimisation problem under study, (ii) the “topology” of the uncertain parameters in the model and (iii) setting the option for “Parametric Envelopes” True or False. Perseus then automatically processes the model and symbolically reformulates the given problem in the internal 4 http://www.sagemath.org/.

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Fig. B.1 PERSEUS structure

required structure. Note, that it suffices for the user to have a version of GAMS with non-commercial solvers.

B.3.2

Solution of the mp-P Problem

Once the original model in GAMS is processed, the algorithms presented in this thesis are followed [21, 22]. The KKT system of the problem is computed and then solved using a routine that involves Gröbner Bases calculations.

Appendix B: PERSEUS: A Prototype Software Implementation

243

Then the output is preprocessed to remove (i) lower dimensional solutions, i.e. solutions that have linear dependence and (ii) solutions that involve purely arithmetic values of variables that qualify them as infeasible, e.g. negative values of Lagrange multipliers or negative values of positive variables. Next, a routine written in Wolfram Mathematica language is used for the computation of CRs and finally the overlapping CRs are identified and processed following the comparison procedure that was described earlier in the thesis. At this point it should be noted that this part of Perseus requires the presence of a version of Mathematica installed in the user’s computing environment, which is the only commercial element in Perseus for the time being. The main bottleneck is the implementation of the a CAD algorithm for the computation of the CRs and the comparison procedure. Implementing a CAD routine can be very time consuming and as such was deemed as out of the scope for this dissertation. Instead, based on the overview given in Sect. B.2 an alternative could have been the use of QEPCAD B but this was rejected because: (i) the development of QEPCAD B has been discontinued with the latest release being since 2012 and (ii) QEPCAD B would limit the platform portability of Perseus as its main library SACLIB runs only in Sun Solaris and Linux workstations.5

B.3.3

The Parametric Envelopes Option

As mentioned in Sect. B.3.1 the third input required by the user is to set the computation of “parametric envelopes” to True or False. In general for the case of non-convex mp-Ps, which is the main topic of the thesis it can happen that in the same parametric space two or more explicit solutions can result in the same objective value. In instances like these, it should be decided whether the decision maker needs to know what alternatives exist at the same objective value or if this is something that is not of high importance. Setting the option to True will result saving all the overlapping solutions with the same objective value while setting the option to False will invoke a routine of Perseus that cuts CRs with the aim of reducing the computational burden, e.g. since CRs in Perseus are processed in a sequential order fashion “cutting off” CRs that are supposed to be examined later on, according to the comparison procedure that will likely eliminate some comparisons and thus reduce the computational time.

B.3.4

The Graphical User Interface (GUI)

As shown in Fig. B.2a preliminary GUI for Perseus was built in order to allow seemless communication of the background functions and for the user to specify the

5 https://www.usna.edu/CS/qepcadweb/B/QEPCAD.html.

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Fig. B.2 PERSEUS’ graphical user interface

file of the optimisation problem under study by means of GAMS file (.gms). Finally, the user needs to specify the uncertainty topology of the mp-P problem.

B.3.5

Ongoing Developments and Future Work

Ongoing developments can be distinguished with regards to two aspects: (i) extension to numerical solution of mp-Ps and (ii) testing more efficient alternative for the solution of the KKT system and the definition of CRs/comparison procedure. • Extension to numerical solution of mp-Ps: Perseus is entirely written in Python and that allows for harnessing all the available libraries of conducting linear algebra computations using graphical processing units (GPUs). The use of parallel computing has been recently investigated by the mp-P community [23] however the use of GPU computing remains an open challenge and can potentially lead to significant speed-up in computations. Ongoing and future algorithmic developments in the field of numerical solution of mp-Ps will be also implemented in Perseus. • Computer algebra developments: Implementing an in-house version of the CAD algorithm is an obvious future direction that can enhance the performance of Perseus, however the computational complexity of CAD operations is by default doubly-exponential with respect to the number of variables which in the case of the presented mp-P algorithm is the number of uncertain parameters. Satisfiabil-

Appendix B: PERSEUS: A Prototype Software Implementation

245

ity Modulo Theories have recently attracted attention from the computer science community and they can serve as a potentially competitive alternative to CAD computations for the symbolic solution of polynomial inequalities [24]. Finally, as of now Perseus uses an implementation of the F4 algorithm of Faugere [25]; implementing the F5 algorithm is another potential extension that can enhance the computational complexity of the mp-P algorithms [26].

References 1. Bauer C, Frink A, Kreckel R (2002) Introduction to the GiNaC framework for symbolic computation within the C++ programming language. J Symb Comput 33(1):1–12 2. Loup U, Abraham E (2011) GiNaCRA: A C++ library for real algebraic computations. In: NASA formal methods symposium. Springer, pp 512–517 3. Redies J (2012) An extension of the GiNaCRA library for the cylindrical algebraic decomposition. Master’s thesis, Rheinisch-Westfalische Technische Hochschule Aachen 4. Moses J (2012) Macsyma: a personal history. J Symb Comput 47(2):123–130 5. Levandovskyy V, Morales JM (2008) Computational d-module theory with singular, comparison with other systems and two new algorithms. In: 21st international symposium on symbolic and algebraic computation. ACM, pp 173–180 6. Van Der Hoeven J (2014) Overview of the mathemagix type system. In: Computer mathematics. Springer, pp 397–425 7. Faugere JC (2010) FGb: a library for computing Gröbner Bases. In: International conference in mathematical software, Lecture notes in computer science, vol 6327. Springer, Berlin, pp 84–87 8. Strzebonski A (2000) Solving systems of strict polynomial inequalities. J Symb Comput, 29(3):471–480 9. Strzebonski A (2006) Cylindrical algebraic decomposition using validated numerics. J Symb Comput 41(9):1021–1038 10. Collins GE, Hong H (1991) Partial cylindrical algebraic decomposition for quantifier elimination. J Symb Comput 12(3):299–328 11. Strzebonski A (2017) CAD adjacency computation using validated numerics. In: International symposium on symbolic and algebraic computation, ISSAC’17. ACM, New York, pp 413–420. ISBN 978-1-4503-5064-8. 12. Yanami H, Anai H (2007) SyNRAC: a MAPLE toolbox for solving real algebraic constraints. ACM Commun Comput Algebra 41(3):112–113 13. Chen C, Maza MM (2016) Quantifier elimination by cylindrical algebraic decomposition based on regular chains. J Symb Comput 75:74–93 14. McCallum S (1988) An improved projection operation for cylindrical algebraic decomposition of three-dimensional space. J Symb Comput 5(1–2):141–161 15. Brown CW (2001) Improved projection for cylindrical algebraic decomposition. J Symb Comput 32(5):447–465 16. Brown CW (2003) QEPCAD B: a program for computing with semi-algebraic sets using CADs. ACM SIGSAM Bull 37(4):97–108 17. Faugere J-C, Gianni P, Lazard D, Mora T (1993) Efficient computation of zero-dimensional gröbner bases by change of ordering. J Symb Comput 16(4):329–344 18. Sturm T (2007) Redlog online resources for applied quantifier elimination. Acta Academiae Aboensis, Ser. B 67(2):177–191

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19. Decker W, Greuel G-M, Pfister G, Schönemann H (2018) Singular 4.1.1 -a computer algebra system for polynomial computations 20. Developers T (2016) Sagemath 21. Charitopoulos VM, Papageorgiou LG, Dua V (2017) Multi-parametric linear programming under global uncertainty. AIChE J 63(9):3871–3895 22. Charitopoulos VM, Papageorgiou LG, Dua V (2017) Nonlinear model-based process operation under uncertainty using exact parametric programming. Engineering 3(2):202–213 23. Oberdieck R, Pistikopoulos EN (2016) Parallel computing in multi-parametric programming. Comput Aid Chem Eng Elsevier 38:169–174 24. Ábrahám E, Abbott J, Becker B, Bigatti AM, Brain M, Buchberger B, Cimatti A, Davenport JH, England M, Fontaine P et al (2017) SC2: satisfiability checking meets symbolic computation. In: 10th international conference on intelligent computer mathematics. Springer, pp 28–43 25. Faugere JC (1999) A new efficient algorithm for computing Gröbner bases (F4). J Pure Appl Algebra 139(1):61–88 26. Faugere JC (2002) A new efficient algorithm for computing Gröbner bases without reduction to zero (f5). In: International symposium on symbolic and algebraic computation ’02. ACM Press

Appendix C

Multi-parametric Linear and Mixed Integer Linear Programming Under Global Uncertainty: Further Results

(See Fig. C.1)

Fig. C.1 Optimal partition of the parametric space for the mp-MILP under global uncertainty example

© Springer Nature Switzerland AG 2020 V. M. Charitopoulos, Uncertainty-aware Integration of Control with Process Operations and Multi-parametric Programming Under Global Uncertainty, Springer Theses, https://doi.org/10.1007/978-3-030-38137-0

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Appendix C: Multi-parametric Linear and Mixed Integer Linear …

Algorithm C.1 Algorithm for mp-MILPs under global uncertainty. Step 1: Input problem (3.25) in a computer algebra system and reformulate it into problem (3.26) by treating the binary variables as parameters bounded between 0-1. Step 2: Formulate the first order KKT conditions of problem (3.26) and solve the resulting square system of equations (3.27) so as to compute the optimisation variables x(y, θ) and the Lagrange multipliers λ(y, θ) as explicit functions of the uncertain parameters, i.e. y, θ. If the solution set of (3.27) is empty then problem (3.25) is infeasible; otherwise go to step 3. Step 3: In order to identify which of these solutions are infeasible evaluate inequalities (3.2)-(3.3) with y and θ treated again as parameters. The candidate solutions that do not satisfy inequalities (3.2)-(3.3) are removed from the list. Step 4: For the remaining candidate solutions fix y to all possible integer combinations and thus compute the optimisation variables and the Lagrange multipliers as explicit functions of the uncertain parameters, i.e. x(θ) and λ(θ). Step 5: Substitute the explicit expressions computed in step 4 into the inequality constraints (3.2)(3.3) and evaluate with a constraint qualification. Compute for each integer candidate solution the parametric region where these inequalities are satisfied. If there exists such a region then add this region to a list as it represents the CR of the corresponding integer candidate solution; otherwise the region is empty and thus the candidate integer solution is infeasible and removed from further consideration. Step 6: For all the CRs computed in step 5, compute whether there exists an overlap where more than one explicit solutions are feasible. If there is no overlap go to step 8; otherwise, for each pair of CRs compute the region of parametric space where the two CRs intersect (CRINT ) and perform a comparison procedure so as to determine the optimal solution in CRINT . Step 7 (Comparison Procedure): Follow the comparison procedure as illustrated in the section “Cylindrical algebraic decomposition and comparison of overlapping CRs” and compute the final non-overlapping CRs. After this step, within each CR only one explicit solution is stored. Step 8: Collect the final CRs and the corresponding explicit solutions and output the overall multiparametric solution.

Algebraic model of multi-objective three stage scheduling problem under global uncertainty The complete scalar model that was employed for the three stage scheduling case study is given by Eqs. (C.1), (C.13) (Tables C.1, C.2, C.3, C.4, C.5, C.6, C.7). min θ4 Cs4 ,j3 + (1 − θ4 )Cs4 ,j1

(C.1)

Cs1 ,j1 − Cs1 ,j2 + θ1 ys1 ,a + 14ys1 ,b + 9ys1 ,c + 11ys1 ,d + θ3 ≤ 0 Cs1 ,j2 − Cs1 ,j3 + 11ys1 ,a + 17ys1 ,b + 5ys1 ,c + 16ys1 ,d ≤ 0 Cs2 ,j1 − Cs2 ,j2 + θ1 ys2 ,a + 14ys2 ,b + 9ys2 ,c + 11ys2 ,d ≤ 0

(C.2) (C.3) (C.4)

Cs2 ,j2 − Cs2 ,j3 + 11ys2 ,a + 17ys2 ,b + 5ys2 ,c + 16ys2 ,d ≤ 0 Cs3 ,j1 − Cs3 ,j2 + θ1 ys3 ,a + 14ys3 ,b + 9ys3 ,c + 11ys3 ,d ≤ 0

(C.5) (C.6)

Cs3 ,j2 − Cs3 ,j3 + 11ys3 ,a + 17ys3 ,b + 5ys3 ,c + 16ys3 ,d ≤ 0 Cs4 ,j1 − Cs4 ,j2 + θ1 ys4 ,a + 14ys4 ,b + 9ys4 ,c + 11ys4 ,d ≤ 0

(C.7) (C.8)

Cs4 ,j2 − θ2 Cs4 ,j3 + 11ys4 ,a + 17ys4 ,b + 5ys4 ,c + 16ys4 ,d ≤ 0

(C.9)

Subject to:

Appendix C: Multi-parametric Linear and Mixed Integer Linear …

249

Table C.1 Critical regions of the mp-MILP under global uncertainty example (i) CRs Mathematical expression ⎧

 ⎧ θ1 = −1 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ −3 ≤ θ2 < −1 ⎨ ⎪ ⎪ ⎪ ⎪ −1 ≤ θ3 ≤ 5  ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ θ = −3 ⎪ ⎪ ⎪ ⎪ ⎩ 2 ⎪ ⎪ −1 < θ1 < 0 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎧⎧ ⎨ ⎪ ⎪ ⎨−5 ≤ θ1 < −1 ⎪ CR1 := ⎪ ⎪ −3 ≤ θ2 < 1 ⎪ ⎪ ⎪⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ 1 < θ3 ⎪ ⎪ ⎪ ⎪ ⎨ θ1 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪θ3 ≤ 5 ⎪⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨−3 < θ2 < −1 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ −1 < θ1 < 0 ⎪ ⎪ θ2 ⎪ ⎪ ⎩ ⎩⎪ ⎩ < θ3 θ1 θ2 +θ1 +1  ⎧ ⎧ ⎧ θ1 ≥ −1 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ θ3 = −1 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨⎨θ1 > θ1 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ 3 ⎪ ⎪ −1 ≤ θ ⎪ ⎪ 2 ⎪⎩−1 < θ ≤ − 1 ⎪ ⎪ ⎪ ⎪  3 ⎪ ⎨ ⎪ 5 ⎪ ⎪ ⎪ ⎪ θ1 < 0 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ θ2 ≤ 5 ⎪ ⎪ ⎪ ⎪ θ ≥ −5 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ 1 ⎪ ⎪ ⎪ ⎩ ⎪ ⎪ ⎪ ⎪ − 1 < θ3 ≤ 5 ⎪ ⎪ ⎪ ⎪ 5 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ θ ≥ θ1 ⎪ ⎪ CR2 := ⎩θ2 > −1 1 3 ⎪ −5 ≤ θ3 < −1 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎧ ⎪ θ ≥ −1 ⎪ ⎪ ⎪ ⎪ ⎪ 1 ⎪ ⎪ θ3 = −1 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨⎨θ1 > θ1 ⎪ θ1 ≤ 0 ⎪ 3 ⎪ −1 ≤ θ2 ⎩ ⎪ 1 ⎪ θ < 0 ⎪ ⎪ ⎪ 2 ⎪ −1 < θ3 ≤ − 5 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ θ1 ≥ −5 ⎪ ⎪ ⎩ ⎩ − 1 < θ3 ≤ 5 5

⎧⎧ ⎪ ⎪ ⎪−5 ≤ θ3 < −1 ⎪ ⎪⎪ ⎨θ2 ≥ −5 ⎪ ⎪ ⎪ ⎪ + θ1 + θ2 ≤ 0 ⎪ ⎪⎪ ⎪3 √ ⎪ ⎪ ⎪⎪ ⎩ ( 129−11) < θ ≤ 2 ⎪ ⎪ 1 ⎨⎧ 2 θ < −1 CR3 := ⎪ ⎪ 3 ⎪ ⎪ ⎪ ⎪ θ ≥ −5 ⎪ ⎪ ⎪ ⎨ 2 ⎪ ⎪ 3 + θ1 + θ2√≤ 0 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪⎪ ⎪0 < θ < ( 129−11) ⎪ ⎪ ⎪ 1 2 ⎪ ⎪ ⎩⎩ θ12 + θ1 + θ3 + 3 > 2θ1 θ3 ⎧ ⎧ −3 ≤ θ2 < −1 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ −1 < θ3 ≤ 1 ⎪ ⎪ θ1 ⎪ ⎪ −5 ≤ θ1 < −1 ⎪ ⎪ −1 ≤ θ2 ≤ 5 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ 3 ⎪ ⎩ 1 < θ3 ≤ ⎪ ⎪ 2θ1 −1 ⎪ ⎧ ⎪ ⎪ ⎪ −1 < θ < 0 ⎪ 1 ⎨ ⎪⎪ ⎪ ⎪ ⎪ ⎪ −5 ≤ θ2 ≤ −3 ⎪ ⎪ ⎪ 3 ⎩ ⎪ ⎪ 2θ −1 ≤ θ3 < −1 ⎪ 1 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨⎧ ⎪0 < θ1 ≤ 51 ⎨ CR4 := ⎪ −3 − θ1 < θ2 < −3 ⎪ ⎪ ⎪ ⎩ 3 ⎪⎪ ⎪ ⎪ 2θ1 −1 ≤ θ3 < 1 ⎪ ⎪ ⎪ ⎪ ⎪ ⎧ ⎪ ⎪ ⎪ ⎪⎨ 51 < θ1 ≤ 2 ⎪ ⎪ ⎪ ⎪ ⎪ ⎩−3 − θ1 < θ2 < −3 ⎪ ⎪ ⎪ ⎪ 5 ≤ θ3 < −1 ⎪ ⎪ ⎪ ⎪ ⎪⎧ ⎪ ⎪ ⎨2 < θ1 ≤ 5 ⎪ ⎪ ⎪ ⎪ ⎪ ⎩⎩−5 ≤ θ2 < −3 −5 ≤ θ3 < −1

Cs1 ,j1 − Cs4 ,j1 + 6ys4 ,a + 9ys4 ,b + 12ys4 ,c + 14ys4 ,d ≤ 0

(C.10)

Cs1 ,j2 − Cs4 ,j2 + θ1 ys4 ,a + 14ys4 ,b + 9ys4 ,c + 11ys4 ,d ≤ 0 Cs1 ,j3 − θ2 Cs4 ,j3 + 11ys4 ,a + 17ys4 ,b + 5ys4 ,c + 16ys4 ,d ≤ 0

(C.11) (C.12)

−Cs,j ≤ 0 ∀s, j

(C.13)

250

Appendix C: Multi-parametric Linear and Mixed Integer Linear …

Table C.2 Critical regions of the mp-MILP under global uncertainty example (ii) CRs

Mathematical expression

⎧⎧θ > −5 ⎪ 3 ⎪ ⎪ ⎨1 ⎪⎪ ⎪ ⎪ ⎪ 3 ≤ θ1 < 1 ⎪ ⎪⎪ ⎪ ⎪θ2 ≥ −2 ⎪ ⎩ ⎪ ⎪ ⎪ ⎪ 5θ1 θ2 + θ1 + θ2 + 5 ≤ 0 ⎪ ⎪ ⎪ ⎧θ = 0 ⎪ ⎪ ⎪ ⎪ 1 ⎪ ⎪ ⎪ ⎪ ⎪ −2 ≤ θ2 < −1 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎧ ⎪ ⎨ ⎪ ⎪ ⎪ 1 + θ1 θ2 < 0 ⎪θ ≤ θ ⎪ ⎪ ⎪ ⎪ 3 2 ⎨ 5 + θ + θ + 5θ θ > 0 ⎪ ⎪ ⎪ 1 2 1 2 ⎪ ⎪ ⎪ ⎪ ⎪ 1 < θ1 ≤ 5 ⎪ 5θ1 θ2 + θ1 + θ2 + 5 < ⎪ ⎪ ⎪ 0 ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ −1 ≤ θ2 1 + θ < 0 ⎪ ⎪ 7 ⎩ ⎪ ⎪ 2 θ1 ⎪ 11 < θ1 < 1 ⎧ ⎪ ⎧ ⎪ ⎪ ⎨ 5θ1 θ2 + 5 < θ1 + θ2 ⎪ ⎪ ⎪ θ1 +θ2 ⎪ ⎪ ⎪ ⎪ 1 0 ⎪ ⎪ ⎪ ⎪ θ1 ≤ 5 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ θ ⎪ ⎪ ⎪−5 ≤ θ ⎩ 1 ≥1 ⎪ 3⎪ ⎪ −2 ≤ θ2 < −1 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨1 + θ1 θ2 > 0 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪1 ≤ θ < 1 ⎪ ⎪ 1 ⎪ ⎩2 ⎪ ⎩ ⎪ 5θ θ + θ + θ2 + 5 < 0 ⎪ ⎪ ⎪ ⎧ 1 2 ⎧ 1 ⎧ ⎪ CR6 := ⎪ ⎪ ⎪ ⎪ 2θ1 ≥ 1 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ 1 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪θ < 1  θ1 + θ2 > 0 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ 1 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ 3θ1 ≥ 1 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ ⎪ 1 + θ3 ≥ 0 ⎪ ⎪ ⎪ ⎪ 5θ1 θ2 + θ1 + θ2 + 5 > 0 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ θ < −1 ⎪ ⎪ 2 ⎪ ⎪ ⎪ ⎪ ⎩ 0 ≤ 2θ1 < 1 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ 2 + θ2 ≥ 0 ⎪⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ (5θ + 1)(5θ ⎪⎪ ⎪ ⎪ ⎪ 1 θ2 + θ1 + θ2 + 5) > 0 ⎪ ⎪ ⎪ ⎪ ⎨1 1 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ 3 ≤ θ1 < 1 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ ⎪ ⎩⎪ ⎪ θ1 +θ2 < θ ⎪ ⎩ ⎪ ⎪ θ1 θ2 +1⎧⎧ 3 ⎪ ⎪ ⎧ ⎪ ⎪ ⎪θ1 = 5 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ θ +5 ⎪⎪ ⎪ ⎪ ⎪ ⎪ ⎪ 2 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪⎪ 5θ2 +1 < θ3 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩θ < − 5 ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ 2 ⎪ ⎪ 13 ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪−1 < θ2 θ1 < 5 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ 5θ1 θ2 +θ1 +θ2 +5 ⎪ ⎪ 1 < θ1 ⎪ ⎪ ⎪ ≤0 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ 5θ1 +1 θ +θ ⎪ ⎪ ⎪ 1 2 0 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩⎩ 1 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ 2 ≤ θ1 < 1 ⎪⎧ ⎪ ⎪ ⎪ ⎪ ⎪ −5 ≤ θ ⎨ ⎪ ⎪ 3 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ −2 ≤ θ2 ≤ −1 ⎩ ⎩ ⎩ 1 < θ1 ≤ 5 ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ θ3 ≤ 5 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨

Appendix C: Multi-parametric Linear and Mixed Integer Linear … Table C.3 Critical regions of the mp-MILP under global uncertainty example (iii) CRs

Mathematical expression ⎧

CR7 :=

CR8 :=

CR9 :=

⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ −3 ≤ θ2 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ 1 1 + CT2p|p 1 translin2 trans Tijp ≤ Tijp ∀i, j, p > 1 translin1 trans + (RZ − 1) · τ Tijp ≥ Tijp max ∀i  = j, CH < p ≤ PH ijp

∀p ∈ CH

(E.25) (E.26) (E.27) (E.28) (E.29) (E.30) (E.31)